1. Introduction

Whispering Gallery Mode (WGM) [1] electromagnetic resonances have long been predicted in a variety of symmetric and asymmetric morphologies [2]. In recent years, optical WGM resonances have been observed, for example, in symmetric [3, 4] and asymmetric [5, 6] dielectric cavities. The dramatically large Q-factors of the WGMs in such cavities have been used to reduce the effective threshold for nonlinear optical processes such as parametric oscillation [7, 8] and Raman lasing [9, 10].

Symmetric, high-Q WGM resonators fabricated from ferroelectric materials such as LiNbO₃ have also been demonstrated [11]. The naturally large nonlinear optical coefficients of LiNbO₃ coupled with its wide optical transparency window make it an excellent candidate material for fabrication of WGM resonators. Because of electro-optic tunability of the resonator spectrum [12] and the frequencies of individual mode families within a given FSR [13], resonators made of ferroelectric material hold particular promise as reconfigurable narrowband filters for optical communications systems [14, 15].

Asymmetric ferroelectric resonators would retain all features of symmetric resonators. But they can also introduce the additional advantage of unstable modes [16] that could be electro-optically coupled to the high-Q modes to allow high speed, band-selective intensity modulation of laser light. In this paper, we present elliptical WGM resonators constructed form LiNbO₃. A model for the behavior of WGMs in elliptical resonators based on the eikonal method is also presented. We discuss the methods of fabrication and the experiments conducted to test the results of the eikonal analysis. We conclude with a discussion of possible uses of electro-optic, high-Q, asymmetric resonators in future devices and experiments.

2. Eikonal Model for Elliptical WGMs

We begin our investigation of elliptical WGM resonators by developing a framework to study their spectral features. The recent successes with utilization of the eikonal method to approximate the eigenfunctions and eigenvalues of WGMs in regions bounded by spheroids [17] suggests that such methods could also be used to understand the spectral features of high-Q elliptical WGM resonators. In this spirit, we follow the example of reference [2].

In the eikonal approximation, WGMs are modeled as a finite number of reflections of rays between the edge of the interior surface of the resonator and a caustic surface confocal with this boundary and sharing its eccentricity. For the type of modes considered here, the lattus rectum of the caustic surface is taken to be either less than or equal to the lattus rectum of the elliptical boundary. Specifically, the solutions to the reduced wave equation in elliptic coordinates

are assumed to be of the form

where A_j and S_j are position dependent amplitude and phase (action) functions, respectively, and k is the eigen wavenumber. Each ray j reflects from the interior surface of the boundary and is tangent to the caustic surface. Upon passing the caustic surface, the phase front S changes curvature much as a Gaussian beam would after passing through the focal plane of a lens.

To determine the resonance condition of WGMs that are azimuthally confined to the plane of the resonator equator, we enforce Bohr-Sommerfeld quantization of the optical path [18]. This assumes that the angular momentum of the mode is constant over the optical path. This is a valid assumption only if the same optical path is followed regardless of the input position, which is experimentally demonstrated below. The phase shift along one round trip about the cavity should equal integer multiples of 2π:

where dσ is a differential element of the round-trip optical path P. For high-Q WGMs that are closely confined to the edge of the resonator, the mode index q is on the order of the ratio of mean radius to wavelength. In this case, the path P asymptotically approaches the perimeter of the caustic surface, which in turn asymptotically approaches the perimeter of the elliptical boundary itself.

Now that the appropriate quantization conditions have been determined, we will model the actual path traveled. For high order WGMs, the optical field intensity is concentrated to within a few microns of the edge surface of the boundary. For resonators that are of comparable size to the ones fabricated for the set of experiments described in this paper, the ratio of the radial mode width to the effective radius of the ellipse is small at all points along the edge. In the regime of critical coupling, the radial mode index q is large (for example, the q index is about 10,000 for the elliptical resonators we used in this experiment.) We assume that the path followed by the rays is along the caustic surface of the congruence of rays, in the limit that the caustic surface perimeter approaches the perimeter of the resonator cross section. Eq. (3) reduces to (adapted from Ref. [2])

E(1,x) is the complete elliptic integral of the second kind. Equation (4) has two parameters that define the geometry of the ellipse: eccentricity ε and semimajor axis length a. Equation (4) can be manipulated to predict the free spectral range (FSR) of an elliptical cavity

where n is the frequency dependant refractive index of the cavity material and c is the speed of light in vacuum.

3. Fabrication

A piece of crystal in the shape of a baseball diamond was cut from a z-cut wafer of congruent LiNbO₃. The lengths of the axes of the diamond were selected to be slightly larger than the target dimensions of the desired ellipse. The edges of this structure were mechanically ground until the sharp edges of the baseball diamond shape were smoothed to a continuous curve resembling an ellipse. At this point, mechanical polishing techniques were applied to shape the edge curvature and polish the edge surfaces [19].

Three optical resonant cavities of elliptical cross section we fabricated from z-cut congruent LiNbO₃ (Fig.1). The z-axis of the material coincides with the vertical axis of the resonator. The eccentricity of the first cavity is 0.57. The lengths of the semimajor and semiminor axes are 4.4 and 3.6 mm, respectively. The second cavity has an eccentricity of 0.71, with semimajor and semiminor axis lengths of 4.6 and 6.5 mm, respectively. The thickness of both resonators is 0.5 mm. The third cavity has an eccentricity of 0.32, with semimajor and semiminor axis lengths of 3.4 and 3.6 mm, respectively.

Design of the resonator geometries is impacted by the coupling mechanism. Evanescent coupling of light into dielectric resonators requires phase matching between the wavenumber in the coupling medium and the wavenumber of the resonant mode. It follows that the refractive index of the coupling medium should be larger than the refractive index of the resonator host material to couple to WGMs. Further, the coupling material should have a comparable optical transparency window to that of the host material to avoid additional round-trip losses. A diamond prism is thus an ideal structure to evanescently couple free space light to resonators constructed from LiNbO₃.

 

Fig. 2. The three elliptical LiNbO₃ resonators referred to in this paper.

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LiNbO₃ is birefringent. The coupling conditions for TE and TM modes are slightly different. For example, to couple to TE modes of the resonator, the free space beam should have an incidence angle of 22° with respect to the plane tangent to the prism-resonator interface. A circular symmetric beam entering the prism at this angle will be elliptical in cross section at the interface plane. For TE modes, the spot size along the polar axis of the resonator is larger than the spot size measured along the azimuthal axis of the resonator by a factor of 2.7. There will be a drop in overall coupling efficiency when the ratio achieved during fabrication is not equal to 2.7.

For circular resonators the critical ratio is fixed during fabrication. The ratio is of course subject to the precision of the fabrication apparatus, knowledge of the refractive indices of the resonator and coupler materials, and knowledge of the wavelength of light being used. In elliptical resonators, the curvature ratio varies along the perimeter as the radius of curvature changes. Realization of a critical curvature ratio at some point along the perimeter is all but guaranteed for every elliptical resonator fabricated using mechanical grinding and polishing.

4. Experiment

Light from a 1550 nm diode laser was evanescently coupled to the resonators through a diamond prism (Fig. 3.) The laser light was scanned over several FSRs of the elliptical resonant cavity. Light reflected from the cavity-prism interface was collected by a 3mm diameter light pipe and sent to an InGaAs photodetector.

One of the assumptions of the approximation used to model WGMs in elliptical resonators is that the modes of the same modal indices will travel the same optical path length regardless of input positions. That is, the FSR of the cavity should be constant as the input position is varied. To test the validity of this assumption, the position of the incident beam, governed by the location of the diamond prism along the cavity perimeter, was varied between semimajor axes in 30 degree steps. It was found that throughout the various input angles, the same frequencies of light are resonant with the cavity and the same FSR was observed (Fig. 4.)

 

Fig. 3. Our experimental setup. CW Laser light scanned over several dozen GHz is evanescently coupled to the elliptical resonators through a diamond prism. The S₁₁ element of the scattering matrix is collected by a 3mm diameter light pipe and sent to an InGaAs photodiode. Some of the laser light is sent through a Fabry-Perot etalon of known separation to form a relative frequency reference.

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Fig. 4. As the position of the coupling prism is shifted around the perimeter of the resonator, the same spectrum is recovered at all points. This confirms the fundamental assumption in the development of the eikonal model—which in turn validates our application of Bohr-Sommerfeld quantization.

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To measure the spectral features predicted with the eikonal analysis, a 1559 nm Fiber laser was coupled into the resonators (Fig. 5.) We measured a coupling efficiency of 97.3% into the third resonator. This is, to our knowledge, the highest coupling efficiency achieved for a diamond prism-coupled WGM resonator. As suggested above, it was found that the coupling efficiency varied when the contact position between the prism and resonator was changed. The FWHM of the mode was measured, and the value was use to calculate the Q-factor of this mode, yielding 1.5 × 10⁸.

Analysis of the spectrum of the resonator shows the FSR predicted by the eikonal model. The measured FSR agrees well with the calculated values from Eq. (5). The results for all three resonators are tabulated in Table 1.

 

Fig. 5. We achieved a coupling efficiency of 97.3% into an elliptical WGM. The inset shows a Lorentzian fit (continuous red curve) to the measured data (blue dots) in the vicinity of optimal coupling. The best fit curve has a linewidth of 1.3 MHz, which corresponds to a quality factor of approximately 1.5 ∙ 10⁸. The amplitude of the fit corresponds to a maximum coupling efficiency of 99.5%.

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Table 1. Comparison of predictions from eikonal model to measured FSRs. C_eff stands for the coupling efficiency of the most efficiently coupled mode.

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Detailed mathematical analysis of the WGMs in elliptical resonators with large size compared to the wavelength of the resonant wave predicts non-degenerate spectra [20]. This prediction is qualitatively evident in the experimental data shown in Fig. 5— the spacing between adjacent modes is comparable to the mode linewidths. More quantitative analysis of the highly non-degenerate spectra will be the subject of further studies.

The coupling efficiency reports the fraction of light transmitted from within the coupling medium to a specific resonator mode and out through an output path symmetric with the input path. Light that couples from the mode to a different output channel, i.e. another cavity mode, is enumerated as additional loss. Thus, the figures reported in Table 1 are directly comparable to the coupling efficiencies achieved using a fiber-taper to couple to silica microsphere WGMs [21]. Fresnel losses at the air-prism interface are not included in this figure. Appropriate anti-reflection coating technology could easily mitigate these effects. Including Fresnel losses and loss associated with instrumentation, the best laser-to-detector transmission efficiency through the resonator remains in excess of 91%.

In reference [21] the inherent limitation in prism coupling to spherical WGM resonators due to coupling to a “large number of output modes” is described. The near unity values reported in this paper do not contradict this assertion. Rather, these observations illustrate that deliberate engineering of an asymmetric geometry lifts the limitations inherent with prism coupling to spherically symmetric WGM resonators.

5. Conclusions

We have shown that WGM resonators with elliptical cross section can exhibit highly coupled, high-Q factor modes in the optical regime. The eikonal approximation of the optical path yields results that are in excellent agreement with measured quantities. We also experimentally verified that the FSR of the modes in elliptical cavities are fixed, and do not vary with the location of the coupling point along the perimeter of the ellipse. The elliptical resonators work exactly as their circularly symmetric counterparts with the advantage of guaranteed coupling efficiency.

The presence of high-Q modes that are efficiently coupled to the input beam is an encouraging sign from the perspective of device physics. Unstable modes that are predicted to exist in elliptical cavities can be coupled to the high-Q modes electro-optically to create a narrow band optical modulator. Explorations of photo-elastic and quantum optical processes in extremely asymmetric boundary conditions are also now a possibility. Ferroelectric domain engineering [22] in elliptical resonators can be used to explore how second order nonlinear optical interactions are affected by asymmetric geometric structure.

The research described in this publication was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.