Fresnel filtering of Gaussian beams in microcavitiesSusumu Shinohara,1,* Takahisa Harayama,2 and Takehiro Fukushima3

1Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Strasse 38, D-01187 Dresden, Germany2NTT Communication Science Laboratories, NTT Corporation, 2-4 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-0237, Japan3Department of Communication Engineering, Okayama Prefectural University, 111 Kuboki, Soja, Okayama 719-1197, Japan

*Corresponding author: susumu@pks.mpg.de

Received December 17, 2010; revised February 7, 2011; accepted February 21, 2011;posted February 23, 2011 (Doc. ID 139800); published March 15, 2011

We study the output from the modes described by the superposition of Gaussian beams confined in the quasi-stadium microcavities. We experimentally observe the deviation from Snell’s law in the output when the incidentangle of the Gaussian beam at the cavity interface is near the critical angle for total internal reflection, providingdirect experimental evidence on the Fresnel filtering. The theory of the Fresnel filtering for a planar interfacequalitatively reproduces experimental data, and a discussion is given on small deviation between the measureddata and the theory. © 2011 Optical Society of AmericaOCIS codes: 140.2020, 140.3410, 140.4780, 230.3990, 230.5750.

When a beam with finite angular spreads is incident to adielectric interface with incident angle near the criticalangle for total internal reflection, it is theoretically shownthat its refracted beam exhibits deviation from Snell’slaw [1,2]. Because of the dependence of Fresnel reflec-tion on incident angle, some angular components aretotally internally reflected, while others far from thecritical incident angle are refracted out, resulting in anangular shift in the far-field emission pattern. This effect,named Fresnel filtering (FF) [1,2], implies the correctionof ray optics in addition to the Goos–Hänchen effect [3]and has recently attracted attention in interpreting emis-sion patterns from microcavities [4–9].In experiments on GaN microlasers with quadrupolar

deformed cavities [4], emission patterns were explainedby the FF effect on the output from “scarred modes,”which are associated with the unstable periodic ray orbit.In case of the scarred modes, however, general proper-ties on the mode profile are not well understood, makingit difficult to quantitatively estimate the FF effect. Con-sidering that the original theory was established forGaussian beams [1,2], it is desirable to experimentallystudy the FF effect for the modes that are well describedby the superposition of Gaussian beams. In this Letter,we study such modes for the quasi-stadium cavities,where Gaussian mode characteristics can be estimatedby the Gaussian-optic theory [10]. We experimentallydemonstrate the deviation from Snell’s law and providean analysis based on the FF theory.Figure 1(a) shows the geometry of the quasi-stadium

cavity [11]. In this Letter, we study microlasers withquasi-stadium shape with cavity widths W ¼ 100 μm,150 μm, 190 μm, and 200 μm, while we fix the cavitylength L ¼ 600 μm. The radius R of curvature of bothcurved edges is fixed as R ¼ L (i.e., confocal resonatorcondition). The devices are fabricated by using ametal-organic-chemical-vapor-deposition-grown gradient-index, separate-confinement-heterostructure, single-quantum-well GaAs=AlxGa1−xAs structure and areactive-ion-etching technique [11].The confocal resonator condition yields that the

Fabry–Perot orbit (bouncing between the boundarypoints A and C) and the ring orbit (reflected at A, B,C, and D) are both stable. In previous studies of the

quasi-stadium microlasers, selective excitation of theFabry–Perot modes and that of the ring modes werestudied in detail [11]. The selective excitation has beensuccessfully demonstrated by patterning the “contactwindow,” where electric currents are injected, alongthe Fabry–Perot orbit or the ring orbit. In this Letter,we consider the devices with the contact windowpatterned along the ring orbit with 5 μm width (see [11]for the details on the device structure and fabricationprocess) and study output from the ring modes asdepicted in Fig. 1(b).

In the ray-optic limit, the output can be predicted bySnell’s law. We denote θi and θe the incident and therefracted angle, respectively, as illustrated in Fig. 1(a).For given L and W , the incident angle is given by

θi ¼ sin−1�

WffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 þW2

p�: ð1Þ

Fig. 1. (Color online) (a) Geometry of the quasi-stadium cavitywith L ¼ 600 μm and W ¼ 150 μm. The radius R of curvature ofboth curved edges is 600 μm. Inside the cavity, the ring orbitreflected at the boundary points A, B, C, and D is shown. (b) Re-sonant mode associated with the ring orbit with the wavelengthλ ≈ 0:856 μm calculated by the Gaussian-optic theory.

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Snell’s law implies the refracted angle θe ¼ sin−1

ðn sin θiÞ, where n is the effective refractive index ofthe cavity, calculated as 3.3 from the structure of thedevices. The ring orbit is confined by total internalreflection when the incident angle is above the criticalangle θc ¼ sin−1ð1=nÞ ≈ 17:64°. In terms of the cavitygeometry, total internal reflection of the ring orbit occursfor W > L=

ffiffiffiffiffiffiffiffiffiffiffiffiffin2

− 1p

≈ 191 μm. Below we compare experi-mentally measured emission patterns with the predic-tions by Snell’s law.In experiments, devices are tested at 25 °C using a

pulsed current with 500 ns width at 1kHz repetition.The strength of an injection current is determined sothat the peak output power exceeds 5mW. Measuredfar-field patterns for the devices with various cavitywidths are shown in Fig. 2 (solid curves). The definitionof the far-field angle ϕ is given in Fig. 2(d). The patternsare normalized so that the maximum intensity becomesunity. We indicate the prediction from Snell’s law byvertical lines.For W ¼ 100 μm (i.e., Δθ ¼ θc − θi ≈ 8:2°) and W ¼

150 μm (i.e., Δθ ≈ 3:6°), we find that the output peaksare in good agreement with the predictions fromSnell’s law. This agreement convinces us the validityof our estimate of the effective refractive index n ¼ 3:3.For W ¼ 190 μm (i.e., Δθ ≈ 0:068°), however, we seeapparent deviation of the experimental data from Snell’slaw. More strikingly, for W ¼ 200 μm (i.e., Δθ ≈ −0:80°),Snell’s law predicts total internal reflection, but theexperimental data show the output peaks aroundϕ ¼ �77°.The observed phenomenon is theoretically understood

by the FF effect. Here we present an analysis based onthe theory by Tureci and Stone [1,2]. First, we assumethat the modes associated with the ring orbit are welldescribed as the superposition of Gaussian beams. Thisis, in fact, justified by the Gaussian-optic theory in theshort-wavelength limit [10]. As the sizes of our cavitiesare quite large compared to the wavelength (i.e., nkR ≈

104), the above assumption is well satisfied. Below, wefocus on the Gaussian beam

Eiðxi; ziÞ ¼E0w0

wðziÞexp

�−

�xi

wðziÞ�

2þ inkzi

�; ð2Þ

with w2ðziÞ ¼ w20 − ið2ziÞ=ðnkÞ scattered at the boundary

point A. The definition of the coordinates ðxi; ziÞ is givenin Fig. 1(a). The beam waistw0, the distance between thebeam waist position and the interface z0, and wavenum-ber k can be estimated by applying the Gaussian-optictheory to the ring orbit. Letting ðα; βÞT the eigenvectorof the stability matrix for a segment of the ring orbit(see [10] for details), we can write w0 ¼ 1=ð ffiffiffiffiffiffi

nkp jαjÞ and

z0 ¼ −Reðαβ�Þ=jαj2.For a given W , we consider the ring mode with the

wavelength λ ≈ 0:856 μm, which corresponds to the lasingwavelength in the experiments. In order to simplifythe calculation, we employ the Dirichlet boundary condi-tion instead of the dielectric boundary condition. Thisapproximation does not affect so much the modal struc-ture inside the cavity when the wavelength is sufficientlyshort. The field intensity distribution of a calculated

mode for W ¼ 150 μm is shown in Fig. 1(b). Fromthe Gaussian-optic calculation, we found that, for thecavities with W ¼ 100–200 μm, the Gaussian beam is

Fig. 2. (Color online) Far-field emission patterns of the quasi-stadium microlasers with (a) W ¼ 100 μm, (b) W ¼ 150 μm,(c) W ¼ 190 μm, and (d) W ¼ 200 μm. The definition of thefar-field angle ϕ is given in the inset of (d). Experimental dataare plotted with solid curves, while the predictions by the FFtheory in dotted curves. The predictions by Snell’s law areindicated by vertical lines.

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characterized by nk ≈ 24 μm−1 and w0 ≈ 5 μm, and thebeam waist is located at the boundary points B and D,

i.e., z0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 þW2

p=2.

Next, we consider how the Gaussian beam is refractedat the boundary point A. In the polar coordinates ðρ;ϕÞ,the refracted electric field has the following asymptoticform in the limit kρ → ∞ [1,2]:

Eeðρ;ϕÞ ≈nkw0E0ffiffiffiffiffiffiffiffiffiffi

2ikρp tðs0ÞGðs0ÞJðϕ; s0Þeikρ; ð3Þ

where s0 ¼ s0ðϕÞ is determined by solving the equationn sinðθi þ δθiðsÞÞ ¼ sinϕ in terms of s with δθi ¼sin−1ðsÞ. tðsÞ is the Fresnel transmission coefficientfor the transverse electric polarization, GðsÞ carries theinformation on the incident Gaussian beam, and Jðϕ; sÞarises from the stationary phase approximation in deriv-ing Eq. (3). These functions are defined as follows:

tðsÞ ¼ 2n cosðΘiðsÞÞcosðΘiðsÞÞ þ n2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisin2 θc − sin2ðΘiðsÞÞ

p ; ð4Þ

GðsÞ ¼ exp

�−

�nkw0

2

�2s2 þ inkz0

ffiffiffiffiffiffiffiffiffiffiffiffi1 − s2

p �; ð5Þ

Jðϕ; sÞ ¼ cosϕffiffiffiffiffiffiffiffiffiffiffiffi1 − s2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2

− sin2 ϕp ; ð6Þ

where ΘiðsÞ ¼ θi þ δθiðsÞ. Using Eq. (3) with nk, w0, andz0 estimated from the Gaussian-optic theory, we calcu-lated theoretical far-field patterns, which are plotted withdotted curves in Fig. 2.When the incident angle θi is far from the critical angle

θc (i.e., for W ¼ 100 μm and W ¼ 150 μm), we find thatthe peak positions of the theoretical curves coincide withthe predictions by Snell’s law. However, the FF effectappears when θi is near θc. For W ¼ 190 μm, we findnoticeable deviation between the peak positions of thetheoretical curve and the prediction by Snell’s law. More-over, for W ¼ 200 μm, where Snell’s law predicts totalinternal reflection, we find the maxima of the theoreticalcurve around ϕ ¼ �82°.When θi ≪ θc, Eq. (3) predicts that the far-field peak

position is almost constant with respect to a change ofthe beam waist w0, while the peak width varies. Fittingthe experimental data for W ¼ 100 μm by Eq. (3) with w0being a free parameter, we found that the peak width isbest reproduced with w0 ≈ 5 μm. This convinces us thatour estimate of w0 from the Gaussian-optic theory isreliable.

The theoretical curves qualitatively explain the shiftsfrom Snell’s law observed in the experimental data,but they underestimate the sizes of the shifts. The devia-tion is 4:9° for W ¼ 190 μm, while 5:0° for W ¼ 200 μm,where for a double peak of W ¼ 190 μm, we comparedthe average of the two peak positions with the peak posi-tion of the theoretical curve.

For θi ≈ θc, Eq. (3) predicts that the far-field peak posi-tion starts to depend on the beam waistw0. This suggestsa possibility that a slight error in estimating w0 shifts thepeak position of Eq. (3). For the experimental data forW ¼ 190 μm and W ¼ 200 μm, we found that, in orderto explain the measured peak positions by Eq. (3), oneneeds to put w0 ≈ 2:5 μm, which deviates too much fromthe estimate from the Gaussian-optic theory. Hence, weconclude that the deviations between the FF theoryand the experimental data cannot be solely attributedto errors in w0.

The FF theory assumes an infinite planar interface,whereas the actual interface has curvature. This canbe another cause for the deviations, although currentlywe lack a theory that quantitatively predicts the effectof curvature. Equation (3) reproduces the experimentaldata very well when θi ≪ θc, and its deviations from theexperimental data are observed only when θi ≈ θc. There-fore, we expect a mechanismmaking the curvature effectprominent especially when θi ≈ θc.

S. Shinohara acknowledges financial support fromDeutsche Forschungsgemeinschaft (DFG) researchgroup 760 “Scattering Systems with Complex Dynamics”and the Deutsche Forschungsgemeinschaft EmmyNoether Program.

References

1. H. E. Tureci and A. D. Stone, Opt. Lett. 27, 7 (2002).2. H. E. Tureci, “Wave chaos in dielectric resonators: asymp-

totic and numerical approaches,” Ph.D. thesis (Yale Univer-sity, 2003), downloadable from http://www.eng.yale.edu/stonegroup/publications/HakanThesis.pdf.

3. F. Goos and H. Hänchen, Ann. Phys. 436, 333 (1947).4. N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and

A. D. Stone, Phys. Rev. Lett. 88, 094102 (2002).5. S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and

C.-M. Kim, Phys. Rev. Lett. 93, 164102 (2004).6. H. Schomerus and M. Hentschel, Phys. Rev. Lett. 96,

243903 (2006).7. E. G. Altmann, G. Del Magno, and M. Hentschel, Europhys.

Lett. 84, 10008 (2008).8. J. Unterhinninghofen and J. Wiersig, Phys. Rev. E 82,

026202 (2010).9. Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shim,

J. Unterhinninghofen, W. Fang, and G. S. Solomon, Phys.Rev. Lett. 105, 103902 (2010).

10. H. E. Tureci, H. G. L. Schwefel, A. D. Stone, and E. E.Narimanov, Opt. Express 10, 752 (2002).

11. T. Fukushima and T. Harayama, IEEE J. Sel. Top. QuantumElectron. 10, 1039 (2004).

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