Whispering-gallery-mode resonator

A whispering-gallery-mode (WGM) resonator is a transparent body with a smooth, rotationally symmetric rim, such as a microsphere, a microtoroid or a polished crystalline disk, in which light travels around the circumference close to the surface, held there by total internal reflection at grazing incidence. The name comes from acoustic whispering galleries, in which sound creeps around a curved wall. Because the light meets no mirrors and the surface can be made extremely smooth, losses are set mainly by material absorption and residual surface scattering, and the quality factor can be very high: about 10⁸ for chip-based silica microtoroids, around 10⁹ for fused-silica microspheres and above 10¹⁰ for polished crystalline disks of calcium or magnesium fluoride. They differ from waveguide ring resonators in that no guiding core is fabricated; the curved outer boundary alone confines the light.

Mode numbers and resonance condition

A WGM of a sphere is labeled by three integers: the angular order $\ell$, close to the number of wavelengths around the equator; the azimuthal order $m$, with $\ell - |m|$ counting the field nodes in the polar direction; and the radial order $q$, counting field maxima along the radius. The fundamental modes ($q = 1$, $\ell = |m|$) are confined to a thin band at the equator. To leading order the resonance condition is

$$2\pi R\, n_{\mathrm{eff}} \approx \ell \lambda,$$

with $R$ the radius and $n_{\mathrm{eff}}$ slightly below the material index, because part of the field lies outside as an evanescent wave. For a silica sphere with $R$ = 100 µm and $n$ = 1.444 at 1550 nm, $\ell \approx 585$. The free spectral range between successive $\ell$ is

$$\mathrm{FSR} = \frac{c}{2\pi R\, n_g},$$

about 325 GHz, or 2.6 nm, taking a group index of 1.47. In a perfect sphere the $2\ell + 1$ values of $m$ are degenerate; a slight eccentricity lifts the degeneracy and fills the spectrum with closely spaced polar families, which is why microsphere transmission spectra look crowded. Toroids and sharp-rimmed disks suppress most of these families.

Q, linewidth and photon lifetime

At 1550 nm (193.4 THz) the loaded linewidth $\nu/Q$ and the photon lifetime $Q/\omega$ are:

For the 100 µm sphere at $Q$ = 10⁹ the finesse, FSR divided by linewidth, is about 1.7 × 10⁶. The intrinsic Q is limited by absorption (in silica, by water adsorbed on the surface and by OH in the glass near 1.4 µm and beyond), by scattering from surface roughness, and, for very small radii, by radiation loss through the curved boundary, which grows rapidly as $R$ approaches a few wavelengths.

Coupling: tapered fiber and prism

Light is coupled in through the evanescent field. A tapered fiber, drawn down to a waist of about 1–2 µm so that its mode extends into the air, is brought within a few hundred nanometers of the rim; the taper diameter is chosen to match the propagation constant of the taper mode to that of the WGM. The transmitted power shows a Lorentzian dip at each resonance, and moving the taper adjusts the coupling rate through under-coupling, critical coupling and over-coupling. Prism coupling is preferred for crystalline disks, and angle-polished fibers and on-chip bus waveguides are also used.

Where they are used

WGM resonators have been used for low-threshold microlasers, Kerr microcombs, Brillouin and Raman lasers, cavity optomechanics and label-free sensing, where a resonance shifts as molecules bind to the surface. Crystalline disks serve as the feedback element for self-injection locking of diode lasers, narrowing their linewidths by several orders of magnitude, and as electro-optic resonators when made of lithium niobate.

Pitfalls

Absorbed power heats the resonator and shifts its resonance through the thermo-optic effect, so a laser scanned across a high-Q mode sees a triangular, asymmetric dip whose width depends on scan direction and speed; Q should be measured at low power or with a fast scan. Backscattering from surface defects couples the clockwise and counterclockwise modes and splits a resonance into a doublet, which is easily mistaken for two transverse modes. Dust and humidity degrade the Q of silica resonators in open air.

Common questions

What is the difference between a whispering-gallery resonator and a ring resonator?

Both are traveling-wave resonators with a round-trip resonance condition. A ring guides light in a fabricated core of finite width, while a WGM resonator confines it only by the outer curved surface, which allows smoother boundaries and higher Q at the cost of a denser mode spectrum.

Why are crystalline WGM resonators higher in Q than silica ones?

Crystalline fluorides have very low absorption in the near infrared and can be polished to sub-nanometer roughness. The resulting intrinsic losses are lower than those of silica, whose surface absorbs water.

References: V. B. Braginsky, M. L. Gorodetsky and V. S. Ilchenko, "Quality-factor and nonlinear properties of optical whispering-gallery modes," Physics Letters A 137, 393 (1989); J. C. Knight, G. Cheung, F. Jacques and T. A. Birks, "Phase-matched excitation of whispering-gallery-mode resonances by a fiber taper," Optics Letters 22, 1129 (1997); D. K. Armani, T. J. Kippenberg, S. M. Spillane and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925 (2003); K. J. Vahala, "Optical microcavities," Nature 424, 839 (2003).