Discrete thin-film thickness-modulated designs: spacing of all possible stopbands.

The spectral spacing of multiple, high-reflectance stopbands is determined analytically for discrete thin-film layer-thickness-modulated designs. Discrete, sinusoidal modulation of layer thickness produces unique multilayer designs for fundamental modulation frequencies within the range 0 < f(m)<or= 0.5. Thickness-modulated designs typically have several high-reflectance stopbands at spectral frequencies that are a function of modulation frequency. All stopbands were verified by the trace of the characteristic matrix for each thickness-modulated design. A universal stopband equation is presented that predicts the relative spectral spacing of all possible stopbands for thickness-modulated designs. Two harmonic components of thickness-modulated designs are defined. The spectral performance (stopband position) is shown graphically for modulation amplitudes 0.25, 0.5, 0.75, 1.0, and 0 < f(m) <or= 0.5.