Bound modes vector wave equations

These vector wave equations are a restatement of Maxwell s equations for an arbitrary refractive index profile. Subject to the requirements that the modal fields are bounded everywhere and decay sufficienfly fasf af large distances from the waveguide, these equations contain all of the information necessary to determine the modal fields and propagation constants of all the guided modes of the waveguide. 

In Chapter 13 we showed how to construct the fields of bound modes on weakly guiding waveguides using simple physical arguments, and then, in Chapter 25, we extended the method to include radiation modes. To complement the physical approach, we now give the formal mathematical derivation using perturbation theory on the vector wave equation. 

The electric and magnetic fields Ej and of a bound mode are source-free solutions of Maxwell s equations, Eq. (30-1), or, equivalently, the vector wave... 

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. 

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. 

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