Perturbation theory for resonances in wave-chaotic resonant cavities

Physica E 9 (2001) 560–563

www.elsevier.nl/locate/physe

Perturbation theory for resonances in wave-chaotic resonant cavities Gregor Hackenbroich ∗ Universitat GH Essen, Fachbereich 7, 45117 Essen, Germany

Abstract We develop a perturbation theory for the lifetime and emission intensity for narrow resonances in asymmetric resonant cavities. In the semiclassical approximation both the inverse lifetime and the emission intensity I () are expressed as sums over the contributions of rays which escape the resonator by refraction. The theory describes the highly directional emission from “bow-tie” modes found in recent experiments on deformed semiconductor microlasers. ? 2001 Elsevier Science B.V. All rights reserved. PACS: 42.25.−p; 05.45.+b; 03.65.Sq Keywords: Microcavity lasers; Wave chaos; Semiclassical theory

1. Introduction Dielectric microcavities are used as high-Q resonators for microlasers and optical spectroscopy [1,2]. The lasing process in such systems is usually carried by whispering gallery modes that propagate close to the boundary of the resonator and have very long lifetimes due to total internal re ection. Recently, experiments [3] on cylindrical semiconductor microlasers with high refractive index for the rst time demonstrated emission from bow-tie modes. These modes are localized at stable periodic orbits shaped as a bow-tie; the lasing emission from these modes was found to be both high-power and highly directional. Ray optics calculations [3] demonstrated that bow-tie modes can ∗

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only exist if the shape of the resonator is deformed from circular or spherical symmetry. The dynamics in such deformed resonators is generally mixed and no analytical results were known except for very small deformations which can then be treated perturbatively [4]. Some time ago, Nockel and Stone [5 –8] noted that such asymmetric resonant cavities (ARCs) are equivalent to quantum billiards with refractive escape and a ray-optics model was developed for the lifetime and emission pattern. Although quite successful for the emission from whispering gallery modes in low-index resonators, the ray model could not account for the bow-tie mode lasing in the high-index microcavity laser. In a recent paper [9] the emission from long-lived states in deformed microlasers was derived perturbatively using an expansion in the inverse resonance lifetime. In zero-order this expansion de nes a closed billiard, and the resonance condition is replaced by an

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G. Hackenbroich / Physica E 9 (2001) 560–563

hermitian eigenvalue problem. It turned out that the formulation [9] of the eigenvalue problem suers from spurious solutions that correspond to bound states satisfying Neumann boundary conditions at the edge of the dielectric. It is the purpose of the present paper to present a new formulation of the perturbation theory that avoids any spurious solutions. As in the paper [9], we nd that the resulting formulae for the emission directionality and the lifetime have a simple physical interpretation in the semiclassical limit nkR0 ¿ 1 (k is the wavenumber of the radiation, R0 is the average radius of the ARC and n the index of refraction). In this limit, the emission intensity can be expressed in terms of refractively escaping rays. For the case of modes associated with stable orbits of the closed system a full analytic semiclassical solution can be found which depends only on the properties of the classical orbit (e.g. its period and stability). We note that even though our work is motivated by studies of optical resonators, most of our theoretical results apply mutatis mutandis to quantum Hamiltonians. 2. Resonance condition We consider a cylindrical dielectric resonator with a convex cross-section of general non-circular shape. The cross-section is parametrized by the distance R() between the origin and a point on the boundary in the -direction. The cylinder is surrounded by air or vacuum. We focus on the transverse-magnetic (TM) modes of the resonator with both the magnetic eld and the propagation vector k perpendicular to the cylinder axis. These modes are of prime interest in the quantum cascade microdisk lasers [3] where selection rules allow light emission only in the TM polarization. For TM modes, the electric eld points in the direction of the cylinder axis, and Maxwell’s equations imply that both the electric eld and its derivative are continous at the boundary. The quantities of main interest for the lasing process are the quasibound states of the resonator. Such states correspond to solutions of the wave equation with no incoming wave. These solutions can only exist for complex frequency ! = nck. The imaginary part = −Im k is the resonance width. Using cylinder coordinates with ez pointing in the direction of the cylinder axis, we nd the quasibound states by expanding

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the electric eld E(r; t) = E(r; )e−ickt ez in a complete set of angular momentum eigenfunctions eim , ∞ P im Am (kr)eim ; (1) E(r; ) = m=−∞

with Am =

(

m Jm (nkr)

for r6R();

m Hm+ (kr)

for r ¿ R():

(2)

Here, Jm denotes a Bessel function and Hm± Hankel functions. The Neumann functions must be excluded from the expansion of the internal eld since they diverge at r = 0. Note that the external eld has only outgoing waves. Solution (1) can be thought of as the limiting case of a wave packet launched in the resonator and decaying to in nity. This is the appropriate description of an emission process such as lasing where the resonant state is generated in the resonator. The resonance condition is found by matching the internal and external elds and their normal derivatives at the boundary of the dielectric. To combine the two matching equations to a single resonance condition, we multiply the matching equation for E by i−‘ H‘− (kR()) exp(−i‘) and the matching equation for the normal derivative @n E by i−‘ exp(−i‘). Integration along the boundary introduces a set of four matrices with the matrix elements Z im−‘ 2 d Cm ()e−i(‘−m) ; (3) (C)‘m = 2 0 m−‘  ‘m = i (CC) 2

Z 0

2

d C‘ ()C m ()e−i(‘−m) ; (4)

where Cm and C m stand for the appropriate Bessel (Hankel) functions or their normal derivatives with arguments nkR() (kR()). For the normal derivative, we use the notation @n [Cm eim ] ≡ Cm0 eim below. Eliminating the coecients m , the resonance condition takes the form of an eigenvalue problem   1 (5) (J 0 ) − (H +0 )(H − H + )−1 (H − J ) |i = 0; n where the coecient vector |i is de ned by expansion coecients (2) of the internal eld. In contrast to the resonance condition derived in Ref. [9] which involved a pair k, k of wavenumbers, the left-hand side of Eq. (5) only depends on the wave number k. The roots of Eq. (5) are the resonant k’s

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of the dielectric. Note that the refractive index enters in this equation both in the argument of the Bessel functions and in the factor 1=n. In the limit of large n → ∞, we can neglect the second term in square brackets and Eq. (5) reduces to the quantization condition for a closed billiard with Neumann boundary conditions. The appearance of bound states re ects the total internal re ection at the boundary of the large-n dielectric. For nite n, we must keep the second term which accounts for the escape in the asymptotic region.

3. Quantum perturbation theory The basic idea of the perturbation theory is to decompose the exact resonance condition into one part describing a closed billiard problem and a second part accounting for the escape. For a narrow resonance with a width much less than the mean resonance spacing,
, the part describing escape can be treated perturbatively. To formalize this idea and identify the closed billiard problem, we note that the resonance condition (5) is equivalent to the integro-dierential equation: Z 1 ˜ ˜ =0 d ˜ K(; )E( ) (6) @n E() − n for the electric eld E() ≡ E(R(); ) at the boundary. We decompose the kernel ˜ = P H +0 ()(H − H + )−1 K(; ) m m‘ m;‘

˜ im−i‘˜ ×H‘− ()e

(7)

into its real and imaginary part, K = K0 + iV , and de ne the “unperturbed” problem by a boundary condition as in Eq. (6) but with K replaced by K0 . The new boundary condition is invariant under complex conjugation. Therefore, the solutions E (0) of the unperturbed problem can be chosen real and yield a vanishing current through the boundary. One easily shows that solutions only exist for real k0 . The unperturbed problem thus represents a closed billiard, however one which incorporates the eect of nite refractive index in the boundary conditions.

The width and the far- eld emission intensity I () of narrow resonances are found by a standard expansion in V . To rst order in V , this yields =

h(0) |V|(0) i h(0) |(@H0 =@k)|(0) i

(8)

and

2 P 0 − (0) ei‘  (H − H + )−1 (H J )  I () = ; 0 ‘m m ‘‘ m;‘;‘0

(9)

where we introduced the matrices H0 ≡ (J 0 ) − (1=n)K0 and V describing the unperturbed problem and the perturbation, respectively, and the eigenvectors |(0) i of H0 with eigenvalue 0. The respective matrix elements are given by V‘m ≡ R ˜ ˜ im−i‘ ˜ m ()e and similarly (with d d ˜ V (; )J ˜ for the matrix ele˜ replaced by K0 (; )) V (; ) ments of K0 . The results of (8) and (9) reduce the calculation of narrow resonances to the quantization of a closed billiard problem. We note that Eqs. (8) and (9) have previously been derived in Ref. [9]. However, the unperturbed problem de ned in the present paper and therefore the eigenvectors |(0) i dier from the results obtained earlier. The main advantage of the present formulation is that it yields no spurious solutions. Numerically, the lifetimes and emission intensities obtained with both formulations do not dier signi cantly for modest deformations of the boundary. In the limit of a circular dielectric, it can be shown that both formulations yield identical results. Further progress is possible by taking advantage of the semiclassical limit. In this limit, the wavelength of the radiation is assumed much smaller than the average radius of the resonator nkR0 1. This condition is satis ed essentially in all optical experiments. In this limit Hankel and Bessel functions may be well represented by the “approximation by tangents” [10] and the matrix elements and angular momentum sums in (8) and (9) can be evaluated in the stationary phase approximation. Evaluating Eq. (9) in the semiclassical limit, we nd that (i) an internal angular momentum m only couples to the angular momentum l which allows the corresponding ray to satisfy Snell’s law (either in transmission or re ection) and (ii) that this ray with angular momentum ‘0 must be emitted into angle 

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Fig. 1. The bow-tie resonance and its width. (a) Husimi projection (color scale – red denotes high intensity) of bow-tie mode at n = 1:5, nkR = 102:0, for quadrupolar defomation  = 0:168 in (; sin ) coordinates, superimposed on the corresponding Poincare surface of section. Inset: Schematic of bow-tie periodic orbit. (b) Comparison of the exact width of the bow-tie resonance (circles and diamonds) with the semiclassical (green line) expression and classical (blue line) prediction. Black circles and red diamond correspond to bow-tie resonances close to nkR0 = 45:0 and nkR0 = 102:0, respectively.

in the far- eld. Using the semiclassical theory developed in Refs. [11,12], we derived a closed analytical result for eigenstates localized at the stable islands of the mixed phase space of an ARC. Below we show results for modes localized at the stable bow-tie periodic orbit depicted in Fig. 1a. In Fig. 1b we plot the width obtained from the semiclassical calculation vs. the refractive index n. The results are compared both with exact data points and a classical result based on the Fresnel re ection coecients for at the bounce points at the boundary. The simple classical model gives the unphysical result = 0 once the bow-tie orbit will be totally internally re ected, whereas our semiclassical model goes beyond the simple ray-optics model and continues to decrease smoothly in good agreement with the numerical data. 4. Conclusion In summary, we have presented a perturbation theory for resonance lifetimes and emission intensities in nonintegrable dielectric resonators. The theory is free from spurious solutions that plagued a previous approach to the problem. The theory has a simple physical interpretation in terms of refractive emission, and gives nontrivial predictions for the lifetimes and emission patterns in asymmetric resonant cavities.

Acknowledgements I gratefully acknowledge support of the Deutsche Forschungsgemeinschaft.

References [1] Y. Yamamota, R.E. Slusher, Phys. Today 46 (1993) 66. [2] M.L. Gorodetsky, V. Ilchenko, J. Opt. Soc. Amer. B 16 (1999) 147. [3] C. Gmachl, F. Capasso, E.E. Narimanov, J.U. Nockel, A.D. Stone, J. Faist, D.L. Sivco, A.Y. Cho, Science 280 (1998) 1493. [4] H.M. Lai et al., Phys. Rev. A 41 (1990) 5187. [5] J.U. Nockel, A.D. Stone, R.K. Chang, Opt. Lett. 19 (1994) 1693. [6] A. Mekis, J.U. Nockel, G. Chen, A.D. Stone, R.K. Chang, Phys. Rev. Lett. 75 (1995) 2682. [7] J.U. Nockel, A.D. Stone, G. Chen, H. Grossman, R.K. Chang, Opt. Lett. 21 (1996) 1609. [8] J.U. Nockel, A.D. Stone, Nature 385 (1997) 47. [9] E.E. Narimanov, G. Hackenbroich, P. Jacquod, A.D. Stone, Phys. Rev. Lett. 83 (1999) 4991. [10] M. A. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972. [11] E.E. Narimanov, A.D. Stone, G.S. Boebinger, Phys. Rev. Lett. 80 (1998) 4024. [12] E.E. Narimanov, A.D. Stone, Phys. D 131 (1999) 220.