A theoretical analysis has revealed that tapered optical waveguides could be useful as white-light whispering-gallerymode (WGM) optical resonators. The compactness and the fixed-narrow-frequency- band nature of the resonances of prior microdisk and microsphere WGM resonators are advantageous in low-power, fixed-narrow-frequency-band applications. However for optical-processing applications in which there are requirements for power levels higher and/or spectral responses broader than those of prior microdisk and microsphere WGM resonators, white-light WGM resonators in the form of optical tapers would be preferable.
In a typical prior microdisk or microsphere WGM resonator, the optical power is concentrated mostly in a small WGM volume, making it necessary to limit the power to a low level in order to minimize undesired nonlinear optical and thermo-optical effects. If one could construct a WGM resonator in which the optical power were spread over a larger volume, then the threshold power level for the onset of undesired nonlinear optical and thermo-optical effects would be higher.
The theoretical analysis was performed for a multimode, axisymmetric, circular-cross-section waveguide having a taper sufficiently smooth and gradual to justify the approximation of adiabaticity. In this approximation, the equation for the dependence of the electromagnetic field upon the axial (longitudinal) waveguide coordinate (z) can be separated from the equation for the dependence upon the radius (r) and the azimuthal angle (ϕ). Electromagnetic modes characterized by high angular momentum (equivalently, large values of the ϕ- dependence quantum number) were considered. The solution of the equation for the axial dependence was found to be an amplitude that varies gradually with z. For a given axial location z, where the outer surface of the waveguide has a radius R(z), the solutions for the radial and azimuthal dependences were found to be WGM modes equivalent to those for a cylinder of radius R(z).
In effect, it was found that the tapered waveguide can be considered to support WGMs propagating along the waveguide axis. It was further found that as the radius tapers down toward the classical critical radius Rc at a classical turnaround axial position zc, the group delay of a WGM increases and the electromagnetic field becomes increasingly concentrated, albeit in an effective mode volume typically much larger than the mode volume in a prior microdisk or microsphere WGM resonator. Thus, it was found that the power density of the electromagnetic field is much less than in a prior microdisk or microsphere WGM resonator and the onset of undesired nonlinearities is shifted to a significantly higher power level. It was found that in the special case of a linear taper, the turning point varies linearly with the frequency of the electromagnetic field, while the resonance quality factor and dispersion remain fixed to first order.
A resonator having these characteristics can be considered a white-light resonator in that it exhibits resonance over a continuous frequency range.