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Multilayer optical filters with random errors

Optical structures consisting of multiple dielectric layers are used as filters in many applications, including wavelength division multiplexing (WDM), in optical networking. The desired frequency response in transmittance or reflectance of the device is obtained by proper design of the thickness and refractive index of each of the different layers. Random deviations from the design parameters will cause the transfer functions to exhibit random departures from the ideal design. Frequently, multilayer devices are targeted for use in large optical networks in which signals will traverse a cascade of such devices. The end-to-end transfer functions seen by these signals (the product of the individual transfer functions) become narrower in bandwidth as the number of transfer functions increases. For a given tolerance on the wavelength drift of each signal laser, the tolerances placed on filter shapes become tighter as the optical network becomes larger, and this can present significant system design and performance issues. Previously, Monte Carlo analysis was used to study random transfer-function errors. This paper discusses an analytical method based on Kronecker products, which is used to determine the statistical errors in transmittance and reflectance of the optical structures. The analysis permits both the in-band and out-of-band statistics to be determined as functions of the statistics of random errors in optical thickness of the different layers.