Fluctuations and Noise Letters Vol. 1, No. 3 (2001) L163–L170 © World Scientific Publishing Company

A MULTIMODE SIMULATION MODEL OF MODE-COMPETITION LOW-FREQUENCY NOISE IN SEMICONDUCTOR LASERS

MOUSTAFA AHMED,• MINORU YAMADA and SALAH ABDULRHMANN Department of Electrical and Electronic Engineering, Kanazawa University, Kanazawa 920-8667, Japan.

Received (12 May 2001) Revised (19 September 2001)

Accepted (21 September 2001)

A multimode model is proposed to simulate the mode-competition low-frequency noise in semicon-ductor lasers taking account of the possible mechanisms of nonlinear gain saturation. A new tech-nique is reported to generate the Langevin noise sources that induce spontaneous-emission fluctua-tions in dynamics of the lasing modes. The model is applied to simulate the intensity noise in Al-GaAs lasers. Agreement of the simulated noise results with the experimental measurements is found. Correlation of the noise characteristics with dynamics of the mode competition is newly introduced. The low-frequency noise is enhanced when the dynamics exhibit mode hopping or a jittering single mode, and is suppressed when the stable single-mode operation is achieved.

Keywords: Semiconductor lasers; multimode simulation; mode competition noise; mode hopping; gain saturation; jittering single mode.

1. Introduction Operation of semiconductor lasers is affected by the mode-competition phenomena that are intrinsically caused by the nonlinear gain saturation effect [1]. The mode competition may enhance the coupling of the side modes with the dominant mode so strongly as to generate mode hopping [2–7]. These phenomena were also proved to induce the optical feedback noise when re-injection of the laser light by optical feedback exists [8]. Analy-sis of noise in semiconductor lasers is commonly performed by applying the approximate small-signal analysis to treat the rate equations including Langevin noise sources that account for the intrinsic fluctuations in the modal photon number and electron number [1,9]. However, this analysis is applicable for cases of small fluctuations compared to the dc-intensity, and its treatment becomes very complicated if a large number of modes ex-ist. Moreover, correspondence between the noise properties and the characteristics of the inducing mode competition in the time domain is missed in that analysis. An alternative method to analyze the noise is to perform numerical integration of the rate equations [5,7]. A typical merit of applying the latter method is to get enough insight of the dynam-

• The author is on leave from the Physics Department, Faculty of Science, Minia University, Egypt.

Ahmed, Yamada and Abdulrhmann

ics of the mode competition and correlate them with the induced noise characteristics. However, reports on satisfactory numerical treatments were limited so far because of two critical problems. The first problem concerns with numerical generation of the Langevin noise sources keeping the correlations of the injected electron number with the photon numbers of the laser modes. In a previous paper, the authors proposed a systematic tech-nique to simulate correlated Langevin noise sources in the numerical calculation for the case of single-mode operation [10]. The second problem is the correct introduction of the cross-saturations on the modal gain, which are seeds of the mode competition.

In this work, we report a new multimode simulation model of the intensity noise by extending the previous technique [10] to the case of multimode operation. The cross-saturations on the modal gain are also taken into account. The simulation results are in good correspondence with the experimental measurements.

2. Multimode Simulation Model The present multimode simulation model is based on numerical integration of the follow-ing rate equations of the photon number Sp of (M+1) longitudinal modes and the injected electron number N:

[ ] pthpq qqpqpppp SGSHDBSA

dt

dS

−+−−= ∑≠ )()(

{ } )()( 20 tFbVN

Va

pp +−−+ ωωξ , Mp ±±±= ,....,2,1,0 and (1)

)(tFeINSA

dtdN

Nsp pp ++−−= ∑

τ . (2)

The mode p=0 is assumed to center the spectral profile of the linear gain Ap [11],

{ }20 )()( ωωξ −−−= pgp bVNN

VaA , (3)

where ξ is the confinement factor of the field into the active region, a and b are material parameters, and Ng is the transparent level of N. ω p is the frequency of mode p. Positive numbers p indicate modes on the higher-photon energy (shorter wavelength) side of the central mode, while negative values apply to modes on the opposite side. Gth is the threshold gain level, and τs is the electron lifetime by the spontaneous emission. Inclusion of the spontaneous emission, which triggers fluctuations in dynamics of the laser modes, into the lasing process is described by the term in the second braces in (1). The saturation mechanisms of the gain of a mode p are included in terms of the coefficients B, Dp(q) and Hp(q) as follows. The coefficient B determines the self-saturation of gain, while the cross saturation by other modes q≠p is given by Dp(q) and Hp(q). The coefficient Dp(q) describes the symmetric dispersion saturation of gain, while Hp(q) describes the asymmetric satura-tion. Concrete analyses of the saturation phenomenon with the density-matrix theory showed that the nonlinear gain coefficients are given by [11,12]

( )scvin

ro

o NNRaVn

B −

=

222

249 ξτ

ε

ω!

! , (4)

A Multimode simulation model of mode-competition low-frequency noise

1)(3

422)( +−

=inqp

qpBD

τωω and (5)

2

2

2

)(

)(231

)(231

)(43

qps

qps

gqp

SVa

SVa

NNVaH

ωωξτ

ωωαξτξ

−+

+

−++−

= . (6)

Rcv is the dipole moment, nr is the refractive index in the active region of volume V, τin is the intraband relaxation time, Ns is an injection level characterizing the gain saturation, S is the total photon number, and α is the linewidth-enhancement factor. This description of the gain saturation indicates that coupling occurs among the modes, which results in mode competition; an increase in the photon number of a mode causes suppression of the gain of the other modes so as to achieve the highest gain and dominate the lasing action. The asymmetric gain saturation functions in promoting the gain of the modes on the long-wavelength side while suppressing the gain of the modes on the shorter side [12,13]. This saturation is one of the possible sources of mode hopping and mode jumping. The influ-ence of the asymmetric gain saturation on the laser operation is enhanced at high injec-tion levels or at large values of the α-factor, as given in (6).

The terms Fp(t) and FN(t) in (1) and (2) are Langevin noise sources and function in inducing instantaneous fluctuations on Sp(t) and N(t) due to the spontaneous emission and the processes of recombination and carrier generation. The sources are originally defined as Poisson random processes, and are well approximated as Gaussian ones with zero mean values. The noise sources are auto- and cross-correlated,

( )')'()( ttVtFtF xyyx −= δ , x,y=p or N , (7)

where Vxy are the variances of the correlations and δ is Dirac’s delta function. In princi-ple, the noise sources on the photon numbers of different modes are uncorrelated, i.e., qppppq VV ,δ= , (8)

Here, δ here is the Kronicher delta. However, The source FN(t) is cross-correlated with the sources Fp(t) because of the mutual interactions between electrons and photons during the lasing process as described in (1) and (2). This property of the cross-correlation repre-sents a fundamental problem when generating the sources using the independent com-puter random number generators.

Equations (1) and (2) are transformed into a new set of equations of the modal photon numbers Sp(t) and a combined quantity of ∑+

p pp tSktN )()( ,

[ ]∑ ∑∑∑

−+−−+

+−−=

+≠p pthpq qqpqpppp

sp ppp pp SGSHDBSAkeINSASkN

dtd

)()(τ

[ ] ∑++−−+

p ppNp tFktFbVNVa )()()( 2

0ωωξ , (9)

in such a way that the superposing noise sources Fp(t) and ∑+p ppN tFktF )()( are uncor-

related. The orthogonality among the sources is artificially achieved by careful setting of the modal coefficient kp as

Ahmed, Yamada and Abdulrhmann

)'()()'()( tFtFktFtF ppppN −= . (10)

These sources can then be generated at each sampling time ti using their auto-variances as

pipp

ip gt

tVtF

∆=

)()( and (11)

N

iNNp ipNip

p ipipiN gt

tVtVtktFtktF

∆

+=+

∑∑

)()()()()()( , (12)

where gp and gN are independent Gaussian random deviates with zero mean values and variances of unity. Furthermore, if these noise sources are imagined to form an (M+2)-dimensional functional space, the noise source FN(t) can be represented in this space by

∑∑ −

+=p ipipp ipipiNiN tFtktFtktFtF )()()()()()( . (13)

Therefore, the simulation can be done by integrating the original rate equations (1) and (2) using (11) and (13) or integrating the transformed equations (1) and (9) using forms (11-12). The variances of the noise sources at a time ti are determined by the mean values of Sp(t) and N(t) at the preceding time ti-1 supposing quasi-steady solutions of (1-2) ( )0≈≈ dtdNdtdS p in the interval 1−−=∆ ii ttt [9,14],

[ ] )(1)(2)( 11 −− += iipipp tNtSVatV ξ

, (14)

)()(12)( 11 −−

+= ∑ i

pip

siNN tNtS

VatV ξ

τ and (15)

[ ][ ]{ })(1)(.)()()( 112

01 −−− ++−+−= ipgippiiNp tSNtSbVtNVatV ωωξ . (16)

The relative intensity noise RIN induced by the mode competition is calculated from the fluctuations in the total photon number StStS −= )()(δ , with S being the mean value of S(t), over a finite time T with the help of the fast Fourier transform (FFT) as

[ ]

=

+= ∫∫ ∫ −∞ 2

02002 )(11)()(11 T j

Tj deS

TSdtdetStS

TSRIN ττδττδδ ωτωτ . (17)

3. Simulation Results In this section, we apply the proposed model to investigate the characteristics of intensity noise in AlGaAs lasers. We count a large number of 31 modes, M=15, to assure accuracy of the simulations. The corresponding 32-rate equations of (1) and (2) are integrated us-ing the fourth-order Runge-Kutta method with time interval of ∆t=20ps over a long pe-riod of T=10µs. Typical values of the parameters used in simulations are: a=2.75x10-

12m3s-1, ξ=0.2, nr=3.59, V=180µm-3, L=300µm, 2cvR =2.8x10-57C2m2, Ng=1.89x108,

Ns=1.53x108, Gth=5.01x1011s-1, τin=0.1ps, τs=2.79ns and α=2.0. The random numbers gp and gN are generated by applying the Box-Mueller algorithm [14] to uniformly distributed

A Multimode simulation model of mode-competition low-frequency noise

random numbers generated by the computer random sources. The RIN is calculated via (17) after the transients of S(t) die away. Ten spectral profiles of RIN are averaged to obtain a reasonable precise statistical spectral profile. The output spectrum of the photon number is determined by averaging the modal photon number Sp(t) over the integration time T.

First we show the results of applying our model in its simplest case of the single mode, i.e, M=0, to simulate the quantum noise without mode competition. The results are plotted in Fig. 1 at a current I of 2.0 of the threshold value Ith. The simulation without gain nonlinearity, i.e., B=0 in (1), are also shown to investigate influence of the nonlinear gain on the characteristics of RIN. The effect of counting the nonlinear gain is found to suppress the RIN data around the peak at the relaxation frequency. However, the low-frequency RIN is less affected. Figure 1 also plots the RIN profile predicted by the small-signal analysis, which was started by Haug [9]. Good correspondence is seen between the simulated noise data and those calculated by the small-signal analysis.

100k 1M 10M 100M 1G 10G

10-16

10-14

10-12

100k 1M 10M 100M 1G 10G

10-16

10-14

10-12

10-10

present model

without gain saturation

small-signal analysis

single-mode modelI = 2.0 Ith

RIN

[Hz-1

]

Frequency [Hz]

single mode

jittering

multimode

mode hopping

stable single mode

I = 1.10 Ith I = 1.28 Ith

I = 1.36 Ith I = 1.80 Ith

RIN

[Hz-1

]

Frequency [Hz]

Fig. 1. The simulated profile of RIN at I=2.0Ith based on a single mode model. The profiles calculated by ignoring the gain saturation and by the small-signal analysis are also plotted.

Fig. 2. The simulated RIN profiles at four distinct operations. The low-frequency RIN is enhanced in the mode-hopping region and is suppressed in the stable single-mode region.

The time-domain simulations indicate that inclusion of both the Langevin noise sources and the asymmetric gain saturation enhances the mode competition to induce instabilities on the laser dynamics and affect the noise characteristics. Four distinct ex-amples of the simulated RIN at currents I=1.10, 1.28, 1.36 and 1.80Ith are shown in Fig. 2. The low-frequency components of the RIN are almost flat (white noise) and coincident with the quantum level when I=1.10 and 1.80Ith, which correspond to multimode and stable single-mode operations, respectively. In the former operation, the instantaneous mode competition is so strong that most of the modes oscillate simultaneously. In the stable single-mode operation, the instabilities induced by coupling fluctuations of the side modes with those of the main mode is so weak that the photon number of each mode fluctuates around its dc-value. Then the induced noise corresponds to quantum noise in this case. On the other hand, the low-frequency RIN is enhanced when I=1.28 and 1.36Ith, which can be understood by examining the corresponding instantaneous variation of the modal photon number Sp(t) shown in Fig. 3. Figure 3(a) shows random hopping of the modes p=0 and -1, which is seen as instantaneous coupling of fluctuations of the photon numbers of the hopping modes as well as mode switching. Moreover, the fluctuations of the mode p=+1 are coupled with those of the hopping modes. Such instabilities promote

Ahmed, Yamada and Abdulrhmann

200 4000.0

0.3

0.6

0.9

1.2

300 350 400 450 5000.0

0.3

0.6

0.9

1.2

(b)(a) mode hopping

jittering single mode

I = 1.28 Ith

p = +1

p = 0 mode p = -1

Mod

al p

hoto

n nu

mbe

r Sp(t

) / S

Time [ns]

I = 1.36 Ith

p = -2p = 0

mode p = -1

Mod

al p

hoto

n nu

mbe

r Sp(t

) / S

Time [ns]

Fig. 3. Simulation results of the modal photon number Sp(t) when (a) I = 1.28Ith and (b) I = 1.36Ith. The figures show the induced mode hopping and jittering single mode. The mode numbers are denoted in the figure.

the RIN level. Figure 3(b) shows another type of unstable operation referred to as “jitter-ing single mode”, which corresponds to the so-called mode partition [2–5]. The instabili-ties induced in the mode dynamics are generated by coupling the fluctuations of the side-modes, p=0 and -2, with those of the main mode, p=-1. The mode competition is not strong enough in this case to bring the coupled modes to instantaneous switching. The term “single mode” is attributed, in this paper, to the high ratio of the photon number between the dominant mode and the strongest side mode, which exceeds 20 dB.

We carry out a large number of simulations to investigate the dependence of the low-frequency RIN on current I. The simulation results at a frequency as low as 120 kHz are plotted in Fig. 4. In order to correlate the simulated results of RIN with the mode dynam-ics, we examine the dynamics and the corresponding state of operation at each current value. We are able to classify the RIN results into four regions as seen in the figure: namely multimode noise, mode-hopping noise, jittering single-mode noise, and stable single-mode noise. Examples of the output spectra of the photon number in these regions are plotted at the top of the figure. Some experimental results obtained for a TJS AlGaAs DH laser to characterize the mode-hopping noise are also plotted in the figure with open circles [15]. The figure indicates good correspondence between the simulated results and the experimental data within the measured range of current. However, the measured RIN attains higher levels, which may be attributed to the noise induced by diffusion of the current from the electrode into the active region in the mechanism of the current injection [15,16]. The peak shown around the threshold current Ith is attributed to the maximum contribution of the spontaneous emission to light amplification, which rapidly decreases above threshold compared with the contribution of the stimulated emission [1]. The noise in this region corresponds to the multimode operation in which the output photon number is distributed almost uniformly among the modes as indicated by spectrum (a). The quan-tum level of RIN is also attained in the stable-single mode region, thth III 8.154.1 << . In this region the dynamics are free from either mode hopping or jittering, and the photon number of the main mode, p=-1, is 20 times higher than that of any side mode as shown in spectrum (d). The RIN level is most enhanced in the mode-hopping region,

thth III 34.112.1 << , because of the instabilities induced by the mode-hopping phenome-non, as discussed before. The peak of the hopping noise is attained when mode jumping

A Multimode simulation model of mode-competition low-frequency noise

1.0 1.2 1.4 1.6 1.810-17

10-15

10-13

10-11

-6 -3 0 3 610-4

10-2

100

-6 -3 0 3 6-6 -3 0 3 6-6 -3 0 3 6

Simulation Experimen

multimode modehopping

jitteringsingle-mode stable

single mode

f ~ 120 kHz

RIN

[Hz-1

]

Injection current I / Ith

(a)

Mode number p

< S

p(t)

> /

S

(d)

(c)

I = 1.10 Ith I = 1.28 Ith I = 1.80 IthI = 1.36 Ith

(b)

Fig. 4. The simulated variation of the low-frequency RIN with the injection current I. Some experimental results are also plotted with open circles. The investigated states of modal operation are denoted, and examples of the time-averaged output spectrum in each state are given in (a)–(d). occurs. That is, the photon number of the hopping side mode, p=-1, balances that of the main mode, p=0. In the other unstable region of jittering single-mode operation,

thth III 54.134.1 << , the RIN decreases with current I. However, the RIN is higher than the quantum noise level because of the generated extra noise. Typical examples of the output spectra in the unstable regions of mode hopping and jittering are given in (b) and (c) of Fig. 4, respectively.

4. Conclusions We reported a new multimode model of simulating the low-frequency intensity noise induced by the mode competition in semiconductor lasers. A new technique is proposed to adopt generation of the Langevin noise sources in the laser rate equations. Correlation of the noise characteristics with the mode-competition dynamics is introduced. The inten-sity noise is enhanced when the laser exhibits mode-hopping or jittering operations.

Acknowledgement The authors would like to thank the Japan Society for Promotion of Science (JSPS) for partly supporting the present work.

Ahmed, Yamada and Abdulrhmann

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