Modes arbitrary waveguides

Fundamental modes of waveguides of arbitrary cross-section 285 13-6 Polarization corrections to the scalar propagation constant 286 13-7 Higher-order modes of circular fibers 287... 

Higher-order modes of waveguides of arbitrary cross-section 289... 

This chapter shows how radiation modes are used to construct the total radiation fields. We first establish the general properties of radiation modes on arbitrary waveguides and then parallel Chapter 13 with a discussion of radiation modes on weakly guiding waveguides. Finally, we give examples of the application of radiation modes to complement the Green s function solutions given in earlier chapters. 

The electric and magnetic fields of a mode of an arbitrary waveguide are expressible in the separable forms of Eqs. (11-3) and (11-6)... 

Relationship with coupled mode equations for arbitrary waveguides... 

We derived the set of coupled local-mode equations for arbitrary waveguides in Section 31-14. In the weak-guidance approximation, the modal fields in the coupling coefficients of Eq. (31-65c) have only transverse components. If we use Table 13-1, page 288, to relate these components to the corresponding normalized solutions of the scalar wave equation of Eq. (33-45), we find with the help of Eq. (33-48b) that... 

These vector wave equations are a restatement of Maxwell s equations for an arbitrary refractive index profile. Subject to the requirements that the modal fields are bounded everywhere and decay sufficienfly fasf af large distances from the waveguide, these equations contain all of the information necessary to determine the modal fields and propagation constants of all the guided modes of the waveguide. 

Thus the modal and ray transit times are equal only when tj - 1. This condition is satisfied only by those rays belonging to modes well above cutoff, i.e. when Vp U, or, equivalently, when 0 < 0c- Hence is inaccurate for arbitrary values of 9. This inaccuracy arises because the ray transit time ignores diffraction effects, which were discussed in Chapter 10. The step-profile planar waveguide is a special case, however, because all diffraction effects can be accounted for exactly by including the lateral shift at each reflection, together with recognizing the preferred ray directions. TWs was carried out in Section 10-6, and for rays, or local plane waves, whose electric field is polarized in the y-direction in Fig. 10-2, leads to the modified ray transit time of Eq. (10-13). If we use Table 36-1 to express 0, and 0(.in terms of U, Vand Wand substitute rj for TE modes from Table 12-2, we find that Eqs. (10-13) and (12-8) are identical since 0 = 0. It is readily verified that the same conclusion holds for TM modes and local plane waves whose magnetic field is polarized in the y-direction of Fig. 10-2. 

First we consider fundamental modes and then higher-order modes. Like the exact propagation constant P, the scalar propagation constant p is largest for fundamental modes. It is convenient to distinguish between fibers of circular cross-section and waveguides of arbitrary cross-section. 

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. 

The fiber or waveguide in Fig. 21-1 has arbitrary refractive-index profile and cross-sectional geometry, and supports modes with the properties described in Chapter 11. Outside of the region occupied by currents, the total fields E and H... 

The description of leaky modes comes from several logical approaches. We begin with a heuristic argument and follow with the detailed mathematical formulation which applies to waveguides of arbitrary cross-section and profile. For clarity, we omit modal subscripts when discussing an individual leaky mode. 

We now examine the properties of leaky modes on nonabsorbing, clad fibers of arbitrary profile and cross-section. Later, in Section 24-20, we briefly discuss the leaky modes of planar waveguides. 

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. 

The results of this section apply to all fundamental and higher-order modes on weakly guiding waveguides of arbitrary cross-section, and parallel the results of Sections 13-5 and 13-8 derived by physical arguments. In addition they apply to the fundamental modes of fibers with circular cross-sections, discussed in the next section. 

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