Optical Interactions in a Dielectric Material with Multiple Perturbations - CaltechTHESIS

where H(I,q,p)=(j/2)(w/c)(nnx e (I,q)Ae(EO) e (I,p)
lm m
J(I,q,p) =(j/4)(w/c)(n(I,q>nx e (I,q)a e (I+ 1 ,p>
lm m
J(I,q,p) = (j/4)(w/c)(nx el (I,q>alm.emand d is a differential operator with respect to r.
Let's look at (4.2.7) in detail. The mode of order I
couples with

eigenmodes of orders I.

I+l, I-1 with

respective phasemismatches. In Section 3.4, we saw that
the rule of

modes coupling is AI = ±1. But here, in

addition to

AI = ±1, we also have a sel:f~-coupling

o.

The

fact

that

one of

the

perturbations

AI=
is

homogeneous reduces to the self coupling with selection
rule

AI= o. (4.2.7) is the basic equation which can be

used to analyze the general homogeneous AEO interaction.
He r e, we d i d n ' t us e a spec i f i c f o r m of opt i c a 1 ,a c o u s t i c
eigenmodes, or interaction geometries. (4.2.7) does not
allow an analytic solution in general because of the
phasemismatch factors. But if the interaction length is

48
large enough so that only two modes couple with each
other. we can obtain an analytic solution. In this case.
the general equation (4.2. 7) reduces to :
dF(0, 1 ) = H(0,1,1)F(0, 1 ) + J(0,1,2)F( 1 • 2 ) exp[jAkr]
dF( 1 • 2 ) = H(1,2,2)F( 1 • 2 ) + J(1,2,1)F(0, 1 ) exp[-jAkr].
(4.2.8)
where

Ak = Ak< 1 • 2 >.

We can introduce new amplitudes
G(0, 1 ) = exp[-H(0,1,1)]F(0, 1 )
G( 1 • 2 ) = exp[-H(1,2,2)]F(l, 2 ).

(4.2.9)

Then (4.2.8) becomes :
dG(0, 1 ) = J(0,1,2)G( 1 • 2 ) exp[j(Ak+q 2-q 1 )r]
dG(l. 2 ) = J(1,2,1)G(0, 1 ) exp[-j(Ak+q 2 -q 1 )r]. (4.2.10)
where

q 1 = H(0,1,1)/j, q 2 = H(1,2,2)/j.

Next let us define the total phasemismatch

AkT as :

As we see in (4.2.11). the total phasemismatch consists
of the phasemismatch due to the AO effect and that due
to the EO effect; i.e •• q 2 - q 1 .
Let us define the phase mismatch due to EO effect as

49

=-(1/2) (2n/ A.) (n< 1 • 2 )>-l

• 8
• e ( 1, 2) r. . e ( 1, 2) E (ext)
11 mJ 1
1Jk m
+(1/2) (2n/A.) (n< 0 • 1 >>-l

·8
·e (0,1)r·. e (0,1)E (ext)
11 mJ 1
1Jk m

(4.2.12)
If we define changes of indices of refraction of modes
1, 2 due to the EO effect as ,
An

An 2

(1/2)(n< 0 • 1 >>-l

·8
·e (0,1>r .. e (0,1)E (ext)
11 mJ 1
1J k m
= (1/2)(n< 1 • 2 >>- 1

·8
·e (L2>r .. e (1,2)E (ext)
11 mJ 1
1J k m

(4.2.13)
then (4.2.12) becomes :
(4.2.14)
The initial condition is given by:
G( 0 ' 1 ) ( 0 )

= 1

, G( 1 ' 2 ) ( 0 )

= O.

(4.2.15)

The solution of (4.2.10) with initial conditions given
by (4.2.15)
G(0, 1 )

is :

= -j(4fJ(0,1,2)f 2 +(AkT) 2 )-ll 2 exp[jAkTrl

X [C+exp(C+r) - c_exp(C_r)J

G( 1 • 2 )

= jJ(0,1,2)*(4fJ(0,1,2)f 2 +(AkT> 2 >- 1 1 2

so
x exp[-1/2j~kTrl2jsin[4JJ(0,1,2) J 2 +(AkT) 2 )-l/ 2 rJ,
(4.2.16)
C± = 1/2[-j~kT ± j(4JJ(0,1,2)J 2 +(AkT) 2 )-l/ 2 J.

where

From (4.2.16),

we see that

the diffracted

light

amplitude is given by, at the length r = L
p(1,2) =-2J(0,1,2)*exp[j(q 2 -1/2AkT)L]
2 1 2
X [4JJ(0,1,2) J +(AkT) J- /
x sin{[4JJ(0,1,2)J 2 +(AkT) 2 J 1 / 2 LJ}.

(4.2.17)

The diffracted light intensity follows :
(5.2.18)
~AO = JJ(0,1,2)LJ 2

where

is the diffraction efficiency

of the AO effect in the absence of the EO effect.
If we compare the diffracted light intensity
formula for AO and homogeneous AEO interactions, we see
that the only difference is the phasemisrnatch. This was
shown

in detail

in

(4.2.12)

and

(4.2.18).

So the

physical interpretation of (4.2.18) can be explained as
in Fig.4.2.1. If we apply an external electric field, we
change the index of refraction. These changes
by

~n

1,

~n

2,

which

are

same

as

are given

(4.2.13).

The

phasernisrnatch for the conventional AO interaction is
given from

Fig.4.2.1 :

51

2 V=e {t)
V=O

FIG 4.2.1

52

Fig 4.2.1:Wave
principle of

the

vector

diagram illustrating the

homogeneous

acousto-electro-optic

interaction.

Ak : Phase mismatch induced by the homogeneous AEO
interaction.

KA

Acoustic wave vector.

53

(4.2.19)
which is the same as (4.2.11).
Then we can directly write down the intensity formula
from that of the conventional AO interaction which gives
exactly the same formula as (4.2.18).
The above derivation of the formula for the
constant AEO interaction using the change of index of
refraction is possible only for homogeneous EO effects.
If the electric field is spatially varying, we need a
two-grating coupled

mode

equation to analyze the

interaction. This subject will be considered in Chapter
5•

4.3. Experimental verification of the homogeneous AEO
interaction
The wavevector diagram of the specific interaction
geometry we choose for the experiment is shown in Fig.
4.3.1.

We considered anisotropic Bragg diffraction. As

is shown in Fig.4.3.1, the acoustic frequency has been
chosen

to

give maximum diffracted

light intensity

without an external voltage. When the external voltage
is applied, we introduce phase mismatch,

and the

intensity of the diffracted light becomes smaller.
The theoretical prediction for this interaction

54

geometry can be done using the general result derived in
Section 4.2.

Let us first consider the theoretical

diffracted light intensity as a function of the applied
external voltage. In

Fig. 4.3.1,

the acoustic wave

travels in the x-direction. and the external voltage is
applied in the y-direction.

Let the height of

the

crystal be h. Then the amplitude of the electric field
for the given voltage Vis
E2 = V/h.

(4.3.1)

Next. let us define the voltage vAEO as

From (4.2.11) and (4.2.13). we see that

vAEO is given

by
vAEO=().h/L)[(l/2)(n(0,1))-18 •8 ·e (0,1)r·· e (0,1)
11 mJ 1
1)2 m

(1/2)(n(1• 2 >)-l8

·8 ·e r. · e

11 mJ 1

(1. 2 >1-l

1)2 m

(4.3.3)

where ). is the wave length of the optical wave and L is
the interaction.
For our case. AkAo = o. This gives the intensity formula
from (4.2.18)
(4.3.4)

55

INPUT OPTICAL WAVE

LIGHT

FI(; LJ. 3.1

56

Fig

4.3.1

Configuration of the interaction geometry

for the AEO light modulation experiment.
ne

index of refraction of the extraordinary
wave.

n0

index of refraction of the ordinary wave.

57

This is the theoretical expression of the diffracted
light intensity as a function of the applied external
voltage for our interaction geometry.
diffraction efficiency is small,

If the AO

as is true for our

experiment, (4.3.4) becomes
(4.3.5)
We see from (4.3.5) that if the applied voltage is equal
to vAEO, the diffracted light intensity is zero. This
shows that the voltage defined in (4.3.3) is the analog
of the half-wave

voltage for a conventional EO

modulator.
For the experimental demonstration of the result
(4.3.5), we designed a Bragg cell with electrodes to
apply the external voltage. The photograph of the device
is shown in Fig. 4.3.2. We chose LiNb0 3 as the crystal
of the Bragg cell. As shown in Fig. 4.3.2, the shear
acoustic wave with polarization in the y-direction is
launched

from

the

transducer glued

on the

(1,0,0)

surface. This acoustic wave propagates in the xdirection, and the center frequency of the acoustic wave
was chosen to be 20 MHz. The velocity of the acoustic
wave is 4.2xlo 5 em/sec. The input impedance of the
transducer was chosen to be 500.

Then two metal

electrodes were evaporated on the (0,1,0) surfaces and

58

59

Fig

4.3.2 : Photograph of the device for the AEOlight

modulation experiment.
Crystal is LiNb0 3 •

60
connected

to

high-voltage power

supply.

The

specifications of the device are summarized in Table
4.3.1.

we used a 5 mW polarized He-Ne laser as a light
source. The input optical wave was polarized in theydirection. propagating at an angle 1.8 deg with respect
to the z-axis. This gives the maximum diffracted light
intensity for the 20 MHz acoustic wave. The above angle
has been calculated.

In the experiment.

the Bragg

rotated

ce 11

and

we illuminated

it to find

the maximum

diffracted light intensity. The angle experimentally
determined

isthe same as thatcalculated.

Also. the

polarization of the diffracted light was measured and it
was in the x-direction. As expected. this interaction
was anisotropic.
polarizer

to

After the Bragg cell.

block

the

undiffracted

we used a

light.

This

decreased the background light and increased the
accuracy of the measurement. A spherical lens with focal
length 60 em was used to focus the output diffracted
optical plane wave on a detector.
we measured the maximum diffracted light intensity
for the 20 MHz acoustic frequency. Then we increased the
external

voltage up to

7.5

kV

and

measured

the

corresponding diffracted light intensities. we found
that at 6 kV the diffracted light intensity was minimum.

Table ~.3.1
Specification of the Bragg cell
• Crystal : LiNb0 3 .
• Size of the crystal : 40-7-12(x-y-z> (mm).
• (0,0,1)surfaces : Polished and A.R. coated.
• Optical wave : 632.8 nm (He-Ne laser).
• Acoustic wave : shear wave with a polarization [0,1,0]
and propagating in the [1,0,0] direction.
velocity is 4.2x10 5 (em/sec).
• Acoustic wave center frequency : 20Z S(Mhz).
• Diffraction efficiency : 10 ('/watt).
• R.F. input power

2 watts.

• (0,1,0) surfaces

metal electrodes.

• Size of the transducer : S-10(y-z> (mm).

62
This gave vAEO

= 6 kV. Then we

intensity and

voltage as

normalized the light

prescribed

in

(4.3.5).

we

plotted the relation. and this experimental result is
shown in Fig.

4.3.3.

In Fig.

4.3.3,

we also drew the

theoretical curve which is given by (4.3.5). As we see,
the experimental result agrees with the theoretical
calculation very well.
we calculated the voltage vAEO for ourexperiment.
using the definition (4.3.3).

As shown in Fig.

4.3.1,

the light propagates near the z-axis. So the index of
refraction is 2.29 for the LiNb0 3 crystal.
From (4.3.3) :
(4.3.6)
where we used r 22

r 1 2.

The wavelength of the He-Ne laser is 632.8 nm. and the
height of the crystal is 7 rom. The interaction length L
is 1 em. and the electro-optic coefficient r 12 for the
low frequency electric field is 6.7xlo- 12 we plug all numbers into (4.3.6), we get

s.s kv.

This

calculation agrees with the experimentally measured
value of vAEO within an error. The acousto-optic
diffraction efficiency measured was about 2 "· We used
the elasto-optic coefficient P 66 because the acoustic
wave with polarization in the y-direction travels in the

rt)

....z

>- ~

ex: ex: >

w Cl..
.... wX

.. ..

I •

tl

LAJ

ct

C\J

<(

..........
t::

.

""IV\

..::::rl!J

.......
LL

64

Fig

4.3.3

Experimental results of an AEO modulator.

65
x-direction and

polarizations of optical waves are in

x- and y-directions.

The value of P 66 for the LiNb0 3

crystal is 0.05. Thus, the diffraction efficiency per
unit acoustic power is given by :

where

is the mass density of the LiNbo 3 , which is

4.7810 kg/m; Va is the acoustic velocity. If we plug

numb e r s

into ( 4 . 3 • 7 ) , we g e t

7 • 5 4fo.

Th i s r e s u 1 t

a1s o

agrees well with the experiment.

4.4. AEO modulator

The most widely used methods for wideband light
modulation are either AO or EO effects. Each type of
modulator has its own strength, and

suffers from its

own distinct limitations. We explain about these in this
section in detail when we compare the AEO

modulator

with AO and EO modulators. When we use AO and EO effects
simultaneously, a new flexibility is introduced with
which we can overcome some of

limitations of

two

individual modulators.
Let me first consider

AO and EO modulators

separately. The AO light modulation has been described
in

Section

4.1.

Within

the

bandwidth of

the

AO

modu 1 a tor, we can neg 1 ect the phase mismatch. Then the

66

modulation function is given from (4.1.4)
(4.4.1)

The diffraction efficiency

is proportional to the

acoustic power. And the acoustic power is given by :
Pa

= v 2 /2R.

(4.4.2)

where Vis the amplitude of the signal voltage, R is the
impedance of the electrical network of the acoustic
transducer.
From (4.4.1)

and

(4.4.2),

we see that the amplitude

modulation function is given by :
Ed

Ein sin [aVl,

(4.4.3)

where a is a constant. If the signal is small. we have :
Ed -

(4.4.4)

Thus. if the nonlinear effect which gives
harmonic and intermodulations is small.

rise

to

we get the

modulation according to (4.4.4). In the following, we
call

(4.4.1)

the modulation

function of

an AO

modulator.
The formula for the EO modulator has been given in
(4.1.6). There are two different types of EO modulators.
One is the longitudinal EO modulator for which the

67

electric field is applied in the direction of the light
propagation. In this case, the electric field is given
by E = V /L, where L is the interaction length.

From

(4.1.6), the modulation function for the longitudinal EO
modulator becomes :
lout = [sin(f(V))J 2

(4.4.5)

where f(V) =(1/2)(w/c)(n< 1 >n< 2 >>-l/ 2 s 1 l.·e mJ·
x I e (1 >r · · e <2 >e IV
and
l.Jk m
ekfield.
If we apply the electric field in the transverse
direction from the light propagation,

we have a

transverse EO modulator. In this case, the electric
field is given by V/h, where h is the height of the
transverse dimension. Then the modulation function is
given by :
lout = [sin(g(V))J 2 ,
where

g (V)

(4.4.6)

(1/2)(w/c)(n< 1 >n< 2 >>-l/ 2 e ll.·s m].

x le (l)r· · e <2 >e (ext) IVL/h
l.Jk m
As we see from (4.4.5), the diffracted light intensity
for the longitudinal EO modulation does not depend on
the interaction length L. But for the transverse EO
modulation, it depends on the ratio L/h, and this gives

68
some flexibility to design a better modulator. Let's
define the half-wave voltages which

give the voltages

required for the full modulation :
f(V)

= rr.V/2V 1f L

g{V) =rr.V/2Vrr.T·

{4.4.7)

From (4.4.5) and (4.4.6) the half-wave voltages are

(4.4.8)
V T = [(1/rr.)(w/c)(n(l>n< 2 >>- 1 12 s
1T

·e

x le (l)r· · e <2 >e (ext)IL/hJ- 1

~Jk

1 ~ mJ

{4.4.9)

In the actual modulation. the signal is biased at Vn/2.
Thus the signal becomes

If we plug {4.4.10) into

If we

use

the

Bessel

{4.4.5) and (4.4.6), we have :

function

identities,

(4.4.11)

becomes :
I = (1/2)[1 + 2J 1 Crr.Vm/Vrr.)sinwmt
+ 2J 3 CnVm/Vn)sin3wmt+ ••• ],

(4.4.12)

where Jn are Bessel functions.
As

we

see

in

(4.4.12)

the

EO modulation

creates

69

nonlinear harmonic frequencies.
AEO light modulation is based on the experimental
curve obtained in Section 4.3 for the homogeneous AEO
interaction. We use one fixed acoustic frequency

which

is the center frequency without the external voltage.
The acoustic power is constant. Then the input signal
for the AEO modulation is the external voltage applied
to the device.Thus. we can think of this AEO modulation
as the hybrid of AO and EO modulations. The modulation
function of the AEO modulator is given by:
(4.4.13)
AEO modulator has some advantages or disadvantages
over

AO and EO modulators.

In the following we compare

various characteristics of the AEO modulator with those
of AO and EO modulators.
BANDWIDTH : The bandwidth of the AO modulator is
limited

by

the

diffraction

efficiency.

For

bandwidth we need a small interaction length.
gives effective. small phasemismatch.
from

(4.1.4).

But

the

larger
which

This can be seen

diffraction

efficiency

is

proportional to the interaction length.Thus. there is a
trade-off

between

the

bandwidth

and

diffraction

efficiency for the AO modulator. For the AEO modulator.
we use only one fixed acoustic frequency with constant

70

acoust icpower. Thus, we can have arbitrary diffraction
efficiency if we choose a large interaction length. The
bandwidth of the AEO modulator is thus same as that of
the EO modulator. The bandwidth of the EO modulator is
given by
(4.4.14)

where RL is the shunting resistance and Cis the crystal
capacitance.
Next, the power needed is proportional to ll.f. Thus, the
practical bandwidth of the AEO modulator is limited
primarily

by

the

maximum

power

supplied

by

the

electrical driving circuit with which the voltage is
applied across

electrodes.

DIFFRACTION EFFICIENCY : As discussed in the above
for

the

AEO

modulator,

the

diffraction

efficiency

depends on the acoustic power supplied by the acoustic
port and the interaction length. Thus,there is no tradeoff between the diffraction efficiency and bandwidth.
Furthermore, the piezoelectric
affect,

unlike the

AO

case,

transducer
the

does

not

bandwidth of the

modulator.Therefore, it can be designed to maximize the
diffraction efficiency by

increasing the transducer

width and thus the interaction length.

71

HALF-WAVE VOLTAGE : Half-wave voltages defined for
EO and AEO modulators are

measures of the voltage

that

gives full modulation. We derived expressions for halfwave

voltages

in

(4.4.9)

for

the

transverse

EO

modulation, and in (4.3.3) for the AEO modulation. From
these expressions we see that the voltage level required
for the AEO modulator is, within a geometricalfactor,
close to unity,

twice that needed in a transverse EO

modulator of the same geometry.
SYSTEM ALIGNING : The modulated light is angularly
separated from the undiffracted light,
anisotropic

AO

EO modulator.

because we used

interactions. Thus, as compared with an
the need for ananalyzer is eliminated.

Furthermore, if we use anisotropic AO diffraction, we
don't need the input polarizer either. The alignment of
an AEO modulator thus (like an AO modulator) is almost
insensitive

to

the

direction

perpendicular

to

the

interaction plane. This is to be compared to the small
numerical aperture,

in both directions,

of an EO

modulator , limited by the natural birefringence. If we
use an analyzer for the anisotropic AO modulator, the
signal -to -noise

ratio

will

be

enhanced, because

polarizations of the diffracted and undiffracted light
are orthogonal,

and

the

analyzer

suppresses the

undiffracted light. The acoustic port can be used to

72

dynamically align the modulator in the plane of the
interaction. by changing the center frequency. and to
compensatefor the intensity of the acousticpower . The
above factors make the system aligning of the AEO
modulator very easy compared with the EO modulator.
EXTENDED BEAM MODULATION : For an AO modulator, the

modulation is done by the travelling acoustic wave. This
means there is an acoustic transit time limit for the AO
modulator. But in the AEO modulator the modulation is
accomplished not by the acoustic wave but by the
voltage. Thus.

an extended collimated optical beam can

be modu 1 a ted.
MODULATION DEPTH

The modulation depth of a

modulator is defined as :
(4.4.15)

For the EO modulator the bias voltage which gives the
largest linear region is Vn/2, because the modulation
function is sin 2 CnV/2Vn>· One of
criteria of the
linearity of the nonlinear modulation function is the
total harmonic distortion (THO). Let's first define THO.
using an arbitrary modulation function E(V(t)). If we
consider a sinusoidal input voltage with bias
V(t) = vb + vd coswt.

(4.4.16)

73

the modulated output follows :

= E(Vb

E(t}

+ Vd

coswt).

(4.4.17}

Now E(t) is a periodic function with period 2n/w and
even in t. Thus. we can expand E(t) as a Fourier cosine
series :
E(t)

= E(n) cosnwt.

(4.4.18}

The total average output power is given by

/E 2 (t)dt
2n/w[(E(0}} 2 + 112'2: (E(n)} 2 ].

(4.4.19}

THD is defined as :

The modulation function of an AEO modulator has no
obvious bias point because of its sinc 2 nature. Also. it
has a second order harmonic. But EO modulator does not
generate a second harmonic. Thus, it is difficult to use
the ratio of harmonics to compare the linearity. This is
the reason why we chose THD to compare the linearities.
First. as a reference point. we calculated the THD of an
EO

modulator,

which

gives

the

harmonic/third harmonic ) 1 "· Then.
we calculated

THD's for

ratio

first

for various biases

different ranges of the

74

modulation voltage.
voltage range,

Finally,

we chose the bias and

which gives the maximum MD

previously fixed THD.

for

the

The result of the computer

calculation shows that the MD for the EO modulator is 48
~.

and

the MD for AEO modulator is 44

with bias 1.38.

This shows that AEO and EO modulators have almost the
same linearity.

4.5. AEO deflector
An optical deflector is a device which can change
the

direction of

light propagation.

Among

many

deflectors AO and EO deflectors are used widely. We can
control the deflection angle electrically for both
deflectors. In this section we concentrate on the AO
deflector. In the AO deflector we change the acoustic
frequency to change the deflection angle. But as we have
seen. if the acoustic frequency deviates from the center
frequency, the deflected light intensity drops. Thus.
the total deflected angle depends on the bandwidth of
the AO device. One of the important figures of merit of
the deflector is the number of resolvable spots. This
quantity NR is defined as :
NR

where

= A.p I 0.. I nW) ,
AP

(4.5.1)

is the total deflected angle inside the

75

crystal, n is the index of refraction, A is the optical
wavelength in vacuum. and W is the beam width of the
optical wave.

In (4.5.1} we assumed a collimated

optical beam with width w. The diffraction-limited angle
of this optical beam is A/nW . Let's first derive a
relation between A~

and the bandwidth Af. The general

wave vector diagram for an AO deflector is shown in Fig.
4.5.1(a}. From the figure we have the following equation
which contains

A~/2

(K + AK} 2 = (2nn 1 /A} 2 + (2nn 2 /A} 2

2(2nn

(4.5.2}

1 /A}(2nn 2 /A}COS(~+A~/2},

AK = 2nAf/Va(Af is the one-sided bandwidth of

where

the device}, n 1 and n 2 are indices

of refraction for

the two polarizations.
From Bragg condition, we have
K2

(2nn 1 /A} 2 +(2nn 2 /A} 2 - 2(2nn 1 /A}(2nn 2 /A}cos(J)}.
(4.5.3}

Here we assume
refraction

A~/2

to be small, so that the index of

does not change

within the angle

we assume AK/K << 1. Then from

A~.

Also

(4.5.2} and (4.5.3}, we

have :
(4.5.4}

76

(a)

(b)

FIG LJ. 5 J

77

Fig

4.5.1:

Wave

vector

diagram

illustrating

phasemismatch compensating AEO deflector.
(a) Ordinary acousto-optic deflector.
(b) Phasemismatch compensated AEO deflector.

the

78

If we use ln 1 -n 2 1/n 1 or n 2
NR

<< 1, (4.5.4) becomes

[W/(Vacos(Ji/2))](2Af),

(4.5.5)

where 2Af is the full bandwidth of the device.

(4.5.5)

shows that the number of resolvable spots is the product
of the bandwidth and the transit time of the acoustic
wave across the optical beam.
One method of increasing the number of resolvable
spots is to increase the bandwidth of the device. As we
have seen in the AEO modulator, there is a trade-off
between the bandwidth and diffraction efficiency.

Thus

it is difficult to increase the bandwidth by decreasing
the interaction length. The full bandwidth of the device
is

determined

by

the

transducer

bandwidth

and

the

phasemismatch introduced by the deviation of the
scanning acoustic wave frequency from the center
frequency. The transducer bandwidth is determined by the
electrical matching network of the transducer.

But the

bandwidth limited by the phasemismatch can be corrected
ifwe change the index of refraction, using the external
voltage. This is just the homogeneous

AEO interaction.

Thus, the AEOdeflector isbased on the

phasemismatch

compensation which manifests in the homogeneous AEO
interaction.
Let's consider the AEO deflector in detail. For

79
each scanning frequency f.

the phasemismatch 4kAo

is introduced as shown in Fig. 4.5.1(b). fc is the
center frequency. The phasemismatch 4kAo is given by :

(4.5.6)
where
The center frequency satisfies
(4.5.7)
From

(4.5.6) and (4.5.7), we have :

AkAO

2 ) - 2 n n cos a I ( n Va) )( f- f c)
+ nA.(f-fc> !n 1 va .
(4.5.8)

( 2 n A. f c I ( n 1Va

From (4.5.8)

we

see that

for

small

deviation of

frequencies, 4kAo is proportional to (f-fc>· The total
phasemismatch of the constant AEO interaction follows
from (4.2.11)
(4.5.9)
If we use (4.5.8) to first order in (f-fc>• we obtain :
4kT

(2nA.fcl - 2nn 2 cosal>
+ (2niA.)((112)n1-1aliamjel (0,1)rijkem(0,1)ek

- (1 I 2)n 2 -1 a 1 iamjel (1,2) rijkem (1,2) ek)VI h .
(4.5.10)

80

Now for each frequency we want AkT =
phasemismatch. and

get

o. Then we get no

the maximum diffracted light

intensity. From (4.5.10) the relation between the
compensating voltage V

and the frequency deviation is

given by :
(4.5.11)
where a

-(2nAfcl - 2nn 2 cos /Cn 1Va))(A/2n)2h
x (n - 1 8 ·8 ·e <0 • 1 >r .. e (O,l)e
11 mJ 1
lJ k m
- n - 1 8 ·8 ·e (1. 2 >r .. e Cl. 2 >e >- 1
11 mJ 1
lJ k m
verify the relation (4.5.11). we did

To

an

experiment with the same device as discussed in Section
4.3. The experimental result is shown in Fig. 4.5.2. The

normalized intensity of the diffracted light obtained
with and without the compensating voltage is plotted as
a function of the acoustic frequency. The compensating
voltage. as a function of the acoustic frequency.

is

plotted in the same figure. From Fig. 4.5.2. we can see
that the bandwidth of the AEO deflector is about 2.5
times larger than that of the AO deflector and was
limited by the electrical bandwidth of the transducer.
Next let's derive the formula of NR

of the AEO

deflector.neglecting the influence of the transducer
bandwidth. The total number NR' is given by :
NR'

a(Af + Vmax F/VAEO) •

(4.5.12)

-Vl

...!::"

Gl

--n

~~

!t::.

. . -0

25

0.0 V

0.2 ~

0.4

0.6

0.8 1-

29

31

~8

10

33

]'

35

..

--12

)(\! c5~

6 ~

~x/~ >

COMPENSATING
VOLTAGE

WITHOUT
VOLTAGE

WITH

"(VOLTAGE

"-x

FREQUENCY (MHz)

27

'-

'=IDEAL RESPONSE

X~

I.O>k "" -~

00
I-'

82

Fig 4.5.2 :

Experimental

compensated

AEO deflector.

results of a phasemismatch

83

where

aAf

NR • Af is the one-sided bandwidth. F is

the whole range of the frequency which compensates the
phasemismatch introduced by the

voltage vAEO • From

(4.5.12) we have :
(4.5.13)
The ratio

F/Af is given by
(4.5.14)

we get the value 2.25 for the above ratio. This finally
gives the number of resolvable spots with compensating
voltage Vmax neglecting transducer bandwidth as :
N '

4.6.

+ 2 2 5V

N (1

max

(4.5.15)

/vAEO)

Novel way of measuring the acoustic transducer

bandwidth
The transducer bandwidth we discussed in Section
4.5 comes from the impedance mismatch of the electrical
network of the transducer. Thus. we can measure this
bandwidth.

analyzing

electrically.
deflector.

the

transducer

network

But from the discussion of the AEO

we can measure . this bandwidth optically.

In

Fig.4.5.2, even if we compensate the phasemismatch. the
output light intensity is not uniform. This is because

84

the acoustic power for scanning frequencies drops due to
the impedance mismatch of the transducer network. Also,
the diffracted light intensity is proportional to the
acoustic power. Thus, if we measure the diffracted light
intensity with the electric field compensation,

we

obtain the transducer bandwidth of the device. As an
example, we see the transducer bandwidth in Fig.4.5.2.

85

5. PBO'l'OREFRACTIVE AEO INTERACTION

5.1. Spatial ABO interaction

In Chapter

4.

we considered

homogeneous AEO

interactions that allows a clear physical interpretation
in terms of the change of index of refraction. ~>~lso. it
is the simplest type of AEO interaction. But to explore
the full potentiality of the AEO interaction. we need to
consider spatially varying electric fields as well.

The

mathematical tool useful in treating this spatial AEO
interaction is

the two-grating coupled mode equation

developed in Section 3.4. In general. we cannot obtain
an analytic solution for the two-grating coupled mode
equation, if there is a phasemismatch between coupled
modes. This is compared with the case of a homogeneous
AEO interaction. For the homogeneous AEO interaction, we
derive

general

diffraction.

solution

in

the

case

of Bragg

For the analysis we choose a specific case

of three mode coupling

in

the next

section,

and

demonstrate the inhomogeneous AEO interaction in Section
5 .4.

We need spatially varying electric fields (i.e .•
gratings ) for the inhomogeneous AEO interaction . The
photorefracti ve
obtaining

effect 9

spatial

is

promising

electro-optic

method

gratings.

If

of
we

86

illuminate

light

intensity

pattern

on

photorefractive crystal. we can generate a corresponding
charge pattern inside the crystal.

This charge pattern

gives an electric field pattern and

EO gratings result

from this field pattern via linear electro-optic effect.
we can also erase this pattern easily.
implement

the

real

time

optical

Thus. we can

signal

processing

system. using the photorefractive AEO interaction.
In the inhomogeneous AEO interaction we have the
freedom to choose arbitrary gratings from different
sources. In this case we may use the nonlinearity of the
interaction of many gratings. This is contrary to the
conventional AO device.

We show a way of using the

intermodulation term in devising a correlator.

5.2. Three-mode photorefractive AEO interaction
The simplest coupling of eigenmodes through two
gratings is three modes coupling. If we assume
Bragg matching.

we can obtain a simple.

perfect
general.

analytical solution. This is interesting in itself. and
furthermore.

we can use

this

correlator.

In this section.

analysis

to devise

we choose a

specific

configuration of three modes and two gratings,
derive

formulae of

and

diffracted light intensities. using

87

the general

two-grating

coupled equation given in

Section 3.4.
Let's consider the wavevector diagram shown in Fig.
5.2.1. One grating is in the near y-direction, and the
other grating

is

in the x-direction.

Thus,

the two

gratings are almost perpendicular to each other. We have
three optical modes. Mode 1 is the incident 1 ight wave
with polarization 1.
photorefractive

This mode 1 interacts with the

grating

diffracted mode 2.

isotropically

Next.

to

give

the

mode 2 interacts with

the

acoustic grating anisotropically as well as with
photorefractive

grating

to

give

the

diffracted

intermodulation light wave(mode 3). We assume that the
phasemismatch between mode 1 and the acoustic grating is
large so that we have only three mode coupling as shown
in the diagram. The coupled mode equation can be written
down from Section 3.4 :

(5.2.1)
where F 1 • F 2 and F 3 are
and

J.l 12

and

J.t 23

are

amplitudes of

optical beams

coupling coefficients;

is a

differential operator with respect to r. As we see in
(5.2.1), we have no direct coupling between

mode 1 and

88

FIG 5.2.1

89

Fig

5.2.1 :

wave vector diagram of a specific

example

of photorefractive AEO interaction.
P.G.: photorefractive grating.
A.G. : acoustic
P.G.

grating

grating.
is

in y-direction.

p: amplitude of the

intermodulation term.

90

mode

3.

This

is

the

consequence of

the

combined

interaction of the two gratings. The initial condition
is given by
F 1 (0) =

F 2 (0) = 0 and F 3 (0)

1,

O.

(5.2.2)

The solution of (5.2.1) with initial conditions (5.2.2)
is easy to obtain and given by

F1 = 1 + £1~121 1<1~121

+ 1~2 3 1 >1

[cos 1 ' 2 r

- 11

F2 = [j~12!(1~1212 + ~~2312>1/21

x [sin 1 ' 2 r 1
F3 = £~12~23•!(1~121

+ 1~231 >1

[cos 1 ' 2 r- 11.

Let's define

diffraction efficiencies

~ 12

and

(5.2.3)
~ 23

as

~12 = <1~121L)2
~23 = <1~231L)2.
Then

(5.2.4)

intensities of

light waves at x = L are

11 = {1 + ~12/(~12 + ~23)

[cos(~12 + ~23)1/2 - 1])2

1 2= [~12 1 <~12
13 =

~23)] [sin(~12 +~23> 112 1 2

[~12~23/(~12 + ~23> 2 1 [cos(~12 + 1'123> 112 -11 2 •
(5.2.5)

91

From (5.2.5) if

23 = o. we have
(5.2.6)

This is the well-known formula of the two modes Bragg
diffraction. If we have ~ 12 = ~ 23 and <~ 12 +~ 23 > 1 1 2 = n;
i.e ••
(5.2.7)

the only nonzero 1 ight wave is mode 3. and the intensity
is 1. This shows that we can transfer all the incident
light energy

into mode 3.

intensity formula of
~ 23 •

and

Also.

if we

look at the

mode 3, it is the product of ~ 12

Thus. we call mode 3 as the intermodulation of

two gratings. For small efficiencies:
~12'~23

<< 1 •

Then we have

(5.2.8)

approximate light intensities

I1 = 1

1 2 = ~12
I 3

This

= 1/4~ 12 ~ 23 •

is

the

incident

usual
light.

(5.2.9)

approximation

of

the

As

in

(5.2.9),

we

see

intermodulation mode is the product of
efficiencies.

undepleted
the

two diffraction

when those efficiencies are small. The

92

ratio

r 3 /r 2

is
(5.2.10}

5.3. Experiment on photorefractive AEO interaction

To verify the analysis in Section 5.2, we performed
experiments using the device described in Section 4.3.
Fortunately. the crystal used for the homogeneous AEO
device was LiNb0 3 • and LiNb0 3 is a photorefractive
material.
The interaction geometry of photorefractive and
acoustic gratings with

three optical modes is as shown

in Fig. 5.2.1. The interaction geometry of the acoustic
wave and optical modes 2 and 3 are the same as that of
the experiment of the constant AEO interaction. The
experimental setup is drawn in Fig. 5.3.1(a}. We used 20
MHz shear acoustic wave. Then we rotated the Bragg cell
to obtain the maximum diffracted light intensity. Next
we used an Ar laser with blue line ( 488 nm > to make
photorefracti ve gratings by interfering two collimated
blue light beams without the acoustic wave. The power of
the laser was 0.7 watt, and the exposure time was 20
minutes. The angle between the interfering beams was 2.6
degrees. Thus we realized the interaction geometry shown
in Fig.

5.2.1.

After writing the photorefractive

94

Fig

5.3.1

Experiment of

the photorefractive

AEO

interaction.
(a) Writing of

the P.G.with two

interfering

beams without A.G.
AEO interaction given by acoustic wave and
the
(b)

photorefractive grating.

Three spots of the incident, first diffracted
and intermodulation lights on the focal plane.

95

grating,

we

used

one

of

the

blue

lights of

two

interfering beams to obtain the first diffracted light
mode 2. Next, we launched the acoustic

wave

to

obtain

intermodulation mode 3. After the Bragg cell we used a
spherical lens with local length 60 em. At the focal
plane we observed three spots corresponding to the three
optical modes
5.3.1(b).

2 and 3,

1,

as illustrated in Fig.

Because the acoustic grating

is perpendicular

to the photorefractive grating, the intermodulation mode
3 is off the line joining the two modes 1,

experiment

the

diffraction

efficiency

2. For our
is

11 12

the

photorefractive diffraction efficiency and it is
constant. But we can change the acoustic power
can

change

and thus

the diffraction efficiency 11 23

The

dependence of 11 23 on the acoustic power is linear.Thus,
the ratio given by (5.2.10) is linear in the acoustic
power. We measure

the ratio (5.2.10) as we increase the

acoustic power. The result is shown in Fig. 5.3.2.
this Figure

In

two scales are arbitrary. For small

acoustic power we have a linear relation. For the large
acoustic power,

however, the relation is not linear.

This comes from the saturation of the r.f.
also measured
condition

amplifier. We

two diffraction efficiencies to check the

(5.2.8).

The

measured

photorefractive

~.

and the maximum

diffraction efficiency was 0.6

96

INTENSilY RATIO

FIG 5.3.2

97

Fig

Experimental

5.3.2

photorefractive AEO
power.

verification

of

the

effect. Pa is the input acoustic

Ex per irnent •

98
acousto-optic diffraction efficiency was 0.2.
Let's calculate the change of index of refraction
induced by the photorefractive grating. LiNbo 3 • which
we used, was not Fe-doped. The diffraction efficiency is
given from the paper by F.

s. Chen, et al. 9 as:

1112 = [sin(n.dnL/(2A.cos(a/2)))] 2 ,

(5.3.1)

where a is the Bragg angle.
For the small diffraction efficiency. we obtain
.dn = (2A.(cos(a/2))11 1 1 2 )/(nL).
In our experiment. A. = 488 nm.

a=

(5.3.2)
2.6 degrees and

11 12

= 0.55%. If we plug these numbers in (5.3.2). we obtain
.An= 2xlo- 6 . This value

is in good agreement with the

result of F.S.Chen, et al.. Also. this is the saturated
value.Next.

we calculate the wave number of the

photorefractive grating. This is an isotropic grating.
Thus we have :

2ksin(a/2)
..., k a,

(5.3.3)

where the approximation is for small a.
If we use a = 2.6 degrees.
The

electric

field

we obtain K = 6x10 3 (cm- 1 >.

induced

by

the

photorefractive

grating can be calculated, using the following equation :

99
An = n 3 r 12 E/2.

(5.3.4)

Now r 12 = 3.4xlo- 12 m/V. we have E = 10 3 V/cm.

5.4.Correlator using the intermodulation mode
The mathematical

definition of one-dimensional

correlation is :

f f(x)h(x-y>*dx

C(y).

(5.4.1)

wecan do this correlation, using the intermodulation
mode of the photorefractive AEO interaction. If the
holographic pattern written

in the photorefractive

crystal is S(x,y), and the acoustic signal delayed in
the same crystal is a(t+x/Va>· then the amplitude of the
intermodulation mode after the crystal is given by :
S(x,y) a(t+x/V a>·

(5.4.2)

We assumed small diffraction efficiencies and used
(5.3.9).

After the crystal we can put a cylindrical

lens with focal length f. Then on the focal plane the
amplitude of the intermodulation mode becomes

f S(x,y)a(t+x/Va> exp[j(xxf/Af)]dx.

(5.4.3)

Thus, the intensity distribution on the focal plane is :

100
I(xf.t.y)

I f S(x.y)a(t+x/Va>exp[jxxf/Af]dxl 2•

(5.4.4)

If we use a pinhole detector at xf = o. we obtain the
correlation between S(x.y) and a(t+x/Va> :
I(xf = o.t.y)

= I /S(x.y)a(t + x/Va>dx 12 •

If we collect all

the intensity.

(5.4.5)

we obtain the

incoherent correlation
C(t.y) = f I(xf.t.y)dxf

= f lsl 2 1a1 2 dx.

(5.4.6)

To demonstrate the correlation by the above method.
we did an auto correlation experiment using the same
device as described in Section 4.3. The bandwidth of the
device is 10 MHz. Thus. we used the simplest pattern
shown in Fig. S.4.1(a).

First. we used a pattern shown

in Fig. S.4.1(a) and two collimated Ar laser beams ( 488
nrn

to write the pattern over

the high-frequency

interference grating inside the crystal. Of course. the
AO device was set before to give the maximum AO
diffraction efficiency at the center frequency 20 MHz.
Next. we generated an electrical signal which exactly
matched the pattern written inside the crystal. when the
signal was delayed by the acoustic wave. The electrical

101

---i6J(a)

--i~

I.BfLsec

(b)
FIG 5 .'L 1

102

Fig 5.4.1 : Experiment of the photorefractive AEO
cor relator.
(a) Input photorefractive pattern. The intensity of
the

left

window

is

twice

that of the right

window.
(b) Electrical signal into the acoustic transducer
which matches the pattern (a).
Oscilloscope trace of the electrical signal.

103
signal was modulated by the center frequency 20 MHz and
shown in Fig. S.4.1(b) and (c). We used a spherical lens
with local length 60 em and collected all the lights of
the intermodulation mode on the focal plane. The
oscilloscope trace of the light intensity is shown in
Fig.S.4.2. This is the incoherent auto correlation of
the pattern shown in Fig. S.4.1(a).

1 04

FIG 5 .4.2

105

Fig 5.4.2 : Oscilloscope trace of the auto-correlation
of the pattern Fig 5.4.1 (a).

106
6. FUTURE RESEARCH

In

previous chapters.

the general

concept of

multiple perturbations applied to the optical signal
processing and optical devices has been developed and
demonstrated using the AEO interaction. The main idea of
this work was that multiple perturbations might give
more

flexibilities

to

play with.

demonstrated successfully in

This

has

Chapters 4 and

the simplest AEO interaction. Thus,

been

s. using

if we consider more

complicated mutiple perturbations.

we may find very

interesting phenomena and can apply these phenomena to
the optical signal processing and devices.
Small effects of the optical interaction between
multiple perturbations is a problem. As we want more
flexibility. this problem becomes more severe. There are
some ways of overcoming this problem in general. First.
we

may

develop

special

susceptibilities of

the

materials

that

interaction

have

large

of multiple

perturbations. People are working on

synthetic organic

materials.

superlattice

liquid

crystals

or

of

semiconductors to obtain large susceptibilities. Another
possibility is to investigate the physical mechanism of
the interaction of multiple perturbations. As an
example.

let's consider AIOHG introduced briefly in

Section 2.1 . As we discussed in Section 2.1. there are

107

two

types

of

interaction.

interactions.
The

One

is

susceptibility of

interaction is very small,

the
this

direct
type

of

and we have no way to

increase the effect except by developing a special
material with a large susceptibility. The second type of
interaction is called induced effect. In this case. we
have two separate phasematching conditions. One is for
the AO interaction.

and the other is for the optical

second harmonic generation. Thus.

if we satisfy

two

phasematching conditions simultaneously, we obtain a
large effect. This has been

demonstrated by Nelson and

Lax 2 • They increased the effect by order of 1000. This
example shows that if we know the physical mechanism of
interactions,

we may enhance the strength of the

interaction. A third way of overcoming

difficulties is

to control the size of the device, so that for the given
value of the susceptibility we can increase
of

multiple perturbations.

This gives

increased strength of the interaction.
wave device is a good example.

amplitudes
an overall

Surface acoustic

In this case.

increase the amplitude of the strain.

we can

Integrated optics

is another example. In this case we may have a large
electric field.

using a small amount of voltage. These

are very interesting areas in which to apply the general
concept of the interaction of multiple perturbations.

108
REFERENCES

1. Murray Sargent III, et al.,Laser Physics (Addison
Wesley, Massachusetts, 1974).
2. D.F.Nelson and M.Lax, Phys.Rev.B 3, 2795 (1971).
3. D.Psaltis,H.Lee and G.Sirat, Appl.Phys.Lett.,46,215

(1985).
4. I.C.Chang, IEEE Trans.Sonics.Ultrason.SU-23,2(1976).
5. J.M.Rouaven,M.G.Ghazaleh,E.Bridoux,and R.Torquet,
J.Appl.Phys,SO,S472(1979).
6.

D.F.Nelsonand P.D.Lazay, Phys.Rev.Lett.17,1187
(1970).

7. A.Yariv and Pochi Yeh, Optical waves in crystals
propagation and control of laser radiation
(Wiley, NY, 1984).
8. N.Uchida and Y.Ohmachi, J.Appl.Phys.40,4692(1969).
9. F.S.Chen,J.T.LaMacchia and D.B.Fraser, Appl.Phys.
Lett. 13,223(1968).