Analysis of radiation power for Fiber Bragg gratings by leaky harmonics

Floquet-Bloch Theory is one of the most rigorous numerical methods to analyze the properties of grating structures. By using Floquet-Bloch theory, the electric and magnetic fields are assumed to consist of an infinite number of space harmonics when an electromagnetic wave propagates in periodic structure of a dielectric waveguide. If a space harmonic is in the fast wave region of the dispersion relationship, the radiation power of gratings occurs. At the first Bragg condition, since all dispersion curves of each harmonic are in the slow wave region of ‘ω-β plot’, no radiation power occurs at the first Bragg condition. The n = −1 harmonic becomes leaky while the resonant condition satisfies the second-order Bragg condition. Therefore, the radiated power can be measured in the second-order gratings of periodic dielectric waveguides and the second-order Fiber Bragg gratings. In this paper, we analyze the radiation power of fiber Bragg gratings caused by leaky harmonics. The field distributions of fiber Bragg gratings are expressed by Floquet-Bloch theory. Note that gratings are in the core region of fiber Bragg gratings. The harmonic fields in the grating region are derived from the Maxwell equation. By matching boundary condition, the expression of the characteristic equation is presented. The effective index of fiber Bragg gratings can be obtained by solving the characteristic equation by Muller method. The dispersion relationship of fiber Bragg gratings and the transmission and reflection spectrum are evaluated in this paper. Moreover, the leaky power efficiency of fiber Bragg gratings are calculated by leaky harmonics. The theoretical and experimental results of radiation power of fiber Bragg gratings are compared and are in good agreement.