Modes scalar wave equation

The optimum profile exponent p, which minimizes the modal dispersion and the difference in the delay of all the modes and maximizes the bandwidth, is expressed as follows on the basis of an analysis of scalar wave equation ... 

There are two exceptional cases when the V, In n terms do not appear in the vector wave equations for the transverse fields. These occur for modes with e j = 0 everywhere on a planar waveguide and on a circularly symmetric fiber, and are called TE modes. In these two special cases, it is indeed true that the transverse electric field ey satisfies the scalar wave equation. Section 33-1, everywhere. 

To summarize, the waveguide structure has polarization properties by virtue of its cross-sectional geometry and refractive-index profile. These effects are built into the vector wave equation, Eq. (11-40), through the V, In ifr terms, which are responsible for the hybrid modes. Ignoring these terms completely disregards polarization properties of the waveguide structure and leads to the scalar wave equation. 

If we substitute for W from inside the back cover, the eigenvalue equations for TE modes in Table 12-2 show that U depends only on V and is independent of A. Thus, U is insensitive to waveguide polarization, as anticipated in Section 11-15, and corresponds to the vanishing of the V,ln n terms in Eqs. (30-21b) and (31-21d) for e and h, respectively. Hence the transverse fields satisfy the scalar wave equation everywhere including the interface. In Section 33-1 we show that any solution of the scalar wave equation must be continuous... 

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 showed above that the modes of weakly guiding waveguides are approximately TEM waves, with fields e = e, h S h, and h, related to e, by Eq. (13-1). In an exact analysis, the spatial dependence of e,(x,y) requires solution of Maxwell s equations, or, equivalently, the vector wave equation, Eq. (1 l-40a). However, when A 1, polarization effects due to the waveguide structure are small, and the cartesian components of e, are approximated by solutions of the scalar wave equation. The justification in Section 13-1 is based on the fact that the waveguide is virtually homogeneous as far as polarization effects are concerned when A 1. As we showed in Section 11-16, these effects... 

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. 

We now consider higher-order modes of fibers with circulariy symmetric cross-sections and profiles If we express in cylindrical polar coordinates r, (j> as in Table 30-1, page 592, there are two separable solutions of Eq. (13-8) for each value of These are 4 = f, (r) cos l(j) and V — F, (r) sin Icj), where / is a positive integer and f, (r) satisfies the equation in Table 13-1. Because of symmetry, any pair of orthogonal x- and y-axes may be chosen as optical axes in the fiber cross-section. It also follows from symmetry that there are four possible directions for e, depending on the particular combination of the two solutions of the scalar wave equation used in Eq. (13-7) [1,2]. This is discussed further in Section 32-7. Hence, for each value of / > 0, there are four modes with the fields shown in Table 13-1. These combinations can also be derived without recourse to symmetry properties using the formal methods of Section 32-6. In general, this representation for is the simplest possible, for reasons explained in Section 32-8. 

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 two fundamental, or HEj j, modes and all other pairs of HEi modes were discussed in Section 13-4. Each mode of a particular pair has a transverse electric field whose direction, or polarization, is parallel to one of an arbitrary pair of orthogonal directions in the fiber cross-section [1], Thus, these modes are uniformly polarized. For convenience we take one mode to be x-polarized and the other y-polarized in Fig. 14-1. There is only one solution of the scalar wave equation for these modes, corresponding to 1 = 0 in Eq. (14-4). The transverse fields, given by Eq. (13-9) and repeated in Table 14-1, ignore all polarization properties of the fiber. For future reference, we give the transformation of the components of these fields from cartesian to polar... 

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. 

There are few clad profiles which lead to closed-form solutions of the scalar wave equation. These include profiles (1) and (p) of Fig. 12-8, which are defined in Table 12-9, p. 270, The fields and eigenvalue equations in Table 12-9 are for TE modes and thus depend on the / = 1 solution of Eq. (14-4). Accordingly, it is straightforward to generalize these solutions to arbitrary values of /. For profile (1) with a uniform core and graded cladding, we deduce that the solution of Eq. (14-4) is [9]... 

On a weakly guiding clad fiber of otherwise arbitrary profile, the fraction of fundamental-mode power residing in the core becomes negligible as F-+ 0. For the step profile, this is clear from the plot of in Fig. 14-3(a). Simultaneously, the fields become nearly uniform over the core, as is evident from the intensity distributions in Fig. 14-3(c), and hence are comparatively insensitive to profile shape. Accordingly, we postulate that the fields depend primarily on the profile volume Q of Eq. (14-42). It then follows from the discussion of the previous section that we can relate the fields of an arbitrary profile, with fiber parameter F, to the known fields of the step profile with fiber parameter Fthrough Eq. (14-46). Thus we replace Fby Fin the small-Fforms for the step-profile fiber listed in Table 14-5. The same conclusion can be derived more formally from the scalar wave equation, as we now show. 

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. 

We begin with a brief review of the weak-guidance approximation for fundamental modes on circular fibers. In Sections 13-2 and 13-4, we showed that the two fundamental modes are virtually TEM waves, with transverse fields that are polarized parallel to one of a pair of orthogonal directions. The transverse field components for the x- and y-polarized HEj j modes are given by Eq. (13-9) relative to the axes of Fig. 14-1. The spatial variation Fq (r) is the fundamental-mode solution (/ = 0) of the scalar wave equation in Table 13-1, page 288. Hence... 

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... 

The derivation of the transverse fields of the two fundamental modes on a weakly guiding, noncircular fiber was described in Section 13-5. These fields are given in Table 16-1 in terms of the solution T(x,y) of the scalar wave equation, which in cartesian coordinates has the form... 

The procedure for constructing higher-order modes of noncircular fibers was established in Section 13-8. For each mode the transverse fields are identical in direction and form to the fundamental-mode fields of Table 16-1, except that F now denotes the appropriate higher-order solution of the scalar wave equation of Eq. (16-3). Only when the fiber cross-section is sufficiently close to circular is this representation inappropriate, as explained in Section 13-9. We quantify this transition in the following section. 

By analogy with the fundamental-mode solution for the elliptical fiber of infinite parabolic profile given in Table 16-1, page 356, we assume that the solution 4 (x, y) of the scalar wave equation of Eq. (16-3) can be approximated by setting [1, 2]... 

We next consider the modes of the elliptical fiber which correspond to the / = 1 modes of the circular fiber. The two solutions of the scalar wave equation for the latter are given by... 

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... 

The efficiency with which beams excite the fundamental modes of circular fibers, with the fields of Eq. (13-9), is of particular interest when the fiber is single moded. In order to account for weakly guiding fibers of otherwise arbitrary profile, when analytical solutions of the scalar wave equation for Fo (r) are not available, we use the Gaussian approximation of Chapter 15. The radial dependence of the fundamental-mode transverse fields is approximated in Eq. (15-2) by setting... 

In Chapter 13 we showed how the bound-mode fields of weakly guiding waveguides can be constructed from solutions of the scalar wave equation. With slight modification, the same procedure applies to the radiation-mode fields as well [4]. However, while the bound modes are approximately TEM waves because j8 = = kn, the radiation modes are not close to being... 

The discussion of bound modes in Section 13-3 applies equally to radiation modes on weakly guiding waveguides, except that the fields are no longer predominantly perpendicular to the waveguide axis. However, the cartesian components of the transverse electric field of Eq. (13-7) are still solutions of the scalar wave equation. Thus, if Vj denotes e j or e j, then... 

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