Abstract

The photonic band structure of a linear array of metallic nanocylinders is calculated using the embedding method. The coupling to the vacuum on either side of the array is treated exactly, allowing the continuum states and plasmon broadening above the light-line to be treated accurately. In addition to the plasmon bands, which broaden at larger cylinder radius, there are two guided modes, with the character of surface plasmon polaritons. These split off the light-line at small wave-vector, becoming almost dispersionless as they enter the plasmon bands. The electric fields associated with the modes are calculated, and their symmetries are discussed.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B 50, 16835–16844 (1994).
    [CrossRef]
  5. T. Ito and K. Sakoda, “Photonic bands of metallic systems. II. Features of surface plasmon polaritons,” Phys. Rev. B 64, 045117 (2001).
    [CrossRef]
  6. A. L. Pokrovsky and A. L. Efros, “Nonlocal electrodynamics of two-dimensional wire mesh photonic crystals,” Phys. Rev. B 65, 045110 (2002).
    [CrossRef]
  7. J. M. Pitarke, J. E. Inglesfield, and N. Giannakis, “Surface-plasmon polaritons in a lattice of metal cylinders,” Phys. Rev. B 75, 165415 (2007).
    [CrossRef]
  8. G. Gantzounis and N. Stefanou, “Propagation of electromagnetic waves through microstructured polar materials,” Phys. Rev. B 75, 193102 (2006).
    [CrossRef]
  9. S. Fan, J. D. Joannopoulos, J. Winn, A. Devenyi, J. C. Chen, and R. D. Meade, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B 12, 1267–1272 (1995).
    [CrossRef]
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    [CrossRef]
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  15. G. Veronis and S. Fan, “Overview of simulation techniques for plasmonic devices,” in Surface Plasmon Nanophotonics, M. L. Brongersma and P. G. Kik, eds. (Springer, 2007), pp. 169–182.
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    [CrossRef]
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    [CrossRef]
  18. J. E. Inglesfield, “A method of embedding,” J. Phys. C 14, 3795–3806 (1981).
    [CrossRef]
  19. J. E. Inglesfield, J. M. Pitarke, and R. Kemp, “Plasmon bands in metallic nanostructures,” Phys. Rev. B 69, 233103 (2004).
    [CrossRef]
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    [CrossRef]
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  22. J. E. Inglesfield, “Embedding at surfaces,” Comp. Phys. Commun. 137, 89–107 (2001).
    [CrossRef]
  23. J. D. Jackson, Classical Electrodynamics (Wiley, 1962).
  24. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders, 1976).
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    [CrossRef]
  26. D. Pines, Elementary Excitations in Solids (Benjamin, 1964).
  27. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

2011

2009

2007

Y. Zhao and Y. Hao, “Finite-difference time-domain study of guided modes in nano-plasmonic waveguides,” IEEE Trans. Antennas Propag. 55, 3070–3077 (2007).
[CrossRef]

D. R. Matthews, H. D. Summers, K. Njoh, S. Chappell, R. Errington, and P. Smith, “Optical antenna arrays in the visible range,” Opt. Express 15, 3478–3487 (2007).
[CrossRef]

J. M. Pitarke, J. E. Inglesfield, and N. Giannakis, “Surface-plasmon polaritons in a lattice of metal cylinders,” Phys. Rev. B 75, 165415 (2007).
[CrossRef]

2006

G. Gantzounis and N. Stefanou, “Propagation of electromagnetic waves through microstructured polar materials,” Phys. Rev. B 75, 193102 (2006).
[CrossRef]

2005

C. Girard, “Near fields in nanostructures,” Rep. Prog. Phys. 68, 1883–1933 (2005).
[CrossRef]

2004

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticles,” Phys. Rev. B 70, 125429 (2004).
[CrossRef]

J. E. Inglesfield, J. M. Pitarke, and R. Kemp, “Plasmon bands in metallic nanostructures,” Phys. Rev. B 69, 233103 (2004).
[CrossRef]

2003

2002

F. J. García-Vidal and L. Martín-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66, 155412 (2002).
[CrossRef]

R. Kemp and J. E. Inglesfield, “Embedding approach for rapid convergence of plane waves in photonic calculations,” Phys. Rev. B 65, 115103 (2002).
[CrossRef]

A. L. Pokrovsky and A. L. Efros, “Nonlocal electrodynamics of two-dimensional wire mesh photonic crystals,” Phys. Rev. B 65, 045110 (2002).
[CrossRef]

S. A. Maier, M. L. Brongersma, P. G. Kik, and H. A. Atwater, “Observation of near-field coupling in metal nanoparticle chains using far-field polarization spectroscopy,” Phys. Rev. B 65, 193408 (2002).
[CrossRef]

2001

T. Ito and K. Sakoda, “Photonic bands of metallic systems. II. Features of surface plasmon polaritons,” Phys. Rev. B 64, 045117 (2001).
[CrossRef]

J. E. Inglesfield, “Embedding at surfaces,” Comp. Phys. Commun. 137, 89–107 (2001).
[CrossRef]

1998

1995

1994

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B 50, 16835–16844 (1994).
[CrossRef]

1981

J. E. Inglesfield, “A method of embedding,” J. Phys. C 14, 3795–3806 (1981).
[CrossRef]

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders, 1976).

Atwater, H. A.

S. A. Maier, M. L. Brongersma, P. G. Kik, and H. A. Atwater, “Observation of near-field coupling in metal nanoparticle chains using far-field polarization spectroscopy,” Phys. Rev. B 65, 193408 (2002).
[CrossRef]

Aussenegg, F. R.

Brongersma, M. L.

S. A. Maier, M. L. Brongersma, P. G. Kik, and H. A. Atwater, “Observation of near-field coupling in metal nanoparticle chains using far-field polarization spectroscopy,” Phys. Rev. B 65, 193408 (2002).
[CrossRef]

Chappell, S.

Chen, J. C.

Devenyi, A.

Efros, A. L.

A. L. Pokrovsky and A. L. Efros, “Nonlocal electrodynamics of two-dimensional wire mesh photonic crystals,” Phys. Rev. B 65, 045110 (2002).
[CrossRef]

Errington, R.

Fan, S.

S. Fan, J. D. Joannopoulos, J. Winn, A. Devenyi, J. C. Chen, and R. D. Meade, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B 12, 1267–1272 (1995).
[CrossRef]

G. Veronis and S. Fan, “Overview of simulation techniques for plasmonic devices,” in Surface Plasmon Nanophotonics, M. L. Brongersma and P. G. Kik, eds. (Springer, 2007), pp. 169–182.

Ford, G. W.

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticles,” Phys. Rev. B 70, 125429 (2004).
[CrossRef]

Gantzounis, G.

G. Gantzounis and N. Stefanou, “Propagation of electromagnetic waves through microstructured polar materials,” Phys. Rev. B 75, 193102 (2006).
[CrossRef]

García-Vidal, F. J.

F. J. García-Vidal and L. Martín-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66, 155412 (2002).
[CrossRef]

Giannakis, N.

J. M. Pitarke, J. E. Inglesfield, and N. Giannakis, “Surface-plasmon polaritons in a lattice of metal cylinders,” Phys. Rev. B 75, 165415 (2007).
[CrossRef]

Girard, C.

C. Girard, “Near fields in nanostructures,” Rep. Prog. Phys. 68, 1883–1933 (2005).
[CrossRef]

Hao, Y.

Y. Zhao and Y. Hao, “Finite-difference time-domain study of guided modes in nano-plasmonic waveguides,” IEEE Trans. Antennas Propag. 55, 3070–3077 (2007).
[CrossRef]

Hochman, A.

Inglesfield, J. E.

J. M. Pitarke, J. E. Inglesfield, and N. Giannakis, “Surface-plasmon polaritons in a lattice of metal cylinders,” Phys. Rev. B 75, 165415 (2007).
[CrossRef]

J. E. Inglesfield, J. M. Pitarke, and R. Kemp, “Plasmon bands in metallic nanostructures,” Phys. Rev. B 69, 233103 (2004).
[CrossRef]

R. Kemp and J. E. Inglesfield, “Embedding approach for rapid convergence of plane waves in photonic calculations,” Phys. Rev. B 65, 115103 (2002).
[CrossRef]

J. E. Inglesfield, “Embedding at surfaces,” Comp. Phys. Commun. 137, 89–107 (2001).
[CrossRef]

J. E. Inglesfield, “The embedding method for electromagnetics,” J. Phys. A 31, 8495–8510 (1998).
[CrossRef]

J. E. Inglesfield, “A method of embedding,” J. Phys. C 14, 3795–3806 (1981).
[CrossRef]

Ito, T.

T. Ito and K. Sakoda, “Photonic bands of metallic systems. II. Features of surface plasmon polaritons,” Phys. Rev. B 64, 045117 (2001).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1962).

Joannopoulos, J. D.

Kemp, R.

J. E. Inglesfield, J. M. Pitarke, and R. Kemp, “Plasmon bands in metallic nanostructures,” Phys. Rev. B 69, 233103 (2004).
[CrossRef]

R. Kemp and J. E. Inglesfield, “Embedding approach for rapid convergence of plane waves in photonic calculations,” Phys. Rev. B 65, 115103 (2002).
[CrossRef]

Kik, P. G.

S. A. Maier, M. L. Brongersma, P. G. Kik, and H. A. Atwater, “Observation of near-field coupling in metal nanoparticle chains using far-field polarization spectroscopy,” Phys. Rev. B 65, 193408 (2002).
[CrossRef]

Krenn, J. R.

Kuzmiak, V.

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B 50, 16835–16844 (1994).
[CrossRef]

Leither, A.

Leviatan, Y.

Ludwig, A.

Maier, S. A.

S. A. Maier, M. L. Brongersma, P. G. Kik, and H. A. Atwater, “Observation of near-field coupling in metal nanoparticle chains using far-field polarization spectroscopy,” Phys. Rev. B 65, 193408 (2002).
[CrossRef]

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

Maradudin, A. A.

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B 50, 16835–16844 (1994).
[CrossRef]

Martín-Moreno, L.

F. J. García-Vidal and L. Martín-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66, 155412 (2002).
[CrossRef]

Matthews, D. R.

Meade, R. D.

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders, 1976).

Njoh, K.

Pincemin, F.

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B 50, 16835–16844 (1994).
[CrossRef]

Pines, D.

D. Pines, Elementary Excitations in Solids (Benjamin, 1964).

Pitarke, J. M.

J. M. Pitarke, J. E. Inglesfield, and N. Giannakis, “Surface-plasmon polaritons in a lattice of metal cylinders,” Phys. Rev. B 75, 165415 (2007).
[CrossRef]

J. E. Inglesfield, J. M. Pitarke, and R. Kemp, “Plasmon bands in metallic nanostructures,” Phys. Rev. B 69, 233103 (2004).
[CrossRef]

Pokrovsky, A. L.

A. L. Pokrovsky and A. L. Efros, “Nonlocal electrodynamics of two-dimensional wire mesh photonic crystals,” Phys. Rev. B 65, 045110 (2002).
[CrossRef]

Quinten, M.

Sakoda, K.

T. Ito and K. Sakoda, “Photonic bands of metallic systems. II. Features of surface plasmon polaritons,” Phys. Rev. B 64, 045117 (2001).
[CrossRef]

Senior, T. B. A.

T. B. A. Senior and J. L. Volakis, Approximate Boundary Conditions in Electromagnetics (The Institution of Electrical Engineers, 1995).

Smith, P.

Stefanou, N.

G. Gantzounis and N. Stefanou, “Propagation of electromagnetic waves through microstructured polar materials,” Phys. Rev. B 75, 193102 (2006).
[CrossRef]

Summers, H. D.

Szafranek, D.

Veronis, G.

G. Veronis and S. Fan, “Overview of simulation techniques for plasmonic devices,” in Surface Plasmon Nanophotonics, M. L. Brongersma and P. G. Kik, eds. (Springer, 2007), pp. 169–182.

Volakis, J. L.

T. B. A. Senior and J. L. Volakis, Approximate Boundary Conditions in Electromagnetics (The Institution of Electrical Engineers, 1995).

Weber, W. H.

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticles,” Phys. Rev. B 70, 125429 (2004).
[CrossRef]

Winn, J.

Zhao, Y.

Y. Zhao and Y. Hao, “Finite-difference time-domain study of guided modes in nano-plasmonic waveguides,” IEEE Trans. Antennas Propag. 55, 3070–3077 (2007).
[CrossRef]

Comp. Phys. Commun.

J. E. Inglesfield, “Embedding at surfaces,” Comp. Phys. Commun. 137, 89–107 (2001).
[CrossRef]

IEEE Trans. Antennas Propag.

Y. Zhao and Y. Hao, “Finite-difference time-domain study of guided modes in nano-plasmonic waveguides,” IEEE Trans. Antennas Propag. 55, 3070–3077 (2007).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. A

J. E. Inglesfield, “The embedding method for electromagnetics,” J. Phys. A 31, 8495–8510 (1998).
[CrossRef]

J. Phys. C

J. E. Inglesfield, “A method of embedding,” J. Phys. C 14, 3795–3806 (1981).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. B

S. A. Maier, M. L. Brongersma, P. G. Kik, and H. A. Atwater, “Observation of near-field coupling in metal nanoparticle chains using far-field polarization spectroscopy,” Phys. Rev. B 65, 193408 (2002).
[CrossRef]

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B 50, 16835–16844 (1994).
[CrossRef]

T. Ito and K. Sakoda, “Photonic bands of metallic systems. II. Features of surface plasmon polaritons,” Phys. Rev. B 64, 045117 (2001).
[CrossRef]

A. L. Pokrovsky and A. L. Efros, “Nonlocal electrodynamics of two-dimensional wire mesh photonic crystals,” Phys. Rev. B 65, 045110 (2002).
[CrossRef]

J. M. Pitarke, J. E. Inglesfield, and N. Giannakis, “Surface-plasmon polaritons in a lattice of metal cylinders,” Phys. Rev. B 75, 165415 (2007).
[CrossRef]

G. Gantzounis and N. Stefanou, “Propagation of electromagnetic waves through microstructured polar materials,” Phys. Rev. B 75, 193102 (2006).
[CrossRef]

F. J. García-Vidal and L. Martín-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66, 155412 (2002).
[CrossRef]

J. E. Inglesfield, J. M. Pitarke, and R. Kemp, “Plasmon bands in metallic nanostructures,” Phys. Rev. B 69, 233103 (2004).
[CrossRef]

R. Kemp and J. E. Inglesfield, “Embedding approach for rapid convergence of plane waves in photonic calculations,” Phys. Rev. B 65, 115103 (2002).
[CrossRef]

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticles,” Phys. Rev. B 70, 125429 (2004).
[CrossRef]

Rep. Prog. Phys.

C. Girard, “Near fields in nanostructures,” Rep. Prog. Phys. 68, 1883–1933 (2005).
[CrossRef]

Other

T. B. A. Senior and J. L. Volakis, Approximate Boundary Conditions in Electromagnetics (The Institution of Electrical Engineers, 1995).

G. Veronis and S. Fan, “Overview of simulation techniques for plasmonic devices,” in Surface Plasmon Nanophotonics, M. L. Brongersma and P. G. Kik, eds. (Springer, 2007), pp. 169–182.

D. Pines, Elementary Excitations in Solids (Benjamin, 1964).

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

J. D. Jackson, Classical Electrodynamics (Wiley, 1962).

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders, 1976).

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Figures (10)

Fig. 1.
Fig. 1.

Cross section of a portion of the array of metallic cylinders, radius ρ, separation a. The solid line outlines the unit cell, with width d. The wave equation is solved explicitly in region I, and region II, consisting of the cylinders and the external region, is replaced by embedding potentials. The plane-wave basis functions are defined in the x direction by parameter D.

Fig. 2.
Fig. 2.

Integrated spectral density n˜elI (plotted on a logarithmic scale) as a function of ω˜ at k˜=0.4 for cylinder radius ρ˜=2.0944, for mmax=4 (dot-dashed line), 8 (dashed line), and 12 (solid line).

Fig. 3.
Fig. 3.

Integrated spectral density n˜elI (plotted on a logarithmic scale) as a function of ω˜ at k˜=0.4 for cylinder radius ρ˜=3.0, for mmax=24 (dashed line), 28 (dotted line), and 30 (solid line).

Fig. 4.
Fig. 4.

Integrated spectral density plotted as a function of k˜ and ω˜ to give band structures. Left-hand figure, ρ˜=2.0944 with mmax=8, right-hand figure ρ˜=2.8 with mmax=14. The arrow indicates the surface plasmon frequency ω˜p/2, and the dashed white line shows the light-line ω˜=k˜. The thin white lines below the light line show modes on a slab of thickness equal to the cylinder diameter.

Fig. 5.
Fig. 5.

Integrated spectral density plotted as a function of k˜ and ω˜ to give band structures. Left-hand figure, ρ˜=3.0 with mmax=24, right-hand figure ρ˜=3.1 with mmax=24. The arrow indicates the surface plasmon frequency ω˜p/2, and the dashed white line shows the light line ω˜=k˜. The thin white lines below the light-line show modes on a slab of thickness equal to the cylinder diameter.

Fig. 6.
Fig. 6.

Real part of electric field in region I for first polariton mode at k˜=0.25, for cylinders with ρ˜=2.0944 and mmax=8. The line of cylinders is along the y axis.

Fig. 7.
Fig. 7.

Real part of electric field in region I for second polariton mode at k˜=0.25, for cylinders with ρ˜=2.0944 and mmax=8. The line of cylinders is along the y axis.

Fig. 8.
Fig. 8.

Integrated spectral density n˜elI as a function of ω˜ for cylinder radius ρ˜=2.8, mmax=14, at k˜=0.12 (solid line), k˜=0.13 (dashed line), and k˜=0.14 (dot-dashed line).

Fig. 9.
Fig. 9.

Real part of electric field in region I for mode at ω˜=0.31, k˜=0.01, for cylinders with ρ˜=2.0944 and mmax=8. The line of cylinders is along the y axis.

Fig. 10.
Fig. 10.

Real part of electric field in region I for mode at ω˜=0.2644, k˜=0.5, for cylinders with ρ˜=2.8 and mmax=14. The line of cylinders is along the y axis.

Tables (1)

Tables Icon

Table 1. Frequencies of Bound States ωι for the Line of Cylinders, with ρ˜=2.0944 and k˜=0.4a

Equations (37)

Equations on this page are rendered with MathJax. Learn more.

1ϵ2Hz=ω2c2Hz,
1ϵIIdrz*2s1ϵIIIIdrHz*2Hz,
1ϵISdrSz*znS1ϵIISdrSHz*HznS,
ω2c2=1ϵIIdrz*·z1ϵII(IIdrHz*2Hz+SdrSHz*HznS)Idr|z|2+IIdr|Hz|2.
1ϵIIHz(rS)nS=SdrSΣ(rS,rS;ω02)Hz(rS)
IIdr|Hz(r)|2=c2SdrSSdrSHz*(rS)Σω02Hz(rS).
ω2c2=1ϵIIdrz*·zSdrSSdrSz*(Σω02Σω02)zIdr|z|2+c2SdrSSdrSz*Σω02z.
z(r)=ihiFi(r),rregionI.
Ah=ω2c2Bh,
Aij=1ϵIIdrFi*·FjSdrSSdrSFi*(Σω02Σω02)Fj
Bij=IdrFi*Fj+c2SdrSSdrSFi*Σω02Fj.
Γ(r,r;ω2/c2)=ijΓij(ω2/c2)Fi*(r)Fj(r),r,rregionI,
l(Ailω2c2Bil)Γlj=δij.
Γ(r,r;ω2/c2)=ιHz,ι*(r)Hz,ι(r)ωι2/c2ω2/c2.
I+IIdr|Hz,ι(r)|2=1,
nel(r,ω)=ιDι*(r)·Eι(r)δ(ωωι),
1cD˙ι=×Hι,
nel(r,ω)=2πωϵ(r)ijFi*(r)·Fj(r)IΓij,rregionI.
nelI(ω)=Idrnel(r,ω).
I+IIdrDι*(r)·Eι(r)=1,
Fi(r)=exp(iki·r),ki=(gxi,k+gyi).
gxi=2πp(i)D,gyi=2πq(i)a,
Fi(rS)=m=exp(im[ϕϕi+π/2])Jm(kiρ),
H(r)=exp(imϕ)Jm(κr),κ2=ω2εc2,
Hz(rS)nS=κm=exp(im[ϕϕi+π/2])Jm(κρ)Jm(κρ)Jm(kiρ).
SdrSΣ(rS,rS;ω02)Fi(rS)=κϵm=exp(im[ϕϕi+π/2])Jm(κρ)Jm(κρ)Jm(kiρ),
SdrSSdrSFi*ΣFj=2πκρϵm=exp(im[ϕiϕj])Jm(κρ)Jm(κρ)Jm(kiρ)Jm(kjρ).
Fi(rS)=exp(i[gxid/2+{k+gyi}y]).
H(r)=exp(i[κx+{k+gyi}y])
κ=ω2/c2(k+gyi)2,
Hz(rS)nS=iκexp(i[gxid/2+{k+gyi}y]),
SdrSSdrSFi*ΣFj=iκaexp(i[gxjgxi]d/2)δq(i),q(j).
SdrSSdrSFi*ΣFj=2iκacos([gxigxj]d/2)δq(i),q(j).
Sij=IdrFi*(r)Fj(r)
nel(r,ω;k)=2πacn˜el(r˜,ω˜;k˜),nelI(ω;k)=a2πcn˜elI(ω˜;k˜).
ϵ(ω)=1ωp2ω(ω+i/τ),
mmaxρkF,

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