300 J. Opt. Soc. Am. B/Vol. 13, No. 2 /February 1996 A. Tyszka-Zawadzka and P. Szczepanski

Influence of mode nonorthogonality on thecorrelation function of the amplitude and of the

intensity fluctuation of a distributed-feedback laser

Anna Tyszka-Zawadzka and Paweł Szczepanski

Institute of Microelectronics and Optoelectronics, WarsawUniversity of Technology, 00-662 Warsaw, ul. Koszykowa 75, Poland

Received November 7, 1994; revised manuscript received August 8, 1995

The influence of mode nonorthogonality on the correlation function of the intensity fluctuation and of theamplitude that determines laser linewidth as well as the coherence of light is investigated. The semiclassicalapproach based on a time-dependent solution of the Fokker–Planck equation is used. Numerical resultsobtained for a distributed-feedback laser with nonvanishing end reflectivity and a complex coupling coefficientreveal the difference between the standard approach (for orthogonal modes) and the more realistic model(mode nonorthogonality included). 1996 Optical Society of America

1. INTRODUCTION

A fundamental source of quantum noise in laser devicesis spontaneous emission from upper-level atoms. In gen-eral, the rate of spontaneous emission, according to thequantum noise theory, is always equal to a signal energyof one excess photon per mode in the laser cavity.

However, more recently the existence of excessspontaneous emission was predicted in gain-guided semi-conductor lasers.1 This prediction was initially contro-versial, as it would seem to violate the accepted principleof quantum noise theory. These difficulties were re-solved by Haus and Kawakami,2 who showed that suchgain or loss systems exhibit correlations of the noiseemission into different propagating modes. This fact isin contrast to the situation with power orthogonal modes,in which the spontaneous emission into different modesis uncorrelated and has an energy of one excess photonper mode in the laser cavity. Siegman3,4 has generalizedthis problem, showing that the correlation of the noiseemission into different modes exists not only in gain-guided laser structures but also in all open-sided laserresonators or optical lens systems and is a consequence ofthe nonpower-orthogonal nature of the transverse modesin such structures (i.e., the non-Hermitian character ofsuch resonator or optical systems). The effect of the cor-relation of the noise emission leads to a transverse excessnoise factor, which depends on the degree of transversefield nonuniformity.

The approach developed by Siegman has been extendedto describe the properties of longitudinal eigenmodes ofthe resonator.5 The nonorthogonality of a longitudinallaser mode is caused by the different longitudinal fielddistributions of the counterrunning waves of the lasermode. The different distributions of the electric field si-multaneously cause the different coupling of the spon-taneous emission from the active medium to the las-ing modes. This effect is important for high-gain laserswith Fabry–Perot resonators, for example, semiconduc-

0740-3224/96/020300-06$06.00

tor lasers. In these kinds of laser the nonuniform elec-tric field distribution is observed when the gain and lossdo not coincide spatially, because of the point losses atthe end mirror caused by the arbitrary mirror transmis-sion. Thus the longitudinal eigenmodes are nonorthog-onal, which leads to a longitudinal excess noise factor.5

The longitudinal excess noise has been also investigatedand measured in bad-cavity gas lasers.6,7

The effect of mode nonorthogonality is also signifi-cant in distributed feedback (DFB) lasers, for which theBragg scattering process providing the optical feedbacksimultaneously causes nonorthogonality of the longitudi-nal eigenmodes (the counterrunning waves of the lasermodes are spatially nonuniform), leading to the longitu-dinal excess noise factor8 – 10 observed experimentally.11

The enhanced spontaneous emission that contributes toan additional widening of the fundamental laser linecan be important when the temporal coherence proper-ties of the laser are of interest, namely, in coherent opti-cal communications, spectroscopy, and optical wavelengthstandards, and also in the injection seeding of pulsedhigh-power unstable resonator lasers.

In general, the theory of laser operation that takesinto account quantum noise requires a stochastic ap-proach, in which the lasing field is treated as a stochas-tic random variable. In the semiclassical formalism thisanalysis usually includes the solution of the appropriateFokker–Planck equation corresponding to a set of cou-pled Langevin equations. By use of this formalism, thestatistical properties of the laser field, such as expec-tation values and the correlation functions representingthe coherence of light, have been investigated in greatdetail.12 – 14 However, in all these theoretical studies or-thogonality of the laser modes has been assumed.

Recently an analysis of the quantum noise in lasersbased on the semiclassical Fokker–Planck equationwas developed to take into account the nonorthogonalnature of the laser modes for DFB lasers.15 In thatanalysis we showed how the steady-state solution of

1996 Optical Society of America

A. Tyszka-Zawadzka and P. Szczepanski Vol. 13, No. 2 /February 1996 /J. Opt. Soc. Am. B 301

the Fokker–Planck equation is modified when modenonorthogonality is included. In particular, for a givenpump parameter the effect of the phase of the end re-flectivity on the mean laser intensity and on the inten-sity fluctuation becomes smaller for stronger couplingstrengths.

However, to calculate the correlation function of theintensity fluctuation and the correlation function of theamplitude that determines the laser linewidth as wellas the light coherence, we require a non-stationary so-lution of the Fokker–Planck equation. The problem ofdetermining time-dependent solutions has been treatedin different ways, for example, by the matrix continuedfraction method or the eigenfunction method.16 How-ever, in all these approaches orthogonality of the lasermodes has been assumed.

In this paper we investigate the effect of modenonorthogonality on the correlation functions that rep-resent the coherence properties of light for single-modeoperation. In our approach, similar to that in Ref. 13,the problem of calculating the correlation functions isreduced to an eigenvalue problem of the Sturm–Liouvilleequation. Using a variational method, we obtain theapproximate eigenfunctions and eigenvalues of theSturm–Liouville equation and calculate the correlationfunctions. To help our analysis we investigate the co-herence properties of light from real DFB lasers withnonvanishing end reflectivities and with complex cou-pling coefficients. These lasers have received increasingattention because of their promising applications as coher-ent sources in optics communication and high-data-ratetransmission nets.

In Section 2 we present the approximate correlationfunctions of the amplitude and of the intensity fluctuationmodified by the excess noise factor. In Section 3 thesesolutions are compared with standard theoretical studies(for orthogonal laser modes). Conclusions are presentedin Section 4.

2. THEORYThe starting point of our theoretical treatment of single-mode laser operation, as in Ref. 15, is the set of Langevinequations for the electric field E, supplemented by the in-troduction of the random Langevin force GsT d correspond-ing to spontaneous emission fluctuations

dEdT

­ sa 2 bjEj2dE 1 GsT d ,

dEp

dT­ sa 2 bjEj2dEp 1 GpsT d , (1)

where a is the net gain coefficient of the laser mode andb is the self-saturation parameter. The noise term GsT dincluding the nonorthogonal nature of the laser modes is acorrelated Gaussian random process, with autocorrelation4KPdsT 2 T 0d, where P is the noise strength for orthog-onal modes and K is the Petermann’s excess noise fac-tor, which depends on the spatial distribution of the lasermodes.1 – 10

The excess noise factor that accounts for a longitudinaleffect is given by5 – 10

K ­Z L

0dzFN szdFp

N szd , (2)

where hFN szdj are the adjoint functions representing a setof eigenfunctions propagating in the opposite or reversedirections along the same system.

One can obtain the adjoint modes hFN szdj from regularmodes hUN szdj:

UN szd ­ C

"RN szdexpsikN zd

SN szdexpsikN zdZ

#(3)

by interchanging the components, i.e.,

FN szd ­ C1

"SN szdexps2ikNzdRN szdexpsikNzd

#, (4)

where kN is the wave vector, C is the normalization con-stant, C1 is the normalization constant determined by thebiorthogonality condition of the adjoint modes,

Z L

0dzUN szdFM szd ­ dNM , (5)

and RN szd and SN szd are the complex amplitudes ofNth counterrunning waves of the Nth longitudinal lasermodes.

In our analysis we investigate DFB lasers. In this casewe can assume that RN szd and SN szd for the Nth DFBmode are proportional to the threshold field distribution,which is a good approximation even for an operation farabove threshold.17 Thus, according to Ref. 18, we have

RN ­ sinh gN sz 1 Ly2d 2 r expsifrdsinh gN sz 2 Ly2d ,

SN ­ 2 sinh gN sz 2 Ly2d 1 r expsifrdsinh gN sz 1 Ly2d ,

(6)

where r expsiwrd ­ R is the complex reflection coeffi-cient and gN denotes the complex propagation constant,which is determined by the transcendental eigenvalueequation18

gN ­ 2ik sinhsgN Ld

3f1 2 r2 exps2ifrdg

f1 2 r expsiwrdexps2gN Ldgf1 2 r expsiwrdexpsgN Ldg,

(7)

where k is the coupling coefficient describing the strengthof the Bragg scattering process.

Substituting Eqs. (6) and (7) into Eq. (2), we obtainthe longitudinal excess noise factor for a DFB laser withnonvanishing end reflectivity as follows9:

K ­

ÉgN

f1 1 r2 exps2iwrdgA 1 r expsiwrdB

É 2

3 jhs1 1 r2d fgb sinhsgaLd 2 ga sinhsgbLdg

1 4r cosswrdfga coshsgaLy2dsinhsgbLy2d

2 gb coshsgbLy2dsinhsgaLy2dgjsgagbd21j2 , (8)

where A ­ gN L coshsgN Ld 2 sinhsgN Ld, B ­ sinhsgN LdcoshsgN Ld 2 gN L, ga ­ gN 1 g

pN , and gb ­ gN 2 g

pN .

302 J. Opt. Soc. Am. B/Vol. 13, No. 2 /February 1996 A. Tyszka-Zawadzka and P. Szczepanski

In the case of a DFB laser with the complex couplingcoefficient k ­ jkjexpsiwd and with vanishing end reflec-tivity, the longitudinal excess noise factor [Eq. (8)] can bewritten in the following way10:

K ­

ÉgN

gN L coshsgN Ld 2 sinhsgN Ld

3gb sinhsgaLd 2 ga sinhsgbLd

gagb

É 2

, (9)

where the propagation constant gN is obtained from thefollowing eigenvalue equation for the complex couplingcoefficient10:

gN ­ 2ik sinhsgNLd . (10)

The analysis of the statistical properties of laser lightrequires solution of the Fokker–Planck equation for thedistribution function W sE, Ep, T d, corresponding to theset of coupled Langevin equations (1). In this equation,in polar coordinates, E ­ r expsifd has the form

≠W s r, f, T d≠T

­ 21r

≠

≠rfsa 2 br2dr2Wg

1 KP

"1r

≠

≠r

√r

≠W≠r

!#1

1r2

≠2W≠f2

.

(11)

For numerical purposes it is convenient to introduce thenormalized variables13,16

r ­4

sb

Pr, t ­

qbP T , a ­

apbP

. (12)

Equation (11) is then transformed into

≠W s r, f, td≠t

­ 21r

≠

≠rfsa 2 r 2dr 2W g

1 K

"1r

≠

≠r

√r

≠W≠r

!#1

1r 2

≠2W≠f2

.

(13)

In our approach the distribution functionW s r, f, td can be expanded into eigenfunctions ofthe Fokker–Planck operator. Inasmuch as W s r, f, tdis a periodic function in f and the time derivative isof the first order, the general solution for W s r, f, td isexpected to be of the form

W s r, f, td ­X

m­0

Xn­2`

Anmfnms rdexpsinfdexps2lnmtd ,

(14)

where lnm are the eigenvalues of Eq. (13). It is clearthat the particular eigenvalue l ­ 0 is associated with thesteady-state solution of the Fokker–Planck equation.15

Putting Eq. (14) into Eq. (13), we find that the eigen-functions fnm satisfy the following Sturm–Liouvilleequation:

≠

≠r

√p

≠fnm

≠r

!2 qnfnm 1 lnmpfnm ­ 0 , (15)

where lnm are the eigenvalues lnm normalized to theexcess noise factor K, lnm ­ lnmyK, and p and qn aregiven by

ps rd ­ r exp

"1K

√r 4

42 a

r 2

2

!#,

qns rd ­

"1K

s2a 2 4r 2d 1n2

r 2

#ps rd . (16)

It has been shown13 that the problem of calculating thecorrelation function of the amplitude fluctuation

gstd ­ k rst 1 tdexpf2ifst 1 tdg rstdexpfifstdgl (17)

and the correlation function of the intensity fluctuation

Rstd ­ ks r 2st 1 td 2 k r 2lds r 2std 2 k r 2ldl (18)

can be reduced to an eigenvalue problem of ordinarysecond-order differential equation (15) with n ­ 0 andn ­ 1. Thus the correlation functions gstd and Rstd takethe following forms:

gstd ­ NX

m­0

"Z `

0r 2f1msrddr

# 2

exps2l1mtd , (19)

Rstd ­ NX

m­1

"Z `

0r 3f0msrddr

# 2

exps2l0mtd , (20)

where N is the normalization constant. To obtainthe eigenvalues and eigenfunctions of Sturm–Liouvilleequation (15) it is convenient to employ a variationalmethod.13 Thus the expression

lnm ­

Z `

0s pfnm

0 2 1 qnfnm2ddrZ `

0pfnm

2dr

, (21)

with the subsidiary conditionR`

0 pfnmfnm0 dr ­ dmm0 , hasto be minimized.

To find the lowest-order eigenvalues l10 and l01 we use,unlike Risken,13 modified trial eigenfunctions f10 and f01

of the following forms:

f10 ­ r exp

"2

r 4

4K1

√aK

1 e

!r 2

2

#, (22)

f01 ­ s1 2 Ar 2dexp

√2

r 4

4K1 a

r 2

2K

!, (23)

where e is the variational parameter and A is determinedby the subsidiary condition,

R`

0 pfn0fn1dr ­ 0. It is

A. Tyszka-Zawadzka and P. Szczepanski Vol. 13, No. 2 /February 1996 /J. Opt. Soc. Am. B 303

Fig. 1. Eigenvalue l10 versus the normalized pump parametera for the amplitudes of the end reflectivity r ­ 0.1 and r ­ 0.5,with the phase of the end reflectivity wr as a parameter. Thecoupling coefficient is kL ­ 0.1. The dashed curve is obtainedfor orthogonal laser modes.

Fig. 2. Eigenvalue l10 as a function of the normalized pumpparameter a for the coupling coefficients kL ­ 0.1 and kL ­ 5,with the phase of the end reflectivity wr as a parameter. Theamplitude of the end reflectivity r ­ 0.1. The dashed curve isobtained for orthogonal modes.

worth noting that eigenfunctions (22) and (23) are re-duced to those obtained by Risken for orthogonal lasermodes, i.e., for K ­ 1. Inserting Eqs. (22) and (23) intoEq. (21), making the substitution v ­ r 2, and calculatingintegrals of the form

Ins yd ­Z `

0vn exp

√2v2

4K1 y

v2K

!dv (24)

by applying the following recurrence relations:

Ins yd ­ 2Ksn 2 1dIn22s yd 1 yIn21s yd ,

I1s yd ­ 2K 1 yI0syd ,

I0s yd ­p

pK exps y2y4Kdf1 1 Fs yy2p

K dg ,

where Fs yd is the error integral, we finally get the ap-proximate eigenvalues

l10 # Minfhsedg ­ Min

"2e

K1

ae2

K21

2e3

K2

1 2

√1 1

e2

K

!I0sa 1 2edI1sa 1 2ed

#, (25)

l01 #

√4

K22

8K

1 8

!I0sadI1sad

fI2sadI0sad 2 I12sadg

. (26)

The correlation function of the amplitude and of theintensity fluctuation can be obtained approximately bythe relations13,16

gstd ø gs0dexps2l10td , (27)

Rstd ø Rs0dexps2l01td , (28)

where the correlation functions for t ­ 0, i.e., gs0d ­ k r 2land Rs0d ­ k r 4l 2 k r 2l2, were obtained for the stationarysolution of the Fokker–Planck equation.15

The expression of the linewidth Dn, defined by

gsT d ­ gs0dexps2DnT d , (29)

is determined by the eigenvalue l10 in the following way:

Dn ­ l10

qbP . (30)

In Section 3 we present the behavior of the eigenvaluesl10 and l01 as a function of the characteristic parametersof the DFB lasers.

3. NUMERICAL RESULTSWe show the effect of nonorthogonality on the nonsta-tionary solution of the Fokker–Planck equation for DFstructures. Figures 1–4 show the influence of the non-vanishing end reflectivity on eigenvalues l10 and l01,which are responsible for the amplitude correlation func-tion (which determines the laser linewidth) and the cor-relation function of the intensity fluctuation, respectively.

Fig. 3. Variation of the eigenvalue l01 with the normalizedpump parameter a for the amplitudes of the end reflectivityr ­ 0.1 and r ­ 0.5, with the phase of the end reflectivity we asa parameter. The coupling coefficient is kL ­ 0.1 The dashedcurve is obtained for orthogonal modes.

304 J. Opt. Soc. Am. B/Vol. 13, No. 2 /February 1996 A. Tyszka-Zawadzka and P. Szczepanski

The dashed curves are obtained for orthogonal lasermodes. The phase of the end reflectivity remarkablyaffects the eigenvalues l10 and l01 and simultaneouslythe coherence properties of the optical field; this is relatedto the fact that the phase of the end reflectivity remark-ably changes the longitudinal field distributions of thecounterrunning waves of the laser mode and simultane-ously causes the different coupling of the spontaneousemission to the lasing mode. As we can see, the effectof the phase of the end reflectivity becomes less impor-tant for stronger coupling strengths; i.e., greater valuesof the normalized coupling coefficient jkLj and of the endreflectivity amplitude r. In this case the electric fieldof the laser mode becomes more uniform, and the excessnoise diminishes. Moreover, with increasing couplingstrengths the eigenvalue l10 decreases. This effect, ingeneral, corresponds to the narrowing laser linewidth,Eq. (30), and the longer coherence time. Furthermore,the eigenvalue l01 decreases (i.e., the correlation of theintensity fluctuation increases) and the laser characteris-tics tend to those obtained for orthogonal laser modes.

In Figs. 5 and 6 the effect of the complex coupling coef-ficient on eigenvalues l10 and l01, respectively, is shown.As we can see, when the phase of the coupling coefficientis equal to zero (i.e., pure index coupling) the eigenval-ues l01 are the lowest and the linewidth Dn, which corre-sponds to the eigenvalue l10, is the smallest. Moreover,an increase of the coupling strength causes the laser char-acteristics to be less sensitive to the phase changes. Itis also worth noting that the difference between two lasermodels (i.e., for orthogonal and nonorthogonal modes) de-creases with increasing coupling strength. This is so be-cause with increasing coupling strength the outcouplinglosses tend to zero (the Q factor of the DFB resonatortends to infinity, and the longitudinal modes become or-thogonal).

In general, the coherence properties of light from a DFBlaser depend strongly on the changes of the geometry, i.e.,the output mirror reflectivity and the coupling strength,of the laser structure. These changes lead to differentdistributions of the electric field and simultaneously to

Fig. 4. Eigenvalue l01 versus the normalized pump parametera for the coupling coefficients kL ­ 0.1 and kL ­ 5, with thephase of the end reflectivity wr as a parameter. The amplitudeof the end reflectivity is r ­ 0.1. The dashed curve is obtainedfor orthogonal laser modes.

Fig. 5. Eigenvalue l10 as a function of the normalized pump pa-rameter a for the coupling strengths jkLj ­ 1 and jkLj ­ 5, withthe phase of the complex coupling coefficient w as a parameter.The dashed curve is obtained for orthogonal modes.

Fig. 6. Eigenvalue l01 versus the normalized pump parametera for the coupling strengths jkLj ­ 1 and jkLj ­ 5, with thephase of the complex coupling coefficient w as a parameter. Thedashed curve is obtained for orthogonal laser modes.

different coupling of the spontaneous emission from theactive medium to the lasing mode.

4. CONCLUSIONSWe have investigated the influence of mode nonorthogo-nality on laser operation. Using the stochastic approachbased on the time-dependent Fokker–Planck equation,we obtained the correlation function of the intensity fluc-tuation and of the amplitude that determines the laserlinewidth as well as the light coherence of distributedfeedback lasers. We have shown that the real parame-ters of the laser structure, the nonvanishing end reflec-tivity and the complex coupling coefficient, remarkablyaffect the coherence properties of light presented by theamplitude correlation function. The effect of the phase ofthe complex coupling coefficient on the coherence of lightbecomes less important for stronger coupling strength,and the difference between two laser models (for orthogo-nal and nonorthogonal modes) diminishes with increasingcoupling strength.

A. Tyszka-Zawadzka and P. Szczepanski Vol. 13, No. 2 /February 1996 /J. Opt. Soc. Am. B 305

ACKNOWLEDGMENTSThe authors thank K. Wodkiewicz, A. Kujawski, andW. Wolinski for stimulating discussions. This work issupported by the foundation for Polish Science.

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