Scalar wave equation normalization

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 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... 

We define n(x, y) as the profile of the composite waveguide, i.e. n = ffi over the core of the first fiber, n = n2 over the core of the second fiber and n = elsewhere, and let P denote the corresponding solution of the scalar wave equation. The second fiber is regarded as a perturbation of the first fiber, in which case the equation satisfied by bi (z) is deduced from Eq. (31-49), by setting fc = 1, and replacing E and by P and Pi/Nj, respectively, in the weak-guidance limit, where Ni is the normalization and we assume nonabsorbing fibers. Hence [4]... 

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... 

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... 

We derived the set of coupled local-mode equations for arbitrary waveguides in Section 31-14. In the weak-guidance approximation, the modal fields in the coupling coefficients of Eq. (31-65c) have only transverse components. If we use Table 13-1, page 288, to relate these components to the corresponding normalized solutions of the scalar wave equation of Eq. (33-45), we find with the help of Eq. (33-48b) that... 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nonrelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that are normally below the relativistic scale, the Berry phase obtained from the Schrodinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

Because the product (3.8) is a probability, it must integrate to 1.0 when all possible outcomes are taken into account. Consequently, the wave functions are multiplied by arbitrary constants (n ),/2 chosen to make this integral come out to 1.0 over the complete range of motion. These are called normalization constants. It is legitimate to multiply solutions to the Schroedinger equation by an arbitrary constant because they are elements of a closed binary vector space. Multiplication of a solution by any scalar yields another element in the space, hence the product of the normalization constant and the wave function (or any other state vector in Hilbert space) also is a solution. 

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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