Part I. Optically induced, ferroelectric

Part I. Optically induced, ferroelectric domain gratings in photorefractive crystals and applications to nonlinear optics. Part II. Self-focusing and self-trapping of optical beams upon photopolymerization and applications to microfabrication - CaltechTHESIS
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Part I. Optically induced, ferroelectric domain gratings in photorefractive crystals and applications to nonlinear optics. Part II. Self-focusing and self-trapping of optical beams upon photopolymerization and applications to microfabrication
Citation
Kewitsch, Anthony S.
(1995)
Part I. Optically induced, ferroelectric domain gratings in photorefractive crystals and applications to nonlinear optics. Part II. Self-focusing and self-trapping of optical beams upon photopolymerization and applications to microfabrication.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/ngzk-1s10.
Abstract
This thesis explores the application of two distinct nonlinear optical phenomena, the
photorefractive effect and photopolymerization, to optically generate microstructures
with feature sizes on the order of optical wavelengths. First, we have found that in
certain photorefractive crystals, the photogenerated space charge field dynamically aligns
ferroelectric domains. This is demonstrated by the observation of Barkhausen noise
linked to the formation of domain gratings. Domain gratings are recorded with spatial
periods on the order of optical wavelengths, which we use for quasi-phase matched
second harmonic generation and holographic data storage.
The second part of this thesis explores the nonlinear optical response accompanying
photopolymerization. In some photopolymers, the crosslinking of polymer chains
induces a significant increase in the index of refraction in the exposed region. This index
perturbation acts as a lens which subsequently focuses down the input light wave. We
observe self-focused and self-trapped optical beams upon photo-induced crosslinking of a
liquid monomer. In the case of self-trapping, the inherent diffraction of the optical beam
is exactly balanced by self-focusing, so the diameter of the beam does not change as it
propagates through the medium. Most importantly, this waveguiding generates solid
polymer microstructures in the illuminated region, which can be used to fabricate micro-electromechanical systems and optical interconnects.
Item Type:
Thesis (Dissertation (Ph.D.))
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California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Yariv, Amnon
Thesis Committee:
Yariv, Amnon (chair)
Cross, Eric
Rossman, George Robert
Grubbs, Robert H.
Defense Date:
16 May 1995
Record Number:
CaltechETD:etd-10152007-130350
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DOI:
10.7907/ngzk-1s10
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Part I: Optically Induced, Ferroelectric Domain Gratings in

Photorefractive Crystals and Applications to Nonlinear Optics

Part II: Self-Focusing and Self-Trapping of Optical Beams Upon

Photopolymerization and Applications to Microfabrication
Thesis by
Anthony S. Kewitsch

In Partial Fulfillment of the Requirements
for the Degree of

Doctor of Philosophy

California Institute of Technology
Pasadena, California
1995
(Submitted May 16, 1995)

In memory of my father and sister

ill

Anthony S. Kewitsch

Acknowledgments

I would first like to thank my thesis advisor, Amnon Yariv, for his support, friendship
and inspiration. The research environment he cultivates is a pleasure and honor to work
in. I would also like to thank Mordechai Segev for his enthusiastic and energetic support.
I deeply appreciate his friendship and guidance. Many members of the Caltech Quantum
Electronics Group have been a pleasure to interact with and learn from, including (in
alphabetical order): Roni Agranat, Gilad Almogy, Gert Cauwenberghs, Lars Eng, Doruk
Engin, Rudy Hofmeister, John Iannelli, John Kitching, Victor Leyva, Bill Marshall,
Chuck Neugebauer, John O’Brien, Sergei Orlov, Volnei Pedroni, George Rakuljic,
Joseph Rosen, Akira Saito, Randy Salvatore, Thomas Schrans, Ali Shakouri, Xiao-lin

Tong, Yuanjian Xu, Shogo Yagi and Min Zhang.

I wish to thank the numerous individuals whom I have had the pleasure to collaborate
with, including Gregory Salamo, Edward Sharp, Ratnakar Neurgaonkar and Terrence
Towe. Several experts who provided valuable insight related to this thesis research
include George Rossman, Eric Cross, Robert Grubbs, Michael Hecht and Yu-Chong Tai.
I thank them for their time and patience. Finally, I would like to thank Paul Davis,
Anthony Siegman and David Jackson for the early opportunities to pursue undergraduate

research under their supervision.

Several Caltech staff members have provided extraordinary assistance, in particular Ali

Ghaffari, Paula Samazon, Reynold Johnson, Jana Mercado, and Larry Begay. I would

also like to thank Tony Stark, Ray Garcia, Bob Paz, and the staff of the interlibrary loan

office.

First and foremost, I would like to thank my parents and sister for their unwavering

support. The lessons in life they provided are the most priceless knowledge of all.

Abstract

This thesis explores the application of two distinct nonlinear optical phenomena, the
photorefractive effect and photopolymerization, to optically generate microstructures
with feature sizes on the order of optical wavelengths. First, we have found that in
certain photorefractive crystals, the photogenerated space charge field dynamically aligns
ferroelectric domains. This is demonstrated by the observation of Barkhausen noise
linked to the formation of domain gratings. Domain gratings are recorded with spatial
periods on the order of optical wavelengths, which we use for quasi-phase matched

second harmonic generation and holographic data storage.

The second part of this thesis explores the nonlinear optical response accompanying
photopolymerization. In some photopolymers, the crosslinking of polymer chains
induces a significant increase in the index of refraction in the exposed region. This index
perturbation acts as a lens which subsequently focuses down the input light wave. We
observe self-focused and self-trapped optical beams upon photo-induced crosslinking of a
liquid monomer. In the case of self-trapping, the inherent diffraction of the optical beam
is exactly balanced by self-focusing, so the diameter of the beam does not change as it
propagates through the medium. Most importantly, this waveguiding generates solid
polymer microstructures in the illuminated region, which can be used to fabricate micro-

electromechanical systems and optical interconnects.

Vii

Preface

This thesis explores the application of two diverse nonlinear optical phenomena, the
photorefractive effect and photopolymerization, to the generation of microstructures with
feature sizes on the order of a micron. First, we explore the use of optical interference
patterns to tailor the ferroelectric domain microstructure of SBN crystals. The dynamic
alignment of ferroelectric domains is observed in response to a photorefractive space
charge field. Chapter one describes the theoretical and experimental background of the
photorefractive effect and its application to record photorefractive holograms. The

material properties of several photorefractive ferroelectrics are discussed in chapter two.

Photorefractive space charge fields modulate the index of refraction by slightly distorting
the position of ions within the crystalline lattice through the linear electrooptic effect
(chapter three). However, we demonstrate that near the ferroelectric/paraelectric phase
transition, materials such as strontium barium niobate (SBN) exhibit hysteresis in the
electrooptic effect (chapter four). That is, ions not only displace slightly, but also
undergo inversion through the center of symmetry. This study demonstrates that in low
coercive field ferroelectrics such as SBN, the dynamic alignment of domains plays an
intimate and previously unappreciated role in the generation of index gratings. This
interpretation is drawn from the experimental observation of Barkhausen noise, a
signature of domain switching, while optically recording domain gratings. This is the

subject of chapter five.

Vill

Chapter six describes one application of this hysteretic electrooptic effect: volume
holographic storage. An important issue in holographic storage is the lifetime of the data
upon readout. Photorefractive holograms are normally destroyed upon reconstruction
unless a thermal fixing stage is applied. However, we show that photorefractive
hologram lifetimes exceed 100 days in SBN crystals, a consequence of ferroelectric

hysteresis. This selective, page-addressable fixing process occurs automatically.

The application of domain microstructures to quasi-phase matched second harmonic
generation is the subject of chapter seven. Optically induced polarization gratings
enhance the second-harmonic conversion efficiency by ensuring that momentum is
conserved during the nonlinear interaction. In fact, we observe that the peak second-
harmonic conversion efficiency is enhanced by a factor of 700 upon recording a domain
hologram. The wavelength response is further tailored by writing an ensemble of
gratings within the same volume. Each grating enhances the second-harmonic generation

at a predetermined wavelength.

The second part of this thesis describes the formation of polymeric microstructures based
on self-focusing and self-trapping of optical beams upon photopolymerization. By
illuminating a photopolymer with UV radiation, the liquid polymer is crosslinked to
produce a solid structure. The photochemistry and polymer chemistry of this process is
presented in chapter eight. For some photopolymers, the index of refraction increases
upon crosslinking. The origin of this index change is discussed analytically in chapter
nine. We show that this index increase enables optical beams to be self-focused or self-
trapped within the polymeric waveguides (chapter ten). This self-trapping balances the
inherent diffraction. We numerically simulate self-focusing and self-trapping, as

described in chapter eleven.

In chapter twelve we demonstrate the use of self-trapping to fabricate very deep
polymeric structures with transverse dimensions on the order of optical wavelengths.
This relaxes the fundamental limit to the depth of focus imposed by diffraction and
enables the microfabrication of structures with extremely large depth-to-width aspect
ratios. The liquid material may be removed after exposure, leaving solidified polymer
only in the original illuminated regions. Polymer microstructures with feature sizes of 10
lum and 1 cm deep have been fabricated. This technique may be instrumental in
fabricating optical filters, optical memory devices, micro-electromechanical systems and

optical interconnects.

Table of Contents
Acknowledgment ..0........-ccsescccesseceesseceseseeesnccesenseesescssceeseeeecsscssscesesesessceesscsseesersseeeees iv
ADSUracCt 2... ec eeececceececsnccsavecccscecerersuscacesscesesecescesscsssseuseecccecnscesesscescassecesesesseesssceceranes vi
Preface .......cccccsecccccccssesecceccccssssoccccccsssecssceccensussssccecesecsenscsssesssesesessseseussacessecsecsccecassenaees Vil
Table of Contents ..........c.ccceecsececccsssseccecscsccccsssccecsscscensvscesccscscorsesssosecscuesssenecesssureesencns x

Part 1: Optically Induced, Ferroelectric Domain Gratings in Photorefractive

Crystals and Applications to Nonlinear Optics

1 Introduction to Photorefractive Nonlinear Optics 1
1.1 Introduction ............eceeseeceeesseeeesececessceeeesneecesncesseseeesessueesseeseessneeessessaeees 1
1.2 Band Transport Theory 0.0... eee cecesseeseeesseeeeseeecsseeessesssesseeseeeseeeee 1
1.3 Diffraction From Volume Phase Holograms ...............:::sscesesseceseeseeees 8
1.4 Permanent Fixing of Photorefractive GratingS...........ee ce eeeeeeeeeeeeeeeees 8
1.5 Fixing by Thermally Assisted Ionic Drift... eects eee cess eeeeee 10
1.6 Fixing by Ferroelectric Polarization Reversal.............cccssssseseeceeeees 13
V.£7 0 SUMMA Y oe eee cece ceeeenecesceceeneessassaascseessseecsneceessessssesneessaserenseees 21
References for chapter ONn@..........s:cscescssceesssseesensseeseessesescesesesesesesseeeeeseneees 23
2 Ferroelectric Materials 27
2.1 Introduction ..........cccsececeseceessececcneeresnececsscesesneeseoseeeesensesseesseeeseeeeseeeaees 27
2.2 Material Properties of Photoferroelectrics ...... cette cece seteetereeeeeeseees 27
2.3. Photoferroelectric PhenOMeNo ............eccsessseseeceseceseceeeseseeneveenereoees 30

2.4 SUMIMALY .0...eeceeeseceeseeeneeceseeseeeessecesseeeseeeeseeerssesssaecesesesssesenecosesesseeseeseene 32

References for Chapter tw0.......ccccssscsecssscsscssesssesssceenesecesesesesseeseeaseeeereeeseenes 33
Ferroelectric Domain Gratings 35
3.1 TNtrOductiOn ......... cs seescsssecessseeceecsesersensseessssoessseneessssesesseassesseseosoeseneees 35
3.2. Theory of the Electrooptic Effect... ccessssesecesessesseeceeseeeseacenes 35
3.3 Ferroelectric Domain Structure ........ eee eceeeeeesseceeeecreeeseeeeneesenseneesees 41
3.4 Transformation of Nonlinear Coefficient Upon Domain Reversal..... 46
3.5 SUMIMALY 000... eesceesceeeceecneeecsseeeseeeeseeeeessssassesasseeessesesassssessssereasensaes 47
References for chapter three... ceecseeeseesecesseseseseeoesnenserecceceeseeeeeseseneneesees 49
Ferroelectric Hysteresis Fixing of Volume Holograms 50
AL [introduction oe eeeeeeecesseseeeecessseaneessesescecesscosasessesaseseserasenesesceeas 50
4.2 High Intensity Fixing ..0... eee ee ececeneeeeceseereceeneerscssaeeessnesesenesenenes 50
4.3 Short Exposure Fixing... ccecccscesesecesseseeeesseeessesensessseeesesseseenes 57
4A SUTIMATY oe cece eeeeeeeeeeeeeeeeneeescereaeeeseeeeneeseacescseeesseseseesseessaeesseseesees 65
References for chapter fur... eee esceesescssseescesseecssesscsesessccsseesseccosrersseneeees 66
Barkhausen Noise 67
5.1 TmtroductiOn .00......scescceeseeseceeceseceeeresseescescsesceseseenscsacesssessessecsetensesenaneese 67
5.2 THEOTY ones eeeeeccceccescccsccencesseceeseeeessessesseeesccecsescessasesesersesssesesenseeasonsess 68
5.3 Domain Nucleation Energy Requirements..............ceceseessesseseeseneees 75
5.4 Experimental Results... cece csessssesceesscessrecsseessesesssesseesesseeeenens 76
5.5 Domain Switching Dynamics... eee eeeeereeseeeeesseceseeseseeneeece 83
5.6 Optical Barkhausen Noise... ceeesessesecseeeecesesessecseseneseseeeesneees 88
5.7 SUMMA Y 00sec eeeeeeeeceeeeeeeeeeneceeeeececesesenseeecueeesaeasanseecessseesseeeeasseneaes 92

Xii

6 Applications of Domain Microstructure to Holographic Data
Storage 96
6.1 Introduction... ee eee eescneneneneccncneesaseseeesesesaeescessseseseecneeesseseeeenseeraes 96
6.2 Volume Holographic Memory ............ccecesssseeeseesceeceneeeceeceeeeeeeeneeses 97
6.3 —_ Electric Field Multiplexing .00..... eee eeeeeeescecneeeteneeeeneeeoesesenseerees 98
6.4 Selective Fixing... cess seeeeseeeseceseeeesecessseeecsesessneeseseeeeseeeessneess 109
B.S SUMMA Y oo. eee eecescceneeesceessecceccercuceceneeesneeessensceeseaeenesaceeseeesenseees 110
References for chapter Six ............:cescesessecesscesseeeseesscesesserssseessesensserecseensanees 112

7 Application of Domain Microstructure to Quasi-Phase Matched Second
Harmonic Generation 114
7.1 Tntroduction .0.... cece esescesssesescecescessesseeseeseseeseeessceneesssesseeeseeesenesesees 114
7.2. Nonlinear Optical Frequency Conversion..............:s:ccssscsssessereeeeeees 114
7.3. Quasi-Phase Matching Using Dynamic Domain Gratings................ 118
74 Experimental Observation of QPM in SBN:75......c.cccecesescsseeeeeees 119
7.5 Experimental Observation of QPM in SBN:61.........ccce eee 126
7.6 Depth of Modulation of Spontaneous Polarization ..............::cceee 132
7.7 SUM occ eee cecescceeseeeseeesnecnecenssenseceseseesaseseeesaeeesseseaeesseeeneeeneeees 135
References for Chapter SCVEM............c:cccsscessscseessessseseseesssseseeeeesesssssaseeeeneees 136

Part 2: Self-Focusing and Self-Trapping of Optical Beams Upon
Photopolymerization and Applications to Microfabrication

8 Photopolymerization 139
8.1 ImtrOduction .........cc:ccsecceececeeesscecesnteeseeeesescseceesneceeseeesseseeseeceseerenseeenees 139
8.2 Photoinitiated Polymerization ..............ceeceeseeeseeessreeesneeeesseeeseneeens 139
8.3 Types of Photoinitiators ...0... tl eeccescscceeseeeceseeereseesseeneeeeeeeseeeeeees 144

8.4 Photosensitized Polymerization ..............:ccccsecceseesseeeteeeneeeneesseneeeeees 145

Xili

8.5 Dipole Resonance Transfer

8.6 Exchange Transfer... ececcesecescecececseeesccesceeseesceecaesereeesneeseeeneeees
8.7 Types of PhotosensitiZers 0... ee eeeseeceecesscesscnesssscsseeerseneesseeeseeees
8.8 —- Chain Propagation... cesssescescesseceesesecsescsecesesessenssoeseeeseeseeaees
8.9 «—- Chain Termination... eee ceeecesseceeecceneceeneceeseeeesneessceseeseesseseeeeees
8.10 Kinetic Model of Polymerization... le cece eeeenceeesereeeeeeeeeene
Sa SS RS) E601 091-1 9 eee
References for chapter eight... cece sseseecseccescesscsssssseeessseseessssoneeseeas

Index Changes Upon Photopolymerization

9.1 ImtrOduction ..........ceeseesceseeesecenececceeesecseseessseeseossessaseseeosesenssenseeseesenes
9.2 Index of Refraction of Polymet........... cc eeesscesesseeseeeeseeeseesseoeesenaees
9.3. Index of Refraction of Photoinitiator ..0..0... eee cess seeeeereceeeeeesaene
9.4 Dynamic Model of Index Change Upon Photopolymerization.........
9.5. Measured Evolution of the Index of Refraction... ceeeeeeeseeeeeee
9.6 SUMIATY 0... eee ee reeeeeceeeneeeteneesersneeesusnecssesssonseacessesesacerssesssseonenssennees
References for chapter mine... eeseesseecseeeeeeeeeeeeessevesceeessaeeeseeeeasaeeegs

Introduction to Self-Trapping and Self-Focusing

10.1
10.2
10.3
10.4
10.5

TMtrOdGUctiOn ........ cece eeceesceneccsceeceseecneseaeeseseseussessesesesesseeseeseeeseesseees
Wave Equation in Inhomogeneous Medium.............c cess eee eeeeees

Critical Waveguide Diameter... eee eeeeeetseceeeceeeesensaeereneeeeeeces

148
149
150
151
153
156
157
159

161
161
161
164
167
168
170
171

XIV

Numerical Simulations of Self-Trapping and Self-Focusing Upon

Photopolymerization 182
T1120 Untroduction ..... eee eeeeeeeeeceseeeseeeseeeeseeeeeeeeaeseeeeesseeensceseeeessecsesesees 182
11.2 Finite Differenced Beam Propagation Method ................ccccccceseeenes 182
11.3 Results of Numerical Simulations... eceeeeceseeeseeessesseessteneeees 184
11.4 — SUMMATY ooo eect neceseeeneesseseseseesescerseseaeesceseeesteseseeneeenseesaees 188
References for chapter leven .............cessescccssesscsccenceeeeeseceeceeeeeesessteeseeseessenees 189

Applications of Self-Trapping and Self-Focusing in Photopolymers 190
12.1 Introduction oo... cece cesseeceseceseeseesesseeseeeeeceseseeceeeseesntessseseseaeens 190
12.2. Application of Self-Focusing to High Resolution

Photolithography oe ecceeeseessecesscceseeeeecesseesteesnseessssessaseesseneaes 190

12.3. Application of Self-Trapping to the Fabrication of Polymeric

MICLOStIUCtUICS......... ee eeeeeseeeeeeesceeseceescecscecneresseeessessseessetersesesseatesses 198
12.4 Data Storage in Corrugated Waveguides ............:ccessccesscseseeenseesseees 202
12.5 Optical Extrusion... le seesseecccenecsseesecceessseeecesneceeesensestseesecaeees 202
12.6 Self-Organized Semiconductor Laser to Fiber Coupling.................. 205
12.7 SUMMA oo ceceeeeseceeeeeeneeeasececeeesceesceeetseeseeensetsceesesteneeesseeaeees 206
References for chapter twelve .............c:ccsscecesscsssecessessceessesnseecesteeesecessncessuers 207

Relevant Publications 210

Chapter One

Introduction to Photorefractive Nonlinear Optics

1.1 Introduction

The photorefractive effect is a nonlinear optical response characterized by a change in the
optical index of refraction proportional to the absorbed optical energy. The photorefractive
effect was first reported in 1966 by Ashkin et al. [1] in ferroelectrics and classified as
“optical damage.” A high intensity visible laser source induced index perturbations that
survived even after the beam was extinguished. The visible laser deposited electronic space
charge along the periphery of the beam, which induced an index change through the linear
electrooptic effect. Soon thereafter, Chen et al. demonstrated that these index changes
could be used to store high resolution, dynamic volume phase holograms [2]. That is,
holograms could be recorded by coherent illumination in the visible spectrum and erased by
uniform visible illumination. The promise of these early observations provided the impetus
for further study of the photorefractive effect and its applications to holographic memories,

optical processors and display devices.
1.2 Band Transport Theory

The dependence of the index change on the optical exposure is described quantitatively by

the band transport theory applied to photorefractive materials by Kukhtaerev et al. [3]. The

material is assumed to consist of non-photoactive, deep acceptor levels, Na, which ionize
some fraction of optically excitable donor levels Ng (figure 1.1) at room temperature. The

material is illuminated with an optical interference pattern:

I(x) =I, + Teik® + c.c. . . 1.1

Conduction band

Wve
©&- © © — ON
E,>eV

Figure 1.1: Photorefractive band model for one photorefractive donor species Ng. Note

that Na + ne = Nj.

In the regions of illumination, electrons are excited from the donor levels Ng, to the
conduction band, where they drift and diffuse until recombining. After many cycles of
excitation and recombination, electrons will tend to accumulate in the dark regions of the
interference fringes, resulting in a modulated density of ionized donors Nj. This produces

a local space charge field (figure 1.2) which is a slightly distorted replica of the optical

intensity pattern. In non-centrosymmetric electrooptic materials, this local electric field
produces an index change through the linear electrooptic effect [4]:
A (5) = lijk Ex . 1.2

n2 ij

rjjk 1s the third-rank linear electrooptic tensor, and Ex is the component of the electric field
along the k direction. This electric field is generated by the modulated density of ionized

donors.

c axis

1 optical

Doptical 7 *

Figure 1.2: Optical interference pattern spatially modulates the density of ionized donors,
producing a space charge field E,, which modulates the index of refraction through
the electrooptic effect.

The electric field can be determined analytically by modeling the charge transport occurring
under the periodic illumination described by equation 1.1. A rate equation describes the

density of excited donor atoms Nj under illumination and in the presence of recombination:

OND(x,t) I(x) _ ne 1.3

ot” NO
a is the absorption cross section, 7 is the quantum efficiency of optical excitation, v is the
frequency of the incident light intensity pattern I(x), ne is the density of electrons in the
conduction band and t, is the recombination time. This produces a photocurrent J,

composed of drift and diffusion terms:

Jx = peneE, + kpTu oe, 1.4

ut is the electron mobility in the conduction band. Note that the photovoltaic charge
transport mechanism is negligible in ferroelectrics such as SBN [5], so it is ignored in this
treatment. The current continuity equation relates this current to the rate of excitation of
ionized donors:

YU, a

The electric field is obtained by integrating the charge density according to Gauss’ law:

JEx
ox

= & (Ng - me - Na). 1.6

The crystal is assumed to be initially charge neutral so the spatial average of the total charge

density p = Nj - ne - Ng is zero:

(P(x))x = 0.

1.7

By solving equations 1.1 through 1.7 for small intensity modulation (i.e., I/Ip << 1), the

steady state, photorefractive space charge field with spatial periodicity Ag is:

1+i—2
p. =_iEs “Ea |i
sc ?
Fal, ..| Eo VU
1 ~a 1 (0) 0]
+ Hi e|

where the diffusion field Eg is defined as

kp T
Fake g

The limiting space charge field Eg is

and the grating vector kg is defined as

2m
Ag

1.8

1.9

1.10

1.11

Eg is the external, uniform electric field parallel to kg. Equation 1.8 predicts the grating

period dependence of the space charge field. Qualitatively, this dependence is illustrated in

figure 1.3. The upper curves depict the diffusion and limiting space charge fields. The

bottom curve indicates the net space charge field given by equation 1.8, whose maximum

value is typically less than 10 kV cnr!.

Electric Field

Figure 1.3: Grating period dependence of photorefractive space charge field.

The band transport equations can be solved in the transient regime, before the space charge
field reaches the steady-state value described by equation 1.8. The transient space charge

field at the fundamental grating component is [6]:
E,¢(t) = Ege [-cos (kyz + 6} + exp (-t/T,) cos (k,z + Wet + 6) . 1.12

The constants of equation 1.12 are:

2.52. 12
_ BotEg 1.13

E,. == E
. I, 5 +Egt+ Eq

where @ is the photorefractive phase shift, defined as the phase difference between the

intensity and index maxima:

Eq + E,) Eq + EZ
= tar! a Eg) Ea + |, 1.14

and

2 2
1444 + Eo
= Fuy _\Pu 1.15
“ed Ea) 4 Hal. Eo Eo”
1+="){1+—S)+—e=2
Eq Eu} Ey Ey
Ey _Eo
q 1440 + Bo
Ey Ey
Tq is the Maxwellian or dielectric relaxation time:
t=, 1.17
d Sa
and
B, = 7A 1.18
LL ke

These relations indicate that the space charge field grows approximately linearly with time
until the onset of saturation. When an external electric field is applied parallel to the grating

vector, the space charge field may “ring” like an underdamped simple harmonic oscillator.

1.3. Diffraction From Volume Phase Holograms

For a medium of thickness L with a sinusoidal index perturbation of amplitude nj, as in

figure 1.2, the diffraction efficiency 1 is obtained by solving the two beam coupled wave

equations for transmission gratings in the symmetric configuration (depicted in figure 1.2

with 0, =9,). If diffraction from a fixed transmission grating is considered in the absence

of self-enhancement [7], the diffraction efficiency [8]:

Ip(L)
= 1.19
1;(0)
is equal to:
1 = exp (- ol | sin? zak | . 1.20
cos 8 i cos 8

a is the absorption coefficient and 06 = 6,=90,;. Note that for weak diffraction, the
diffraction efficiency is proportional to the square of the index modulation n,. Therefore,
for a linear electrooptic material, the diffraction efficiency is proportional to the square of
the photorefractive space charge field. In chapter five, the space charge field is computed

from the measured diffraction efficiency using equation 1.20.

1.4 Permanent Fixing of Photorefractive Gratings

Volume phase holograms have been stored in a wide range of materials: silver halide
photographic emulsions, dichromated gelatin films, photosensitive glass, photorefractive
polymers, organic materials and ferroelectric crystals [9]. Each of these materials poses a
unique set of technological problems when attempting to record optical interference patterns

as permanent index gratings. These issues include dimensional stability, optical sensitivity,

scattering, shrinkage after development, dynamic range (i.e., maximum index change), and

the hologram lifetime.

Two common ferroelectric oxide crystals for recording volume phase holograms are
LiNbO3 and Sr,Baj_,Nb20¢ (SBN:x). When these materials are doped to optimize their
photoconductive properties, optical interference patterns redistribute charge among traps to
generate local electronic space charge fields which are replicas of optical interference
patterns (but generally shifted in phase and spatially distorted because of electron/hole
diffusion). An index change typically arises from the spatially modulated space charge
field through the linear electrooptic effect. The complex microscopic details of this process
are absorbed within a measured electrooptic tensor rj, which relates the index change to the
space charge field. These index gratings are electronic in origin, so they are bleached by
the same optical field that reconstructs them. This dynamic property is extremely useful for
real time holography applications; however, for applications such as holographic data

storage, permanent index gratings are desirable.

The goal of hologram fixing techniques in ferroelectrics is to transform the grating
composed of a spatially modulated density of optically active species (electrons, holes) into
a modulated density of optically inactive species (ion cores, vacancies, defects). This has
been achieved with varying degrees of success in LINDO3, LiTaO3, Ba,NaNbs50 15, SBN,
KNbO3, KLTN, and BaTi03. The common fixing stage in these oxides is the conversion
of a dynamic grating composed of electrons and/or holes into its replica composed of
stable, displaced ions. Upon reconstruction of the hologram, the electronic grating is
partially erased and the ionic grating persists. These ions are typically optically inactive and
have a relatively low mobility. The obstacle to fixing is the simultaneous requirement that
ions must be mobile enough to redistribute under the influence of local space charge fields,

yet be resistant to optical erasure. In addition, mobile charge tends to redistribute to

minimize electric fields within the crystal, leading to compensation of the revealed, fixed

grating. This effect reduces the net diffraction efficiency from the grating.

One technique to achieve fixing is to uniformly heat the crystal during or after the hologram
recording process so that ions can drift under the influence of electronic space charge
fields. A second technique uses the hysteresis associated with the reversal of ferroelectric
domains to provide memory even after the electronic space charge fields are diminished.
This hysteresis may be increased to a significant level by applying external electric fields or
by recording near the ferroelectric Curie temperature. In contrast to a ferromagnetic
memory, in which the magnetization can be reversed locally on a surface by a magnetic
recording head, periodic electric fields can modulate the polarization locally throughout the

entire volume of the ferroelectric.

1.5. Fixing by Thermally Assisted Ionic Drift

Fixing by ionic drift was first demonstrated in LiNbO3 in 1971 by Amodei and Staebler
[10]. They discovered that by heating the crystal to 100-200 °C either during or after the
hologram exposure and by subsequently cooling, a permanent grating 180 degrees out of
phase with the electronic grating was generated. They proposed that at elevated
temperatures, the electronic space charge field generated by optically excitable ions causes
optically inactive ions or vacancies to drift under the force of the local space charge grating
to compensate the local fields. The electronic grating remains trapped in the absence of
light and at the elevated temperatures because electrons recombine at sufficiently deep traps
(> eV). When the crystal is cooled to room temperature in the dark, the ionic grating is
rendered immobile. The electronic grating is subsequently depleted by uniform

illumination, causing the diffraction efficiency to vanish when the electronic and ionic

grating are equal in magnitude and opposite in sign. Further erasure of the electronic

grating reveals the fixed ionic grating.

Thermal fixing has been demonstrated in LINbO3 samples of various doping combinations,
such as Cu, Fe, Mn, Si (see, for example, [11] and references therein) as well as
LiTaO3:Fe [12] and BagNaNbs0 5:(Fe,Mo) [10], Similar thermal fixing results under an
external ac electric field have been reported in the non-ferroelectric Bi,2TiO9 [13], Several
subsequent studies have followed Amodei and Staebler's original work to determine the
ionic species responsible for the fixed grating. It is believed that the drift of protons are
primarily responsible for fixing in LINbO3 [14]. The 1.1 eV activation energy of OH™ ions
[15] agrees with the those values (1.1 eV [16] to 0.93 eV [17]) determined from fixed
hologram lifetimes, based on an Arrhenius type thermal activation process. The role of
protons is further supported by the change in the OH” infrared absorption band in thermally
fixed regions [18]. The OH" is believed to enter the crystal lattice during crystal growth
[14]. Subsequent proton migration induced by the space charge field takes place at about
140 °C, well below the temperature at which electron modulation in deep traps is thermally
degraded (= 180 °C) [14], At room temperature, the probability of ion redistribution by an
Arrhenius type activation process is exceeding low (10-17). The dark storage time is
estimated to be approximately 10° years [19]. A comprehensive discussion of experimental

results related to thermal fixing in LINbO3 may be found in reference [17].

Fixed diffraction efficiencies as large as 32% have been obtained by recording the
electronic grating at the elevated fixing temperature [20]. This allows mobile protons to
continuously neutralize the electronic space charge field. Since the recording beams remain
on, the electronic grating continues to grow until it reaches the limiting space charge value.
This technique is effective at grating periods larger than the Debye screening length (i. e.,

in the diffusion limited regime of the photorefractive band transport model). A large

enhancement in fixing efficiency is produced by short circuiting the crystal while exposing,
then developing the grating under open circuit conditions. The largest fixed diffraction

efficiency reported in LiNbO; is 98% for a reflection spectral filter [21].

» Z
me Z
Ay & _ —
<= OO < © be ;
© 5 > 25 >
© @-
(d) P Z, average
Potdp| ,

Figure 1.4: Ionic drift fixing method: (a) Generation of spatially modulated electronic
space charge field, which induces an index grating through the distortion dp. (b)
Upon heating, ions drift to compensate the electronic space charge field. (c) After
cooling, crystal is uniformly illuminated to erase the electronic grating and reveal
180 degree out of phase, permanent ionic grating. (d) The permanent modulation
of the polarization arising from the net space charge field (ionic - electronic)
produces the index grating.

The ionic drift fixing method is illustrated schematically in figure 1.4. In this case the
initial buildup of the space charge grating is described by the conventional band transport
equations, with the addition of the photovoltaic charge transport mechanism. An electronic

space charge grating is recorded under spatially periodic illumination at a sufficiently low

temperature so that ions are immobile (figure 1.4(a)). Sp is the change in polarization due
to a change in position of the transition metal ion under the influence of the space charge
field. This polarization change by definition produces the electrooptic index grating. This
is discussed in detail in chapter three. Following electronic hologram recording, the crystal
is heated to a temperature at which the ions are mobile while electrons remain immobile in
the dark. At elevated temperatures ions compensate the electronic space charge field by
drift (b). The net space charge field is diminished. The crystal is cooled and subsequently
the hologram is uniformly illuminated, partially erasing the electronic grating and revealing
the out-of-phase ionic grating (c). The permanent index modulation arises from the net
space charge field (ionic - electronic) perturbing the lattice polarization (d). This
polarization grating caused by ionic drift (not domain switching) is 180 degrees out of
phase with the initial electronic index grating. Note that the space charge field and
polarization as depicted in figure 1.4 are spatially averaged on a scale much larger than a

unit cell, yet smaller than the optical wavelength of interest.

1.6 Fixing by Ferroelectric Domain Reversal

Fixing by ferroelectric domain switching is based on the almost binary displacement of
ferroelectric ions associated with the spontaneous polarization, rather than the drift of
hydrogen impurities. That is, the electrooptic grating exhibits hysteresis. The ions
responsible for the electrooptic index change not only distort slightly, but also switch to
produce inverted domains. Figure 1.5 illustrates an idealized ferroelectric hysteresis curve
for a voltage applied across the c faces of the crystal. P,; is the spontaneous or saturated
polarization, P, is the remnant polarization in the absence of an applied field, and E, is the
coercive field. The coercive field is that field required to invert the spontaneous
polarization within half the volume. This hysteresis diagram motivates the fixing

methodology only on the most basic level. The electronic space charge field AE creates a

replica modulation of the local spontaneous polarization, APaynamic- Upon optical erasure
of the electronic space charge field, a remnant polarization modulation APyemnant persists.
This produces the fixed hologram. The goal of ferroelectric hysteresis fixing methods is to
maximize the memory. Note that the polarization hysteresis diagram represents a
macroscopic polarization. Since the photorefractive space charge field modulation has a
period of microns, which is much greater than the unit cell, the macroscopic picture is
applied to a small region in the crystal corresponding to a single grating period. There are
two issues to keep in mind when applying this diagram. First, the scale of the hysteresis
curve may depend on the size of the domain region under consideration. Second, figure
1.5 is experimentally measured with a voltage source which provides an effectively infinite
amount of charge. The photo induced space charge fields provide only limited charge, so
charge compensation requirements at domain walls will be a limiting factor in the

polarization modulation depth (see chapter three).

P, aa
AP dynamic r| / nN P emnant

c > E

Figure 1.5: Typical polarization hysteresis curve.

Evidence for polarization fixing can be traced back to the early work on optical damage, a
phenomenon now synonymous with the photorefractive effect yet actually arising from a
variety of photoferroelectric phenomenon. For instance, Ashkin et al. [1] observed
antiparallel domains in BaTiO3 following the illumination of an initially poled crystal.
Rudyak [22] reported the optical generation of needle shaped 180 degree domains in
LiNbO3:Fe following illumination with a single beam at 440 nm, at room temperature.
Kovalevich et al. [23] observed a distortion of the crystal structure in the region of
illumination in LiINbO3:Fe by the method of x ray diffraction. Following optical erasure,
the strain of 0.03% disappears. Apparently, this strain indicates the coupling of the space

charge field to the lattice constant through the piezoelectric effect.

These experimental observations suggest that the interaction of light and domains is indeed
a common phenomenon shared by many photoferroelectrics. To date, however, efficient
ferroelectric domain gratings have been reported only in La doped lead zirconate-titanate
(PLZT), SBN and BaTiO3. The results may be classified into the following representative
classes: (1) cooling through a ferroelectric phase transition, (2) electrical fixing and (3)

photoassisted domain switching. These techniques will be described in this order.

Micheron and Trotier [24] reported on the formation of domain replicas of electronic
volume holograms in SBN:75 single crystals while cooling through the paraelectric-
ferroelectric phase transition (T, = 45 °C in their sample). They proposed that in the
paraelectric phase, local ionic displacements cancel the photoinduced electronic space
charge field. This is expected in the vicinity of the ferroelectric phase transition, where the
low frequency dielectric constant is extremely large (~10,000). Upon cooling to the
ferroelectric phase, the orientation of the nucleating domains is influenced by the light-

induced space charge field, to produce a permanent domain grating. Similar fixing results

were obtained by illuminating with very high optical energy densities (100 W cm-2),

apparently because of optical heating.

Electrical fixing of volume holograms in both SBN:75 and BaTiO3 in the head-to-head
domain configuration has been demonstrated during the early work on volume holographic
memories [25-27] and has been confirmed by more recent work [28, 29]. In principle, this
technique has the desirable characteristic that the crystal need not be heated as in thermal
fixing and the process can be performed in a relatively short time (~ 1 second) with low
intensity optical beams. The electrical field fixing threshold is nearly equal to the average
coercive field E, = - 970 V cm! (negative values represent depoling fields) and the
minimum fixing time is practically limited by the maximum current imax supplied by the

voltage source [26]:

tmin = 2 Qy/imax » 1.21

where Q, = A P,, A is the area of the electrode and P,is the remnant polarization.
Typically, tmin is 10 msec. A fixed efficiency of 52% was achieved with a fixing pulse of
1.25 kV cm! applied for 0.5 seconds. This domain grating was stable for more than 10
hours during continuous reconstruction with a 1 W cmr? reference beam. The domain
pattern is unstable when the fixing voltage is too large compared to the coercive voltage or
when the fixing pulse is applied for a time much longer than tip, because significant
depolarization of the entire crystal bleaches out the grating modulation. The fixed hologram

can be erased by applying a field much larger than the coercive field.

In these experiments the grating vector is oriented nearly parallel to the c axis, so the space
charge field is parallel to the c axis. The applied depoling field biases the space charge field

near the negative coercive field, increasing the ferroelectric hysteresis. As apparent from

the hysteresis diagram (figure 1.5), the slope dP/dE is a maximum at this point, so

significant polarization modulation is expected in this region.

The method to generate ferroelectric domain gratings by space charge fields is illustrated
schematically in figure 1.6. The crystal is exposed to spatially periodic illumination (figure
1.6(a)). A depoling field may be applied parallel to the c axis to assist in the domain
reversal process after exposure. The internal field can then be estimated to be the sum of
the external and space charge field. The initial electronic space charge field (b) induces an
index change through the lattice distortion 6p. This distortion, unlike the case of ionic drift
fixing, also consists of dipoles switching orientations, so a dynamic domain grating is also
present. This observation is described in detail in chapter five. Under uniform illumination
(c), the electronic space charge field is partially depleted. The polarization perturbation
relaxes, but ferroelectric hysteresis prevents the switched dipoles from returning to their

pre-exposure locations.

Figure 1.6(a) depicts the resulting inverted microdomains in the regions of negative space
charge field, which produce a spatially averaged remnant polarization modulation as shown
in figure 1.6(d). The polarization modulation is biased about the spontaneous polarization.
Note that the remnant polarization grating is in phase with the initial electronic grating.
Thus the spatial phase shift Ad between adjacent maxima of the two index gratings is zero.
Clearly, the same ionic displacements are responsible for both the electrooptic index change
and the ferroelectric polarization. This phase relationship is apparent upon reconstruction
of the hologram (figure 1.7) while the dynamic grating is optically erased. If the two
gratings are in phase (Ad = 0), the diffraction efficiency does not pass through zero upon
reconstruction, but decays to the remnant value. However, if the dynamic and remnant
gratings are out of phase (Ad = 7) and the dynamic grating is stronger than the remnant

grating, then the diffraction efficiency will go through zero when the two gratings are equal

in strength but opposite in sign, and will thereafter grow as the dynamic grating is further
extinguished. Note that this latter case corresponds to the revealing characteristic exhibited
following ionic fixing, where the electronic grating is compensated by an ionic grating of

opposite sign and smaller magnitude during the heating stage.

Photoexcited,

redistributed electron
= Inverted
microdomain
(0) Ee
Lex PX
op \ foe X
(c) E,

(d) x, average

P.-Sp | — NN

Figure 1.6: Polarization fixing by photoinduced space-charge fields: (a) Experimental
setup (b) Generation of spatially modulated electronic space-charge field and
resulting index grating caused by distortion 5p. (c) Uniform illumination partially
erases electronic space charge field; however, a remnant polarization modulation dp
persists due to ferroelectric hysteresis. (d) Remnant polarization modulation
responsible for fixed domain grating.

Diffraction Efficiency

Readout Time

Figure 1.7: Revealing characteristics, depending on relative phase, Ad, of electronic and
fixed index gratings.

Another method to fix gratings based on ferroelectric hysteresis is called photoassisted
domain switching. This effect plays a dominant role in the formation of ferroelectric
domain gratings in configurations where the photoinduced space charge field is
perpendicular to the c axis (figure 1.8). In this geometry external electric fields are
essential to reverse domains, since the photorefractive space charge fields have a negligible

projection along the c axis, the poling direction.

(a)
Cc
. Photoexcited, redistributed
electron
q Inverted microdomain
(b) Ee

E,, oe NOS ~\ o> x

P, ry NL

Figure 1.8: Polarization fixing by photoassisted domain reversal. (a) Experimental setup
and overview (b) Light intensity pattern modulates the crystal coercive field. (c)
An external depoling electric field reduces the spontaneous polarization more
significantly along grating planes with low coercive field. As in figure 1.6(d), this
remnant polarization modulation produces a fixed domain grating.

The photoassisted domain switching mode of electrical fixing was first demonstrated in
photoconductive PLZT ceramics by Smith and Land [30]. They found that uniform
illumination assisted both the poling and depoling of the sample. That is, the coercive field
was reduced in the illuminated regions. The regions of maximum illumination were
completely depoled following the simultaneous exposure and application of a depoling
field. A photograph was stored in the poled plate by imaging it onto the ceramic while a
depoling electric field was simultaneously applied. The depoled regions produce
transmittance minima because the formation of 71° and 109° domain walls enhances the
light scattering, creating a negative of the original image. This device was used as a

primitive spatial light modulator.

Photoassisted domain switching has also been observed in SBN:60 by Kahmann et al.
[31]. In agreement with Smith and Land, they observed depolarization in the illuminated
regions following the application of a negative fixing pulse. Because antiparallel domain
walls are relatively narrow in SBN, they do not scatter light as strongly as PLZT. The
grating is instead revealed by beam coupling topography [32]. Similar observations of

polarization gratings in SBN:75 appear to be related to this effect [33, 34].

In regions of illumination, the field required to invert domains is reduced, due to the
presence of photoexcited charge which screens the ferroelectric dipole interaction and
provides charge to screen the depolarization fields generated upon domain inversion. This
lowers the local coercive field in the illuminated regions. Domain reversal is expected in
the illuminated regions, and the polarization modulation is exactly out of phase with the

intensity pattern.

1.7 Summary

The mechanisms of dynamic hologram grating formation in photorefractives is described
by the band transport model. To develop these gratings into permanent replicas, a fixing
process must be applied. Fixing has been achieved in a large number of ferroelectric
crystals over the past twenty five years. The only technique of practical importance thus far
is ionic fixing in LiNbO3. However, there are several promising techniques in SBN based
on ferroelectric domain reversal. A fixed grating has been observed in SBN:75 following
the application of a depoling electric field along the c axis. This external applied field was
believed to convert the photorefractive space charge field into a permanent modulation of
the spontaneous polarization. As will be described later, ferroelectric domains
automatically orient under the photorefractive space charge even in the absence of an

applied field. The presence of this dynamic domain grating has been unappreciated in the

early work on photorefractives, despite the fundamental theoretical and practical

implications.

References for chapter one

[1] A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein,
K. Nassau, Appl. Phys. Lett. 9, 72-74 (1966).

[2] F. S. Chen, J. T. LaMacchia, D. B. Fraser, Appl. Phys. Lett. 13, 223-225 (1968).

[3] N. V. Kukhtaerev, M. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetsky,
Ferroelectrics 22, 949 (1979).

[4] J. PF. Nye, Physical Properties of Crystals Their Representations by Tensors and
Matrices. (Oxford University Press, Oxford, 1964).

[5] B. I. Sturman, V. M. Fridkin, The Photovoltaic and Photorefractive Effects in
Noncentrosymmetric Materials. G. W. Taylor, Ed., Ferroelectricity and Related

Phenomena (Gordon and Breach, Philadelphia, 1992), vol. 8.

[6] P. Yeh, Introduction of Photorefractive Nonlinear Optics., Wiley Series in Pure and

Applied Optics (Wiley, New York, 1993).

[7] M. Segev, A. Kewitsch, A. Yariv, G. Rakuljic, Appl. Phys. Lett. 62, 907-909
(1993).

[8] H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

[9] H. M. Smith, Ed., Holographic Recording Materials , vol. 20 (Springer-Verlag,
Berlin, 1977).

[10] J. J. Amodei, D. L. Staebler, Appl. Phys. Lett. 18, 540-542 (1971).

[11] R. A. Rupp, Appl. Phys. B 41, 153 (1986).

[12] E. Kratzig, R. Orlowski, Appl. Phys. 15, 133 (1978).

[13] S. W. McCahon, D. Rytz, G. C. Valley, M. B. Klein, B. A. Wechsler, Appl. Opt.
28, 1967-1969 (1989).

[14] R. Miiller, L. Arizmendi, M. Carrascosa, J. M. Cabrera, Appl. Phys. Lett. 60, 3212-

3214 (1992).

[15] W. Bollmann, H. J. Stohr, Phys. Status Solidi (a) 39, 477 (1977).

[16] D. L. Staebler, J. J. Amodei, Ferroelectrics 3, 107-113 (1972).

[17] R. Matull, R. A. Rupp, J. Phys. D: Appl. Phys. 21, 1556-1565 (1988).

[18] H. Vormann, G. Weber, S. Kapphan, E. Kratzig, Solid State Commun. 40, 543

(1981).

[19] D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, Appl. Phys. Lett. 26, 182-
184 (1975).

[20] R. Miiller, M. T. Santos, L. Arizmendi, J. M. Cabrera, J. Phys. D: Appl. Phys. 27,
241-246 (1994).

[21] V. Leyva, G. A. Rakuljic, B. O’Conner, Appl. Phys. Lett. 65, 1079-1081 (1994).
[22] V. M. Rudyak, Soviet Physics Uspekhi 13, 461-479 (1971).

[23] V. I. Kovalevich, L. A. Shuvalov, T. R. Volk, Phys. Stat. Sol. (a) 45, 249-252
(1978).

[24] F. Micheron, J. C. Trotier, Ferroelectrics 8, 441-442 (1974).

[25] F. Micheron, G. Bismuth, Appl. Phys. Lett. 20, 79-81 (1972).

[26] F. Micheron, G. Bismuth, Appl. Phys. Lett. 23, 71-72 (1973).

[27] J. B. Thaxter, M. Kestigian, Appl. Opt. 13, 913-924 (1974).

[28] Y. Qiao, S. Orlov, D. Psaltis, R. R. Neurgaonkar, Opt. Lett. 18, 1004-1006 (1993).

[29] R. S. Cudney, J. Fousek, M. Zgonik, P. Giinter, M. H. Garrett, D. Rytz, Appl.
Phys. Lett. 63, 3399-3401 (1993).

[30] W. D. Smith, C. E. Land, Appl. Phys. Lett. 20, 169-171 (1972).
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[32] F. Kahmann, R. Matull, R. A. Rupp, J. Seglins, Europhys. Lett. 13, 405 (1990).

[33] M. Horowitz, A. Bekker, B. Fischer, Appl. Phys. Lett. 62, 2619-2621 (1993).

[34] M. Horowitz, A. Bekker, B. Fischer, Opt. Lett. 18, 1964-1966 (1993).

Chapter Two

Ferroelectric Materials

2.1 Introduction

Fixing has been most thoroughly studied in three common photorefractive ferroelectrics:
LiNbO3, SBN:75 and BaTiO3. Samples doped to maximize the density of ionized donors
exhibit dramatic photorefractive responses. In addition, they display a broad range of
optical and ferroelectric properties, which is responsible for the rich diversity of fixing
phenomena reported. The two main classes of fixing are based on (1) the drift of hydrogen
impurities and (2) the switching of the spontaneous polarization. Some defining material

properties of relevance to these fixing techniques will be highlighted in this chapter.
2.2 Material Properties of Photoferroelectrics

LiNbO3 is the primary material displaying the ionic drift thermal fixing mechanism.
Photorefractive charge transport in lithium niobate is dominated by the photovoltaic effect
[1]. Photovoltaic fields can exceed 100 kVcm"!. Lithium niobate possesses an extremely

large coercive field (3 10° V cm!) [2] and spontaneous polarization (70 wC cm-) [3] . It

exhibits a second order ferroelectric-paraelectric phase transition at the Curie temperature of
1200 °C, so the domain structure is relatively stable even at elevated temperatures.
Therefore, the low frequency dielectric constant is relatively small at room temperature (€33
= 32). Lithium niobate belongs to the 3m symmetry class, with two orientations of the

spontaneous polarization (+180°) [3].

The perovskite BaTiO3 belongs to the 4mm symmetry class and undergoes three successive
first order ferroelectric phase transitions at 120, 5 and -90 °C. It possesses 4 possible
orientations of polarization ( +90°, +180°) and a strong electrooptic effect (r42 = 1600 pm
V-!) at room temperature. In contrast to LiNbOs, it has a large low frequency dielectric

constant, €33 = 3600 [3].

SBN was first identified as a ferroelectric by Francombe [4]. It belongs to the 4mm
symmetry class but exhibits a disordered tungsten bronze structure [5]. SBN:75 is highly
electrooptic (133 ~ 1400 pm V-!) with only two allowed spontaneous polarization directions
( 180°). The remnant polarization is 15 to 25 uC cm?. The large low frequency
dielectric constant (€33 = 3400) reduces the limiting space charge field (equation 1.10), so
the net photorefractive space charge field is typically less than 1 kV cm™!. The optical
quality of SBN generally suffers from growth striations, while LiNbO3 and BaTiO;

possess relatively good optical quality.

Nb5+, Tat, Tit+

Figure 2.1: Octahedral cage surrounding the transition metal ion, a characteristic unit of
ferroelectric oxides. The spatially varying space charge field repolarizes the lattice,
introducing a perturbation Sp in the local polarization.

SBN:75 is a relaxor [6, 7] ferroelectric characterized by a diffuse phase transition and a low
coercive field (of the order of 1 kV cm:!). Microdomains of polar phase exist above T,,
while below T, microdomains coalesce into macrodomains. The tungsten bronze structure
of SBN exhibits disorder in the occupancy of Ba and Sr along the five-fold and four-fold
tunnels in the structure [8]. These microscopic compositional fluctuations induce local
strain fields which cause a diffuse phase transition. That is, the dielectric constant shows a
pronounced broadening at the Curie temperature, as each small micro-polar region has a

different T,.

In the ferroelectric phase, the Nb>+, Sr2+ and Ba?+ ions occupy one of two stable positions
along the c axis (figure 2.1). In the paraelectric phase the Nb, Sr, and Ba ions are centrally
located between the two stable positions [9]. A ferroelectric domain consists of a region of
homogeneous polarization or coherent displacement of these ions. The energy barrier
separating the two equivalent polarization orientations scales with the volume of the
microscopic domain, so a typical microdomain of characteristic length on the order of 10
nm thermally fluctuates between polarization states, analogous to a superparaelectric [8].
These thermal fluctuations are believed to freeze out as competing interactions (ferroelectric
and antiferroelectric ordering) frustrate the fluctuations. One implication of these glassy

properties is an enhanced domain grating lifetime below the freezing temperature.
2.3 Photoferroelectric Phenomena

Fixing techniques encompass several photoferroelectric phenomena in addition to the
photorefractive effect [10]. Some relevant examples are photoassisted domain switching
and the photodomain effect. These phenomena occur in a special class of ferroelectrics
which are also photoconductive, hence the term photoferroelectric. These materials share
several properties in common with semiconductors (e.g. optically excitable impurity
centers), yet are distinct from semiconductors because of their relatively low values of

electronic mobility (~ 1 cm? s"! V-!) and relatively large band gaps (> 3 eV).

Photoferroelectric phenomena are dealt with theoretically by including the electronic and
lattice contributions to the thermodynamic free energy. By minimizing this free energy, the

equilibrium configuration of the system can be determined. The effect of optical

illumination on the ferroelectric properties, such as the spontaneous polarization, Curie
temperature and domain structure, can then be predicted [10]. Needless to say, the
expression for the free energy is extremely complicated [3] and is still a topic of research.
Qualitatively, optically excited electrons partially screen the Coulomb interaction among

lattice ions, from which the long range ferroelectric order originates.

Numerous observations of photoferroelectric phenomena have been reported in the
literature. For instance, the external electric fields required to pole and depole SBN
decrease under illumination [11]. Accordingly, light increases the number of Barkhausen
pulses [12, 13] observed in the ferroelectric SbSI [14] under applied fields. These pulses
are current transients measured across the crystal as domains invert, which are described in
detail in chapter six. The kinetics of domain nucleation and growth can be deduced [15]
from the shape, duration, amplitude and number of these current pulses. This
phenomenon, in which light reduces the coercive field, is called photoassisted domain

switching.

Uniform illumination alters the ferroelectric structure in the absence of external fields. For
instance, light reduces the tetragonal to cubic phase transition temperature by 2.6 °C in
BaTiO; [16]. Illumination of SbSI at a wavelength of strong optical absorption accelerates
the transition from a monodomain to a polydomain state [17]. This latter phenomenon is
called the photodomain effect and is attributed to screening of the ferroelectric polarization

by nonequilibrium, optically excited carriers.

The photorefractive effect discussed in chapter one is perhaps the most familiar example of
a photoferroelectric effect. In the case of a ferroelectric oxide, as represented in figure 2.1,
a space charge field directed along the c axis perturbs the position of the transition metal ion
within the oxygen octahedral cage, introducing a polarization perturbation 5p. The position
of this ion along the c axis contributes to the remnant polarization, so there is a close
relationship between the space charge fields and the magnitude and sign of the unit cell
polarization. This polarization perturbation alters the index of refraction by typically 10° to

10-3 through the electrooptic effect.

2.4 Summary

Fixing techniques in ferroelectrics encompass a broad range of photoferroelectric
phenomenon. Simplified models to describe the dielectric response of these materials upon
illumination have been developed. These models assist in a qualitative understanding of
domain hysteresis fixing, but quantitative predictions of these highly nonlinear dielectrics

are still highly speculative.

References for chapter two

[1] B. I. Sturman, V. M. Fridkin, The Photovoltaic and Photorefractive Effects in
Noncentrosymmetric Materials. G. W. Taylor, Ed., Ferroelectricity and Related
Phenomena (Gordon and Breach, Philadelphia, 1992), vol. 8.

[2] S. H. Wemple, M. Di Domenico, Appl. Phys. Lett. 12, 209 (1968).

[3] M. E. Lines, A. M. Glass, Principles and Applications of Ferroelectrics and Related
Materials. (Clarendon Press, Oxford, 1977).

[4] M. H. Francombe, Acta Cryst. 13, 131 (1960).

[5] L. Bursill, P. Lin, Philos. Mag. B 54, 157 (1986).

[6] A. S. Bhalla, R. Guo, L. E. Cross, G. Burns, F. H. Dacol, R. R. Neurgaonkar, J.
Appl. Phys. 71, 5591 (1992).

[7] W. H. Huang, D. Viehland, R. R. Neurgaonkar, J. Appl. Phys. 76, 490-496 (1994).

[8] L. E. Cross, Ferroelectrics 76, 241 (1987).

[9] P. B. Jamieson, S. C. Abrahams, J. L. Bernstein, J. Chem. Phys. 48, 5048 (1968).

[10] V. M. Fridkin, Photoferroelectrics. M. Cardona, P. Fuldeand H.-J. Queisser, Eds.,

Springer Series in Solid State Sciences (Springer-Verlag, Berlin, 1979), vol. 9.

[11] F. Kahmann, R. Pankrath, R. A. Rupp, Opt. Comm. 107, 6-10 (1994).

[12] H. Barkhausen, Physikalische Zeitschrift 20, 401 (1919).

[13] V. M. Rudyak, Soviet Physics Uspekhi 13, 461-479 (1971).

[14] A. A. Bogomolov, V. V. Ivanov, V. M. Rudyak, Sov. Phys.-Cryst. 14, 894-896
(1970).

[15] A. G. Chynoweth, Phys. Rev. 110, 1316-1332 (1958).

[16] T. R. Bolk, A. A. Grekov, N. A. Kosonogov, A. I. Rodin, V. M. Fridkin, Sov.
Phys.-Cryst. 16, 198-200 (1971).

[17] V. M. Fridkin, I. I. Groshik, V. A. Lakhovizkaya, M. P. Mikhailov, V. N. Nosov,
Appl. Phys. Lett. 10, 354-356 (1967).

Chapter Three
Ferroelectric Domain Gratings

3.1 Introduction

Photoferroelectric phenomena encompass a broad range of effects which induce structural
and index of refraction changes in these materials. These effects are difficult to model
quantitatively from first principles, so realistic simplifying approximations are adopted. In
the context of optical diffraction from domain holograms, the linear electrooptic
contribution to the index change is dominant and the domains are treated as ideal dipoles of
homogeneous polarization. A simple theoretical framework for the optical properties of

domain gratings will be developed based on these two approximations.
3.2. Theory of the Electrooptic Effect

Photorefractive diffraction gratings are produced by a modulated space charge field that
alters the index of refraction through the electrooptic effect. The index change created by
the total polarization field may be expressed as a linear electrooptic effect or equivalently as
a quadratic electrooptic effect "biased" by the remnant polarization [1]. The tensor relation

for the Fourier component of the index change at the grating vector Kg is:

a(t (kg) = 2 gijxi(Pok AP(Kg) ; 3.1
n2 ij

where gjj,1 is the quadratic electrooptic tensor, Po is the average remnant polarization and
AP (Kg) is the Fourier component of the static (@ = 0) polarization modulation at ky. This
polarization modulation may be generated by a static space charge field AE(Kg), for

instance. The modulation of the polarization AP (k,) is expressed as the Fourier transform

of the spatial variation of polarization AP (r):

AP(k) = —1 AP(r) ei k¥ dr , 3.2
aia

where the polarization perturbation includes three contributions:

AP(r) _ Apoptical excitation(y.) + Apcharge transport.) + Apiertoelectric distortion) . 3.3
These three terms may be estimated analytically from:

AP(r) => | (Rj+r'- r)Ap;(Ri+r'-r) dr’ , 3.4

where the sum is over the N lattice sites i, r' is the position relative to the address of the ith

lattice site Rj, and pj; is the total charge density of the atoms associated with the lattice site i.

The somewhat arbitrary decomposition of equation 3.3 makes three contributions to index
changes explicit. The first term on the right represents the change in dipole moment of
donors and acceptors upon optical excitation. The second term represents the change in
polarization arising from both electron transport (due to drift, diffusion, pyroelectric or
photovoltaic mechanisms producing photorefractive space charge fields) and ionic transport

(typically due to drift induced by electronic space charge fields). The charges undergo

transport between distant lattice sites (a distance on the order of the grating spacing,
typically ~ zm). The third term represents the change in lattice polarization arising from the
displacement of ions. In this case the center of charge is displaced only slightly within the
unit cell from the unperturbed position (typically << 1 A). The electric dipole generated by
the first two terms polarizes the lattice through this third term. Note that the polarization is
understood to be spatially averaged on a scale much larger than a unit cell, yet smaller than

the optical wavelength of interest.

The standard electrooptic effect described by equation 3.1 treats only that polarization
perturbation corresponding to the third term of equation 3.3. That is, measurements of the
electrooptic effect are performed by applying an external electric field to the crystal and
measuring the resulting net index change. The external electric field will modify the
crystalline polarization by slightly displacing the positions of ions. As long as negligible
current from the external source flows through the crystal, the drift of electrons across the
crystal will also be negligible. Therefore, the electrooptic effect will not include the direct
index changes resulting from electron and ion optical excitation or drift; namely, the first

two terms of equation 3.3.

The electrooptic effect describes the interaction of a low frequency electric field E° and an

optical field E®, generating an optical polarization AP®-° that is shifted in phase relative to

the incident optical wave:
Ap™© = yO2poB2, 3.5

The proportionality constant is a nonlinear susceptibility y°° defined as

gq Poe

3 . 3.6
JE OE [pe RP

To determine the coefficient y°-° responsible for the linear electrooptic effect, the changes
AP due to optical frequency electric fields E® must be considered. Electron clouds in the
vicinity of ion cores will feel very strong attractive Coulombic forces with the nucleus.
These electron orbitals will be weakly perturbed by the optical field, which is typically
much weaker than the ion core attraction. Electrons which spend a greater fraction of their
time far from the core and experience a screened core potential will exhibit stronger

polarizability at optical frequencies.

Bonds can be separated into two archetypes: purely covalent or ionic [2]. The character of
bonds between elements with different electronegativities falls between these two extremes.
The stronger the covalent character of the bond, the greater fraction of the time the electron
will be found between the ion cores where the core attraction is significantly weaker.
Therefore, covalent bonding orbitals are more highly polarizable than ionic bonding orbitals
under moderate optical fields. A familiar example is that purely covalent crystals (i. e.,

diamond) have larger indices of refraction than ionic crystals (i. e., NaCl).

The nonlinear optical properties of the ferroelectric niobates arise from the distorted NbO¢
octahedron [1]. The contributions to the linear and nonlinear optical polarizabilities can be
determined by decomposing the crystal bond-by-bond into individual dipole radiators. The
highly ionic Li-O bond susceptibility [3] makes a negligible contribution, since the valence
electrons are tightly bound to the oxygen. For LiNbO; and SBN, it is a good
approximation to consider only the Nb-O bond contribution to the nonlinear polarizability
(similarly for BaTi03 and LiTaQ3, the dominant contribution to the susceptibility arises

from the covalent Ti-O and Ta-O bonds [3]). This contribution to the electrooptic effect can

be estimated by considering the dependence of the linear optical susceptibility of the Nb-O

bond on rotation and stretch [4, 5].

Pr (t)=2 Be €)

(a) inversion symmetry

Pel}= (0 +4? )e €)+ (e -¥ }e? €)

PitjJ=aMe(t)+a%e?(t) pr t)=yPVe t)- Ye? 6)

(b) no inversion symmetry

Figure 3.1: Microscopic model of the origin of nonlinear optical susceptibility in typical
ferroelectric oxides.

When the crystal is in the paraelectric phase, the niobium ion is equidistant from the oxygen
atoms along the c axis (figure 3.1). The nonlinear polarizability from these two bonds is

the vector sum of the individual bonds. As indicated in figure 3.1(a), despite the individual

bond polarizability dependence on both even and odd powers of the optical field e(t), only
the even order terms to the otal nonlinear polarizability p, remain. In contrast, in the
ferroelectric phase (figure 3.1(b)), the Nb ionic displacement breaks the symmetry of the
two bonds along the c axis, so the total polarizability depends on both even and odd
powers of the electric field. The dependence of p; on the square of the field e2(t) is

manifest in the linear electrooptic effect and second harmonic generation.

From this microscopic viewpoint, the displacement of the Nb ion along the c axis changes
the polarization AP (Kg) and induces an index change through the electrooptic effect. Note
that the distortion of this Nb-O bond length also produces the ferroelectric polarization in
materials such as LINbO3 and SBN. Thus, the ferroelectric and nonlinear optical properties
are intimately related. The decomposition of equation 3.3 on the atomic bonding level links
the third term, describing ferroelectric distortion, to the electrooptic effect in these
materials. This association is fundamental to understanding the relationship between
nonlinear optical and ferroelectric properties, and as will be discussed later, the distinction

between dynamic and remnant ferroelectric domain gratings.

The origins of index gratings induced by internal space charge fields are microscopically
more complicated than the phenomenon described by the electrooptic effect, in which
external fields rather than space charge densities modulate the index of refraction. Indeed,
from equation 3.3 there are two other contributions. The first term represents the change of
dipole of the ions upon optical excitation, the excited state dipole moment being an
example. This makes a negligible contribution to the steady state index change because the
density of electrons in the conduction band at any instant in time is very small relative to the
cumulative charge redistribution that occurs upon the generation of steady state
photorefractive space charge fields. This is true at moderate optical intensities (< kW cmr2)

in the highly resistive ferroelectric oxides described here.

The second term describes the change in polarization due to the optical redistribution of
electrons among deep traps. By definition, since these photorefractive traps are deep, the
electrons are strongly bound to their host atom, and as a consequence we assume that these
trapping sites make a small contribution to the nonlinear susceptibility relative to the highly
covalent Nb-O bonds. Note that in a strained, centrosymmetric crystal this contribution can
play an important role through a Jahn-Teller distortion [6]. However, the materials under
consideration possess a strong linear electrooptic effect, so the second term is neglected.
The third term also has a contribution from the motion of ions (such as protons in the case
of LiNbO; thermal fixing) to new lattice sites by drift induced by electronic space charge

fields. These ions compensate the electronic grating to reduce the net internal field.

The first two contributions of equation 3.3 polarize their surroundings and perturb the
lattice polarization through the third term to produce an index change. In effect, the
photogenerated space charge fields and ionic space charge fields can be treated as external
fields, which modify the index of refraction through the standard electrooptic effect. This
justifies a description of the index change arising from a modulation of the spontaneous

polarization in terms of the linear electrooptic effect.

3.3 Ferroelectric Domain Structure

The ferroelectric domain structure is intimately linked to the nonlinear optical properties of
the material. From the microscopic description of the electrooptic effect, it is apparent that
an electrooptic grating corresponds to a modulation of the ferroelectric polarization. As
described latter, this modulation may correspond not only to small perturbations of the Nb-

O bond length, but to actual switching between the two stable polarization orientations.

From electrostatic considerations, the potential and field of a polarized object is equivalent

to that field produced by a bound volume charge density pp:

and a bound surface charge density op

Op =P-n, 3.8

where P(r) is the macroscopic polarization averaged over a volume much larger than a unit
cell, and Ni is the unit normal vector to the surface of the crystal. The bound charge can not
be removed from a domain interface (where P changes); it is permanently attached to the
ions responsible for the spontaneous polarization. In contrast to this bound charge, free
charge can be optically excited and redistributed, producing photorefractive space charge
fields, for instance. Note that this free charge spends the great majority of its time trapped,
since the recombination time (~ 10 ps) in oxide ferroelectrics such as SBN is extremely
short (in this sense, the term free charge is misleading). If sufficient mobile charge is
available, the bound charge is compensated by an equal and opposite free charge density p¢
and surface charge density of. Compensating charge is electrostatically trapped at domain
walls for which the divergence of the polarization P is non-zero (i.e. head-to-head rather

than antiparallel).

Compensating electric charge plays an important role in minimizing depolarization fields at
domain walls in ferroelectrics, analogous to the closure requirements of ferromagnetic
domains. To analyze charge compensation at domain walls consider the simplest case of a
normal ferroelectric phase in which there are only two allowed polarization directions (0

and 180 degrees to the c axis). The polarization within an individual domain is assumed to

be homogeneous. Figure 3.2(a) illustrates an idealized volume grating configuration in
which the grating vector is parallel to the c axis. The free carrier density required to
compensate the non-zero divergence of the polarization for a domain grating composed of
head-to-head domain walls is given by

Neomp = 2 3.9
q Ag

and the carrier density required to produce a field equal to the coercive field is

2 1 € Ecoercive ; 3.10
q Ag

Neoercive =

Ag is the grating period, q is the charge of an electron, and € is the dielectric permittivity.
For grating periods of 1 micron, P, equal to 25 UC cm and Epoercive equal to 1 kV cm!
(typical for SBN:75), the compensating charge density required on the domain walls for
bipolar modulation (AP = 2 Ps) is Ncomp ~ 2.5x10!8 cmr3, and the coercive charge density
is 10!7 cm3. Therefore, significant charge densities are required to completely screen
depolarization fields at head-to-head domain walls and canted domain walls making small
angles with the c axis. Thermally ionized shallow acceptors and illumination in the intrinsic
or extrinsic bands contribute to these screening fields. In general, the average modulation
AP may be much smaller than the spontaneous polarization, which reduces the

compensating charge requirements described by equation 3.9.

(a) (b)

Figure 3.2: (a) Head-to-head and (b) canted domain wall configuration.

The divergence of the polarization at a domain wall is reduced by a factor of sin 6 for
canted domain walls, where 8 is the angle between the direction of spontaneous
polarization and the domain interface (figure 3.2(b)). Therefore, for 8 = 0, no charge is
required at the wall for perfect bipolar modulation of the spontaneous polarization. These
antiparallel walls are expected to be relatively stable. For this domain microstructure,
charge compensation occurs only at the crystal c faces. These electrostatic considerations
influence the shape and size of individual ferroelectric domains. Domains in SBN are
dagger-like in shape and oriented along the c axis, which reduces depolarization fields at
the head-to-head domain interface. A grating composed of these dagger-like domains is

depicted graphically in figure 3.3.

C axis

Reversed Macrodomain

Hologram Writing Beams Reversed Microdomains

Figure 3.3. Model of glassy domain grating consisting of inverted microdomains and
macrodomains in regions of negative space charge field.

| l=x=a
Pe | y 2=y=a
Cc 3=7=C

Figure 3.4: Shaded box indicates reflection plane through which spontaneous polarization
is reversed upon domain inversion.

3.4 Transformation of Nonlinear Optical Tensor Under Domain Inversion

The signs of the second order nonlinear optical tensor elements dj, depend on the
orientation of the ferroelectric domain. By spatially modulating the ferroelectric
polarization, the nonlinear coefficients are also modulated. In general, the inversion of a
third rank tensor is not a trivial undertaking. Therefore, for illustrative purposes, consider
the relevant example of SBN with nonzero elements d33 = d333, d3; = 311 and dy5 =
d31 = 413. The orientation of the crystalline axes is illustrated in figure 3.4. The
nonlinear media is assumed to be lossless in the wavelength region of interest to satisfy
Kleinman’s conjecture [7] . Before inversion, the induced optical polarizability at the

second harmonic is:

P3 = d333 E3 E3 + d3y1 E; Ey 3.11

Py = dy13 Ey E3 +31 E3 Ey . 3.12

If the ferroelectric domain is inverted, the induced optical polarization and field along the c
axis changes sign, because these quantities are now referenced to a new coordinate system:

P3 —-P3, P} >P; , E3 >-E3, E; ~E;. Substituting this change of variables into 3.11 and

3.12, the optical polarizability at the second harmonic for an inverted domain is:
-P3 = d433 (-E3)(-E3) + d3,; Ey, E: 3.13
P, =dji3 E1 (-Es) +413; (-Es)Ei 3.14

The primes on the nonlinear coefficients dj;, indicate that the domain is inverted.

Rearranging coefficients, equations 3.11 and 3.12 become:
P3 = (-dy33) E3 Bs + (51) 1 Ey 3.15
Pi =(-dyy3)E1 Es +(dy3))E3 Br 3.16

By comparing equations 3.11, 3.15 and 3.12 and 3.16, it is apparent that domain inversion
changes the sign of all three nonlinear optical coefficients d33, d3,, djs. This fact will be
used in chapter seven to select the proper crystalline orientation for optimal second

harmonic generation.
3.5 Summary
The modulation of the spontaneous polarization in domain gratings modulates both the

index of refraction through the linear electrooptic effect and the nonlinear optical

susceptibility. This provides two methods of studying and utilizing this optically induced

microstructure: optical diffraction and quasi-phase matched second harmonic generation.

This first application, ferroelectric hysteresis fixing, is described in chapter four.

References for chapter three

[1] M. DiDomenico Jr., S. H. Wemple, J. Appl. Phys. 40, 720 (1969).

[2] J. C. Phillips, Bonds and Bands in Semiconductors. A. M. Alperand, A. S. Nowick,
Eds., Materials Science and Technology (Academic Press, New York, 1973).

[3] C. R. Jeggo, G. D. Boyd, J. Appl. Phys. 41, 2741-2743 (1970).

[4] C. Shih, A. Yariv, Phys. Rev. Lett. 44, 281-284 (1980).

[5] C. Shih, A. Yariv, J. Phys. C: Solid State Phys. 15, 825-846 (1982).

[6] R. Hofmeister, A. Yariv, S. Yagi, A. Agranat, Phys. Rev. Lett. 69, 1459-1462
(1992).

[7] D. A. Kleinman, Phys. Rev. 126, 1977 (1962).

Chapter Four

Ferroelectric Hysteresis Fixing of Volume
Holograms

4.1 Introduction

This chapter will focus on hologram fixing based on ferroelectric domain reversal. This
fixing process is the manifestation of hysteresis in the electrooptic effect near the
ferroelectric/paraelectric phase transition. This is demonstrated in SBN:75, at a temperature
20° C below the phase transition. First, high intensity fixing using optical heating to assist
the domain switching process is described [1, 2]. Next, short exposure fixing using a dark
development stage is demonstrated [3]. The lifetimes of these remnant domain gratings are

found to be highly temperature dependent.
4.2 High Intensity Fixing

The crystals used in this experiment are Cr doped SBN:75 single crystals, 6x6x6 mm,
grown at the Rockwell International Science Center. Sample A has facets cut normal to the
principal axes, whereas the axes of sample B are rotated 45° about the y axis in the x-c
plane. The crystals are poled by application of a dc electric field of 5 kV cm:! while
uniformly heated to 80 °C in a high dielectric-strength oil bath. With the field on, the

samples are cooled at a rate of 0.5 °C min! to 25 °C. The poling field is then removed.

Note that the 45° cut of sample B makes the task of poling difficult; hence, the diffraction

efficiencies for this sample are very small.

The experimental setup is depicted in Figure 4.1. During the writing process, A; and A»
are the signal and reference beams, respectively. During the coherent reconstruction with
A2, By, and By» are the reconstructed signal and remnant reference beams, respectively.
Since the holograms are sufficiently weak, the diffraction efficiency is defined as B,/B3, to
include reflection and absorption losses (absorption coefficient 1.9 cm-!). The
transmission holograms are written in the image plane at 488 nm, with equal angles of

incidence (8 = 16°). The experiments are performed at or near room temperature (25 - 30

°C) in the ferroelectric phase.

Photodetector 633 Band-Pass Filter

Image Shutter

Figure 4.1: Experimental setup for domain fixing.

The fixing process occurs simultaneously with the dynamic hologram writing process; no
thermal development cycle or electrical fixing pulse is required. However, increases in
incident intensity, temperature (while well below T, = 56 °C), or write time all tend to

enhance the diffraction efficiency of the fixed hologram. The hologram is composed of a

dynamic component, with a characteristically fast response time, and a fixed component,
with a lifetime that is several orders of magnitude larger than the dynamic grating. This
dependence of relaxation rate on exposure level was first reported by Thaxter and Kestigian
[4], who measured a fixed component with a lifetime of several minutes. A significant
fixed component is obtained simply by writing holograms with an incident intensity greater
than 1 W cm? for 2 minutes or more, an exposure in excess of 100 J cm-2. For a total
intensity of signal and reference beams of 100 mW cnr”, only a significant dynamic grating
is present. By increasing the intensity to 4 W cm”, a significant remnant hologram is

recorded automatically.

3 UJ T LJ
oS
2.51 4
> °
= 0
@ 25 ° 4
oO mle)
ea] 1 5 r 5°? 4

rele)
5 0°
ret 1 L of? 4
_—
OQ °
S °
oo
oe 0.5 r 5000000" 7
a °
@) goo ' !

0 05 I %t5 2
Time (hours)

Figure 4.2: Diffraction efficiency during high intensity writing process.

The diffraction efficiency of the remnant hologram grows by a factor of 25 from the initial
value of 0.1 % during the writing process of 2 hours (figure 4.2). While it is exceedingly

difficult to maintain fringe stability during these long write times, the diffraction efficiency

increases for approximately 20 hours, until saturation occurs at a maximum diffraction
efficiency of 3%. This long exposure optically fatigues the crystal by increasing the
volume of charge compensated microdomains. This reduces the viscosity of the dipoles so
domains will subsequently align more rapidly and readily under the influence of the space
charge field. Response times of hours are uncharacteristically slow for photorefractive

phenomenon and suggest the presence of a slow domain relaxation phenomenon.

The monatonic increase in fixed diffraction efficiency occurs because domains continuously
orient to produce bound charge in opposition to the electronic space charge. The net
internal field, arising from sum of the free and bound charge, is then reduced. Since the
spatially modulated charge density decreases, charge transport by diffusion is also reduced.
Diffusion limits the build-up of space charge for grating periods larger than the Debye

screening length, given by:

fy =2E,/ kel 4.1
q VoNe

The space charge field in figure 1.3 attains a maximum for a grating period equal to 4.

For SBN:75 4 is of the order of 1 um. Domain reversal allows the electronic grating to
exceed the diffusion limit and reach the limiting space charge value at grating periods
greater than the Debye length, given by the curve labeled E, in figure 1.3. Nevertheless,
the typical fixed diffraction efficiency is a modest 3%, because a large fraction of domain
reversal is random, which reduces the linear electrooptic coefficients responsible for

diffraction.

The hologram written at high exposure levels is automatically fixed during the writing
stage. Prior to the curves of figure 4.3, a hologram is recorded for 100 seconds at either

0.1, 1, 2, or 3 W cm total incident intensity. Upon readout with the same intensity, the

strength of the remnant hologram is found to increase in proportion to the incident intensity

(figure 4.3).

0.8 ? qT T J

S : :

3 0.67. ;

aS) r

m 0.4t

fom)

Ao) -

SI 0.2+

ra Li 1.0 ei meri
O 0.1 i l 1
100 120 140 160 180 200

Time (sec)

Figure 4.3: Revealing of remnant domain grating following 100 second exposure for
writing and readout intensities 0.1, 1, 2, and 3 W cnr?.

N io.)
go"
it

Diffraction Efficiency (%)

=)
om)
wf
al

Time (days)

Figure 4.4: Lifetime of remnant domain grating during continuous readout with a high-
intensity beam at 488 nm (I= 1 W cm).

Figure 4.4 illustrates the long term decay of a fixed hologram written at room temperature
with a total intensity of 4 W cmr? for 1 hour. Extrapolating this curve, the hologram is
expected to survive at least several hundred days during continuous readout at 25 °C. By
cooling the crystal to below 20 °C, the domain grating is permanently frozen-in. The
lifetime is enhanced for readout with a weak plane wave, which minimizes undesirable
domain reversal due to the accumulation of space charge on the periphery of a focused
reference beam and local heating which reduces the polarization viscosity. A fixed page
written for 15 minutes has been monitored intermittently for over 1 month without any

degradation in image quality.

These high intensity recordings are performed in initially poled crystals. Following a
continuous optical exposure for two days, the macroscopic spontaneous polarization as
measured from the electrooptic coefficient decreased by a factor of 10. This severe

- depolarization results from optical exposures on the order of 10° J cm-2.

Diffraction Efficiency
(arbitrary units)

0 10 20 30 40
Time (sec)

Figure 4.5: Recording of domain grating in canted wall configuration; (a) poled crystal (b)
optically depoled crystal following exposure.

Domain gratings may also be optically recorded in the canted wall configuration discussed
in reference to figure 3.3(b). This is demonstrated by recording a domain grating at an
angle of 8 = 7 degrees to the c axis, so the charge compensation requirements at domain
walls are significantly reduced. The total diffraction efficiency while writing the grating is
illustrated in figure 4.5. Curve (a) illustrates the diffraction efficiency while the first
grating was written. Following this recording, the half-wave voltage, which is inversely
proportional to the effective linear electrooptic coefficient rp, increased by a factor of 3 in
the region where the hologram was recorded. Thus, the macroscopic polarization in the
region of the hologram has decreased by 3. This is the reason why the diffraction
efficiency of curve (a) in figure 4.5 decreases after 4 seconds of exposure. A second
hologram was subsequently recorded in the same volume and the diffraction efficiency (b)

decreased dramatically because of the depolarization of the crystal.

In this canted domain grating configuration, with ky nearly perpendicular to c, the crystal
rapidly depolarizes in the region of illumination. More rapid depolarization is indeed
expected because of the reduced compensating charge requirements for canted domain
gratings. By re-poling the crystal with 3 kV cm:! for approximately 1 minute, the crystal is
refreshed and a third hologram recorded in the same volume displays the same diffraction
efficiency as curve (a). In this geometry the space charge field is nearly perpendicular to
the c axis, so the photorefractive space charge fields do not play a central role in domain
reversal. These gratings are produced by a combination of optical heating and

photoassisted domain reversal.

4.3. Short Exposure Fixing

While high intensities enhance the fixing efficiency, low intensities and short optical

exposures can also produce fixed holograms of significant diffraction efficiency if the

recording stage is followed by a dark development stage. The longer the photorefractive
space charge field is applied in the dark, the greater the ferroelectric hysteresis or memory.
The short exposure hologram fixing technique isolates the role of the space charge field
from thermo-optic effects and random depolarization, which play a dominant role in high
intensity fixing. Obviously, a fixing technique that significantly depoles the crystal during
recording is of little value if multiple holograms are to be stored, because the reduction of
the average electrooptic coefficient leads to weak diffraction. The short exposure technique
reduces this random depolarization. Nevertheless, the primary utility of this technique is
that it provides a means to characterize the domain fixing technique. The dependence of the

domain modulation on the photorefractive space charge field alone can be determined.

The amplitude of this periodic space charge field is controlled by interfering two plane
waves, each of 50 mW cm intensity, for a variable amount of time (1 to 5 seconds). The
space charge field amplitude increases with exposure until saturation occurs after 5
seconds. The short exposure is followed by a dark development stage of typically minutes,
during which time the domains stabilize in their reversed orientation. The development
stage is effective because the space charge fields persist in the dark due to the low dark
conductivity, which is verified in separate experiments by monitoring the decay of uniform
pyroelectric fields in the dark. Following a dark development stage of 10 minutes, the
dynamic polarization grating is erased with a 1 W cm” non-Bragg matched plane wave
(488 or 515 nm) and the remnant diffraction efficiency is monitored with a low intensity,
Bragg matched He-Ne probe. This remnant diffraction corresponds to the fixed or remnant
domain hologram.

The amplitude of the spatially periodic electric field can be estimated from the measured
diffraction efficiency of the grating during Bragg-matched reconstruction. In the symmetric

transmission grating configuration with the grating vector k, parallel to the c axis, the

amplitude of the net space charge field, for small diffraction efficiencies and small half

angle of the writing beams, is from equation 1.20:

2 e& Li2V 1 (kg)
7 L n3 133 ,

4.2

Esc(Kg) =

where k, and ky are the wave vectors of the signal and reference plane waves in the
medium, a is the absorption at the reconstruction wavelength 4 in vacuum, L the
interaction length between the readout beam and the volume grating, n, the unperturbed
index of refraction of the medium, r33 the dominant electrooptic coefficient for SBN:75 and
YN the Bragg matched diffraction efficiency defined as I,/I), where I, and I, are the
intensities of the diffracted and incident He-Ne beams, respectively. The HeNe is used to
readout the hologram rather than the Ar-ion because the SBN sample exhibits negligible

absorption at red wavelengths, reducing the buildup of fanning noise.

The remnant polarization modulation exhibits a linear dependence on the initial field
modulation (figure 4.6(a)). This is expected since the photogenerated field lies in the small
field, linear region of the hysteresis loop. The polarization modulation depicted in figure
4.6 is expressed in terms of an equivalent bound charge field to permit a quantitative
comparison to the photorefractive space charge field. The fixing efficiency, defined as the
ratio of the bound charge field to the initial space charge field, is typically 1%. These light
induced domain patterns are produced in the absence of external electric fields or thermal
effects, thus providing conclusive evidence that the photogenerated space charge fields

alone induce the periodicity in the ferroelectric polarization.

(a) (b)
T 25 t 7 r

ne
aS

vs)

ad
nN

2°
an

Bound Charge Field (V cm’!)

Bound Charge Field (V cm”)

0 1 L L 1 1 I
0 15 30 45 60 0 200 400 600 800
Initial Space Charge Field (V cm’) Dark Development Time (min)

Figure 4.6: Short exposure fixing (a) Dependence of remnant polarization grating
(expressed in terms of an equivalent bound charge field) on initial space charge field
(b) Dependence of remnant polarization grating on dark development time.

The hysteresis is further enhanced by increasing the time duration of the dark development
stage and/or applying an external depoling field during the development stage. For
instance, a one second depoling field pulse of 3 kV cm”! applied after the optical exposure
increases the fixed diffraction efficiency by a factor of two. Increasing the dark
development time from 2 to 200 minutes increases the diffraction efficiency of fixed
holograms by more than a factor of 100, as illustrated in figure 4.6(b). For this set of
experimental conditions (grating period = 4 um, T = 25 °C), the remnant polarization
modulation saturates at approximately 0.02 % of the spontaneous polarization. These
experiments confirm a general characteristic of ferroelectrics: the induced polarization
change increases in proportion to the product of the depoling space charge field and the
time the field is applied (until the onset of saturation) according to the empirical

relationship:

AP = AP}! - exp al 4.3

A thermodynamic study of glassy domain reversal offers some further interesting
conclusions. Although it is conventional to apply Gibb's free energy equilibrium
arguments to evaluate the stability of domains, the stable glassy domain configuration may
not be the equilibrium configuration. In fact, a non-equilibrium domain grating may be
stable below a freezing temperature where the glassy polarization phase is inherently non-
ergodic. At these temperatures the viscosity of the dipoles increases dramatically and
freezes-in the polarization modulation. Therefore, one experimental approach to evaluate
the thermodynamics of this process is to study the temperature dependence. Figure 4.7(a)
illustrates the thermally activated erasure of the domain grating at 35 °C. The temporal
response of the decay follows a t-8 law (as is common in spin glasses), were B is a
measure of the polarization viscosity [5]. The decay of the polarization grating was
monitored continuously for 104 to as long as 7x105 seconds. The lifetime is defined as the
time in which the polarization modulation amplitude decays to 10% of its value at t = 1
second, to avoid the unphysical singularity of the power law at t= 0. The lifetime of the
grating at 20 °C extends well beyond measurable times, yet the onset of freezing was
readily apparent from the absence of grating decay after only 4000 seconds. By fitting at
curve to the decay, the lifetime is estimated to be 1015 seconds. The uncertainty of this
value, while very large, does not influence the fit parameters significantly because of the
divergence of the lifetime near the freezing temperature renders it relatively insensitive to

uncertainties in lifetime.

The hologram lifetime data is summarized in figure 4.7(b). Data indicated by triangles
correspond to measured lifetimes, the circle to an extrapolated lifetime. Each data point

corresponds to two independent yet identical experiments. The functional fit (i) is the

Arrhenius relation for the characteristic time t for thermal excitation out of a potential well
of depth E, for a system at temperature T:

tii ci
1 exp fe . 4.4

For a system that exhibits a polarization freezing temperature T;(typical of glassy
ferroelectrics), the Vogel-Fulcher law [6-8] is a more accurate empirical relationship

between the decay time (i. e. the lifetime of the remnant domain hologram) and the

temperature:
11 -E, 4.5
tT To larrny| ,

where T is greater than or equal to T;, the dipolar freezing temperature. The freezing
behavior is readily apparent for temperatures below 30 °C in figure 4.7(b), where the
inverse lifetimes asymptotically approach zero. Fit (ii) illustrates the Vogel-Fulcher fit,
with Ts = 19 °C, E, = 5 meV and T, = 1 sec, and displays excellent agreement with the
data. This indicates that polarization gratings can be permanently frozen-in at sufficiently

low temperatures.

(b)
38S oo

(ii)

1000 T! (K")

1 n 3 . ] 5 Ce SE Pa SS SO
05 4000 8000 10° 10° 10°
Time (sec) t (sec)

Polarization Modulation Amplitude
(normalized)

Figure 4.7: (a) Power law decay of remnant polarization modulation amplitude at 35 °C.
(b) Lifetime of remnant polarization modulation amplitude for different
temperatures. Curve fit (i) describes an Arrhenius and (ii) a Vogel-Fulcher thermal

activation process.

The magnitude of the remnant polarization modulation shows a temperature resonance for
recording stages conducted between 30 to 37 °C (figure 4.8). The weak response at low
temperatures is due to the large viscosity of the glassy polarization. On the other hand, at
temperatures above 35 °C, the decrease of the electrooptic coefficient and the thermal
scrambling of the domain grating conspire to reduce the diffraction efficiency. A two-stage
fixing process, in which the domain grating formation is thermally assisted at elevated
temperatures (30 to 35 °C) and then frozen-in at slightly lower temperatures (25 to 20 °C),

enhances both the grating modulation and lifetime.

0.06 T T T T T

oO
BB

Diffraction Efficiency (%)

20 30 40 50
Temperature (°C)

Figure 4.8: Resonance in the diffraction efficiency of remnant polarization grating for
holograms recorded between 30 and 37 °C.

The mechanism of optically induced domain reversal by space charge fields is an example
of the macro-to-microdomain transformation observed under external electric fields [4].
External depoling fields in the presence of illumination have reportedly produced a
microdomain state in SBN [9]. In the experiments described in this chapter, the periodic
space charge is the source of the depoling field. By cooling the crystal from above the
Curie temperature to the freezing temperature Ts, in the absence of an applied field, the
ferroelectric is in a zero-field-cooled state in which long range ferroelectric order is absent.
However, by cooling under an applied poling field, long range order can be frozen-in
below Ts. This is conventionally called the field-cooled state. Depoling electric fields E,
applied parallel to the c axis, lower the local freezing temperature through the deAlmedia-

Thouless relationship [6, 10]:

TE) = TA0) E (EP 4.6

where A is a constant defined as A= kpT/p, and p, is the freezing dipole moment. The
photogenerated space charge field is the source of this applied field. The periodic field
spatially modulates the freezing temperature. In those regions where the freezing
temperature is perturbed downward, the relaxation rate of the glassy polarization to the
unpoled state increases dramatically. Therefore, in regions where the space charge field is
parallel to the c axis, the long range order (or frozen-in polarization) will "thaw out" and

reduce the polarization locally.

A microphotograph of the domain grating recorded using high intensity fixing is illustrated
in figure 4.9. This photograph is obtained by illuminating the crystal with an
extraordinarily polarized plane wave at 488 nm. The index perturbation causes the incident
beam to waveguide along the grating planes. This produces the intensity contrast evident in
figure 4.9. This technique has been analyzed in detail by Rupp [11]. The crystal is initially
uniformly poled and in the field-cooled state (figure 4.9, left). A permanent domain grating
with a period of 10 microns is subsequently recorded (center) and then electrically erased
(right) with 3000 V cm for 10 seconds under weak illumination. The intensity maxima
correspond to grating planes of larger index relative to the minima. The SBN crystal is

negative uniaxial, so the dark fringes correspond to regions of depolarization.

Figure 4.9: Microphotometrically revealed domain grating with period of 10 microns in
SBN:75. left: poled crystal; center: domain grating; right: electrically erased
grating.

4.4 Summary

Optically induced domain reversal enables light to modulate the symmetry of a bulk crystal
over distances on the order of optical wavelengths. The role of the space charge field in the
local domain reversal is isolated by performing low intensity experiments in the absence of
optical heating or external fields. A qualitative thermodynamic description of the fixing
process based on spin glasses elucidates the kinetics of domain reversal. These kinetics are
studied in greater detail by measuring Barkhausen current noise, as discussed in the next

chapter.

References for chapter four

[1] A. S. Kewitsch, M. Segev, A. Yariv, R. R. Neurgaonkar, Opt. Lett. 18, 1262-1264
(1993).

[2] A. S. Kewitsch, A. Yariv, M. Segev, in The Photorefractive Effect D. Nolte, Ed.
(Kluwer Academic, New York, 1995).

[3] A. S. Kewitsch, M. Segev, A. Yariv, G. J. Salamo, T. W. Towe, E. J. Sharp, R. R.
Neurgaonkar, Phys. Rev. Lett. 73, 1174-1177 (1994).

[4] J. B. Thaxter, M. Kestigian, Appl. Opt. 13, 913-924 (1974).

[5] K. Binder, A. P. Young, Rev. Mod. Phys. 58, 801 (1986).

[6] D. D. Viehland, Dissertation, The Pennsylvania State University (1991).

[7] H. Vogel, Phys. Z. 22, 645 (1921).

[8] G. J. Fulcher, J. Am. Cer. Soc. 8, 339 (1925).

[9] M. Horowitz, A. Bekker, B. Fischer, Appl. Phys. Lett. 62, 2619-2621 (1993).

[10] J. R. de Almedia, D. J. Thouless, J. Phys. A. 11, 983 (1978).

[11] R. A. Rupp, Appl. Phys. B 41, 153 (1986).

Chapter Five

Barkhausen Noise

5.1 Introduction

The dynamics of domain grating formation are readily studied by measuring the
displacement current or Barkhausen noise across the crystal as domains invert [1]. These
measurements isolate the role of photorefractive space charge fields from thermal effects in
the generation of ferroelectric domain gratings. They also demonstrate that the domain
grating has a large dynamic component displaying a time response similar to the

photorefractive space charge field.

The Barkhausen effect was first observed in the early work on ferromagnetic domain
reversal [2]. Subsequently, electrical Barkhausen noise was linked to ferroelectric domain
switching in barium titanate [3, 4]. As a domain inverts, the changing electric dipole
induces charge on the surface electrodes. This time varying charge produces a current

transient which is conventionally called a Barkhausen jump.

Barkhausen currents have been measured in numerous photorefractive ferroelectrics under
external fields. These jumps have been observed in LiNbO3 under intense illumination and
at elevated temperatures, and were used as evidence for a domain switching contribution to

hologram fixing in this material [5]. Current transients in BaTiO3 during electrical fixing

[6, 7] have also been observed. However, these experiments did not isolate the noise
caused by both external fields and heating from that noise originating from photoinduced

space charge fields. That is the purpose of this chapter.

5.2 Theory

The early measurements and analyses of Barkhausen noise are generally restricted to
crystals in the form of thin c plates (~100 um thickness). Domains typically nucleate below
an electrode evaporated on the crystal surface and subsequently grow to the other electrode
[8]. The crystal is also assumed to be non-conducting, appropriate for crystals such as
BaTiO3 in the dark. However, to investigate photoinduced domains, measurements are
performed under illumination and in the presence of mobile charge. It is necessary to relate
the volume of the inverted domains deep within the illuminated region of the crystal to the
measured Barkhausen current noise. This relationship is established by drawing on the
electrostatic theory of a polarized dielectric. The central theoretical problem is to relate the
measured short circuit current across the crystal to the free and bound charges moving
within the crystal. To simplify this analysis, the electrodes across the ferroelectric crystal
are treated as infinite conducting sheets short-circuited by an ideal current meter i(t) (figure
5.1). Only the center of the crystal is illuminated, so charge carriers do not flow directly
from the illuminated region to the electrodes. Thus, the induced current is the displacement
current € dE/ot alone. A quasi-electrostatic approach is adopted since the frequencies of
interest are below 100 KHz. Thus, currents induced by the magnetic fields associated with

moving charges are ignored.

i(t) v

Actual Domain Model Domain
> as
eo %, / 4
she

Figure 5.1: Barkhausen noise current theoretical model.

Both free and bound charge will contribute to the displacement current measured across the
crystal. For example, the free electronic charge is photoexcited and spatially redistributed
under illumination. By using the method of images from electrostatics (figure 5.2) [9], an
analytical expression is derived for the charge Qa and Qg induced on the conducting plates
by the electric field of a point charge qg embedded in a medium with a low frequency

dielectric constant €:
Qa =- ar, Qp = - at F . 5.1

ais the distance from electrode A to the point charge and b is the distance from electrode B

to the point charge, and a+b = L, the distance between the electrodes. From the above

analysis, it is apparent that the displacement current generated within the crystal under

illumination is composed of a free charge component, even in the absence of domain

reversal:
rs) oa

i(t) = —(Qp - Qa) =- qe 2— . 5.2
x "Lat

da/dt gives the component of the free charge velocity parallel to the surface normal of the
electrodes. In general, this free charge contribution to the displacement current arises from
the transport of free charge due to drift, diffusion, and the photovoltaic effect. Note that

the current given by equation 5.2 is independent of the low frequency dielectric constant e€.

og 2n(a+b) +2b
oq

On
DAAAAAANS AN POSS SSS
™~ b
eq 0 Pr

a ~~,
oq

oq + - 2n(a+b) +2b

Figure 5.2: Model of displacement current by method of images.

A second source of current noise in addition to the motion of photoexcited electronic charge
is domain switching. The inversion of ferroelectric domains creates characteristically sharp
current transients due to the collective displacement of ferroelectric ions. This distinguishes
domain switching from slowly varying space charge effects. As an inverted domain
nucleates and subsequently grows, bound charge is created, generating current noise. To
relate the measured current across the crystal to the volume of the inverted domain(s)
associated with each current spike, the charge Qa and Qz induced on the conducting plates
as a function of the location and size of the inverted, ellipsoidal domain is computed. The
domain is treated theoretically as charges q and -q separated by a distance d, as in figure
5.1. Note that d is slightly smaller than the physical domain length. The charge is
distributed along the entire domain wall for which P, is not parallel to the interface. To
simplify the analysis, the charge is treated as if it were all localized near the foci of the

ellipsoidal domain. The induced charge at the electrodes is then:
=-qi = qa

where d is the distance between the opposite centers of total charge q and -q along the c
axis. The induced charge as given by equation 5.3 is independent of the location of the
domain within the crystal and independent of €. In practice, the charge induced by domains
near the edges of the finite electrodes does depend on position. This effect is neglected
because the crystal is illuminated only at the center in our experiments. The current across
the crystal is computed by differentiating the induced charge on the electrodes with respect

to time:

oq +}. 5.4

q is composed of both the bound charge qp and free charge qs. The free charge is generated

upon optical illumination and compensates the bound charge at the domain walls:

Qp = APA, =2PygA., 5.5

where A, is the cross sectional area indicated in figure 5.1. Py is the average polarization
of the individual domain, typically larger than the macroscopic spontaneous polarization.
By substituting equation 5.5 into equation 5.4, the transient displacement current due to

domain switching is:

dInA, ,dInd din Pa) 4 2 (a Sat a

ar ae” a eS) 5-6

i(t) = r Pa A, d
The first term of equation 5.6 represents the sideways growth of domains (perpendicular to
c axis), the second term represents nucleation and forward growth (parallel to c axis), and
the third term represents the change in the individual dipole moment (arising from heating,
for example). The last two terms describe the free charge compensation of bound charge.
They are present in photoconductors because illumination and thermal excitation provide

mobile compensating charge.

The subtle temporal sequence of events in the formation of the space charge field, lattice
distortion, domain switching and free charge compensation can be revealed by examining
the transients in displacement current. Upon writing a domain grating with photorefractive
space charge fields, the domains may invert when the field is larger than the local coercive
field. However, upon inversion, significant depolarization fields equal to 2 Pg/e are
generated at the head-to-head domain walls. For the resulting domain configuration to be
stable, the depolarization fields must also be screened. In fact, the screening of

depolarization fields requires significantly more charge (on the order of Pg/eE, ~ 10 times

more, where Pg ~ P, is the spontaneous polarization and E, the coercive field) than the
initial charge required to establish the coercive field. The primary source of this
compensating charge is expected to be the photorefractive space charge. However,
additional charge may be provided by thermally or optically excited carriers unrelated to the
original spatially periodic photorefractive space charge field, or at pre-existing domain
walls. The important role of this later contribution is supported by the observation of

increased domain switching in optically fatigued crystals

The depth of spatial modulation of the spontaneous polarization is highly dependent on the
optical exposure. This effect results from accelerated free charge compensation and
screening of the ferroelectric dipole interaction [10] in the presence of illumination. The
response time of compensation is the dielectric relaxation time Tgje, for diffusion dominated
charge transport (i. e., for grating periods > Debye screening length, which is on the order
of um). Under illumination with light of intensity I,

tie = 5.7
diel Ga +l Oph

where 6, is the dark conductivity and 6,, is the photoconductivity. For optical intensities
of 1 W cm” at 45 °C, Tgiei is 8 seconds for Cr doped SBN:75, while in the dark, Tgjey is
effectively infinite because of the extremely low dark conductivity. These decay times are
measured by monitoring the decay of uniform pyroelectric fields due to conduction through
the crystal (figure 5.3). The pyroelectric fields are measured with a Trek Inc. non-contact
electrostatic voltmeter with an input impedance of 10!4 ohms. Thus, at high optical
intensities, the conductivity is sufficient to screen out some fraction of Barkhausen noise

events.

10} No Illumination 4

Pyroelectric Field (kV cm!)

Time (sec)

Figure 5.3: Decay of pyroelectric fields in the dark and under illumination. The dielectric
relaxation time of Cr doped SBN:75 at 45 °C is 8 sec at I, = 1 W cm”.

The volume of the inverted region is estimated by integrating the total charge Q;.; under a
current spike. The current event is assumed to correspond to the switching of a single

ellipsoidal domain of volume V:

Qtor L
v-—. .
6 Py 5.8

Equation 5.8 also assumes that the compensating free charge exhibits a buildup time
(proportional to the dielectric relaxation time of equation 5.7) which is much slower than
the domain switching transient. This is a good assumption for intensities below 100 W
cm, a regime where Barkhausen noise frequency components greater than the inverse
dielectric relaxation time are unscreened by photoexcited charge. For 10 W cm, this

lower cutoff frequency is estimated to be 1.25 Hz from equation 5.7.

5.3 Domain Nucleation Energy Requirements:

The energy decrease upon reorientation of spontaneous polarization P, in an electric field E

is given by:

AW, = -2 P,-E AV, 5.9

where AV is the volume of the inverted domain. The increase in surface energy (including

ferroelastic coupling and depolarization fields) is:

AW, =6 AA, 5.10

where AA is the change in surface area of the domain wall with surface energy density o.
To simplify the analysis, the domain is treated as a cylinder of radius r and height h.

Then the total change in crystal energy upon domain nucleation is:

AW =2nrho-2P,Enrh. 5.11

The equilibrium domain radius ry, assuming a typical domain aspect ratio h/r = 10, is:

_26 5.12
‘O"3 PE’

This relates the size of the inverted domain to the surface energy, spontaneous polarization

and external field.

Faraday Cage

488 nm |

writing beams

—_—a
—© “ tN
ae 633 nm
O* readout beam
Photodetector

Figure 5.4: Experimental setup to measure Barkhausen current noise.

5.4 Experimental Results

The experimental setup to measure the diffraction efficiency and current noise is
schematically represented in figure 5.4. The experiments are performed on room
temperature poled, Cr-doped SBN:75 and SBN:61 single crystals. The c faces of the
crystal are electroded with silver paint and thermally annealed. To maintain reproducibility,
the saturated spontaneous polarization is periodically restored by thermallizing the domain
structure ( heat treating at 350 to 450 °C in air) and subsequently electrically poling with 10
KV cm! at room temperature. The current noise is measured by the input preamplifier of a
Stanford Research Systems SR830 Lock-in amplifier (1 KQ input impedance) or by an
oscilloscope (1 MQ input impedance, 500 MHz bandwidth). A Tektronix 2440 digital
storage oscilloscope is used for high temporal resolution studies, and a computer data-

acquisition system is used for low temporal resolution studies.

The volume holograms are recorded with two collimated TEMo, single frequency (488 nm)
beams (diameter 1.5 mm). The holographic grating (typical period 1.1 wm) is
simultaneously reconstructed with a counter-propagating, 50 mW cnr?, 633 nm HeNe
laser to monitor the diffraction efficiency. The relative polarization of the two recording
beams is mechanically adjusted by rotating a half wave plate in the optical path of one of the
beams. The photorefractive space charge field can then be tailored without changing the
total optical intensity. This is essential to isolate the Barkhausen noise generated by space

charge fields from that generated by optical heating.

Current (nA)

Diffraction

Time (ms)

Figure 5.5: Noise in current (50 Hz) induced by oscillating space charge field (modulation
in diffraction ~ 0.5%). The current lags diffraction by 0.78 ms (SBN:61, I, ~ 1 W

cm-2, T = 22 °C, thermal steady state).

The first contribution to the displacement current arises from the moving space charge
directly. By inducing a periodic phase shift between the two recording beams at 50 Hz, the

spatial phase of the interference pattern is modulated by ~ 20°. 50 Hz noise in the current

is observed (figure 5.5). The noise in diffraction efficiency occurs at twice the modulation
frequency. This effect is apparent in both SBN:61 and SBN:75. At these low current
levels, a fraction of the noise current may also result from microdomain alignment in glassy
ferroelectrics such as SBN. Since the microdomains are believed to possess extremely
small volumes (~100 nm3), the current induced by inverting microdomains would also

produce an apparently smoothly varying signal.

A non-steady-state current has also been reported in the photoconductors BSO and BGO
upon temporal modulation of the space charge field [11, 12]. In these experiments, the
displacement current is believed to be negligible compared to the charge transport across the
crystal and through the external circuit. The entire crystal is illuminated, so indeed the high
photoconductivity of these materials facilitates charge transport from the crystal bulk to the
electrodes. In contrast, in the experiments of this chapter the displacement current is

measured because only the center of the crystal is illuminated.

oO
=]
OQ
Z,
3 i 1 i] 1 1
25 75 125 175

Time (sec)

Figure 5.6: Pyroelectric and Barkhausen current due to optical heating (SBN:75, I, ~ 20
W cm”, Tambient = 22 °C, prior to thermal steady state).

The second and most interesting contribution to the noise current is unambiguously linked
to the moving bound charge created upon domain switching. Figure 5.6 illustrates the
noise generated across the crystal during the initial stage of intense optical illumination (~
20 W cm?) with two orthogonally polarized recording beams. One beam is extraordinary,
the other ordinary, so the electric field vectors of the beams remain orthogonal as they
propagate through the uniaxial crystal. At this stage the beams do not produce a spatially
modulated electron density. Ulumination begins at t = 0, and the peak pyroelectric current
reaches a value of 250 WA on the time scale of ms. The current decays exponentially with a
time constant of 400 ms. Figure 5.6 illustrates the subsequent decay, the vertical scale

magnified to reveal individual Barkhausen jumps.

If the polarizations of the two beams instead have a component parallel to one another,
producing a photorefractive grating, the noise current does not change significantly. This
indicates that most of the current noise during the initial stage of hologram recording with
high intensity beams is thermal in origin. Local heating due to non-uniform illumination
generates local pyroelectric fields. These fields cause domains to invert and produce
Barkhausen current spikes (described by the first two terms of equation 5.6). Heating also
reduces the intrinsic polarization of the individual domains (described by the third term of
equation 5.6) to produce a smoothly varying pyroelectric current. Current jumps are
superimposed on the exponential decay of the pyroelectric current until thermal steady state

is reached (figure 5.6).

Figure 5.7 Etched a face of crystal following high intensity optical exposure

The rapid increase in current during the first seconds of exposure and the subsequent
exponential decay is consistent with the expected pyroelectric current generated along the c
axis by optical heating. The heating partially depolarizes the crystal in the region of intense
illumination. This depolarization is revealed by etching the entrance a face of the SBN
crystal (figure 5.7) for 15 minutes in a concentrated solution ( 37% ) of HCI at room
temperature. The origin of domain contrast is believed to be the selective etching of domain
walls. Walls possess a high concentration of trapped electronic charge that enhances the
rate of etching. This is analogous to the contrast mechanism in BaTiOs3, in the sense that

+c domains (charge compensated by "free" electrons) etch at a faster rate than -c domains

(electrons depleted) [13]. The sample was subsequently viewed under a transmission
optical microscope using polarized, incoherent light. The circular etched region
corresponds to the 1.5 mm beam diameter. The spatially periodic domain grating is not

apparent, but has been revealed within the bulk by optical means in figure 4.9.

Barkhausen jumps arising from the spatially periodic space charge field alone are also
observed. The role of the space charge field in domain switching is isolated from thermal
effects by maintaining a constant illumination level while tailoring the modulation depth of
the interference pattern. First, the crystal is illuminated with two beams of orthogonal
optical fields and total intensity of 1 W cm? for 10 minutes. The noise current up to this
point is thermal in origin. After thermal steady state is reached, the electric field vector of
one of the writing beams is then rotated by 90°, establishing the interference pattern. A
periodic space charge grating builds up at t = 0, as illustrated in the upper trace of figure
5.8. This data reveals the initiation of Barkhausen current events (lower trace) upon
hologram exposure. The peak magnitude of the space charge field is ~ 50 V/cm, as
estimated from the measured diffraction efficiency at t = 2 seconds (for r33 = 1000 pm V"!,
appropriate for a partially depoled crystal). These measurements are repeated with the
grating vector perpendicular to the c axis, and the Barkhausen noise transients then vanish.

In this geometry, the space charge fields are unable to orient domains periodically.

Diffraction Efficier

Time (ms)

Figure 5.8: Noise in current (bottom) and diffraction efficiency in % (top) as
photorefractive space charge grating is recorded (SBN:75, Ip ~ 1 W cm, T = 30
°C, thermal steady state).

The domain grating rapidly follows the photoinduced space charge field on the time scale of
the photorefractive buildup time. This is labeled the dynamic domain grating, in contrast to
the more typical remnant domain grating used to permanently fix domain holograms. The
dynamic domain grating relaxes immediately upon the removal of the space charge field.
Ferroelectric hysteresis provides memory so that a fraction of the dynamic domain grating
survives as a remnant grating. It is important to note that the dynamic domain grating is an
inextricable part of the photorefractive effect in SBN. Fundamentally, domain fixing is a

manifestation of hysteresis in the electrooptic effect.

5.5 Domain Switching Dynamics

The microscopic mechanisms of domain reversal can be deduced from the form of the
current transients. Barkhausen current jumps have been classified into two groups [3] in
the early literature: The first type are dome shaped, with a transient characterized by a
gentle rise and fall. The second type are spikes, with a very sharp leading edge (duration
of 10 us) followed by an exponential decay to zero. Jumps of the first type appear for all
values of the switching field, while jumps of the second type typically appear under strong
fields. Both types have smaller average dimensions in aged rather than restored crystals
because of domain wall pinning. According to Rudyak [14], the first type of jump is
expected during the relatively slow sideways growth of domains. The sharp leading edge
of the second type of pulse is associated with a fast nucleation event, and the subsequent
exponential decay is associated with the growth of the domain [3]. The velocity of
- sideways domain wall motion (parallel to an a axis) is believed to be slower than the
forward velocity of domain walls (parallel to a c axis) because of the large depolarization
fields established in the former case. The rate of sideways domain growth is expected to
increase with optical intensity because of the availability of mobile charge to screen these

depolarization fields.

Single domain switching events of both the first and second type are observed under
illumination. The first type, characterized by a gradual rise and fall of 5 ms duration, are
associated with the slow sideways growth of a domain (figure 5.9(a)). The second type
have durations of typically 1 ms and rise times on the order of 10 Us (figure 5.9(b)). The
sharp rise is attributed to domain nucleation and the subsequent decay to domain growth.
This rise time is in agreement with measurements of single domain events in BaTiO; [3].

Temporal and spatial correlation of domain switching events is also present in SBN. The

majority of optically induced current transients in an unfatigued SBN sample display

multiple sharp peaks occurring over a period of 100 ms (figure 5.9(c)).

(a)

Current (nA)

Time (ms)

(b)

Current (nA)

Time (ms)

85
(c)

Current (nA)

Time (ms)

Figure 5.9:
Barkhausen current events: single domain events of (a) type I and (b) type II, and

From equation 5.8 the volume of an individual inverted domain associated with a typical
current spike in SBN:75 (1 ms duration, 10 nA height and Py ~ P, ~ 20 uC cnr? ) is 7x10*
uum3. This volume increases dramatically (by about two orders of magnitude) when
multiple domain switching events occur simultaneously. At ambient temperatures, current
spikes with peak heights as large as 100 nA have been observed in SBN:75, while in the
SBN:61 sample the spikes are typically 5 to 10 times smaller in amplitude.

(%)

Diffraction Efficiency

Time (sec)

Figure 5.10: Intensity dependence of diffraction efficiency noise (SBN:75, Tambient = 22
°C, prior to thermal steady state).

Diffraction Efficiency (%)
1)

(nA)

von)

Noise Current

1 | L it i

20 40 60 80
Time (sec)

iS)
=)

Figure 5.11: Long-time temporal correlation of current and diffraction efficiency noise
(SBN:75, I, ~ 20 W cm, Tambient = 22 °C, prior to thermal steady state).

5.6 Optical Barkhausen Noise

In addition to the strong current noise upon high intensity recording, a previously
unreported intensity dependent noise in diffraction efficiency is also apparent (figure 5.10).
Above intensities of 1 W cm, the noise in diffraction efficiency increases dramatically.
To deduce the origin of this effect, the current and diffraction efficiency are measured
simultaneously (figure 5.11). The degree of correlation between these two signals is
computed numerically. The correlation coefficient function between the diffraction d(t) and

the Barkhausen current i(t) is defined as [15]:

Ria(T) - Hibta 5.13
v|Ri(0) - u?|[Raa(O) - 13

Pia(T) =

where pj and Ug denote the mean values of current and diffraction, respectively, and the

cross-correlation function Rjg(t) is defined as

Rig() = limp 5.0 + | i(t) d(t-t) dt. 5.14

The autocovariance functions for current R;;(t) and diffraction Rgg(t) are defined as:

Rj(t) = limy_,.. t | i(t) i(t-t) dt 5.15
and
Rag(t) = limy_s.. + | d(t) d(t-t) dt. 5.16

These correlation integrals may be evaluated in the frequency domain if the data is
stationary and ergodic. The noise current is low pass filtered prior to digital sampling to
prevent aliasing. This retains frequency components lower than approximately 50 Hz,
which comprise the dominant contribution to the signal in an electrically restored crystal.
The sample time window is 16 seconds, beginning at t = 64 seconds in figure 5.11. This
data consists of 2048 points sampled every 8 ms. The calculated correlation coefficient
function is illustrated in figure 5.12. The large correlation peak at tT = 0 indicates that the
noise in current and diffraction is correlated for simultaneous events. The series of smaller

peaks between t = 0 and 3 seconds indicates a weaker correlation between an initial

diffraction peak and later current transients.

— 0.6

0.3

| | —-

i | bo

-10 0 10

Cross Correlation Time T (sec)

Figure 5.12: Correlation coefficient function of current and diffraction efficiency (for data
from t = 64 to 80 seconds in figure 5.11).

The origin of the noise correlation is two-fold. The first source of noise correlation arises
from domain switching induced by a changing space charge field, as already discussed in
reference to figure 5.8. In this case, a change in diffraction efficiency triggers a domain
switching event. For instance, for the first correlated noise event in figure 5.13, the space
charge field perturbation induces domain switching approximately 200 ms later. This class
of events display smooth jumps in diffraction efficiency, followed at some later time by
sharp jumps in current. The temporal relationship is apparent in figure 5.12 by the delayed
sequence of correlation peaks between t = 0 and 3 seconds. This figure indicates that if it
were possible to apply a delta function in diffraction efficiency to the crystal, a series of
noise spikes in current would be observed, whose probability of occurrence decays within
a characteristic time of about 3 seconds. That is, figure 5.12 is the impulse response of the
ferroelectric distortion under periodic space charge fields. A primary source of this noise is
the instability in the spatial phase and amplitude of the optical interference pattern during
- optical heating. These noise events vanish once thermal steady-state is reached. However,
if the optical interference pattern is subsequently perturbed, a sequence of current spikes

will be generated as the new domain grating is recorded.

Diffraction Efficiency (%)

Time (s)

Figure 5.13: Two characteristic noise events in current and diffraction efficiency. The first
event is characteristic of a fluctuation in the space charge field initiating domain
reversal. The second event is characteristic of noise in diffraction efficiency
induced by domain switching (SBN:75, Ip ~ 8 W cn, Tambient = 22 °C, prior to
thermal steady state).

Secondly, a large fraction of correlated noise events in current and diffraction occur
simultaneously. In this case, the Barkhausen current jump is initiated before the space
charge field has changed, so the noise in diffraction efficiency is triggered by domain
switching (e. g., the second event of figure 5.13). Accordingly, this class of events is not
initiated by noise in the optical interference pattern. They are represented by a strong
correlation peak ( piq = 0.55 at t = 0 in figure 5.12) and characterized by abrupt transients
in both diffraction efficiency and current. This noise also vanishes after thermal steady
state is reached, indicating that the domain switching responsible for noise in diffraction
efficiency is initiated by random depolarization due to optical heating, rather than periodic
space charge fields. This is expected, since the domain switching arising from optical
heating is much more significant than that induced by the space charge fields. The relative

contributions are apparent by noting the much stronger Barkhausen noise in figure 5.11

compared to figure 5.8. Note that this interpretation regarding noise triggering is not
proven by the limited sample of noise events presented here. However, a large number of
individual noise correlation events in the spirit of figure 5.13 have been observed,

providing unequivocal evidence.

This second source of noise correlation is primarily the result of the displacement current
noise perturbing the free charge transport during dynamic hologram formation. This is
analogous to the noise in dynamic beam coupling induced by a fluctuating external applied
field. Indeed, the photorefractive band transport model predicts that moderate electrical
noise fields across the grating volume will introduce noise in the dynamic beam coupling.
Once thermal steady state is reached, this noise current driven by non-uniform optical
heating vanishes. In addition, the domain volume passes through the paraelectric phase for
a short period of time upon switching between the two ferroelectric orientations, which
results in a large transient index change (estimated to be ~ 0.02 at 25 °C from Ref. [16]).
The entire domain volume becomes an efficient scatterer of light for a fraction of the
switching time. In a separate experiment, the beam fanning during these high intensity
exposures is monitored. An identical correlation in the total intensity of the fanning beam

and the noise current is observed.

5.7 Summary

Barkhausen current noise associated with ferroelectric domain reversal is induced by
photorefractive space charge fields. A strong correlation between the current and
diffraction efficiency noise under high intensity optical exposures is present. Furthermore,
the rapid buildup of current noise upon hologram recording indicates that a dynamic
domain grating is formed. This characterization technique isolates the primary factors

responsible for optically induced domain reversal; namely, high intensities and space

charge fields.

References for chapter five

[1] A. S. Kewitsch, A. Saito, A. Yariv, M. Segev, R. R. Neurgaonkar, JOSA B , in press
(1995).

[2] H. Barkhausen, Physikalische Zeitschrift 20, 401 (1919).
[3] A. G. Chynoweth, Phys. Rev. 110, 1316-1332 (1958).
[4] V. M. Rudyak, Soviet Physics Uspekhi 13, 461-479 (1971).

[5] V. I. Kovalevich, L. A. Shuvalov, T. R. Volk, Phys. Stat. Sol. (a) 45, 249-252
(1978).

[6] R. S. Cudney, J. Fousek, M. Zgonik, P. Giinter, M. H. Garrett, D. Rytz, Appl. Phys.
Lett. 63, 3399-3401 (1993).

[7] R. S. Cudney, J. Fousek, M. Zgonik, P. Giinter, M. H. Garrett, D. Rytz, Phys. Rev.
Lett. 72, 3883-3886 (1994). |

[8] R. C. Miller, Phys. Rev. 111, 736-739 (1958).
[9] J. D. Jackson, Classical Electrodynamics. (Wiley, New York, 1975).

[10] V. M. Fridkin, Photoferroelectrics. M. Cardona, P. Fulde, H.-J. Queisser, Eds.,

Springer Series in Solid State Sciences (Springer-Verlag, Berlin, 1979), vol. 9.

[11] M. P. Petrov, I. A. Sokolov, S. I. Stepanov, G. S. Trofimov, J. Appl. Phys. 68,
2216-2225 (1990).

[12] I. A. Sokolov, S. I. Stepanov, JOSA B 10, 1483-1488 (1993).
[13] J. A. Hooton, W. J. Merz, Phys. Rev. 98, 409-413 (1955).

[14] V. M. Rudyak, A. Y. Kudzin, T. V. Panchenko, Sov. Phys.- Solid State 14, 2112-

2113 (1973).

[15] J. S. Bendat, A. G. Piersol, Engineering Applications of Correlation and Spectral

Analysis. (Wiley-Interscience, New York, 1993).

[16] A. S. Bhalla, R. Guo, L. E. Cross, G. Burns, F. H. Dacol, R. R. Neurgaonkar, J.

Appl. Phys. 71, 5591 (1992).

Chapter Six

Application of Domain Microstructure to
Holographic Data Storage

6.1 Introduction

Two critical technical issues in a permanent holographic data storage system are high
efficiency fixing and a convenient multiplexing scheme. One drawback of the ionic fixing
technique in LiNbO; is a single development stage at elevated temperatures in which all
space charge holograms are simultaneously fixed. While this is an acceptable method of
production for a pre-mastered holographic ROM, it makes it difficult to incrementally add
data to the memory. Each fixing process requires heating of the entire hologram volume.
For applications such as a holographic random access memory, the ability to add and
update fixed holograms in a rapid manner is highly desirable. Domain gratings have this

desirable property of being both permanent and updatable at room temperature [1].

Multiple holograms are accessed within the volume by angle or wavelength multiplexing.
Over a limited tuning range, wavelength multiplexing can also be accomplished with a
single wavelength source by tuning the Bragg law with an applied electric field [2]. This
chapter will describe work on both domain fixing and electric field multiplexing for

holographic data storage systems.

Spatial Light Lens Photorefractive
Modulator Crystal

<—_f—_> <—f—>

Figure 6.1: Theoretical storage capacity of a typical volume holographic memory system.

6.2 Volume Holographic Memory

The motivation for storing data in the form of volume holograms is the potential data
storage capacity and rapid, parallel access. The classic expression for the memory capacity
of a volume hologram is derived by summing the independent plane wave modes within a
cavity the size of the crystal [3, 4]. This predicts a data capacity of 1 TByte cnr? (figure
6.1). This number corresponds to the number of independent sinusoidal gratings that can
be stored in the crystal, which is equivalent to the number of bits. The throughput of data

in a holographic memory is potentially very high because an entire image may be read out

in parallel. For a binary image with 1000 by 1000 pixels, the number of bits that may
accessed in parallel is 1 Mb. The performance goals of a volume holographic random
access memory are ambitious: 1 TB capacity, 50 GBps data rate, 10 msec page write, and

100 nsec access time [5].

Angular [4, 6], wavelength [7] and phase-coded [8, 9] multiplexing are the three common
techniques for storing multiple holograms in a common volume. These techniques exploit

the dependence of the Bragg condition on the angle and wavelength of the writing beams:

k,= k)-k2 , 6.1

where k, = 2n/Agis the magnitude of the grating vector, k, = ky = 2mn/A are the
magnitudes of the reference and signal beam wave vectors, respectively, Ag is the grating
period, n is the index of refraction and A is the wavelength in vacuum. Angle multiplexing
typically requires mechanical or acoustooptic beam deflectors. Wavelength multiplexing
requires a tunable, narrow linewidth laser source. It is desirable to eliminate some of these
requirements to reduce the system complexity. One alternative, called electric field

multiplexing, offers some potential design flexibility.

6.3 Electric Field Multiplexing

Wavelength multiplexing can be implemented by employing a single wavelength source
while tuning the index of refraction in the medium. For instance, an external electric field
will change the index through the electrooptic effect. This concept has been demonstrated

in SBN:75 [2] and LiNbO3 [10-12]. For ferroelectric crystals, the index change is due to a

combination of the electrooptic, elastooptic, and piezoelectric effects. Table 6.1 lists

different first order field-induced effects which play an important role in multiplexing.

Effect: Response Pairs:
electrooptic electric field, index
elastooptic strain, index
piezoelectric stain, electric field
photoelastic stress, index

Table 6.1: Optical and electrical effects in pyroelectric crystals [13].

If a dc electric field is applied across the crystal, both the electrooptic and piezoelectric
effects contribute to Bragg detuning. In fact, a simple relation can be derived to illustrate
the formal equivalence between electric field and wavelength multiplexing. This analysis is
restricted to the case of counter-propagating signal and reference beams, both normally
incident on the crystal. The Bragg condition becomes Ag = A/2n. The selectivity of the
hologram is related to the angular, wavelength, or index detuning required to move off the
Bragg diffraction peak. For reflection holograms, the Bragg wavelength and index
selectivity are maximal. If the configuration displays inherently high selectivity, then the
magnitude of the multiplexing field is reduced. This allows more holograms to be

electrically accessed for a given maximum electric field.

The electric field required to detune from Bragg diffraction peak can be determined

analytically. Differentiating the Bragg condition at constant temperature T and mechanical

100

stress O, a relation for the change AA required to maintain Bragg matching under a field

induced change An[E] and AA,[E] is obtained:

AX. AnfE.A] _ Aste] 6.2
1d Ag

This result applies to the counter-propagating, reflection grating configuration. Since the
index of refraction in the crystal depends on the electric field and wavelength, the field

induced index perturbation is:

An[E,A] = An{E] + doiM Ai.
dx 6.3

Substituting equation 6.3 into equation 6.2, the relationship for the change in wavelength

equivalent to a field induced change of the index of refraction and grating period is:

Ak. 1 _{ AnfE] , ASE)

n (1-Nndwaal 2 A, 6.4

In a non-centrosymmetric crystal, the field induces an index change through a combination
of the electrooptic, elastooptic and piezoelectric effects [13] and may rotate the principal
axes by a field dependent angle. Consider a simple yet common case, in which the optical
and de fields are parallel to the field-induced principal axes. In this configuration the
principal axes do not rotate under applied fields. The change in index along a principal axis

is [13]:

101

7 [1k Ext Pim dkm Ex] , 6.5

Any[Ex] = -

where rik, Pim; dkm are the electrooptic, elastooptic and piezoelectric tensors, respectively,
and | = (ij), m = (np) in contracted notation. The field also induces a strain in the crystal,

described by the piezoelectric effect:

— = = din Ex. 6.6

In practice, both the clamped and unclamped electrooptic coefficients are measured. In a
clamped crystal, the strain is by definition zero, so the second term on the right of equation
6.5 is zero. The clamped electrooptic coefficient therefore corresponds to r,. Most often,
the unclamped electrooptic coefficient is measured, which is actually equal to r, + Pim dkm-

The unclamped value is typically about twice the clamped value.

The electrooptic, piezoelectric and elastooptic effects combine to produce a total field-

induced detuning AA:
An = d 38 tr B dim Ex] + dum E 6.7
( 7 an/an) 7 lik k+ Pim dim Ex] + dim Ex | - .
- Mn

Therefore, multiplexing is achieved by varying the indices of refraction and the lattice

parameters. The material parameters are dependent not only on E but also on the

102

temperature T and mechanical stress o. A relation analogous to equation 6.4 is obtained
upon replacing E by T or o for the cases of "temperature" or "stress" multiplexing. For

constant E and o:

An 1 _{ AnlT] | AASIT) -
Ad = L .
x (1-Anda/da)l Ae
while for constant T and E:
~ (1-4 dn/da) Ag

The following discussion will focus field multiplexing under constant temperature and

stress.

To simplify the analysis of electric field multiplexing described by equation 6.4, the index
dispersion dn/dA is neglected relative to n/A. This results in an error of approximately 20%

[14]. The second term on the right of equation 6.4, the strain term, is on the order of An/n.

A scale factor €g is defined as:

An{E] , AAg{E] _ £ An{E] 6.10
D A. “mn?

where €p is assumed to be a slowly varying function of the field E in the range of

O< led < 10 [15], depending on the sign and magnitudes of the linear electrooptic and

103

piezoelectric coefficients. An analogous scale factor is defined for the temperature

dependence:

AnfT] , AAT] _ , AnfT]

6.11
n Ag n

where AA, = o AT (a is the thermal expansion coefficient). For the stress dependence:

6.12

An{o] , AAg[o] _ gE An[o]
n + A = Go yh >

where the first term on the left is a piezo-optical contribution and the second term is the
strain. Then the index selectivity or index half-power bandwidth, defined as lAnlpwiny, for

constant T and 6, is related to the spectral half-power bandwidth of a Bragg peak A Alpin

[16] by equations 6.4 and 6.10:

lAnnbven ~ 1 [A Awan 1
n Ee OA Es

onl aa

6.13

where L is the thickness of the crystal. The above expression relates the Bragg selectivity

for field multiplexing to the familiar result for wavelength multiplexing. Since the grating
period for two plane waves in this configuration is Ag = A/2n, the minimum index change

Anmin = 2 Antewen between adjacent holograms written at the same wavelength is

approximately

104

The temperature and stress must be stabilized to within:

AT << Anim (aay
Sr 6.15
and
Ao << SA nmin (day .
3 6.16

In practice, the stabilization of stress and temperature can be readily achieved by proper
mounting of the medium. When these stability conditions are met, the number of

holograms Nholograms that can be multiplexed electrically is:

Anmax = GD. AB max 6.17

Nholograms =
Omin

To maximize Nnolograms, two primary requirements must be satisfied. The material must
have at least one large electrooptic coefficient that induces a significant index change for an
electric field (1) along kg to produce index gratings and (2) along ei, e2 (the polarization
vectors of the reference and signal) for electrically biasing the index of refraction. In

addition, there are three conditions of secondary importance: it is desirable that the external

105

field does not rotate (3) the optical field vectors of the incident beams or (4) the principal
axes. Failure to satisfy conditions (3) and (4) will reduce the beam coupling because the
polarization of the two beams will no longer remain parallel and in the desired direction of
condition (2) as they propagate through the crystal. Finally, (5) the material should have a

large piezoelectric tensor component dm of the proper sign such that Eg is maximized

The counter-propagating reflection grating geometry is optimal for electric field
multiplexing. This configuration has two primary advantages. First, the counter-
propagating configuration exhibits inherently high wavelength selectivity [7]. Second, it
maintains a strong index dependence in the Bragg law. That is, a given change of index
does not induce as large a change in the grating vector for oblique incidence. In fact, for
the transmission geometry with equal angles of incidence, the index dependence of the

Bragg condition disappears:

Nevertheless, the Bragg condition retains its field dependence A,[E] even in this case

through the piezoelectric term.

For typical parameters ( A, = .5 pm, L = 1 cm, extraordinarily polarized signal and
reference) and estimating Eg = €; ~ 1.25, the minimum index separation Anpjn between

independent holograms is 4x10°5. For a material such as SBN:75 with dn/dT = 2.5x10-4

[9], the temperature must be stable to within AT << 0.16 °C at 25 °C. Since r33 is the only

significant electrooptic coefficient in SBN:75, the optimal geometry is a compromise to

106

simultaneously satisfy both conditions (1) and (2) ( figure 6.2). This configuration also
satisfies conditions (3) and (4). The total index tuning range in the “optimized”

configuration is approximately Anmax = 0.0025, half of Anmax = 0.005 [17] for a material

such as SBN:75. Then the number of holograms that can be electrically multiplexed about

A, in SBN:75 is approximately 60. Obviously, this number is difficult to achieve if
holograms are stored as domain gratings, since the applied fields will tend to erase the fixed

holograms.

a Reference c

0 < y l ext

Signal k,

e& +

Figure 6.2: Optimum configuration for electric-field multiplexing in SBN:75, with
dominant electrooptic coefficient r33.

107

kK,

| k,
Reference e, —— Signal
| e,
| SA
E |
| ok

Figure 6.3: Transmission grating configuration for electric field multiplexing.

Non-standard, 45° cut crystals are required to implement electric field multiplexing in the
counter-propagating configuration. Instead, a regular cut SBN:75 has been selected to
experimentally demonstrate electric field multiplexing in the configuration of figure 6.3.
This configuration prevents condition (1) from being satisfied in the counter-propagating
geometry, since r33 is the only large electrooptic coefficient. As a tradeoff, the

transmission grating configuration with 01,;, = 90°, 62,:, = 60° and the external field

applied parallel to the c axis is chosen.

A single hologram at 0 V cm! has been switched off its Bragg peak with a field increment
as low as 500 V cm! and recovered when returning to 0 V cm:!. The field increment for

the optimum configuration is expected to be 100 V cm!, from equation 6.7. In addition,

108

two fixed holograms have been recorded (angular bandwidth of images ~ 4 mrad), one
written with no external field (figure 6.4(a)) and another written with a field of 2000 V
cm! (figure 6.4(b)). Care must be taken to prevent this applied field from being screened
by optically excited charges while recording the holograms. The holograms are selectively
addressed by re-applying the field at which they were written. As seen in the figures, the
two holograms exhibit little crosstalk. Their fixed diffraction efficiency is approximately
1%. The holograms have also been de-multiplexed by varying the angle of the
reconstruction beam. These fixed holograms are domain gratings, so after approximately

twenty switching cycles, the hologram quality degraded dramatically.

(a) (b)

Figure 6.4: Experimental demonstration of electric field multiplexing of two fixed
holograms (a) 0 V cm:!, (b) 2000 V cnr!.

109

6.4 Selective Fixing

An important application of domain fixing is holographic data storage. Domain gratings
possess an inherent advantage that they can be individually and selectively fixed [1]. This
property is desirable in applications such as holographic random access memories or
reconfigurable optical interconnects, yet it is difficult to implement using conventional
thermal fixing techniques in LINbO3. Figure 6.5 shows the selective overwriting of one of
three fixed, angle multiplexed holograms. The reference beams of each hologram are
separated by A@ = 0.01 rad. The crystal is not thermally stabilized during writing, so the
Bragg peaks of the holograms are thermally broadened by a factor of 10. The three
holograms, labeled (a)-(c), were individually written and automatically fixed with 4 W cm-2
for 30 min each. The index modulation of holograms at other addresses is nevertheless
slightly degraded during the fixing process. Proper exposure scheduling during the
selective overwriting process will minimize the degradation of adjacent fixed holograms.
However, as is apparent in figure 6.5, the dynamic range of the recording media (i.e., the
maximum index perturbation) is insufficient to perform several recording cycles. The
hologram strength is close to the inherent noise floor arising from index inhomogeneities
and scattering from domain interfaces. The dynamic range of the photorefractive crystal
can be improved by optimizing the density of photorefractive traps, given by Ngt in chapter

one.

110

(b)

Figure 6.5: Selective, page addressable fixing of three angle multiplexed volume
holograms (a)-(c). The poor image quality is the result of scattering from growth
striations and domain walls.

6.5 Summary

Volume holograms can be electrically multiplexed. Based on a simple theory, we estimate
that approximately 50 image carrying holograms are field addressable about each reference
beam wavelength and angle. Alternately, electric fields may be applied to fine-tune the
Bragg condition and compensate for wavelength drift of the reference beam, for instance.

Individual holograms have been selectively fixed by ferroelectric hysteresis and a primitive

111

image storage device has been implemented. These techniques have potential benefits in a
holographic memory, optical interconnect, or optical correlator. However, major progress
must still be achieved to improve the optical quality and dynamic range of these holographic

materials.

112

References for chapter six

[1] A. S. Kewitsch, M. Segev, A. Yariv, R. R. Neurgaonkar, Opt. Lett. 18, 1262-1264
(1993).

[2] A. S. Kewitsch, M. Segev, A. Yariv, R. R. Neurgaonkar, Opt. Lett. 18, 534-536
(1993).

[3] T. Jannson, Optica Applicata TX, 169-177 (1979).

[4] P. J. van Heerden, Appl. Opt. 2, 393-400 (1963).

[5] R. G. Zech, Optics & Photonics News, August, 16-25 (1992).

[6] F. H. Mok, M. C. Tackitt, H. M. Stoll, Opt. Lett. 16, 605-607 (1991).

[7] G. A. Rakuljic, V. Leyva, A. Yariv, Opt. Lett. 17, 1471-1473 (1992).

[8] C. Denz, G. Pauliat, G. Rosen, Opt. Comm. 85, 171-176 (1991).

[9] Y. Taketomi, J. E. Ford, H. Sasaki, J. Ma, Y. Fainman, S. H. Lee, Opt. Lett. 16,
1774-1776 (1991).

[10] S. I. Stepanov, A. A. Kamshilin, M. P. Petrov, Sov. Tech. Phys. Lett. 3, 36-38

(1977).

[11] M. P. Petrov, S. I. Stepanov, A. A. Kamshilin, Ferroelectrics 21, 631-633 (1978).

113

[12] M. P. Petrov, S. I. Stepanov, A. A. Kamshilin, Optics Communications 29, 44-47
(1979).

[13] J. F. Nye, Physical Properties of Crystals Their Representations by Tensors and
Matrices. (Oxford University Press, Oxford, 1964).

[14] E. L. Venturini, E. G. Spencer, P. V. Lenzo, A. A. Ballman, J. Appl. Phys. 39,

343-344 (1968).

[15] E. Nakamura, Ed., Landolt-Bérnstein Ferroelectric Oxides, New Series , vol. III/28a

(Springer-Verlag, Berlin, 1990).
[16] H. Kogelnik, Bell Sys. Tech. J. 48, 2909-2947 (1969).

[17] J. B. Thaxter, M. Kestigian, Appl. Opt. 13, 913-924 (1974).

114

Chapter Seven

Applications to Domain Microstructure to Quasi-
Phase Matched Second Harmonic Generation

7.1 Introduction

Optical second harmonic generation is a process in which an optical wave at frequency @
induces a wave at frequency 2@ within a nonlinear optical medium. Classically, the
medium is thought of as a collection of dipole radiators driven by the incident optical field.
The polarizability of individual bonds typically exhibits a nonlinear dependence on the
driving field. This bond charge oscillates not only at @, but also at the higher harmonics.
Under a condition called phase matching, these radiators each emit a field which adds
constructively to efficiently produce light at a particular harmonic. This phase matching
condition only occurs at particular frequencies and in certain materials. A technique to
achieve phase matching at any optical wavelength is desired. This chapter describes a new

way to phase match in the visible spectrum using dynamic ferroelectric domain gratings.
7.2 Nonlinear Optical Frequency Conversion

The nonlinear optical response of a medium is described by an expansion of the induced

optical polarization P in powers of the electric field amplitude E [1]:

115

Pi(t) = €oXjEj + dix EjEx + 4XijEjExEs t+... . 74

The term djj,Ej;E, describes first order optical nonlinearities, typically the dominant
nonlinear term in non-centrosymmetric materials. First order nonlinearities include second
harmonic generation, sum/difference frequency generation, optical rectification and the
electrooptic effect. Theoretically, these processes are described by the interaction of a

monochromatic wave:

EP!) =Re{E}” eient)= 1(pP" ciorts EP" ero), 7.2

with a medium whose response is given by equation 7.1. The electrooptic effect described
in chapter two can equivalently be expressed as

Q 0,0, Q

pe = & iit EVE, . 7.3

Here © is a low frequency electric field, typically in the range of dc to MHz. Alternately,
both fields can be optical. This interaction mixes the two waves to produce new waves at

the sum and difference frequencies. Sum/difference frequency generation takes the form:

PS = die BPE? 7.4
J>

The degenerate case of this process is second harmonic generation:

p20 = x dO OEPES 7.5
J:

116

k2o
2ko Ak

Figure 7.1: Momentum conservation relation for second harmonic process. Ak is the
wavevector mismatch.

Second harmonic generation will be the focus of this chapter. The coupled wave equation
describing the evolution of the second harmonic optical beam as it passes through the

nonlinear medium is given by [1]:

d29 = [E20 fp0(] exp (i Ak). 16

Integrating across the crystal length z = 0 to L for an input plane wave E®,

20,0,0\2
3/2 q2 (220°) 12

sinc{Ak L/2) . 7.7

p20 =? (pe? (fH
n3

A \€&

To obtain significant second harmonic power it is necessary that the wavevector mismatch

Ak << 2, for L on the order of mm. This requires that 2 k® = k*®, or equivalently

n®=n2®. This phase matching condition is equivalent to a momentum conservation
relation for the nonlinear interaction. Figure 7.1 depicts the momentum conservation

relation graphically.

When the wavevector mismatch is zero, the second harmonic power grows quadratically

with distance as it propagates through the medium, according to equation 7.7.

117

p2o _ q2,2. 7.8

In general, the mismatch is nonzero because of normal dispersion at optical wavelengths.

In this case, the mismatch can be compensated for by providing an additional momentum
Ak, = Ak. This is achieved by periodically poling the crystal. Fundamentally, a soft

phonon mode (@ = 0, k = Ak.) associated with the spontaneous polarization of the crystal

provides the additional momentum.

For a square wave modulation of the spontaneous polarization, the second harmonic power
still grows quadratically as in equation 7.8; however, the nonlinear optical coefficient is
replaced by an effective coefficient, dere. For the case of perfect square wave modulation of

the spontaneous polarization,

devrad—2.. .
eff ds 7.9

For the case of sinusoidal modulation of the spontaneous polarization, the nonlinear

coefficient is reduced by an additional factor of 1/4:

dere = 7.10

iA.

an:

Since the photorefractive space charge fields are approximately sinusoidal, the modulation
of the spontaneous polarization averaged over a fraction of the grating period is also

expected to be sinusoidal.

118

7.3, Quasi-Phase Matching Using Dynamic Domain Gratings

Quasi-phase matching (QPM) [2, 3] is a technique to periodically compensate for the phase
mismatch between the fundamental and second harmonic beams. This technique relaxes
the stringent phase matching requirements based on birefringence [4] or modal dispersion
in waveguides [5]. As in the previous section, QPM can be achieved by periodically poling
a ferroelectric crystal so that the relevant nonlinear coefficients for SHG are spatially
modulated with a period equal to twice the coherence length. For co-linear beams, the
coherence length is 4, =A/4 (n?®-n®), where A is the wavelength in vacuum of the
fundamental beam, n® is the index of refraction at the fundamental wavelength and n2® is
the index at the second harmonic. In the absence of periodic domain inversion, the
coherence length is the maximum effective crystal length that contributes to the second
harmonic power. Coherence lengths exceeding a few mm are needed in practice for
efficient conversion. In ferroelectric crystals such as LiNbO3 and Sr75Ba.25Nb20¢
(SBN:75), the coherence length is on the order of a few microns for second harmonic
generation in the visible; consequently, the non-phase matched conversion is negligibly
small despite the relatively large nonlinear coefficients. For these and many other highly
nonlinear yet dispersive materials , QPM is a means of increasing the effective interaction

length to promote efficient frequency conversion.

QPM by periodic poling has been demonstrated during growth [6], by indiffusion [7, 8],
by applying external electric fields [9, 10], electron beams [11], or SiOz masks [12].
Typically, only a very narrow frequency band is converted efficiently and it is difficult or
impossible to tune the center frequency dynamically, except over a small range using
temperature or angle tuning. Fabrication constraints often restrict the region of polarization

modulation to a thin layer on the surface or to thin waveguides rather than throughout the

119

entire volume of the crystal. These periodic domain gratings are typically static and

permanent.

7.4 Experimental Observation of QPM in SBN:75

We use dynamic domain gratings to demonstrate tunable QPM-SHG in SBN [13, 14].
This technique uses photoinduced space charge fields alone to modulate the ferroelectric
polarization of the crystalline unit cell [15-17], rather than photoinduced screening of
external electric fields [10]. Figure 7.2 illustrates the experimental setup for writing
dynamic domain gratings and simultaneously generating the second harmonic. A tunable,
mode-locked Ti-sapphire laser with 2 ps transform-limited pulses and 250 W peak power is
frequency doubled within the Cr doped, 45 ° cut SBN:75 crystal. The fundamental beam is
loosely focused with a 20 cm lens to a 100 micron beam diameter at the center of the SBN
crystal, producing a peak fundamental intensity of 0.8 MW cm. The second harmonic is
isolated from the fundamental with an infrared blocking filter, then collected and focused
onto a photomultiplier tube (PMT). The far field intensity profile of the second harmonic is
viewed simultaneously with a 240 by 240 element CCD array. The signal from the PMT is
synchronously detected using a lock-in amplifier with a nominal 400 Hz reference signal

generated by the chopper controller.

The average power of the Ti-sapphire laser is monitored with a photodetector and used to
correct the second harmonic data for variations in the pump power during the wavelength
scan (by dividing the second harmonic signal by the square of the pump power). The pulse
duration (2 ps) is relatively constant over the entire wavelength range (880-990 nm). A 1.3
Watt argon-ion laser at 514.5 nm writes a domain grating throughout the crystal volume,
and a third, non-Bragg matched beam erases the gratings. Both the argon-ion and Ti-

sapphire lasers are extraordinarily polarized.

120

Argon-Ion Laser

Autocorrelator
Computer 2 + Amplifier/
Data Acquisition Filter
Lock-in .
Amplifier Pye Detector
IR Filter - _
hw |. <=--- Ti-Sapphire
PMT beef =f |
eS Computer 1+ | |Temperature Chopper
& p
5 Frame Grabber Controller Controller
oO
oO

Figure 7.2: Experimental setup for quasi-phase matched second harmonic generation using
dynamic domain gratings.

The orientation of the crystalline axes of the SBN:75 crystal shown in figure 7.2 must be
selected so that both the grating vector and polarization of the fundamental beam have a
significant projection along the c axis. Since the space charge field is used to invert
domains, a component of the grating vector along the poling axis is required. Similarly,
the fundamental beam must have a significant projection of its optical electric field along the
c axis to induce a large polarization at the second harmonic frequency. The magnitude of

the polarization at the second harmonic perpendicular to k®, the fundamental wave vector,

1S:

p2© — 2 (2 dyssin20 cos © + d33 cos?0 + d3;sin2@ cos @) E® E®, TAL

121

where 6 is the angle of both P?© and E® relative to the c axis. This function is plotted in
figure 7.3. Although P2® attains a maximum value at 0 = 0 degrees, this orientation

prevents the space charge field from having a significant component parallel to the c axis.
However, P2® remains significant for 6 as large as 45 degrees. 6 = 45 degrees is selected
as a convenient tradeoff. By symmetry arguments, inverting the orientation of a
ferroelectric domain changes the sign of all three nonlinear coefficients (see chapter two).
Thus the effective nonlinear coefficient dere = (2 d5 + d33 + d31)/29/2 is modulated in sign
by the polarization grating. The grating vector ky = 1/é is oriented parallel to k® and k?® to

maximize the spatial overlap of the fundamental and second harmonic field profiles.

P? (arbitrary units)

0 20 40 60 80
0 (degrees)

Figure 7.3: Dependence of quasi-phase matched second harmonic power on the crystal
configuration of SBN:75.

122

Enhancement in 2°

457.25 457.5
Wavelength (nm)

Enhancement in 2°
Oo

0 02 04 06 O08
Wavelength Scan (nm)

Figure 7.4: (a) QPM tuning curve centered on 457.25 nm with a full width half maxima of
0.175 nm (b) Multiple quasi-phase matching peaks generated by writing several
gratings sequentially at the same location within the crystal.

The QPM second harmonic enhancement peak can be shifted across the available tuning
range, from 440 to 495 nm, by writing gratings with periods ranging from 3.7 to 3.0

microns. Figure 7.4(a) illustrates a typical QPM peak centered on 457.25 nm. The full

123

width half maximum (FWHM) of the QPM peak is calculated using reference [18] in
addition to the measured index dispersion data [19]. A FWHM of 0.2 nm is predicted for a
crystal of thickness 4.25 mm, in close agreement with the measured FWHM of 0.175 nm
from figure 7.4(a). This observation confirms that the grating is uniform over the entire
4.25 mm propagation distance. The slight discrepancy is accounted for by differences in

the index dispersion of different crystal samples.

An ensemble of gratings with different periods can be written in the same or different
locations of the crystal to tailor the spectral response of the SHG and can be rendered
permanent by cooling and/or increasing the writing beam intensity. Figure 7.4b illustrates
a series of QPM peaks written sequentially in the same volume, with the strongest
enhancement peak corresponding to the last grating written. The spectral response can be
tailored by writing several gratings (either simultaneously or consecutively) with different
periods. This figure also demonstrates that an enhancement of a factor of 17 above the
background can be achieved by writing dynamic domain gratings. Typical single pass
conversion efficiencies obtained experimentally are of the order of 0.01%. The SHG
enhancement will increase further with higher intensity writing beams or with a lower
intensity fundamental beam. While the latter option will decrease the conversion efficiency,
both options will increase the fringe visibility of the optical interference pattern and enhance
the polarization modulation of the grating. The uniform fundamental beam is incoherent
with the two writing beams so it reduces the modulation index of the intensity grating. The
degree of reduction depends on the absorption of the crystal at the fundamental wavelength.
As a direct consequence the photoinduced space charge field and the polarization

modulation are reduced by a factor of the modulation index.

124

LOTIIT

LOTT YLT TTI
ERATE
i LUT LTT TLL TT Thy }
TUT EL ARRSER TAY
LON py, LMT (AL) aN ee, LTT]
LHL as {] RY
Ty UL if p

Figure 7.5: Quasi-phase matched, second harmonic far field intensity profile (TEMoo
mode, angular beam divergence 6, y on the order of 1 mrad).

The second harmonic beam exits the crystal in a well collimated, symmetric TEMo9 mode
(figure 7.5) collinear with the fundamental. No change in second harmonic power was
measured while scanning the fundamental beam across the face of the crystal and
maintaining the relative orientation, confirming that the polarization modulation extends

well into the bulk of the crystal.

The rapid enhancement of second harmonic power during the hologram recording stage
further supports the existence of dynamic polarization gratings discussed earlier in the
context of the Barkhausen noise measurements. Figure 7.6 illustrates the time evolution of
this enhancement while writing a QPM polarization grating with a total intensity of 1 W
cm? and a spatial period of 3.5 microns. The transients in the peak enhancement result
from the decay and build-up of the grating as the optical interference pattern shifts due to
vibration, air currents and temperature changes. The build-up time constant of the QPM

peak is approximately inversely proportional to intensity and is consistent with the

125

photorefractive grating build-up time for SBN:75. At 1 W cm? the second harmonic build-

up time is 0.5 seconds, and at 0.33 W cm it is 1.5 seconds.

erase |

NH B&O fH WN
oe ¥

Enhancement in 12°
ay

50. 100 150 200
Time (sec)

Figure 7.6: Time response of the buildup and decay of second harmonic enhancement
while centered on a QPM peak at 457 nm.

At second harmonic wavelengths removed several nanometers from the center of the QPM
spectral peak, the second harmonic power is diminished by typically a factor of two rather
than enhanced. The time evolution of the second harmonic power is then opposite to that
of figure 7.6. This is consistent with the notion that the background pseudo-random
domains which contribute to a broadband enhancement of the second harmonic [10]
become ordered due to the space charge field, so the second harmonic at wavelen gths other

than the QPM peak is reduced when the grating is formed.

The decay of the second harmonic enhancement upon uniform illumination indicates that a
significant number of domains aligned at kg reorient, causing the domain grating to decay.
As indicated in figure 7.6, a small remnant enhancement A in the second harmonic is
permanent, corresponding to the fixed domain grating reported earlier. The lifetime of this

permanent grating is highly dependent on temperature but is relatively independent of the

126

intensity of the erasing beams. Its strength can be increased by applying a fixing technique
[17]. At lower temperatures (T < 20 °C) the non-ergodicity characteristic of SBN's glassy
polarization state enables the domain orientation to be permanently frozen-in, while at
higher temperatures (T > 45 °C) thermal fluctuations of the microdomain orientation

randomize the grating [17].

7.5 Experimental Observation of QPM in SBN:61

Domain gratings were recorded in SBN:61 and the temperature dependence of QPM-SHG
was analyzed [14]. Space charge field induced QPM-SHG can be achieved even in the
paraelectric phase, where the nonlinear optical susceptibility is expected to vanish. In
addition, an optically fatigued crystal displays significantly stronger QPM-SHG than an
electrically poled crystal. A strong broad-band second harmonic enhancement occurs in
optically depoled crystals, as observed earlier [10, 20] with external electric fields applied
along the c axis. In fact, this broad-band enhancement can attain values several orders of

magnitude larger than the QPM enhancement.

The tunable, mode-locked Ti-sapphire laser is frequency doubled within the 45° cut, Ce
doped SBN:61 crystal (4.5x4.5x5.5 mm), with a ferroelectric-paraelectric phase transition
at 75 °C. The fundamental infrared beam is focused with a 20 cm lens to a 60 micron beam
diameter, producing a peak fundamental intensity of 17 MW cm at the beam waist. A 2 to
6.6 Watt argon-ion laser at 514.5 nm records a domain grating throughout the entire crystal
volume. To maintain the poled state at elevated temperatures, a field of 8000 V cm! is

applied across the electrodes between optical exposures.

The spatial profile of the second harmonic wave is related to the ferroelectric domain

structure through a Fourier transform relation [21-23], The angular spectrum of this wave

127

indicates the periodicities of the spontaneous polarization or nonlinear optical susceptibility.
Upon recording a domain grating in an initially poled crystal, a collimated QPM second
harmonic beam is observed (figure 7.7), in addition to a non-phase matched streak. This
streak exits the crystal at an angular displacement of approximately 10 degrees below the
QPM spot. The origin of this angular walk-off between the two distinct second harmonic
beams is the 45 degree inclination of the c axis relative to horizontal. The broad extent of
the streak along the x axis and the narrow extent in the y direction indicates that domains
are extremely narrow in diameter, while being elongated along the c axis. Typical
dimensions of microdomains are tens of nanometers in diameter to 1 4m in length. Figure
7.8 depicts the resulting range of grating vectors generated by the randomly distributed
microdomains, represented as a multiplicity of grating vectors with different magnitudes.
The momentum conservation relation for the three photon-one phonon interaction is
2k®° +k, = k2®, where k® is the fundamental wavevector and k2® is the second harmonic
wavevector. The second harmonic far field profile will be directed to a spot (upon
interacting with grating periodicity k/’™) and a streak (upon interacting with multiplicity of
grating periodicities k§eak), For a random distribution of microdomains of diameter 10

nm, the distribution of k$ireak is also random up to grating vectors as large as 6 108 m-!.

128

screen

Figure 7.7: Experimental setup: Optical waves k, and k, record dynamic domain

grating, k® is the fundamental beam, and the second harmonic waves exit
the crystal to form both a spot k?®,,,, and a streak k?®,,,.., in the far field.

129

k”
ellipsoid

2k°+k, =k™,
2 k® + Kyandom = k Rak

Figure 7.8: k space diagram illustrating origin of angular displacement of second
harmonic spot (QPM contribution) and streak (random domain contribution).

130

This total second harmonic power in the streak can attain a value two orders of magnitude
larger than the QPM spot, if the random depolarization of the crystal is more severe than the
periodic depolarization. The random depolarization arises because each time a QPM grating
is recorded, inverted microdomains form along those grating planes in which the space
charge field is directed opposite to the spontaneous polarization. Instabilities in the spatial
phase of the optical interference pattern ( arising from temperature changes, for instance )
tend to uniformly depole the crystal over time. Thus, the volume of uniformly distributed,
inverted microdomains continues to increase as multiple holograms are exposed. In
contrast, the domain modulation at the periodicity of the hologram compensates the
instantaneous photorefractive space charge field. This spontaneous polarization modulation
depth is small in comparison to the broadband distribution of periodicities arising from the
accumulation of microdomains. The second harmonic conversion efficiency is enhanced
by a factor of 1000 ( to 1% ) in an optically fatigued crystal, above the conversion in a
poled crystal. Long term fringe stability (greater than hours) is necessary to maximize the
periodic domain modulation while minimizing the uniform depolarization. In practice, this

level of stability is extremely difficult to achieve.

Periodic depolarization is induced by the photorefractive space charge field to produce a
narrow band SHG enhancement. This effect displays strong temperature dependence in
SBN:61. By heating the crystal to within 20 °C of the ferroelectric phase transition, we
find that the QPM enhancement, or equivalently the modulation of the spontaneous
polarization, attains a maximum. Furthermore, the glassy ferroelectric properties of SBN
is manifest in the observation of QPM-SHG above the ferroelectric phase transition
temperature (figure 7.9). The origin of this effect is the alignment of micropolar regions in
the paraelectric phase under the influence of the space charge field. In fact, micropolar

regions are believed to exist up to temperatures as large as 300 °C [24]. However, the

131

quasi-phase matching enhancement observed here becomes extremely weak at temperatures

in excess of 85 °C.

Go
GN

NN KN Ww
oN ©
T T T

Second Harmonic
Peak Power (mW)
oO

1 1 1. 1 i vl

50 55 60 65 70 75 80 85 90

Crystal Temperature CC)

Figure 7.9: Temperature dependence of quasi-phase matching peak enhancement for
gratings written with total intensity of 1 W cm-? at 514.5 nm. Note the existence of

QPM-SHG above the Curie temperature T, = 75 °C.

The second harmonic power is dramatically enhanced by recording gratings in optically
fatigued crystals; that is, crystals which have been exposed to significant optical energy ( >
104 J cm ) at 514.5 nm (figure 7.10). The formation of random microdomains in the
illuminated regions reduces the macroscopic spontaneous polarization by a factor of 3 to 10
(as indicated by the degradation of the electrooptic coefficient). The pre-existence of
microdomains facilitates the subsequent domain grating formation. The small volume of
microdomains (< 100 nm>) lowers the energy required to invert an individual dipole.
Furthermore, the reduction of electrical Barkhausen noise at this stage in the ferroelectric
history of the crystal [25] suggests that the primary mechanism for depolarization is the

sideways motion of domain walls under the action of the space charge field, rather than

reversed domain nucleation.

132

25
Optically Fatigued
“x

Electrically Poled

Second Harmonic
Peak Power (mW)

0 0.5 1 1.5 2
Wavelength Scan (nm)

Figure 7.10: Enhancement of second harmonic peak in an optically fatigued crystal by high
intensity exposure for ~ 3 hours, 1 W cm. The noise features superimposed on
peak are due to fluctuations in the interference grating.

Unlike earlier measurements performed in SBN:75 [13], a measurable remnant
enhancement in second harmonic conversion efficiency is not observed in SBN:61.
Apparently, the ferroelectric hysteresis for exposures and readout conducted at the same
temperature (30 to 90 °C) is negligible in this material. To produce a significant remnant
enhancement, the crystal must be cooled following the optical exposure to freeze-in a

fraction of the dynamic domain grating [17].

7.6 Depth of Modulation of Spontaneous Polarization

The second harmonic power arising from a single coherence length ( & = 1.12 um ) slice of
the crystal is predicted to be 0.1 mW from published values of the nonlinear optical
susceptibilities [26]. The measured background second harmonic is ~ 1 mW, suggesting

that the conversion efficiency is enhanced by the broad-band random domain contribution.

133

This power is centered on the QPM spot, and does not include the power contribution from
the second harmonic streak. From the second harmonic power at the QPM peak (figure
7.11) the depth of periodic modulation of the spontaneous polarization is estimated. For
instance, the measured peak second harmonic power of 30 mW translates into an effective
crystal length of 40 coherence lengths. The spontaneous polarization is equivalently
modulated by 1 %, so the effective nonlinear optic susceptibility induced by the space
charge field is 0.06 pm V-!. To attain conversion efficiencies of several percent at these
fundamental beam intensities (17 MW cm), the spontaneous polarization must be
modulated by at least 20 %. The inset of figure 7.11 indicates the predicted second
harmonic power as a function of the polarization modulation, expressed as a percentage of

the room temperature spontaneous polarization.

vv
oO

pet
ws

Second Harmonic Peak Power (mW)

rw Mirch, ten

0 n
463.4 464.4 465.4

Second Harmonic Wavelength (nm)

Figure 7.11: Typical quasi-phase matched tuning curve in SBN:60 at 60 °C for 3.25
micron grating written with a total intensity of 1 W cm-?. Maximum conversion
efficiency is 0.01 %. Inset: calculated second harmonic peak power as a function
of the percent modulation of the spontaneous polarization (for a 17 MW cm2
fundamental plane wave).

134

The modulation of the spontaneous polarization achieved during the relatively short optical
exposures conducted here (typically less than 15 minutes) is limited by the mobile charge
available to compensate the depolarization fields at head-to-head domain walls. The free
carrier density required to compensate the non-zero divergence of the polarization, for a
grating composed of domain walls whose surface normal is at 45 degrees to the c axis, is

given by:

Neomp =V2 ZAP , 7.12
q Ag

where AP is the polarization modulation amplitude at spatial periodicity A, and q is the
charge of an electron. The carrier density required to produce a field equal to the coercive

field is:

212 1 € Eooercive 7.13
q Ag

N coercive —

where q is the charge of an electron and € is the low frequency dielectric constant. For P,
equal to 25 UC cm, Eooercive equal to 1 kV cm! (typical for SBN:61 at 45 °C) and a
grating period of 3.25 micron, the compensating charge density required on the domain
walls for bipolar modulation (AP = 2 Ps) is Ncomp ~ 10!8 cmr3, and the coercive charge
density is 4x10!6 cm-3. The charge should be localized about the domain walls, so
significantly higher charge densities may be required (by a factor of 10). In the
experiments described here, the small effective modulation of the spontaneous polarization
( 1% ) indicates that the spatially periodic photoinduced space charge density is ~ 10!6 cm
to ~ 10!” cmr3, depending on the degree of localization of the compensating charge. This

suggests that further improvement in the QPM second harmonic powers can be attained by

135

tailoring the effective density of photorefractive traps to increase the space charge field

without strongly increasing the absorption at the second harmonic wavelength.

7.7. Summary

The rapid enhancement in second harmonic power upon recording phase matching gratings
further demonstrates that domain gratings exhibit a large dynamic component. Domain
gratings can be recorded above the ferroelectric-paraelectric phase transition due to the
glassy ferroelectric nature of SBN. The QPM-SHG is significantly enhanced by recording
in optically fatigued rather than electrically poled crystals. The narrow band second
harmonic is significantly weaker than the broadband second harmonic enhancement arising
from the microdomain structure. The relatively small QPM-SHG conversion efficiencies
are a consequence of the limited compensating charge available to screen the depolarization
fields. Therefore, the primary effort to increase the degree of domain modulation should be
directed at increasing the effective density of photorefractive traps through optimization of

the dopant chemistry.

136

References for chapter seven

[1] A. Yariv, Quantum Electronics (Saunders College Publishing, Philadelphia, 1991).

[2] J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 127, 1918
(1962).

[3] S. Somekh, A. Yariv, Opt. Comm. 6, 301 (1972).

[4] J. A. Giordmaine, Phys. Rev. Lett. 8, 19 (1962).

[5] P. K. Tien, Appl. Opt. 10, 2395 (1971).

[6] N. Ming, J. Hong, D. Feng, J. Mater. Sci. 17, 1663 (1982).

[7] C. J. van der Poel, J. D. Bierlein, J. B. Brown, S. Colak, Appl. Phys. Lett. 57, 2074
(1990).

[8] E. J. Lim, M. M. Fejer, R. L. Beyer, Electron. Lett. 25, 174 (1989).

[9] A. Feisst, P. Koidl, Appl. Phys. Lett. 47, 1125 (1985).

[10] M. Horowitz, A. Bekker, B. Fischer, Appl. Phys. Lett. 62, 2619-2621 (1993).

[11] H. Ito, C. Takyu, H. Inabe, Electron. Lett. 27, 1221 (1991).

137

[12] J. Webjérn, F. Laurell, G. Arvidsson, IEEE Photon. Technol. Lett. 1, 1989 (1989).

[13] A. S. Kewitsch, M. Segev, A. Yariv, G. J. Salamo, T. W. Towe, E. J. Sharp, R. R.
Neurgaonkar, Appl. Phys. Lett. 64, 3068-3070 (1994).

[14] A. S. Kewitsch, T. W. Towe, G. J. Salamo, A. Yariv, M. Zhang, M. Segev, E. J.
Sharp, R. R. Neurgaonkar, Appl. Phys. Lett. 66, 1865-1867 (1995).

[15] A. S. Kewitsch, M. Segev, A. Yariv, R. R. Neurgaonkar, Opt. Lett. 18, 1262-1264
(1993).

[16] A. S. Kewitsch, M. Segev, A. Yariv, R. R. Neurgaonkar, Jpn. J. Appl. Phys. 32,
5445-5546 (1993).

[17] A. S. Kewitsch, M. Segev, A. Yariv, G. J. Salamo, T. W. Towe, E. J. Sharp, R. R.
Neurgaonkar, Phys. Rev. Lett. 73, 1174-1177 (1994).

[18] M. M. Fejer, G. A. Magel, D. H. Jundt, R. L. Byer, J. Quant. Elect. 28, 2631
(1992).

[19] E. L. Venturini, E. G. Spencer, P. V. Lenzo, A. A. Ballman, J. Appl. Phys. 39,
343-344 (1968).

[20] M. Horowitz, A. Bekker, B. Fischer, Appl. Phys. Lett. 65, 679-681 (1994).

[21] R. C. Miller, Phys. Rev. 134, A1313 (1962).

138

[22] I. Freund, Phys. Rev. Lett. 21, 1404 (1968).

[23] G. Dolino, Phys. Rev. B 6, 4025 (1972).

[24] A. S. Bhalla, R. Guo, L. E. Cross, G. Burns, F. H. Dacol, R. R. Neurgaonkar, J.

Appl. Phys. 71, 5591 (1992).

[25] A. S. Kewitsch, A. Saito, A. Yariv, M. Segev, R. R. Neurgaonkar, JOSA B , in

press (1995).

[26] Yamada, Landolt-Bérnstein New Series II. T. Mitsuiand E. Nakamura, Eds.,

Ferroelectric Oxides (Springer Verlag, Berlin, 1990), vol. 28a.

139

Chapter Eight

Photopolymerization

8.1 Introduction

The process of linking individual monomer molecules into polymer chains is called
polymerization. These chains can be joined along the backbone to form a crosslinked
solid. This reaction is ubiquitous in the commercial production of plastics from precursors
based on liquid monomers and oligomers. Polymerization and crosslinking are typically
achieved by thermal initiation or photoinitiation. This discussion will focus on
photoinitiation and on the nonlinear optical phenomenon arising upon photopolymerization.
The chemistry of photopolymerization is described in detail in numerous texts [1-8]. This

chapter will provide a brief review of photochemistry and polymerization.
8.2 Photoinitiated Polymerization

Light induced radical chain polymerization is a chain reaction initiated by UV generated
reactive species. Most monomers do not produce initiating species with sufficiently high
yield upon UV exposure, so a photosensitive initiator is introduced. Typical UV curable
materials are made of at least two components: (1) a photosensitive initiator system that

effectively absorbs the incident light and readily splits into reactive components and (2) a

140

monomer/oligomer containing at least two unsaturated bonds that will generate the polymer

network.

The first step in the photoinitiation process is the absorption of a photon by a suitable
chromophore and the promotion of a valence electron to an unoccupied molecular orbital.
For a one photon absorption process, the energy E of the photon must be at least as large as
the energy difference between the ground and the first excited singlet states of the molecule.

Since most photoinitiators have a AE value above 350 kJ mol-!, they will absorb strongly

at wavelengths below 340 nm [9].

The electronic states of molecules are conveniently characterized in terms of the total
electronic spin. A state with all spins paired is called a singlet state; one unpaired spin is a
doublet state; two unpaired spins is a triplet state. While there are exceptions (e.g. Oz), in
most ground-state molecules all electrons are paired (i.e., in the singlet state) [1]. A
valence electron of the photoinitiator molecule is typically promoted to an excited singlet
state upon absorption of a suitable photon. Excited singlet molecules are short-lived
species (10-9 to 10-6 s) that decay by fluorescence, intersystem crossing and

phosphorescence, or energy transfer.

One deactivation pathway of particular importance in photopolymerization is the
intersystem crossing of the excited singlet state S; to the excited triplet state T, [10]. The
excited triplet state is in turn deactivated by a radiative process or by a nonradiative
intersystem crossing to the ground state. The energy level diagram of this process is
depicted in the left half of figure 8.1, where the reactant is a photoinitiator. The right-hand
side of this diagram will be discussed in the context of photosensitized reactions in section

8.4.

141

Energy
S1 rT
| A ;
| fT,
“_
— |
Ti | Va
5 |
s €|
& 5 |
oO A
1 ©
5 3|
cond ify
a -
"Bb 2|
a] 7
a 5
Sg ——___—_— | So

Reactant Sensitizer

Figure 8.1: Energy level diagram typical for singlet excitation (solid lines) versus triplet
sensitization (dashed lines).

Because of the long lifetime of the triplet state and the presence of two unpaired electrons,
excited triplet states are more likely to undergo bond cleavage to form long lived radicals.
The cleavage reaction is particularly important because it is responsible for the production
of free radicals which will initiate either the degradation of an existing polymer or the
crosslinking of a functionalized prepolymer. The bond breaking reaction can occur only if
the photon energy exceeds the bond dissociation energy of one of the covalent bonds in the

polymer. The bond dissociation energy of various bonds are summarized in table 8.1.

142

Bond Broken: Bond Dissociation Energy
(einstein kJ mol-1)

C-H 380 to 420

O-H 420

C-OH 380

C-C 340

C-O 320

C-Cl 320

Table 8.1: Typical bond dissociation energies [10].

Symbol: Chemical Species:

R- Initiator Radical

Mn Terminated Polymer of m Monomeric Units
Mn: Polymer Radical

PI Photoinitiator

PS Photosensitizer

ps* Excited State Photosensitizer

C Chain Terminating Species

Table 8.2: Photopolymerization nomenclature.

The photoinitiators used in this study are based on the mechanism of photofragmentation.
The absorption of a photon by a molecule promotes an electron from a bonding to an

antibonding molecular orbital. Upon fragmentation, radicals are produced by the scission

of a covalent bond:

143

PI+hv > R}-+Ro . 8.1

The nomenclature is described in table 8.2. For photofragmentation, the rate of radical
generation at a particular location within the polymer depends on the extinction coefficient

€,, the local concentration of the photoinitiator [PI] and the local light intensity I [5]:

APU =e, [PT 8.2

Assuming the two radicals R;-, R- are identical, the rate of initiation is then given by

aR J, 4Pu 2.3

In a liquid solution, the effective radical yields upon photodissociation are reduced by cage
recombination. The solvent surrounding the initiator molecule at the moment of scission
forms a molecular cage which must rearrange before the initiator fragments can physically
separate. The cage keeps the fragments together for some time during which they may
readily recombine. Only a fraction of the radicals produced in the scission event become
available for reaction outside the cage [3, 11, 12]. The cage effect is responsible for the

strong viscosity dependence of the initiation efficiency, for instance.

The probability of radical recombination in the solvent cage depends on the multiplicity of
the excited state. An excited singlet state dissociates into radicals with antiparallel spins.
Their recombination is spin allowed and may occur within the time of a single nuclear
vibration (10-15 s). However, radicals formed from an excited triplet state have parallel

spins and can recombine to a singlet state only after spin inversion, which requires a time

on the order of 100 ns [11]. Therefore, excited triplet states play the dominant role in

photoinitiation.

144

8.3. Types of Photoinitiators

There are three primary classes of photoinitiators [7] used in industrial applications. In

addition to the photocleavage mechanism described above, additional mechanisms are

based on photoinduced hydrogen abstraction and photogeneration of acids.

materials are summarized below.

il.

ill.

Photocleavage (Benzoin Ethers, Benzil Ketal, Acetophenones,
Phosphine Oxide, Titanocene) This class includes aromatic
carbonyl compounds that undergo a bond scission when exposed to
UV light. The most commonly employed photoinitiators are
benzoin ethers, which are highly efficient radical sources.

Hydrogen abstraction (Benzophenone or related compounds with
Amine, Thioxanthone, Camphorquinone, Bisimidazole) Aromatic
ketones do not undergo fragmentation upon optical excitation.
Instead, they “abstract” or borrow a hydrogen atom from another H-
donor molecule. Tertiary amines often play the role of hydrogen
donors.

Cationic photoinitiators (Aryldiazonium salts, Arylsulfonium and
Aryliodonium salts, Ferrocenium salts) Cations form free acids
under illumination, which are efficient initiators for the
polymerization of epoxy monomers by opening the epoxy ring to
form chains. Chain propagation is initiated either by another cation

or by a strong electophile, for example a Lewis acid (BF3) or a

145

protonic Brgnsted acid. Oxygen has no effect on cationic
polymerization because the acids are oxygen insensitive. This is an

important advantage for applications such as UV curable thin resists.

8.4 Photosensitized Polymerization

The response of the photopolymers can be shifted to the visible and infrared spectrum by
using the electron-transfer reaction common among dyes. This is typically called
photosensitized rather than photoinitiated polymerization. To achieve singlet energy
transfer with a photosensitizer that absorbs at a longer wavelength than the reactant would
require endothermic energy transfer, which is energetically improbable. However, it is
often possible to find a sensitizer whose singlet excited state lies below that of the reactant,
but whose triplet level lies above the triplet level of the reactant (figure 8.1). While in the
context of section 8.3 the reactant was the photoinitiator, that is not necessarily the case for

photosensitized polymerization.

Most sensitization processes in organic photochemistry are based on the following reaction:
So (reactant) + T*(sensitizer) —> T"(reactant) + So (sensitizer) . 8.4
Since the overall spin of the system is conserved, the transfer process is spin allowed and
occurs efficiently, as long as the transfer is exothermic ( Ey (sensitizer) > Ey (reactant) ).

A photosensitizer such as a dye typically absorbs the optical energy,

PS + hv, > PS* 8.5

146

and subsequently transfers an electron to another molecule. Excimers and exciplexes are
two types of molecular excitations that play an important role in the process of electron
transfer. Excimers are formed by pairs of aromatic molecules that do not significantly
interact in the ground state, yet become weakly bonded in the excited state. This bonding
occurs between an excited state molecule and a ground state molecule of the same species.
The origin of this bonding is the change of orbital symmetry that accompanies excitation
and leads to orbital overlap and hence to bonding between the two systems. In the ground
state the two molecules repel each other upon approaching within the distance of their van
der Waals radii, but in the excited state the attractive force of the orbital overlap creates a
potential energy well that leads to an energetically stable excimer state. The depth of the
well is the excimer binding energy. This is the energy decrease relative to the energy of the

excited state of an isolated (monomeric) molecule [3].

If two molecules differ significantly in their electron affinities, so that one may be regarded
as an electron donor and the other as an electron acceptor, then the bonding process
between these two molecules is accompanied by a partial transfer of charge. In this case
the interaction is stronger than in conventional excimers and the transient excited species is
called an exciplex (excited complex). Exciplexes lead to the complete transfer of an

electron from one molecule to another and to the formation of radical ions [3].

Electron transfer in photosensitized reactions is often mediated by an exciplex. There are
two prototypical sensitization mechanisms. In the first, an excited sensitizer PS* functions
as an electron acceptor. In the presence of a suitable electron donor D, the reduction of
PS* produces the donor cation radical D+, which may be transformed into a species capable
of initiation. It is also possible for excited sensitizers to act as electron donors for a suitable

ground state electron acceptor A. The reduced acceptor A- may then produce radical

147

initiators upon further reaction. In practice both types of sensitization have been observed.

The first type is called photoreducible dye sensitization [13]:

PS*+D—3PS +D*, 8.6

DtoR.. 8.7

The second type of sensitization by electron transfer employs a photooxidizable dye:

PS*+A—PSt+A° , 8.8

The electron donor D and the electron acceptor A are sometimes referred to as activators or
coinitiators. Typical classes of activators are amines (primarily tertiary amines), sulfinates,
enolates, carboxylates, and organometallics [13]. Often the photoinitiator compounds

described in the previous section also serve as effective activators or coinitiators.

Dye sensitization can occur not only by electron transfer, but also by energy transfer.
Energy transfer is possible when two molecules are in close proximity. The molecule that
carries excitation energy is distinguished as the energy donor (D), the other molecule as the

energy acceptor (A). Energy transfer is accomplished by the interaction

D*+A>5D+A", 8.10

where the * denotes that the molecule is in an excited electronic state. Energy transfer is an

electronic process that occurs on the time scale of 10-!> seconds. It is energetically

148

favorable when the excitation energy of D* is more than or equal to that of A*. Two
distinct coupling mechanisms are important in this process: coulombic or dipolar

interaction (Forster) and electron exchange or orbital interaction (Dexter) [3].
8.5 Dipole Resonance Transfer

Energy transfer by coulomb interaction is a dipole radiation effect. The electronic
transitions of two molecules couple in a manner similar to two oscillating dipoles. The
radiation field produced by one molecule is absorbed by the other [14, 15]. The rate of

energy transfer for this type of coupling is given by [3]:

2,.2
keq(coulombic) = AED 8.11

TAD

where kept is the rate constant, Wa and Up are the transition dipole moments of the
absorption (A-A”*) and fluorescence transition (D*-D), and rap is the separation between the
two molecular centers at the moment of interaction. This expression can be written in terms

of measured quantities [16]:

2 D ~
ker = 8.8 x 102589) | wen, 8.12
nt rot J, v4

where n is the index of refraction, dg(D) is the quantum yield of fluorescence of the donor,
w is the fluorescence lifetime of the donor in the absence of the acceptor and Kk? is a
geometric factor. In fluid systems, where the rotational relaxation time is shorter than the
lifetime of the excited state, Khas a value of 0.67, while in a rigid medium it is 0.457 [3,
16]. fp(v) is the fluorescence spectrum of the donor normalized to unity, €a(v) is the

absorption spectrum of the donor (not normalized), and r is the separation between the two

149

molecules. In dipole resonance transfer the rate of energy transfer depends on the
fluorescence intensity and on the fluorescence lifetime of the donor, as well as on the
spectral overlap between the fluorescence of the donor and the absorbance of the acceptor.
Because the energy transfer is carried by the electric field, Forster transfer does not require
diffusional encounters between molecules, so it is relatively independent of solution

viscosity.

8.6 Exchange Transfer

Exchange transfer is an alternate means of energy transfer between molecules that arises
from the physical interaction between orbitals of the donor and acceptor (figure 8.2). The
exchange mechanism requires orbital overlap between the donor and acceptor upon
collisions. Exchange transfer occurs on almost every molecular encounter if the process is
exothermic (Ep« - Ea» < 0) [3]. The exchange transfer is less sensitive to changes in spin
multiplicity than resonance transfer, so it is dominant in triplet-to-singlet energy transfer

[17].

Antibonding

= Bondin
—o—_o— g
OO —o-—-O—
—O—o—-
D* A

Figure 8.2: Energy transfer by electron exchange as represented by a Htickel orbital
diagram [3].

150

8.7. Types of Photosensitizers

Photosensitizers typically exhibit the following properties [3]:

i. They have a high rate of intersystem crossing from S; to T; and
consequently a high quantum yield oy of triplet formation.

ii. The energy difference between the singlet excited state S, and the
triplet excited state T, (called singlet-triplet splitting) must be small.

iii. The triplet lifetime should be long to increase the probability of
energy transfer between sensitizer and reactant.

iv. They should absorb strongly in a spectral region where the reactant
does not absorb.

Vv. They must be soluble in the reaction medium (solvent or polymer

matrix).

The majority of common sensitizers are either ketones or contain the carbonyl group. The
nz* character of the excited states (triplet and singlet) of aromatic ketones favors
intersystem crossing, high quantum yields, as well as small singlet-triplet splitting [3]. The
nz* configuration is formed by promoting an electron from a nonbonding n orbital on
oxygen into an antibonding m* orbital that is delocalized over the aromatic 1 system.
Charge delocalization produces a partial positive charge on oxygen that may interact with

the electron rich C-H bond typical of the polymer backbone.

Several classes of photoreducible dyes have been identified: xanthene (eosin, fluorescein,
Rose Bengal), thiazine (methylene blue, thionine), and acridinium dyes (acriflavin). The
absorption maximum is centered on 460 nm for acridinium, 565 nm for xanthene, and 645

151

of light absorbing photoproducts (discoloration) and the destruction of the original

chromophores (bleaching) responsible for absorption.

Photocrosslinking has been achieved with wavelengths as long as 632.8 nm, by using a
dye such as methylene blue and toluenesulfinic acid. The dye is thought to be reduced by
the sulfinate ion to form a radical which initiates acrylate polymerization [18, 19]. This
reaction proceeds very slowly (1-2 days) in the absence of illumination. The initiating
system is quite sensitive to illumination. The required exposure is as low as 0.6 mJ cm”?.

Therefore, photopolymerization can be induced by visible diode, He-Ne and Ar-ion lasers.

8.8 Chain Propagation

A dominant mechanism for the growth of polymer chains upon illumination is radical chain
polymerization. The initiation of the polymer chain begins with the reaction of a radical

with a monomer unit:
R-+M-—> M.. 8.13

Once the monomer is radicalized, a chain reaction proceeds and the monomer grows into a
polymer chain. This stage is called radical chain propagation, and is represented by the

process:

The addition of a monomer molecule to the growing polymer chain can be represented by
the conventional expression for the rate of a second-order reaction [5]. Since the only

source of polymer molecules is chain propagation, and since neither polymer molecules nor

152

polymer radicals migrate significantly during the course of the reaction, the polymer radical

concentration [P | is given approximately by:

ar | KM] [My] . 8.15
where
[P-]= Y [M,-] . 8.16

K, is the second-order rate constant, [M] the concentration of monomer, and [Mi i] the
local concentration of polymer radicals of chain length 3. Monomer molecules diffuse from
regions of high monomer concentration according to the diffusion equation, including a

monomer consumption term on the right [5]:

Ay Da [Ml KeiMl S [My]. 8.17
Dm is the monomer diffusivity in the polymer matrix, which itself is a function of time
because the polymer becomes more viscous upon crosslinking. Mass transport by
convection or turbulence is ignored. The rate of chemical reactions in viscous media is
influenced by the rate of diffusion of reagents towards one another. Radical polymerization
reaction kinetics are diffusion controlled. Propagating polymer chains become very large
and lose their mobility during polymerization. However, monomers continue to diffuse
more or less freely. In the polymer matrix, the diffusivities of small molecules range from
10-6 to 10-19 cm? s-!, while the diffusivity of the polymer species ranges from 10-!! to

10-13 cm? s-! [5].

153

8.9 Chain Termination

In theory one absorbed photon may polymerize the entire monomer solution, since one
radical R * can join an infinite number of monomers. In practice, this reaction is terminated
at some intermediate stage by chain termination agents C. This limits the mechanical
strength of the crosslinked polymer, for instance. The mechanism of chain termination is
an important factor in the spatial resolution of polymeric microstructures. A quantitative
theory is essential to model this process and address issues such as resolution, contrast,
and index of refraction change. The elementary results which comprise this section are

based primarily on chapter five of the text by Krongauz [5].

Chain termination is represented by the following process:

C+Mn 73 C-+Mnp . 8.18
Oxygen inhibition is a common mechanism for chain termination. It leads to a threshold
energy for photopolymerization (an induction period) and a reduced rate of polymerization

or crosslinking. This occurs by two processes. The photoinitiator radical may be

deactivated by oxygen:

R- + O02 > ROO. , 8.19

or the polymer chain may be terminated by oxygen directly:

02+ Mm: 73 M,OO- . 8.20

154

Oxygen inhibition has the desirable effect of scavenging the diffusing photoinitiator, which
improves the spatial contrast between cured and uncured regions. In the absence of
oxygen, chain termination is dominated by radical occlusion. The radicals are trapped at
locations where no reaction partners (monomers) are available. Radical-radical chain
termination is usually ignored because the mobility of growing polymer chains is orders of

magnitude lower than that of monomer molecules.

The oxygen triplet ground state exhibits a high reactivity with excited states and free
radicals because of oxygen’s biradical nature. It deactivates both singlet and triplet excited

states by an energy transfer process:
S* +0, 38405 , 8.21
T° +0, 9T+0, . 8.22

In air saturated systems, effective scavenging by oxygen occurs only if the radical lifetime
is larger than 10-7 s, which is typically the case in polymers. Otherwise, the lifetime of the
radical is not sufficient to come into contact with oxygen. If a chain transfer agent C and
oxygen are free to diffuse within the polymer matrix, then the chain-termination rate is

expressed as:

all {Ky [Oz] + Keric]} ¥ [Mj] . 8.23
j=l

where Ky and Kcr are the second order rate constants of the chain termination by Oy and
by a chain terminating agent, respectively, and [P] is the concentration of terminated

polymer. The fraction of radicals that succeed in initiating polymerization is denoted by fj:

155

K, [MI

fi =

where K, is the rate constant for radical initiation, the reaction described by equation 8.13.

By comparing equation 8.23 and 8.15, it is apparent that chain propagation dominates the
early stages of the photopolymerization reaction when the concentration of monomer is
high, while chain termination dominates the latter stages of the reaction when the

concentration of polymer radical is high.

The diffusivity of oxygen, Dy, is higher than that of the monomer because of its smaller
size. Therefore, polymerization would not occur if oxygen solubility in the film were high.
Fortunately, the usual photopolymer oxygen solubility is quite low (< 3 10° mol cm) [5].

The concentration of oxygen and the chain terminating agent are given by diffusion

equations:

102 al _ pV [Oy] - Kz [0] SIM | 8.25

and

a = DerV" [C] - Ker [C] YIM ;]. 8.26

Oxygen can be scavenged from the polymer by reacting with tertiary amines. Amines are
very effective as hydrogen donors and consume oxygen in a chain reaction. This reaction

is able to consume up to 12 molecules of oxygen with 12 amine molecules, with the loss of

156

only one radical M- [20]. Tertiary amines are commonly added to the photopolymer

system as antioxidants.

Several mechanisms reduce the rate of photopolymerization. Additives exploiting these

mechanisms are called photostabilizers. There are five primary types [9]:

i. UV screeners: prevent absorption of incident light by the polymer
ii. UV absorbers: compete with the polymer for absorption of light
iii. Quenchers: deactivate the excited states formed in the polymer

Iv. Radical scavengers: react with alkyl, alkoxy, and peroxy radicals

Vv. Peroxide decomposers: suppress the chain branching process

The latter three types of photostabilizers may be added to the photopolymer system to
reduce unwanted polymerization in dark regions due to radical diffusion. While this
improves the spatial resolution of the photopolymer system, it also reduces the light

sensitivity.

8.10 Kinetic Model of Polymerization

The local concentration of the polymer can be determined as a function of time by modeling
the processes of initiation, propagation and termination. The goal is to ultimately determine
the index of refraction from the polymer concentration as a function of space and time.
This requires the simultaneous solution of the chemical rate equations, the diffusion
equations, and the wave equation in an inhomogeneous medium. In chapter nine a
relatively simple, approximate expression for the index evolution is used, which is solved

numerically in chapter eleven.

157

For a more exact treatment of the kinetics of polymerization, the evolution of the polymer

radical concentration [P | must be evaluated. The evolution in time is given by [5]:

AP = 2o4 - {Kp [O2]+Ker[C]} P- . 8.27

The initiator, monomer and polymer absorb light within the medium. Since chemical
reactions and diffusion change the distribution of the absorbing species, the local light

intensity also changes upon illumination:

1 - {en [PI] +em{ [P] +[P-]+[M]+[M-] dX €[Q] . 8.28
where [Q,] is the concentration of the kth component of the reactive mixture and € is the
extinction coefficient of the Ath component. The extinction coefficient €y of the monomer
and polymer are assumed to be identical. These equations can be solved numerically to
determine the local concentration of polymer at any time. The relative concentration of
polymer is typically linearly proportional to the density. We can then model the time
evolution of the index of refraction exactly by relating the density change to the index of

refraction change. This will be the subject of future studies.

8.11 Summary

Photopolymerization is a technique of great practical and commercial importance. This
reaction is also of interest in nonlinear optics because dramatic and. permanent index
changes are generated with low optical exposures. As the monomer crosslinks, the density

of polymer increases and the index tends to increase. However, the change in absorption

158

upon illumination also introduces large index changes. The factors which determine the

index change are described in the next chapter.

159

References for chapter eight

[1] T. H. Lowry, K. S. Richardson, Mechanism and Theory in Organic Chemistry.
(Harper & Row, New York, 1987).

[2] G. Odian, Principles of Polymerization. (Wiley, New York, 1981).

[3] A. Reiser, Photoreactive Polymers The Science and Technology of Resists. (Wiley,

New York, 1989).

[4] D. Volman, G. S. Hammond, K. Gollnick, Eds., Advances in Photochemistry , vol.
13 (Wiley, New York, 1986).

[5] V. V. Krongauz, A. D. Trifunac, Eds., Processes in Photoreactive Polymers
(Chapman & Hall, New York, 1995).

[6] M. Koizumi, S. Kato, N. Mataga, T. Matsuura, Y. Usui, Photosensitized Reactions.

(Kagakudojin Publishing Co., Kyoto, 1978).

[7] C.-H. Chang, A. Mar, A. Tiefenthaler, D. Wostratzky, in Handbook of Coatings
Additives L. J. Calbo, Ed. (Marcel Dekker, Inc., 1992), vol. 2, pp. 1-43.

[8] N. S. Allen, Ed., Photopolymerisation and Photoimaging Science and Technology

(Elsevier Applied Science, London, 1989).

[9] C. Decker, in Handbook of Polymer Science and Technology N. P. Cheremisinoff,
Ed. (M. Dekker, New York, 1989), vol. 3, pp. 541-608.

160

[10] B. Ranby, J. Rabek, Photodegradation, Photooxidation and Photostabilization of
Polymers. (Wiley, London, 1975).

[11] J. Franck, E. Rabinowitch, Trans. Faraday Soc. 30, 120 (1934).

[12] R. M. Noyes, J. Am. Chem. Soc. 77, 2042 (1955).

[13] D. F. Eaton, in Advances in Photochemistry D. Volman, G. Hammond, K. Gollnick,
Eds. (Wiley, New York, 1986), vol. 13, pp. 427-487.

[14] T. Forster, Ann. Phys. 2, 55 (1953).

[15] T. Forster, Discuss. Faraday Soc. 27, 7 (1959).

[16] M. Z. Maksimov, I. B. Rotman, Opt. Spectrosc. 12, 237 (1962).

[17] P. J. Wagner, Creat. Detect. Excited State 1A, 174 (1977).

[18] Jenny, J. Opt. Soc. Am. 60, 1155 (1970).

[19] J. A. Jenny, J. Opt. Soc. Am. 61, 1116 (1971).

[20] R. F. Bartholomew, R. S. Davidson, J. Chem. Soc. Chem. Commun. , 2347 (1971).

161

Chapter Nine

Index Changes Upon Photopolymerization

9.1 Introduction

Permanent changes in the optical index of refraction as large as 0.3 may be induced by
photochemical reactions, with typical values equal to 0.04. For instance, crosslinking
increases the density of the polymer. The change in index of refraction is typically linearly
proportional to the fractional density change. Photopolymerization provides a tool to
permanently tailor the optical and structural properties of the material on the microscopic
scale. The application of this technique to self-focusing and self-trapping optical beams is

of particular interest.
9.2 Index of Refraction of Polymer

The Lorentz-Lorenz relation relates the index of refraction of an isotropic material to the

optical polarizability P per unit volume:

m-1lo4gp. 9.1
n27+2 3

This relation holds for relatively high densities of the liquid and solid phase, and is accurate

to within a few percent for nonpolar molecules in the transparent region of the optical

162

spectrum [1]. The polarizability may be expressed in terms of the polarizability a; per

molecule i. For an ideal mixture of j components, with no interactions among components,

=3rd “es Si, 9.2

where Mj is the molecular weight, p; is the density and N; is the number of molecules in

each component j. The refractivity of component j is defined as 4nNj0,/3. Equation 9.2

addresses three basic ways to change the index of refraction:

1. The polarizabilities of the individual molecules 0; can be altered by
chemical reactions.
li. The density of the components can be changed by altering the

molecular structure. For instance, photopolymerization changes
molecular packing and induces substantial density changes.

iii. The relative concentrations of the various components of the sample
can be altered. This requires the addition or removal of material. To
get a large refractive index change, one must alter the relative
concentrations of components that have significantly different

indices of refraction (e.g., remove uncured liquid from sample).

Accurate estimates of the molecular polarizability based on measured bond polarizabilities
have been computed. For example, in diacrylate and triacrylate monomers, the electronic
polarizability decreases upon polymerization because a double bond is replaced by two
single bonds. As described in chapter three, the highly covalent double bond has more
bond charge whose position will be perturbed by the optical field. However,
polymerization results in a substantial increase in density, since van der Waals interactions

between molecules are replaced by covalent bonds which draw individual monomer units

163

together into a solid polymer network. The majority of monomer-polymer systems share
the property of giving polarizability and density contributions to the index change of

opposite signs, yet in many the density change is dominant.

A density change Ap produces an index change [2]:

(n2 + 2) (n2 - 1) Ap 93

An = 6n p

For n = 1.5, this gives An = 0.59 Ap/p. The largest density changes are associated with
changes in molecular packing. This is maximized for reactions of small molecules, in
which the reactive bonds constitute a larger fraction of the molecule. For example, the
density change upon polymerization of a relatively small molecule such as methyl
methacrylate (M = 100) produces a fractional density change Ap/p of 0.254 (see table 9.1).
Methyl methacrylate exhibits a fractional index change An/n upon polymerization of
+0.052. Photochemical reactions are unlikely to produce density changes in excess of
Ap/p = 0.5, corresponding to a An of 0.3. By removing the uncured liquid photopolymer
using a solvent, it is possible to achieve an index change as large as 0.7 between the solid

and air, since the index of typical organic solids is in the range of n ~ 1.4 to 1.7.

164

Monomer: MW Monomer Polymer Ap/p
Density Density
Acrylonitrile 53 0.880 1.35 0.534
Vinyl] chloride 62.5 0.919 1.41 0.534
Methyl acrylate 68 0.952 1.32 0.387
Vinyl acetate 86 0.934 1.19 0.274
Methy] methacrylate 100 0.940 1.179 0.254
Styrene 104 0.909 1.068 0.175
n-propyl methacrylate 142 0.902 1.049 0.163
Cyclohexyl methacrylate 168 1.030 1.142 0.109

Table 9.1: Fractional density change upon photopolymerization. The density change data
is compiled from Ref.[3].

The index of refraction of polymers before and after curing is not typically measured.
However, extensive work has been performed to characterize the volume shrinkage of
polymers upon crosslinking [3-6]. From equation 9.3, the index change can be deduced
from the volume shrinkage. Acrylates exhibit more shrinkage than epoxies. The volume
shrinkage is reduced in a system for which the polymerization propagates through ring
opening, such as epoxies, because the unfolding of the ring counteracts the increase in
density upon crosslinking. Therefore, epoxies typically have shrinkage factors of less than

5%.

9.3 Index of Refraction of Photoinitiator

The mass fraction of the photoinitiator relative to the polymer/monomer background can

range from more than 1 to typically less than a 0.01. High concentrations of photoinitiator

165

are feasible if the absorption exhibits photobleaching at the curing wavelength and if the
photoinitiator is sufficiently soluble in the polymer liquid. The light intensity as a function
of depth in a bleachable photoinitiator is [7]:

Iy
T= FTTexpcK nlp t)(1-expluco™)] ’ oA

where I, is the incident intensity (erg cm-2 sec!), . is the absorption coefficient (cm?
mol!), cy is the initial concentration of the absorber (mol cm-3), and « is the photochemical
efficiency of the reaction. Figure 9.1 illustrates the theoretical light intensity as a function
of propagation depth z at different times t. Individual layers of polymer as thin as the

absorption length may cured, a distance which may be less than an optical wavelength [8].

Intensity
to
il
Tl
bo
co
iT
oo
ay
wat

0 1 2 3 4 5
Penetration (microns)

Figure 9.1: Theoretical intensity distribution at different times in a medium which exhibits
photobleaching. The extinction coefficient is 0.1 microns.

The contribution to the index change arising from photobleaching is related to the
absorption change through the Kramers-Kronig relations [9]. For wavelengths longer than

the center wavelength of the optical transition (for instance, 475 nm for CGI 784

166

photoinitiator), the index increases upon photobleaching (figure 9.2). Therefore, for the
Ar-ion optical beam at 488 nm or 514.5 nm, the structural index change upon solidification
and the index change due to photobleaching are both positive. Alternately, operating at
wavelengths below the center wavelength results in a reduced index change which can be

either positive or negative upon exposure.

Absorption a

~_ etore exposure

after photobleaching

Index of Refraction n

decreases above Wo

Seed

n increases above O,

Figure 9.2: Index change due to photobleaching of photoinitiator.

167

The intrinsic absorption of the monomer/polymer typically is not photobleachable. In fact,
the ring structure of epoxies results in strong absorption by the polymer in the UV
spectrum, leading to very small cure depths. Polymers are classified according to their
intrinsic absorption in table 9.2. The near UV cutoff which distinguishes the two classes

of materials is approximately 290 nm.

Polymers not absorbing in Polymers absorbing at

near UV near UV

Polyolefins Polyphenoxy
Polyacrylics Polyethersulfones
Polystyrenes Polycarbonates
Poly(viny! chloride) Polyphenylene oxide
Polydienes Aromatic polyesters
Polyesters Aromatic polyurethanes
Aliphatic polyamides

Aliphatic polyurethanes

Aliphatic polyethers

Table 9.2: Absorption characteristics of some common polymers [10].

9.4 Dynamic Model of Index Change Upon Photopolymerization

To model the time evolution of the refractive index, the shrinkage or density change is

taken to be proportional to the degree of polymerization:

Ap _g_(Phi 9.5

168

where B is a constant indicating the degree of volume shrinkage. Using equation 9.5 in
addition to the rate equations for [P] discussed in chapter eight, the local index of refraction

can be determined at all times.

0.04 T T T 1
Uo= 2.68 mJ cm-2

Index Change
ro)
NO

diacrylate
0.05 wt% photoinitiator

0 i it al L
0 5 10 15 20 25

Exposure (mJ cm”)

Figure 9.3: Measured change in index of refraction (at 589.3 nm) of the diacrylate
photopolymer under UV exposure at 325 nm. The curve fit is given by equation

9.6 with U, (in energy units) equal to 2.68 mJ cmr?.

9.5 Measured Evolution of the Index of Refraction

The index variation with time is expected to display three distinct responses. During the
first stage, at small times or small exposures, the polymer chains are still small (< 10
monomer units). The bonding of two small chains induces only a small density and index
of refraction change. The photoinitiation efficiency is also small during the early stage of
the exposure because most of the radicals are scavenged by the highly reactive oxygen

radicals [11]. This corresponds to the initial flat induction period in figure 9.3 for

169

exposures less than 7.5 mJ cm-*. During the second stage, larger polymer chains are
formed, so the crosslinking between two adjoining large molecules occurs at several sites.
This induces a more rapid increase in the index. The third stage exhibits saturation of the
index change, occurring near the end of crosslinking. The polymer chains have reached
their maximum length (typically > 10,000 units) and the efficiency of photopolymerization
is reduced because of the cage effect, in which the large viscosity restricts the movement of

radicals [11].

The complex index of refraction n = n +n’ is composed of a real part representing the
index of refraction and an imaginary part representing absorption. The real and imaginary
parts of the index can be solved numerically from equation 9.5 assuming that the index
change due to photobleaching of the absorption is relatively small. An analytical solution to
the chemical rate equations does not exist. However, experimental observations, such as

that presented in figure 9.3, indicate that the index response can approximated by:

An(x,y,z,t) = Ang f - exp nal a . 9.6

The expression for absorption photobleaching is approximately:

An'(x,y,z,t) = Ang exp = | Pa . 9.7

OJo

Note that the index and absorption display different critical exposures Uy and Us,
respectively, which in general depend on time because of the dramatic increase in polymer
viscosity during chain growth. The absorption change occurs very rapidly, since the
characteristic time of optical excitation and bond cleavage of the photoinitiator molecule is

on the order of 10-!5 sec. However, the rate of index change generated by chain

170

propagation and crosslinking is determined by the diffusion time of the radical within the
viscous polymer network to an appropriate monomer unit. In fact, from the measured
index time response, we can infer that the lifetime of the radicals successful in initiating
polymerization is approximately 250 ms for low monomer conversion. This lifetime for
high monomer conversion (i.e., when the index change is 0.9 of the maximum index
change) is in excess of 30 seconds. Note that equation 9.6 neglects the induction period
apparent for small exposures in figure 9.3, because the exposure threshold is reduced in
our experiments by minimizing oxygen scavenging of photogenerated radicals. The curve
fit of equation 9.6 for constant critical exposure Up displays good correlation with the data
presented in figure 9.3, for exposures below 30 mJ cm-2. Beyond this exposure, the

viscosity increases dramatically, so Up and U, both depend strongly on time.
9.5 Summary

The index change upon photopolymerization is large enough to produce dramatic nonlinear
optical effects such as self-focusing and self-trapping of optical beams. The time response
of the index change is an important material consideration. The index must change rapidly
enough to begin to guide the optical beam before a large number of photogenerated radicals
are produced outside the waveguide. The index change displays an absorbed energy

dependence similar to the photorefractive effect.

171

References for chapter nine

[1] M. V. Klein, T. E. Furtak, Optics. (Wiley, New York, 1986).

[2] W. J. Tomlinson, E. A. Chandross, in Advances in Photochemistry J. J. N. Pitts, G.
S. Hammond, K. Gollnick, Eds. (Wiley, New York, 1985), vol. twelve, pp. 201-281.

[3] J. S. Arney, L. Sanders, R. Hardesty, J. of Imaging Sci. and Tech. 38, 262-268
(1994).

[4] J. d. Boer, R. J. Visser, G. P. Melis, Polymer 33, 1123-1126 (1992).

[5] P. Karrer, S. Corbel, J. C. Andre, D. J. Lougnot, J. Polymer Sci.:Part A: Polymer
Chem. 30, 2715-2723 (1992).

[6] A. M. Duclos, J. Y. Jézéquel, J. C. André, J. Photochem. Photobiol. A: Chem. 70,
285-299 (1993).

[7] J. H. C. Kessler, J. of Phys. Chem. 71, 2736-2737 (1966).

[8] N. Biihler, Chimia 47, 375-377 (1993).

[9] A. Yariv, Quantum Electronics. (Wiley, New York, 1989).

[10] C. Decker, in Handbook of Polymer Science and Technology N. P. Cheremisinoff,
Ed. (M. Dekker, New York, 1989), vol. 3, pp. 541-608.

172

[11] G. Odian, Principles of Polymerization. (Wiley, New York, 1981).

173

Chapter Ten

Introduction to Self-Trapping and Self-Focusing

10.1 Introduction

Self-trapping and self-focusing of optical beams based on the Kerr effect are familiar
phenomena extensively reported in the literature. Self-focusing describes the formation of
a light-induced lens within the nonlinear medium. Self-trapping occurs when this self-
focusing exactly balances diffraction, so the beam maintains a constant diameter as it passes
through the medium. The conditions under which an optical beam is self-trapped due to the
Kerr effect were first described by Ciao et al. [1]. In a Kerr medium, the index of

refraction is dependent on the optical field amplitude E through the relation:
N=no+n7E? . 10.1

For media with positive Kerr constants n2, the index increases in regions of high optical
intensity. This induces waveguiding of the original optical beam, which balances
diffraction. For instance, if the beam diameter is D, the beam expands by diffraction in a

homogeneous media with an angular divergence given by

~ 1.22-A_ | 10.2
0 aD 0

174

where A is the wavelength of light in vacuum. If the critical angle for total internal
reflection at the interface between the bright and dark regions of the transverse intensity
profile is greater than @ given by equation 10.2, then spreading by diffraction is absent.
The transverse intensity profile of the beam will remain constant, independent of the
propagation distance. By definition this is called self-trapping. In general, the diameter of
a trapped beam may be slightly modulated as if the waveguiding were due to a periodic
sequence of convex lenses; however, the average diameter along the propagation direction

is constant. For small angles, the beam will be trapped for powers P greater than:

_ ED? ngE*e 5 (1 95 2c
p= 2D" Po” >(1. Van 10.3

For typical materials, the critical power for trapping is believed to be within one or two
orders of magnitude of 106 W. As we will see later, no critical power is needed for self-
; trapping in photopolymers, because the index change depends on optical exposure rather

than intensity.

The Fresnel! diffraction equation for the electric field amplitude E with wavevector k, in a
Kerr medium with small index modulation ny, is:
dE , dE

2ik + 52° 2k?nonJE’E = 0. 10.4

Note that we are dealing with trapping in one transverse direction only (i.e., x) to simplify
the analysis. These results will readily generalize to trapping in two dimensions. The

trapped solution (i.e., dE/dz =0) to equation 10.4 is:

E,(x) = E,(0) sech*Tx , 10.5

175

where

r = 5 Bako E,0) . 10.6

Closed form analytical solutions to equation 10.4 demand overly restrictive assumptions.
This dilemma will be even more apparent for trapping in photopolymers. For instance,
equation 10.4 ignores the effect of saturation of the index of refraction. A more realistic
analysis requires a numerical simulation. Reference [2] presented a numerical analysis for
beam propagation in a Kerr like medium with a saturable, intensity dependent refractive

index. The modified Fresnel equation used in this study is:

2 2. 2
QE OE , 2k"nongH"E _ 9. 10.7
Oz 9x2 «1+ (B/E?

2ik
Equation 10.7 possesses solutions which exhibit both self-focusing and trapping. It is
common for the beam diameter to oscillate and produce periodic foci. The distance

between adjacent foci is given by:

1/4
(P/Pex) 5° 10.8
[(P/P,,)¥? ~ 1)"

Z,= Tk ag

where

(as/a9)? = 0.273 (P/PEs) . 10.9

As the beam propagates, the refractive index near the axis rises until saturation. The

resultant induced “convex lens” is flat in the center and therefore tends to focus incoming

176

(still nearly parallel) rays into a ring. This produces a characteristic ring-like transverse

intensity distribution, rather than the typical TEM, profile.

The filaments of optical damage observed experimentally in inorganic crystals have been
explained both by a self-trapping and by a transient self-focusing mechanism. Shen [3]
reported that small scale filaments were composed of a continuous series of focal spots.
These observations are consistent with a model based on a time varying focal length.
Lugovoi and Prokhorov [4] also suggested that in some situations filaments are simply
tracks of moving foci, in accordance with the time variation of the input laser intensity
during an individual pulse. Shen’s observations of a 50 micron diameter beam producing
filaments of 10 micron diameter in toluene and 5 microns in CS2 support this moving foci
picture. The finite length of the filament is attributed to the depletion of the incoming laser
power arising from stimulated backward Raman and Brillouin scattering. For a certain

laser power P the focal spot appears at the self-focusing distance [5]

Ze = =-—_—_—_>———— . 10.10
(Pe)! - 1

P,, is the critical power for saturation of the Kerr effect. The position of the focal spot is
then a function of the input laser power, which moves as the power within the pulse

changes.

Recently, optical beams have been self-trapped using the Kerr effect in sodium vapor [6]
and a planar glass waveguide [7]. Self-trapping has also been achieved at low optical
intensities by the photorefractive effect, in which diffraction is balanced by two wave

mixing among the spatial frequency components of the input beam [8-10].

177

10.2 Wave Equation in Inhomogeneous Medium

The photopolymerization reaction leads to a significant increase in the index of refraction.
This change in index may exceed by orders of magnitude that usually associated with the
Kerr effect. The irreversible change in index of refraction of a photopolymer is a function
of absorbed optical energy rather than intensity. While Kerr solitons require a critical
intensity of MW cm for trapping, no critical power is needed for self-trapping in
photopolymers. Self-trapping arises from a fundamentally different mechanism than Kerr
spatial solitons. However, like the Kerr effect, photopolymerization is a local

phenomenon, for the local index change depends on the local light intensity.

The propagation of light in a photopolymer is described by the scalar, three-dimensional,

paraxial wave equation for the optical electric field envelop E in an inhomogeneous

medium:
2ikno 37 + 342 * gy? [n(x,y,z) -ng]E=0 > 10.11

where n and ng are the spatially modulated and unperturbed complex indices of refraction,
respectively, k = 2 1/A is the wavevector for a paraxial beam propagating along the z

direction and A is the wavelength in vacuum. The index of refraction and absorption
described by n depend on optical exposure, time and position x, y, z, where the real part

is:

n=n,+An, 10.12a

and the imaginary part is:

178

n’=nj+An’. 10.12b

The expressions for An’ and An” given by equations 9.6 and 9.7 permit a solution to the

wave equation in the photopolymer at all times.
10.3 Critical Waveguide Diameter

The guidance condition for an optical waveguide of radius r depends on the index change
between the core and cladding. In this case the solid polymer is the core and the liquid
photopolymer is the cladding. In general, the inequality relating the radius to the index

change An’ is [11]:

kr (2n,An)? >2 | 10.13

For a typical An’ of 0.04, the smallest stable diameter of the self-trapped beam is predicted
to be 0.6 um. Therefore, if a TEM,, Ar-ion beam at 488 nm initiates polymerization, the
minimum transverse feature size is on the order of a micron. The actual diameter of the
waveguide is expected to be slightly larger than the wavelength because of the time lag of

the index change upon illumination and because of scattering of light within the waveguide.
10.4 Waveguide evolution

Some fundamental qualitative predictions regarding the evolution of the light-induced
waveguide can be deduced, based on the measured index of refraction dynamics presented
in figure 9.3. Figure 10.1 presents a qualitative model of the evolution of the optical
waveguide upon photopolymerization. At early times and low exposures, an induction

period causes exposure thresholding. The very small index change increases

179

approximately quadratically. Polymerization is initiated first in regions of intense
illumination, so the dimension of the waveguide is smaller that the optical beam. The lens-
like index profile focuses the central part of the beam. At later times, the index increases
linearly with exposure, and the index distribution is a faithful replica of the transverse
intensity profile. Self-focusing is remains strong. For large exposures, the index change
begins to saturate at the center of the waveguide. The index is nearly uniform across the
beam, and self-focusing becomes very weak. At this stage, the waveguide becomes
essentially a step-index fiber. The cladding layer is composed of the unpolymerized
monomer. If the optical beam is strongly confined to the core, the diameter of the
waveguide will not increase significantly upon continued exposure. Of course, for very
large exposures the entire liquid volume will polymerize due to the scatter of light and the
diffusion of free radicals beyond the illuminated region. The index can also begin to
decrease after saturation, as UV illumination begins to break bonds on the polymer
backbone itself. While this is common in positive photoresists, we have not observed this

effect in diacrylate and triacrylate photopolymers.

180

Intensity n
Dinax
r E
0 0 Ey
DOmax
Quadratic
7 r
. Dmax
Linear
6 r

Saturation

Figure 10.1: Three regimes of waveguide evolution upon photopolymerization

10.5. Summary

The propagation of light in a photopolymer is described by the nonlinear wave equation in
an inhomogeneous medium. Both the index of refraction and absorption depend on optical
exposure, time and position. Analytical solutions of the wave equation are overly
simplistic and not described here. However, some qualitative predictions regarding the
evolution of the light-induced waveguide can be drawn based an experimentally observed
index dynamics. A numerical simulation of the propagation of a beam through the polymer

is presented in chapter eleven.

181

References for chapter ten

[1] R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479-482 (1964).
[2] J. H. Marburger, E. Dawes, Phys. Rev. Lett. 21, 556-558 (1968).

[3] M. M. T. Loy, Y. R. Shen, Phys. Rev. Lett. 22, 994-997 (1969).

[4] V. N. Lugovoi, A. M. Prokhorov, JETP Letters 7, 117-119 (1968).

[5] P. L. Kelley, Phys. Rev. Lett. 15, 1005-1008 (1965).

[6] J. E. Bjorkholm, A. Ashkin, Phys. Rev. Lett. 32, 129-132 (1974).

[7] J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E.
Leaird, E. M. Vogel, P. W. E. Smith, Opt. Lett. 15, 471-473 (1990).

[8] M. Segev, B. Crosignani, A. Yariv, B. Fischer, Phys. Rev. Lett. 68, 923-926 (1992).

[9] J. G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. D.
Porto, E. J. Sharp, R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533-536 (1993).

[10] B. Crosignani, M. Segev, D. Engin, P. D. Porto, A. Yariv, G. Salamo, JOSA B 10,
446-453 (1993).

[11] D. Marcuse, Theory of Dielectric Optical Waveguides. (Academic, New York, 1974).

182

Chapter Eleven

Numerical Simulations of Self-Trapping and Self-
Focusing Upon Photopolymerization

11.1 Introduction

Analytical solutions to the inhomogeneous wave equation with indices of refraction given
by equations 9.6 and 9.7 could be obtained only for overly simplistic approximations.
Consequently, we analyzed the beam propagation numerically. The propagation of the
optical wave through the photopolymer is computed by solving the wave equation in an
inhomogeneous medium, where the index of refraction evolves as in chapter nine. The
time rate of change in the index of refraction is much slower than the period of optical
oscillations, so the wave evolution can be solved at a particular time by using the index
profile from the earlier time-step. This optical intensity profile is used to compute the
exposure at every point in the medium, and the index of refraction is then updated before
the next time-step. This iterative process is repeated to simulate both the temporal and

spatial evolution of the optical beam in the nonlinear polymer medium.
11.2 Finite Differenced Beam Propagation Method

The evolution of the electric field amplitude and phase of the optical beam is computed by

using the beam propagation method (BPM). BPM is typically implemented using a finite

183

difference rather than fast-Fourier transform method [1, 2]. The addition of a transparent
boundary condition [3, 4] prevents reflections at the edge of the grid and allows the
boundaries of the solution to be placed closer to the physical region of interest. This

maximizes the transverse spatial resolution.

The finite differenced wave equation for propagation in the z direction and one transverse

dimension x is computed by the symmetric finite difference scheme [1]:

- a Ej.4(z+Az) + b Ei(z+Az) - a Ejyy(z+Az) 11.1
=a Ej.;(z) +c E(z) + a Ejy41(z) .

The coefficients of the scalar electric field E; are defined as:

a = —Az 7 11.2
2{Ax)
b= ae 2 n2(z+Az) - 3) + 2 j kono, 11.3
Ax
and
c= Tag titan) 8) +2 kato 11.4
Ax

The wavefront at any position z+Az is completely determined by the wavefront at position
z. This allows the solution to be solved iteratively in the z direction. Note that the index of
refraction and absorption depend on position (x,z) and time, so the expressions b and c

also depend both on position and time.

184

Equations 11.1-11.4 are a system of linear equations that can be solved efficiently by the
tridiagonal algorithm [5]. This is apparent by expressing equation 11.1 as the matrix
equation A-u = b, where u is a vector composed of the discretized electric field amplitudes
E; at a particular z slice. The right-hand side of equation 11.1 can be evaluated at the input
z= 0 to determine b. By inverting the tridiagonal matrix A, the electric field amplitude can
be solved at each subsequent z. The value of the electric field at the boundary is computed

according to the transparent boundary condition.

11.3 Results of Numerical Simulations

The evolution of the optical intensity profile is computed using the evolution of the index
given by equation 9.6. The parameters of this function are determined by a least squares
curve fit to figure 9.3. We ignore the induction period because the oxygen level in the self-
trapping experiments is dramatically lower than that of the index measurement. Typical
photopolymer compositions used in these experiments have low optical absorption

associated with the photoinitiator (0.48 cm! at 325 nm), so photobleaching is also ignored.

Figure 11.1 illustrates a numerical simulation of the intensity profile of the self-trapped,
one dimensional Gaussian beam in a photopolymer liquid. The gray scale of the simulation
is linearly related to the optical intensity. The waist of the beam is located at z=0, the input
surface of the photopolymer. The beam radius at the input is equal to two wavelengths and
the propagation distance is 103 wavelengths. Note that the horizontal scale is magnified by
a factor of ten compared to the vertical scale, to improve transverse resolution. The left-
most simulation illustrates the evolution of the beam through the homogeneous
photopolymer before any index changes are induced. The time-steps from left to right are
in units of 5 te. te is anormalized parameter chosen to be equal to that time to reach the gel

point at the location of maximum intensity (z=0, r=0) of the optical beam. The gel point

185

corresponds to the 1-1/e point on the index of refraction change versus exposure curve.
Self-trapping of the optical beam over a distance of 500 wavelengths is apparent after an
exposure of 15 to, the right-most figure. The oscillations in the beam diameter evident at 15
t- appear to be a common phenomenon for self-trapped beams. Indeed, this effect has also
been predicted for Kerr spatial solitons [6]. The background absorption length is 103
wavelengths, so self-trapping terminates in approximately this distance because the

exposure remains extremely weak.

These simulations are highly sensitive to numerical precision and the accuracy of the
transparent boundary condition. In left-most simulation in figure 11.1, the beam intensity
is extremely weak after propagating 200 wavelengths because of the strong diffraction.
Beyond this propagation distance, numerical precision becomes inadequate since the
intensity decreases as the inverse square of the propagation distance for the unguided beam.
As a consequence, the on-axis intensity is slightly lower than the off-axis intensity, an
unphysical result. This computing artifact is only apparent in the regions of extremely

weak optical intensity, which lie outside the problem area of interest.

These calculations also reveal self-focusing for the same beam and material parameters.
Figure 11.2 illustrates the focusing of a Gaussian beam with its waist at z = 0, for a
propagation distance of 100 wavelengths. Again, the input beam radius is 2 wavelengths.
The top figure illustrates the diffraction of a Gaussian beam in the homogeneous liquid
photopolymer before crosslinking. The bottom figure reveals strong self-focusing after 15
te. The minimum beam diameter is approximately a factor of 2 smaller than the input waist

after propagating approximately 20 wavelengths.

186

Figure 11.1: Numerical simulations of beam propagation in photopolymer in timesteps
equal to 5 t.. Leftmost beam: t = 0, rightmost beam: t = 15 t,. The vertical scale is
10° wavelengths, horizontal scale is 10 wavelengths.

187

Figure 11.2: Numerical simulation of self-focusing upon photopolymerization. Top: t = 0,
bottom: t = 15 t.. The vertical scale is 100 wavelengths, horizontal scale is 10
wavelengths. The periodic focusing is indicative of self-trapping.

188

11.4 Summary

Numerical simulations of beam propagation upon photopolymerization for typical material
parameters display self-focusing and self-trapping. The diameter of the self-trapped beam
oscillates with propagation distance, similar to the Kerr soliton results described in chapter
ten. These results agree qualitatively with the experimental demonstration to be discussed

in chapter twelve.

189

References for chapter eleven

[1] Y. Chung, N. Dagli, JEEE J. Q.E. 26, 13351-1339 (1990).

[2] L. Sun, G. L. Yip, Opt. Lett. 18, 1229-1231 (1993).

[3] G. R. Hadley, Opt. Lett. 16, 624-626 (1991).

[4] G. R. Hadley, JEEE J. Q. E. 28, 363-370 (1992).

[5] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes
in C-The Art of Scientific Computing. (Cambridge University Press, Cambridge, 1992).

[6] J. H. Marburger, E. Dawes, Phys. Rev. Lett. 21, 556-558 (1968).

190

Chapter Twelve

Applications of Self-Trapping and Self-Focusing in
Photopolymers

12.1 Introduction

Self-focusing and self-trapping of optical beams overcome the diffraction effects which
become significant for optical beams with diameters on the order of an optical wavelength.
The waveguide associated with the self-trapped beam forms microscopic filaments from
which polymeric microstructures can be fabricated. This technique has applications to high
resolution photolithography, microstereolithography and micromachining. These

applications will be described in detail in this chapter.
12.2 Application of Self-Focusing to High Resolution Photolithography

The primary obstacle to the continued increase in integrated circuit density is the limitation
on the resolution and depth of focus in the photolithography stage. The smallest feature

SiZ€ Xin that can be projected by a coherent imaging system is

12.1

*min = 5 ox

and the depth of focus (DOF) is:

191

DOF = —A— , 12.2
2(N.A.P

where A is the wavelength of the illumination and N.A. is the numerical aperture. The
most direct route to smaller feature sizes is to reduce the wavelength from the i-line
standard of today (365 nm) to excimer laser wavelengths (248 or 193 nm). The N.A. is
typically 0.5, so the feature size is on the order of the exposure wavelength. State-of-the-
art semiconductor fabrication facilities in the year 2005 are forecasted to use a 0.1 um
process, at which point the resolution and depth of focus constraints of optical lithography
become severe. Therefore, techniques to extend the performance of optical lithography into

this regime are of particular technological and economic significance.

One proven method to improve the resolution is based on a contrast enhancement layer
(CEL) composed of a photobleachable dye overcoated on the photoresist [1-11]. Upon
projecting a photolithography mask onto the CEL, the regions exposed to the highest
intensity bleach first, while those parts of the layer that receive the lowest intensity are
absorbed. The effect of the photobleachable layer is to increase the contrast of the projected
image by introducing exposure thresholding. Extensive work has been done to develop
materials for contrast enhancement based on bleachable dyes, which has been incorporated

into a 0.25 um i-line process [3].

Improvements in the resolution and the depth of focus may be attained by exploiting the
nonlinear optical property of self-focusing upon photopolymerization [12]. When a liquid
photopolymer is illuminated with UV radiation, the exposed regions crosslink to form a
solid with an index of refraction typically 0.1 to 0.04 larger than the liquid. This exposure
dependent index of refraction leads to self-focusing. Therefore, a standard photoresist may

be overcoated with a resolution enhancement layer (REL) that self-focuses the light (figure

192

12.1) before developing the photoresist. The addition of a REL to present and future
lithography processes has the potential of reducing the feature size to half an exposure

wavelength using existing steppers and photoresists, a factor of two improvement in

resolution.

UV
Tilumination

High Molecular
Mass Polymer

Solubilized
Photoresist

Liquid
Monomer |
Photoresist

Silicon
Substrate

Figure 12.1: Resolution enhancement layer for photolithography.

A numerical simulation of the propagation of a single illuminated line through a REL
(figure 11.2) indicates that the feature size may be reduced by more than a factor of two
upon propagating 20 wavelengths in the medium. Theoretical predictions based on the
measured index changes upon photopolymerization suggest that the optimal REL thickness

for a 0.3 um process is 2 to 3 um. If the waveguiding is strong enough, the focused

193

optical beam becomes trapped, dramatically extending the depth of focus. For instance,
figure 11.2 shows that the beam diameter remains smaller than its input waist while
propagating over a distance of 100 wavelengths. We believe that the addition of a
photobleachable dye to the REL will reduce the oscillations in the beam diameter and trap

the beam at a smaller diameter.

Figure 12.2: Approximate model of self-focusing layer index profile by a quadratic
function of radius from center of optical beam.

The self-focusing can be dealt with analytically by treating the index profile within the
medium as quadratic. For a input beam of large diameter (representative of an individual

pixel, for example), the index profile is

ny (1) = no Aah 12.3

For an input beam of smaller diameter, the index profile is

Ans

n2(r) = No f a 12.4

194

In both instances, the effective focal length is a function of the propagation length of the

medium and the index change An [13]:

fol 1 . 12.5
An sin An L

For the example of figure 12.2, Ang is greater than An;, so fj > f2. This leads us to the
fortuitous condition that the self-focal length decreases with beam diameter. The self-focal

lengths for different feature sizes and index changes are summarized in figure 12.3.

iw)

1 um feature size

0 0a fir _
0 0.02 0.04 0.06 0.08 0.1

Self-Focusing Length (11m)
=)

Index Change

Figure 12.3: Self-focal length as a function of minimum feature size and index change
across profile of optical beam.

There are several requirements on the photopolymer system for a practical REL. First, the
rate of crosslinking within the self-focusing layer must be rapid enough that the light is
focused before the underlying photoresist is exposed. This implies that the exposure

necessary to induce self-focusing should be smaller than the optical energy required to

195

expose the photoresist. Alternately, two different wavelengths can be used, first to create
the lens, next to expose the photoresist. Second, the addition of a photobleachable dye
within the REL will enhance and stabilize the focusing effect. Third, since the excitation
rate of radicals is typically larger than the rate of crosslinking, the light source should be
intensity modulated to allow the material to crosslink partially before the next exposure
pulse. Fourth, the viscosity of the monomer should be sufficiently high that the radicals do
not diffuse into the unexposed regions during the time scale of the optical exposure. Fifth,
the sensitivity of the REL must be high enough that wafer throughput is not sacrificed.
Finally, the REL must be easily removed by a suitable solvent. While these requirements
are numerous, the REL is essentially a thick negative photoresist, so the majority of these

issues have already been addressed in a different context.

Some candidate monomers for resolution enhancement layers are ethylene, isobutylene,
acrylonitrile and vinyl chloride. The distinctive feature of thes materials is a large density
upon photopolymerization. Their chemical structure is described in table 12.1.
Acrylonitrile has be photoinitiated by xanthene dyes (fluorescein, Rose Bengal) in the
presence of reducing agents (phenylhydrazine, ascorbic acid) and oxygen. The amount of
dye required was very small, 0.1% of the weight of the monomer, yet the quantum yield of
monomer consumption was in excess of 4000 monomer units per photon [14]. The
addition of bleachable dyes may further enhance the resolution, as demonstrated with

contrast enhancement layers.

196

Monomer Name: Monomer: Polymer:
ethylene CH =CH, ;
—CH,—CH, —|

CH, T
isobutylene CH, =¢ cu, é-

CH; CH,
acrylonitrile CH» =CH -CN —CH, —H —

CN
vinyl chloride CH, =CN —Cl —CH, -CH _
Cl

Table 12.1: These low molecular weight monomers [15] typically undergo a relatively
large density change upon photopolymerization, and for this reason are well suited
as resolution enhancement layers in the presence of a suitable photoinitiator species
(e.g., Irgacure 369).

197

4 Q

as 5

ne} g

4 3

E c= Liquid Photopolymer

a) —
[4 UV Sensitive
Screen
_t Quartz Cell on 3-Axes

Argon or Nitrogen Line Translation Stage

Figure 12.4: Experimental setup for photopolymerization.

Figure 12.5: Self-focusing upon photopolymerization. Leftmost: photopolymer at t = 0;
Rightmost: photopolymer at t = 30 seconds.

198

We have demonstrated self-focusing in a liquid diacrylate monomer in the experimental
configuration of figure 12.4. The input beam is a 5 mW, TEM,, Gaussian mode at 325
nm from a HeCd laser, with a diameter at the input face of 100 um. The samples are
polymerized within a 1x1x5 cm quartz cuvette filled with a diacrylate photopolymer. Prior
to exposure, nitrogen is bubbled through the liquid to liberate the material of oxygen

scavengers. All experiments are conducted at ambient temperature and humidity.

Figure 12.5 illustrates the evolution of a self-focused beam upon photopolymerization.
The vertical scale is 2 mm, the horizontal scale is 500 um. The illuminated regions
correspond to the visible florescence emitted by the photopolymer. The left-most figure
illustrates the beam profile at t=0 inside the liquid photopolymer. The time increases from
left to right. The right-most figure is taken after approximately 30 seconds of continuous
illumination with 2 mW incident intensity. The position of the beam waist moves closer to
the input plane as photopolymerization proceeds, a manifestation of self-focusing.
However, the beam waist increases rather than decreases. To improve the self-focusing,
the intensity of the input beam should be modulated to allow the slow index time response

to keep up with the rapid photoexcitation time response.

12.3 Application of Self-Trapping to the Fabrication of Polymeric Microstructures

We have used self-trapping to fabricate solid optical fiber cores as long as 3 cm with
diameters of 10 Um to 1 mm. The input beam is a 5 mW, TEM,, Gaussian mode at 325
nm from a HeCd laser (figure 12.4). The samples are grown within a 1x1x5 cm quartz
cuvette filled with either a diacrylate or triacrylate photopolymer. Prior to exposure,
nitrogen or argon is bubbled through the liquid to liberate the material of oxygen
scavengers. Exposures are conducted in regions of the liquid several mm below the

polymer-air interface, to minimize the effects of subsequent oxygen indiffusion .

199

Figure 12.6 shows a series of polymeric fibers of 10 1m diameter and several mm length
still submerged within the liquid photopolymer. The optical contrast in the photograph is a
result of the large index perturbation in the region of the crosslinked solid. Several minutes
after the exposure, the fibers bend downward under the force of gravity, because the solid
fibers are more dense than the liquid surroundings. Each fiber is exposed 30 times for 1/16
second, with a delay of 1 to 5 seconds between each successive exposure. The continued
crosslinking within the waveguide is visually apparent for several second after the UV
illumination is blocked. This observation has been used to tailor the delays between UV
exposures to optimize the self-focusing process. Regular arrays of fibers of 200 im
diameter and 10 mm length have been fabricated in an epoxy-triacrylate photopolymer, as
illustrated in figure 12.7. Each post was formed using a single 1/8 second exposure, so

trapping was not as dramatic. Hence, the diameter of the waveguides are significantly

larger.

Figure 12.6: Series of fibers grown by self-trapping the UV curing beam. The nominal
diameter is 10 um.

200

Figure 12.7: Array of fibers of 200 um diameter and 10 mm length.

The direction of growth of these fibers can be controlled by changing the intensity
distribution of the optical beam at the input. Any perturbation from radial symmetry across
the intensity profile of the beam leads to the filamentation and break-up of the optical beam.
This break-up can be used to produce optical fiber “Y” junctions, for instance. In addition,
by introducing an asymmetry in the intensity profile, the fiber can be steered as it is grown,

because one side of the waveguide will induce stronger self-focusing to deflect the beam.

An array of fiber-like structures can be fabricated to produce a three-dimensional
microlattice. The fabrication technique described here is ideal for producing high aspect
ratio structures with dimensions on the order of optical wavelengths, and index changes as

large as 0.7 between air and the polymer. A primitive three-dimensional lattice of fiber-like

201

filaments can be fabricated using this technique (figure 12.9). The diameter of each solid

filament is 50 um, spaced by approximately 500 um.

Figure 12.8: Three-dimensional fiber lattice microstructure.

Figure 12.9: Three-dimensional fiber lattice with 50 4m diameter filaments fabricated
within a photopolymer filled cuvette.

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12.4 Data Storage in Corrugated Waveguides

By modulating the diameter of these waveguides periodically, the reflectivity spectrum of
the fiber can be tailored. The theory of dielectric waveguides with modulated diameters has
been developed by Marcuse [16-18]. Forward waveguide modes of a particular
wavelength A are coupled to backward waveguide modes by modulating the waveguide
diameter every 1/2n. By introducing many different spatial frequency components, (as in a
chirped distributed feedback grating) information can be encoded in the optical reflectivity
spectrum of each fiber. The presence of a particular periodicity corresponds, say, to a bit
“1”, and the absence to a bit “O”. Each bit will be read out by measuring the amount of
light reflected from the fiber at a particular wavelength address. A tunable laser can read

out data from a multitude of adjacent waveguides simultaneously.

This memory architecture has the advantages inherent in holographic storage systems (e.g.,
fast, parallel access, high information density 10! bits cm-3). In addition, it is unique
because the memory is permanent and the dynamic range is extremely large. The index
difference obtained once the uncured polymer is removed is ~0.5, compared to an index
change of 10-3 in ferroelectric photorefractive materials. It can potentially be applied to a

memory system analogous to a multi-level CD ROM with a terabyte memory capacity.

12.5 Optical Extrusion

The two primary tools for micromachining in silicon are sacrificial layers and anisotropic
etching. These techniques enable microstructures as thick as a hundred microns to be
fabricated. A complementary tool to produce deep structures in silicon is ion beam milling.
In addition, thick microstructures are fabricated in polymers, metals, and ceramics using

the LIGA process [19-24]. This tool uses synchrotron radiation lithography to produce

203

precise masters of three-dimensional microstructures with very high depth-to-width aspect
ratios. A subsequent electroforming stage converts the polymer structure into a metal

master, which can serve as a mold for additional replication.

A new tool for micromachining deep structures based on optical self-trapping is called
optical extrusion [12]. Traditional deep etch techniques such as LIGA project a collimated
beam of x-rays or ions onto a proximity mask whose feature size is typically microns, so
diffraction effects are negligible. However, undesirable diffraction occurs as the
illumination wavelength approaches the feature size. That is, for a high resolution mask

imaged by coherent light at wavelength A in an optical system of numerical aperture N.A.,

the depth of focus Zax is:

a 12.6
2(N.A.?

Zmax =
The depth of focus is typically only a micron for UV illumination at 365 or 248 nm. To
produce a deep structure using UV illumination, the part must normally be built up layer by
layer as in stereolithography [25, 26], where the layer thickness is less than the depth of

focus.

An important exception to this depth of focus constraint occurs when imaging in nonlinear
optical materials such as photopolymers. Strong optical nonlinearities are present upon
photopolymerization. As UV illumination impinges on a photosensitized liquid monomer,
it produces free radicals which initiate polymerization and crosslinking of the monomer to
form a solid. The index of refraction of the solid is typically 0.04 larger than the index of
the liquid, so strong optical waveguiding is induced. This guides the beam through the

liquid without diffraction, leaving a solid structure in its path. The diameter of the beam is

204

maintained as it propagates through the liquid, providing a new tool to create thick

microstructures.

A simple implementation of optical extrusion is illustrated in figure 12.10. A reservoir of
liquid photopolymer is placed over a standard photolithography mask, which is back
illuminated by a collimated source of UV illumination (a He-Cd laser at 325 nm or a
mercury i-line lamp at 365 nm). The absorption is chosen to be sufficiently strong that
only a thin layer of photopolymer will cure at any one time. However, as the absorption of
this layer bleaches, the light is guided to the next layer where the same image is exposed.
As this process continues, a polymeric structure whose cross section is a replica of the
original mask is extruded. Because this is an additive process, other structures can be
subsequently woven-in by projecting masks in different directions. Finally, the uncured
monomer is removed by an appropriate solvent, leaving only the desired part. Like the
LIGA process, these parts may be converted into complementary metal structures by a

subsequent electroforming stage.

These extruded parts have many applications to microfabrication. Optical fiber coupling
segments can be grown from the facets of laser arrays to facilitate fiber coupling. Optical
interconnects, x-ray optics and indeed any structure produced using the LIGA technique
can be fabricated. The primary commercial advantage of optical extrusion over LIGA is the
use of a standard UV light bulb rather than a synchrotron radiation source, albeit at the

expense of a more complicated photopolymer system.

UV source
Mercury i-line

Opaque mask

Thick polymeric structures "extruded" from
back illuminated mask surface

Figure 12.10: Optically extruding thick, polymeric microstructures by using waveguiding
of UV illumination to overcome mask diffraction.

12.6 Self-Organized Semiconductor Laser to Fiber Coupling

Efficient coupling of the highly divergent beams from semiconductor lasers to fibers is a
major technical hurdle. By selecting a photosensitizer/photoinitiator system with strong
absorption at the laser wavelength (typically in the near IR, 0.7 to 1.5 mm), fibers can be
grown from the laser output coupler automatically, guided by the output beam of the laser
itself. In principle, this allows very efficient , self-aligned, tappered, and high numerical

aperture fiber coupling to lasers or indeed any device.

206

12.7 Summary

In summary, we have demonstrated theoretically and experimentally that optical beams are
self-focused and self-trapped upon photopolymerization. Self-trapping is used to fabricate
polymeric fibers of 10 um diameter and 10 mm length. These filaments are used to directly
fabricate polymeric microstructures in polymers. Fibers have been grown in different
directions to “weave” three-dimensional lattices. In principle, this technique provides an
alternative to using x-rays or ion beams to deep etch structures in polymers. Self-focusing
in photoresist systems may enhance the resolution of optical photolithography for
microelectronics. This nonlinear optical phenomenon provides a novel tool to permanently

tailor the optical and structural properties of the polymer on a microscopic scale.

207

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210

Chapter Thirteen

Relevant Publications

Self-Focusing and Self-Trapping of Optical Beams Upon Photopolymerization, A.
Kewitsch, A. Yariv, to be published.

Permanently Fixed Volume Phase Gratings in Ferroelectrics, A. Kewitsch, A. Yariv, M.

Segev, Chapter 9, The Photorefractive Effect, Ed. D. Nolte, (Kluwer Academic,
1995).

Optically Induced Quasi-Phase Matching in Strontium Barium Niobate, A. Kewitsch, G.
Salamo, T. Towe, A. Yariv, M. Tong, M. Segev, E. Sharp, R. Neurgaonkar,
Appl. Phys. Lett. 66, 1865 (1995).

Optical and Electrical Barkhausen Noise Induced While Recording Ferroelectric Domain
Holograms, A. Kewitsch, A. Saito, M. Segev, A. Yariv, R. Neurgoankar,
accepted to JOSA B (1995).

Ferroelectric Domain Gratings Induced by Photorefractive Space Charge Fields, A.
Kewitsch, M. Segev, A. Yariv, G. Salamo, T. Towe, E. Sharp, R. Neurgoankar,
Phys. Rev. Lett. 73, 1174 (1994).

211

Tunable Quasi-Phase Matching Using Dynamic Ferroelectric Domain Gratings Induced by
Photorefractive Space-Charge Fields, A. Kewitsch, M. Segev, A. Yariv, G.
Salamo, T. Towe, E. Sharp, R. Neurgoankar, Appl. Phys. Lett. 64, 3068 (1994).

Controlled, Periodic Domain Formation in Strontium Barium Niobate Induced by Volume
Holograms, A. Kewitsch, M. Segev, A. Yariv, R. Neurgoankar, Jpn. J. Appl.
Phys. 32, 5445 (1993).

Selective, Page-Addressable Fixing of Volume Holograms in Strontium Barium Niobate,

A. Kewitsch, M. Segev, A. Yariv, R. Neurgoankar, Opt. Lett. 18, 1262 (1993).

Electric-field Multiplexing/Demultiplexing of Volume Holograms in Photorefractive Media,
A. Kewitsch, M. Segev, A. Yariv, R. Neurgoankar, Opt. Lett. 18, 534 (1993).

Self-Enhanced Diffraction From Fixed Photorefractive Gratings During Coherent
Reconstruction, M. Segev, A. Kewitsch, A. Yariv, G. Rakuljic, Appl. Phys. Lett.

62, 907 (1993).

Electric Field Multiplexing, Selective Fixing, Overwriting and Erasure of Volume
Holograms, and Quasi-Phase Matched Second Harmonic Generation Using
Optically Induced Domain Gratings, A. Kewitsch, M. Segev, A. Yariv, U.S.
Patent Application No. 08/037,076, January 1993.

Self-Focusing and Self-Trapping of Optical Beams Upon Photopolymerization, A.

Kewitsch, A. Yariv, US Patent Application pending.