Scalar wave equation orthogonality

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 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... 

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... 

We note that Eq. (20-11) can be derived directly from the scalar wave equation. In the weak-guidance approximation, the fields ey in Eq. (20-la) satisfy the scalar wave equation. Consequently, the above result follows directly from the orthogonality condition of Eq. (33-5b). 

We showed how to determine the radiation modes of weakly guiding waveguides in Sections 25-9 and 25-10, starting with the transverse electric field e, which is constructed from solutions of the scalar wave equation. However, unlike bound modes, the corresponding magnetic field h, of Eq. (25-23b) does not satisfy the scalar wave equation. This means that the orthogonality and normalization of the radiation modes differ in form from that of the bound modes in Table 13-2, page 292, as we now show. 

Radiation inodes of the scalar wave equation 33-7 Orthogonality and normalization 33-8 Leaky modes... 

The radiation field of the scalar wave equation can be represented by the continuum of scalar radiation modes discussed above, or by a discrete summation of scalar leaky modes and a space wave. This is clear by analogy with the discussion of vector radiation and leaky modes for weakly guiding waveguides in Chapters 25 and 26. Scalar leaky modes have solutions P of Eq. (33-1) below their cutoff values when P becomes complex. Many of the properties of bound modes derived in this chapter also apply to leaky modes. For example, the orthogonality condition of Eq. (33-5a) applies to leaky modes, provided only that the cross-sectional area A. is replaced by the complex area A of Section 24-15 to ensure that the line integral of Eq. (33-4) vanishes. 

In this appendix, we summarize the basic properties of the characteristic solutions to the scalar and vector wave equations. We recall the integral and orthogonality relations, and the addition theorems under coordinate rotations and translations. 

The first equation is scalar, and has a wave solution with velocity Vi = -J c /p). This is the longitudinal wave of eqn (6.7). It is sometimes called an irrotational wave, because V x u = 0 and there is no rotation of the medium. The second equation is vector, and has two degenerate orthogonal solutions with velocity v = s/(cu/p)- These are the transverse or shear waves of eqn (6.6) the degenerate solutions correspond to perpendicular polarization. They are sometimes called divergence-free waves, because V u = 0 and there is no dilation of the medium. Waves in fluids may be considered as a special case with C44 = 0, so that the transverse solutions vanish, and C = B, the adiabatic bulk modulus. 

Equation of quantum state. The Dirac bra-ket formalism of quantum mechanics. Representation of the wave-momentum and coordinates. The adjunct operators. Hermiticity. Normal and adjunct operators. Scalar multiplication. Hilbert space. Dirac function. Orthogonality and orthonormality. Commutators. The completely set of commuting operators. 

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