Abstract

This paper considers the two-dimensional electromagnetic scattering from periodic array of circular cylinders in which some cylinders are removed, and presents a formulation based on the recursive transition-matrix algorithm (RTMA). The RTMA was originally developed as an accurate approach to the scattering problem of a finite number of cylinders, and an approach to the problem of periodic cylinder array was then developed with the help of the lattice sums technique. This paper introduces the concept of the pseudo-periodic Fourier transform to the RTMA with the lattice sums technique, and proposes a spectral-domain approach to the problem of periodic cylinder array with defects.

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References

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  1. S. A. Rinne, F. García-Santamaría, and P. V. Braun, “Embedded cavities and waveguides in three-dimensional silicon photonic crystals,” Nat. Photonics 2, 52–56 (2007).
    [CrossRef]
  2. J. Ouellette, “Seeing the future in photonic crystals,” Ind. Phys.  7, 14–17 (2001).
  3. Ch. Kang and S. M. Weiss, “Photonic crystal with multi-hole defect for sensor applications,” Opt. Express 16, 18188–18193 (2008).
    [CrossRef] [PubMed]
  4. A. V. Giannopoulos, J. D. Sulkin, Ch. M. Long, J. J. Coleman, and K. D. Choquette, “Decimated photonic crystal defect cavity lasers,” IEEE J. Sel. Top. Quantum Electron.  17, 1693–1694 (2011).
    [CrossRef]
  5. W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).
  6. D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
    [CrossRef]
  7. H. Roussel, W. C. Chew, F. Jouvie, and W. Tabbara, “Electromagnetic scattering from dielectric and magnetic gratings of fibers — a T-matrix solution,” J. Electromagn. Waves Appl.  10, 109–127 (1996).
    [CrossRef]
  8. K. Watanabe and K. Yasumoto, “Two-dimensional electromagnetic scattering of non-plane incident waves by periodic structures,” Prog. Electromagn. Res.  PIER 74, 241–271 (2007).
    [CrossRef]
  9. K. Watanabe and Y. Nakatake, “Spectral-domain formulation of electromagnetic scattering from circular cylinders located near periodic cylinder array,” Prog. Electromagn. Res. B 31, 219–237 (2011).
  10. K. Watanabe, J. Pištora, and Y. Nakatake, “Rigorous coupled-wave analysis of electromagnetic scattering from lamellar grating with defects,” Opt. Express 19, 25799–25811 (2011).
    [CrossRef]
  11. K. Watanabe, J. Pištora, and Y. Nakatake, “Coordinate transformation formulation of electromagnetic scattering from imperfectly periodic surfaces,” Opt. Express(to be published).
    [PubMed]
  12. N. A. Nicorovici and R. C. McPhedran, “Lattice sums for off-axis electromagnetic scattering by gratings,” Phys. Rev. E 50, 3143–3160 (1994).
    [CrossRef]
  13. K. Yasumoto and K. Yoshitomi, “Efficient calculation of lattice sums for free-space periodic Green’s function,” IEEE Trans. Antennas Propag.  47, 1050–1055 (1999).
    [CrossRef]
  14. C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989).

2011 (3)

A. V. Giannopoulos, J. D. Sulkin, Ch. M. Long, J. J. Coleman, and K. D. Choquette, “Decimated photonic crystal defect cavity lasers,” IEEE J. Sel. Top. Quantum Electron.  17, 1693–1694 (2011).
[CrossRef]

K. Watanabe and Y. Nakatake, “Spectral-domain formulation of electromagnetic scattering from circular cylinders located near periodic cylinder array,” Prog. Electromagn. Res. B 31, 219–237 (2011).

K. Watanabe, J. Pištora, and Y. Nakatake, “Rigorous coupled-wave analysis of electromagnetic scattering from lamellar grating with defects,” Opt. Express 19, 25799–25811 (2011).
[CrossRef]

2008 (1)

2007 (2)

S. A. Rinne, F. García-Santamaría, and P. V. Braun, “Embedded cavities and waveguides in three-dimensional silicon photonic crystals,” Nat. Photonics 2, 52–56 (2007).
[CrossRef]

K. Watanabe and K. Yasumoto, “Two-dimensional electromagnetic scattering of non-plane incident waves by periodic structures,” Prog. Electromagn. Res.  PIER 74, 241–271 (2007).
[CrossRef]

2001 (1)

J. Ouellette, “Seeing the future in photonic crystals,” Ind. Phys.  7, 14–17 (2001).

1999 (1)

K. Yasumoto and K. Yoshitomi, “Efficient calculation of lattice sums for free-space periodic Green’s function,” IEEE Trans. Antennas Propag.  47, 1050–1055 (1999).
[CrossRef]

1996 (1)

H. Roussel, W. C. Chew, F. Jouvie, and W. Tabbara, “Electromagnetic scattering from dielectric and magnetic gratings of fibers — a T-matrix solution,” J. Electromagn. Waves Appl.  10, 109–127 (1996).
[CrossRef]

1994 (2)

N. A. Nicorovici and R. C. McPhedran, “Lattice sums for off-axis electromagnetic scattering by gratings,” Phys. Rev. E 50, 3143–3160 (1994).
[CrossRef]

D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
[CrossRef]

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989).

Braun, P. V.

S. A. Rinne, F. García-Santamaría, and P. V. Braun, “Embedded cavities and waveguides in three-dimensional silicon photonic crystals,” Nat. Photonics 2, 52–56 (2007).
[CrossRef]

Chew, W. C.

H. Roussel, W. C. Chew, F. Jouvie, and W. Tabbara, “Electromagnetic scattering from dielectric and magnetic gratings of fibers — a T-matrix solution,” J. Electromagn. Waves Appl.  10, 109–127 (1996).
[CrossRef]

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).

Choquette, K. D.

A. V. Giannopoulos, J. D. Sulkin, Ch. M. Long, J. J. Coleman, and K. D. Choquette, “Decimated photonic crystal defect cavity lasers,” IEEE J. Sel. Top. Quantum Electron.  17, 1693–1694 (2011).
[CrossRef]

Coleman, J. J.

A. V. Giannopoulos, J. D. Sulkin, Ch. M. Long, J. J. Coleman, and K. D. Choquette, “Decimated photonic crystal defect cavity lasers,” IEEE J. Sel. Top. Quantum Electron.  17, 1693–1694 (2011).
[CrossRef]

Felbacq, D.

García-Santamaría, F.

S. A. Rinne, F. García-Santamaría, and P. V. Braun, “Embedded cavities and waveguides in three-dimensional silicon photonic crystals,” Nat. Photonics 2, 52–56 (2007).
[CrossRef]

Giannopoulos, A. V.

A. V. Giannopoulos, J. D. Sulkin, Ch. M. Long, J. J. Coleman, and K. D. Choquette, “Decimated photonic crystal defect cavity lasers,” IEEE J. Sel. Top. Quantum Electron.  17, 1693–1694 (2011).
[CrossRef]

Jouvie, F.

H. Roussel, W. C. Chew, F. Jouvie, and W. Tabbara, “Electromagnetic scattering from dielectric and magnetic gratings of fibers — a T-matrix solution,” J. Electromagn. Waves Appl.  10, 109–127 (1996).
[CrossRef]

Kang, Ch.

Long, Ch. M.

A. V. Giannopoulos, J. D. Sulkin, Ch. M. Long, J. J. Coleman, and K. D. Choquette, “Decimated photonic crystal defect cavity lasers,” IEEE J. Sel. Top. Quantum Electron.  17, 1693–1694 (2011).
[CrossRef]

Maystre, D.

McPhedran, R. C.

N. A. Nicorovici and R. C. McPhedran, “Lattice sums for off-axis electromagnetic scattering by gratings,” Phys. Rev. E 50, 3143–3160 (1994).
[CrossRef]

Nakatake, Y.

K. Watanabe and Y. Nakatake, “Spectral-domain formulation of electromagnetic scattering from circular cylinders located near periodic cylinder array,” Prog. Electromagn. Res. B 31, 219–237 (2011).

K. Watanabe, J. Pištora, and Y. Nakatake, “Rigorous coupled-wave analysis of electromagnetic scattering from lamellar grating with defects,” Opt. Express 19, 25799–25811 (2011).
[CrossRef]

K. Watanabe, J. Pištora, and Y. Nakatake, “Coordinate transformation formulation of electromagnetic scattering from imperfectly periodic surfaces,” Opt. Express(to be published).
[PubMed]

Nicorovici, N. A.

N. A. Nicorovici and R. C. McPhedran, “Lattice sums for off-axis electromagnetic scattering by gratings,” Phys. Rev. E 50, 3143–3160 (1994).
[CrossRef]

Ouellette, J.

J. Ouellette, “Seeing the future in photonic crystals,” Ind. Phys.  7, 14–17 (2001).

Pištora, J.

K. Watanabe, J. Pištora, and Y. Nakatake, “Rigorous coupled-wave analysis of electromagnetic scattering from lamellar grating with defects,” Opt. Express 19, 25799–25811 (2011).
[CrossRef]

K. Watanabe, J. Pištora, and Y. Nakatake, “Coordinate transformation formulation of electromagnetic scattering from imperfectly periodic surfaces,” Opt. Express(to be published).
[PubMed]

Rinne, S. A.

S. A. Rinne, F. García-Santamaría, and P. V. Braun, “Embedded cavities and waveguides in three-dimensional silicon photonic crystals,” Nat. Photonics 2, 52–56 (2007).
[CrossRef]

Roussel, H.

H. Roussel, W. C. Chew, F. Jouvie, and W. Tabbara, “Electromagnetic scattering from dielectric and magnetic gratings of fibers — a T-matrix solution,” J. Electromagn. Waves Appl.  10, 109–127 (1996).
[CrossRef]

Sulkin, J. D.

A. V. Giannopoulos, J. D. Sulkin, Ch. M. Long, J. J. Coleman, and K. D. Choquette, “Decimated photonic crystal defect cavity lasers,” IEEE J. Sel. Top. Quantum Electron.  17, 1693–1694 (2011).
[CrossRef]

Tabbara, W.

H. Roussel, W. C. Chew, F. Jouvie, and W. Tabbara, “Electromagnetic scattering from dielectric and magnetic gratings of fibers — a T-matrix solution,” J. Electromagn. Waves Appl.  10, 109–127 (1996).
[CrossRef]

Tayeb, G.

Watanabe, K.

K. Watanabe and Y. Nakatake, “Spectral-domain formulation of electromagnetic scattering from circular cylinders located near periodic cylinder array,” Prog. Electromagn. Res. B 31, 219–237 (2011).

K. Watanabe, J. Pištora, and Y. Nakatake, “Rigorous coupled-wave analysis of electromagnetic scattering from lamellar grating with defects,” Opt. Express 19, 25799–25811 (2011).
[CrossRef]

K. Watanabe and K. Yasumoto, “Two-dimensional electromagnetic scattering of non-plane incident waves by periodic structures,” Prog. Electromagn. Res.  PIER 74, 241–271 (2007).
[CrossRef]

K. Watanabe, J. Pištora, and Y. Nakatake, “Coordinate transformation formulation of electromagnetic scattering from imperfectly periodic surfaces,” Opt. Express(to be published).
[PubMed]

Weiss, S. M.

Yasumoto, K.

K. Watanabe and K. Yasumoto, “Two-dimensional electromagnetic scattering of non-plane incident waves by periodic structures,” Prog. Electromagn. Res.  PIER 74, 241–271 (2007).
[CrossRef]

K. Yasumoto and K. Yoshitomi, “Efficient calculation of lattice sums for free-space periodic Green’s function,” IEEE Trans. Antennas Propag.  47, 1050–1055 (1999).
[CrossRef]

Yoshitomi, K.

K. Yasumoto and K. Yoshitomi, “Efficient calculation of lattice sums for free-space periodic Green’s function,” IEEE Trans. Antennas Propag.  47, 1050–1055 (1999).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron (1)

A. V. Giannopoulos, J. D. Sulkin, Ch. M. Long, J. J. Coleman, and K. D. Choquette, “Decimated photonic crystal defect cavity lasers,” IEEE J. Sel. Top. Quantum Electron.  17, 1693–1694 (2011).
[CrossRef]

IEEE Trans. Antennas Propag (1)

K. Yasumoto and K. Yoshitomi, “Efficient calculation of lattice sums for free-space periodic Green’s function,” IEEE Trans. Antennas Propag.  47, 1050–1055 (1999).
[CrossRef]

Ind. Phys (1)

J. Ouellette, “Seeing the future in photonic crystals,” Ind. Phys.  7, 14–17 (2001).

J. Electromagn. Waves Appl (1)

H. Roussel, W. C. Chew, F. Jouvie, and W. Tabbara, “Electromagnetic scattering from dielectric and magnetic gratings of fibers — a T-matrix solution,” J. Electromagn. Waves Appl.  10, 109–127 (1996).
[CrossRef]

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

Nat. Photonics (1)

S. A. Rinne, F. García-Santamaría, and P. V. Braun, “Embedded cavities and waveguides in three-dimensional silicon photonic crystals,” Nat. Photonics 2, 52–56 (2007).
[CrossRef]

Opt. Express (2)

Phys. Rev. E (1)

N. A. Nicorovici and R. C. McPhedran, “Lattice sums for off-axis electromagnetic scattering by gratings,” Phys. Rev. E 50, 3143–3160 (1994).
[CrossRef]

Prog. Electromagn. Res (1)

K. Watanabe and K. Yasumoto, “Two-dimensional electromagnetic scattering of non-plane incident waves by periodic structures,” Prog. Electromagn. Res.  PIER 74, 241–271 (2007).
[CrossRef]

Prog. Electromagn. Res. B (1)

K. Watanabe and Y. Nakatake, “Spectral-domain formulation of electromagnetic scattering from circular cylinders located near periodic cylinder array,” Prog. Electromagn. Res. B 31, 219–237 (2011).

Other (3)

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).

K. Watanabe, J. Pištora, and Y. Nakatake, “Coordinate transformation formulation of electromagnetic scattering from imperfectly periodic surfaces,” Opt. Express(to be published).
[PubMed]

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989).

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Figures (8)

Fig. 1
Fig. 1

Periodic circular cylinder array with defects.

Fig. 2
Fig. 2

Convergence of the field intensities at (x,y) = (0,±d) for a line-source excitation. The dotted curves are obtained by the trapezoidal scheme, and the solid curves are obtained by the Gauss-Legendre scheme applying to split subintervals. (a) Convergence on the number of sample points L (b) Convergence on the truncation order K.

Fig. 3
Fig. 3

Field intensities near a periodic cylinder array with defects for the line-source excitation. (a) TM-polarization (b) TE-polarization.

Fig. 4
Fig. 4

Scattering pattern for the line-source excitation.

Fig. 5
Fig. 5

Numerical results of reciprocity test for the line-source excitation.

Fig. 6
Fig. 6

Convergence of the field intensities at (x,y) = (0,±d) for the plane-wave incidence. The dotted curves are obtained by the trapezoidal scheme, and the solid curves are obtained by the Gauss-Legendre scheme applying to split subintervals. (a) Convergence on the number of sample points L (b) Convergence on the truncation order K.

Fig. 7
Fig. 7

Field intensities near a periodic cylinder array with defects for the plane-wave incidence. (a) TM-polarization (b) TE-polarization.

Fig. 8
Fig. 8

Scattering pattern of the residual field ψ(s,d) for the plane-wave incidence.

Equations (40)

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(g(Z)(x,y))n=Zn(ksρ(x,y))einϕ(x,y)
ρ(x,y)=x2+y2
ϕ(x,y)=arg(x+iy)
g(Z)(xxq,yyq)=G(Z)(xrxq,yryq)g(J)(xxr,yyr)
(G(Z)(x,y))n,m=Znm(ksρ(x,y))ei(nm)ϕ(x,y).
ψ(i)(x,y)=g(J)(xqd,y)taq(i)
ψ(s)(x,y)=q𝒟cg(H(1))(xqd,y)taq(s)
ψ(x,y)=g(J)(xqd,y)t [aq(i)+r𝒟c\{q}G(H(1))((qr)d,0)tar(s)]+ g(H(1))(xqd,y)taq(s)
aq(s)=T[aq(i)+r𝒟c\{q}G(H(1))((qr)d,0)tar(s)].
(T)n,m={δn,mζsJn(ksa)Jn(kca)ζcJn(ksa)Jn(kca)ζcHn(1)(ksa)Jn(kca)ζsHn(1)(ksa)Jn(kca)for TM-polarizationδn,mζcJn(ksa)Jn(kca)ζsJn(ksa)Jn(kc)aζsHn(1)(ksa)Jn(kca)ζcHn(1)(ksa)Jn(kca)for TE-polarization
f¯(x;ξ)=qf(xqd)eiqdξ
f(x)=1kdkd/2kd/2f¯(x;ξ)dξ
ψ¯(i)(x;ξ,y)=g(J)(x,y)ta¯(i)(ξ)
ψ¯(s)(x;ξ,y)=qg(H(1))(xqd,y)ta¯(s)(ξ)eiqdξ
a¯(i)(ξ)=qaq(i)eiqdξ
a¯(s)(ξ)=q𝒟caq(s)eiqdξ.
aq(f)=1kdkd/2kd/2a¯(f)(ξ)eiqdξdξ
M(p)(ξ)a¯(s)(ξ)+1kdkd/2kd/2M(d)(ξ,ξ)a¯(s)(ξ)dξ=a¯(i)(ξ)+1kdkd/2kd/2c(d)(ξ,ξ)a¯(i)(ξ)dξ
M(p)(ξ)=T1L(ξ)
M(d)(ξ,ξ)=c(d)(ξ,ξ)L(ξ)
L(ξ)=q\{0}G(H(1))(qd,0)teiqdξ
c(d)(ξ,ξ)=q𝒟eiqd(ξξ)
a¯(s)(ξ)=a¯(s,p)(ξ)+a¯(s,d)(ξ)
a¯(s,p)(ξ)=M(p)(ξ)1a¯(i)(ξ).
M(p)(ξ)a¯(s,d)(ξ)+1kdkd/2kd/2M(d)(ξ,ξ)a¯(s,d)(ξ)dξ=1kdkd/2kd/2(c(d)(ξ,ξ)a¯(i)(ξ)M(d)(ξ,ξ)a¯(s,p)(ξ))dξ.
l=1LMl,la¯(s,d)(ξl)=bl
Ml,l=δl,lM(p)(ξl)+wlkdM(d)(ξl,ξl)
bl=1kdkd/2kd/2(c(d)(ξl,ξ)a¯(i)(ξ)M(d)(ξl,ξ)a¯(s,p)(ξ))dξ.
(a¯(s,d)(ξ1)a¯(s,d)(ξL))=(M1,1M1,LML,1ML,L)1(b1bL).
ψ(i)(x,y)=H0(1)(ksρ(xx0,yy0)).
(a¯(i)(ξ))n=q2dβq(ξ)eiαq(ξ)x0{(iαq(ξ)βq(ξ)ks)neiβq(ξ)y0fory0>a(iαq(ξ)+βq(ξ)ks)neiβq(ξ)y0fory0<a
αq(ξ)=ξ+qkd
βq(ξ)=ks2αq(ξ)2
bl=l=1Lwlkd(c(d)(ξl,ξl)a¯(i)(ξl)M(d)(ξl,ξl)a¯(s,p)(ξl)).
σ(xp,yp;xq,yq)=|ψ(xp,yp;xq,yq)ψ(xq,yq;xp,up)||ψ(xp,yp;xq,yq)|
ψ(i)(x,y)=ei(kxx+kyy)
a¯(i)(ξ)=kda0(i)qδ(ξkxqkd)
a¯(s,p)(ξ)=kdM(p)(kx)1a0(i)qδ(ξkxqkd)
bl=(c(d)(ξl,kx)IM(d)(ξl,kx)M(p)(kx)1)a0(i)
(a0(i))n=ein(θ(i)+π2).

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