quantum noise in vertical-cavity surface-emitting lasers with polarization competition

Bava et al. Vol. 16, No. 11 /November 1999 /J. Opt. Soc. Am. B 2147

Quantum noise in vertical-cavity surface-emittinglasers with polarization competition

Gian Paolo Bava and Laura Fratta

Dipartimento di Elettronica and Istituto Nazionale di Fisica della Materia, Politecnico di Torino,Corso Duca degli Abruzzi 24, 10129 Turin, Italy

Pierluigi Debernardi

Cespa-Consiglio Nazionale delle Ricerche and Istituto Nazionale di Fisica della Materia, Politecnico di Torino,Corso Duca degli Abruzzi 24, 10129 Turin, Italy

Received April 1, 1999; revised manuscript received June 25, 1999

The polarization competition in vertical-cavity surface-emitting lasers with two nearly degenerate modes isanalyzed with the aim of studying their noise properties and, in particular, amplitude fluctuations and spectrallinewidth. The coupling between the two modes is attributed to carriers, different spin populations, andstructural anisotropies, including the consequences of the electro-optic effect that results from current injec-tion. The model is based on quantum equations of motion with Langevin noise sources. Results for the noisespectra in the regions where mode competition is stronger are reported and discussed. © 1999 Optical Societyof America [S0740-3224(99)00511-1]

OCIS codes: 140.5960, 250.7260, 270.2500, 270.6570, 230.5590.

1. INTRODUCTIONVertical-cavity surface-emitting lasers (VCSEL’s) haverecently been shown to be promising devices because theyhave such advantages compared with conventional edge-emitting lasers as high integrability, low threshold cur-rent, large modulation bandwidth, and array features.Another important advantage of VCSEL’s is the almostperfect symmetry of their emitted beams owing to the cir-cular geometry of the device about the axis of the resona-tor. Owing to such transverse symmetry, VCSEL devicesideally support two perfectly degenerate modes that cor-respond to orthogonal polarizations that can lasesimultaneously.1–5 However, in realistic structures thedegeneracy between the two modes is removed by differ-ent mechanisms, that are related, for instance, to slightgeometrical asymmetries, anisotropies in the structure,strain, or optical anisotropies in the mirrors.5 Experi-ments have in fact shown that the polarization of the las-ing modes depends strongly on small changes, such as intemperature and injection current, in the system param-eters.

Various theoretical models have been proposed to ex-plain the observed stability or bistability of the polariza-tion state of light. Polarization switching as the injectioncurrent is increased has been described as a consequenceof gain–loss anisotropy or temperature-related effects6 or,as proposed in Refs. 1 and 3, as a consequence of fre-quency anisotropy owing to birefringence in the opticalcavity. According to a detailed analysis presented in Ref.7, the simultaneous contribution of all three of these ef-fects seem best to describe the mechanism for polariza-tion stability. In fact, when the gain–loss anisotropy andtemperature-related effects are introduced into the SanMiguel model,1 a comparison between the polarization

0740-3224/99/112147-11$15.00 ©

fluctuations predicted by it with the corresponding experi-mental measurements can provide a quantitative evalua-tion of the gain–loss and frequency anisotropy effects andof their dependence on temperature.

In the model presented here, based in part on the SanMiguel description, two independent reservoirs ofelectron–hole pairs with different spin are present in thesemiconductor quantum-well active region, each charac-terized by a carrier density and coupled by spin relaxationprocesses. The competition between the two polarizationmodes is based mainly on the coupling between the spinthat characterizes the electron–hole pair recombinationand the polarization state of the corresponding emittedphoton. The further coupling provided by the anisotro-pies and asymmetries of the system also contributes tothe competition between the modes and is added to themodel in a phenomenological way. The model thereforeconsists of a set of quantum-mechanical equations of mo-tion for the different polarization field operators and forthe carrier density operators with different spin. Eachequation includes a proper Langevin noise term becausethe main purpose of the present research is to study thetwo-polarization behavior of a VCSEL as it relates tonoise performance.

Noise in microcavity semiconductor lasers has beenwidely studied in recent years,8,9 mainly with reference tosingle-mode operation. Peculiar characteristics havebeen found, such as the possibility of generating lightwith sub-shot-intensity noise when the laser is quietlypumped. Moreover, the noise behavior of multimodesemiconductor lasers has also been extensively investi-gated by several authors10,11 but in simple treatments inwhich the different modes are coupled only through thetotal carrier density (which influences gain and spontane-

1999 Optical Society of America

2148 J. Opt. Soc. Am. B/Vol. 16, No. 11 /November 1999 Bava et al.

ous emission). A simple model12 was recently developedto estimate the noise performances in the specific case ofVCSEL’s, in which the competition between the two po-larized modes plays a fundamental role; here the twomodes are considered to be coupled also by cross-saturation effects, but the fundamental coupling mecha-nism described in the San Miguel model is not taken intoaccount. More-refined models in which this couplingmechanism is included were presented in Refs. 7 and 13;however, not all the noise sources were taken into accountand only specific working conditions that greatly simplifythe problem were considered.

In this paper we study the noise properties of a two-polarization VCSEL device, including all the noisesources starting from the microscopic quantum equationsof the system. In particular, we are concerned with theevaluation of the output photon number fluctuation spec-tra and of the spectral linewidth.

In Section 2 we introduce the Langevin equations ofmotion of the model for the field and carrier operators.On the basis of Ref. 14, only the difference in theconduction-band spin populations is included, because thevalence-band spin relaxation is much faster. The opticalgain and the spontaneous emission, both global and in thelasing modes, are computed self-consistently: For the de-vice that we consider, an airpost laser with a multiple-quantum-well active region, a complete model is pre-sented in Ref. 15. The effects of small geometricalasymmetries and anisotropies are introduced partly phe-nomenologically and partly by inclusion of the depen-dence of the mirror’s optical response on the current in-jection (electro-optical effect).

In Section 3 the stable operating conditions of the de-vice are determined and discussed; a map of the stablestationary solutions of the system equations in the planedefined by pump and birefringence is presented.

In Section 4 the coupling between internal and externalfields through the mirrors is studied. In this study weevaluate the contributions of internal noise, externalvacuum field fluctuations, and their relative interferenceto the output noise of the device. In particular, the noisebehavior is analyzed in terms of photon number fluctua-tions and of spectral linewidth of the emitted light. Nu-merical results that show how the behavior of the outputspectra varies as the stable operating conditions arechanged are reported.

In Appendix A the diffusion coefficients that are rela-tive to the Langevin noise sources that appear in themodel equations are computed, starting from the funda-mental diffusion coefficients known from literature.

2. MODELThe model adopted in this paper deals with the interac-tion between a semiconductor multiple quantum well andthe two almost degenerate optical modes in a VCSEL; tothis purpose, to simplify the computations, only the low-est conduction band and the highest heavy-hole valenceband are assumed to contribute to the laser process, andthe carrier conservation of momentum in the electron–hole transitions is assumed.

The system is described by several operators: for thefields al , for the electron–hole polarization per unit vol-ume ph , and for the conduction- and valence-band carrierdensity nc,vh ; the last two quantities are defined in theactive volume Va . The subscript l (51, 2) refers to thetwo almost degenerate cavity modes, and the subscripth 5 ↑, ↓ refers to the spin state of the carriers. The op-erators nc,vh and ph depend implicitly on k, where thespin dependence is explicitly indicated because it plays amajor role in this model.1

The carrier–field interaction coefficient is defined by

glh 5 ~\vcv2 /4v l!

1/2El • dh , (1)

where El is the electric field of the l cavity mode with an-gular frequency v l normalized to the unit stored energy,\vcv is the transition energy between conduction and va-lence bands for a given k-vector, and dh is the correspond-ing dipole matrix element for each spin state h : dh

5 Rk( x 1 ish y), where Rk is the elementary dipole ma-trix element16 and x and y the two unit vectors in the de-vice’s transverse cross section. sh has the meaning sh

5 11 when h 5 ↑ and sh 5 21 when h 5 ↓.The Heisenberg equations of motion for the system op-

erators are deduced in a standard way; by also includingphenomenological damping terms and proper Langevinnoise sources (L),17 one obtains

dal

dt5 2iv lal 2 ~i/\! E

Va

dV (k,h

glh* ph1 2 1/2g llal

2 1/2g lmam 1 Lal,

dph

dt5 ivcvph 2 ~i/\! (

lal

1glh* ~nch 2 nvh! 2 gpph

1 Lph,

dnch

dt5 2~i/\! (

l~ glh phal 2 glh* al

1ph1! 1 Lc 2 Rch

1 gc~ fch 2 nch! 1 gcs~nch82 nch! 1 Lnch

,

dnvh

dt5 1~i/\! (

l~ glh ph al 2 glh* al

1ph1! 1 Lv 2 Rvh

1 gv~ fvh 2 nvh! 1 gvs~nvh82 nvh! 1 Lnvh

.

(2)

In Eqs. (2) g lm are the elements of a 2 3 2 matrix g5 gu 1 g i , which includes the effect of dissipative pro-cesses both through the output coupling mirror (gu ) andthrough other losses (g i ) inside the resonator and causedby the back mirror; the other nondissipative effects are al-ready accounted for by the different mode frequencies v l .In the equations for carriers, the terms Lc,v are the pumpdistributions and Rc,vh are the nonradiative and sponta-neous recombinations in all the nonlasing modes. Thefastest relaxation time of the system is the one for the po-larization (gp ); of the same order of magnitude are therelaxation processes that are due to intraband scattering(gc,v ) that bring electrons and holes to quasi-equilibrium


distributions fc,vh for each spin configuration. FollowingRef. 1, direct interaction (gc,vs ) between populations withdifferent spins h8 and h is also included. With respect tothese spin relaxation coefficients, it has been observed14

that the corresponding time constants are much biggerthan those of polarization and carriers (;2 orders of mag-nitude, that is, 1/gcs of the order of 1–10 ps) but at thesame time much smaller than the carrier recombinationrate. It also turns out that gvs @ gcs , which allows us tointroduce a strong simplification by setting gvs 5 `, so

nv↑ 5 nv↓ 5 nv . (3)

The spin populations in the conduction band can insteadbe linearized about their average value because they dif-fer only slightly, owing to the large spin relaxation con-stant:

nch 5 nc 1 sh ndnc

dN, (4)

where the overbar stands for the expectation value,

nc 5 1/2~nc↑ 1 nc↓! (5)

is the average carrier distribution,

N 5 EVa

dV (k

~nc↑ 1 nc↓! (6)

is the operator for the total number of carriers, and

n 5 EVa

dV (k

~nc↑ 2 nc↓! (7)

is the operator for the total spin population difference.In our model we consider charge neutrality, so the totalnumber of holes is equal to the total number of electrons.

To simplify the treatment, in what follows, we assumethat the carriers are uniformly distributed within the ac-tive volume, as a consequence of carrier diffusion.

All the processes are assumed to be Markovian, so ingeneral

^L~t !L1~s !& 5 2Dd ~t 2 s !, (8)

where 2D is the diffusion coefficient. It turns out18,19

that

2Dam1al

5 gml* nth ,

2Damal1 5 gml~1 1 nth!,

2Dph ph1 5 2gpnch~1 2 nv! 5 X 1 sh nx,

2Dph1ph

5 2gpnv~1 2 nch! 5 Y 1 sh ny,

2Dnc,vh nc,vh5 Lc,v 1 Rc,vh ,

2Dnc,vh nc,vh85 gc,vs (

hnc,vh~1 2 nc,vh!, (9)

where nth 5 @exp(\v /KT) 2 1#21 is the number of ther-mal photons, which is negligible at optical frequencies;moreover, the assumption that the mean carrier densitync,v follows quasi-equilibrium Fermi–Dirac distributionsfc,v allows us to introduce the following symbols, whichgreatly simplify the notation:

X 5 2gp~1 2 fv!fc ,

Y 5 2gp~1 2 fc!fv ,

x 5 2gp~1 2 fv!d fc

d N,

y 5 22gpfv

d fc

d N. (10)

It is convenient to introduce slowly varying quantities byassuming a common angular frequency of reference, v:

am 5 Am exp~2ivt !,

ph 5 Ph exp~ivt ! (11)

and adiabatically eliminate the polarization:

Ph 51

\(m

gmh*

Dk~nch 2 nv!Am

1 1 iLph

Dkexp~2ivt !,

(12)

where

Dk 5 ~vcv 2 v! 1 igp . (13)

When inserting Eq. (12) into Eqs. (2) and integratingthe field equations over Va , we encounter fundamentalquantities of the type

EVa

dVgmh* glh

5 Mml2 E

Va

dV~Emx 2 sh iEmy!~Elx 1 sh iEly!

5 Mml2 ~Cml 1 sh iCml!, (14)

where

Mml2 5

\Rk2vcv

2

4«aAvmv l

.\Rk

2vcv2

4«av5 M2, (15)

«a is the active region permittivity, and we have assumedthat v . vm . v l because the proper modes are nearlydegenerate. In Eq. (14) the quantities

Cml 5 EVa

«a~EmxElx 1 EmyEly!dV,

Cml 5 EVa

«a~EmxEly 2 EmyElx!dV. (16)

are confinement factors for the electric field in a general-ized sense. By assuming that the two ideally degeneratemodes in a cylindrically symmetrical structure can be su-perimposed by a 90° rotation about the z axis, we easily

find that C11 5 C22 5 C12 5 2C21 5 C and C12

5 C215 C11 5 C22 5 0.To describe the behavior of the macroscopic system it is

convenient to consider a set of equations for the macro-scopic operators A1 , A2 , N, and n obtained by summingover k system (2); taking into account the preceding con-siderations, we obtain


dA1

dt5 @i~v 2 v1!

1 1/2~G 2 g11!#A1 2 1/2~g12 2 ign !A2 1 LA1,

dA2

dt5 @i~v 2 v2!

1 1/2~G 2 g22!#A2 2 1/2~g21 1 ign !A1 1 LA2,

dN

dt5

I

q2 R 2 Gr~A1

1A1 1 A21A2!

2 igrn~A11A2 2 A2

1A1! 1 LN ,

dn

dt5 2gnn 2 grn~A1

1A1 1 A21A2!

2 iGr~A11A2 2 A2

1A1! 1 Ln , (17)

where I is the injected useful current, R is the recombina-tion rate, which also includes the spontaneous emission inthe two modes considered, and gn 5 dR/dN 1 2gcs is then relaxation constant. Moreover, we have introduced thecomplex intensity gain:

G 5 Gr 1 iGi 5 24i (k

M2C

\2Dk*~ fc 2 fv! (18)

and the quantity gn, which introduces a coupling betweenthe modes connected to the spin population difference; gis given by

g 5 gr 1 igi 5 24i (k

M2C

\2Dk*

dfc

dN. (19)

The macroscopic Langevin noise forces in Eqs. (17) are

obtained in a self-consistent way from the microscopicquantum system [Eqs. (2)], and their expressions are de-rived in Appendix A. In this way we can properly ac-count for the effect of the noise introduced by the polar-ization ( ph) and n; for this reason we have to begin froma microscopic description to study the macroscopic sys-tem.

3. STATIONARY OPERATING CONDITIONSThe solution of Eqs. (17) is in general complicated; twosimpler limit situations are one in which only one mode

lases and the other mode is only noise (g12 5 0) and onein which the two modes are frequency locked. Here weconsider the second situation, a selection that is justifiedbecause the two modes are nearly degenerate (their fre-quency spacing is of the order of a few gigahertz accordingto experimental measurements) and characterized bysimilar decay constants. Moreover, the regime of single-frequency operation is important for several applications.

We obtain the stationary solution of Eqs. (17) by ne-glecting the noise sources and setting d/dt 5 0. It is con-venient to split the modulus and the phase of the fields byintroducing the following approximation:

A 5 S A1

A2D 5 Fexp~if1 1 iDf1!

0

0

exp~if2 1 iDf2!G

3 FM1 1 DM1

M2 1 DM2G

5 exp@i~fD 1 DfD!#~M 1 DM!, (20)

where fD is a diagonal matrix of elements f1 and f2 .Inasmuch as one of the two mode phases can always bearbitrarily set to zero, under stationary conditions onlythe phase difference w 5 f2 2 f1 has to be determined.Henceforth we shall therefore deal with a reduced prob-lem of five real equations and introduce the vectorialquantity of expectation values a 0

T 5 (M1 , M2 , w, N, n).The operator that corresponds to a0 is

a 5 a0 1 Da. (21)

For the expectation values a0 , the following equationholds:

da0

dt5 L~a0! 5 0, (22)

with

L~a0! 5 S ~L 1 d!M1 2 @~gr 1 1/2 gin !cos w 1 ~g i 2 1/2 grn !sin w#M2

~L 2 d!M2 2 @~gr 2 1/2 gin !cos w 1 ~g i 2 1/2 grn !sin w#M1

S 1 @~M2 /M1! 1 ~M1 /M2!#~g i 2 1/2 grn !cos w 1 $@~M2 /M1! 1 ~M1 /M2!#gr 1 @~M2 /M1! 2 ~M1 /M2!#1/2 gin%sin w

I

q2 R 2 Gr~M1

2 1 M22! 1 2grnM1M2 sin w

2gnn 2 grn~M12 1 M2

2! 1 2GrM1M2 sin w

D ,

(23)

where d 5 1/4(g22 2 g11), L 5 1/2(Gr 2 g), and g5 1/2(g22 1 g11); we also set g12 5 gr 1 ig i 5 g21* , andS 5 v1 2 v2 is related to birefringence. Moreover, bothof the gains, G and g, and R depend on carrier number N.

Some examples of numerical results for the stable op-erating conditions are now presented. We refer to a poststructure (2-mm diameter) that consists of anAl0.18Ga0.82As l cavity with a 3-GaAs quantum well (each7 nm thick) active region. The operating wavelength is;0.87 mm. Except for the transverse rod geometry, thestructure is the same as that considered in Ref. 5. The


bottom Bragg mirror consists of 25.5Al0.18Ga0.82As/AlAspairs, and the upper one contains 20 pairs, the outputcoupling loss is 62.7 ns21, and the overall losses are g5 95 ns21. To solve Eq. (23) we have linearized all thequantities near Nth defined by Gr 5 g; unlike in othermodels, the dependence of g on N has also been accountedfor.

An important constituent in the model for the anisotro-pies in VCSEL’s where current flows in the mirror-dopedlayers is the electro-optic effect in the structure, as waspointed out by van Exter et al.5 This fact introduces ageneral dependence on the injected current of the mirrors’optical response. It turns out that this kind of anisotropyinduces a preference in the polarization of the outputlight in the crystallographic direction @110# that in ournotation corresponds to mode 1. We have taken this ef-fect into account by introducing a dependence on thepump current of the system parameters. In particular,the birefringence

F 5S

2p5 F0 2 0.067P @GHz# (24)

is split into a fixed contribution F0 , which can account forother structural effects and in what follows will be consid-ered a parameter, and a term that is proportional to thenormalized pump level P 5 I/Ith 2 1. We obtained thecoefficient of P in Eq. (24) directly by using Eqs. (8), and(13) of Ref. 5 and relating the voltage drop in the mirrorsto P. In a similar way, the loss difference is expressed as

d 5 ~3.5 1 1.2P !1023 @ns21#, (25)

where we have evaluated the coefficient of P by includingthe P depencence of the refractive indices of the Bragg re-flectors in the computation of the mirror losses; the con-stant value accounts for the threshold voltage. Thequantity d turns out to be of the same order of the corre-sponding parameter in the model of Ref. 4; the couplingconstant between the two resonant modes is assumed realand with gr 5 0.1 ns21.

The term R for the carriers includes spontaneous re-combinations, computed self-consistently in Ref. 15, and anonradiative term with a relaxation constant of 0.75 ns21.The spin relaxation constant is gcs 5 37 ns21, followingRefs. 4 and 14, and consequently the n relaxation con-stant is gn 5 75 ns21. The intraband polarization relax-ation parameter gp is assumed to be 2 3 104 ns21.

In Fig. 1 the stability regions in the plane P –F0 areshown. The result is similar to the analogous results re-ported in Ref. 4. The output signal is elliptically polar-ized, with low ellipticity (nearly linear polarization) al-most everywhere; this fact is in agreement with somerecent experimental measurements, and in the presentmodel it is connected to the introduction of the parameterg12 , which does not appear in other models. In what fol-lows, we use the dominant component to characterize themode, so, when speaking of mode 1, we mean an ellipticalmode with a dominant component M1 , and similarly forM2 . In the white regions of Fig. 1 mode 1 is stable, andmode 2 is stable in the black areas; at the border betweenthe two regions (near F0 5 0), M1 . M2 , and crossingthis line there is a smooth transition between mode 1 and

mode 2. This smooth transition is related to the param-eter g12 , which in other papers is related to a misalign-ment between the birefringence and the dichroism axes;this phenomenon was also found in the research reportedin Ref. 4 where this misalignment was introduced.

The characteristics of the output polarization for threevalues of F0 are shown in Fig. 2, where the ellipse axisratio and the tilting angle are shown related to P; one canobserve that the ellipse rotates 90° and that near a tiltingangle of 45° the axis ratio tends to infinity, which corre-sponds to linear polarization. In this operating conditionit turns out that w 5 F 5 0, as can be obtained from Eq.(26).

In the hatched areas of Fig. 1 both mode 1 and 2 coexist(bistable operation). The thin shaded areas at the bor-ders of the hatched regions correspond to higher-ellipticity solutions (mode 1 dominates on the left, mode 2on the right); these solutions correspond to the elliptically

Fig. 1. Stability regions of two-mode operation in the F0 –Pplane. White region, mode 1; black areas, mode 2; hatched ar-eas, both modes stable. At the borders of the hatched regionsthe small shaded areas correspond to modes with higher elliptic-ity.

Fig. 2. Characteristics of the elliptical polarization (tiltingangle and axis ratio) in the smooth transition region betweenmodes 1 and 2 versus P for three values of F0 : continuouscurves; F0 5 0.08 GHz; dotted curves, F0 5 0.3 GHz; dashedcurves, F0 5 0.5 GHz.


polarized solution that is also present in the San Miguelmodel.3 Notice that mode 1 (with lower losses) is alwayspresent up to a certain pump level; mode 2 appears forlow values of the pump level owing to the low-loss differ-ence, and therefore a pump logarithmic scale is adopted.

For comparison, in Fig. 3 we present a result that isanalogous to Fig. 1 but for which the dependence of bire-fringence and loss difference on pump level is neglected.Notice that the most important difference is related to thedependence of birefringence on the pump level, which inFig. 1 induces a clockwise rotation of the stability regionswith respect to Fig. 3.

4. OUTPUT COUPLING AND NOISESPECTRAThe Hamiltonian that describes the interaction betweenthe fields internal to the resonator (A) and the externalfields (A0) is of the form17

H 5 i\~A1KA0 2 A01K* TA!, (26)

where A0 is a two-component vector that is related to thetwo-mode propagation in the waveguide (for instance, anoptical fiber) coupled to the laser modes under analysisand K is the 2 3 2 matrix of the output mirror’s couplingcoefficients. Following standard procedure,17 one canshow that the elements of the K matrix are related to theelements of the gu matrix, which describes the outputlosses, by the relation

KK* T 5 gu 5 Fg11

g21

g12

g22G

u, (27)

from which we read that g21u* 5 g12u

.By direct inspection of the Hamiltonian [Eq. (26)], we

can obtain the relation between the internal fields andthe externally radiated fields s, defined such that F5 s1s is the output photon flux. The exchange of en-ergy is assumed to consist of the simultaneous creation ofa quantum in the internal system and the correspondingannihilation of a quantum in the external system and viceversa. Therefore the complex amplitude of the output ra-

Fig. 3. Same as Fig. 1, but here the electro-optic effect has beenneglected.

diated field can be directly deduced from the second termof H; if also the reflected vacuum field fluctuations19 aretaken into account, the complex amplitude reads as

s 5 S s1

s2D 5 K* TA 2 AR0fe , (28)

where R0 is the mirror reflectivity coefficient as seen fromoutside the cavity and fe is the vacuum field fluctuationvector.

Both the spectral line and the fluctuations of the outputphoton flux DF about the stationary state F0 are relatedthrough Eq. (28) to the internal field fluctuations. Weobtain the dynamics of the internal fluctuations Da by lin-earizing system (25) about its stationary solution a0s:

d~Da!

dt5 JDa 1 f, (29)

where J is the Jacobian of L evaluated in the stationaryvalue a0s and f is the vector of the Langevin noisesources: f T 5 (LM1

, LM2, Lw , LN , Ln), where

LMm5 1/2@LAm

exp~2ifm! 1 exp~1ifm!LAm1#,

Lfm5 ~1/2iMm!@LAm

exp~2ifm! 2 exp~1ifm!LAm1#,

Lw 5 Lf22 Lf1

(30)

and we assume quiet pumping operation.8

It is crucial to put into evidence in Eq. (29) the contri-bution to the noise that is due to the incoming vacuumfield fluctuations from the output mirror to account forthe interference at the output between the external fluc-tuations reflected from the mirror and the noise emergingfrom the laser. To this purpose the Langevin noisesources of Eqs. (17) have been split into two terms, as fol-lows:

LA 5 LA8 1 Kfe , (31)

and, as a consequence, the noise vector of system (29) be-comes f 5 fi 1 Kife 1 Ki* f e

1T . Ki 5 QK follows directlyfrom K when the change from the complex amplitude tothe modulus and phase variables is performed and theconstant of proportionality Q is defined as

QT 51

2exp~2iw1!

3 F 1

0

0

exp~2iw!

i1

M1

2iexp~2iw!

M2

0

0

0

0G . (32)

In Fourier space (V), differential equations (29) becomean algebraic system that can be formally inverted to pro-vide the following expression for the fluctuation vector:

Da~V! 5 L~V!f~V!, (33)

where

L 5 ~iVD 2 J !21; (34)


VD is a 5 3 5 diagonal matrix of elements V. The corre-lation relations of the noise sources in the frequency do-main are defined as

^f~V!f 1~V8!& 5 2p2Dd ~V 2 V8!,

^fe~V!fe1~V8! 5 2pId ~V 2 V8!,

^fe1~V!fe~V8!& 5 ^fe~V!fe~V8!&

5 ^f e1~V!f e

1~V8!& 5 0, (35)

where 2D is the diffusion coefficient matrix (derived inAppendix A) and I is the 2 3 2 unit matrix.

To evaluate the photon flux spectrum it is convenient tointroduce the first-order approximation of Eq. (20) for A:

A . exp~ifD!M 1 exp~ifD!DM 1 i exp~ifD!MDDf,(36)

where DM and Df are the modulus and phase fluctuationvectors. Inserting expression (36) into Eq. (28) and ne-glecting the second-order noise terms yields the followingexpression for the photon flux fluctuations DF about thestationary state F0 5 MTGM:

DF 5 MTGDM 1 DM1GM 1 iMT~GDf 2 Df1G!M

2 AR0@MT exp~2ifD!Kfe 1 f eTK* T exp~ifD!M#,

(37)

where G 5 exp(2ifD)gu exp(ifD).Equation (37) is greatly simplified if the following prop-

erties are taken into account:

MTGDM 1 DM1GM 5 2MT Re~G!DM,

MT~GDf 2 Df1G!M 5 M1M2~G12 2 G21!Dw, (38)

and the photon flux fluctuations read simply as

DF 5 RDm 2 AR0~S* Tf e 1 f e1S!, (39)

where R 5 @2MT Re(G),22M1M2 Im(G12)# is a 1 3 3 linevector, S 5 K* T exp(ifD)M is a 1 3 2 column vector, andDm is the fluctuation vector Da defined in Eq. (33), whichwe have reduced to its first three components by takinginto account only the first three rows of the matrix L (L8).

The photon flux spectrum SDF , defined by

2pd ~V 2 V8!SDF~V! 5 ^DF~V!DF1~V8!&, (40)

is obtained from expressions (39), (35), and (33) and isgiven by

SDF~V! 5 RL82DL8* TRT 1 R0MTGM

2 2AR0R Re@L8Q exp~ifD!G#M. (41)

It is interesting that, both in SDF and in the system equa-tions, the mirror coupling coefficients K appear in theform KK* T, so the four parameters g lm are actually theonly relevant parameters for the system. In the threeterms of Eq. (41) we can distinguish the contributions tothe total noise spectrum that are due to the noise emerg-ing from the device, to the noise reflected from the outputmirror, and to the interference between the externalvacuum field fluctuations and the internal fluctuationsthat are caused by outside fluctuations. The last contri-

bution is negative, and in the case of suppressed pumpnoise it allows us to obtain amplitude squeezing. Thisfact is well known in single-mode operation19; in thepresent case the result depends strongly on the polariza-tion competition, as will now be discussed in the presen-tation of the numerical results.

In what follows, we present the results for noise char-acteristics in the transition regions between the twomodes where the competition is stronger and the proper-ties are more interesting and different from those insingle-mode operation. We refer to the normalized pho-ton flux spectrum (NPFS), defined as

NPFS 5SDF~0 !

F0. (42)

In Fig. 4 the dependence of the NPFS on P for three val-ues of F0 is shown. The curves refer to the same situa-tions already examined in Fig. 2 for the static operation;the only new parameters are gu 5 62 ns21 and du , whichare defined by analogy to g and d but refer to the outputmirror. We have computed du by taking into account theelectro-optic effect in the output mirror structure; it has apump dependence of the type of Eq. (25) that is the sum ofdu and db induced by the back mirror. As regards g12 ,two cases have been considered in which it is due only tothe output mirror or only to the back mirror.

Far from the transition regions the NPFS tends to thevalue of single mode operation, because in fact the polar-ization is almost linear, as defined by the dominant mode.In Fig. 4 the single-mode characteristic is related to thelower envelope of the curves, with an asymptotic limit atincreasing P of 1 2 g iiu /g ii . From the figure it appearsthat, with increasing P, the NPFS can achieve values lessthan 1, that is, sub-shot-noise operation. In the point ofthe transition region where the polarization is linear,NPFS assumes the single-mode value; between these twoconditions (linear and nearly linear polarizations) themode competition and the noise correlations give an in-creased value of NPFS that in this way presents twomaxima adjacent to the center of the transition region.

The behavior of NPFS versus F0 for two values of P isshown in Fig. 5; the upper curves refer to the lower P

Fig. 4. NPFS versus P for three values of F0 ; the values of F0and the types of curve are the same as in Fig. 2.


value, and only the modes that show a smooth transitionare considered. In this case the influence of the off-diagonal terms of gu is illustrated by two examples; thecontinuous curves refer to a gr that is due only to the backmirror, whereas for the dashed curves gr is attributedcompletely to the output coupling mirror. From the pointof view of the fields inside the cavity the two situationsare identical; however, in the latter case there is one morecoupling mechanism that influences the external fields,yielding greater noise reduction.

With respect to the output spectral line, because we aredealing with a two-component output signal it must befirst defined. We can, for example, measure the spec-trum I(V) of the signal s by using a beat with an ellip-tically polarized probe signal sP

T 5 (sP1 , sP2) of afrequency that is close to the laser operating frequency.Inasmuch as the amplitude of the beat signal, Ab5 sP1s1 1 sP2s2 , depends on the complex amplitudesof the polarization components of the two signals, themeasurement of the spectral line is strongly affected bythe particular choice of probe parameters sP1 and sP2 ofthe experimental apparatus.

The intensity spectral distribution I(V), defined as

I~V! 5 E dt exp~2iVt !^Ab~t !Ab1~0 !&, (43)

in the matrix formalism of this model can be expressed as

I~V! 5 E dt exp~2iVt !(i, j

^~sb~t !sb1~0 !!i, j&, (44)

where, using Eqs. (28) and (20), we set matrix sb to besb 5 PDK* TMDFD exp(iw)c [ M exp(iw)c , where PD ,MD , and FD are three diagonal matrices of diagonal ele-ments sP , M, and exp(f1) and exp(f2), respectively;exp(iw)c is the column vector of the phase noise contribu-tions. Obviously, matrices (21) and (23), and conse-quently vector f and matrix (34), must be generalizedsuch that the phases are treated separately. In definingM we have as usual assumed that far enough abovethreshold the fluctuations of the field intensities are neg-

Fig. 5. NPFS versus F0 for two values of P. Continuouscurves, gr is due completely to the back mirror; dashed curves, gris due completely to the output mirror.

ligible with respect to phase fluctuations. Moreover, wehave verified that the reflected external field fluctuationsthat appear in Eq. (28) make a constant contribution tothe spectrum and are therefore neglected in the linewidthevaluation.

Inserting the definition of sb into Eq. (44), we can writethe spectral line in the compact form

I~V! 5 (i, j

MCij~V!M, (45)

where only the matrix elements of C remain to be evalu-ated:

Cij~V! 5 E dt exp~2iVt !^exp$i@f i~t ! 2 f j~0 !#%&. (46)

Considering that we are dealing with Gaussian noise pro-cesses, we can write expression (46) in the equivalentform17:

Cij~V! 5 E dt exp~2iVt !exp$21/2^@f i~t ! 2 f j~0 !#2&%.

(47)

In a first approximation the mean value at the exponentis a linear function of time:

^@f i~t ! 2 f j~0 !#2&

51

2pE @^f i~v!f i* ~v!& 1 ^f j~v!f j* ~v!&

2 2 Re~^f i~v!f j* ~v!&exp~ivt !!]dv

. a ijt, (48)

so Cij(V) functions are Lorentzian curves of widths a ij ,respectively. The terms ^f i(v)f j* (v) are deduced di-rectly from generalized equation (33):

^f i~v!f i* ~v!& 5 ~Lf9 2DfLf9 !ij , (49)

where the subscript f indicates that we are dealing withmatrices L and D in which the individual phase contribu-tions are accounted for separately; the two primes indi-cate that we consider only the third and fourth rows of L.

Fig. 6. Spectral linewidth DfPout versus P for three values ofF0 ; the values of F0 and the types of curve are the same as inFig. 2.


The dependence on P of spectral linewidth Df is shownin Fig. 6 for the same conditions examined in Fig. 4; it isgiven, as usual, as the product DfPout . To make thiscomputation we assumed a local oscillator linearly polar-ized along the direction of the main axis of the output sig-nal polarization ellipse. The same general consider-ations already introduced for the preceding figures alsoapply to the spectral line, and obviously the horizontallimit corresponds to single-mode operation.

5. CONCLUSIONSThis study has been concerned with the evaluation of thephoton flux spectrum and of the spectral linewidth of thelight emitted by a VCSEL. The model adopted is basedon a set of quantum Langevin equations in which the twonearly degenerate modes are coupled by the carrier den-sity, by the effects of the different spin populations, andby the anisotropies of the system including the electro-optic effect in the mirrors. The effect of the two-modecompetition on the properties of the device noise has beeninvestigated. As a result, it has been found that the re-gions near the transition from one mode to the other,where the amplitudes of the two modes inside the cavitybecome comparable, are critical for the noise behavior:Both the amplitude noise and the spectral linewidth tendto grow above the single-mode operation value, exceptonly at the exact transition point in which the polariza-tion is linear and the single-mode operation is thereforerecovered. Away from the transition regions the spectralbehavior is very similar to that of single-mode lasers and,for values of the pump current high above threshold,below-shot-noise operation is still achieved.

APPENDIX ABecause of the change of variables [Eqs. (11)] and of theadiabatic elimination of the polarization [Eq. (12)], theLangevin noise terms in Eqs. (17) have the following ex-pressions:

LAm5 Lam

exp~ivt ! 21

\E

Va

dV (kh

gmh*

Dk*Lph

1 exp~ivt !,

LN 5 EVa

dV (k

~Lnc↑8 1 Lnc↓

8 !,

Ln 5 EVa

dV (k

~Lnc↑8 2 Lnc↓

8 !, (A1)

where

Lnch8 5 Lnch

11

\ FLphexp~2ivt !(

m

gmh

DkAm

1 Lph

1 exp~ivt !(m

gmh*

Dk*Am

1G (A2)

is the Langevin term for nc after the elimination of Ph .

In the evaluation of the output noise spectrum an im-portant parameter is the diffusion coefficient matrix forthe quantities inside the resonator; in an explicit form itreads as

D 5 F DM1M1

DM1M2*

DM1w*

DM1N*

DM1n*

DM1M2

DM2M2

DM2w*

DM2N*

DM2N*

DM1w

DM2w

Dww

DwN*

Dwn*

DM1N

DM2N

DwN

DNN

DNn*

DM1n

DM2n

Dwn

DNn

Dnn

G .

(A3)

Only the diagonal elements, which are the diffusion coef-ficients of the different variables with themselves, arereal and positive; all the other elements are in generalcomplex. It appears that the elements of the matrix canbe classified into three categories: diffusion coefficientsbetween quantities related to fields, diffusion coefficientsbetween quantities related to fields and carriers, and dif-fusion coefficients between quantities related to carriers.

All the diffusion coefficients related to fields can be de-rived from

DMmMl5 1/4@DAmAl

1 exp~iw lm! 1 DAm1Al

exp~2iw lm!#,

DMmfl5

1

4iMl@2DAmAl

1 exp~iw lm!

1 DAm1Al

exp~2iw lm!#,

DMmw 5 DMmw22 DMmw1

,

Dww 5DM1M1

M12 1

DM2M2

M22 2

DM1M2

M1M22

DM2M1

M1M2, (A4)

where w lm 5 f l 2 fm and the fundamental blocks are

2DAm1Al

5 2Dam1a1

1 EVa

dV (kh

gmh glh*

\2uDku2 2Dph ph1

5 2Dam1al

1 (kh

M2

\2uDku2 @~XCml 2 inxCml!#,

2DAmAl1 5 2Damal

1 1 EVa

dV (kh

gmh* glh

\2uDku2 2Dph1 ph

5 2Damal1 1 (

kh

M2

\2uDku2 @~YCml 1 inyCml!#.

(A5)

If we define the spontaneous-emission rate in a lasingmode as

S 5 (kh

2gpM2C

\2uDku2 ~1 2 fv!fc (A6)

and the corresponding contribution that is due to spin dif-ference (as for G and g) as

s 5 (kh

2gpM2C

\2uDku2 ~1 2 fv!d fc

dN, (A7)


it is easily verified that

(kh

M2C

\2uDku2 ~X 1 Y ! 5 2S 2 Gr ,

(kh

M2C

\2uDku2 ~X 2 Y ! 5 Gr ,

(kh

M2C

\2uDku2 ~x 1 y ! 5 2s 2 gr ,

(kh

M2C

\2uDku2 ~x 2 y ! 5 gr . (A8)

So we finally obtain

2DMmMl5 1/4$gml exp~iw lm!

1 cos w lm@Cml~2S 2 Gr! 2 ingrCml#

1 i sin w lm@2GrCml 1 in~2s 2 gr!Cml#%,

2DMmfl5

1

4iMl$2gml exp~iw lm!

1 cos w lm@CmlGr 2 in~2s 2 gr!Cml#

1 i sin w lm@2~2S 2 Gr!Cml 1 ingrCml#%.

(A9)

All the diffusion coefficients between fields and carrierscan be obtained from

^@LAmexp~2ifm! 6 exp~ifm!LAm

1#~Lnc↑ 6 Lnc↓!&,(A10)

where the basic diffusion coefficients are

2DAmnchexp~2ifm!

5 2(l

gmh* glh Ml exp~iw lm!(k

2Dph1 ph

\2uDku2 ,

2DAm1nch

exp~ifm!

5 2(l

gmh glh* Ml exp~2iw lm!(k

2Dph ph1

\2uDku2 .

(A11)

Proceeding similarly as before, we obtain

2DMlN5 21/2$CMl~2S 2 Gr! 2 nMmClm@igr cos wml

1 ~2s 2 gr!sin wml#%,

2DMln5 21/2$2ClmMm@~2S 2 Gr!sin wml

1 iGr cos wml# 1 nCMl~2s 2 gr!%,

2DflN5 2~1/2iMl!$2GrCMl 1 nClmMm@gr sin wml

1 i~2s 2 gr!cos wml#%,

2Dfln5 2~1/2iMl!$ClmMm@Gr sin wml

1 i~2S 2 Gr!cos wml# 2 nCMlgr%. (A12)

The remaining diffusion coefficients are related to car-riers:

DNN 5 EVa

dV (k

~Dnc↑nc↑8 1 Dnc↓nc↓

8 1 2Dnc↑nc↓8 ,

Dnn 5 EVa

dV (k


8 2 2Dnc↑nc↓8 !,

DNn 5 EVa

dV (k


8 !. (A13)

The basic blocks in this case are

Dnch nch8 5 Dnch nch

1 ~Dph1 ph

1 Dph ph1!

3 (ml

gmh glh*

\2uDku2MmMl exp~iwml!,

Dnch nch88 5 Dnch nch8

, (A14)

so, finally,

2DNN 5 R 1 C~2S 2 Gr!~M12 1 M2

2!

2 2nCM1M2~2s 2 gr!sin w

1 2gcsn2 (

kS dfc

dND 2

,

2Dnn 5 R 1 C~2S 2 Gr!~M12 1 M2

2!

2 2nCM1M2~2s 2 gr!sin w

1 2gcs (k

~2fc 2 f c2!,

2DNn 5dRdN

n 1 nC~2s 2 gr!~M12 1 M2

2!

2 2CM1M2~2S 2 Gr!sin w. (A15)

ACKNOWLEDGMENTSThis study was carried out within the framework of theEuropean Strategic Programme for R & D in InformationTechnology Long-Term Research project entitled Ad-vanced Quantum Information Research and with the fi-nancial support of Progetto Finalizzato Materiali e Dis-positivi per l’Elettronica dello Stato Solido II by theItalian Consiglio Nazionale delle Richerche.

P. Debernardi’s e-mail address is [email protected].

REFERENCES1. M. San Miguel, Q. Feng, and J. V. Maloney, ‘‘Light-

polarization dynamics in surface-emitting semiconductorlasers,’’ Phys. Rev. A 52, 1728–1739 (1995).

2. M. Travagnin, M. P. van Exter, A. K. Jansen van Doorn,


and P. Woerdman, ‘‘Role of optical anisotropies in the po-larization properties of surface-emitting semiconductor la-sers,’’ Phys. Rev. A 54, 1647–1660 (1996).

3. J. Martin-Regalado, F. Prati, M. San Miguel, and N. B.Abraham, ‘‘Polarization properties of vertical-cavitysurface-emitting lasers,’’ IEEE J. Quantum Electron. 33,765–783 (1997).

4. M. Travagnin, ‘‘Linear anisotropies and polarization prop-erties of vertical cavity surface emitting semiconductor la-sers,’’ Phys. Rev. A 56, 4094–5005 (1997).

5. M. P. van Exter, A. K. Jansen van Doorn, and J. P. Woerd-man, ‘‘Electro-optic effect and birefringence in semiconduc-tor vertical-cavity lasers,’’ Phys. Rev. A 56, 845–853 (1997).

6. K. D. Choquette, D. A. Richie, and R. E. Leibenguth, ‘‘Tem-perature dependence of gain-guided vertical-cavity surfaceemitting laser polarization,’’ Appl. Phys. Lett. 64, 2062–2064 (1994).

7. H. F. Hofmann and O. Hess, ‘‘Quantum noise and polariza-tion fluctuations in vertical-cavity surface-emitting lasers,’’Phys. Rev. A 56, 868–876 (1997).

8. Y. Yamamoto, S. Machida, and O. Nilsson, ‘‘Amplitudesqueezing in a pump-noise suppressed laser oscillator,’’Phys. Rev. A 34, 4025–4042 (1986).

9. Y. Yamamoto, S. Machida, and G. Bjork, ‘‘Microcavity semi-conductor laser with enhanced spontaneous emission,’’Phys. Rev. A 44, 657–668 (1991).

10. D. Marcuse, ‘‘Computer simulation of laser photon fluctua-tions: theory of single cavity laser,’’ IEEE J. QuantumElectron. QE-20, 1139–1148 (1984).

11. M. Yamada, ‘‘Variation of intensity noise and frequencynoise with spontaneous emission factor in semiconductorlasers,’’ IEEE J. Quantum Electron. 30, 1511–1519 (1994).

12. J. L. Vey, K. Auen, and W. Elsaesser, ‘‘Quantum noise prop-erties of vertical cavity surface emitting lasers: theory andexperiment,’’ Phys. Status Solidi B 206, 427–436 (1998).

13. M. P. van Exter, M. B. Willemsen, and J. P. Woerdman,‘‘Polarization fluctuations in vertical-cavity semiconductorlasers,’’ Phys. Rev. A 58, 4191–4205 (1998).

14. T. C. Damen, L. Vina, J. E. Cunningham, J. Shah, and L. J.Sham, ‘‘Subpicosecond spin relaxation dynamics of excitonsand free carriers in GaAs quantum wells,’’ Phys. Rev. Lett.67, 3432–3435 (1991).

15. G. P. Bava and P. Debernardi, ‘‘Spontaneous emission insemiconductor microcavity post lasers,’’ IEE Proc. Optoelec-tron. 145, 37–42 (1998).

16. M. Asada and Y. Suematsu, ‘‘Density-matrix theory ofsemiconductors lasers with relaxation model. Gain andgain-suppression in semiconductor lasers,’’ IEEE J. Quan-tum Electron. QE-21, 434–442 (1985).

17. W. Chow, S. Koch, and M. Sargent, Semiconductor LaserPhysics (Springer-Verlag, Berlin, 1994).

18. H. Haug, ‘‘Quantum mechanical theory of fluctuations andrelaxation in semiconductor lasers,’’ Z. Phys. 200, 57–68(1967).

19. Y. Yamamoto and N. Imoto, ‘‘Internal and external fieldfluctuations of laser oscillator. I. Quantum mechanicalLangevin treatment,’’ IEEE J. Quantum Electron. QE-22,2032–2042 (1986).

quantum noise in vertical-cavity surface-emitting lasers with polarization competition

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