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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 pair with identical propagation constants (different from the first pair) has different attenuation coefficients. The four attenuation coefficients may have either a positive value, representing power leaking out of a waveguide mode or a negative value, representing power from an external source leaking into a mode. Moreover, a mode with a positive (negative) attenuation before the second Bragg condition, has a negative (positive) attenuation after the second Bragg condition. Radiation near the second-order Bragg condition of a periodic waveguide typically occurs at an angle perpendicular or nearly perpendicular to the propagation direction of the waveguide because the scattering centers have a period equal to or close to the period of the longitudinal propagation constant of the mode. In this paper, stable numerical solutions for the modes of periodic dielectric structures are developed using Floquet-Bloch theory. One primary focus in this paper illustrates a unique method of analyzing such modes using eigenvectors forming a Hilbert space, allowing for expansion of arbitrary vectors and their derivatives used for calculations such as that involving group velocities. The dimension of the vector space is determined by the number of space harmonics used in the solution of the Floquet-Bloch equations. The accuracy of the numerical solutions is affected by the number of space harmonics; as that number is increased, the number of waveguide partitions must be increased to maintain a given accuracy.

Waveguide Characteristics Near the Second Bragg Condition