Abstract
Sturm-sequence properties are reported for recurrence dispersion functions
of periodic waveguide arrays. According to these properties, the number of
sign changes in the recurrence sequence at modal cutoff equals the number
of zeros of the dispersion function of the array. A generalized differential
form of this equality is developed, which applies along any path in the parameter
space. It allows deriving explicit analytical design rules for these arrays.
The derived rules reveal the possibility of supporting a specific number of
TE or TM modes, which is independent of coupling conditions. Even under strong
coupling, it is shown that the zeros of the consecutive recurrence dispersion
functions are interlaced. A recurrence zero-search algorithm employs this
interlacing in resolving closely-spaced zeros of the dispersion functions
of both symmetric and asymmetric arrays. The algorithm is applied with the
derived rules in maximizing phase and group birefringence of single-mode silicon-on-insulator
(SOI) waveguides. Two strip-loaded SOI waveguides are designed with phase
and group birefringence of 1.03 and 1.64 at a free-space wavelength of 1.55 $\mu{\hbox {m}}$.
© 2009 IEEE
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