Abstract

We numerically performed wave dynamical simulations based on the Maxwell–Bloch (MB) model for a quadrupole-deformed microcavity laser with spatially selective pumping. We demonstrate the appearance of an asymmetric lasing mode whose spatial pattern violates both the x- and y-axes mirror symmetries of the cavity. Dynamical simulations revealed that a lasing mode consisting of a clockwise or counterclockwise rotating-wave component is a stable stationary solution of the MB model. From the results of a passive-cavity mode analysis, we interpret these asymmetric rotating-wave lasing modes by the locking of four nearly degenerate passive-cavity modes. For comparison, we carried out simulations for a uniform pumping case and found a different locking rule for the nearly degenerate modes. Our results demonstrate a nonlinear dynamical mechanism for the formation of a lasing mode that adjusts its pattern to a pumped area.

© 2017 Chinese Laser Press

1. INTRODUCTION

Since a close analogy between optical microcavities and open dynamical billiards was pointed out [1], intensive investigations have been carried out for ray-chaotic optical cavities, whose typical shape is a deformation of a circle [25]. Combination of total internal reflection and ray chaos is expected to achieve a low-threshold and directional-emission microcavity laser [6,7].

Recently, Aung et al. experimentally demonstrated that threshold current reduction and directional emission can be simultaneously achieved by forming an appropriate selective pumping area in the quadrupole-deformed microcavity laser [8]. In this cavity, there are resonant modes that are localized along twin periodic orbits with the shape of the double triangle consisting of upward-pointing and downward-pointing triangles [see Fig. 1(a)]. In the experiments reported in Ref. [8], the selective pumping area was formed along one of the two triangle orbits, and directional emission was observed. The assumption of the existence of a lasing mode that localizes along the sole triangle orbit plays a key role in explaining the experimental data in Ref. [8]. With this assumption, the directional emission can be explained by the mechanism of unstable-manifold-guided emission [911] via dynamical tunneling [1214], and the threshold current reduction can be attributed to the fact that the triangle orbit is confined by total internal reflection. However, because this lasing mode violates the mirror symmetry of the quadrupole cavity, it cannot be simply regarded as a passive-cavity mode.

 

Fig. 1. (a) Double-triangle orbits in the quadrupole-deformed cavity. (b) Spatial selective pumping (yellow region) along the upward-pointing triangle orbit (red lines).

Download Full Size | PPT Slide | PDF

A theoretical explanation is needed on how selective pumping leads to the formation of the asymmetric lasing mode, which does not have a direct counterpart in the passive-cavity modes. It is the purpose of this paper to numerically demonstrate the existence of the asymmetric lasing mode and theoretically reveal its appearance mechanism. We perform wave dynamical simulations based on the Maxwell–Bloch (MB) model [3,15], which fully takes into account the nonlinear interaction between the light field and a gain medium described by a two-level atom system (i.e., the optical Bloch equations [16]). We numerically reproduce the appearance of the asymmetric lasing mode when the selective pumping condition is adopted, and explain it by the locking of four nearly degenerate modes associated with the double-triangle orbits. Dynamical simulations reveal that clockwise (CW) and counterclockwise (CCW) rotating-wave states are stable stationary solutions of the MB model. For comparison, we perform simulations for a uniform pumping case, and found a different locking rule for the nearly degenerate modes.

2. MODEL

A. MB Model

The MB model is a set of wave dynamical equations describing the nonlinear interaction between the light field and a gain medium [3,15,17]. For transverse-magnetic (TM) polarization, it is given by

2t2(Ez+4πεPz)=c2n22Ez2βtEz,
Pz=N(ρ+ρ*)κ,
tρ=iω0ρiκWEzγρ,
tW=2iκEz(ρρ*)γ(WW),
where Ez(x,y), Pz(x,y), and W(x,y) are the electric field, polarization, and population inversion, respectively, n=n(x,y) is the refractive index, and ε=n2 is the permittivity. The constant β is introduced phenomenologically to describe the background uniform absorption, ρ(x,y) is the microscopic polarization, N the atomic number density, κ the coupling strength for the light–matter interaction, ω0 the transition frequency of the two-level gain medium, and W(x,y) the pumping strength parameter. γ and γ are the transversal and longitudinal relaxation rates, respectively. In the study reported in this paper, we numerically solve Eq. (1) by the finite-difference time-domain method and Eqs. (3) and (4) by the Euler method.

B. Quadrupole-Deformed Cavity

In the MB model, the effect of the cavity shape appears through the refractive index function n(x,y). We fix the cavity shape to be the quadrupole-deformed cavity defined in the polar coordinates (r,θ) by

r(θ)=r0[1+ϵcos(2θ)],
where r0 is the size parameter, and the deformation parameter ϵ is fixed at 0.09 throughout the paper. The refractive indices inside and outside the cavity are fixed at nin=3.3 (GaAs) and nout=1 (air), respectively. In this paper, we focus our attention on the periodic orbits with the shape of the double triangle, consisting of upward-pointing and downward-pointing stable triangle orbits as shown in Fig. 1(a).

In the experiments by Aung et al. [8], the effective refractive index of the cavity is 3.67, and the emitted light is transverse-electric polarized, which are different from our setting. However, we believe that our claims presented in this paper are qualitatively applicable to the microcavity laser studied by Aung et al.

C. Resonant Modes for the Passive Cavity

The resonant modes for the passive cavity are the solutions of Eq. (1) with Pz0 and β=0. That is, they are the eigensolutions of the following Helmholtz equation:

(2+n2ω2c2)ψ(x,y)=0,
where Ez=Re[ψ(x,y)exp(iωt)]. Because we assume TM polarization, we impose that both the wave function and its normal derivative are continuous at the cavity boundary. The eigenfrequency ω takes a complex value with a negative imaginary part, because Eq. (6) is solved with the outgoing wave condition at infinity, i.e., ψeiωr/c/r as r. Because the quadrupole cavity has mirror symmetries with respect to both the x and y axes, the resonant modes are divided into four symmetry classes, i.e.,
ψab(x,y)=aψab(x,y),
ψab(x,y)=bψab(x,y),
where a,b{+,} are parity indices.

For the double-triangle orbits, the associated resonant modes can be shown to have fourfold near degeneracy by a symmetry argument [18]. Figure 2 shows an example of the four nearly degenerate modes with Reω/ω01, where the even–even (ee) and odd–odd (oo) modes constitute a closer pair, whereas the even–odd (eo) and odd–even (oe) modes form another closer pair. Here we used the scaled eigenfrequency Reω/ω0, where ω0=2.2×1015 is the transition frequency of the two-level gain medium tuned to fit the frequencies of the nearly degenerate modes. We numerically obtained the eigenfrequencies and eigenfunctions by the boundary element method [19].

 

Fig. 2. Intensity distributions of the resonant modes for a passive quadrupole-deformed cavity with refractive index 3.3. The modes are four nearly degenerate modes associated with the double-triangle orbits. The double-triangle orbits (red and green lines) are superposed, and the intensities outside the cavity are plotted in log scale. (a) Even–even mode with scaled frequency Reω/ω0=1.0008278. (b) Even–odd mode with Reω/ω0=0.999085. (c) Odd–even mode with Reω/ω0=0.999075. (d) Odd–odd mode with Reω/ω0=1.0008277.

Download Full Size | PPT Slide | PDF

Because of the relatively small wavenumbers, the intensity localization along the triangle orbits is less obvious in Fig. 2. However, by investigating the Husimi distributions [20] of the wave functions, we could identify the intensity localization along the double-triangle orbits. Figure 3 shows the upper-half phase space for the ray dynamics [24], where the phase space is spanned by (s,sinϕ) with s and ϕ being the arc length along the cavity boundary and the incident angle for a ray orbit, respectively. In Fig. 3, the islands of stability corresponding to the upward-pointing and downward-pointing triangle orbits are shown by red and green points, respectively. In Fig. 3, we also superpose the Husimi distribution for the resonant mode eo, whose wave function pattern is shown in Fig. 2(b). In the Husimi distribution, we can see high intensities near the islands of stability. We also checked that modes similar to those in Fig. 2 appear regularly in the ω plane with a constant modal spacing corresponding to the optical path length of the triangle orbit.

 

Fig. 3. Phase space of the ray dynamics for the quadrupole-deformed cavity. The islands of stability corresponding to the upward-pointing and downward-pointing triangle orbits are indicated by red and green points, respectively. The critical line for total internal reflection is indicated by a line at sinϕ=1/3.3. Husimi distribution for the eo mode shown in Fig. 2(b) is superposed.

Download Full Size | PPT Slide | PDF

As can be seen in Fig. 3, the double-triangle orbits are located above the critical line for total internal reflection. The emission of a well-confined mode in a ray-chaotic cavity can be explained by the chaos-assisted emission (CAE) mechanism [1214]. As Aung et al. have shown [8], the CAE mechanism for the quadrupole cavity results in strong emissions at the far-field angles θ50°, 130°, 230°, and 310°.

Figure 4 shows the eigenfrequency distribution of the resonant modes in the complex ω/ω0 plane, where Imω/ω0 represents the decay rate of the mode. The modes encircled by the green circle are the four nearly degenerate modes associated with the double-triangle orbits (the ee and oo modes are almost on top each other, and so are the eo and oe modes). The longitudinal modal spacing for the double-triangle modes is estimated to be (ΔReω)/ω00.0244 from the optical path length of the triangle orbit. For the numerical simulations reported in this paper, we only consider the cases where a single set of the four nearly degenerate modes has positive gain.

 

Fig. 4. Distribution of the complex eigenfrequencies ω scaled by ω0, where ω0 is the gain center parameter. The four nearly degenerate modes associated with the double-triangle orbits are encircled by a green circle (the ee and oo modes are almost on top of each other, and so are the eo and oe modes). The modes indicated by filled circles (•) have positive linear gain [i.e., satisfying Eq. (9)] for the selective pumping with W=1.0×103, whereas those indicated by crosses (×) do not satisfy Eq. (9).

Download Full Size | PPT Slide | PDF

In the eigenfrequency distribution, we can see that some modes are aligned at Im(ω/ω0)0.003. This decay rate value was found to agree with that estimated by the Fresnel-coefficient-weighted ray simulation [3] for chaotic orbits located near the critical line for total internal reflection in the phase space. Thus, these modes are associated with the chaotic orbits.

3. NUMERICAL RESULTS OF THE MB MODEL SIMULATION

A. Positive Linear Gain Condition for a Resonant Mode

The condition for a resonant mode to have positive linear gain is derived in the limit of a single-mode approximation ignoring modal couplings [15]:

2πNκ2Wn2γReωs(Reωsω0)2+γ2>Imωs+β,
with
W=WDdxdy|ψ(x,y)|2Θ(x,y)Ddxdy|ψ(x,y)|2,
where ωs (sN) is the eigenfrequency of a passive-cavity mode, D denotes the area inside the cavity, and Θ(x,y) represents a characteristic function that takes 1 inside the pumped area (i.e., the area along the upward-pointing triangle orbit) and takes 0 otherwise. We note that this condition depends on the wave function ψ(x,y) through Eq. (10), and that the gain center is controlled by the parameter ω0. This condition is useful for estimating which passive-cavity modes have the potential to contribute to a lasing state for given gain center ω0 and pumping strength W.

For the simulations reported in this paper, we fixed the parameter values as follows: r0=2.027μm, β=8.8×1012s1, ω0=2.2×1015s1, γ=13.2×1012s1, γ=6.6×1012s1, and Nκ2=0.55Js1cm3. As mentioned in Section 2.C, the value of ω0 is chosen so that the double-triangle modes become the nearest to the gain center. The pumping strength parameter W is dimensionless, and it is varied in the simulations.

B. Selective Pumping Simulations

Selective pumping in two-dimensional microcavity lasers has been experimentally demonstrated in Refs. [8,13,2125], and theoretically studied in Refs. [2630]. The study reported in this paper numerically demonstrates a nonlinear dynamical mechanism for the formation of a lasing mode that adjusts its pattern to a pumped area.

Figures 5(b) and 6 show the results of the selective pumping simulation when the pumping strength is set at W=1.0×103, and the initial distribution of the electric field Ez is given by a Gaussian distribution, as shown in Fig. 5(a). For selective pumping with W=1.0×103, the number of the modes that satisfy the positive linear gain condition, Eq. (9), turned out to be 10, including the four nearly degenerate modes. In Fig. 4, these 10 modes are indicated by filled circles (•).

 

Fig. 5. Electric field intensity distributions. (a) An initial condition for the MB model simulation. (b) Time-averaged pattern of the stationary lasing state of the MB model for the selective pumping case with W=1.0×103. The intensity outside the cavity is plotted in log scale. The boundary of the pumped area is indicated by yellow lines.

Download Full Size | PPT Slide | PDF

 

Fig. 6. Results of the MB model simulation for the selective pumping case with W=1.0×103. (a) Time evolution of the total light intensity inside the cavity. (b) Power spectrum of the electric field for the stationary lasing regime. The peak frequency is around ω/ω0=0.9988.

Download Full Size | PPT Slide | PDF

Figure 6(a) shows the time evolution of the total light intensity inside the cavity, where we can observe the formation of a stationary lasing state after a transient. Figure 6(b) shows the power spectrum of the time series of the electric field at a certain point in the cavity taken in the stationary regime. From this power spectral data, we can confirm the single-mode lasing. The corresponding time-averaged electric field intensity pattern is shown in Fig. 5(b), where the average was taken over the time interval of T4×103×(2π/ω0). This pattern clearly shows that the stationary lasing state is a CW rotating wave, which violates mirror symmetries with respect to both the x and y axes. Because the MB system with the triangle-orbit pumping pattern is invariant under the transformation xx, a CCW rotating wave [obtained by reversing the x axis in Fig. 5(b)] is also a solution of the MB model.

C. Interpretation of the Stationary Rotating-Wave States of the MB Model by the Resonant Modes

It has been numerically demonstrated for the Schrödinger–Bloch (SB) model that the frequencies of nearly degenerate modes can be locked as a result of a nonlinear modal interaction, and the locking phenomenon results in the appearance of asymmetric emission patterns [3134]. The SB model is an approximation of the MB model, where the slowly varying envelope approximation of the field variables is adopted for reducing a numerical computation cost.

The power spectrum in Fig. 6(b) shows a single-mode lasing, under the condition of preferential excitation of the four nearly degenerate modes. This result suggests that the locking occurs for these nearly degenerate modes. For confirming this, we computed the superposition of the resonant-mode eigenfunctions. First, we found that intensity localization along the upward-pointing triangle orbit can be reproduced by taking the superpositions of two different parity modes as follows:

ξψee+ψeo,
ηψoe+ψoo.

We plot the intensity distributions of ξ(x,y) and η(x,y) in Figs. 7(a) and 7(b), respectively. In the same manner, the intensity localization along the downward-pointing triangle orbit can be obtained by ξψeeψeo and ηψoeψoo. Secondly, we found that the CW and CCW rotating-wave states can be reproduced by taking the superpositions of ξ(x,y) and η(x,y) as follows:

ΨCWξ+iη=(ψee+ψeo)+i(ψoe+ψoo),
ΨCCWξiη=(ψee+ψeo)i(ψoe+ψoo).

 

Fig. 7. Intensity distributions of the superpositions of the resonant-mode wave functions. The triangle orbit is indicated by red lines, and the intensities outside the cavities are plotted in log scale. (a) ξ=ψee+ψeo. (b) η=ψoe+ψoo. (c) ΨCW=ξ+iη=(ψee+ψeo)+i(ψoe+ψoo). (d) ΨCCW=ξiη=(ψee+ψeo)i(ψoe+ψoo).

Download Full Size | PPT Slide | PDF

The intensity distributions of ΨCW(x,y) and ΨCCW(x,y) are plotted in Figs. 7(c) and 7(d), respectively. Comparing Figs. 5(b) and 7(c), we can confirm that the CW rotating-wave state of the MB model can be very well reproduced by the superposition of the four nearly degenerate double-triangle modes.

By performing simulations for various initial conditions and pumping strengths, we numerically found that the CW and CCW rotating-wave states are stable solutions of the MB model. Because our selective pumping pattern is symmetric with respect to the y axis, when we prepare the initial field distributions symmetric with respect to the y axis, we obtained standing-wave stationary states that obey the same symmetry. We checked that these standing-wave stationary states can be reproduced by the resonant-mode superpositions ξ and η shown in Figs. 7(a) and 7(b). However, we numerically found that these standing-wave states are dynamically unstable stationary solutions [31].

For selective pumping, we found the lasing threshold to be W2.8×104. Just above the threshold, we already observed the rotating-wave state localized along the triangle orbit. This appears to be reasonable, because the lasing mode needs to adjust its pattern along the pumped area so that it can be excited.

The fact that the lasing mode at the threshold already localizes along the triangle orbit yields the discrepancy between the actual lasing threshold and the positive linear gain condition Eq. (9). In the estimation of W by Eq. (10), the resonant-mode wave function ψ(x,y) does not localize only along the upward-pointing triangle orbit. For the resonant modes ψee, ψeo, ψoe, and ψoo, Eq. (9) predicts the positive linear gain thresholds to be W=3.14×104, 3.32×104, 3.40×104, and 3.12×104, respectively, which are all larger than the actual lasing threshold. When we used ξ(x,y)±iη(x,y) instead of ψ(x,y) for the estimation of W by Eq. (10), we found that the positive linear gain threshold is around 2.4×104, which is closer to the actual lasing threshold. For a more accurate prediction of the lasing threshold, incorporating the effect of the polarization term (thus the effect of selective pumping) in the linear Helmholtz equation [26,2830] is expected to be effective, which is, however, beyond the scope of this paper.

D. Uniform Pumping Simulation

To compare with the selective pumping case, we carried out simulations when the cavity is uniformly pumped. We used the same parameter values as in the selective pumping case, except for the pumping strength, which was set at W=3.0×104. For this W value, the number of the modes satisfying the positive linear gain condition, Eq. (9), turned out to be 11, including the four nearly degenerate double-triangle modes.

Figures 8 and 9 show the results of the MB model simulation with uniform pumping. Figures 8(a) and 8(b), respectively, show the time evolution of the total light intensity inside the cavity and the power spectrum of the electric field for the stationary lasing regime. In the spectral data, we can identify two dominant peaks at ω/ω0=0.9990 and ω/ω0=1.0012 and also observe their beat oscillation in the stationary regime of the time-series data (we note that the ratio of the peak heights depends on the position where the time-series data are acquired). These data suggest that the stationary state consists mainly of the two modes for W=3.0×104. For higher pumping strengths (i.e., W1.0×101), we did not observe the tendency for the two modes to be locked.

 

Fig. 8. Results of the MB model simulation for the uniform pumping case with W=3.0×104. (a) Time evolution of the total light intensity inside the cavity. (b) Power spectrum of the electric field for the stationary lasing regime. The frequencies of the primary and secondary peaks are ω/ω00.9990 and ω/ω01.0012, respectively.

Download Full Size | PPT Slide | PDF

 

Fig. 9. Time-averaged pattern of the stationary lasing state of the MB model for the uniform pumping case with W=3.0×104. The intensity outside the cavity is plotted in log scale.

Download Full Size | PPT Slide | PDF

We show in Fig. 9 the time-averaged electric field intensity pattern for the stationary lasing regime, where we can see a CW rotating-wave state different from the one for the selective pumping case. By investigating the superposition of the resonant-mode wave functions, we found that the pattern in Fig. 9 can be reproduced by ψeo+iψoe and ψee+iψoo. Their patterns are shown in Figs. 10(a) and 10(b), both of which show CW rotating-wave patterns. We note that the eo and oe modes have very close eigenfrequencies, and so do the ee and oo modes (see the caption of Fig. 2 for their eigenfrequencies). The scaled eigenfrequency Reω/ω0 of the former pair is around 0.99908, whereas that for the latter is around 1.000828. These values closely correspond to the peak positions in the spectral data, which provides a strong support for our interpretation that each of the two lasing modes is the locked state of a nearly degenerate pair.

 

Fig. 10. Intensity distribution of the superpositions of the resonant-mode wave functions. (a) ψeo+iψoe. (b) ψee+iψoo. The intensities outside the cavity are plotted in log scale.

Download Full Size | PPT Slide | PDF

For uniform pumping, we found the lasing threshold to be W2.0×104, which is smaller than the threshold for selective pumping. This result seems to be natural, because the lasing modes are different between the two cases, and for selective pumping, the double-triangle modes are partially pumped.

4. CONCLUSIONS

We numerically demonstrated the lasing of a triangle orbit mode in the quadrupole-deformed microcavity laser with spatial selective pumping along the periodic orbit. We used the MB model to describe the nonlinear interaction between the light field and a gain medium. The nonlinear interaction is essential for the existence of the triangle orbit lasing mode, as there is no corresponding resonant mode for the passive cavity because of the x- and y-axes mirror symmetries of the quadrupole cavity. By a passive-cavity mode analysis, we concluded that the asymmetric lasing mode can be interpreted as the locked state of the four nearly degenerate modes associated with the double-triangle orbits. In view of our theoretical results, the experimental study by Aung et al. [8] is considered to be a very nice illustration of the nonlinear dynamical effect on the formation of asymmetric lasing modes. Although our results presented here are for γγ, we numerically confirmed that the locking phenomena of the four nearly degenerate modes were similarly observed for γγ (e.g., γ=102 and γ=105).

We numerically confirmed that the CW and CCW rotating-wave triangle orbit modes are stable solutions for the MB model, whereas standing-wave ones are unstable. It would be an interesting future problem to examine if a CW or CCW rotating-wave stationary state can be obtained even when multiple longitudinal modes are involved in lasing.

Another open issue is the detailed theoretical mechanism for the nonlinear interaction of the four different parity resonant modes. Our comparison between the selective and uniform pumping cases revealed that the modal interaction mechanism does depend on the pumping pattern. Elucidating its mechanism would be useful for better controlling lasing modes through selective pumping.

Regarding device optimization for threshold current reduction by selective pumping, it might be effective to adopt an active–passive structure, where the non-pumped cavity area is made of a passive material so as to suppress material absorption.

Funding

Waseda University Grant for Special Research Projects (2017B-197).

Acknowledgment

The authors thank Mr. Shunya Sekiguchi for his support in ray-dynamical simulations. T. H. is grateful to Prof. Li Ge for fruitful discussions.

REFERENCES

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

2. H. G. L. Schwefel, H. E. Tureci, A. D. Stone, and R. K. Chang, “Progress in asymmetric resonant cavities: Using shape as a design parameter in dielectric microcavity lasers,” in Optical Microcavities, K. Vahala, ed. (World Scientific, 2004), pp. 415–495.

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

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

5. X.-F. Jiang, C.-L. Zou, L. Wang, Q. Gong, and Y.-F. Xiao, “Whispering-gallery microcavities with unidirectional laser emission,” Laser Photon. Rev. 10, 40–61 (2016). [CrossRef]  

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

7. J. Wiersig, J. Unterhinninghofen, Q. H. Song, H. Cao, M. Hentschel, and S. Shinohara, “Review on unidirectional light emission from ultralow-loss modes in deformed microdisks,” in Trends in Nano- and Micro-cavities, O. Kwon, B. Lee, and K. An, eds. (Bentham Books, 2011), pp. 109–152.

8. N. L. Aung, L. Ge, O. Malik, H. E. Türeci, and C. F. Gmachl, “Threshold current reduction and directional emission of deformed microdisk lasers via spatially selective electrical pumping,” Appl. Phys. Lett. 107, 151106 (2015). [CrossRef]  

9. 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]  

10. S.-Y. Lee, J.-W. Ryu, T.-Y. Kwon, S. Rim, and C.-M. Kim, “Scarred resonances and steady probability distribution in a chaotic microcavity,” Phys. Rev. A 72, 061801(R) (2005). [CrossRef]  

11. S. Shinohara, T. Harayama, H. E. Türeci, and A. D. Stone, “Ray-wave correspondence in the nonlinear description of stadium-cavity lasers,” Phys. Rev. A 74, 033820 (2006). [CrossRef]  

12. V. A. Podolskiy and E. E. Narimanov, “Chaos-assisted tunneling in dielectric microcavities,” Opt. Lett. 30, 474–476 (2005). [CrossRef]  

13. 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]  

14. J. Yang, S.-B. Lee, S. Moon, S.-Y. Lee, S. W. Kim, T. T. A. Dao, J.-H. Lee, and K. An, “Pump-induced dynamical tunneling in a deformed microcavity laser,” Phys. Rev. Lett. 104, 243601 (2010). [CrossRef]  

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

16. R. Loudon, The Quantum Theory of Light (Oxford University, 2000).

17. 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]  

18. T. E. Tureci, H. G. L. Schwefel, A. D. Stone, and E. E. Narimanov, “Gaussian-optical approach to stable periodic orbit resonances of partially chaotic dielectric micro-cavities,” Opt. Express 10, 752–776 (2002). [CrossRef]  

19. J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A 5, 53–60 (2003). [CrossRef]  

20. M. Hentschel, H. Schomerus, and R. Schubert, “Husimi functions at dielectric interfaces: Inside–outside duality for optical systems and beyond,” Europhys. Lett. 62, 636–642 (2003). [CrossRef]  

21. T. Fukushima, T. Harayama, P. Davis, P. O. Vaccaro, T. Nishimura, and T. Aida, “Ring and axis mode lasing in quasi-stadium laser diodes with concentric end mirrors,” Opt. Lett. 27, 1430–1432 (2002). [CrossRef]  

22. G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett. 83, 1710–1712 (2003). [CrossRef]  

23. T. Fukushima and T. Harayama, “Stadium and quasi-stadium laser diodes,” IEEE J. Sel. Top. Quantum Electron. 10, 1039–1051 (2004). [CrossRef]  

24. M. Choi, T. Tanaka, T. Fukushima, and T. Harayama, “Control of directional emission in quasistadium microcavity laser diodes with two electrodes,” Appl. Phys. Lett. 88, 211110 (2006). [CrossRef]  

25. S. F. Liew, L. Ge, B. Redding, G. S. Solomon, and H. Cao, “Pump-controlled modal interactions in microdisk lasers,” Phys. Rev. A 91, 043828 (2015). [CrossRef]  

26. L. I. Deych, “Effects of spatial nonuniformity on laser dynamics,” Phys. Rev. Lett. 95, 043902 (2005). [CrossRef]  

27. T.-Y. Kwon, S.-Y. Lee, M. S. Kurdoglyan, S. Rim, C.-M. Kim, and Y.-J. Park, “Lasing modes in a spiral-shaped dielectric microcavity,” Opt. Lett. 31, 1250–1252 (2006). [CrossRef]  

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

29. L. Ge, O. Malik, and H. E. Türeci, “Enhancement of laser power-efficiency by control of spatial hole burning interactions,” Nat. Photonics 8, 871–875 (2014). [CrossRef]  

30. L. Ge, “Selective excitation of lasing modes by controlling modal interactions,” Opt. Express 23, 30049–30056 (2015). [CrossRef]  

31. 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]  

32. S. Sunada, T. Harayama, and K. S. Ikeda, “Nonlinear whispering-gallery modes in a microellipse cavity,” Opt. Lett. 29, 718–720 (2004). [CrossRef]  

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

34. S. Shinohara, S. Sunada, T. Harayama, and K. S. Ikeda, “Mode expansion description of stadium-cavity laser dynamics,” Phys. Rev. E 71, 036203 (2005). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385, 45–47 (1997).
    [Crossref]
  2. H. G. L. Schwefel, H. E. Tureci, A. D. Stone, and R. K. Chang, “Progress in asymmetric resonant cavities: Using shape as a design parameter in dielectric microcavity lasers,” in Optical Microcavities, K. Vahala, ed. (World Scientific, 2004), pp. 415–495.
  3. T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photon. Rev. 5, 247–271 (2011).
    [Crossref]
  4. H. Cao and J. Wiersig, “Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87, 61–111 (2015).
    [Crossref]
  5. X.-F. Jiang, C.-L. Zou, L. Wang, Q. Gong, and Y.-F. Xiao, “Whispering-gallery microcavities with unidirectional laser emission,” Laser Photon. Rev. 10, 40–61 (2016).
    [Crossref]
  6. J. Wiersig and M. Hentschel, “Combining directional light output and ultralow loss in deformed microdisks,” Phys. Rev. Lett. 100, 033901 (2008).
    [Crossref]
  7. J. Wiersig, J. Unterhinninghofen, Q. H. Song, H. Cao, M. Hentschel, and S. Shinohara, “Review on unidirectional light emission from ultralow-loss modes in deformed microdisks,” in Trends in Nano- and Micro-cavities, O. Kwon, B. Lee, and K. An, eds. (Bentham Books, 2011), pp. 109–152.
  8. N. L. Aung, L. Ge, O. Malik, H. E. Türeci, and C. F. Gmachl, “Threshold current reduction and directional emission of deformed microdisk lasers via spatially selective electrical pumping,” Appl. Phys. Lett. 107, 151106 (2015).
    [Crossref]
  9. 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]
  10. S.-Y. Lee, J.-W. Ryu, T.-Y. Kwon, S. Rim, and C.-M. Kim, “Scarred resonances and steady probability distribution in a chaotic microcavity,” Phys. Rev. A 72, 061801(R) (2005).
    [Crossref]
  11. S. Shinohara, T. Harayama, H. E. Türeci, and A. D. Stone, “Ray-wave correspondence in the nonlinear description of stadium-cavity lasers,” Phys. Rev. A 74, 033820 (2006).
    [Crossref]
  12. V. A. Podolskiy and E. E. Narimanov, “Chaos-assisted tunneling in dielectric microcavities,” Opt. Lett. 30, 474–476 (2005).
    [Crossref]
  13. 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]
  14. J. Yang, S.-B. Lee, S. Moon, S.-Y. Lee, S. W. Kim, T. T. A. Dao, J.-H. Lee, and K. An, “Pump-induced dynamical tunneling in a deformed microcavity laser,” Phys. Rev. Lett. 104, 243601 (2010).
    [Crossref]
  15. T. Harayama, S. Sunada, and K. S. Ikeda, “Theory of two-dimensional microcavity lasers,” Phys. Rev. A 72, 013803 (2005).
    [Crossref]
  16. R. Loudon, The Quantum Theory of Light (Oxford University, 2000).
  17. 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]
  18. T. E. Tureci, H. G. L. Schwefel, A. D. Stone, and E. E. Narimanov, “Gaussian-optical approach to stable periodic orbit resonances of partially chaotic dielectric micro-cavities,” Opt. Express 10, 752–776 (2002).
    [Crossref]
  19. J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A 5, 53–60 (2003).
    [Crossref]
  20. M. Hentschel, H. Schomerus, and R. Schubert, “Husimi functions at dielectric interfaces: Inside–outside duality for optical systems and beyond,” Europhys. Lett. 62, 636–642 (2003).
    [Crossref]
  21. T. Fukushima, T. Harayama, P. Davis, P. O. Vaccaro, T. Nishimura, and T. Aida, “Ring and axis mode lasing in quasi-stadium laser diodes with concentric end mirrors,” Opt. Lett. 27, 1430–1432 (2002).
    [Crossref]
  22. G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett. 83, 1710–1712 (2003).
    [Crossref]
  23. T. Fukushima and T. Harayama, “Stadium and quasi-stadium laser diodes,” IEEE J. Sel. Top. Quantum Electron. 10, 1039–1051 (2004).
    [Crossref]
  24. M. Choi, T. Tanaka, T. Fukushima, and T. Harayama, “Control of directional emission in quasistadium microcavity laser diodes with two electrodes,” Appl. Phys. Lett. 88, 211110 (2006).
    [Crossref]
  25. S. F. Liew, L. Ge, B. Redding, G. S. Solomon, and H. Cao, “Pump-controlled modal interactions in microdisk lasers,” Phys. Rev. A 91, 043828 (2015).
    [Crossref]
  26. L. I. Deych, “Effects of spatial nonuniformity on laser dynamics,” Phys. Rev. Lett. 95, 043902 (2005).
    [Crossref]
  27. T.-Y. Kwon, S.-Y. Lee, M. S. Kurdoglyan, S. Rim, C.-M. Kim, and Y.-J. Park, “Lasing modes in a spiral-shaped dielectric microcavity,” Opt. Lett. 31, 1250–1252 (2006).
    [Crossref]
  28. L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: Generalization and analytic results,” Phys. Rev. A 82, 063824 (2010).
    [Crossref]
  29. L. Ge, O. Malik, and H. E. Türeci, “Enhancement of laser power-efficiency by control of spatial hole burning interactions,” Nat. Photonics 8, 871–875 (2014).
    [Crossref]
  30. L. Ge, “Selective excitation of lasing modes by controlling modal interactions,” Opt. Express 23, 30049–30056 (2015).
    [Crossref]
  31. 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]
  32. S. Sunada, T. Harayama, and K. S. Ikeda, “Nonlinear whispering-gallery modes in a microellipse cavity,” Opt. Lett. 29, 718–720 (2004).
    [Crossref]
  33. S. Sunada, T. Harayama, and K. S. Ikeda, “Multimode lasing in two-dimensional fully chaotic cavity lasers,” Phys. Rev. E 71, 046209 (2005).
    [Crossref]
  34. S. Shinohara, S. Sunada, T. Harayama, and K. S. Ikeda, “Mode expansion description of stadium-cavity laser dynamics,” Phys. Rev. E 71, 036203 (2005).
    [Crossref]

2016 (1)

X.-F. Jiang, C.-L. Zou, L. Wang, Q. Gong, and Y.-F. Xiao, “Whispering-gallery microcavities with unidirectional laser emission,” Laser Photon. Rev. 10, 40–61 (2016).
[Crossref]

2015 (4)

L. Ge, “Selective excitation of lasing modes by controlling modal interactions,” Opt. Express 23, 30049–30056 (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]

N. L. Aung, L. Ge, O. Malik, H. E. Türeci, and C. F. Gmachl, “Threshold current reduction and directional emission of deformed microdisk lasers via spatially selective electrical pumping,” Appl. Phys. Lett. 107, 151106 (2015).
[Crossref]

S. F. Liew, L. Ge, B. Redding, G. S. Solomon, and H. Cao, “Pump-controlled modal interactions in microdisk lasers,” Phys. Rev. A 91, 043828 (2015).
[Crossref]

2014 (1)

L. Ge, O. Malik, and H. E. Türeci, “Enhancement of laser power-efficiency by control of spatial hole burning interactions,” Nat. Photonics 8, 871–875 (2014).
[Crossref]

2011 (1)

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

2010 (3)

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. Yang, S.-B. Lee, S. Moon, S.-Y. Lee, S. W. Kim, T. T. A. Dao, J.-H. Lee, and K. An, “Pump-induced dynamical tunneling in a deformed microcavity laser,” Phys. Rev. Lett. 104, 243601 (2010).
[Crossref]

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

2008 (1)

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

2006 (4)

M. Choi, T. Tanaka, T. Fukushima, and T. Harayama, “Control of directional emission in quasistadium microcavity laser diodes with two electrodes,” Appl. Phys. Lett. 88, 211110 (2006).
[Crossref]

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]

T.-Y. Kwon, S.-Y. Lee, M. S. Kurdoglyan, S. Rim, C.-M. Kim, and Y.-J. Park, “Lasing modes in a spiral-shaped dielectric microcavity,” Opt. Lett. 31, 1250–1252 (2006).
[Crossref]

S. Shinohara, T. Harayama, H. E. Türeci, and A. D. Stone, “Ray-wave correspondence in the nonlinear description of stadium-cavity lasers,” Phys. Rev. A 74, 033820 (2006).
[Crossref]

2005 (6)

L. I. Deych, “Effects of spatial nonuniformity on laser dynamics,” Phys. Rev. Lett. 95, 043902 (2005).
[Crossref]

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

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

S.-Y. Lee, J.-W. Ryu, T.-Y. Kwon, S. Rim, and C.-M. Kim, “Scarred resonances and steady probability distribution in a chaotic microcavity,” Phys. Rev. A 72, 061801(R) (2005).
[Crossref]

V. A. Podolskiy and E. E. Narimanov, “Chaos-assisted tunneling in dielectric microcavities,” Opt. Lett. 30, 474–476 (2005).
[Crossref]

S. Shinohara, S. Sunada, T. Harayama, and K. S. Ikeda, “Mode expansion description of stadium-cavity laser dynamics,” Phys. Rev. E 71, 036203 (2005).
[Crossref]

2004 (3)

2003 (4)

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A 5, 53–60 (2003).
[Crossref]

G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett. 83, 1710–1712 (2003).
[Crossref]

M. Hentschel, H. Schomerus, and R. Schubert, “Husimi functions at dielectric interfaces: Inside–outside duality for optical systems and beyond,” Europhys. Lett. 62, 636–642 (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]

2002 (2)

1997 (1)

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

Aida, T.

An, K.

J. Yang, S.-B. Lee, S. Moon, S.-Y. Lee, S. W. Kim, T. T. A. Dao, J.-H. Lee, and K. An, “Pump-induced dynamical tunneling in a deformed microcavity laser,” Phys. Rev. Lett. 104, 243601 (2010).
[Crossref]

Aung, N. L.

N. L. Aung, L. Ge, O. Malik, H. E. Türeci, and C. F. Gmachl, “Threshold current reduction and directional emission of deformed microdisk lasers via spatially selective electrical pumping,” Appl. Phys. Lett. 107, 151106 (2015).
[Crossref]

Ben-Messaoud, T.

Cao, H.

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

S. F. Liew, L. Ge, B. Redding, G. S. Solomon, and H. Cao, “Pump-controlled modal interactions in microdisk lasers,” Phys. Rev. A 91, 043828 (2015).
[Crossref]

J. Wiersig, J. Unterhinninghofen, Q. H. Song, H. Cao, M. Hentschel, and S. Shinohara, “Review on unidirectional light emission from ultralow-loss modes in deformed microdisks,” in Trends in Nano- and Micro-cavities, O. Kwon, B. Lee, and K. An, eds. (Bentham Books, 2011), pp. 109–152.

Chang, R. K.

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]

G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett. 83, 1710–1712 (2003).
[Crossref]

H. G. L. Schwefel, H. E. Tureci, A. D. Stone, and R. K. Chang, “Progress in asymmetric resonant cavities: Using shape as a design parameter in dielectric microcavity lasers,” in Optical Microcavities, K. Vahala, ed. (World Scientific, 2004), pp. 415–495.

Chern, G. D.

G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett. 83, 1710–1712 (2003).
[Crossref]

Choi, M.

M. Choi, T. Tanaka, T. Fukushima, and T. Harayama, “Control of directional emission in quasistadium microcavity laser diodes with two electrodes,” Appl. Phys. Lett. 88, 211110 (2006).
[Crossref]

Chong, Y. D.

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

Dao, T. T. A.

J. Yang, S.-B. Lee, S. Moon, S.-Y. Lee, S. W. Kim, T. T. A. Dao, J.-H. Lee, and K. An, “Pump-induced dynamical tunneling in a deformed microcavity laser,” Phys. Rev. Lett. 104, 243601 (2010).
[Crossref]

Davis, P.

Deych, L. I.

L. I. Deych, “Effects of spatial nonuniformity on laser dynamics,” Phys. Rev. Lett. 95, 043902 (2005).
[Crossref]

Fukushima, T.

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, T. Tanaka, T. Fukushima, and T. Harayama, “Control of directional emission in quasistadium microcavity laser diodes with two electrodes,” Appl. Phys. Lett. 88, 211110 (2006).
[Crossref]

T. Fukushima and T. Harayama, “Stadium and quasi-stadium laser diodes,” IEEE J. Sel. Top. Quantum Electron. 10, 1039–1051 (2004).
[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. Fukushima, T. Harayama, P. Davis, P. O. Vaccaro, T. Nishimura, and T. Aida, “Ring and axis mode lasing in quasi-stadium laser diodes with concentric end mirrors,” Opt. Lett. 27, 1430–1432 (2002).
[Crossref]

Ge, L.

S. F. Liew, L. Ge, B. Redding, G. S. Solomon, and H. Cao, “Pump-controlled modal interactions in microdisk lasers,” Phys. Rev. A 91, 043828 (2015).
[Crossref]

L. Ge, “Selective excitation of lasing modes by controlling modal interactions,” Opt. Express 23, 30049–30056 (2015).
[Crossref]

N. L. Aung, L. Ge, O. Malik, H. E. Türeci, and C. F. Gmachl, “Threshold current reduction and directional emission of deformed microdisk lasers via spatially selective electrical pumping,” Appl. Phys. Lett. 107, 151106 (2015).
[Crossref]

L. Ge, O. Malik, and H. E. Türeci, “Enhancement of laser power-efficiency by control of spatial hole burning interactions,” Nat. Photonics 8, 871–875 (2014).
[Crossref]

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

Gmachl, C. F.

N. L. Aung, L. Ge, O. Malik, H. E. Türeci, and C. F. Gmachl, “Threshold current reduction and directional emission of deformed microdisk lasers via spatially selective electrical pumping,” Appl. Phys. Lett. 107, 151106 (2015).
[Crossref]

Gong, Q.

X.-F. Jiang, C.-L. Zou, L. Wang, Q. Gong, and Y.-F. Xiao, “Whispering-gallery microcavities with unidirectional laser emission,” Laser Photon. Rev. 10, 40–61 (2016).
[Crossref]

Harayama, T.

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]

S. Shinohara, T. Harayama, H. E. Türeci, and A. D. Stone, “Ray-wave correspondence in the nonlinear description of stadium-cavity lasers,” Phys. Rev. A 74, 033820 (2006).
[Crossref]

M. Choi, T. Tanaka, T. Fukushima, and T. Harayama, “Control of directional emission in quasistadium microcavity laser diodes with two electrodes,” Appl. Phys. Lett. 88, 211110 (2006).
[Crossref]

S. Shinohara, S. Sunada, T. Harayama, and K. S. Ikeda, “Mode expansion description of stadium-cavity laser dynamics,” Phys. Rev. E 71, 036203 (2005).
[Crossref]

S. Sunada, T. Harayama, and K. S. Ikeda, “Multimode lasing in two-dimensional 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]

S. Sunada, T. Harayama, and K. S. Ikeda, “Nonlinear whispering-gallery modes in a microellipse cavity,” Opt. Lett. 29, 718–720 (2004).
[Crossref]

T. Fukushima and T. Harayama, “Stadium and quasi-stadium laser diodes,” IEEE J. Sel. Top. Quantum Electron. 10, 1039–1051 (2004).
[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. Fukushima, T. Harayama, P. Davis, P. O. Vaccaro, T. Nishimura, and T. Aida, “Ring and axis mode lasing in quasi-stadium laser diodes with concentric end mirrors,” Opt. Lett. 27, 1430–1432 (2002).
[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]

M. Hentschel, H. Schomerus, and R. Schubert, “Husimi functions at dielectric interfaces: Inside–outside duality for optical systems and beyond,” Europhys. Lett. 62, 636–642 (2003).
[Crossref]

J. Wiersig, J. Unterhinninghofen, Q. H. Song, H. Cao, M. Hentschel, and S. Shinohara, “Review on unidirectional light emission from ultralow-loss modes in deformed microdisks,” in Trends in Nano- and Micro-cavities, O. Kwon, B. Lee, and K. An, eds. (Bentham Books, 2011), pp. 109–152.

Ikeda, K. S.

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

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

S. Shinohara, S. Sunada, T. Harayama, and K. S. Ikeda, “Mode expansion description of stadium-cavity laser dynamics,” Phys. Rev. E 71, 036203 (2005).
[Crossref]

S. Sunada, T. Harayama, and K. S. Ikeda, “Nonlinear whispering-gallery modes in a microellipse cavity,” Opt. Lett. 29, 718–720 (2004).
[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]

Jiang, X.-F.

X.-F. Jiang, C.-L. Zou, L. Wang, Q. Gong, and Y.-F. Xiao, “Whispering-gallery microcavities with unidirectional laser emission,” Laser Photon. Rev. 10, 40–61 (2016).
[Crossref]

Johnson, N. M.

G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett. 83, 1710–1712 (2003).
[Crossref]

Kim, C.-M.

T.-Y. Kwon, S.-Y. Lee, M. S. Kurdoglyan, S. Rim, C.-M. Kim, and Y.-J. Park, “Lasing modes in a spiral-shaped dielectric microcavity,” Opt. Lett. 31, 1250–1252 (2006).
[Crossref]

S.-Y. Lee, J.-W. Ryu, T.-Y. Kwon, S. Rim, and C.-M. Kim, “Scarred resonances and steady probability distribution in a chaotic microcavity,” Phys. Rev. A 72, 061801(R) (2005).
[Crossref]

Kim, S. W.

J. Yang, S.-B. Lee, S. Moon, S.-Y. Lee, S. W. Kim, T. T. A. Dao, J.-H. Lee, and K. An, “Pump-induced dynamical tunneling in a deformed microcavity laser,” Phys. Rev. Lett. 104, 243601 (2010).
[Crossref]

Kneissl, M.

G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett. 83, 1710–1712 (2003).
[Crossref]

Kurdoglyan, M. S.

Kwon, T.-Y.

T.-Y. Kwon, S.-Y. Lee, M. S. Kurdoglyan, S. Rim, C.-M. Kim, and Y.-J. Park, “Lasing modes in a spiral-shaped dielectric microcavity,” Opt. Lett. 31, 1250–1252 (2006).
[Crossref]

S.-Y. Lee, J.-W. Ryu, T.-Y. Kwon, S. Rim, and C.-M. Kim, “Scarred resonances and steady probability distribution in a chaotic microcavity,” Phys. Rev. A 72, 061801(R) (2005).
[Crossref]

Lee, J.-H.

J. Yang, S.-B. Lee, S. Moon, S.-Y. Lee, S. W. Kim, T. T. A. Dao, J.-H. Lee, and K. An, “Pump-induced dynamical tunneling in a deformed microcavity laser,” Phys. Rev. Lett. 104, 243601 (2010).
[Crossref]

Lee, S.-B.

J. Yang, S.-B. Lee, S. Moon, S.-Y. Lee, S. W. Kim, T. T. A. Dao, J.-H. Lee, and K. An, “Pump-induced dynamical tunneling in a deformed microcavity laser,” Phys. Rev. Lett. 104, 243601 (2010).
[Crossref]

Lee, S.-Y.

J. Yang, S.-B. Lee, S. Moon, S.-Y. Lee, S. W. Kim, T. T. A. Dao, J.-H. Lee, and K. An, “Pump-induced dynamical tunneling in a deformed microcavity laser,” Phys. Rev. Lett. 104, 243601 (2010).
[Crossref]

T.-Y. Kwon, S.-Y. Lee, M. S. Kurdoglyan, S. Rim, C.-M. Kim, and Y.-J. Park, “Lasing modes in a spiral-shaped dielectric microcavity,” Opt. Lett. 31, 1250–1252 (2006).
[Crossref]

S.-Y. Lee, J.-W. Ryu, T.-Y. Kwon, S. Rim, and C.-M. Kim, “Scarred resonances and steady probability distribution in a chaotic microcavity,” Phys. Rev. A 72, 061801(R) (2005).
[Crossref]

Liew, S. F.

S. F. Liew, L. Ge, B. Redding, G. S. Solomon, and H. Cao, “Pump-controlled modal interactions in microdisk lasers,” Phys. Rev. A 91, 043828 (2015).
[Crossref]

Loudon, R.

R. Loudon, The Quantum Theory of Light (Oxford University, 2000).

Malik, O.

N. L. Aung, L. Ge, O. Malik, H. E. Türeci, and C. F. Gmachl, “Threshold current reduction and directional emission of deformed microdisk lasers via spatially selective electrical pumping,” Appl. Phys. Lett. 107, 151106 (2015).
[Crossref]

L. Ge, O. Malik, and H. E. Türeci, “Enhancement of laser power-efficiency by control of spatial hole burning interactions,” Nat. Photonics 8, 871–875 (2014).
[Crossref]

Moon, S.

J. Yang, S.-B. Lee, S. Moon, S.-Y. Lee, S. W. Kim, T. T. A. Dao, J.-H. Lee, and K. An, “Pump-induced dynamical tunneling in a deformed microcavity laser,” Phys. Rev. Lett. 104, 243601 (2010).
[Crossref]

Narimanov, E. E.

Nishimura, T.

Nöckel, J. U.

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

Park, Y.-J.

Podolskiy, V. A.

Redding, B.

S. F. Liew, L. Ge, B. Redding, G. S. Solomon, and H. Cao, “Pump-controlled modal interactions in microdisk lasers,” Phys. Rev. A 91, 043828 (2015).
[Crossref]

Rex, N. B.

Rim, S.

T.-Y. Kwon, S.-Y. Lee, M. S. Kurdoglyan, S. Rim, C.-M. Kim, and Y.-J. Park, “Lasing modes in a spiral-shaped dielectric microcavity,” Opt. Lett. 31, 1250–1252 (2006).
[Crossref]

S.-Y. Lee, J.-W. Ryu, T.-Y. Kwon, S. Rim, and C.-M. Kim, “Scarred resonances and steady probability distribution in a chaotic microcavity,” Phys. Rev. A 72, 061801(R) (2005).
[Crossref]

Ryu, J.-W.

S.-Y. Lee, J.-W. Ryu, T.-Y. Kwon, S. Rim, and C.-M. Kim, “Scarred resonances and steady probability distribution in a chaotic microcavity,” Phys. Rev. A 72, 061801(R) (2005).
[Crossref]

Sasaki, T.

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]

Schomerus, H.

M. Hentschel, H. Schomerus, and R. Schubert, “Husimi functions at dielectric interfaces: Inside–outside duality for optical systems and beyond,” Europhys. Lett. 62, 636–642 (2003).
[Crossref]

Schubert, R.

M. Hentschel, H. Schomerus, and R. Schubert, “Husimi functions at dielectric interfaces: Inside–outside duality for optical systems and beyond,” Europhys. Lett. 62, 636–642 (2003).
[Crossref]

Schwefel, H. G. L.

Shinohara, S.

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]

S. Shinohara, T. Harayama, H. E. Türeci, and A. D. Stone, “Ray-wave correspondence in the nonlinear description of stadium-cavity lasers,” Phys. Rev. A 74, 033820 (2006).
[Crossref]

S. Shinohara, S. Sunada, T. Harayama, and K. S. Ikeda, “Mode expansion description of stadium-cavity laser dynamics,” Phys. Rev. E 71, 036203 (2005).
[Crossref]

J. Wiersig, J. Unterhinninghofen, Q. H. Song, H. Cao, M. Hentschel, and S. Shinohara, “Review on unidirectional light emission from ultralow-loss modes in deformed microdisks,” in Trends in Nano- and Micro-cavities, O. Kwon, B. Lee, and K. An, eds. (Bentham Books, 2011), pp. 109–152.

Solomon, G. S.

S. F. Liew, L. Ge, B. Redding, G. S. Solomon, and H. Cao, “Pump-controlled modal interactions in microdisk lasers,” Phys. Rev. A 91, 043828 (2015).
[Crossref]

Song, Q. H.

J. Wiersig, J. Unterhinninghofen, Q. H. Song, H. Cao, M. Hentschel, and S. Shinohara, “Review on unidirectional light emission from ultralow-loss modes in deformed microdisks,” in Trends in Nano- and Micro-cavities, O. Kwon, B. Lee, and K. An, eds. (Bentham Books, 2011), pp. 109–152.

Stone, A. D.

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

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]

S. Shinohara, T. Harayama, H. E. Türeci, and A. D. Stone, “Ray-wave correspondence in the nonlinear description of stadium-cavity lasers,” Phys. Rev. A 74, 033820 (2006).
[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]

G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett. 83, 1710–1712 (2003).
[Crossref]

T. E. Tureci, H. G. L. Schwefel, A. D. Stone, and E. E. Narimanov, “Gaussian-optical approach to stable periodic orbit resonances of partially chaotic dielectric micro-cavities,” Opt. Express 10, 752–776 (2002).
[Crossref]

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

H. G. L. Schwefel, H. E. Tureci, A. D. Stone, and R. K. Chang, “Progress in asymmetric resonant cavities: Using shape as a design parameter in dielectric microcavity lasers,” in Optical Microcavities, K. Vahala, ed. (World Scientific, 2004), pp. 415–495.

Sunada, S.

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

S. Shinohara, S. Sunada, T. Harayama, and K. S. Ikeda, “Mode expansion description of stadium-cavity laser dynamics,” Phys. Rev. E 71, 036203 (2005).
[Crossref]

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

S. Sunada, T. Harayama, and K. S. Ikeda, “Nonlinear whispering-gallery modes in a microellipse cavity,” Opt. Lett. 29, 718–720 (2004).
[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]

Tanaka, T.

M. Choi, T. Tanaka, T. Fukushima, and T. Harayama, “Control of directional emission in quasistadium microcavity laser diodes with two electrodes,” Appl. Phys. Lett. 88, 211110 (2006).
[Crossref]

Tureci, H. E.

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]

G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett. 83, 1710–1712 (2003).
[Crossref]

H. G. L. Schwefel, H. E. Tureci, A. D. Stone, and R. K. Chang, “Progress in asymmetric resonant cavities: Using shape as a design parameter in dielectric microcavity lasers,” in Optical Microcavities, K. Vahala, ed. (World Scientific, 2004), pp. 415–495.

Tureci, T. E.

Türeci, H. E.

N. L. Aung, L. Ge, O. Malik, H. E. Türeci, and C. F. Gmachl, “Threshold current reduction and directional emission of deformed microdisk lasers via spatially selective electrical pumping,” Appl. Phys. Lett. 107, 151106 (2015).
[Crossref]

L. Ge, O. Malik, and H. E. Türeci, “Enhancement of laser power-efficiency by control of spatial hole burning interactions,” Nat. Photonics 8, 871–875 (2014).
[Crossref]

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]

S. Shinohara, T. Harayama, H. E. Türeci, and A. D. Stone, “Ray-wave correspondence in the nonlinear description of stadium-cavity lasers,” Phys. Rev. A 74, 033820 (2006).
[Crossref]

Unterhinninghofen, J.

J. Wiersig, J. Unterhinninghofen, Q. H. Song, H. Cao, M. Hentschel, and S. Shinohara, “Review on unidirectional light emission from ultralow-loss modes in deformed microdisks,” in Trends in Nano- and Micro-cavities, O. Kwon, B. Lee, and K. An, eds. (Bentham Books, 2011), pp. 109–152.

Vaccaro, P. O.

Wang, L.

X.-F. Jiang, C.-L. Zou, L. Wang, Q. Gong, and Y.-F. Xiao, “Whispering-gallery microcavities with unidirectional laser emission,” Laser Photon. Rev. 10, 40–61 (2016).
[Crossref]

Wiersig, J.

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

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

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A 5, 53–60 (2003).
[Crossref]

J. Wiersig, J. Unterhinninghofen, Q. H. Song, H. Cao, M. Hentschel, and S. Shinohara, “Review on unidirectional light emission from ultralow-loss modes in deformed microdisks,” in Trends in Nano- and Micro-cavities, O. Kwon, B. Lee, and K. An, eds. (Bentham Books, 2011), pp. 109–152.

Xiao, Y.-F.

X.-F. Jiang, C.-L. Zou, L. Wang, Q. Gong, and Y.-F. Xiao, “Whispering-gallery microcavities with unidirectional laser emission,” Laser Photon. Rev. 10, 40–61 (2016).
[Crossref]

Yang, J.

J. Yang, S.-B. Lee, S. Moon, S.-Y. Lee, S. W. Kim, T. T. A. Dao, J.-H. Lee, and K. An, “Pump-induced dynamical tunneling in a deformed microcavity laser,” Phys. Rev. Lett. 104, 243601 (2010).
[Crossref]

Zou, C.-L.

X.-F. Jiang, C.-L. Zou, L. Wang, Q. Gong, and Y.-F. Xiao, “Whispering-gallery microcavities with unidirectional laser emission,” Laser Photon. Rev. 10, 40–61 (2016).
[Crossref]

Zyss, J.

Appl. Phys. Lett. (3)

N. L. Aung, L. Ge, O. Malik, H. E. Türeci, and C. F. Gmachl, “Threshold current reduction and directional emission of deformed microdisk lasers via spatially selective electrical pumping,” Appl. Phys. Lett. 107, 151106 (2015).
[Crossref]

G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett. 83, 1710–1712 (2003).
[Crossref]

M. Choi, T. Tanaka, T. Fukushima, and T. Harayama, “Control of directional emission in quasistadium microcavity laser diodes with two electrodes,” Appl. Phys. Lett. 88, 211110 (2006).
[Crossref]

Europhys. Lett. (1)

M. Hentschel, H. Schomerus, and R. Schubert, “Husimi functions at dielectric interfaces: Inside–outside duality for optical systems and beyond,” Europhys. Lett. 62, 636–642 (2003).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

T. Fukushima and T. Harayama, “Stadium and quasi-stadium laser diodes,” IEEE J. Sel. Top. Quantum Electron. 10, 1039–1051 (2004).
[Crossref]

J. Opt. A (1)

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A 5, 53–60 (2003).
[Crossref]

J. Opt. Soc. Am. B (1)

Laser Photon. Rev. (2)

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

X.-F. Jiang, C.-L. Zou, L. Wang, Q. Gong, and Y.-F. Xiao, “Whispering-gallery microcavities with unidirectional laser emission,” Laser Photon. Rev. 10, 40–61 (2016).
[Crossref]

Nat. Photonics (1)

L. Ge, O. Malik, and H. E. Türeci, “Enhancement of laser power-efficiency by control of spatial hole burning interactions,” Nat. Photonics 8, 871–875 (2014).
[Crossref]

Nature (1)

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

Opt. Express (2)

Opt. Lett. (4)

Phys. Rev. A (6)

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]

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

S.-Y. Lee, J.-W. Ryu, T.-Y. Kwon, S. Rim, and C.-M. Kim, “Scarred resonances and steady probability distribution in a chaotic microcavity,” Phys. Rev. A 72, 061801(R) (2005).
[Crossref]

S. Shinohara, T. Harayama, H. E. Türeci, and A. D. Stone, “Ray-wave correspondence in the nonlinear description of stadium-cavity lasers,” Phys. Rev. A 74, 033820 (2006).
[Crossref]

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

S. F. Liew, L. Ge, B. Redding, G. S. Solomon, and H. Cao, “Pump-controlled modal interactions in microdisk lasers,” Phys. Rev. A 91, 043828 (2015).
[Crossref]

Phys. Rev. E (2)

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

S. Shinohara, S. Sunada, T. Harayama, and K. S. Ikeda, “Mode expansion description of stadium-cavity laser dynamics,” Phys. Rev. E 71, 036203 (2005).
[Crossref]

Phys. Rev. Lett. (5)

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]

L. I. Deych, “Effects of spatial nonuniformity on laser dynamics,” Phys. Rev. Lett. 95, 043902 (2005).
[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]

J. Yang, S.-B. Lee, S. Moon, S.-Y. Lee, S. W. Kim, T. T. A. Dao, J.-H. Lee, and K. An, “Pump-induced dynamical tunneling in a deformed microcavity laser,” Phys. Rev. Lett. 104, 243601 (2010).
[Crossref]

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

Rev. Mod. Phys. (1)

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

Other (3)

H. G. L. Schwefel, H. E. Tureci, A. D. Stone, and R. K. Chang, “Progress in asymmetric resonant cavities: Using shape as a design parameter in dielectric microcavity lasers,” in Optical Microcavities, K. Vahala, ed. (World Scientific, 2004), pp. 415–495.

J. Wiersig, J. Unterhinninghofen, Q. H. Song, H. Cao, M. Hentschel, and S. Shinohara, “Review on unidirectional light emission from ultralow-loss modes in deformed microdisks,” in Trends in Nano- and Micro-cavities, O. Kwon, B. Lee, and K. An, eds. (Bentham Books, 2011), pp. 109–152.

R. Loudon, The Quantum Theory of Light (Oxford University, 2000).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1.
Fig. 1. (a) Double-triangle orbits in the quadrupole-deformed cavity. (b) Spatial selective pumping (yellow region) along the upward-pointing triangle orbit (red lines).
Fig. 2.
Fig. 2. Intensity distributions of the resonant modes for a passive quadrupole-deformed cavity with refractive index 3.3. The modes are four nearly degenerate modes associated with the double-triangle orbits. The double-triangle orbits (red and green lines) are superposed, and the intensities outside the cavity are plotted in log scale. (a) Even–even mode with scaled frequency Reω/ω0=1.0008278. (b) Even–odd mode with Reω/ω0=0.999085. (c) Odd–even mode with Reω/ω0=0.999075. (d) Odd–odd mode with Reω/ω0=1.0008277.
Fig. 3.
Fig. 3. Phase space of the ray dynamics for the quadrupole-deformed cavity. The islands of stability corresponding to the upward-pointing and downward-pointing triangle orbits are indicated by red and green points, respectively. The critical line for total internal reflection is indicated by a line at sinϕ=1/3.3. Husimi distribution for the eo mode shown in Fig. 2(b) is superposed.
Fig. 4.
Fig. 4. Distribution of the complex eigenfrequencies ω scaled by ω0, where ω0 is the gain center parameter. The four nearly degenerate modes associated with the double-triangle orbits are encircled by a green circle (the ee and oo modes are almost on top of each other, and so are the eo and oe modes). The modes indicated by filled circles (•) have positive linear gain [i.e., satisfying Eq. (9)] for the selective pumping with W=1.0×103, whereas those indicated by crosses (×) do not satisfy Eq. (9).
Fig. 5.
Fig. 5. Electric field intensity distributions. (a) An initial condition for the MB model simulation. (b) Time-averaged pattern of the stationary lasing state of the MB model for the selective pumping case with W=1.0×103. The intensity outside the cavity is plotted in log scale. The boundary of the pumped area is indicated by yellow lines.
Fig. 6.
Fig. 6. Results of the MB model simulation for the selective pumping case with W=1.0×103. (a) Time evolution of the total light intensity inside the cavity. (b) Power spectrum of the electric field for the stationary lasing regime. The peak frequency is around ω/ω0=0.9988.
Fig. 7.
Fig. 7. Intensity distributions of the superpositions of the resonant-mode wave functions. The triangle orbit is indicated by red lines, and the intensities outside the cavities are plotted in log scale. (a) ξ=ψee+ψeo. (b) η=ψoe+ψoo. (c) ΨCW=ξ+iη=(ψee+ψeo)+i(ψoe+ψoo). (d) ΨCCW=ξiη=(ψee+ψeo)i(ψoe+ψoo).
Fig. 8.
Fig. 8. Results of the MB model simulation for the uniform pumping case with W=3.0×104. (a) Time evolution of the total light intensity inside the cavity. (b) Power spectrum of the electric field for the stationary lasing regime. The frequencies of the primary and secondary peaks are ω/ω00.9990 and ω/ω01.0012, respectively.
Fig. 9.
Fig. 9. Time-averaged pattern of the stationary lasing state of the MB model for the uniform pumping case with W=3.0×104. The intensity outside the cavity is plotted in log scale.
Fig. 10.
Fig. 10. Intensity distribution of the superpositions of the resonant-mode wave functions. (a) ψeo+iψoe. (b) ψee+iψoo. The intensities outside the cavity are plotted in log scale.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

2t2(Ez+4πεPz)=c2n22Ez2βtEz,
Pz=N(ρ+ρ*)κ,
tρ=iω0ρiκWEzγρ,
tW=2iκEz(ρρ*)γ(WW),
r(θ)=r0[1+ϵcos(2θ)],
(2+n2ω2c2)ψ(x,y)=0,
ψab(x,y)=aψab(x,y),
ψab(x,y)=bψab(x,y),
2πNκ2Wn2γReωs(Reωsω0)2+γ2>Imωs+β,
W=WDdxdy|ψ(x,y)|2Θ(x,y)Ddxdy|ψ(x,y)|2,
ξψee+ψeo,
ηψoe+ψoo.
ΨCWξ+iη=(ψee+ψeo)+i(ψoe+ψoo),
ΨCCWξiη=(ψee+ψeo)i(ψoe+ψoo).

Metrics