Bilinear transformed reflectance of materials with a spatially changing refractive index.
New approximate analytical expressions in integral form for the reflectance of materials with graded refractive-index profiles are presented. The reflectance of material containing a buried Gaussian refractive-index profile is simulated by these analytical expressions and compared with the reflectance computed by the matrix method. The two methods agree to within 0.7% in the root-mean-square sense. The analytical expressions, however, are 1 to 2 orders of magnitude faster to compute. The reflectance of a buried parabolic refractive-index profile is also investigated, and results from the proposed theory are compared with those from the WKB approximation. Also, in this instance it is shown that the integral formulation is superior to the WKB approximation at long wavelengths. As a practical example, the measured reflectance of semi-insulating GaAs implanted with protons is compared with theoretical calculations by use of the proposed formulas.
Aizenberg
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Swart
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Lacquet
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