Theorem Concerning Reflection From a Plane Stratified Medium

Let a plane electromagnetic wave of wavelength λ be incident in free space at an angle of incidence i upon a medium that is stratified in horizontal planes. Let the squared complex refractive index be an analytic function of height, and consider height as a complex variable. By sliding the contour map of the analytic function parallel to the imaginary height axis a family of refractive index profiles is obtained along the real height axis. Let one member of the family correspond to a nonabsorbing medium, and let another typical member of the family be obtained by displacing the contour map of the analytic function a distance b in the direction of the negative imaginary height axis. Then the amplitude reflection coefficient of the stratified medium is exp and the phase change on reflection is independent of b . This is true regardless of the precise form of the analytic function specifying the refractive index profiles. A function with singularities may be used, provided that the range of b is restricted to use only a region of the complex plane where the function is analytic.