Abstract

An improved method for the calculation of light–matter interaction that appears with the light propagation through multilayer periodically corrugated structures consisting of any dielectric or absorptive media is reported. The method is based on the differential formalism for a system of Maxwell’s equations when the boundary conditions are simplified by the introduction of a curvilinear nonorthogonal coordinate system. The solution for electromagnetic fields was written in the form of the superposition of partial plane waves. The obtained method essentially reduces computation time and increases accuracy compared with the Chandezon method. For a numerical demonstration of the proposed method, calculation of long-range surface plasmon polaritons was performed. The presented method can be enhanced for calculations of light propagation through thin absorptive films with various periodic profiles at both film interfaces.

© 2008 Optical Society of America

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References

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  1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolf, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-669 (1998).
    [CrossRef]
  2. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Mareno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002).
    [CrossRef] [PubMed]
  3. S. Collin, F. Pardo, R. Teissier, and J.-L. Pelonard, "Efficient light absorption in metal-semiconductor-metal nanostructures," Appl. Phys. Lett. 85, 194-196 (2004).
    [CrossRef]
  4. M. S. Anderson, "Locally enhanced Raman spectroscopy with an atomic force microscope," Appl. Phys. Lett. 76, 3130-3132 (2000).
    [CrossRef]
  5. A. Otto, "Excitation of surface plasma waves in silver by the method of frustrated total reflection," Z. Phys. 216, 398-410 (1968).
    [CrossRef]
  6. E. Kretschmann, "Die Bestimmung optischer Konstanten von Metallen durch Oberflachenplasmaschwingungen," Z. Phys. 241, 313-324 (1971).
    [CrossRef]
  7. N. L. Dmitruk and A. V. Korovin, "Generalized analytical model for the calculation of light transmittance through a thin conducting film," Thin Solid Films 484, 382-388 (2005).
    [CrossRef]
  8. H. Raether, Surface Plasmons (Springer-Verlag, 1988).
  9. O. J. F. Martin and N. B. Piller, "Electromagnetic scattering in polarizable backgrounds," Phys. Rev. E 58, 3909-3915 (1998).
    [CrossRef]
  10. F. Tiogo, A. Marvin, V. Celli, and N. R. Hill, "Optical properties of rough surfaces: general theory and the small roughness limit," Phys. Rev. B 15, 5618-5626 (1977).
    [CrossRef]
  11. D. L. Mills, "Interaction of surface polaritons with periodic surface structures; Rayleigh waves and gratings," Phys. Rev. B 15, 3097-3118 (1977).
    [CrossRef]
  12. J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, "Multicoated gratings: a differential formalism applicable in the entire optical region," J. Opt. Soc. Am. 72, 839-846 (1982).
    [CrossRef]
  13. L. Li, "Multilayer-coated diffraction gratings: differential method of Chandezon revisited," J. Opt. Soc. Am. A 11, 2816-2828 (1994).
    [CrossRef]
  14. A. Ye. Poyedinchuk, Yu. A. Tuchkin, N. P. Yashina, J. Chandezon, and G. Granet, "C-method: several aspects of spectral theory of gratings," Electromagn. Waves 59, 113-149 (2006).
    [CrossRef]
  15. W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, "Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings," Phys. Rev. B 54, 6227-6244 (1996).
    [CrossRef]
  16. C.-H. Liao, C.-M. Lee, Y.-T. Cheng, J.-S. Shyu, and W.-K. Su, "Resonant properties of long-range plasmons in an arbitrary multilayer structure," Jpn. J. Appl. Phys., Part 1 38, 5938-5944 (1999).
    [CrossRef]
  17. D. Sarid, "Long-range surface-plasma waves on very thin metal films," Phys. Rev. Lett. 47, 1927-1930 (1981).
    [CrossRef]
  18. P. B. Johnson and R. W. Christy, "Optical constants of the noble metal," Phys. Rev. B 6, 4370-4379 (1972).
    [CrossRef]
  19. E.D.Palic, ed., Handbook of Optical Constants of Solids (Academic, 1985).

2006 (1)

A. Ye. Poyedinchuk, Yu. A. Tuchkin, N. P. Yashina, J. Chandezon, and G. Granet, "C-method: several aspects of spectral theory of gratings," Electromagn. Waves 59, 113-149 (2006).
[CrossRef]

2005 (1)

N. L. Dmitruk and A. V. Korovin, "Generalized analytical model for the calculation of light transmittance through a thin conducting film," Thin Solid Films 484, 382-388 (2005).
[CrossRef]

2004 (1)

S. Collin, F. Pardo, R. Teissier, and J.-L. Pelonard, "Efficient light absorption in metal-semiconductor-metal nanostructures," Appl. Phys. Lett. 85, 194-196 (2004).
[CrossRef]

2002 (1)

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Mareno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002).
[CrossRef] [PubMed]

2000 (1)

M. S. Anderson, "Locally enhanced Raman spectroscopy with an atomic force microscope," Appl. Phys. Lett. 76, 3130-3132 (2000).
[CrossRef]

1999 (1)

C.-H. Liao, C.-M. Lee, Y.-T. Cheng, J.-S. Shyu, and W.-K. Su, "Resonant properties of long-range plasmons in an arbitrary multilayer structure," Jpn. J. Appl. Phys., Part 1 38, 5938-5944 (1999).
[CrossRef]

1998 (2)

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

O. J. F. Martin and N. B. Piller, "Electromagnetic scattering in polarizable backgrounds," Phys. Rev. E 58, 3909-3915 (1998).
[CrossRef]

1996 (1)

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, "Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings," Phys. Rev. B 54, 6227-6244 (1996).
[CrossRef]

1994 (1)

1982 (1)

1981 (1)

D. Sarid, "Long-range surface-plasma waves on very thin metal films," Phys. Rev. Lett. 47, 1927-1930 (1981).
[CrossRef]

1977 (2)

F. Tiogo, A. Marvin, V. Celli, and N. R. Hill, "Optical properties of rough surfaces: general theory and the small roughness limit," Phys. Rev. B 15, 5618-5626 (1977).
[CrossRef]

D. L. Mills, "Interaction of surface polaritons with periodic surface structures; Rayleigh waves and gratings," Phys. Rev. B 15, 3097-3118 (1977).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, "Optical constants of the noble metal," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

1971 (1)

E. Kretschmann, "Die Bestimmung optischer Konstanten von Metallen durch Oberflachenplasmaschwingungen," Z. Phys. 241, 313-324 (1971).
[CrossRef]

1968 (1)

A. Otto, "Excitation of surface plasma waves in silver by the method of frustrated total reflection," Z. Phys. 216, 398-410 (1968).
[CrossRef]

Appl. Phys. Lett. (2)

S. Collin, F. Pardo, R. Teissier, and J.-L. Pelonard, "Efficient light absorption in metal-semiconductor-metal nanostructures," Appl. Phys. Lett. 85, 194-196 (2004).
[CrossRef]

M. S. Anderson, "Locally enhanced Raman spectroscopy with an atomic force microscope," Appl. Phys. Lett. 76, 3130-3132 (2000).
[CrossRef]

Electromagn. Waves (1)

A. Ye. Poyedinchuk, Yu. A. Tuchkin, N. P. Yashina, J. Chandezon, and G. Granet, "C-method: several aspects of spectral theory of gratings," Electromagn. Waves 59, 113-149 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Jpn. J. Appl. Phys., Part 1 (1)

C.-H. Liao, C.-M. Lee, Y.-T. Cheng, J.-S. Shyu, and W.-K. Su, "Resonant properties of long-range plasmons in an arbitrary multilayer structure," Jpn. J. Appl. Phys., Part 1 38, 5938-5944 (1999).
[CrossRef]

Nature (London) (1)

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

Phys. Rev. B (4)

P. B. Johnson and R. W. Christy, "Optical constants of the noble metal," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, "Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings," Phys. Rev. B 54, 6227-6244 (1996).
[CrossRef]

F. Tiogo, A. Marvin, V. Celli, and N. R. Hill, "Optical properties of rough surfaces: general theory and the small roughness limit," Phys. Rev. B 15, 5618-5626 (1977).
[CrossRef]

D. L. Mills, "Interaction of surface polaritons with periodic surface structures; Rayleigh waves and gratings," Phys. Rev. B 15, 3097-3118 (1977).
[CrossRef]

Phys. Rev. E (1)

O. J. F. Martin and N. B. Piller, "Electromagnetic scattering in polarizable backgrounds," Phys. Rev. E 58, 3909-3915 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

D. Sarid, "Long-range surface-plasma waves on very thin metal films," Phys. Rev. Lett. 47, 1927-1930 (1981).
[CrossRef]

Science (1)

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Mareno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002).
[CrossRef] [PubMed]

Thin Solid Films (1)

N. L. Dmitruk and A. V. Korovin, "Generalized analytical model for the calculation of light transmittance through a thin conducting film," Thin Solid Films 484, 382-388 (2005).
[CrossRef]

Z. Phys. (2)

A. Otto, "Excitation of surface plasma waves in silver by the method of frustrated total reflection," Z. Phys. 216, 398-410 (1968).
[CrossRef]

E. Kretschmann, "Die Bestimmung optischer Konstanten von Metallen durch Oberflachenplasmaschwingungen," Z. Phys. 241, 313-324 (1971).
[CrossRef]

Other (2)

H. Raether, Surface Plasmons (Springer-Verlag, 1988).

E.D.Palic, ed., Handbook of Optical Constants of Solids (Academic, 1985).

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Figures (4)

Fig. 1
Fig. 1

Schematic of a multilayered corrugated structure with the same interface profiles described by periodic function ξ ( x , y ) . Here x ̂ , y ̂ , and z ̂ are Cartesian coordinate unit vectors.

Fig. 2
Fig. 2

Dispersion of self-modes for SPPs at “glass–Au” interfaces: long range (dashed curve), short range (dotted curve), and without coupling at both Au interfaces.

Fig. 3
Fig. 3

Angular dependencies (for 632.8 nm incident light wavelength) for all (solid curve) and zero (dashed curve) diffraction orders reflected from glass with embedded Au film ( 50 nm thickness) and all diffraction orders reflected from glass with embedded very thick Au film (dotted curve). The 1D sinusoidal grating parameters are 30 nm relief depth and 600 nm period.

Fig. 4
Fig. 4

Spectral dependencies (for a 30° incidence angle) for all (solid curve) and zero (dashed curve) diffraction orders reflected from glass with embedded Au film and all diffraction orders reflected from glass with embedded very thick Au film (dotted curve). The other parameters are the same as in Fig. 3.

Tables (1)

Tables Icon

Table 1 Eigenvalues for Air Calculated in Curvilinear Coordinates (Corresponding to 1D Sinusoidal Grating) for 30° Incidence Angle and 632.8 nm Wavelength

Equations (32)

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{ x 1 = x x 2 = y x 3 = z ξ ( x , y ) } ,
x 1 , x 2 , x 3 ν = e ̃ ν ( ρ ) e i ( κ ρ + σ k n ( κ ) ( x 3 z n 1 + ξ ( ρ ) ) ) ,
[ E n ( ρ , x 3 ) H n ( ρ , x 3 ) ] = σ d κ ( 2 π ) 2 [ e ̃ n , σ , s , κ ( ρ ) e ̃ n , σ , p , κ ( ρ ) ε n e ̃ n , σ , p , κ ( ρ ) ε n e ̃ n , σ , s , κ ( ρ ) ] c ̂ n σ ( κ ) e i ( κ ρ + σ k n ( κ ) ( x 3 z n 1 + ξ ( ρ ) ) ) ,
e ̃ n , σ , s , κ ( ρ ) = e n , σ , s , κ , e ̃ n , σ , p , κ ( ρ ) = e n , σ , p , κ + κ ε n ξ ( ρ ) ρ .
e i β ξ ( ρ ) = m Λ m ( β ) e i G m ρ ,
ξ ( ρ ) ρ e i β ξ ( ρ ) = m G m β Λ m ( β ) e i G m ρ ,
Ψ ̂ n ( ρ , x 3 ) = σ , m d κ ( 2 π ) 2 e i ( κ + G m ) ρ M n , m σ ( κ , x 3 ) c ̂ n σ ( κ ) ,
M n , m σ ( κ , x 3 ) [ z × κ σ k n ( κ ) [ κ 2 G m k n ( κ ) 2 κ ] ε n σ k n ( κ ) [ κ 2 G m k n ( κ ) 2 κ ] ε n z × κ ] Λ m ( σ k n ( κ ) ) e i σ k n ( κ ) ( x 3 z n 1 ) κ .
Ψ ̂ n + 1 ( ρ , z n ) Ψ ̂ n ( ρ , z n ) = 0 ( n = 1 , 2 , , N 1 ) .
σ , m d κ e i ( κ + G m ) ρ ( 2 π ) 2 [ M n + 1 , m σ ( κ , z n ) c ̂ n + 1 σ ( κ ) M n , m σ ( κ , z n ) c ̂ n σ ( κ ) ] = 0 .
σ , m [ M n + 1 , m σ ( κ G m , z n ) c ̂ n + 1 σ ( κ G m ) M n , m σ ( κ G m , z n ) c ̂ n σ ( κ G m ) ] = 0 .
c ̂ 1 ( + ) ( κ ) = ( 2 π ) 2 δ ( κ κ i ) [ cos θ α sin θ α ] ,
Ψ ̂ i ( ρ , x 3 ) = m e i ( κ i + G m ) ρ M 1 , m ( + ) ( κ i , x 3 ) [ cos θ α sin θ α ] .
c ̂ n σ ( κ ) = m ( 2 π ) 2 δ ( κ κ i G m ) c ̂ n , m σ .
σ , m [ M n + 1 , m m σ ( κ i + G m , z n ) c ̂ n + 1 , m σ M n , m m σ ( κ i + G m , z n ) c ̂ n , m σ ] = δ n , 1 M i , m ( + ) ( κ i , 0 ) [ cos θ α sin θ α ] .
Ψ ̂ n ( ρ , u ) = σ , m , m e i ( κ i + G m ) ρ M n , m m σ ( κ i + G m , u ) c ̂ n , m σ .
σ , m [ Λ m m ( σ k n + 1 ( κ i + G m ) ) c s n + 1 , m σ Λ m m ( σ k n ( κ i + G m ) ) Γ n ( κ i + G m ) σ c s n , m σ ] = δ n , 1 Λ m ( k i ( κ i ) ) cos θ α ,
σ , m σ [ k n + 1 ( κ i + G m ) Ξ n + 1 σ ( m , m ) c s n + 1 , m σ k n + 1 ( κ i + G m ) Ξ n σ ( m , m ) Γ n ( κ i + G m ) σ c s n , m σ ] = δ n , 1 k i ( κ i ) Ξ 1 ( + ) ( m , 0 ) cos θ α ,
σ , m [ ε n + 1 Λ m m ( σ k n + 1 ( κ i + G m ) ) c p n + 1 , m σ ε n Λ m m ( σ k n ( κ i + G m ) ) Γ n ( κ i + G m ) σ c p n , m σ ] = δ n , 1 ε 1 Λ m ( k i ( κ i ) sin θ α ,
σ , m σ [ k n + 1 ( κ i + G m ) ε n + 1 Ξ n + 1 σ ( m , m ) c p n + 1 , m σ k n ( κ i + G m ) ε n Ξ n σ ( m , m ) Γ n ( κ i + G m ) σ c p n , m σ ] = δ n , 1 k 1 ( κ i ) ε 1 Ξ 1 σ ( m , 0 ) sin θ α ,
Γ n ( κ ) e i k n ( κ ) ( z n z n 1 ) ,
Ξ n σ ( m , m ) = ( 1 κ i + G m G m m k n ( κ i + G m ) 2 ) Λ m m ( σ k n ( κ i + G m ) ) .
S N = S 3 N 3 R e ( E 1 * H 2 E 2 * H 1 ) 1 + ( ξ ( ρ ) x ) 2 + ( ξ ( ρ ) y ) 2 ,
T = S cell ( S N ) d ρ S cell ( S i N ) d ρ ,
× E = i H ,
× H = i ε n E ,
x , y , z ν = e ν e i ( κ ρ + σ k n ( κ ) ( z z n 1 ) ) ,
e n , σ , s , κ = z ̂ × n , e n , σ , p , κ = κ z ̂ σ k n ( κ ) n ε n ,
E n ( R ) = ν c ν ν = σ , α d κ ( 2 π ) 2 c n , σ , α , κ e n , σ , α , κ e i K n σ ( κ ) R .
H n ( R ) = ε n σ , α d κ ( 2 π ) 2 c n , σ , α , κ ( 1 ) α e n , σ , α ¯ , κ e i K n σ ( κ ) R ,
[ E n ( R ) H n ( R ) ] = σ d κ ( 2 π ) 2 [ e n , σ , s , κ e n , σ , p , κ ε n e n , σ , p , κ ε n e n , σ , s , κ ] c ̂ n σ ( κ ) e i K n σ ( κ ) R ,
c ̂ n σ ( κ ) [ c n , σ , s , κ c n , σ , p , κ ] .

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