Refractive-index profile approximation

It is well known that the dispersion in the optical fibers is divided into three parts, modal dispersion, material dispersion, and waveguide dispersion. In the case of the SI POF, the modal dispersion is so large that the other two dispersions can be approximated to be almost zero. However, the quadratic refractive-index distribution in the GI POF can dramatically decrease the modal dispersion. We have succeeded in controlling the refractive-index profile of the GI POF to be almost a quadratic distribution by the interfacial-gel polymerization technique (2). Therefore, in order to analyze the ultimate bandwidth characteristics of the GI POF in this paper the optimum refractive index profile is investigated by taking into account not only the modal dispersion but also the material dispersion. 

Refractive-Index Profile. The refractive-index profile was approximated by the conventional power law. The output pulse width from the GI POF was calculated by the Wentzel-Kramers-Brillouin (WKB) method (10) in which both modal and material dispersions were taken into account as shown in Equations (3), (4), and (5). Here, aintemodai cTintramodai, and CTtotai signify the root mean square pulse width due to the modal dispersion, intramodal (material) dispersion, and both dispersions, respectively. 

In principle matrix methods analogous to the ones discussed in the section about neutron reflectivity can be applied to calculate ellipsometric angles for an arbitrary refractive index profile (Lekner 1987) and analytical approximations have also been developed (Charmet and de Gennes 1983). In practice the use of ellipsometry to obtain fine details of the structure of interfaces at the level of tens of angstrom units is likely to be difficult and to require extreme care. 

This method is certainly not the easiest one for the calculation of the reflectivity properties of a single homogenous layer at the interface between two bulk phases, but is advantageous, if the layer has an inner structure, e.g. a continuously varying refractive index normal to the interfaces. For just a few refractive index profiles it is possible to calculate Vp and exactly, but in most cases approximations must be used A continuously varying refractive index profile is subdivided into many thin layers. This method uses matrices to relate the electrical field and its derivative in between adjacent layer and matrices to account for changes of the phase in each layer. In every layer the refractive index is assumed to be constant. Obviously this approximation gets the better the finer the subdivision is. 

Up to now the only simplification is the discretization of the continuous refractive index profile. The next step is a reduction of the computational effort by using a Taylor-approximation up to the second order in the phaseshift 5 . This approximation is valid if the discrete layer thicknesses are much smaller than the wavelength of the light in other words a sufficiently high number of layers is required. 

A power-law index profile approximation is a well-known method of analyzing the RIP of graded-index (GI) MMFs [2]. In this approximation, the refractive index distribution of a GI MMF is described by... 

The refractive index profiles of GI POFs can be approximated by the following power law [78] ... 

Here we examine a second case of practical importance for which the transit time can be evaluated approximately. We assume that the change in the nonuniformity along the fiber is so slow that the refractive-index profile is virtually uniform, i.e. independent of z, over the distance Zp required for a ray to undergo a half-period, i.e. Zp fln(r,z)/flz 1. An example of a slowly varying fiber is shown in Fig. 5-2. The nonuniformities may be arbitrarily large, e.g. a tapered fiber whose core radius at one end is many multiples of the radius at the other end, provided the taper angle is everywhere small. 

In Chapter 11 we discussed the fundamental properties of modes on optical waveguides. The vector fields of these modes are solutions of Maxwell s source-free equations or, equivalently, the homogeneous vector wave equations. However, we found in Chapter 12 that there are few known refractive-index profiles for which Maxwell s equations lead to exact solutions for the modal fields. Of these the step-profile is probably the only one of practical interest. Even for this relatively simple profile the derivation of the vector modal fields on a fiber is cumbersome. The objective of this chapter is to lay the foundations of an approximation method [1,2], which capitalizes on the small... 

The slowly varying fiber in Fig. 19-1 (a) has the z-dependent refractive-index profile n x,y,z). To construct its local mode fields, we approximate the fiber by the series of cylindrical sections in Fig. 19-1 (b) [1]. The profile is independent... 

Fig. 19-1 (a) A nonuniform fiber varies along its length and has refractive-index profile n (x,y,z) and (b) the approximate model is a series of sections, where denotes the center of each section and dz is the length of a particular section. 

Consider a single-mode, elliptical fiber whose refractive-index profile rotates along its length, as shown in Fig. 19-2. We recall from Section 13-5 that in the weak-guidance approximation one fundamental mode of the cylindrically symmetric, elliptical fiber is plane polarized with its transverse electric field parallel to the x-axis in Fig. 19-2(a) and has propagation constant The other fundamental mode s field is parallel to the y-axis... 

The two fibers in Fig. 29-1 have core refractive-index profiles (x, y) and 2 (x, y), referred to a common set of axes, and a uniform cladding of index n i. For convenience we assume both fibers are single moded in isolation of each other, and, within the weak-guidance approximation, we work with the x-polarized fundamental modes. Hence if iVj and N2 are the normalizations of Table 13-2, page 292, we set... 

When each fiber in isolation has a Gaussian refractive-index profile, we can determine the coupling coefficient using the Gaussian approximation of Chapter 15 for the fundamental modes. The profile for the composite waveguide is given in terms of the radial coordinates of Fig. 27-1 (b) by... 

Let n(x, y) and n(x, y, z) be the refractive-index profiles of the uniform and nonuniform waveguides. Within the weak-guidance approximation, the field (x, >, z) of the nonuniform waveguide satisfies the three-dimensional scalar wave equation, which is expressible as... 

Recently, another discovery regarding the coextmsion process was reported.To obtain the maximum bandwidth, the refractive index profile must be optimized. The general refractive index profile of the GI POF can be approximated by the following power law " ... 

The theoretical approach is based on the solution to the mixed type linear/nonlinear generalized Schrodinger equation for spatiotemporal envelope of electrical field with account of transverse spatial derivatives and the transverse profile of refractive index. In the quasi-static approximation, this equation is reduced to the linear/nonlinear Schrodinger equation for spatiotemporal pulse envelope with temporal coordinate given as a parameter. Then the excitation problem can be formulated for a set of stationary light beams with initial amplitude distribution corresponding to temporal envelope of the initial pulse. 

Microscope slides (72 mm x 25 mm x 0.8 mm approximately) and coverslips should be of the best optical quality and of the correct thickness for obtaining quality images (usually 0.17 mm). In the case of counting a particular fiber, an assumption has to be made about the cross-sectional profile of the fiber type chrysotile is usually assumed to be cylindrical in shape, while the amphiboles are considered to have a thickness-to-width ratio of 1.6 1. Discrimination of asbestos fibers using the morphology and refractive index in such cases is aided by prior identification of the fiber types present in the bulk material using PLM or other methods. 

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