Analytical expressions, modeling, and
simulations of intensity and frequency
fluctuations in directly modulated
semiconductor lasers
Samy Ghoniemy, MEMBER SPIE
Carleton University
Systems and Computer Department
1125 Colonel By Drive
Ottawa, Ontario K1S 5B6
Canada
E-mail: ghoniemy@sce.carleton.ca
Leonard MacEachern
Carleton University
Department of Electronics
1125 Colonel By Drive
Ottawa, Ontario K1S 5B6
Canada
Samy Mahmoud
Carleton University
Systems and Computer Department
1125 Colonel By Drive
Ottawa, Ontario K1S 5B6
Canada
Abstract. Analytical expressions for the intensity and frequency/phase
noise of single-mode semiconductor lasers based on quantummechanical rate equations are derived. Correlated photons, electrons,
and phase Langevin noise sources and their auto and cross-correlation
relations are presented, along with a novel self-consistent normalized
laser model that includes the laser’s correlated noise sources. A symbolically defined device (SDD) is constructed using the proposed normalized
model and implemented in Agilent’s advanced design system (ADS)
CAD tool. Dynamic laser characteristics are predicted using the SDD
implementation for 1300-nm InGaAsP/InP lasers. The results of time domain dynamic simulations of photons, carriers, optical output power, and
phase—with and without the effects of noise—are presented. Simulation
results are used to show the effects of random noise on both the phase
and optical power output of semiconductor lasers. Simulation results are
analyzed to demonstrate the resonance frequency shift dependence on
the bias current levels, the relation between the frequency response and
the bias current, and the dependence of the laser line width broadening
on the frequency fluctuations. Comparison between the presented results and other published results (simulations and measurements) show
good agreement while achieving simulation time enhancement. The suitability of the proposed models for the study and characterization of the
performance of complete systems in both circuit and system simulations
is examined. © 2004 Society of Photo-Optical Instrumentation Engineers.
[DOI: 10.1117/1.1628692]
Subject terms: semiconductor laser modeling; quantum noise; intensity noise;
phase noise; laser simulations.
Paper 030101 received Feb. 25, 2003; revised manuscript received Jun. 13,
2003; accepted for publication Jun. 16, 2003.
1
Introduction
The high-frequency digital and analog modulation characteristics of semiconductor lasers have been the subject of
intense research over the past few years. The desire for
high-speed data transmission, including that of video
signals,1–3 and the need for broadband communication,4 –7
have been driving forces. Attaining a large modulation
bandwidth while still maintaining a low cost is often an
essential requirement in these applications.8 –11 Laser intensity noise12–14 and nonlinear distortion15–19 are the main
factors limiting the dynamic range in laser-based optical
communication links. Wavelength modulation 共frequency
noise or chirp兲 may result in dispersion penalties, especially
in long distance optical links.9 Sophisticated mathematical
models of laser noise have been introduced,20–26 but they
are not generally suitable for both device and systems
simulations as proposed in Refs. 27 and 28. Analytical laser
noise models based on small signal approximations to the
laser rate equations driven by Langevin noise sources appear in the literature,20,23,24,29,30 but the reported models do
not include dynamic 共instantaneous兲 fluctuation informa224
Opt. Eng. 43(1) 224–233 (January 2004)
tion, and they are not reliable for large signal fluctuation
analysis. Several sophisticated computer simulations based
on direct numerical integration of the laser rate equations,
including the effect of Langevin noise sources, have been
introduced,22,31,32 but the cross-correlation relations between carriers, photons, and phase have been largely neglected in these models. Circuit models12,13,33 have been
used for the prediction of relative intensity noise 共RIN兲, but
again, the cross-correlation relations were neglected, and
the resulting models are not generally suitable for system
simulations.
In this work, an extended analytical study for the autoand cross-correlation relations between the laser noise
sources is given. A modified model including these expressions is proposed. The proposed model is suitable for both
circuit and system simulations, and can predict laser characteristics of interest. In Sec. 2, the theory of quantum
noise generation is discussed. The derived semiconductor
laser rate equations driven by noise sources are discussed in
Sec. 3, and in Sec. 4, the derived analytical expressions for
auto- and cross-correlation relations between carriers, pho-
0091-3286/2004/$15.00
© 2004 Society of Photo-Optical Instrumentation Engineers
Ghoniemy, MacEachern, and Mahmoud: Analytical expressions . . .
tons, and phase are presented. The derived expressions provide a theoretical base for the expressions presented by
Ahmed, Yamada, and Saito34 and are a generalization of
Marcuse’s noise analysis.21 In Sec. 5 the proposed normalized laser model including the noise fluctuations and their
auto- and cross-correlation relations, is presented. A selfconsistent technique for numerically generating the correlated noise sources is proposed in Sec. 6. The SDD implementation of the enhanced laser model, including the effect
of the noise is described in Sec. 7. The proposed SDD
model is used in Sec. 8 to simulate a laser’s dynamic characteristics using realistic parameters for 1.3-␮m InGaAsP/
InP lasers.
2
Quantum Theory of Laser and Noise
Generation
Representing a semiconductor laser as a homogenously
broadened laser is perhaps the most accurate method for
modeling its behavior, but the complexity of such a model
makes it cumbersome.21 Simplified models in which the
laser is modeled using a set of energy levels are normally
used. The Hamiltonian provides a suitable framework for
the study of semiconductor lasers, because electrons, photons, and the interaction between them are all represented
by energy operators. The effect of the heat bath contributions can also be included, which is an important consideration since electron and photon heat baths are the origin of
the laser noise that causes output fluctuations.21,35
Based on Heisenberg’s equation of motion and quantum
mechanics theory, each laser variable can be represented by
a corresponding quantum mechanical variable.
The analysis in this work starts with the results of the
analysis given by Sargent, Sgully, and Lamb,35 in which
they derived a system of coupled differential equations representing the time evolution of the quantum mechanical
operators in a three-level laser system, as given in Eqs. 共1兲,
共2兲, and 共3兲.
dl 1
⫽I 1 ⫺ ␥ 1 l 1 ⫹ıg 兵 ⌳ c l 21 exp关 i 共 ⍀⫺ ␻ 兲 t 兴 ⫺hc其 ⫹F 1 ,
dt
共1兲
dl 21
⫽⫺ ␥ l 21⫹ıg 关共 l 1 ⫺l 2 兲 兵 ⌳ a exp关 i 共 ⍀⫺ ␻ 兲 t 兴 其 兴 ⫹F l , 共2兲
dt
a
1
d⌳
⫽⫺
⌳ a ⫺ıg 共 N 兵 l 21 exp关 i 共 ⍀⫺ ␻ 兲 t 兴 其 兲 ⫹F ph ,
dt
2␶p
共3兲
where l 1 and l 2 are the occupancy operators of level 1 and
2, l 21 is the electron polarization operator, I 1 is the pump
operator, ⌳ a and ⌳ c are the photon annihilation and creation operators, ␥ 1 and ␥ are the decay rates of level 1 and
the polarization phase, ␶ p is the photon lifetime, g is the
gain constant, ␻ is the angular transition frequency between
levels 1 and 2, ⍀ is the angular photon frequency, F 1 , F l ,
and F ph are the Langevin noise sources for level 1 electron
polarization and photons, l and l * are the operators of the
electron dipole polarization and its conjugate, and N is the
total number of electrons in the laser cavity.
2.1 Langevin Equation and Einstein Relations
The strength of the Langevin sources 共variance兲 depends on
the laser parameters. The general form of the Langevin
equation is,35
d␰共 t 兲
⫽A 共 t 兲 ⫹F 共 t 兲 ,
dt
共4兲
where ␰ (t) is any operator, A(t) is the function representing the system, and F(t) is a random Langevin function
whose autocorrelation relation is given by the Markoffian
approximation 具 F ␮ (t)F ␯ (t ⬘ ) 典 ⫽2A ␮ ␯ ␦ (t⫺t ⬘ ), where A ␮ ␯
is the magnitude of the fluctuations peaked at about t⫽t ⬘
that corresponds to the maximum correlation between ␰ (t)
and ␰ (t ⬘ ), while 具 典 indicates an ensemble average over the
heat bath.21 Taking the average of Langevin noise in Eq. 共4兲
and allowing that the average of the product ␰ (t)F(t ⬘ ) vanishes for t⬍t ⬘ , the generalized Einstein relation is given
by,
2A ␮ ␯ ⫽⫺ 具 A ␮ ␰ ␯ 典 ⫺ 具 ␰ ␮ A ␯ 典 ⫹
d
具␰ ␰ 典.
dt ␮ ␯
共5兲
Equation 共5兲 is used to determine the variance of the
Langevin function F(t) in Eq. 共4兲. Hence, Eq. 共5兲 will be
the basis for evaluating the variance of the laser noise
sources.
2.2 Autocorrelation of the Atomic Noise Operators
Based on the analysis by Sargent, Sgully, and Lamb,35 the
correlation of the atomic operators F 1 (t), F l (t), and
F ph (t) appearing in Eqs. 共1兲, 共2兲, and 共3兲 can be written as,
具 F 1共 t 兲 F 1共 t ⬘ 兲 典 ⫽
1
共 具 I 典 ⫹ ␥ 1 具 l 1 典 兲 ␦ 共 t⫺t ⬘ 兲 ,
N 1
共6兲
具 F l *共 t 兲 F l t ⬘ 兲 典 ⫽
1
共 具 I 典 ⫹ 共 2 ␥ ⫺ ␥ 1 兲 具 l 1 典 兲 ␦ 共 t⫺t ⬘ 兲 ,
N 1
共7兲
具 F ph * 共 t 兲 F ph 共 t ⬘ 兲 典 ⫽
p̄ th
␦ 共 t⫺t ⬘ 兲 ,
␶ ph
共8兲
具 F ph 共 t 兲 F ph * 共 t ⬘ 兲 典 ⫽
1
共 p̄ ⫹1 兲 ␦ 共 t⫺t ⬘ 兲 .
␶ ph th
共9兲
For semiconductor lasers at room temperature, the average
thermal photon number is zero, hence p̄ th can be substituted by zero in Eqs. 共8兲 and 共9兲.
3 Noise Driven Laser Rate Equations
Coupling the atomic system to the field system by adding
the electric-dipole perturbation energy to the Hamiltonian35
using the density matrix method36 yields the laser rate
equations. As discussed in Refs. 21 and 35, the atomic relaxation frequency ⌳ a (t) can be treated as a constant in the
atomic equation of motion, allowing the time rate of change
of l 1 ,l 2 and l 12 to be neglected. Further simplifications are
made by neglecting the frequency components in the noise
Optical Engineering, Vol. 43 No. 1, January 2004
225
Ghoniemy, MacEachern, and Mahmoud: Analytical expressions . . .
operators that are greater than the relaxation frequency,
while still considering enough bandwidth to properly include the noise of the atomic operators with respect to the
slowly varying field ⌳ a (t). Based on the previous above
approximations, and using the impulse response simplification from Eq. 共10兲, the solution of Eq. 共2兲 is as shown in
Eq. 共11兲.
␦ 共 t⫺t ⬘ 兲 ⫽ 21 关 ␥ ⫾ı 共 ␻ ⫺⍀ 兲兴 exp关 ␥ ⫾ı 共 ␻ ⫺⍀ 兲 兩 t⫺t ⬘ 兩 兴 , 共10兲
d ␾ ␤ 2g 2
⫽
共 C⫺C̄兲 L⫹F ␾ 共 t 兲 ,
dt
2 ␥
where F C(t),F P(t), and F ␾ (t) are the electrons, photons,
and phase Langevin noise operators, respectively, as given
by Eqs. 共18兲, 共19兲, and 共20兲.
2g 2
F C共 t 兲 ⫽F 1 共 t 兲 ⫹
LC⫺igN exp关 ı 共 ␻ ⫺⍀ 兲 t 兴
␥
l 21⫽⌼ 兵 ıg 共 l 1 ⫺l 2 兲 ⌳ a exp关 ı 共 ␻ ⫺⍀ 兲 t 兴 其
⫹
冕
t
⫺⬁
⫻
exp关 ␥ 共 ␶ ⫺t 兲兴 F l 共 ␶ 兲 d ␶ .
共11兲
Substitution of Eq. 共11兲 into Eq. 共1兲 followed by multiplication by N gives the electron rate equation as,
再
2
冎
2
共12兲
where ⌼⫽1/( ␥ ⫹ı( ␻ ⫺⍀) is the complex line shape factor,
C⫽Nl 1 is the number of electrons, Ctr ⫽Nl 2 is the transparency carrier number, I p ⫽NI 1 is the total pump operator,
and the photon number operator P⫽⌳ c ⌳ a . L( ␻ ⫺⍀)
⫽1/2␥ (⌼⫹⌼ * )⫽ ␥ 2 / 关 ␥ 2 ⫹( ␻ ⫺⍀) 2 兴 is the Lorentzian
coefficient. Substituting Eq. 共11兲 into Eq. 共3兲 yields,
1
d⌳ a
⫽⫺
⌳ a ⫹g 2 共 C⫺Ctr 兲 ⌼⌳ a ⫹G 共 t 兲 ,
dt
2␶p
共13兲
in which G(t) is the new noise operator.
The photon rate equation given by Eq. 共14兲 is derived by
multiplying the photon annihilation rate equation in Eq.
共13兲 by the photon creation operator ⌳ c , applying the differentiation rule of multiple operators, and then substituting
from the definitions of P, C, Ctr , and L,
dP
P 2g 2
2g 2
⫽⫺ ⫹
LC⫹F P共 t 兲 .
共 C⫺Ctr 兲 LP⫹
dt
␶p
␥
␥
共14兲
The electric field phase variation can be included in the
density matrix using quantum mechanics, but semiclassical
theory is easier and suffices for this purpose.35 Starting with
Maxwell’s equations, using the analysis given by Refs. 24
and 34 through 37, the wave equation may be solved using
the definition of the electric field inside the laser cavity, as
given in Eq. 共15兲. The electric field amplitude and phase
rate equations may then be derived as given in Eqs. 共16兲
and 共17兲.
1
E 共 z,t 兲 ⫽ Ã ␺ 共 z 兲 exp共 ⫺2ı ␲ f t 兲 exp关 ⫺ı ␾ 共 t 兲兴 ⫹cc,
2
再
共15兲
226
冎
Optical Engineering, Vol. 43 No. 1, January 2004
t
⫺⬁
exp关 ␥ 共 ␶ ⫺t 兲兴 ⌳ a F *
l 共 ␶ 兲 d ␶ ⫹cc,
冕
t
⫺⬁
exp关 ␥ 共 ␶ ⫺t 兲兴 ⌳ a F l* 共 ␶ 兲 d ␶ ⫹cc
⫽2QÃRe兵 U 共 t 兲 exp共 ⫺i ␾ 兲 其 ,
F ␾共 t 兲 ⫽
1
Ã
共16兲
共18兲
2g 2
LC⫹ıgN exp关 ı 共 ␻ ⫺⍀ 兲 t 兴
␥
Im兵 U 共 t 兲 exp共 ⫺ı ␾ 兲 其 .
共19兲
共20兲
In Eqs. 共15兲–共20兲, Ã is the amplitude of the electric field, ␾
is the optical phase, and Q⫽2␧/ប ␻ .
4
Auto- and Cross-Correlation Relations
The random noise functions given by Eqs. 共18兲, 共19兲, and
共20兲 are assumed to be Gaussian random variables representing the Langevin noise sources related to carriers, photons, and phase, respectively, with zero mean.
具 F C共 t 兲 典 ⫽ 具 F P共 t 兲 典 ⫽ 具 F ␾ 共 t 兲 典 ⫽0.
共21兲
As discussed in Sec. 2.2, the laser noise sources are functions of the atomic noise operators and are correlated by
their variances. Consequently, the Langevin noise sources
given by Eqs. 共18兲, 共19兲, and 共20兲 are cross-correlated.
These cross-correlation relations were neglected in Refs.
12, 23, 24, 29, and 38, but the alternative approach presented here allows for a mathematical treatment of these
important terms. Mathematical manipulations of the autoand cross-correlated noise sources are cumbersome, but a
suitable approximation of these terms can be chosen.
Markoffian approximations,35 provide a convenient means
of representing these terms, as the auto- and crosscorrelation functions of the noise sources may be written in
terms of Dirac delta functions, as given by,
具 F i 共 t 兲 F i 共 t ⬘ 兲 典 ⫽D ii ␦ 共 t⫺t ⬘ 兲
具 F i 共 t 兲 F j 共 t ⬘ 兲 典 ⫽RD i D j ␦ 共 t⫺t ⬘ 兲 ,
1 2g 2
dà 共 t 兲 ⫺1
⫽
à 共 t 兲 ⫹
共 C⫺Ctr 兲 L Ã 共 t 兲
dt
2␶p
2 ␥
⫹Re兵 U 共 t 兲 exp共 ⫺ı ␾ 兲 其 ,
冕
a
F P共 t 兲 ⫽⌳ a * F Ph ⫹F *
Ph ⌳ ⫺
⫻
dC
2g
2g
⫽I p ⫺ ␥ 1 ⫹
L C⫺
L共 C⫺Ctr 兲 P⫹F C共 t 兲 ,
dt
␥
␥
共17兲
共22兲
in which D i and D j are the variances of F i (t) and F j (t),
respectively, and R is the correlation coefficient. Applying
this rule to our case yields,
Ghoniemy, MacEachern, and Mahmoud: Analytical expressions . . .
具 F P共 t 兲 F P共 t ⬘ 兲 典 ⫽D PP␦ 共 t⫺t ⬘ 兲
具 F ␾ 共 t 兲 F ␾ 共 t ⬘ 兲 典 ⫽D ␾␾ ␦ 共 t⫺t ⬘ 兲
具 F C共 t 兲 F C共 t ⬘ 兲 典 ⫽D CC␦ 共 t⫺t ⬘ 兲
具 F P共 t 兲 F C共 t ⬘ 兲 典 ⫽D PC␦ 共 t⫺t ⬘ 兲
具 F P共 t 兲 F ␾ 共 t ⬘ 兲 典 ⫽0
具 F C共 t 兲 F ␾ 共 t ⬘ 兲 典 ⫽D C␾ ␦ 共 t⫺t ⬘ 兲
冎
.
共23兲
Using Eq. 共10兲, the heat bath averages of operators
具 ABCD 典 ⫽ 具 AB 典具 CD 典 ⫹ 具 AC 典具 BD 典 ⫹ 具 AD 典具 BC 典 , 21 and
using the derivations given in Appendix A in Sec. 10, the
calculated variances of F P(t), F ␾ (t), and F C(t) are given
as,
再 冉
具 F C共 t 兲 F C共 t ⬘ 兲 典 ⫽ I P ⫹ ␥ 1 ⫹
⫹
冊
2g 2
L Cs
␥
冎
2g 2
L共 Cs ⫹Ctr 兲 Ps ␦ 共 t⫺t ⬘ 兲
␥
⫽D CC␦ 共 t⫺t ⬘ 兲 ,
再
Ps 2g 2
⫹
LNCs
具 F P共 t 兲 F P共 t ⬘ 兲 典 ⫽
␶ ph
␥
共24兲
冎
2g 2
2g 2
LNCs ⫹
L共 Cs ⫹N tr 兲 Ps
␥
␥
共25兲
冎
⫽D PC␦ 共 t⫺t ⬘ 兲 .
共26兲
The relation between the steady-state photon number probability in terms of the transition probabilities of emission
and absorption and the amplitude of the electric field34,35 is
given by
⫽
再
Ps ⫹1
in case of emission 共 up transition兲
P
in case of absorption 共 down transition兲 .
s
共27兲
Combining Eqs. 共15兲, 共17兲, and 共20兲, with Eq. 共27兲 gives
具 F ␾ 共 t 兲 F ␾ 共 t ⬘ 兲 典 ⫽ 具 F P共 t 兲 F P共 t ⬘ 兲 典 /4共 Ps ⫹1 兲 2
⫽
D PP
␦ 共 t⫺t ⬘ 兲
4 共 Ps ⫹1 兲 2
⫽D ␾␾ ␦ 共 t⫺t ⬘ 兲 ,
共28兲
具 F C共 t 兲 F ␾ 共 t ⬘ 兲 典 ⫽ 具 F P共 t 兲 F C共 t ⬘ 兲 典 /2共 P s ⫹1 兲
⫽
1⫹Prn
冉
⫺
Pn
␶ ph
T⫺T r Eq
⫺
To
kT
n
兲
共 Cn ⫺Ctr
dCn 共 t 兲
I eff共 t 兲
⫺
⫽
n
dt
qV c Cth ␶ e ␶ ph 共 1⫺Ctr
兲
冊册
C2n ⫹F Pn 共 t 兲 ,
冑
Pn
1⫹Prn
⫺
Cn
⫹F Cn 共 t 兲 ,
␶e
共31兲
共29兲
共32兲
where F Pn (t), F Cn (t), F ␾ n (t) are the normalized photon,
carrier, and phase Langevin noise sources, Pn
n
⫽P␶ e /(⌫Cth ) , Cn ⫽C/Cth , Ctr
⫽Ctro /Ctho , C̄n ⫽C̄/Cth , Prn
⫽Pn /Psn , Psn ⫽ ␶ e Ps /⌫Cth , and Ps ⫽␧ o n̄n g /⌫ ␻ o ប I s . P
represents the intracavity photon number, C is the active
region carrier number, Ctho is the threshold carrier number
at room temperature, Ctro is the transparency carrier number at room temperature, C̄ is the mean of the carrier number, B o is the radiative recombination coefficient at 300 K,
␶ e and ␶ ph are the carrier and photon lifetimes, respectively,
and ␤ sp is the spontaneous emission factor. V c is the volume of the active region, n̄ and n g represent the effective
mode index and the group refractive index, respectively,
ប⫽h/2␲ 共Planck’s constant兲, and I s is the saturation intensity. E is the activation energy for the spontaneous radiative
recombination. T is the active region temperature, T r is the
equilibrium temperature without injection, and T o is the
characteristic temperature of the active region. ␤ is the linewidth enhancement factor. G n ⫽⌫ v g 㜷, in which ⌫ is the
mode confinement factor 共or the filling factor兲, v g is the
group velocity, and 㜷 is the differential gain coefficient.
6
D PP
␦ 共 t⫺t ⬘ 兲
4 共 Ps ⫹1 兲 2
⫽D C␾ ␦ 共 t⫺t ⬘ 兲 ,
冋
冑
Pn
d ␾ n共 t 兲 ␤
⫽ G n 共 Cn ⫺C̄n 兲 ⫹F ␾ n 共 t 兲 ,
dt
2
⫻ ␦ 共 t⫺t ⬘ 兲
QÃ 2
n
兲
共 Cn ⫺Ctr
dPn 共 t 兲
⫽
n
dt
␶ ph 共 1⫺Ctr 兲
共30兲
⫽D PP␦ 共 t⫺t ⬘ 兲 ,
再
5 Laser Model Including the Noise Fluctuations
Equations 共12兲, 共14兲, and 共17兲 may be modified by equating
兵 (2g 2 / ␥ )L(C⫺Ctr ) 其 with the modified gain formulation
given in Ref. 28, 兵 ␥ 1 ⫹(2g 2 / ␥ )L其 ⫽1/␶ e , and using I p
⫽I eff(t)/(qVc), in which I eff(t) is the effective current injected into the active region.
The normalized laser model given in Ref. 39 may be
modified by adding the phase rate equation and the effects
of the noise sources, leading to the normalized modified
noise-driven laser model, given by,
⫹ ␤ sp B o Ctho ␶ e exp
2g 2
⫹
L共 Cs ⫹Ctr 兲 Ps ␦ 共 t⫺t ⬘ 兲
␥
具 F P共 t 兲 F C共 t ⬘ 兲 典 ⫽⫺
where Cs and Ps are the carrier and photon steady-state
numbers. Relations in Eqs. 共23兲, 共28兲, and 共29兲 are the auto
and cross-correlation relations for the carrier and photon
noise sources.
Noise Generation Technique
The Langevin noise operators F C(t), F P(t), and F ␾ (t) in
Eqs. 共12兲, 共14兲, and 共17兲 are described by Eqs. 共18兲, 共19兲,
and 共20兲, and their variance and covariance are given by
Eqs. 共24兲, 共25兲, 共26兲, 共28兲, and 共29兲. The stochastic funcOptical Engineering, Vol. 43 No. 1, January 2004
227
Ghoniemy, MacEachern, and Mahmoud: Analytical expressions . . .
Fig. 1 (a) Schematic symbol for the four port SDD model. (b) HP-ADS schematic diagram for the
experimental setup. The diode symbol shown represents the hierarchically defined SDD shown in (a).
tions F Pn (t) and F ␾ n (t) given in Eqs. 共30兲 and 共31兲 may be
generated by a set of discrete-time Gaussian white noise
sources that satisfy a multidimensional distribution with a
probability density function pd f (x), given as,
pdf共 x兲 ⫽
1
关共 2 ␲ 兲 n 储 K储 兴 1/2
exp关 ⫺ 共 xគ ⫺ ␮ 兲 T K⫺1 共 xគ ⫺ ␮ 兲 /2兴 ,
共33兲
in which x is a length n vector, K is the n⫻n covariance
matrix, ␮ is the mean value vector, and the superscript T
indicates the matrix transpose. The covariance matrix K is
a diagonal matrix with diagonal elements RDii x i that represent the variance in the number of events affecting i in a
time interval ⌬t. The diagonal elements can be evaluated
using Eqs. 共24兲, 共25兲, 共26兲, 共28兲, and 共29兲. The set of jointly
Gaussian random variables represents the changes in photons, carriers, and phase induced by the Langevin operators
during the integration time step ⌬t. To obtain a suitable
event rate induced by the Langevin noise operators, the
variables should be divided by the time interval between
any two successive sampling times ⌬t.
Based on this discussion, the proposed general selfconsistent technique for generating these stochastic functions is presented as follows.
• Generate three different white Gaussian random variable xP , x␾ , and xC .
• Generate the Langevin photons, phase, and the new
variable RVn noise operators as:
FPn 共 t 兲 ⫽
F ␾ n共 t 兲 ⫽
D PP兩 n
冑⌬t
8 Simulation Results
Realistic laser diode parameters obtained from Refs. 12 and
24 for InGaAsP semiconductor lasers and used in the simulations are listed in Table 1. The proposed model shown in
Fig. 1共a兲 was used as part of the experimental setup shown
in Fig. 1共b兲 to predict L-I characteristics, dc, and instantaneous values of carriers, photons, power, and phase, and to
study the effect of thermal heating and the leakage current
on these values.28,39 Also, laser performance characteristics
such as modulation response, relaxation frequency, and
modulation bandwidth were studied.27,41
The fixed simulation time steps were chosen to ensure
white Gaussian noise generation while preserving reasonable simulation time. The largest time step used was ⌬t
xP ,
D ␾␾ 兩 n
冑⌬t
7 Enhanced SDD Model Implementation
A semiconductor laser model, as defined by Eqs. 共30兲, 共31兲,
and 共32兲, was implemented using symbolically defined devices 共SDDs兲 in Agilent ADS. The three-port SDD laser
model proposed in Ref. 39 was enhanced by adding an
extra port for implementing the phase rate equation. The
implemented model is suitable for both small and large
signal simulations including transient and steady-state
simulations in the time and frequency domains. Once
implemented, an SDD may be used as part of a much larger
system simulation.40 Figure 1共a兲 shows the schematic symbol for the four-port SDD used to implement the normalized modified laser rate equations, while the experimental
setup used for time domain simulations is illustrated in Fig.
1共b兲.
x␾
and
n
F RVn
共 t 兲⫽
冉
D n ⬘n ⬘兩 n
⌬t
冊
1/2
xC .
共34兲
• Generate the carriers noise source using F Cn (t)
n
n
⫽F RVn
(t)⫺(RPC
F Pn ⫹RCn ␾ F ␾ n ).
The 兩 n means the normalized form of the operator, and RPn ,
and RCn ␾ are the normalized cross-correlated operators.
228
Optical Engineering, Vol. 43 No. 1, January 2004
Fig. 2 Simulated evolution of photon and carrier densities and their
dependency temperature.
Ghoniemy, MacEachern, and Mahmoud: Analytical expressions . . .
Table 1 Device parameters for an InGaAsP semiconductor buried hetrostructure (SI-BH).
Parameter
definition
Symbol
and Value
Parameter
definition
Symbol
and Value
Active region length
L ⫽250 ␮ m
Active region width
W ⫽2 ␮ m
Active region thickness
d ⫽0.2 ␮ m
Ctro ⫽1⫻1024m ⫺3
Operating wavelength
␭⫽1.3 ␮ m
Transparency carrier
density
Reflectivity of facets
R 1⫽ R 2⫽0.3246
Mode confinement factor
⌫⫽0.3
Effective mode index
n̄ ⫽3.5
Group refractive index
n g ⫽4
Line width enhancement factor
␤ ⫽5
Differential gain coefficient
㜷⫽2.5⫻10⫺20 m 2
Nonradiative recombination rate
A nr ⫽1⫻108 S ⫺1
Radiative recombination
coefficient
B o ⫽1⫻10⫺16 m 3 / S
Dipole relaxation time
␶ in⫽0.1⫻10⫺12 S
Hole interband relaxation
time
␶ v ⫽0.07⫻10⫺12 S
Active region characteristic
temperature
T o ⫽70K
⫺41
Auger recombination coefficient
C a ⫽3⫻10
Dipole moment
d m ⫽9⫻10⫺29 mC
Electrons interaband relaxation
time
␶ c ⫽0.3⫻10
Spontaneous emission factor
␤ sp ⫽10⫺4
6
m /S
⫺12
S
⫽5 ps, which corresponds to a 200-GHz noise spectrum.
This bandwidth is much larger than the laser’s relaxation
oscillation frequency 共6 GHz at a 60-mA bias level27兲.
Figure 2 illustrates the simulated laser carrier and photon densities, and their dependence on the leakage and thermal effects. Noise was not included in this simulation. As
shown in the figure, the laser response is strongly dependent on the temperature. Beyond a critical temperature, the
laser ceases to operate correctly due to the reduction in
carrier mobility at extreme temperatures.
8.1 Carrier, Photon, and Power Results
The time evolution of carrier number, photon number, and
optical power output—with and without the effect of the
noise—including the start-up transient period, is shown in
Figs. 3共a兲, 3共b兲, and 3共c兲. The corresponding steady-state
condition is shown in Figs. 3共d兲, 3共e兲, and 3共f兲. From Fig.
3, it is readily seen that adding the noise operators causes a
fluctuation of the carriers, photons, and power around their
steady-state values during and after the transient period.
The effect of noise is less visible during the the start-up
transient, due to the relative scale of the transient conditions and noise. The steady-state solution as shown Figs.
3共d兲, 3共e兲, and 3共f兲 agrees with the small signal results using the perturbation theory given by Ref. 24.
8.2 Intensity, and Phase/Frequency Noise
Laser intensity noise is characterized by relative intensity
noise 共RIN兲. RIN is defined as the ratio of the spectral
density of the fluctuated power to the steady-state mean
square power. RIN may significantly degrade an optical
link’s carrier-to-noise ratio 共CNR兲, so it is important to
study its effect on the link performance.
Phase noise leads to a frequency shift, which is interpreted as laser line width broadening. Phase noise,
FNS ␾ . ( ␻ ), is defined by the spectral density of the frequency noise. Phase noise is an important concern in coherent optical communication systems and interferometer
sensors. The magnitude of the noise depends on a laser’s
structure and material. Noise is also strongly dependent the
laser’s operating characteristics.
Simulations of frequency noise and relative intensity
noise were performed using the symbolically defined laser
model. Figure 4 shows the instantaneous frequency fluctuations, including transient in Fig. 4共a兲 and steady state in
Fig. 4共b兲, with and without including the noise operators.
As shown in this figure, the frequency noise is high during
the transient period, then it rapidly dampens and fluctuates
around the steady state with a nearly constant rate.
Figure 5共a兲 shows the simulated RIN spectra. From this
figure, the laser RIN is maximum near the threshold, which
is also around the relaxation oscillation frequency 共6 GHz兲.
RIN drops off from a peak located just above this frequency. The RIN varies approximately between ⫺160 and
⫺140 dB/Hz. As shown in Fig. 5共a兲, the RIN is nearly flat
共white兲 at low frequencies above 10 kHz. Below 10 kHz,
RIN has a 1/f characteristic. Figure 5共b兲 shows the phase
noise character of a laser diode. The frequency noise spectrum is relatively flat in the low-frequency period, then it
peaks at the relaxation oscillation frequency. Comparison
of Fig. 5共a兲 with Fig. 5共b兲 demonstrates that the flat range
of the frequency noise spectrum is larger than that of the
intensity noise spectrum, but the qualitative behavior is
similar in both cases. Frequency noise simulations at very
low frequencies are necessary for line width ⌬ f characterization, as line width is defined as FNS ␾ . (0)/(2 ␲ ). The
results shown in Fig. 5共b兲 are useful for determining the
laser’s line width. Generally, the line width decreases as the
laser power increases. However, the line width is found to
saturate to a value in the range 1 to 10 MHz at a power
level above 10 mW.24 At a 60-mA bias level, the line width
is ⌬ f ⫽13.5 MHz, which is comparable to published
values.12,24
9
Conclusion
Theoretical investigation and modeling of the quantum
noise in semiconductor lasers using the theory of density
Optical Engineering, Vol. 43 No. 1, January 2004
229
Ghoniemy, MacEachern, and Mahmoud: Analytical expressions . . .
Fig. 3 Time evolution of the (a) carrier number, (b) photon number, and (c) power with the effect of
noise operators (solid lines) and without the effect of noise operators (light lines) including transient
(d), (e), and (f) during steady state.
Fig. 4 Time evolution of the phase with the effect of noise operators (solid lines) and without the effect
of noise operators (light lines) (a) including transient (b) during steady state.
230
Optical Engineering, Vol. 43 No. 1, January 2004
Ghoniemy, MacEachern, and Mahmoud: Analytical expressions . . .
Fig. 5 Laser diode noise (a) relative intensity noise spectra (RIN) and frequency dependance of the
amplitude noise (b) phase/frequency noise spectra.
matrix in quantum electronics is presented. A general normalized semiconductor laser model, including effects of
temperature, leakage current, and noise, is demonstrated.
SDD realization of these models in Agilent ADS is discussed. A novel self-consistent technique for relating and
generating correlated photons, carriers, and phase noise
sources is presented. Photon density, carrier density, optical
power, and phase are simulated for various operating conditions. Noise effects are analyzed through the study of
RIN, FNS, and line width ⌬ f . The RIN and the FNS are
seen to have the same qualitative behavior for the simulation conditions and laser parameters considered. Based on
the simulated results, temperature, leakage, and noise effects are important considerations in models intended for
analog optical communication simulations. The proposed
model provides a critical block for small and large signal
simulations in both time and frequency domains.
10
⫽ 具 F C共 t 兲 F ␾ 共 t 兲 典具 F C共 t ⬘ 兲 F ␾ 共 t ⬘ 兲 典 ⫹ 具 F C共 t 兲 F C共 t ⬘ 兲 典具 F ␾ 共 t 兲
⫻ 共 t 兲 F ␾ 共 t ⬘ 兲 典 ⫹ 具 F C共 t 兲 F ␾ 共 t ⬘ 兲 典具 F ␾ 共 t 兲共 t 兲 F C共 t ⬘ 兲 典
⫽ 共 D CCD ␾␾ ⫺2D C2 ␾ 兲 ␦ 共 t⫺t ⬘ 兲 .
共36兲
Assuming a new random variable equals the sum of Eqs.
共35兲 and 共36兲 with zero mean, gives
具 RVn 典 ⫽ 具 F P共 t 兲 F C共 t 兲 F P共 t ⬘ 兲 F C共 t ⬘ 兲 典 ⫹ 具 F C共 t 兲 F ␾ 共 t 兲 F C共 t ⬘ 兲 F ␾ 共 t ⬘ 兲 典
⫽ 具 F P共 t 兲 F C共 t 兲 F P共 t ⬘ 兲 F C共 t ⬘ 兲 典 ⫹ 具 F C共 t 兲 F ␾ 共 t 兲 F C共 t ⬘ 兲 F ␾ 共 t ⬘ 兲 典
⫽0.
共37兲
Appendix A: Independent Carrier Noise
Source Generation
One may conclude from a study of the noise sources given
in Eqs. 共18兲, 共19兲, and 共20兲 that F P and F ␾ (t) are not correlated. Therefore, they can be represented by two independent Gaussian noise sources with zero mean and variances,
given by 具 F P(t)F P(t ⬘ ) 典 , and 具 F ␾ (t)F ␾ (t ⬘ ) 典 respectively.
F C(t) is correlated with both F P and F ␾ (t), where their
cross-correlation relations are given by 具 F P(t)F C(t ⬘ ) 典 , and
具 F C(t)F ␾ (t ⬘ ) 典 , respectively. Therefore, F C(t) cannot be
represented by an independent Gaussian random variable.
In this appendix, we convert the cross-correlation relations between the carrier noise source, and both the photon
and phase noise sources, to alternative relations involving
independent Gaussian random variables with zero mean.
Application of the rule of the heat bath average of operators, as given in Sec. 4, assuming the noise sources as
operators, yields the following,
After algebraic manipulations and substitution of Eqs. 共35兲
and 共36兲 into Eq. 共37兲, 具 RVn 典 is given as,
具 RVn 典 ⫽ 共 D CC⫹RPCD PC⫹RC␾ D C␾ 兲 ␦ 共 t⫺t ⬘ 兲
⫽D n ⬘ n ⬘ ␦ 共 t⫺t ⬘ 兲 ,
⫽ 具 F P共 t 兲 F C共 t 兲 典具 F P共 t ⬘ 兲 F C共 t ⬘ 兲 典 ⫹ 具 F P共 t 兲 F P共 t ⬘ 兲 典
⫻ 具 F C共 t 兲 F C共 t ⬘ 兲 典 ⫹ 具 F P共 t 兲 F C共 t ⬘ 兲 典具 F C共 t 兲 F P共 t ⬘ 兲 典
共35兲
共38兲
in which 具 RVn 典 is the auto-correlation of the new random
variable RVn, RPC⫽⫺D PC /D PP is the cross-correlation
coefficient between F P(t) and F C(t), RC␾ ⫽⫺D C␾ /D ␾␾ is
the cross-correlation coefficient between F C and F ␾ (t) , and
D n ⬘ n ⬘ is the variance of the new random variable RVn.
RVn can be simplified and given as
RVn⫽ 共 F C⫹RPCF P⫹RC␾ F ␾ 兲 .
具 F P共 t 兲 F C共 t 兲 F P共 t ⬘ 兲 F C共 t ⬘ 兲 典
2
⫽ 共 D CCD PP⫺2D PC
兲 ␦ 共 t⫺t ⬘ 兲 ,
具 F C共 t 兲 F ␾ 共 t 兲 F C共 t ⬘ 兲 F ␾ 共 t ⬘ 兲 典
共39兲
Equation 共39兲 represents the new random variable with zero
mean and variance 具 RVn 典 g, and it is orthogonal to the
random variables representing photons and phase noise
sources.
Optical Engineering, Vol. 43 No. 1, January 2004
231
Ghoniemy, MacEachern, and Mahmoud: Analytical expressions . . .
Acknowledgment
This research work was supported by National Capital
Institute of Telecommunications 共NCIT兲.
27.
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Samy Ghoniemy is currently working toward the PhD degree at the Department of
Systems and Computer Engineering, Carleton University, Canada. Previous degrees were attained at MTC, Egypt (BSc in
1990 and MASc in 1996). His current research is focused on optical analog transmission, laser modeling, and laser predistortion techniques.
Leonard MacEachern earned his PhD
(Waterloo) and PEng (APENS), and is currently an assistant professor with the Department of Electronics, Carleton University, Canada. Previous degrees were
attained at Acadia University (BSc in 1990)
and the Technical University of Nova
Scotia (BEng in 1993, MASc in 1996).
From 1996 to 2000, he conducted his PhD
work as an NSERC scholar and in-house
researcher at Mitel Semiconductor in the
areas of integrated rf complementary metal oxide semiconductor
(CMOS) transceiver topologies, rf CMOS device modeling, and the
design of low-power and low-voltage rf circuitry. His current research
is focused on electro-optical interfaces, laser modeling, and laser
predistortion techniques.
Ghoniemy, MacEachern, and Mahmoud: Analytical expressions . . .
Samy Mahmoud obtained the MEng and
PhD degrees in electrical engineering from
Carleton University in 1971 and 1974, respectively. He joined the Faculty of Engineering at Carleton University in 1975,
where he served as Chair of the Department of Systems and Computer Engineering from 1988 to 1998. At present he is the
Dean of the Faculty of Engineering and Design at Carleton University. His main academic research and professional interests
are in the areas of mobile and personal communication systems,
broadband networks, radio-over-fiber and distributed computing
Systems. He has directed several large research and development
projects in these areas, involving joint university/industry collaborations. He recently led a major initiative to establish the National
Capital Institute of Telecommunications (NCIT), a joint research organization involving several large international companies in the
telecommunications and computer industries, leading university researchers, and scientists and engineers from two major Canadian
Government research laboratories (CRC and NRC). He has published more than 150 archival and conference papers in the past 20
years in the areas of wireless communications, speech coding and
transmission, and broadband networks, and supervised more than
80 masters and 30 doctoral theses in these research areas.
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233