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The increase of resolution by the use of microspheres is related to the use of evanescent waves satisfying complex Snell's law with complex trigonometric functions related to the incident and refracted angles, while the refractive indices are real. The evanescent waves are obtained in addition to initial propagating waves satisfying the ordinary Snell's law. The lateral spatial wave vectors of the evanescent waves, which include information on the object fine structures, are converted at the microsphere surface to smaller wave vectors. Due to the reduction in the magnitudes of the spatial wave vectors of the evanescent waves, they become propagating waves including the fine structures which are recovered in the image plane.
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Y. Ben-Aryeh (2016) studied this question.
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