Validity of separating group‐velocity dispersion of photonic‐crystal fibers into material and waveguide components

Abstract

where a is the core radius, n 1 and n 2 are the refractive indices (n 2 Ͻ n 1 ) of the core and cladding, respectively, and is the free-space wavelength of light. As V increases with decreasing wavelength, a step-index fiber propagating a single mode at a certain wavelength may not do so at a lower wavelength, as the condition for single-mode propagation may be violated. If the difference between n 1 and n 2 decreases with decreasing wavelength, the increase of V may be counteracted to satisfy the single-mode condition at all wavelengths. This is the principle behind the "endlessly single-mode" photonic-crystal fiber [1].Research on propagation characteristics, such as the singlemode condition and group-velocity dispersion (GVD) in stepindex fibers, has been carried out at a nominal value of the refractive index of silica. This is perfectly justified for such fibers, as the core and cladding refractive indices differ by a constant but have identical dispersion characteristics. Similarly, research on photonic-crystal fibers has also been conducted for a nominal value of the refractive index of silica [1][2][3]. The purpose of this paper is to examine whether this approximation is valid. For single-mode step-index fibers, GVD can be expressed as the sum of contributions arising from material and waveguide dispersion. The validity of this approach for the analysis of photonic-crystal fibers is also examined. For simplicity, the basic effective-index approximation is used, even though this technique is known to produce results of limited accuracy. However, the use of this analysis technique does provide useful means of gaining insight. The concept of "endlessly single-mode" photonic-crystal fiber in fact arose from consideration of the effective refractive index [1]. SINGLE-MODE CONDITION FOR A PHOTONIC-CRYSTAL FIBERA photonic-crystal fiber is made of undoped fused silica with a cladding implemented by means of a hexagonal array of air holes running along the length of the fiber, with a missing central hole serving as the core. The fiber can be considered to have a silica core with a cladding of effective refractive index n 2eff . To obtain the effective index, the wave equation is solved for the propagation constant of a wave propagating in an infinitely extended silica medium with a periodic hexagonal array of air holes [1][2][3][4]. The propagation constant divided by the free-space propagation constant (k ϭ 2/) gives the effective index.Step-index fiber theory can then be used, taking the refractive index of the core as the refractive index of pure silica and refractive index of the cladding as n 2eff .Although accurate methods of calculating effective index have been reported [3,4], we use the simple procedure in [1], where the scalar-wave equation for the weakly guiding approximation is used together with the circular approximation of the boundary of hexagonal cells forming the periodic structure of holes. As in [1], the pitch (distance between centers of nearest holes) is taken as ⌳ ϭ 2.3 m and the ratio of ho...