Orthogonality leaky modes

We emphasize that leaky modes do not obey the power orthogonality of Eq. (11-13). This is because x h is not an analytic function and thus cannot be continued into the complex plane as is necessary for the above generalization. 

As we now have orthogonality relations and normalization expressions for leaky modes, results which were derived for bound modes in earlier chapters can simply be extended to apply to leaky modes. These include the perturbation expressions of Chapter 18, the modal amplitudes due to illumination in Chapter 20, and the excitation and scattering effects of current sources in Chapters 21 to 23. We give an example of leaky-mode excitation by a source in Section 24—23. 

Sammut, R. and Snyder A. W. (1976) Leaky modes on a dielectric waveguide-orthogonality and excitation. Appl. Opt., 15, 1040-4. 

The first application of the conjugated form of the reciprocity theorem demonstrates orthogonality of bound, radiation and leaky modes of nonabsorbing waveguides. Consider two modes propagating in the forward direction with propagation constants Pj nd The subscripts jand k may refer to two different bound modes, but, in the case... 

For the leaky modes of Chapter 24, orthogonality is expressed by Eq. (31-14) if A is replaced by the complex cross-section A, as discussed in Section 24-15. 

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. 

The fully leaky guided mode technique makes use of a thin 3 pm) HAN sample between glass plates with a refractive index lower than both the principal indices of the nematic. A p or 5 polarized He-Ne laser beam, with the electric vector in the plane of incidence or orthogonal to it, respectively, incident in an appropriate range of incident angles / (see... 

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