Side-mode gain in semiconductor lasers

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J. Opt. Soc. Am. B/Vol. 9, No. 8/August 1992

Sargentet al.

Side-mode gain in semiconductor lasers Murray Sargent III and Stephan W Koch Optical Sciences Center, Universityof Arizona, Tucson, Arizona 85721 Weng W Chow

Division 2531, Sandia National Laboratories,Albuquerque, New Mexico 87185 Received July 8, 1991; revised manuscript received November 25, 1991

Side-mode gain and coupling coefficients in semiconductor laser media are calculated with the use of a multimode Fourier technique valid both for mode spacings that are small compared with the carrier-carrier relaxation rates and for spacings comparable with these rates as predicted by the Boltzmann theory of Binder et al. [Phys. Rev. B 45, 1107(1992)]. The medium is described by a free-carrier model that provides for carrierprobability pulsations around quasi-equilibrium Fermi-Dirac values. We find that population pulsations play just as important a role as spectral hole burning for mode spacings comparable with the intraband relaxation rates. For the carrier-carrier relaxation rates of Binder et al., side-mode gain is predicted to be smaller than the main-mode gain, leading to single-mode operation. However, for somewhat smaller intraband relaxation rates, side-mode gain is readily found that exceeds the single-mode gain, which would encourage multimode operation. In addition, we find that the gain and coupling coefficient spectra are sensitive to the k dependence of the carrier-carrier relaxation rates and might provide a useful way to measure these rates. We are also able to explain the asymmetric side-mode gain spectra for small beat frequencies in terms of the rapid decrease of the quasi-equilibrium Fermi-Dirac electron distribution just above the gain region.

1.

INTRODUCTION

A major question in semiconductor laser physics is why and when the semiconductor laser oscillates with multiple modes. This problem was of major interest in the 1960's with traditional laser models and underwent a major resurgence of interest in the 1980's.' About this time the problem acquired substantial importance in semiconductor laser diodes. Treatments include those of Yamada and Suematsu2 and Kazarinov et al.,3 which are both based on two-level atom physics with appropriate modifications to fit experimental data. Reference 3 pointed out the importance of population pulsations in coupling modes placed on either side of an oscillating mode, a process well known in this context owing to Lamb laser theory.4 5 As with Refs. 2 and 4, the treatment used third-order perturbation theory, although theories for an arbitrarily intense oscillating mode were available.67 The decay times chosen were aimed at understanding multimode phenomena for small (but not too small!) diode lasers, so that the intermode beat frequencies were 100 GHz or more. For longer lasers obtainable with external cavities, the modal beat frequencies are smaller, and the interband relaxation times can play a role in the coupling of laser modes. Agrawal' treated this case with a medium whose gain is linearly proportional to the total carrier density, and he treated the short-laser case with a model similar to that of Ref. 3. Mukai and Saitoh9 studied the small beatfrequency case both experimentally and theoretically, using a linear-gain model. While these simple phenomenological theories certainly reveal important operational trends, their very simplicity can conceal the underlying semiconductor physics. In particular, they require that high-level quantities such as the gain be fitted from experimental

data, whereas one would like to be able to de0740-3224/92/081288-11$05.00

scribe the laser behavior using only low-level constants such as temperature, effective masses, and electric dipole matrix elements. Toward this end the multiwave mixing theory of Sargent et al.'" and the third-order laser theory of Zhou et al." are based on quasi-equilibrium Fermi-Dirac distributions and include a carrier-densitydependent renormalization of the band-gap energy. In contrast to the claim in Ref. 3 that lasing takes place below the band gap in the Urbach tail, the semiconductor laser always operates above the actual band gap, which, however, is renormalized and lies below the zero-density band gap. As in the linear-gain model of Refs. 8 and 9, the theories assume that the total carrier density pulsates at the modal beat frequencies and thereby couples the modes analogously to the individual k-dependent pulsations modeled approximately by an inhomogeneously broadened two-level model. However, the details and the level of the

calculation are significantly different from those in Refs. 8 and 9 in that, instead of using a phenomenological gain that is linearly proportional to the carrier density, we use a gain that results from quasi-equilibrium FermiDirac distributions that effectively breathe, driven by the

field variations through the total carrier density and therefore through the carrier chemical potentials. This more fundamental approach permits one to explain, for example, the asymmetric side-mode gain spectra in terms of the rapid change of the electron quasi-equilibrium Fermi-Dirac distribution at the chemical potential ILe [see the discussion following Eq. (54) below]. The side-mode

gain and coupling coefficients are determined by a Fourier expansion technique highly developed for two-level atoms.'2 The analysis also includes results appropriate for pump-probe spectroscopy, modulation spectroscopy, and nondegenerate phase conjugation in semiconductors. The present paper uses a unified free-carrier theory to C 1992 Optical Society of America

Vol. 9, No. 8/August 1992/J. Opt. Soc. Am. B

Sargentet al.

1289

lar to those for inhomogeneously broadened two-level me2 6 dia subjected to an arbitrarily intense wave. " We find

that population pulsations sharpen the spectral holes

6'3 l

@(k)

|

I,

w

Fig. 1. Spectrum of a three-mode field.

v, and v2

V3

V3

P2

Waves with frequencies

are usually taken to be weak (nonsaturating), while the

wave is permitted

to be arbitrarily

intense.

treat the complete range of pulsation frequencies appropriate for the slow variations found in long external mirror lasers all the way up to the frequencies that one finds in small lasers. A complete many-body theory would include a series of effects due to Coulomb interactions between the charge carriers, namely, band-gap renormalization, Coulomb enhancement of the gain spectra, rapid carrier 3 5 relaxation, and dynamic screening. - The appropriate integrodifferential equations incorporating these Coulomb effects are intricate and can require extensive computer computations for their analysis. In this paper we take a less ambitious approach that nevertheless uses re5 sults from the many-body calculations' and in principle goes substantially beyond the earlier studies described above. In addition to the terms in the free-carrier models of Refs. 10 and 11, we include an exponential

relaxation

approximation, using electron and hole decay rates ye and yh, respectively, predicted by the many-body Boltzmann analysis of Ref. 15. These decay constants generalize the single phenomenological intraband decay-time approach used, for example, in Refs. 3, 8, and 15. The computed

decay rates differ significantly from the best-fit rates used in the simpler two-level theories, perhaps because making those theories match the relatively complicated physics requires different choices. In particular, according to Ref. 15, an effective intraband T, is computed to have the same order of magnitude

as

T2

[see Eqs. (7) and

(32) below], instead of being an order of magnitude smaller as used in Ref. 3. This change substantially modifies the relative roles of spectral hole burning and population pulsations, here each having roughly equal influence on the mode coupling. The present theory of the medium is also more general than that of Refs. 3, 8, and 9 in that

we effectively expand the solutions about quasi-equilibrium Fermi-Dirac distributions, which makes the theory sensitive to basic parameters such as temperature and carrier effective masses and permits a description of slow and fast population pulsations as well as spectral hole burning. Since Ref. 10 found that modulation of the band-gap renormalization contributed relatively little to the mode-coupling coefficients, we neglect these contributions in the present paper. For simplicity, we also ignore the effects of transverse variations of the field and longitudinal spatial hole burning. We find side-mode gain and coupling coefficients that reduce to those of Ref. 10 for small beat frequencies, but, for beat frequencies comparable with the fast carriercarrier scattering rates, the coefficients have forms simi-

burned by the strong mode and double their depth. This suppresses modes close to the oscillating mode (a point noted in Ref. 3) while encouraging the oscillation of modes further away. Section 2 calculates the side-mode polarization of the medium in a way that is general enough to treat the full range of possible modal beat frequencies, from gigahertz to terahertz and beyond. Section 3 presents the sidemode coupled-mode equations with explicit coefficients both for beat frequencies that are small compared with intraband relaxation times and for those comparable with these times. Section 4 gives numerical illustrations and a discussion of the results.

2. SIDE-MODE GAIN COEFFICIENTS We consider a semiconductor medium subjected to an arbitrarily

intense mode and one or two nonsaturating

side

modes. We assume that the saturating-mode intensity is constant throughout the interaction region and ignore transverse variations. We label the side modes by the indices 1 and 3 and the saturator mode by 2, as shown in Fig. 1. We consider a multimode plane-wave electric field of the form E(z, t) = /2 E ,, (t)exp[i (K,,z - v,,t)] + c.c.,

(1)

where the mode amplitudes T,,(t) are, in general, complex and K, are the wave propagation vectors. For simplicity, we take mode functions that are appropriate for a unidirectional ring laser. The mode index equals 1, 2, or 3. The field [Eq. (1)] induces the complex polarization

P(z, t) /2 1 gP,(t)exp[i(Kz - vt)] + c.c.,

(2)

where 91,,(t) is a complex polarization coefficient that yields index and absorption/gain characteristics for the side-mode and saturator waves. In general, the polarization P(z, t) has other components, but we are interested only in those given by Eq. (2). In particular, strong wave interactions induce components not only at the frequencies Vl, V2, v3 but at v, ± i (V2- v,) as well, where j is an integer. In this linear side-mode approximation, other pairs of modes placed symmetrically about v2 does not interact with the (v,, V3) pair, and hence the present theory describes the onset of general multimode operation. The problem reduces to determining the side-mode polarization 91(t), which drives the side-mode amplitude n according to the equation of motion

- v)% -[-- i(fl,, =vQp, 2e

L2Q,,

(3)

where e is the permittivity of the host medium, Q,,is the cavity Q for the nth mode, and fQn is the passive cavity frequency of the nth mode. One might guess that the side-mode polarization is simply the sum over k of a sidemode Lorentzian multiplied by a probability difference saturated by the saturator wave. However, additional

1290

Sargent et al.

J. Opt. Soc. Am. B/Vol. 9, No. 8/August 1992

contributions enter owing to population pulsations. Specifically, the nonlinear probabilities respond to the super-

position of the modes to give pulsations at the beat frequency A = V2 - VI. Since we suppose that the side mode does not saturate, the pulsations occur only at ±A, a point proved below. These pulsations act as modulators (or as Raman shifters), puttingsidebands onto the medium's response to the v2 mode. One of these sidebands falls precisely at v1, yielding a contribution to the side-mode absorption coefficient. The other sideband influences the polarization of the side mode placed symmetrically on the other side of the strong mode, namely, at the frequency V2 + V2 -

V1.

In this section we derive the nonsaturating-side-mode polarization. This is determined by the off-diagonal element of a two-band density matrix whose elements are functions of the carrier k vector k. We write the equations of motion for the density matrix in the limit in which electron and hole relaxation times 1/ye and 1 /yh, respectively, and the dipole dephasing time y 1 l are much less than other relaxation times and times for which the electric-field mode amplitudes vary appreciably. We suppose that the yv (a = e, h) are functions of the momen-

tum k as given by a Boltzmann equation

analysis.'5

For these cases the density-matrix equations of Ref. 10 become

Pc(k) = -[id) + y(k)]p,,(k) + PV',(z, t) X [e(k) + nh(k) -1], he(k) = Ae(k)- Ynrnfe(k) -Fne(k)nh(k)

(4) -

X [nh(k) - fh(k)] - (icvpvc + c.c.),

(6)

where the subscripts c and v refer to the conduction and valence bands, respectively, and the dipole decay rate y(k) is given by

~

(7)

in which ki is a wave vector corresponding to the probewave frequency. The simple relaxation terms -Ya(k 2 ) [na(k) - fa(k)] for a = e or a = h are an approximation to the Boltzmann collision equation of the form

dt~k) dtou

= FIn(k,na) [1 - na(k)

-

(we drop this below, too).- The transition given by

(k, n.)n.(k), (8)

which has the property that the lk vanishes, i.e., the scattering changes the k dependences of the carriers but not the total carrier density. To preserve this feature in the relaxation-time approximation used in Eqs. (5) and (6), we replace ya(k) by ya(k2 ). This choice, rather than that of a varying k dependence, makes no perceptible difference in Figs. 5 and 7 of this paper, presumably because the ya vary little over the region for mode beat frequencies of the order of yh or less, which is the approximate width of the population-pulsation factor 9;. The na dependence in the F functions of Eq. (8) is a functional dependence; integrals over various na terms are involved in the scatter-

energy Iio(k) is

h(k) + g + 8 sg(N),

ico(k)= ee(k) +

(9)

where the electron and hole energies are given by ea(k) =

i2k2 /2ma (a = e, h),

(10)

and mh are the electron and hole effective masses, eg is the zero-field band-gap energy, 38g(N) is the carrierdensity- (N-) dependent band-gap renormalization shift (we assume that our detunings are relative to the unmodulated renormalized band gap, which can be determined by a more detailed theory). If we set ,3 = 1/kB T where kB is Boltzmann's constant and T is the absolute temperature, then we define fe(k) and fh(k), the electron and hole Fermi-Dirac distributions, respectively, by me

fa..)

exp{j9[sa(k)

-

1

+

=

A.e]} + 1(a=eh)

(11)

where iAe and h are the electron and hole chemical potentials, respectively. The carrier density N is given by the closure relation

N(t)= V'flne(k) = V-'Enh(k) k

k

= V-1E fe(k) = V-1E fh(k),

(5)

fh(k) = Ah(k)- Ynrnh(k) - nh(k)nh(k) - yh(k2)

*

conduction-band carriers and holes for the valence-band carriers. The elements Pcv, ne, and nh also depend on z and t, but for typographical simplicity we display only k

y(k 2 )

X [e(k) - fe(k)] - (ircvp.c + c.c.),

y(k) = 2 [ye(k)+ Yh(k)]

ing processes. In general, we use electrons for the

k

k

(12)

where Vis the volume of the semiconductor. Since lk is proportional to V, the volume cancels out, leaving a number density. For a given N, Ae and /Lh are determined by Eq. (12) through lookup tables or by an approximation formula. 4 The pump rates Ae and Ah correspond to the injection of carriers by a current density J into zero-field Fermi-Dirac distributions fa0 according to the value Aa(k) = 77Jfa0(k)/edNa 0

(a = e, h),

(13)

where -qis the efficiency with which the injected carriers reach the active region, e is the charge of an electron, d is the thickness of the active region, and Na0 = V 1 kfa0 (k). The nonradiative decay constant for the electron and hole probabilities is given by Ynr,and the radiative recombination rate constant is 1. The largest decay rate constants are e and h, which model the attempt of carrier-carrier and carrier-phonon scattering to drive the carrier distributions toward the Fermi-Dirac values fe(k) and fh(k), respectively. The dipole decay constant y is approximately the same value. To derive a self-consistent set of equations, we need to find the carrier number density N in the presence of the various pump and decay processes. To this end, we sum Eq. (5) over the momentum k to find

N=

A-YnrN -

where A = -qJ/ed.

r

- F ne(k)nh(k) V k tOrcPvc(k)+ c.c. ,

(14)

Vol. 9, No. 8/August 1992/J. Opt. Soc. Am. B

Sargent et al.

The interaction energy matrix element 7,,(k, z, t) for

Vc_(k,z,t) =-

-

2i

,(t)exp[i(Kz - vat)],

(15)

where the electric dipole matrix element p varies slowly over the range of k values that interact. As in Ref. 10, we determine the response of the medium to this multimode field by Fourier analyzing the polarization component p,,(k) of the density matrix, the number probabilites na, the probability difference D, the total carrier density N, and the quasi-equilibrium Fermi-Dirac functions [a. Specifically, we write p,,(k) = exp[i(Kiz

-

X exp{im[(K2 - Ki)z - At]}, na(k) = 2,naj(k)exp{ij[(K i

2

- KI)z

-

(16)

At]}

N(t)

=

fa(k, z, t) =

2

dj(k, z, t)

X exp{ij[K 2 - Ki)z - At]},

(18)

ENj exp{ij[(K2 - Ki)z - At]},

(19)

E

j

fj(k)exp{ij[(K2

- At]} (a

Kz

-

e, h). (20)

We approximate the faq(k) by the expansions fa[I_.(No +

AN)]

AN atj 1la

+ f.aI.L.(No) [

+ AN

fJo[1.(No)1

IaULI-a'-'u'J

Equation (23) is simply the single-mode density-matrix element in which we include a subscript 2 to specify the saturator wave and have factored out the rapidly varying time-space factor exp(iK 2 z - iv 2t). We calculate the dc probability Fourier components neO and nhO saturated by the saturator wave %2alone. To simplify the equations of motion for ne and nh, we note that the yeand yh terms dominate the other decay processes by several orders of magnitude, causing na to be nearly equal to [a.

Eqs. (5) and (6) by

Hence we approximate

he = ke + yee - yene - (ircvpvc+ c.c.),

nr (a = e, h).

Ya' Y a +

(26)

(27)

Here we have neglected the small I recombination term, although, with some small N dependence, its effects can be lumped into ya' In principle, we could approximate Fnenh in the nh equation by Fnefh and in the Ah equation by leading to a small carrier-dependent difference bernhfe, tween the two decay rates. With these small contributions neglected, Eqs. (25) and (26) have the same form as the inhomogeneously broadened two-level system and can be solved in similar ways. The major difference is the closure relation (12), which determines the carrier number density N and introduces important carrier-density pulsation contributions. Since the yeand Yh contributions and F cannot be neglected cancel out in the N equation, Ynr for this equation, and in fact they play crucial roles. Substituting Eqs. (15)-(17) and (20) into Eq. (25), we have the following for the j = 0 term: Ye'fneO+ [i(;A%2 /2hi)p 2 * + c-c-],

0 = Xe + YefeO -

aaN afa a, 0

c.c.),

(25)

where the decay constants ye' and Yh' are nearly equal to ye and yh, respectively, and are given by

(a = e, h), (17)

D(k, z, t) ne(k, z, t) + nh(k, z, t) - 1 =

(24)

- VA ]

gn = 1/[Y + i)

+ ih = Ah + Yhfh- Yh'nh - (itVcvpPc

>pm+i(k)

vt)]

Lorentzian

approxi-

the field of Eq. (1) is given in the rotating-wave mation by

1291

which gives = fao[a(No)]+ Njga,

(21)

Ae e + YefeO kf%2/IiIl 22do . n-e - k= yJ7o - Jp2 _

where g,. is given by

=a± fao aNo a

fao(1

V

-

faa)

gfa 0 (1 -

aO)

(22)

We substitute the expansions (16)-(20) into the densitymatrix equations of motion (4)-(6) and identify coefficients of common exponential frequency factors. In the approximation in which %I and Z 3 do not saturate, only p i, P2, and p 3 occur in the polarization expansion (16) and onlyj = 0, +1 appear in the probability expansions (17)Calculating the coefficient of exp(iK2z - iv2t) for (19).3 the saturating mode by neglecting the nonsaturatingside-mode fields, we find

-iV2P2= -(i

yye'

Ye'

Here 22 is the saturator dimensionless Lorenzian: 2 2 £2 = Y /[Y + (9 -V2)2]

Xh A

+

YhI

hfhO _ J7I2/I 2 yyh'

I2d

(30)

Adding Eqs. (28) and (30), we find Xh + Yhfho I222do

+ ye fo

YhI

YeI

i(P'62/2)do,

Ae + YefeO + Ah + Yhfho

which gives

Yh'

Ye

P2 = -i(/2)%292do,

(29)

The W and 3 contributions are ignored, since we assume that the side modes do not saturate. Similarly, the dc probability component nhO is given by

do

+ Y)P2 -

(28)

1 + I222

(23)

where, for typographical simplicity, we do not write the k dependence, and 92 is the n = 2 case of the complex

-

feo + fho - 1

1 + I222

(31)

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J. Opt. Soc. Am. B/Vol. 9, No. 8/August 1992

Sargent et al.

where the dimensionless saturator intensity is

Proceeding with the probability-pulsation terms ne, -1, and d- 1 , we substitute the expansions (15)-(17) into Eq. (25) (ignoring the small time dependences introduced by the eg terms in the sin) and identify coefficients of exp(iAt) to find

2

(32)

12 = 9. 0 T1 T 2 ,

in which RO is the Rabi frequency lp62 /il, T2 is the dipole decay time (-') given by Eq. (7), and T is the population-difference decay time

nh,-i,

iAlne,-i = yefe,-i - ye'ne,-1 +

i(p/2h)

X 061P2* + Z2P3* - %2*P1==

(1

1+)

The intensity [Eq. (32)] is usually less than unity, since the ytaare larger than the Rabi frequency for typical laser operation. The semiconductor would break down for average internal cw field intensities I greater than approximately 6 MW/cm2. Writing I as

3*p2)

From Eq. (26) we see that the hole probability component nh,-l obeys the same equation with e > h. Solving for d- = ne,-1+ nh,-l, we obtain d-1 =

efe,-l

+

Ye' + iA

Yh fh,-1 Yfh + iA

+ i(P)T9i(A)(%1P 2 * + 62P3* - %2*P1 -

2

1 2=1 ha, 2 = ceO I= -ceE - ) 2 2 P

c3P2)

(34)

where the high-frequency dimensionless population-

we find the Rabi frequency

pulsation factor 9;(A) is defined by 0to =

/

(35)

2o 2

/ 1 1 + 2T1 y' + iA

9;

We express our frequencies in millielectron volts (meV). For this, note that imev= 10OOA/eand p = er, where e is the magnitude of the electron charge in coulombs. This gives

d- =

_ __=

1 yh' +

.1*-

(40)

iA,

Here 9;(A) is normalized such that 9(0) = 1. Substitut-

ing the polarizations [Eqs. (23), (37), and (38)] into Eq. (39) and using the fact that di = d-1 *, we obtain

dop'T,

-lefe,-I yhhfh,-i Ye A + Y yle + iA 1h' + iA

2i 2h2

+ 2b2*)c19c62*+ (2

+ 263*)A2513*]

1

1 + -I2 2 9(A)Y(R1+ 93*) =

~~~

1000rcV 2.998 X 8.85 x 10-5)

imevo

=

~2

1/2

2.75 x 105X rvI.

(36)

3 x 10-8 cm and I= 6 MW/cm2, this gives 'mev00 2 meV. We will see shortly that for laser excitations the a are several times larger than this (see For r

Our calculation is self-consistent, since only do and dj can take nonzero values from pi, P2, and p 3, and vice versa. Note that Eq. (41) differs from the usual inhomogeneously broadened probability-pulsation component d 1 in that it includes Fermi-Dirac contributions fe,-i + fh,-1. In the limit in which both R 0 and Al 0, which then yields an asymmetric side-mode gain, with larger gain for v < V2.

Except for the appearance of Ekdo instead of fd 0 W(Q)/ (1 + I2 E2), these formulas have the same form as those found for inhomogeneously broadened two-level systems (see Chaps. 8 and 9 of Ref. 12) and can be evaluated in the inhomogeneously broadened limit (IHBL) by contour integration. Since this limit reveals some of the features of the large-A case with substantially less computational effort, we give those formulas here and illustrate them along

with numerical evaluations of Eqs. (57) and (58) in

-(rF i) ( -

= -

-

ial)Y

-

[ray - AB - iA

X [Ige()

I)-Y(V ) [ge(w)+gh()] 2 -

+ rF'i)8i(w

- vi)DT(v2 - V1 )

+ g(()].

The asymmetric part of this expression is proportional to AB, = (v2 - vl)51, where g weights

A > 0 preferentially

in the k, i.e., the gain for v1 < v2 is favored. Still more fundamentally, ge peaks near Isle because this is where the Fermi-Dirac distribution fe changes most rapidly as a function of N. Since h < 0 for bulk GaAs, this energy region is above the gain/absorption crossover given by the total chemical potential I.e + h, and hence all the values of 2 within the gain region lead to the same qualitative asymmetry. This phenomenon is also the origin of the relatively large index of refraction in this energy region. Hence our free-carrier model is able to give a microscopic explanation of the asymmetric side-mode gain, while the phenomenological theories must base their explanation on the linewidth-enhancement factor, which is fitted from experimental data. We note that the asymmetry is enhanced by the Vs- factor in the bulk density-of-states

function, but the asymmetry exists even without this factor as in a quantum-well laser. In Eq. (54) the X implicitly includes a phase-mismatch factor exp[2i(K2 - K) r]. Since in the present cavity configuration the K are all parallel, this mismatch tends to be negligible. If it is significant, the Xn terms in Eqs. (51) and (52) average to zero, leaving behind simple Beer's law equations for the probe waves. For perfect phase matching [(P%2/I) 2 = Ro2] in this quasi-equilibrium limit, the gain coefficient al of Eq. (55) can be written in terms of the Xi of Eq. (54) as al = ai. + Xi,

(55)

where the incoherent gain coefficient, i.e., that neglecting the response to the coherent pump-probe fringe, is given by

Section 4. Specifically, from Eqs. (5.52), (9.28), and (9.48) of Ref. 12, we have

al = ainc+ aoh, (59) where the incoherent and coherent contributions are, respectively,

ainc= ao'(v)i 1 aCoh =-2ao'(V2)I2

y(v + ; ,,2 '

XL[(y'+ ) (y' Lv'('+ v) 24

_ )

+

v)

(60)

V

32

(P + ) ( - v)1

/3( + v)

(61)

18

'e, tIh

12

6

0

0

100

200

300

Fig. 3. gain /-

Ye (solid curves), y/h (dashed curves), and delta-function (feo + fho - 1) (dotted-dashed curves) versus reducedmass energy sin meV for carrier densities N = 2 X 1018,4 X 1018,

and 8 X 1018cm- 3 in order of decreasing y and increasing gain. The e and h values are taken from Ref. 15. The carrier masses are me = 1.28m and mh = 4.53m, respectively, where m is the reduced mass, and the temperature is T = 300 K.

Vol. 9, No. 8/August 1992/J. Opt. Soc. Am. B

Sargent et al.

1295

= 1.05443 X 10-34 J s = 6.5817 x 10-13 meV s. In Eq. (65), ao' has units of inverse (Planck's constant)/2,7r =

3.5

seconds. In the figures we multiply by h in units of meV s, so that al and Xl are also given in meV,which give convenient values for plotting. We also choose a total carrier density of N = 4 x 3 1018 cm- . The cavity Q is given by

3 2.5 2 Refa,)

(66)

Q = 27rL/fA = vLn/cf, 1.5

where L is the cavity length, A is the wavelength, and f is the fractional loss per pass." For a 250-/.tm laser with f = 0.33 and n = 3.5, this gives v/Q cf/nL = 1.14 X 1012s-1 or 0.075 meV,which is less than the gains given in the figures, even when a reasonable mode-filling factor is included. Hence a self-consistent clamped value of

0.5 0

20

0

120

100

80 60 V, - g

40

140

3.5

Fig. 4. IHBL probe absorption al (solid curve) and incoherent part ai- (dashed curve) (in meV) versus probe detuning Vi - g (in MeV) carrier density N = 4 x 1018, carrier masses m = 1.28m and mh = 4.53m, respectively, temperature pump Rabi energy 2 meV (which corresponds to T = 300 K, 3

3 2.5

5.9 MW/cm ), pump detuning 82 =-59 meY and average electron and hole decay constants Iiy, = 12.52 meV and hyh = 9.22 meV,

respectively. These y values are those given by Fig. 3 for the probe tuned to the pump wave (v, = v2).

2 Refa,) 1.5

the complex frequency is v = y + iA, and the complex frequency factor is ,82 = V(V

(62)

+ YI29;).

Similarly, the coupling coefficient Xi can be evaluated (1/2)aoy 2 I2 9(A)(y + v) X =-'1(y' + O)

as 0-

(see Section 6-6 of Ref. 12) ->

k

2 X 4

3 (270- fJ0

2

k 2 dk = (2m/

IVds

(64)

3

where the exciton Bohr radius and Rydberg energy are

2.5

_R =

2

40

60 80 v - g

100

120

140

3.5

1s27r -i13 f 'V dJ, eRas 0 given by a = 112e/me2 and

20

Fig. 5. Inhomogeneously broadened probe absorption al (solid curve) and incoherent part ai- (dashed curve) versus probe detuning v - eg. All the parameters are the same as in Fig. 4, except that the integrals over e are done numerically.

fJ

27r

0

(63)

To calculate the multiplicative factor ao' we note first that

V

0.5

/2mao 2 , respectively.

2

Including the vp2'/2hie lead factor in Eq. (57) (for exRefal)

ample), we obtain

1.5

a

= 2- e(72a,383/2)

-sg)

+ fe(AV - Eg)- 1],

2

[fe(Vl

-

s) (65)

where, as usual, we give all energies consistently in millielectron volts, e.g., Ivl, se, and SR. In the figures in Section 4 we take the electric dipole matrix element g = e X 3 X 10-3 cm, where the charge of the electron is e = 1.6 X 10-9 C, the exciton Bohr radius is ao = 1.243 X 10-6 cm, the exciton Rydberg energy is SR = 4.2 meV, the band gap is sg = 1426.2 meV, the permittivity of the medium is e = n2E0,the index of refraction is n = 3.5, the permittivity of free space is eO = 8.85 X 10-14 F/cm, and

0.5 0 0

20

40

80

60 L

-

100

120

140

g

Fig. 6. Inhomogeneously broadened probe absorption al (solid curve) and incoherent part ainC(dashed curve) versus probe detuning Vi - eg. All the parameters are the same as in Fig. 4, except that the integrals over e are done numerically, with vy(k) and yh(k) taken from the values in Fig. 3.

1296

J. Opt. Soc. Am. B/Vol. 9, No. 8/August 1992

Sargent et al.

2.28

(m is the reduced mass), temperature T = 300 K,

2.24

pump Rabi energy hiRo = 2 meV (which corresponds to 5.9 MW/cm 3 ), and pump detuning ha2 = -59 meV All the frequencies and the coefficients are given in meV In Figs. 4-6 we plot the side-mode gain coefficient versus the probe detuning above the band gap, using the N = 4 x 1018 cm- 3 values of Fig. 3 in three approximations. Figure 4 uses the IHBL [Eq. (59)] for the probe absorption a (solid curve) and the incoherent part aic (dashed curve) and average values of ye and h given by

Re(a,)

ye =

2.2

12.52 meV and yh = 9.22 meV, respectively.

These y

values are those given by Fig. 3 for the probe tuned to the pump wave (l = v2 ). Figures 5 and 6 plot the same curves, except that in Fig. 6 the integrals over are done 3.5

2.16 V

-

g

3

Fig. 7. Blowup of Fig. 5 including strong-mode gain a2 (dotteddashed curve).

2.5 2.32 2 Re(a,) 1.5 2.28 Reaol)

0.5 0

2.24

0

20

40

60

80

100

120

140

V - g

2.2

_ 40

Fig. 9. IHBL probe absorption al (solid curve) inchoherent part ainc (dashed curve), and strong-mode gain a 2 (dotted-dashed curve) versus probe detuning v - g. All the parameters are the same as in Fig. 4, except for the smaller decay constants hlye= 50

60 L -

Fig. 8.

70

80

hyh = 3 meV;

g

Blowup of Fig. 6 including strong-mode gain a2 (dotted-

3.5

N corresponding to saturated gain = loss would give a smaller value of N We defer a precise comparison to a future investigation.

3

dashed curve).

4.

NUMERICAL ILLUSTRATIONS

Reference 15 reports the values of the carrier-carrier relaxation rates ye and h for a variety of situations. We summarize some of those results in Fig. 3 for use in our multimode calculations. Specifically, Fig. 3 plots ye (solid

curves), Yh (dashed curves), and delta-function gain s(feo + fho - 1) (dotted-dashed curves) versus the reduced-mass energy in meV for several carrier densities. We see that the hole constant h does not vary appreciably across the gain region, but the ye decreases roughly linearly in this region. Some of the following figures use these decay constants, and some use smaller values. All the figures use a carrier density of N = 4 x 10l8 cm-3, a main-mode Rabi frequency of 2 meV,carrier masses m = 1.28m and mh = 4.53m, respectively

2.5 2 Re(aj) 1.5

0.5 0

0

20

40

60 U -

80

100

120

140

g

Fig. 10. Inhomogeneously broadened probe absorption a, (solid curve), incoherent part ain, (dashed curve), and strong-mode gain a2 (dotted-dashed curve) versus probe detuning v - g. All the parameters are the same as in Fig. 9, except that the integrals over are done numerically.

Vol. 9, No. 8/August 1992/J. Opt. Soc. Am. B

Sargent et al. l

l

0.1

0

, , , , ,.

I

-

-0.1I I

Re{X1 ) -0.2

I

I

I I

-0.3

Iy

~~~~~~~ -0.4 -

-0.5

20

30

,

40

50

I 60 VL'Eg

1(10

70 90 80

100

90

80

70

versus probe detuning

Fig. 11. Real parts of a,0 h and Xi (in meV v, - sg (in meV) for the parameters in F ig. 9.

The curves are

3.1

,

,

, -E

strong-mode gain a 2 (dotted-dashed curves) for reference. We see that, for the large ratio of decay rates to the Rabi frequency, no side-mode gain exceeds or equals the strongmode gain, a condition needed for side-mode growth. We also see that the k dependence of the y/ and hence of y(k) leads to decidedly different asymmetries from those predicted by constant ya. This may provide a good test for the more general k-dependent ya theory. For smaller decay rates, excess side-mode gain is readily predicted by the theory, and side-modes could oscillate. For example, Fig. 9 plots al, ainceand a 2 (dotted-dashed curve) versus probe detuning V1 - a, for the same parameters as in Fig. 4, except for smaller decay constants ye = yh = 3 meV Figure 10 plots the same curves, except that the integrals over s were done numerically. We see that, for these values and models, side modes could oscillate quite easily. Comparing ain and a1 , we see that the effects of population pulsations are just as big as the power-broadened

spectral hole burned by the strong mode.

They cause modes within -Yaof the strong mode to be suppressed, while modes farther

similar and nearly symmetric about the p'ump frequency v2 .

, 3

2.9 2.7 2.5 2.3 2.1

1297

away have increased

gain.

In the linear side-mode approximation used here, population pulsations redistribute the gain and do not change the value obtained from integration over v1. The a 2 curves show that the strong mode itself does not perceive the hole that it burns. In the onset of multimode laser operation, population pulsations induced by a weak side mode (at frequency v1 in Fig. 1) and a strong mode (at frequency V2 ) scatter energy between those modes and also between the strong mode and the weak mode (at frequency v3) symmetrically placed on the other side of the strong mode. This second scattering couples the weak modes as described by the coupledmode equations (51) and (52). Figure 11 plots the real parts of the corresponding population-pulsation contributions al h and X1versus probe detuning V1 - sg for the parameters in Fig. 9. The curves are similar and nearly symmetric about the pump frequency v2. Figure 12 plots the real parts of ainc (dashed curve), a 2 (dotted-dashed curve), a1 - xi, a 1, and a 1 + Xi (solid curves in order of increasing dip depth) versus probe detuning V1 - sg in meV for the parameters in Fig. 9. The al ± Xi values are the approximate eigenvalues for dual-side-mode buildup described in Eqs. (51) and (52). In particular, the a1 + X1value shows increased side-mode suppression in the immediate vicinity of V2 and increased dual-side-mode gain outside this region. 0

1.9 I

1.7

45

50

55

60

65

,

| 70

, 75

V - 9

Fig. 12. Real parts of aoh and Xi (in meV versus probedetuning vi - sg (in meV) for the parameters in IPig.9. The curves are similar and nearly symmetric about the ppumpfrequencyV2.

numerically for the average values ye = 12.52 meV and Yh = 3.22 meV and that in Fig. 7 ye(k) and yh(k)are taken from the values in Fig. 3. In all the cases the decay processes are sufficiently fast to limit severely the extent of hole burning and population pulsations. Note that the IHBL overestimates the gain considerably. This is because the large decay rates give Lorentzian linewidths with large tails that sample regions of lesser gain, while the IHBL uses the single value at s = hv1. As Figs. 9 and 10 below show, the IHBL approximation is noticeably better for smaller yVa,as one would expect.

In this connec-

tion the Lorentzian line shapes themselves are an approximation to the true line shapes, which decay more rapidly, so that the true values should lie somewhere between the numerical and IHBL curves. Figures 7 and 8 zoom in on Figs. 5 and 6, respectively, to give a closer look at the effects of hole burning and

population pulsations. In addition, they include the

5.

CONCLUSIONS

The onset of multimode operation in semiconductor laser diodes is determined by a number of factors, such as transverse variations and saturation by the fluorescence of modes below threshold, in addition to the mechanisms dealt with in this paper. What we see here is that, acrelaxation model, cording to a simple carrier-carrier population pulsations play as important a role as spectral hole burning in the side-mode gain and coupling coefficients for intermode beat frequencies comparable with the intraband carrier-carrier scattering rates. By using consistently computed decay constants ye and yh, we incorporate some of the features of a more comprehensive Boltzmann theory. In a unified way the theory also ap-

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J. Opt. Soc. Am. B/Vol. 9, No. 8/August 1992

plies to mode spacings that are small compared with the carrier-carrier relaxation rates, such as those that might be encountered in external mirror lasers. We are able to explain the asymmetric side-mode gain spectra for such beat frequencies in terms of the rapid decrease of the quasi-equilibrium Fermi-Dirac electron distribution just above the gain region. We have neglected the renormalization of the electric dipole interaction energy of the semiconductor Bloch equations, which leads to changes in the gain spectra,'7 ',8 particularly in quantum-well lasers. We have also neglected the saturation of the total carrier density by the spontaneous emission of modes below threshold. As such, the present theory cannot give a precise prediction of the onset of side-mode oscillation in a semiconductor laser, but it is computationally simple and predicts some of the main features of mode coupling in semiconductor lasers without having to fit macroscopic parameters such as the linear gain and the linewidthenhancement factor.

Sargent et al. Sargent III, D. A. Holm, and M. S. Zubairy, Phys. Rev. A 31, 3112 (1985). 2. M. Yamada and Y Suematsu, J. Appl. Phys. 52, 2653 (1981). 3. R. F. Kazarinov, C. H. Henry, and R. A. Logan, J. Appl. Phys. 53, 4631 (1982). 4. W E. Lamb, Jr., Phys. Rev. 134, A1429 (1964).

5. M. Sargent III, M. 0. Scully, and W E. Lamb, Jr., Laser Physics (Addison-Wesley, Reading, Mass., 1977).

6. T. Fu and M. Sargent III, Opt. Lett. 4, 366 (1979). 7. R. W Boyd, M. G. Raymer, P. Narum, and D. Harter, Phys. Rev. A 24, 411 (1981). 8. G. P. Agrawal, J. Opt. Soc. Am. B 5, 147 (1988). This refer-

ence also includes many citations of earlier literature. 9. T. Mukai and T. Saitoh, IEEE J. Quantum Electron. 26, 865 (1990), which includes references to earlier studies of these authors. 10. M. Sargent III, F. Zhou, and S. W Koch, Phys. Rev. A 38, 4673 (1988). 11. F. Zhou, M. Sargent III, and S. W Koch, Phys. Rev. A 41, 463 (1990).

12. P. Meystre and M. Sargent III, Elements of Quantum Optics (Springer-Verlag, Heidelberg, 1990). 13. M. Lindberg and S. W Koch, Phys. Rev. B 38, 3342 (1988).

14. H. Haug and S. W Koch, Quantum Theory of the Optical and

ACKNOWLEDGMENTS This study was supported in part by the U.S. Army Research Office, in part by the U.S. Air Force Office of Scientific Research, in part by the U.S. Office of Naval Research, in part by Sandia National Laboratories, and in

part by the National Science Foundation. We thank R. Binder, K. Henneberger, M. Lindberg, A. Paul, and D. Scott for helpful discussions.

REFERENCES 1. For a brief review of the early theories, see the introductory

section to the quantum multiwave mixing paper: M.

Electronic Properties of Semiconductors(WorldScientific,

Singapore, 1990). 15. R. Binder, D. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W Koch, Phys. Rev. B 45, 1107 (1992). 16. M. Sargent III and P. E. Toschek, Appl. Phys. 11, 107 (1976). 17. W W Chow, S. W Koch, and M. Sargent III, IEEE J. Quan-

tum Electron. 26, 1052 (1990). 18. W W Chow, S. W Koch, M. Sargent III, and C. Ell, Appl. Phys.

Lett. 58, 328 (1991).