Abstract

A novel method derived from the source-model technique is presented to solve the problem of scattering of an electromagnetic plane wave by a two-dimensional photonic crystal slab that contains an arbitrary defect (perturbation). In this method, the electromagnetic fields in the perturbed problem are expressed in terms of the field due to the periodic currents obtained from a solution of the corresponding unperturbed problem plus the field due to yet-to-be-determined correction current sources placed in the vicinity of the perturbation. Appropriate error measures are suggested, and a few representative structures are presented and analyzed to demonstrate the versatility of the proposed method and to provide physical insight into waveguiding and defect coupling mechanisms typical of finite-thickness photonic crystal slabs.

© 2004 Optical Society of America

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References

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  1. E. Yablonovitch, T. J. Gmitter, K. M. Leung, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950–1957 (1989).
    [CrossRef] [PubMed]
  2. J. D. Joannoppulos, R. D. Meade, J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N. J., 1995).
  3. H. Mosallaei, Y. Rahmat-Samii, “Periodic bandgap and effective dielectric materials in electromagnetics: characterization and application in nanocavities and waveguides,” IEEE Trans. Antennas Propag. 51, 549–563 (2003).
    [CrossRef]
  4. S. G. Johnson, A. Mekis, S. Fan, J. D. Joannopoulos, “Molding the flow of light,” Comput. Sci. Eng. 3, 38–47 (2001).
    [CrossRef]
  5. E. Centeno, D. Felbacq, “Guiding waves with photonic crystals,” Opt. Commun. 160, 57–60 (1999).
    [CrossRef]
  6. G. Tayeb, D. Maystre, “Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities,” J. Opt. Soc. Am. A 14, 3323–3332 (1997).
    [CrossRef]
  7. L. C. Botten, A. A. Asatryan, T. N. Langtry, T. P. White, C. M. de Sterke, R. C. McPhedran, “Semianalytic treatment for propagation in finite photonic crystals waveguides,” Opt. Lett. 28, 854–856 (2003).
    [CrossRef] [PubMed]
  8. A. Boag, B. Z. Steinberg, “Narrow-band microcavity waveguides in photonic crystals,” J. Opt. Soc. Am. A 18, 2799–2805 (2001).
    [CrossRef]
  9. K. Busch, S. F. Mingaleev, A. G. Martin, M. Schillinger, D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, 1233–1256 (2003).
    [CrossRef]
  10. Y. Leviatan, A. Boag, A. Boag, “Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies—theory and numerical solutions,” IEEE Trans. Antennas Propag. 36, 1722–1734 (1988).
    [CrossRef]
  11. A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
    [CrossRef]
  12. A. Ludwig, Y. Leviatan, “Analysis of bandgap characteristics of two-dimensional periodic structures by using the source-model technique,” J. Opt. Soc. Am. A 20, 1553–1562 (2003).
    [CrossRef]
  13. E. Centeno, D. Felbacq, “Characterization of defect modes in finite bidimensional photonic crystals,” J. Opt. Soc. Am. A 16, 2705–2712 (1999).
    [CrossRef]

2003 (4)

H. Mosallaei, Y. Rahmat-Samii, “Periodic bandgap and effective dielectric materials in electromagnetics: characterization and application in nanocavities and waveguides,” IEEE Trans. Antennas Propag. 51, 549–563 (2003).
[CrossRef]

K. Busch, S. F. Mingaleev, A. G. Martin, M. Schillinger, D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, 1233–1256 (2003).
[CrossRef]

L. C. Botten, A. A. Asatryan, T. N. Langtry, T. P. White, C. M. de Sterke, R. C. McPhedran, “Semianalytic treatment for propagation in finite photonic crystals waveguides,” Opt. Lett. 28, 854–856 (2003).
[CrossRef] [PubMed]

A. Ludwig, Y. Leviatan, “Analysis of bandgap characteristics of two-dimensional periodic structures by using the source-model technique,” J. Opt. Soc. Am. A 20, 1553–1562 (2003).
[CrossRef]

2001 (2)

S. G. Johnson, A. Mekis, S. Fan, J. D. Joannopoulos, “Molding the flow of light,” Comput. Sci. Eng. 3, 38–47 (2001).
[CrossRef]

A. Boag, B. Z. Steinberg, “Narrow-band microcavity waveguides in photonic crystals,” J. Opt. Soc. Am. A 18, 2799–2805 (2001).
[CrossRef]

1999 (2)

1997 (1)

1989 (1)

E. Yablonovitch, T. J. Gmitter, K. M. Leung, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950–1957 (1989).
[CrossRef] [PubMed]

1988 (2)

Y. Leviatan, A. Boag, A. Boag, “Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies—theory and numerical solutions,” IEEE Trans. Antennas Propag. 36, 1722–1734 (1988).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
[CrossRef]

Asatryan, A. A.

Boag, A.

A. Boag, B. Z. Steinberg, “Narrow-band microcavity waveguides in photonic crystals,” J. Opt. Soc. Am. A 18, 2799–2805 (2001).
[CrossRef]

Y. Leviatan, A. Boag, A. Boag, “Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies—theory and numerical solutions,” IEEE Trans. Antennas Propag. 36, 1722–1734 (1988).
[CrossRef]

Y. Leviatan, A. Boag, A. Boag, “Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies—theory and numerical solutions,” IEEE Trans. Antennas Propag. 36, 1722–1734 (1988).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
[CrossRef]

Botten, L. C.

Busch, K.

K. Busch, S. F. Mingaleev, A. G. Martin, M. Schillinger, D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, 1233–1256 (2003).
[CrossRef]

Centeno, E.

de Sterke, C. M.

Fan, S.

S. G. Johnson, A. Mekis, S. Fan, J. D. Joannopoulos, “Molding the flow of light,” Comput. Sci. Eng. 3, 38–47 (2001).
[CrossRef]

Felbacq, D.

Gmitter, T. J.

E. Yablonovitch, T. J. Gmitter, K. M. Leung, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950–1957 (1989).
[CrossRef] [PubMed]

Hermann, D.

K. Busch, S. F. Mingaleev, A. G. Martin, M. Schillinger, D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, 1233–1256 (2003).
[CrossRef]

Joannopoulos, J. D.

S. G. Johnson, A. Mekis, S. Fan, J. D. Joannopoulos, “Molding the flow of light,” Comput. Sci. Eng. 3, 38–47 (2001).
[CrossRef]

Joannoppulos, J. D.

J. D. Joannoppulos, R. D. Meade, J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N. J., 1995).

Johnson, S. G.

S. G. Johnson, A. Mekis, S. Fan, J. D. Joannopoulos, “Molding the flow of light,” Comput. Sci. Eng. 3, 38–47 (2001).
[CrossRef]

Langtry, T. N.

Leung, K. M.

E. Yablonovitch, T. J. Gmitter, K. M. Leung, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950–1957 (1989).
[CrossRef] [PubMed]

Leviatan, Y.

A. Ludwig, Y. Leviatan, “Analysis of bandgap characteristics of two-dimensional periodic structures by using the source-model technique,” J. Opt. Soc. Am. A 20, 1553–1562 (2003).
[CrossRef]

Y. Leviatan, A. Boag, A. Boag, “Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies—theory and numerical solutions,” IEEE Trans. Antennas Propag. 36, 1722–1734 (1988).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
[CrossRef]

Ludwig, A.

Martin, A. G.

K. Busch, S. F. Mingaleev, A. G. Martin, M. Schillinger, D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, 1233–1256 (2003).
[CrossRef]

Maystre, D.

McPhedran, R. C.

Meade, R. D.

J. D. Joannoppulos, R. D. Meade, J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N. J., 1995).

Mekis, A.

S. G. Johnson, A. Mekis, S. Fan, J. D. Joannopoulos, “Molding the flow of light,” Comput. Sci. Eng. 3, 38–47 (2001).
[CrossRef]

Mingaleev, S. F.

K. Busch, S. F. Mingaleev, A. G. Martin, M. Schillinger, D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, 1233–1256 (2003).
[CrossRef]

Mosallaei, H.

H. Mosallaei, Y. Rahmat-Samii, “Periodic bandgap and effective dielectric materials in electromagnetics: characterization and application in nanocavities and waveguides,” IEEE Trans. Antennas Propag. 51, 549–563 (2003).
[CrossRef]

Rahmat-Samii, Y.

H. Mosallaei, Y. Rahmat-Samii, “Periodic bandgap and effective dielectric materials in electromagnetics: characterization and application in nanocavities and waveguides,” IEEE Trans. Antennas Propag. 51, 549–563 (2003).
[CrossRef]

Schillinger, M.

K. Busch, S. F. Mingaleev, A. G. Martin, M. Schillinger, D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, 1233–1256 (2003).
[CrossRef]

Steinberg, B. Z.

Tayeb, G.

White, T. P.

Winn, J. N.

J. D. Joannoppulos, R. D. Meade, J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N. J., 1995).

Yablonovitch, E.

E. Yablonovitch, T. J. Gmitter, K. M. Leung, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950–1957 (1989).
[CrossRef] [PubMed]

Comput. Sci. Eng. (1)

S. G. Johnson, A. Mekis, S. Fan, J. D. Joannopoulos, “Molding the flow of light,” Comput. Sci. Eng. 3, 38–47 (2001).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

H. Mosallaei, Y. Rahmat-Samii, “Periodic bandgap and effective dielectric materials in electromagnetics: characterization and application in nanocavities and waveguides,” IEEE Trans. Antennas Propag. 51, 549–563 (2003).
[CrossRef]

Y. Leviatan, A. Boag, A. Boag, “Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies—theory and numerical solutions,” IEEE Trans. Antennas Propag. 36, 1722–1734 (1988).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Phys. Condens. Matter (1)

K. Busch, S. F. Mingaleev, A. G. Martin, M. Schillinger, D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, 1233–1256 (2003).
[CrossRef]

Opt. Commun. (1)

E. Centeno, D. Felbacq, “Guiding waves with photonic crystals,” Opt. Commun. 160, 57–60 (1999).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

E. Yablonovitch, T. J. Gmitter, K. M. Leung, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950–1957 (1989).
[CrossRef] [PubMed]

Radio Sci. (1)

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model,” Radio Sci. 23, 612–624 (1988).
[CrossRef]

Other (1)

J. D. Joannoppulos, R. D. Meade, J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N. J., 1995).

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Figures (15)

Fig. 1
Fig. 1

General problem of TMy plane-wave scattering by a two-dimensional photonic crystal slab with a local perturbation.

Fig. 2
Fig. 2

Simplified problem of TMy plane-wave scattering by a one-layer photonic crystal slab locally perturbed.

Fig. 3
Fig. 3

Simulated equivalences for (a) exterior region, Rext, (b) perturbed cylinder, Rint(0), and (c) first cylinder to the right of the perturbed cylinder, Rint(1).

Fig. 4
Fig. 4

Slab configuration taken for the calculation of the convergence error for R=3.

Fig. 5
Fig. 5

Convergence error versus frequency for various values of the range parameter R.

Fig. 6
Fig. 6

Normalized power flow density over a plane located just above the photonic crystal slab with a vertical line of cylinders removed.

Fig. 7
Fig. 7

Normalized power flow density over a plane located just below the photonic crystal slab with a vertical line of cylinders removed.

Fig. 8
Fig. 8

Normalized power flow density over a plane located above the slab for various values of the range parameter R.

Fig. 9
Fig. 9

Normalized power flow density over a plane located above the slab with (a) slightly tilted line of cylinders removed and (b) considerably tilted line of cylinders removed.

Fig. 10
Fig. 10

Normalized power flux through a plane located above the slab versus frequency, for the case of an unperturbed slab and a slab with a cavity-like perturbation created by removal of a cylinder.

Fig. 11
Fig. 11

Normalized power flux through a plane located above the slab versus frequency, for a cavity embedded in a five- and a seven-layer slab.

Fig. 12
Fig. 12

Normalized power flux through a plane located above the slab versus frequency, for several different incident angles.

Fig. 13
Fig. 13

Normalized power flux through a plane located above the slab versus frequency, for a cavity-like perturbation created by the reduction of the radius a of one cylinder in the middle layer.

Fig. 14
Fig. 14

Normalized power flux through a plane located above the slab versus frequency, for a cavity-like perturbation created by increasing the radius a of one cylinder in the middle layer.

Fig. 15
Fig. 15

Resonant frequency versus permittivity of the perturbed cylinder for the cases of a slab with three, five, and seven layers obtained by using the suggested numerical method and for the case of an infinite bulk obtained by using the Wannier-function approach.9

Equations (44)

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Jjq=eyIjqδ(x-xjq)δ(z-zjq),j=1, 2,, Nq.
Jext=kJext(k)=kjJjext(k),
k[n^(i)×Es(Jext(k))]-n^(i)×Eint(i)(Jint(i))
=-n^(i)×Einc on S(i),i=0, ±1,
k[n^(i)×Hs(Jext(k))]-n^(i)×Hint(i)(Jint(i))
=-n^(i)×Hinc on S(i),i=0, ±1,
Jjext=eyIjextδ(z-zjext)exp[-jkxinc(x-xjext)]×n=-f(x-xjext-ndx),j=1, 2,, Next,
Jjint=eyIjintδ(x-xjint)δ(z-zjint),
j=1, 2,, Nint,
M=-dx/2dx/2f(x)exp(-jkxincx)dx.
JjextkJjext(k).
Jjq=eyIjq(x-xjq)δ(z-zjq),j=1, 2,, Nq,
Ijext(k)=Mexp[-jkxinc(xjext(k)-xjext)]Ijext,
Ijint(i)=Ijint.
k[n^(i)×Es(Jext(k))]-n^(i)×Eint(i)(Jint(i))
=-n^(i)×Einc on S(i),i=0, ±1,,
k[n^(i)×Hs(Jext(k))]-n^(i)×Hint(i)(Jint(i))
=-n^(i)×Hinc on S(i),i=0, ±1,,
Nq=Nq(xjq, zjq)=(xjq, zjq),j=1, 2,, Nq,
q{ext(k), int(i)}fori, k0.
J˜q=Jq-Jq,q{ext(k), int(i)}fori, k0Jq,q{ext(k), int(i)}fori, k=0,
J˜jq=eyI˜jqδ(x-xjq)δ(z-zjq),j=1, 2,, Nq,
q{ext(k), int(i)}fori, k=0, ±1, ±2,,
I˜jq=Ijq-Ijq,q{ext(k), int(i)}fori, k0Ijq,q{ext(k), int(i)}fori, k=0.
k[n^(i)×Es(J˜ext(k)+(1-δk0)Jext(k))]
-n^(i)×Eint(i)(J˜int(i)+(1-δi0)Jint(i)) 
=-n^(i)×Einc on S(i),i=0, ±1,,
k[n^(i)×Hs(J˜ext(k)+(1-δk0)Jext(k))]
-n^(i)×Hint(i)(J˜int(i)+(1-δi0)Jint(i)) 
=-n^(i)×Hinc on S(i),i=0, ±1,,
δmn=1,m=n0,mn.
k[n^(i)×Es(J˜ext(k))]-n^(i)×Eint(i)(J˜int(i))
=-n^(i)×Einc-k0[n^(i)×Es(Jext(k))]+(1-δi0)n^(i)×Eint(i)(Jint(i))
on S(i),i=0, ±1,,
k[n^(i)×Hs(J˜ext(k))]-n^(i)×Hint(i)(J˜int(i))
=-n^(i)×Hinc-k0[n^(i)×Hs(Jext(k))]+(1-δi0)n^(i)×Hint(i)(Jint(i))
on S(i),i=0, ±1, .
S(i)=S(i)n^(i)=n^(i),i0.
n[n^(i)×Es(J˜ext(k))]-n^(i)×Eint(i)(J˜int(i))=n^(i)×Es(Jext(0))on S(i),i0-n^(i)×Einc-k0[n^(i)×Es(Jext(k))]on S(i),i=0
k[n^(i)×Hs(J˜ext(k))]-n^(i)×Hint(i)(J˜int(i))=n^(i)×Hs(Jext(0))on S(i),i0-n^(i)×Hinc-k0[n^(i)×Hs(Jext(k))]on S(i),i=0
Eint(i)(Jint(i))=Eint(i)(J˜int(i))+Eint(i)(Jint(i)),i0Eint(i)(J˜int(i)),i=0.
Es(Jext)=kEs(J˜ext(k))+k0Es(Jext(k)).
ΔI(R)=(Next(-1)+Next(1))-1jI˜jext(-1)(R)-I˜jext(-1)(Rmax)I˜jext(-1)(Rmax)2+I˜jext(1)(R)-I˜jext(1)(Rmax)I˜jext(1)(Rmax)21/2,
ΔP=2DSzupper(x)dx-DSzlower(x)dxDSzupper(x)dx+DSzlower(x)dx,

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