Multimode lasing in wave-chaotic semiconductor microlasers

Alexander Cerjan,1 Stefan Bittner,2 Marius Constantin,2 Mikhail Guy,3

Yongquan Zeng,4 Qi Jie Wang,4 Hui Cao,2 and A. Douglas Stone2, ∗

1Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA2Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA3 Yale Center for Research Computing, Yale University, New Haven Connecticut 06520

4Center for OptoElectronics and Biophotonics, School of Electrical and Electronic Engineering and Photonics Institute,Nanyang Technological University, 639798 Singapore

We investigate experimentally and theoretically the lasing behavior of dielectric microcavity laserswith chaotic ray dynamics. Experiments show multimode lasing for both D-shaped and stadium-shaped wave-chaotic cavities. Theoretical calculations also find multimode lasing for different shapes,sizes and refractive indices. While there are quantitative differences between the theoretical lasingspectra of the stadium and D-cavity, due to the presence of scarred modes with anomalously highquality factors, these differences decrease as the system size increases, and are also substantiallyreduced when the effects of surface roughness are taken into account. Lasing spectra calculationsare based on Steady-State Ab Initio Laser Theory, and indicate that gain competition is not sufficientto result in single-mode lasing in these systems.

I. INTRODUCTION

There has been a great deal of interest in the proper-ties of dielectric microcavity lasers or resonators basedon quasi-two-dimensional cavities, for which differentboundary shapes can generate chaotic, mixed or regu-lar ray dynamics [1], with corresponding implications forthe resonant wave solutions and lasing modes of suchcavities. We will use the term wave-chaotic cavity to re-fer to cavity shapes for which the ray dynamics based onspecular reflection of rays at the boundary and neglectingrefractive escape satisfy standard definitions of chaotic orpartially chaotic dynamics [2]. The motivation for thiswork are recent experimental results and theoretical ar-guments concerning semiconductor microlasers with fullychaotic ray dynamics (i.e., no stable periodic orbits orquasi-periodic Kolmogorov–Arnold–Moser, or KAM or-bits), and in this article we will only treat cavity shapeswith either fully chaotic ray dynamics or with fully inte-grable ray dynamics (such as the ellipse). It should beemphasized that we do not concern ourselves with thetemporal dynamics, which for semiconductor lasers canbe unstable or chaotic for a number of reasons [3], andin the theoretical work presented below we confine our-selves to the study of single- or multimode steady-statesolutions of the laser equations.

The earliest dielectric cavity lasers were microdisklasers with whispering gallery type high-Q modes con-fined by total internal reflection [4]. Not long after thesefirst microdisk lasers were demonstrated, the idea of de-forming the resonator boundary to non-circular shapeswas introduced [5, 6] to explore the implications of fullyor partially chaotic ray dynamics on the wave solutionsand lasing properties. A major focus of this earlier workwas on obtaining directional emission from smoothly de-

∗ Corresponding author. Email: douglas.stone@yale.edu

formed cavity shapes, and on using properties of thephase space of the ray dynamics to predict and controlthe emission patterns [7–9]. In addition, many of theseearly studies considered laser cavities much larger thanthe wavelength for which one could not resolve individuallasing lines, motivating a statistical treatment via ray es-cape models. However when wave-chaotic microcavitieswith sizes of the order of the wavelength were fabricatedand studied [10, 11], the lasing spectrum was generallyfound to be multimode [12, 13]. The nature of their lasingmodes and their relationship to passive cavity modes andthe classical ray dynamics have been studied extensively[1].

Motivated by wave-chaotic and random microcavitylasers the Steady State Ab Initio Laser Theory (SALT)was developed starting in 2006 [14–19], an approachwhich can be employed for arbitrary complex geometriesand yields the active mode spectra of microcavity lasers.SALT is based on the Stationary Inversion Approxi-mation (SIA), which becomes reasonable for sufficientlysmall cavities, as emphasized first by Fu and Haken [20].The quantitative validity of SALT for microlasers in theappropriate regime has been confirmed by comparisonwith full time-dependent solutions of the two-level andmulti-level semiclassical laser equations [21]. The the-ory is designed to describe multimode steady-state las-ing and takes gain competition and gain saturation intoaccount to all orders, within the SIA. SALT and a fur-ther ”single-pole approximation”, known as SPA-SALT,has been used to study how spatially selective pumpingcan be used to control the lasing spectra of wave-chaotic,random and circular microlasers [17].

More recently, wave-chaotic cavity lasers or randomlasers have been proposed and demonstrated by severalof the authors as novel bright sources emitting spatiallyincoherent light for imaging and microscopy applications[22–25]. For these applications, wave-chaotic GaAs D-cavity lasers with sizes of the order of 102 − 103µm andpulsed electrical pumping were tested and compared to

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circular disk and Fabry-Perot cavity lasers fabricated ina similar manner [22]. Speckle contrast measurements ofthe D-laser emission indicated NM ∼ 102 − 103 lasingmodes with distinct spatial profiles, many orders of mag-nitude greater than the number of spatial modes found intraditional lasers such as Fabry-Perot broad-area lasers ofcomparable surface area. The incoherent superposition ofso many different spatial modes results in low spatial co-herence which suppresses coherent artifacts. It was thusargued that wave-chaotic microlasers were particularlygood devices for generating highly-multimode, spatiallyincoherent lasing emission, since they exhibit many spa-tially distinct modes with similar Q-factors, and theirmodes have speckle-like intensity distributions filling theentire cavity, utilizing the entire available gain medium.

Very different lasing behavior was found by Sunadaet al. in other recent experiments with stadium-shapedGaAs microlasers of a somewhat smaller size. In Ref. [26]it was found that the stadium microlasers transitionedfrom multimode lasing when pumped with very shortpulses to single-mode lasing for pump pulses longer than100 µs. Experiments in the steady-state regime [27] withcontinuous wave (cw) pumping showed single-mode lasingfor the wave-chaotic stadium lasers, whereas multimodelasing was found for elliptical microlasers (which have in-tegrable ray dynamics). The authors argued, based onnumerical calculations, that the modes of wave-chaoticresonators overlap so strongly in space that their crossgain-saturation results in single-mode lasing, whereaswhispering gallery modes with different radial quantumnumbers in the elliptic resonator have small enough over-lap to allow multimode lasing. Subsequent theoreticaland numerical work indicated that single-mode lasingmay be typical for wave-chaotic microlasers in certainparameter regimes [28]. These two sets of experiments(Refs. [26, 27] and Ref. [22]) raise fundamental questionsabout the lasing dynamics of wave-chaotic semiconductormicrolasers, and the aim of this article is to elucidate theinfluence of different experimental parameters and phys-ical mechanisms on the number of lasing modes giventhat the microlasers used in the two sets of experimentsare very similar, though not identical. In particular, weconsider the effects of mode competition, the size, refrac-tive index and surface roughness of the cavities, as well asnon-universal features of different wave-chaotic resonatorgeometries.

In the first part of the article we present new ex-perimental measurements of the lasing spectra of bothD-cavity and stadium-shaped wave-chaotic GaAs micro-lasers with time resolution in order to address the ques-tions raised above. In the second part of the paper wepresent the results of SPA-SALT calculations that shedlight on the role of mode competition in wave-chaotic cav-ity lasers. It should be noted that the microlaser cavitiesused in experiments are too large to be directly simu-lated either by time domain or frequency domain (e.g.,steady-state) approaches. Moreover, SALT loses quanti-tative validity when the lasing spectra becomes too dense

100 µm(a)

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Figure 1. Top view SEM images of (a) a D-cavity with radiusR = 200 µm and (b) a stadium cavity with L = 238 µm.(c, d) Perspective SEM images of the curved sidewall of a D-cavity, highlighting its verticality and low surface roughness.

with increasing cavity size. Hence our calculations do notallow quantitative modeling, but provide evidence andphysical insight into the role of wave chaos and modecompetition in multimode lasing.

II. EXPERIMENTAL RESULTS

We investigated edge-emitting GaAs quantum wellsemiconductor lasers fabricated from a commercial epi-wafer (Q-Photonics QEWLD-808) by photolithographyand dry etching (see Ref. [29] for details of the fabrica-tion process). Microlasers in the shape of a D-cavity, astadium and an ellipse were created. Scanning-electronmicroscope (SEM) images of two cavities are shown inFigs. 1(a, b), respectively. The dry-etching process en-sured vertical sidewalls [see Fig. 1(c)] and a low degreeof surface roughness [see Fig. 1(d)].

The D-cavity shape is a circle with radius R from whicha segment has been cut off R/2 away from the center asshown in the inset of Fig. 2(a). The stadium cavitiesconsidered in this article consist of a square with sidelength L to which two semicircles with radius L/2 areattached as shown in the inset of Fig. 2(b). The ray-dynamics of both cavity types is fully chaotic [30, 31].The ellipse cavities we investigated have an aspect ratioof b/a = 2 where a, b are the two diameters [see inset ofFig. 2(c)]. Their ray dynamics is integrable.

The cavities were mounted on a large Cu block whichacted as a passive heat sink and pumped electrically with2 to 500 µs-long pulses by a diode driver (DEI ScientificPCX-7401). All experiments were performed at ambienttemperature. The laser emission from the cavities wascollected by an objective and transmitted to an imag-ing spectrometer (Acton SP300i) with a multimode fiberbundle. An intensified CCD camera (ICCD, Andor iStar

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DH312T-18U-73) attached to the spectrometer was usedto measure the evolution of the emission spectrum duringa pulse with microsecond resolution. The experimentalsetup and measurement procedures are described in moredetail in Ref. [29].

The lasing spectra integrated over 2 µs-long pulses fora D-cavity, a stadium and an ellipse are shown in Fig. 2.All three cavities have approximately the same area of25, 300 µm2 and hence the same resonance density. Thespectra for 500 mA pump current show multimode lasingwith about 20 peaks for all three geometries. The emis-sion has transverse electric (TE) polarization with theelectric field parallel to the cavity plane. Similar resultswere obtained for cavities with two times larger linear di-

Cavity 〈Ith〉 (mA) 〈jth〉 (A cm−2)

D-cavity, R = 100 µm 125± 6 494.6

Stadium, L = 119 µm 100± 2 395.5

Ellipse, a = b/2 = 127 µm 68± 4 269.6

Table I. Average threshold currents 〈Ith〉 and correspondingthreshold current densities 〈jth〉 for cavities with the samesize as those presented in Fig. 2.

mension. The actual number of lasing modes cannot bedetermined from the spectra as our spectrometer cannotresolve closely-spaced lasing modes due to its finite spec-tral resolution. Nevertheless, the appearance of multiplepeaks in the emission spectrum clearly evidences multi-mode lasing in D-cavities and stadia of this size.

The threshold currents vary very little between cavi-ties of the same type and size, but depend on the cavitygeometry. An overview of the average threshold currents〈Ith〉 for the three cavity shapes shown in Fig. 2 is givenin Table I. The stadium cavities have somewhat lowerthresholds than the D-cavities. The ellipse cavities haveclearly lower thresholds than the two wave-chaotic cav-ities. We would, however, expect orders of magnitudelower lasing thresholds for ellipse cavities due to the ex-istence of whispering gallery modes (WGMs) with ultra-high Q-factors. We attribute the only moderate differ-ence to the small, but not negligible surface roughness.These results motivate a detailed study how the cavitygeometry and surface roughness determine the passivemode quality factors presented further below.

For a detailed understanding of the multimode lasingdynamics, it is important to take into account thermaleffects. The cavities heat up quickly during current injec-tion, which leads to a red-shift of the emission spectrumand changes of the active lasing modes during the pumppulse [29]. The temperature and hence the emission spec-trum gradually stabilizes, however, over the course oflonger pulses. A D-cavity with R = 200 µm and a sta-dium with L = 238 µm were pumped with 500 µs-longpulses at 800 mA pump current. The threshold currentsare Ith = 270 mA (jth = 267.1 A cm−2) for the D-cavityand Ith = 230 mA (jth = 227.4 A cm−2) for the stadium,respectively. Both cavities have approximately the samearea of 101, 100 µm2. Excerpts of the spectrochrono-grams of the D-cavity and the stadium measured with1 µs time resolution are shown in Figs. 3(a, b), respec-tively. After more than 400 µs, the lasers have stabi-lized so well that the emission spectrum does not changeover the course of 10 µs. It has also been verified thatthe lasers do not exhibit any fast, nanosecond time-scaledynamics [29]. Thus, a quasi-steady state of the lasingdynamics is reached. Even though thermal equilibriumis not reached due to the lack of active cooling, the re-maining thermally-induced fluctuations of the emissionspectrum with a time scale longer than 10 µs are sev-eral orders of magnitude slower than the intrinsic dy-namical time scales of semiconductor lasers. Hence, a

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Figure 3. Spectrochronogram of (a) a D-cavity with R = 200 µm and (b) a stadium with L = 238 µm during a 500 µs-longpulse measured with 1 µs time resolution. The emission spectra show no discernible change over the course of 10 µs. The pumpcurrent was 800 mA in both cases. (c) Emission spectrum of the D-cavity during the time interval 460–461 µs and (d) of thestadium during the time interval 450–451 µs, showing multiple lasing peaks.

steady-state model such as SALT is appropriate to inves-tigate the interactions of the lasing modes via the activemedium in this regime.

The spectra of both D-cavity and stadium lasers atany given time exhibit multiple lasing peaks as shownin the spectra in Figs. 3(c, d). It should be noted thatthe actual number of lasing modes is higher than thenumber of peaks observed in the spectra due to the finiteresolution of the imaging spectrometer. Both stadiumand D-cavities evidently always exhibit multimode lasingas proven by the presence of several peaks in the spectra.Furthermore, the typical number of lasing peaks is similarfor both cavity geometries, and there is no qualitativedifference between the lasing spectra of D-cavities andstadia.

The similarities between D-cavity and stadium lasersare attributed to the common features of wave-chaoticmodes in these resonators. However, the lasing modestypically correspond to the highest-Q modes, which mayhave different characteristics than the majority of the res-onance spectrum in wave-chaotic cavities. Specifically, itis known that wave-chaotic microlasers can exhibit lasingon localized scar modes [32, 33], which can have higherQ-factors than expected from statistical analysis of wave-chaotic lasers [34, 35]. Such effects will vary with thecavity shape and size and may be affected significantlyby surface roughness. Therefore, a detailed analysis ofgeometry-specific properties of wave-chaotic microlasersas well as the effect of surface roughness on the lasing

dynamics is needed to shed further light on the experi-mental results.

III. THEORETICAL STUDY

The experimental results presented above are consis-tent with earlier work by our group on wave-chaotic D-cavity lasers [22], and provide more insight into the timeevolution of the lasing spectra. They confirm the qual-itatively different behavior of the lasers we have stud-ied from those reported in references [26, 27], and thesedifferent results present a challenge to obtain a consis-tent understanding of the lasing behavior of wave-chaoticsemiconductor lasers. As noted earlier, there is no possi-bility to simply perform ab initio integration of the laserequations for two-dimensional complex cavities with asize of over 100 µm. The largest cavities to be treatedby brute force integration are two orders of magnitudesmaller in linear dimensions [36–38] and thus have afour order of magnitude smaller resonance density thanthe experimental microlasers. Our partially analytic ap-proach based on SALT can do somewhat better, provid-ing results for resonators on a 60µm scale, but with somecaveats described below. Thus there is at present noabsolutely rigorous computational method [39] to decidethe validity of the conjecture in Refs. [26–28] that fullywave-chaotic cavities are intrinsically single-mode lasersin steady-state.

5

However, one can use existing methods for such lasersat smaller scale to analyze the physical processes whichare known to determine the number of lasing modes.Multimode lasing is known to result from the interplaybetween gain saturation, which tends to clamp the gainonce a single mode starts lasing, and spatial hole burning,which refers to the spatial non-uniformity of gain satura-tion in standing wave cavities. Near the intensity minimaof a lasing mode there is only a small field stimulatingemission and the local inversion (gain) is not saturated,and therefore continues to increase with increasing pump-ing in such regions. This allows new, lower-Q modes withdifferent intensity minima or, more generally, weaker spa-tial overlap to reach threshold and begin lasing. For ex-ample, in conventional stable resonators with Gaussianmodes, higher order transverse modes have significant in-tensity away from the optical axis of the resonator, andcan thus exploit distinct regions of the gain medium tolase in addition to the fundamental transverse mode.

In wave-chaotic cavities, resonances tend to fill theentire resonator with a speckle-like intensity distribu-tion. In Ref. [27] it was proposed that significant spa-tial overlap of the intensity distributions was the originof single-mode lasing: wave-chaotic modes always over-lap so strongly that a single mode clamps the gain inspite of spatial fluctuations of its intensity distribution.The spatially extended nature of the intensity distribu-tions was argued to be the relevant difference betweena wave-chaotic shape, such as the stadium, and a non-wave-chaotic shape with similar aspect ratio, such as theellipse, which showed multimode lasing. This conjectureapparently contradicts the numerical results in Ref. [22]for D-cavity lasers, which found eight modes lased in a D-cavity with area≈ 63 µm2. Here we extend the numericalstudies of Ref. [22] to study in more detail the conjecturethat strong gain clamping and cross-saturation lead tosingle-mode lasing in wave-chaotic microlasers.

The method we use, Steady-State Ab-Initio Laser The-ory (SALT), is an approach specifically developed tostudy microcavity lasers with complex geometries, andwas used in earlier works on microcavity lasers [14–17, 21, 40]. Not only was SALT developed to treat com-plex 2D (and in principle 3D) cavities, it was also de-signed to deal with multimode lasing and spatial holeburning quantitatively. SALT is a semiclassical theoryand does not include quantum fluctuation effects, whichare not relevant to this study. The version of SALT mostrelevant to the current work, and its limitations, are de-scribed in detail in [17].

A. Review of Steady-State Ab-Initio Laser Theory

The SALT equations for steady-state multimode las-ing are derived from the semiclassical laser equations byneglecting time-dependent non-linear terms in the equa-tions, which drive oscillations in the inversion. Theyare a set of non-linear wave equations to be solved self-

consistently with purely outgoing boundary conditions,and have the form[(

∇×∇×)−(εc(x) +

γ⊥D0(x)

ωµ − ωa + iγ⊥

× 1

1 +∑NL

ν Γν |Ψν |2

)ω2µ

c2

]Ψµ(x) = 0, x ∈ C .

(1)

They determine the number of lasing modes, their opti-cal field distributions, Ψµ(x), and lasing frequencies, ωµ.The approximation used to derive the SALT equations isthat the inversion density (which only appears implicitlyin the SALT equations) is time-independent. The inputsto the SALT equations are the dielectric function of thepassive cavity, εc(x), the dephasing rate of the polariza-tion, γ⊥, the atomic transition frequency, ωa, and the ex-ternal pump profile, D0(x). For two-level atoms the gaincurve is Lorentzian and centered at ωa with width γ⊥.The Stationary Inversion Approximation (SIA) requiresthat γ⊥ and the typical frequency spacing between lasingmodes, ∆ω, are much larger than the population relax-ation rate, γ‖. The latter condition becomes harder tomeet in the highly-multimode regime and for larger lasercavities. In the regime where the SIA holds, excellentagreement is found between SALT and full integration ofthe semiclassical laser equations [21].

Here we have written the SALT equations as scalarequations since we consider modes with transverse mag-netic (TM) polarization, for which Ψν corresponds to thez-component of the electric field. (We have calculated Qvalue distributions for the (TE) modes and find quali-tatively similar behavior to that described below). Thedielectric function is 1 outside of the cavity region C.The number of non-trivial purely outgoing solutions in-creases by one at each lasing threshold as the pump D0

is increased starting from zero. The non-linear denom-inator represents the saturable gain susceptibility, andenforces self-saturation and cross-saturation of the gainin a spatially-varying manner, which takes into accountspatial hole-burning exactly. The Lorentzian gain factorof mode ν is Γν = γ2⊥/[(ων − ωa)2 + γ2⊥]. Ψ and D0

are written in dimensionless form in terms of the naturalunits of the electric field, ec = ~√γ‖γ⊥/(2g), and the in-

version, dc = ~γ⊥/(4πg2) (g is the dipole matrix elementof the transition).

Two main types of algorithms have been developedto solve the SALT equations [14, 17, 41]. The first ap-proach, which is the basis for the algorithm used in thiswork, expands the solution of the SALT equations in acomplete set of biorthogonal outgoing wave functions at agiven frequency, known as the Threshold Constant Flux(TCF) states. One of these TCF functions is the ex-act solution of the semiclassical equations at the lasingthreshold, denoted as the threshold lasing mode (TLM);the others take into account the change in the spatialpattern of the mode and its non-linear frequency shiftabove threshold. While this solution method is relativelyefficient compared to FDTD [40], it is still quite com-

6

putationally expensive when applied to 2D wave-chaoticcavities, so we use two further approximations to SALTwhich enable us to treat larger laser cavities.

B. Single Pole Approximation - SPA-SALT

The first one, the “Single Pole Approximation” (SPA-SALT) [17], assumes that the field distribution and fre-quency of each lasing mode are fixed to their values atthreshold, as given by the TLMs. Hence as the pumpis increased only the overall amplitudes of the modeschange and need to be determined. Moreover, sinceSALT neglects the beating of lasing modes, the phasesof the modes do not enter the solution, and one can re-duce the full SALT equations to the following equationsfor the intensities, Iµ, of each mode as a function of thepump,

D0

Dµ0

− 1 =∑ν

ΓνχµνIν , (2)

χµν =

∫d2rΨ2

µ|Ψν |2. (3)

Here, the Dµ0 are the non-interacting thresholds of the

Threshold Lasing Modes (TLM), which reflect their Q-factors and their proximity to the gain center, but neglecteffects of gain competition. The coefficients χµν representmodal overlaps. In general they have a small imaginarypart for high-Q modes, which we will neglect by usingonly the real part, χµν ≈ Re[χµν ]. We rewrite theseequations as

D0

Dµ0

− 1 =∑ν

AµνIν , (4)

Aµν = Γνχµν . (5)

The matrix Aµν represents the cross-saturation interac-tion of all pairs of lasing modes for a given value of thepump power D0. Equation (4) is nominally linear, how-ever the set of lasing modes to include at each pump valueis not known, and is determined implicitly by the non-linear interactions contained in the Aµν . The thresholdof the first mode to turn on is given directly by the TLMcalculation, but subsequent mode thresholds are deter-mined by the constraint that all intensities fulfill Iν ≥ 0.Hence we must search at each pump value for the largestset of modes which yields positive semi-definite values forthe Iν , and only then invert Eq. (4) using the appropriatematrix Aµν . Between modal thresholds, when the matrixAµν is fixed, the non-zero Iν vary linearly with pump andare given by

Iµ = cµD0 − bµ, (6)

cµ =

N∑ν=1

(A−1)µνDν

0

, (7)

bµ =

N∑ν=1

(A−1)µν . (8)

The solution over the full pump range of interest has akink at each of the thresholds Dµ

0,int, where the subscriptint denotes the threshold in the presence of modal inter-actions. These interacting thresholds are given by [17]

DN0,int = DN

0

[1 +

N∑ν=1

ANν(cνDN0,int − bν)

]. (9)

This equation follows from the condition that as the N th

mode turns on its intensity passes through zero fromnegative values; the matrix A and the constants bν , cνthen change appropriately above this pump threshold.This approach has been validated by comparing numeri-cal results between the SPA-SALT and full SALT meth-ods [17].

C. Resonance SPA-SALT

The necessary first stage of a SPA-SALT calculationis the calculation of the TLMs and their non-interactingthresholds and frequencies. This can be done by trackingcavity resonances and quasi-modes as the gain increasesuntil they reach the real axis [17, 18, 40]. To avoid thisstep, recently Cerjan et al. [42] proposed an analytic ap-proximation for the evolution of poles as the pump in-creases, so that standard codes for calculating passivecavity modes such as COMSOL can be used. For high-Qmodes, the lasing modes will differ little from the passivecavity modes (within the cavity), and thus the passivecavity modes can replace the TLMs in SPA-SALT. Wecall the resulting method resonance SPA-SALT; a work-ing code for this method is available for download [43].This approach was used in our earlier work on D-cavitylasers and was compared with results from full SALT,finding reasonable agreement. We will use this methodin the current work to allow us to consider even largercavities and explore more of the parameter space of inter-est. Resonance SPA-SALT expresses the non-interactingthresholds needed for SPA-SALT in terms of the complexfrequencies of the passive cavity resonances

Dµ0 =

∣∣∣(Re[ωµ]− ωa + iγ⊥γ⊥

)(ω2µ − Re[ωµ]2

Re[ωµ]2.)∣∣∣ . (10)

The lasing frequencies ωµ are approximated by the realpart of the passive cavity resonance frequencies, and las-ing modes within the cavity by the passive cavity modefield distributions.

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IV. RESULTS

As noted, the number of lasing modes will be deter-mined ultimately by the passive cavity mode Q-factors,the width, γ⊥, and center, ωa, of the gain curve, and bythe non-linear interactions between modes due to gaincompetition/saturation. The Q-factor distribution de-pends only on the passive cavity geometry and refractiveindex; examples are shown in Fig. 4. For resonance SPA-SALT the non-interacting thresholds follow immediatelyfrom Eq. (10). The effects of mode competition are as-sessed by analysis of the SPA-SALT lasing equations andtheir predictions. We will explore both aspects in thefollowing section.

A. Lasing in Wave-Chaotic Cavities and Q-FactorDistributions

We begin by examining the Q-factor distributions oftwo resonator geometries, the stadium and the D-cavity,that each have completely chaotic ray dynamics [30]. Thestadium consists of a rectangle capped by two semicir-cles on two opposite sides and is shown in the inset ofFig. 2(b). Its aspect ratio is defined as the ratio betweenthe length L of the rectangle and the radius R of thesemicircles, ρS = L

R . Our simulation results are for theaspect ratio ρS = 2, for which the ray dynamics showsthe highest degree of chaos, i.e., the highest Lyapunovexponents [30]. The D-cavity, also known as a cut circle,consists of a circle with a part cut off along a single chordas shown in the inset of Fig. 2(a). We define the cut pa-rameter as is the ratio of the distance from the center ofthe circle to the chord, d, and the radius of the circle, R,as ρD = d

R . In our simulations we use ρD = 0.5, whichalso maximizes the degree of chaos of its ray dynamics[31].

The simulation results for a stadium with 2L = 10 µmand a D-cavity with R = 4.2 µm, which have approxi-mately the same area, and refractive index n = 3.5 areshown in Fig. 4. Note that in later sections we showresults for larger stadium cavities, with a long axis aslarge as 2L = 60 µm. Figure 4(a) shows the Q-factorsof the resonances as a function of the wavelength λ in awavelength window containing 1000 modes for each res-onator geometry. The stadium resonances are indicatedby black circles whereas the D-cavity resonances are indi-cated by red stars. The center wavelength of the windowis chosen to be 1 µm, which also corresponds to the cen-ter of the gain curve for the lasing calculations below,λa = 2πc/ωa. The width of the gain curve used in thesesimulations is γ⊥ = 50 nm. This is considerably smallerthan wavelength window in which we compute the reso-nances, ∆λFW = 280 nm. Hence the simulation resultsinclude all the high-Q resonances relevant to lasing.

Figure 4(b) shows the distribution of the Q-factors ofthe stadium. The bulk of the distribution shows reso-nances with Q-factors less than 2000. However, there

are many outliers with higher Q-factors, and the insetshows the spatial patterns of the first two lasing modes.The first lasing mode is based on the highest-Q resonance(Q = 7096), which is localized on an Unstable PeriodicOrbit known as the double diamond orbit, a phenomenonknown as scarring [44]. An important aspect is that allthe reflections of the double diamond orbit have incidenceangles equal to 45◦ and are thus contained by Total In-ternal Reflection (TIR) for n = 3.5. The second lasingmode is based on a resonance with Q = 3263 and has amore uniformly distributed spatial pattern. It does nothave the second-highest Q-factor (or second-lowest non-interacting lasing threshold) as naively expected, so itsorder in the lasing turn-on sequence is due to its weakercompetition with the first lasing mode compared to otherpotential lasing modes. Figure 4(c) shows the distribu-tion of the Q-factors for the D-cavity, which does nothave any significant high-Q outliers. The inset showsthe spatial patterns of the first two lasing modes. Thefirst lasing mode is based on the highest-Q resonance(Q = 1567), which does not show a strong localizationpattern. However, the second lasing mode is based on aresonance with much lower Q-factor, Q = 1100, which ismoderately scarred by the triangle orbit. The incidenceangles of the three reflections of the triangle orbit are47.0◦ and 21.5◦, respectively, and thus this orbit is alsocontained by TIR for n = 3.5.

Since gain competition does not influence the thresh-old of the first lasing mode, the ratio of 4.5 between thehighest-Q resonances of the stadium and the D-cavityshould lead to a substantially lower lasing threshold forthe stadium compared to the D-cavity. Taking into ac-count the wavelength of the highest-Q modes with re-spect to the gain center yields an about 4.6 times lowerthreshold for the stadium compared to the D-cavity.However, experimentally the stadia had an only 1.25times lower lasing threshold (see Table I). The similar-ity of the measured thresholds for stadia and D-cavitiescould, however, be explained by the fact that the high-Qscarred modes in the stadium do not exist in much largercavities or when surface roughness is added as shown fur-ther below.

The Q-factor distributions shown here are representa-tive of our results using different wavelength windows andcentral wavelengths. Moreover our wavelength window ischosen large enough to contain typical high-Q resonances.For example, the free spectral range (FSR) of modes lo-calized on the shortest periodic orbits (with length `)

in each shape is λFSR = λ2

2n` = 28.5 nm for the sta-dium and λFSR = 22.7 nm for the D-cavity. The high-Q scarred modes correspond to even longer orbits andshorter FSRs, so they are always contained for differentcentral wavelengths λa.

8

Figure 4. (a) Calculated spectrum (Q-factors and wavelengths of passive cavity modes) for the stadium (black circles) andD-cavity (red stars) resonators. The spectra contain 200 resonances each in a wavelength window centered at λ = 1 µm. Thestadium data shows many resonances with higher Q-factors than the typical resonances, whereas the D-cavity data does nothave any significant outliers. (b) Distribution of the Q-factors of 1000 resonances for the stadium resonator and the (c) D-cavityresonator. The insets show the spatial field distributions of the resonances that support the first two lasing modes. The greenlines highlight the scarring by the Double Diamond Unstable Periodic Orbit and the Triangle Unstable Periodic Orbit two ofthe modes.

Figure 5. (a) Resonance SPA-SALT results for the lasing intensities of the stadium resonator as a function of the pump strength.There are two lasing modes that turn on within a factor of 10 of the first lasing threshold. The lasing interactions prevent moremodes from turning on in this interval. (b) Resonance SPA-SALT results for the lasing intensities of the D-cavity resonator asa function of the pump strength. There are eight lasing modes that turn on within a factor of 10 of the first lasing threshold.The lasing interactions are not strong enough to prevent the first seven lasing modes from turning on within a factor of 2.0 ofthe first lasing threshold. The gain spectrum width used in the simulations is 50 nm.

B. Lasing and Modal Interactions

Performing SPA-SALT calculations with the resonancedata shown in Fig. 4, we obtain the thresholds, num-ber of lasing modes and mode intensities as a functionof pump for a stadium and a D-cavity laser with thesame area. Consistent with our expectations and theresults of Ref. [22], both the stadium and the D-cavitylasers exhibit multimode lasing, even with this small sizeand the full inclusion of gain competition through cross-

saturation in the calculations. However, the rather dif-ferent Q-factor distributions of the two wave-chaotic cav-ities lead to quantitatively different behavior as shown inFig. 5. For the stadium resonator, only two modes startlasing within a factor of 10 of the first lasing threshold,whereas for the D-cavity resonator there are eight lasingmodes within a factor of 10 of its first lasing threshold.As already noted, the lasing threshold for the stadium ofthis size, shape and index is ∼ 4.6 times lower than forthe D-cavity laser, but here we are comparing the num-

9

Resonator n Lasing Modes Gain Clamping Limit

Stadium 3.5 2 6

D-cavity 3.5 8 8

Stadium 3.0 8 9

D-cavity 3.0 14 14

Stadium 2.5 5 5

D-cavity 2.5 4 6

Table II. Number of lasing modes within a factor of 10 of thefirst lasing threshold and number of lasing modes when gainclamping sets in for cavities with different refractive indices.

ber of lasing modes within the same relative range ofpump for the two shapes. As we increase the normalizedpump strength beyond 10 we find that a few more modesturn on for the stadium, reaching six modes. But be-yond a certain pump strength no additional lasing modesturn on; this phenomenon is known as “gain clamping”and has already been reported before in wave-chaotic res-onators [16, 17].

Thus our results show that cross-saturation stronglylimits the number of lasing modes in wave-chaotic lasers,but does not lead to single-mode lasing at high relativepump values in any case we have studied. If one modeturns on very early because of its anomalously high Q-factor, then it is able to saturate the gain substantiallybefore other modes are close to threshold, leading tofewer modes lasing, as in the stadium. For the refrac-tive index n = 3.5 considered so far the stadium hassuch outlier modes and the D-cavity does not, leadingthe latter to have more lasing modes. However this effectdepends on the refractive index, and Table II shows thatfor n = 2.5 the stadium has more lasing modes within theinitial factor of 10 of relative pump values. The differentnumbers of lasing modes that we find here for stadiumand D-cavity lasers result from non-universal effects ofscar modes on short periodic orbits with anomalouslyhigh Q-factors in the tails of the Q-factor distributions.Such effects cannot be described by statistical theories[45, 46]. In the following we explain the different numberof lasing modes for stadium and D-cavity lasers by de-tailed analyses of the cross-gain saturation. Furthermore,we show that the non-universal effects due to high-Q scarmodes become weaker for larger cavities and when sur-face roughness is added.

C. Cross-Gain Saturation

The non-universal variation of the number of lasingmodes led us to examine the interaction coefficients tosee if the results can be explained by cross-saturation.SALT shows us that the quantity which represents modalinteractions between lasing modes is the SALT interac-

tion coefficient χµν , which we define as

χµν =

∣∣∣∣∫CdxΨµ(x)Ψµ(x)|Ψν(x)|2∫

CdxΨµ(x)Ψµ(x)|Ψµ(x)|2

∣∣∣∣ . (11)

Note that mode µ starts lasing before mode ν and thatthe denominator of Eq. (11) normalizes the interactioncoefficient by dividing it by the self-interaction χµµ ofthe mode that started lasing first. Furthermore, we haveomitted the Lorentzian factor Γµ in Aµν , which dependson the relative location of the gain center.

We calculated the SPA-SALT interaction coefficientsfor all pairs of modes that start lasing up to the onset ofgain clamping. Note that the coefficients describe boththe interaction between a pair of modes where both of themodes are lasing, as well as the effect of a lasing modeon second mode before the second one turns on. The val-ues shown for the stadium resonator, Fig. 6(a), and theD-cavity, Fig. 6(b), show that there is significant modalinteraction and that it has an important effect. For in-stance, for the stadium, lasing mode #4 does not turnon immediately after the first lasing mode as it wouldhave done in the absence of interactions because of itsrelatively strong interaction with mode #1 (comparedto other modes). Indeed, as shown in the insets, theirelectric field spatial patterns are very similar, leading tostrong cross-saturation of mode #4 by mode #1. Notethat these are the two highest-Q modes in the distri-bution and that they are both strongly scarred by thedouble diamond periodic orbit. This demonstrates thatour results on lasing modes and thresholds are consis-tent with the expectations for spatial-hole burning andcross-saturation effects. For both the stadium and theD-cavity the interactions influence significantly the orderin which modes start to lase.

In the case of the D-cavity, the interaction coefficientsare more uniform, but there is a strong hole-burning in-teraction between modes #1 and #6. Even though thesemodes are not strongly scarred by a single periodic orbit,the insets show a non-uniform spatial pattern as well as astrong similarity of their electric field intensity distribu-tions. In fact there is a weak pseudo-caustic structure inthe D-cavity modes which we will discuss elsewhere [47].It appears that the main effect of the absence of high-Qscarred modes in the D-cavity is to allow more modesto lase simultaneously. The D-cavity hence appears tohave a Q-factor distribution closer to that expected foran ideal wave-chaotic cavity, as described in Ref. [46].

D. Sub-Threshold Intensities

Not only does spatial-hole burning and gain satura-tion control the order in which lasing modes turn on,even more importantly it limits their total number. SPA-SALT allows to study the interaction with modes belowthreshold, which determine when and if a given modeturns on. As noted above, the interacting threshold for a

10

Figure 6. Normalized SALT interaction coefficients [Eq. 11] for (a) stadium and (b) D-cavity. (a) For the stadium, theinteraction between modes #1 and #4 is higher than the interaction between modes #1 and the other lasing modes, #2, 3, 5, 6,due to their strong spatial overlap (as shown in the inset, top mode #1, bottom mode #4). (b) For the D-cavity, the interactionsare overall more uniform than for the stadium. Modes which have similar spatial patterns form two groups: modes #1, 6, 7and modes #2, 3, 4, 5, 8. The insets show the spatial distribution of the electric field for the lasing mode #1 (top) and lasingmode #6 (bottom). This pair has the highest SALT interaction coefficient for the D-cavity. The cavities have identical index(n = 3.5) and identical surface area, corresponding to a stadium length of 2L = 10 µm.

Figure 7. Resonance SPA-SALT results for the sub-threshold intensities of the (a) stadium and the (b) D-cavity as a functionof the pump. The black dashed lines show the intensity variation neglecting modal interactions and intersect the x axis atthe non-interacting thresholds. The colored lines show the sub-threshold intensities of the same modes when including theeffects of interactions. The resonance SPA-SALT approximations ensure that the sub-threshold intensities are linear betweenadjacent thresholds, with discontinuities of both their value and slope at the interacting thresholds, which are marked by thecolored arrows. The dashed vertical lines serve as a guide for the eye and connect the values of the sub-threshold intensitiesimmediately before and after a new threshold is reached. The gain curve width used in the simulations is 50 nm. The cavitieshave identical index (n = 3.5) and identical surface area, corresponding to a stadium length of 2L = 10 µm.

11

given mode N corresponds to the pump value at which itsintensity passes through zero. So just below this thresh-old, if we expand the size of the (N −1)× (N −1) matrixAµν , we find that in addition to the positive intensities ofthe N − 1 lasing modes, SPA-SALT predicts that modeN has a small negative intensity, approaching zero withpositive slope as the pump increases. While this negativeintensity is unphysical, its distance below zero, combinedwith its slope, is a measure of the proximity of the modefrom threshold. Hence we introduce here sub-threshold“intensity” plots to analyze further the effect of modalinteractions on modes below threshold.

As noted above, we compute the sub-threshold nega-tive intensities by enlarging the SPA-SALT matrix Aµνof the lasing modes given in Eq. (4), at a given valueof the pump, with each of the sub-threshold modes ofinterest. Between the lasing thresholds these intensi-ties vary continuously and linearly as well as the abovethreshold modal intensities. As one of the sub-thresholdmodes reaches zero intensity and turns on, it is addedto the physical matrix Aµν of the lasing modes. Whenthis happens, all other sub-threshold modes experience a(typically negative) change of their slope and intensity.This behavior contrasts with the above threshold, physi-cal modes, which must have continuous intensities as thenumber of modes increases, although they also have adiscontinuity of their slope at the thresholds.

This threshold effect is calculated separately for eachof the non-lasing modes as a way of characterizing themodes that are suppressed from lasing, resulting in aplot of the type shown in Fig. 7. It shows the evolutionof the “negative intensities” of several non-lasing modesas the various lasing modes turn on. The arrows on thetop of the figure mark the thresholds of the lasing modesat which the (negative) values of the intensities of thesub-threshold modes have a discrete jump. Note that allof the intensity jumps as well as the slope changes arenegative, indicating that each new lasing mode typicallyreduces the gain for all other modes. Since the evolutionof the sub-threshold intensities is linear between thresh-olds, if the intensity slope of a mode turns negative, thismode will never turn on, no matter how strong the pumpbecomes (barring the very rare, but not forbidden eventthat another mode turning on increases its gain, whichis not observed in Fig. 7). When all sub-threshold modeshave negative slopes, no further mode can turn on andwe have reached the gain-clamping regime.

Figure 7(a) shows the sub-threshold intensities for thestadium. The black dashed lines show the behavior ofthe intensities in the absence of interactions; for this casenine lasing modes would turn on within a factor of 3.5 ofthe first threshold. However, once the first lasing modeturns on, the modal interactions decrease the values andslopes of the sub-threshold intensities of the other modesso only one more mode turns on. When the second modeturns on, most of the remaining modes obtain negativeslopes, so that they can never turn on. Among the modesshown, only the cyan and blue modes in the upper right

part of the plot can still turn on, but they get pushedup to thresholds that are many times higher than theirnon-interacting values which are off the range of the plot.

Figure 7(b) shows a similar plot of the sub-thresholdintensities for the D-cavity. For this case twelve lasingmodes would turn on within a factor of 2.0 of the firstthreshold in the absence of any interactions. While thethreshold of the first mode immediately pushes some ofthe modes down in intensity, none acquire a negativeslope, and the three other modes close to threshold turnon almost immediately, with three more modes turningon within a factor of 2.0 of the first threshold. Only thenare the cumulative interactions sufficiently strong to keepother sub-threshold modes from lasing. This plot clearlyshows the effect of the D-cavity having high-Q modesthat are closer in their Q-factors compared to the sta-dium with its outliers in the Q-factor distribution: thefirst mode does not have enough intensity to suppressother modes before they turn on. However there are stillsufficiently strong interaction effects to keep a number ofother modes from lasing and push others to much higherthresholds. Thus, as discussed in Ref. [22], the total num-ber of lasing modes in steady-state is a function of boththe Q-factor distribution of the passive cavity modes andthe gain competition interactions in the active cavity.

Whereas the results presented so far were computedfor resonators with refractive index n = 3.5 (near the ex-perimental value of n ≈ 3.37), the lasing behavior showssignificant dependence on the refractive index, as sum-marized in Table II. In particular, the few-mode lasingbehavior observed for the stadium with n = 3.5 is notrobust against a change of the refractive index, and fordifferent simulation parameters we see significantly morelasing modes. For the stadium with n = 2.5 there arefive lasing modes within a factor of 10 of the first lasingthreshold, and eight lasing modes in the same range forn = 3.0. Moreover, none of the simulation results showsingle mode lasing. We conclude that while the numberof lasing modes depends on the specific value of the in-dex of refraction of the resonator, multimode lasing isobserved for all physically relevant refractive indices.

E. Dependence on the Size of the Wave-ChaoticResonators

Our simulation results for a stadium with long axis2L = 10 µm (area 44.6 µm2) and a D-cavity of equalarea with n = 3.5 predict a significant difference be-tween the absolute lasing thresholds of the stadium andthe D-cavity lasers due to the ∼ 4.5 larger Q-factor ofthe highest-Q mode in the stadium compared to that ofthe D-cavity. This difference, if it could be extrapolatedto the larger cavities studied experimentally, would beclearly observable, whereas only a ∼ 1.25 lower thresh-old for the stadium was observed. Naively one mightexpect the Q-factors of scarred modes (as well as of allother mode types) to increase linearly with the linear

12

Figure 8. Dependence of the Q-factor distributions on the size of the stadium resonator. (a) 2L = 10 µm. The inset showsthe electric field distribution of the highest-Q mode. (b) 2L = 20 µm. The inset shows the normalized standard deviationof the three Q-factor distributions as a function of cavity size. (c) 2L = 60 µm. The inset shows the spatial distribution ofthe electric field of the highest-Q mode. Notice the qualitative decrease in electric field compared to the highest-Q mode for2L = 10 µm. For panels (a), (b) and (c), the red dashed vertical line marks the Q-factor of the long living ray trajectories.(d) Size dependence of Q-factors. Indicated are the highest Q-factor, QMAX (black solid line), the average of the ten highestQ-factors, 〈Q〉10 (blue solid line), and the average of all Q-factors, 〈Q〉ALL (green solid line). The linear extrapolations of Qmax

and 〈Q〉all from their values at 2L = 10 µm are shown as black and green solid lines, respectively. The red dashed line indicatesthe extrapolation of Qmax according to square-root scaling.

dimensions of the cavity since the Q-factor is given bythe ratio of energy stored in the resonator, which is pro-portional to the area, by the energy radiated during oneoscillation period, which is proportional to the circumfer-ence of the cavity. This simple scaling argument assumesthat the general structure of a certain mode type doesnot depend on the ratio of cavity size and wavelength.It is, however, known that the scarred electric field of aperiodic orbit in a given mode field distribution tends todecrease in the semiclassical limit, i.e., when the cavitysize becomes much larger than the wavelength [48]. Fur-thermore, the Q-factors of scar modes can depend sensi-tively on interference effects and hence the ratio of cavitysize and wavelength [12]. We therefore studied the evolu-tion of the Q-factor distributions of stadium and D-cavity

resonators with the cavity size to see if the Q-factors ofhigh-Q scar modes scale linearly with the cavity size, aswell as how the ratio between the highest Q-factors ofthe distributions evolves.

We studied stadia with linear dimensions two and sixtimes larger than those in Fig. 4, with the same refrac-tive index n = 3.5. The three histograms in Fig. 8 showthe Q-factor distributions for stadia with increasing size,where the long axis 2L equals to 10 µm, 20 µm and 60 µm,respectively. The insets show the electric field distribu-tion of the corresponding highest-Q modes. We observethat most of the high-Q modes have a Q-factor increas-ing with size, but only sub-linearly. In addition, the Q-factor distributions narrow at both the high- and low-Qtails, comprised in the stadium by scarred modes (highQ-

13

factors) and Bouncing Ball modes (low Q-factors). Thedashed red vertical lines in the distributions mark thevalue of the average Q-factor of the most long-lived tra-jectories obtained from ray tracing simulations [2, 49, 50].As expected, for each distribution this value is higherthan the mean of the distribution but lower than a signif-icant number of the high-Q modes. The inset of Fig. 8(b)shows the decrease in the normalized standard deviationof the three distributions, summarizing the narrowing ofthe distribution with increasing size. Examining in de-tail the field distributions of the highest-Q modes, [insetsof Fig. 8(a) and Fig. 8(c)], we notice a significant reduc-tion in scarred electric field intensity as the linear sizeof the resonator increases, i.e., the electric field intensityenhancement along the double-diamond orbit is smallerfor the larger stadium.

Figure 8(d) summarizes the evolution of the Q-factorvariations with size: we plot the highest Q-factors,QMAX , the average Q-factor of the ten most long-livedmodes, 〈Q〉10, as well as the average Q-factors of the en-tire distribution, 〈Q〉ALL, for stadia with long axis equalto 10 µm, 20 µm and 60 µm, respectively. The dashedgreen line shows that the average Q-factor of the en-tire distribution 〈Q〉ALL increases almost linearly withthe linear size of the resonator. In contrast, the highestQ-factors clearly increase sub-linearly (compare to theblack dashed line), and have a scaling behavior closer toa square root law (dashed red line). These results agreewith theoretical work that suggests that the effects ofscarring decrease as the size of the resonator increases[48, 51] and that the tail of anomalously high Q-factorsfrom scarred modes shrinks as well [35]. In analogouscalculations for the D-cavity comparing the results forsizes R = 4.2 µm and R = 8.4 µm (not shown), theaverage Q-factor 〈Q〉ALL scales almost linearly with theresonator size as well. However, the highest-Q modes arenot scarred as strongly as in the case of the stadium,and while their Q-factors also increase sub-linearly, thereis a smaller difference between the highest-Q modes andthe rest of the distribution. We thus conclude that thenon-universal effects of scars from short periodic orbitsbecome less important for larger cavities, leading to asimilar lasing thresholds for stadium and D-cavity as ob-served in the experimental data.

F. Effects of Surface Roughness

Another effect that can result in more similar lasingthresholds for cavities of different shape is surface rough-ness. Scattering at the rough boundary can affect thefield distributions and Q-factors in particular of scarredmodes. To include this effect in our study, we performedadditional simulations in which we randomly modifiedthe smooth boundary of the 2D geometries for three res-onator geometries: the stadium, the D-cavity and an el-liptical cavity, which has integrable ray dynamics. Weintroduced random variations in the direction normal to

the ideal boundary, with a variation over length scalesfrom 0.3 µm to 10 µm (see Ref. [52] for the explicit defi-nition of the deformed boundary). The surface roughnessis quantified by the variance of the deviation in normaldirection, r, from the ideal boundary position, r0. Inour simulations the standard deviation of the boundary

deformation is σ = E[(r−r0)2]r0

. Based on our estimate ofthe surface roughness visible in SEM images of the ex-perimental devices, we used σ = 30 nm. The effects ofperturbing the resonator boundary in this way are pre-sented in Fig. 9, which shows the Q-factor distributionsfor the stadium, D-cavity and ellipse with surface rough-ness. We notice that the Q-factor distribution for thestadium with roughness no longer features high-Q out-liers as we suspected. The Q-factor distributions of thestadium and D-cavity resonators are much more similarwith surface roughness than for a smooth boundary, es-pecially concerning the high-Q tail of the distributions.Still, the average Q-factor for the stadium is still higherthat of the D-cavity.

The field distributions of the high-Q modes are also af-fected as shown in the insets of Fig. 9, and they deviatesignificantly from those for smooth boundaries. There-fore the SALT interaction coefficients and thus modecompetition are affected as well by the roughness. Ta-ble III summarizes the number of lasing modes within afactor of 10.0 of the first lasing threshold and the evo-lution of the Q-factors as a function of cavity size forstadium, D-cavity and ellipse. When including surfaceroughness, doubling the size leads to a roughly two-foldincrease of the number of lasing modes, i.e., a linear in-crease with the cavity size. Note that the simulated el-lipse has identical refraction index (n = 3.5) and surfacearea (A = 44.6 µm) as the other two resonators, and theaspect ratio b/a = 2.

Reference [27], in which single-mode lasing forstadium-shaped semiconductor microlasers was found,also reported experiments and simulations for ellipticalcavities. The ellipse has integrable ray dynamics, and thewhispering gallery modes of a dielectric ellipse resonatorcan be labeled with radial and azimuthal quantum num-bers. The experiments in [27] found multimode lasingfor elliptical microlasers in contrast to the single-modelasing of the stadium lasers. It is challenging to modelthese experiments since an ideal ellipse resonator has ex-tremely high-Q whispering gallery modes, which wouldlead to many orders of magnitude lower thresholds thanobserved experimentally. We attribute this to effects suchas surface roughness, inaccurate shape fabrication, andintrinsic absorption that reduce the Q-factors of actualresonators. We simulated elliptical microlasers with sur-face roughness equal to that used for the wave-chaoticresonators. We found that the rough ellipse cavities al-ways showed multimode lasing, with a number of modescomparable to, but smaller than for the wave-chaotic cav-ities, as shown in Table III. The highest Q-factor forthe rough ellipse was four times higher than that for therough stadium, and the mode showed a more regular field

14

Figure 9. Q-factor distributions for the (a) stadium, (b) D-cavity and (c) ellipse resonators with surface roughness. Eachdistribution contains 1000 resonances centered at λ = 1.0 µm. The random boundary roughness in each of the simulations isdrawn from the same distribution, and the standard deviation of the boundary in the normal direction is σ = 30 nm. Theinsets show the spatial patterns of the modes with the highest Q-factors for each distribution. The scarring effect observed forthe highest-Q modes of cavities with smooth boundaries do not persist for this level of surface roughness. The cavities haveidentical index (n = 3.5) and identical surface area, corresponding to a stadium length of 2L = 10 µm.

distribution despite the surface roughness. This indicatesthat an integrable cavity shape, like the ellipse or the cir-cle, has resonance properties which are distinct from thewave-chaotic cavities, even in the presence of substantialsurface roughness.

Resonator QMAX 〈Q〉10 LM

Stadium SR 2L = 10 µm 1833 1524 9

D-cavity SR 2R = 8.4 µm 1034 969 11

Ellipse SR 2b = 10.6 µm 7662 5282 6

Stadium SR 2L = 20 µm 2852 2724 15

D-cavity SR 2R = 16.8 µm 2083 1895 22

Ellipse SR 2b = 21.2 µm 16495 13800 14

Table III. Highest Q-factor (QMAX) and the average of the10 highest Q-factors (〈Q〉10), as well as the number of LasingModes (LM) within a factor of 10.0 of the first lasing thresholdfor the stadium, D-cavity and ellipse with surface roughness.The refractive index is n = 3.5 and the geometrical parame-ters are chosen such that the the different geometries in thetop half of the table have the same area, as do the ones in thebottom half of the table.

V. DISCUSSION AND CONCLUSION

We present an experimental and theoretical study con-cerning the question if wave-chaotic semiconductor mi-crolasers in cw operation generally exhibit multimodelasing, as was found for experiments with pulsed pump-ing in Ref. [22] and steady-state numerical simulations inRef. [42], or single-mode lasing as found in experimentswith cw pumping and theoretical studies in Refs. [26–28].

Our new experimental results, presented in the firstpart of this paper, can be summarized as follows. We

find multimode lasing for both stadium and D-cavitylasers in qualitative agreement with our earlier resultson the D-cavity [22]. Time-resolved measurements forlong pump pulses show fewer lasing peaks at a given timecompared to integration over a whole pulse, but we donot observe a consistent reduction of the number of ac-tive lasing modes towards a single mode over the course oflonger pulses as was found in Ref. [26]. The time-resolvedmeasurements showed that the spectra were stable overtime scales very long compared to other time scales inthe system, so we believe our results are representativeof the steady-state lasing properties. The devices studiedin the current experiments have lasing thresholds compa-rable to those studied in Ref. [27], thus it is unlikely thata different degree of surface roughness is responsible forour observation of multimode lasing in contrast to single-mode lasing in the experiments of Refs. [26, 27].

The new theoretical and simulation results presentedhave the following implications in our view. Non-universal effects, such as lasing on scarred modes, be-come already weak in wave-chaotic cavities with dimen-sions well below the size of the experimental cavities, andcannot explain single-mode lasing in wave-chaotic micro-lasers. SALT theory, which takes into account gain com-petition and spatial hole burning very accurately, neverpredicts single-mode lasing for wave-chaotic resonatorspumped well above threshold. Hence gain saturationand spatial overlap of modes alone do not explain theexperimental results found in Ref. [27]. Hence the re-maining theoretical uncertainties are in the dynamics ofthe gain medium, which is neglected in SALT. It is wellknown that for nearly degenerate lasing modes, a fre-quency locking effect can occur above threshold due tothe population dynamics, reducing the number of lasingmodes [53]. FDTD simulations of wave-chaotic lasers bythe Harayama and coworkers [28, 54] appear to show such

15

merging of modes as the pump is increased, but these ef-fects are found at unphysically high relative pumps, andat large values of the population relaxation rate γ‖. Ifthere is a regime where single-mode lasing is favored forwave-chaotic lasers, it must be due to subtle dynamicaleffects such as these, which go beyond standard spatialhole-burning and gain competition. We currently lackthe theoretical and computational tools to answer defini-tively this question and we did not explicitly consider cer-tain effects that might impact the dynamics, such as car-rier diffussion. Our experimental results indicate that if

there is such a regime, it is not realized in all wave-chaoticsemiconductor lasers, and needs to be better character-ized experimentally.

ACKNOWLEDGMENTS

This work was supported by AFOSR Award No.FA9550-16-1-0416.

The authors thank T. Harayama, T. Fukushima, S.Sunada, S. Shinohara, M. Guy and K. Kim.

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