# A plane wave of E 0 is obliquely incident on a four-layersystem located between free space on the…

A plane wave of E0 is obliquely incident
on a four-layersystem located between free space on the incident side and an
infinite dielectric region on the transmit side. Using the coordinate system of
Figure 5.19 (assuming the vertical direction in this figure to be the
x-direction), the incident plane wave is given by  where _I
, _I are the relative permeability and permittivity,
respectively, of the incident region and _I is the
incident angle of the plane wave.

(a) Using general layer thicknesses (i.e., s_,
_ = 1, 2, 3, 4) and material parameters (i.e., relative
permeability
»

A plane wave of E0 is obliquely incident
on a four-layersystem located between free space on the incident side and an
infinite dielectric region on the transmit side. Using the coordinate system of
Figure 5.19 (assuming the vertical direction in this figure to be the
x-direction), the incident plane wave is given by  where _I
, _I are the relative permeability and permittivity,
respectively, of the incident region and _I is the
incident angle of the plane wave.

(a) Using general layer thicknesses (i.e., s_,
_ = 1, 2, 3, 4) and material parameters (i.e., relative
permeability __ and permittivity, __,
_ = 1, 2, 3, 4), write Maxwells equations in component form in
any region for the problem under consideration.

(b) Put the resulting equations of (a) in state variable
form for the EM field components that are parallel to the interfaces of the
multilayer system and solve for the eigenfunctions associated with the state
variable equations.

(c) Using the eigenfunction solutions of (b), find the
general form of the EM fields in each region, expressed in terms of forward and
backward, obliquely, traveling plane waves of unknown amplitude.

(d) Assuming that the transmit region has material
parameters _T, _T, by matching EM
boundary conditions at all material interfaces, formulate matrix equations from
which the EM fields in all regions of the system may be found.

(e) Form a ladder matrix equation, which relates the EM
field amplitudes of the plane waves of (b) in the layer that is adjacent
to the transmit side (call this layer Region 4, assumed to have width s4)
to the EM field amplitudes of the plane waves in the layer that is adjacent to
the incident side (call this layer Region 1, assumed to have width s1).

(f) Using the ladder matrix found in (e), matching EM
boundary conditions, formulate a reduced matrix equation from which the EM
fields in all regions of the system may be found.

(g) Picking specific layer thicknesses (i.e., s_,
_ = 1, 2, 3, 4) and lossless layer material parameters (i.e.,
relative permeability __ and permittivity, __,
_ = 1, 2, 3, 4) of your choice, solve numerically all equations
formulated in (a)(f). Present numerical results of (d), (e), and (f).

(h) For your numerical example, verify that the conservation
of power holds in the system.

»

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