Dispersion Information for Photonic Fiber Modes from CUDOS Simulations

The CUDOS MOF Utilities are a software package for simulating microstructured optical fibers (or MOF’s, also known as photonic crystal fibers) using the multipole method of Fourier-Bessel expansions in a Microsoft Windows environment. The code can search for an electromagnetic mode of a specified symmetry, and having identified such a mode, determine its dispersion properties. One of the key outputs is a table of the real and imaginary parts of the effective index of refraction as a function of the free-space wavelength, that is 2πc/ω, the speed of light divided by the mode’s frequency. From this table one can immediately obtain the phase and group velocities, the dispersion parameter (spread of wave-packet with distance), the Poynting flux loss coefficient (from the imaginary part of the refractive index), and the full dispersion diagram (frequency versus mode wave number). Examples of MOF’s in the literature are used to illustrate the output and present some new results. 1. Dispersion Formulas The purpose of this note is to clarify and illustrate some of the essential output from the CUDOS software. In particular one can use the output table of the refractive index as a function of free-space wavelength to extract important dispersion information about a confined mode in the fiber. The CUDOS software and documentation is available in Reference [1], and has references to the original literature. The software is based on a method which uses a multipole expansion centered on each cylindrical hole of a fiber to enforce boundary conditions and then matches the local and global expansions to determine the expansion coefficients [2,3]. A key aspect of the current public version of the software is that it makes use of the circularity of the inclusions. It can deal with fibers with either a solid core surrounded by holes, or a hollow core, again surrounded by holes. The multipole method has frequency, ω, as an input parameter. It should be noted that the actual input file calls for the user to supply the starting frequency for the mode search in the form of the free-space wavelength, λ = 2πc/ω. Equally important, although not explicitly stated in the code’s documentation, the output tabulates the effective index of refraction as a function of free-space wavelength, in other words as a function of frequency. Recall that the effective refractive index neff is defined as the ratio of the mode’s propagation constant (wave-number) kz in the fiber divided by the free-space wave number k = 2π/λ = c/ω. From the code’s tabulation of refractive index versus free-space wavelength, several important parameters are immediately obtained. The phase velocity of a mode is defined by eff p n c v / = . (1) A particular subset of modes that are of interest for relativistic particle acceleration are the so-called “speed of light” modes (SOL) with vp = c [4]. These are characterized by neff = 1, and the ability to tune the fiber’s refractive index to a desired value by the appropriate choice of hole geometry is one of the chief attributes of photonic crystal fibers. The group velocity is given by the derivative vg = dω/dkz. It can be obtained by inspection of the slope of the dispersion diagram of ω versus kz = neff k or by a finite-difference derivative in the neff vs λ output table. Noting that ω = ck = ckz/neff , the following two equivalent formulas for the group velocity are useful. The first is