Propagation constant leaky modes

The leaky-mode fields have the same functional form as the bound-mode fields, but the modal parameters and propagation constants are complex. 

So far we have presented all the properties necessary for understanding the physical attributes of leaky modes, and for determining their excitation. All these properties, though, require a knowledge of the complex propagation constant for each leaky mode. The solution of the eigenvalue equation for values of F below cutoff must, in general, be performed numerically. Only for weakly leaky modes when V is close to cutoff are analytical solutions available [13]. We now consider examples. 

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... 

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