Bloch theorem in cylindrical coordinates and its application to a Bragg fiber

We study the description of the Bloch theorem in the cylindrically periodic structure and we show that it is valid in a good approximation except for the neighborhood of the origin. The cylindrical Bloch theorem is applied to the analysis of a Bragg fiber, a kind of photonic bandgap fibers. It is confirmed that the above exception does not prevent us from using the theorem for a practical use. We show a procedure to obtain the propagation constant and electromagnetic fields of the TE mode in the Bragg fiber. In the present scheme, electromagnetic fields are genuinely treated as cylindrical waves, namely, Bessel functions without the asymptotic expansion. Then we can obtain a clear mathematical description of the Bragg fiber and investigate its physical properties. The present theory gives a criterion to design generalized Bragg fibers.