Waveguide Characteristics Near the Second Bragg Condition

We show that in an optical waveguide with no material losses at wavelengths near the second-order Bragg condition, there exists two pairs of modes. One pair has identical propagation constants but have different attenuation coefficients, while a second (different from first pair) and coefficients. The four coefficients may either positive value, representing power leaking out mode or negative external source into mode. Moreover, (negative) before (positive) after condition. Radiation condition periodic typically occurs angle perpendicular nearly to direction because scattering centers period equal close longitudinal constant In this paper, stable numerical solutions for modes dielectric structures are developed using Floquet-Bloch theory. primary focus paper illustrates unique method analyzing such eigenvectors forming Hilbert space, allowing expansion arbitrary vectors their derivatives used calculations as involving group velocities. dimension vector space is determined by number harmonics solution equations. accuracy affected harmonics; increased, partitions must be increased maintain given accuracy.