Step-profile waveguides fibers

A waveguide is said to be multimoded or overmoded if V> when many bound modes can propagate. At the opposite extreme, when V is sufficiently small so that only the two polarization states of the fundamental mode can propagate, the waveguide is said to be single-moded. For example, the step-profile, circular fiber is single-moded when V < 2.405, as we show in Section 12-9. 

Step-profile planar waveguide 12-20 Step-profile uniaxial fiber... 

The step-profile waveguide has a core of uniform refractive index n, surrounded by a cladding of uniform refractive index n, which is assumed unbounded. Thus the only variation in profile is a step, or jump, discontinuity at the core-cladding interface in Fig. 11-1 (a). This profile has exact analytical solutions for the modal fields on planar waveguides, circularly symmetric fibers and elliptical fibers. 

The construction of ray paths within the core of the step-profile waveguides of Chapters 1 and 2 is based on straight-line trajectories, which are solutions of the ray-path equation of Eq. (1-18) in a uniform medium. When the core is graded, the cartesian component equations of the ray-path equation follow directly, as in Eqs. (1-19) and (2-49). Here we derive the corresponding component equations in directions defined by the cylindrical polar coordinates (r, 0, z) of Fig. 2-1, for application to fibers with graded profiles n(r) in Chapter 2, and, by simple generalization, to slowly varying fibers with profiles n(r, z) in Chapter 5. 

In Section 1-5 we defined the ray transit time t as the time taken for a ray to propagate distance z along a waveguide. For the step-profile fiber, we deduce from Eqs. (1-14), (2-10) and (2-11) that... 

For handy reference we have included definitions of all quantities relevant to the description of propagation on step-profile fibers and planar waveguides inside the front cover. 

The general shape of the ray paths can be deduced from the paths of Fig. 1-8 for a graded-prolile planar waveguide and the paths of Fig. 2-2 for the step-profile fiber. Assuming that the paths are confined to the core, they have the characteristic forms shown in Fig. 2-4. Meridional rays cross the fiber axis... 

We emphasize that, in general, the transit times depend on both P and /,and reduce to the corresponding expressions for planar waveguides only in the case of meridional rays. However, transit times are independent of /, i.e. independent of skewness, for certain profiles, including the step and clad power-law profiles, as we show below. We also recall from Section 1-9 that graded profiles tend to equalize transit times compared to the step profile. Unlike planar waveguides, though, there is no known profile for which complete equalization of all ray transit times on a fiber is possible. 

The approximations in these equations are for weakly guiding fibers and paraxial rays. Hence the optimum profile is close to parabolic. The pulse width is a factor of A/8, or dl/16, times that for the step profile, in Eq. (3-3), and is therefore considerably reduced. Since l/tj is one measure of the informationcarrying capacity of a waveguide, we deduce that capacity is increased by a factor of 8/A, or 16/0. We plot tj of Eq. (3-7) as the normalized time ctjzn o against q, corresponding to the solid curve in Fig. 3-3, for A = 0.01 or 0c = 0.14. There is a cusp at q pt, which means that ray dispersion is very sensitive to small variations about qopt- For example, when q = opt the pulse width increases by a factor of nearly 10. The normalized pulse width for a step profile with the same value of A is included for comparison. 

We showed in Section 2-13 that the transit time for a step-profile fiber is independent of the cross-sectional geometry. Consequently Eqs. (3-2) and (3-3) give the ray dispersion for step-profile fibers of arbitrary cross-section. We also found in Section 2-13 that the ray transit time for the noncircular, clad power-law profiles of Eq. (2-55) is identical to the transit time for the symmetric, clad power-law profiles in Table 2-1, page 40, i.e. dependent on only. Thus Eqs. (3-8) and (3-9) also give the optimum profile and minimum pulse spread for those noncircular profiles [5], which includes the clad parabolic-profile fiber of elliptical cross-section. In other words, ray dispersion on step-profilefibers of arbitrary cross-section and clad power-law profilefibers of noncircular cross-section is also given by the corresponding solutions for planar waveguides. 

The core and cladding indices of the step-profile fiber are nco(- ) and respectively, when the materials are dispersive. We showed in Section 2-5 that the ray transit time in this situation is identical to the planar-waveguide expression of Eq. (1-17), which involves the group index g of Eq. (1-16). By analogy with the derivation in Section 3-1, we deduce that the pulse spread is given by... 

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... 

Fig. 14-10 (a) The distortion parameter for the fundamental modes of clad power-law profiles and (b) the values of the fiber parameter at which waveguide dispersion vanishes, the q= oo line corresponding to the value for the step profile. 

The composite waveguide consists of two identical, step-profile fibers of core radius P = Pi = P2 in Fig. 18-4 and center-to-center separation d. In isolation the fibers are single moded. Within each core n = and n = in the surrounding medium, while = co over the core of the first fiber and n, = n elsewhere. 

Consider a composite waveguide consisting of two step-profile fibers of core radii p and p + 5p, and common core and cladding indices and n, . Using the notation of the previous section, the difference — 2 in fundamental-mode propagation constants is given by — of Eq. (18-13a), and C follows from Eq. (18-42), assuming the fibers are well separated. Hence... 

Example Analytical solution for the step-profile fiber 24-19 Example Numerical solution for the step-profile fiber 24-20 Example Step-profile planar waveguide... 

Optical Waveguide Theory Step-profile fiber... 

The attenuation of leaky rays on multimode fibers is discussed in Section 7-1. For higher-order modes, it is intuitive that there should be good agreement between the leaky-mode and corresponding leaky-ray attenuation coefficients. We discuss this agreement both qualitatively and quantitatively in Section 36-11 for step-profile planar waveguides and fibers. 

The radiation-mode fields are solutions of the same equations satisfied by the bound-mode fields, so that whenever an exact solution exists for bound modes, a corresponding solution for radiation modes exists. We showed in Chapter 12 that, for waveguides with arbitrary variation in profile, there are few known profiles for which exact solutions of Maxwell s equations can be obtained analytically. Even in these cases, the expressions for the radiation-mode fields are generally more complex than those for the bound-mode fields. In the following section we consider the step-profile fiber. The radiation-mode fields of the step-profile planar waveguide can be derived similarly. 

Fig. 36-1 Local plane wave decomposition of the modal fields within (a) the core of a step-profile planar waveguide and (b) the core cross-section of a step-profile fiber.

Table 36—1 Ray and modal parameters for step-profile fibers and planar waveguides. Parameters are defined at the front and back of the book.

The weak-guidance approximation, described in Chapter 13, greatly simplifies the determination of the modal fields of optical waveguides, because it depends on solutions of the scalar wave equation, rather than on vector solutions of Maxwell s equations. For circular fibers, with an arbitrary profile, the scalar wave equation must normally be solved by purely numerical methods. We discussed the few profiles that have analytical solutions in Chapter 14. These solutions, including those for profiles of practical interest such as the step and clad power-law profiles, are given in terms of special functions or by series expansions, which usually necessitate tables or numerical evaluation to reveal the physical attributes of the modes. 

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