Abstract

By applying a scattering-wave theory, the electromagnetic response of an arbitrary array of multiple slits perforated on a metallic film and filled with different slit dielectric materials can be studied in an analytical way. Here, the wavelength-dependent splitting of a light beam into two by asymmetrically filled slits in a metal film using intraslit and interslit dual-wave interferences is fully explored. We consider a triple-slit structure perforated on a gold film, where the middle slit is used for the surface-plasmon (SP) excitation by a narrow Gaussian beam while the two side slits are used for the detection of a transmitted SP wave propagated from the middle opaque slit either at a particular wavelength or at double that wavelength, respectively. For this proposed simple structure, we show that only one of the two side observation slits can be in a passing state for a particular wavelength, but the other blocked slit will change to a passing state at double that wavelength with a specific design for the slit depth, slit dielectric, and interslit distance in the deep subwavelength regime. In this sense, SP mediated light transmission becomes wavelength sensitive in our model, and a single light beam can be separated into two according to its wavelength in the transverse direction parallel to the array. This provides us with a unique way for direct optical reading in the near-field region using a nonspectroscopic approach.

© 2012 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. G. Gumbs and D. H. Huang, Properties of Interacting Low-Dimensional Systems (Wiley-VCH, 2011), Chaps. 4 and 5.
  3. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010) [and references therein].
    [CrossRef]
  4. T. W. Ebbsen, 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]
  5. 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]
  6. T. Thio, H. F. Ghaemi, H. J. Lezec, P. A. Wolff, and T. W. Ebbesen, “Surface-plasmon-enhanced transmission through hole arrays in Cr films,” J. Opt. Soc. Am. B 16, 1743–1748 (1999).
    [CrossRef]
  7. J. C.-C. Chang, Z.-P. Yang, D. H. Huang, D. A. Cardimona, and S.-Y. Lin, “Strong light concentration at the subwavelength scale by a metallic hole-array structure,” Opt. Lett. 34, 106–108 (2009).
    [CrossRef]
  8. J. C.-C. Chang, Y. D. Sharma, Y.-S. Kim, J. A. Bur, R. V. Shenoi, S. Krishna, D. H. Huang, and S.-Y. Lin, “A surface plasmon enhanced infrared photodetector based on InAs quantum dots,” Nano Lett. 10, 1704–1709 (2010).
    [CrossRef]
  9. A. A. Maradudin, ed., Light Scattering and Nanoscale Surface Roughness (Springer Science+Business Media, 2007).
  10. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005) [and references therein].
    [CrossRef]
  11. F. de Leon-Perez, G. Brucoli, F. J. Garcia-Vidal, and L. Martin-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” New J. Phys. 10, 105017 (2008).
    [CrossRef]
  12. B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Transmission of light through a thin metal film with periodically and randomly corrugated surfaces,” J. Opt A: Pure Appl. Opt. 8, S191–S207 (2006).
    [CrossRef]
  13. E. Laux, C. Genet, T. Skauli, and T. W. Ebbesen, “Plasmonic photon sorters for spectral and polarimetric imaging,” Nat. Photonics 2, 161–164 (2008).
    [CrossRef]
  14. L. D. Wellems, D. H. Huang, T. A. Leskova, and A. A. Maradudin, “Nanogroove array on thin metallic film as planar lens with tunable focusing,” Phys. Lett. A 376, 216–220 (2012).
    [CrossRef]
  15. L. D. Wellems, D. H. Huang, T. A. Leskova, and A. A. Maradudin, “Optical spectrum and electromagnetic-field distribution at double-groove metallic surface gratings,” J. Appl. Phys. 106, 053705 (2009).
    [CrossRef]
  16. F. López-Tejeira, F. J. Garcia-Vidal, and L. Martin-Moreno, “Scattering of surface plasmons by one-dimensional periodic nanoindented surfaces,” Phys. Rev. B 72, 161405 (2005).
    [CrossRef]
  17. T. A. Leskova, A. A. Maradudin, and I. Novikov, “Impedance boundary conditions for a metal film with a rough surface,” Appl. Opt. 38, 1197–1212 (1999).
    [CrossRef]
  18. A. A. Maradudin and A. Sentenac, “The impedance boundary condition for a periodically corrugated metal surface,” Solid State Commun. 84, 159–163 (1992).
    [CrossRef]
  19. H. Lochbihler and R. Depine, “Highly conducting wire gratings in the resonance region,” Appl. Opt. 32, 3459–3465 (1993).
    [CrossRef]
  20. D. Crouse, “Numerical modeling and electromagnetic resonant modes in complex grating structures and optoelectronic device applications,” IEEE Trans. Electron Devices 52, 2365–2373 (2005).
    [CrossRef]
  21. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]
  22. L. D. Wellems and D. H. Huang, “Near-field light focusing by a slit array in a planar metal film with nonuniform slit dielectric material,” Am. J. Phys. 80, 122–132 (2012).
    [CrossRef]
  23. T. López-Rios, D. Mendoza, F. J. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in Lamellar metallic gratings,” Phys. Rev. Lett. 81, 665–668 (1998).
    [CrossRef]
  24. F. J. Garcia-Vidal, J. Sánchez-Dehesa, A. Dechelette, E. Bustarret, T. López-Rios, T. Fournier, and B. Pannetier, “Localized surface plasmons in Lamellar metallic gratings,” J. Lightwave Technol. 17, 2191–2195 (1999).
    [CrossRef]
  25. V. M. Serdyuk, “Diffraction of a plane electromagnetic wave by a slot in a conducting screen of arbitrary thickness,” Tech. Phys. 50, 1076–1083 (2005).
    [CrossRef]
  26. D. H. Huang, G. Gumbs, P. M. Alsing, and D. A. Cardimona, “Nonlocal mode mixing and surface-plasmon-polariton-mediated enhancement of diffracted terahertz fields by a conductive grating,” Phys. Rev. B 77, 165404 (2008).
    [CrossRef]
  27. D. H. Huang, G. Gumbs, and S.-Y. Lin, “Self-consistent theory for near-field distribution and spectrum with quantum wires and a conductive grating in terahertz regime,” J. Appl. Phys. 105, 093715 (2009).
    [CrossRef]

2012 (2)

L. D. Wellems, D. H. Huang, T. A. Leskova, and A. A. Maradudin, “Nanogroove array on thin metallic film as planar lens with tunable focusing,” Phys. Lett. A 376, 216–220 (2012).
[CrossRef]

L. D. Wellems and D. H. Huang, “Near-field light focusing by a slit array in a planar metal film with nonuniform slit dielectric material,” Am. J. Phys. 80, 122–132 (2012).
[CrossRef]

2010 (2)

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010) [and references therein].
[CrossRef]

J. C.-C. Chang, Y. D. Sharma, Y.-S. Kim, J. A. Bur, R. V. Shenoi, S. Krishna, D. H. Huang, and S.-Y. Lin, “A surface plasmon enhanced infrared photodetector based on InAs quantum dots,” Nano Lett. 10, 1704–1709 (2010).
[CrossRef]

2009 (3)

L. D. Wellems, D. H. Huang, T. A. Leskova, and A. A. Maradudin, “Optical spectrum and electromagnetic-field distribution at double-groove metallic surface gratings,” J. Appl. Phys. 106, 053705 (2009).
[CrossRef]

D. H. Huang, G. Gumbs, and S.-Y. Lin, “Self-consistent theory for near-field distribution and spectrum with quantum wires and a conductive grating in terahertz regime,” J. Appl. Phys. 105, 093715 (2009).
[CrossRef]

J. C.-C. Chang, Z.-P. Yang, D. H. Huang, D. A. Cardimona, and S.-Y. Lin, “Strong light concentration at the subwavelength scale by a metallic hole-array structure,” Opt. Lett. 34, 106–108 (2009).
[CrossRef]

2008 (3)

D. H. Huang, G. Gumbs, P. M. Alsing, and D. A. Cardimona, “Nonlocal mode mixing and surface-plasmon-polariton-mediated enhancement of diffracted terahertz fields by a conductive grating,” Phys. Rev. B 77, 165404 (2008).
[CrossRef]

E. Laux, C. Genet, T. Skauli, and T. W. Ebbesen, “Plasmonic photon sorters for spectral and polarimetric imaging,” Nat. Photonics 2, 161–164 (2008).
[CrossRef]

F. de Leon-Perez, G. Brucoli, F. J. Garcia-Vidal, and L. Martin-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” New J. Phys. 10, 105017 (2008).
[CrossRef]

2006 (1)

B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Transmission of light through a thin metal film with periodically and randomly corrugated surfaces,” J. Opt A: Pure Appl. Opt. 8, S191–S207 (2006).
[CrossRef]

2005 (4)

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005) [and references therein].
[CrossRef]

F. López-Tejeira, F. J. Garcia-Vidal, and L. Martin-Moreno, “Scattering of surface plasmons by one-dimensional periodic nanoindented surfaces,” Phys. Rev. B 72, 161405 (2005).
[CrossRef]

V. M. Serdyuk, “Diffraction of a plane electromagnetic wave by a slot in a conducting screen of arbitrary thickness,” Tech. Phys. 50, 1076–1083 (2005).
[CrossRef]

D. Crouse, “Numerical modeling and electromagnetic resonant modes in complex grating structures and optoelectronic device applications,” IEEE Trans. Electron Devices 52, 2365–2373 (2005).
[CrossRef]

1999 (3)

1998 (3)

T. López-Rios, D. Mendoza, F. J. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in Lamellar metallic gratings,” Phys. Rev. Lett. 81, 665–668 (1998).
[CrossRef]

T. W. Ebbsen, 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]

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]

1993 (1)

1992 (1)

A. A. Maradudin and A. Sentenac, “The impedance boundary condition for a periodically corrugated metal surface,” Solid State Commun. 84, 159–163 (1992).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Alsing, P. M.

D. H. Huang, G. Gumbs, P. M. Alsing, and D. A. Cardimona, “Nonlocal mode mixing and surface-plasmon-polariton-mediated enhancement of diffracted terahertz fields by a conductive grating,” Phys. Rev. B 77, 165404 (2008).
[CrossRef]

Baumeier, B.

B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Transmission of light through a thin metal film with periodically and randomly corrugated surfaces,” J. Opt A: Pure Appl. Opt. 8, S191–S207 (2006).
[CrossRef]

Brucoli, G.

F. de Leon-Perez, G. Brucoli, F. J. Garcia-Vidal, and L. Martin-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” New J. Phys. 10, 105017 (2008).
[CrossRef]

Bur, J. A.

J. C.-C. Chang, Y. D. Sharma, Y.-S. Kim, J. A. Bur, R. V. Shenoi, S. Krishna, D. H. Huang, and S.-Y. Lin, “A surface plasmon enhanced infrared photodetector based on InAs quantum dots,” Nano Lett. 10, 1704–1709 (2010).
[CrossRef]

Bustarret, E.

Cardimona, D. A.

J. C.-C. Chang, Z.-P. Yang, D. H. Huang, D. A. Cardimona, and S.-Y. Lin, “Strong light concentration at the subwavelength scale by a metallic hole-array structure,” Opt. Lett. 34, 106–108 (2009).
[CrossRef]

D. H. Huang, G. Gumbs, P. M. Alsing, and D. A. Cardimona, “Nonlocal mode mixing and surface-plasmon-polariton-mediated enhancement of diffracted terahertz fields by a conductive grating,” Phys. Rev. B 77, 165404 (2008).
[CrossRef]

Chang, J. C.-C.

J. C.-C. Chang, Y. D. Sharma, Y.-S. Kim, J. A. Bur, R. V. Shenoi, S. Krishna, D. H. Huang, and S.-Y. Lin, “A surface plasmon enhanced infrared photodetector based on InAs quantum dots,” Nano Lett. 10, 1704–1709 (2010).
[CrossRef]

J. C.-C. Chang, Z.-P. Yang, D. H. Huang, D. A. Cardimona, and S.-Y. Lin, “Strong light concentration at the subwavelength scale by a metallic hole-array structure,” Opt. Lett. 34, 106–108 (2009).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Crouse, D.

D. Crouse, “Numerical modeling and electromagnetic resonant modes in complex grating structures and optoelectronic device applications,” IEEE Trans. Electron Devices 52, 2365–2373 (2005).
[CrossRef]

de Leon-Perez, F.

F. de Leon-Perez, G. Brucoli, F. J. Garcia-Vidal, and L. Martin-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” New J. Phys. 10, 105017 (2008).
[CrossRef]

Dechelette, A.

Depine, R.

Ebbesen, T. W.

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010) [and references therein].
[CrossRef]

E. Laux, C. Genet, T. Skauli, and T. W. Ebbesen, “Plasmonic photon sorters for spectral and polarimetric imaging,” Nat. Photonics 2, 161–164 (2008).
[CrossRef]

T. Thio, H. F. Ghaemi, H. J. Lezec, P. A. Wolff, and T. W. Ebbesen, “Surface-plasmon-enhanced transmission through hole arrays in Cr films,” J. Opt. Soc. Am. B 16, 1743–1748 (1999).
[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]

Ebbsen, T. W.

T. W. Ebbsen, 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]

Fournier, T.

Garcia-Vidal, F. J.

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010) [and references therein].
[CrossRef]

F. de Leon-Perez, G. Brucoli, F. J. Garcia-Vidal, and L. Martin-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” New J. Phys. 10, 105017 (2008).
[CrossRef]

F. López-Tejeira, F. J. Garcia-Vidal, and L. Martin-Moreno, “Scattering of surface plasmons by one-dimensional periodic nanoindented surfaces,” Phys. Rev. B 72, 161405 (2005).
[CrossRef]

F. J. Garcia-Vidal, J. Sánchez-Dehesa, A. Dechelette, E. Bustarret, T. López-Rios, T. Fournier, and B. Pannetier, “Localized surface plasmons in Lamellar metallic gratings,” J. Lightwave Technol. 17, 2191–2195 (1999).
[CrossRef]

García-Vidal, F. J.

T. López-Rios, D. Mendoza, F. J. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in Lamellar metallic gratings,” Phys. Rev. Lett. 81, 665–668 (1998).
[CrossRef]

Genet, C.

E. Laux, C. Genet, T. Skauli, and T. W. Ebbesen, “Plasmonic photon sorters for spectral and polarimetric imaging,” Nat. Photonics 2, 161–164 (2008).
[CrossRef]

Ghaemi, H. F.

T. Thio, H. F. Ghaemi, H. J. Lezec, P. A. Wolff, and T. W. Ebbesen, “Surface-plasmon-enhanced transmission through hole arrays in Cr films,” J. Opt. Soc. Am. B 16, 1743–1748 (1999).
[CrossRef]

T. W. Ebbsen, 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]

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]

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, 6779–6782 (1998).
[CrossRef]

Gumbs, G.

D. H. Huang, G. Gumbs, and S.-Y. Lin, “Self-consistent theory for near-field distribution and spectrum with quantum wires and a conductive grating in terahertz regime,” J. Appl. Phys. 105, 093715 (2009).
[CrossRef]

D. H. Huang, G. Gumbs, P. M. Alsing, and D. A. Cardimona, “Nonlocal mode mixing and surface-plasmon-polariton-mediated enhancement of diffracted terahertz fields by a conductive grating,” Phys. Rev. B 77, 165404 (2008).
[CrossRef]

G. Gumbs and D. H. Huang, Properties of Interacting Low-Dimensional Systems (Wiley-VCH, 2011), Chaps. 4 and 5.

Huang, D. H.

L. D. Wellems, D. H. Huang, T. A. Leskova, and A. A. Maradudin, “Nanogroove array on thin metallic film as planar lens with tunable focusing,” Phys. Lett. A 376, 216–220 (2012).
[CrossRef]

L. D. Wellems and D. H. Huang, “Near-field light focusing by a slit array in a planar metal film with nonuniform slit dielectric material,” Am. J. Phys. 80, 122–132 (2012).
[CrossRef]

J. C.-C. Chang, Y. D. Sharma, Y.-S. Kim, J. A. Bur, R. V. Shenoi, S. Krishna, D. H. Huang, and S.-Y. Lin, “A surface plasmon enhanced infrared photodetector based on InAs quantum dots,” Nano Lett. 10, 1704–1709 (2010).
[CrossRef]

L. D. Wellems, D. H. Huang, T. A. Leskova, and A. A. Maradudin, “Optical spectrum and electromagnetic-field distribution at double-groove metallic surface gratings,” J. Appl. Phys. 106, 053705 (2009).
[CrossRef]

D. H. Huang, G. Gumbs, and S.-Y. Lin, “Self-consistent theory for near-field distribution and spectrum with quantum wires and a conductive grating in terahertz regime,” J. Appl. Phys. 105, 093715 (2009).
[CrossRef]

J. C.-C. Chang, Z.-P. Yang, D. H. Huang, D. A. Cardimona, and S.-Y. Lin, “Strong light concentration at the subwavelength scale by a metallic hole-array structure,” Opt. Lett. 34, 106–108 (2009).
[CrossRef]

D. H. Huang, G. Gumbs, P. M. Alsing, and D. A. Cardimona, “Nonlocal mode mixing and surface-plasmon-polariton-mediated enhancement of diffracted terahertz fields by a conductive grating,” Phys. Rev. B 77, 165404 (2008).
[CrossRef]

G. Gumbs and D. H. Huang, Properties of Interacting Low-Dimensional Systems (Wiley-VCH, 2011), Chaps. 4 and 5.

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Kim, Y.-S.

J. C.-C. Chang, Y. D. Sharma, Y.-S. Kim, J. A. Bur, R. V. Shenoi, S. Krishna, D. H. Huang, and S.-Y. Lin, “A surface plasmon enhanced infrared photodetector based on InAs quantum dots,” Nano Lett. 10, 1704–1709 (2010).
[CrossRef]

Krishna, S.

J. C.-C. Chang, Y. D. Sharma, Y.-S. Kim, J. A. Bur, R. V. Shenoi, S. Krishna, D. H. Huang, and S.-Y. Lin, “A surface plasmon enhanced infrared photodetector based on InAs quantum dots,” Nano Lett. 10, 1704–1709 (2010).
[CrossRef]

Kuipers, L.

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010) [and references therein].
[CrossRef]

Laux, E.

E. Laux, C. Genet, T. Skauli, and T. W. Ebbesen, “Plasmonic photon sorters for spectral and polarimetric imaging,” Nat. Photonics 2, 161–164 (2008).
[CrossRef]

Leskova, T. A.

L. D. Wellems, D. H. Huang, T. A. Leskova, and A. A. Maradudin, “Nanogroove array on thin metallic film as planar lens with tunable focusing,” Phys. Lett. A 376, 216–220 (2012).
[CrossRef]

L. D. Wellems, D. H. Huang, T. A. Leskova, and A. A. Maradudin, “Optical spectrum and electromagnetic-field distribution at double-groove metallic surface gratings,” J. Appl. Phys. 106, 053705 (2009).
[CrossRef]

B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Transmission of light through a thin metal film with periodically and randomly corrugated surfaces,” J. Opt A: Pure Appl. Opt. 8, S191–S207 (2006).
[CrossRef]

T. A. Leskova, A. A. Maradudin, and I. Novikov, “Impedance boundary conditions for a metal film with a rough surface,” Appl. Opt. 38, 1197–1212 (1999).
[CrossRef]

Lezec, H. J.

T. Thio, H. F. Ghaemi, H. J. Lezec, P. A. Wolff, and T. W. Ebbesen, “Surface-plasmon-enhanced transmission through hole arrays in Cr films,” J. Opt. Soc. Am. B 16, 1743–1748 (1999).
[CrossRef]

T. W. Ebbsen, 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]

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]

Lin, S.-Y.

J. C.-C. Chang, Y. D. Sharma, Y.-S. Kim, J. A. Bur, R. V. Shenoi, S. Krishna, D. H. Huang, and S.-Y. Lin, “A surface plasmon enhanced infrared photodetector based on InAs quantum dots,” Nano Lett. 10, 1704–1709 (2010).
[CrossRef]

D. H. Huang, G. Gumbs, and S.-Y. Lin, “Self-consistent theory for near-field distribution and spectrum with quantum wires and a conductive grating in terahertz regime,” J. Appl. Phys. 105, 093715 (2009).
[CrossRef]

J. C.-C. Chang, Z.-P. Yang, D. H. Huang, D. A. Cardimona, and S.-Y. Lin, “Strong light concentration at the subwavelength scale by a metallic hole-array structure,” Opt. Lett. 34, 106–108 (2009).
[CrossRef]

Lochbihler, H.

López-Rios, T.

F. J. Garcia-Vidal, J. Sánchez-Dehesa, A. Dechelette, E. Bustarret, T. López-Rios, T. Fournier, and B. Pannetier, “Localized surface plasmons in Lamellar metallic gratings,” J. Lightwave Technol. 17, 2191–2195 (1999).
[CrossRef]

T. López-Rios, D. Mendoza, F. J. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in Lamellar metallic gratings,” Phys. Rev. Lett. 81, 665–668 (1998).
[CrossRef]

López-Tejeira, F.

F. López-Tejeira, F. J. Garcia-Vidal, and L. Martin-Moreno, “Scattering of surface plasmons by one-dimensional periodic nanoindented surfaces,” Phys. Rev. B 72, 161405 (2005).
[CrossRef]

Maradudin, A. A.

L. D. Wellems, D. H. Huang, T. A. Leskova, and A. A. Maradudin, “Nanogroove array on thin metallic film as planar lens with tunable focusing,” Phys. Lett. A 376, 216–220 (2012).
[CrossRef]

L. D. Wellems, D. H. Huang, T. A. Leskova, and A. A. Maradudin, “Optical spectrum and electromagnetic-field distribution at double-groove metallic surface gratings,” J. Appl. Phys. 106, 053705 (2009).
[CrossRef]

B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Transmission of light through a thin metal film with periodically and randomly corrugated surfaces,” J. Opt A: Pure Appl. Opt. 8, S191–S207 (2006).
[CrossRef]

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005) [and references therein].
[CrossRef]

T. A. Leskova, A. A. Maradudin, and I. Novikov, “Impedance boundary conditions for a metal film with a rough surface,” Appl. Opt. 38, 1197–1212 (1999).
[CrossRef]

A. A. Maradudin and A. Sentenac, “The impedance boundary condition for a periodically corrugated metal surface,” Solid State Commun. 84, 159–163 (1992).
[CrossRef]

Martin-Moreno, L.

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010) [and references therein].
[CrossRef]

F. de Leon-Perez, G. Brucoli, F. J. Garcia-Vidal, and L. Martin-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” New J. Phys. 10, 105017 (2008).
[CrossRef]

F. López-Tejeira, F. J. Garcia-Vidal, and L. Martin-Moreno, “Scattering of surface plasmons by one-dimensional periodic nanoindented surfaces,” Phys. Rev. B 72, 161405 (2005).
[CrossRef]

Mendoza, D.

T. López-Rios, D. Mendoza, F. J. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in Lamellar metallic gratings,” Phys. Rev. Lett. 81, 665–668 (1998).
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Novikov, I.

Pannetier, B.

F. J. Garcia-Vidal, J. Sánchez-Dehesa, A. Dechelette, E. Bustarret, T. López-Rios, T. Fournier, and B. Pannetier, “Localized surface plasmons in Lamellar metallic gratings,” J. Lightwave Technol. 17, 2191–2195 (1999).
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T. López-Rios, D. Mendoza, F. J. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in Lamellar metallic gratings,” Phys. Rev. Lett. 81, 665–668 (1998).
[CrossRef]

Raether, H.

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

Sánchez-Dehesa, J.

F. J. Garcia-Vidal, J. Sánchez-Dehesa, A. Dechelette, E. Bustarret, T. López-Rios, T. Fournier, and B. Pannetier, “Localized surface plasmons in Lamellar metallic gratings,” J. Lightwave Technol. 17, 2191–2195 (1999).
[CrossRef]

T. López-Rios, D. Mendoza, F. J. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in Lamellar metallic gratings,” Phys. Rev. Lett. 81, 665–668 (1998).
[CrossRef]

Sentenac, A.

A. A. Maradudin and A. Sentenac, “The impedance boundary condition for a periodically corrugated metal surface,” Solid State Commun. 84, 159–163 (1992).
[CrossRef]

Serdyuk, V. M.

V. M. Serdyuk, “Diffraction of a plane electromagnetic wave by a slot in a conducting screen of arbitrary thickness,” Tech. Phys. 50, 1076–1083 (2005).
[CrossRef]

Sharma, Y. D.

J. C.-C. Chang, Y. D. Sharma, Y.-S. Kim, J. A. Bur, R. V. Shenoi, S. Krishna, D. H. Huang, and S.-Y. Lin, “A surface plasmon enhanced infrared photodetector based on InAs quantum dots,” Nano Lett. 10, 1704–1709 (2010).
[CrossRef]

Shenoi, R. V.

J. C.-C. Chang, Y. D. Sharma, Y.-S. Kim, J. A. Bur, R. V. Shenoi, S. Krishna, D. H. Huang, and S.-Y. Lin, “A surface plasmon enhanced infrared photodetector based on InAs quantum dots,” Nano Lett. 10, 1704–1709 (2010).
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Skauli, T.

E. Laux, C. Genet, T. Skauli, and T. W. Ebbesen, “Plasmonic photon sorters for spectral and polarimetric imaging,” Nat. Photonics 2, 161–164 (2008).
[CrossRef]

Smolyaninov, I. I.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005) [and references therein].
[CrossRef]

Thio, T.

T. Thio, H. F. Ghaemi, H. J. Lezec, P. A. Wolff, and T. W. Ebbesen, “Surface-plasmon-enhanced transmission through hole arrays in Cr films,” J. Opt. Soc. Am. B 16, 1743–1748 (1999).
[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).
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T. W. Ebbsen, 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]

Wellems, L. D.

L. D. Wellems and D. H. Huang, “Near-field light focusing by a slit array in a planar metal film with nonuniform slit dielectric material,” Am. J. Phys. 80, 122–132 (2012).
[CrossRef]

L. D. Wellems, D. H. Huang, T. A. Leskova, and A. A. Maradudin, “Nanogroove array on thin metallic film as planar lens with tunable focusing,” Phys. Lett. A 376, 216–220 (2012).
[CrossRef]

L. D. Wellems, D. H. Huang, T. A. Leskova, and A. A. Maradudin, “Optical spectrum and electromagnetic-field distribution at double-groove metallic surface gratings,” J. Appl. Phys. 106, 053705 (2009).
[CrossRef]

Wolff, P. A.

T. Thio, H. F. Ghaemi, H. J. Lezec, P. A. Wolff, and T. W. Ebbesen, “Surface-plasmon-enhanced transmission through hole arrays in Cr films,” J. Opt. Soc. Am. B 16, 1743–1748 (1999).
[CrossRef]

T. W. Ebbsen, 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]

Yang, Z.-P.

Zayats, A. V.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005) [and references therein].
[CrossRef]

Am. J. Phys. (1)

L. D. Wellems and D. H. Huang, “Near-field light focusing by a slit array in a planar metal film with nonuniform slit dielectric material,” Am. J. Phys. 80, 122–132 (2012).
[CrossRef]

Appl. Opt. (2)

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D. Crouse, “Numerical modeling and electromagnetic resonant modes in complex grating structures and optoelectronic device applications,” IEEE Trans. Electron Devices 52, 2365–2373 (2005).
[CrossRef]

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D. H. Huang, G. Gumbs, and S.-Y. Lin, “Self-consistent theory for near-field distribution and spectrum with quantum wires and a conductive grating in terahertz regime,” J. Appl. Phys. 105, 093715 (2009).
[CrossRef]

L. D. Wellems, D. H. Huang, T. A. Leskova, and A. A. Maradudin, “Optical spectrum and electromagnetic-field distribution at double-groove metallic surface gratings,” J. Appl. Phys. 106, 053705 (2009).
[CrossRef]

J. Lightwave Technol. (1)

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

B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Transmission of light through a thin metal film with periodically and randomly corrugated surfaces,” J. Opt A: Pure Appl. Opt. 8, S191–S207 (2006).
[CrossRef]

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

Nano Lett. (1)

J. C.-C. Chang, Y. D. Sharma, Y.-S. Kim, J. A. Bur, R. V. Shenoi, S. Krishna, D. H. Huang, and S.-Y. Lin, “A surface plasmon enhanced infrared photodetector based on InAs quantum dots,” Nano Lett. 10, 1704–1709 (2010).
[CrossRef]

Nat. Photonics (1)

E. Laux, C. Genet, T. Skauli, and T. W. Ebbesen, “Plasmonic photon sorters for spectral and polarimetric imaging,” Nat. Photonics 2, 161–164 (2008).
[CrossRef]

Nature (1)

T. W. Ebbsen, 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]

New J. Phys. (1)

F. de Leon-Perez, G. Brucoli, F. J. Garcia-Vidal, and L. Martin-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” New J. Phys. 10, 105017 (2008).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (1)

L. D. Wellems, D. H. Huang, T. A. Leskova, and A. A. Maradudin, “Nanogroove array on thin metallic film as planar lens with tunable focusing,” Phys. Lett. A 376, 216–220 (2012).
[CrossRef]

Phys. Rep. (1)

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005) [and references therein].
[CrossRef]

Phys. Rev. B (4)

F. López-Tejeira, F. J. Garcia-Vidal, and L. Martin-Moreno, “Scattering of surface plasmons by one-dimensional periodic nanoindented surfaces,” Phys. Rev. B 72, 161405 (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, 6779–6782 (1998).
[CrossRef]

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

D. H. Huang, G. Gumbs, P. M. Alsing, and D. A. Cardimona, “Nonlocal mode mixing and surface-plasmon-polariton-mediated enhancement of diffracted terahertz fields by a conductive grating,” Phys. Rev. B 77, 165404 (2008).
[CrossRef]

Phys. Rev. Lett. (1)

T. López-Rios, D. Mendoza, F. J. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in Lamellar metallic gratings,” Phys. Rev. Lett. 81, 665–668 (1998).
[CrossRef]

Rev. Mod. Phys. (1)

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010) [and references therein].
[CrossRef]

Solid State Commun. (1)

A. A. Maradudin and A. Sentenac, “The impedance boundary condition for a periodically corrugated metal surface,” Solid State Commun. 84, 159–163 (1992).
[CrossRef]

Tech. Phys. (1)

V. M. Serdyuk, “Diffraction of a plane electromagnetic wave by a slot in a conducting screen of arbitrary thickness,” Tech. Phys. 50, 1076–1083 (2005).
[CrossRef]

Other (3)

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

G. Gumbs and D. H. Huang, Properties of Interacting Low-Dimensional Systems (Wiley-VCH, 2011), Chaps. 4 and 5.

A. A. Maradudin, ed., Light Scattering and Nanoscale Surface Roughness (Springer Science+Business Media, 2007).

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Figures (8)

Fig. 1.
Fig. 1.

Illustration for a z-direction slit array (brown) that extends in the y direction, where zj and 2j are the center position and the width of the jth slit with j=0,±1,,±N. The regions at the left- and right-hand sides of the slits are denoted as region I and region III, respectively, with real dielectric constants ϵL and ϵR. The region for the slit array is denoted as region II, and slits are filled with medium having a dielectric constant κj (real or complex) for j=0,±1,,±N. The depth of slits in the x direction is 2d, and ϵM(ω) represents the dielectric function of the metal film containing slits. A Gaussian beam is incident on the slit array from the left side with an incident angle θ0 and at a center position z=zG. The incident wave number is ϵLk0 and β0 is the incident wave vector along the z direction.

Fig. 2.
Fig. 2.

Contour plot of |Hy(x,z)|2 for p-polarization normal incidence (from upper surface) with θ0=0°, where nL=1 and nR=4.5. In our calculations, we set the parameters as follows: 1=0=1=ζ/4, κ1=1, κ0=1+30i, κ1=2, z1/ζ=1.75, z0=zG=0, z1/ζ=2, d=(7/8)ζ, g=60, and λ0=ζ, where ζ=0.588μm.

Fig. 3.
Fig. 3.

Plot for calculated Tj=(1/2j)jjdz|Hy(d+δ,zzj)|2 for j=1 (blue solid curve) and 1 (red dashed curve) as a function of slit depth d/ζ for p-polarization normal incidence as in Fig. 2, where nL=1, nR=4.5, δ=ζ/2 and the vertical black dashed lines indicate the positions determined by d/ζ=2/8, 4/8, 6/8, 8/8. In our calculations, we set the parameters as follows: 1=0=1=ζ/6, κ1=1, κ0=1+30i, κ1=2, z1/ζ=1.75, z0=zG=0, z1/ζ=2, g=60, and λ0=ζ, where ζ=0.588μm.

Fig. 4.
Fig. 4.

Plot for calculated Tj=(1/2j)jjdz|Hy(d+δ,zzj)|2 for j=1 (blue solid curve) and 1 (red dashed curve) as a function of slit depth d/ζ for p-polarization normal incidence as in Fig. 2, where nL=1, nR=4.5, δ=ζ/2. In our calculations, we set the parameters as follows: 1=0=1=ζ/6, κ1=1, κ0=1+30i, κ1=2, z0=zG=0, z1/ζ=2, g=60, and λ0=ζ, where ζ=0.588μm. Here, we chose z1/ζ=1.5 [in (a)] and z1/ζ=2 [in (b)], respectively.

Fig. 5.
Fig. 5.

Contour plot of |Hy(x,z)|2 for p-polarization normal incidence (from upper surface) with θ0=0°, where nL=1 and nR=4.5. In our calculations, we set the parameters as follows: 1=0=1=ζ/4, κ1=1, κ0=1+30i, κ1=2, z1/ζ=1.75, z0=zG=0, z1/ζ=2, d=(7/8)ζ, g=60, and λ0=2ζ, where ζ=0.588μm.

Fig. 6.
Fig. 6.

Plot for calculated Tj=(1/2j)jjdz|Hy(d+δ,zzj)|2 for j=1 (blue solid curve) and 1 (red dashed curve) as a function of slit depth d/ζ for p-polarization normal incidence as in Fig. 5, where nL=1, nR=4.5, δ=ζ/2 and the vertical black dashed lines indicate the positions determined by d/ζ=2/8, 4/8, 6/8, 8/8. In our calculations, we set the parameters as follows: 1=0=1=ζ/6, κ1=1, κ0=1+30i, κ1=2, z1/ζ=1.75, z0=zG=0, z1/ζ=2, g=60, and λ0=2ζ, where ζ=0.588μm.

Fig. 7.
Fig. 7.

Contour plot of |Hy(x,z)|2 for p-polarization normal incidence (from upper surface) with θ0=0°, where nL=1 and nR=4.5. In our calculations, we set the parameters as follows: 1=0=1=ζ/4, κ1=1, κ0=1+30i, κ1=2, z1/ζ=1.75, z0=zG=0, z1/ζ=2, d=(7/8)ζ, g=60, and λ0=ζ, where ζ=0.588μm. In this case, we set ηL=ηR=0 for a PEC.

Fig. 8.
Fig. 8.

Contour plot of |Hy(x,z)|2 for p-polarization normal incidence (from upper surface) with θ0=0°, where nL=1 and nR=4.5. In our calculations, we set the parameters as follows: 1=0=1=ζ/4, κ1=4, κ0=1+30i, κ1=2, z1/ζ=1.75, z0=zG=0, z1/ζ=2, d=(7/8)ζ, g=60, and λ0=2ζ, where ζ=0.588μm.

Equations (45)

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u(x,z)x|x=d0={ηLu(x,z)|x=d0left surfacezj+j<z<zj+1j+1ϵLκju(x,z)x|x=d+0middle slit|zzj|<jηLu(x,z)|x=d0right surfacezj1+j1<z<zjj,
u(x,z)x|x=d+0={ηRu(x,z)|x=d+0left surfacezj+j<z<zj+1j+1ϵRκju(x,z)x|x=d0middle slit|zzj|<jηRu(x,z)|x=d+0right surfacezj1+j1<z<zjj.
u(x,z)z|z=zj±j=0for all slits|x|d.
u(I)(x,z)=k0ϵL0dβk1(β)[Gs(β)cos(βz)+iGa(β)sin(βz)]eik1(β)(x+d)k0ϵL0dβk1(β)[As(β)cos(βz)+iAa(β)sin(βz)]eik1(β)(x+d),
Gs(β)=[Gn(β)+Gp(β)]cos(βzG)i[Gn(β)Gp(β)]sin(βzG),Ga(β)=[Gn(β)Gp(β)]cos(βzG)i[Gn(β)+Gp(β)]sin(βzG),
Gp(β)=gk1(β)2πϵLk0exp[g2(β+β0)24]Θ(nLk0|β|),Gn(β)=gk1(β)2πϵLk0exp[g2(ββ0)24]Θ(nLk0|β|).
u(III)(x,z)=k0ϵR0dβk2(β)[Bs(β)cos(βz)+iBa(β)sin(βz)]eik2(β)(xd),
u(II)(x,z)=k0jΘ(j|zzj|)n{κjσsnj[asnjeiσsnj(x+d)bsnjeiσsnj(xd)]×cos[ξsnj(zzj)]+iκjσanj[aanjeiσanj(x+d)banjeiσanj(xd)]sin[ξanj(zzj)]},
0dβ{[Gs(β)+As(β)]cos(βz)+i[Ga(β)+Aa(β)]sin(βz)}=iηLϵL0dβk1(β){[Gs(β)As(β)]cos(βz)+i[Ga(β)Aa(β)]sin(βz)}
0dβ{[Gs(β)+As(β)]cos(βz)+i[Ga(β)+Aa(β)]sin(βz)}=jΘ(j|zzj|)n[(asnj+bsnjeiσsnj2d)cos[ξsnj(zzj)]+i(aanj+banjeiσanj2d)sin[ξanj(zzj)]]
0dβ[Bs(β)cos(βz)+iBa(β)sin(βz)]=iηRϵR0dβk2(β)[Bs(β)cos(βz)+iBa(β)sin(βz)]
0dβ[Bs(β)cos(βz)+iBa(β)sin(βz)]=jΘ(j|zzj|)n[(bsnj+asnjeiσsnj2d)cos[ξsnj(zzj)]+i(banj+aanjeiσanj2d)sin[ξanj(zzj)]]
As(β)+Gs(β)=n{jjπ[(asnj+bsnje2iσsnjd)Qsnj(β)cos(βzj)i(aanj+banje2iσanjd)Qanj(β)sin(βzj)]}iηL0dβk1(β)[Ps(β,β)+Ws(β,β)][Gs(β)As(β)]+ηL0dβk1(β)[Pc(β,β)+Wc(β,β)][Ga(β)Aa(β)],
Aa(β)+Ga(β)=n{jjπ[i(asnj+bsnje2iσsnjd)Qsnj(β)sin(βzj)+(aanj+banje2iσanjd)Qanj(β)cos(βzj)]}ηL0dβk1(β)[Pc(β,β)+Wc(β,β)][Gs(β)As(β)]iηL0dβk1(β)[Pa(β,β)+Wa(β,β)][Ga(β)Aa(β)],
Bs(β)=n{jjπ[(bsnj+asnje2iσsnjd)Qsnj(β)cos(βzj)i(banj+aanje2iσanjd)Qanj(β)sin(βzj)]}iηR0dβk2(β)[Ps(β,β)+Ws(β,β)]Bs(β)+ηR0dβk2(β)[Pc(β,β)+Wc(β,β)]Ba(β),
Ba(β)=n{jjπ[i(asnj+bsnje2iσsnjd)Qsnj(β)sin(βzj)+(aanj+banje2iσanjd)Qanj(β)cos(βzj)]}ηR0dβk2(β)[Pc(β,β)+Wc(β,β)]Bs(β)iηR0dβk2(β)[Pa(β,β)+Wa(β,β)]Ba(β).
χnκjσsnjϵL(asnjbsnje2iσsnjd)=0dβk1(β)[Gs(β)As(β)]Qsnj(β)cos(βzj)+i0dβk1(β)[Ga(β)Aa(β)]Qsnj(β)sin(βzj),
χnκjσanjϵL(aanjbanje2iσanjd)=i0dβk1(β)[Gs(β)As(β)]Qanj(β)sin(βzj)+0dβk1(β)[Ga(β)Aa(β)]Qanj(β)cos(βzj),
χnκjσsnjϵR(asnje2iσsnjdbsnj)=0dβk2(β)Bs(β)Qsnj(β)cos(βzj)+i0dβk2(β)Ba(β)Qsnj(β)sin(βzj),
χnκjσanjϵR(aanje2iσanjdbanj)=i0dβk2(β)Bs(β)Qanj(β)sin(βzj)+0dβk2(β)Ba(β)Qanj(β)cos(βzj),
asnj=Yjn(2)e2iσsnjdYjn(1)2isin(2σsnjd),bsnj=e2iσsnjdYjn(2)Yjn(1)2isin(2σsnjd),aanj=Xjn(2)e2iσanjdXjn(1)2isin(2σanjd),banj=e2iσanjdXjn(2)Xjn(1)2isin(2σanjd).
Gs(β)+As(β)=iηL0dβk1(β)[Ps(β,β)+Ws(β,β)][Gs(β)As(β)]+ηL0dβk1(β)[Pc(β,β)+Wc(β,β)][Ga(β)Aa(β)]+0dβk2(β)[Ts(1)(β,β)Bs(β)+iTs(2)(β,β)Ba(β)]0dβk1(β){Ts(3)(β,β)[Gs(β)As(β)]+iTs(4)(β,β)[Ga(β)Aa(β)]}i0dβk2(β)[Rs(1)(β,β)Ba(β)+iRs(2)(β,β)Bs(β)]+i0dβk1(β){Rs(3)(β,β)[Ga(β)Aa(β)]+iRs(4)(β,β)[Gs(β)As(β)]},
Ga(β)+Aa(β)=ηL0dβk1(β)[Pc(β,β)+Wc(β,β)][Gs(β)As(β)]iηL0dβk1(β)[Pa(β,β)+Wa(β,β)][Ga(β)Aa(β)]i0dβk2(β)[Ta(1)(β,β)Bs(β)+iTa(2)(β,β)Ba(β)]+i0dβk1(β){Ta(3)(β,β)[Gs(β)As(β)]+iTa(4)(β,β)[Ga(β)Aa(β)]}+0dβk2(β)[Ra(1)(β,β)Ba(β)+iRa(2)(β,β)Bs(β)]0dβk1(β){Ra(3)(β,β)[Ga(β)Aa(β)]+iRa(4)(β,β)[Gs(β)As(β)]},
Bs(β)=iηR0dβk2(β)[Ps(β,β)+Ws(β,β)]Bs(β)+ηR0dβk2(β)[Pc(β,β)+Wc(β,β)]Ba(β)+0dβk2(β)[Ns(1)(β,β)Bs(β)+iNs(2)(β,β)Ba(β)]0dβk1(β){Ns(3)(β,β)[Gs(β)As(β)]+iNs(4)(β,β)[Ga(β)Aa(β)]}i0dβk2(β)[Ms(1)(β,β)Ba(β)+iMs(2)(β,β)Bs(β)]+i0dβk1(β){Ms(3)(β,β)[Ga(β)Aa(β)]+iMs(4)(β,β)[Gs(β)As(β)]},
Ba(β)=ηR0dβk2(β)[Pc(β,β)+Wc(β,β)]Bs(β)iηR0dβk2(β)[Pa(β,β)+Wa(β,β)]Ba(β)i0dβk2(β)[Na(1)(β,β)Bs(β)+iNa(2)(β,β)Ba(β)]+i0dβk1(β){Na(3)(β,β)[Gs(β)As(β)]+iNa(4)(β,β)[Ga(β)Aa(β)]}+0dβk2(β)[Ma(1)(β,β)Ba(β)+iMa(2)(β,β)Bs(β)]0dβk1(β){Ma(3)(β,β)[Ga(β)Aa(β)]+iMa(4)(β,β)[Gs(β)As(β)]}.
Qsnj(β)=1jjjdzcos(ξsnjz)cos(βz)=sinc[(βξsnj)j]+sinc[(β+ξsnj)j],
Qanj(β)=1jjjdzsin(ξanjz)sin(βz)=sinc[(βξanj)j]sinc[(β+ξanj)j],
Ps(β,β)=|zNN|cos(βz)cos(βz)dz+zN+Ncos(βz)cos(βz)dz=π[δ(ββ)+δ(β+β)]|zNN|2{sinc[(β+β)|zNN|]+sinc[(ββ)|zNN|]}zN+N2{sinc[(β+β)(zN+N)]+sinc[(ββ)(zN+N)]},
Pa(β,β)=|zNN|sin(βz)sin(βz)dz+zN+Nsin(βz)sin(βz)dz=π[δ(ββ)δ(β+β)]+|zNN|2{sinc[(β+β)|zNN|]sinc[(ββ)|zNN|]}+zN+N2{sinc[(β+β)(zN+N)]sinc[(ββ)(zN+N)]},
Pc(β,β)=|zNN|sin(βz)cos(βz)dz+zN+Nsin(βz)cos(βz)dz12{cos[(β+β)|zNN|]β+βcos[(ββ)|zNN|]ββ}+12{cos[(β+β)(zN+N)]β+βcos[(ββ)(zN+N)]ββ},
Ws(β,β)=j=NN1zj+jzj+1j+1cos(βz)cos(βz)dz=12j=NN1{(zj+1j+1)(sinc[(ββ)(zj+1j+1)]+sinc[(β+β)(zj+1j+1)])(zj+j)(sinc[(β+β)(zj+j)]+sinc[(ββ)(zj+j)])},
Wa(β,β)=j=NN1zj+jzj+1j+1sin(βz)sin(βz)dz=12j=NN1{(zj+1j+1)(sinc[(ββ)(zj+1j+1)]sinc[(β+β)(zj+1j+1)])+(zj+j)(sinc[(β+β)(zj+j)]sinc[(ββ)(zj+j)])},
Wc(β,β)=j=NN1zj+jzj+1j+1sin(βz)cos(βz)dz=12j=NN1{cos[(ββ)(zj+j)]ββ+cos[(β+β)(zj+j)]β+βcos[(ββ)(zj+1j+1)]ββcos[(β+β)(zj+1j+1)]β+β}.
Yjn(1)=σsnjϵLχnκj0dβk1(β)Qsnj(β)×{[Gs(β)As(β))]cos(βzj)+i[Ga(β)Aa(β)]sin(βzj)},
Yjn(2)=σsnjϵRχnκj0dβk2(β)Qsnj(β)[Bs(β)cos(βzj)+iBa(β)sin(βzj)],
Xjn(1)=σanjϵLχnκj0dβk1(β)Qanj(β)×{[Ga(β)Aa(β)]cos(βzj)+i[Gs(β)As(β)]sin(βzj)},
Xjn(2)=σanjϵRχnκj0dβk2(β)Qanj(β)[Ba(β)cos(βzj)+iBs(β)sin(βzj)].
[Ts(1)(β,β)Ts(2)(β,β)Ts(3)(β,β)Ts(4)(β,β)]=1iπn,j[σsnjjχnκjsin(2σsnjd)]Qsnj(β)Qsnj(β)×[ϵRcos(βzj)cos(βzj)ϵRcos(βzj)sin(βzj)ϵLcos(2σsnjd)cos(βzj)cos(βzj)ϵLcos(2σsnjd)cos(βzj)sin(βzj)],
[Rs(1)(β,β)Rs(2)(β,β)Rs(3)(β,β)Rs(4)(β,β)]=1iπn,j[σanjjχnκjsin(2σanjd)]Qanj(β)Qanj(β)×[ϵRsin(βzj)cos(βzj)ϵRsin(βzj)sin(βzj)ϵLcos(2σanjd)sin(βzj)cos(βzj)ϵLcos(2σanjd)sin(βzj)sin(βzj)],
[Ta(1)(β,β)Ta(2)(β,β)Ta(3)(β,β)Ta(4)(β,β)]=1iπn,j[σsnjjχnκjsin(2σsnjd)]Qsnj(β)Qsnj(β)×[ϵRsin(βzj)cos(βzj)ϵRsin(βzj)sin(βzj)ϵLcos(2σsnjd)sin(βzj)cos(βzj)ϵLcos(2σsnjd)sin(βzj)sin(βzj)],
[Ra(1)(β,β)Ra(2)(β,β)Ra(3)(β,β)Ra(4)(β,β)]=1iπn,j[σanjjχnκjsin(2σanjd)]Qanj(β)Qanj(β)×[ϵRcos(βzj)cos(βzj)ϵRcos(βzj)sin(βzj)ϵLcos(2σanjd)cos(βzj)cos(βzj)ϵLcos(2σanjd)cos(βzj)sin(βzj)],
[Ns(1)(β,β)Ns(2)(β,β)Ns(3)(β,β)Ns(4)(β,β)]=1iπn,j[σsnjjχnκjsin(2σsnjd)]Qsnj(β)Qsnj(β)×[ϵRcos(2σsnjd)cos(βzj)cos(βzj)ϵRcos(2σsnjd)cos(βzj)sin(βzj)ϵLcos(βzj)cos(βzj)ϵLcos(βzj)sin(βzj)],
[Ms(1)(β,β)Ms(2)(β,β)Ms(3)(β,β)Ms(4)(β,β)]=1iπn,j[σanjjχnκjsin(2σanjd)]Qanj(β)Qanj(β)×[ϵRcos(2σanjd)sin(βzj)cos(βzj)ϵRcos(2σanjd)sin(βzj)sin(βzj)ϵLsin(βzj)cos(βzj)ϵLsin(βzj)sin(βzj)],
[Na(1)(β,β)Na(2)(β,β)Na(3)(β,β)Na(4)(β,β)]=1iπn,j[σsnjjχnκjsin(2σsnjd)]Qsnj(β)Qsnj(β)×[ϵRcos(2σsnjd)sin(βzj)cos(βzj)ϵRcos(2σsnjd)sin(βzj)sin(βzj)ϵLsin(βzj)cos(βzj)ϵLsin(βzj)sin(βzj)],
[Ma(1)(β,β)Ma(2)(β,β)Ma(3)(β,β)Ma(4)(β,β)]=1iπn,j[σanjjχnκjsin(2σanjd)]Qanj(β)Qanj(β)×[ϵRcos(2σanjd)cos(βzj)cos(βzj)ϵRcos(2σanjd)cos(βzj)sin(βzj)ϵLcos(βzj)cos(βzj)ϵLcos(βzj)sin(βzj)].

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