View metadata, citation and similar papers at core.ac.uk brought to you by CORE

provided by Digital.CSIC

Polarization and Transverse Mo de Selection in Quantum Well

VerticalCavity SurfaceEmitting Index and Gainguided

Devices

J MartnRegalado S Balle and M San Miguel

Instituto Mediterraneo de Estudios Avanzados IMEDEA CSICUIB and Departament de Fsica

Universitat de les Il les Balears E Palma de Mal lorca Spain

A Valle and L Pesquera

Instituto de Fsica de Cantabria CSICUC Facultad de Ciencias E Santander Spain

Abstract

We study p olarization switching and transverse mo de comp etition in Vertical

Cavity Surface Emitting Lasers in the absence of temp erature eects We use

a mo del that incorp orates the vector nature of the eld saturable disp er

sion dierent carrier p opulations asso ciated with dierent magnetic sublevels of

the conduction and heavy hole valence bands in quantum well media spinip

relaxation pro cesses and cavity birefringence and dichroism We consider b oth

indexguided and gainguided VCSELs and we nd that spinip dynamics and

linewidth enhancement factor are crucial for the selection of the p olarization state

corresp onding to a given injection current For indexguided VCSELs the eect

of spatial hole burning on the p olarization b ehavior within the fundamental mo de

regime is discussed For gainguided VCSELs transverse mo de and p olarization

selection is studied within a MaxwellBlo ch approximation which includes eld

diraction and carrier diusion switching is found in the funda

men tal mo de regime The rst order transverse mo de starts lasing orthogonally

p olarized to the fundamental mo de At larger currents p olarization co existence

with several active transverse mo des o ccurs

I INTRODUCTION

Vertical Cavity Surface Emitting Lasers VCSELs present sp ecial optical and electrical

prop erties singlelongitudinal mo de emission lowthreshold current lowdivergence circular out

put b eam simplicity for building D arrays etc which make these devices extremely attractive

for several applications such as opticalb er and freespace communications optical information

storage laser printers etc They also oer a numb er opp ortunities to study fundamental asp ects

of semiconductor laser dynamics Among the latter the issue of laser p olarization state selection

and the comp etition of p olarization state and transverse mo de prole are particularly interesting

Polarization selection in conventional edgeemitting lasers is largely determined by the geometry

of the laser but the cylindrical symmetry of the VCSEL gives in principle no preference for

any p olarization direction in the plane of the wafer This makes very relevant the question of

p olarization selection by the nonlinear gain dynamics Such a question was examined at large

in the early days of laser physics where preference for linear or circular p olarization for several

gas lasers was understo o d in terms of symmetries asso ciated with dierent lasing transitions

Pattern formation in the transverse prole of a laser b eam has received considerable attention in

the general framework of nonlinear dynamics VCSELs are interesting in this asp ect b ecause

they easily have a rather large Fresnel numb er whichfavors the app earance of transverse struc

tures Polarization control and transverse mo de control are two linked practical problems

which call for a b etter fundamental understanding of the physical mechanisms of p olarization

and transverse mo de selection

Light emitted from VCSELs is typically linearly p olarized with the vector eld oriented

along one of two orthogonal crystal axis which are p erp endicular to the emission direction The

two linearly p olarized mo des have also slightly dierent emission frequencies GHz

due to the birefringence of the crystal Even though several VCSELs display stable linear

p olarization emission at any value of the applied current many of these devices show

a p olarization instability a relative abrupt switch in the orientation of the optical eld known

as Polarization Switching PS This o ccurs as the injection current is increased further ab ove

threshold Emission in b oth linearly p olarized mo des with dierent emission frequencies

as well as in b oth linearly p olarized mo des with the same emission frequency elliptically

p olarized have b een rep orted These p olarization phenomena o ccur close to threshold

within the fundamental transverse mo de regime For higher injection currents they may be

accompanied by changes in the emission prole due to the excitation of higher order transverse

mo des The onset of the rst order transverse mo de o ccurs for an applied current which strongly

dep ends on the dimensions of the strip e contact In addition it is commonly observed that

the rst order transverse mo de starts lasing orthogonally p olarized to the fundamental mo de

An explanation oered to p olarization state selection in VCSELs is based on the dierent

rates of the thermal wavelength shift of the gain curve and the cavity combined

with the uncontrolled strain present in the growth pro cess whichmay cause VCSEL anisotropies

birefringence and dichroism As a consequence of the birefringence of the crystal the two

linearly p olarized states exp erience slightly dierent material gains in suchaway that the mo de

with higher gain to loss ratio dominates at threshold When the injected current is increased

ab ove the lasing threshold selfheating of the active region pro duces a faster red shift of the gain

peak relative to the cavity resonances so that the gain dierence between the linearly p olarized

mo des changes Within this framework i stable p olarization emission o ccurs when the gain

dierence favors the same p olarization mo de at any currentvalue ii PS is exp ected when there

is an exchange of the relative gain of the two mo des iii co existence of b oth linearly p olarized

states occurs if the gain dierence is to o small

Several p oints are worth noting in the ab ove explanation In the rst place since the fre

quency splitting b etween orthogonal linearly p olarized states is often very small as compared to

the width of the gain curve b elow or of the order of GHz as compared to nm or more

y be minute so PS in the fundamental mo de should be resp ectively the gain dierences ma

dicult to observe Second it do es not explain the preference for linearly p olarized emission

in front of ie circularly p olarized light in VCSELs with very small anisotropies Finally it

predicts stable p olarization emission if thermal eects are minimized on the device while PS

is a well known phenomenon in gas lasers in which thermal eects are not imp ortant In fact

recent exp eriments in gainguided VCSELs by indicate that PS is still observed when the active

region temp erature is kept constant by using fast pulse low duty cycle op eration Such PS

do es not t in the scheme just discussed but it is consistent with the mechanisms invoked for

PS in the mo del intro duced in Ref SFM mo del

The SFM mo del takes into account the vector nature of the laser eld saturable disp ersion

asso ciated with the linewidth enhancement factor and the carrier dynamics asso ciated with

the dierent magnetic sublevels of the conduction and heavy hole valence bands in a quantum

well It predicts the preference of quantum well material for linearly p olarized emission due to

the nite coupling via spinip relaxation pro cesses between the carrier p opulations involved

in the two natural circularly p olarized emission channels For a p erfect isotropic cavity the

linearly p olarized laser lightis randomly oriented in the plane of the active region while for an

anisotropic cavity the cavity anisotropies x the direction of linearly p olarized emission A

detailed study of the stability of the solutions of the SFM mo del incorp orating cavity anisotropies

reveals that the p olarization state selection within the fundamental mo de regime in VCSELs is

governed by the coupling b etween the two carrier p opulations in the presence of amplitudephase

coupling linewidth enhancement factor As the injection current is scanned ab ove the

lasing threshold stable p olarization emission p olarization switching p olarization co existence

or elliptically p olarized emission can b e predicted dep ending on the VCSEL anisotropies

The studies in were based on rateequations for the amplitudes of the two circularly

p olarized mo des disregarding spatial eects However SpatialHoleBurning and carrier dif

fusion may have an imp ortant role in the interplay between p olarization and transverse mo de

selection Two dierent situations can b e distinguished according to the transverse structure of

the VCSEL In indexguided devices the mo dal proles and frequencies are determined by the

builtin refraction index distribution of the VCSEL thus allowing for a description in terms of

mo dal amplitudes Instead for gainguided devices a mo dal description is not strictly

valid b ecause the eld prole is fully determined by the carrier distribution which in turn is

dynamically coupled to the eld through stimulated emission

In this pap er we consider an extension of the SFM mo del to incorp orate spatial eects in

both indexguided and gainguided VCSELs Concerning indexguided VCSELs p olarization

prop erties have already b een analyzed in Ref by using a mo del that includes birefringence

and spatial hole burning eects In this pap er we incorp orate saturable disp ersion and p olariza

tion sensitive dynamics of the carriers in semiconductor quantum well media and we restrict our

analysis to the lowest order transverse mo de Spatial eects and transverse mo de comp etition in

gainguided VCSELs are incorp orated within a spatially continuous two dimensional approach

whichtakes into account light diraction carrier diusion and frequency dep endent gain in the

context of a MaxwellBlo chtyp e approximation for a quantum well laser A partial account

of the main results of this mo del was rep orted in Ref

The pap er is organized as follows In Sect II we summarize the SFM mo del which is based

on the energy band structure of quantumwell VCSELs and we review the main results for p olar

ization selection and switching in the rateequation description for the fundamental transverse

mo de In Sect III we study p olarization stability and PS within the fundamental transverse

mo de for a weakly indexguided birefringentVCSEL In this case birefringence induces dierent

connement factors for the linearly p olarized mo des As a consequence the linearly p olarized

mo des exp erience dierent gains which leads to p olarization selection at threshold Polariza

tion switching is observed as the curren t is increased due to the combined eect of saturable

disp ersion and spin dynamics In Sect IV we discuss a continuous twodimensional mo del for

gainguided lasers in the context of MaxwellBlo ch equations The spatiotemp oral and p olar

ization prop erties of widearea gainguided VCSELs are discussed in Sect V We rst study the

p olarization prop erties during the turnon of the VCSEL Polarization stability and PS within

the fundamental transverse mo de are shown as the current is scanned for two situations where

the relative gain dierence between the linearly p olarized mo des is dierent New p olarization

instabilities are observed at higher injection currents related to the onset of higher order mo des

The rst order transverse mo de always starts lasing orthogonally p olarized to the fundamental

one At larger currents p olarization co existence with several active transverse mo des o ccurs

Weshow that these results are sensitive to the strength of coupling b etween carrier p opulations

asso ciated with dierent circular p olarizations of light spin relaxation rate A summary of our

results and conclusions is given in Sect VI

II LIGHT POLARIZATION DYNAMICS IN VCSELS

The mo del prop osed in Ref to describ e p olarization dynamics in a VCSEL takes into

account the coupling of the vector eld amplitude to the dominant allowed optical transitions



see Fig corresp onding to the spin sublevels of the conduction and heavy hole

 

valence bands in surface emitting quantum well semiconductor lasers Transitions to the light

hole valence band are neglected In a rst approximation we consider a twolevel typ e



of approach in which transitions only o ccur at the center of the band gap Consideration of

lasing transitions at dierent wavenumb ers leads to band mixing between the heavy and light

hole band which are nondegenerate at the center of the band gap A detailed analysis of

these eects indicates that they do not pro duce qualitative changes in the results that we will

describ e The dynamical variables considered are the twoslowly varying circularly p olarized

comp onents of the scaled vector optical eld F and the scaled carrier p opulations of the

two emission channels D which are coupled via spinip relaxation pro cesses Dening

the normalized total carrier number as D D D and the dierence in the carrier

 

numbers of the two magnetic sublevels as d D D the equations appropriate for

 

narrow contact single mo de VCSELs op erating at constant active region temp erature read

F F i F i D dF i F

o a p 

 

D D D djF j D djF j

e e  e 

 

d d D djF j D djF j

s e  e 

where is the eld decay rate is the linewidth enhancement factor is a change of



the frequency reference frame is the decay rate of the total carrier p opulation and is the

e

injection current normalized to threshold

The parameter is the decay rate of the carrier p opulation dierence through spinip

s

relaxation pro cesses Polarization sp ectroscopy measurements of spin relaxation times in



quantum wells estimate the value of the spinip decay time in tens of picoseconds

s

This gives an order of magnitude of the value of this parameter of the theory However precise

determination of eective spinip relaxation rate in dierent op erating conditions of a VCSEL

is still lacking Indirect evidence of the coupling between D and D as implied by might

  s

be given by anticorrelations of the two indep endent p olarization comp onents in RIN sp ectra

An indirect estimation of in the context of the predictions of this mo del has also b een

s

rep orted by measuring the magneto elipticity of the VCSEL output light in the presence of a weak

magnetic eld Also in the context of this mo del there are indications that the value of the

spinip relaxation rate can be obtained from birefringency measurements of the sp ontaneous

emission optical sp ectra as well as from p olarization uctuations sp ectra of the VCSEL

output Measurements of could also be obtained in exp eriments with optically pump

s

VCSELs with circularly p olarized pumps

It is worth noting that the equation for the p opulation dierence cannot be adiabatically



eliminated since the spin decay time is intermediate between the decay rate of the

s

electric eld and the one of the total carrier p opulation D As we will show next the carrier

variable d plays an imp ortant role in the p olarization prop erties of VCSELs In particular the

preference for linearly p olarized light arises even in the absence of anisotropies from the



nite value of In the limit of innitely fast spin relaxation and in the

s

s

absence of cavity anisotropies there is no preference for linearly or circularly p olarized light

emission Finallywe recall that the physics of the usual semiconductor laser rate equations for

 

the intensity of a linearly p olarized mo de given by I jE j jE j is recovered by setting

 

d in Eqs

The meaning of the VCSEL anisotropies is most clearly displayed by rewriting Eq in the

p p

linear basis F F F F iF F

x   y  

F F i F i DF idF

x a x o p x x y

F F i F i DF idF

y a y o p y y x

   

F F F F D D jF j jF j id

y x e x y

x y

   

jF j iD F F F F d d djF j

y y x s e x

x y

The frequency splitting b etween the linearly p olarized mo des is given by birefringence The

p

strength of their anisotropic gainlosses is given by dichroism VCSEL anisotropies have

a

been intro duced phenomenologically in the mo del assuming for simplicity that the principal

axes for birefringence and dichroism coincide so that b oth are diagonalized by the same basis

states The axis mismatch between and yields elliptically instead of linearly p olarized

a p

light emission the ellipticity dep ending on the angle between the linear anisotropies

Rep orted measurements of elliptically p olarized emission in VCSELs show that the ellipticityis

very small which indicates that the axes of anisotropy are nearly aligned

A thorough discussion of the p olarization states found in the rate equation mo del given by

Eqs is given in Ref Here we just review the main results The mo del reveals

the existence of dierent p olarization states i two orthogonal linearly p olarized mo des with

th

dierent thresholds D and dierent emission frequencies given by

i

i

v

u

th

u

D

a

x

i t th

t x

F e D

x

x

th

D

x

v

u

th

u

D

a

y

i t th

y

t

e D F

y

y

th

D

y

th th

where and d d Notice that wehave dened x y p olarization as

x p a y

x y

the mo de with high low frequency for p ositive but this denition is arbitrary and do es not

p

inuence the results see Ref ii elliptically p olarized states where there is co existence

of two frequency degenerated linear p olarization states but with dierent power strengths and

iii unstable states of circularly p olarized light which only exist in the absence of birefringency

and dichroism

The linear stability of the linearly p olarized solutions for a xed value of the factor is

studied in terms of twocontrol parameters commonly measured in PS exp eriments the injection

current normalized to the threshold current and the frequency splitting b etween the linearly

p olarized mo des induced by birefringence We recall that typical p olarized lightcurrent

p

measurements corresp ond to vertical scannings in diagrams since the frequency splitting

p

is a xed characteristic of a given VCSEL which is nearly indep endent of the injected current

The results of the linear stability analysis are summarized in Fig for two general situations

in which the relative gaintoloss ratio b etween the mo des is dierent

Fig a corresp onds to the stability diagram for VCSELs where the gain dierence favors

the yp olarized mo de at threshold The yp olarized mo de is stable on the left and

a

below the dashed line The xp olarized mo de is stable on the right and ab ove the solid line

The stability diagram is divided into several regions there are zones in which only one mo de

is stable a zone in which none of the linearly p olarized mo des is stable and a bistable domain

Stable yp olarized emission o ccurs close to threshold for any birefringence value For VCSELs

where an abrupt yx switching o ccurs as the dashed line is crossed during the

p s

current scanning The PS o ccurs by destabilization of the mo de with the higher gainto

loss ratio in favor of the weaker mo de In addition the switching current linearly dep ends on

the gainloss anisotropy

 

sw a

s p

th p s e

Such a dep endence is consistent with recent exp erimental results in gainguided VCSELs op er

ated at constant temp erature of the active medium

For gain dierence favoring xp olarization at any injection current the x p olarized

a

mo de is stable b elow the solid line while yp olarization is stable inside the zone b ounded by the

dashed curve see Fig b Regions of linearly p olarized monostability bistability and total

instability are again obtained Stable x p olarized emission is exp ected close ab ove threshold for

any birefringence value For VCSELs in which x y PS can o ccur if the injection

p s

current is increased suciently so that the solid line is crossed The switching current is given

by

 

s sw

p a



th a p e

p

This switching is not abrupt Rather it o ccurs through one of two frequencydegenerate orthog

onal orientations stable elliptically p olarized states Consequently elliptically p olarized light

can be understo o d as intermediate states reached in the destabilization by a steady bifurcation

of the linearly p olarized solution as the current is increased Hysteresis in the switching current

as rep orted in some exp erimental observations is also predicted as the injection current is

raised and lowered

We recall that the stability diagrams in Fig are a consequence of the combined eect of

saturable disp ersion asso ciated with the factor and of the spin dynamics asso ciated with the

nite value of This statement is checked by studying the stability diagrams in the limits

s

or In any of these limits the only region that survives is the bistable one and

s

laser emission o ccurs in the lowest threshold mo de at any current value The fact that PS

will never o ccur in this situation indicates that the p olarization state selection in VCSELs is

largely determined by b oth the AMFM coupling through the factor and the carrier p opulation

coupling of the natural circularly p olarized emission channels via spinip relaxation pro cesses

III POLARIZATION DYNAMICS OF INDEXGUIDED VERTICALCAVITY

SURFACE EMITTING LASERS

Studies of p olarization state selection in indexguided VCSELs have been rep orted by Val le

et al For these devices there is a mo dal gain dierence b etween transverse mo des

with orthogonal p olarization as a consequence of the combined eect of birefringence and spatial

hole burning Within this framework p olarization switching within the fundamental transverse

mo de as well as p olarization switching in higher order transverse mo des is attributed to spatial

hole burning However the role of a nite spinip relaxation rate together with the factor

were disregarded

In this section we extend the mo del in to account for spin dynamics and AMFM coupling

whic h we have shown in the previous section to be imp ortant mechanisms in the selection of

the p olarization state in VCSELs We consider a cylindrically symmetric weak indexguided

VCSEL Birefringence is taken into accountby assuming that the core refraction index in the x

direction is smaller than in the y direction while the cladding is the same in b oth

directions We will consider an index step n smaller than Then the appropriate mo des

are the LP mo des Here we treat the case of comp etition between the two p olarization

mn

states of the fundamental transverse mo de LP Several mechanisms of selection of this mo de



have b een discussed in the literature The mo del can if required b e extended to account

for comp etition b etween p olarization states of higher order transverse mo des However the

p olarization b ehavior due to spatial hole burning eects is similar in b oth the fundamental and

multitransverse mo de op eration regimes

Spatial eects can b e incorp orated in Eqs by considering carrier diusion the optical

mo de transverse proles and the carrier transverse distribution Due to the cylindrical symmetry

of the VCSEL structure a cylindrical co ordinate system is appropriate to describ e spatial eects

In the basis of linearly p olarized mo des the transverse optical eld can b e written in the following

form

it

Er tE t r x E t r y e cc

x x y y

where r and r are the proles for the x and y mo des They are obtained by solving

x y

the Helmholtz equation These proles are slightly dierent due to birefringency As the y

mo de has greater core refraction index lower emission frequency it is b etter conned than the

mo dex The equations describing the optical elds E ti x y the total carrier distribution

i

N rt and the dierence of the carrier distributions asso ciated with the dierence magnetic

band sublevels nrt read

s

q

q

m m

E i E g iE g i E N n t N n t

x x y a p x  

x xy

s

q q

m m

E i E g iE g i E i N n t N n t

y y x a p y  

y yx

      

N j tC r Dr N N jE j jE j inE E E E

t e x y x y x y

 x y y x

      

iN E E E E n n Dr n njE j jE j

x y x y t s e x y

y y x  x

m

where g ti x y is the mo dal gain normalized to the threshold gain

i

R





N r t r rdr

 i

m

R

g



i



r rdr

i



m

and g t is given by

ij

R



nrt r r rdr

i j



m

R

g t



ij



r rdr

i



Note that the mo dal gains for the x and y mo des are dierent due to the dierent optical

mo de proles However we neglect the material gain dierence since the frequency splitting is

very small as compared to the width of the gain curve The total injected current is uniformly

distributed across a circular disc contact of diameter and the prefactor j t allows for the

generation of current ramps Sp ontaneous emission pro cesses are mo deled by the terms

taken as complex Gaussian white noise sources of zero mean value and correlated in time In

the noise terms the carrier distribution is integrated over the active region of radius a

R R

a a

N rtrdr nrtrdr

 

N t nt

 

a a

The numerical integration of Eqs is p erformed with an integration step in the



radial direction of m and an integration time step of ps The b oundary condition for

the carrier distribution is taken as N tnt The initial conditions corresp ond to

the stationary solution in the state b elow threshold with The parameters values

  

involved are ns ns and D cm s Dierent values of the

e

spinip relaxation rate and of the index step n will be considered to study the combined

s

eect of spin dynamics and spatial hole burning on the p olarization prop erties of indexguided

VCSELs When n increases the dierence b etween the optical transverse proles for the x and

y mo des decreases b ecause the ratio of birefringence to index step decreases Then the mo dal

gain anisotropy also decreases and spatial hole burning eects have a smaller impact

We rst consider the case without external gainloss anisotropy with a small bire

a

 

GH z ns and n The current is injected fringence ns

s p

using a circular disc contact of diameter m equal to the diameter of the active region

The inset of Fig shows the stability diagram for calculated as in Fig spatial ef

a

fects are not taken into account Just ab ove threshold and for small values of the birefringence

there is bistability of the two linearly p olarized mo des However the yp olarized mo de is al

ways selected at threshold in the numerical simulations This selection is due to the mo dal gain

anisotropybetw een the fundamental mo des induced by birefringence Since at low currentvalues

the carriers accumulate near the center of the active region the mo dal gain of the b etter con

ned p olarization the ymo de is always larger than that of the orthogonal p olarization mo de

Fig shows the p olarized LI characteristics obtained by linearly increasing the applied current

from I I to I I in ns After the selection of the yp olarization at threshold

th th

combined eect of spin dynamics a yx PS is observed at I I This PS is due to the

th

and factor since the mo de with less mo dal gain is selected when the current is increased

see Fig however the switching current is higher than predicted by Eqs b ecause of

spatial eects

In order to force emission in the xp olarization at threshold we intro duce an external



gainloss anisotropy favoring xp olarization ns We also consider a higher bire

a

 

fringence ns around GHz and a higher spinin relaxation rate ns

p s

In this situation even though the mo dal gain for the b etter conned ymo de is higher than for

the x mo de the total net gain for the x mo de is higher than for the ymo de thus favouring x

p olarization at threshold see Fig This situation corresp onds to the stability diagram shown

in the inset of Fig which has b een calculated by neglecting spatial eects as in Sect I I From

the inset a x y PS is exp ected at I I However this switching is delayed by spatial

th

eects since the mo dal gain of the b etter conned y mo de decreases when the injection current

increases The spatial hole burnt near the center of the active region in the carrier prole is

deep er which leads to an accumulation of carriers towards the cladding region and hence to

a decrease in the mo dal gain of the b etter conned p olarization However for the parameters

considered in our previous case equal radius for the contact and the cavity and an index step

n the delay in the PS is rather small and the switching occurs at I I The

th

delay increases for smaller n and larger radius of the disc contact even though the sweep rate

across the bifurcation is not changed Fig shows the LI characteristics for a device with a

contact diameter of m and an index step of Similar results are obtained when the

contact and the cavity have the same radius but an eective diusion length of m for the

charge carriers is considered to takeinto account current spreading Since extra carriers are

injected in the cladding region spreading eects help spatial hole burning in creating a carrier

prole concentrated in the near cladding region

The LI characteristics in Fig is obtained by linearly increasing the applied current from

I I to I I in ns At threshold the VCSEL switchson in the x p olarized

th th

mo de b ecause of the eect of the external gainloss anisotropy Due to the spin dynamics and

though the linewidth enhancement factor an abrupt xy PS is observed at I I even

th

mo dal gain of the b etter conned y p olarization is smaller than that of the x p olarization due

to spatial hole burning

IV TWOLEVEL MAXWELLBLOCH MODEL FOR GAINGUIDED

VERTICALCAVITY SURFACE EMITTING LASERS

In this section we study the spatiotemp oral prop erties of gainguided proton implanted

VCSELs The SFM mo del can b e extended to account for spatial eects in the transverse plane

by considering in Eqs the optical eld diraction carrier diusion and the transverse

dep endence of the variables F D d and the injection current The continuous description

includes in a natural wayavery high numb er of transverse mo des In addition one has to takeinto

account that the dynamics of several transverse mo des in VCSELs evolves in time scales of the

order of the inverse of their frequency dierence typically GHz Hence to prop erly describ e

such a mo de comp etition we should include in the mo del an equation incorp orating the material

susceptibility providing a frequencydep endent gain and disp ersion In this pap er we do this in

a simple approximation for the material p olarization of the laser medium by considering it as a

set of twolevel coupled systems one for each of the two indep endent p olarization comp onents

of the eld Whitin this framework the system can be describ ed in terms of two sets of

semiclassical MaxwellBlo ch equations coupled through spin relaxation pro cesses which describ e

the dynamics of the carrier densities in each spin channel the slowlyvarying amplitudes SVA

of the circularly p olarized optical elds and their asso ciated material dip ole densities see ie

After an appropriate rescaling of the dynamical variables the equations describing the spatio

temp oral and p olarization prop erties of a singlelongitudinal mo de gainguided VCSEL read



c



E x y t i E P i r E i E

t a p 





n

e

q



P x y t i P g N N nE N n x y t

t   N 

  

N x y t j tC x y N Dr N E P E P c c

t e  

  

  

nx y t n Dr n E P E P c c

t s  

  

where P are the SVA of the material dip ole densities corresp onding to the left and right

circularlyp olarized optical elds E is the detuning of the carrier fre

mat 

quency from the material dip ole frequency N N is the total carrier

mat 

distribution referred to the transparency value N and n is the dierence of the carrier distri



butions asso ciated with E and E resp ectively Optical diraction and carrier diusion are

 



taken into account in a longitudinal mean eld approximation by the transverse Laplacian r



terms in Eq and Eq resp ectively In our gainguided situation diraction leads

to the selection of the transverse mo de proles and the asso ciated frequencies The total

injected current I is assumed to be uniformly distributed across a circular region of diame

ter VCSEL contact dening the transverse current density distribution C x y while the

prefactor j t allows for the generation of current ramps Sp ontaneous emission pro cesses are

mo deled by the terms taken as complex Gaussian white noise sources of zero mean value

and correlated in space and time in the usual way see ie Ref The parameters and

a

are related to the linear dichroism and birefringence of the VCSEL resp ectively but notice

p

that as discussed in the next section they do not directly corresp ond to and b ecause of

a p

the nonlinear susceptibility Other parameters in the equations are the index of refraction n

e

the carrier frequency corresp onding to the emission wavelength cn the dierential

e

gain g the carrier diusion constant D and the sp ontaneous emission factor We recall that

N

neglecting the transverse dep endence Eqs are recovered after the adiabatic elimination

q

of Eq using the scaled variables D g N N d g n and F E

N  N e

We next discuss the limitations of the twolevel MaxwellBlo ch mo del for semiconductor

lasers The activematerial dip ole densities P are sources of the optical elds providing both

material gain and changes in the background refraction index of the system disp ersion through

the complex electrical susceptibility The complex nonlinear susceptibility resulting from our

macroscopic nonlinear p olarization is obtained byFourier transform of Eq In the frequency

domain it reads

P N E

o



g N N

N 

i E







where we have taken into account that n in the steady state for linearly p olarized light

The real part of the complex susceptibility yields the gain sp ectrum which in the TwoLevel

approach has a symmetric Lorentzian prole The disp ersion sp ectrum asso ciated with the

imaginary part of the complex susceptibility is antisymmetric The gain peak is lo cated at the

dip ole resonance frequency where disp ersion vanishes It is also worth noting that the

mat

ratio of the carrierinduced changes in the imaginary and the real parts of the susceptibility

which would corresp ond to Henrys factor in semiconductor lasers is given by



N



N 

where is the emission frequency referred to the carrier frequency Hence the detuning in the

twolevel mo del plays the role of the parameter in semiconductor lasers Besides providing

AMFM coupling the eect of the detuning inatwolevel mo del is twofold i it determines

the change in refraction index induced by carrier density for p ositive negative detuning an

increase in carrier density is accompanied by a decrease increase in the index of refraction thus

yielding carrierinduced index antiguiding guiding and opp osing reinforcing the gainguiding

mechanism for the eld and ii since higher order mo des have always higher frequencies than

the fundamental one the sign of the detuning aects the stability of the fundamental mo de for

negative detuning the gain of the fundamental mo de is always the highest while for p ositive

detuning a higher order mo de closer to the gain p eak can exist Typical values of the

detuning in twolevel lasers are while for semiconductor lasers

Such large values of as compared to those of in twolevel lasers imply that the system must

be articially enforced to op erate very far away from the gain p eak resonance

From the ab ove discussion w e see that the TwoLevelMo del do es not account for some

prop erties of a semiconductor medium which include i a strongly asymmetric gain sp ectrum

ii an op eration wavelength close to the gain p eak and iii large factor in the vicinity of

the gain p eak However an accurate description of the material p olarization in semiconductors

is really complicated since radiative transitions o ccur between band states with uneven tem

p erature dep endent o ccupation moreover the band structure and transition probabilities of the

system can b e aected bymanyb o dy eects like bandgap shrinkage and Coulomb enhancement

Several microscopic mo dels have b een formulated for the calculation of the material susceptibil

ity but their high complexity makes dicult to interpret the dynamical features of the

system when aiming to identify the physical dominant mechanisms In addition there is no sim

ple way to write timedep endent dynamical equations since the nonlinear frequency dep endent

susceptibility is given For this reason there is ongoing research towards developing simpler

mo dels which incorp orate the main results of the microscopic theories in a phenomenological

way

Having sp elled out clearly the diculties of mo deling the nonlinear dynamics of semicon

ductors gainguided VCSELs we use here a TwoLevel mo del in order to provide the system

with a frequency dep endent susceptibility gain and disp ersion It incorp orates in good com

parison with exp erimental measurements a prop er description of the spatiotemp oral dynamics

of semiconductor gainguided lasers which are strongly dominated by transverse mechanisms as

the mo dal gain the carrier diusion and optical eld diraction We cho ose to op erate

the VCSEL on the negative detuning side of the gain sp ectrum negative in order to preserve

the observed prop erty that higher order transverse mo des have typically lower gain that the

fundamental one We note that the eects of such a negative value which leads to carrier

inducedguiding instead of antiguiding o ccur in real VCSELs due to thermal eects since

the index distribution dep ends on b oth the carrier distribution which leads to carrierinduced

indexantiguiding through the factor and the temp erature prole in the active region asso ci

ated to the current distribution The latter leads to thermally induced indexguiding Typical

   

values for these mechanisms are dndN m and dndT K

Thus both mechanisms act in opp osite directions so the carrierinduced index antiguiding can

b e comp ensated or even overcome by the thermallyinduced guiding From measurements of the

wavefront curvature in gainguided Quantum Well VCSELs it is shown that thermalguiding is

stronger than carrierantiguiding

V POLARIZATION AND TRANSVERSE MODE DYNAMICS OF GAINGUIDED

VERTICALCAVITY SURFACE EMITTING LASERS

In this Section the p olarization and transverse mo de dynamics of gainguided VCSELs is

studied by numerical integration of Eqs The integration scheme we follow has four

steps i the diraction and diusion terms are calculated by Fast Fourier Transform FFT

from the previous integration step initially from noise initial conditions ii the complex

spatial distributions for the sp ontaneous emission in b oth material p olarizations are generated

iii the spatial distributions for the optical elds E the carrier densities N and n and the

material p olarizations P are up dated via the Euler metho d for the integration of sto chastic

dierential equations and iv in order to obtain the lightintensity LI characteristics the

injected current is up dated by increasing j t and the scheme is rep eated for the next time

step We point out that the choice of the time integration step is strongly inuenced by the

optical diraction term since otherwise the algorithm b ecomes numerically unstable In order

to achieve a compromise between spatial resolution equation stability and computation time



we use x grid p oints for a transverse width of x m and an integration time step of



ps

Several things are worth noticing from the numerical scheme On the one hand the calcula

tion of the Laplacian terms by the FFT metho d requires the use of p erio dic b oundary conditions

for the dynamical variables However such a problem is solved by considering a transverse in

tegration region much wider than the injection area so that the dynamical variables decay to

oiding spurious reen values of the order of the noise level at the integration b oundaries thus av

tering waves On the other hand since the thermal resp onse of the VCSEL is around s

and the carriereld interaction is around ns the use of short current ramps ns to generate

the LI characteristics ensures that the measurements are taken in a quasisteady situation while

there is no thermally induced red shift of b oth the cavity mo de and gain sp ectrum



The equations of the mo del are rescaled so that jE x y j i x y represents a magnitude

i o o

prop ortional to the photon number going out from the VCSEL at the point x y of the

o o

transverse plane The total emitted power is calculated by integrating the optical intensity

transverse distribution for each p olarization and assuming a quantum eciency of and a

m eective cavity length The rest of the parameter values involved in the equations are

     

ns ps ns ns N m

 e s 

   

nm n g m ps and D cm s

e N

At this p oint it is imp ortant to discuss the mechanisms which can lead to gainloss

anisotropies for the linearly p olarized mo des in the present mo del These are the exter

nal gainloss anisotropies mo dal gain anisotropies and material gain anisotropy External

anisotropies which are describ ed by can have dierent origins they can be either inten

a

tional ie intro duced in either the cavity geometry or in the gain medium

or unintentional ie due to the imp erfections in the fabrication pro cess Mo dal gain

dierences are due to the dierent overlap of the optical mo de proles and the material gain

distribution This eect is naturally included in the mo del The evaluation of this source of gain

anisotropy in gainguided VCSELs and generally in gainguided lasers is a dicult task since

the transverse mo de proles are intimately related to the carrier density transverse distribution

which strongly dep ends on the carrier diusion the leakage current and on the p ower distribu

tion of the optical mo de spatial hole burning Finallythe material gain anisotropy is related

to b oth the birefringence and the material gain sp ectrum since the two linearly p olarized mo des

are frequency split they also have slightly dierent material gain co ecients

An approximate estimation of the material gain anisotropyintheVCSEL can b e p erformed

rewriting Eqs in the frequency domain neglecting the transverse terms and consider

ST ST th

ing the steady state carrier p opulation values for linearly p olarized lightn N N

i

These equations in the linear basis read

i E i E P i E

i i i a p i

 th

i P i P g N N E

i  i  N  i

i

where the negative p ositive sign stands for the i xy p olarized TEM mo de Replacing



Eq into Eq we end up with the fundamental mo de emission frequencies

p a p a

y x

 

and the carrier threshold values for each linearly p olarized mo de



i 

th

i x y N N

a 

i



g

N

where the p ositive negative sign stands for x y p olarization Ab ove threshold the carrier

th th th

density remains clamp ed to its threshold value N smallest value of N and N which

x y

will dep end on the particular choice of the and values Then the material gain for each

a p

linearly p olarized mo de with asso ciated frequency is provided by the real part of the nonlinear

i

susceptibility

 th

g N No

N

g g i x y

i i



i 

Hence the material gain dierence between the linearly p olarized mo des is

g g g

y x

while the real frequency splitting b etween the linearly p olarized mo des is

x y p a



The devices we study are circular contact VCSELs with dierent diameters a xed value of

and dierent values The rst device we consider VCSEL A has a diameter of m

p a

and wechose and For these parameters see the inset in Fig

p e a y x

th th th

GH z so N N N As a consequence the material gain favors yp olarization at

y x

threshold g In addition the threshold current calculated from the threshold

e

carrier density is I mA

th

Fig shows the p olarization dynamics during the laser turnon The laser has a prebias

current of I and is abruptly biased to I at t Initially the laser suers a

th th

delay of ns in the switchon time b ecause the prebias current is b elow threshold After the

switchon the intensity in b oth p olarizations shows the typical relaxation oscillations the p ower

in the yp olarized mo de b eing larger than in the xp olarized mo de The reason is that during

the switchon the total carrier density overcomes the threshold carrier density for each linearly

p olarized mo de and therefore the mo dal gain for b oth p olarizations overcomes its threshold

value However ns after the switchon the output power in the xTEM mo de go es to zero



th

since the total carrier density is reaching N x y As a consequence the x p olarized mo de

y

switches o and the yp olarization mo de remains as exp ected A similar behavior has b een

exp erimentally found in Ref but in this case the relaxation oscillations are more damp ed

b ecause the pumping conditions prebias current of I and biased to I Nevertheless the

th th

characteristic time for the selection of the p olarization state is in go o d agreement with the one

found numerically

Fig shows the p olarized LI characteristics for VCSEL A obtained by linearly increasing the

applied current from I I to I I in ns The total output p ower emitted by the

th th

VCSEL increases linearly with the current and stable yp olarized emission in the fundamental

transverse mo de is observed up to I I This p olarization b ehavior is predicted in Fig a

th

for low birefringences as it is the case For current values larger than I higher order

th

mo des start lasing in b oth p olarizations Here we observe a general feature of the p olarization

instabilities seen in several VCSELs the rst order transverse mo de starts lasing orthogonally

p olarized to the fundamental mo de

The insets in Fig show the instantaneous transverse neareld proles at the indicated



currents the plotted area corresp onds to a square of m During fundamental mo de

op eration the width of the yp olarized Gaussian mo de is ab out of the current contact

and slightly increases with increasing current The op olarization mo de only emits amplied

sp ontaneous emission from the entire VCSEL contact In the multitransverse mo de regime the

insets in Fig show that while the yp olarized comp onent still consists on a single lob e with

the peak moving around the VCSEL contact as the current increases the xp olarized beam

prole dynamically adopts several shap es i two lob es oscillating in time around the center

utshap e at I I iii an ocentered lob e of the VCSEL at I I ii a doughn

th th

distribution at I I

th

In order to force x p olarized emission at threshold we should intro duce an external gainloss

anisotropy to overcome the material gain anisotropy In addition to increase the current

a

range where fundamental mo de op eration o ccurs wehave to decrease the diameter of the pump

region For these reasons we consider VCSEL B with a contact diameter of m and the

following parameters values and The threshold current is now lower

p e a e

I mA b ecause of the smaller active region volume The real frequency splitting is

th

th th

GH z so However the threshold carrier density remains clamp ed to N N

y x

x

b ecause of the negative value of Hence even though the material gain for the fundamental

a

yp olarized mo de is higher than for the xp olarized mo de g the total net

e

gain for the x p olarized mo de is higher than that for the yp olarized mo de

Fig shows the LI characteristics for VCSEL B At threshold the VCSEL switcheson in

the xp olarized TEM mo de b ecause of the eect of the external anisotropy For increasing



current an abrupt xy PS is observed at I I while the mo de prole do es not change

th

The switching current dep ends on the value of ie for PS o ccurs at I I

a a e th

For VCSEL B the single mo de regime extends up to I to b e compared with VCSEL A for

th

which the onset of higher order mo des o ccurs at I For this reason weexpectthat PS will

th

disapp ear in wider VCSELs with the same parameter values The rst order transverse mo de

is orthogonally p olarized to the fundamental one as for VCSEL A During the multitransverse

mo de regime the xp olarized total power increases almost linearly while the yp olarized total

power saturates Such general b ehavior corresp onds to the scenario found in

The mo dal b ehavior of the VCSEL emission can be obtained by integrating Eqs

at a xed current value instead of using a current ramp Fig shows the optical sp ectra and

the transverse mo de proles obtained at four dierent injection current values These sp ectra

are equivalent to those obtained by a FabryPerot interferometer with a free sp ectral range of

GH z and a frequency resolution of GH z At I I the p olarized sp ectrum in

th

Fig a shows that the laser mainly emits in the x TEM mo de However the orthogonal



p olarization shows a strongly suppressed p eak dB while the frequency dierence is b elow

the frequency resolution so it may corresp ond to elliptically p olarized light Beyond the switch

ing current at I I laser emission o ccurs in the yp olarized Gaussian mo de Fig b

th

For increasing current I I two transverse mo des the TEM and the TEM

th  

co exist but with dierent p olarizations Fig c The onset of the TEM can be explained



through the change in its mo dal gain due to the comp etition b etween spatial hole burning and

carrier diusion but an explanation of why it is orthogonally p olarized to the fundamen

tal mo de requires invoking p olarizationdep endent mechanisms Also notice that b ecause of the

twolevel nonlinear susceptibility mo del there is a small red shift of the mo dal frequencies for

increasing current whereas a blue shift is exp ected for a b etter description of semiconductor

susceptibility

Fig d shows the sp ectrum at I I Several transverse mo des are active in each

th

p olarization It is remarkable that the two linear p olarizations cho ose to op erate in mo des of

dierent parity Even order mo des are yp olarized a dominant TEM mo de and some



strongly suppressed second order mo des and while xp olarized mo des have odd order

proles TEM and TEM mo des In opp osition to the cases shown in Fig ac

 

where the total intensity emitted in each p olarization is constant at this currentvalue the output

of the laser oscillates in time The totalx p olarized p ower is mo dulated at twice the b eat note of

the x p olarized rstorder transverse mo des GH z Perio dic mo dulation at twice the b eat

frequency is also observed in the total yp olarized power but in this case as a consequence of

the nonlinear coupling b etween the two linearly p olarized eld comp onents and the total carrier

p opulation

For even larger current values I I co existence of several transverse mo des in both

th

linear p olarizations is observed in the optical sp ectrum Fig b The time dep endence of the

total x and yp olarized output powers shown in Fig a is as previously describ ed for I

I but a fast additional mo dulation ripples in b oth linearly p olarized p owers is observed

th

This fast mo dulation corresp onds to twice the b eat notes b etween the fundamental mo de and the

frequency nondegenerated rst order mo des and GH z resp ectively Although the

total emitted p ower has a similar b ehavior for b oth p olarizations the spatiotemp oral dynamics

is dieren t The yp olarized b eam keeps a singlelob e distribution almost Gaussian with the

peak slightly moving around the center of the VCSEL contact In the orthogonal p olarization

the b eam prole oscillates p erio dically it do es not twist b etween two p ositions where the b eam

consists on twolob es oriented along the diagonals of the x y plane see the insets in Fig and

where the maximum power is emitted Between these two p ositions we observe doughnutlike

emission Similar spatiotemp oral b ehavior is observed in the case of Fig d

Wehave also studied the relevance of the coupling mechanism b etween the circularly p olarized

emission channels in the transverse and p olarization prop erties of gainguided VCSELs by taking

a fast spinip relaxation rate In these conditions there is very fast mixing of

s e

the carrier p opulation between the two channels so that n quickly relaxes to zero for linearly

p olarized light Hence the two linearly p olarized elds are coupled to a single carrier p opulation

N The most relevant dierence with resp ect to previous cases is that now the dynamics is

not sensitive to the p olarization In either VCSEL A or B p olarization stability is observed

in the fundamental mo de regime and transverse mo des start lasing in the p olarization of the

fundamental mo de For VCSEL A the onset of higher order mo des o ccurs at I that is

th

similar to the value found when the spinip dynamics is not to o fast see Fig However

the fundamental mo de regime extends to I I for VCSEL B as evidenced from the LI

th

characteristics in Fig These features indicate that the selection of a particular transverse

mo de do es not only dep end on the mo dal gain when the spinip dynamics is not to o fast

Therefore we infer that physical mechanisms asso ciated with the spinip relaxation rate are

crucial in determining the transverse and p olarization prop erties of gainguided VCSELs

VI CONCLUSIONS

Our results show that physical mechanisms asso ciated with the spin dynamics are crucial in

determining the p olarization b ehavior of b oth indexguided and gainguided VCSELs A nite

spinip relaxation rate together with the linewidth enhancement factor are crucial in determin

ing the transverse and p olarization state of the VCSEL Polarization switching from the less

to the better conned p olarization in the fundamental mo de of indexguided VCSELs is not

observed when these mechanisms are not taken into account As compared to the rate equation

description the lo cation of the b oundary b etween the dierent regions in the stability diagram of

the linearly p olarized fundamental mo de is mo died by spatial eects For gainguided devices

wehaveusedaTwoLevel mo del which prop erly describ es most p olarization and spatiotemp oral

prop erties of real VCSELs The time necessary for the selection of the p olarization state during

a turnon event is in go o d agreement with the exp erimental observations For LightIntensity

measurements either p olarization stability or p olarization switching o ccur within the fundamen

tal mo de regime dep ending on the VCSEL anisotropies In addition the onset of the rst order

transverse mo de o ccurs in a p olarization orthogonal to the lasing fundamental transverse mo de

as observed in the exp eriments Finallywe notice that the o ccurence of p olarization switching

in the fundamental mo de may b e inhibited by the excitation of higher order transverse mo des

The authors wish to thank Prof NB Abraham Prof JJ Ro cca Dr F Prati Dr JLA

Chilla and Dr GHM van Tartwijk for helpful discussions and comments We acknowledge

nancial supp ort from CICYT Spain pro jects TIC and PB and Europ ean

Union HCM Grant Gigapico CHRX CT

REFERENCES

TE Sale Vertical Cavity Surface Emitting Lasers Wiley New York

CJ ChangHasnain Vertical Cavity Surface Emitting Lasers in Semiconductor Lasers

Past Present and Future Ed GP Agrawal AIP Press New York

D Lenstra On the theory of p olarization eects in gas lasers Phys Repts

Sp ecial issue Chaos Solitons and Fractals Ed L Lugiato

H Li T L Lucas J G McInerney and R A Morgan Transverse mo des and patterns of

electrically pump ed verticalcavity surfaceemitting semiconductor lasers Chaos Solitons

and Fractals

C J ChangHasnain M Orestein A Von Lehmen L T Florez J P Harbison and N G

Stoel Transverse mo de characteristics of verticalcavity surfaceemitting lasers Appl

Phys Lett C J ChangHasnain J P Harbison G Hasnain A C Von

Lehmen L T Florez and N G Stoel Dynamics p olarization and transverse mo de

characteristics of verticalcavity surfaceemitting lasers IEEE J Quantum Electron

A K Jansen van Do orn M P van Exter and J P Wo erdman Eects of transverse

anisotropyon VCSEL sp ectra Electron Lett

AK Jansen van Do orn MP van Exter and JP Wo erdman Appl Phys Lett

MPvan Exter AK Jansen van Do orn and JPWo erdman The electrooptic eect and

birefringence in semiconductor verticalcavity lasers to b e published in Phys Rev A

K D Cho quette and R E Leib enguth Control of verticalcavity laser p olarization with

anisotropic transverse cavity geometries IEEE Phot Tech Lett

K D Cho quette D A Richie and R E Leib enguth Temp erature dep endence of gain

guided verticalcavity surface emitting laser p olarization Appl Phys Lett

K D Cho quette R PSchneider Jr K L Lear and R E Leib enguth Gaindep endent

p olarization prop erties of VerticalCavity lasers IEEE J of Sel Topics in Quantum Elec

tron

JH Ser YG Ju JH Shin and YH Lee Polarization stabilization of verticalcavity

surface emitting lasers by inscription of ne metalinterlaced gratings Appl Phys Lett

Z G Pan S Jiang M Dagenais R A Morgan K Ho jima M T Asom and R E

Leib enguth Optical injection induced p olarization bistability in verticalcavity surface

emitting lasers Appl Phys Lett

S Jiang Z Pan M Dagenais R A Morgan and K Ko jima Highfrequency p olarization

selfmo dulation in verticalcavit y surfaceemitting lasers Appl Phys Lett

M Travagnin M Pvan Exter A K Jansen van Do orn and JPWo erdman Polarization

behavior of surfaceemitting semiconductor lasers in an axial magnetic eld Opt Commun

M P van Exter A K Jansen van Do orn and J P Wo erdman Transverse anisotropy

and mo de structure in VCSELs in Physics and Simulation of Optoelectronic Devices IV

WW Chow and M Osinski eds SPIE Bellingham

U Fiedler G Reiner PSchnitzer and KJ Eb eling Top surfaceemitting verticalcavity

laser dio des for Gbs data transmission IEEE Phot Tech Lett

Brusenbach Polarization Switching J MartnRegalado J L A Chilla J J Ro cca and P

in VerticalCavity Surface Emitting Lasers observed at Constant Active Region Temp era

ture Appl Phys Lett issue

M San Miguel Q Feng J V Moloney Lightp olarization dynamics in surfaceemitting

semiconductor lasers Phys Rev A

J MartinRegalado M San Miguel N B Abraham F Prati Polarization Switching in

Quantum Well VerticalCavity Surface Emitting Lasers Optics Letters

J MartnRegalado M San Miguel NB Abraham and F Prati Polarization Prop erties

of Vertical Cavity Surface Emitting Lasers IEEE J Quantum Electronics

J MartinRegalado M San Miguel N B Abraham and F Prati Polarization State

Selection and Switching in VCSELs in Physics and Simulation of Optoelectronic Devices

IVW W Chow and M Osinski eds SPIE Bellingham

C Serrat N B Abraham M San Miguel R Vilaseca and J MartinRegalado Polariza

tion dynamics in avertical cavity laser with an axial magnetic eld Phys Rev A Rapid

Comm R

A Valle K A Shore and L Pesquera Polarization selection in birefringentvertical cavity

J of Lightwave Tech surface emitting lasers IEEE

R Mueller AKhler A Valle J Sarma and K A Shore Eects of Spatial HoleBurning

eects on p olarization dynamics in edgeemitting and vertical cavity surface emitting laser

dio des Semiconductor Science and Technology

J MartnRegalado S Balle and M San Miguel Polarization and Transverse Mo de Dy

namics of GainGuided VerticalCavity Surface Emitting Lasers Opt Lett

L Fratta F Prati and M Travagnin Band mo del for light p olarization in QWVCSELs

unpublished

T C Damen L Vina J E Cunningham J Shah and L J Sham Subpicosecond spin

ett relaxation dynamics of excitons and free carriers in GaAs quantum wells Phys Rev L

Optical orientation edited by F Meier and B P Zachachrenya Amsterdam North Holland

Publishing Company

D C Kilp er D G Steel R Craig and D R Scifre Polarizationdep endent noise in photon

number squeezed light generation by quantumwell lasers Opt Lett

MP van Exter RFM Hendriks and JP Wo erdman Physical insight into the VCSEL

p olarization rate equations unpublished

H van der Lem and D Lenstra Saturation induced frequency shift in the noise sp ectrum

of a birefringentVertical Cavity Surface Emitting Laser unpublished

H F Hofmann and O Hess Quantum noise and p olarization uctuations in Vertical

Cavity Surface Emitting Lasers Phys Rev A

AK Jansen van Do orn MPvan Extern AM van der Lee and JPWo erdman Coupled

mo de description for the p olarization state of a verticalcavity semiconductor laser Phys

Rev A

H van der Lem Polarization eects in a vertical cavity surface emitting laser Masters

Thesis

H Kawaguchi I S Hiday at Y Takahashi and Y Yamayoshi Pitchfork bifurcation p olar

ization bistabilityinverticalcavity surface emitting lasers Electron Lett

A Valle L Pesquera and K A Shore Polarization behavior of birefringent multi

transverse mo de VerticalCavity Surface Emitting Lasers IEEE Phot Tech Lett

A Valle L Pesquera and K A Shore Polarization and transverse mo de selection in

birefringent VerticalCavity Surface Emitting Lasers in Physics and Simulation of Opto

electronic Devices V W W Chow and M Osinski eds SPIE Bellingham

A Valle J Sarma and K A Shore Dynamics of Transverse Mo de Comp etition in Vertical

Cavity Surface Emitting Laser Dio des Optics Comm p A Valle J

Sarma and K A Shore Spatial Hole Burning Eects on the dynamics of Vertical Cavity

Surface Emitting Laser Dio des IEEE J Quantum Electron

YA Wu G S Li R F Nabiev K D Cho quette C Canneau and CJ ChangHasnain

Singlemo de passive antiguide vertical cavity surface emitting laser IEEE J of Selected

Topics in Quantum Electron

R A Morgan G D Guth M W Focht M T Asom L E Rogers and S E Callis

Transverse mo de control of verticalcavity topsurface emitting lasers IEEE Phot Tech

Lett

CH Chong and J Sarma Lasing mo de selection in Vertical Cavity Surface Emitting

Lasers IEEE Phot Tech Lett

N K Dutta Analysis of current spreading carrier diusion and transverse mo de guiding

in surface emitting lasers J Appl Phys

J MartinRegalado S Balle and N B Abraham Spatiotemp oral dynamics of gain

ys IEEE J Quantum Electron J Martn guided semiconductor laser arra

Regalado S Balle and NB Abraham Mo deling spatiotemp oral dynamics of gainguided

multistrip e and broad area lasers IEE Proceedings Optoelectronics

J MartnRegalado GHM van Tartwijk S Balle and M San Miguel Control and

Pattern Stabilization in BroadArea Lasers by Optical Feedback Phys Rev A

C H Henry Theory of the linewidth enhancement factor of semiconductor lasers IEEE

J Quant Electron

P K Jakobsen J V Moloney AC Newell and R A Indik Spacetime dynamics of

widegainsection lasers Phys Rev A

F Stern Band tail mo del for optical absortion and the mobility edge in amorphous silicon

Phys Rev B F Stern Calculated sp ectral dep endence of gain in excited

GaAs J Appl Phys

H Haug and S W Ko ch Semiconductor laser theory with manyb o dy eects Phys Rev

A p W W Chow S W Ko ch and M Sargent I I I SemiconductorLaser

Physics SpringerVerlag New York

S Balle Eective twolevelmo del with asymmetric gain for laser dio des Optics Comm

p

J Yao G P Agrawal P Gallion and C M Bowden Semiconductor laser dynamics

beyond the rateequation approximation Optics Comm p

J Manning R Olshansky and CB Su The carrierinduced index change in AlGaAs and

m InGaAsP dio de lasers IEEE J Quant Electron

JS Blakemore Semiconducting and other ma jor prop erties of gallium arsenide J Appl

Phys R

M San Miguel and R Toral Sto chastic eects in physical systems in Instabilities and

nonequilibrium structures VI E Titap egui and W Zelle eds Kluewer

G Hasnain K TaiLYang Y H Wang RJ Fischer J D Wynn B Weir N K Dutta

and A Y Cho Performance of gainguided surface emitting lasers with semiconductor

distributed Bragg reectors IEEE J Quantum Electron

J W Scott B J Thib eault C J Mahon L A Coldren and F H Petters High mo du

lation eciency of intracavity contacted vertical cavity lasers Appl Phys Lett

D Taub er G Wang R S Geels J C Bowers and L A Coldren Large and small

signal dynamics of verticalcavity surfaceemitting lasers Appl Phys Lett

K D Cho quette K L Lear R E Leib enguth and M T Asom Polarization mo dulation

of cruciform verticalcavity laser dio des Appl Phys Lett

K Iga Engineered p olarization control of GaAsAlGaAs T Mukaihara F Koyama and

surfaceemitting lasers by anisotropic stress from elliptical etched substrate hole IEEE

Phot Tech Lett

A ChavezPirson H Ando H Saito and H Kanbe Polarization prop erties of a vertical

cavity surface emitting laser using a fractional layer sup erlattice gain medium Appl Phys

Lett

T Ohtoshi T Kuro da A Niwa and S Tsuji Dep endence of optical gain on crystal

orientation in surfaceemitting lasers with strained quantum wells Appl Phys Lett

J P Wo erdman A K Jansen van Do orn and M P van Exter Symmetry breaking in

VerticalCavity Semiconductor Lasers to b e published in Laser Physics

DV Kuksenkov H Temkin S Swirhun Polarization instability and relative intensity

noiseinvertical cavity surface emitting lasers Appl Phys Lett

D Vakhsho ori J D Wynn G J Zydzik R E Leib enguth M T Asom K Ko jima

and R Morgan Topsurface emitting lasers with V threshold voltage and the eect of

erse mo de op eration and eciencies Appl Phys Lett spatial hole burning on their transv

FIGURES

FIG Four level mo del for p olarization dynamics in QWVCSELs

FIG Stability diagrams of the linearly p olarized solutions for a y favored and b

a e

  

xfavored The rest of the parameters are ns ns ns and

a e e s

FIG a LI characteristic for the linear p olarizations The stability diagram obtained from

Sect I I app ears in the inset b Mo dal gain versus injection current Curves related to x y p olariza

tion are plotted with solid dashed line

  

ns n and FIG Same as in Fig but for ns ns

p s a

m

FIG Turnon event for VCSEL A The inset shows the MaxwellBlo chtwolevel gain sp ectrum

and the lo cation of the linearly p olarized TEM mo des for the set of parameters chosen



FIG VCSEL A LI characteristic for the linearly x solid and y dashed p olarizations

FIG Same as in Fig but for VCSEL B

FIG VCSEL B Optical sp ectra of the linearly p olarized eld comp onents E solid and E

x y

dashed for xed current values indicated in Fig by arrows a I I b I I c

th th

I I d I I

th th

FIG VCSEL B a Time evolution of the total emitted power and b optical sp ectrum at

I I Solid dashed line stands for the x y p olarized eld comp onent

th

FIG Fast spinip relaxation rate Same as in Fig but for

s e