Perturbation approach to resonance shifts of whispering-gallery modes in a dielectric microsphere as a probe of a surrounding medium

A first-order perturbation theory similar to the one widely used in quantum mechanics is developed for transverse-electric and transverse-magnetic photonic resonance modes in a dielectric microsphere. General formulas for the resonance frequency shifts in response to a small change in the exterior refractive index and its radial profile are derived. The formulas are applied to three sensor applications of the microsphere to probe the medium in which the sphere is immersed: a refractive-index detector, an adsorption sensor, and a refractive-index profile sensor. When they are applied to a uniform change in the refractive index in the surrounding medium, the formulas give the same results that one would obtain from the exact resonance equations for the two modes. In the application to adsorption of a thin layer onto the sphere surface, the results are identical to the first-order terms in the exact formulas obtained for the adsorption layer. In the last-named example, a scheme is proposed for instantaneous measurement of the refractive-index profile near the sphere's surface.