For TE modes, we know that
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(1) |
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(2) |
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(3) |
where
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and when normalized
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For TM modes, we know that
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(4) | |
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(5) | |
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(6) | |
and when normalized
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The excitation amplitudes are
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(7) |
where ![]() ![]()
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(8) |
Hence
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(9) |
Taking the integral gives
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(10) |
where ![]() ![]() ![]()
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(11) |
since at large m,n the wave number is imaginary as all TM modes are cutoff modes. Likewise
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(12) |
as all the modes are cutoff for ![]() ![]()
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(13) |