Self-consistent multimode lasing theory for complex or random lasing media

Hakan E. Türeci*Institute of Quantum Electronics, ETH Zurich, 8093 Zurich, Switzerland

A. Douglas Stone and B. CollierDepartment of Applied Physics, P. O. Box 208284, Yale University, New Haven, Connecticut 06520-8284, USA

Received 30 May 2006; published 31 October 2006

A semiclassical theory of single and multimode lasing is derived for open complex or random media usinga self-consistent linear response formulation. Unlike standard approaches which use closed cavity solutions todescribe the lasing modes, we introduce an appropriate discrete basis of functions which describe also theintensity and angular emission pattern outside the cavity. This constant flux CF basis is dictated by the Greenfunction which arises when formulating the steady state Maxwell-Bloch equations as a self-consistent linearresponse problem. This basis is similar to the quasibound state basis which is familiar in resonator theory andit obeys biorthogonality relations with a set of dual functions. Within a single-pole approximation for the Greenfunction the lasing modes are proportional to these CF states and their intensities and lasing frequencies aredetermined by a set of nonlinear equations. When a near threshold approximation is made to these equations ageneralized version of the Haken-Sauermann equations for multimode lasing is obtained, appropriate for opencavities. Illustrative results from these equations are given for single and few mode lasing states, for the caseof dielectric cavity lasers. The standard near threshold approximation is found to be unreliable. Applications towave-chaotic cavities and random lasers are discussed.

DOI: 10.1103/PhysRevA.74.043822 PACS numbers: 42.55.Ah, 42.65.Sf, 42.55.Zz, 42.55.Sa

I. INTRODUCTION

A long-standing problem in laser theory is the formulationof a model for lasing which correctly treats the openness ofthe lasing medium and/or cavity and the nonlinearity of thecoupled matter-field equations. This mathematical challengehas become of great relevance with the current high interestin complex or random lasers, for which the mode geometryis not fixed by the placement and orientation of mirrors andone typically has multimode lasing behavior. In this casemany spatially complex internal modes contribute to the ex-ternal emission pattern and we currently lack any theory topredict the directionality of emission patterns and to under-stand and predict the output power as a function of pumpstrength. One class of systems of particular interest in thisregard are dielectric cavity lasers with complex and some-times chaotic ray dynamics 1,2; another class is the so-called “random” lasers 3,4 in which light undergoes diffu-sive motion within the gain medium. In the former case thelaser will typically have relatively high-Q modes whose las-ing properties are nonetheless determined by complicatedcompetition between spatially complex modes. The latterpart of this work will focus on the dielectric cavity casealthough the formalism we are developing should be usefulfor both random lasers and for certain conventional lasers inwhich mode competition is important. We believe the currentapproach solves the problem of treating the openness of thecavity in the simplest possible manner, and makes clear theconnection between the resonances or quasibound states ofthe passive cold cavity and the lasing modes of the activecavity. In particular if a periodic or multiperiodic solution of

the Maxwell-Bloch MB equations exists, then the lasingmodes and frequencies are determined by a set of self-consistent integral equations the kernel of which is the Greenfunction of the linear problem with outgoing wave boundaryconditions. When the modes are relatively high Q a simpleapproximation to this Green function implies that the spatialmodes are given by a certain set of states which we refer toas constant flux CF states. The effects of spatial-hole burn-ing are then described by interactions between these modes,which unlike previous theories, are modes which exist in allspace and can thus be used to predict output power and emis-sion patterns from complex or random two- and three-dimensional laser cavities. Throughout this work we employthe semiclassical description of lasing implied by the MBequations, so the effects of field quantization, such as quan-tum fluctuations, are not included.

The modal description of lasing assumes that the laser isin a regime in which the steady-state electric field in thepumped medium Ex , t which we will henceforth refer toas the “cavity” has a finite number of frequencies and henceby Fourier transform can be seen as a sum of nonlinearmodes Ex ,. It is now well known that lasers can havechaotic temporal dynamics which cannot be described by afinite number of frequencies 5–8. For clarity we wish toemphasize that we are not treating lasers which exhibit dy-namical chaos in this sense; the “chaotic lasers” we are treat-ing are those with complex modal patterns which can berelated to the chaotic motion of light rays in the geometricoptics limit. A term for this type of laser consistent withusage in the field of quantum chaos is “wave chaotic.” Thetemporal dynamics we are treating is conventional multi-mode MM lasing, typical of the vast majority of lasers.There have been a number of interesting experiments forwhich the emission patterns from deformed nonspherical or*Electronic address: tureci@phys.ethz.ch

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cylindrical dielectric cavity lasers have not been explainablein terms of conventional “whispering gallery” modes, butinstead required careful analysis of different modal patternswith different ray-optical interpretations 9–16. Furthermoreseveral experiments have found a dramatic variation of theoutput power for a given pump strength with the shape of thelaser cavity 9,10. Our current formalism is proposed to de-scribe and predict the results of such experiments, which areinadequately treated by standard approaches.

There are several methods to predict or describe lasingmodes and emission in this case of multiperiodic steady statesolutions. The simplest approach is to work with the “coldcavity” CC; i.e., to find the electromagnetic solutions forthe Helmholtz equation describing light in the cavity neglect-ing gain and nonlinear effects. If the lasing modes have high-Q values then the nonlinear solutions are often very similarspatially to these CC modes 17,18 and from the experimen-tally observed MM lasing frequency spectrum one can asso-ciate the lasing state with a sum of cold cavity modes or asingle mode if only one frequency is present. Within this CCapproximation a further approximation is to treat the cavityas closed and work with Hermitian modes of a perfectlyconfining cavity. The Gaussian modes of Fabry-Perot reso-nators shown in textbooks are solutions of this type.

Another more common approach within the CC approxi-mation is to approximate the lasing modes by the quasibound(QB) states or resonances of the cavity. These states are de-fined as the solutions of the linear Maxwell wave equationin the absence of gain which satisfy the boundary condi-tions at the cavity boundary and only have outgoing waves atinfinity. This outgoing wave boundary condition can only besatisfied for discrete complex values of k=q− i with 0 which means that the outgoing spherical waves willgrow as expr as r→. Such modes are not Hermitian andare not orthogonal to one another though a modified or-thogonality relation can be defined in separable non-Hermitian problems 19,20. The quantity determines theQ value of the resonance which can then be used in formu-lating the laser theory. The Fox-Li method 21 dating fromthe early days of laser theory is one technique for findingsuch quasibound modes; there are a number of modern meth-ods for doing this as well 1,22–25. This theoretical ap-proach has been widely applied to analyze post factor themodes of wave-chaotic or random lasers see 1 for refer-ences. Its obvious drawback is that the method is not pre-dictive. First, a larger set of QB modes is found and a subset,the lasing modes, are chosen as those which “look” like theexperimental results. Furthermore, there is no means withinthis approach to understand and predict the output power ofthe laser.

The standard approach to go beyond the CC approxima-tion is to use the simplest semiclassical lasing theory, de-scribing a pumped medium of two-level atoms coupled tolight within a cavity, known as the Maxwell-Bloch equations.Our theory below is based on analysis of these MB equa-tions. The MB equations can be analyzed to find single-frequency solutions, usually approximated by single modesof the cold cavity. One then finds that it is the highest-Q coldcavity mode which lases first and that its frequency is shiftedfrom the cold cavity frequency towards the atomic transition

frequency but not by a large amount if the Q of the cavity issufficiently high 5,14,17,26. The spatial field distribution istypically assumed unchanged from the CC mode which canbe approximated by the closed cavity mode or the corre-sponding CC resonance. The complication in the theorycomes when one attempts to describe multimode lasing. Firstthe general nonlinear MB equations cannot be solved ana-lytically using a modal expansion of the electric field. Exactnumerical solution of the equations can be done for somecases and is useful 18; however, this becomes computation-ally intractable in the short wavelength limit of interest hereand it is difficult to extract qualitative physical ideas fromsuch an approach. A nice method due to Haken 27 datesback to the early days of laser theory; one limit of our theoryis an extension of this approach. The Haken method expandsthe electric field which solves the MB equations as a sum ofCC modes and writes an equation of motion for the ampli-tude of each mode which can be reduced to a constrainedlinear equation for the modal intensities in the near thresholdapproximation. The major drawback of the Haken approachis that it is formulated in terms of the modes of the idealclosed cavity and calculates only the internal field intensitiesof these modes; thus without some ad hoc assumptions itdoes not predict output power or directional emission pat-terns. This is done to exploit the orthogonality of closedcavity modes which simplifies the mathematical description.For comparison to our results below, the Haken near thresh-old MM lasing equations for the steady-state electric fieldintensities I are

1 −

D0=

gAI, 1

where ,D0 ,g are the cavity decay rate, pumpstrength, and laser gain profile, respectively in appropriatelyscaled units and

A = d3xx2x2 2

is a matrix which describes interactions between CC modesx ,x. The complexity of modal solutions in spacethus determines the lasing amplitudes through the propertiesof this matrix. This equation was derived by Haken andSauermann in 1963 to describe the effects of spatial holeburning: the fact that lasing modes deplete the inversion in aspatially varying manner which can then allow many modesto lase in steady-state when the pump is sufficiently strong. Astatistical analysis of certain aspects of these equations forthe case of closed cavities described by random matrixtheory was performed by Misirpashaev and Beenakker sometime ago 28.

Note that while Eq. 1 appears to be a simple inhomoge-neous linear equation for the I, it cannot be trivially in-verted to yield these intensities since we have the additionalconstraint that I0,∀. The equation thus needs to besolved by an iterative procedure. In a recent work 29 wedescribe this procedure and used it along with some ad hocassumptions to calculate multimode output intensities for de-

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formed dielectric cylinder lasers, making contact with theexperimental results of Ref. 9.

In the current work we present a formalism which gives ageneralization of Eq. 1, and which avoids these ad hocassumptions. The basic innovation of the approach is to for-mulate the solution of the MB equations as a self-consistentlinear response problem and write this solution in terms ofthe Green function of the wave equation with the boundarycondition of constant outgoing flux at infinity. Using thisGreen function the lasing solutions and frequencies are de-termined by an integral equation which can be used to de-scribe multimode lasing in both high-Q and low-Q cavities,and in random media acting as a cavity. To describe suchcomplex cavity modes it is useful to represent this Greenfunction in terms of a new set of linear eigenfunctions whichsatisfy the wave equation with real wave vector but purelyoutgoing boundary conditions; we refer to these functions asconstant flux CF states. They are similar to the resonantsolutions of the cold cavity in that they have a complex wavevector inside the lasing medium, but they differ in that theyhave a real wave vector outside the medium and hence givea well-defined field at infinity. Like the quasibound states,the CF states are not orthogonal, and the Green functioninvolves both the CF states and their biorthogonal partnerswhich represent states of constant incoming flux from infin-ity. The use of biorthogonal pairs of resonator states is wellestablished in resonator theory, and it is well known that thetwo states correspond to different directions of propagationthrough the apparatus see Ref. 30, pp. 847–857. Our CFstates are just a generalization of this idea. The new featurehere is that we represent the non-linear lasing mode in termsof these biorthogonal functions. Moreover we show that asimple “single-pole” approximation to the Green functionimplies that the lasing modes are proportional to the CFstates and leads to a generalization of the Haken-Sauermannmulti-mode lasing equations in terms of CF states. Iterativesolutions of these equations predict the multimode lasingstates both inside and outside the cavity and hence provide apredictive theory of output power and directional emissionfrom complex cavities.

The paper is organized as follows. In Sec. II we derive theself-consistent integral equation for the lasing modes assum-ing that a multiperiodic solutions exists. To do this we intro-duce a Green function for the inhomogeneous wave equationwithin the cavity. In Sec. III we discuss the boundary condi-tions on this Green function and its spectral representation.We show that in order to satisfy the outgoing wave boundaryconditions this spectral representation is expressed in termsof two sets of biorthogonal functions the CF states. Theapproximation corresponding to standard multimode lasingtheory is to approximate the Green function by the contribu-tion of the single-pole nearest to the lasing frequency. Thisimplies that the lasing modes are proportional to a single CFstate. We discuss examples of CF states for a slab one di-mensional 1D, cylinder 2D, and deformed cylindrical di-electric cavity. In Sec. IV we derive the multimode lasingequations which follow from this approximation, leading tolinear and nonlinear generalizations of the Haken-Sauermannequations, valid for the open cavity. In Sec. V we discussiterative methods for solving these equations for the modal

intensities and the lasing frequencies and then present solu-tions for the single and two mode cases, beyond the standardHaken-Sauermann near threshold approximations. Our re-sults suggest that the near threshold approximation intro-duces significant error and greatly overestimates the numberof lasing modes at a given pump power. We summarize ourresults in Sec. VI and provide further detail on CF states inthe Appendix.

II. DERIVATION OF THE SELF-CONSISTENTSEMICLASSICAL LASER EQUATIONS

Following the standard semiclassical laser theory 5,27,we start with the Maxwell-Bloch equations MB in the form

2E −1

c2 E =4

c2 P+ + P+* , 3

P+ = − ia + P+ +g2

iED , 4

D = D0 − D −2

iEP+* − P+ . 5

This is a set of nonlinearly coupled spatiotemporal partialdifferential equations for the electric field amplitude Ex , t,the macroscopic polarization Px , t=nag 12x , t+g* 21x , t= P++ P−, and the inversion Dx , t=na 22x , t− 11x , t. Here, P+= P−*=nag 12. The pa-rameters entering the equations are as follows: D0 is the ex-ternal pump strength, g is the dipole moment matrix element, and are phenomenological damping constants for thepolarization and the inversion, respectively, and c is thespeed of light. Note that E, D, and P are real valued fields.

We will make the following assumptions:i We focus here on a scalar field E defined in two space

dimensions x= x ,y. For instance, in the case of a dielectriccylinder laser with arbitrary cross section, E will denote the zcomponent of the electric or magnetic field for kz=0 modessee Ref. 1 for the justification of this model in the case ofa dielectric cylinder laser.

ii We assume a uniform, homogeneously broadenedatomic medium with density of atoms na, atomic transitionfrequency a, and the quantum mechanical density matrix ijx , t.

We decompose P= PL+ PNL, into a linear component PLfor instance the nonresonant response of the substrate mate-rial, and a nonlinear resonant component PNL due to the gainmedium, here taken to be the uniformly distributed set oftwo-level atoms. Let

PLx,t = dtx,t − tEx,t . 6

Decomposing the fields P= PL+ PNL and E as

Fx,t = F+x,te−iat + F−x,teiat, 7

where F= E , PL , PNL, and using the slowly varying enve-lope approximation and the rotating wave approximation,Eq. 3 becomes

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2e +a

2

c2 e + 4pL +2ia

c2 e + 4pL = −4a

2

c2 pNL.

8

The envelope equations corresponding to Eqs. 3–5 be-come

tn2e =i

2aa

2n2 + c22e + 2iapNL, 9

pNL = − pNL +g2

ieD , 10

D = D0 − D −2

iepNL

* − pNLe* . 11

Here, pNLx , t= PNL+ x , t, pLx , t= PL

+x , t, ex , t=E+x , t,and n2 stands for the convolution operator dtn2x , t− t.The time-dependent index of refraction nx , t is related tox , t by n2x , t=1+4x , t. In the derivation of theseequations we have assumed that the residual time depen-dence of the envelopes is much slower than a and that a and dropped the terms proportional to e , pNL ,apNL , pL.

The properties of the index of refraction nx , t of thebackground medium will depend on the particular complexlasing system under investigation; for dielectric cavity lasersit will be constant in the cavity and unity outside, for a ran-dom laser it will vary randomly in space within some regionand then fall off to unity at the edges of the medium. For thewave-chaotic dielectric cavity the “randomness” comes fromthe scattering at an irregularly shaped boundary. In what fol-lows the background index of refraction can be either con-stant in space within a boundary, or varying in space withinsome boundary and then uniform outside, allowing for treat-ment of both cases. We will also replace pNL by p henceforth.

We assume now that the fields px , t and ex , t are mul-tiperiodic in time

px,t =

pxe−it, ex,t =

xe−it 12

in the steady state. In contrast to previous approaches5,26,31–34, we leave the spatial functions px and xto be unknown functions and to be the unknown lasingfrequencies to be determined. Such a solution with a finitenumber of discrete frequencies a “multimode” lasing solu-tion is only possible if the inversion is approximately sta-tionary 35. Henceforth, we will assume this to be the case.Note also that this solution has zero linewidth, hence weneglect any line broadening effects due to quantum fluctua-tions which is not captured within the semiclassical lasertheory, including the additional quantum broadening arisingfrom the nonorthogonality of the modes Petermann factor36. From Eq. 10, the ansatz equation 12, and the timeindependence of the inversion, we obtain

px =g2

i

Dx− i +

x 13

and

Dx D0

1 +1

ec2

gx2. 14

Here, g=2 / 2+

2 is the gain profile and ec

= /2g gives the typical electric field scale. Henceforthwe will measure the lasing modes and polarization,x , px in units of ec. We can then rewrite Eq. 9 inthe form

2iatn2 + a

2n2 + c22ex,t = − 4a2px,t . 15

Substituting the results for px into Eq. 15, the pumpedatomic medium becomes equivalent to a multiperiodic forc-ing term on the right side of the Eq. 15, given by

− 4a2px,t = − 4a

2g2

i

D0

1 +

gx2

1

− i +

xe−it, 16

where the polarization which is the source for ex , t=xe−it is itself a nonlinear function of . ThusEq. 15 gives a self-consistent set of equations for the modeamplitudes x and lasing frequencies .

To derive from Eq. 15 an integral equation for eachx, we first Laplace transform Eq. 15 to the frequencydomain

a + 2n2x,ex, + c22ex, = − 4a2px, .

17

Here, n2x ,=1+4x ,, the Laplace transforms are de-fined by p= 1

20dtei+itpt, and we only retain the in-

finitesimal imaginary part of the frequency, , when it isneeded for convergence. Note that the real frequency vari-able corresponds to the residual time dependence of px , tafter removing the rapid variation at frequency a, andhence we have used a

2+2aa+2.Initially we treat the right-hand side of Eq. 17 as a given

source and introduce a Green function for the inhomoge-neous equation, which ensures that the solution ex , sat-isfies the appropriate boundary conditions, i.e., that the fieldhas only outgoing contributions. Assuming for the momentthat such a Green function exists we can immediately write aformal solution to Eq. 15 within the cavity

ex, =

x2i − + i

= −4a

2

c2 cavity

dxGx,xpx, . 18

The integral on the right-hand side of Eq. 18 is only overthe cavity because the source pumped medium is zero out-side of the cavity. The cavity Green function satisfies theequation

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2 +n2

c2 + a2Gx,x = x − x 19

which formally inverts Eq. 17 to yield Eq. 18 for theelectric field component ex , for x within the cavity. Out-side the cavity the wave equation changes and a differentGreen function would be needed to invert the equation.However as the cavity Green function determines the solu-tion on the boundary of the cavity, one can just use the out-going wave boundary condition to fix the solution outside.This will be done through our definition of the CF statesbelow.

Both sides of Eq. 18 involve sums over the modes weighted by the factor 1 /2i−+ i which will givethe multiperiodic time dependence of the solution when it isinverse Laplace transformed. The analytic structure of therelevant Green function can be shown to produce no addi-tional harmonic dependence in time as t→ so that it ispossible to equate the residues of these real poles to yield anintegral equation for each of the modes x,

x = i4a

2g2D0

c2− i +

cavity

dxGx,xx

1 +

gx2. 20

This set of nonlinear integral equations for the lasing modesis completely general assuming multiperiodic solutions ex-ists. If the modes of the cavity are sufficiently high Q that theimaginary part of their frequency is much smaller than thetypical mode spacing then one can introduce an approxima-tion for the Green function of Eq. 20 which leads to gen-eralizations of the conventional multimode lasing equationssee below. However, it is possible to work directly with Eq.20 when, as in a random “diffusive” cavity, the assumptionof well-separated high-Q modes is not correct. In this casedifferent approximations to the Green function, similar tothose made for disordered electronic systems, would be moreappropriate and would yield composite lasing modes gener-ated from many very broad cold cavity resonances. We willnot pursue this direction in the current work, but wish topoint out the feasibility of such an approach.

III. BIORTHOGONAL CONSTANT FLUX STATES

A. General definition and properties

To express the lasing solutions x as an expansion ina set of linear cavity modes we now introduce a spectralrepresentation of the Green function Gx ,x of the form

Gx,x = m

mx,m* x,

mkk + ka2 − km + ka2, 21

where k= /c, ka=a /c, mk is a normalization constantdefined in Eq. 28 below and the functions in the numeratorare chosen so that G satisfies the outgoing wave boundaryconditions. It is here that the non-Hermitian nature of these

boundary conditions requires a change from the standard ap-proach utilizing closed cavity Hermitian modes. We neededto introduce in Eq. 21 two sets of functions which are bior-thogonal 37 mx ,, mx ,. The set mx , sat-isfies the eigenvalue equation

− 2mx, = n2x,km + ka2mx, , 22

defined in the cavity, D. While the Green function of Eq.21 is only used to find the electric field inside the cavityand would give an incorrect answer for the electric fieldoutside, one can define the functions m outside the cavityas well. Outside the cavity they satisfy the free wave equa-tion with the fixed external wave vector, ka+k not ka+km,

− 2mx, = ka + k2mx, , 23

with the outgoing wave boundary conditions expressed in ddimensions by

rd−1/2mr → , eika+kr, 24

equivalently it can be expressed as a linear homogeneousboundary condition on the function mxrd−1/2mr inthe form d ln /dr= ika+k as r→.

Thus each m satisfies a different differential equation in-side and outside the cavity but are connected by the continu-ity conditions

mD− = mD+, nmD− = nmD+, 25

at the boundary of the cavity here D denotes the boundaryof D and n is the normal derivative on D 39. This choiceguarantees that in the approximation developed below, inwhich each lasing mode is proportional to a single m, theelectric field will also be continuous at the dielectric bound-ary, satisfy the free wave equation outside, and have onlyoutgoing components at infinity. The spectral representation,Eq. 21, is only meaningful for x ,x within the cavity andon its boundary. We will focus on the typical case in whichone can impose the outgoing wave boundary condition di-rectly on the cavity boundary as in the slab and cylinderexamples given below. In this case the eigenvalues km

+ka2 are determined by linear homogeneous boundary con-ditions on a finite region, and are complex due to the com-plexity of the logarithmic derivative at the boundary.

Note however that this condition differs subtly from thecondition defining the quasibound QB states, for which thecomplex eigenvalue km itself would appear in the logarithmicderivative at the boundary, and hence in the external outgo-ing wave. As a result, as noted earlier, the QB states growexponentially at infinity and carry infinite flux outwards. Thestates m we have just defined have real wave vector at in-finity and carry constant flux; hence we refer to them asconstant flux CF states.

The CF states have appeared naturally from the conditionthat the Green function of Eq. 18 satisfy the correct radia-tion boundary condition; this allows us to formulate a lasingtheory valid for an open cavity in terms of these states. In thesimplest approximation, to be developed below, the lasingmodes are just single CF states those above threshold for agiven pump power, each scaled by an overall intensity fac-

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tor determined by solving Eq. 20, or its near-threshold ap-proximation, which turns out to be a generalization of theHaken-Sauermann equations. Note that CF states have com-plex wave vector inside the cavity, and are amplified whiletraveling in the gain medium, but real wave vector and con-served flux outside as one expects for lasing modes see Fig.2 below. We show in the Appendix that for high-Q modesthe CF wave vector inside the medium is very close to that ofthe corresponding QB state, justifying the use of QB states tocalculate the Q value of lasing modes, as is often done.

As mentioned already, due to the non-Hermitian nature ofthe associated eigenvalue problem we need to introduce asecond set of functions, which satisfy different boundaryconditions, in order to represent the Green function of Eq.21. The functions mx , satisfy

− 2mx, = n2x,*km + ka2mx, 26

with the continuity conditions 25 but with the incomingwave boundary condition

rd−1/2mr → , e−ika+kr. 27

It can easily be shown 37 that km2 = km

2 * and mx ,=m

* x ,.The key properties which motivates the use of the two

sets of functions in the spectral representation is that theysatisfy biorthogonality with respect to the following innerproduct

mn = D

dxn2x,m* x,nx, = mmn,

28

and form a complete basis within the cavity 37

n2x,m

1

mmx,m

* x, = x − x . 29

The normalization factor m can be set equal to unity for anyspecific choice of the frequency , but it is useful to retain itexplicitly in some contexts to denote that the normalizationdepends on . Note that we have here the usual Hermitianinner product, except that it is defined between eigenfunc-tions living in adjoint spaces. These pairs of functions witheigenvalues related by complex conjugation are called bior-thogonal pairs or partners.

The boundary condition 27 clearly has the meaning thatthe adjoint CF states have constant incoming flux at infinity.For these states one can view the source as being at infinityand emitting radiation which impinges on the dielectric me-dium and then decays inwards at the same rate as the CFstates grow while moving outwards from the origin. In otherwords the dielectric medium acts as an amplifying mediumfor outgoing CF states and as an absorbing medium for in-going adjoint CF states.

B. CF states and multimode lasing

The importance of CF states becomes clear when we con-sider the self-consistent integral equations for the lasing

modes 20 using the CF spectral representation of the Greenfunction. We can write Eq. 21 in the form

Gx,x =c2

2am

mx,m* x,

m − m30

where we have used the fact that a ,m. As alreadynoted, the km=m /cqm− im are complex but for thehigh finesse cavities their imaginary parts, m, will be smallcompared to the typical spacing between their real parts.Therefore, for the case of cavities with high finesse, theGreen function is going to be multiply-peaked as a functionof which is real around the values cqm. The mth termin the spectral sum is large when =cqmwm, proportionalto 1/m; whereas the other terms will have denominators atleast as large as q, the mean spacing between the real partof the eigenfrequencies which is roughly the same as themode spacing of the closed cavity. This spacing is constantin 1D, and only decreases as some power of the parameterkR R is the typical linear size of the resonator in higherdimensions, whereas the imaginary quantities m can be ex-ponentially small in this parameter. For typical dielectriccavity resonators there are many high-Q modes for whichmq and the Green function will be dominated by thesingle nearest CF pole at =wm. It is thus natural to intro-duce the single-pole approximation to the CF Green functionin which we replace the full Green function near the lasingfrequencies by the single term involving the nearest pole inthe spectral representation 40.

Thus we assume that the possible lasing frequencies are inone-to-one correspondence with the real parts of the CF ei-genvalues wm=cqm. We write m+m

− im m

0. In the single-pole approximation, the lasing modes aregiven by x=am

mx and are just proportional to single

CF states. Note that not all CF states will lase; the solution,Eq. 20, will find that many of the coefficients am

are zero;the remainder will give the intensities of the various lasingmodes. Thus, as noted above, the lasing modes at this levelof approximation are identical to a subset of the CF statesup to an overall scale factor. We will work within this ap-proximation for the remainder of this article, so we can nowadopt a simpler notation, dropping the subscript m: am

→a ,mx→x.

Having just shown that CF states are the correct functionsto describe solutions for high-Q lasing modes of open reso-nators, before we discuss the derivation and solution of thelasing equations for their intensities and frequencies, we willexamine some examples of CF states and relate them to themore familiar QB states. Henceforth we focus on the case ofdielectric cavity lasers with uniform index

C. Examples of CF states

1. One-dimensional CF state

The simplest example of a set of CF states which also hasthe virtue of being essentially exactly solvable are the statesof a semiinfinite slab laser see Fig. 1. Note that this ex-ample is also a crude model for an edge-emitting semicon-ductor laser cavity.

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Consider a dielectric medium of index n=n0 which is uni-form and infinite in the y ,z directions and extends from x=0 to x=a in the x direction. At x=0 it is terminated in aperfectly reflecting mirror and from x=a to there isvacuum n=1. To clarify the difference between the CFstates and the usual cold cavity resonances, first let us con-sider the quasibound states of such a resonator. These aredefined as the solutions of

− x2mx = n0

2km2 mx 31

and the boundary condition which can be moved to the in-

terface xmx=a= ikmmx=a. The complex eigenfre-quencies of this problem are the solution of

tann0kma = − in0 32

which can be explicitly solved to yield

n0kma = m + 1/2 − i1

2lnn0 + 1

n0 − 1 n0qm − i0a ,

33

where m=1,2 , . . . ,. Note that the imaginary part of thewave vector is always negative, as it should be, and in thiscase is constant, corresponding to the fixed transmissivity ofa dielectric interface of index n0 at normal incidence. Sincethese resonances are outgoing at infinity, each QB state var-

ies as e+ikmxe+0x and grows exponentially at infinity asmentioned in the general discussion of QB states in the In-troduction.

Now consider the CF states; inside the medium they sat-isfy exactly the same equation and the continuity conditionsat the dielectric interface are the same as well, but the CFstate outside the dielectric satisfies the wave equation with afixed external wave vector

− x2mx = k2mx , 34

and with the corresponding outgoing wave boundary condi-tion, xmx=a= ikmx=a. This defines a family of basisstates depending on the choice of the real external frequency=ck. These eigenvalues satisfy the equation

tann0kma = − in0km

k. 35

In the Appendix it is shown that this is a well-defined eigen-value problem for any value of k and the eigenvalues arealways complex with negative imaginary part, similar to theusual QB states. But again, unlike the QB states, since k isreal, these states simply oscillate at infinity as eikx and carrya constant outgoing flux.

One can see that if k is chosen to be equal to qm, the realpart of the mth QB state wave vector, then the two eigen-

value conditions are almost the same in this vicinity. Notehowever that for every choice of the external wave vector kthere is an infinite set of CF eigenvalues. It is shown in theAppendix that from that infinite set there is one mth CF statewith a wave vector very close to that of the correspondingQB state for k= qm. Specifically, the difference between thewave vector of the mth QB state and the corresponding CFstate is given by

n0qm − qma =n00

k; n0m − 0a =

fn0ka2 , 36

where fn0 is a number of order unity depending on theindex, n0. Hence this difference is small in the semiclassicallimit, ka1. Moreover both the CF and QB states are of theform A sinkmx within the dielectric, so if their complexwave vectors are close, then the CF and QB states are almostidentical in the medium. Thus we see that the standard coldcavity resonances are almost equal to the lasing mode, butnot exactly equal. Because the CF states are sines of complexwave vector they will oscillate and increase in amplitudewithin the medium Fig. 2.

n>1

FIG. 1. Schematic of a dielectric slab cavity of index n0 andlength a with perfectly reflecting mirror at the origin.

0 0.5 1 1.5

x/a0

1

2

3

| ϕ (

x)| 2

0 0.5 1 1.5x/a

0

0.5

1

1.5

2

2.5

3

| ϕ(x

)| 2

(a)

(b)

FIG. 2. Color online Solutions of the QB and CF state eigen-value equations for a dielectric slab cavity with n0=2. a A QB

state at kma=11.51917306−0.53647930i. b A CF state at kma=11.55535804−0.53100809i with external wave vector ka=11.51917306. Since the eigenvalues are almost the same the twosolutions are almost identical within the cavity and show large am-plification, but they differ qualitatively outside as the QB intensitygrows exponentially while the CF intensity remains constant as re-quired for a true lasing solution.

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As noted in the general discussion, these CF states are notorthogonal when integrated within the medium. It is there-fore useful to define adjoint CF states which satisfy the com-plex conjugate differential equation, and with the incomingboundary condition at infinity, xmx=a=−ikmx=a. It is

easy to show that the eigenvalues km and eigenfunctions mof this problem are the complex conjugates of the CF eigen-values km, and eigenfunctions m. In the Appendix it is con-firmed that these functions satisfy the orthogonality relation0

adxn02mxn

*x=0adxn0

2 sinn0kmxsinn0knx=mmn

and the normalization constant m is calculated. The bior-thogonality of these functions follows from the boundaryconditions at x=a.

2. Cylindrical CF states

The next case of interest is the case of a circular 2D orcylindrical 3D dielectric resonator of uniform index n0 andradius R. The CF eigenvalues can be found by applying theconditions Eqs. 24 and 25 for the circle 2D and cylinder3D, solutions uniform in z direction. Solutions can be la-beled by their good angular momentum index in the z direc-tion, M, leading to a countably infinite sequence of eigenval-ues for each value of M which we will label by m. Thestates take the form

mMr,, = JMn0kmre±iM, r R ,

JMn0kmRHM

+ kmRHM

+ kre±iM, r R 37

and the CF eigenvalue condition is found to be

JM n0kmRHM+ kR

JMn0kmRHM+kR

=1

n0

k

km. 38

Note that this equation differs from that defining the QBstates for this problem simply in that the corresponding QBeigenvalue condition would have km appearing in the argu-ments of HM

+ , the external solutions, as well as in the argu-ments of JM, the internal solutions.

As for the slab resonator discussed above, if we choose=cqm, the real part of a specific QB state frequency, thenwe will find one CF eigenvalue close to the complex QBstate eigenvalue, as long as the QB state itself has smallimaginary frequency high Q. A comparison of two suchcorresponding states is given in Fig. 3. Further analysis ofthe 2D case is given in the Appendix.

3. CF states for general shapes

The CF boundary conditions for a general dielectric bodyof arbitrary shape and uniform index of refraction n0 are stilleasily formulated in terms of outgoing spherical waves atinfinity, since the object will appear pointlike at arbitrarylarge distances. We simply require that the CF states satisfythe wave equation with index n0 within the medium and thatfar away they have the form in 2D

mr → , = fm1 r

eikr. 39

In order to satisfy this boundary condition it will be neces-sary to solve the wave equation within the body by somemethod and continue it sufficiently far outside to a sphere orcircle enclosing the body at which radius the interior solutioncan be connected to a superposition of outgoing waves withthe wave-vector k. The situation is simplified if the dielectricbody has a smooth shape and is not too far from spherical, inwhich case a Rayleigh Bessel expansion can be imple-mented within the cavity and can be matched to outgoingHankel functions directly on the boundary specified by R

mr, = M

MJMn0kmreiM, r R ,

M

MHM+ kreiM, r R . 40

The continuity boundary conditions on the boundary can beeasily cast into a secular equation and the eigenvalue condi-

0 0.5 1 1.5 2r/a

0

0.01

0.02

0.03

0.04

| ϕ(r

)| 2

0 0.5 1 1.5 2r/a

0

0.01

0.02

0.03

0.04

| ϕ(r

)| 2

(a)

(b)

FIG. 3. Color online Solutions of the QB and CF state eigen-value equations for a cylindrical dielectric resonator with n0=1.3and angular momentum quantum number M =40. a A QB state at

kmR=71.17700357−0.74936509i. b A CF state at kmR=71.19539070−0.74853902i with external wave vector kR=71.17700357. Again the two solutions are almost identical withinthe cavity; outside the QB solution grows while the CF solutiondecays as 1/r to conserve flux in 2D. The amplification of thesolutions within the cavity is less obvious since both solutions havethe usual peak at the caustic at r /a0.45; the decay away from thispeak is actually slower going outwards then it would be for the coldcavity due to the amplification.

TÜRECI, STONE, AND COLLIER PHYSICAL REVIEW A 74, 043822 2006

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tion replaced by a singularity condition. The method is thenidentical to that described in Ref. 1 for finding the QBstates of such an asymmetric resonant cavity ARC exceptthat the matrix S which appears from the matching condi-tions is slightly different. Two examples from implementingthis method for CF states are shown in Fig. 4 below.

IV. GENERALIZED MULTIMODE LASING EQUATIONS

A. Nonlinear multimode equations

We now return to the formulation of the multimode lasingequations within the single-pole approximation to the Greenfunction. The outgoing lasing field ex , is only nonzero atthe lasing frequencies; we approximate the Green functionin the vicinity of = by the term arising from the nearestpole, i.e., Gx ,x =c2x ,

* x , /2a−. Here and henceforth we will assume the

CF modes to be properly normalized. Equation 20 nowtakes the form

a1 − iD01

− + i

− i +

cavity

dx

* x,x,

1 +

ga2x,2 = 0

41

where D0 is the scaled pumping rate given by

D0 =2ag2

D0. 42

We are looking for lasing solutions to these equations, i.e.,solutions where at least one a0 and positive. Since a

=0,∀ is always a solution for the steady-state equations,once we reach a pump level at which nonzero solutions exist,we must check their stability.

This can only be done by writing time-dependent equa-tions for the CF state amplitudes at analogous to thestandard modal expansions and determining the effect of asmall deviation from the coexisting solutions with a=0 anda0. Previous work based on the standard theory finds35 that whenever a solution with a0 exists, the a=0solutions are unstable to small deviations, which then flow tothe finite amplitude solution with the largest number of las-ing modes. We assume this property for the current work andwill explore this stability issue in subsequent work. Rulingout the identically zero solutions, and defining Ia2, thegeneral nonlinear multimode problem can be cast in the form

1 +i− + i− i +

D0

= FI , 43

where

FI 1 − cavity

dx

* x,x,

1 +

gIx,2,

44

and we must look for the solution with the maximum numberof nonzero positive I. Note that the CF states x in thedefinition of F are understood to be evaluated for =

and normalized to unity at this frequency. We will refer tothe full nonlinear form of the self-consistent equation as thenonlinear Haken-Sauermann NLHS equations; these equa-tions can be treated numerically and we will present resultsfor the single- and two-mode cases below.

B. Generalized Haken-Sauermann equations

To treat the multimode case more simply and to makecontact with the standard Haken-Sauermann treatmentwe now make the near threshold approximation for F

expanding the denominator of the integrand as 1/ 1

(a)

(b)

FIG. 4. Color online False color plots of CF solutions for adielectric cavity with a quadrupolar deformation R=R01+ cos 2 at =0.16 and an index of refraction n=3.3 at externalwave vector kR0=20.0. left A bouncing ball mode at kmR0

=20.48417472−0.09782270i and right a bowtie mode at kmR0

=19.71417046−0.040257141i. Modes of the bowtie type werefound to lase in the experiments of Ref. 9.

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+gax21−gax2. This leads tothe cubic nonlinearity assumed in standard texts5,26. Wethen arrive at the generalized linear Haken-SauermannGHS equations for the modal intensities,

1 +i− + i− i +

D0

=

gAI, 45

where the overlap matrix A is given by

A = dx* xxx2. 46

In the limit in which and the frequency shift

0 we recover the standard form of the HS equation Eq. 1above, with the important difference that the overlap matrixA is differently defined; it is complex and involves theintegral of three outgoing CF states and one biorthogonalincoming state, whereas the usual HS equations A involvesthe squared moduli of two closed cavity states and is purelyreal. This new form allows us to determine iteratively thelasing frequency shifts in the multimode case, whereas thereal approximation does not. A second difference is that inthe usual HS equations the cavity widths are put in byhand at this point, as the closed cavity states have zero width.In our approach the widths appear automatically as theimaginary parts of the complex wave vectors of the CFstates. Finally we note that in this formulation the theory ofinteracting lasing modes can be seen to have some similarityto the theory of interacting electrons in quantum dots, as theoverlap matrix A is similar to a mesoscopic two-body ma-trix element. When the resonator involved has complex e.g.,wave chaotic modes these matrix elements will fluctuateand depend sensitively on external parameters such as shapeand pump profile. Concepts from random matrix theory andsemiclassical quantum mechanics may be useful in analyzingthese interactions; an example of this approach using theusual near threshold HS equations is the work of Misirpa-shaev and Beenakker 28.

V. SOLUTION OF THE GENERALIZED HS EQUATIONS

A. Iterative method

Both the nonlinear form 20 and linear form 45 of thegeneralized HS equations require a self-consistent iterativesolution since the function FI in its exact and linearizedform depends on the unknown lasing frequencies as wellas the unknown intensities I. Recall that the quantities

and which appear explicitly in the NLHS and GHS equa-tions are the difference between the lasing frequency andthe nearest CF state complex frequency ; thus both ofthese quantities are determined by the real lasing frequen-cies. Therefore an initial ansatz for the lasing frequencies isrequired in order to begin the root search to determine theI. Conceptually the linear and nonlinear cases are solvedby similar iterative schemes, the only difference being that inthe linear case GHS the root search becomes equivalent toinverting a linear system. In the linear case the solution mustobserve the constraint that I0 while in the nonlinear case

the positive roots, if they exist, have to be determined asI=0 is always a solution.

At least two approaches are possible. One can start withthe N quasibound state frequencies which have real partswithin a linewidth of a and use Re

qbqb as the

initial iterate for the real lasing frequencies. A second ap-proach is to linearize the eigenvalue equation around theatomic frequency a, assuming that the imaginary parts ofthe CF frequencies do not change rapidly so that one can usethe imaginary parts determined by solving the CF eigenvaluecondition at a as a starting point for the iteration. Each ofthese approaches has advantages and can lead to a tractablescheme, as we will discuss below in the context of specificexamples. We will describe the first approach here.

We first separate the real and imaginary parts of Eq. 45,

1 −1

D0

+ = ReF 47

− + = D0 ImF , 48

where we have dropped the tilde on D0. Our initial guess forthe lasing frequencies is

0=qb. The quantities

, , ,F in Eq. 47 are all functions of the true las-ing frequencies . However we will approximate them bytheir values at the

0: let 0=Re

0−0,

0

=−Im0, F=F

0F0; Eq. 47 then be-

comes

1 −1

D0

0

0 + 0 = ReF

0I . 49

This equation can be solved for the modal intensities I0 by

varying the pump power and looking for positive roots fordifferent trial sets of lasing modes. In practice one easilyfinds the threshold for the first mode to lase and then in-creases the pump in steps above this.

Having determined the intensities, Eq. 47 yields an ex-pression for the lasing frequencies by substituting =

0

+, =0−, =

0, and solving for ,

=

0 + D0 ImF0 −

00

+ 0 . 50

These equations yield the updated lasing frequencies 1

which can be reinserted into the equations to obtain higheriterations. We should note that Eqs. 49 and 50 are correctto O and O

2 , respectively, and we have neglectedterms of O / which can be shown to be small in theshort wavelength limit. Thus Eqs. 49 and 50 are expectedto converge rapidly with iteration number. In very recentwork we have developed a somewhat different iterativemethod which treats the lasing frequency and intensity of thesame footing and which is found to converge rapidly 41.

The equation for the frequency shifts is a generalizationof well-known results for the single-mode case and theclosed cavity. First note that for the closed cavity the func-tion F is real and does not enter the imaginary part of theHS equations; hence the pump strength drops out. In addi-tion, in a standard treatment the real frequency shift

0 is

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assumed to be zero since one begins with the cold cavityfrequency, so one obtains

=−

00

+ 0 . 51

This gives the well-known result that when the cavity widthis large compared to the atomic relaxation rate one obtains=−

0→=0, i.e., lasing at the atomic frequency a;when

0 the cavity mode is much narrower thanthe atomic linewidth one obtains =−

0 /0

→0=Re

qb, i.e., lasing at the cold cavity fre-quency. Our generalization shows that openness of the cavityintroduces the pump strength into these equations and im-plies a change of the lasing frequencies with pump strength.This is a prediction of our theory which can in principle betested experimentally.

B. Single-mode solutions

1. Near-threshold behavior

The simplest solution of Eqs. 49 and 50 is the casewhere the electric field is oscillating at a single frequency. Then, we can solve Eqs. 49 and 50 straightforwardlyfor I and

I =1

g0A

0 1 −

0

D0 . 52

With uniform pumping this equation implies that the firstmode to lase and the only one in the single-mode approxi-mation is indeed the mode with the highest Q, correspond-ing to the CF state with the smallest value of its imaginarywave vector . Of course, in a realistic scenario the lasertypically will not remain single mode for the whole range ofpump rates D0 and other modes can begin to oscillate. Asnoted above, unlike the closed cavity single-mode theory, forwhich the lasing frequency is independent of pump strength,in this case we find

=

0 + D0g0ImA

0 I

+ 0 . 53

The first term in the numerator gives the “center-of-mass”formula for the lasing frequency discussed above, but thesecond term is proportional to I and demonstrates that thereis a frequency pulling or pushing effect even in the single-mode case which depends on the pump strength and on theimaginary part of the inverse mode volume, A

−1 . The gener-alization to multi-mode lasing is trivial here with the factorImAI simply replaced by ImAI cf. Eq. 45.

2. Far from threshold behavior

To generalize the above results to arbitrary pump rates wesimply replace in Eqs. 49 and 50 F by its exact expres-sion

F = 1 − cavity

dx

* xx1 + gx2I

, 54

where it is understood that the CF states are calculated at theiteration frequency =

i.In Fig. 5 we show an example of the proposed iteration

scheme for a dielectric resonator of circular cross section.There are several observations we can make here. First of all,the intensity calculated from the near-threshold theorygreatly underestimates the actual intensity at higher pumpstrengths D0. As we shall further discuss below in Sec. V C,this will have important consequences for mode competitionin the multimode regime as the nonlinear thresholds will besubstantially different from those predicted by the near-threshold approximation. Second, as seen in Fig. 5b wenow obtain a frequency shift that is power dependent, with anonlinear dependence on the lasing intensities. This depen-dence is stronger the leakier the lasing mode. Third, the exactformula leads to an approximately linear dependence of themodal intensity I on D0. This linear dependence is termed“saturation” in laser texts because it is much slower than thenear threshold rise, but it is still a much stronger dependencethan predicted by the near threshold HS theory.One can obtain a naive approximation for I byreplacing 1+gx2I in the denominator withgx2I, then

I = Reg

D0 55

where

0 0.25 0.5 0.75 10

100

200

300

400

500

600

700

I

D0

0 0.25 0.5 0.75 10

0.2

0.4

0.6

0.8

1

1.2x 10

−3

Ω

D0

(a) (b)

FIG. 5. Color online Single-mode single-pole solutions for themodal intensity vs pump strength D0 for a circular dielectric reso-nator with an index of refraction n=2.0 and gain center at a

=50.0. a Single-mode solution for the intensity I of the mode withan angular momentum quantum number M =40 lying closest to thegain center. Dashed line near-threshold solution; solid line, exactsolution for modal intensities I; dot-dashed line, far-from thresholdapproximation discussed in the text plotted only for I120. bLasing frequency as a function of the pump strength D0. Notethat is measured from its value at the lasing threshold given by0=49.6668 and the shift is towards the gain center frequency.

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= cavity

dxe2ix 56

where x=−iArgx. For comparison, we plot theresult of this expression in Fig. 5b dot-dashed line. Wesee that this formula is qualitatively correct but overesti-mates the modal intensity.

C. Two-mode solutions

The numerical effort required to solve the linear and non-linear generalized HS equations for a nontrivial case, e.g., adielectric cavity with fifty modes underneath the gain curve,is substantial. Moreover the interesting questions requirevarying the pump strength and the shape of the cavity. Inearlier work we have done extensive calculations along theselines using the near threshold HS equations for the closedcavity and then using reasonable but uncontrolled approxi-mations to evaluate the modal intensities outside the cavity29. The current generalized formalism gives us a method toevaluate emission directionality and output intensities cor-rectly and without any approximations beyond the usual onesallowing a multimode solution. It also presents a formulationof the lasing equations valid away from the threshold, usingthe general function, FI, defined in Eq. 44 above. Astriking shortcoming of the near threshold approximationthat we found in our earlier work 29 is that it appears tooverestimate greatly the number of lasing modes comparedto the experimental observations 9. Here we will notpresent realistic calculations on complex cavities using ourgeneralized formalism; we defer such calculations to futurework. Instead we present a simple comparison of the near-threshold and nonlinear lasing solutions for the two-modecase, which confirms that the widely used near thresholdapproximation can introduce a large quantitative error in thelasing thresholds and hence overestimates greatly the numberof lasing modes for a given pump power in the complexdielectric cavity lasers we are studying. We solve Eq. 43for two interacting modes of the dielectric cylinder othermodes which might also lase are neglected. In Figs. 6 and 7we show the results. For mode 1, with the lower threshold,the near-threshold and nonlinear calculation give the samethreshold as they must D00.11, and then deviate substan-tially well above threshold as we already saw in the previoussection. Mode 2, which would have a threshold of D00.18 without mode competition has a threshold of D00.25 in the near-threshold approximation, including the ef-fects of mode competition, but has a correct threshold, takingnonlinear effects into account, of D00.41. Hence we seethat in the large interval 0.25D00.41 the near-thresholdtheory predicts that two modes would be lasing and withcomparable intensity, while the full theory predicts only onelasing mode. Moreover, because of this delay, when mode 2does begin to lase, its intensity lags mode 1 by an order ofmagnitude in contrast to the prediction of the near thresholdanalysis. We expect these discrepancies to be enhanced ifthere are many competing modes and many fewer modes tobe lasing at a given pump power than predicted by the near-threshold theory. These issues will be explored in detail infuture work.

VI. SUMMARY AND CONCLUSIONS

We have reformulated semiclassical lasing theory to treatopen cavities, with particular attention to complex and ran-dom lasers, for which the output power and directional emis-sion patterns are not trivially found from knowledge of theinternal modal intensities. We begin from the assumption thata steady-state multiperiodic lasing solution exists and isstable. A key idea is to formulate the theory in terms of theself-consistent linear response to the polarization described

0 0.25 0.50

50

100

150

200

250

D0

I 1

0 0.25 0.50

5

10

15

20

D0

I 2

(a) (b)

FIG. 6. Color online A single-pole two-mode solution for themodal intensities as a function of pump strength D0 for a circulardielectric resonator with an index of refraction n=2.0 and gain cen-ter at a=20.0. We have chosen =50 using effectively a flat gaincurve. The two modes have the angular momentum quantum num-bers M1=20 and b M2=18, respectively. The modal intensities I1

and I2 are given in a and b. Solid lines denotes the exact solu-tion, while we plot the near threshold approximate solution indashed lines. The noninteracting thresholds are given by Dth

1

=0.1069 and Dth2=0.1767 marked with vertical arrow in b. By

definition modal interactions cannot change the first threshold,which is thus the same in the near-threshold and exact solutions.However interactions dramatically modify subsequent thresholds; inthis case the second threshold is increased to Dth

2=0.2464 in thenear threshold approximation, but actually should be increasedmuch more, to Dth

2=0.4139, based on the exact solution. This in-dicates that even taking modal interactions into account through thenear-threshold approximation is inadequate and likely greatly over-estimates the number of lasing modes at a given pump power.

0 0.2 0.4 0.6 0.8 1−6

−5

−4

−3

−2

−1

0

1

2x 10

−3

D0

∆Ωµ

FIG. 7. Color online The corresponding laser frequency varia-tion with pump power for the example in Fig. 6 as a function ofpump strength D0, for the exact solution. The frequencies of the twomodes are measured with respect to their values at their respectivelasing thresholds given by 1=19.8584 full line and 2

=20.3764 dashed line. Note that since the gain center is a

=20.0 both modes are pulled towards the gain center.

TÜRECI, STONE, AND COLLIER PHYSICAL REVIEW A 74, 043822 2006

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by an outgoing Green function. This yields a set of self-consistent integral equations for the lasing modes which canbe solved by various means. The approach which is closestto conventional multimode lasing theory is to write a spectralrepresentation of this Green function in terms of outgoingstates with constant flux. For high Q cavities these states aresimilar but distinct from the usual resonances or quasiboundstates; they satisfy useful biorthogonality relations with ad-joint functions. The Green function is then approximated bya single pole near each lasing frequency implying the lasingmode is just an intensity I times a single CF state. Generaland near-threshold equations then follow for these intensitiesI and for the lasing frequencies . The near thresholdequations are generalizations of the well-known Haken-Sauermann equations derived for a closed cavity. Theseequations can account for the effects of mode competitionand spatial hole burning which are important but difficult totreat in complex cavity lasers. An iterative scheme was de-scribed for solving these equations and some illustrative re-sults were given for the single-mode and two-mode cases.The one- and two-mode solutions indicate that the near-threshold approximation substantially overestimates thenumber lasing modes well above threshold and underesti-mates their output power.

In our view this approach clarifies the longstanding ques-tion of how to treat rigorously open cavities within semiclas-sical lasing theory. It now remains to apply this formalism tocases of interest: dielectric microcavity lasers of interestingshape, cavities with fully chaotic ray motion, and lasing fromrandom media.

ACKNOWLEDGMENTS

We thank Harald Schwefel, Takahisa Harayama, SusumuShinohara, and Stefan Rotter for useful conversations. Thiswork was supported by NSF Grant No. DMR 0408636.

APPENDIX: PROPERTIES OF THE CONSTANT FLUXBASIS

In this Appendix we will explore the constant flux CFstates, as defined in Sec. III C 1. We investigate the analyti-cally tractable cases of the 1D dielectric slab laser and the2D cylindrical laser, and argue that the qualitative features ofthe solutions for these simple geometries should also hold formore general shapes. Here we focus on three properties ofthe CF basis. First, that for each high-Q QB state of thecavity there exists a single CF state with similar real andimaginary part and hence very similar behavior within thecavity where they satisfy the same differential equation withjust a slightly shifted eigenvalue. Second, we prove that allthe CF states have eigenvalues with negative imaginaryparts, corresponding to amplification of the wave within thecavity. Third, we confirm explicitly the biorthogonality of thetwo outgoing and incoming CF bases within the cavity fora general geometry.

1. Dielectric slab cavity

As stated in Sec. III C 1, for a semi-infinite dielectric“slab” resonator the QB and CF eigenvalue equations are

tann0kma = − in0 A1

and

tann0kma = − in0qm − im

k , A2

where km qm− im and kmqm− im are the complex QBand CF wave numbers, a is the length of the slab, n0 is thepossibly complex uniform index of refraction, and k is thewave number corresponding to the external Fourier fre-quency. As in the main text we use tildes to denote quantitiesassociated with the QB states. We also define the dimension-less quantities xm=n0qma, ym=−n0ma, and z=n0ka. Usingtrigonometric identities, we can expand both equations intoreal and imaginary parts,

cot qm1 − tanh2 mcot2 qm + tanh2 m

= 0, A3

tanh mcot2 qm + 1cot2 qm + tanh2 m

= − n0, A4

and

cot xm1 − tanh2 ymcot2 xm + tanh2 ym

=n0ym

z, A5

tanh ymcot2 xm + 1cot2 xm + tanh2 ym

= −n0xm

z. A6

In the method used to solve the multimode laser equationspresented in this paper, the Green’s function is approximatedby a single pole, i.e., by a single CF state in the spectralrepresentation of Eq. 30. We are therefore especially inter-ested in finding solutions to the CF eigenvalue equation withreal part close to the wave vector, k, appearing as a param-eter in the CF eigenvalue equation. Furthermore, since weexpect the real part of the QB mode frequencies to provide afirst approximation to the actual lasing frequencies, we lookfor solutions to the CF eigenvalue equations with the exter-nal Fourier frequency set equal to the real part of a quasi-bound mode frequency. Equations A3–A6 suggest that,for each QB mode with a sufficiently short wavelength, acorresponding CF solution exists with approximately the

same eigenvalue, since in this limit x=n0kma is large. In thatcase the right-hand side of Eq. A5 is small so that Eq. A5is nearly identical to Eq. A3, and for qmx Eq. A6 ap-proaches Eq. A4.

The solution to Eqs. A3 and A4 was given in the textby Eq. 33, and Eqs. A5 and A6 can be solved fortanh ym and cot xm,

tanh ym

=− cot2 xm + 1 ± cot2 xm + 12 − 4n0xm/z2 cot2 xm

2n0xm/z,

A7

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cot xm

=1 − tanh2 ym ± 1 − tanh2 ym2 − 4n0ym/z2 tanh2 ym

2n0ym/z.

A8

Note that in order to self-consistently choose solutions toEqs. A7 and A8 which are close to QB solutions, we mustselect the minus signs in both equations. We now expand Eq.A5 in the small difference qm=qcf −qqb,

xm1 − 1/n02

1/n02 −

n0

zym. A9

Assuming that ym is, at most, first order in the small param-eter z−1, we can replace ym with m, leading to

xmn02 − 1 −

n0

zm. A10

Or, using the expression for 0 from Eq. 33 and the defi-nitions of xm and z,

qm lnn0 + 1/n0 − 1

2n02 − 1

1

n0a2k. A11

We see the difference between real part of the CF resonanceand the external frequency is small when 1 the externalfrequency is set near the real part of a QB resonance and 2the semiclassical parameter z−1 is small. To find an approxi-mation for m, as well as to justify the approximation used inderiving Eq. A11 that ym m, we next expand Eq. A7 tosecond order in qm

−1 and x−1. After some manipulation weobtain

m −1

2n02 − 12 lnn0 + 1

n0 − 11 +

lnn0 + 1/n0 − 12

1

n0a3k2 =fn0ka2

1

a, A12

which is the result quoted in Sec. III C 1. Thus we see thatqm is first order and m is second order in z−1. Figure 8shows that the approximation used here agrees well with thenumerically evaluated results for the difference between theQB and the nearest CF wave vector.

2. Cylindrical dielectric cavity

It is well known that for a cylindrical cavity, there existwhispering gallery mode solutions to the QB eigenvalueequation that have very small widths, m. Moreover, it isthese modes which will dominate lasing emission from di-electric cylinder lasers. In this section we show that thereexist corresponding solutions to the CF eigenvalue equationsclose to these QB solutions. Specifically, the difference be-tween the CF and QB lifetimes is proportional to a smallfactor times the QB lifetimes; hence these CF and QB stateshave almost identical lifetimes.

The QB eigenvalue equation for a cylindrical resonator is

d

drJMn0kmR/JMn0kmR =

d

drHM

+kmR/HM+kmR .

A13

Within the regime of interest where the modes are welltrapped and the index of refraction is not too small, we canapproximate the Hankel and Bessel functions appearing inthis equation by the Debye approximations 38. Define

tanh = 1− kmR /M2 and tan = n0kmR /M2−1. Interms of these variables, the Debye approximations are

JMkmR =eMtanh −

2M tanh , A14

YMkmR = −e−Mtanh −

1

2M tanh

, A15

JMn0kmR =cosM tan − M − /4

1

2M tanh

, A16

FIG. 8. The difference between the CF and QB eigenvalues fora 1D dielectric slab resonator, as a function of the real part of theQB eigenvalue. a The fractional difference between the real partof the eigenvalues, and b the fractional difference between thecomplex parts. In both plots, the triangles represent the actual dif-ference, found numerically, which is compared to the linear ap-proximation, Eq. A12, derived in the text and represented by thesolid curves.

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YMn0kmR =sinM tan − M − /4

1

2M tanh

. A17

Note that in the regime in which these approximations arevalid, we can ignore the r dependence outside the exponentwhen we evaluate its derivatives, so that the eigenvalueequation is approximately

tan tan tanh e + 2ie−

e − 2ie− , A18

where we have defined =Mtan −− /4 and =Mtanh −. By expanding in the small quantity m andneglecting e compared to e− which is valid for welltrapped modes, we can derive an approximate expressionfor m,

m M tanh

n02a2qmn0

2 − 1e2. A19

A similar, but not identical, result was obtained by Nöckel23, where it was pointed out that the small factor e2 couldbe interpreted as the exponential decay factor due to tunnel-ing.

The eigenvalue equation for CF modes in a cylindricalresonator has the same form as Eq. A13 above for QBstates except that the functions , , , and appearing inthe CF equation are evaluated at different arguments. Explic-

itly, and are evaluated at kmR= kmR+kmR, whereas and are evaluated at the external Fourier frequency, whichwe have set equal to qm. We can then expand in the smallquantities kmR and m. To first order, the imaginary part ofthis equation yields

m m

Mn02 − 1tanh

, A20

where it is understood that tanh is evaluated at tanh

= 1− kmR /M2. Note that in the semiclassical regime whis-pering gallery modes have a high M value, so the differencebetween the lifetimes for the CF and QB mode is a smallfactor times the already small QB lifetime m see Fig. 9.

3. General argument that Im„km…=−m0

We were able to show above explicitly that for the CFeigenvalue closest to the corresponding QB state eigenvalue,m0 for both the 1D slab laser and the 2D cylindrical laser.Here we show that this result holds generally for all CFeigenvalues and for arbitrary geometries. This is importantbecause the exact CF Green function involves all CF eigen-values and it would be unphysical for it to have poles in thepositive imaginary plane.

The equation satisfied by the CF states is

− 2mx = n02km

2 mx inside ,

k2mx outside . A21

We define a vector flux as fm= 12i m

* m−mm* . The

outgoing wave boundary conditions, as expressed in the

main text by Eq. 24, imply that if we integrate the radialcomponent of f over increasingly larger spheres S or circlesin the 2D case with origin within the cavity, the value of thisintegral approaches a positive constant proportional to k:

S

f · dA → k m2d . A22

Note that · f vanishes outside the cavity, whereas inside it isequal to −n0

2mm* Imkm

2 =2n02m2qmm. By using the diver-

gence theorem and the CF equation we can also write thisintegral as

S

f · dA = IntS

· fdV = 2n02qmm

cavity

m2dx .

A23

Comparing Eqs. A22 and A23 we can see that in order tosatisfy Eq. A22, which follows from the outgoing waveboundary condition, we must have m0.

4. Biorthogonality of incoming and outgoing CF states

Consider a solution to the CF eigenvalue equation mwith eigenvalue km, along with an adjoint solution n witheigenvalue kn

* for a cavity with a uniform index of refraction

FIG. 9. a The log of the imaginary part of the QB resonanceeigenvalue for a cylindrical resonator with n0=2, plotted against thereal part of the eigenvalue; triangles represent numerical resultscompared to the analytic approximation of Eq. A19. b The frac-tional difference between the CF and QB lifetimes vs the real partof the QB eigenvalue for the same resonator. Triangles are numeri-cal results, solid line the approximation of Eq. A20. We haveplotted the sequence corresponding to angular momentum M =50.

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n0. Using the fact that these two functions satisfy the eigen-value equations 22 and 26 respectively, we have

n02kn

2 − km2

cavity

dxn*m =

cavity

dxn*2m − m2n

*

= boundary

n* m

− m n* · dA , A24

where the last integral is over the surface of the cavityboundary. Outside the cavity we have · n

*m−m n*

=0, so we can in fact replace the integral over the cavityboundary with an integral over a large sphere S enclosing thecavity. We can then invoke the outgoing and/or incomingwave boundary conditions to evaluate the surface integralexplicitly:

S

n* m − m n

* · dA

→ S

ikn*m − ikmn

* · dA = 0. A25

For the case knkm, Eqs. A24 and A25 therefore imply

cavity

dxn02m

* n = 0. A26

It should be noted that, although the CF functions are definedto exist both inside and outside the cavity, the inner productis defined by an integral over the cavity interior only. Thebiorthogonality of the CF states is therefore determined com-

pletely by their behavior within the cavity and at its bound-ary; we merely use the behavior of the CF functions at in-finity to show that the surface term in Eq. A24 vanisheswhenever the CF states satisfy the outgoing and/or incomingboundary conditions.

We can check the biorthogonality of the CF states explic-itly for the case of the dielectric slab cavity, and also calcu-late the normalization factor m defined in Eq. 28. Inthe cavity interior we have

nx = sinn0knx A27

and

m* = sinn0kmx . A28

We can then write the inner product for the case mn as

0

a

n02 sinn0knxsinn0kmxdx

= n02cosn0knacosn0kma

2n0

tann0kna − tann0kmakn − km

−tann0kna + tann0kma

kn + km = 0. A29

Where we have used the eigenvalue equation A2. The nor-malization is

0

a

n02 sin2n0kmxdx =

an02

21 −

sin 2n0kma2n0kma = m .

A30

1 H. E. Tureci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone,Prog. Opt. 47, 75 2005.

2 H. G. L. Schwefel, H. E. Tureci, A. D. Stone, and R. K. Chang,in Optical Processes, edited by K. J. Vahala World Scientific,Singapore, 2004.

3 H. Cao, Waves Random Media 13, R1 2003.4 H. Cao, J. Phys. A 38, 10497 2005.5 H. Haken, Light North-Holland, Amsterdam, Netherlands,

1985, Vol. 2.6 L. A. Lugiato, Phys. Rep. 219, 293 1992.7 F. T. Arecchi, S. Boccaletti, and P. Ramazza, Phys. Rep. 318, 1

1999.8 S. Sunada, T. Harayama, and K. S. Ikeda, Phys. Rev. E 71,

046209 2005.9 C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D.

Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 15561998.

10 S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist,and E. Gornik, IEEE J. Quantum Electron. 36, 458 2000.

11 C. Gmachl, E. E. Narimanov, F. Capasso, J. N. Baillargeon,and A. Y. Cho, Opt. Lett. 27, 824 2002.

12 N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and

A. D. Stone, Phys. Rev. Lett. 88, 094102 2002.13 S. B. Lee, J. H. Lee, J. S. Chang, H. J. Moon, S. W. Kim, and

K. An, Phys. Rev. Lett. 88, 033903 2002.14 T. Harayama, T. Fukushima, S. Sunada, and K. S. Ikeda, Phys.

Rev. Lett. 91, 073903 2003.15 G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M.

Kneissl, and N. M. Johnson, Appl. Phys. Lett. 83, 17102003.

16 V. A. Podolskiy, E. Narimanov, W. Fang, and H. Cao, Proc.Natl. Acad. Sci. U.S.A. 101, 10498 2004.

17 T. Harayama, P. Davis, and K. S. Ikeda, Phys. Rev. Lett. 90,063901 2003.

18 T. Harayama, S. Sunada, and K. S. Ikeda, Phys. Rev. A 72,013803 2005.

19 P. T. Leung, S. Y. Liu, and K. Young, Phys. Rev. A 49, 30571994.

20 E. S. C. Ching, P. T. Leung, A. M. van den Brink, W. M. Suen,S. S. Tong, and K. Young, Rev. Mod. Phys. 70, 1545 1998.

21 A. Fox and T. Li, Proc. IRE 48, 1904 1960.22 A. B. Manenkov, IEE Proc.: Optoelectron. 141, 287 1994.23 J. U. Nöckel, Ph.D. thesis, Yale University 1997 ubpub-

lished.

TÜRECI, STONE, AND COLLIER PHYSICAL REVIEW A 74, 043822 2006

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24 M. Hentschel and K. Richter, Phys. Rev. E 66, 056207 2002.25 J. Wiersig, J. Opt. A, Pure Appl. Opt. 5, 53 2003.26 M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics

Addison-Wesley, Reading, MA, 1974.27 H. Haken and H. Sauermann, Z. Phys. 173, 261 1963.28 T. S. Misirpashaev and C. W. J. Beenakker, Phys. Rev. A 57,

2041 1998.29 H. E. Tureci and A. D. Stone, Proc. SPIE 5708, 255 2005.30 A. E. Siegman, Lasers University Science Books, Mill Valley,

CA, 1986.31 L. A. Lugiato, G. L. Oppo, J. R. Tredicce, L. M. Narducci, and

M. A. Pernigo, J. Opt. Soc. Am. B 7, 1019 1990.32 K. Staliunas, M. F. H. Tarroja, and C. O. Weiss, Opt. Commun.

103, 69 1993.33 P. Mandel, C. Etrich, and K. Otsuka, IEEE J. Quantum Elec-

tron. 29, 836 1993.34 V. Zehnle, Phys. Rev. A 57, 629 1998.35 H. Fu and H. Haken, Phys. Rev. A 43, 2446 1991.36 M. V. Berry, J. Mod. Opt. 50, 63 2003.

37 P. M. Morse and H. Feshbach, Methods of Theoretical Physics,Part I McGraw-Hill, New York, NY, 1953.

38 M. Abramovitz and I. A. Stegun, Handbook of MathematicalFunctions Dover, New York, 1972.

39 These continuity conditions strictly hold for instance for theTM polarized high-Q modes of a cylindrical structure with anarbitrary cross section; the CF modes would then represent theaxial component of the electric field. In the most general case,one has to resort to a spinor formalism for CF modes where theBCs will be of mixed type see, for instance, Ref. 1. Theformalism laid out here can easily be generalized to accountfor more complicated continuity conditions on D.

40 We note that this approximation breaks down for degenerate orquasidegenerate poles. In such a situation, one has to take intoaccount the contribution from more than one pole and a con-sistent multipole theory can be derived leading to correct treat-ment of collective frequency locking phenomena.

41 H. E. Türeci, A. Douglas Stone, and L. Ge unpublished.

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