Notes 6

1
ECE 6341
Spring 2009
Prof. David R. Jackson ECE Dept.
Notes 45
Notes 42
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Strip Grating
(Infinite Periodic Structure)
Scattering from a 1-D array of metal strips
(metal-strip grating)
3
Strip Grating (cont.)
Incident field at interface
From symmetry,
4
Strip Grating (cont.)
The strip currents are periodic, except for a
uniform progressive phase shift.
The scattered field should have this same
property.
5
Strip Grating (cont.)
Denote
Then
Hence
(periodic function)
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Strip Grating (cont.)
Assume a complex Fourier series representation
We then have
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Strip Grating (cont.)
Denote
We then have
For z gt 0
For z gt 0
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Strip Grating (cont.)
We can write this as
For z gt 0
For z gt 0
"Floquet waves"
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Strip Grating (cont.)
Magnetic field
We then have
(The field is TMy)
(dont need this equation)
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Strip Grating (cont.)
For z gt 0
For z lt 0
Boundary condition at z 0
Note we only need to satisfy this over one
strip, since the BC is then automatically
satisfied over the other strips.
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Strip Grating (cont.)
Hence
Multiply both sides by and then
integrate over the period.
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Strip Grating (cont.)
Examine the integral
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Strip Grating (cont.)
Hence we have
or
where
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Strip Grating (cont.)
Hence
or
Next, we enforce the EFIE on the n 0 strip.
The EFIE is then automatically satisfied on the
other strips.
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Strip Grating (cont.)
The total electric field on the interface is
EFIE
16
Strip Grating (cont.)
Hence
Introduce basis functions
so
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Strip Grating (cont.)
We then have
Introduce testing function
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Strip Grating (cont.)
We then have
or
or
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Strip Grating (cont.)
Hence
or
Define
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Strip Grating (cont.)
We can then write
or
This is an M ? M matrix equation for the unknown
coefficients cm.
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Strip Grating (cont.)
Approximate solution M 1
Choose
(accurate for narrow strips)
(Galerkins method with a single Maxwell basis
function.)
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Strip Grating (cont.)
or
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Strip Grating (cont.)
Hence we have
which becomes
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Strip Grating (cont.)
Since the Bessel function is an even function, we
have
or
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Strip Grating (cont.)
Recall that
Hence
Note The summation index has been changed to q
to avoid confusion with n.
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Strip Grating (cont.)
Hence
For z gt 0
For z gt 0
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Strip Grating (cont.)
Grating Waves
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Strip Grating (cont.)
To avoid grating waves (waves that propagate)
or
The Floquet waves with n gt 1 must then also be
cutoff.
Set n 1
This will always be satisfied if
since
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Strip Grating (cont.)
Note the same conclusion results from using n
-1.
so