Abstract

We study image enhancement of transmitted optical fields though plasmon film lenses with smooth surfaces, random rough surfaces, and sinusoidal rough surfaces. We show that the silver film lens with subwavelength roughness provides image enhancement by reducing the spatial width of the image while preserving the image integrity. Images of multiple sources at random positions are enhanced, are spatial translational invariant for subwavelength spacing, and the quality of the enhanced images is preserved. Two numerical methods, namely, the method of moments and the T-matrix method, are used to calculate the theoretical results providing a validity check of the accuracy of the solutions. The analytic small perturbation method to arbitrary orders is utilized to demonstrate image enhancement effects in the spectral domain. The role of surface plasmon polaritons (SPPs) in plasmon imaging is explained and it is shown that the surface with subwavelength roughness enhances images by suppression of the SPPs.

© 2011 Optical Society of America

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  1. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314(2005).
    [CrossRef]
  2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [CrossRef] [PubMed]
  3. N. Fang and X. Zhang, “Imaging properties of a metamaterials superlens,” Appl. Phys. Lett. 82, 161–163 (2003).
    [CrossRef]
  4. X. S. Rao and C. K. Ong, “Amplification of evanescent waves in a lossy left-handed material slab,” Phys. Rev. B 68, 113103(2003).
    [CrossRef]
  5. V. A. Podolskiy and E. E. Narimanov, “Near-sighted superlens,” Opt. Lett. 30, 75–77 (2005).
    [CrossRef] [PubMed]
  6. S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
    [CrossRef]
  7. D. O. S. Melville, R. J. Blaikie, and C. R. Wolf, “Submicron imaging with a planar silver layer,” Appl. Phys. Lett. 84, 4403–4405(2004).
    [CrossRef]
  8. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).
  9. N. Fang, Z. Liu, T.-J. Yen, and X. Zhang, “Experimental study of transmission enhancement of evanescent waves through silver films assisted by surface plasmon excitation,” Appl. Phys. A 80, 1315–1325 (2005).
    [CrossRef]
  10. M. Schøler and R. J. Blaikie, “Simulations of surface roughness effects in planar superlenses,” J. Opt. A Pure Appl. Opt. 11, 105503 (2009).
    [CrossRef]
  11. Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7, 403–408 (2007).
    [CrossRef] [PubMed]
  12. S. Durant, Z. Liu, J. M. Steele, and X. Zhang, “Theory of the transmission properties of an optical far-field superlens for imaging beyond the diffraction limit,” J. Opt. Soc. Am. B 23, 2383–2392(2006).
    [CrossRef]
  13. J. Q. Bagley, B. Wu, and L. Tsang, “Electromagnetic fields and modal excitations on a thin silver film,” J. Opt. Soc. Am. A 26, 2362–2367 (2009).
    [CrossRef]
  14. L. Tsang and B. Wu, “Electromagnetic fields of Hertzian dipoles in layered media of moderate thickness including the effects of all modes,” IEEE Antennas Wirel. Propag. Lett. 6, 316–319(2007).
    [CrossRef]
  15. J. Q. Bagley and L. Tsang, “Image enhancement using rough surface effects in plasmon materials,” to be presented at the IEEE International Symposium on Antennas and Propagation, Spokane, Wash., USA, 3–8 July 2011.
  16. L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves, Vol.  2 of Numerical Simulations(Wiley-Interscience, 2001).
  17. M. A. Demir and J. T. Johnson, “Fourth- and higher-order small-perturbation solution for scattering from dielectric rough surfaces,” J. Opt. Soc. Am. A 20, 2330–2337 (2003).
    [CrossRef]

2009

M. Schøler and R. J. Blaikie, “Simulations of surface roughness effects in planar superlenses,” J. Opt. A Pure Appl. Opt. 11, 105503 (2009).
[CrossRef]

J. Q. Bagley, B. Wu, and L. Tsang, “Electromagnetic fields and modal excitations on a thin silver film,” J. Opt. Soc. Am. A 26, 2362–2367 (2009).
[CrossRef]

2007

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7, 403–408 (2007).
[CrossRef] [PubMed]

L. Tsang and B. Wu, “Electromagnetic fields of Hertzian dipoles in layered media of moderate thickness including the effects of all modes,” IEEE Antennas Wirel. Propag. Lett. 6, 316–319(2007).
[CrossRef]

2006

2005

V. A. Podolskiy and E. E. Narimanov, “Near-sighted superlens,” Opt. Lett. 30, 75–77 (2005).
[CrossRef] [PubMed]

N. Fang, Z. Liu, T.-J. Yen, and X. Zhang, “Experimental study of transmission enhancement of evanescent waves through silver films assisted by surface plasmon excitation,” Appl. Phys. A 80, 1315–1325 (2005).
[CrossRef]

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

2004

D. O. S. Melville, R. J. Blaikie, and C. R. Wolf, “Submicron imaging with a planar silver layer,” Appl. Phys. Lett. 84, 4403–4405(2004).
[CrossRef]

2003

N. Fang and X. Zhang, “Imaging properties of a metamaterials superlens,” Appl. Phys. Lett. 82, 161–163 (2003).
[CrossRef]

X. S. Rao and C. K. Ong, “Amplification of evanescent waves in a lossy left-handed material slab,” Phys. Rev. B 68, 113103(2003).
[CrossRef]

M. A. Demir and J. T. Johnson, “Fourth- and higher-order small-perturbation solution for scattering from dielectric rough surfaces,” J. Opt. Soc. Am. A 20, 2330–2337 (2003).
[CrossRef]

2002

S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

2000

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

Ao, C. O.

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves, Vol.  2 of Numerical Simulations(Wiley-Interscience, 2001).

Bagley, J. Q.

J. Q. Bagley, B. Wu, and L. Tsang, “Electromagnetic fields and modal excitations on a thin silver film,” J. Opt. Soc. Am. A 26, 2362–2367 (2009).
[CrossRef]

J. Q. Bagley and L. Tsang, “Image enhancement using rough surface effects in plasmon materials,” to be presented at the IEEE International Symposium on Antennas and Propagation, Spokane, Wash., USA, 3–8 July 2011.

Blaikie, R. J.

M. Schøler and R. J. Blaikie, “Simulations of surface roughness effects in planar superlenses,” J. Opt. A Pure Appl. Opt. 11, 105503 (2009).
[CrossRef]

D. O. S. Melville, R. J. Blaikie, and C. R. Wolf, “Submicron imaging with a planar silver layer,” Appl. Phys. Lett. 84, 4403–4405(2004).
[CrossRef]

Demir, M. A.

Ding, K.-H.

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves, Vol.  2 of Numerical Simulations(Wiley-Interscience, 2001).

Durant, S.

Fang, N.

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7, 403–408 (2007).
[CrossRef] [PubMed]

N. Fang, Z. Liu, T.-J. Yen, and X. Zhang, “Experimental study of transmission enhancement of evanescent waves through silver films assisted by surface plasmon excitation,” Appl. Phys. A 80, 1315–1325 (2005).
[CrossRef]

N. Fang and X. Zhang, “Imaging properties of a metamaterials superlens,” Appl. Phys. Lett. 82, 161–163 (2003).
[CrossRef]

Johnson, J. T.

Kong, J. A.

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves, Vol.  2 of Numerical Simulations(Wiley-Interscience, 2001).

Lee, H.

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7, 403–408 (2007).
[CrossRef] [PubMed]

Liu, Z.

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7, 403–408 (2007).
[CrossRef] [PubMed]

S. Durant, Z. Liu, J. M. Steele, and X. Zhang, “Theory of the transmission properties of an optical far-field superlens for imaging beyond the diffraction limit,” J. Opt. Soc. Am. B 23, 2383–2392(2006).
[CrossRef]

N. Fang, Z. Liu, T.-J. Yen, and X. Zhang, “Experimental study of transmission enhancement of evanescent waves through silver films assisted by surface plasmon excitation,” Appl. Phys. A 80, 1315–1325 (2005).
[CrossRef]

Maradudin, A. A.

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

Melville, D. O. S.

D. O. S. Melville, R. J. Blaikie, and C. R. Wolf, “Submicron imaging with a planar silver layer,” Appl. Phys. Lett. 84, 4403–4405(2004).
[CrossRef]

Narimanov, E. E.

Ong, C. K.

X. S. Rao and C. K. Ong, “Amplification of evanescent waves in a lossy left-handed material slab,” Phys. Rev. B 68, 113103(2003).
[CrossRef]

Pendry, J. B.

S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

Pikus, Y.

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7, 403–408 (2007).
[CrossRef] [PubMed]

Podolskiy, V. A.

Raether, H.

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

Ramakrishna, S. A.

S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

Rao, X. S.

X. S. Rao and C. K. Ong, “Amplification of evanescent waves in a lossy left-handed material slab,” Phys. Rev. B 68, 113103(2003).
[CrossRef]

Schøler, M.

M. Schøler and R. J. Blaikie, “Simulations of surface roughness effects in planar superlenses,” J. Opt. A Pure Appl. Opt. 11, 105503 (2009).
[CrossRef]

Schultz, S.

S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

Schurig, D.

S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

Smith, D. R.

S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[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).
[CrossRef]

Steele, J. M.

Sun, C.

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7, 403–408 (2007).
[CrossRef] [PubMed]

Tsang, L.

J. Q. Bagley, B. Wu, and L. Tsang, “Electromagnetic fields and modal excitations on a thin silver film,” J. Opt. Soc. Am. A 26, 2362–2367 (2009).
[CrossRef]

L. Tsang and B. Wu, “Electromagnetic fields of Hertzian dipoles in layered media of moderate thickness including the effects of all modes,” IEEE Antennas Wirel. Propag. Lett. 6, 316–319(2007).
[CrossRef]

J. Q. Bagley and L. Tsang, “Image enhancement using rough surface effects in plasmon materials,” to be presented at the IEEE International Symposium on Antennas and Propagation, Spokane, Wash., USA, 3–8 July 2011.

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves, Vol.  2 of Numerical Simulations(Wiley-Interscience, 2001).

Wolf, C. R.

D. O. S. Melville, R. J. Blaikie, and C. R. Wolf, “Submicron imaging with a planar silver layer,” Appl. Phys. Lett. 84, 4403–4405(2004).
[CrossRef]

Wu, B.

J. Q. Bagley, B. Wu, and L. Tsang, “Electromagnetic fields and modal excitations on a thin silver film,” J. Opt. Soc. Am. A 26, 2362–2367 (2009).
[CrossRef]

L. Tsang and B. Wu, “Electromagnetic fields of Hertzian dipoles in layered media of moderate thickness including the effects of all modes,” IEEE Antennas Wirel. Propag. Lett. 6, 316–319(2007).
[CrossRef]

Xiong, Y.

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7, 403–408 (2007).
[CrossRef] [PubMed]

Yen, T.-J.

N. Fang, Z. Liu, T.-J. Yen, and X. Zhang, “Experimental study of transmission enhancement of evanescent waves through silver films assisted by surface plasmon excitation,” Appl. Phys. A 80, 1315–1325 (2005).
[CrossRef]

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).
[CrossRef]

Zhang, X.

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7, 403–408 (2007).
[CrossRef] [PubMed]

S. Durant, Z. Liu, J. M. Steele, and X. Zhang, “Theory of the transmission properties of an optical far-field superlens for imaging beyond the diffraction limit,” J. Opt. Soc. Am. B 23, 2383–2392(2006).
[CrossRef]

N. Fang, Z. Liu, T.-J. Yen, and X. Zhang, “Experimental study of transmission enhancement of evanescent waves through silver films assisted by surface plasmon excitation,” Appl. Phys. A 80, 1315–1325 (2005).
[CrossRef]

N. Fang and X. Zhang, “Imaging properties of a metamaterials superlens,” Appl. Phys. Lett. 82, 161–163 (2003).
[CrossRef]

Appl. Phys. A

N. Fang, Z. Liu, T.-J. Yen, and X. Zhang, “Experimental study of transmission enhancement of evanescent waves through silver films assisted by surface plasmon excitation,” Appl. Phys. A 80, 1315–1325 (2005).
[CrossRef]

Appl. Phys. Lett.

D. O. S. Melville, R. J. Blaikie, and C. R. Wolf, “Submicron imaging with a planar silver layer,” Appl. Phys. Lett. 84, 4403–4405(2004).
[CrossRef]

N. Fang and X. Zhang, “Imaging properties of a metamaterials superlens,” Appl. Phys. Lett. 82, 161–163 (2003).
[CrossRef]

IEEE Antennas Wirel. Propag. Lett.

L. Tsang and B. Wu, “Electromagnetic fields of Hertzian dipoles in layered media of moderate thickness including the effects of all modes,” IEEE Antennas Wirel. Propag. Lett. 6, 316–319(2007).
[CrossRef]

J. Mod. Opt.

S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

J. Opt. A Pure Appl. Opt.

M. Schøler and R. J. Blaikie, “Simulations of surface roughness effects in planar superlenses,” J. Opt. A Pure Appl. Opt. 11, 105503 (2009).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Nano Lett.

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7, 403–408 (2007).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rep.

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

Phys. Rev. B

X. S. Rao and C. K. Ong, “Amplification of evanescent waves in a lossy left-handed material slab,” Phys. Rev. B 68, 113103(2003).
[CrossRef]

Phys. Rev. Lett.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

Other

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

J. Q. Bagley and L. Tsang, “Image enhancement using rough surface effects in plasmon materials,” to be presented at the IEEE International Symposium on Antennas and Propagation, Spokane, Wash., USA, 3–8 July 2011.

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves, Vol.  2 of Numerical Simulations(Wiley-Interscience, 2001).

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Figures (11)

Fig. 1
Fig. 1

Geometry of the plasmon superlens with rough surfaces.

Fig. 2
Fig. 2

(a) Spatial image field for the smooth surface lens ( λ 0 = 351 nm , d = 0.1 λ 0 ) showing agreement of the MoM (black crosses) and the T-matrix method (red circles) with SIP numerical integration (green solid curve), and compared to the field at the same distance ( Δ z = 0.3 λ 0 ) due to a line source in free space. (b) Corresponding image field spectra for the smooth surface lens and the line source in free space.

Fig. 3
Fig. 3

(a) Spatial image field and (b) image field spectra for a moderate random rough surface lens ( l = 0.1 λ 0 ) compared to a small random rough surface lens ( l = 0.01 λ 0 ).

Fig. 4
Fig. 4

Spatial image field of two identical sources (spaced Δ x = 0.6216 λ 0 apart) for a moderate random rough surface lens ( l = 0.1 λ 0 ) compared to a small random rough surface lens ( l = 0.01 λ 0 ). Vertical red dotted lines indicate the source spacing.

Fig. 5
Fig. 5

(a) Spatial image field and (b) image field spectra for a periodic surface lens of moderate height and period ( h = 0.03 λ 0 / 2 , P = 0.33 λ 0 ) compared to small height and period ( h = 0.01 λ 0 / 2 , P = 0.1 λ 0 ).

Fig. 6
Fig. 6

Spatial image field of two identical sources (spaced Δ x = 0.6216 λ 0 apart) for a periodic surface with (a) moderate height and period ( h = 0.03 λ 0 / 2 , P = 0.33 λ 0 ) and (b) small height and period ( h = 0.01 λ 0 / 2 , P = 0.1 λ 0 ); (c) the sources are shifted by Δ x = 0.1589 λ 0 and (d) the amplitude of source on the right is doubled.

Fig. 7
Fig. 7

SPM first-order term of the transmitted field spectra for the rough surface lens with small height ( h = 0.01 λ 0 / 2 ) for two different surface periods P.

Fig. 8
Fig. 8

SPM fourth-order term of the transmitted field spectra for the rough surface lens with small height ( h = 0.01 λ 0 / 2 ) for two different surface periods P.

Fig. 9
Fig. 9

Image field spectra for the smooth surface lens showing contributions of the SDP, LSPP, and SSPP when ε 2 = ε 0 .

Fig. 10
Fig. 10

Spatial image field for smooth surface lens showing contributions of the SDP, LSPP, and SSPP to the image field when ε 2 = ε 0 .

Fig. 11
Fig. 11

Image field spectra for the rough surface lens with small height and period ( h = 0.01 λ 0 / 2 , P = 0.1 λ 0 ) compared to the smooth surface lens when ε 2 = ε 0 .

Tables (1)

Tables Icon

Table 1 Image Spatial Width b

Equations (48)

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ψ inc ( r ¯ ) + S 0 d s 0 n ^ 0 · [ ψ 0 ( r ¯ ) ¯ g 0 ( r ¯ , r ¯ ) g 0 ( r ¯ , r ¯ ) ¯ ψ 0 ( r ¯ ) ] = { ψ 0 ( r ¯ ) r ¯ Ω 0 0 r ¯ Ω 0 ,
S 0 d s 0 n ^ 0 · [ ψ 0 ( r ¯ ) ¯ g 1 ( r ¯ , r ¯ ) ρ 10 g 1 ( r ¯ , r ¯ ) ¯ ψ 0 ( r ¯ ) ] S 1 d s 1 n ^ 1 · [ ψ 2 ( r ¯ ) ¯ g 1 ( r ¯ , r ¯ ) ρ 12 g 1 ( r ¯ , r ¯ ) ¯ ψ 2 ( r ¯ ) ] = { ψ 1 ( r ¯ ) r ¯ Ω 1 0 r ¯ Ω 1 ,
S 1 d s 1 n ^ 1 · [ ψ 2 ( r ¯ ) ¯ g 2 ( r ¯ , r ¯ ) g 2 ( r ¯ , r ¯ ) ¯ ψ 2 ( r ¯ ) ] = { ψ 2 ( r ¯ ) r ¯ Ω 2 0 r ¯ Ω 2 ,
g p ( r ¯ , r ¯ ) = i 4 H 0 ( 1 ) ( k p | r ¯ r ¯ | ) ,
ρ 10 = { ε 1 / ε for     TM 1 for     TE ,
ρ 12 = { ε 1 / ε 2 for     TM 1 for     TE .
g p ( x , z ; x , z ) = { m i k p z m L e i k p z m | z z | cos ( k x m x ) cos ( k x m x ) for     m   odd m i k p z m L e i k p z m | z z | sin ( k x m x ) sin ( k x m x ) for     m   even ,
k x m = m π L ,
g ( x 0 , z 0 ; x , z ) = L / 2 L / 2 d x [ a 0 ( x ) ( d f 0 ( x ) d x x + z ) g ( r ¯ , r ¯ ) g ( r ¯ , r ¯ ) b 0 ( x ) ] r ¯ S 0 r ¯ Ω 0 .
0 = L / 2 L / 2 d x [ a 0 ( x ) ( d f 0 ( x ) d x x + z ) g 1 ( r ¯ , r ¯ ) ρ 10 g 1 ( r ¯ , r ¯ ) b 0 ( x ) ] r ¯ S 0 r ¯ Ω 0 L / 2 L / 2 d x [ a 2 ( x ) ( d f 1 ( x ) d x x + z ) g 1 ( r ¯ , r ¯ ) ρ 12 g 1 ( r ¯ , r ¯ ) b 2 ( x ) ] r ¯ S 1 r ¯ Ω 0 ,
0 = L / 2 L / 2 d x [ a 0 ( x ) ( d f 0 ( x ) d x x + z ) g 1 ( r ¯ , r ¯ ) ρ 10 g 1 ( r ¯ , r ¯ ) b 0 ( x ) ] r ¯ S 0 r ¯ Ω 2 L / 2 L / 2 d x [ a 2 ( x ) ( d f 1 ( x ) d x x + z ) g 1 ( r ¯ , r ¯ ) ρ 12 g 1 ( r ¯ , r ¯ ) b 2 ( x ) ] r ¯ S 1 r ¯ Ω 2 .
0 = L / 2 L / 2 d x [ a 2 ( x ) ( d f 1 ( x ) d x x + z ) g 2 ( r ¯ , r ¯ ) g 2 ( r ¯ , r ¯ ) b 2 ( x ) ] r ¯ S 1 r ¯ Ω 2 ,
a q ( x ) = { n = a q n cos ( k x n x ) for     n   odd n = a q n sin ( k x n x ) for     n   even ,
b q ( x ) = { n = b q n cos ( k x n x ) for     n   odd n = b q n sin ( k x n x ) for     n   even ,
[ A m n B m n 0 0 A 1 m n ρ 10 B 1 m n A 2 m n ρ 12 B 2 m n A 3 m n ρ 10 B 3 m n A 4 m n ρ 12 B 4 m n 0 0 A 5 m n B 5 m n ] [ a 0 n b 0 n a 2 n b 2 n ] = [ E m 0 0 0 ] ,
E m = { e i k z m z 0 cos ( k x m x 0 ) for     m   odd e i k z m z 0 sin ( k x m x 0 ) for     m   even ,
ψ t ( x , z ) = m e i k 2 z m z [ i e i k 2 z m d k 2 z m L ( n A t m n a 2 n n B t m n b 2 n ) ] { cos ( k x m x ) m   odd sin ( k x m x ) m   even .
g p ( x , z ; x , z ) = i 4 π d k x e i k x ( x x ) + i k p z | z z | k p z ,
ψ inc ( x , z ) = d k x e i k x x i k z z ψ ˜ inc ( k x ) .
ψ ˜ inc ( k x ) + i 4 π k z S 0 d x ( a ( x ) ( i k x d f 0 ( x ) d x + i k z ) b ( x ) ) e i k x x e i k z f 0 ( x ) = 0 ,
i 4 π k 1 z S 0 d x ( a ( x ) ( i k x d f 0 ( x ) d x i k 1 z ) ρ 10 b ( x ) ) e i k x x e i k 1 z f 0 ( x ) i 4 π k 1 z S 1 d x ( i k 1 z a 1 ( x ) ρ 12 b 1 ( x ) ) e i k x x e i k 1 z d = 0 ,
i 4 π k 1 z S 0 d x ( a ( x ) ( i k x d f 0 ( x ) d x + i k 1 z ) ρ 10 b ( x ) ) e i k x x e i k 1 z f 0 ( x ) i 4 π k 1 z S 1 d x ( i k 1 z a 1 ( x ) ρ 12 b 1 ( x ) ) e i k x x e i k 1 z d = 0 ,
i 4 π k 2 z S 1 d x ( i k 2 z a 1 ( x ) b 1 ( x ) ) e i k x x e i k 2 z d = 0 ,
e ± i k p z f 0 ( x ) = 1 + n = 1 ( ± i k p z ) n ( f 0 ( x ) ) n n ! ,
m = 0 [ A ( m ) ( k x ) + i k z B ( m ) ( k x ) ] = 2 ψ ˜ inc ( k x ) k x k z n = 0 ( i k z ) n n ! d k x m = 0 A ( m ) ( k x ) F D ( n + 1 ) ( k x k x ) n = 1 ( i k z ) n n ! d k x m = 0 [ A ( m ) ( k x ) + i k z B ( m ) ( k x ) ] F ( n ) ( k x k x ) ,
m = 0 [ A ( m ) ( k x ) i ρ 10 k 1 z B ( m ) ( k x ) e i k 1 z d A 1 ( m ) ( k x ) + i ρ 12 k 1 z e i k 1 z d B 1 ( m ) ( k x ) ] = k x k 1 z n = 0 ( i k 1 z ) n n ! d k x m = 0 A ( m ) ( k x ) F D ( n + 1 ) ( k x k x ) n = 1 ( i k 1 z ) n n ! d k x m = 0 [ A ( m ) ( k x ) i ρ 10 k 1 z B ( m ) ( k x ) ] F ( n ) ( k x k x ) ,
m = 0 [ A ( m ) ( k x ) + i ρ 10 k 1 z B ( m ) ( k x ) e i k 1 z d A 1 ( m ) ( k x ) i ρ 12 k 1 z e i k 1 z d B 1 ( m ) ( k x ) ] = k x k 1 z n = 0 ( i k 1 z ) n n ! d k x m = 0 A ( m ) ( k x ) F D ( n + 1 ) ( k x k x ) n = 1 ( i k 1 z ) n n ! d k x m = 0 [ A ( m ) ( k x ) + i ρ 10 k 1 z B ( m ) ( k x ) ] F ( n ) ( k x k x ) ,
m = 0 [ A 1 ( m ) ( k x ) i k 2 z B 1 ( m ) ( k x ) ] = 0.
m = 0 ψ ˜ 2 ( m ) ( k x ) = i k 2 z e i k 2 z d m = 0 B 1 ( m ) ( k x ) = e i k 2 z d m = 0 A 1 ( m ) ( k x ) .
ψ t ( x , z ) = d k x e i k x x i k 2 z z ψ ˜ 2 ( k x ) .
ψ ˜ t ( k x , z ) = 1 2 π d x ψ t ( x , z ) e i k x x + i k 2 z z .
f 0 ( x ) = 2 h sin ( 2 π x / P ) ,
ψ t ( x , z ) = i 2 π 0 d k x cos ( k x x ) 2 ε 1 e i k z z 0 e i k 2 z z ε 1 k z + ε k 1 z T 12 T M e i ( k 1 z k 2 z ) d 1 + R 01 T M R 12 T M e 2 i k 1 z d ,
k x LSPP = ( 1.1282 + 0.0269 i ) k 0 , k x SSPP = ( 2.7628 + 0.6249 i ) k 0 .
A m n = L / 2 L / 2 d x e i k z m f 0 ( x ) { [ k x m d f 0 ( x ) d x sin ( k x m x ) + i k z m cos ( k x m x ) ] cos ( k x n x ) m   odd , n   odd [ k x m d f 0 ( x ) d x sin ( k x m x ) + i k z m cos ( k x m x ) ] sin ( k x n x ) m   odd , n   even [ k x m d f 0 ( x ) d x cos ( k x m x ) + i k z m sin ( k x m x ) ] cos ( k x n x ) m   even , n   odd [ k x m d f 0 ( x ) d x cos ( k x m x ) + i k z m sin ( k x m x ) ] sin ( k x n x ) m   even , n   even ,
B m n = L / 2 L / 2 d x e i k z m f 0 ( x ) { cos ( k x m x ) cos ( k x n x ) m   odd , n   odd cos ( k x m x ) sin ( k x n x ) m   odd , n   even sin ( k x m x ) cos ( k x n x ) m   even , n   odd sin ( k x m x ) sin ( k x n x ) m   even , n   even ,
A 1 m n = L / 2 L / 2 d x e i k 1 z m f 0 ( x ) { [ k x m d f 0 ( x ) d x sin ( k x m x ) i k 1 z m cos ( k x m x ) ] cos ( k x n x ) m   odd , n   odd [ k x m d f 0 ( x ) d x sin ( k x m x ) i k 1 z m cos ( k x m x ) ] sin ( k x n x ) m   odd , n   even [ k x m d f 0 ( x ) d x cos ( k x m x ) i k 1 z m sin ( k x m x ) ] cos ( k x n x ) m   even , n   odd [ k x m d f 0 ( x ) d x cos ( k x m x ) i k 1 z m sin ( k x m x ) ] sin ( k x n x ) m   even , n   even ,
B 1 m n = L / 2 L / 2 d x e i k 1 z m f 0 ( x ) { cos ( k x m x ) cos ( k x n x ) m   odd , n   odd cos ( k x m x ) sin ( k x n x ) m   odd , n   even sin ( k x m x ) cos ( k x n x ) m   even , n   odd sin ( k x m x ) sin ( k x n x ) m   even , n   even ,
A 2 m n = e i k 1 z m d L / 2 L / 2 d x e i k 1 z m f 1 ( x ) { [ k x m d f 1 ( x ) d x sin ( k x m x ) i k 1 z m cos ( k x m x ) ] cos ( k x n x ) m   odd , n   odd [ k x m d f 1 ( x ) d x sin ( k x m x ) i k 1 z m cos ( k x m x ) ] sin ( k x n x ) m   odd , n   even [ k x m d f 1 ( x ) d x cos ( k x m x ) i k 1 z m sin ( k x m x ) ] cos ( k x n x ) m   even , n   odd [ k x m d f 1 ( x ) d x cos ( k x m x ) i k 1 z m sin ( k x m x ) ] sin ( k x n x ) m   even , n   even ,
B 2 m n = e i k 1 z m d L / 2 L / 2 d x e i k 1 z m f 1 ( x ) { cos ( k x m x ) cos ( k x n x ) m   odd , n   odd cos ( k x m x ) sin ( k x n x ) m   odd , n   even sin ( k x m x ) cos ( k x n x ) m   even , n   odd sin ( k x m x ) sin ( k x n x ) m   even , n   even ,
A 3 m n = L / 2 L / 2 d x e i k 1 z m f 0 ( x ) { [ k x m d f 0 ( x ) d x sin ( k x m x ) + i k 1 z m cos ( k x m x ) ] cos ( k x n x ) m   odd , n   odd [ k x m d f 0 ( x ) d x sin ( k x m x ) + i k 1 z m cos ( k x m x ) ] sin ( k x n x ) m   odd , n   even [ k x m d f 0 ( x ) d x cos ( k x m x ) + i k 1 z m sin ( k x m x ) ] cos ( k x n x ) m   even , n   odd [ k x m d f 0 ( x ) d x cos ( k x m x ) + i k 1 z m sin ( k x m x ) ] sin ( k x n x ) m   even , n   even ,
B 3 m n = L / 2 L / 2 d x e i k 1 z m f 0 ( x ) { cos ( k x m x ) cos ( k x n x ) m   odd , n   odd cos ( k x m x ) sin ( k x n x ) m   odd , n   even sin ( k x m x ) cos ( k x n x ) m   even , n   odd sin ( k x m x ) sin ( k x n x ) m   even , n   even ,
A 4 m n = e i k 1 z m d L / 2 L / 2 d x e i k 1 z m f 1 ( x ) { [ k x m d f 1 ( x ) d x sin ( k x m x ) + i k 1 z m cos ( k x m x ) ] cos ( k x n x ) m   odd , n   odd [ k x m d f 1 ( x ) d x sin ( k x m x ) + i k 1 z m cos ( k x m x ) ] sin ( k x n x ) m   odd , n   even [ k x m d f 1 ( x ) d x cos ( k x m x ) + i k 1 z m sin ( k x m x ) ] cos ( k x n x ) m   even , n   odd [ k x m d f 1 ( x ) d x cos ( k x m x ) + i k 1 z m sin ( k x m x ) ] sin ( k x n x ) m   even , n   even ,
B 4 m n = e i k 1 z m d L / 2 L / 2 d x e i k 1 z m f 1 ( x ) { cos ( k x m x ) cos ( k x n x ) m   odd , n   odd cos ( k x m x ) sin ( k x n x ) m   odd , n   even sin ( k x m x ) cos ( k x n x ) m   even , n   odd sin ( k x m x ) sin ( k x n x ) m   even , n   even ,
A 5 m n = L / 2 L / 2 d x e i k 2 z m f 1 ( x ) { [ k x m d f 1 ( x ) d x sin ( k x m x ) i k 2 z m cos ( k x m x ) ] cos ( k x n x ) m   odd , n   odd [ k x m d f 1 ( x ) d x sin ( k x m x ) i k 2 z m cos ( k x m x ) ] sin ( k x n x ) m   odd , n   even [ k x m d f 1 ( x ) d x cos ( k x m x ) i k 2 z m sin ( k x m x ) ] cos ( k x n x ) m   even , n   odd [ k x m d f 1 ( x ) d x cos ( k x m x ) i k 2 z m sin ( k x m x ) ] sin ( k x n x ) m   even , n   even ,
B 5 m n = L / 2 L / 2 d x e i k 2 z m f 1 ( x ) { cos ( k x m x ) cos ( k x n x ) m   odd , n   odd cos ( k x m x ) sin ( k x n x ) m   odd , n   even sin ( k x m x ) cos ( k x n x ) m   even , n   odd sin ( k x m x ) sin ( k x n x ) m   even , n   even .
A t m n = L / 2 L / 2 d x e i k 2 z m f 1 ( x ) { [ k x m d f 1 ( x ) d x sin ( k x m x ) i k 2 z m cos ( k x m x ) ] cos ( k x n x ) m   odd , n   odd [ k x m d f 1 ( x ) d x sin ( k x m x ) i k 2 z m cos ( k x m x ) ] sin ( k x n x ) m   odd , n   even [ k x m d f 1 ( x ) d x cos ( k x m x ) i k 2 z m sin ( k x m x ) ] cos ( k x n x ) m   even , n   odd [ k x m d f 1 ( x ) d x cos ( k x m x ) i k 2 z m sin ( k x m x ) ] sin ( k x n x ) m   even , n   even ,
B t m n = L / 2 L / 2 d x e i k 2 z m f 1 ( x ) { cos ( k x m x ) cos ( k x n x ) m   odd , n   odd cos ( k x m x ) sin ( k x n x ) m   odd , n   even sin ( k x m x ) cos ( k x n x ) m   even , n   odd sin ( k x m x ) sin ( k x n x ) m   even , n   even .

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