Scalar wave equation propagation constant

We next consider a waveguide with a nonuniform refractive-index profile n = n(x, y). The propagation constant now depends on the orientation of the electric field, and the modes are no longer TEM waves. In general the modal fields are not solutions of the scalar wave equation but obey the vector wave... 

We emphasize that in Eq. (13-8), P denotes the propagation constant for the scalar wave equation, as distinct from the exact propagation constant P for the vector wave equation. In Section 33-1 we show that any solution of the scalar wave equation and its first derivatives are continuous everywhere. Together with the requirement that be bounded everywhere, this property leads to an eigenvalue equation for the allowed values of p. 

The fundamental and HEi , modes of a fiber of circular cross-section are formed from the scalar wave equation solution with no azimuthal variation. Hence in Eq. (13-8) depends only on the radial position r. There is no perferred axis of symmetry in the circular cross-section. Thus, in this exceptional case, the transverse electric field can be directed so that it is everywhere parallel to one of an arbitrary pair of orthogonal directions. If we denote this pair of directions by x- and y-axes, as in Fig. 12-3, then there are two fundamental or HEi , modes, one with its transverse electric field parallel to the x-direction, and the other parallel to the y-direction. The symmetry also requires that the scalar propagation constants of each pair of modes are equal. 

The expressions in Table 13-2 for the group velocity and distortion parameter are given in terms of solutions of the scalar wave equation. Given the polarization correction dfi to the scalar propagation constant, we can write down higher-order corrections to these expressions. This is facilitated by first defining the mode parameter U associated with the scalar propagation constant... 

The corrections SPi to the scalar propagation constant are given in Table 14-1 in terms of /j and I2. In the numerator of each expression, the derivative d//d J is the Dirac delta function 3(R — 1), as explained in Section 14-6, and the integral in the denominator is given in Table 14-6. This leads to the expressions for SPi and the corresponding SUt in the same table. There is no correction for the TEo modes, whose fields satisfy the scalar wave equation exactly. 

The fundamental modes of the infinite parabolic profile fiber have a Gaussian spatial variation it is the exact solution of the scalar wave equation. Thus, the essence of the Gaussian approximation is the approximation of the fundamental-mode fields of an arbitrary profile fiber by the fundamentalmode fields of some parabolic profile fiber, the particular profile being determined from the stationary expression for the propagation constant in Table 15-1. Clearly this approach can be generalized to apply to higher-order modes, by fitting the appropriate solution for the infinite parabolic profile [9]. 

The simplest example of a noncircular waveguide is the planar waveguide of Chapter 12, whose modes are either TE or TM, as explained in Section 11-16. For each TE mode the electric field lies in the cross-section and is uniformly polarized. Consequently the weak-guidance solution is identical to the exact solution for the field ey and the propagation constant. Both satisfy the scalar wave equation of Eq. (12-16), and examples with analytical solutions are given in Table 12-7, page 264. Within the weak-guidance approximation the... 

When the perturbed and unperturbed fibers are weakly guiding, the variation in both profiles n and n is small. The modal fields can then be constructed from solutions of the scalar wave equation, as described in Chapter 13. If T and jS denote the unknown solution and propagation constant for the perturbed fiber. 

The propagation constants associated with the two fundamental-mode solutions of the scalar wave equation are generally distinct because of the... 

We derived the exact solution of the scalar wave equation for the double parabolic profile in Section 16-8. The propagation constants for the fundamental modes are given implicitly by the eigenvalue equations of Eq. (16-35). If the normalized separation is sufficiently large to satisfy d/p > it can be readily verified that the... 

In order to correct the scalar propagation constant, we first derive a relationship between the scalar field S, and propagation constant P, derived from the scalar wave equation, and the exact field e and propagation constant p derived from the vector wave equation. Thus, from Eqs. (32-15) and (32-3) we have... 

We recall from Chapters 11 and 12 that modes with = 0 everywhere are TE modes. Table 32-1 shows that the term V, -e, in Eq. (32-22) is proportional to on weakly guiding waveguide. Consequently V,-e, = 0, and the scalar propagation constant is identical to the exact propagation constant. This is in keeping with Chapter 12, where we showed that the exact TE mode fields are derivable from the scalar wave equation. 

When the cross-section of the waveguide is noncircular, there is only one solution V of the scalar wave equation for each discrete value of the scalar propagation constant in Eq. (13-8). The direction of e, as expressed by Eq. (13-7), then takes the general form... 

The combination of solutions of the scalar wave equation for the transverse fields of weakly guiding fibers of circular cross-section are given in Table 13-1, page 288. As we showed in the previous section, these combinations can be derived using perturbation theory. In this section we show how the combinations can be deduced using only symmetry arguments [2]. We start with the four vector solutions constructed from the solutions of the scalar wave equation with the common propagation constant P, and denote them by... 

We discussed higher-order modes of fibers with nearly circular cross-sections in Section 13-9, and showed that the transverse electric field must take the forms given at the bottom of Table 13-1, page 288. These forms are in terms of the solutions of the scalar wave equation of Eq. (32-37) and unknown constants a+ and a. When the cross-section is exactly circular, the two solutions of Eq. (32-37) have the same propagation constant. However, if the cross-section is only near to circular, the two solutions of the scalar wave equation have similar but distinct propagation constants P and p , as discussed in Section 13-8. 

We showed in Section 13-3 that the cartesian components of the transverse electric field are solutions of the scalar wave equation. If denotes either component of Eq. (13-7), and P is the scalar propagation constant, then... 

In Section 11-13 we showed that the exact propagation constant is given explicitly in terms of integrals over the vector modal fields. Here we derive the analogous expression for the scalar propagation constant in terms of scalar solutions of the scalar wave equation. Starting with Eq. (33-1), we multiply by P and integrate over the infinite cross-section to obtain... 

Starting with the scalar wave equation, we can derive a reciprocal relation between modes of different waveguides. Let T and fi be the field and propagation constant of a... 

There are few known refractive-index profiles which have closed-form solutions of the scalar wave equation, as discussed in Section 14-8, and even fewer profiles which have analytical expressions for the propagation constants as well However, in the case of the infinite power-law profiles on circular fibers, we can derive closed-form expressions for... 

The local propagation constant can be regarded as a modified propagation constant which accounts for the bend. In other words, we can determine the field of the bent fiber approximately, by solving the scalar wave equation for the straight fiber, but with p replaced by p. In cylindrical polar coordinates (r, (j>, z), this substitution from Eq. (36-50) into Eq, (14—4), together with a rearrangement of the equation, leads to... 

If we are to account for waveguide polarization properties in the propagation constant, we must add a correction dp to the scalar propagation constant p.To determine Sp exactly we would have to solve the vector wave equation. However, the V, Inn term on the right of Eq. (ll-40a) is small for weakly guiding waveguides, so we use simple perturbation methods in Section 32-4. From Eq. (32-24) we have... 

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