Abstract

For a fully chaotic two-dimensional (2D) microcavity laser, we present a theory that guarantees both the existence of a stable single-mode lasing state and the nonexistence of a stable multimode lasing state, under the assumptions that the cavity size is much larger than the wavelength and the external pumping power is sufficiently large. It is theoretically shown that these universal spectral characteristics arise from the synergistic effect of two different kinds of nonlinearities: deformation of the cavity shape and mode interaction due to a lasing medium. Our theory is based on the linear stability analysis of stationary states for the Maxwell–Bloch equations and accounts for single-mode lasing phenomena observed in real and numerical experiments of fully chaotic 2D microcavity lasers.

© 2017 Chinese Laser Press

1. INTRODUCTION

Universality is a key concept in quantum chaos study. We can find a common feature in a quantum system if the corresponding classical system exhibits fully chaotic dynamics [13]. The most representative example is the Bohigas–Giannoni–Schmit conjecture concerning universal spectral fluctuations of quantized fully chaotic systems [411]. Universality in quantum chaos has also been observed for electron transport in mesoscopic devices [12,13].

Because of an analogy between the classical-quantum and the ray-wave correspondence, quantum chaos theory can be directly applied to “wave-chaotic” systems, which exhibit chaotic dynamics in the ray-optic limit [1425]. A representative example is a two-dimensional (2D) microcavity laser, whose resonant modes can be viewed as those of quantum billiards [26,27]. By a quantum-chaos approach, the emission patterns of 2D microcavity lasers with various cavity shapes have been successfully explained and predicted [1427]. However, there is another important aspect of 2D microcavity lasers that cannot be elucidated only by quantum chaos theory, that is, nonlinear interactions among resonant modes due to a laser gain medium. From the viewpoint of universality, it is of interest to uncover how this additional nonlinear effect manifests itself depending on the chaoticity or integrability of underlying ray dynamics inside a cavity.

A recent experimental study of semiconductor 2D microcavity lasers has demonstrated that single-mode lasing is achieved with a stadium-shaped (i.e., fully chaotic) cavity, while multimode lasing with an elliptic (i.e., nonchaotic) cavity [28,29]. This drastic difference was attributed to the difference of spatial modal patterns between the stadium and elliptic cavities. It was numerically shown that for the stadium cavity, an arbitrary low-loss modal pair has a significant spatial overlap, while for the elliptic cavity, there exist low-loss modal pairs whose spatial overlaps are small [29]. This means that any lasing mode of the stadium cavity tends to strongly interact with the other modes, while multiple lasing modes can coexist for the elliptic cavity because of small interactions among them.

It is important that all of the various fully chaotic stadium-type cavities of different aspect ratios and sizes studied in the experiments of Ref. [29] have shown single-mode lasing while all of various integrable elliptic cavities have multimode lasing. Therefore, it was conjectured that single-mode lasing is universal for fully chaotic cavity lasers [29]. In order to ascertain this universality, it is important to further examine it experimentally and numerically for various 2D cavities. Such studies are important because they would add evidence of the universality. However, these pieces of evidence cannot directly reveal the insight of the universality. In past studies of universalities not only in quantum chaos but also in second-order phase transitions and critical phenomena, establishment of a theory that verifies a universality was the most difficult, challenging, and important task [711,30].

For a theoretical verification of the conjecture of universal single-mode lasing, it is necessary to evaluate the stability of a stationary-state solution of a full nonlinear model such as the Maxwell–Bloch equations [3134]. Stability analysis for one-dimensional lasers in a low pumping regime just above the lasing threshold has been established by Lamb [3537]. It also has been applied to 2D microcavity lasers and explained the spontaneous symmetry breaking of a lasing pattern [32,38]. However, Lamb’s perturbation theory becomes invalid for a high pumping regime where the universal single-mode lasing can be observed because his perturbation theory expresses the population inversion by a power series expansion of lasing modes.

In this paper, we introduce a different expansion method for the population inversion in the Maxwell–Bloch equations that is applicable to a high pumping regime. Furthermore, we explicitly derive the stability matrix for stationary-state solutions, which describes the interactions among a huge number of lasing modes. The matrix elements turn out to be greatly simplified under the assumptions that the cavity size is much larger than the wavelength and the external pumping power is sufficiently large. Moreover, the eigenvalues of this matrix can be analytically evaluated by applying a theorem in linear algebra. This enables us to theoretically show that for a fully chaotic 2D microcavity laser, at least one single-mode lasing state is stable, while all multimode lasing states are unstable. This result provides a theoretical ground for universal single-mode lasing in fully chaotic 2D microcavity lasers.

2. FUNDAMENTAL EQUATIONS

For modeling the microcavity, we assume that it is wide in the xy-directions and thin in the z-direction. This allows us to separate the electromagnetic fields into transverse-magnetic (TM) and transverse-electric (TE) modes. Here we focus only on TM modes, where the electric field vector is expressed as E=(0,0,Ez). We also assume that the atoms in the lasing medium have spherical symmetry and two energy levels. The relaxation due to the interaction with the reservoir can be described phenomenologically with decay constants γ for the microscopic polarization ρ and γ for the population inversion W. We also need to include phenomenologically the effect of the external energy injected into the lasing medium by the pumping power W.

By applying the slowly varying envelope approximation, one can reduce the Maxwell equation as follows:

tE˜=i2(xy2+1)E˜αLE˜+2πNκϵρ˜,
where E˜ and ρ˜ are respectively the slowly varying envelopes of the z-component of the electric field and the microscopic polarization, N is the number density of the atoms, κ is the coupling strength, ϵ is the permittivity, and αL represents the losses describing absorption inside the cavity. In the above, space and time are made dimensionless by the scale transformation ((ninωs/c)x, (ninωs/c)y)(x,y), tωst, respectively, where nin denotes the effective refractive index inside the cavity and ωs is the oscillation frequency of the fast oscillation part of the electric field. In the same way, we have the equation for the electric field outside the cavity,
nout2nin2tE˜=i2(xy2+nout2nin2)E˜,
where nout denotes the refractive index outside the cavity. For the boundary condition at infinity, we adopt the outgoing wave condition.

The optical Bloch equations are also transformed to the following form:

tρ˜=γ˜ρ˜iΔ0ρ˜+κ˜WE˜,
tW=γ˜(WW)2κ˜(E˜ρ˜*+E˜*ρ˜),
where the dimensionless parameters are defined as follows: γ˜γ/ωs, γ˜γ/ωs, Δ0[ω0ωs]/ωs, and κ˜κ/ωs has the dimension of the inverse of the electric field and ω0 is the transition frequency of the two-level atoms.

A theoretical method to obtain stationary-state solutions of Eqs. (1)–(4) has been developed as “steady-state ab initio laser theory (SALT)” [33,34,3945]. However, the existence of a stationary-state solution does not always mean its experimental observability. That is, a stationary-state solution must be stable so that it can be experimentally observed, especially when the experiment is performed with continuous-wave operation, where the long-term dynamical effect is expected to be important. Such a dynamical stability is not explicitly incorporated in the SALT approach.

In the following, we carry out the stability analysis of a stationary-state solution for the Maxwell–Bloch equations. Although the equations for the stability analysis are very complicated in general, they turn out to be greatly simplified thanks to full chaoticity and the short wavelength limit. By applying the analysis to a 2D microcavity laser where the ray dynamics are fully chaotic, we show that at least one single-mode lasing state is stable and all of the multimode lasing states are unstable when the size of the cavity is much larger than the wavelength and the pumping power is sufficiently large.

3. DYNAMICS OF ALMOST STATIONARY LASING STATES

We assume that near a stationary state the light field and polarization can be expressed as follows:

E˜=iEi(t)eiΔitUi(x,y),
ρ˜=iρi(t)eiΔitVi(x,y),
where Δi represents the lasing oscillation frequency. Note that the lasing mode i depends on the pumping power and can be a fusion of several modes that coalesce by frequency-locking and separate into individual lasing modes with different frequencies as the pumping power decreases [28,38]. Ui is supposed to be normalized.

Then, from Eq. (1), we obtain

dEi(t)dt+jidEj(t)dteiΔijtUij=[i(Δi+12)(αL+γ˜ii)]Ei(t)+ji{[i(Δj+12)αL]Uijγ˜ij}Ej(t)eiΔijt+2πNκεjeiΔijtρj(t)DUi*(x,y)Vj(x,y)dxdy,
where Δij denotes the frequency difference between modes j and i, i.e., ΔijΔjΔi and Uij is defined by the inner product UijDUi*(x,y)Uj(x,y)dxdy, and it will be shown later that γ˜ii is related to the flux of the light field intensity from inside to outside the cavity through the cavity edge and γ˜ij is defined as follows:
γ˜iji2DdxdyUi*(x,y)2Uj(x,y).
In the above, D denotes the area inside the cavity.

From Eq. (3), we have

ρj(t)Vj(x,y)=κ˜Wγ˜iΔ0jEj(t)Uj(x,y).
Therefore, when the light field is almost stationary, from Eqs. (4) and (9), one can express the population inversion W by the light field amplitudes Ei and the spatial patterns Ui, i.e.,
W=W/{1+[ij2κ˜2EiEj*UiUj*eiΔjit(γ˜+iΔ0j)(γ˜iΔji)+c.c.]}.
The conventional approach to treat the nonlinear terms in W is to perturbatively expand the right-hand side of Eq. (10) for small light field amplitudes [35,36]. This method is only applicable to just above the lasing threshold and cannot correctly describe the case when the external pumping power W is very large. In the following, we present a different approach applicable to the high-pumping cases. Note that our method can be applied to semiconductor lasers in the same way as the conventional approach [37].

We introduce the dimensionless quantities L and C related to the total intensity and mode interference, respectively, as follows:

L(x,y)1+mam|Um|2,
where am denotes the dimensionless light field intensity of the mode m weighted by the Lorentzian gain g(Δm)γ˜/(γ˜2+Δ0m2), i.e., am(4κ˜2/γ˜)g(Δm)|Em|2, and
C(x,y)l,jlj2κ˜2ElEj*UlUj*eiΔjlt(γ˜+iΔ0j)(γ˜iΔjl)+c.c.
Then the denominator of the term in the right-hand side in Eq. (10) is expressed as L+C=L(1+C/L). The basic idea of our approach is to expand it in the power series of C/L under the condition |C|/|L|<1 almost everywhere in the cavity. From Eqs. (7), (9), and (10), we obtain
dEi(t)dt+jidEj(t)dteiΔijtUij=[i(Δi+12)(αL+γ˜ii)]Ei(t)ji{[i(Δj+12)αL]Uijγ˜ij}Ej(t)eiΔijt+ξWkeiΔiktEkγ˜iΔ0k×DdxdyUi*UkL(x,y){1C(x,y)L(x,y)+[C(x,y)L(x,y)]2},
where ξ2πNκκ˜/ϵ. Since we are focusing on the vicinity of the stationary state of the slowly varying envelope, dEi(t)/dt is very small. Therefore, we can assume Ei(t)eεit where εi|Δij| for all i and j (ji). When Eq. (13) is integrated over t, the second terms on both sides have the coefficients of 1/(εjiΔij) while the first terms are of 1/εi. Consequently, the contributions of the terms concerning fast oscillations like the second terms are much smaller than those of the first terms. Accordingly, one can ignore the terms oscillating faster than eiΔikt.

By ignoring the terms oscillating faster than eiΔikt, Eq. (13) is reduced to

dEidt[i(Δi+12)(αL+γ˜ii)]Ei+ξWEiγ˜iΔ0iDdxdy|Ui|2L(x,y)ξWEiDdxdy|Ui|2[L(x,y)]2kki2κ˜2|Ek|2|Uk|2(γ˜iΔ0k)(γ˜iΔki)×(1γ˜+iΔ0k+1γ˜iΔ0i).
Therefore, we obtain
d|Ei|2dt=ddt(EiEi*)=Ei*dEidt+EidEi*dt=(2αL+γ˜ii+γ˜ii*)|Ei|2+2ξWg(Δi)Ddxdy[L(x,y)]2×|Ei|2|Ui|2{L(x,y)k,ki2κ˜2g(Δk)g(ΔiΔk)×[2γ˜+(ΔiΔ0)(ΔiΔk)/γ˜+(ΔiΔk)(Δi+Δk2Δ0)/γ˜]|Ek|2|Uk|2},
where g(ΔiΔk) is a Lorentzian defined as g(ΔiΔk)γ˜/[γ˜2+(ΔiΔk)2]. If Δi is far from Δ0, the second term of the second line does not contribute because g(Δi) almost vanishes. Therefore, we assume |ΔiΔ0|γ˜. Since the terms concerning Δk contribute in L(x,y) if Δk is as close to Δ0 as Δi due to the Lorentzian g(Δk), we obtain
L(x,y)1+k,|ΔkΔi|γ˜4κ˜2γ˜g(Δk)|Ek|2|Uk|2.
Because of the Lorentzian g(ΔiΔk), only the terms whose Δk values are close to Δi such that |ΔiΔk|γ˜ contribute to the sum over k in Eq. (15). Accordingly, we have g(ΔiΔk)1/γ˜ and
2γ˜(ΔiΔ0)(ΔiΔk)/γ˜+(ΔiΔk)(Δi+Δk2Δ0)/γ˜.
Consequently, we obtain
d|Ei|2dt=(2αL+γ˜ii+γ˜ii*)|Ei|2+2ξWg(Δi)Ddxdy[L(x,y)]2|Ei|2|Ui|2×[1+k,|ΔkΔi|γ˜4κ˜2γ˜g(Δk)|Ek|2|Uk|2k,ki,|ΔkΔi|γ˜4κ˜2γ˜g(Δk)|Ek|2|Uk|2]=(2αL+γ˜ii+γ˜ii*)|Ei|2+2ξWg(Δi)Ddxdy[L(x,y)]2|Ei|2|Ui|2×[1+4κ˜2γ˜g(Δi)|Ei|2|Ui|2].
Therefore, we finally obtain the equation for the time evolution of the light field intensity Ii|Ei|2 of the lasing mode i,
dIidtSiIi,
where Si denotes the balance of the loss, gain, and saturation of the mode i, and is defined as
Si2(αL+γi)+2ξWg(Δi)Ddxdy|Ui|2Li(x,y)[L(x,y)]2,
and Li(x,y) is related to the dimensionless light field intensity of the mode i, i.e., Li(x,y)1+ai|Ui|2. γi is derived by applying Green’s theorem to (γ˜ii+γ˜ii*) and represents the rate of the flux of the light field intensity going outside the cavity through the cavity edge for the lasing mode i:
γii4Dds(Ui*UinUiUi*n),
where /n is a normal derivative on the cavity edge. It is important to note that Eq. (19) is derived without any assumption of small intensities, and hence it can be applied to the strongly pumped regimes.

4. STABILITY ANALYSIS OF STATIONARY LASING STATES

A. Stability Matrix

The light field intensities that make the right-hand side of Eq. (19) vanish correspond to stationary-state solutions. The stability of a stationary-state solution is evaluated by the time evolution of the small displacements δIi from the intensities Is,i for the stationary-state subject to the differential equation dδI/dt=M˜δI. Here the displacement vector is defined by δI(δI1δI2δIn)T and the matrix M˜ is given by M˜PMP1, and

Mij{Sj+2ξWbj2Ddxdy|Uj|4[L(x,y)]2}δij4ξWbibjDdxdy|Ui|2|Uj|2Li(x,y)[L(x,y)]3,
where Pdiag(|E1|,,|EN|) and bi[aig(Δi)]1/2=2κ˜g(Δi)|Ei|/γ˜1/2. For simplicity, we introduced the matrix P and assumed that its inverse matrix P1 exists. However, |Ei|1 in P1 always cancels out |Ei| in P. Therefore, M˜ is always well-defined and the following discussion is valid irrespective of the existence of P1.

B. Single-Mode Lasing States

From Eq. (19), one can see that the fixed point (0,,0,Is,j,0,,0), which satisfies Sj=0, corresponds to the single-mode lasing state of the mode j whose light field intensity is equal to Is,j=|Es,j|2. Then, the matrix M˜ for this fixed point becomes a diagonal matrix, i.e.,

M˜jj=2ξWbj2Ddxdy|Uj|4Ls,j2<0,
and for ij,
M˜ii=2ξWg(Δi)Ddxdy|Ui|2Ls,iLs,j2Ls,j2Ls,i,
where Ls,i is related to the light intensity of the single-mode lasing corresponding to the fixed point (0,,0,Is,i,0,,0), i.e., Ls,i1+as,i|Ui|2 and as,i(4κ˜2/γ˜)g(Δi)Is,i. From Eq. (23), one can see that the single-mode lasing state of the mode j is stable in the direction (00δIj00)T.

It is important to note that Ls,i(j) contains the spatial pattern of the mode i(j). As shown in the Section 5.B, if the spatial pattern |Ui|2 overlaps with |Uj|2 inside the cavity, M˜ii can be negative. Then, the single-mode lasing state of the mode j can be stable in the direction (00δIi00)T.

C. Multimode Lasing States

From Eq. (19), one can see that a multimode lasing state corresponds to the solutions Is,i of the simultaneous equations Si=0 (i=1,2,,N). The number N of the lasing modes that have nonzero light field intensities is an arbitrary natural number more than 1 because Ii=0 always satisfies the stationary-state condition for Eq. (19). Then, from Eq. (22), we have

Mii=2ξWbi2Ddxdy|Ui|4Li(x,y)[L(x,y)]3[L(x,y)Li(x,y)2],
Mij=4ξWbibjDdxdy|Ui|2|Uj|2Li(x,y)[L(x,y)]3.
Consequently, if and only if all of the eigenvalues of the N×N matrix M˜(PMP1) are negative, this multimode lasing state is stable.

5. FULLY CHAOTIC 2D MICROCAVITY LASERS

A. Spatial Patterns of Stationary Lasing Modes

For bounded chaotic systems, theories on quantum ergodicity have shown that the probability density of finding a quantum particle in a small area whose state is described by the eigenfunction of a quantized fully chaotic system approaches a uniform measure as the energy of the particle increases and the wavelength becomes shorter [4651].

For open chaotic mapping systems, it has been shown that the long-lived eigenstates tend to be localized on the forward trapped set of the corresponding classical dynamics as the wavelength decreases [5256]. This tendency can be considered as a manifestation of quantum ergodicity in open systems.

A similar tendency has also been numerically observed for chaotic 2D microcavities [29], where resonance wave functions of low-loss modes are supported by the forward trapped set of the corresponding ray dynamics with Fresnel’s law [57,58] in the short wavelength limit. Because of this property, the overlap of wave functions between an arbitrary pair of low-loss modes takes a large value [29]. Therefore, we assume that the spatial patterns of the wave functions for low-loss modes are similar to each other and expressed as |Ui(x,y)|2=[1+εi(x,y)]|U0(x,y)|2, which implies

Ddxdyεi(x,y)|U0(x,y)|2=0,
because of the normalization for all of the spatial patterns of the lasing modes. We assume that the fluctuation εi(x,y) is so small almost everywhere and random that |εi(x,y)|U0(x,y)|2|1 and Ddxdyεi(x,y)0. We also assume Ddxdyεi(x,y)|U0(x,y)|20.

B. Stability of Single-Mode Lasing States

In the case of a fully chaotic 2D microcavity, one can apply the wave function property in Eq. (27) to evaluate M˜ii in Eq. (24) whose sign determines the stability of the single-mode lasing state of the mode j in the direction (00δIi00)T. Indeed, M˜ii is reduced for the first order of εi(x,y)|U0(x,y)|2 as follows:

M˜ii=2ξWg(Δi)Sdxdy(1+εi)|U0|2×as,i(1+εi)as,j2(1+εj)2|U0|2as,j2(1+εj)2(1+εi)as,i|U0|6|U0|2=2ξWg(Δi)as,j2as,iSdxdy{[1+O(εi(x,y)|U0(x,y)|2)]as,i|U0|2[1+O(εj(x,y)|U0(x,y)|2)]as,j2}2ξWg(Δi)as,j2as,i(as,iSdxdy1|U0|2Aas,j2),
where S denotes the support of |U0(x,y)|2 and A is its area.

If and only if M˜ii is negative, the single-mode lasing state of the mode j is stable in the direction (00δIi00)T. Therefore, from Eq. (28), we obtain the stability condition for the single-mode lasing of the mode j,

(as,jA)2>as,iASdxdy1A2|U0|2.
Accordingly, the single-mode lasing state of the mode j is stable when its intensity is large enough to satisfy Eq. (29) for all of the other mode i values. Note that the integral in Eq. (29) is estimated to be approximately unity because |U0|2 can be approximated to be A. Consequently, the mode that has the largest single-mode intensity is stable, and hence there always exists one stable single-mode lasing state at least.

C. Instability of Multimode Lasing States

Next, we show that all of the multimode lasing states are unstable under the assumptions that all of the spatial patterns Ui of the single-mode lasing states are similar to each other, and the pumping power W is very high, which implies that the light field intensities are very large.

According to the assumption in Section 5.A for the spatial patterns of the lasing modes in a fully chaotic microcavity, we obtain

Li(x,y)(1+εi)ai|U0|2,
L(x,y)m=1N(1+εm)am|U0|2
for the support S of |U0|2 and we assumed ai1. Then, the matrix elements in Eqs. (25) and (26) can be expressed by using the fluctuations εi(x,y). Therefore, from the properties assumed for εi(x,y), we obtain for the first order of εi(x,y)|U0(x,y)|2,
Mii4ξAWbi2aiatot3(1atot2ai),
Mij4ξAWbibjaiatot3,
where atotm=1Nam.

It is important to note that M˜(N) can be factorized as M˜(N)=4ξAW/(atot3)PQBRBP1, where Qdiag(a1,,aN), Bdiag(b1,,bN), and the diagonal elements of R are given by Rii1atot/(2ai), while the off-diagonal elements Rij=1. Then, as is explained in Appendix A, all of the eigenvalues of M˜(N) are equal to those of M(N) defined as

M(N)4ξAWatot3diag(a1,,aN)×BRBdiag(a1,,aN).
All of the eigenvalues of M(N) are real because it is a real symmetric matrix. According to the theorem of linear algebra, the number of the negative eigenvalues of M(N) is equal to that of the sign changes of the sequence {1,|M(1)|,|M(2)|,,|M(N)|}, where the minor determinants of M(N) are given as explained in Appendix A,
|M(k)|=(2ξAWatot2)k(1i=1k2aiatot)i=1kbi2.
Since the term (1i=1k2ai/atot) decreases monotonically for k and equals 1 when k=N, the above sequence of the minor determinants of M(N) changes the sign once. Accordingly, M˜ has one negative eigenvalue and (N1) positive eigenvalues, which means the fixed point corresponding to the multimode lasing state is an unstable saddle point.

6. SUMMARY AND DISCUSSION

By introducing an expansion method different from conventional theories [3537] for the population inversion in the Maxwell–Bloch equations and evaluating the eigenvalues of a stability matrix describing the interactions among a huge number of lasing modes, we theoretically showed that in a fully chaotic 2D microcavity laser, at least one single-mode lasing state is stable, while all multimode lasing states are unstable, when the external pumping power is sufficiently large and the cavity size is much larger than the wavelength to the extent that γ˜|Δij|, where Δij is the difference between the adjacent lasing frequencies. This result provides a theoretical ground for recent experimental observations of universal single-mode lasing in fully chaotic 2D microcavity lasers [28,29].

It is important to note that the theory presented in this paper should be applied to explain the observation of single-mode lasing in the experiments of continuous-wave pumping cases. Generally, the lifetime of an unstable multimode lasing state can be much longer than a pulse width. Thus, for a pulsed operation, it is likely that the collapse of an unstable multimode lasing state cannot be achieved within a pulse width, even if the size of a fully chaotic 2D microcavity is sufficiently large, and multimode lasing is observed and universal single-mode lasing seems to disappear [59,60]. In this case, as the pulse width is increased, the number of lasing modes decreases [28]. Multimode lasing in a fully chaotic 2D microcavity can also be observed when the condition γ˜|Δij| is not satisfied [61,62]. This condition for multimode lasing coincides with that derived for one-dimensional lasers [63].

The theory presented in this paper cannot give the threshold pumping power for single-mode lasing, but it is useful for understanding the single-mode lasing mechanism. In addition, it does not take into consideration these phenomena that might affect lasing characteristics such as the thermal effect. However, according to previous studies [28,29], we can at least say that the threshold for single-mode lasing is achievable in real experiments. It is of interest to further demonstrate the predicted single-mode lasing experimentally for various fully chaotic cavities. It is also important to elucidate experimentally, numerically, and theoretically how multimode lasing states in a low pumping regime and/or in a small cavity change into a single-mode lasing state as the pumping power and/or the size of the cavity are increased.

APPENDIX A

Let us suppose that x is the eigenvector corresponding to the eigenvalue λ of the matrix Mdiag(B1,,BN)Cdiag(A1,,AN), where the diagonal element of the matrix C is defined as CiiCi and every off-diagonal element is equal to 1, that is,

Mx=diag(B1,,BN)Cdiag(A1,,AN)x=λx.
The left-hand side of Eq. (A1) is rewritten as follows:
diag(B1,,BN)Cdiag(A11/2,,AN1/2)×diag(A11/2,,AN1/2)diag(B11/2,,BN1/2)×diag(B11/2,,BN1/2)x=diag(B1,,BN)Cdiag(B11/2,,BN1/2)×diag(A11/2,,AN1/2)diag(B11/2,,BN1/2)×diag(A11/2,,AN1/2)x.
On the other hand, the right-hand side of Eq. (A1) is rewritten as follows:
λdiag(A11/2,,AN1/2)diag(A11/2,,AN1/2)×diag(B11/2,,BN1/2)diag(B11/2,,BN1/2)x=λdiag(A11/2,,AN1/2)diag(B11/2,,BN1/2)×diag(B11/2,,BN1/2)diag(A11/2,,AN1/2)x.
Therefore, we have
diag(B1,,BN)Cdiag(B11/2,,BN1/2)×diag(A11/2,,AN1/2)x=λdiag(A11/2,,AN1/2)diag(B11/2,,BN1/2)x,
where xdiag(B11/2,,BN1/2)diag(A11/2,,AN1/2)x. Operating diag(B11/2,,BN1/2)diag(A11/2,,AN1/2) to both sides of Eq. (A4) yields
diag(A11/2,,AN1/2)diag(B11/2,,BN1/2)C×diag(B11/2,,BN1/2)diag(A11/2,,AN1/2)x=λx.
From Eq. (A5), one can see that the eigenvalue of the matrix M defined as
Mdiag(A11/2,,AN1/2)diag(B11/2,,BN1/2)C×diag(B11/2,,BN1/2)diag(A11/2,,AN1/2)
is equal to λ. Since Mij=Cij(AiAjBiBj)1/2 and C is a real symmetric matrix, M is a real symmetric matrix. The determinant of M is given as follows:
|M|=|diag(A11/2,,AN1/2)||diag(B11/2,,BN1/2)||C|×|diag(B11/2,,BN1/2)||diag(A11/2,,AN1/2)|=(i=1NAiBi)(i=1N(Ci1))(1+i=1N(Ci1)1).

Funding

Waseda University (2017B-197).

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4. O. Bohigas, M. J. Giannoni, and C. Schmit, “Characterization of chaotic quantum spectra and universality of level fluctuation laws,” Phys. Rev. Lett. 52, 1–4 (1984). [CrossRef]  

5. G. Casati, F. Valz-Gris, and I. Guarneri, “On the connection between quantization of nonintegrable systems and statistical theory of spectra,” Lett. Nuovo Cimento Soc. Ital. Fis. 28, 279–282 (1980). [CrossRef]  

6. M. V. Berry, “Quantizing a classically ergodic system: Sinai’s billiard and the KKR method,” Ann. Phys. (N.Y.) 131, 163–216 (1981). [CrossRef]  

7. M. V. Berry, “Semiclassical theory of spectral rigidity,” Proc. R. Soc. London Ser. A 400, 229–251 (1985). [CrossRef]  

8. M. Sieber, “Leading off-diagonal approximation for the spectral form factor for uniformly hyperbolic systems,” J. Phys. A 35, L613–L619 (2002). [CrossRef]  

9. S. Müller, S. Heusler, P. Braun, F. Haake, and A. Altland, “Periodic-orbit theory of universality in quantum chaos,” Phys. Rev. Lett. 93, 014103 (2004). [CrossRef]  

10. S. Heusler, S. Müller, A. Altland, P. Braun, and F. Haake, “Periodic-orbit theory of level correlations,” Phys. Rev. Lett. 98, 044103 (2007). [CrossRef]  

11. M. Sieber and K. Richter, “Correlations between periodic orbits and their role in spectral statistics,” Phys. Scr. T90, 128–133 (2001). [CrossRef]  

12. R. A. Jalabert, H. U. Baranger, and A. D. Stone, “Conductance fluctuations in the ballistic regime: a probe of quantum chaos?” Phys. Rev. Lett. 65, 2442–2445 (1990). [CrossRef]  

13. C. M. Marcus, A. J. Rimberg, R. M. Westervelt, P. F. Hopkins, and A. C. Gossard, “Conductance fluctuations and chaotic scattering in ballistic microstructures,” Phys. Rev. Lett. 69, 506–509 (1992). [CrossRef]  

14. J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385, 45–47 (1997). [CrossRef]  

15. C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998). [CrossRef]  

16. S.-B. Lee, J.-H. Lee, J.-S. Chang, H.-J. Moon, S. W. Kim, and K. An, “Observation of scarred modes in asymmetrically deformed microcylinder lasers,” Phys. Rev. Lett. 88, 033903 (2002). [CrossRef]  

17. S.-Y. Lee, S. Rim, J. W. Ryu, T. Y. Kwon, M. Choi, and C.-M. Kim, “Quasiscarred resonances in a spiral-shaped microcavity,” Phys. Rev. Lett. 93, 164102 (2004). [CrossRef]  

18. H. G. L. Schwefel, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-Messaoud, and J. Zyss, “Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers,” J. Opt. Soc. Am. B 21, 923–934 (2004). [CrossRef]  

19. V. A. Podolskiy, E. Narimanov, W. Fang, and H. Cao, “Chaotic microlasers based on dynamical localization,” Proc. Natl. Acad. Sci. USA 101, 10498–10500 (2004). [CrossRef]  

20. J. Wiersig, “Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities,” Phys. Rev. Lett. 97, 253901 (2006). [CrossRef]  

21. J. Wiersig and M. Hentschel, “Combining directional light output and ultralow loss in deformed microdisks,” Phys. Rev. Lett. 100, 033901 (2008). [CrossRef]  

22. E. Bogomolny, R. Dubertrand, and C. Schmit, “Trace formula for dieletric cavities: I. General properties,” Phys. Rev. E 78, 056202 (2008). [CrossRef]  

23. S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, T. Sasaki, and E. E. Narimanov, “Chaos-assisted directional light emission from microcavity lasers,” Phys. Rev. Lett. 104, 163902 (2010). [CrossRef]  

24. Q. Song, L. Ge, B. Redding, and H. Cao, “Channeling chaotic rays into waveguides for efficient collection of microcavity emission,” Phys. Rev. Lett. 108, 243902 (2012). [CrossRef]  

25. R. Sarma, L. Ge, J. Wiersig, and H. Cao, “Rotating optical microcavities with broken chiral symmetry,” Phys. Rev. Lett. 114, 053903 (2015). [CrossRef]  

26. H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87, 61–111 (2015). [CrossRef]  

27. T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photon. Rev. 5, 247–271 (2011). [CrossRef]  

28. S. Sunada, T. Fukushima, S. Shinohara, T. Harayama, and M. Adachi, “Stable single-wavelength emission from fully chaotic microcavity lasers,” Phys. Rev. A 88, 013802 (2013). [CrossRef]  

29. S. Sunada, S. Shinohara, T. Fukushima, and T. Harayama, “Signature of wave chaos in spectral characteristics of microcavity lasers,” Phys. Rev. Lett. 116, 203903 (2016). [CrossRef]  

30. M. Fisher, “The renormalization group in the theory of critical behavior,” Rev. Mod. Phys. 46, 597–616 (1974). [CrossRef]  

31. T. Harayama, P. Davis, and K. S. Ikeda, “Stable oscillations of a spatially chaotic wave function in a microstadium laser,” Phys. Rev. Lett. 90, 063901 (2003). [CrossRef]  

32. T. Harayama, S. Sunada, and K. S. Ikeda, “Theory of two-dimensional microcavity lasers,” Phys. Rev. A 72, 013803 (2005). [CrossRef]  

33. H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006). [CrossRef]  

34. H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007). [CrossRef]  

35. W. E. Lamb, “Theory of an optical maser,” Phys. Rev. A 134, A1429–A1450 (1964). [CrossRef]  

36. M. Sargent III, M. O. Scully, and W. E. Lamb Jr., Laser Physics (Addison-Wesley, 1974).

37. M. Sargent III, “Theory of a multimode quasiequilibrium semiconductor laser,” Phys. Rev. A 48, 717–726 (1993). [CrossRef]  

38. T. Harayama, T. Fukushima, S. Sunada, and K. S. Ikeda, “Asymmetric stationary lasing patterns in 2D symmetric microcavities,” Phys. Rev. Lett. 91, 073903 (2003). [CrossRef]  

39. H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008). [CrossRef]  

40. L. Ge, R. J. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16895–16902 (2008). [CrossRef]  

41. H. E. Türeci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009). [CrossRef]  

42. L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010). [CrossRef]  

43. A. Cerjan and A. D. Stone, “Steady-state ab initio theory of lasers with injected signals,” Phys. Rev. A 90, 013840 (2014). [CrossRef]  

44. S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, and A. D. Stone, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014). [CrossRef]  

45. A. Pick, A. Cerjan, D. Liu, A. W. Rodriguez, A. D. Stone, Y. D. Chong, and S. G. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015). [CrossRef]  

46. A. I. Shnirelman, “Ergodic properties of eigenfunctions,” Usp. Mat. Nauk 29, 181–182 (1974).

47. Y. Colin de Verdie’re, “Ergodicité et fonctions propres du laplacien,” Commun. Math. Phys. 102, 497–502 (1985). [CrossRef]  

48. S. Zelditch, “Uniform distribution of eigenfunctions on compact hyperbolic surfaces,” Duke Math. J. 55, 919–941 (1987). [CrossRef]  

49. B. Helffer, A. Martinez, and D. Robert, “Ergodicité et limite semiclassique,” Commun. Math. Phys. 109, 313–326 (1987). [CrossRef]  

50. S. Zelditch and M. Zworski, “Ergodicity of eigenfunctions for ergodic billiards,” Commun. Math. Phys. 175, 673–682 (1996). [CrossRef]  

51. A. Bäcker, R. Schubert, and P. Stifter, “Rate of quantum ergodicity in Euclidean billiards,” Phys. Rev. E 57, 5425–5447 (1998). [CrossRef]  

52. H. Schomerus and J. Tworzydlo, “Quantum-to-classical crossover of quasibound states in open quantum systems,” Phys. Rev. Lett. 93, 154102 (2004). [CrossRef]  

53. S. Nonnenmacher and M. Zworski, “Fractal Weyl laws in discrete models of chaotic scattering,” J. Phys. A 38, 10683–10702 (2005). [CrossRef]  

54. J. P. Keating, M. Novaes, S. D. Prado, and M. Sieber, “Semiclassical structure of chaotic resonance eigenfunctions,” Phys. Rev. Lett. 97, 150406 (2006). [CrossRef]  

55. D. L. Shepelyansky, “Fractal Weyl law for quantum fractal eigenstates,” Phys. Rev. E 77, 015202(R) (2008). [CrossRef]  

56. M. Novaes, “Resonances in open quantum maps,” J. Phys. A 46, 143001 (2013). [CrossRef]  

57. T. Harayama and S. Shinohara, “Ray-wave correspondence in chaotic dielectric billiards,” Phys. Rev. E 92, 042916 (2015). [CrossRef]  

58. E. G. Altmann, “Emission from dielectric cavities in terms of invariant sets of the chaotic ray dynamics,” Phys. Rev. A 79, 013830 (2009). [CrossRef]  

59. B. Redding, A. Cerjan, X. Huang, M. L. Lee, A. D. Stone, M. A. Choma, and H. Cao, “Low-spatial coherence electrically-pumped semiconductor laser for speckle-free full-field imaging,” Proc. Natl. Acad. Sci. USA 112, 1304–1309 (2015). [CrossRef]  

60. M. Choi, S. Shinohara, and T. Harayama, “Dependence of far-field characteristics on the number of lasing modes in stadium-shaped InGaAsP microlasers,” Opt. Express 16, 17544–17559 (2008).

61. A. Cerjan, B. Redding, L. Ge, S. F. Liew, H. Cao, and A. D. Stone, “Controlling mode competition by tailoring the spatial pump distribution in a laser: a resonance-based approach,” Opt. Express 24, 26006–26015 (2016). [CrossRef]  

62. S. Sunada, T. Harayama, and K. S. Ikeda, “Multimode lasing in fully chaotic cavity lasers,” Phys. Rev. E 71, 046209 (2005). [CrossRef]  

63. H. Fu and H. Haken, “Multifrequency operation in a short-cavity standing-wave laser,” Phys. Rev. A 43, 2446–2454 (1991). [CrossRef]  

References

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  1. H.-J. Stöckmann, Quantum Chaos: An Introduction (Cambridge University, 1999).
  2. F. Haake, Quantum Signatures of Chaos (Springer, 2000).
  3. K. Nakamura and T. Harayama, Quantum Chaos and Quantum Dots (Oxford University, 2004).
  4. O. Bohigas, M. J. Giannoni, and C. Schmit, “Characterization of chaotic quantum spectra and universality of level fluctuation laws,” Phys. Rev. Lett. 52, 1–4 (1984).
    [Crossref]
  5. G. Casati, F. Valz-Gris, and I. Guarneri, “On the connection between quantization of nonintegrable systems and statistical theory of spectra,” Lett. Nuovo Cimento Soc. Ital. Fis. 28, 279–282 (1980).
    [Crossref]
  6. M. V. Berry, “Quantizing a classically ergodic system: Sinai’s billiard and the KKR method,” Ann. Phys. (N.Y.) 131, 163–216 (1981).
    [Crossref]
  7. M. V. Berry, “Semiclassical theory of spectral rigidity,” Proc. R. Soc. London Ser. A 400, 229–251 (1985).
    [Crossref]
  8. M. Sieber, “Leading off-diagonal approximation for the spectral form factor for uniformly hyperbolic systems,” J. Phys. A 35, L613–L619 (2002).
    [Crossref]
  9. S. Müller, S. Heusler, P. Braun, F. Haake, and A. Altland, “Periodic-orbit theory of universality in quantum chaos,” Phys. Rev. Lett. 93, 014103 (2004).
    [Crossref]
  10. S. Heusler, S. Müller, A. Altland, P. Braun, and F. Haake, “Periodic-orbit theory of level correlations,” Phys. Rev. Lett. 98, 044103 (2007).
    [Crossref]
  11. M. Sieber and K. Richter, “Correlations between periodic orbits and their role in spectral statistics,” Phys. Scr. T90, 128–133 (2001).
    [Crossref]
  12. R. A. Jalabert, H. U. Baranger, and A. D. Stone, “Conductance fluctuations in the ballistic regime: a probe of quantum chaos?” Phys. Rev. Lett. 65, 2442–2445 (1990).
    [Crossref]
  13. C. M. Marcus, A. J. Rimberg, R. M. Westervelt, P. F. Hopkins, and A. C. Gossard, “Conductance fluctuations and chaotic scattering in ballistic microstructures,” Phys. Rev. Lett. 69, 506–509 (1992).
    [Crossref]
  14. J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385, 45–47 (1997).
    [Crossref]
  15. C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
    [Crossref]
  16. S.-B. Lee, J.-H. Lee, J.-S. Chang, H.-J. Moon, S. W. Kim, and K. An, “Observation of scarred modes in asymmetrically deformed microcylinder lasers,” Phys. Rev. Lett. 88, 033903 (2002).
    [Crossref]
  17. S.-Y. Lee, S. Rim, J. W. Ryu, T. Y. Kwon, M. Choi, and C.-M. Kim, “Quasiscarred resonances in a spiral-shaped microcavity,” Phys. Rev. Lett. 93, 164102 (2004).
    [Crossref]
  18. H. G. L. Schwefel, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-Messaoud, and J. Zyss, “Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers,” J. Opt. Soc. Am. B 21, 923–934 (2004).
    [Crossref]
  19. V. A. Podolskiy, E. Narimanov, W. Fang, and H. Cao, “Chaotic microlasers based on dynamical localization,” Proc. Natl. Acad. Sci. USA 101, 10498–10500 (2004).
    [Crossref]
  20. J. Wiersig, “Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities,” Phys. Rev. Lett. 97, 253901 (2006).
    [Crossref]
  21. J. Wiersig and M. Hentschel, “Combining directional light output and ultralow loss in deformed microdisks,” Phys. Rev. Lett. 100, 033901 (2008).
    [Crossref]
  22. E. Bogomolny, R. Dubertrand, and C. Schmit, “Trace formula for dieletric cavities: I. General properties,” Phys. Rev. E 78, 056202 (2008).
    [Crossref]
  23. S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, T. Sasaki, and E. E. Narimanov, “Chaos-assisted directional light emission from microcavity lasers,” Phys. Rev. Lett. 104, 163902 (2010).
    [Crossref]
  24. Q. Song, L. Ge, B. Redding, and H. Cao, “Channeling chaotic rays into waveguides for efficient collection of microcavity emission,” Phys. Rev. Lett. 108, 243902 (2012).
    [Crossref]
  25. R. Sarma, L. Ge, J. Wiersig, and H. Cao, “Rotating optical microcavities with broken chiral symmetry,” Phys. Rev. Lett. 114, 053903 (2015).
    [Crossref]
  26. H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87, 61–111 (2015).
    [Crossref]
  27. T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photon. Rev. 5, 247–271 (2011).
    [Crossref]
  28. S. Sunada, T. Fukushima, S. Shinohara, T. Harayama, and M. Adachi, “Stable single-wavelength emission from fully chaotic microcavity lasers,” Phys. Rev. A 88, 013802 (2013).
    [Crossref]
  29. S. Sunada, S. Shinohara, T. Fukushima, and T. Harayama, “Signature of wave chaos in spectral characteristics of microcavity lasers,” Phys. Rev. Lett. 116, 203903 (2016).
    [Crossref]
  30. M. Fisher, “The renormalization group in the theory of critical behavior,” Rev. Mod. Phys. 46, 597–616 (1974).
    [Crossref]
  31. T. Harayama, P. Davis, and K. S. Ikeda, “Stable oscillations of a spatially chaotic wave function in a microstadium laser,” Phys. Rev. Lett. 90, 063901 (2003).
    [Crossref]
  32. T. Harayama, S. Sunada, and K. S. Ikeda, “Theory of two-dimensional microcavity lasers,” Phys. Rev. A 72, 013803 (2005).
    [Crossref]
  33. H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
    [Crossref]
  34. H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
    [Crossref]
  35. W. E. Lamb, “Theory of an optical maser,” Phys. Rev. A 134, A1429–A1450 (1964).
    [Crossref]
  36. M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, 1974).
  37. M. Sargent, “Theory of a multimode quasiequilibrium semiconductor laser,” Phys. Rev. A 48, 717–726 (1993).
    [Crossref]
  38. T. Harayama, T. Fukushima, S. Sunada, and K. S. Ikeda, “Asymmetric stationary lasing patterns in 2D symmetric microcavities,” Phys. Rev. Lett. 91, 073903 (2003).
    [Crossref]
  39. H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
    [Crossref]
  40. L. Ge, R. J. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16895–16902 (2008).
    [Crossref]
  41. H. E. Türeci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009).
    [Crossref]
  42. L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
    [Crossref]
  43. A. Cerjan and A. D. Stone, “Steady-state ab initio theory of lasers with injected signals,” Phys. Rev. A 90, 013840 (2014).
    [Crossref]
  44. S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, and A. D. Stone, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
    [Crossref]
  45. A. Pick, A. Cerjan, D. Liu, A. W. Rodriguez, A. D. Stone, Y. D. Chong, and S. G. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
    [Crossref]
  46. A. I. Shnirelman, “Ergodic properties of eigenfunctions,” Usp. Mat. Nauk 29, 181–182 (1974).
  47. Y. Colin de Verdie’re, “Ergodicité et fonctions propres du laplacien,” Commun. Math. Phys. 102, 497–502 (1985).
    [Crossref]
  48. S. Zelditch, “Uniform distribution of eigenfunctions on compact hyperbolic surfaces,” Duke Math. J. 55, 919–941 (1987).
    [Crossref]
  49. B. Helffer, A. Martinez, and D. Robert, “Ergodicité et limite semiclassique,” Commun. Math. Phys. 109, 313–326 (1987).
    [Crossref]
  50. S. Zelditch and M. Zworski, “Ergodicity of eigenfunctions for ergodic billiards,” Commun. Math. Phys. 175, 673–682 (1996).
    [Crossref]
  51. A. Bäcker, R. Schubert, and P. Stifter, “Rate of quantum ergodicity in Euclidean billiards,” Phys. Rev. E 57, 5425–5447 (1998).
    [Crossref]
  52. H. Schomerus and J. Tworzydlo, “Quantum-to-classical crossover of quasibound states in open quantum systems,” Phys. Rev. Lett. 93, 154102 (2004).
    [Crossref]
  53. S. Nonnenmacher and M. Zworski, “Fractal Weyl laws in discrete models of chaotic scattering,” J. Phys. A 38, 10683–10702 (2005).
    [Crossref]
  54. J. P. Keating, M. Novaes, S. D. Prado, and M. Sieber, “Semiclassical structure of chaotic resonance eigenfunctions,” Phys. Rev. Lett. 97, 150406 (2006).
    [Crossref]
  55. D. L. Shepelyansky, “Fractal Weyl law for quantum fractal eigenstates,” Phys. Rev. E 77, 015202(R) (2008).
    [Crossref]
  56. M. Novaes, “Resonances in open quantum maps,” J. Phys. A 46, 143001 (2013).
    [Crossref]
  57. T. Harayama and S. Shinohara, “Ray-wave correspondence in chaotic dielectric billiards,” Phys. Rev. E 92, 042916 (2015).
    [Crossref]
  58. E. G. Altmann, “Emission from dielectric cavities in terms of invariant sets of the chaotic ray dynamics,” Phys. Rev. A 79, 013830 (2009).
    [Crossref]
  59. B. Redding, A. Cerjan, X. Huang, M. L. Lee, A. D. Stone, M. A. Choma, and H. Cao, “Low-spatial coherence electrically-pumped semiconductor laser for speckle-free full-field imaging,” Proc. Natl. Acad. Sci. USA 112, 1304–1309 (2015).
    [Crossref]
  60. M. Choi, S. Shinohara, and T. Harayama, “Dependence of far-field characteristics on the number of lasing modes in stadium-shaped InGaAsP microlasers,” Opt. Express 16, 17544–17559 (2008).
  61. A. Cerjan, B. Redding, L. Ge, S. F. Liew, H. Cao, and A. D. Stone, “Controlling mode competition by tailoring the spatial pump distribution in a laser: a resonance-based approach,” Opt. Express 24, 26006–26015 (2016).
    [Crossref]
  62. S. Sunada, T. Harayama, and K. S. Ikeda, “Multimode lasing in fully chaotic cavity lasers,” Phys. Rev. E 71, 046209 (2005).
    [Crossref]
  63. H. Fu and H. Haken, “Multifrequency operation in a short-cavity standing-wave laser,” Phys. Rev. A 43, 2446–2454 (1991).
    [Crossref]

2016 (2)

S. Sunada, S. Shinohara, T. Fukushima, and T. Harayama, “Signature of wave chaos in spectral characteristics of microcavity lasers,” Phys. Rev. Lett. 116, 203903 (2016).
[Crossref]

A. Cerjan, B. Redding, L. Ge, S. F. Liew, H. Cao, and A. D. Stone, “Controlling mode competition by tailoring the spatial pump distribution in a laser: a resonance-based approach,” Opt. Express 24, 26006–26015 (2016).
[Crossref]

2015 (5)

T. Harayama and S. Shinohara, “Ray-wave correspondence in chaotic dielectric billiards,” Phys. Rev. E 92, 042916 (2015).
[Crossref]

B. Redding, A. Cerjan, X. Huang, M. L. Lee, A. D. Stone, M. A. Choma, and H. Cao, “Low-spatial coherence electrically-pumped semiconductor laser for speckle-free full-field imaging,” Proc. Natl. Acad. Sci. USA 112, 1304–1309 (2015).
[Crossref]

A. Pick, A. Cerjan, D. Liu, A. W. Rodriguez, A. D. Stone, Y. D. Chong, and S. G. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
[Crossref]

R. Sarma, L. Ge, J. Wiersig, and H. Cao, “Rotating optical microcavities with broken chiral symmetry,” Phys. Rev. Lett. 114, 053903 (2015).
[Crossref]

H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87, 61–111 (2015).
[Crossref]

2014 (2)

A. Cerjan and A. D. Stone, “Steady-state ab initio theory of lasers with injected signals,” Phys. Rev. A 90, 013840 (2014).
[Crossref]

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, and A. D. Stone, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

2013 (2)

M. Novaes, “Resonances in open quantum maps,” J. Phys. A 46, 143001 (2013).
[Crossref]

S. Sunada, T. Fukushima, S. Shinohara, T. Harayama, and M. Adachi, “Stable single-wavelength emission from fully chaotic microcavity lasers,” Phys. Rev. A 88, 013802 (2013).
[Crossref]

2012 (1)

Q. Song, L. Ge, B. Redding, and H. Cao, “Channeling chaotic rays into waveguides for efficient collection of microcavity emission,” Phys. Rev. Lett. 108, 243902 (2012).
[Crossref]

2011 (1)

T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photon. Rev. 5, 247–271 (2011).
[Crossref]

2010 (2)

S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, T. Sasaki, and E. E. Narimanov, “Chaos-assisted directional light emission from microcavity lasers,” Phys. Rev. Lett. 104, 163902 (2010).
[Crossref]

L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[Crossref]

2009 (2)

H. E. Türeci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009).
[Crossref]

E. G. Altmann, “Emission from dielectric cavities in terms of invariant sets of the chaotic ray dynamics,” Phys. Rev. A 79, 013830 (2009).
[Crossref]

2008 (6)

M. Choi, S. Shinohara, and T. Harayama, “Dependence of far-field characteristics on the number of lasing modes in stadium-shaped InGaAsP microlasers,” Opt. Express 16, 17544–17559 (2008).

D. L. Shepelyansky, “Fractal Weyl law for quantum fractal eigenstates,” Phys. Rev. E 77, 015202(R) (2008).
[Crossref]

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[Crossref]

L. Ge, R. J. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16895–16902 (2008).
[Crossref]

J. Wiersig and M. Hentschel, “Combining directional light output and ultralow loss in deformed microdisks,” Phys. Rev. Lett. 100, 033901 (2008).
[Crossref]

E. Bogomolny, R. Dubertrand, and C. Schmit, “Trace formula for dieletric cavities: I. General properties,” Phys. Rev. E 78, 056202 (2008).
[Crossref]

2007 (2)

S. Heusler, S. Müller, A. Altland, P. Braun, and F. Haake, “Periodic-orbit theory of level correlations,” Phys. Rev. Lett. 98, 044103 (2007).
[Crossref]

H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
[Crossref]

2006 (3)

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[Crossref]

J. Wiersig, “Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities,” Phys. Rev. Lett. 97, 253901 (2006).
[Crossref]

J. P. Keating, M. Novaes, S. D. Prado, and M. Sieber, “Semiclassical structure of chaotic resonance eigenfunctions,” Phys. Rev. Lett. 97, 150406 (2006).
[Crossref]

2005 (3)

S. Nonnenmacher and M. Zworski, “Fractal Weyl laws in discrete models of chaotic scattering,” J. Phys. A 38, 10683–10702 (2005).
[Crossref]

S. Sunada, T. Harayama, and K. S. Ikeda, “Multimode lasing in fully chaotic cavity lasers,” Phys. Rev. E 71, 046209 (2005).
[Crossref]

T. Harayama, S. Sunada, and K. S. Ikeda, “Theory of two-dimensional microcavity lasers,” Phys. Rev. A 72, 013803 (2005).
[Crossref]

2004 (5)

S. Müller, S. Heusler, P. Braun, F. Haake, and A. Altland, “Periodic-orbit theory of universality in quantum chaos,” Phys. Rev. Lett. 93, 014103 (2004).
[Crossref]

S.-Y. Lee, S. Rim, J. W. Ryu, T. Y. Kwon, M. Choi, and C.-M. Kim, “Quasiscarred resonances in a spiral-shaped microcavity,” Phys. Rev. Lett. 93, 164102 (2004).
[Crossref]

H. G. L. Schwefel, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-Messaoud, and J. Zyss, “Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers,” J. Opt. Soc. Am. B 21, 923–934 (2004).
[Crossref]

V. A. Podolskiy, E. Narimanov, W. Fang, and H. Cao, “Chaotic microlasers based on dynamical localization,” Proc. Natl. Acad. Sci. USA 101, 10498–10500 (2004).
[Crossref]

H. Schomerus and J. Tworzydlo, “Quantum-to-classical crossover of quasibound states in open quantum systems,” Phys. Rev. Lett. 93, 154102 (2004).
[Crossref]

2003 (2)

T. Harayama, T. Fukushima, S. Sunada, and K. S. Ikeda, “Asymmetric stationary lasing patterns in 2D symmetric microcavities,” Phys. Rev. Lett. 91, 073903 (2003).
[Crossref]

T. Harayama, P. Davis, and K. S. Ikeda, “Stable oscillations of a spatially chaotic wave function in a microstadium laser,” Phys. Rev. Lett. 90, 063901 (2003).
[Crossref]

2002 (2)

S.-B. Lee, J.-H. Lee, J.-S. Chang, H.-J. Moon, S. W. Kim, and K. An, “Observation of scarred modes in asymmetrically deformed microcylinder lasers,” Phys. Rev. Lett. 88, 033903 (2002).
[Crossref]

M. Sieber, “Leading off-diagonal approximation for the spectral form factor for uniformly hyperbolic systems,” J. Phys. A 35, L613–L619 (2002).
[Crossref]

2001 (1)

M. Sieber and K. Richter, “Correlations between periodic orbits and their role in spectral statistics,” Phys. Scr. T90, 128–133 (2001).
[Crossref]

1998 (2)

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref]

A. Bäcker, R. Schubert, and P. Stifter, “Rate of quantum ergodicity in Euclidean billiards,” Phys. Rev. E 57, 5425–5447 (1998).
[Crossref]

1997 (1)

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385, 45–47 (1997).
[Crossref]

1996 (1)

S. Zelditch and M. Zworski, “Ergodicity of eigenfunctions for ergodic billiards,” Commun. Math. Phys. 175, 673–682 (1996).
[Crossref]

1993 (1)

M. Sargent, “Theory of a multimode quasiequilibrium semiconductor laser,” Phys. Rev. A 48, 717–726 (1993).
[Crossref]

1992 (1)

C. M. Marcus, A. J. Rimberg, R. M. Westervelt, P. F. Hopkins, and A. C. Gossard, “Conductance fluctuations and chaotic scattering in ballistic microstructures,” Phys. Rev. Lett. 69, 506–509 (1992).
[Crossref]

1991 (1)

H. Fu and H. Haken, “Multifrequency operation in a short-cavity standing-wave laser,” Phys. Rev. A 43, 2446–2454 (1991).
[Crossref]

1990 (1)

R. A. Jalabert, H. U. Baranger, and A. D. Stone, “Conductance fluctuations in the ballistic regime: a probe of quantum chaos?” Phys. Rev. Lett. 65, 2442–2445 (1990).
[Crossref]

1987 (2)

S. Zelditch, “Uniform distribution of eigenfunctions on compact hyperbolic surfaces,” Duke Math. J. 55, 919–941 (1987).
[Crossref]

B. Helffer, A. Martinez, and D. Robert, “Ergodicité et limite semiclassique,” Commun. Math. Phys. 109, 313–326 (1987).
[Crossref]

1985 (2)

Y. Colin de Verdie’re, “Ergodicité et fonctions propres du laplacien,” Commun. Math. Phys. 102, 497–502 (1985).
[Crossref]

M. V. Berry, “Semiclassical theory of spectral rigidity,” Proc. R. Soc. London Ser. A 400, 229–251 (1985).
[Crossref]

1984 (1)

O. Bohigas, M. J. Giannoni, and C. Schmit, “Characterization of chaotic quantum spectra and universality of level fluctuation laws,” Phys. Rev. Lett. 52, 1–4 (1984).
[Crossref]

1981 (1)

M. V. Berry, “Quantizing a classically ergodic system: Sinai’s billiard and the KKR method,” Ann. Phys. (N.Y.) 131, 163–216 (1981).
[Crossref]

1980 (1)

G. Casati, F. Valz-Gris, and I. Guarneri, “On the connection between quantization of nonintegrable systems and statistical theory of spectra,” Lett. Nuovo Cimento Soc. Ital. Fis. 28, 279–282 (1980).
[Crossref]

1974 (2)

M. Fisher, “The renormalization group in the theory of critical behavior,” Rev. Mod. Phys. 46, 597–616 (1974).
[Crossref]

A. I. Shnirelman, “Ergodic properties of eigenfunctions,” Usp. Mat. Nauk 29, 181–182 (1974).

1964 (1)

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. A 134, A1429–A1450 (1964).
[Crossref]

Adachi, M.

S. Sunada, T. Fukushima, S. Shinohara, T. Harayama, and M. Adachi, “Stable single-wavelength emission from fully chaotic microcavity lasers,” Phys. Rev. A 88, 013802 (2013).
[Crossref]

Altland, A.

S. Heusler, S. Müller, A. Altland, P. Braun, and F. Haake, “Periodic-orbit theory of level correlations,” Phys. Rev. Lett. 98, 044103 (2007).
[Crossref]

S. Müller, S. Heusler, P. Braun, F. Haake, and A. Altland, “Periodic-orbit theory of universality in quantum chaos,” Phys. Rev. Lett. 93, 014103 (2004).
[Crossref]

Altmann, E. G.

E. G. Altmann, “Emission from dielectric cavities in terms of invariant sets of the chaotic ray dynamics,” Phys. Rev. A 79, 013830 (2009).
[Crossref]

An, K.

S.-B. Lee, J.-H. Lee, J.-S. Chang, H.-J. Moon, S. W. Kim, and K. An, “Observation of scarred modes in asymmetrically deformed microcylinder lasers,” Phys. Rev. Lett. 88, 033903 (2002).
[Crossref]

Bäcker, A.

A. Bäcker, R. Schubert, and P. Stifter, “Rate of quantum ergodicity in Euclidean billiards,” Phys. Rev. E 57, 5425–5447 (1998).
[Crossref]

Baranger, H. U.

R. A. Jalabert, H. U. Baranger, and A. D. Stone, “Conductance fluctuations in the ballistic regime: a probe of quantum chaos?” Phys. Rev. Lett. 65, 2442–2445 (1990).
[Crossref]

Ben-Messaoud, T.

Berry, M. V.

M. V. Berry, “Semiclassical theory of spectral rigidity,” Proc. R. Soc. London Ser. A 400, 229–251 (1985).
[Crossref]

M. V. Berry, “Quantizing a classically ergodic system: Sinai’s billiard and the KKR method,” Ann. Phys. (N.Y.) 131, 163–216 (1981).
[Crossref]

Bogomolny, E.

E. Bogomolny, R. Dubertrand, and C. Schmit, “Trace formula for dieletric cavities: I. General properties,” Phys. Rev. E 78, 056202 (2008).
[Crossref]

Bohigas, O.

O. Bohigas, M. J. Giannoni, and C. Schmit, “Characterization of chaotic quantum spectra and universality of level fluctuation laws,” Phys. Rev. Lett. 52, 1–4 (1984).
[Crossref]

Braun, P.

S. Heusler, S. Müller, A. Altland, P. Braun, and F. Haake, “Periodic-orbit theory of level correlations,” Phys. Rev. Lett. 98, 044103 (2007).
[Crossref]

S. Müller, S. Heusler, P. Braun, F. Haake, and A. Altland, “Periodic-orbit theory of universality in quantum chaos,” Phys. Rev. Lett. 93, 014103 (2004).
[Crossref]

Cao, H.

A. Cerjan, B. Redding, L. Ge, S. F. Liew, H. Cao, and A. D. Stone, “Controlling mode competition by tailoring the spatial pump distribution in a laser: a resonance-based approach,” Opt. Express 24, 26006–26015 (2016).
[Crossref]

B. Redding, A. Cerjan, X. Huang, M. L. Lee, A. D. Stone, M. A. Choma, and H. Cao, “Low-spatial coherence electrically-pumped semiconductor laser for speckle-free full-field imaging,” Proc. Natl. Acad. Sci. USA 112, 1304–1309 (2015).
[Crossref]

R. Sarma, L. Ge, J. Wiersig, and H. Cao, “Rotating optical microcavities with broken chiral symmetry,” Phys. Rev. Lett. 114, 053903 (2015).
[Crossref]

H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87, 61–111 (2015).
[Crossref]

Q. Song, L. Ge, B. Redding, and H. Cao, “Channeling chaotic rays into waveguides for efficient collection of microcavity emission,” Phys. Rev. Lett. 108, 243902 (2012).
[Crossref]

V. A. Podolskiy, E. Narimanov, W. Fang, and H. Cao, “Chaotic microlasers based on dynamical localization,” Proc. Natl. Acad. Sci. USA 101, 10498–10500 (2004).
[Crossref]

Capasso, F.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref]

Casati, G.

G. Casati, F. Valz-Gris, and I. Guarneri, “On the connection between quantization of nonintegrable systems and statistical theory of spectra,” Lett. Nuovo Cimento Soc. Ital. Fis. 28, 279–282 (1980).
[Crossref]

Cerjan, A.

A. Cerjan, B. Redding, L. Ge, S. F. Liew, H. Cao, and A. D. Stone, “Controlling mode competition by tailoring the spatial pump distribution in a laser: a resonance-based approach,” Opt. Express 24, 26006–26015 (2016).
[Crossref]

B. Redding, A. Cerjan, X. Huang, M. L. Lee, A. D. Stone, M. A. Choma, and H. Cao, “Low-spatial coherence electrically-pumped semiconductor laser for speckle-free full-field imaging,” Proc. Natl. Acad. Sci. USA 112, 1304–1309 (2015).
[Crossref]

A. Pick, A. Cerjan, D. Liu, A. W. Rodriguez, A. D. Stone, Y. D. Chong, and S. G. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
[Crossref]

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, and A. D. Stone, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

A. Cerjan and A. D. Stone, “Steady-state ab initio theory of lasers with injected signals,” Phys. Rev. A 90, 013840 (2014).
[Crossref]

Chang, J.-S.

S.-B. Lee, J.-H. Lee, J.-S. Chang, H.-J. Moon, S. W. Kim, and K. An, “Observation of scarred modes in asymmetrically deformed microcylinder lasers,” Phys. Rev. Lett. 88, 033903 (2002).
[Crossref]

Chang, R. K.

Cho, A. Y.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref]

Choi, M.

M. Choi, S. Shinohara, and T. Harayama, “Dependence of far-field characteristics on the number of lasing modes in stadium-shaped InGaAsP microlasers,” Opt. Express 16, 17544–17559 (2008).

S.-Y. Lee, S. Rim, J. W. Ryu, T. Y. Kwon, M. Choi, and C.-M. Kim, “Quasiscarred resonances in a spiral-shaped microcavity,” Phys. Rev. Lett. 93, 164102 (2004).
[Crossref]

Choma, M. A.

B. Redding, A. Cerjan, X. Huang, M. L. Lee, A. D. Stone, M. A. Choma, and H. Cao, “Low-spatial coherence electrically-pumped semiconductor laser for speckle-free full-field imaging,” Proc. Natl. Acad. Sci. USA 112, 1304–1309 (2015).
[Crossref]

Chong, Y. D.

A. Pick, A. Cerjan, D. Liu, A. W. Rodriguez, A. D. Stone, Y. D. Chong, and S. G. Johnson, “Ab initio multimode linewidth theory for arbitrary inhomogeneous laser cavities,” Phys. Rev. A 91, 063806 (2015).
[Crossref]

L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[Crossref]

Colin de Verdie’re, Y.

Y. Colin de Verdie’re, “Ergodicité et fonctions propres du laplacien,” Commun. Math. Phys. 102, 497–502 (1985).
[Crossref]

Collier, B.

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[Crossref]

Davis, P.

T. Harayama, P. Davis, and K. S. Ikeda, “Stable oscillations of a spatially chaotic wave function in a microstadium laser,” Phys. Rev. Lett. 90, 063901 (2003).
[Crossref]

Dubertrand, R.

E. Bogomolny, R. Dubertrand, and C. Schmit, “Trace formula for dieletric cavities: I. General properties,” Phys. Rev. E 78, 056202 (2008).
[Crossref]

Esterhazy, S.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, and A. D. Stone, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

Faist, J.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref]

Fang, W.

V. A. Podolskiy, E. Narimanov, W. Fang, and H. Cao, “Chaotic microlasers based on dynamical localization,” Proc. Natl. Acad. Sci. USA 101, 10498–10500 (2004).
[Crossref]

Fisher, M.

M. Fisher, “The renormalization group in the theory of critical behavior,” Rev. Mod. Phys. 46, 597–616 (1974).
[Crossref]

Fu, H.

H. Fu and H. Haken, “Multifrequency operation in a short-cavity standing-wave laser,” Phys. Rev. A 43, 2446–2454 (1991).
[Crossref]

Fukushima, T.

S. Sunada, S. Shinohara, T. Fukushima, and T. Harayama, “Signature of wave chaos in spectral characteristics of microcavity lasers,” Phys. Rev. Lett. 116, 203903 (2016).
[Crossref]

S. Sunada, T. Fukushima, S. Shinohara, T. Harayama, and M. Adachi, “Stable single-wavelength emission from fully chaotic microcavity lasers,” Phys. Rev. A 88, 013802 (2013).
[Crossref]

S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, T. Sasaki, and E. E. Narimanov, “Chaos-assisted directional light emission from microcavity lasers,” Phys. Rev. Lett. 104, 163902 (2010).
[Crossref]

T. Harayama, T. Fukushima, S. Sunada, and K. S. Ikeda, “Asymmetric stationary lasing patterns in 2D symmetric microcavities,” Phys. Rev. Lett. 91, 073903 (2003).
[Crossref]

Ge, L.

A. Cerjan, B. Redding, L. Ge, S. F. Liew, H. Cao, and A. D. Stone, “Controlling mode competition by tailoring the spatial pump distribution in a laser: a resonance-based approach,” Opt. Express 24, 26006–26015 (2016).
[Crossref]

R. Sarma, L. Ge, J. Wiersig, and H. Cao, “Rotating optical microcavities with broken chiral symmetry,” Phys. Rev. Lett. 114, 053903 (2015).
[Crossref]

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, and A. D. Stone, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

Q. Song, L. Ge, B. Redding, and H. Cao, “Channeling chaotic rays into waveguides for efficient collection of microcavity emission,” Phys. Rev. Lett. 108, 243902 (2012).
[Crossref]

L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[Crossref]

H. E. Türeci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009).
[Crossref]

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[Crossref]

L. Ge, R. J. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16895–16902 (2008).
[Crossref]

H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
[Crossref]

Giannoni, M. J.

O. Bohigas, M. J. Giannoni, and C. Schmit, “Characterization of chaotic quantum spectra and universality of level fluctuation laws,” Phys. Rev. Lett. 52, 1–4 (1984).
[Crossref]

Gmachl, C.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref]

Gossard, A. C.

C. M. Marcus, A. J. Rimberg, R. M. Westervelt, P. F. Hopkins, and A. C. Gossard, “Conductance fluctuations and chaotic scattering in ballistic microstructures,” Phys. Rev. Lett. 69, 506–509 (1992).
[Crossref]

Guarneri, I.

G. Casati, F. Valz-Gris, and I. Guarneri, “On the connection between quantization of nonintegrable systems and statistical theory of spectra,” Lett. Nuovo Cimento Soc. Ital. Fis. 28, 279–282 (1980).
[Crossref]

Haake, F.

S. Heusler, S. Müller, A. Altland, P. Braun, and F. Haake, “Periodic-orbit theory of level correlations,” Phys. Rev. Lett. 98, 044103 (2007).
[Crossref]

S. Müller, S. Heusler, P. Braun, F. Haake, and A. Altland, “Periodic-orbit theory of universality in quantum chaos,” Phys. Rev. Lett. 93, 014103 (2004).
[Crossref]

F. Haake, Quantum Signatures of Chaos (Springer, 2000).

Haken, H.

H. Fu and H. Haken, “Multifrequency operation in a short-cavity standing-wave laser,” Phys. Rev. A 43, 2446–2454 (1991).
[Crossref]

Harayama, T.

S. Sunada, S. Shinohara, T. Fukushima, and T. Harayama, “Signature of wave chaos in spectral characteristics of microcavity lasers,” Phys. Rev. Lett. 116, 203903 (2016).
[Crossref]

T. Harayama and S. Shinohara, “Ray-wave correspondence in chaotic dielectric billiards,” Phys. Rev. E 92, 042916 (2015).
[Crossref]

S. Sunada, T. Fukushima, S. Shinohara, T. Harayama, and M. Adachi, “Stable single-wavelength emission from fully chaotic microcavity lasers,” Phys. Rev. A 88, 013802 (2013).
[Crossref]

T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photon. Rev. 5, 247–271 (2011).
[Crossref]

S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, T. Sasaki, and E. E. Narimanov, “Chaos-assisted directional light emission from microcavity lasers,” Phys. Rev. Lett. 104, 163902 (2010).
[Crossref]

M. Choi, S. Shinohara, and T. Harayama, “Dependence of far-field characteristics on the number of lasing modes in stadium-shaped InGaAsP microlasers,” Opt. Express 16, 17544–17559 (2008).

S. Sunada, T. Harayama, and K. S. Ikeda, “Multimode lasing in fully chaotic cavity lasers,” Phys. Rev. E 71, 046209 (2005).
[Crossref]

T. Harayama, S. Sunada, and K. S. Ikeda, “Theory of two-dimensional microcavity lasers,” Phys. Rev. A 72, 013803 (2005).
[Crossref]

T. Harayama, P. Davis, and K. S. Ikeda, “Stable oscillations of a spatially chaotic wave function in a microstadium laser,” Phys. Rev. Lett. 90, 063901 (2003).
[Crossref]

T. Harayama, T. Fukushima, S. Sunada, and K. S. Ikeda, “Asymmetric stationary lasing patterns in 2D symmetric microcavities,” Phys. Rev. Lett. 91, 073903 (2003).
[Crossref]

K. Nakamura and T. Harayama, Quantum Chaos and Quantum Dots (Oxford University, 2004).

Helffer, B.

B. Helffer, A. Martinez, and D. Robert, “Ergodicité et limite semiclassique,” Commun. Math. Phys. 109, 313–326 (1987).
[Crossref]

Hentschel, M.

S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, T. Sasaki, and E. E. Narimanov, “Chaos-assisted directional light emission from microcavity lasers,” Phys. Rev. Lett. 104, 163902 (2010).
[Crossref]

J. Wiersig and M. Hentschel, “Combining directional light output and ultralow loss in deformed microdisks,” Phys. Rev. Lett. 100, 033901 (2008).
[Crossref]

Heusler, S.

S. Heusler, S. Müller, A. Altland, P. Braun, and F. Haake, “Periodic-orbit theory of level correlations,” Phys. Rev. Lett. 98, 044103 (2007).
[Crossref]

S. Müller, S. Heusler, P. Braun, F. Haake, and A. Altland, “Periodic-orbit theory of universality in quantum chaos,” Phys. Rev. Lett. 93, 014103 (2004).
[Crossref]

Hopkins, P. F.

C. M. Marcus, A. J. Rimberg, R. M. Westervelt, P. F. Hopkins, and A. C. Gossard, “Conductance fluctuations and chaotic scattering in ballistic microstructures,” Phys. Rev. Lett. 69, 506–509 (1992).
[Crossref]

Huang, X.

B. Redding, A. Cerjan, X. Huang, M. L. Lee, A. D. Stone, M. A. Choma, and H. Cao, “Low-spatial coherence electrically-pumped semiconductor laser for speckle-free full-field imaging,” Proc. Natl. Acad. Sci. USA 112, 1304–1309 (2015).
[Crossref]

Ikeda, K. S.

S. Sunada, T. Harayama, and K. S. Ikeda, “Multimode lasing in fully chaotic cavity lasers,” Phys. Rev. E 71, 046209 (2005).
[Crossref]

T. Harayama, S. Sunada, and K. S. Ikeda, “Theory of two-dimensional microcavity lasers,” Phys. Rev. A 72, 013803 (2005).
[Crossref]

T. Harayama, T. Fukushima, S. Sunada, and K. S. Ikeda, “Asymmetric stationary lasing patterns in 2D symmetric microcavities,” Phys. Rev. Lett. 91, 073903 (2003).
[Crossref]

T. Harayama, P. Davis, and K. S. Ikeda, “Stable oscillations of a spatially chaotic wave function in a microstadium laser,” Phys. Rev. Lett. 90, 063901 (2003).
[Crossref]

Jalabert, R. A.

R. A. Jalabert, H. U. Baranger, and A. D. Stone, “Conductance fluctuations in the ballistic regime: a probe of quantum chaos?” Phys. Rev. Lett. 65, 2442–2445 (1990).
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Johnson, S. G.

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Equations (42)

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tE˜=i2(xy2+1)E˜αLE˜+2πNκϵρ˜,
nout2nin2tE˜=i2(xy2+nout2nin2)E˜,
tρ˜=γ˜ρ˜iΔ0ρ˜+κ˜WE˜,
tW=γ˜(WW)2κ˜(E˜ρ˜*+E˜*ρ˜),
E˜=iEi(t)eiΔitUi(x,y),
ρ˜=iρi(t)eiΔitVi(x,y),
dEi(t)dt+jidEj(t)dteiΔijtUij=[i(Δi+12)(αL+γ˜ii)]Ei(t)+ji{[i(Δj+12)αL]Uijγ˜ij}Ej(t)eiΔijt+2πNκεjeiΔijtρj(t)DUi*(x,y)Vj(x,y)dxdy,
γ˜iji2DdxdyUi*(x,y)2Uj(x,y).
ρj(t)Vj(x,y)=κ˜Wγ˜iΔ0jEj(t)Uj(x,y).
W=W/{1+[ij2κ˜2EiEj*UiUj*eiΔjit(γ˜+iΔ0j)(γ˜iΔji)+c.c.]}.
L(x,y)1+mam|Um|2,
C(x,y)l,jlj2κ˜2ElEj*UlUj*eiΔjlt(γ˜+iΔ0j)(γ˜iΔjl)+c.c.
dEi(t)dt+jidEj(t)dteiΔijtUij=[i(Δi+12)(αL+γ˜ii)]Ei(t)ji{[i(Δj+12)αL]Uijγ˜ij}Ej(t)eiΔijt+ξWkeiΔiktEkγ˜iΔ0k×DdxdyUi*UkL(x,y){1C(x,y)L(x,y)+[C(x,y)L(x,y)]2},
dEidt[i(Δi+12)(αL+γ˜ii)]Ei+ξWEiγ˜iΔ0iDdxdy|Ui|2L(x,y)ξWEiDdxdy|Ui|2[L(x,y)]2kki2κ˜2|Ek|2|Uk|2(γ˜iΔ0k)(γ˜iΔki)×(1γ˜+iΔ0k+1γ˜iΔ0i).
d|Ei|2dt=ddt(EiEi*)=Ei*dEidt+EidEi*dt=(2αL+γ˜ii+γ˜ii*)|Ei|2+2ξWg(Δi)Ddxdy[L(x,y)]2×|Ei|2|Ui|2{L(x,y)k,ki2κ˜2g(Δk)g(ΔiΔk)×[2γ˜+(ΔiΔ0)(ΔiΔk)/γ˜+(ΔiΔk)(Δi+Δk2Δ0)/γ˜]|Ek|2|Uk|2},
L(x,y)1+k,|ΔkΔi|γ˜4κ˜2γ˜g(Δk)|Ek|2|Uk|2.
2γ˜(ΔiΔ0)(ΔiΔk)/γ˜+(ΔiΔk)(Δi+Δk2Δ0)/γ˜.
d|Ei|2dt=(2αL+γ˜ii+γ˜ii*)|Ei|2+2ξWg(Δi)Ddxdy[L(x,y)]2|Ei|2|Ui|2×[1+k,|ΔkΔi|γ˜4κ˜2γ˜g(Δk)|Ek|2|Uk|2k,ki,|ΔkΔi|γ˜4κ˜2γ˜g(Δk)|Ek|2|Uk|2]=(2αL+γ˜ii+γ˜ii*)|Ei|2+2ξWg(Δi)Ddxdy[L(x,y)]2|Ei|2|Ui|2×[1+4κ˜2γ˜g(Δi)|Ei|2|Ui|2].
dIidtSiIi,
Si2(αL+γi)+2ξWg(Δi)Ddxdy|Ui|2Li(x,y)[L(x,y)]2,
γii4Dds(Ui*UinUiUi*n),
Mij{Sj+2ξWbj2Ddxdy|Uj|4[L(x,y)]2}δij4ξWbibjDdxdy|Ui|2|Uj|2Li(x,y)[L(x,y)]3,
M˜jj=2ξWbj2Ddxdy|Uj|4Ls,j2<0,
M˜ii=2ξWg(Δi)Ddxdy|Ui|2Ls,iLs,j2Ls,j2Ls,i,
Mii=2ξWbi2Ddxdy|Ui|4Li(x,y)[L(x,y)]3[L(x,y)Li(x,y)2],
Mij=4ξWbibjDdxdy|Ui|2|Uj|2Li(x,y)[L(x,y)]3.
Ddxdyεi(x,y)|U0(x,y)|2=0,
M˜ii=2ξWg(Δi)Sdxdy(1+εi)|U0|2×as,i(1+εi)as,j2(1+εj)2|U0|2as,j2(1+εj)2(1+εi)as,i|U0|6|U0|2=2ξWg(Δi)as,j2as,iSdxdy{[1+O(εi(x,y)|U0(x,y)|2)]as,i|U0|2[1+O(εj(x,y)|U0(x,y)|2)]as,j2}2ξWg(Δi)as,j2as,i(as,iSdxdy1|U0|2Aas,j2),
(as,jA)2>as,iASdxdy1A2|U0|2.
Li(x,y)(1+εi)ai|U0|2,
L(x,y)m=1N(1+εm)am|U0|2
Mii4ξAWbi2aiatot3(1atot2ai),
Mij4ξAWbibjaiatot3,
M(N)4ξAWatot3diag(a1,,aN)×BRBdiag(a1,,aN).
|M(k)|=(2ξAWatot2)k(1i=1k2aiatot)i=1kbi2.
Mx=diag(B1,,BN)Cdiag(A1,,AN)x=λx.
diag(B1,,BN)Cdiag(A11/2,,AN1/2)×diag(A11/2,,AN1/2)diag(B11/2,,BN1/2)×diag(B11/2,,BN1/2)x=diag(B1,,BN)Cdiag(B11/2,,BN1/2)×diag(A11/2,,AN1/2)diag(B11/2,,BN1/2)×diag(A11/2,,AN1/2)x.
λdiag(A11/2,,AN1/2)diag(A11/2,,AN1/2)×diag(B11/2,,BN1/2)diag(B11/2,,BN1/2)x=λdiag(A11/2,,AN1/2)diag(B11/2,,BN1/2)×diag(B11/2,,BN1/2)diag(A11/2,,AN1/2)x.
diag(B1,,BN)Cdiag(B11/2,,BN1/2)×diag(A11/2,,AN1/2)x=λdiag(A11/2,,AN1/2)diag(B11/2,,BN1/2)x,
diag(A11/2,,AN1/2)diag(B11/2,,BN1/2)C×diag(B11/2,,BN1/2)diag(A11/2,,AN1/2)x=λx.
Mdiag(A11/2,,AN1/2)diag(B11/2,,BN1/2)C×diag(B11/2,,BN1/2)diag(A11/2,,AN1/2)
|M|=|diag(A11/2,,AN1/2)||diag(B11/2,,BN1/2)||C|×|diag(B11/2,,BN1/2)||diag(A11/2,,AN1/2)|=(i=1NAiBi)(i=1N(Ci1))(1+i=1N(Ci1)1).

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