# This problem concerns using the state variable method to determine the EM fields that result when a.

This problem concerns using the state variable method to
determine the EM fields that result when a plane wave, possessing arbitrary
polarization, is obliquely incident on a uniform bianisotropic layer located
between free space on the incident side and an infinite dielectric region on
the transmit side. The EM fields of the bianisotropic system are defined by the
relations of Ref. , namely, are lossy, in general, nonzero complex
constant dyadic quantities. Figure 4.10 shows the general layer system geometry
assuming that an arbitrarily polarized plane wave is obliquely incident on
»

This problem concerns using the state variable method to
determine the EM fields that result when a plane wave, possessing arbitrary
polarization, is obliquely incident on a uniform bianisotropic layer located
between free space on the incident side and an infinite dielectric region on
the transmit side. The EM fields of the bianisotropic system are defined by the
relations of Ref. , namely, are lossy, in general, nonzero complex
constant dyadic quantities. Figure 4.10 shows the general layer system geometry
assuming that an arbitrarily polarized plane wave is obliquely incident on the
system and that characterize the Region 2 material
parameters of the system. The analysis of this problem is to be carried by
formulating state variable and associated matrix equations to solve Maxwells
equations and then using this formulation to solve a specific numerical
example. Complete the following parts to perform the just stated analysis.

(a) Assume that the oblique incident plane wave is given
mathematically by

Assume that the wave vector values kx and kz
are known and given, and that the incident plane waves polarization is
specified by known, given values of ExI = SxI
and EzI = SzI. From Maxwells equations and
the assumed known valuefind the other field components of the
incident plane wave. Find the form of the reflected and transmitted plane waves
that propagate in Regions 1 and 3, respectively.

(b) In Region 2, write out Maxwells curl equations
in dimensionless form lettingusing dimensionless dyadics defined by

(c) Let all EM field components in the Region 2 material
layer be proportional to the factor exp(since an incident plane wave possessing
this factor is incident on the layer, and phase matching must occur at the
interfaces of the slab). Substitute these EM field expressions into Maxwells
normalized equations, and find and write the resulting equations.

(d) From the resulting six equations of (c), simplify these
equations to a set of four equations involving only the transverse electric and
magnetic field components (i.e., -, -components)
by expressing the longitudinal electric and magnetic fields in terms of the
transverse electric and magnetic field components. Put the resulting four
equations in the state variable form

Why, possibly, is it useful to reduce the set of six
equations to a set of four equations involving only tangential field
components?

(e) Solve the state variable equation of (d) for the
eigenfunctions of Region 2 and express the general EM field solution of Region
2 in terms of these eigenfunctions.

(f) By matching EM boundary conditions, using the Regions 1,
2, and 3 solutions of (a) and (e), formulate the final matrix equation from
which all of the unknowns of the system may be determined.

(g) Using the formulation described in (a)(f), solve the
following numerical example assuming: in Region 1, _1 =
1.3 and _1 = 1.8, the incident plane wave has the amplitude SxI
= 1 (V/m), SzI = 0.9 (V/m), kx = sin(_I
) cos(I ), and kz
= sin(_I ) sin(I ), where _I
= 35°, I = 65°; in Region 3,
_3 = 1.9 and _3 = 2.7; in Region 2, we take the
layer thickness and we consider a complicated numerical
example where all material parameters of are taken to be nonzero and are given by

(h) Make sample plots of the EM fields in Regions 1, 2, and
3.

»

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