Abstract

The excitation of surface plasmons by subwavelength slits in metal films is studied using a rigorous diffraction model. It is shown that the plasmon is launched by a slit in antiphase with the incident magnetic field. This is true independent of slit width and of the metal used. Using this phase information, maxima and minima in transmission are explained in the case of two and more slits.

© 2006 Optical Society of America

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References

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  1. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer Verlag, 1988).
  2. V. M. Agranovich, Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces (North-Holland, 1982).
  3. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
    [CrossRef]
  4. Q. Cao and P. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057403 (2002).
    [CrossRef] [PubMed]
  5. P. Lalanne, J. P. Hugonin, and J. C. Rodier, "Theory of surface plasmon generation at nanoslit apertures," Phys. Rev. Lett. 95, 263902 (2005).
    [CrossRef]
  6. 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]
  7. 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]
  8. H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. ’t Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
    [CrossRef] [PubMed]
  9. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, "Transmission of light through slit apertures in metallic films," Opt. Express 12, 6106-6121 (2004).
    [CrossRef] [PubMed]
  10. H. 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]
  11. M. M. J. Treacy, "Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings," Phys. Rev. B 66, 195105 (2002).
    [CrossRef]
  12. H. F. Schouten, T. D. Visser, D. Lenstra, and H. Blok, "Light transmission through a subwavelength slit: Waveguiding and optical vortices," Phys. Rev. E 67, 036608 (2003).
    [CrossRef]
  13. J. Gaspar-Armenta, R. García-Llamas, and J. Durán-Favela, "Electromagnetic near and far fields from the interaction between surface plasmons and a surface defect in a thin metallic film," Phys. Rev. B 73, 155412 (2006).
    [CrossRef]
  14. A. Degiron and T. W. Ebbesen, "The role of localized surface plasmon modes in the enhanced transmission of periodic subwavelength apertures," J. Opt. A: Pure Appl. Opt. 7S90-S96 (2005).
    [CrossRef]
  15. W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, "Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film," Phys. Rev. Lett. 92, 107401 (2004).
    [CrossRef] [PubMed]
  16. J.-P. Berenger, "A perfectly matched absorbing layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185 - 200 (1994).
    [CrossRef]
  17. C. M. Rappaport, "Perfectly matched absorbing boundary conditions based on anistropic lossy mapping of space," IEEE Microw. Guid. Wave Lett. 5, 90-92 (1995).
    [CrossRef]
  18. H. J. Eom, Wave Scattering Theory (Springer, 2001), Chap. 3.
    [CrossRef]
  19. J. M. Brok and H. P. Urbach, "Extraordinary transmission through 1, 2, and 3 holes in a perfect conductor, modeled by a mode expansion technique," Opt. Express 14, 2552-2572 (2006).
    [CrossRef] [PubMed]
  20. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (Springer, 1999), Chap. 6.
  21. A. Nesci, R. Dändliker, and H. P. Herzig, "Quantitative amplitude and phase measurement by use of a heterodyne scanning near-field optical microscope," Opt. Lett. 26, 208 - 210 (2001).
    [CrossRef]

2006 (3)

2005 (4)

A. Degiron and T. W. Ebbesen, "The role of localized surface plasmon modes in the enhanced transmission of periodic subwavelength apertures," J. Opt. A: Pure Appl. Opt. 7S90-S96 (2005).
[CrossRef]

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]

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. ’t Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

P. Lalanne, J. P. Hugonin, and J. C. Rodier, "Theory of surface plasmon generation at nanoslit apertures," Phys. Rev. Lett. 95, 263902 (2005).
[CrossRef]

2004 (3)

2003 (1)

H. F. Schouten, T. D. Visser, D. Lenstra, and H. Blok, "Light transmission through a subwavelength slit: Waveguiding and optical vortices," Phys. Rev. E 67, 036608 (2003).
[CrossRef]

2002 (2)

Q. Cao and P. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057403 (2002).
[CrossRef] [PubMed]

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

2001 (1)

1998 (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 391, 667-669 (1998).
[CrossRef]

1995 (1)

C. M. Rappaport, "Perfectly matched absorbing boundary conditions based on anistropic lossy mapping of space," IEEE Microw. Guid. Wave Lett. 5, 90-92 (1995).
[CrossRef]

1994 (1)

J.-P. Berenger, "A perfectly matched absorbing layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185 - 200 (1994).
[CrossRef]

’t Hooft, G. W.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. ’t Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Alkemade, P. F. A.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. ’t Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Barnes, W. L.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, "Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film," Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

Berenger, J.-P.

J.-P. Berenger, "A perfectly matched absorbing layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185 - 200 (1994).
[CrossRef]

Blok, H.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. ’t Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

H. F. Schouten, T. D. Visser, D. Lenstra, and H. Blok, "Light transmission through a subwavelength slit: Waveguiding and optical vortices," Phys. Rev. E 67, 036608 (2003).
[CrossRef]

Brok, J. M.

Cao, Q.

Q. Cao and P. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057403 (2002).
[CrossRef] [PubMed]

Dändliker, R.

Degiron, A.

A. Degiron and T. W. Ebbesen, "The role of localized surface plasmon modes in the enhanced transmission of periodic subwavelength apertures," J. Opt. A: Pure Appl. Opt. 7S90-S96 (2005).
[CrossRef]

Devaux, E.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, "Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film," Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

Dintinger, J.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, "Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film," Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

Dubois, G.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. ’t Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Durán-Favela, J.

J. Gaspar-Armenta, R. García-Llamas, and J. Durán-Favela, "Electromagnetic near and far fields from the interaction between surface plasmons and a surface defect in a thin metallic film," Phys. Rev. B 73, 155412 (2006).
[CrossRef]

Ebbesen, T. W.

A. Degiron and T. W. Ebbesen, "The role of localized surface plasmon modes in the enhanced transmission of periodic subwavelength apertures," J. Opt. A: Pure Appl. Opt. 7S90-S96 (2005).
[CrossRef]

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, "Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film," Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

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

Eliel, E. R.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. ’t Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

García-Llamas, R.

J. Gaspar-Armenta, R. García-Llamas, and J. Durán-Favela, "Electromagnetic near and far fields from the interaction between surface plasmons and a surface defect in a thin metallic film," Phys. Rev. B 73, 155412 (2006).
[CrossRef]

Gaspar-Armenta, J.

J. Gaspar-Armenta, R. García-Llamas, and J. Durán-Favela, "Electromagnetic near and far fields from the interaction between surface plasmons and a surface defect in a thin metallic film," Phys. Rev. B 73, 155412 (2006).
[CrossRef]

Gbur, G.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. ’t Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Ghaemi, H. F.

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

Herzig, H. P.

Hugonin, J. P.

P. Lalanne, J. P. Hugonin, and J. C. Rodier, "Theory of surface plasmon generation at nanoslit apertures," Phys. Rev. Lett. 95, 263902 (2005).
[CrossRef]

Kuzmin, N.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. ’t Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Lalanne, P.

P. Lalanne, J. P. Hugonin, and J. C. Rodier, "Theory of surface plasmon generation at nanoslit apertures," Phys. Rev. Lett. 95, 263902 (2005).
[CrossRef]

Q. Cao and P. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057403 (2002).
[CrossRef] [PubMed]

Lenstra, D.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. ’t Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

H. F. Schouten, T. D. Visser, D. Lenstra, and H. Blok, "Light transmission through a subwavelength slit: Waveguiding and optical vortices," Phys. Rev. E 67, 036608 (2003).
[CrossRef]

Lezec, H.

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 391, 667-669 (1998).
[CrossRef]

Mansuripur, M.

Moloney, J. V.

Murray, W. A.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, "Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film," Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

Nesci, A.

Rappaport, C. M.

C. M. Rappaport, "Perfectly matched absorbing boundary conditions based on anistropic lossy mapping of space," IEEE Microw. Guid. Wave Lett. 5, 90-92 (1995).
[CrossRef]

Rodier, J. C.

P. Lalanne, J. P. Hugonin, and J. C. Rodier, "Theory of surface plasmon generation at nanoslit apertures," Phys. Rev. Lett. 95, 263902 (2005).
[CrossRef]

Schouten, H. F.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. ’t Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

H. F. Schouten, T. D. Visser, D. Lenstra, and H. Blok, "Light transmission through a subwavelength slit: Waveguiding and optical vortices," Phys. Rev. E 67, 036608 (2003).
[CrossRef]

Thio, T.

H. 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]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 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]

Urbach, H. P.

Visser, T. D.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. ’t Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

H. F. Schouten, T. D. Visser, D. Lenstra, and H. Blok, "Light transmission through a subwavelength slit: Waveguiding and optical vortices," Phys. Rev. E 67, 036608 (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 391, 667-669 (1998).
[CrossRef]

Xie, Y.

Zakharian, A. R.

IEEE Microw. Guid. Wave Lett. (1)

C. M. Rappaport, "Perfectly matched absorbing boundary conditions based on anistropic lossy mapping of space," IEEE Microw. Guid. Wave Lett. 5, 90-92 (1995).
[CrossRef]

J. Comput. Phys. (1)

J.-P. Berenger, "A perfectly matched absorbing layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185 - 200 (1994).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

A. Degiron and T. W. Ebbesen, "The role of localized surface plasmon modes in the enhanced transmission of periodic subwavelength apertures," J. Opt. A: Pure Appl. Opt. 7S90-S96 (2005).
[CrossRef]

Nature (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 391, 667-669 (1998).
[CrossRef]

Opt. Express (5)

Opt. Lett. (1)

Phys. Rev. B (2)

J. Gaspar-Armenta, R. García-Llamas, and J. Durán-Favela, "Electromagnetic near and far fields from the interaction between surface plasmons and a surface defect in a thin metallic film," Phys. Rev. B 73, 155412 (2006).
[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 (1)

H. F. Schouten, T. D. Visser, D. Lenstra, and H. Blok, "Light transmission through a subwavelength slit: Waveguiding and optical vortices," Phys. Rev. E 67, 036608 (2003).
[CrossRef]

Phys. Rev. Lett. (4)

Q. Cao and P. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057403 (2002).
[CrossRef] [PubMed]

P. Lalanne, J. P. Hugonin, and J. C. Rodier, "Theory of surface plasmon generation at nanoslit apertures," Phys. Rev. Lett. 95, 263902 (2005).
[CrossRef]

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. ’t Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, "Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film," Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

Other (4)

H. J. Eom, Wave Scattering Theory (Springer, 2001), Chap. 3.
[CrossRef]

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer Verlag, 1988).

V. M. Agranovich, Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces (North-Holland, 1982).

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (Springer, 1999), Chap. 6.

Supplementary Material (1)

» Media 1: AVI (1023 KB)     

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Figures (7)

Fig. 1.
Fig. 1.

The structures used (a) A gold film with thickness t with a single slit with width W suspended in air. (b) A gold film with 10nm thick titanium coating on a glass substrate in air with a single slit. (c-d) As a-b but with two slits.

Fig. 2.
Fig. 2.

Amplitude and phase of surface plasmon excited when a TM polarized plane wave (λ=800nm) is incident on the air-gold-air structure (FEM sym), the air-gold-titaniumglass structure (FEM asym) and the analytically calculated PEC film in air (PEC). (left) Varying the slit width W for constant film thickness t=200nm. (right) Varying the film thickness t while keeping the slit width constant at W=200nm.

Fig. 3.
Fig. 3.

(1.0 MB) Movie of real magnetic field (Hy ) at a gold film with two slits 15.3µm apart. Only the slit at x=0nm is illuminated with a TM polarized spot focused in the x-direction only. The field at the not-illuminated slit is shown. The plasmon on the lower surface is weak because of a 10nm thick titanium layer. Shown is the real magnetic field Hy .

Fig. 4.
Fig. 4.

Schematic view of two transmission mechanisms for a plasmon excited at one slit interacting with another slit. (a) The interference of excited surface plasmons with the incident field. (b) The interference of plasmons excited at different slits.

Fig. 5.
Fig. 5.

The scattered field |Hysca | for two slits illuminated by a TM plane wave (λ=800nm) with slit distance d corresponding to (top) a minimum in transmission at d=14.8µm and (bottom) a maximum in transmission at d=15.3µm. Outside the two slit region, constructive interference of plasmons is visible in the top image and destructive interference of plasmons is visible in the bottom image.

Fig. 6.
Fig. 6.

Normalized transmission for two slits illuminated by a TM plane wave (λ=800nm) as function of the separation of the slits and for three slit widths. The left plot corresponds to the air-gold-titanium-glass structure and the right plot to the air-gold-air structure. The circles indicate the maxima and minima as predicted by the plasmon phase obtained from the single slit experiment.

Fig. 7.
Fig. 7.

Normalized transmission for a structure with 2, 3, 4, 8, 16 slits and a periodic structure with W=200nm and t=200nm illuminated by a plane wave (λ=800nm). The bold lines correspond to calculations with W=200nm and t=350nm. Differences between the plot of the periodic structure and the 16 slit structure mainly originate because fewer points are calculated for the 16 slit structure.

Equations (9)

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

k sp = 2 π λ sp = ε m ε d ε m + ε d ,
H y 0 ( x , z ) = exp ( i k 0 z ) + exp ( i k 0 z ) ,
H y s ( x , z ) = 1 2 π H ̂ y s ( k x ) exp ( i k x x + i k z z ) d k x
H y t ( x , z ) = 1 2 π H ̂ y t ( k x ) exp ( i k x x + i k z ( z + t ) ) d k x .
H y d ( x , z ) = A 0 exp ( i k 0 z ) + B 0 exp ( i k 0 z ) . ( inside the slit )
H y s ( x , 0 ) = 2 ( A 0 B 0 ) ( 2 π k 0 x ) sin ( k 0 W 2 ) .
2 π d λ sp + φ sp up = 2 π m , ( m = 1 , 2 , 3 , . . . ) ,
2 π d λ sp π ( 2 m 1 ) , ( m = 1 , 2 , 3 , . . . ) ,
2 π d λ sp = 2 π m , ( m = 1 , 2 , 3 , . . . ) .

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