Analytic solution for cascaded long-period fiber gratings

The analytic expression for the transmission spectrum of cascaded long-period fiber gratings is presented in a closed form. When several identical gratings are cascaded in-series with a regular distance, the transmission spectrum is revealed to have a series of regularly spaced peaks, suitable for multi-channel filters. The analytic solution is obtained by diagonalizing the transfer matrix of each grating unit that is composed of a single grating and a grating-free region between adjacent gratings. The spectrum of the device is simply described with the number of cascaded gratings and a single parameter that has the information of the phase difference between the modes. With the derived equation, the spectral behaviors of the proposed device are investigated. The intensity of each peak can be controlled by adjusting the strength of a single grating. The separation between adjacent gratings determines the spacing between the peaks. The finesse of the peaks can be increased by cascading more gratings. The derived analytic results are compared with the known results of paired gratings and phase-shifted gratings.