Solutions to the Vector Wave Equations

We showed in Chapter 3 that a physically realizable time-harmonic electromagnetic field (E, H) in a linear, isotropic, homogeneous medium must satisfy the wave equation 

Suppose that, given a scalar function xp and an arbitrary constant vector c, we construct a vector function M  

Therefore, M satisfies the vector wave equation if is a solution to the scalar wave equation 

We may also write M = — c X V, which shows that M is perpendicular to c. Let us construct from M another vector function 

Therefore, M and N have all the required properties of an electromagnetic field they satisfy the vector wave equation, they are divergence-free, the curl of M is proportional to N, and the curl of N is proportional to M. Thus, the problem of finding solutions to the field equations reduces to the comparatively simpler problem of finding solutions to the scalar wave equation. We shall call the scalar function ip a generating function for the vector harmonics M and N the vector c is sometimes called the guiding or pilot vector. 

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The independent solutions to the vector wave equations can be constructed as [215]... 

The Green s tensor can be treated as the solution of the vector wave equation with the right-hand side given, according to formula (13.86), by the product of the... 

The circularly symmetric fiber of Fig. 12-3 has refractive-index profile n(r), and the modal fields have the separable form of Eq. (12-10). If we examine Eq. (30-15), then there are consistent solutions of the vector wave equation provided the fields are independent of (p and e, = e = h = 0. These are TE mode solutions, and satisfies Eq. (30-15b), which in normalized form reduces to... 

The problem is cast in spherical coordinates in order to ensure the surface of the particle coincides with one of the coordinate surfaces, making it much easier to impose the boundary conditions correctly. Solutions to the scalar wave equation are then used to derive two vector fields, linear combinations of which satisfy Maxwell s equations. Particular integrals are found by requiring continuity of the field components across the particle surface. The scattered field at large distances from the particle is then evaluated, leading to explicit expressions for particle cross sections and the single scattering phase function. 

The reason for the intractability of the anisotropic sphere scattering problem is the fundamental mismatch between the symmetry of the optical constants and the shape of the particle. For example, the vector wave equation for a uniaxial material is separable in cylindrical coordinates that is, the solutions to the field equations are cylindrical waves. But the bounding surface of the... 

Now, from any function that satisfies (29), can be formed three independent vectors that satisfy the vector wave equation (28) [42], They are traditionally signified by L = grad (v(/), P = curl (v(/a), and T = curl curl (vj/a), where a is an arbitrary constant vector. Thus, to find solutions to (21), we express the vector B as a linear superposition of the three vectors... 

As in the scalar case (equation (13.66)), we can conclude that, using Green s tensor G of the vector wave equation, one can find the solution to this equation with an arbitrary right-hand side F (r, t), as the convolution of the Green s tensor G with the function F (r, t), i.e.,... 

The cumbersome procedure of generating the Landau expansion from the underlying mean-field theory has recently been made unnecessary because the problem of obtaining a solution to the mean-field equations with any desired symmetry and without further approximation was solved by Matsen and Schick [77]. All functions of position are expanded in a complete set of states that possess the desired symmetry, so that one is left with the equivalent self-consistent equations expressed in terms of the coefficients in the expansion. These equations can be solved numerically to whatever accuracy is possible with the guarantee that the solution has the desired symmetry. In this way solutions with up to 450 different wave vectors have been obtained, providing accuracy to very large incompatibilities / or, equivalently, to very low temperatures. 

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... 

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 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. 

It is convenient to describe normalization of solutions of the scalar wave equation in terms of the vector normalization N in Table 11-1, page 230. By repeating the above argument, it follows that... 

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... 

Thus, the discrete values of P for the bound inodes of Eq. (33-1) are replaced by a continuum of values for P(Q). We explained in Chapter 25 why it is more convenient to work with the radiation mode parameter Q, which is defined inside the back cover. We are also reminded that both the electric and magnetic transverse fields, e, and h, of the vector bound modes of weakly guiding waveguides are solutions of the scalar wave equation. However, only e Q) of the vector radiation modes satisfies the scalar wave equation, as we showed in Chapter 25. 

An approximate solution to the null-field equations can be obtained by approximating the surface fields e-, and h, by the complete set of regular vector spherical wave functions for the interior domain (or the interior wave... 

We shall adopt Eqs. (9-510) and (9-511) as the covariant wave equation for the covariant four-vector amplitude 9ttf(a ) describing a photon. The physically realizable amplitudes correspond to positive frequency solutions of Eq. (9-510), which in addition satisfy the subsidiary condition (9-511). In other words the admissible wave functions satisfy... 

The first group of methods then manipulate a very small subset of vector elements Vi at a time, and a direct method continually updates the affected elements rj. Such methods are collectively known as relaxation methods, and they are primarily used in situations where, for each elements >< to be changed, the set of affected rjt and the matrix elements Ay, are immediately known. This applies in particular to difference approximations and also to the so-called Finite Element Method for obtaining tabular descriptions of the wave function, i.e. a list of values of the wave function at a set of electron positions. (Far some reason, such a description is commonly referred to as a numerical solution to the Schrodinger equation). Relaxation methods have also been applied to the Cl problem in the past, but due to their slow convergence they have been replaced by analytical methods. Even for numerical problems, the relaxation methods are slowly yielding to analytical methods. 

This is clearly a Beltrami equation, but what is more amazing is that the field result (88) describes a solution to the free-space Maxwell equations that, in contrast to standard PWS, the electric (E0) and magnetic (Bo) vectors are parallel [e.g., Eo x Bo = 0, where Eo x Bo = i(E0 A Bo)], the signal (group) velocity of the wave is subluminal (v < c), the field invariants are non-null, and as (91) clearly shows, this wave is not transverse but possesses longitudinal components. Moreover, Rodrigues and Vaz found similar solutions to the free-space Maxwell equations that describe a superluminal (v > c) situation [71]. 

To introduce some of these ideas, let s begin with a square lattice, 18, defined by the translation vectors ai and a2. Suppose there is an H Is orbital on each lattice site. It turns out that the Schrodinger equation in the crystal factors into separate wave equations along the x and y axes, each of them identical to the one-dimensional equation for a linear chain. There is a kx and a ky, the range of each is 0 < kx, ky < itla (a = ai = a2 ). Some typical solutions are shown in 19. 

The wave function has the same amplitude at equivalent positions in each unit cell. Thus, the full electronic structure problem is reduced to a consideration of just the number of electrons in the unit cell (or half that number if the electronic orbitals are assumed to be doubly occupied) and applying boundary conditions to the cell as dictated by Bloch s theorem (Eq. 4.14). Each unit cell face has a partner face that is found by translating the face over a lattice vector R. The solutions to the Schrodinger equation on both faces are equal up to the phase factor exp(zfe R), determining the solutions inside the cell completely. 

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