Optics Communications 113 (1994) 133-143

High-6 optical whispering-gallery microresonators:

M.L. Gorodetsky, V.S. Ilchenko

Abstract

High-Q optical whispering-gallery modes TE(TM),„„, in spheroidal dielectric microresonators are shown to be equivalent to processing circular modes TE(TM )//,. with the rate of precession depending on the value of the residual resonator nonsphericity. This approach gives a natural interpretation to the characteristic two-lobe emission patterns of these modes with prism couplers. Spatial recognition of different modes on the basis of their emission patterns facilitates optimal coupling to selected modes. which is of interest for applications. Experimental results on the observation of two-lobe emission patterns and imaging of internal mode field distributions are presented. The feasibility of a single-prism transient filter and an optical delay line based on processing modes is discussed.

Presently, all experimental efforts have been fo

cused at reaching the highest possible Q and feasibility demonstrations of some of the proposed applications. No specific studies of mode spectra in high-Q solid-state spherical microresonators have been reported so far, nor have emission properties of different modes with existing prism couplers been investigated. Meantime, ultimate results in applications (especially in cavity QED experiments) require optimal coupling of the probe beams to selected given modes.

0030-4018/94/S07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0030-4018 (94 )00494-3

continuous total internal reflection from the inner surface of the sphere. Extension of this interpretation on other modes l+ m having wider field distribution is not so evident. In a perfect sphere, all modes with the same / and different m are frequency-degenerate. In real resonators, even a small residual nonspheric-ity removes this degeneracy and complicates the identification of modes by fitting spectral fragments in the frequency domain '.

Light can be inserted into the resonator by means of a coupling prism put at a small distance 0.1-U from the resonator surface. The probe beam is focused on the "touching point" under an angle providing total internal reflection, and the light tunnels through the small gap between the prism and the resonator. It is natural to expect that after making a number of orbits inside the sphere, the light will be re-emitted into the prism under the same angle and in the same plane of incidence in which the input beam has come. In the experiment, however, a different situation can be observed (see Sect. 3): the light matching in frequency to an observable high-0 mode, can be re-emitted in a different plane, as if the light orbit moved inside the sphere. This effect is analogous to the recently observed "precession of morphology-dependent resonances" (MDR) in resonance scattering of pulsed plane-wave laser radiation on deformed droplets of aerosols [12]. The authors ofRef. [12] interpreted this effect in terms of group angular velocity of MDRs, considered as a kind of wave packets inside the sphere. While traveling along the precessing orbital trajectory, MDR sequentially emits (through radiative leakage) light into upper and lower half-space divided by the symmetry plane of the droplet.

In this paper, we focus our attention on the following questions:

(i) If a similar effect is observed in solid spheroidal microresonators under monochromatic irradiation, what is the relation between the observed resonances and the classical electrodynamic solutions for E,Htmq modes? Is it possible to use symmet-

' Spectral identification of modes requires a precise knowledge of the resonator geometry, or at least, a very high level of sphericity when l+ m modes remain degenerate. Schiller and Byer [ 9 ] reported the frequency identification of optical WG modes with different /and q in a unique high-precision geometry (A/40) silica sphere 4 cm in diameter.

(ii) What are the emission patterns of these modes with prism coupler and how can these patterns be used for their identification?

(iii) How may all these phenomena be interpreted in terms of quasi-geometrical approximation?

The most popular approach for the description of spectra in dielectric spheres, Mie theory, gives good results in the analysis of elastic scattering of plane waves on relatively small spheres (up to few tens of wavelengths) with a rare spectrum of internal modes of moderate radiative Q. In whispering-gallery microresonators, only special modes having maximum Q are of interest, and practically all types of such modes (including those with ultra-high radiative Q in relatively large spheres) become accessible only with the help of prism couplers. In this context, a straightforward approach based on classical electro-dynamical solutions is more adequate, and a special analysis of resonator coupling to radiative modes in the prism is required.

The problem ofe.m. oscillation in dielectric sphere with refraction index ”r is solved explicitly. The mode field distribution is expressed via Debye potentials V(r,^v) [13],

ML. Gorodetsky, V.S. llchenlw / Optics Communications 113(1994) 133-143

where Y|,„=e""^VP^m (cosi9) are angular spherical functions. As a solution of a three-dimensional boundary-value problem, the field distribution is characterized by three indexes /, m, and q, where in denotes the number of field maxima in the "equator" plane (t?=0): q is the (number ofzeroes+ 1) of the Bessel function ji{kn^r} inside the resonator; /is the mode number (approximately equal to the number of wavelengths packed along the circumference of the resonator I” I'n.RnJ'k).

For the analysis of mode spectra in degenerate spherical resonators, only the radial part in (2) is essential as it determines the characteristic equation for resonant frequencies. In this article we pay special attention to the angular functions Y,^, because they determine the excitation and emission properties as well as the frequency splitting of l+ m modes in the spheroidal resonators.

If the characteristic dimensions are large enough, it is fruitful to use the quasiclassical approach. The simplest configuration is characteristic for the modes with l=m. In case of large /, such mode can be interpreted as a "gaussian" beam

(3)

having l/q internal reflections from the surface during one revolution inside the sphere.

Let us show that an arbitrary spherical function can be modeled as a result of precession of inclined fundamental circular mode.

We shall obtain this result, first, by assuming that the resonator is slightly aspherical. Let it have the shape of a stretched or oblate spheroid (ellipsoid of

In this way one can speak about precession with the rate determined by the value of eccentricity. If the length of the intracavity wave packet is longer than the full period of the precession (this is secured in case of monochromatic excitation and high Q of modes), the phase of precession becomes uncertain, just like the phase of optical wave bouncing between two mirrors in Fabry-Perot resonator. As a result, a stationary field distribution is formed. Below we show that under the condition of small eccentricity, the resulting field is in the first order of approximation identical to the undisturbed spherical harmonic (K/,„). As soon as the resonance condition (provided by the completion of complex spiral trajectory) is different from that of a single plane orbit, the frequency of each precessing resonance will be different from the initial frequency of degenerate K/„, modes.

ML. Gorodetsky, V-S. Ilchenko / Optics Communications 113 (1994) 133-143

ogous to the quantization of the angular momentum projection L: in quantum mechanics.

The frequency shifts for precessing modes can be estimated from the change in the perimeter length for the inclined ellipse. Simple calculations yield in the first order of approximation:

^=±⁽-^-P=±^e-^2(p-^ w 4 4/²

where a positive sign stands for oblate spheroid, and negative sign for a stretched one. This simple estimate agrees well with the earlier reported calculations [14] by means of perturbation theory specially modified for open systems. The only small difference is that our expression for \u”/w contains /² instead of /(/+ 1) in denominator. This discrepancy can be easily explained by the fact that a geometrical interpretation does not take into account the uncertainty of angular momentum for fundamental circular mode translating from nonzero cross-section of the beam:

The frequency shift (5) of "precessing" modes means that they are no longer degenerate with respect to m.

Indeed, the stationary field in the given point may be represented as a superposition of fields produced by inclined circular mode Yi/ during different phases of its precession, taken with the phase factor exp(iwAa). Here m is an azimuthal wavenumber describing the precession. If the eccentricity is small (single-orbit angular shift Ao'“27r/w), the sum can be substituted by the integral:

(6)

Note that the coefficient before the integral in (8) has only two sharp resonances cos /?= m/!, in accordance with the geometrical consideration.

Hence the precessing circular mode is shown to be identical to the /, m eigenmode with a shifted frequency. The frequency shift, according to (7), is connected with the phase velocity of precession by the simple relation Q= (u)/„,—fc”//)/w.

Expression (8) illustrates the intrinsic connection between arbitrary spherical harmonics and circular modes. The absolute rate of precession is not essential in this representation, because in the high-Q limit, all resonators are nonspherical. Continuing the quantum-mechanical analogy, we can recall that in the case of a long relaxation time of atomic energy levels, even a small external field is sufficient to produce Zeeman or Stark splitting and to give a natural reference for the definition of the momentum projection operator L^. (It is also worth mentioning that the quasi-geometrical formalism of precessing spherical modes has much in common and may be useful for the quasi-classical description of "circular Rydberg atoms" and "Kepler wave packets" that are now extensively investigated in connection with cavity QED and quantum state preparation experiments [15].)

M.L. Gorodetsky. V.S. Ilchenko / Optics Communications 113(1994) 133-143 3. Emission patterns of resonator with prism coupler

Fig. 2. Coupling to whispering-gallery modes in dielectric spheroid. The input beam couples to the coplanar fundamental circular mode Yii. When rotating around the spheroid symmetry axis (coinciding with the stem axis), the processing Yu mode (i) forms the Yi^, mode field distribution and (ii) produces two gaussian beams in the coupling prism.

Fig. 1. Schematic of prism coupling to whispering-gallery modes.

( (1+m)\\ J (/-w)!!' '"l^+m^r^ [(/-w-l)!!'

X(l+/^²/2w²)exp(2a7/•*)

Resonators were formed on silica stems 30 um in diameter (Fig. 6a). During the formation of a sphere by fusing in flame, solid stem collar produces axi-symmetrical perturbation of uniform surface tension field. As a result, the final shape is different from perfect sphere, and in the area of naturally defined equator plane (big circle normal to stem axis), the surface can be approximated by ellipsoid of revolution. According to the results of optical microscope observations, the eccentricity of spheroids in our experiment did not exceed the value f < 0.01 (this estimate translates from the measured difference between ellipsoid semiaxes(l±0.3)%).

gether with a reflected part of the input beam are observed on the screen. For a measurement of the Q-factor, a photodiode is put instead of the screen at optimal position, and the resonance bandwidth is measured by standard technique, with the probe laser frequency swept by the piezo-drive of one of its mirrors. With the change of separation between the resonator and the prism, the observed quality-factor changes between the loaded quality-factor Q^ (zero gap) and intrinsic quality-factor Qo (the asymptotic value obtained during the gradual increase of the gap). In our experiment, Qo varied between (6-8) xl 0^s for q=\ modes and (3-6) Xl O⁷ for "deeper" modes q= 15-20. (The index q was estimated from the value of insertion angle 6>o, see formula (10);

modes with higher q were not accessible due to the limited aperture of the insertion lens at the given position.)

For the observation of "clean" output patterns of modes, we used an independently translated second (outcoupler) prism on the other side of the resonator (schematic diagram in Fig. 4). The output beam fits the aperture of the insertion lens and goes out in backward direction in the form of two parallel beams corresponding to the two angular lobes of the emission pattern. The output is side-reflected by the mirror and passes through the second lens, which reproduces the emission pattern of the given mode with

Fig. 4. Optical setup for observation of "clean" emission patterns ofWG modes and direct imaging of mode field distributions in the resonator-prism coupling zone.

the prism. The latter is observed on the second screen. Fig. 4b presents a typical "clean" emission pattern for a mode with /- m = 4, obtained by this technique.

The setup shown in Fig. 4 allowed also to obtain a real image of the mode field configuration in the "prism-resonator" contact zone. This was achieved by the slight pulling of the insertion lens away from the prism-resonator assembly so that the microscope configuration emerged. The real image of the mode field was formed on the screen with a magnification of about 1000 (image size about 1 cm). The corresponding images, for different modes are presented in Fig. 5. They reproduce fragments of the corresponding i9-angle distributions P" (cosi?) seen "through the gaussian window" formed by the prism-resonator contact zone. Fig. 5c obtained for the 1—mwlO mode illustrates the appropriateness of the cosine approximation (6) for the latitudinal field distribution in the contact zone, for the case of large /- m.

And finally, we directly observed and photographed field localization zones of different modes on the resonator surface. In a partially loaded resonator (Q~ 1 X 10⁷), manual adjustment of the laser frequency was enough to trace temperature-induced mode-frequency detunings and preserve single-mode excitation, with the visual tuning control by output emission spots. With the given power of the laser ~ 1 mW, exposure times about 1 minute were sufficient to photograph through the microscope the speckle pictures produced by weak surface scattering of light in the area occupied by the e.m. field of whispering-gallery modes (Fig. 6).

Identification of modes in our experiments was made by fitting the observed emission patterns and field distributions to the results of calculations in Sects. 2, 3.

The experimental results presented in Sect. 3 illustrate the fruitfulness of the precession approach in predicting the properties of high-Q optical whispering-gallery modes in slightly-aspherical solid dielectric microresonators. Although the notion of precession is only conventionally applicable to the description of stationary fields, it gives a clear physical interpretation to the origin of the observed phenomena: (i) removal of the spatial and frequency degeneracy of WG modes in a real spheroidal resonator;

(ii) excitation ofTE(TM);„,„ modes by the inclined input beams; (iii) emission of the modes in form of two gaussian beams in the coupler prism.

In this paper, we limited ourselves by a description of the emission and coupling properties of "monochromatic" WG modes. However, the precession property puts forward new perspectives for the application of WG mode resonators with prism couplers as spectral devices.

First, the presence of two-lobe emission patterns facilitates an easy realization of the single-prism reentrant narrow-band optical filter on the basis ofWG microresonator. Instead of using two prisms as input and output ports, it is possible to couple the input beam to one of the emission lobes, and the second one would serve as the output. It is evident that the total loss of such filter can be made as small as 3 dB (the full inserted input power equally divided between two output beams) and smaller, if a critical coupling ql= Qo for the selected mode is achieved.

Another possibility can be realized if we further load the resonator until the loaded resonance covers a group of non-degenerate "inclined" modes. In this case, the resonator may serve as a delay line for optical pulses, with the delay time equal to half of the precession period. Since the precession frequency can vary in a wide range between a few hundred kilohertz (single mode bandwidth) and up to 10'° Hz and above (by choosing different resonator size, eccentricity and insertion angles), optical delay times of the order lO^-lO"¹⁰ s may be therefore available.

This work was supported in part by the International Science Foundation, and by the Quantum Op-

tics Lab, California Institute of Technology. Authors are grateful to V.B. Braginsky and S.P. Vyatchanin for helpful discussion.

As we could not find expression (8) in the literature we give here its short formal derivation.

sin0cos0=sin;?cos(<”-o') cos P - cos;? sin ft,

sin0sin0=sin i?sin(y”—a) ,

Hence

r//(0(a,/?),<”(a,J8))=^(cos0)exp(i/0)

=(2/-1)!! (sinffcos((p-a)cosp

-cos^sin/P+isin^sin^-a!))'. (A.2)

142 M.L. Gorodetsky, V.S. Ilchenko / Optics Communications 113(1994) 133-143

Fig. 6. Optical microresonator and speckle photographs of different modes formed by residual surface scattering (sphere diameter 250 \im): (a) WG microresonator under external illumination; (b) /-w=0, (c) /-w=2, (d) l—m”70.

•у +i sm'Qsmif/)'dy, (A.4)

where К is normalization coefficient. It can be shown Substituting (A.2) into (A.3) and making substitu- that the integral in (A.4) satisfies the following dif-tion у=(/)-а, we have ferential equation:

д1_ _ / cos P- m тр^ sin^

^/=^;sin^(/-"^l)/?(l+cos/?)"^lF(l9,^г)) . (А.5)

expression (A.4) will not depend on/?. In formal way putting ft= -л/2 in (A.4) and comparing resulting equation with the well known Heine integral

Yt^,(p)=(-l)'~'" 2д/! sin^-^W+cos/?)"¹

X [ e""" Yit(6, ф) da . (A.8)

Comm. 107 (1994)41. [ 6 ] Т. Ваег, Optics Lett. 12(1987) 392;