For the intermediate chromium layer, a linear combination of two Hankel functions
has to be used.
1] are Bessel and Hankel functions
of orders zero and one correspondingly.
This approach is based on Helmholtz's equation in cylindrical coordinates, for which the solution is expressed in terms of Bessel and Hankel functions
in the radial direction and a Fourier expansion in the angular direction.
rho]m] [absolute value of [rho] - [rho]l]) are Hankel functions
of second kind and
Using Equations (20)-(21), the identities  and the following recursive relation when differentiating the Bessel and Hankel functions
the ith, jth derivatives for the matrix entries can be derived
By assuming that [beta]r >> 1 (this entails that the observation domain is only at few wavelengths from the source domain) and by resorting to the asymptotic form of the Hankel functions
for large argument, the relationship [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCI.
it is possible to derive the modified asymptotic expansions for large z corresponding to the Hankel functions
(and hence to the standard Bessel functions [J.
n] are Hankel functions
of the first kind and nth order.
The general solution is a linear combination of radially outward and inward propagating waves expressed as Hankel functions
The fields within the horns can be expressed in terms of cylindrical TE wave functions which include Hankel functions
(TM waves are not supported by SIW structures).
Recall now the Graf's addition formula for Hankel functions
A8) and the asymptotic expressions for the Hankel functions