Abstract

We characterize the modes with complex wavenumber for both longitudinal and transverse polarization states (with respect to the mode traveling direction) in three dimensional (3D) periodic arrays of plasmonic nanospheres, including metal losses. The Ewald representation of the required dyadic periodic Green’s function to represent the field in 3D periodic arrays is derived from the scalar case, which can be analytically continued into the complex wavenumber space. We observe the presence of one longitudinal mode and two transverse modes, one forward and one backward. Despite the presence of two modes for transverse polarization, we notice that the forward one is “dominant” (i.e., it contributes most to the field in the array). Therefore, in case of transverse polarization, we describe the composite material in terms of a homogenized effective refractive index, comparing results from (i) modal analysis, (ii) Maxwell Garnett theory, (iii) Nicolson-Ross-Weir retrieval method from scattering parameters for finite thickness structures (considering different thicknesses, showing consistency of results), and (iv) the fitting of the fields obtained through HFSS simulations. The agreement among the different methods justifies the performed homogenization procedure in case of transverse polarization.

© 2011 OSA

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  38. P. P. Ewald, “Die Berechnung optischer und elektrostatischer Gitterpotentiale,” Ann. Phys. (Berlin) 369(3), 253–287 (1921).
  39. K. E. Jordan, G. R. Richter, and P. Sheng, “An efficient numerical evaluation of the Green's function for the Helmholtz operator on periodic structures,” J. Comput. Phys. 63(1), 222–235 (1986).
  40. A. Kustepeli and A. Q. Martin, “On the splitting parameter in the Ewald method,” IEEE Microw. Guid. Wave Lett. 10(5), 168–170 (2000).

2011

2009

2008

R. Sainidou and G. F. de Abajo, “Plasmon guided modes in nanoparticle metamaterials,” Opt. Express 16(7), 4499–4506 (2008).

G. Lovat, P. Burghignoli, and R. Araneo, “Efficient evaluation of the 3-D periodic Green's function through the Ewald method,” IEEE Trans. Microw. Theory Tech. 56(9), 2069–2075 (2008).

2007

I. Stevanoviæ and J. R. Mosig, “Periodic Green's function for skewed 3-D lattices using the Ewald transformation,” Microw. Opt. Technol. Lett. 49(6), 1353–1357 (2007).

R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers,” Radio Sci. 42(6), RS6S21 (2007).

A. Alù and N. Engheta, “Three-dimensional nanotransmission lines at optical frequencies: A recipe for broadband negative-refraction optical metamaterials,” Phys. Rev. B 75(2), 024304 (2007).

M. G. Silveirinha, “Generalized Lorentz-Lorenz formulas for microstructured materials,” Phys. Rev. B 76(24), 245117 (2007).

2006

A. Alù and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B 74(20), 205436 (2006).

A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express 14(4), 1557–1567 (2006).

2004

V. A. Markel, V. N. Pustovit, S. V. Karpov, A. V. Obuschenko, V. S. Gerasimov, and I. L. Isaev, “Electromagnetic density of states and absorption of radiation by aggregates of nanospheres with multipole interactions,” Phys. Rev. B 70(5), 054202 (2004).

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1 Pt 2), 016608 (2004).

2002

D. R. Smith, S. Schultz, P. Markoscaron, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).

2000

A. Kustepeli and A. Q. Martin, “On the splitting parameter in the Ewald method,” IEEE Microw. Guid. Wave Lett. 10(5), 168–170 (2000).

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62(23), 15299–15302 (2000).

1998

P. Myun-Joo, P. Jongkuk, and N. Sangwook, “Efficient calculation of the Green's function for the rectangular cavity,” IEEE Microw. Guid. Wave Lett. 8(3), 124–126 (1998).

A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37(22), 5271–5283 (1998).

1997

A. H. Boughriet, C. Legrand, and A. Chapoton, “Noniterative stable transmission/reflection method for low-loss material complex permittivity determination,” IEEE Trans. Microw. Theory Tech. 45(1), 52–57 (1997).

1994

1986

K. E. Jordan, G. R. Richter, and P. Sheng, “An efficient numerical evaluation of the Green's function for the Helmholtz operator on periodic structures,” J. Comput. Phys. 63(1), 222–235 (1986).

1974

W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE 62(1), 33–36 (1974).

1972

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).

1970

A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas. 19(4), 377–382 (1970).

1921

P. P. Ewald, “Die Berechnung optischer und elektrostatischer Gitterpotentiale,” Ann. Phys. (Berlin) 369(3), 253–287 (1921).

Albani, M.

Alitalo, P.

S. Steshenko, F. Capolino, P. Alitalo, and S. Tretyakov, “Effective model and investigation of the near-field enhancement and subwavelength imaging properties of multilayer arrays of plasmonic nanospheres,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(1 Pt 2), 016607 (2011).

Alù, A.

A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84(7), 075153 (2011).

A. Alù and N. Engheta, “Three-dimensional nanotransmission lines at optical frequencies: A recipe for broadband negative-refraction optical metamaterials,” Phys. Rev. B 75(2), 024304 (2007).

A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express 14(4), 1557–1567 (2006).

A. Alù and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B 74(20), 205436 (2006).

Araneo, R.

G. Lovat, P. Burghignoli, and R. Araneo, “Efficient evaluation of the 3-D periodic Green's function through the Ewald method,” IEEE Trans. Microw. Theory Tech. 56(9), 2069–2075 (2008).

Biswas, R.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62(23), 15299–15302 (2000).

Boughriet, A. H.

A. H. Boughriet, C. Legrand, and A. Chapoton, “Noniterative stable transmission/reflection method for low-loss material complex permittivity determination,” IEEE Trans. Microw. Theory Tech. 45(1), 52–57 (1997).

Burghignoli, P.

G. Lovat, P. Burghignoli, and R. Araneo, “Efficient evaluation of the 3-D periodic Green's function through the Ewald method,” IEEE Trans. Microw. Theory Tech. 56(9), 2069–2075 (2008).

Campione, S.

Capolino, F.

Chapoton, A.

A. H. Boughriet, C. Legrand, and A. Chapoton, “Noniterative stable transmission/reflection method for low-loss material complex permittivity determination,” IEEE Trans. Microw. Theory Tech. 45(1), 52–57 (1997).

Chen, X.

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1 Pt 2), 016608 (2004).

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).

de Abajo, G. F.

Djurisic, A. B.

Elazar, J. M.

El-Kady, I.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62(23), 15299–15302 (2000).

Engheta, N.

A. Alù and N. Engheta, “Three-dimensional nanotransmission lines at optical frequencies: A recipe for broadband negative-refraction optical metamaterials,” Phys. Rev. B 75(2), 024304 (2007).

A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express 14(4), 1557–1567 (2006).

A. Alù and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B 74(20), 205436 (2006).

Ewald, P. P.

P. P. Ewald, “Die Berechnung optischer und elektrostatischer Gitterpotentiale,” Ann. Phys. (Berlin) 369(3), 253–287 (1921).

Fainman, Y.

Fructos, A. L.

Gerasimov, V. S.

V. A. Markel, V. N. Pustovit, S. V. Karpov, A. V. Obuschenko, V. S. Gerasimov, and I. L. Isaev, “Electromagnetic density of states and absorption of radiation by aggregates of nanospheres with multipole interactions,” Phys. Rev. B 70(5), 054202 (2004).

Grzegorczyk, T. M.

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1 Pt 2), 016608 (2004).

Ho, K. M.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62(23), 15299–15302 (2000).

Isaev, I. L.

V. A. Markel, V. N. Pustovit, S. V. Karpov, A. V. Obuschenko, V. S. Gerasimov, and I. L. Isaev, “Electromagnetic density of states and absorption of radiation by aggregates of nanospheres with multipole interactions,” Phys. Rev. B 70(5), 054202 (2004).

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).

Jongkuk, P.

P. Myun-Joo, P. Jongkuk, and N. Sangwook, “Efficient calculation of the Green's function for the rectangular cavity,” IEEE Microw. Guid. Wave Lett. 8(3), 124–126 (1998).

Jordan, K. E.

K. E. Jordan, G. R. Richter, and P. Sheng, “An efficient numerical evaluation of the Green's function for the Helmholtz operator on periodic structures,” J. Comput. Phys. 63(1), 222–235 (1986).

Karpov, S. V.

V. A. Markel, V. N. Pustovit, S. V. Karpov, A. V. Obuschenko, V. S. Gerasimov, and I. L. Isaev, “Electromagnetic density of states and absorption of radiation by aggregates of nanospheres with multipole interactions,” Phys. Rev. B 70(5), 054202 (2004).

Kong, J. A.

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1 Pt 2), 016608 (2004).

Kustepeli, A.

A. Kustepeli and A. Q. Martin, “On the splitting parameter in the Ewald method,” IEEE Microw. Guid. Wave Lett. 10(5), 168–170 (2000).

Legrand, C.

A. H. Boughriet, C. Legrand, and A. Chapoton, “Noniterative stable transmission/reflection method for low-loss material complex permittivity determination,” IEEE Trans. Microw. Theory Tech. 45(1), 52–57 (1997).

Lomakin, V.

Lovat, G.

G. Lovat, P. Burghignoli, and R. Araneo, “Efficient evaluation of the 3-D periodic Green's function through the Ewald method,” IEEE Trans. Microw. Theory Tech. 56(9), 2069–2075 (2008).

Mackowski, D. W.

Majewski, M. L.

Markel, V. A.

V. A. Markel, V. N. Pustovit, S. V. Karpov, A. V. Obuschenko, V. S. Gerasimov, and I. L. Isaev, “Electromagnetic density of states and absorption of radiation by aggregates of nanospheres with multipole interactions,” Phys. Rev. B 70(5), 054202 (2004).

Markoscaron, P.

D. R. Smith, S. Schultz, P. Markoscaron, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).

Martin, A. Q.

A. Kustepeli and A. Q. Martin, “On the splitting parameter in the Ewald method,” IEEE Microw. Guid. Wave Lett. 10(5), 168–170 (2000).

Mesa, F.

Mosig, J. R.

I. Stevanoviæ and J. R. Mosig, “Periodic Green's function for skewed 3-D lattices using the Ewald transformation,” Microw. Opt. Technol. Lett. 49(6), 1353–1357 (2007).

Myun-Joo, P.

P. Myun-Joo, P. Jongkuk, and N. Sangwook, “Efficient calculation of the Green's function for the rectangular cavity,” IEEE Microw. Guid. Wave Lett. 8(3), 124–126 (1998).

Nicolson, A. M.

A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas. 19(4), 377–382 (1970).

Obuschenko, A. V.

V. A. Markel, V. N. Pustovit, S. V. Karpov, A. V. Obuschenko, V. S. Gerasimov, and I. L. Isaev, “Electromagnetic density of states and absorption of radiation by aggregates of nanospheres with multipole interactions,” Phys. Rev. B 70(5), 054202 (2004).

Pacheco, J.

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1 Pt 2), 016608 (2004).

Pustovit, V. N.

V. A. Markel, V. N. Pustovit, S. V. Karpov, A. V. Obuschenko, V. S. Gerasimov, and I. L. Isaev, “Electromagnetic density of states and absorption of radiation by aggregates of nanospheres with multipole interactions,” Phys. Rev. B 70(5), 054202 (2004).

Rakic, A. D.

Richter, G. R.

K. E. Jordan, G. R. Richter, and P. Sheng, “An efficient numerical evaluation of the Green's function for the Helmholtz operator on periodic structures,” J. Comput. Phys. 63(1), 222–235 (1986).

Ross, G. F.

A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas. 19(4), 377–382 (1970).

Sainidou, R.

Salandrino, A.

Sangwook, N.

P. Myun-Joo, P. Jongkuk, and N. Sangwook, “Efficient calculation of the Green's function for the rectangular cavity,” IEEE Microw. Guid. Wave Lett. 8(3), 124–126 (1998).

Schultz, S.

D. R. Smith, S. Schultz, P. Markoscaron, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).

Sheng, P.

K. E. Jordan, G. R. Richter, and P. Sheng, “An efficient numerical evaluation of the Green's function for the Helmholtz operator on periodic structures,” J. Comput. Phys. 63(1), 222–235 (1986).

Shore, R. A.

R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers,” Radio Sci. 42(6), RS6S21 (2007).

Sigalas, M. M.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62(23), 15299–15302 (2000).

Silveirinha, M. G.

M. G. Silveirinha, “Generalized Lorentz-Lorenz formulas for microstructured materials,” Phys. Rev. B 76(24), 245117 (2007).

Smith, D. R.

D. R. Smith, S. Schultz, P. Markoscaron, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).

Soukoulis, C. M.

D. R. Smith, S. Schultz, P. Markoscaron, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62(23), 15299–15302 (2000).

Steshenko, S.

S. Campione, S. Steshenko, and F. Capolino, “Complex bound and leaky modes in chains of plasmonic nanospheres,” Opt. Express 19(19), 18345–18363 (2011).

S. Steshenko, F. Capolino, P. Alitalo, and S. Tretyakov, “Effective model and investigation of the near-field enhancement and subwavelength imaging properties of multilayer arrays of plasmonic nanospheres,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(1 Pt 2), 016607 (2011).

Stevanoviæ, I.

I. Stevanoviæ and J. R. Mosig, “Periodic Green's function for skewed 3-D lattices using the Ewald transformation,” Microw. Opt. Technol. Lett. 49(6), 1353–1357 (2007).

Tretyakov, S.

S. Steshenko, F. Capolino, P. Alitalo, and S. Tretyakov, “Effective model and investigation of the near-field enhancement and subwavelength imaging properties of multilayer arrays of plasmonic nanospheres,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(1 Pt 2), 016607 (2011).

Vallecchi, A.

Van Orden, D.

Weir, W. B.

W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE 62(1), 33–36 (1974).

Wu, B.-I.

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1 Pt 2), 016608 (2004).

Yaghjian, A. D.

R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers,” Radio Sci. 42(6), RS6S21 (2007).

Ann. Phys. (Berlin)

P. P. Ewald, “Die Berechnung optischer und elektrostatischer Gitterpotentiale,” Ann. Phys. (Berlin) 369(3), 253–287 (1921).

Appl. Opt.

IEEE Microw. Guid. Wave Lett.

A. Kustepeli and A. Q. Martin, “On the splitting parameter in the Ewald method,” IEEE Microw. Guid. Wave Lett. 10(5), 168–170 (2000).

P. Myun-Joo, P. Jongkuk, and N. Sangwook, “Efficient calculation of the Green's function for the rectangular cavity,” IEEE Microw. Guid. Wave Lett. 8(3), 124–126 (1998).

IEEE Trans. Instrum. Meas.

A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas. 19(4), 377–382 (1970).

IEEE Trans. Microw. Theory Tech.

G. Lovat, P. Burghignoli, and R. Araneo, “Efficient evaluation of the 3-D periodic Green's function through the Ewald method,” IEEE Trans. Microw. Theory Tech. 56(9), 2069–2075 (2008).

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Figures (14)

Fig. 1
Fig. 1

3D periodic array of metallic nanospheres embedded in a homogeneous medium with permittivity ε h . The radius of each nanosphere is r; and a, b and c are the periodicities along x-, y- and z-direction, respectively.

Fig. 2
Fig. 2

Dispersion diagram for T-pol. (a) Real part and (b) imaginary part of the wavenumber k z = β z +i α z , only for modes whose power flow is toward the positive z-direction, i.e., α z 0 . The black dotted curves show the behavior of each mode in the lossless case, i.e., when setting γ=0 in the silver constitutive relation.

Fig. 3
Fig. 3

Trajectories of modal wavenumbers in the complex k z plane for T-pol, with respect to (a) the z-periodicity c, and (b) the host wavenumber k . Notice that in (b), crossing the vertical black dash-dotted line at “ 1 ” and “1” means crossing the light line β z =k . Arrows indicate direction of increasing frequency. The black dotted curves in (a) show the modal wavenumber trajectories in the ideal lossless case: notice the four-quadrant symmetry in this case, which is broken when accounting for silver losses.

Fig. 4
Fig. 4

As in Fig. 2, for L-pol.

Fig. 5
Fig. 5

As in Fig. 3, for L-pol.

Fig. 6
Fig. 6

Comparison between the wavenumbers k z = β z +i α z of the Mode 1, T-pol, in the three structures in Table 1 versus frequency, normalized to their respective z-periodicity c. (a) Real part and (b) imaginary part.

Fig. 7
Fig. 7

As in Fig. 6, for L-pol.

Fig. 8
Fig. 8

Comparison between the figure of merit F of the Mode 1 in the three structures in Table 1 versus frequency. (a) Transversal and (b) longitudinal polarization.

Fig. 9
Fig. 9

Magnitude of transmission, reflection and absorption coefficients for a stack of 10 layers of Structure I (only a transverse cut in the xz plane of a 3D array is shown). Results obtained by using the SDA and HFSS are in good agreement.

Fig. 10
Fig. 10

Effective refractive index retrieved by three different methods: Mode analysis n eff = k z / k 0 , Maxwell Garnett (MG) using Mie polarizability in Eq. (2), and the NRW method in Eq. (8) from HFSS simulations of 10 layers.

Fig. 11
Fig. 11

Effective refractive index retrieved by using NRW from HFSS for different number of layers (4, 6 and 10), and by mode analysis.

Fig. 12
Fig. 12

As in Fig. 11, but using R and T computed through SDA.

Fig. 13
Fig. 13

Field in 10 layers of plasmonic nanospheres. Comparison between the HFSS full-wave field (Simulation) and the fitting result at 745 THz (i.e., large Re[ n eff ] and low Im[ n eff ] ).

Fig. 14
Fig. 14

As in Fig. 13, at 875 THz (i.e., low Re[ n eff ] and low Im[ n eff ] ).

Tables (1)

Tables Icon

Table 1 3D Periodic Array Parameters

Equations (19)

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p= α ee E loc ,
α ee = 6πi ε h ε 0 k 3 a 1 = 6πi ε h ε 0 k 3 m r ψ 1 ( m r kr ) ψ 1 ( kr ) ψ 1 ( kr ) ψ 1 ( m r kr ) m r ψ 1 ( m r kr ) ξ 1 ( kr ) ξ 1 ( kr ) ψ 1 ( m r kr ) ,
E 0 loc ( r 0 , k B )= E inc ( r 0 )+ G ¯ ( r 0 , r 0 , k B ) p 0 ,
G ¯ ( r, r 0 , k B )= n G ¯ ( r, r n ) e i k B d n
G ¯ ( r, r )= e ikR 4π ε h ε 0 [ ( k 2 R + ik R 2 1 R 3 ) I ¯ ( k 2 R + 3ik R 2 3 R 3 ) R ^ R ^ ],
A ¯ ( k B ) p 0 = α ee E inc ( r 0 ), A ¯ ( k B )= I ¯ α ee G ¯ ( r 0 , r 0 , k B ).
F= β z α z .
n eff =± cos 1 ( 1 R 2 + T 2 2T ) k 0 t + 2πq k 0 t ,
E y ( n )= E + e i k z ( n 1 2 )c + E e i k z ( n 1 2 )c ,
G ¯ ( r, r 0 , k B )= 1 ε h ε 0 [ k 2 I ¯ + ] G ( r, r 0 , k B ),
G ( r, r 0 , k B )= 1 4π n( 0,0,0 ) e ik R n R n e i k B d n ,
G ¯ ( r, r 0 , k B )= G ¯ spectral ( r, r 0 , k B )+ G ¯ spatial ( r, r 0 , k B )
G spectral ( r 0 , r 0 , k B )= 1 abc n e γ n 2 4 E 2 γ n 2 ,
G spatial ( r 0 , r 0 , k B )= 1 8π n( 0,0,0 ) e i k B d n R n f( R n ) + f ( 0 )2ik 8π ,
f( R n )= e ik R n erfc( β )+ e +ik R n erfc( β + ).
E= [ π 2 ( 1/ a 2 +1/ b 2 +1/ c 2 ) a 2 + b 2 + c 2 ] 1/4 .
G spectral ( r 0 , r 0 , k B )= 1 abc n ( k B + k n )( k B + k n ) e γ n 2 4 E 2 γ n 2 ,
G spatial ( r 0 , r 0 , k B )= 1 8π n( 0,0,0 ) e i k B d n F ¯ spatial,n + 1 8π f ( 0 )+2i k 3 3 I ¯ ,
F ¯ spatial,n =( f ( R n ) R n 2 f( R n ) R n 3 ) I ¯ +( f ( R n ) R n 3 f ( R n ) R n 2 + 3f( R n ) R n 3 ) R ^ n R ^ n ,

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