Abstract

Electromagnetic plane waves, incident on and reflecting from a dielectric-conductor interface, set up a standing wave in the dielectric with the B-field adjacent to the conductor. It is shown here how the harmonic time variation of this B-field induces an E-field and a conduction current J c within the skin depth of a real metal; and that at frequencies in the visible and near-infrared range, the imaginary term σ i of the complex conductivity ̃ σ=σ r+i σ i dominates the optical response. Continuity conditions of the E-field through the surface together with the in-quadrature response of the conductivity determine the phase relation between the incident E-M field and J c. If slits or grooves are milled into the metal surface, a displacement current in the dielectric gap and oscillating charge dipoles at the structure edges are established in quadrature phase with incident field. These dipoles radiate into the aperture and launch surface waves from the edges. They are the principle source of light transmission through the apertures.

© 2008 Optical Society of America

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References

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  1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature (London) 424, 824-830 (2003).
    [CrossRef] [PubMed]
  2. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, "Nano-optics of surface plasmon polaritons," Phys. Reports-Review Section of Phys. Lett. 408, 131-314 (2005).
  3. E. Ozbay, "Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions," Science 311, 189-193 (2006).
    [CrossRef] [PubMed]
  4. C. Genet and T. W. Ebbesen, "Light in tiny holes," Nature 445, 39-46 (2007).
    [CrossRef] [PubMed]
  5. F. J. G. de Abajo, "Colloquium: Light scattering by particle and hole arrays," Rev. Mod. Phys. 79, 1267-1290 (2007).
    [CrossRef]
  6. M. M. J. Treacy, "Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings," Phys. Rev. B 66, 195,105 (2002).
    [CrossRef]
  7. H. J. Lezec and T. Thio, "Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays," Opt. Express 12, 3629-3651 (2004).
    [CrossRef] [PubMed]
  8. H. T. Liu and P. Lalanne, "Microscopic theory of the extraordinary optical transmission," Nature 452, 728-731 (2008).
  9. A. R. Zakharian, M. Mansuripur, and J. V. Moloney, "Transmission of light through small elliptical apertures," Opt. Express 12, 2631-2648 (2004).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  11. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, "Transmission of light through a periodic array of slits in a thick metallic film," Opt. Express 13, 4485-4491 (2005).
    [CrossRef] [PubMed]
  12. N. Engheta, "Circuits with light at nanoscales: Optical nanocircuits inspired by metamaterials," Science 317, 1698-1702 (2007).
    [CrossRef] [PubMed]
  13. J. Weiner, "Phase shifts and interference in surface plasmon polariton waves," Opt. Express 16, 950-956 (2008).
    [CrossRef] [PubMed]
  14. G. Gay, O. Alloschery, B. V. de Lesegno, J. Weiner, and H. J. Lezec, "Surface wave generation and propagation on metallic subwavelength structures measured by far-field interferometry," Phys. Rev. Lett. 96, 213,901 (2006).
    [CrossRef]
  15. D. Pacifici, H. J. Lezec, H. A. Atwater, and J. Weiner, "Quantitative determination of optical transmission through subwavelength slit arrays in Ag films: Role of surface wave interference and local coupling between adjacent slits," Phys. Rev. B 77, 115411 (2008).
    [CrossRef]
  16. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, 1995). Chap. 3, pp. 109-120
  17. C. Kittel, Introduction to Solid State Physics, 7th ed. (John Wiley & Sons Inc, New York, 1996). Chap. 6, p. 150,
  18. N. W. Ashcroft and N. D. Mermin, Solid State Physics, (Thomson Learning, 1976). Chap. 1, p. 10
  19. P. B. Johnson and R. W. Christy, "Optical Constants of Noble Metals," Phys. Rev. B 6, 4370-4379 (1972).
    [CrossRef]
  20. E. Palik and G. Ghosh, eds., The Electronic Handbook of Optical Constants of Solids (Academic, New York, 1999).
  21. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons Inc, 1999).
  22. R. Ruppin, "Electromagnetic energy density in a dispersive and absorptive material," Phys. Lett. A 299, 309-312 (2002).
    [CrossRef]
  23. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, "Transmission of light through periodic arrays of sub-wavelength slits in metallic hosts," Opt. Express 14, 6400-6413 (2006).
    [CrossRef] [PubMed]
  24. G. Leveque, O. J. F. Martin, and J. Weiner, "Transient behavior of surface plasmon polaritons scattered at a subwavelength groove," Phys. Rev. B 76, 155,418 (2007).
    [CrossRef]
  25. B. Ung and Y. L. Sheng, "Optical surface waves over metallo-dielectric nanostructures: Sommerfeld integrals revisited," Opt. Express 16, 9073-9086 (2008).
    [CrossRef] [PubMed]

2008

H. T. Liu and P. Lalanne, "Microscopic theory of the extraordinary optical transmission," Nature 452, 728-731 (2008).

D. Pacifici, H. J. Lezec, H. A. Atwater, and J. Weiner, "Quantitative determination of optical transmission through subwavelength slit arrays in Ag films: Role of surface wave interference and local coupling between adjacent slits," Phys. Rev. B 77, 115411 (2008).
[CrossRef]

J. Weiner, "Phase shifts and interference in surface plasmon polariton waves," Opt. Express 16, 950-956 (2008).
[CrossRef] [PubMed]

B. Ung and Y. L. Sheng, "Optical surface waves over metallo-dielectric nanostructures: Sommerfeld integrals revisited," Opt. Express 16, 9073-9086 (2008).
[CrossRef] [PubMed]

2007

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

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

N. Engheta, "Circuits with light at nanoscales: Optical nanocircuits inspired by metamaterials," Science 317, 1698-1702 (2007).
[CrossRef] [PubMed]

G. Leveque, O. J. F. Martin, and J. Weiner, "Transient behavior of surface plasmon polaritons scattered at a subwavelength groove," Phys. Rev. B 76, 155,418 (2007).
[CrossRef]

2006

E. Ozbay, "Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions," Science 311, 189-193 (2006).
[CrossRef] [PubMed]

Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, "Transmission of light through periodic arrays of sub-wavelength slits in metallic hosts," Opt. Express 14, 6400-6413 (2006).
[CrossRef] [PubMed]

G. Gay, O. Alloschery, B. V. de Lesegno, J. Weiner, and H. J. Lezec, "Surface wave generation and propagation on metallic subwavelength structures measured by far-field interferometry," Phys. Rev. Lett. 96, 213,901 (2006).
[CrossRef]

2005

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, "Nano-optics of surface plasmon polaritons," Phys. Reports-Review Section of Phys. Lett. 408, 131-314 (2005).

Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, "Transmission of light through a periodic array of slits in a thick metallic film," Opt. Express 13, 4485-4491 (2005).
[CrossRef] [PubMed]

2004

2003

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature (London) 424, 824-830 (2003).
[CrossRef] [PubMed]

2002

M. M. J. Treacy, "Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings," Phys. Rev. B 66, 195,105 (2002).
[CrossRef]

R. Ruppin, "Electromagnetic energy density in a dispersive and absorptive material," Phys. Lett. A 299, 309-312 (2002).
[CrossRef]

1972

P. B. Johnson and R. W. Christy, "Optical Constants of Noble Metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Alloschery, O.

G. Gay, O. Alloschery, B. V. de Lesegno, J. Weiner, and H. J. Lezec, "Surface wave generation and propagation on metallic subwavelength structures measured by far-field interferometry," Phys. Rev. Lett. 96, 213,901 (2006).
[CrossRef]

Atwater, H. A.

D. Pacifici, H. J. Lezec, H. A. Atwater, and J. Weiner, "Quantitative determination of optical transmission through subwavelength slit arrays in Ag films: Role of surface wave interference and local coupling between adjacent slits," Phys. Rev. B 77, 115411 (2008).
[CrossRef]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature (London) 424, 824-830 (2003).
[CrossRef] [PubMed]

Christy, R. W.

P. B. Johnson and R. W. Christy, "Optical Constants of Noble Metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

de Abajo, F. J. G.

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

de Lesegno, B. V.

G. Gay, O. Alloschery, B. V. de Lesegno, J. Weiner, and H. J. Lezec, "Surface wave generation and propagation on metallic subwavelength structures measured by far-field interferometry," Phys. Rev. Lett. 96, 213,901 (2006).
[CrossRef]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature (London) 424, 824-830 (2003).
[CrossRef] [PubMed]

Ebbesen, T. W.

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

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature (London) 424, 824-830 (2003).
[CrossRef] [PubMed]

Engheta, N.

N. Engheta, "Circuits with light at nanoscales: Optical nanocircuits inspired by metamaterials," Science 317, 1698-1702 (2007).
[CrossRef] [PubMed]

Gay, G.

G. Gay, O. Alloschery, B. V. de Lesegno, J. Weiner, and H. J. Lezec, "Surface wave generation and propagation on metallic subwavelength structures measured by far-field interferometry," Phys. Rev. Lett. 96, 213,901 (2006).
[CrossRef]

Genet, C.

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

Johnson, P. B.

P. B. Johnson and R. W. Christy, "Optical Constants of Noble Metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Lalanne, P.

H. T. Liu and P. Lalanne, "Microscopic theory of the extraordinary optical transmission," Nature 452, 728-731 (2008).

Leveque, G.

G. Leveque, O. J. F. Martin, and J. Weiner, "Transient behavior of surface plasmon polaritons scattered at a subwavelength groove," Phys. Rev. B 76, 155,418 (2007).
[CrossRef]

Lezec, H. J.

D. Pacifici, H. J. Lezec, H. A. Atwater, and J. Weiner, "Quantitative determination of optical transmission through subwavelength slit arrays in Ag films: Role of surface wave interference and local coupling between adjacent slits," Phys. Rev. B 77, 115411 (2008).
[CrossRef]

G. Gay, O. Alloschery, B. V. de Lesegno, J. Weiner, and H. J. Lezec, "Surface wave generation and propagation on metallic subwavelength structures measured by far-field interferometry," Phys. Rev. Lett. 96, 213,901 (2006).
[CrossRef]

H. J. Lezec and T. Thio, "Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays," Opt. Express 12, 3629-3651 (2004).
[CrossRef] [PubMed]

Liu, H. T.

H. T. Liu and P. Lalanne, "Microscopic theory of the extraordinary optical transmission," Nature 452, 728-731 (2008).

Mansuripur, M.

Maradudin, A. A.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, "Nano-optics of surface plasmon polaritons," Phys. Reports-Review Section of Phys. Lett. 408, 131-314 (2005).

Martin, O. J. F.

G. Leveque, O. J. F. Martin, and J. Weiner, "Transient behavior of surface plasmon polaritons scattered at a subwavelength groove," Phys. Rev. B 76, 155,418 (2007).
[CrossRef]

Moloney, J. V.

Ozbay, E.

E. Ozbay, "Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions," Science 311, 189-193 (2006).
[CrossRef] [PubMed]

Pacifici, D.

D. Pacifici, H. J. Lezec, H. A. Atwater, and J. Weiner, "Quantitative determination of optical transmission through subwavelength slit arrays in Ag films: Role of surface wave interference and local coupling between adjacent slits," Phys. Rev. B 77, 115411 (2008).
[CrossRef]

Ruppin, R.

R. Ruppin, "Electromagnetic energy density in a dispersive and absorptive material," Phys. Lett. A 299, 309-312 (2002).
[CrossRef]

Sheng, Y. L.

Smolyaninov, I. I.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, "Nano-optics of surface plasmon polaritons," Phys. Reports-Review Section of Phys. Lett. 408, 131-314 (2005).

Thio, T.

Treacy, M. M. J.

M. M. J. Treacy, "Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings," Phys. Rev. B 66, 195,105 (2002).
[CrossRef]

Ung, B.

Weiner, J.

D. Pacifici, H. J. Lezec, H. A. Atwater, and J. Weiner, "Quantitative determination of optical transmission through subwavelength slit arrays in Ag films: Role of surface wave interference and local coupling between adjacent slits," Phys. Rev. B 77, 115411 (2008).
[CrossRef]

J. Weiner, "Phase shifts and interference in surface plasmon polariton waves," Opt. Express 16, 950-956 (2008).
[CrossRef] [PubMed]

G. Leveque, O. J. F. Martin, and J. Weiner, "Transient behavior of surface plasmon polaritons scattered at a subwavelength groove," Phys. Rev. B 76, 155,418 (2007).
[CrossRef]

G. Gay, O. Alloschery, B. V. de Lesegno, J. Weiner, and H. J. Lezec, "Surface wave generation and propagation on metallic subwavelength structures measured by far-field interferometry," Phys. Rev. Lett. 96, 213,901 (2006).
[CrossRef]

Xie, Y.

Zakharian, A. R.

Zayats, A. V.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, "Nano-optics of surface plasmon polaritons," Phys. Reports-Review Section of Phys. Lett. 408, 131-314 (2005).

Nature

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

H. T. Liu and P. Lalanne, "Microscopic theory of the extraordinary optical transmission," Nature 452, 728-731 (2008).

Nature (London)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature (London) 424, 824-830 (2003).
[CrossRef] [PubMed]

Opt. Express

Phys. Lett.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, "Nano-optics of surface plasmon polaritons," Phys. Reports-Review Section of Phys. Lett. 408, 131-314 (2005).

Phys. Lett. A

R. Ruppin, "Electromagnetic energy density in a dispersive and absorptive material," Phys. Lett. A 299, 309-312 (2002).
[CrossRef]

Phys. Rev. B

M. M. J. Treacy, "Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings," Phys. Rev. B 66, 195,105 (2002).
[CrossRef]

G. Leveque, O. J. F. Martin, and J. Weiner, "Transient behavior of surface plasmon polaritons scattered at a subwavelength groove," Phys. Rev. B 76, 155,418 (2007).
[CrossRef]

P. B. Johnson and R. W. Christy, "Optical Constants of Noble Metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

D. Pacifici, H. J. Lezec, H. A. Atwater, and J. Weiner, "Quantitative determination of optical transmission through subwavelength slit arrays in Ag films: Role of surface wave interference and local coupling between adjacent slits," Phys. Rev. B 77, 115411 (2008).
[CrossRef]

Phys. Rev. Lett.

G. Gay, O. Alloschery, B. V. de Lesegno, J. Weiner, and H. J. Lezec, "Surface wave generation and propagation on metallic subwavelength structures measured by far-field interferometry," Phys. Rev. Lett. 96, 213,901 (2006).
[CrossRef]

Rev. Mod. Phys.

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

Science

E. Ozbay, "Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions," Science 311, 189-193 (2006).
[CrossRef] [PubMed]

N. Engheta, "Circuits with light at nanoscales: Optical nanocircuits inspired by metamaterials," Science 317, 1698-1702 (2007).
[CrossRef] [PubMed]

Other

E. Palik and G. Ghosh, eds., The Electronic Handbook of Optical Constants of Solids (Academic, New York, 1999).

J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons Inc, 1999).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, 1995). Chap. 3, pp. 109-120

C. Kittel, Introduction to Solid State Physics, 7th ed. (John Wiley & Sons Inc, New York, 1996). Chap. 6, p. 150,

N. W. Ashcroft and N. D. Mermin, Solid State Physics, (Thomson Learning, 1976). Chap. 1, p. 10

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Figures (6)

Fig. 1.
Fig. 1.

Basic layout showing metal layer with two slits of length l and width d separated by distance p. Dimensions are those typical of experiments, for example, in Refs. [14], [15] with l~10-20 µm, d~50-100 nm, pspp. Metal layer thickness ~200–300 nm with skin depth δ⋍25 nm. Plane waves incident normal to the top surface are polarized in TM mode, penetrate to depth δ and reflect from the surface.

Fig. 2.
Fig. 2.

Incident light on a structured metal surface. Panel (A) shows a numerical simulation of the standing wave field set up by the incident and reflected light in TM mode as indicated in Fig. 1. Dark blue rectangle with two slits is a silver layer. Above the surface and in the slit gaps B-field amplitude is color coded with red maximum and blue minimum. Note that the B-field predominates near the surface. Panel (B) is an enlarged schematic of the interface region, showing the optical half-cycle in which the B-field points in the +y direction, the skin depth δ, the distance p between slits of width d, and the induced charges at the top edges of the slits.

Fig. 3.
Fig. 3.

Left and right panels show the real and imaginary parts respectively of the permittivity for Ag metal over the petahertz frequency range. Plotted blue triangles are from [19], red circles, [20]. Black curve is the harmonic oscillator model (HOM), Eq. 24, with ω p =1.4×1016 rad s-1 and Γ=1×1014 rad s-1. The model agrees well with data over the range 1≤ω≤6×1015 rad s-1.

Fig. 4.
Fig. 4.

(A) Orientation of E-M fields propagating along +z. (B) Orientation of E-M fields propagating along -z. (C) Orientation of E-M fields for normal reflection at x-y plane of incidence. Applying the continuity conditions upon reflection at the surface reverses the direction of the E-field and leaves the B-field orientation unchanged, consistent with the Poynting vector condition S=1/µ 0 (E×B).

Fig. 5.
Fig. 5.

Panel (A): Front view of incident plane wave setting up a standing wave at the surface of a smooth, featureless metal slab. Red amplitude indicates B-field maxima, blue shows E-field maxima. Panel (B) Diagram of the field metal surface shown in panel (A) for an optical half-cycle with E I pointing along +x and B I pointing along +y. The skin depth in z is indicated as δ, and p is some arbitrary length along x. The area A 1= and contour C 1 are used for the integration of Eq. (36). The conduction current induced in the metal, J c , is in phase with the incident E-field.

Fig. 6.
Fig. 6.

Panel (A): Standing plane wave above the two-slit structure, showing transmission through the slits at some arbitrary slit separation. Red (blue) color shows maximum B-field (E-field) amplitudes. Note that propagating fields within the slits set up Fabry-Perot-like cavities, the characteristics of which depend on the slit width, metal film thickness, and metal permittivity. The simulation shown here is for 100 nm slit in a silver film of thickness ⋍350 nm. Panel (B): A diagram of the area between the slits separated by λ spp, showing the two superimposed standing-wave B-field amplitudes, one from the incident plane wave, the other from the spp standing wave. The E-field of the spp standing wave is indicated as a thick black arrow (E spp) pointing from positive to negative charge at the two slit edges. The induced current density in the metal from the plane standing wave is denoted J pw and from the spp standing wave J spp.

Tables (1)

Tables Icon

Table 1. Comparison of dielectric constants for silver derived from two frequently cited data sets and the damped harmonic oscillator model (HOM), ω=2.0×1015 s-1. The real and imaginary terms of the index of refraction η,κ are equivalent to n 1,n 2 used here.

Equations (45)

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

· E = 0
· B = 0
× E = B t Faraday Maxwell
× B = μ ε E t + μ J c Ampre - Maxwell
2 E = μ ε 2 E t 2 + μ σ E t
2 B = μ ε 2 B t 2 + μ σ B t
E m ( r , t ) = E ˜ 0 m e i ( k ˜ m · r ω t )
B m ( r , t ) = B ˜ 0 m e i ( k ˜ m · r ω t )
k ˜ m 2 = μ ε ˜ ω 2 + i μ σ ˜ ω
k ˜ m = k m 1 + i k m 2 ε ˜ = ε r + i ε i σ ˜ = σ r + i σ i
k ˜ m 2 = ( μ ε r ω 2 μ σ i ω ) + i ( μ ε i ω 2 + μ σ r ω )
k m 1 2 k m 2 2 = ( μ ε r ω 2 μ σ i ω )
2 k m 1 k m 2 = ( μ ε i ω 2 + μ σ r ω )
k m 1 2 k m 2 2 = μ 0 ε 0 ω 2 ( ε ' σ i ε 0 ω ) = k 0 2 ( ε ' σ i ε 0 ω ) = β 2
2 k m 1 k m 2 = μ 0 ε 0 ω 2 ( ε + σ r ε 0 ω ) = k 0 2 ( ε + σ r ε 0 ω ) = γ 2
k m 1 = ± β 2 1 ± 1 + ( γ β ) 4
k m 2 = ± β 2 1 ± 1 + ( γ β ) 4
k ˜ m = k 0 n ˜ = k 0 ( n 1 + i n 2 )
k m 1 = k 0 ε 1 2 1 ± 1 + ( ε 2 ε 1 ) 4
k m 2 = k 0 ε 1 2 1 ± 1 + ( ε 2 ε 1 ) 2
ε 1 = ( ε ' σ i ε 0 ω )
ε 2 = ( ε + σ r ε 0 ω )
ω p 2 = e 2 N e m e ε 0
J c = σ ˜ E = σ ˜ ε 0 ε 0 E
σ ˜ ε 0 = Γ ( Γ 2 ω p 2 + ω 2 ω p 2 ) + i ω ( Γ 2 ω p 2 + ω 2 ω p 2 )
σ ˜ ε 0 Γ ( ω 2 ω p 2 ) + i ω ( ω 2 ω p 2 ) = Γ ω p 2 ω 2 + i ω p 2 ω
ε 1 = ( ε ' ω p 2 ε 0 ω 2 ) ε 2 = ( ε + Γ ω p 2 ε 0 ω 3 )
E ˜ 0 I + E ˜ 0 R = E ˜ 0 T
B ˜ 0 I B ˜ 0 R = B ˜ 0 T = 1 c ( E ˜ 0 I E ˜ 0 R ) = k ˜ m ω E ˜ 0 T
k ˜ m = k m 1 + i k m 2 = k m e i φ and k m = k m 1 2 + k m 2 2 with φ = tan 1 k m 2 k m 1
B 0 T E 0 T = e i ( φ BT φ ET ) = k m ω e i φ
n ˜ = n 1 + i n 2 = k ˜ m k 0 = c ω ( k m 1 + i k m 2 )
E ˜ 0 T = ( 2 1 + n ˜ ) E ˜ 0 I = 2 n ˜ ( 1 1 n ˜ + 1 ) E ˜ 0 I = 2 n ˜ ( 1 1 n ˜ + 1 n ˜ 2 ) E ˜ 0 I 2 n ˜ E ˜ 0 I
φ = tan 1 k m 2 k m 1 π 2 φ BT φ ET π 2 and E 0 T E 0 I 2 k 0 k m e i π 2
B 0 T 2 B 0 I
B 0 T E 0 T = B 0 ( i ) E 0 ( i ) 2 B 0 I ( 2 E 0 I n 2 )
< u > d = 1 2 [ 1 2 ε 0 E 0 ( i ) 2 + 1 2 1 μ 0 B 0 ( i ) 2 ]
< u > d = 1 2 ε 0 E 0 2 incident [ 2 n 2 2 + 2 ]
A 1 ( × E T ( 1 ) ) · n A 1 d A 1 = C 1 E T ( 1 ) · d s 1 = A 1 ( B T ( 0 ) t ) · n A 1 d A 1
E T ( 1 ) = i ω B T ( 0 ) k m 2 = i c 2 B I n 2 = i 2 E I n 2
J c = σ ˜ E T ( 1 ) σ i 2 E I n 2
B T ( 1 ) = μ 0 σ i E T ( 1 ) n 2 k 2 m μ 0 σ i 2 E I n 2 2 k 0 ( ω p ω ) 2 2 E I n 2 2 c
J p = J slit = ε 0 ε slit E slit t
E slit p = i ( ω p ω ) 2 2 E I ε slit n 2
B slit p = i μ 0 σ i 2 E I n 2 k slit = i ( ω p ω ) 2 2 E I n 2 n slit c

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