Abstract

An analytical theory of extraordinary optical transmission (EOT) through realistic metallic screens perforated by a periodic array of subwavelength holes is presented. The theory is based on our previous work on EOT through perfect conducting screens and on the surface impedance concept. The proposed theory is valid for the complete frequency range where EOT has been reported, including microwaves and optics. A reasonably good agreement with electromagnetic simulations is shown in all this frequency range. We feel that the proposed theory may help to clarify the physics underlying EOT and serve as a first step to more accurate analysis.

© 2010 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. 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(15), 6779-6782 (1998).
    [CrossRef]
  3. M. Sarrazin, and J-P. Vigneron, "Optical properties of tungsten thin films perforated with a bidimensional array of subwavelength holes," Phys. Rev. E 68, 016603 (2003)
    [CrossRef]
  4. M. Beruete, M. Sorolla, I. Campillo, J. S. Dolado, L. Martín-Moreno, J. Bravo-Abad, and F. J. García-Vidal, "Enhanced millimeter-wave transmission through subwavelength hole arrays," Opt. Lett. 29(21), 2500-2502 (2004).
    [CrossRef] [PubMed]
  5. M. Sarrazin, and J-P. Vigneron, "Light transmission assisted by Brewster-Zennek modes in chromium films carrying a subwavelength hole array," Phys. Rev. B 71, 075404 (2005)
    [CrossRef]
  6. R. F. CollinField Theory of Guided Waves (Edt. IEEE Press, New York, 199,1 2nd Ed.)
  7. J. B. Pendry, L. Martín-Moreno, and F. J. García-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305,847-848 (2004).
    [CrossRef] [PubMed]
  8. M. M. J. Treacy, "Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings," Phys. Rev. B 66, 195105 (2002)
    [CrossRef]
  9. F. J. Garía de Abajo, R. Gómez-Medina, and J. J. Sáenz, "Full transmission through perfect conductor subwavelength hole arrays," Phys. Rev. E 72, 016608 (2005)
    [CrossRef]
  10. C. C. Chen, "Transmission of Microwave Through Perforated Flat Plates of Finite Thickness," IEEE Trans. Microwave Theory Tech. 21(1), 1-6 (1973).
    [CrossRef]
  11. F. Medina, F. Mesa, and R. Marqués, "Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective," IEEE Trans. Microwave Theory Tech. 56,3108-3120 (2008).
    [CrossRef]
  12. F. Medina,J. A. Ruiz-Cruz, F. Mesa, M. Rebollar, J. R. Montejo-Garai, and R. Marqués "Experimental verification of extraordinary transmission without surface plasmons," Appl. Phys. Lett. 95, 071102 (2009).
    [CrossRef]
  13. R. Marqués, F. Mesa, L. Jelinek, and F. Medina, "Analytical theory of extraordinary transmission through metallic diffraction screens perforated by small holes," Opt. Express 17(7), 5571-5579 (2009).
    [CrossRef] [PubMed]
  14. Sergei Tretyakov, Analytical Modeling in Applied Electromagnetics (Edt. Artech House 2003).
  15. R. Gordon, "Bethe’s aperture theory for arrays," Phys. Rev. A 76, 053806 (2007).
    [CrossRef]
  16. E. A. Coronado and G. C. Schatz, "Surface plasmon broadening for arbitrary shape nanoparticles: A probability approach," J. Chem. Phys. 119, 3926-3934 (2003).
    [CrossRef]

2009

F. Medina,J. A. Ruiz-Cruz, F. Mesa, M. Rebollar, J. R. Montejo-Garai, and R. Marqués "Experimental verification of extraordinary transmission without surface plasmons," Appl. Phys. Lett. 95, 071102 (2009).
[CrossRef]

R. Marqués, F. Mesa, L. Jelinek, and F. Medina, "Analytical theory of extraordinary transmission through metallic diffraction screens perforated by small holes," Opt. Express 17(7), 5571-5579 (2009).
[CrossRef] [PubMed]

2008

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

2007

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

2005

M. Sarrazin, and J-P. Vigneron, "Light transmission assisted by Brewster-Zennek modes in chromium films carrying a subwavelength hole array," Phys. Rev. B 71, 075404 (2005)
[CrossRef]

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

2004

2003

M. Sarrazin, and J-P. Vigneron, "Optical properties of tungsten thin films perforated with a bidimensional array of subwavelength holes," Phys. Rev. E 68, 016603 (2003)
[CrossRef]

E. A. Coronado and G. C. Schatz, "Surface plasmon broadening for arbitrary shape nanoparticles: A probability approach," J. Chem. Phys. 119, 3926-3934 (2003).
[CrossRef]

2002

M. M. J. Treacy, "Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings," Phys. Rev. B 66, 195105 (2002)
[CrossRef]

1998

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(15), 6779-6782 (1998).
[CrossRef]

1973

C. C. Chen, "Transmission of Microwave Through Perforated Flat Plates of Finite Thickness," IEEE Trans. Microwave Theory Tech. 21(1), 1-6 (1973).
[CrossRef]

Beruete, M.

Bravo-Abad, J.

Campillo, I.

Chen, C. C.

C. C. Chen, "Transmission of Microwave Through Perforated Flat Plates of Finite Thickness," IEEE Trans. Microwave Theory Tech. 21(1), 1-6 (1973).
[CrossRef]

Coronado, E. A.

E. A. Coronado and G. C. Schatz, "Surface plasmon broadening for arbitrary shape nanoparticles: A probability approach," J. Chem. Phys. 119, 3926-3934 (2003).
[CrossRef]

Dolado, J. S.

Ebbesen, T. W.

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(15), 6779-6782 (1998).
[CrossRef]

García-Vidal, F. J.

Garía de Abajo, F. J.

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

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(15), 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]

Gómez-Medina, R.

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

Gordon, R.

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

Grupp, D. E.

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(15), 6779-6782 (1998).
[CrossRef]

Jelinek, L.

Lezec, H. J.

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(15), 6779-6782 (1998).
[CrossRef]

Marqués, R.

F. Medina,J. A. Ruiz-Cruz, F. Mesa, M. Rebollar, J. R. Montejo-Garai, and R. Marqués "Experimental verification of extraordinary transmission without surface plasmons," Appl. Phys. Lett. 95, 071102 (2009).
[CrossRef]

R. Marqués, F. Mesa, L. Jelinek, and F. Medina, "Analytical theory of extraordinary transmission through metallic diffraction screens perforated by small holes," Opt. Express 17(7), 5571-5579 (2009).
[CrossRef] [PubMed]

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

Martín-Moreno, L.

Medina, F.

F. Medina,J. A. Ruiz-Cruz, F. Mesa, M. Rebollar, J. R. Montejo-Garai, and R. Marqués "Experimental verification of extraordinary transmission without surface plasmons," Appl. Phys. Lett. 95, 071102 (2009).
[CrossRef]

R. Marqués, F. Mesa, L. Jelinek, and F. Medina, "Analytical theory of extraordinary transmission through metallic diffraction screens perforated by small holes," Opt. Express 17(7), 5571-5579 (2009).
[CrossRef] [PubMed]

F. Medina, F. Mesa, and R. Marqués, "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,J. A. Ruiz-Cruz, F. Mesa, M. Rebollar, J. R. Montejo-Garai, and R. Marqués "Experimental verification of extraordinary transmission without surface plasmons," Appl. Phys. Lett. 95, 071102 (2009).
[CrossRef]

R. Marqués, F. Mesa, L. Jelinek, and F. Medina, "Analytical theory of extraordinary transmission through metallic diffraction screens perforated by small holes," Opt. Express 17(7), 5571-5579 (2009).
[CrossRef] [PubMed]

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

Montejo-Garai, J. R.

F. Medina,J. A. Ruiz-Cruz, F. Mesa, M. Rebollar, J. R. Montejo-Garai, and R. Marqués "Experimental verification of extraordinary transmission without surface plasmons," Appl. Phys. Lett. 95, 071102 (2009).
[CrossRef]

Pendry, J. B.

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

Rebollar, M.

F. Medina,J. A. Ruiz-Cruz, F. Mesa, M. Rebollar, J. R. Montejo-Garai, and R. Marqués "Experimental verification of extraordinary transmission without surface plasmons," Appl. Phys. Lett. 95, 071102 (2009).
[CrossRef]

Ruiz-Cruz, J. A.

F. Medina,J. A. Ruiz-Cruz, F. Mesa, M. Rebollar, J. R. Montejo-Garai, and R. Marqués "Experimental verification of extraordinary transmission without surface plasmons," Appl. Phys. Lett. 95, 071102 (2009).
[CrossRef]

Sáenz, J. J.

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

Sarrazin, M.

M. Sarrazin, and J-P. Vigneron, "Light transmission assisted by Brewster-Zennek modes in chromium films carrying a subwavelength hole array," Phys. Rev. B 71, 075404 (2005)
[CrossRef]

M. Sarrazin, and J-P. Vigneron, "Optical properties of tungsten thin films perforated with a bidimensional array of subwavelength holes," Phys. Rev. E 68, 016603 (2003)
[CrossRef]

Schatz, G. C.

E. A. Coronado and G. C. Schatz, "Surface plasmon broadening for arbitrary shape nanoparticles: A probability approach," J. Chem. Phys. 119, 3926-3934 (2003).
[CrossRef]

Sorolla, M.

Thio, T.

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(15), 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]

Treacy, M. M. J.

M. M. J. Treacy, "Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings," Phys. Rev. B 66, 195105 (2002)
[CrossRef]

Vigneron, J-P.

M. Sarrazin, and J-P. Vigneron, "Light transmission assisted by Brewster-Zennek modes in chromium films carrying a subwavelength hole array," Phys. Rev. B 71, 075404 (2005)
[CrossRef]

M. Sarrazin, and J-P. Vigneron, "Optical properties of tungsten thin films perforated with a bidimensional array of subwavelength holes," Phys. Rev. E 68, 016603 (2003)
[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.

F. Medina,J. A. Ruiz-Cruz, F. Mesa, M. Rebollar, J. R. Montejo-Garai, and R. Marqués "Experimental verification of extraordinary transmission without surface plasmons," Appl. Phys. Lett. 95, 071102 (2009).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

C. C. Chen, "Transmission of Microwave Through Perforated Flat Plates of Finite Thickness," IEEE Trans. Microwave Theory Tech. 21(1), 1-6 (1973).
[CrossRef]

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

J. Chem. Phys.

E. A. Coronado and G. C. Schatz, "Surface plasmon broadening for arbitrary shape nanoparticles: A probability approach," J. Chem. Phys. 119, 3926-3934 (2003).
[CrossRef]

Nature (London)

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. Express

Opt. Lett.

Phys. Rev. A

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

Phys. Rev. B

M. Sarrazin, and J-P. Vigneron, "Light transmission assisted by Brewster-Zennek modes in chromium films carrying a subwavelength hole array," Phys. Rev. B 71, 075404 (2005)
[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(15), 6779-6782 (1998).
[CrossRef]

M. M. J. Treacy, "Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings," Phys. Rev. B 66, 195105 (2002)
[CrossRef]

Phys. Rev. E

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

M. Sarrazin, and J-P. Vigneron, "Optical properties of tungsten thin films perforated with a bidimensional array of subwavelength holes," Phys. Rev. E 68, 016603 (2003)
[CrossRef]

Science

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

Other

Sergei Tretyakov, Analytical Modeling in Applied Electromagnetics (Edt. Artech House 2003).

R. F. CollinField Theory of Guided Waves (Edt. IEEE Press, New York, 199,1 2nd Ed.)

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Figures (5)

Fig. 1.
Fig. 1.

Metallic 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. The screen has a finite thickness t.

Fig. 2.
Fig. 2.

Transmission coefficient in decibels versus normalized frequency (where f w = c/a is the Wood’s anomaly frequency and c is the speed of light in vacuum) of the structure shown in Fig. 1 for a = 30 mm (f w ≈ 10 GHz), t = a/10 and different values of b. Solid lines correspond to the analytical model and dotted lines correspond to data from CST. In the main plot the metal is aluminium modeled by a finite conductivity (σ = 37.8 × 106 S/m). In the inner plots the transmission of screens made of aluminium and copper (σ = 59.6 × 106 S/m) with b = a/6 is compared.

Fig. 3.
Fig. 3.

Transmission coefficient in decibels versus normalized frequency for a = 300 μm (f w = c/a ≈ 1 THz), b = a/6 and different values of t. Solid lines correspond to the analytical model and dotted lines correspond to data from CST. In the main plot the metal is copper modeled by a finite conductivity (σ = 59.6 × 106 S/m). In the inner plots the transmission of screens made of aluminium (σ = 37.8 × 106 S/m) and copper with t = a/20 is compared.

Fig. 4.
Fig. 4.

Transmission coefficient in decibels versus normalized frequency for a = 1 μm (f w = c/a ≈ 300 THz), t = a/10 and different values of b. Solid lines correspond to the analytical model and dotted lines correspond to data from CST. In the main plot the metal is silver modeled by a Drude-Lorentz permittivity (ω p = 2π × 2175 THz and f c = 2π × 4.35 THz). In the inner plots the transmission of screens made of silver and copper (ω p = 2π × 1914 THz and f c = 2π × 8.34 THz) with b = a/4 is compared.

Fig. 5.
Fig. 5.

Transmission coefficient in decibels versus normalized frequency for a = 1 μm (f w = c/a ≈ 300 THz), b = a/6 and different values of t. Solid lines correspond to the analytical model and dotted lines correspond to data from CST. In the main plot the metal is silver modeled by a Drude-Lorentz permittivity (ω p = 2π × 2175 THz and f c = 2π × 4.35 THz). In the inner plots the transmission of screens made of silver and copper (ω p = 2π × 1914 THz and f c = 2π × 8.34 THz) with t = a/10 is compared.

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

E y = 1 + R + n = 1 , m = 0 N , M C nm , R TE f nm ( x , y ) + n = 1 , m = 0 N , M C nm , R TM f nm ( x , y )
E y + = T + n = 1 , m = 0 N , M C nm , T TE f nm ( x , y ) + n = 0 , m = 1 N , M C nm , T TM f nm ( x , y )
H x = Y 0 ( 1 R ) n = 1 , m = 0 N , M Y 2 n , 2 m TE C nm , R TE f nm ( x , y ) n = 0 , m = 1 N , M Y 2 n , 2 m TM C nm , R TM f nm ( x , y )
H x + = Y 0 T n = 1 , m = 0 N , M Y 2 n , 2 m TE C nm , T TE f nm ( x , y ) + n = 0 , m = 1 N , M Y 2 n , 2 m TM C nm , T TM f nm ( x , y ) ,
Y 2 n , 2 m TE = k 0 z nm k 0 Y 0 and Y 2 n , 2 m TE = k 0 k 0 z nm Y 0 ,
k 0 , z nm = k 0 2 ( 2 n π a ) 2 ( 2 m π a ) 2
E x = n , m = 1 N , M ( m n C nm , R TE n m C nm , R TM ) g nm ( x , y ) 0
E x + = n , m = 1 N , M ( m n C nm , T TE n m C nm , T TM ) g nm ( x , y ) 0 ,
{ 2 z 2 + k s 2 } E y 0 ; μ H x = E y z .
( E y + ( x , y ) + E y ( x , y ) E y + ( x , y ) E y ( x , y ) ) ( Z s , 1 0 0 Z s , 2 ) ( H x + ( x , y ) H x ( x , y ) H x + ( x , y ) + H x ( x , y ) )
Z s , 1 = [ 1 + cos ( k s t ) ] i sin ( k s t ) Y s and Z s , 2 = i sin ( k s t ) [ 1 + cos ( k s t ) ] Y s ,
wg [ ( E y + + E y ) Z s , 1 ( H x + H x ) ] f nm dS = h [ ( E y + + E y ) Z s , 1 ( H x + H x ) ] f nm dS
h [ ( E y + + E y ) Z s , 1 ( H x + H x ) ] dS = wg [ ( E y + + E y ) Z s , 1 ( H x + H x ) ] dS
wg [ ( E y + E y ) Z s , 2 ( H x + + H x ) ] f nm dS = h [ ( E y + E y ) Z s , 2 ( H x + + H x ) ] f nm dS
h [ ( E y + E y ) Z s , 2 ( H x + + H x ) ] dS = wg [ ( E y + E y ) Z s , 2 ( H x + + H x ) ] dS
( 1 δ n 0 ) ( 1 Z s , 1 Y 2 n , 2 m TE ) C nm , T TE + ( 1 δ n 0 ) ( 1 Z s , 1 Y 2 n , 2 m TE ) C nm , R TE
+ ( 1 δ m 0 ) ( 1 Z s , 1 Y 2 n , 2 m TM ) C nm , T TM + ( 1 δ m 0 ) ( 1 Z s , 1 Y 2 n , 2 m TM ) C nm , R TM
= 2 ( 1 δ n 0 δ m 0 ) ( 1 + R + T ) + 2 ( 2 δ n 0 δ m 0 ) Z s , 1 Y 0 ( 1 R T )
( 1 δ n 0 ) ( 1 Z s , 2 Y 2 n , 2 m TE ) C nm , T TE ( 1 δ n 0 ) ( 1 Z s , 2 Y 2 n , 2 m TE ) C nm , R TE
+ ( 1 δ m 0 ) ( 1 Z s , 2 Y 2 n , 2 m TM ) C nm , T TM ( 1 δ m 0 ) ( 1 Z s , 2 Y 2 n , 2 m TM ) C nm , R TM
= 2 ( 2 δ n 0 δ m 0 ) ( 1 R + T ) 2 ( 2 δ n 0 δ m 0 ) Z s , 2 Y 0 ( 1 R + T ) .
C nm , T TM m 2 n 2 C nm , T TE and C nm , R TM m 2 n 2 C nm , R TE
Z h , 1 = [ 1 + cos ( k h t ) ] i sin ( k h t ) Y h and Z h , 2 = i sin ( k h t ) [ 1 + cos ( k h t ) ] Y h ,
h [ ( E y + + E y ) Z h , 1 ( H x + H x ) ] dS 0
h [ ( E y + E y ) Z h , 2 ( H x + + H x ) ] dS 0 ,
( 1 + T + R ) + Z h , 1 Y 0 ( 1 + T + R ) + [ n = 1 , m = 0 N , M C nm , T TE + C nm , R TM Z h , 1 Y 2 n , 2 m TE ( C nm , T TM + C nm , R TM )
+ n = 0 , m = 1 N , M C nm , T TM + C nm , R TM Z h , 1 Y 2 n , 2 m TM ( C nm , T TM + C nm , R TM ) ] sin c ( nπb a ) sin c ( mπb a ) = 0
( 1 + T R ) Z h , 2 Y 0 ( 1 + T R ) + [ n = 1 , m = 0 N , M C nm , T TE + C nm , R TE Z h , 2 Y 2 n , 2 m TE ( C nm , T TE C nm , R TE )
+ n = 0 , m = 1 N , M C nm , T TM C nm , R TM Z h , 2 Y 2 n , 2 m TM ( C nm , T TM C nm , R TM ) ] sin c ( nπb a ) sin c ( mπb a ) = 0
ε i σ ω ε 0
ε ε 0 ( 1 ω p 2 ω ( ω i f c ) ) ,
f c = f c ( 1 + l e 2 t )

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