Various structures of periodic layers in a planar geometry are used as elements of optical filters in resonators, and as quasi-optical reflection systems of integrated optics [Kogelnik & Li, 1966; Hodgson & Weber, 1997; Ramo et al., 1994; Hummel & Guenther, 1995].

Nowadays difierent optical systems with the use of microspheres as one of the important elements are under wide investigation. The use of microspheres provides a possibility to achieve very narrow resonant lines [Vassiliev et al., 1998]. The basic regime for the operation of these micro-spheres is a whispering gallery mode (WGM) for a microsphere with a radius of order 20 100 ?m. Extreme great values of a Q factor in a narrow frequency range are observed [Vassiliev et al., 1998; Ilchenko et al., 1998], but in a general case it is desirable to get the spherical resonators to possess high quality Q factors beyond WGM. Such a regime may be reached in a microsphere of a submicron size in a low spherical mode regime. An application of the well-known idea on coating by quarter-wave layers gives a possibility of a sharp increase of the Q factor of such a system by up to Q ~ 10 ⁹ as in a WGM.

The layered systems with a spherical geometry are quite complex and electromagnetic oscillations have been investigated rather completely only for an asymptotically great number of lossless layers [Brady et al., 1993]. In [Li et al., 2001],...