The analogy between the transmission spectrum of a cascaded long-period grating (LPG) and the diffraction pattern of a multislit is presented. Both of them can be expressed with the exact same equation. Only the phase parameters that describe the interference are different. The spectrum of the cascaded LPG and/or a single LPG has been derived into a single analytic equation by diagonalizing the matrix that describes the system. The diagonalizing method that gives the analytic solution of the cascaded device is introduced in detail. Experimentally and theoretically, it is presented that the device composed of N equally spaced identical LPGs has a series of regularly spaced main-loss peaks enveloped by a slowly varying curve determined by the spectrum of the single LPG. Between the adjacent main peaks, there exists N-2 side peaks. With the multislit analogy, the complicated mode coupling in an LPG and/or cascaded LPGs can be explained with the interference among the beams coupled by the series of gratings from an undepleted core mode into a cladding mode. The recoupling back to the core mode does not need to be considered but embedded in the phases of the coupled beams. The analogy holds for any strength of gratings. The phase parameter is proved to be the eigenvalue of the system unit matrix composed of an LPG and the grating-free region between LPGs. The spectral properties of the cascaded LPG are analyzed in detail. As the diffraction pattern of the multislit is enveloped by the curve determined by the shape of a single slit, the interference of the cascaded LPG is enveloped by the curve determined by the specification of a single LPG. The fringe-enveloping curve is also derived in a closed analytic form. As the finesse of the multislit becomes high as the number of slits increases, the finesse of the cascaded LPG increases with the number of the LPGs. The free spectral range decreases with the separation between elements in both cases.
Byeong Ha Lee and Un-Chul Paek, "Multislit Interpretation of Cascaded Fiber Gratings," J. Lightwave Technol. 20, 1750- (2002)