Correcting Ray Optics at Curved Dielectric Microresonator Interfaces: Phase-Space Unification of Fresnel Filtering and the Goos-Hänchen Shift

Abstract

We develop an amended ray optics description for reflection at the curved dielectric interfaces of optical microresonators which improves the agreement with wave optics by about one order of magnitude. The corrections are separated into two contributions of similar magnitude, corresponding to ray displacement in independent quantum phase space directions, which can be identified with Fresnel filtering and the Goos-Hänchen shift, respectively. Hence we unify two effects which only have been studied separately in the past.PACS numbers: 05.45. Mt, 42.55.Sa Over the recent years it has become feasible to design optical microresonators that confine photons by means of dielectric interfaces into a small spatial region not larger than a few micrometers [1,2,3]. Two promising lines of research are the amplification of photons by stimulated emission in active media, which yields lasing action [1,2,3,4,5,6,7,8,9,10,11], and the generation and trapping of single photons which can be used as carriers of quantum information [12]. These applications require integration of several components and interfacing with electronics, which are best realized in twodimensional resonator geometries where the main in-and out-coupling directions are confined to a plane, and can be selected via the (asymmetric) resonator geometry. Furthermore, because of the requirements of mode selection, these applications favor microresonators of mesoscopic dimensions, with size parameters kL = O(100) − O(1000) (where L is the linear size, k = 2π/λ is the wavenumber and λ is the wavelength) which quickly puts these systems out of the reach of numerical simulations. On the other hand, ray-optics predictions of the intricate resonator modes [4,6,9,13,14,15,16,17] can deviate substantially from experimental observations [5,7] and theoretical predictions [5,11,15,16].The purpose of this paper is to develop an amended ray optics (ARO) description which still idealizes beams as rays, but incorporates corrections of the origin and propagation direction of the reflected ray. We identify these corrections by utilizing quantum-phase space representations of the incident and reflected beam [18] and relate them to the recently discovered Fresnel filtering effect [19] and the long-known Goos-Hänchen shift [20]. These two effects have only been discussed separately in the past (for applications to microresonators see, e.g., Refs. [5,11,21,22]), and their complementary nature has not been realized. Moreover, their uniform analysis for all angles of incidence is known to pose considerable technical challenges [19,23,24,25]. In the phasespace representation, the Fresnel filtering and Goos-Hänchen corrections are simply determined by the position of maximal phase-space density. For the prototypical case of a Gaussian • is close to the critical angle χ ′ = 41.8• . Conventional ray optics predicts that the beam is specularly reflected at the point of incidence. In this paper we use phase-space representations to obtain a more accurate reflection law, which accounts for (i...

Inferring periodic orbits from spectra of simply shaped microlasers

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“…with ξ = 0 for odd p and ξ = 1/2 for even p. x ′ and y ′ in (26) are coordinates of the point symmetric of (x, y) with respect to the inversion on the edge of the pentagon. In the coordinate system when the pentagon edge passes through the origin (as in Fig.…”

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“…The GHS is a lateral shift of totally reflected beams along the optical interface [48], i.e., the points of incidence and reflection do not coincide. In the case of the FF [49][50][51], partial waves with angles of incidence below the critical angle for total internal reflection are (partially) refracted out of the cavity, leading to a shift ∆χ of the partial waves between the incident and outgoing angles -i.e., a violation of Snell's law. In the short-wavelength limit λ → 0 the GHS and FF disappear leading to the standard ray dynamics of geometric optics.…”

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“…Inspired by the doubtless advantages of the ray model, such as its easy implementationand its conceptual success, our objective will be to identify semiclassical correctionssuch that the resulting (adjusted) ray model can quantitatively better capturethe wave properties of the system. Following the last example in the previous Section where deviations between the ray and wave behaviour are clearly visible in a single, near-critical reflection, we analyze this situation in some more detail [22]. Our means of choice are Husimi functions at dielectric interfaces.…”

“…5(b). Their magnitude depends on the wavenumber and is typically several degrees [22], with the Goos-Hänchen correction being the larger.…”