Abstract

A scheme is described for calculating the scattering parameters of patterned conductive films in waveguide. The films can have non-uniform, non-isotropic, and non-local sheet impedances. Once the scattering parameters are known, they can be combined with the scattering parameters of paths, dielectric slabs, and waveguide steps to build up models of complicated components comprising patterned films in profiled lightpipes and cavities. It is then straightforward to calculate the Stokes fields of the total reception pattern, the natural optical modes to which the component is sensitive, the Stokes fields of the individual natural modes, and the spatial state of coherence. The method is demonstrated by modeling an absorbing pixel in a length of shorted multimode waveguide. The natural optical modes change from being those of the waveguide to those of a free-space pixel as the size of the absorber is reduced.

© 2010 Optical Society of America

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  1. P. J. B. Clarricoats and A. D. Olver, Corrugated Horns for Microwave Antennas (Peter Peregrinus Ltd, 1984).
    [CrossRef]
  2. R. Padman and J. A. Murphy, “Radiation patterns of scalar lightpipes,” Infrared Phys. 31, 441–446 (1991).
    [CrossRef]
  3. J. A. Murphy and R. Padman, “Radiation patterns of few-moded horns and condensing lightpipes,” Infrared Phys. 31, 291–299 (1991).
    [CrossRef]
  4. A. Wexler, “Solution of waveguide discontinuities by modal analysis,” IEEE Trans. Microwave Theory Tech. 15, 508–517 (1967).
    [CrossRef]
  5. A. D. Olver, P. J. B. Clarricoats, A. A. Kishk, and L. Shafai, Microwave Horns and Feeds (IEEE Press, 1994) Chp. 4.
    [CrossRef]
  6. R. C. Hall, R. Mittra, and K. M. Mitzner,“Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
    [CrossRef]
  7. Chen-To Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE Press, 1994).
  8. S. Withington and C. N. Thomas, “Optical theory of partially coherent thin-film energy-absorbing structures for power detectors and imaging arrays,” J. Opt. Soc. Am. A 26, 1382–1392 (2009).
    [CrossRef]
  9. C. N. Thomas and S. Withington, “Electromagnetic simulations of the partially coherent optical behavior of resistive film TES detectors,” in 21st International Symposium on Space Terahertz Technology (2010), in press.
  10. S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 3, Chap. 3.
  11. D. T. Chuss, E. J. Wollack, S. Harvey Moseley, S. Withington, and G. Saklatvala, “Diffraction considerations for planar detectors in the few-mode limit,” Publ. Astron. Soc. Pac. 120, 430–438 (2008).
    [CrossRef]
  12. C. N. Thomas, S. Withington, D. T. Chuss, E. J. Wollack, and S. Harvey Moseley, “Modeling the intensity and polarization response of planar bolometric detectors,” J. Opt. Soc. Am. 27, 1219–1231 (2010).
    [CrossRef]
  13. S. Withington and J. A. Murphy, “Modal analysis of partially coherent submillimetre-wave quasioptical systems,” IEEE Trans. Antennas Propag. 46, 1651–1659 (1998).
    [CrossRef]

2010 (2)

C. N. Thomas and S. Withington, “Electromagnetic simulations of the partially coherent optical behavior of resistive film TES detectors,” in 21st International Symposium on Space Terahertz Technology (2010), in press.

C. N. Thomas, S. Withington, D. T. Chuss, E. J. Wollack, and S. Harvey Moseley, “Modeling the intensity and polarization response of planar bolometric detectors,” J. Opt. Soc. Am. 27, 1219–1231 (2010).
[CrossRef]

2009 (1)

2008 (1)

D. T. Chuss, E. J. Wollack, S. Harvey Moseley, S. Withington, and G. Saklatvala, “Diffraction considerations for planar detectors in the few-mode limit,” Publ. Astron. Soc. Pac. 120, 430–438 (2008).
[CrossRef]

1998 (1)

S. Withington and J. A. Murphy, “Modal analysis of partially coherent submillimetre-wave quasioptical systems,” IEEE Trans. Antennas Propag. 46, 1651–1659 (1998).
[CrossRef]

1994 (2)

Chen-To Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE Press, 1994).

A. D. Olver, P. J. B. Clarricoats, A. A. Kishk, and L. Shafai, Microwave Horns and Feeds (IEEE Press, 1994) Chp. 4.
[CrossRef]

1991 (2)

R. Padman and J. A. Murphy, “Radiation patterns of scalar lightpipes,” Infrared Phys. 31, 441–446 (1991).
[CrossRef]

J. A. Murphy and R. Padman, “Radiation patterns of few-moded horns and condensing lightpipes,” Infrared Phys. 31, 291–299 (1991).
[CrossRef]

1989 (1)

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 3, Chap. 3.

1988 (1)

R. C. Hall, R. Mittra, and K. M. Mitzner,“Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
[CrossRef]

1984 (1)

P. J. B. Clarricoats and A. D. Olver, Corrugated Horns for Microwave Antennas (Peter Peregrinus Ltd, 1984).
[CrossRef]

1967 (1)

A. Wexler, “Solution of waveguide discontinuities by modal analysis,” IEEE Trans. Microwave Theory Tech. 15, 508–517 (1967).
[CrossRef]

Chuss, D. T.

C. N. Thomas, S. Withington, D. T. Chuss, E. J. Wollack, and S. Harvey Moseley, “Modeling the intensity and polarization response of planar bolometric detectors,” J. Opt. Soc. Am. 27, 1219–1231 (2010).
[CrossRef]

D. T. Chuss, E. J. Wollack, S. Harvey Moseley, S. Withington, and G. Saklatvala, “Diffraction considerations for planar detectors in the few-mode limit,” Publ. Astron. Soc. Pac. 120, 430–438 (2008).
[CrossRef]

Clarricoats, P. J. B.

A. D. Olver, P. J. B. Clarricoats, A. A. Kishk, and L. Shafai, Microwave Horns and Feeds (IEEE Press, 1994) Chp. 4.
[CrossRef]

P. J. B. Clarricoats and A. D. Olver, Corrugated Horns for Microwave Antennas (Peter Peregrinus Ltd, 1984).
[CrossRef]

Hall, R. C.

R. C. Hall, R. Mittra, and K. M. Mitzner,“Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
[CrossRef]

Harvey Moseley, S.

C. N. Thomas, S. Withington, D. T. Chuss, E. J. Wollack, and S. Harvey Moseley, “Modeling the intensity and polarization response of planar bolometric detectors,” J. Opt. Soc. Am. 27, 1219–1231 (2010).
[CrossRef]

D. T. Chuss, E. J. Wollack, S. Harvey Moseley, S. Withington, and G. Saklatvala, “Diffraction considerations for planar detectors in the few-mode limit,” Publ. Astron. Soc. Pac. 120, 430–438 (2008).
[CrossRef]

Kishk, A. A.

A. D. Olver, P. J. B. Clarricoats, A. A. Kishk, and L. Shafai, Microwave Horns and Feeds (IEEE Press, 1994) Chp. 4.
[CrossRef]

Kravtsov, Yu. A.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 3, Chap. 3.

Mittra, R.

R. C. Hall, R. Mittra, and K. M. Mitzner,“Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
[CrossRef]

Mitzner, K. M.

R. C. Hall, R. Mittra, and K. M. Mitzner,“Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
[CrossRef]

Murphy, J. A.

S. Withington and J. A. Murphy, “Modal analysis of partially coherent submillimetre-wave quasioptical systems,” IEEE Trans. Antennas Propag. 46, 1651–1659 (1998).
[CrossRef]

R. Padman and J. A. Murphy, “Radiation patterns of scalar lightpipes,” Infrared Phys. 31, 441–446 (1991).
[CrossRef]

J. A. Murphy and R. Padman, “Radiation patterns of few-moded horns and condensing lightpipes,” Infrared Phys. 31, 291–299 (1991).
[CrossRef]

Olver, A. D.

A. D. Olver, P. J. B. Clarricoats, A. A. Kishk, and L. Shafai, Microwave Horns and Feeds (IEEE Press, 1994) Chp. 4.
[CrossRef]

P. J. B. Clarricoats and A. D. Olver, Corrugated Horns for Microwave Antennas (Peter Peregrinus Ltd, 1984).
[CrossRef]

Padman, R.

R. Padman and J. A. Murphy, “Radiation patterns of scalar lightpipes,” Infrared Phys. 31, 441–446 (1991).
[CrossRef]

J. A. Murphy and R. Padman, “Radiation patterns of few-moded horns and condensing lightpipes,” Infrared Phys. 31, 291–299 (1991).
[CrossRef]

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 3, Chap. 3.

Saklatvala, G.

D. T. Chuss, E. J. Wollack, S. Harvey Moseley, S. Withington, and G. Saklatvala, “Diffraction considerations for planar detectors in the few-mode limit,” Publ. Astron. Soc. Pac. 120, 430–438 (2008).
[CrossRef]

Shafai, L.

A. D. Olver, P. J. B. Clarricoats, A. A. Kishk, and L. Shafai, Microwave Horns and Feeds (IEEE Press, 1994) Chp. 4.
[CrossRef]

Tai, Chen-To

Chen-To Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE Press, 1994).

Tatarskii, V. I.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 3, Chap. 3.

Thomas, C. N.

C. N. Thomas, S. Withington, D. T. Chuss, E. J. Wollack, and S. Harvey Moseley, “Modeling the intensity and polarization response of planar bolometric detectors,” J. Opt. Soc. Am. 27, 1219–1231 (2010).
[CrossRef]

C. N. Thomas and S. Withington, “Electromagnetic simulations of the partially coherent optical behavior of resistive film TES detectors,” in 21st International Symposium on Space Terahertz Technology (2010), in press.

S. Withington and C. N. Thomas, “Optical theory of partially coherent thin-film energy-absorbing structures for power detectors and imaging arrays,” J. Opt. Soc. Am. A 26, 1382–1392 (2009).
[CrossRef]

Wexler, A.

A. Wexler, “Solution of waveguide discontinuities by modal analysis,” IEEE Trans. Microwave Theory Tech. 15, 508–517 (1967).
[CrossRef]

Withington, S.

C. N. Thomas and S. Withington, “Electromagnetic simulations of the partially coherent optical behavior of resistive film TES detectors,” in 21st International Symposium on Space Terahertz Technology (2010), in press.

C. N. Thomas, S. Withington, D. T. Chuss, E. J. Wollack, and S. Harvey Moseley, “Modeling the intensity and polarization response of planar bolometric detectors,” J. Opt. Soc. Am. 27, 1219–1231 (2010).
[CrossRef]

S. Withington and C. N. Thomas, “Optical theory of partially coherent thin-film energy-absorbing structures for power detectors and imaging arrays,” J. Opt. Soc. Am. A 26, 1382–1392 (2009).
[CrossRef]

D. T. Chuss, E. J. Wollack, S. Harvey Moseley, S. Withington, and G. Saklatvala, “Diffraction considerations for planar detectors in the few-mode limit,” Publ. Astron. Soc. Pac. 120, 430–438 (2008).
[CrossRef]

S. Withington and J. A. Murphy, “Modal analysis of partially coherent submillimetre-wave quasioptical systems,” IEEE Trans. Antennas Propag. 46, 1651–1659 (1998).
[CrossRef]

Wollack, E. J.

C. N. Thomas, S. Withington, D. T. Chuss, E. J. Wollack, and S. Harvey Moseley, “Modeling the intensity and polarization response of planar bolometric detectors,” J. Opt. Soc. Am. 27, 1219–1231 (2010).
[CrossRef]

D. T. Chuss, E. J. Wollack, S. Harvey Moseley, S. Withington, and G. Saklatvala, “Diffraction considerations for planar detectors in the few-mode limit,” Publ. Astron. Soc. Pac. 120, 430–438 (2008).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

R. C. Hall, R. Mittra, and K. M. Mitzner,“Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
[CrossRef]

S. Withington and J. A. Murphy, “Modal analysis of partially coherent submillimetre-wave quasioptical systems,” IEEE Trans. Antennas Propag. 46, 1651–1659 (1998).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

A. Wexler, “Solution of waveguide discontinuities by modal analysis,” IEEE Trans. Microwave Theory Tech. 15, 508–517 (1967).
[CrossRef]

Infrared Phys. (2)

R. Padman and J. A. Murphy, “Radiation patterns of scalar lightpipes,” Infrared Phys. 31, 441–446 (1991).
[CrossRef]

J. A. Murphy and R. Padman, “Radiation patterns of few-moded horns and condensing lightpipes,” Infrared Phys. 31, 291–299 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

C. N. Thomas, S. Withington, D. T. Chuss, E. J. Wollack, and S. Harvey Moseley, “Modeling the intensity and polarization response of planar bolometric detectors,” J. Opt. Soc. Am. 27, 1219–1231 (2010).
[CrossRef]

J. Opt. Soc. Am. A (1)

Publ. Astron. Soc. Pac. (1)

D. T. Chuss, E. J. Wollack, S. Harvey Moseley, S. Withington, and G. Saklatvala, “Diffraction considerations for planar detectors in the few-mode limit,” Publ. Astron. Soc. Pac. 120, 430–438 (2008).
[CrossRef]

Other (5)

P. J. B. Clarricoats and A. D. Olver, Corrugated Horns for Microwave Antennas (Peter Peregrinus Ltd, 1984).
[CrossRef]

C. N. Thomas and S. Withington, “Electromagnetic simulations of the partially coherent optical behavior of resistive film TES detectors,” in 21st International Symposium on Space Terahertz Technology (2010), in press.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 3, Chap. 3.

Chen-To Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE Press, 1994).

A. D. Olver, P. J. B. Clarricoats, A. A. Kishk, and L. Shafai, Microwave Horns and Feeds (IEEE Press, 1994) Chp. 4.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

(a) Multimode flared lightpipe, metal-mesh filter, thin-film bolometer, and integrating cavity. (b) Metal-mesh filter and thin-film bolometer array in a rectangular box.

Fig. 2
Fig. 2

Section of waveguide of length L containing resistive planes at positions z k . The cell has ports at z = 0 and z = L .

Fig. 3
Fig. 3

Signal flow graph of a terminated two-port.

Fig. 4
Fig. 4

Eigenvalue spectra of a square absorbing film in a shorted waveguide having dimensions 10 × 10 wavelengths. The film extends the whole way across the waveguide. The different plots correspond to backshort positions of (a) 0.2, (b) 0.3, (c) 0.4, and (d) 0.5 wavelengths.

Fig. 5
Fig. 5

Eigenvalue spectra of square absorbing films in a shorted waveguide having dimensions 10 × 10 wavelengths. The top plot is for absorbers measuring (a) 4 × 4 , (b) 2 × 2 , and (c) 1 × 1 wavelengths. The bottom plot shows the 15 lowest-order terms for absorbers measuring 0.5 × 0.5 (crosses) and 0.25 × 0.25 (stars) wavelengths.

Fig. 6
Fig. 6

Stokes fields of the reception pattern of a 4 × 4 wavelength absorbing pixel in a 10 × 10 wavelength waveguide. Top left, Stokes I; top right, Stokes Q; bottom left, Stokes U; bottom right, Stokes V; all normalized to their own peak values. The relative peak values are I (100%), Q (6.5%), U(0.35%), and V (0.1%).

Fig. 7
Fig. 7

Stokes fields of the reception pattern of a 0.5 × 0.5 wavelength absorbing pixel in a 10 × 10 wavelength waveguide. Top left, Stokes I; top right, Stokes Q; bottom left, Stokes U; bottom right, Stokes V; all normalized to their own peak values. The relative peak values are I (100%), Q (9.0%), U(2.8%), and V (0.06%).

Fig. 8
Fig. 8

Stokes I reception pattern of a 0.5 × 0.5 wavelength absorbing pixel in a 10 × 10 wavelength waveguide. The distance between the pixel and the reference plane in the waveguide is varied. Top left, zero wavelengths; top right, 1.0 wavelength; bottom left, 2.0 wavelengths; bottom right, 3.0 wavelengths. Each plot is normalized to its own peak value with the relative peak values being 100%, 61%, 10%, and 5%.

Fig. 9
Fig. 9

Stokes I reception patterns of various arrays in a 10 × 10 wavelength waveguide. The pixel sizes are as follows: top left, 0.5 wavelengths; top right, 1.0 wavelengths; bottom left, 1.5 wavelengths; bottom right 2.0 wavelengths.

Fig. 10
Fig. 10

Left, Stokes I field, and right, Stokes Q field of two absorptive strips in a waveguide. The strips are displaced laterally and longitudinally, such that as the position of the backshort is changed, the absorption in the films can be turned on and off differentially. All of these images correspond to the front reference plane of the component.

Equations (70)

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S = [ S 11 S 12 S 21 S 22 ] .
[ E k i ( r t ) + E k s ( r t ) ] tangential = S k Z ̿ k s ( r t , r t ) J k s ( r t ) d 2 r t ,
[ E k i ( r t ) + E k s ( r t ) ] tangential = Z ̿ k s ( r t ) J k s ( r t ) ,
E k t ( r t ) = Z ̿ k s ( r t ) J k s ( r t ) k S k G ̿ k k t ( r t | r t ) J k s ( r t ) d 2 r t ,
U n ± ( r t , z ) = U n ( r t ) exp [ ± i β n z ] ,
G ̿ k k t ( r t | r t ) = 1 2 n Z n U n ( r t ) exp [ ± i β n z k ] U n ( r t ) exp [ i β n z k ] ,
G ̿ k k t ( r t | r t ) = 1 2 n Z n exp [ ± i β n ( z k z k ) ] U n ( r t ) U n ( r t ) .
G ̿ k k t ( r t | r t ) = 1 2 n Z n U n ( r t ) U n ( r t ) .
E k t ( r t ) = k S k L ̿ k k ( r t | r t ) J k s ( r t ) d 2 r t ,
L ̿ k k ( r t | r t ) = Z ̿ k s ( r t ) δ k k δ ( r t r t ) G ̿ k k t ( r t | r t ) .
E k t ( r t ) = n a k n U n ( r t ) ,
J k s ( r t ) = m h k m U m ( r t ) ,
n a k n U n ( r t ) = k m h k m S k L ̿ k k ( r t | r t ) U m ( r t ) d 2 r t .
S U n ( r t ) U m ( r t ) d 2 r t = δ n m ,
a k n = k m h k m S k S k U n ( r t ) L ̿ k k ( r t | r t ) U m ( r t ) d 2 r t d 2 r t ,
L k n , k m = S k S k U n ( r t ) L ̿ k k ( r t | r t ) U m ( r t ) d 2 r t d 2 r t ,
a k n = k m h k m L k n , k m k , n .
y = L x .
L = F Σ G .
L 1 = G Σ 1 F .
x = L 1 y
L k n , k m = δ k k Z n m , k s + 1 2 Z n exp [ ± i β n ( z k z k ) ] δ n m ,
Z n m , k s = S k U n ( r t ) Z ̿ k s ( r t ) U m ( r t ) d 2 r t
Z n m , k s = S k S k U n ( r t ) Z ̿ k s ( r t , r t ) U m ( r t ) d 2 r t d 2 r t ,
E 0 ( r t ) = k m h k m S k G ̿ 0 k t ( r t | r t ) U m ( r t ) d 2 r t ,
E 0 ( r t ) = 1 2 k m h k m Z m exp [ + i β m z k ] U m ( r t ) .
E 0 = m b m U m ( r t ) ,
b m = 1 2 k h k m Z m exp [ + i β m z k ] .
( S 11 ) m n = b m a n ,
S 11 = Z 1 S 11 Z .
E K ( r t ) = m a K m U m ( r t ) + k m h k m S k G ̿ K k t ( r t | r t ) U m ( r t ) d 2 r t ,
E K ( r t ) = m a K m U m ( r t ) 1 2 k m h k m Z m exp [ + i β m ( L z k ) ] U m ( r t ) .
E K ( r t ) = m c m U m ( r t ) ,
c m = a K m 1 2 k h k m Z m exp [ + i β m ( L z k ) ] .
( S 21 ) m n = c m a n ,
S 21 = Z 1 S 21 Z .
L n m = Z n m s + 1 2 Z n δ n m ,
Z n m s = S S U n ( r t ) Z ̿ s ( r t ) U m ( r t ) d 2 r t d 2 r t .
Z n m s = S R s ( r t ) U n ( r t ) U m ( r t ) d 2 r t .
b = 1 2 Z L 1 a ,
S 11 = 1 2 Z L 1 .
S 11 = 1 2 Z 1 Z L 1 Z = 1 2 Z L 1 Z .
S 21 = S 12 T = I 1 2 Z L 1 = I + S 11 ,
S 21 = S 12 T = I 1 2 Z L 1 Z .
L n m = ( R s + 1 2 Z n ) δ n m ,
( S 11 ) n n = ( S 22 ) n n = 1 2 ( Z n R s + Z n 2 ) ,
( S 21 ) n n = ( S 12 ) n n = 1 1 2 ( Z n R s + Z n 2 ) = R s R s + Z n 2 ,
b = S 11 a + S 12 [ I + Γ L S 22 + ( Γ L S 22 ) 2 + ( Γ L S 22 ) 3 + ] Γ L S 21 a ,
b = S 11 a + S 12 [ I Γ L S 22 ] 1 Γ L S 21 a ,
Γ = S 11 + S 12 [ I Γ L S 22 ] 1 Γ L S 21 .
S 11 = A 11 + A 12 [ I B 11 A 22 ] 1 B 11 A 21 ,
S 21 = B 21 [ I A 22 B 11 ] 1 A 21 ,
S 12 = A 12 [ I B 11 A 22 ] 1 B 12 ,
S 22 = B 22 + B 21 [ I A 22 B 11 ] 1 A 22 B 12 .
P i = 1 2 Tr [ a a ] = 1 2 Tr [ A ] ,
P r = 1 2 Tr [ b b ] = 1 2 Tr [ B ] ,
P r = 1 2 Tr [ S A S ] ,
P a = 1 2 Tr [ A ] 1 2 Tr [ S A S ] = 1 2 Tr [ A ] 1 2 Tr [ S S A ] = 1 2 Tr { [ I S S ] A } .
P a = 1 2 Tr [ K A ] .
K = W Λ W ,
P a = 1 2 Tr [ W Λ W A ] = 1 2 Tr [ Λ W A W ] .
K = I Γ Γ .
1 2 A = I h ν d ν e h ν k T 1 .
1 2 B = 1 2 S A S + 1 2 N ,
1 2 B = I h ν d ν e h ν k T 1 ,
N = 2 ( I S S ) h ν d ν e h ν k T 1 ,
U n m TM = ( ε 0 n ε 0 m a b ) 1 2 1 k c , n m [ k x , n cos ( k x , n x ) sin ( k y , m y ) x ̂ + k y , m sin ( k x , n x ) cos ( k y , m y ) y ̂ ] ,
U n m TE = ( ε 0 n ε 0 m a b ) 1 2 1 k c , n m [ k y , m cos ( k x , n x ) sin ( k y , m y ) x ̂ k x , n sin ( k x , n x ) cos ( k y , m y ) y ̂ ]
ε 0 n = { 1 if n = 0 2 if n > 0 } .
( Γ L ) n m = exp [ + i β n m 2 L ] δ n m ,

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