Abstract

First introduced by Sossi,¹ the Fourier transform technique is a powerful tool to design optical filters involving a variable refractive-index layer. A relationship between the transmittance or reflectance and Fourier transform of the refractive-index profile can be established with various degrees of approximation. If the transmittance is specified for a range of wavelengths, the inversion of the Fourier transform yields a variable refractive-index profile. The designer can choose to approximate this profile by a series of homogeneous layers or to keep it as such. The Fourier transform of the logarithmic derivative of the refractive-index profile involves an amplitude term Q(σ) and a phase factor F(σ) where σ is the wavenumber. It is worth emphasizing that this relationship is an approximation. Workers have established several expressions for the Q-function in terms of the transmittance or reflectance. We show that even for the best of the Q-functions, the Fourier transform remains an approximation. The technique is applied to the case of a quarterwave stack such as a mirror (HL)^pH, and it is shown how the mirror characteristics expected from the Fourier transform technique differ from the actual values obtained using conventional exact matrix calculations.

© 1988 Optical Society of America