Abstract

We introduce a solution based on the source-model technique for periodic structures for the problem of electromagnetic scattering by a two-dimensional photonic bandgap crystal slab illuminated by a transverse-magnetic plane wave. The proposed technique takes advantage of the periodicity of the slab by solving the problem within the unit cell of the periodic structure. The results imply the existence of a frequency bandgap and provide a valuable insight into the relationship between the dimensions of a finite periodic structure and its frequency bandgap characteristics. A comparison shows a discrepancy between the frequency bandgap obtained for a very thick slab and the bandgap obtained by solving the corresponding two-dimensionally infinite periodic structure. The final part of the paper is devoted to explaining in detail this apparent discrepancy.

© 2003 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. D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
    [CrossRef]
  4. L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
    [CrossRef]
  5. B. Gralak, S. Enoch, G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am. A 17, 1012–1020 (2000).
    [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. A. Boag, Y. Leviatan, A. Boag, “Analysis of diffraction from echellete gratings using a strip current model,” J. Opt. Soc. Am. A 6, 543–549 (1989).
    [CrossRef]
  8. Y. Leviatan, A. Boag, A. Boag, “Generalized fomulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies—theory and numerical solutions,” IEEE Trans. Antennas Propag. 36, 1722–1734 (1988).
    [CrossRef]
  9. Y. Leviatan, “Analytic continuation considerations when using generalized formulation for scattering problems,” IEEE Trans. Antennas Propag. 38, 1259–1263 (1990).
    [CrossRef]
  10. C. Hafner, The Generalized Multidipole Technique for Computational Electromagnetics (Artech House, Norwood, Mass., 1990).
  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]

2001 (1)

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

2000 (1)

1997 (1)

1994 (1)

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

1990 (1)

Y. Leviatan, “Analytic continuation considerations when using generalized formulation for scattering problems,” IEEE Trans. Antennas Propag. 38, 1259–1263 (1990).
[CrossRef]

1989 (2)

A. Boag, Y. Leviatan, A. Boag, “Analysis of diffraction from echellete gratings using a strip current model,” J. Opt. Soc. Am. A 6, 543–549 (1989).
[CrossRef]

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 fomulations 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.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Boag, A.

A. Boag, Y. Leviatan, A. Boag, “Analysis of diffraction from echellete gratings using a strip current model,” J. Opt. Soc. Am. A 6, 543–549 (1989).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of diffraction from echellete gratings using a strip current model,” J. Opt. Soc. Am. A 6, 543–549 (1989).
[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]

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

Botten, L. C.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

de Sterke, C. M.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Enoch, S.

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]

Gralak, B.

Hafner, C.

C. Hafner, The Generalized Multidipole Technique for Computational Electromagnetics (Artech House, Norwood, Mass., 1990).

Joannoppulos, J. D.

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

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.

Y. Leviatan, “Analytic continuation considerations when using generalized formulation for scattering problems,” IEEE Trans. Antennas Propag. 38, 1259–1263 (1990).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of diffraction from echellete gratings using a strip current model,” J. Opt. Soc. Am. A 6, 543–549 (1989).
[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]

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

Maystre, D.

McPhedran, R. C.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Meade, R. D.

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

Nicorovici, N. A.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Tayeb, G.

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]

IEEE Trans. Antennas Propag. (2)

Y. Leviatan, A. Boag, A. Boag, “Generalized fomulations 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, “Analytic continuation considerations when using generalized formulation for scattering problems,” IEEE Trans. Antennas Propag. 38, 1259–1263 (1990).
[CrossRef]

J. Opt. Soc. Am. A (3)

Phys. Rev. E (1)

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

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]

Pure Appl. Opt. (1)

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

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 (2)

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

C. Hafner, The Generalized Multidipole Technique for Computational Electromagnetics (Artech House, Norwood, Mass., 1990).

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Figures (13)

Fig. 1
Fig. 1

General problem of TM plane-wave scattering by a two-dimensional photonic crystal slab.

Fig. 2
Fig. 2

Simulated equivalence for the exterior region.

Fig. 3
Fig. 3

Simulated equivalence for the ith cylinder.

Fig. 4
Fig. 4

PBC slab configuration investigated.

Fig. 5
Fig. 5

Normalized reflected power versus frequency for various number of layers and for θinc=0°.

Fig. 6
Fig. 6

Normalized reflected power versus frequency for various number of layers and for (a) θinc=45° and (b) θinc=81°.

Fig. 7
Fig. 7

Normalized reflected power versus θinc for different frequencies: (a) ωd/(2πc)=0.366 and (b) ωd/(2πc)=0.3.

Fig. 8
Fig. 8

Band-structure diagram obtained for the corresponding infinite periodic structure.

Fig. 9
Fig. 9

Product of the normalized reflected power over all incident angles compared with the band-structure diagram.

Fig. 10
Fig. 10

(a) Two-dimensional band-structure diagram for the first band. (b) The projection of the band-structure diagram of the first band on the kx axis.

Fig. 11
Fig. 11

Intersection of the projections on the kx axis of the band-structure diagram and the light cone reduced to the first Brillouin zone, and the resultant effective bandgap widening.

Fig. 12
Fig. 12

Comparison of the frequency range of total reflection for all incident angles obtained from the graph of the product of the normalized reflected power over all incident angles and the effectively widened bandgap displayed in the band-structure diagram.

Fig. 13
Fig. 13

Normalized transmitted power for an evanescent incident wave with a given kx0inc depicted to the right of the graph of the projection on the kx axis of the first and the second bands of the band-structure diagram.

Equations (41)

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Einc=eyexp[-j(kxincx+kzincz)],
Hinc=1kIηI (-exkzinc+ezkxinc)exp[-j(kxincx+kzincz)],
Jj(0)=eyIj(0)δ(z-zj(0))exp[-jkxinc(x-xj(0))]×n=-f(x-xj(0)-ndx),j=1, 2,, N(0).
Jj(i)=eyIj(i)δ(x-xj(i))δ(z-zj(i)),j=1, 2,, N(i).
n^(i)×(Es-E(i))=-n^(i)×EinconS(i),
i=1, 2,, N, 
n^(i)×(Hs-H(i))=-n^(i)×HinconS(i),
i=1, 2,,N, 
Ejs=eyEjys=-eykIηIIj(0)2n=-n=ankzn×exp{-j[kxn(x-xj(0))+kzn|z-zj(0)|]}.
Hjs=exHjxs+ezHjzs=exsgn(z-zj(0))Ij(0)2n=-n=an×exp{-j[kxn(x-xj(0))+kzn|z-zj(0)|]}-ezIj(0)2n=-n=kxnankznexp{-j[kxn(x-xj(0))+kzn|z-zj(0)|]}.
kxn=kxinc+2πndx,
kzn=kI2-kxn2,
an=1dx-dx/2dx/2f(x)expj 2πndx xdx.
Ej(i)=eyEjy(i)=-eykIIηIIIj(i)4 H0(2)(kIIρj(i)).
Hj(i)=exHjx(i)+ezHjz(i)=[ex(z-zj(i))-ez(x-xj(i))]×kIIIj(i)4jρj(i) H1(2)(kIIρj(i)).
ρj(i)=(x-xj(i))2+(z-zj(i))2
[Z]I=V,
[Z]
=[Ze(0),S(1)][Ze(1),S(1)][Ze(0),S(2)][Ze(2),S(2)]00[Ze(0),S(N)][Ze(N),S(N)][Zh(0),S(1)][Zh(1),S(1)][Zh(0),S(2)][Zh(2),S(2)]00[Zh(0),S(N)][Zh(N),S(N)],
  
I=I(0)I(1)I(2)I(N),
V=VeS(1)VeS(2)VeS(N)VhS(1)VhS(2)VhS(N).
Es±(x, z)=eyn=-Fn±exp[-j(kxnx±kznz)],
Hs±(x, z)=exkIηIn=-kznFn±exp[-j(kxnx±kznz)]+ezkIηIn=-kxnFn±exp[-j(kxnx±kznz)],
Fn±=-ωμIan2kznj=1N(0)Ij(0)exp[j(kxnxj(0)±kznzj(0))].
Pz+=n1kIηIRe(kzn*)exp[2z Im(kzn)]|Fn++δn0|2,
Pz-=n-1kIηIRe(kzn*exp[-2z Im(kzn)]×{|Fn-|2-2j Im[Fn-δn0exp(-2jkznz)]-|δn0|2}),
δn0=1n=00n0.
Pz+=Pztrans,
Pz-=Pzinc-Pzref,
Pztrans=n1kIηIRe(kzn*)exp[2z Im(kzn)]|Fn++δn0|2,
Pzref=n1kIηIRe(kzn*)exp[-2z Im(kzn)](|Fn-|2),
Pzinc=1kIηIRe(kz0*exp[-2z Im(kz0)]×{2j Im[F0-exp(-2jkz0z)]+1}).
ΔEbc(i)=|n^(i)×(Es+Einc-E(i))|onS(i)|Einc|max,
i=1, 2,, N, 
ΔHbc(i)=|n^(i)×(Hs+Hinc-H(i))|onS(i)|Hinc|max,
i=1, 2,, N.
ΔP=|Pzinc-Pzref-Pztrans||Pzinc|.
GP=(kxinc, ω)||kxinc|ω/c.
GE1=(kxinc, ω)||kxinc|>ω/cand|[kxinc]red|ω/c.
GE2=(kxinc, ω)||kxinc|>ω/cand|[kxinc]red|>ω/c.

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