Coupling from tapered fibers and polished half-block couplers into the high-Q whispering gallery modes of microsphere resonators is investigated analytically. Numerous formulas are derived to predict the external coupling Q values, and intrinsic whispering gallery loss, for arbitrary structures, and for any sphere mode. Phase-mismatch due to the differences in propagation constants between input and sphere modes is taken into account. These formulas are strictly mechanical once a simple characteristic equation is solved which relates the spherical mode orders to the resonant wave vector. Results are in very good agreement with values that are calculated by different numerical methods.
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