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 pair with identical
propagation constants (different from the first pair) and have 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.