Bound modes waveguides

In Part II, we pay particular attention to single-mode waveguides that propagate one bound mode only-called the fundamental mode-since they are of great practical interest, both to the field of optical communications [4-6] and to the study of visual photoreceptors of animals [7,8], At this point, we anticipate possible confusion associated with the nomenclature. When we refer to a waveguide that is single moded, we mean that it can propagate only the two polarization states of the fundamental mode. The adjective bound is usually omitted from this description. 

If the waveguide is nonabsorbing, so that the refractive index n(x, y) is real, we show in Section 30-4 that, among various delineations, it is always possible to choose the field components of each bound mode to satisfy the convention... 

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. 

The exact number of bound modes which propagate on a waveguide can be found by counting the discrete solutions of the eigenvalue equation. In general this is a cumbersome procedure, but simple expressions are available for multimode waveguides with F > 1. Examples are given in Sections 36-12 and 36-13. 

Table 11-1 Properties of bound modes. Summary of bound-mode properties, dropping the subscript j when only one mode is involved. A nonabsorbing waveguide is assumed so that e, h, can be taken to be real and e, are then imaginary. The modal amplitude coefficient, a, is defined by Eq. (11-2).

To summarize this part of the chapter, we have laid out inside the back cover physical quantities and waveguide parameters defining light propagation on a waveguide, together with bound-mode parameters. Table 11-1 summarizes the properties of bound modes. 

The basic properties of bound modes on optical waveguides were given in the previous chapter. In this chapter we display these properties explicitly for those few profiles which have exact solutions of Maxwell s equations. Our primary objective is to derive analytical expressions for the modal vector fields, which contain all the polarization properties of the waveguide discussed in Section 11-16. We pay particular attention to fundamental modes, since these... 

The total number of bound modes which can propagate on the step-profile planar waveguide is most simply found by counting solutions of the eigenvalue equations. Since the cutoff values of V in Fig. 12-1 are tc/2 apart and there are two polarization states for each value of cutoff, we deduce that M( = Int(4 VIn), where Int denotes the largest integer nor exceeding 4 VIn. Hence the number of modes depends only on V. 

The waveguide is weakly guiding if Mj-o - ci equivalently, if the profile height is small, i.e. 1. The dependence of C/ on A in the TM mode eigenvalue equations of Table 12-2 is then slight. A further consequence is that the range of propagation constant values for bound modes, kn < kn, ... 

It is harder to count the number of bound mode solutions for the fiber, than it is for the planar waveguide, because the roots of the eigenvalue equations in Table 12-4(a) are now not uniformly spaced, even at cutoff. However, we show in Section 36-13 that, provided the fiber is multimoded with 1, My = Int the smallest integer just exceeding F /2. 

We start by recalling from Eq. (11-6) that the electric and magnetic fields E and H of individual bound modes of a waveguide are expressible as... 

We saw in Chapter 11 that the propagation constant )5 of a bound mode is given by an eigenvalue equation, provided the waveguide parameter Kis above the cutoff value for that mode. If we now investigate this solution for values... 

The total electromagnetic fields of an optical waveguide can be expressed as the sum of two components. One component, expressible as a summation over bound modes, describes the spatial steady state, where energy is guided indefinitely along a nonabsorbing waveguide. The second component is the radiation field, which describes the spatial transient. In Chapter 24 we... 

Radiation modes, like bound modes, obey nearly all of the general properties presented in Chapter 11. As we show below, the essential difference between bound and radiation modes is that there is no eigenvalue equation for radiation modes because of the relaxation of the boundary condition that the fields are evanescent as r -> oo. Furthermore, at large distances from the waveguide, their fields are oscillatory, or wavelike, and do not have the evanescent behavior of the bound-mode fields [1, 2],... 

The discrete set of bound-mode propagation constants means that the fields of the waveguide in the spatial steady state are given by the finite sum over all bound modes in Eq. (11-2). In contrast, each radiation and evanescent mode can take any of the continuum of propagation-constant values given in Table 25-1, and thus an integration over all values of is necessary. However, like bound modes, the total radiation field requires a summation over the subscript j of Eq. (25-1) to account for the transverse fields of different modes. However, rather than use / as the continuum variable, when P is real for radiation modes and imaginary for evanescent modes, we use instead the modal parameter Q, defined below, in order to simplify the notation. We take / to be the positive root of the inverse relation whence... 

In Section 11-4, we showed that each bound mode is orthogonal to every other bound mode and to all radiation modes. The orthogonality properties of radiation modes are derived in Section 31-3. On a nonabsorbing waveguide, the th and kth forward-propagating radiation modes obey... 

Given a source of excitation, either at the waveguide endface or within the waveguide, the modal amplitudes aj Q) of radiation modes are found by analogy with the bound-mode amplitudes aj. 

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. 

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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