Properties of Fresnel's Interface Reflection Coefficients

Let denote an interface Fresnel reflection coefficient for - or -polarized light at an oblique angle of incidence and denote the same coefficient at normal incidence. Then it can be shown that is an analytic function of that depends parametrically on the angle of incidence . Here , where and are the normal-incidence amplitude reflectance and phase shift.
The figure shows the trajectories of the -reflection coefficient in the complex plane as the angle of incidence increases from 0 (normal incidence) to 90° (grazing incidence) for and . Each curve originates on the semicircle with radius with a constant and ends at the point , moving counterclockwise.

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R. M. A. Azzam, "Transformation of Fresnel's Interface Reflection and Transmission Coefficients between Normal and Oblique Incidence," Journal of the Optical Society of America, 69(4), 1979 pp. 590-596.
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