Abstract

A vector integral–differential equation to describe electromagnetic wave propagation in nanowaveguides and photonic crystals containing thin metal layers is developed. Exact solution of the equation is obtained with the Galerkin method, taking into account the complex dielectric permittivity of metals in the optical range. A simple method for finding complex effective refractive indices for low-loss waveguide structures is developed and proved. Surface plasmon-polariton waves are simulated in the structures under consideration.

© 2014 Chinese Laser Press

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  32. V. A. Kuznetsov and A. M. Lerer, “Dispersion characteristics of dielectric waveguides on substrates,” Radio Eng. Electron. Phys. 29, 53–58 (1984).
  33. G. A. Kalinchenko and A. M. Lerer, “Investigations of dielectric gratings using electrodynamic models based on volume integral equations,” J. Commun. Technol. Electron. 48, 1221–1227 (2003).
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    [CrossRef]

2013

2012

A. M. Lerer, “Theoretical investigation of 2D periodic nanoplasmon structures,” J. Commun. Technol. Electron. 57, 1151–1159 (2012).
[CrossRef]

2011

2010

2009

2008

J. Guo and R. Adato, “Control of 2D plasmon-polariton mode with dielectric nanolayers,” Opt. Express 16, 1232–1237 (2008).
[CrossRef]

T. Kim, J. J. Ju, S. Park, M.-S. Kim, S. K. Park, and M.-H. Lee, “Chip-to-chip optical interconnect using gold long-range surface plasmon polariton waveguides,” Opt. Express 16, 13133–13138 (2008).
[CrossRef]

A. V. Krasavin and A. V. Zayats, “Three-dimensional numerical modeling of photonic integration with dielectric-loaded SPP waveguides,” Phys. Rev. B 78, 045425 (2008).
[CrossRef]

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef]

2007

T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75, 245405 (2007).
[CrossRef]

R. Buckley and P. Berini, “Figures of merit for 2D surface plasmon waveguides and application to metal stripes,” Opt. Express 15, 12174–12182 (2007).
[CrossRef]

2006

2005

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef]

L. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express 13, 6645–6650 (2005).
[CrossRef]

2004

M. Hochberg, T. Baehr-Jones, C. Walker, and A. Scherer, “Integrated plasmon and dielectric waveguides,” Opt. Express 12, 5481–5486 (2004).
[CrossRef]

R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A 21, 2442–2446 (2004).
[CrossRef]

C.-O Cho, Y.-G. Roh, Y. Park, J.-S. I, H. Jeon, B.-S. Lee, H.-W. Kim, Y.-H. Choe, M. Sung, and J. C. Woo, “Towards nano-waveguides,” Appl. Phys. 4, 245–249 (2004).
[CrossRef]

D. K. Gramotnev and D. F. P. Pile, “Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface,” Appl. Phys. Lett. 85, 6323–6325 (2004).
[CrossRef]

2003

G. A. Kalinchenko and A. M. Lerer, “Investigations of dielectric gratings using electrodynamic models based on volume integral equations,” J. Commun. Technol. Electron. 48, 1221–1227 (2003).

2002

I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66, 035403 (2002).
[CrossRef]

2000

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000).
[CrossRef]

1987

E. Yablonovich, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef]

1984

V. A. Kuznetsov and A. M. Lerer, “Dispersion characteristics of dielectric waveguides on substrates,” Radio Eng. Electron. Phys. 29, 53–58 (1984).

1982

V. A. Kuznetsov and A. M. Lerer, “Dispersion characteristics of rectangular dielectric waveguide,” Radio Eng. Electron. Phys. 27, 24–27 (1982).

Adato, R.

Agrawal, G. P.

Arbel, D.

D. Arbel and M. Orenstein, “W-shaped plasmon waveguide for silicon based plasmonic modulator,” in LEOS Annual Meeting (2006), pp. 262–263.

Baehr-Jones, T.

Bankov, S. E.

S. E. Bankov, Electomagnitnye kristally (FIZMATLIT, 2010) [Russian].

Berini, P.

R. Buckley and P. Berini, “Figures of merit for 2D surface plasmon waveguides and application to metal stripes,” Opt. Express 15, 12174–12182 (2007).
[CrossRef]

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000).
[CrossRef]

Bernussi, A. A.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1968).

Bozhevolnyi, S. I.

J. Gosciniak, V. S. Volkov, S. I. Bozhevolnyi, L. Markey, S. Massenot, and A. Dereux, “Fiber-coupled dielectric-loaded plasmonic waveguides,” Opt. Express 18, 5314–5319 (2010).
[CrossRef]

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef]

T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75, 245405 (2007).
[CrossRef]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef]

Brongersma, M. L.

Buckley, R.

Catrysse, P. B.

Cho, C.-O

C.-O Cho, Y.-G. Roh, Y. Park, J.-S. I, H. Jeon, B.-S. Lee, H.-W. Kim, Y.-H. Choe, M. Sung, and J. C. Woo, “Towards nano-waveguides,” Appl. Phys. 4, 245–249 (2004).
[CrossRef]

Choe, Y.-H.

C.-O Cho, Y.-G. Roh, Y. Park, J.-S. I, H. Jeon, B.-S. Lee, H.-W. Kim, Y.-H. Choe, M. Sung, and J. C. Woo, “Towards nano-waveguides,” Appl. Phys. 4, 245–249 (2004).
[CrossRef]

Degiron, A.

Dellagiacoma, C.

Dereux, A.

Devaux, E.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef]

Ebbesen, T. W.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef]

Farhat, M.

García-Vidal, F. J.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef]

Golovacheva, E. V.

E. V. Golovacheva, A. M. Lerer, and N. G. Parkhomenko, “Diffraction of electromagnetic waves of optical range on a metallic nanovibrator,” Moscow Univ. Phys. Bull. 66, 5–11 (2011).
[CrossRef]

Gosciniak, J.

Gramotnev, D. K.

D. K. Gramotnev and D. F. P. Pile, “Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface,” Appl. Phys. Lett. 85, 6323–6325 (2004).
[CrossRef]

Grave de Peralta, L.

Guo, J.

Hameed, O.

Han, Z.

Hattori, H. T.

He, S.

Heikal, A. M.

Hochberg, M.

Holmgaard, T.

T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75, 245405 (2007).
[CrossRef]

Holtz, M.

I, J.-S.

C.-O Cho, Y.-G. Roh, Y. Park, J.-S. I, H. Jeon, B.-S. Lee, H.-W. Kim, Y.-H. Choe, M. Sung, and J. C. Woo, “Towards nano-waveguides,” Appl. Phys. 4, 245–249 (2004).
[CrossRef]

Jeon, H.

C.-O Cho, Y.-G. Roh, Y. Park, J.-S. I, H. Jeon, B.-S. Lee, H.-W. Kim, Y.-H. Choe, M. Sung, and J. C. Woo, “Towards nano-waveguides,” Appl. Phys. 4, 245–249 (2004).
[CrossRef]

Joannopoulus, J. D.

J. D. Joannopoulus and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 1995).

Ju, J. J.

Kalinchenko, G. A.

G. A. Kalinchenko and A. M. Lerer, “Wideband all-dielectric diffraction grating on chirped mirror,” J. Lightwave Technol. 28, 2743–2749 (2010).
[CrossRef]

G. A. Kalinchenko and A. M. Lerer, “Investigations of dielectric gratings using electrodynamic models based on volume integral equations,” J. Commun. Technol. Electron. 48, 1221–1227 (2003).

Kim, H.-W.

C.-O Cho, Y.-G. Roh, Y. Park, J.-S. I, H. Jeon, B.-S. Lee, H.-W. Kim, Y.-H. Choe, M. Sung, and J. C. Woo, “Towards nano-waveguides,” Appl. Phys. 4, 245–249 (2004).
[CrossRef]

Kim, M.-S.

Kim, T.

Krasavin, A. V.

A. V. Krasavin and A. V. Zayats, “Three-dimensional numerical modeling of photonic integration with dielectric-loaded SPP waveguides,” Phys. Rev. B 78, 045425 (2008).
[CrossRef]

Krishnan, A.

Kuznetsov, V. A.

V. A. Kuznetsov and A. M. Lerer, “Dispersion characteristics of dielectric waveguides on substrates,” Radio Eng. Electron. Phys. 29, 53–58 (1984).

V. A. Kuznetsov and A. M. Lerer, “Dispersion characteristics of rectangular dielectric waveguide,” Radio Eng. Electron. Phys. 27, 24–27 (1982).

Lasser, T.

Lee, B.-S.

C.-O Cho, Y.-G. Roh, Y. Park, J.-S. I, H. Jeon, B.-S. Lee, H.-W. Kim, Y.-H. Choe, M. Sung, and J. C. Woo, “Towards nano-waveguides,” Appl. Phys. 4, 245–249 (2004).
[CrossRef]

Lee, M.-H.

Lerer, A. M.

A. M. Lerer, “Theoretical investigation of 2D periodic nanoplasmon structures,” J. Commun. Technol. Electron. 57, 1151–1159 (2012).
[CrossRef]

E. V. Golovacheva, A. M. Lerer, and N. G. Parkhomenko, “Diffraction of electromagnetic waves of optical range on a metallic nanovibrator,” Moscow Univ. Phys. Bull. 66, 5–11 (2011).
[CrossRef]

G. A. Kalinchenko and A. M. Lerer, “Wideband all-dielectric diffraction grating on chirped mirror,” J. Lightwave Technol. 28, 2743–2749 (2010).
[CrossRef]

G. A. Kalinchenko and A. M. Lerer, “Investigations of dielectric gratings using electrodynamic models based on volume integral equations,” J. Commun. Technol. Electron. 48, 1221–1227 (2003).

V. A. Kuznetsov and A. M. Lerer, “Dispersion characteristics of dielectric waveguides on substrates,” Radio Eng. Electron. Phys. 29, 53–58 (1984).

V. A. Kuznetsov and A. M. Lerer, “Dispersion characteristics of rectangular dielectric waveguide,” Radio Eng. Electron. Phys. 27, 24–27 (1982).

Liu, L.

Maradudin, A. A.

I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66, 035403 (2002).
[CrossRef]

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Norstand Reinhold, 1972).

Markey, L.

Martin, O.

Martín-Moreno, L.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef]

Massenot, S.

Meade, R. D.

J. D. Joannopoulus and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 1995).

Mock, J.

Moreno, E.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef]

Novikov, I. V.

I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66, 035403 (2002).
[CrossRef]

Obayya, S. S. A.

Orenstein, M.

D. Arbel and M. Orenstein, “W-shaped plasmon waveguide for silicon based plasmonic modulator,” in LEOS Annual Meeting (2006), pp. 262–263.

Pannipitiya, A.

Park, S.

Park, S. K.

Park, Y.

C.-O Cho, Y.-G. Roh, Y. Park, J.-S. I, H. Jeon, B.-S. Lee, H.-W. Kim, Y.-H. Choe, M. Sung, and J. C. Woo, “Towards nano-waveguides,” Appl. Phys. 4, 245–249 (2004).
[CrossRef]

Parkhomenko, N. G.

E. V. Golovacheva, A. M. Lerer, and N. G. Parkhomenko, “Diffraction of electromagnetic waves of optical range on a metallic nanovibrator,” Moscow Univ. Phys. Bull. 66, 5–11 (2011).
[CrossRef]

Pile, D. F. P.

D. K. Gramotnev and D. F. P. Pile, “Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface,” Appl. Phys. Lett. 85, 6323–6325 (2004).
[CrossRef]

Premaratne, M.

Rodrigo, S. G.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[CrossRef]

Roh, Y.-G.

C.-O Cho, Y.-G. Roh, Y. Park, J.-S. I, H. Jeon, B.-S. Lee, H.-W. Kim, Y.-H. Choe, M. Sung, and J. C. Woo, “Towards nano-waveguides,” Appl. Phys. 4, 245–249 (2004).
[CrossRef]

Rukhlenko, I. D.

Scherer, A.

Selker, M. D.

Smith, D.

Smith, D. R.

Stockman, M. I.

Sung, M.

C.-O Cho, Y.-G. Roh, Y. Park, J.-S. I, H. Jeon, B.-S. Lee, H.-W. Kim, Y.-H. Choe, M. Sung, and J. C. Woo, “Towards nano-waveguides,” Appl. Phys. 4, 245–249 (2004).
[CrossRef]

Volkov, V. S.

J. Gosciniak, V. S. Volkov, S. I. Bozhevolnyi, L. Markey, S. Massenot, and A. Dereux, “Fiber-coupled dielectric-loaded plasmonic waveguides,” Opt. Express 18, 5314–5319 (2010).
[CrossRef]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef]

Walker, C.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1968).

Woo, J. C.

C.-O Cho, Y.-G. Roh, Y. Park, J.-S. I, H. Jeon, B.-S. Lee, H.-W. Kim, Y.-H. Choe, M. Sung, and J. C. Woo, “Towards nano-waveguides,” Appl. Phys. 4, 245–249 (2004).
[CrossRef]

Yablonovich, E.

E. Yablonovich, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef]

Zayats, A. V.

A. V. Krasavin and A. V. Zayats, “Three-dimensional numerical modeling of photonic integration with dielectric-loaded SPP waveguides,” Phys. Rev. B 78, 045425 (2008).
[CrossRef]

Zia, R.

Appl. Phys.

C.-O Cho, Y.-G. Roh, Y. Park, J.-S. I, H. Jeon, B.-S. Lee, H.-W. Kim, Y.-H. Choe, M. Sung, and J. C. Woo, “Towards nano-waveguides,” Appl. Phys. 4, 245–249 (2004).
[CrossRef]

Appl. Phys. Lett.

D. K. Gramotnev and D. F. P. Pile, “Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface,” Appl. Phys. Lett. 85, 6323–6325 (2004).
[CrossRef]

J. Commun. Technol. Electron.

A. M. Lerer, “Theoretical investigation of 2D periodic nanoplasmon structures,” J. Commun. Technol. Electron. 57, 1151–1159 (2012).
[CrossRef]

G. A. Kalinchenko and A. M. Lerer, “Investigations of dielectric gratings using electrodynamic models based on volume integral equations,” J. Commun. Technol. Electron. 48, 1221–1227 (2003).

J. Lightwave Technol.

J. Opt. Soc. Am. A

Moscow Univ. Phys. Bull.

E. V. Golovacheva, A. M. Lerer, and N. G. Parkhomenko, “Diffraction of electromagnetic waves of optical range on a metallic nanovibrator,” Moscow Univ. Phys. Bull. 66, 5–11 (2011).
[CrossRef]

Opt. Express

L. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express 13, 6645–6650 (2005).
[CrossRef]

A. Degiron and D. R. Smith, “Numerical simulations of long-range plasmons,” Opt. Express 14, 1611–1625 (2006).
[CrossRef]

R. Buckley and P. Berini, “Figures of merit for 2D surface plasmon waveguides and application to metal stripes,” Opt. Express 15, 12174–12182 (2007).
[CrossRef]

J. Guo and R. Adato, “Control of 2D plasmon-polariton mode with dielectric nanolayers,” Opt. Express 16, 1232–1237 (2008).
[CrossRef]

T. Kim, J. J. Ju, S. Park, M.-S. Kim, S. K. Park, and M.-H. Lee, “Chip-to-chip optical interconnect using gold long-range surface plasmon polariton waveguides,” Opt. Express 16, 13133–13138 (2008).
[CrossRef]

J. Gosciniak, V. S. Volkov, S. I. Bozhevolnyi, L. Markey, S. Massenot, and A. Dereux, “Fiber-coupled dielectric-loaded plasmonic waveguides,” Opt. Express 18, 5314–5319 (2010).
[CrossRef]

A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express 18, 6191–6204 (2010).
[CrossRef]

M. Hochberg, T. Baehr-Jones, C. Walker, and A. Scherer, “Integrated plasmon and dielectric waveguides,” Opt. Express 12, 5481–5486 (2004).
[CrossRef]

C. Dellagiacoma, T. Lasser, O. Martin, A. Degiron, J. Mock, and D. Smith, “Simulation of complex plasmonic circuits including bends,” Opt. Express 19, 18979–18988 (2011).
[CrossRef]

M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express 19, 22029–22106 (2011).
[CrossRef]

Phys. Rev. B

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000).
[CrossRef]

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Figures (8)

Fig. 1.
Fig. 1.

Structures under consideration.

Fig. 2.
Fig. 2.

Solid curves represent dispersion characteristics and effective propagation length for the metal waveguide shown in the inset. The red curves correspond to b = 20 nm , black to 15 nm. The dashed curves depict analytical solutions for thin-film waveguides.

Fig. 3.
Fig. 3.

Dispersion characteristics and effective propagation length for the metal waveguide shown in the inset. The red curves correspond to W = 500 nm ; green, 900 nm; and black, infinity.

Fig. 4.
Fig. 4.

Dispersion characteristics and effective propagation length for the metal waveguide shown in the inset. All dimensions are in nanometers.

Fig. 5.
Fig. 5.

Dispersion characteristics for the metal waveguide shown in the inset.

Fig. 6.
Fig. 6.

Dispersion characteristics of waves propagating at different angles to the axis x in all-dielectric PC [Fig. 1(c)]. Black solid curves correspond to φ = 0 ° , green to φ = 10 ° , red to φ = 12 ° , and blue to φ = 14 ° . The dashed curves depict the result for φ = 0 ° obtained by Ansoft HFSS commercial software. All dimensions are in nanometers.

Fig. 7.
Fig. 7.

Normalized losses (top) and dispersion characteristics (bottom) for PC made of perforated silver film placed over a dielectric substrate [Fig. 1(c)]. Waves propagate at the angle φ = 0 ° . Green symbols correspond to zero harmonics, blue to 1 st harmonics. Red solid curve corresponds to nonperforated film.

Fig. 8.
Fig. 8.

Dispersion characteristics for PC made of silver cylinders placed on a two-layer dielectric structure [Fig. 1(d)]. The dielectric layer thickness is b = 100 nm on the upper graph and b = 150 nm on the lower graph. The red symbols refer to cylinders of 70 nm diameter, black to 90 nm. A wave propagates at the angle φ = 0 ° .

Tables (3)

Tables Icon

Table 1. Complex Refractive Index Obtained with Eqs. (5) and (6) for E-Wave Propagating on the Boundary of Half-Infinite Silver and Dielectric Layers

Tables Icon

Table 2. Complex Refractive Index Obtained with Eqs. (5) and (6) for E-Wave Propagating in Vacuum-Silver Film-Dielectric Structure

Tables Icon

Table 3. Effective Refractive Index and Effective Propagation Length Obtained with Volume Integral Method and Full-Vectorial Finite Difference Method for Linear Oblique and Curved Interfaces for E-Wave Propagating in Rectangular Gold Groove

Equations (16)

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D r ( x , y , z ) τ = p = q = s = 1 3 V exp [ i ( α p x ¯ + β q y ¯ ) ] g ˜ r s ( z , z ) D s ( z ) d v , r = 1 , 2 , 3 , x , y , z V ,
x ¯ = x x , y ¯ = y y , D r ( x , y , z ) = E r ( x , y , z ) τ ( x , y , z ) , τ ( x , y , z ) = ε b ( x , y , z ) / ε n ( z ) 1 ,
α p = 2 p π d x + k n cos φ , β p = 2 q π d y + k n sin φ ,
D r ( x , y , z ) = l = N φ N φ m = 1 N r X lmn r V lmn ( x , y , z ) ,
V lmn ( x , y , z ) = exp ( i l φ ) J l ( ζ m ( l ) r ( z ) ) Z n ( z ) ,
x = a x ( z ) r sin φ , y = a y ( z ) r cos φ .
ρ p q = ( a x ( z ) α p ) 2 + ( a y ( z ) β p ) 2 > R max 1 ,
D r ( y , z ) = μ = 0 N y 1 v = 0 N z X μ v r V μ v ( y , z ) ,
V μ v ( y , z ) = Y μ v ( y ) σ v ( 1 ) ( z ) , Y μ v ( y ) = C μ v P μ ( y y ¯ v l v ) ,
Y ˜ μ v ( β q ) = ( i ) μ J μ + 1 / 2 ( β q l v ) ( β q l v ) 1 / 2 exp ( i β q y ¯ v ) .
β ( ε i ε ) β ( ε ) i ε β ( ε ) ( ε ) 2 β ( ε ) / 2 .
ε β ( ε ) β ( ε ) β ( ε ε ) , ( ε ) 2 β ( ε ) β ( ε + ε ) + β ( ε ε ) 2 β ( ε ) .
β ( ε i ε ) β ( ε ) i [ β ( ε ) β ( ε ε ) ] [ β ( ε + ε ) + β ( ε ε ) 2 β ( ε ) ] .
n = β k = n 1 2 n Ag 2 n 1 2 + n Ag 2 ,
L 0 = 1 Im ( β ) = λ 2 π n .
d n cos φ / λ = 0.5 ,

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