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 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]

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]

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, 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.

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]

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, 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]

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).
[CrossRef]

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).
[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]

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).
[CrossRef]

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]

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]

Wellems, L. D.

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]

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]

J. Appl. Phys. (2)

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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