Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators

Abstract

The theory of surface-second-harmonic generation in a dielectric microsphere using whispering-gallery modes ͑WGMs͒ is developed. The second-order nonlinearity is restricted to the surface of the sphere. The coupling coefficients for a coupled-mode theory are derived and conditions for double resonance and phase matching are discussed for TE and TM polarizations. We demonstrate that phase matching of WGMs amounts to conservation of the angular momentum of the electromagnetic mode while at the same time we obtain an analytical expression for the coherence length.

Nonlinear and Quantum Optics with Whispering Gallery Resonators

“…Analysis of nonlinear optical processes dynamics in phasematched WGM resonators has been reported for the SHG [107,213,256,317,335] and OPO [213,[253][254][255][256]336], as well as for the sum-frequency generation (SFG) [320,321,337,338] and difference-frequency generation (DFG) [339]. Usually this analysis is carried out in terms of coupled ODEs such as given by equation (25).…”

“…Further selection rules are discussed in [317][318][319][320] 11 . In particular, it is shown [318][319][320] that in large resonators the overlap integral s 123 factors into the radial and angular parts as a very good approximation.…”

Naturally Phase Matched Second Harmonic Generation in a Whispering Gallery Mode Resonator

“…In a spherical WGMR, a mode is defined by a set of polar, azimuthal, and radial numbers fL; m; qg. Assuming that the WGMR radius R is much larger than the optical wavelength , such that there is no coupling disturbance, and that the WGMs are near equatorial, we can reduce the electric fields in both polarizations to a scalar eigenfunction [14] with a scaling factor E 0 :…”

Wide-range cyclic phase matching and second harmonic generation in whispering gallery resonators

“…14 For phase-matching, momentum conservation in bulk material transforms to the conservations of angular momentum in WGM resonators, expressed via selection rules on the mode numbers of the modes involved. 15 In particular, they require that the azimuthal mode number m has to be conserved.…”