Increase of resolution by use of microspheres related to complex Snell's law and phase contrast interferometer

The increase of resolution by the use of microspheres is related to the use of evanescent waves, satisfying complex Snell law where the trigonometric functions of the incident and refracted angles are complex trigonometric functions, related to the incident and refracted angles, while the refractive indices are real. The evanescent waves are obtained in addition to propagating waves satisfying the ordinary Snell law. Measurements with high resolutions of phase objects like those of semi-transparent biological tissues, is described by an optical system composed of a combination of the microsphere with interferometer.