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Frequency chirp and spectral dynamics in semiconductor lasers
Citation
Feng, Jing
(1997)
Frequency chirp and spectral dynamics in semiconductor lasers.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/mahd-9t42.
Abstract
A study of the effects of the longitudinal distribution of optical intensity and carrier density on the static and dynamic characteristics of semiconductor lasers has been performed. Through a self-consistent way, a static model for above threshold operation of a single mode distributed feedback (DFB) laser is developed by calculating the longitudinal optical intensity and carrier density distribution. A dynamic model for large signal modulation of the DFB laser is also presented based on time-dependent coupled-mode equation for electric traveling waves in the laser. The spatial hole burning (SHB) has been analyzed in a quarter wavelength shifted DFB laser and a conventional DFB laser.
A small-signal model is developed by including the optical intensity and carrier density distributions. Expressions are derived for the intensity modulation and resonance frequency, the frequency chirp and FM modulation, and the linewidth enhancement factor. Theoretical analysis of the frequency chirp in the DFB lasers has been used to support our experimental results. The model has led us to a new understanding of frequency chirp in DFB lasers and discovery of the ultra small chirp lasers.
The spectral dynamics and high speed response of uncooled DFB lasers have been studied. The most distinguished element differentiating the uncooled DFB lasers from uncooled FP lasers is that in uncooled DFB lasers; the wavelength detuning plays an important role in determining their spectral and high speed characteristics at high temperatures. Comparing with lasers lasing at gain peak, the DFB lasers with large negative wavelength detune could have better high speed performance at room temperature, but they might have higher threshold current. We can achieve optimum performance of uncooled DFB lasers by choosing wavelength detuning properly based on the laser applications.
Item Type:
Thesis (Dissertation (Ph.D.))
Degree Grantor:
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:
Unknown, Unknown
Defense Date:
6 August 1996
Record Number:
CaltechETD:etd-06032005-160031
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DOI:
10.7907/mahd-9t42
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Frequency Chirp and Spectral Dynamics

in Semiconductor Lasers
Thesis by

Jing Feng

In Partial Fulfillment of the Requirements
for the Degree of

Doctor of Philosophy

California Institute of Technology
Pasadena, California
1997

(Defended August 6, 1996)

Jing Feng

ili

To My Parents and Jing

Acknowledgments

I am very grateful to my advisor Professor Yariv for his guidance, patience, and
assistance during my graduate career. I have been greatly inspired by his scientific vision
and keen physical insight and benefited by his abilities to provide the necessary resources.
It truly has been a privilege for me to work with talented students and research fellows in
his group. The success of my research can be largely contributed to the open, intellectually
stimulating and creative atmosphere Professor Yariv has created and maintained in his
laboratories. His encouragement and trust in my abilities have been inspiring me over the
years and will always be appreciated.

Special thanks to Professor T.R. Chen who provided most of the lasers for my
research and collaborated with me on many projects. I would also like to thank Professor
Amir Sa'ar, who introduced me to the experimental world of semiconductor quantum
structure devices and taught me device fabrication and measurement techniques. I am
grateful to Dr. Bin Zhao, who had spent many days to help me with the experimental set-up
for frequency chirp measurement as well as Drs. Lars Eng, Ali Shakouri and Nao Kuze
who grew the MBE samples for my experiments.

I would like to express my appreciation to Drs. John Jannelli and Thomas Schrans
who closely worked with me on different research projects over the years and with whom I
can always consult. Many thanks go to Yuhua Zhang, Drs. Randy Salvatore, Volnei

Pedroni, Gert Cauwenberghs, Gilad Almogy, John O'Brien, Xiaolin Tong, Dan

Provenzano, Yuanjian Xu, Bill Marshall, Matthew McAdams, and Reginald Lee. Thanks
are also extended to many other current and former members of the group.

I would like to express my appreciation to Jana Mercado, whose administrative
assistance was always appreciated very much. The appreciation also goes to Ali Ghaffari,
Kevin Cooper, and Paula Samazan.

I owe a great debt to the people who made the experimental work possible. In
particular I am grateful to Larry Begay who had skillfully constructed many parts for my
experiment. Without his help, it is impossible to imagine that my research could be so
fruitful.

Beyond the boundaries of Caltech, I would like to thank Dr. Joe Paslaski, who came
to Caltech to help me to adjust the photodiode for my frequency chirp experiment, and
Norman Kwong for his expertise in high speed measurement.

This thesis is dedicated to my father, mother, and my wife Jing, whose love,
support, expectation, and faith in me have always inspired me for excellence and success.

Without their love and support, none of this would have been possible.

Abstract

A study of the effects of the longitudinal distribution of optical intensity and carrier density
on the static and dynamic characteristics of semiconductor lasers has been performed.
Through a self-consistent way, a static model for above threshold operation of a single
mode distributed feedback (DFB) laser is developed by calculating the longitudinal optical
intensity and carrier density distribution. A dynamic model for large signal modulation of
the DFB laser is also presented based on time-dependent coupled-mode equation for electric
traveling waves in the laser. The spatial hole burning (SHB) has been analyzed in a quarter
wavelength shifted DFB laser and a conventional DFB laser.

A small-signal model is developed by including the optical intensity and carrier
density distributions. Expressions are derived for the intensity modulation and resonance
frequency, the frequency chirp and FM modulation, and the linewidth enhancement factor.
Theoretical analysis of the frequency chirp in the DFB lasers has been used to support our
experimental results. The model has led us to a new understanding of frequency chirp in
DFB lasers and discovery of the ultra small chirp lasers.

The spectral dynamics and high speed response of uncooled DFB lasers have been
studied. The most distinguished element differentiating the uncooled DFB lasers from
uncooled FP lasers is that in uncooled DFB lasers; the wavelength detuning plays an
important role in determining their spectral and high speed characteristics at high

temperatures. Comparing with lasers lasing at gain peak, the DFB lasers with large

vil

negative wavelength detune could have better high speed performance at room
temperature, but they might have higher threshold current. We can achieve optimum
performance of uncooled DFB lasers by choosing wavelength detuning properly based on |

the laser applications.

Vili

Contents

1 Introduction

1.1 Optical communication systems

1.2 Single longitudinal mode semiconductor lasers

1.3. Outline of the thesis

Theoretical Model in Distributed Feedback Lasers

2.1 Introduction

2.2 Coupled-mode equations in DFB lasers
2.2.1 Effective index approximation for waveguide modes
2.2.2 Coupled-mode equations

2.3 F-matrix method

2.4 Gain, effective index

2.5 Carrier and photon density

2.6 The lasing condition in DFB lasers

Numerical Simulation of Distributed Feedback Lasers

3.1 Introduction

3.2 Threshold simulation of DFB lasers

3.3. Above-threshold simulation

20-

-28

3.4

Large signal modeling

4 Small Signal and Spectral Characteristics of Distributed Feedback

Lasers

4.1

4.2

4.3

4.4

4.5

Introduction

Solving the coupled-mode equations by Green's Function
4.2.1 Green's function for coupled-mode equations
4.2.2 Green's function solution for electric field

4.2.3 Small signal rate equations

Amplitude modulation

4.3.1 Expression for AM modulation in DFB lasers
4.3.2 Resonance frequency in single section DFB lasers
4.3.3 High speed response in two-section DFB laser
Linewidth, effective linewidth enhancement factor

Conclusion

5 Small Frequency Chirp Distributed Feedback Lasers

5.1

5.2

5.3

Introduction

Simple relation between intensity modulation and frequency chirp
Propagation of optical signals in fibers

5.3.1 Time-bandwidth product of chirped pulses

5.3.2 Transmission of chirped pulses over single-mode fibers

77.

5.4 Theoretical foundation of the frequency chirp reduction in two-section DFB
lasers 86
5.5 Measurement of the frequency chirp in two-section DFB lasers under small
signal modulation 91
5.5.1 Experiment No. 1: Frequency chirp in 1.55um two-section DFB laser 91
5.5.2 Experiment No. 2: Frequency chirp in 1.3m InGaAsP/InP
quantum well two-section DFB laser 97

5.6 Measurement of the frequency chirp in two-section DFB lasers under large

signal modulation 103
5.7 Control of the frequency chirp in DFB lasers 112
5.8 Conclusion 122

6 Spectral Dynamics and High Speed Performance of Uncooled

Distributed Feedback Lasers 126
6.1 Introduction 126
6.2 Gain spectrum 129

6.2.1 Gain expression ; 129.
6.2.2 Collisional dephasing time 131
6.2.3 Gain spectrum in detuned DFB lasers 132
6.3 DC characteristics of uncooled DFB lasers 134
6.3.1 Threshold current density - 134

6.3.2 Carrier leakage in quantum well lasers - ; 136

6.4

6.5

6.6

6.7

6.3.3 Internal loss in quantum well lasers

6.3.4 External differential quantum efficiency

Gain spectrum of uncooled DFB laser

6.4.1 The measurement of gain spectrum

6.4.2 The influence of the corrugation grating on the measured gain spectra
Linewidth enhancement factor, linewidth of uncooled DFB laser

6.5.1 Measurement of @ factor

6.5.2 Calculation of @ factor in uncooled DFB lasers

6.5.3 Linewidth of uncooled DFB lasers

High speed response of uncooled DFB lasers

Conclusion

137

138

139

140

141

150

153

156

159

165

List

1.1
1.2
2.1
2.2
2.3
3.1
3.2

3.3

xii

of Figures

A simple intensity-modulated optical fiber communication system
A coherent heterodyne optical fiber communication system
Schematic diagram of DCPBH DFB laser

F-matrix for DFB laser

General DFB laser structure

Algorithm to solve lasing condition

Algorithm for self-consistency above threshold

The photon, electron density and effective index distribution in a

quarter wave shifted DFB laser above threshold ( Rj=Rp=0.3)

3.4

The photon, electron density and effective index distribution in a DFB

laser above threshold ( Ry=0.1, Rp=0.7)

3.5

The photon density distribution in quarter wave shifted DFB lasers

with different coupling coefficients above threshold ( Re=Rp=0.3)

3.6

Optical response of A/4 phase-shifted DFB laser with pulse current

injection

xiii

3.7 Frequency chirp of 4/4 phase-shifted DFB laser modulated with pulse

function for : (a) e=10°!7cm3, and 10-!6cm"3, (b) @=2.5, and 5
4.1 The calculated 7 parameter for DFB lasers

4.2 The resonance frequency of a two-section DFB laser versus square root

of the optical power for various I, the injection current into section two

4.3 The damping rate of the DFB laser versus the optical power for various

Ip
4.4 A measured linewidth as function of P-! in DFB laser

4.5 The measured linewidth of a 1.55pm two-section DFB laser at different

current ratios

4.6 The calculated linewidth of the two-section DFB laser at different current

ratio of the two electrodes for various assumed values for facet phases

4.7 The calculated threshold and lasing wavelength of the DFB laser

4.8 The longitudinal photon distribution in the DFB laser

4.9 The calculated effective linewidth enhancement factor of the 1.554m
DFB laser

5.1 A two-section distributed feedback semiconductor laser

5.2 A 1.55u1m two-section DFB laser used in experiment

5.3 (a) The measured spectrum width of 1.55~1m two-section DFB laser

(b) The calculated chirp of the two-section DFB laser |

69
70
71

xX1V

5.4 Calculated photon and carrier densities of the 1.55pm two-section DFB

laser

5.5 (a) Calculated Ca(z) and (b) calculated N;(z), N7(z) *Im(Cy(z))

5.6 Experimental setup for measuring FM/AM response in DFB lasers

5.7 A typical output intensity from Fabry-Perot interferometer

5.8 The measured B/m of 1.3m InGaAsP/InP QW two-section DFB laser

5.9 The phase difference 02-0; as a function of the current density ratio
iofi; from the measured FM spectrum of the two-section DFB laser under AM

modulation

5.10 Measured threshold of the 1.3m two-section DFB laser (L;=400um,

L2=100pm) for various current ratios

5.11 Experimental setup for the measurement of the chirp of two-section

DFB laser under large signal modulation

5.12 The measured frequency chirp of the two-section DFB laser

5.13. The calculated frequency chirp for the two-section DFB laser
5.14 | Optical pulse propagation in optical fibers

5.15 Time-averaged power spectra from directly modulated DFB lasers

5.16 The measured time-averaged optical power spectrum of a two-section

DFB laser at 5 GHz modulation for various injection current distributions

-98

101

102

104

106

107

110

113

114

116

5.17 The calculated spectral density of the optical field of a single mode laser

under 5 GHz modulation

5.18 Calculated photon density and the product of Nj(z) and Im{Cy(z)} in

the two-section DFB laser

5.19 Calculated phase difference 02—@; of the directly modulated two-

section DFB laser

6.1 Optical gain of 1.3 InGaAsP material as function of wavelength for

various carrier densities

6.2 The calculated optical gain of 1.3 InGaAsP material as function of

wavelength at various temperatures for carrier density N-=6x10/8cm-3

6.3 Calculated gain of 1.3um InGaAsP material as function of injection

carrier density for various wavelength detunes

6.4 Measured L-I of a 1.3um InGaAsP/InP QW DFB lasers at various

temperatures

6.5 The spectrum of a 1.3m InGaAsP/InP QW DEB laser at 20°C
6.6 The spectrum of the DFB laser at 40°C

6.7 The spectrum of the DFB laser at 60°C

6.8 The spectrum of the DFB laser at 80°C

6.9 The spectrum of the DFB laser below threshold for A=1290 nm-1315

119

120

121

129

132

133

136

143

143°

144

145

XV1

6.10 The spectrum of the DFB laser below threshold for A=1315 nm-1340

6.11 The estimated modal gain of the DFB laser at 20°C
6.12. The estimated modal gain of the DFB laser at 40°C
6.13 The estimated modal gain of the DFB laser at 60°C
6.14 The estimated modal gain of the DFB laser at 80°C

6.15 Measured wavelength detune of the 1.3m InGaAsP/InP QW DFB laser

at different temperatures

6.16 Calculated normalized gain deviation versus normalized deviation

6.17 Measured B/m of 1.3um InGaAsP/InP QW DEB laser as function of

modulation frequency at different temperatures

6.18 Calculated linewidth enhancement factor of 1.3m DFB laser as

function of temperature
6.19 The experimental setup for linewidth measurement

6.20 Measured linewidth of 1.3m InGaAsP/InP QW DEB laser vs. the

inverse of optical power at 20°C, 40°C, 60°C, and 80°C

6.21 The calculated dg/dN ¢g as function of temperature in DFB lasers with

various wavelength detunes

6.22 (a) The measured resonance frequency as function of square root of

optical power in 1.3m DFB laser (AAp=-5 nm) for various temperatures

145

146

147

148

149

152

155

156

158

160

163

Xvii

6.22 (b) The measured damping rate as function of optical power in 1.3m

DFB laser 163

6.23 (a) The measured resonance frequency as function of square root of
optical power in 1.294m DFB laser (AAp=-10 nm) for various temperatures
(b) The measured damping rate as function of optical power in 1.294m

DEB laser | 164

Chapter 1

Introduction

Continuous progress in semiconductor lasers has enhanced the performance of high speed
and long haul optical fiber communication systems. The development of the erbium doped
fiber amplifier (EDFA) [1-3] also greatly extends the transmission distance of optical signals
in fibers. The fastest transmitter has been reported to reach modulation bandwidth of 40
GHz [4]. However, as current digital communication systems approach data rates of 10
Gbits/s, without dispersion compensation, the transmission distance of the optical signals
from directly modulated single longitudinal mode (SLM) lasers is severely limited by large
laser chirp, typically on the order of 1 A for large signal modulation. Numerous research
efforts are pursuing various approaches for alleviating this problem, such as using external
modulators, dispersion compensated fibers [5-6], optical phase conjugation [7-9], and pre-

chirped laser pulses [10-12].

In addition to this limitation, the laser transmitters have to satisfy the following
conditions to be suitable for long haul, high speed optical communications.
For CATV applications:

® Low threshold current

® Single longitudinal mode (SLM) operation

® Narrow linewidth

® Low relative intensity noise (RIN)

1.1 Optical communication systems

Optical communication systems differ in principle from other communication systems only

in the frequency range of the carrier wave. The optical carrier frequency is typically ~100

THz, compared with the microwave carrier frequency of ~ 1-10 GHz. A communication
system consists of a transmitter, a communication channel, and a receiver. Optical
communication systems can be classified into two broad categories: guided and unguided.
In guided lightwave systems, the optical beam emitted by the transmitter remains spatially
confined. This is achieved by using optical fibers. In unguided optical communication
systems, the optical beam emitted by the transmitter spreads in space. In the case of
atmospheric propagation, the signal in the unguided systems can deteriorate considerably
by scattering from free particles. So, for unguided communication systems, the
requirement for the transmitter is quite different from that in guided communication

systems. This thesis does not consider unguided optical communication systems.

= —_——< ] ve A

Single-mode optical fiber
Laser diode Photodiode

Figure 1.1: A simple intensity-modulated optical fiber communication system.

Fig. 1.1 shows a simple optical fiber communication system, in which the laser diode
converts electrical signals into optical signals, which are transmitted by an optical fiber and
received by a photodiode. The laser output power is modulated either through direct
injection current modulation or through an external modulator. This system is relatively

simple in terms of detection schemes.

Local oscillator laser

i LowG,

rad

o— ‘ ——O
Oe Single-mode optical fiber wom
Laser ,diode Photodiode

Figure 1.2: A coherent heterodyne optical fiber communication system.

There is a great interest in more sophisticated optical fiber communication systems
and especially in coherent optical fiber communication systems. These systems make use of
the fact that light is not only characterized by its power but also by its emission frequency
or phase. A simple coherent optical fiber system is sketched in Fig. 1.2. The modulated
injection current now not only yields a modulation of the optical power, but also of the
optical emission frequency. Therefore, the laser diode in Fig. 1.2 will emit a frequency-
modulated signal (for digital signals this is denoted as frequency-shift-keying = FSK). At
the réceiver, the signal will be coherently added to the optical signal of a local oscillator
laser. At the photodiode, a beat signal between transmitter laser and local oscillator laser is
created and spectrum of the photodiode current peaks at the beat frequency. Therefore,
according to the optical frequency modulation of the transmitter, the photodiode current is

frequency modulated and may be demodulated by conventional electronic circuits.

1.2 Single longitudinal mode semiconductor lasers

The most widely used SLM lasers are distributed feedback (DFB) lasers, which use a
frequency dependent feedback from a corrugation grating (Bragg grating) close to the active
layer in the axial direction, to obtain SLM operation above threshold. In such structures an
optical wave traveling in the + z -direction is successively reflected by the grating into a
wave traveling in the -z-direction and vise versa. The opposing traveling waves are
strongly coupled with each other due to the existence of the grating. As a result of the
coupling between the waves, the optical intensity is not uniform over the whole laser
length. Through stimulated emission the optical intensity will induce a nonuniformity in the
carrier density distribution and the associated gain and refractive index. The laser
characteristics become power dependent through the nonuniform longitudinal distribution
of optical power and carrier density.

There have been intensive efforts to improve the material parameters such as
linewidth enhancement factor and differential gain for achieving low chirp and high speed
lasers [13-16]. Many of these technical achievements are based on joint progress in the
material growth technologies and theoretical understanding of quantum well structures.
The use of molecular beam epitaxy (MBE) [17] and metal organic chemical vapor
deposition (MOCVD) [18-19] to grow ultra thin layers (on the order of ten atomic layers)

also paved the way for the development of new types of semiconductor lasers.

To study spectral dynamics and high speed response of semiconductor lasers, most
models consider the oversimplified uniform cavity. The complete optical cavity structure is
lumped into one parameter, the photon lifetime. One can see from above arguments that it is
needed to develop a model, which also considers the longitudinal distributions and their
power dependence, for DFB laser transmitters in high speed and long haul optical

communication.

1.3 Outline of the thesis

This thesis begins with the investigation of the effects of the longitudinal distributions on
the laser characteristics. A comprehensive model for SLM DFB lasers will be developed,
where the longitudinal distribution of the power and carrier density will be taken into
account in a self-consistent way. Chapter 2 describes the static theory which calculates the
power and carrier density distributions in a self-consistent manner. The interaction between
the power distribution and carrier density distribution as well as gain and refractive index-
distributions will be discussed. Chapter 3 introduces the computer algorithms in our
calculation of static and dynamic characteristics of DFB lasers. A small-signal dynamic
theory is described which includes the longitudinal power and carrier density distributions

as well as structural effects.

In Chapter 5, frequency chirp and its measurement in complex devices is analyzed for
several two-section DFB lasers with supporting experimental results. A new approach to
reduce frequency chirp is described for nonuniform current injection in the DFB lasers.

Lastly, the spectral dynamic and high speed response characteristics of uncooled DFB

lasers with wavelength detune are studied theoretically and experimentally. ~

Reference

[1] E. Desurvire, J.R. Simpson, and P.C. Becker, Opt. Lett. 12, 888 (1987)

[2] R.J. Mears, L. Reekie, J.M. Jauncy, and D.N. Payne, Electron. Lett. 23, 1026 (1987)
{3] Y. Kimura, K. Suzuki, and M. Nakazawa, Electron. Lett. 25, 1656 (1989)

[4] S. Weisser, E.C. Larkin, K. Czotscher, W. Benz, J. Daleiden, I. Esquivias, J.
Eleissnser, J.D. Ralston, and B. Romero, JEEE Phot. Tech. Lett. 8, 608 (1996)

[5] C. Lin, H. Kogelnik, and L.C. Cohen, Opt. Lett. 5, 476 (1980)

{6] H. Izadpanah, C. Lin, J.L. Gimlett, A.J. Anots, D.W. Hall, and D. K. Smith,
Electron. Lett. 28, 1469 (1992)

{7] A. Yariv, D. Fekete, and D. Pepper, Opt. Lett. 4, 52 (1979)

[8] R.A. Fisher, B-.R. Suydam, and D. Yevick, Opt. Lett. 8, 611 (1983)

[9] S. Watanable, T. Naito, and T. Chikama, IEEE Phot. Tech. Lett. 5, 92 (1993)

[10] K. Kishino, S. Aoki, and Y. Suematsu, IEEE J. Quantum Electron., QE-18, 343
(1982)

[11] T.L. Koch, and R.C. Alferness, IEEE J. Lightwave Technol. LT-3, 800 (1985)

[12] N. Henmi, T. Saito, and T. Ishida, JEEE J. Lightwave Technol. LT-10, 1706
(1994)

[13] Y. Arakawa, and A. Yariv, IEEE J. Quantum Electron., QE-21, 1666 (1985)

[14] L.F. Lester, S.S. O'Keefe, W.J. Schaff, and L.F. Eastman, Electron. Lett. 28, 383
(1992)

[15] K. Uomi, S. Sasaki, T. Tsuchiya, H. Nakano and N. Chinone, JEEE Phot. Tech.
Lett. 2, 229 (1990)

[16] H. Yasaka, R. Iga, Y. Noguchi, and Y.Yoshikuni, JEEE Phot. Tech. Lett. 4, 825
(1992)

{17] A.Y. Cho, J. Vac. Sci. 8, 31 (1971)

[18] R.D. Dupuis, and P.D. Dapkus, Appl. Phys. Lett. 31, 466 (1977)

[19] R.D. Dupuis, P.D. Dapkus, N. Holonyak, E.A. Rezek, and R. Chin Appl. Phys.

Lett. 32, 295 (1978)

Chapter 2

Theoretical Model in Distributed

Feedback Lasers

2.1 Introduction

In conventional Fabry-Perot (FP) semiconductor lasers, the feedback is provided by facet
reflections whose magnitude remains the same for all longitudinal modes. The only
longitudinal mode discrimination in the FP lasers is from gain spectrum. However, since
gain spectrum is usually much wider than the longitudinal mode. spacing, the mode
discrimination is poor. A frequency-dependent feedback [1] is one way to improve the
mode selectivity. A distributed feedback laser has either periodic index perturbation (for
index-coupled), or gain (loss) perturbation (for gain-coupled) along the lasing axis. The
feedback is provided by Bragg scattering, which couples the forward- and backward-

propagating waves. Mode selectivity of the DFB lasers results from the Bragg condition.

The Bragg condition states that coherent coupling between the counter-propagating waves

occurs only for wavelengths such that the grating period [2]

A= mAl2ner (2.1)
where A is the free-space wavelength and m is an integer.

Due to the coupling between the counter-propagating waves, the optical intensity ina
DFB laser is not uniform, which is known as spatial hole burning (SHB) [3-4]. This
nonuniformity in the optical intensity increases with increasingly stronger gratings.
Through stimulated emission the optical intensity will induce a nonuniformity in the
electron density, which results nonuniformity in gain and refractive index distributions. In
this chapter, we will introduce coupled-mode equations and discuss the solving of the
coupled-mode equations with F-matrix method. In order to get nonuniform optical
intensity, the DFB laser is divided into many small segments and F-matrix method [5] is
used to numerically solve the coupled-mode equations. The threshold condition for single
longitudinal mode (SLM) DFB laser is developed [6-7]. The longitudinal dependence of the

two counter-propagating waves and electron density is solved in a self-consistent way.

2.2 Coupled-mode equations in DFB lasers

2.2.1 Effective index approximation for waveguide modes

Figure 2.1 Schematic diagram of DCPBH DFB laser.

We start from the time-independent wave equation

V°E+ &(x,y,z) kfE = 0 (2.2)

where kg=ayc, @ is the mode frequency. € is a periodic function of z. It is useful to write

E(X,V,Z)=E (x,y) + AE(X,Y,Z) (2.3)

where the dielectric perturbation Age is nonzero only over the grating region whose
thickness is equal to the corrugation depth. € (x,y) is the average value of €. The dielectric

perturbation can be written as

jAOW,Y.2)

@ (2.4)

Ae(x,y,z) = n2(x,y,z)-ng(x,y) -

where n?(x,y,Z) includes the grating and waveguide, né(x,y) is the index of the
waveguide, and AO(x,y,z) is due to gain or loss grating.
In the absence of a grating (Ae =0 ) and external pumping, the general solution of

(2.2) has the form

E(x,y,z) =x U(x,y) [F exp(iBz)+R exp(-iBz)] (2.5)
where X is the unit vector along the junction plane (for the TE mode) and the field

distribution corresponding to a specific waveguide mode is obtained by solving

PU&y) , PUY)
ax? oy?

+ [Exy) ki-B°JU@y) = 0 (2.6)
for a given device structure, and where B is the mode-propagation constant. For
simplicity, we assume that the device supports only the fundamental waveguide mode. To
obtain an exact solution of the two-dimensional equation (2.6) is a difficult task. It is
essential to make certain simplifying assumptions whose nature and validity vary from
device to device. The effective index approximation [8-10] has been frequently used in the
problems of semiconductor lasers. Instead of solving the two-dimensional wave equation,

the problem is split into two one-dimensional parts whose solutions are relatively easy to .

obtain. The physical motivation behind the effective index approximation is that the

dielectric constant often varies slowly in the lateral x direction compared to its variation in

the transverse y direction. The U(x,y) is thus approximated by

U(x,y) = Ox) Py) (2.7)

and (2.6) is rewritten as

1 Ox), 1 OPO). yy) 2-827 =0 2.8
Gs) x2 “Woy ay2 1 OM MP T=0- ™”

In the effective index approximation, we have

1 oy)

3 2 R2 _

Yo) ay? + [e(x,y) ko -Bex)] = 0 (2.9)
1 9°92) . 1B2y) -B21=0 2.10
ga) dx? + [Bégx) -B) (2.10)

For a given laser structure, (2.9) and (2.10) can be used to obtain the transverse and lateral
modes, respectively.

E (x,y) includes background (real) refractive index np (y) which is constant for each

layer, and small perturbation \SE (x,y)| « ney) due to the loss and contribution of

external pumping.
E (x,y) = nj(y) + 5€ (x,y) (2.11)

2.2.2 Coupled-mode equations

In presence of the index or gain (loss) grating in single-mode DFB laser, ‘there is some /p

for which B = wo = Bo. Then the electric field is rewritten as

E(x,y,z, 0) =x U(x,y)F(z,0) (2.12)
and the transverse electric field is

F(z,t) = E*(z) exp[-iBoz+i(@pt+A@)t] + E-(z) exp[+iBoz+i(@p+A@)t] (2.13)
where E+, E> are the complex field amplitudes of the forward and backward waves,
respectively, and Wo is the reference frequency. Bo = 27/Apg for lo =2, where Ag is the
Bragg wavelength. A@=@ - Wp, @ is the lasing frequency. The amplitudes E+, E- are

described by the following coupled-mode equations [2,11,12]:

) 5 +i 4
> =iKE™ - (10 +iA@/Vg -g +Qlloss JE (2.14)
“3p = - (16 +i 0/ Vz -g +Qos¢ ) (2.15)

where vz is the group velocity of light in the active region, K is the coupling coefficient, g
is the optical gain, Qos, is the internal loss, and 6 = B - Bo with B as propagation
constant. The coupled-mode equations can be derived from Maxwell's equations using the
slowly-varying amplitude approximations. The detailed derivations can be found in [2, 12].
(2.14), (2.15) have not included the spontaneous emission coupled into the lasing
mode. This can be solved by adding terms in (2.14), (2.15) to account for spontaneous

emission, and the coupled-mode equations with spontaneous emission [7] are

a =ikE™ - (id +iAO/ Vg -2 +Q[os5 E+ + TT (2.16)
OE” . + 1) . - -
3S =iKE* - (16 +iA@/Vg -g +Qjoss JE” + T . (2.17)

T+ are the terms due to the spontaneous emission coupled into the lasing mode [7], and

Tte “2 Et (2.18)
le*)

where J,, is a constant related to spontaneous emission. Tt are written here as

contributions to the gain and are nonlinear terms.

2.3 F-matrix method

PPAR PALIT FFI IMF IF IFILL IOP III

. UJ LJ ¥ LI ' . id

0 4 4 6 ya L

Figure 2.2: F-matrix for DFB laser.

The coupled-mode equations with boundary conditions form an eigenvalue problem with z-
dependent coefficients in the differential equations. The electric waves have to be solved
numerically. The coupled-mode equations are solved by using the transfer matrix or F-
matrix [5] approach, where the DFB laser is divided into many small segments in which the

carrier and photon density, optical gain, and effective index are assumed to be constants...

From coupled-mode equations, the electric fields at the output interface of one
segment are related to the input fields as
( E*(z+Az) \=/( Fir Fy2 ye)
. — - 2.19
E(z+Az) Fy) Fo 7 SE(2) (2.19)

where F’ is the F-matrix[5] and its elements are

Fii=l [cosh(yAz) +i sinh(yAz)] (2.20)

Fy7 =if y sinh(yAz)] (2.21)

Fy, =-if ¥ sinh(yAz)] (2.22)

Fo = [cosh yAz)-i9 sinh(yAz)] (2.23)

where

E = exp[(gip-8ip) Az/2] (2.24)

D = -(Aa/v,+ 6)+i(g-Qloss) - 1 (gsp+85p) (2.25)

P= [i(A@/Vg+5)-(g-Cioss) + K? (2.26) .
zZt+Az

Sip = 2 | dz'I|E*(2 (2.27)

The Sip describe the spontaneous emission effects. The F -matrix depends on the carrier

density at z via the spontaneous emission, the spatial hole burning, and nonlinear gain

suppression.

The propagation equation for the F -matrix is

E*(z))
E (2)

E*(0)
E(0)

= F(z;.1)99*F (21) F (0) (2.28)

which is used to calculate the electric field and photon intensity distribution in a DFB laser.

2.4 Gain, effective index

Conventionally, if the lasing wavelength of a DFB laser is at gain peak, the material gain of

the active region is assumed to be linearly dependent on the electron density

8m (N) =a [N-Niy] (2.29)
where a is the material differential gain, and N;, is the transparency density. However, in
a DFB laser with wavelength detune, e.g., the Bragg wavelength is designed to be away
from the gain peak, the lasing wavelength is not at the gain peak and the material gain as a
function of carrier density has to be calculated numerically. By choosing the Bragg ,
wavelength shorter than that of the gain peak, a larger differential gain can be achieved (see
detail in Chapter 6). In a wavelength-detuned DFB laser [13], the gain is also dependent
on the detune. For carrier density near threshold, the material gain in the DFB laser can be

approximated as

gat (N) =analN-Nj(AA)] . | (2.30)

Here AA is the wavelength detune.

The modal gain is given by [13]

TP ag[N-N, (AA
gt (N) = “ae Te )

(2.31)

where I is the confinement factor and € is the nonlinear gain suppression coefficient.
With considering the SHB in the DFB laser, the photon and carrier densities are not

uniform. The gain can be rewritten as

(2.32)
[ 1+éS(z) ]

The effective refractive index is a combination of n2,{z), which is the value with no

injection carriers, and Ang [N(z)] , which is the result of the injection carriers.

Neff (2) =Nepf(z) + Aner [N(Z)] (2.33)
The linear part of the carrier induced index change is often related to the change in the

carrier density with the following relation

Aneg (N) = - aaa . (2.34) .

@ is the linewidth enhancement factor for the active layer, and A is the vacuum

wavelength. The effective refractive index is written as

Neg (Z) SNe) - aad (z) (2.35)
v(

It will be seen later that, without current modulation, the index dependence on the carrier
density results in the shift of the lasing wavelength. With modulation, the index change

due to the carrier density variation is the major source for frequency chirp in single mode

DFB lasers.

2.5 Carrier and photon density

The carrier recombination will be considered only with respect to spontaneous emission
and nonradiative transitions. If the stimulated recombination is not considered, one may set

up a simple rate equation for electron density in the active layer:

dN — J. R(N 2.36
dt ed ) 259)

with recombination rate being represented by R(N), the injection current density by J.
R(N) may be split into the spontaneous emission term R,,(N), non-radiative term Ry,(N)

and Auger recombination term [14-18]

RW)=Rsp(N)+RnN)t+Rauger(N) (2.37)
The spontaneous emission rate is proportional to the product of the number of occupied
states in the conduction band (~n) and the number of vacant electron states in the valence

band (~p). The spontaneous emission term may be written as

Rsp(N)=Bnp | (2.38)

where B is a recombination coefficient. To some extent the recombination coefficient B
depends on the doping level and decreases with increasing injected carrier density. It is

strongly temperature dependent. B has been measured at room temperature and found to be

of order of [19-22]

B~=0.3 ... 2.0x10°cm3/s (2.39)
for both GaAs and InGaAsP devices. For undoped material we have n=p=N and the

spontaneous emission is

Rsp(N)=BN? (2.40)
which is often denoted as bimolecular recombination.

The non-radiative term R,,(N) is

Rur(N)= AnrN = (2.41)
T is the lifetime of monomolecular recombination.

For long-wavelength semiconductor lasers, however, the Auger process is generally
the predominated nonradiative mechanism. The Auger recombination process involves four
particle (electron and hole) states. In the process, the energy released during the electron-
hole recombination is transferred to another electron (or hole), which excites to a high
energy state in the band. In low doping material we have n=p=N and _ the Auger

recombination may be approximately written as

Rauger(N) = CN? (2.42)

where the Auger recombination coefficient C is of the order of

C=1...3x10%cm/s (2.43)
for lightly doped InGaAsP.
In DFB lasers, the carrier and photon density are non-uniform, and the rate equation

for carrier density can be written as

dN(Z,t) _ J(z,t) _ N(z,t) -BN(z,t)* _ CM(z,t)? _ aanlN(z,t)-Nir(AA)]

VoS(Z,t
dt ed T [ 1+eS(z,t) ] eS(Z,t)

(2.44)

and photon density distribution S(z) 1s related to the forward and backward waves by

2 - 2
S(z) « CE*(@)| 7+ EI) (2.45)
In (2.44), the stimulated recombination, the last term is also included, and under DC bias,

the steady state for carrier density is obtained by setting dN/dt=0

J(z) _ N(z) | 2_ 3 aaalN(z)-N;A(AA)] _
at BN(z)* - CN(z) Teste) | vpS(Z) =O (2.46)

Normally, the photon density distribution is assumed initially. Then, the carrier density and
photon density distributions are solved self-consistently by satisfying the lasing condition.
The detailed description about solving for the carrier density and photon density

distributions is in Chapter 3.

2.6 The lasing condition in DFB lasers

MARR WONMMMOM™X™ OOo

7s FE

p (z) p (z)

RAR RARRRARRRRAARDRAR FARRAR EFSF FFEFRFIT

0 L

Figure 2.3: General DFB laser structure.

Taking some point z somewhere along the cavity, as shown in Fig. 2.3, the cavity is then

divided by two parts, which can be replaced by the effective complex reflectivities, p1(z)

and 99(z) ; p1(z) represents the reflectivity of the left-hand part, and p2(z) the right-hand
_ part. Those reflectivities depend on the frequency @._ and the injected current J. The lasing

condition is that the round-trip gain for p1(z) *p2(z) should be equal to one:

p1(z) e#f1@)po(z) ef92(2) = 1 | (2.47)
here Pi(Z), P2(z), $1(2), $2(z) are the amplitudes and phases of the complex reflection

pi(z) ,p2(z) .

The 91(z) and p2(z) are calculated by means of the propagating matrices an ol),

FR(o,D) of the left-hand and right-hand parts of the cavity. The 91(z) ; p2(z) are defined

as
~ E(z)
= 2.48
Pil2) = =F G) (2.48)
and
~ E*(z)
= 2.49
p22) = = () (2.49)
The boundary conditions are
E“(L)= rp Et(L) (2.50)
and
E*(0)=ryE (0). (2.51)

Here rf and rh are the front and back complex facet reflectivities, respectively, and they are

given by

rp =VRy el%> (2.52)
and

rp=vR¢ el 9 (2.53)

Ry and R, are front and back facet reflectivities for optical power. dy and @» are the
phases of the grating at the facets.

The electric fields are derived by forward propagation

rf
= F(z-Az)***F (Az) F (0)

| Ef (2)
Ej (z)

and backward propagation

| Er@) | _ Flctdcwerttndyre)
ER(z) °
rb

We can get expressions for p1(z) ,p2(z) [6,7,12] as

pi(z) = _ Fout (Fiji
r¢F21+(F1)2
po(z) = _Py(Fr)11-(Fr)21

ro(Fp)12-(FR)22

(2.54)

(2.55)

(2.56)

(2.57)

The lasing condition has to be solved numerically to determine the lasing wavelength and

the threshold current. Above threshold, the lasing condition is used to calculate the output

optical power for a given injection current.

The power level in the laser is set by the output power Phu from the front facet with

Phu Ir) IE (0)

and the output power P?, from the back facet is then

P= (14P) (ETL)

where the fields E+ have been normalized in mW!7.

(2.58)

(2.59)

Reference

[1] H. Kogelnik, and C.V. Shank, Appl. Phys. Lett. 18, 152 (1971)

[2] D. Marcuse, Theory of Dielectric Optical Waveguide (Academic Press, 1991), ch.3

[3] J. I. Kinoshita, and K. Matsumoto, IEEE J. Quantum Electron., QE-24, 2160 (1988)
[4] H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, and H. Imai, /EEE J.
Quantum Electron., QE-23, 804 (1987)

[5] M. Yamada, and K. Sakuda, Appl. Opt. 26, 3474 (1987)

[6] J.E.A. Whiteaway, G.H. Thompson, A.J. Collar, and C. J. Armistead, IEEE J.
Quantum Electron., QE-25, 1261 (1989)

[7] P. Vankwikelberge, G. Morthier, and R. Baets, JEEE J. Quantum Electron., QE-26,
1728 (1990)

[8] R.M. Knox, and P.P. Toulios, Proc. MRI Symposium on Submillimeter Waves, (ed.,
J. Fox, Brooklyn, N.Y., Polytechnic Press, 1970)

[9] W. Streifer and E. Kaon, Appl. Opt. 18, 3724 (1979)

[10] J. Buus, JEEE J. Quantum Electron., QE-18, 1083 (1982)

[11] A. Yariv, Optical Electronics, 4th Editon (Holt, Rinehart and Winton, New York,
1991), ch.13

[12] T. Schrans, Ph.D. Thesis, California Institute of Technology, Pasadena, Calif., 1994
[13] H. Nishimoto, M. Yamaguchi, I. Mito, and K. Kobayashi, JEEE J. Lightwave
Technol. LT-5, 1399 (1987)

[14] A.R. Beattie, and P.T. Landsberg, Proc. R. Soc. London Ser. A. 249, 16 (1959)
[15] Y. Horikoshi, and Y. Furukawa, Jpn. J. Appl. Phys. 18, 809 (1979)

[16] G.H. Thompson, and G.D. Henshall, Electron. Lett. 16, 42 (1980)

[17] N.K. Dutta, and R.J. Nelson, Appl. Phys. Lett. 38, 407( 1981)

[18] A. Sugimura, JEEE J. Quantum Electron., QE-17, 627 (1981)

[19] R. Slshansky, C.B. Su, J. Manning, and W. Powazinik, JEEE J. Quantum Electron.
QE-20, 838 (1984)

[20] A. P. Mozer, S. Hausser, and M.H. Pikuhn, JEEE J. Quantum Electron. QE-21,
719 (1985)

[21] H.C. Casey, Jr., and F. Stern, J. Appl. Phys. 47, 631 (1976)

[22] R.J. Nelson, and R.G. Sobers, J. Appl. Phys. 49, 6103 (1978)

Chapter 3

Numerical Simulation of Distributed

Feedback Lasers

3.1 Introduction

A computer model for solving the photon and carrier density in DFB lasers is developed in
this chapter. The longitudinal dependence of the two counter-propagating waves and
electron density is included in a self-consistent way through the dependence of the electron
density on the optical intensity and the dependence of the gain and the refractive index on
the electron density. The model can be applied to any combination of active and passive
sections including distributed feedback from index or loss gratings and reflections from
facets or the interfaces between sections. Each section can have z-dependent index and

loss coefficients, effective refractive index representing waveguide structure, and current

injection. The material gain and the refractive index are assumed to be linear functions of
carrier density as described in (2.31) and (2.35).

The algorithms for solving the lasing condition and above threshold [1-12] are
presented in this chapter. Characteristics of two semiconductor lasers are discussed. For
DFB lasers under large signal modulation, a dynamic model based on the time-dependent
coupled-mode equation is developed. The model can be used to calculate the frequency

chirp and pulse shape from DFB lasers under large signal modulation.

3.2 Threshold simulation of DFB lasers

With lasing condition (2.47) and the expressions of complex reflectivities for left- and
right-hand parts, we can solve the threshold problem for DFB lasers. The laser is divided
into many small segments in which the carrier density and refractive index are assumed to
be constant. The current complex reflectivities can be calculated by using the F-matrix
[13] method. The procedure is shown in Fig. 3.1. For given device and material
parameters, the carrier density distribution can be calculated below threshold if the injection
current profile is given. Then the round-trip gain is found at a given frequency @, and we
scan the frequency around the Bragg frequency @p,agg to find the longitudinal modes for
which the phase of the round-trip gain is equal to zero. The mode with maximum round-
trip gain is to lase, so we adjust the current to make the round-trip gain equal to one. After
the round-trip gain is equal to one, we get the threshold current and lasing wavelength.
Each time the current is changed, the modes are readjusted to have a round-trip gain phase
of zero. This iteration goes until the current with a maximum modal round-trip gain of one

is found.

Algorithm to Solve Lasing Condition

eee ener eee eee ee ee ee eee eee cence eee ce ee

Enter device and material parameters

we) Give a current profile

Calculate carrier density N(z)

Calculate F matrix

Calculate p1(@) and p2(@) at given w

Scan @ around Wbragg and find the modes
@i which satisfy arg(pi p2) =0

Calculate Ri=l p1(0di) p2(0i)|
and find maximum Rmax

Ith, (ase

Figure 3.1: Algorithm to solve lasing condition.

3.3 Above-threshold simulation

Above threshold the electron density is determined by the current injection, spontaneous
emission and stimulated emission as expressed in (2.46). The stimulated emission depends
on the photon density. The photon density and electric fields depend on the electron
distribution through the effective index and modal gain. The field distribution will therefore
depend on the photon density and has to be solved self-consistently. The algorithm for
above threshold problems is shown in Fig. 3.2. The injection current is chosen as the
independent variable and remains fixed to determine the output optical power and photon
density distribution. With keeping the current profile fixed, the photon density and electric
field distribution are calculated self-consistently to satisfy the lasing condition (2.47). If the
calculated photon distribution differs from the assumed photon distribution, a new guess
for the photon distribution is made. This new distribution can be either the newly calculated
photon distribution or the average of the calculated and the previously assumed photon
distribution. Only when the calculated photon distribution differs from the assumed one by
less than a predetermined error is the photon distribution found. The rate equation (2.46) is
used to find the electron density. During this iteration, only one mode is considered, and a
final check has to be performed on other longitudinal modes once the photon distribution
converges. If a longitudinal mode is found with a round-trip gain larger than 1, the iteration
Starts over on this new mode. If the new mode turns out to be the maximum round-trip

gain mode, it is assumed that a mode switch has occurred.

Algorithm for Self-Consistency above Threshold

Give a current level

Ith, @1 <_—___—_

aa Assume photon density distribution Pin(z)

Calculate carrier density distribution N(z)

Calculate F matrix

Calculate p1(@1) and p2(a1)

Scan around @) and find the new 1
which satisfies arg(p1p2) =0

Calculate Rmax=| pi(@1) 92(a@1)|=1?

Set Pin(z) =Pout(z)

\] Yes

Calculate P(z)=Pout(z)

[Pout(z) -Pin(z)I/ Pin(z)

Ves

Get @:, P(z), AViinewidth, eff, AVchirp

Is there any other mode @m
whose R(@m )> R(a1 )?

Yes

V No

Results

Set OH=Om

Figure 3.2: Algorithm for self-consistency above threshold.

A quarter wave shifted DFB laser has been calculated above threshold. The photon
density, electron density and effective index distribution are shown in Fig. 3.3. The laser
is 300um long and with coupling constant KL=1. The phases of facet reflectivities are
assumed to be zero, and it can be seen that the spatial hole burning is strong at the center of
the laser for the quarter wave shifted DFB laser (Fig. 3.3 ). Due to the spatial hole burning
effect, the carrier density and effective index distributions are affected by nonuniform
photon density distribution. Another commonly used DFB structure has asymmetric facet
reflectivities. In our example, we use a DFB laser with Re=0.1 and Rp=0.7. The DFB
lasers with equal reflectivity at both ends generally have two modes lasing. The photon
density, electron density and effective index distributions of the asymmetric DFB laser
have been calculated and shown in Fig. 3.4. The asymmetric DFB lasers are used to
generate one lasing mode by suppressing one of the two potential lasing modes. For high
temperature operation of single longitudinal mode (SLM) DFB lasers, the asymmetrical
structure has been frequently used to suppress the Fabry-Perot modes in the lasers. The |
front facet is usually anti-reflection (AR) coated with R;<5% and the back facet high-
reflection (HR) coated with Rs~70%. In Fig. 3.5, we show the calculated photon
distributions of three 4/4-shifted DFB lasers with different coupling constants. The
coupling constant of a DFB laser has a strong effect on the spatial hole burning. The DFB

laser with larger coupling constant has bigger nonuniformities in its photon and electron

distributions. For A/4-shifted DFB laser with high «KL values, it loses its SLM operation
when the output power reaches a certain power level. So, low coupling (kL~1) DFB lasers

are good candidates for high power SLM operations.

“ 0
FE isk E
2 F J
= IS5E 4
> OE
_ 1.2 B 4
< _ =
3 5 7
0.9 4
s c 4
2 Y 5
= 0.6 & a
= Fist tipper lira er tap bi bt
0 50 100 150 200 250 = 300
z (um)
a 2.68 er TT TTTTT TITY)? Tt Tht dy a) a) pa) |] th) Ud
= ; j i | I t :
- 2.676 4
= 2.666 4
P2656 4
n . =
5 : ;
a 2.646 4
fw C 4
2 2.63 fh 4
i c 7
5 2.62 Esprrtirritir i Mires tar ti diri4
0 50 100 150 200 250 300
z (um)
3.2089 ee ee 13
3.2683 E 4
3.2682 4
3.2681 Evrae tirirtiriitipip tii rtrd
0 50 100 150 =—.200 250 300

z (um)

Figure 3.3: The photon density, electron density and effective index distributions in a
quarter wave shifted DFB laser above threshold. ( Rr=Rp=0.3)

“ 1.8 Lirrrypgygri TT TET perrrprrrrvypyry
3 7
: 1.6 4
Fon q
= 1.4
2 F :
A 5 J
5 an q
2 J J
= O.8 Cxnestrrartrrirtririertirrirtir iti d
Aa 0 50 100 150 200 250 300
z (um)
“a Prove prrrryprreryprrrryprrr repre
EF _2.89
Z 2.79
but
vo
5 pertrrrartiarrzrtlry rr lip pp ty yyy
0 50 100 150 200 250 300
z (um)
3.267
3.26 Fee pet ryt rrT)
vr r 7

3.2676

; :
3.2675 mm ‘i a Lt it l ji tit | duit | ii. | ae a | ul
0 50 100) = 150.200 250 300
zZ (um)

Figure 3.4: The photon density, electron density and effective index distributions in a
DFB laser above threshold. ( R=0.1, R,=0.7)

2.5 & TT | Pryrprryry | 1rT es | Terry | TTT? a
or: oN :
5 r kL=0.5 /
cs { 5 . Sof, 4
s ~~ a “. “eg
= . «KL=2-" 7
= 0.5 a ro 4

@) F Lt I Littl i LELE l Leet ii Lisi i Ler 4
0 50 100 150 200 250 300

z (um)

Figure 3.5: The photon density distributions in quarter wave shifted DFB lasers

with different coupling coefficients above threshold. ( Rs=Rp=0.3)

3.4 Large signal modeling

For single-mode DFB laser, the transverse electric field can be represented by [1,14]

F(z,t) = E+(z,t) exp[-iBoz+i(@p+A@)t] + E-(z,t) exp[+iBoz+il@p+Aa@)t] (3.1)
where Et, E> are the complex field amplitudes of the forward- and backward-waves,
respectively, and Wg is the reference frequency. By = 2m/Ao, where Ag is the Bragg
wavelength. Aw=@ - Wp, @ is the lasing frequency. For a DFB laser under large signal
modulation, the amplitudes Et, E- are described by the following coupled traveling rate

equations [1]:

1 OE*(z,t) 4, OEM)
Vg or dz

=iKE“(z,t) - (16 +iA@/Vg -g+ oss JE*(z,) + Tt

(3.2)

4 OE (et) OE) _

. + : . _ _
Vo or Oz iKE *(z,t) - (10 +iA@/Ve -8 +Qoss JE (z,) + T

(3.3)
The variation in amplitudes is accounted by assuming that in any local sections the forward

and backward fields grow or decay at the same rate. Hence, in each segment of the laser,

u(z,t) = —2F G2 E(t)

- 3.4)
Vo Ev(z,t) ot Vo E*(z,t) of

where {J is real and assumed to be uniform over any segment but varies from segment to

segment. By introducing a new dynamic gain distribution

G(z,H=2(Z,t)-U,t) , (3.5)

we can give new quasi-steady coupled-mode equations

ofa =iKE “(z,t) - (16 +1A@/ Vg -G +O os5 )E*(z,t) + T* (3.6)
: ee =iKE *(z,t) - (i5 +iA@/ Vp -G +QJo55 JE (zt) + T (3.7)

Photon density S(z) is proportional to |E*(z)I?+1E-(z)I?_ so that one can also write

I OS(z,t)

(3.8)
Qv—S(z.t) 4

L(z,0) =

Given that is known, then the coupling-mode equations (3.6) and (3.7) can be solved as
in the above threshold simulation. The operations in the large mode modeling are outlined

as the following:

1) Switching at a given injection current above threshold, we start to solve the photon
S(z,t=0) and field E*(z,t=0), carrier density S(z,t=0) distributions above threshold as
we described in Section 2.3. The u=0 at t=0.

2) Calculating the parameters: G(z,t=0) from (2.31), refractive index nep{z,t=0) from
(2.35).

3) Finding the lasing mode ay at t=0.

4) At t>0, we solve field and photon distributions from (2.54), (2.44). We get uw by

u(z,t) = S(z,t)-S(z,t-At)

(3.9)
2Vg At [(S(z,1)+S(z,t-At)|/2

Here S(z,t) is the calculated photon density at present and S(z,t-At) is the photon
density at previous time t-At. We take this predicted value back to step 2) and repeat
this process until the errors between the two iterations are less than a preselected limit.
5) For t+At>0, we get N(t+At) from calculating dN/dt for the carrier density rate

equation (2.44) at t

N(e+AN) = No +4 | (3.10)

then go back to step 2) until the desired time length is reached.

The model here is used to analyze 4/4 phase-shifted DFB laser with AR coating at
both facets, e.g., Re=Rp=0.05. At conditions of large signal modulation, the laser is
initially biased just above the threshold and modulated with a pulse function. The injection
current as a function of time is shown in Fig. 3.6. The dynamic response of the DFB
laser can be obtained from the model (Fig. 3.6). The transient response shows damped
oscillations as expected. Fig. 3.6 also shows the dynamic response of a laser with larger
nonlinear gain coefficient €=10-!®cm3, of which the oscillations are significantly reduced to
almost a single pulse. The frequency chirp of the DFB laser is shown in Fig. 3.7a with
€=10-!7cm3 and e=10-!®cm3. The frequency change is caused by variations of the
refractive index that is related to the change of the carrier density as given (2.34). Due to
the strong SHB in A/4 phase-shifted DFB lasers, the nonuniformity of the refractive index
is serious, which contributes extra chirp in addition to the chirp caused by uniform change

of the refractive index. We write the frequency chirp Av (in Fig. 3.7) as AA=-Av A2/c. As

we will show in Chapter 5, a simple formula based on conventional rate equation can be

used to describe the frequency chirp and it is given as

d In S(t) n ES(t)

7 ty (3.11)

A V(t) = tn

The first term represents the transient chirp, which is proportional to changing rate of
output power. The second term represents the adiabatic chirp, which is proportional to the
power level and nonlinear gain coefficient. From Fig. 3.7, it is obvious to see that the
transient chirp is a result of the modulated optical power. For large €, the adiabatic chirp
becomes important. In Fig. 3.7a, the laser with larger € has larger chirp, which comes
mostly from adiabatic chirp. From (3.11), linewidth enhancement factor is found to be a
determining factor for the chirp. In Fig. 3.7b, we calculated the chirp of DFB lasers with
a=2.5 and 5, respectively. The laser with a= 5 has larger chirp as predicted in (3.11). For
multisection DFB lasers, (3.11) is not valid to describe the chirp and an expression for
frequency chirp, which is based on rate equations with considering SHB, has to be used.

We will derive the expression in Chapters 4 and 5.

Figure 3.6: Optical response of 4/4 phase-shifted DFB laser with pulse current
injection. The nonlinear gain suppression coefficient is chosen to be e=10-!7cm},

and 107!6¢m3,

2, Ln ] roe 1 T a { a ae | } TOT ry

0 E e-10' em? 3

oat 4 2

2. 4

J °F :

106. (a) 4

12 Es Lu | Lie l L Lui l Lda l it 4

0 700 1400 2100 2800 3500
Time (ps)

1 rrr yt votT 1 TT fT | For oT 1 om | ro

c (b) 4

4 = Lut I i Lut im it | Ld | il 11]

0 700 1400 2100 2800 3500
Time (ps)

Figure 3.7: Frequency chirp of 4/4 phase-shifted DFB laser modulated with pulse

function for: (a) €=10-!7cm3, and 10-!6cm3, (b) a=2.5, and 5.

Reference

[1] P. Vankwikelberge, G. Morthier, and R. Baets, IEEE J. Quantum Electron., QE-26,°
1728 (1990)

[2] J.E.A. Whiteaway, G.H. Thompson, A.J. Collar, and C. J. Armistead, IEEE J.
Quantum Electron., QE-25, 1261 (1989)

[3] K.O. Hill, and A. Wantanbe, Appl. Opt. 14, 950 (1975)

[4] H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, and H. Imai, JEEE J.
Quantum Electron., QE-23, 804 (1987)

[5] Y. Nakano, T. Okmatani, and T. Tada, J 1th IEEE International Semiconductor Laser
Conf. H2 (1982)

[6] G.P. Agrawal, and A.H. Bobeck, IEEE J. Quantum Electron., QE-24, 2407 (1988)
[7] M. Okai, S. Tsuji, and N. Chinone, EEE J. Quantum Electron., QE-25, 1314 (1989)
[8] M. Usami, and S. Akiba, IEEE J. Quantum Electron., QE-25, 1245 (1989)

[9] K. Kkuchi, and H. Tomofuji, IEEE J. Quantum Electron., QE-26, 1717 (1990)

[10] M.G. Davis, and R.F. O'Dowd, IEEE Phot. Tech. Lett. 3, 603 (1991)

[11] I. Orfanos, T. Sphicopoulos, A. Tsigopoulos, and C. Caroubalos, JEEE J. Quantum

Electron., QE-27, 946 (1991)

[12] T. Yamanaka, S. Seki, and K. Yokoyama, [EEE Phot. Tech. Lett. 3, 610 (1991)
[13] M. Yamada, and K. Sakuda, Appl. Opt. 26, 3474 (1987)

[14] L.M. Zhang and J-E. Carroll, JEEE J. Quantum Electron., QE-28, 604 (1992)

Chapter 4

Small Signal and Spectral
Characteristics of Distributed

Feedback Lasers

4.1 Introduction

Small signal analysis in Fabry-Perot lasers is based on the conventional rate equations [1-
2]. In the lasers, the photon and carrier densities are assumed to be uniform. The -
assumption holds well as long as the facet reflectivities are greater than 0.2 [3]. However,
In DFB lasers, the longitudinal spatial hole burning (LSHB) has been found to be an
influence factor on linewidth, frequency chirp, and high speed performance. A Green's
function method has been used to solve the coupled-mode equations in DFB lasers when

the LSHB is also considered.

Pure AM modulation or FM modulation of single section semiconductor lasers is
very difficult due to the link of the phase of the lasing mode to the intensity by the @ factor.
It is found that two-section DFB lasers can be used to achieved chirpless AM[4]
modulation and very high efficiency FM modulation [5].

The early theory [6-7] for two section semiconductor lasers was based on the rate
equations for a closed resonator [8], in which the carrier density and the optical photon
density distributions are assumed to be uniform in each section. Kuneztsov [9] then
introduced the effective length in DFB lasers to calculate the wavelength tuning and
frequency chirp in two-section DFB lasers. In his theory, the optical power distribution is
assumed to be uniform in the lasers. Tromborg [10] proposed the Green's function method
to solve the electrical field distribution in DFB lasers with considering the LSHB and

derived the rate equations for optical power and instantaneous lasing frequency.

4.2 Solving the coupled-mode equations by Green's

Function

4.2.1 Green's function for coupled-mode equations
In single-mode DFB laser, the transverse electric field can be represented by [11]
E(z) = E*(z) exp[-iBz]+ E-(z) exp[+iBz] (4.1)

The coupled mode equations (2.14) and (2.15) can be written in vectorial form as [12]

dZ = M(z) Z(z) + P(z) (4.2)
dz
with
_ | Et(z)
L(z) = EO (4.3)
P(z) = | (4.4)

P is the noise source due to spontaneous emission. Zr(z) and Z;(z) are assumed to be
two solutions of the coupled mode equations for P=0 and are satisfying the boundary
conditions on the left (Z_(z)) or on the right (Zp(z)). Green's function is defined as the

solution to

dG. -M(z)G(z,z’) + | 6(2-2') | (4.5)
dz -6(Z-Z')

satisfying all boundary conditions. Solving the equation gives [12]

T/.1
G(z,2') ={ Z; (z)ZR(z') Z

ZR(ZLZE(Z) —-z>7z'
The Green's function solution to (4.2) is given by

Z(z) aie G(z,z')P(z")dz' (4.7)

with the Wronskian W which is defined as

W =Z,(z)4 Ze(z)-Ze(z) 4 Zz)
dz dz

=-2iB[ER(ZEL(z)-ER(Z)EL(z) | (4.8)

It can be shown that W is z independent from (4.7 )

ay = (M\\+M22)W = 0 (4.9)
zZ

We can define frequency domain electrical field E(@, z) by

E(@,z)=| E(z,t) e!@'dt (4.10)

The field F (z, t) is given as
E(z,t) = E*(z,t) exp[-iBz+i(@p+A@)t] + E-(z,t) exp[+iPz+i(@pt+A@)t] (4.11)

The solution to (4.1) is given by

WIB@IE(@,z) = | G(z,2')folz')dz' (4.12)

Here Green's function is rewritten from (4.6) as

G(z,2') = Z(z) Zr(z') O(Z-z')-Zr(z) ZZ) O(z'-z) (4.13)

©(z) is the Heaviside step function and W is given by (4.8). f@ is noise source in
frequency domain.
From (4.12) and in the limit of a vanishing force (f=0), there can only be a nonzero

solution, when W=0. For the stationary case with de current injection, the lasing modes

are the states for which

WI[B(z)] = 0 (4.14)
is fulfilled with the rate equation for the carrier density (2.46). The equation (4.14) is
equivalent to the lasing condition (2.47). From (4.8), Zr is proportional to Z,, and from

(4.12), E(@,z) is proportional to Z, or Zp.

4.2.2 Green's function solution for electric field
Following Henry [13], we define a mode amplitude Aq by
E(@,z) =Aw Zr (Z) (4.15)

and the corresponding envelope function A(t) by

A(N=} Agek® ida (4.16)

When the driving force F40 or with modulation, the Wronskian is not exactly zero, and
B(z) fluctuates around its stationary value. In order to derive a rate equation for the
envelope function A(t), we expand the Wronskian W/(z)] around the stationary solution

Boz). The B(z) depends on frequency , carrier density, and photon density. Hence,

=B(z)-B(7)=2 Pi w-@) +2 AN +2.
AB(z)=B(z)- Bo(z) 5 Oo)+s AN+. AS (4.17)

where AN=N-No, AS=S-So.

The first-order expansion of the Wronskian can be written as

W[B(z)]=WIBo(z)]+ | 3 GS 5m abe 4 z' (4.18)

By (4.12) and (4.15), the mode amplitude satisfies

WIB(z)]Ao= | ZR(Z')feolZ')dz' (4.19).

With (4.19) and making an inverse Fourier transform, we get the rate equation for A(t)

l l
Cy(z)AM(z,t) dz + | C5(z)AS (z,t) dz] + Fa)

d =
4 = AO i ,

(4.20)
The rate equations for the optical power and the phase of the optical field can be derived
from (4.20) if we write
A(t) = 1P@ eiPt (4.21)
and the rate equations are [10, 12]

l l

Re{Cn(z)} AN(z,t) dz ‘| Re{Cs(z)}AS(z,1) dz] +R + Fp(t)

(4.22)

40-1] Im{Cy(z) }AN(z,t) dz ‘| Im{ Cs(z)}AS(z,t) dz] +F9(2)
0 0

(4.23)

Here the weight functions Cy(z) and Cs(z) are defined by

1 ‘
Cx = jE") E(z) Ez) / | E*(z') E(z’) Fe) dz} (X=N,S) (4.24)

R-

_ The noise source is given by

1 , oo
278 J,
do 0

4.2.3 Small signal rate equations
For small signal modulation at frequency f,,=@,,/27, the dynamic variables current density
J(z,t), carrier density N(z,t), and photon density S(z,t) can be written as the sum of their

stationary distributions and the variations due to RF modulation

J(z,t) = Jo(z)hye!Omt (4.26a)
S(z,t) = So(z)+S0(z)5S(@mer mt (4.26b)
P(t) = Po+PodS( Qe? Ont (4.26c)
M(z,t) = No(z)+Ni@e!Omt (4.26d)

Here Jo(z), No(z), and So(z) are the stationary distributions in DFB lasers.

The small signal rate equations can be gotten from (2.44) and (4.22)

415S( Omer] =-I;, 5S(On)e'On'+ [ | Re{Cy(z)} 6N(z,f) dz + Fp()/Po

(4.27)

d6N(z,t) _ (Zt) a ils Nil, 5
i od - Fy(z) 6N(z,t) - +650(Z yy°8 6S(z,t)

(4.28)

where

. | F
I's(z) -=R /So(z) - | Re{Cs(z)}So(z) dz (4.29)
, 4 Ie dGIdN So(z)
Ty (2) =O +V¢ [e590 (4.30)
Here T, (z) is defined as
1 1 -2BN(z) - 3CN(z)2 (4.31)

Ts(Z)

-and R is the spontaneous emission rate.

4.3 Amplitude modulation

4.3.1 Expression for AM modulation in DFB lasers

The frequency response of DFB lasers can be obtained from (4.26), (4.27), and (4.28)

| dz Cy(z) Jy/[eG Ont I (Z)]
5S(@,,) = 22 —, 4.32
(Om) —— D(@m) ““
where
D( Om) - (Or(Z)+Wm-iT R(Z))(OR(Z)- Om til R(Z)) dz, (4.33)
0 iOnt+In(Z)

and the local resonance frequency @ r(z), and the damping rate IR(z) are

OR(z) = 21 Cy(z)vga(2)So(z) - [Dn - 's]7/4, (4.34)

55
and
Tr(2) = Un(2)+ (22, (4.35)

where 9(z) is the local optical gain, and I, is the optical confinement factor for the lasing
mode. The resonance response of DFB lasers occurs when D(@,,) is minimized. To find
the resonance frequency @p and the damping rate Tz of DFB lasers, the numerical
calculation of (4.32) (with (4.34) and (4.35)) has to be carried out. Because p(z) and
Ip(z) are functions of photon density, So(z), and electric fields E+, E-, @e and Ip of

DEB lasers depend on the photon density distributions.

4.3.2 Resonance frequency in single section DFB lasers
In the low output power limit, the damping rate is much smaller than the resonance

frequency and the small signal response (4.32) can be simplified as [14]

2 4
8m)“ OR (4.36)
8500) @eo) +4T ROM
and
Op = | 2 Cr(z)vga(z)So(z) az (4.37)

In single section DFB lasers, the photon density distributions do not change
dramatically as long as optical output powers are not very large. The resonance frequency

can be approximated as

Op = aveg So (4.38)

and we define 7) as

, 2 i Cy(z)g(z)So(z)az
= — . (4.39)
ae g So
Here So is the average photon density, g is the average optical gain, a is the differential
gain. 7 =1 for lasers with uniform photon densities. For DFB lasers, 7 #1 and has to be
numerically calculated. 7 has been calculated for several typical DFB lasers and is shown
as a function of optical power in Fig. 4.1. It can be seen that 7 depends on optical power.

| Therefore, the resonance frequencies of DFB lasers are expected to slightly depart from the

linear function of ~/ So at low optical power.

1.3

1.2

9 to 14 ;
0.0 2.0 4.0 6.0 8.0 10.0
P(mw)

Figure 4.1: The calculated’ parameter for DFB lasers:
(a) AR/HR coated DFB laser with KL=2, Ry=0.05, Rp=0.95
(b) As-cleaved DFB laser with KL=2, R=0.33, Ry=0.33

4.3.3 High speed response in a two-section DFB laser

. A two-electrode 1.3 um InGaAsP/InP DFB laser is fabricated for our experiment to study
the effects of nonuniform photon distribution on the high speed response of DFB lasers.
The laser has two sections, 400m and 100um long, with an electric isolation of 1.4 kQ.
The front facet is AR coated and the rear facet is left as-cleaved. The DFB laser oscillates at
a single wavelength of A=1.32um with at least 35 dB of side-mode-suppression-ratio
(SMSR) under CW operation, and it has the coupling coefficient KL=3. The two sections
are pumped with DC current, with the longer one also AC modulated. The éJ(z,t) can be

written as

Je! Ont Oa(zp={ for O

0 for L1

The expression of the frequency response of the two-section DFB laser can be written as

Ly
dz Cn(Z) Jue Ont I n(Z)]

S(O) | na . (4.41)

The only difference between (4.32) and (4.41) is that the integration length in the

~ numerator has been changed. They both represent the same frequency response.

f (GHz)

| i
0.5 0.8 1.1 1.4
P'?(m wi”)

Figure 4.2: The resonance frequency of a two-section DFB laser versus
square root of the optical power for various Ip, the injection current into

section two.

18 pee yo ype tt
---F--- OMA a

1.6 | —a& -5mA ;
—@ -10mA

. --M@--20mA ‘ ye

T/20 (GHz)

* ®
*» a
\ KY
. x
“se aa “
* . 7
Poecclotibirprbirirtiri tis

0.4 Ersiortiiiitiiritiriiitiris

0 0.5 1 1.5 2
P(mW)

NMLitt

1.8

1.6

1.4.

1.2

0.8

0.6

0.4

Figure 4.3: The damping rate of the DFB laser versus the optical power

for various I>,

: The high frequency modulation of the DFB laser is carried out by a standard
experimental arrangement, including an S-parameter test set and a network analyzer. The
resonaioe frequency and damping rate are obtained from the data filling [15] according to
(4.38). Fig. 4.2 and Fig. 4.3 show the measured resonance frequency and damping rate as
functions of gptical power. The measurement is conducted for the output optical power P =
0.23, 0.4, 0.84, 1.3, and 2 mW for various injection currents. At each optical power
level, the current into section one, J;, is adjusted to maintain the same output power while
‘In is chosen to be 0 mA, 5 mA, 10 mA, 15 mA and 20 mA. Fig. 4.2 shows the resonance
frequencies for different injection currents, which differ as much as 0.7 GHz. The
difference between resonance frequencies at several optical power levels is the result of the
dependence of the resonance frequencies on the distributions of the photon density and the
traveling electric fields in the DFB laser, as given in (4.32)-(4.35). We calculate 7 in
(4.39) to analyze the effect of the nonuniform photon density on the resonance frequencies
of the DFB laser. The relatively large value of f, at low P, when Jp = 0 mA, is predicted
by the calculated 7 . The photon density is much larger in the long section than in the short
. section for J, = 0 mA, and according to (4.39), we found that 7 = 1.2. At high P, the
spatial hole burning effect is serious in the long section, and 7 ~ 0.85 is found. The
gradually reducing n explains why, as we increase optical power, the rate of increase of
the resonance frequency is slowed at high P when no current is in the short section (Fig.
4.2). For I, = 20 mA, the photon density is much larger in the short section than that in

the long section, and the calculated Cy(z) shows that it is very small in the short section at

low P. From (4.39), we found that 7 ~ 0.9. The small Cy(z) contributes the relatively
_ small f, at low P (Fig. 4.2) for I= 20 mA. At high P, Cy(z) is more uniform along the
lasing axis and 7 ~=1.1, which results in a relative large f, when [2 = 20 mA (Fig. 4.2).
The damping rate, Ip , is obtained from the frequency response of the two-section
DFB laser (Fig. 4.3). Like the resonance frequency f,, [rp is dependent on the photon
density distribution in the laser. The variation of Trp becomes larger at high optical power.
It is believed that at high optical power, the damping rate is dominated by the contribution
of gain suppression, the second term in (4.29). The contribution of gain suppression to the
damping rate as seen in (4.29), is the integration of photon density So(z) with function
Cy(z). That results in the larger difference of Ip at high optical power shown in Fig.
4.3. In DFB lasers, the damping rate is influenced more strongly by the photon density
distribution than is the resonance frequency, since Ip increases linearly with So while f,

increases linearly with V'So .

4.4 Linewidth, effective linewidth enhancement
factor — .

The statistical properties of the complex noise functions are given [10,14]

= R O(t-t') (4.42)
= 2 RP O(t-t') (4.43)
FOF o(t)>= —& &1-¢ 4.44
OP, (t-t') (4.44)

If the frequency noise (normally ~ GHz) is much broader than the linewidth, the white

noise approximation is used [16]

Av = | <59(1)59(t+)> dt
20

-0o (4.45)
= <16@(0)I">
where
500) = F (0) - reg @ PO + | K(z’) Fy(z',0) dz’ (4.46)
2Po |,
K(z) =tpr(z) [Im{ Cn(z)}-OegRe{ Cy(z)}] (4.47)

The noise spectra F4(0), Fs(0), and Fx(O) can be assumed white noise [12] and the

linewidth is obtained as

Av= AVsp+AVnN +Avns (4.48)

The linewidth consists of three parts, the spontaneous emission linewidth AVsp, the

. electron noise component Avyn and the stimulated emission component AVys. They are

given by
— R
AVsp dal “ef? (4.49)
AVnn = i | K?(z') Dyn(z')dz' (4.50)
Avs = - 2 Ott | K (z') Dns(z')dz' (4.51)

Here Dyn and Dygare the diffusion parameters and are given in [10].

We introduce the effective linewidth enhancement factor [10]

Im( { (Cul2) Tele) Este) Soe)-Ca(2)Su(2)) dz)
Oegt = 2 (4.52)
Re { (Cul2) Te(2) E842) Se)-Ca(2)S0(0) dz )

Here R is the spontaneous emission rate, Ig the total photon number in the cavity, Tr, the
carrier lifetime, and R,;, the local stimulated emission rate. Og reduces approximately to
material @ if the photon density distribution is uniform. The spontaneous emission
linewidth (4.49) is the modified Schawlow-Townes formula with o,g replacing material

a, and it is the dominant part in (4.48). In most of cases, AVyy and AVys are much

smaller than AVsp and we neglect AVyy and AVyg in our calculation of linewidth. Fig.
4.4 shows a measured linewidth of a single section DFB laser. The measured linewidth has
nearly a linear relation with P-/ for moderate optical power. This is predicted in the
expression (4.49) and conventional linewidth expression. There are several possible
mechanisms for the rebroadening of linewidth at high powers, such as pumping noise and
stimulated emission noise (AVyy, and AVws), which become dominant once the
spontaneous emission noise AVs, is small enough at high powers. At low frequency,
-Avwn and Avys can have a 1/f component, which is dominant [17-20]. The reduced
side-mode-suppression-ratio (SMSR) at high powers in DFB lasers is also a candidate for

linewidth rebroadening [21].

' DFB @20°C
12 a
F | —¥—C 7
_ c :
ee j
S 6E 3
=) r ;
a 45 J
2k 3
fo) ee ee
0 0.5 1.5 2 2.5

p! (mW ’)

Figure 4.4: A measured linewidth as function of P-! in DFB laser.

A two-electrode 1.55,1m InGaAsP/InP DFB laser has been used to measure effects of
LSHB on linewidth. The laser has cleaved facets and the two sections are 106m and
138m long with an electric isolation SkQ between them. The DFB laser lases at a single
wavelength of A=1.55ym with at least 35 dB of SMSR. It has a threshold of 40 mA when

uniformly pumped. We get KL=1 from the measured stopband width.

A time delay self-heterodyne detection scheme is used in our measurement of laser
linewidth. The laser is isolated with a 70 dB isolator. The laser beam is modulated with an
acoustooptic (A-O) modulator at 85MHz. The fundamental beam from the A-O modulator
is focused into two kilometers of single mode optical fiber. It is joined by the first order
beam at a fiber coupler. One output of the coupler is received by an Ortel 2610 detector.
The output of the detector is amplified and then sent to a spectrum analyzer where the
linewidth is measured. The other output of the coupler is sent to a power monitor for line

adjustment..

The linewidth at different current injection has been measured. Linewidth at various
current ratios is shown in Fig. 4.5 for total current J, of 5|0mA, 60mA and 70mA. It can
be seen that there are certain current ratios at which the linewidth is very small. For
I,=50mA, 60mA, two minima are smaller than the linewidth of uniform injection
(I;/12=0.77). The minima will shift if the total current increases. The linewidth is found

to be very large for either small or big current ratios (1j/I2<0.2 or Ij/Iz7>2.5).

35 rerprrrryprreryprrrr
1 =70mA 3

70 repereryprrrryrrr ty 40 rerprrrryrrrryrrs
I = 50mA 1 = 60mA

‘waren
ao a
ou

THUP TUTTE Ty Ty

cla l ruil
(MHz)
Nh
ao

LIEU OBES EEE? Ue ORE RE
tpt rbe eb ee ts eh

0 pei bed o poe de de ef pistrrerisers te srs
0.0 1.0 2.0 3.0 4.0 0.0 1.0 2.0 3.0 4.0 0 0.5 1.0 1.5 2.0
J, LAL, L/L,

BEUIN EEE E EL

Figure 4.5: The measured linewidth of two-section DFB laser at different current

ratio. The total injection current is

(a) ly = 50mA (b) Ip = 60mA (c) It = 70 mA.

%.0 0.5 1.0 1.5 2.0 2.5

L/I
12

20 ia eee eT Tg
§=90 og
156 4
GC r I =1.5 I :
= i ]
—10Fr -
2 i _ J
0) ee ee eee ee ee ee eee
0.0 0.5 1 1.5 2.0 2.5 3.0 3

mo fF O

>= 4 5° o=

I =1.51
t th

TUTPTTT yr ery reap rrr perrparrgare

I =2I

t th
Ee.
0 0.5 10 15 2.0 2.5

1/1

1 2
BEBE U MORE BEERS COR EE EE
- ° ® =0°
$=135 >.=0

Are
- t th

0.00.5 1.01.5 2.0 2.5 3.0 3.5
I /l

1 2

Figure 4.6: The calculated linewidth of the two-section DFB laser at different

current ratio of the two electrodes for various assumed values for facet phases.

35 pr rrr rrrrrpererperrrprrr Ts a 0
F ¢ =135 ,=0 1
i + -
SOF Ae
; + -20
L >
< 25 7 4-30 £
5 4-40 3
— 20 7 E
L 4 - ~
F 50 S:
15 F-
C —> + -60
10 i aa rar Ps ee -70
0.0 05 1.0 1.5 2.0 2.5 3.0 3.5
I /T,

Figure 4.7: The calculated threshold and lasing wavelength of the DFB laser for

facet phases $,=135°, dr=0°.

Photon Density (10° * cm’? )

Figure 4.8: The longitudinal photon distribution in the DFB laser for facet

phases @,=135°, dr=0°. The current ratio is assumed to be 1)/I2=0.33,

0.77, and 3.3.

35 - Lay La | rrie i' TCUr | arag | ada Le |

28 & Lg@44-1mA I /1,=0.33 4

14 4

0] 7 LLL | LLLi L, Lit | LLEiL | Lit ut
0 50 100 150 200 250

z (um)

35 = rd | aed | Rie 1 aere | Trait 4

28 I =44.1mA 1/.=0.77

C total 1°2 3

216 4

7E 4

0 ct L Ll LELi Ly Lui j Lit il LEI ud
0 50 100 150 200 250

Z (ym)

35 r Trg |] Tire Prag | gitge | Tere 4

28 & -

21 F :

14 E :

7 I =44.1mA IA =3.3

- total 12 4

0 Popes haart tarp tare sle iil
0 50 100 150 200 250

zZ (um)

4.4

eff

2.8

Dot----l = 21. J
t th
Triangle---|, = 1.5 ld

00 05 10 15 20 2.5 30 3.5

L/L,

Figure 4.9: The calculated effective linewidth enhancement factor of the

DFB laser for facet phases @,=135°, @p=0°.

To understand the experiment results, the above threshold DFB model (Chapter 3)
which includes LSHB is used to calculate the linewidth of the DFB laser. We neglect the
linewidth broadening contributed by the noise from carrier diffusion. Because the facet
phases are difficult to determine from the experiment, several values of them have been
assumed in our calculation of the linewidth (Fig. 4.6). In the calculation, the width
between two electrodes is chosen as 10um, the carrier lifetime 2ns, the differential gain
3x 10-16 cm 2, the waveguide confinement factor 0.2, and the loss from absorption and
scattering 40 cm-!, The linewidth is obtained at J,=1.5 I,, and I, = 2 J,,, where I, is the
threshold current with uniform current injection. It is found that the linewidth is small for
either I/Ip=0.2-0.3 or Ij/17=2.5-3.5. At those ratios, the small linewidth has also been
measured. The DFB laser is found to be multimode for either Jj/I7<0.2 or Ij/I2>3.5. This
may explain the extremely large linewidth we obtained in the measurement. The calculated
linewidth is sensitive to the facet phases. It seems that the linewidth in Fig. 4.6 is the result

closest to the measurement.

The threshold current and lasing wavelength are simulated in Fig. 4.7. The
threshold current doesn't change very much with the current ratio. It is found that mode

switching occurs for those current ratios at which the linewidth is very large.

To examine that the SHB is important in linewidth broadening, the photon
distribution is shown in Fig. 4.8 for I;/Iz=0.33, 0.77, and 3.3. The linewidth at

I)/Ip=0.33 and 3.3 corresponds to the minima in Fig. 4.6 (@;=135°, @r=0°). Fig. 4.8

shows that LSHB is serious under uniform injection (I;/l2=0.77). From (4.49), the
influence of SHB on the linewidth comes from its influence on Oege and R, which are
dependent on the photon distribution. The total photon number J, is proportional to the
output power and the function of the injection current. The effective linewidth
enhancement factor Og is shown in Fig. 4.9. The a is very much affected by LSHB
and can change greatly. It is large when the LSHB is serious (Fig. 4.8). Large og will

result in large linewidth even if the material @ is small. The a, is also found to be

nearly independent on the total injected current.

4.5 Conclusion

In this chapter a small signal model for SLM DFB lasers including LSHB is presented.
The solution of the coupled-mode equations has been solved by the Green's function
method. The rate equations for the photon density and the phase of the optical field in DFB
lasers have been derived. The model can be used above threshold in a self-consistent way
if the photon and carrier density distributions are calculated by including LSHB.

The effects of nonuniform photon density distribution on the high frequency response
of DFB lasers are studied. The resonance frequency and damping rate of a two-section
DFB laser is measured as the functions of optical power for various current injections. At
the same optical power level, the resonance frequency and damping rate are different for

different photon density distributions. We have also measured the linewidth of a 1.55 pm

two-section DFB laser. It is found that by adjusting the current ratio between two
electrodes, the linewidth can be minimized. Our theoretical analysis of the linewidth of the
laser gives similar results to the experiment. It shows that LSHB does play an important

role in the broadening of the linewidth.

~ 75

Reference

[1] H. Statz, and G. deMars, Quantum Electronics (Ed. by C. H. Towns, Columbia
University Press, New York, 1960), p. 530

2) K.M. Lau, and A. Yariv, IEEE J. Quantum Electron., QE-21, 121 (1985)

[3] J. B. Moreno, J. Appl. Phys. 48, 4152 (1977)

[4] J. Feng, T. R. Chen, and A. Yariv, (to be published)

[5] M. Kuznetsov, A. E. Willner, and I. P. Kaminov, Appl. Phys. Lett. 55, 1826 (1989)
[6] R.J. Lang, and A. Yariv, IEEE J. Quantum Electron., QE-21, 1683 (1985)

[7] R.J. Lang, and A. Yariv, IEEE J. Quantum Electron., QE-22, 436 (1986)

[8] A. Yariv, Quantum Electronics, 3rd ed., ch. 11 (1989)

[9] M. Kuznetsov, JEEE J. Quantum Electron., QE-24, 1837 (1988)

~ [10] B. Tromborg, H. Oslesen, and X. Pan, IEEE J. Quantum Electron., QE-27, 178
(1991)

[11] P. Vankwikelberge, G. Morthier, and R. Baets, JEEE J. Quantum Electron., QE-26,
1728 (1990)

[12] T. Schrans, Ph.D. Thesis, California Institute of Technology, Pasadena, Calif., 1994

[13] C.H. Henry, IEEE J. Lightwave Technol. LT-4, 288 (1986)

76 |
a ; ,
[14] J. Feng, T. R. Chen, and A. Yariv, Appl. Phys. Lett. 67, 3706 (1995)
[15] P.A. Morton, T. Tanbunek, R.A. Logan, A.M. Sergent, P.F. Sciortino, and D.L.
Coblentz, IEEE Phot. Tech. Lett. 4, 133 (1992)
[16] A. Yariv, Quantum Electronics, 3rd ed., ch. 21 (1989)
[17] K. Kikuchi, Electron. Lett. 24, 1001 (1988)
[18] K. Kikuchi, IEEE J. Quantum Electron., QE-25, 684 (1989)
[19] M. Okai, T. Tsuchiya, K. Uomi, N. Chinone, and T. Harada, JEEE Phot. T. ech: Lett.
3, 427 (1991)
[20] M. Okai, T. Tsuchiya, K. Uomi, N. Chinone, and T. Harada, IEEE Phot. Tech. Lett.

4, 526 (1992)

[21] X. Pan, H. Oslesen, and B. Tromborg, Electron. Lett. 26, 1074 (1990)

Chapter 5

Small Frequency Chirp Distributed

Feedback Lasers

5.1 Introduction

Until most recently, the understanding of frequency chirp in single mode semiconductor
lasers has been very limited. It is widely known that undesirable spectral broadening of
single mode DFB lasers often exists when the lasers are under amplitude current
modulation (AM) [1-5]. It is believed that the frequency chirp is due to the index variation
~ caused by injection current modulation [6-7], and the axial variation in carrier density
attendant upon the longitudinal photon density variation (spatial hole burning) [8-9]. A
simple relation between optical output power and optical frequency was derived, in which a
linewidth enhancement factor (o parameter) links the frequency chirp and the instantaneous
optical power when the lasers directly modulated [6]. Means for reducing the frequency

chirp have included the use of laser with a smaller linewidth enhancement factor of the

78
a .
active material [10]. and methods which reduce the longitudinal spatial hole burning
(LSHB) [11-12]. The second approach involves fabrication of either complex laser
structures which introduce gain coupling or quarter-wavelength-shift three-section DFB
lasers. With those methods, DFB lasers have chirps about 1A (which corresponding to
12.5 GHz chirp for 1.55 um DFB lasers). In high bit rate data transmission in optical fiber
over long distance, low chirp DFB lasers are urgently needed.

We discover that ultra small chirp ("zero chirp") could be realized in two-section
DFB lasers [13]. The reduction of the chirp is due to the effects of carrier density
fluctuations in the two sections on the chirp partially compensating each other [14]. The
axial carrier density is not uniform in DFB lasers due to the LSHB. In DFB lasers the
contribution of local carrier density variation to the frequency chirp is described by an
axially varying function (Cy(z) function) which depends on the distribution of the
traveling fields in the lasers, rather than by the @ parameter in conventional theory for

uniform photon density lasers. The difference between DFB lasers and uniform photon

density lasers makes it possible to fabricate a "zero chirp" DFB laser with a multisection

_ Structure.

In this chapter, we will show that changing the distribution of the photon density
along the laser cavity can affect the sign and magnitude of the frequency chirp [15]. By
changing the injection currents into the two sections independently, we have a major
reduction in frequency chirp in 1.3m lasers and the production of either positively (chirp

increases with time) or negatively (chirp decreases with time) chirped signals. We will

also report on the measurement of the frequency chirp of a two-section DFB laser under
large signal modulation. The unique advantage for using two-section DFB lasers is that the

frequency chirp can be reduced even when LSHB is not completely suppressed.

5.2 Simple relation between intensity modulation

and frequency chirp

In this section we derive a simple relation between the intensity modulation and frequency
chirp in single longitudinal mode semiconductor lasers assuming a uniform photon
distribution. We consider the conventional rate equation for the average photon density in

an SLM semiconductor laser

&8 = (ry,g(N) - Lys + Pol! (5.1)
t Tsp

Tsp
where I” is the mode confinement factor, g(N) , the optical gain, t,,, the cavity photon
lifetime, Tp», the spontaneous lifetime, and Bsp» the fraction of spontaneous emission into
the lasing mode.

We need the frequency shift from some operating point gg, Sg, and lasing
frequency Vo. Without invoking any small signal approximation, we let g(t)=g9+Ag(t)

and S(t)=So9+AS(t). We adopt the approximation g(N)=a(N-N,). (5.1) is rewritten as

vel Ag(t) _ aS , AS() BN | 4 Bsp | (5.2)

ot So Tsp AV eT sp

(5.2) is then reformulated in terms of the complex effective refractive index,
Neff = n, att n, jp With positive na representing gain
Ag(th= 42 And A (5.3)

and we also employ the linewidth enhancement factor which is

— .__eff | __ eff
a=- aN / aN (5.4)

The a@ factor here is assumed to be not strongly carrier density dependent. Then we can

say that Ang = Any |

The laser frequency chirp Av associated with An, g iS taken to first order as

Av A" (5.5)
Vv ?
Nog

With (5.2), we then obtain

_ glam) AS) BN Bsp
A vt) = | a *S@S0 tm) \! * S@avaty ©-6)

For system application it is easily verified by inserting numbers in (5.6) that, apart from

very low optical powers, we obtain [7]

A v(t) = won sO . (5.7)

With considering the gain suppression, the gain is given as

g(N, S) =a (N-N,) (1-€S) (5.8)

Then (5.1) is rewritten with (5.8) and we can rederive the expression for frequency chirp

_. as [6]

wry = (Ams , S(t)
A v(2) #| SO aa (5.9)

This is a very useful relation. For a given intensity modulation, it predicts the related
frequency chirp. However, the analysis is oversimplified, since a uniform photon density
is assumed. In DFB lasers, due to the nonuniformity of photon density and carrier density
(spatial hole burning), (5.9) does not accurately describe the frequency chirp. The rate
equations (4.22) and (4.23) for photon density and the phase of optical field in DFB

lasers have to be used to solve the frequency chirp.

5.3 Propagation of optical signals in fibers

5.3.1 Time-bandwidth product of chirped pulses
_ In order to estimate the spectral width for a given pulse shape S(t), one has to consider the

field with respect to amplitude and phase by, e.g., considering the complex field amplitude

E(t) =vS(@ expGo@) (5.10)
@(t) is related to S(t) according to (5.9) by considering

“OO = 2m (V-Vn) (5.11)

For a first-order estimation we shall neglect the influence of nonlinear gain, carrier
. inhomogeneity and spontaneous emission. Thus, (5.9) yields

AK) _ qgdinSO
dt 2 dt

(5.12)

Using (5.12), the relation between the complex amplitude E(t) and the photon density S(t)

is obtained as

E(t) = [S(e) |"? (5.13)
We are going to consider a Gaussian pulse shape which has normalized photon

density as

S() = exp (-(tl/%)*) (5.14)

with a pulse width (full width half maximum)

t=2Vin2 T%} (5.15)
(5.12) yields a linear chirp for this Gaussian pulse:

dt) 2
i= t/t) (5.16)

- After performing the Fourier transform, the Gaussian pulse also yields a Gaussian shape

spectrum

Ip() = |E(o)? ~ exp(-(@/Q,)") (5.17)

with

Qy =(1/T) Vi+o2 (5.18)

The spectrum of the pulse is obtained as the Fourier transform of (5.14)

. Bo) | E(t) exp(-iot) dt) (5.19)

The spectral width (full width half maximum) Av with respect to the optical frequency is

related to Q, as

Av = 2Vin2 Q, /21 (5.20)
. finally yielding for the time-bandwidth product of Gaussian pulses [16]

Av At=0.44 Q, %| = 0.44 Vi+02 (5.21)

5.3.2 Transmission of chirped pulses over single-mode fibers

If a single-mode fiber of length L is used as the transmission medium, the transmission of
field to the output field is described by a propagation factor exp( -iBL) with the propagation
constant $B. Since the optical pulse only comprises frequency components close to the

center frequency @p, the propagation constant 8 may be expanded around @=ap, yielding
B(@) = Bo + Bi (@-0o) +5 Br(@-ao)" +-- (5.22)

The parameters B; and fp are related to the refractive index n and its derivatives through

the relations
By = Ln +d = 1 (5.23)
Boal 9.dn +odn ~ A> d’n (5.24)
Cl d@ dw! 2c? qd)?

Optical pulses broaden with the distance of propagation due to the group velocity

, . dv
dispersion of the fiber io #0 . We define the dispersion parameter
p=%h1 _ 2m g,~- A d’n (5.25)
dx 2 Cc d 2

Most of the installed fiber base in the world today has a zero dispersion at A = 1.33 um
and a D = 17 psec/nm-km at A = 1.55 um.

The input and output pulses are described by their slowly varying complex field
amplitudes according to (5.10) with E(t)=E;,(t) for the input pulse and E(t)=E,,,t) for the
output pulse or their respective Fourier transforms Ej,(@) and Egy(@) as in (5.19)
yielding

Eou(@) = exp(-iPoL) exp(-iBiLo) exp(i A DLO) En(@) (6.26)

' After applying the Fourier-transform relationships to Ej,(@) and Eoy@), the relation
between the input and output fields, E;,(t) and E,,,(t), is obtained. Therefore, the related
input and output pulse shape in the optical power Pj,(t) ~ |Ein(? and Pot) ~ Eoudt)? are
- obtained. For input/output relationship the phase term exp(-iPpL) is irrelevant and the term
exp(-iB;Lq@) just corresponds to a time delay of B;L between the input and output pulses
so that the change in pulse shape between the input and output pulses is solely described by
the term exp[i (A?/4nc)DL@*]. The formalism described above can be applied to any shape
of the input pulse. In this chapter, we restrict discussion to the transmission of Gaussian

pulses.

After applying the Fourier transform to an input Gaussian pulse, which is described

by (5.14), one obtains Ej,(@). (5.26) then yields E,,,(@) from which the inverse Fourier

transform E,,,(t) is obtained. For the Gaussian input, the output is also Gaussian,

Poult) ~ lEouXOP ~ exp(-U(t BL) (L))) (5.27)

with

[LP = D4L2(A7/ 2nc ) +(B+aD2L a) 2m)

(5.28)

-%L) becomes very large for large tj as well as very small t — 0. The minimum output

pulse width is obtained for

%=V14+Q? D*L A7/ 21 (5.29)

yielding % —> To, min

(CL) Prin =2 (Y1+02+0) D7L A7/ 21 | (5.30)

With (5.15) and (5.30), the output pulse width is given as

1/2
Atmin(L) =2 Vin 2. Tmin(L) = Heo £0 Av D2L A2Ic 631)
1+aQ- .

Numerous research efforts are pursuing various approaches for alleviating the
broadening of optical pulses in fibers. These include:
1. Using two fibers in each link with opposite dispersion DL; = - D2L2 to renarrow
broadened pulses,

2. Using mid-span optical phase conjugation to renarrow the pulses,

. 86

3. ~ Eliminating chirp from lasers or generating pre-chirped laser pulses.
We will focus our effort on reducing the laser chirp, which, comparing with other
methods, could be the most efficient and economic way to overcome the broadening of the

pulses in fibers.

5.4 Theoretical foundation of the frequency chirp

reduction in two-section DFB lasers

The optical field in DFB laser is described by the coupled-mode equations (2.14) and
(2.15). The solution of the equations is obtained through a spatial integration of Green's
function weighted with driving force[17-18]. The rate equations for the optical power and

the phase of the optical field are given in Chapter 4 as

l l

£ P(t) =2P@) ({ Re{Cn(z)}AN(z,t) dz | Re{Cs(z)}AS(z,t) dz] +R + Fp(t)
0 0 a
(5.32)
l Il
60-1] Im{ Cn(@)}AN(z,t) dz ‘| Im{Cs(z) }AS(z,t) dz] +F g(t) (5.33)
0 0

where

Cx = 5E*@E@) Bo! ( E*(2') E(z') Be ae’ (5.34)
a 0X 0 d@

and for a single-mode DEB laser, the transverse electric field can be represented by

E(z,1) = E*(z,t) exp[-ifz+i(@p+A@)t] + E-(z,t) exp[+ifPz+i(@p+Aa)t] (5.35)
where X= N,S. Here R is the spontaneous emission coupled into lasing mode, and S , the
photon density. E+(z), E-(z) are the forward and backward propagating electric fields in
the laser, and B is the propagation constant of the lasing mode. AN(z,t), AS(z,t) are the
deviations of carrier and photon densities from their stationary distributions under DC
bias. Fp and Fy are Langevin noise sources related to spontaneous emission. The

derivatives of B are

OB _ dys 98
deb aeeT (5.36a)
OB __ j_&

S 2 1+eS (9.360)
op _ 1 fj og
Jo = ve +350 (5.37)

and the modal gain g is given as

a INGt-Nil

5.38
[ 1+eS(z,t) ] 638)

8g (z,1) =

where v, is the group velocity, € is the nonlinear gain suppression coefficient, and a is the
differential gain.

The evolution of carrier density distribution is given by the rate equation

-BN(z,t)* - CN(z,t)? - a INGNerl_, S(z,t) (5.39)

[ 1+eS(z,t) ]

AN(z,t)_ Jt) N(z,t)
dt ._ ed T

In this equation, the nonlinear gain suppression the last term and the Auger recombination
rate ( CN3 ) are included. S is the photon density, and T is the carrier lifetime.
For small signal modulation, the dynamic variables J(z,t), N(z,t), and S(z,t) can be

written as the sum of their stationary distributions and the variations due to RF modulation

J(z,t) = Jo(z)+J1(z.t) (5.40a)
S(z,0) = So(z)+So(z)S1() (5.40b)
N(z,t) = No(z)+N1(z,0) (5.40c)

Here Jo(z), No(z), and So(z) are the stationery distributions in the laser. The dynamic

frequency @(t)=Qjgset+ dat). Combining (5.32), (5.33), (5.39) and (5.40), the frequency

chirp of the laser is obtained
l l
60 (t) = [ Im{ Cn) }N1(Z,t) dz + suo] Im{ Cs(Z)}So(z) dz (5.41)
0 0

The Langevin noise source Fp and Fg are neglected in the case when the laser bias
- current is modulated. A temperature dependent adiabatic chirp should be included in (5.41)
if the modulation frequency is less than 10 MHz. The first term in (5.41) is known as
transient chirp, which is proportional to modulation current; the second term is adiabatic
chirp, which depends on the bias current. In our case, the transient chirp is about two

orders of magnitude larger than the adiabatic chirp.

Figure 5.1: A two-section distributed feedback semiconductor laser.

The imaginary part of Cy(z) is the contribution of the local carrier density variation
Nj(z) to the frequency chirp. Contrary to its expression in the case of a laser with uniform
photon density, where it is a spatially independent constant proportional to a, Im{Cy(z)}
strongly depends on the distribution of traveling fields E+(z), E-(z) and can vary greatly.
| A small @ DFB laser could have very large chirp because Cp(z) is more sensitive to the
distribution of E+(z ) and E-(z). For a two-section DFB laser, if one of the sections is RF
modulated, the carrier density variation N;(z) has two contributions, one from injected
modulation current, the other due to carrier depletion of photon density variation. The
"zero" chirp of two-section DFB lasers can be realized when one section contributes

positive chirp and the other one, equal amount of negative chirp, which is expressed as

is +l.

[ Im{ Cn(z)}N1(z) dz + } Im{ Cn(z)}N1(z) dz =0 (5.42)

#0 h
To calculate the chirp in multisection DFB lasers, a comprehensive computer
simulation program introduced in Chapter 3 has been used. The program includes the
nonuniformity of optical intensity and carrier density in lasers, which can compute
threshold, linewidth and frequency chirp of multisection index coupling and gain coupling

distributed feedback lasers.

5.5 Measurement of the frequency chirp in two-

| section DFB lasers under small signal modulation

5.5.1 Experiment No. 1: Frequency chirp in 1.55m_ two-section

DFB laser

CY)

a) Jia Jitj ei rQ +

Li=138um L2=106um

Figure 5.2: A 1.55ym two-section DFB laser used in experiment. The laser has

two sections of 138m and 106m long, respectively.

A two-electrode 1:55pm InGaAsP/InP DFB laser is fabricated. The laser has one cleaved
facet and one AR coated, and the two sections are 106m and 138um long with an
electric isolation of 1kQ. The detailed laser structure has been described in Section 4.4.
The two sections are pumped with DC currents while one of them is also RF modulated.
The laser output is fed into an optical fiber after an isolator of more than 45 dB isolation.
The output from the fiber is collimated and sent to a Burleigh RC-140 Fabry-Perot with a
resolution of 0.2 GHz and a free spectral range of 15 GHz. The output from the Fabry-
Perot is incident on a Ge detector, and the time average spectral density of the optical
field of the laser is displayed on an oscilloscope.

The spectral width (frequency chirp) versus the current ratio J>/T; is measured at a
modulation frequency of 50 MHz and an RF current of 4mA. The total current into the
laser is kept at 70 mA. The result is shown in Fig. 5.3. When the long section is RF
modulated, the spectral width of laser has a minimum for /;=20mA (Jp/I;=2.5). If the
short section is RF modulated, the minimum of the chirp occurs for 17=15mA (I2/1;=3.7).
The chirp increases dramatically when the injected currents are changed from conditions
_ where the chirp is at its minimum. The minima in Fig. 5.3 are less than the resolution of
the Fabry-Perot. The behavior of the measured spectral width will be the same as long as
the modulation frequency is much smaller than the oscillation frequency of the laser.

The photon density and carrier density distributions at minimum chirp bias are shown
in Fig. 5.4. Carrier density is close to uniform in each of the sections due to the different

pumpings in two sections. But the photon density shows strong nonuniformity and varies

| 93
from 6x1018 cm-3 to 14x10!8 cm-3. Cy(z) and N)(z) for the DFB laser are also calculated
and shown in Fig. 5.5 and Fig. 5.6. The product of Im{Cy(z)} and Nj(z), which gives
the chirp form (5.41), can be obtained and is plotted in Fig. 5.6. When the chirp is at its
minimum, it can be seen that N;(z) in the section two is out phase with that in the section

one, and the integration of the Im{Cy(z)} *Ni(z) along the laser axis is very close to zero.

This verifies. the minimum chirp condition given by (5.42).

10°

Settion oné RF modulated ! (a)
Section two RF modulated

Av (MHz/mA)
2,

1 I Litt
10 0 1 2 4 5
1/1
2 #1
10'°
EOSection one RF modulated =|
- 4Section two RF modulated (b) q
10°

pol

10°

Av (Hz/mA)

Figure 5.3: (a) The measured spectrum width of 1.55u1m two-section DFB laser
(L1=106um, L2=138pm), at various ratios, with bias current Ij+I1,=70mA.

(b) The calculated chirp of the two-section DFB laser. The bias current
I,+Ip=1.75In. I, is the calculated threshold current of the DFB laser uniformly

pumped.

a PUPUPPerrypsreryprrraeparrr 5 Q
— 14h 4 t
2 of 44h

5 4 i]
= 12 pay green 7 *
— 5 4 “4 Qu.
> : 7g
_ 10/ : 2.
3 [onnonesenen ne ( 424
SF set i +e
= 3 1,¢
2 r a
= 6F |
faut Pes bore pda rir iitiiiid g a

0 50 100 150 200 250
z(um)

Figure 5.4: Calculated photon and carrier densities of the two-section DFB laser
with the 1st section RF modulated. The bias current J7+J,=1.75I,, and current

ratio IJ; =3.9, which correspond to the minimum chirp condition in (5.42).

80 Gey 377300
a hn
8 60L ~,
Ss : 4250 4
40 s
= ME =

x : 1200
oO 206 =
> - =
[ 150}

Re of.
-20 Cossidaseiterer tips dies 100
0 50 100 150 200 250
z(Um)

ale SAMA MAA RAE SY 10 ey
2 0.05 [acral 5
Fi . ae 2
Le —
~ a a
~ r 2
= oof 0 2
nN
7_~ a e —
> -0.05 ee i a
. C “sen” =
s —

00.1 Conriborestirri tii ties -10

Figure 5.5: (a) Calculated Cy(z) and (b) calculated N;(z), Ni(z) eIm(Cy(z)) at the

minimum chirp condition

50
z(m)

in Fig. 5.2.

100 150 200 250

_ 97

5.5.2 Experiment No. 2: Frequency chirp in 1.3m
7 InGaAsP/InP quantum well two-section DFB laser

A two-electrode 1.32um strained quantum well InGaAsP-InP DFB laser is used. The
laser has cleaved facets, and the two sections are 375m and 250um long with an electric
isolation 1kQ. The DFB laser has at least 40 dB of SMSR under CW condition. It has a
threshold of 18 mA when uniformly pumped. The coupling coefficient is calculated at
kL=3. The light output is collimated and divided by a beamsplitter. One beam is sent to a
Coherent No. 240 Fabry-Perot interferometer where FM spectrum is measured, and the
other beam is sent to an Ortel 2516A photodiode to determine the intensity modulation

index m (see Fig. 5.6). The laser is optically isolated with minimum 60 dB isolation.

Photodiode

Isolator

Fabry-Perot
Interferometer

Figure 5.6: Experimental setup for measuring FM/AM response in DFB lasers.

Fig. 5.7 shows a typical measured FM spectrum from the Fabry-Perot

interferometer. The phase difference between the AM and FM modulations depends on the

modulation frequency @,, and optical power. Normally, the FM spectrum is asymmetrical
- at modulation frequency much smaller than the resonant frequency. As the @, approaches
the resonance frequency, the FM modulation is 7/2 out phase with the AM response and
the FM spectrum has equal sidebands around the lasing frequency.
The optical field E(t) can be represented as
E(t) = Eo[1+m Sin{(@nt)+ 01} ]!“*exp[i{ @jt-B Cos( @pt+ 62)} ]. (5.43)

Here f is the phase modulation index. We write the AM modulation, and FM or frequency

chirp as
p(t) = m Sin(@,t+ 81) (5.44)
dat) = BO,» Sin( Dpt+ 92) (5.45)

Time base (200uUsec/div.)

Figure 5.7: Atypical output intensity from Fabry-Perot interferometer.

To determine the phase modulation index B and the @2—0;, we directly measure the
. intensity modulation index m from AC output of the Ortel 2516A photodiode with an HP
8565A spectrum analyzer and observe the FM spectrum on the Fabry-Perot interferometer
(Fig. 5.7). The heights of the central peak and the first order FM sidebands are given by

[19-20] |

oP=s2(p) +m? J i (B) sin?( 6-61) (5.46)

|E( cot Om °=J>(B) +m[Jo(B)+J5(B)V4 +m2Jo(B) J2(B) cos[2(O>-01)/2

(5.47)
tmJ\(B) (Jo(B)+J2(B)] cos(@-1)

where J, is the nth-order Bessel function of the first kind. The B and the @2—0; can be
solved with (5.46) and (5.47).

The measurement is conducted at modulation frequencies f,,=@,,/27 =250 MHz, 500
MHz, 800 MHz and 1.5 GHz. The injection current distribution is changed by adjusting
the DC currents into the two sections and the optical power is kept at P = 0.8 mW
throughout the measurement. In Fig. 5.8 we show the values of B/m as function of the
injection current distribution. We use current density ratio i2/ij=I2Lj/1)L2, where J; and Ip
are the DC currents into the two sections. The uniform pumping condition corresponds to
i/ij=1. From Fig. 5.8, we see that as the current density ratio i2/i; is increasing, the B/m
has large drops at i2/i7=1.3 (at 250 MHz), 1.7 (at 500 MHz) and 1.85 to 2.3 (at 800
MHz). The f/m decreases approximately by one order of magnitude. It is expected that the

condition for ‘zero chirp' (5.42), is approximately satisfied at those injection current

100

distributions. For’ fin = 1 .5 GHz, the modulation frequency is close to the resonant
frequency, and the f/m is much less sensitive to the optical density distribution and
expected to be closed to o/2 [21-22]. That is why we have much smaller reduction of B/m
at 1.5 GHz modulation by a factor of less than 2 from Fig. 5.8d.

The phase difference @2—@; is also calculated from the observed FM spectrum
according to (5.46) and (5.47), and the result is shown in Fig. 5.9. The 02-0; is less
than 1/2 for small current ratio i2/i1, which corresponds to |E(@+@,,)|>1E(@-@,)|. As i2/i]
‘increases until the chirp reaches its minimum, the 02-0; approaches 1/2. If i2/i;

increases further, the 0,—0, is larger than n/2 and |E(@+@,,)|<|E(@—-@,,)I.

101

PTTeTrPrpP erry rey
f =500MHz 7

0 pirrtresrtirirtisirtsiss

rerprbiezprrttisrrrtisr1

“0.0 0.5 1.0 1.5 2.0 2.5 do 10 2.0 3.0 4.0

i/i, ii,
Ad A 10 Gere pee
: (c) f, =800MHz J r (d) £,=1500MHz 1
Bib 1 =] |
co J GF 4
0.1 a oo ] beeew deers Pa
0.0 10 20 3.0 40 0.0 1.0 2.0 3.0 4.0 5.0

Ji, ifi,

Figure 5.8: The measured f/m of 1.3m InGaAsP/InP QW two-section DFB
laser (L1=375um, L2=250um) for various current density ratios i2/i;. The two-
section DFB laser is under AM modulation: (a) fin = 250 MHz, (b) ff, = 500 MHz,
(c) fin = 800 MHz, (d) fin = 1.5 GHz.

102

a»
3.14 Gee 3.14 Geet
: f =250MHz 4 L. f = 00MHz-
© 1.57 [™ 71.57
a a
7) re re ee 0 rercrtlarritirzarsrtirins
0.0 05 1.0 15 20 25 0.0 10 2.0 30 4.0
i/i i/i
21 eT
3.14 a 3.14 peepee
; f =800MHz - f =1500MHz -
71.57 {— F 1.57 Pad
e i | [Oeeeeee 1
5 pore ti pa ba Pl rrp ttrrrrttisrrrtizperrtiryys
0.0 10 20 3.0 4.0 0.0 1.0 2.0 3.0 4.0 5.0
i fi, i fi,

Figure 5.9: The phase difference 02-0; as a function of the current density ratio in/ij

from the measured FM spectrum of the two-section DFB laser under AM modulation.

103

5.6 Measurement of the frequency chirp in two-

section DFB lasers under large signal modulation

If the photon density is uniform, the frequency chirp can be expressed simply as the
combination of the "transient" chirp, which is proportional to the time derivative of the
photon density in lasers, and "adiabatic" chirp, which is proportional to the photon
density. The chirp in DFB lasers has to be calculated numerically due to the longitudinal
| photon density variations in the lasers. Numerical simulations of the frequency chirp of
one-section DFB lasers have been proposed by several groups [8, 23]. We study the
frequency chirp of multisection DFB lasers with a model based on the time-dependent
coupled mode wave equations (3.6) and (3.7). The model explains the basic observed
features of the frequency chirp of the two-section DFB lasers.

A two-electrode 1.3 um InGaAsP/InP DFB laser has been used in the experiment.
The laser structure has been given in Section 5.5. The two sections are 400um and 100um
long with an electric isolation of 1.4 kQ. The threshold current for section one is measured
~ as a function of the injection current in section two and is shown in Fig. 5.10. As the
current in section two increases, the active material of section two becomes less absorptive
and, eventually, increasingly amplifying. The threshold for section one will thus
gradually decrease. The peak in the figure is the stage when the laser switches from one

single mode to another mode.

104

35 A RE

= Z J
set
By a ‘
=} - 4
0g [ 1
o0 a _
2A 2b a
en 5 4
Ss f
15 rn re re ee Pe eee ee

0 5 #10 15 20 25 30 35
I, (mA)

Figure 5.10: Measured threshold of the 1.3m two-section DFB laser

(L}=400um, L2=100pm) for various current ratios.

The two sections are pumped with DC currents while the long one is also AC
- modulated. In all measurements, section one is biased above threshold. The laser output is
collimated and divided by a beamspliter into two beams. One beam is focused onto an Ortel
2516A high speed photodiode, while the other .beam is focused on a SPEX 1870
monochromator slit with a resolution of 0.5 A to observe the modulated light spectrum.
The monochromator detection system consists of a Ge detector and a lock-in amplifier. The

wavelength of the monochromator is scanned at a low speed of 0.1 Als so that the envelope

-105—

of the dynamic wavelength shift is recorded. The full width at half maximum of the
optical spectrum is measured to evaluate the frequency chirp. The output of the photodiode
is fed toa sampling oscilloscope where the time waveforms for the modulated light are
observed. The section one is biased at J;/I,,=1.8 for I,=0, and the optical power is then
maintained at that level for various J2 values by changing the current 17. The optical
modulation index m is monitored by the sampling oscilloscope and fixed at 100 percent by

adjusting the AC microwave power (typically 10 dBm) into the section one (see Fig. 5.11).

106

UOneTNpoU eusis eB] apun Josey g.jC] woH}Oes-om} Jo diryo oy} Jo JuouLomsesur oy} Joy dyes yeyuourtIodxY :[1"¢ omn3ty

adoodsoji9sO
Buryjdwes

Jayndwoy

apolpojoud
V9LS2 [SHO

saddoyuy

sodury |

Ul-4907

y+ JoyeEWwOJyYOOUOW

10}]B19USE)
jeuBbls

40]9919q

Jo} }1;)dswesg

oa] seig

107

2 Perrperrrprerryprreryprerrys

1.8

1.6

An (A)

1.4

1.2

Crryprrryprrryprpere

1 i 3 | a | { = oe I | Oe ee 3 | Ltd
0.0 5.0 10.0 15.0 20.0 25.0
r (mA)

Figure 5.12: The measured frequency chirp of the two-section DFB laser with
optical modulation index M=100%; average optical power is kept at the same

level as that of J 7=1.81t, and Iz=0. It, is the threshold current for section

one when J>=0.

108

The frequency chirp is measured at a modulation frequency of 5 GHz, and the result
is shown in Fig. 5.12. A minimum nears 1.2A when J) is in the range of 14 mA to 18
mA. We find the frequency chirp to be strongly dependent on the modulation frequency.
The chirp of the laser is also measured at modulation frequencies of 3 GHz and 2 GHz.
The frequency chirp at those modulation frequencies is much smaller than that at 5 GHz.
The features of the frequency chirp at those frequencies can not be obtained because for
some of the cases the chirp is less than the resolution of the monochromator. It is observed
that the distortion of output optical waveform becomes serious when either the modulation
current or the modulation frequency increases. The distortion is a result of the relaxation
oscillation which is accompanied by a large frequency chirp [9, 24].

Taking into account the spatial hole burning along the laser axis, we calculate the
complex reflectivities of a two-section DFB laser by means of the propagator F matrices
[25]. The dynamics of the laser under high frequency, large single modulation is
calculated with the coupled mode equations (3.6), (3.7) and the rate equation (5.39).

The frequency chirp of the two-section laser using this numerical simulation is
_ shown in Fig. 5.13. The modulation frequency is set to be 5 GHz, and the optical
modulation index at 100% with the average optical power kept a constant. The parameters
used in the calculation are: a linewidth enhancement factor of 4.8, a nonlinear gain
suppression coefficient of 10-!7cm3, a spontaneous emission factor of 10-4, a carrier life-
time of 2ns, and a photon lifetime of 1ps. In Fig. 5.13, we observe that the calculated

frequency chirp also has a minimum value of 1.43 A when Iz=7.3mA. For values of I,

109

less than 1 mA or greater than 14 mA, we have difficulty with the convergence of the
solution for the dynamic lasing condition because the photon density distribution is very
asymmetrical. We find that the calculated chirp hardly changes when other values of the
phases of the facet reflectivities are used in the calculation. In our calculation, the phases
are assumed to be zero. The basic features and the range of values of the numerical

analysis are in agreement with the observation.

110

1.6 t 1 LJ T i | ' T i i Ly v ' | qT

1.5

1.4

1.3

An (A)

1.2

1.1

rrpcetotr ris rrrlipriti esta

1 Ps ee
4.0 8.0 12.0 16.0
I (mA)

Figure 5.13: The calculated frequency chirp for the two-section DFB laser
used in the experiment with optical modulation index M=100%, average
optical power kept the same. The peak to peak modulation current is about 10

mA.

To explain the reduction of the chirp in two-section DFB laser, we use the average
photon densities in each section and the mode frequency shift Aq due to average gain

change. AG ; in the two sections is then given by [25]

Aon = A (L#AG +L 2AG2) / (LEELA) (5.48)

othe

R-

Here A is the material constant of the active layer, and we define the effective lengths of

the sections as

LY = (0g/2) sn (5.49)

To maintain oscillation, the gain change in one section must be compensated by an
opposite change in the other section (AG;, AG2 have the opposite values), and Lf
depends on both the gain (or injection current) [25] and the wavelength detune 6=A-
Abrage Where Apyage is the Bragg wavelength. By adjusting the current distribution in the
DFB laser, the effective lengths can be changed significantly. In theory, the frequency
chirp Aq can be reduced to zero if we can change the effective lengths by adjusting the

current distribution so that Le AGi+L of AG? equals zero. However, when taking spatial

hole burning into account, a finite frequency chirp Aq results.

112

5.7 Control of the frequency chirp in DFB lasers

Chirped pulses have been used for pulse compression in optical fibers as a compensation
for chromatic dispersion of the fiber [26-28]. To achieve pulse compression, positively
chirped or blue-shift chirped (frequency increases with time) pulses are sought in order to
compensate the fiber dispersion due to the negative chromatic dispersion (B2<0) of fiber at
long wavelength (> 1.3m). Negatively chirped or red-shift chirped (frequency decreases
with time) pulses, however, will broaden after traveling in any distance of optical fiber
(Fig. 5.14), The time-averaged power spectrum of single section DFB lasers under high
speed modulation has been measured by several groups [2-3]. In Fig. 5.15, we show two
time-averaged power spectra of directly modulated SLM DFB lasers. One is negatively
chirped, the other is positively chirped. Unfortunately, for most directly modulated single
section DFB lasers, the spectra are found to be associated with predominantly negative
frequency chirp. To get the blue-shift chirped optical pulses, an FM DFB laser and an

external intensity modulator had to be used [28].

- 113

Pulse Compression in Optical Fiber

Positively chirped Pulse compression Negatively chirped
pyise pulse
Dispersive Fiber f.<0 Dispersive Fiber =———,
time tine >
time
. . Pulse
Negatively chirped
® pulse me broadening
DE
PT
Dispersive Fiber B:<0
> —>
time time

Figure 5.14: Optical pulse propagation in optical fibers.

114

2.5 I T ( ; 1.5 I t T
Ad | (b)
ist a a 7
aL :
0.5 — -
0.5L =
i 0 | | ]
13183 13185 13187 13185 13187 13188
~% (um) A (um)

Figure 5.15: Time-averaged power spectra from directly modulated DFB lasers;

(a) Negatively chirped spectrum (b) Positively chirped spectrum.

We show in this section the control of frequency chirp of a two-section DFB laser by
nonuniform current injection and the achievement of blue-shift optical signals from the
two-section DFB laser under direct current modulation. The 1.3 um InGaAsP/InP DFB

. laser has two sections of 400um and 100pm long. Through a measurement of the time-
averaged power spectrum of the laser at 5 GHz modulation with fixed optical modulation
index, the changing frequency chirp is demonstrated for various distributions of
longitudinal photon density. The detailed experimental procedure has been described in

Section 5.6.

115

2.667 +

1.333

13183 13185 13187

1.5

r (A)

0.5

L=14mA

13185 13187 13188
dX (A)

1 =4mA (b)

13183 13185 13187
a (A)
| |
=10 (d)
3h —

13185 13187 13188
12 d (A)

1=20mA

0.8

0.4

l l
13185 13187 13188

r (A)

116

Figure 5.16: The measured time-averaged optical power spectrum of a two-
. section DFB laser at 5 GHz modulation for various injection current distributions.
The optical modulation index M=100%, and average optical power is kept at the

same level.

The spectrum of the laser is measured for various Ip, the injection current into
section two. The results are shown in Fig. 5.16 (a)-(f). For low injection current Ip (Fig.
5.16(a),(b)), the spectrum is similar to those of single section DFB lasers and is
characterized by a predominantly negative chirp. As the Ip increases to J2=8mA, the
spectrum becomes nearly symmetrical about the lasing frequency (Fig. 5.16(c)), and for Ip
larger than 10mA, the laser is mostly positively chirped (Fig. 5.16(d)-(f).

The field spectrum of the current modulated laser can be calculated and it is a
combination of AM and FM modulations. The time-averaged power spectrum is

proportional to the spectral density of the laser field E(t), which is given by [29-30]

Si(@) = 4 | Cx) e-i@t dt (5.50)

Loe)

and autocorrelation function is written as

CA) = lim +

dt (5.51)
T—

| TP _n* (4-0) Eptc.c. >
-T/2 4

The optical field E(t) can be represented as

- 117

E(t) = Eo{1+m Sin{(@mt)+ 01} }!2*exp[i{ at-B Cos( @nt+ 92) }]. (5.52)

Here m is the intensity modulation index, B is the phase modulation index, and @ is the

lasing frequency without modulation. We write the AM and FM or frequency chirp as

p(t)= m Sin( Wnt 0)), (5.53)

OA t)- O; = BO, Sin( Oyt+ 02). (5.54)

Combining (5.50), (5.51) with (5.52), the Sz(@) can be calculated for several values
of 02-6; and is shown in Fig. 5.17. The Sg(@) has a negative chirp for 02—0) = 7, anda
positive chirp for @2—0;~ 0. This is consistent with the FM spectrum of small signal
modulation derived previously [31]. In the case of small signal modulation (m « 1), the
Sx(@) is symmetrical when 02-0; =2/2. However, we found that the symmetry of the
Sp(@) occurs when @2~6; is somewhere between 7/2 and 7, depending on the intensity
modulation index m and phase modulation index P.

The instantaneous frequency deviation 5@(t) is calculated from (5.41) and AM
modulation can be got from (5.32). The phase difference between FM and AM, 62-6;
can be obtained. For small J 2, the injected current in section two, @2—0;, is approximately
equal to 1, which gives the red-shifted spectrum (as shown in Fig. 5.17b), and is
consistent with the observed spectrum of small J, (Fig. 5.16(a),(b)). The value of 02-0;
sharply decreases to zero for Jz >10 mA, which results in the blue-shifted spectrum at

large Iz (Fig. 5.16(d)-(f)). The calculated 02-6; is shown in Fig. 5.19.

“118

The frequency deviation 5a@(t) involves, according to (5.41), a spatial integration of
the imaginary’ part of Cy(z) and N (2), the AC component of local carrier density. The z-
dependent N;(z) and Im{Cy(z)} in DFB lasers determine the phase of 5a(t). In the two-
section DFB laser, by adjusting the injection current distribution, distributions No(z),
So(z), the function Cy(z), and N;(z) can be controlled. This results in a control of the
phase difference 92-0; and thus the characteristics of the frequency chirp. Qualitatively,
when the photon density in section one is much smaller than that in section two (Fig.
5.18b), N (2) is in phase with the modulation current in the longer section (section one)
where the injected modulation current is the dominant contribution. This, according to
(5.41), results in a positive chirp (Fig. 5.18b). Our calculation also shows that in most of
single section DFB lasers, the phase difference @2—0; is close to 2, which according to
the above analysis, results in a red-shifted chirp consistent with the observed spectra of

single section DFB lasers[2-3].

119

a) 9-0-0

rrr
—_

0.8
0.6

0.4

0.2

Optical Power (Normalized)

-20-15-10 -5 0 5 10 15 20
Af (GHz)
1 PEeCPPPesrpereepeern pare rgrrregpeerryears
0.8 0-8, =" (b)

Optical Power (Normalized)

030-15 -10 -5 0 5 10 15 20
Af (GHz)

Figure 5.17: The calculated spectral density of the optical field of a single mode
laser under 5 GHz modulation. The intensity modulation index and phase

modulation index are set atm =1, B=3.

120

1.5
x z
nd
= 0.0 &
ce 8
= =
= No
Penney
“1.5
0 100 200 300 400 500
zZ (um)
10
_ Prem
= eS
z ~
Jeon
= 0.0 >.
“= 8
Ss) 5
ee 1
s —
Patan

-10

0 100 200 300 400 500
z (um)

Figure 5.18: Calculated photon density and the product of Nj(z) and Im{Cy(z)} in
the two-section DFB laser for injection current in section two as (a) I2=4 mA, (b)

lh=16mA .

121 —

we
ETEQPPaUperrryperreyrrerpare

Or pa bs TorREQR GSMS RE picts wiles

i—]
= PPUOpuerere

5 10 15 20 25
Ir (mA)

Figure 5.19: Calculated phase difference 02-0; of the directly modulated two-
section DFB laser used in the experiment for various />, the injection current into

the section two.

122.

5.8 Conclusion

In this chapter, we have demonstrated chirp reduction and chirp control in DFB lasers by
controlling the longitudinal distribution of the photon density. This changes the dynamics of
spatial hole burning, and consequently, laser chirp. The expression for chirp in DFB lasers
has been derived based on the rate equations with considering the LSHB. The nonuniform
photon and carrier density distributions have been obtained from the coupled-mode equations
by using the Green's function method. The relation between the phase and intensity of the
lasing mode involves not only the @ parameter, but also the distribution of electric traveling
fields in the DFB laser. It is found that, unlike the case of a single section laser, the measured
chirp does not depend directly on the @ parameter but rather on the traveling field ©
distributions in multisection DFB lasers. This is very different from the case of chirp in
uniform photon density lasers. By fabricating two-section DFB lasers and changing the
injection current into the two sections independently, we can adjust the photon density
distribution and change the chirp characteristics. The reduction of the chirp by a factor of ten
_ has been realized, and we also achieved blue-shift optical signals from the two-section DFB
laser under direct current modulation. The unique advantage for using two-section DFB lasers

is that frequency chirp: can be reduced even when spatial hole burning is not suppressed.

123

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126 —

: Chapter 6

Spectral Dynamics and High Speed
Performance of Uncooled

Distributed Feedback Lasers
6.1 Introduction

In most of the present laser transmitters, thermoelectric coolers are required to keep the
laser operating temperature constant. The thermoelectric cooler adds to the costs of the
transmitter, and its long-term reliability is also a concern. Uncooled laser transmitters are
~ cheaper and more reliable than thermoelectric cooled laser transmitters because of their
simplicities in laser packaging. It is desirable to have uncooled semiconductor lasers that
can perform well in an extreme temperature environment. Extensive research efforts [1-10]
have been made to improve the temperature characteristics of lasers, and there are a lot of
progresses in achieving high temperature operation and small deteriorating external

differential quantum efficiency over extensive temperature range. It is believed that the poor

127

temperature characteristics of conventional lasers are partly due to Auger recombination
[11] in the low bandgap material and partly due to poor electron confinement resulting from
| the small conduction band offset (AE, = 0.4 AE,) of the InGaAsP/InP lasers [1]. It had
been ‘proposed that the laser threshold, Auger recombination, and intervalence band
absorption can be reduced by using the combination of biaxial strain and quantum
confinement to reduce the in-plane hole effective mass [12-13]. The Al,Ga,In,_,_,As/InP
material system, for its large conduction band offset (AE, = 0.72 AE,) [14-15], has been
chosen to provide a strong electron confinement. The design of high temperature lasers in
terms of the quantum well number and mirror loss is quite different from that of low
threshold current lasers [1,16]. To avoid gain saturation at high temperature operation, the
optimum operating point should be designed by using the gain curve of the highest
operating temperature at the expense of a slightly high threshold current at room
temperature.

For both analog CATV applications and long haul, high speed digital
communications, uncooled SLM DFB lasers are very attractive because of their low costs.
However, most research efforts now have been devoted to uncooled Fabry-Perot (FP)
lasers. In this chapter, we intend to investigate the high temperature performance of single
mode distributed feedback (DFB) lasers. The methods are basically the same on minimizing
the change of threshold current and slope efficiency both in uncooled FP lasers and in
uncooled DFB lasers. Additionally for uncooled SLM DFB lasers, it is essential to maintain

single mode operation and good SMSR at high temperatures.

128

: in this chapter, we first review the gain profile of semiconductor materials at high
temperature and DC characteristics (such as threshold current and external differential
quantum efficiency) of uncooled DFB lasers. Then we are going to give our studies on the
spectral dynamics and high speed response of uncooled DFB lasers with wavelength
detune (e.g., the lasing wavelength is different from that at the gain peak). In wavelength-
detune uncooled DFB lasers, the gain is not as a linear function of the injected carrier
density as that in FP lasers, which lase at the gain peak. The gain, as a function of injected
carrier density, has to be numerically calculated for certain wavelength detune. DFB lasers
with negative wavelength detunes, which lase at the short wavelength away from the gain
peak, have relatively larger differential gain. The lasers with negative wavelength detunes
have smaller linewidth enhancement factors and better high speed responses than those
lasers without wavelength detune. But the negative wavelength-detune DFB lasers have
slightly higher threshold currents. Also, we will show that the high speed response of
the DFB lasers with large negative detune deteriorates faster over high temperature. To
achieve best performance of uncooled DFB lasers, it is necessary to design wavelength

detune carefully for specific applications.

129 —

6.2 Gain spectrum

6.2.1 Gain expression
The temperature dependence of the band gap of 1.3:m InGaAsP lattice-matched to InP has

been obtained from measurements of the wavelength of InGaAsP/InP lasers. E,(T) is an

almost linear function of temperature and is given by the experimental relation [17]

E,(T) = E92 -3.610-(T'- 300) (eV ) (6.1)

Here ER is the bandgap energy at 300°K.

300 an ELS aa

Nc=5 e18 7

250] — — -Ne=4el8 4

neeeeeeee Ne=3 e18 :

200 4

= 150 2

= 7

~ 100 s

£ \

eS 50 —

e / eons :

0 / q

-50 ] : 3

-100 I. I. i | { i 1. 4 Ley 1 1 I. | Lnnovcll Lull. :
1.1 1.15 1.2 1.25 1.3

A (um).

Figure 6.1: Optical gain of 1.3 InGaAsP material as function of

wavelength for various carrier densities.

130

The modal gain is given by [18]

g(E.N)=T AE | A(E\p{(E) fetfi-l) —Alt__dE
: Aon 2 (At) +(E-E)” (6.2)

E is the photon energy, T is the collisional dephasing time, and p is the density of states

function. A(E) includes the transition dipole moment and the polarization modification
factor for the dipole moment in the QW structure. f, and f, are quasi-Fermi distribution

functions for electrons and holes, which are given as

_ 1
"EE
kT (6.3a)
In= E o
land Manel Ol
exe kT J (6.3b)

where Ef, EF are the quasi-Fermi energy levels for electrons and holes, respectively.
Typical gain curves for three different injected carrier densities are shown in Fig. 6.1.
From numerical calculation of the optical gain (6.2), the peak gain is found to vary almost
linearly with the injected carrier density, and can be approximately expressed by
greak(N,T) = A(T) [N- Ni(T)] (6.4)
where A is the differential gain, and N,, is the transparency carrier density. A and N,, are
temperature dependent. The gain expression (6.4) has been used extensively in FP lasers.
However, if the lasing wavelength of a DFB laser is detuned from that of gain peak, we
have to use (6.2) to calculate the gain at the detuned wavelength, and the gain can not be

described by (6.4) [19].

131

6.2.2. Collisional dephasing time

It can be seen ‘that the collisional dephasing time t is important to determine the gain
profile. Theoretical calculations show that major mechanisms contributing to the collisional
dephasing rate are: carrier-carrier scattering [20], carrier phonon scattering [21], and alloy
scattering [22] > and their temperature dependencies have been well known. It is difficult to
predict theoretically the temperature dependence of T since the three major mechanisms
have comparable magnitude in the temperature range of our interest. The temperature
dependence of tT has been measured through measuring photoluminescence spectrum and
comparing it with the theoretically calculated spectrum. The experimental temperature

dependence of 7 has the following formula [23]:

- T -To i
xT) wep To

(6.5)
where 1, §, Ty and T; have to be determined through experiment.

Quasi-Fermi level EX, EP do not change much as temperature changes from 20°C to 120°C,
so carriers spread over a larger energy range in conduction and valence bands as we can
see from quasi-Fermi distributions (6.3). The gain curves over different temperatures have
been calculated in Fig. 6.2. At the same injected carrier density, gain becomes smaller at
high temperature. Decrease of 1(T) at high temperature makes g(A) broader and reduces
its peak height (Fig. 6.2). That gain peak moves to long wavelength at high temperature is

due to the reduction of bandgap energy according to ( 6.1).

132 —

1 1.1 1.2 1.3
A (um)

Figure 6.2: The calculated optical gain of 1.3 InGaAsP material as

function of wavelength at various temperatures for carrier density

N,=6x1 018cm-3.

6.2.3 Gain spectrum in detuned DFB lasers

In FP lasers, the oscillation wavelength is around the peak of the gain curve. Accordingly,
the peak gain (6.4) is considered to describe laser characteristics, and the differential gain A
is one of the important factors determining high speed performance of the lasers. In DFB
lasers, the lasing wavelength is mainly determined by the grating pitch. Normally, the

Bragg wavelength of a DFB laser sets at a shorter wavelength than that of the gain peak

[19]. The laser is also referred to as negatively detuned. In a detuned DFB laser, the gain
as a function of injection carrier density has to be numerically calculated. The gain as a
function of injected carrier density is shown in Fig. 6.3 for various detuned wavelengths.
In comparison, the peak gain is also plotted. Because the gain distribution (Fig. 6.1)
moves to the shorter wavelength as the injection carrier density increases, the differential
gain is larger at the wavelength which is shorter than that at the gain peak (Fig. 6.1 and

Fig. 6.3). Therefore, the negatively detuned DFB lasers are expected to result in a higher

133 ©

resonance frequency through the increase in differential gain.

500

400

300

200

100

Gain (cm)

-100

-200

— - - 1.330 pm
Gain Peak

Figure 6.3: Calculated gain of 1.3m InGaAsP material as function of

NIbLiprerdie rer lpi lay

3 4 5 6
N, (10 8 om® )

injection carrier density for various wavelength detunes.

134

| 6.3 DC characteristics of uncooled DFB lasers

High temperature performance of uncooled lasers suffers from Auger recombination in the
low bandage materials and poor electron confinement resulting in the small conduction

band offset ( AE,=0.4AE, for Ga,In;_,AsyP}_,/InP materials). The increase of intervalence

absorption reduces the external differential quantum efficiency.

6.3.1 Threshold current density

The threshold current density in typical long wavelength lasers is taken as [11]

Jin = Any+BnZ, +Cn3, (6.6)
where the first term is the monomolecular nonradiative recombination, and the second term
is the bimolecular radiative current density. The third term is the Auger recombination in
which an electron and a hole recombine across the bandgap and, instead of emitting a
photon, they excite a third carrier into the valence or conduction band. The recombination

‘coefficients themselves, A, B and C, also have a temperature dependence.
We assume that Auger is an activated process and the Auger recombination coefficient

is given by [24]

C = Co exp(-E,/kT) (6.7)

where E,, is the Auger activation energy, which is determined by the band structure.

(135 —

The radiative recombination coefficient, B, is assumed to vary with temperature as for an
ideal QW with [25]

peel
7. (6.8)

The threshold carrier density, n,,

dependence of the threshold carrier density is taken as [26]

nap cc TY? (6.9)
due to nonideal factors such as carriers occupying higher subbands, carrier spillover into
the barrier material, or intervalence band absorption, which all may increase the temperature
sensitivity of the threshold beyond linearity.

It has been shown that the monomolecular nonradiative processes depend weakly on
the temperature, and therefore it is ignored.

The temperature dependence of a semiconductor laser is usually described by the Tp
parameter, which is given by the experimental relation

Ty = din)
dT (6.10)

136 |

Peryprrrprrrprrepree

Tarn a ee

0 20 40 60 80 100
I (mA)

Figure 6.4: Measured L-I of 1.3m InGaAsP/InP QW DFB lasers at

various temperatures.

6.3.2 Carrier leakage in quantum well lasers
Taking into account Auger recombination in the active region, free-carrier absorption in the
active region and waveguide, and current leakage to the cladding barrier, total current

_through a laser includes the following components [27]:

J; total active +), werd, Augert J, leakage=J majority+ J} leakage (6.11)
where Jyctiye and J,,, represent radiative recombination current in the active region and
waveguide, respectively. J4,¢r is nonradiative Auger recombination current in the active
region and Jjegkage is a current of minority carrier overflowing from the waveguide to the

cladding layer. Jiggkage is essentially a current of electrons leaking to the p-cladding layer

137

and eventually recombining on the boundary with the highly doped p-contact layer. A
higher effective mass of holes results in a quasi-Fermi level position close to the valence
band edge of the active layer, and hole leakage current may be effectively neglected. The
first three terms i in (6.11) are calculated according to standard theory to give Jngjority. The
analytical approximation for leakage current can be obtained by analytically solving the

diffusion-drift equation while assuming constant electric field [27-28]

Jteaka = qDin W/L kT y’ cothy | 1+ (kL 4kT
ow . 2qEo 2qEo “oqEo (6.12)

where

Eo = ! V; majority+—PJ) leakage)
GMpPo Dp (6.13)

6.3.3 Internal loss in quantum well lasers

Applying the linear relationship between the internal loss of the QW and temperature, we

have [29]
off = 0+ (T-T1) (6.14)
where
yedae /dT (6.15)

The total internal loss is

o=N 2M O24 SCH (6.16)

138

6.3.4 External differential quantum efficiency

The power emitted by stimulated emission is expressed by [30]

(ln) i
Pe hv (6.17)

The external differential quantum efficiency is defined as the ratio of the photon output rate

that results from an increase in the injection rate to the increase in the injection rate

_ d(Po/hv)

In DFB lasers, the external differential quantum efficiency can be derived from the
measured or calculated L-I characteristics. A simplified expression for "ex can be obtained
by writing the threshold gain for DFB lasers as

Sth= %m + O (6.19)
with @; as the internal loss and a, as the so-called cavity loss accounting for the power
leaving the DFB laser cavity. a@,, may be obtained from numerical calculation of the
threshold gain in DFB lasers [31-34]. For a typical DFB laser with kL=2, we have

On Ll 95 (6.20)

_In order to compare this cavity loss with the mirror of an FP laser, @,, is found by

recalling the expression for g,, in the FP laser

0, = 1/2L In(1/R)R>) . (6.21)
In this sense, the cavity loss a,, L may be related to an effective reflectivity R=0.14 (for
R=R)=R,).

With considering leakage current, we write the output power of a DFB laser as

139

R-

P _@ Tl leakage) nNihv Amn
oe Olin + 06; (6.22)

and the external differential quantum efficiency is

dl leakage ) Am
dl Amt Oj (6.23)

Nex=(Ni
With (6.16), the temperature dependence of 7, can be obtained as

dT eaka 7)] Om(T)
Nex(T)=(Ni - a)
dl Om(T)+N 2 [on+T-T) +05 6.24)

where 7; is the internal quantum efficiency, and it shows little temperature dependence in

the temperature range of our interest.

6.4 Gain spectrum of uncooled DFB laser

Several 1.3m InGaAsP/InP strained quantum well (QW) DFB lasers have been
fabricated. The lasers are AR (<5%) coated at front facets and HR (>70%) coated at back
- facets. AR/HR coating is essential for the DFB lasers to maintain SLM operation at high
temperatures. This is specially important for wavelength-detune DFB lasers. The
wavelength-detune DFB lasers lase away from the gain peak, and at high temperatures the
FP modes are normally much stronger than the main mode of the DFB lasers if the front
facet is not appropriately AR coated. AR/HR coating will suppress the FP modes to keep

the SLM operation of DFB lasers. The spectrum of a 1.3m InGaAsP/InP QW DFB laser

140

has been measured at various temperatures (Fig. 6.5-Fig. 6.8). The laser is negatively
detuned and the wavelength detune is AAp=Mase- Again peak ~ [Onm at room temperature.
| It can be seen from Fig. 6.5-Fig. 6.8 that the laser keeps its SLM to 80°C. The lasing
wavelength of the laser is 1290 nm at room temperature and only increases to 1295 nm at

80°C.

6.4.1 The measurement of gain spectrum

The gain spectra can be calculated by measuring the FP mode spectra below threshold bias
current. We show here the spectrum (Fig. 6.9 and Fig. 6.10) of the DFB laser for I=9.5
mA (below threshold) at room temperature, and the threshold current is about 10.6 mA.
The modal gain of lasers can be calculated from modulation depth of the measured spectra,
which is given by Hakki and Paoli as [35]

g(E) = 1 In 1 VV-1 | +Qoss
L \WR,R, VV+1

(6.25)

V = Emax
Pin (6.26)

where Rj and R, are the respective reflectivities of the front and rear facets. The P,,,,, and
Pnin donate the FP peaks and valleys. When the gain spectra are estimated from the
measured spectra by using (6.6), the results are | shown in Fig. 6.11-Fig. 6.14. The
calculated gain from spectra is not accurate at the wavelength around the Bragg condition

due to the influence of the corrugation grating in the lasers.

141
~ From Fig. 6.11-Fig. 6.14, we get the wavelength detune of the DFB laser at various
temperatures, which is shown in Fig. 6.15. It can be seen that the detune increases greatly
with temperature. The increase of the wavelength detune is largely due to the shift of the

gain peak. The gain peak moves to long wavelength as we increase the substrate

temperature of the laser (Fig. 6.11-Fig. 6.14).

6.4.2 The influence of the corrugation grating on the measured
gain spectra

In measuring the DFB laser gain spectra below threshold, the corrugation grating, formed
within the laser cavity, has effects on the FP mode spectra. The gain spectra we calculated
from (6.25) are only valid for the FP mode spectra. The gain spectra (Fig. 6.11-Fig. 6.14)
are a combination of the FP spectra and additional loss generated by the grating. Using F-
matrix [36] method, the loss due to the grating can be numerically calculated. Fig. 6.16
shows the gain deviation of the estimated normalized gain as a function of normalized

deviation. The normalized gain deviation is defined as [37]

AG = Ges -Grp
Grp (6.27)

where Gprp and Grp denote the modal gains for DFB and FP lasers and are given by

GprB =f; material -Qoss- Ugrating (6.28)

Grp = F'gmaterial -Oloss- mirror (6.29)

The normalized deviation is given as

142 —

AB =B- Borage (6.30)
where B is the propagation constant at reference wavelength A and Bg,ag, is the
propagation constant at the Bragg wavelength.

The gain deviation is large when the normalized deviation is close to zero, e.g., the
reference wavelength is close to the lasing wavelength. This indicates that the estimated
gain for the DFB laser is quite different from that for the FP laser near the Bragg
wavelength and we can find that there are sharp changes around the lasing wavelength from
the estimated gain spectra (Fig. 6.11-Fig. 6.14). Away from the lasing point (IABLI>20),

the estimated gain is only slightly different from its accurate value.

143

28:16:45 SEP 8, 1995

-20.08 dBo NKR #1,NUL_ 8.9 on
~HY ldBo -37.93 dD |
18,08 dB/G1U . ‘
MARKER. a
9 bm
-37.03 dB
iis una y
START 1264.4 no STOP 1360.8 na
*RB @.4 nm YB 288 Hz ST=6.6 sec

Figure 6.5: The spectrum of a 1.3m InGaAsP/InP QW DEB laser at 20°C.

19:18:47 SEP #8, 1995

L -13.94 dBa HKR #1,NVL -12.4 om
eSENS| ~-8@ [dBm 24.43
18.88 dB/qry
MARKER 4
-ic. om
29.498 dB
naan inl .
“START 1260.0 ne STOP 1360.0 na
*RB .1 nm VB 288 Hz , $T26.5 sec

Figure 6.6: The spectrum of the DFB laser at 40°C.

144

(@)] 19:39:84 SEP 8, 1995

RL -78.08 dBm HKR #1aHUL 18.8 nm
RSENS] -09 [dba ~28. 85 dB
2.08} dB/D1V ‘
RES BH WAVELENGITH

&.1 on

STAR 1268.8 nm STOP 1368.4 nm
*RB @.4 nm VB 1@8 Hz =§$ §T=18 sec

Figure 6.7: The spectrum of the DFB laser at 60°C.

19:47:58 SEP @, 1995

~70,.82 dBa NKR #1,HVL_ 22.6 nm

S$] -85 [dbs “17.41 cB
2.08) dB/DIV P
MARKER «
ec.5| no
-17.71 dB

“START (260.0 ne STOP 1360.0 ne
*RB G.1 no UB 188 Hz «6S ST=18 sec

Figure 6.8: The spectrum of the DFB laser at 80°C.

145

(@) 23:15:49 SEP 6, 1995

RL -73.58 dBn MKR #1 WUL 1318.34 nn
S| -S8|qBe 72.498 dBm
1.08) dB/D]V 4

RES BH WAYELENGITH
.1 fe

START 1040, 00 ne TOP ah na
*RB @.1 nn VB 18 Hz = S $Ta27 sec

Figure 6.9: The spectrum of the DFB laser below threshold for A=1290 nm-1315 nm.

23:26:42 SEP 6, 1995
RL_-76.0@ dBe MKR #1 WVL 1315.75 no
eit: ~94 |dBa 76.459 dBo

1. BG] dB/DqV

ES BW WAVELENGTH

—-,-

VT
START 1415.48 no TOP 1390.48 no
RB @.1 ne VB 18 Hz § ST=28 sec

Figure 6.10: The spectrum of the DFB laser below threshold for A=1315 nm-1340 nm.

Gain (a.u.)

1290

Figure 6.11: The estimated modal gain of the DFB laser at 20°C.

Gain (a.u.)

-100

1290

Figure 6.12: The estimated modal gain of the DFB laser at 40°C.

146

Trrypreryprreprrrperryere
eon

20°C

1300 1310 1320 1330 1340
A (nm)

orery

rorgeie Lr a ae |
{ L
ar

40 °C

riptliyrreretrprper taupe stars i

w rirtirrtirrtisrtls the aptpre dui

1300 1310 1320

X (nm)

1330 1340

147

100 TLSLELIE GLALSESL EE GAELS DLL

68 °C ee

Gain (a.u.)

oO
TVTPRUPpe repr ep rreyprrrpreayprre

-60
1290 1300 1310 1320 1330 1
dX (nm)

Figure 6.13: The estimated modal gain of the DFB laser at 60°C.

100 Tr Tryp rr repr rrr pr, prrrye
80
60
40

BB°C ..

Gain (a.u.)
oO

Oo
i ee

-60 repr typyprrterta sas Lap ps Lt

1300 1310 1320 1330 1340 1
dX (nm)

OD Liprbsrir darters dea ta

Figure 6.14: The estimated modal gain of the DFB laser at 80°C.

148

Wavelength Detune

-r’ _ (nm)
lasing
= nm e)
o1 [o>] o1

gain peak

ssp tnpos dees i gs has iggy
0 20 40 60 80 100 120
T ( C)

Figure 6.15: Measured wavelength detune of the 1.3m InGaAsP/InP

QW DEFB laser at different temperatures.

GAIN DEVIATION [%]

149

50 T T T T T T T T
f J
0 _
-—50 1 l ! i i i 1 i t
~50 50

NORMALIZED DEVIATION ABL

Figure 6.16: Calculated normalized gain deviation versus normalized

deviation.

150

»-

6.5 Linewidth enhancement factor, linewidth of

uncooled DFB laser

6.5.1 Measurement of a factor

We have a direct relation (5.9) between photon density and lasing frequency [38, 39, 40]

V-Vy, = d(inS) + es
4n \ dt Tph

(6.31)

If a small signal sinusoidal modulation of the photon number S is assumed according to
S = (S)+ Re(AS exp(i@nt) ), (6.32)

here |ASI « ~~and @,, is denoted to the modulation frequency. The frequency~~

modulation may be written as

v= (Vv) + Re(Av exp(i@mt) ) (6.33)

and (6.31) yields

AVIAS = —2(i@m+ Wz)
47S) ° (6.34)

with

It is often more convenient to relate the FM-modulation index
B=RTAV/On| (6.36)

to the AM-modulation index

151

m=|ASAS] (6.37)

(6.38)

In Fig. 6.17, the B/m generally decreases with increasing modulation frequency and
approaches to a/2 at high frequency as predicted by (6.38). The measured linewidth
enhancement factor is found to be a@ = 6.6 (20°C), 4 (40°C), 4 (60°C), and 5.6 (80°C),
respectively. The @ is smaller at high temperatures than that at room temperature. The
reason for smaller @ at high temperatures, as we can see in the next section, is due to the
increase of wavelength detune at high temperature (Fig. 6.15). By carefully examining the
Fig. 6.17, we can see that not all the points satisfy (6.38). That is because in DFB lasers,
the exact relation between FM and AM modulations should be derived from rate equations
(4.22) and (4.23) which include the LSHB. Although the B/m can be approximately
described by (6.38), for some modulation frequencies, the FM modulation is strongly

influenced by LSHB and does not satisfy (6.38).

152

rrree prt Pt rfp prerre

1 porertr pap Li pt

0 0.5 1 1.5

f (GHz)

Nii pebe ppt lir ay bare te

Figure 6.17: Measured B/m of 1.3um InGaAsP/InP QW DFB laser as

function of modulation frequency at different temperatures.

153

6.5.2 Calculation of a factor in uncooled DFB lasers

The effective linewidth enhancement factor in a semiconductor laser generally consists of
the interband transition component and free carrier component, which is due to the plasma
effect of injected carriers in active layer and the optical confining region.

The @ factor can be written as [41,42]

O=Qinterband+ Ofree carrier (6.39)

The Q,rerhand iS the interband transition component and can be defined as

Oxi" aye"
Olnterba = aN F) N

(6.40)
where 7/""(N)=y}"'+i y!"" is the optical susceptibility due to interband transition. The
Ojnterband Has to be calculated at lasing frequency of a DFB laser which is normally
detuned from the gain peak.

The real and imaginary parts of the optical susceptibility are given by [42,43]

X#EN)=-T>, AME) pf (BE) (fetfi-1) Wt_—dE (6.41)
i=in J (Alt)?+(E-E)

r |
XAEN=-TY, | AWE) pf (E) (fetfi-l) —-EE— dE 6.42)
iin J (Alt)°+(E-E)

Eis the photon energy, T is the collisional dephasing time, f, and f, are quasi-Fermi
distribution functions for electrons and holes, and p is the density of states function. A(E)
includes the transition dipole moment and the polarization modification factor for the dipole

moment in the quantum well structure.

154

»-

The interband Can be calculated as a function of temperature by using (6.34) and is
shown in Fig.'6.18. The calculated @;,,,¢rpanq decreases with temperature due to the
increase of wavelength detune (Fig. 6.15). The DFB laser with large negative wavelength
detune has larger differential gain than the one lasing at gain peak (Fig. 6.3), which may
contribute to the decline Of Oinrerband at high temperatures. The Q,¢¢ carrier iS due to free
carriers in the active layer and barrier layers. It increases with temperature and does not
change greatly with wavelength. Combining the contribution from interband transition and
that-from free carrier, the @ parameter reaches its smallest value around 60°C, beyond
which Qinterband Changes slowly and the @ parameter could increase again due to Ope,

carrier’ Lis may explain our measured results of @ parameter (Fig. 6.17).

155

Calculated Linewidth Enhancement Factor due to
Interband Transition in 1.3um DFB laser

6.5

oO

interband -

5.5

10

4.5 A ree ee ee ee

20 40 60 80 100 120

Temperature (°C)

Figure 6.18: Calculated linewidth enhancement factor of 1.3m DFB laser
as function of temperature. The wavelength detune used in the calculation

is from the experimental results (Fig. 6.15).

156

he

6.5.3 Linewidth of uncooled DFB lasers

69 Acoustooptic Detector

tL Isolator Modulator ' Th
DFB] ( | ( |
t 2 Km

Figure 6.19: The experimental setup for linewidth measurement.

HP Spectrum
Analyzer

A time delay self-heterodyne detection scheme is used in our measurement of laser
linewidth. The laser is isolated and an acoustooptic (A-O) modulator is used to modulate
the laser at 85 MHz. The experimental setup for linewidth measurement is shown in Fig.
6.19, and measured linewidth for the 1.3m 10 QW DFB laser is in Fig. 6.20. We can see
that the linewidth is generally proportional to P-/ and there is enhancement of linewidth at
large optical power for low temperatures. The possible causes for the enhancement have
been mentioned in Chapter 4. Due to decreasing external differential quantum efficiency at
high temperatures, we haven't injected sufficient current into the laser to observe the

enhancement of linewidth in order to avoid damaging the laser.

157

he

From Fig. 6.20, the smallest linewidth is obtained as 1.8 MHz at 20°C, 1.5 MHz at
40°C, 1 MHz at 60°C, and 1.2 MHz at 80 °C, respectively. From expression (4.50) for
linewidth in DFB lasers and the measured a parameter (Fig. 6.17), it is not difficult to
understand that, in a DFB laser with wavelength detuning, the linewidth could be smaller at

high temperatures.

Av (MHz)

158

: | ° DFB@40°C

1 ist. J

11°F c :

= 106 4

: 2 c :

4 5L J

: 0 leeds

0 05 1 15 2 25 0 05 1 #15 2
P! (mw?) P’ (mw’)
DFB@60°C DFB@80°C

20

15

10

Av (MHz)

rititirtertsiirtinpr tiers

Ober tirrs tera rteeer tiie

0 1 2 3 4

ol
Oo
-_
O> Crserius

4 1 2 3 4 5

Figure 6.20: Measured linewidth of 1.34m InGaAsP/InP QW DFB laser

vs. the inverse of optical power at 20°C, 40°C, 60°C, and 80°C.

159

6.6 High speed response of uncooled DFB lasers

In low optical power limit, we have simplified expression (4.38) for resonance frequency

in an uncooled DFB laser as [44]

@2(T) =H ve [dg(T)/dN] a(T) So (6.43)
Here the differential gain dg/dN and threshold gain g are both dependent on temperature.
In FP lasers, the high speed performance deteriorates at high temperature due to the
reduction of the differential gain. In DFB lasers, wavelength detune plays an important
role in the high speed response of the lasers. Due to increasing detune over high
temperature, the differential gain does not drop as fast as that at the gain peak. We calculate
the product of the differential gain and threshold gain for several detuned DFB lasers with
increasing temperature. The DFB lasers lase at 1.29um, 1.30pm, 1.31pm, and they are
detuned at AAp=Ajase-Again peak=-20nm, -10nm, and -3nm, respectively. The 1.29m
laser with the largest detune among the three lasers has the largest [dg/dN g] at room
temperature, and the 1.31,1m laser, which is the least detuned, has the smallest [dg/dN g].
When we increase temperature, the [dg/dN g] of 1.29 um laser has the fastest dropping
rate, while the [dg/dN g] of 1.31 [um laser only changes slightly. This means that for high
temperature application of uncooled DFB lasers, we have to optimize the wavelength

detune to achieve the best high speed performance over extensive temperature range.

160

3000

2800

2600

2400 PS

L a *« A,

(10 “8cm)

a re
2200% ~

cena
2000 rirtr,,,IT,,,g,;,;i,tyuy

300 320 340 360 380 400
T @ K)

g (dg/dN)

Figure 6.21: The calculated dg/dN °g as function of temperature in DFB

lasers with various wavelength detunes (AAp=Mase~Again peak=-20nm for

1.29um laser, -10nm for 1.3m laser, and -3nm for 1.31pm laser,

respectively).

Two DEB lasers (1.29.m and 1.30 um) have been fabricated to compare their high

speed performances at high temperature. The DFB lasers have wavelength detune of

AAp=-10 nm (for 1.29m laser), and -5 nm (for 1.30pm laser).

The measured high speed response of 1.3 um laser is shown in Fig. 6.22 for various

temperatures. The resonance frequency is approximately to be

fr(GHz) = 21P(m wr") for all temperatures. (6.44)

161

From (6.43), the resonance frequency can be expressed as

fp2API2 | (6.45)
where
A=C Vee dg/dN g (6.46)

Here C is a constant dependent on the facet reflectivity of the laser.
From the experimental results, the [dg/dN g] changes little with increase of temperature,
which agrees with our calculation of 1.31um DFB laser (Fig. 6.22), which has detune of
AAp=-3 nm.

The measurement of 1.29um DFB laser is shown in Fig. 6.23. The resonance

frequency has relation with optical power as

fr(GHz) =3.51P(mW"”) at 20°C. (6.47)

But A drops from 3.5 GHz/mW!7(20°C) to 2.4 GHz/mW! at 80°C. Because 1.291m
laser has larger wavelength detune of 10 nm, [dg/dN g] will drop as temperature increases
(Fig. 6.23). The reduction of A is predicted in our calculation of detuned DFB lasers (Fig.
6.21). .

It is desirable to achieve fast DFB lasers by using a large detune. From gain curves
for various detunes (Fig. 6.3), in order to get threshold gain, we have to inject more
current to reach threshold for larger negative wavelength detune. Also, as we have shown

in our measurement, the high speed response of the DFB lasers with large negative detune

162

deteriorates faster over high temperature. To achieve the best performance of uncooled

DFB lasers, it is necessary to design wavelength detune carefully for specific applications.

| 163

7F i Sh i
—™ =5E 4
Ne _ c
Ss] 3 :
CO 46 :
iz," 3 5
i er
0 1 2 3 4
P'”? (mw 1”)
3.5
3.0
2 2.5
2.0
cis
1.0
0.54
0.0
0 2 4 6 8 1012

P (mW)

Figure 6.22: (a) The measured resonance frequency as a function of
square root of optical power in 1.3m DFB laser (AAp= -5 nm) for
various temperatures; (b) The measured damping rate as function of

optical power in 1.3m DFB laser.

164

1) PEreerereacereeeets ts cere ee ee

0 1 2 3
pi”? (mw 1/2)

pel

Figure 6.23: (a) The measured resonance frequency as a function of
square root of optical power in 1.2914m DFB laser (AAp=-10 nm) for
various temperatures; (b) The measured damping rate as function of

optical power in 1.29um DFB laser.

165

6.7 Conclusion

In this chapter a detailed study of spectral dynamics and high speed response of uncooled
DFB lasers has been presented. The most distinguished element differentiating the
uncooled DFB lasers from uncooled FP lasers is that in uncooled DFB lasers, the
wavelength detune plays an important role in determining their spectral and high speed
characteristics at high temperatures. In uncooled DFB lasers, the linewidth actually
decreases with increasing temperature due to the increasing wavelength detune at high
temperature. The increase of the wavelength detune is mostly contributed by the shift of the
gain peak to the longer wavelength at high temperatures. To achieve the best high speed
performance of uncooled DFB lasers, the wavelength detune has to be chosen properly.
Choosing a large negative wavelength detune could enhance the high speed performance of
an uncooled DFB laser at room temperature. But the penalties for the DFB laser are a
slightly higher threshold current at room temperature and fast deteriorating high frequency
response at high temperatures. To achieve optimum performance of uncooled DFB lasers,
it is necessary to design wavelength detune carefully based on their specific applications
and requirements for threshold current and high speed performance. A computer model
for the uncooled DEB lasers has been used to calculate high speed responses of two lasers
with different wavelength detunes. By including LSHB and wavelength detune, the model
can give us a satisfactory explanation for the results from our high speed measurement of

the uncooled DFB lasers.

| 166

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