Abstract

In this letter, the problem of extraordinary (ET) transmission of electromagnetic waves through opaque screens perforated with subwavelength holes is addressed from an analytical point of view. Our purpose was to find a closed-form expression for the transmission coefficient in a simple case in order to explore and clarify, as much as possible, the physical background of the phenomenon. The solution of this canonical example, apart from matching quite well with numerical simulations given by commercial solvers, has provided new insight in extraordinary transmission as well as Wood’s anomaly. Thus, our analysis has revealed that one of the key factors behind ET is the continuous increase of excess electric energy around the holes as the frequency approaches the onset of some of the higher-order modes associated with the periodicity of the screen. The same analysis also helps to clarify the role of surface modes –or spoof plasmons–in the onset of ET.

© 2009 Optical Society of America

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References

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  1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-669 (1998).
    [CrossRef]
  2. H. A. Bethe,"Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944).
    [CrossRef]
  3. H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through subwavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
    [CrossRef]
  4. D. E. Grupp, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, and T. Thio, "Crucial role of metal surface in enhanced transmission through subwavelength apertures," Appl. Phys. Lett. 77, 1569-1571 (2000).
    [CrossRef]
  5. L. Martın-Moreno, F. J. Garc’ıa-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, D. E. Grupp, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
    [CrossRef] [PubMed]
  6. M. Beruete, M. Sorolla, I. Campillo, J. S. Dolado, L. Mart’ın-Moreno, J. Bravo-Abad, and F. J. Garcıa-Vidal, "Enhanced millimeter-wave transmission through subwavelength hole arrays," Opt. Lett. 29, 2500-2502 (2004).
    [CrossRef] [PubMed]
  7. J. A. Porto, F. J. Garcıa-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
    [CrossRef]
  8. F. J. Garcıa-de-Abajo, R. Gomez-Medina, and J. J. Saenz, "Full transmission through perfect-conductor subwavelength hole arrays," Phys. Rev. E 72, 016608 (2005).
    [CrossRef]
  9. J. B. Pendry, L. Martın-Moreno, and F. J. Garcıa-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004).
    [CrossRef] [PubMed]
  10. A. P. Hibbins, M. J. Lockyear, I. R. Hooper, and J. R. Sambles, "Waveguide arrays as plasmonic metamaterials: transmission below cutoff," Phys. Rev. Lett. 96, 073904 (2006)
    [CrossRef] [PubMed]
  11. F. Medina, F. Mesa, and R. Marques, "Equivalent circuit model to explain extraordinary transmission," in IEEE MTT-S Int. Microw. Symp. Dig., Atlanta, GA, 213-216, June 2008.
  12. F. Medina, F. Mesa, and R. Marques, "Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective," IEEE Trans. Microwave Theory Tech. 56, 3108-3120 2008.
    [CrossRef]
  13. C. Genet and T. W. Ebbesen, "Light in tiny holes," Nature 445, 39-46 (2007).
    [CrossRef] [PubMed]
  14. F. J. Garcıa-de-Abajo, "Colloquium: Light scattering by particle and hole arrays," Rev. Mod. Phys. 79, 1267-1290 (2007).
    [CrossRef]
  15. R. Gordon, "Bethe’s aperture theory for arrays," Phys. Rev. A 76, 053806 (2007).
    [CrossRef]
  16. J. D. Jackson, Classical Electrodynamics, Edt. (Wiley, New York, 1999), 3rd Ed.

2008 (1)

F. Medina, F. Mesa, and R. Marques, "Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective," IEEE Trans. Microwave Theory Tech. 56, 3108-3120 2008.
[CrossRef]

2007 (3)

C. Genet and T. W. Ebbesen, "Light in tiny holes," Nature 445, 39-46 (2007).
[CrossRef] [PubMed]

F. J. Garcıa-de-Abajo, "Colloquium: Light scattering by particle and hole arrays," Rev. Mod. Phys. 79, 1267-1290 (2007).
[CrossRef]

R. Gordon, "Bethe’s aperture theory for arrays," Phys. Rev. A 76, 053806 (2007).
[CrossRef]

2006 (1)

A. P. Hibbins, M. J. Lockyear, I. R. Hooper, and J. R. Sambles, "Waveguide arrays as plasmonic metamaterials: transmission below cutoff," Phys. Rev. Lett. 96, 073904 (2006)
[CrossRef] [PubMed]

2005 (1)

F. J. Garcıa-de-Abajo, R. Gomez-Medina, and J. J. Saenz, "Full transmission through perfect-conductor subwavelength hole arrays," Phys. Rev. E 72, 016608 (2005).
[CrossRef]

2004 (2)

2001 (1)

L. Martın-Moreno, F. J. Garc’ıa-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, D. E. Grupp, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

2000 (1)

D. E. Grupp, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, and T. Thio, "Crucial role of metal surface in enhanced transmission through subwavelength apertures," Appl. Phys. Lett. 77, 1569-1571 (2000).
[CrossRef]

1999 (1)

J. A. Porto, F. J. Garcıa-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

1998 (2)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-669 (1998).
[CrossRef]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through subwavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
[CrossRef]

1944 (1)

H. A. Bethe,"Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944).
[CrossRef]

Beruete, M.

Bethe, H. A.

H. A. Bethe,"Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944).
[CrossRef]

Bravo-Abad, J.

Campillo, I.

Dolado, J. S.

Ebbesen, T. W.

C. Genet and T. W. Ebbesen, "Light in tiny holes," Nature 445, 39-46 (2007).
[CrossRef] [PubMed]

L. Martın-Moreno, F. J. Garc’ıa-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, D. E. Grupp, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

D. E. Grupp, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, and T. Thio, "Crucial role of metal surface in enhanced transmission through subwavelength apertures," Appl. Phys. Lett. 77, 1569-1571 (2000).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-669 (1998).
[CrossRef]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through subwavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
[CrossRef]

Garcia-de-Abajo, F. J.

F. J. Garcıa-de-Abajo, "Colloquium: Light scattering by particle and hole arrays," Rev. Mod. Phys. 79, 1267-1290 (2007).
[CrossRef]

F. J. Garcıa-de-Abajo, R. Gomez-Medina, and J. J. Saenz, "Full transmission through perfect-conductor subwavelength hole arrays," Phys. Rev. E 72, 016608 (2005).
[CrossRef]

Garcia-Vidal, F. J.

M. Beruete, M. Sorolla, I. Campillo, J. S. Dolado, L. Mart’ın-Moreno, J. Bravo-Abad, and F. J. Garcıa-Vidal, "Enhanced millimeter-wave transmission through subwavelength hole arrays," Opt. Lett. 29, 2500-2502 (2004).
[CrossRef] [PubMed]

J. B. Pendry, L. Martın-Moreno, and F. J. Garcıa-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004).
[CrossRef] [PubMed]

L. Martın-Moreno, F. J. Garc’ıa-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, D. E. Grupp, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

J. A. Porto, F. J. Garcıa-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Genet, C.

C. Genet and T. W. Ebbesen, "Light in tiny holes," Nature 445, 39-46 (2007).
[CrossRef] [PubMed]

Ghaemi, H. F.

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through subwavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-669 (1998).
[CrossRef]

Gordon, R.

R. Gordon, "Bethe’s aperture theory for arrays," Phys. Rev. A 76, 053806 (2007).
[CrossRef]

Grupp, D. E.

L. Martın-Moreno, F. J. Garc’ıa-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, D. E. Grupp, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

D. E. Grupp, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, and T. Thio, "Crucial role of metal surface in enhanced transmission through subwavelength apertures," Appl. Phys. Lett. 77, 1569-1571 (2000).
[CrossRef]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through subwavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
[CrossRef]

Hibbins, A. P.

A. P. Hibbins, M. J. Lockyear, I. R. Hooper, and J. R. Sambles, "Waveguide arrays as plasmonic metamaterials: transmission below cutoff," Phys. Rev. Lett. 96, 073904 (2006)
[CrossRef] [PubMed]

Hooper, I. R.

A. P. Hibbins, M. J. Lockyear, I. R. Hooper, and J. R. Sambles, "Waveguide arrays as plasmonic metamaterials: transmission below cutoff," Phys. Rev. Lett. 96, 073904 (2006)
[CrossRef] [PubMed]

Lezec, H. J.

L. Martın-Moreno, F. J. Garc’ıa-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, D. E. Grupp, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

D. E. Grupp, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, and T. Thio, "Crucial role of metal surface in enhanced transmission through subwavelength apertures," Appl. Phys. Lett. 77, 1569-1571 (2000).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-669 (1998).
[CrossRef]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through subwavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
[CrossRef]

Lockyear, M. J.

A. P. Hibbins, M. J. Lockyear, I. R. Hooper, and J. R. Sambles, "Waveguide arrays as plasmonic metamaterials: transmission below cutoff," Phys. Rev. Lett. 96, 073904 (2006)
[CrossRef] [PubMed]

Marques, R.

F. Medina, F. Mesa, and R. Marques, "Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective," IEEE Trans. Microwave Theory Tech. 56, 3108-3120 2008.
[CrossRef]

Martin-Moreno, L.

M. Beruete, M. Sorolla, I. Campillo, J. S. Dolado, L. Mart’ın-Moreno, J. Bravo-Abad, and F. J. Garcıa-Vidal, "Enhanced millimeter-wave transmission through subwavelength hole arrays," Opt. Lett. 29, 2500-2502 (2004).
[CrossRef] [PubMed]

J. B. Pendry, L. Martın-Moreno, and F. J. Garcıa-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004).
[CrossRef] [PubMed]

L. Martın-Moreno, F. J. Garc’ıa-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, D. E. Grupp, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

Medina, F.

F. Medina, F. Mesa, and R. Marques, "Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective," IEEE Trans. Microwave Theory Tech. 56, 3108-3120 2008.
[CrossRef]

Mesa, F.

F. Medina, F. Mesa, and R. Marques, "Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective," IEEE Trans. Microwave Theory Tech. 56, 3108-3120 2008.
[CrossRef]

Pellerin, K. M.

L. Martın-Moreno, F. J. Garc’ıa-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, D. E. Grupp, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

D. E. Grupp, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, and T. Thio, "Crucial role of metal surface in enhanced transmission through subwavelength apertures," Appl. Phys. Lett. 77, 1569-1571 (2000).
[CrossRef]

Pendry, J. B.

J. B. Pendry, L. Martın-Moreno, and F. J. Garcıa-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004).
[CrossRef] [PubMed]

L. Martın-Moreno, F. J. Garc’ıa-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, D. E. Grupp, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

J. A. Porto, F. J. Garcıa-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Porto, J. A.

J. A. Porto, F. J. Garcıa-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Sambles, J. R.

A. P. Hibbins, M. J. Lockyear, I. R. Hooper, and J. R. Sambles, "Waveguide arrays as plasmonic metamaterials: transmission below cutoff," Phys. Rev. Lett. 96, 073904 (2006)
[CrossRef] [PubMed]

Sorolla, M.

Thio, T.

L. Martın-Moreno, F. J. Garc’ıa-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, D. E. Grupp, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

D. E. Grupp, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, and T. Thio, "Crucial role of metal surface in enhanced transmission through subwavelength apertures," Appl. Phys. Lett. 77, 1569-1571 (2000).
[CrossRef]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through subwavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-669 (1998).
[CrossRef]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-669 (1998).
[CrossRef]

Appl. Phys. Lett. (1)

D. E. Grupp, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, and T. Thio, "Crucial role of metal surface in enhanced transmission through subwavelength apertures," Appl. Phys. Lett. 77, 1569-1571 (2000).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

F. Medina, F. Mesa, and R. Marques, "Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective," IEEE Trans. Microwave Theory Tech. 56, 3108-3120 2008.
[CrossRef]

Nature (1)

C. Genet and T. W. Ebbesen, "Light in tiny holes," Nature 445, 39-46 (2007).
[CrossRef] [PubMed]

Nature (London) (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-669 (1998).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

H. A. Bethe,"Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944).
[CrossRef]

Phys. Rev. A (1)

R. Gordon, "Bethe’s aperture theory for arrays," Phys. Rev. A 76, 053806 (2007).
[CrossRef]

Phys. Rev. B (1)

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through subwavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
[CrossRef]

Phys. Rev. E (1)

F. J. Garcıa-de-Abajo, R. Gomez-Medina, and J. J. Saenz, "Full transmission through perfect-conductor subwavelength hole arrays," Phys. Rev. E 72, 016608 (2005).
[CrossRef]

Phys. Rev. Lett. (3)

A. P. Hibbins, M. J. Lockyear, I. R. Hooper, and J. R. Sambles, "Waveguide arrays as plasmonic metamaterials: transmission below cutoff," Phys. Rev. Lett. 96, 073904 (2006)
[CrossRef] [PubMed]

L. Martın-Moreno, F. J. Garc’ıa-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, D. E. Grupp, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[CrossRef] [PubMed]

J. A. Porto, F. J. Garcıa-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Rev. Mod. Phys. (1)

F. J. Garcıa-de-Abajo, "Colloquium: Light scattering by particle and hole arrays," Rev. Mod. Phys. 79, 1267-1290 (2007).
[CrossRef]

Science (1)

J. B. Pendry, L. Martın-Moreno, and F. J. Garcıa-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004).
[CrossRef] [PubMed]

Other (2)

F. Medina, F. Mesa, and R. Marques, "Equivalent circuit model to explain extraordinary transmission," in IEEE MTT-S Int. Microw. Symp. Dig., Atlanta, GA, 213-216, June 2008.

J. D. Jackson, Classical Electrodynamics, Edt. (Wiley, New York, 1999), 3rd Ed.

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Figures (5)

Fig. 1.
Fig. 1.

Perfect conductor screen perforated with square holes: front view (a) and two lateral cuts (b). Front (c) and lateral (d) views of the structure unit cell or equivalent waveguide. (e) Equivalent circuit for the discontinuity in the waveguide. It has been assumed that t → 0.

Fig. 2.
Fig. 2.

Transmission coefficient of the structure shown in Fig. 1 for different values of the ratio a/b versus the ratio (fW f)/fW, where fW = c/a is the Wood’s anomaly frequency, with c being the light velocity in free space. Solid Lines correspond to data from (6). Dotted lines correspond to data from CST. For comparison purposes, the numerical value for the ET frequency provided in [15] for a/b = 7.07 (i.e. holes covering a 2% of the total area) is shown with an arrow

Fig. 3.
Fig. 3.

Equivalent circuit for the computation of the frequency of excitation of surface waves with k = 2π/a.

Fig. 4.
Fig. 4.

Equivalent circuit for the structure of Figs. 1(a)-(b) with finite thickness (t ≠ 0).

Fig. 5.
Fig. 5.

Transmission coefficient of the structure shown in Fig.1 with b = a/6 and t = 0, a/7.

Tables (1)

Tables Icon

Table 1. Normalized resonance and extraordinary transmission frequencies, (fW f)/fW , for the cases studied in Fig. 2.

Equations (16)

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E y ( x , y ) = T + n = 1 N A n 0 TE f n 0 ( x , y ) + m = 1 M A 0 m TM f 0 m ( x , y ) + n , m = 1 N , M ( A n m TE + A n m TM ) f n m ( x , y ) ,
E x ( x , y ) = n , m = 1 ( m n A n m TE n m A n m TM ) g n m ( x , y )
w E y f n m d x d y = h E y f n m d x d y h E y d x d y = w E y d x d y = a 2 T ,
A n 0 TE 2 T ; A 0 m TM 2 T ; A n m TE + A n m TM 4 T .
A n m TE 4 T n 2 n 2 + m 2 ; A n m TM 4 T m 2 n 2 + m 2 .
b 2 Y 0 ( 1 T ) = a b T n = 1 N 2 n π Y 2 n , 0 TE sin ( n π b a ) + a b T m = 1 M 2 m π Y 0 , 2 m TM sin ( m π b a )
+ a 2 T n , m = 1 N , M 4 n m π 2 ( Y 2 n , 2 m TE n 2 n 2 + m 2 + Y 2 n , 2 m TM m 2 n 2 + m 2 ) sin ( n π b a ) sin ( m π b a ) ,
Y 2 n , 2 m TE = i Y 0 ( n λ a ) 2 + ( m λ a ) 2 1
Y 2 n , 2 m TM = i Y 0 / ( n λ a ) 2 + ( m λ a ) 2 1 .
Y 0 ( 1 T ) = T ( i ω C 2 1 i ω 2 L )
B C = ω C / 2 = 1 i { a b m = 1 M 2 Y 0,2 m TM m π sin ( m π b a )
+ ( a b ) 2 n , m = 1 N , M 4 Y 2 n , 2 m TM n m π 2 m 2 n 2 + m 2 sin ( n π b a ) sin ( m π b a ) }
B L = 1 2 ω L = 1 i { a b n = 1 M 2 Y 0,2 m TE n π sin ( n π b a )
+ ( a b ) 2 n , m = 1 N , M 4 Y 2 n , 2 m TE n m π 2 m 2 n 2 + m 2 sin ( n π b a ) sin ( m π b a ) } .
i ω C ( ω ) 1 i ω L ( ω ) + 2 R 0 = 0
P = 1 2 I 2 Y ' = 1 2 ( Y TE 10 ) 1 b a I 2

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