Abstract

The classic differential method is applied for modeling the diffraction of light from two-dimensional photonic crystals that consist of dielectric cylindrical objects. Special attention is paid to mutual interpenetration of consecutive layers. Two algorithms for dealing with a stack of repetitive layers are discussed, namely, the eigenvalue technique and the S-matrix algorithm. Their advantages and limitations are analyzed, and times required for their implementation are compared.

© 2000 Optical Society of America

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References

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  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef] [PubMed]
  2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
    [CrossRef] [PubMed]
  3. K. Ho, C. Chan, C. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
    [CrossRef] [PubMed]
  4. P. Villeneuve, Sh. Fan, J. Joannopoulos, “Microcavities in photonic crystals,” in Microcavities and Photonic Bandgaps, J. Rarity, C. Weisbuch, eds. (Kluwer Scientific, Dordrecht, The Netherlands, 1996), pp. 133–151.
    [CrossRef]
  5. Z. Zhang, S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
    [CrossRef] [PubMed]
  6. R. Meade, K. Brommer, A. Rappe, J. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10,961–10,964 (1991).
    [CrossRef]
  7. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  8. 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]
  9. D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
    [CrossRef]
  10. F. Wijnands, J. Pendry, P. Roberts, P. Bell, L. Moreno, F. Garcia-Vidal, “Numerical method for calculating spontaneous emission rate near a surface using Green’s functions,” in Microcavities and Photonic Bandgaps, J. Rarity, C. Weisbuch, eds. (Kluwer, Scientific, Dordrecht, The Netherlands, 1996), pp. 299–308.
    [CrossRef]
  11. P. Dansas, N. Paraire, “Fast modeling of photonic bandgap structures by use of a diffraction-grating approach,” J. Opt. Soc. Am. A 15, 1586–1598 (1998).
    [CrossRef]
  12. F. Montiel, M. Neviere, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
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  13. E. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997).
  14. E. Popov, B. Bozhkov, D. Maystre, J. Hoose, “Integral method for echelles covered with lossless or absorbing thin dielectric layers,” Appl. Opt. 38, 47–55 (1999).
    [CrossRef]
  15. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
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  16. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
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1999 (1)

1998 (1)

1997 (1)

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]

1996 (1)

1994 (2)

1991 (1)

R. Meade, K. Brommer, A. Rappe, J. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10,961–10,964 (1991).
[CrossRef]

1990 (2)

K. Ho, C. Chan, C. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Z. Zhang, S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[CrossRef] [PubMed]

1987 (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

1982 (1)

Bell, P.

F. Wijnands, J. Pendry, P. Roberts, P. Bell, L. Moreno, F. Garcia-Vidal, “Numerical method for calculating spontaneous emission rate near a surface using Green’s functions,” in Microcavities and Photonic Bandgaps, J. Rarity, C. Weisbuch, eds. (Kluwer, Scientific, Dordrecht, The Netherlands, 1996), pp. 299–308.
[CrossRef]

Bozhkov, B.

Brommer, K.

R. Meade, K. Brommer, A. Rappe, J. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10,961–10,964 (1991).
[CrossRef]

Chan, C.

K. Ho, C. Chan, C. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Dansas, P.

Fan, Sh.

P. Villeneuve, Sh. Fan, J. Joannopoulos, “Microcavities in photonic crystals,” in Microcavities and Photonic Bandgaps, J. Rarity, C. Weisbuch, eds. (Kluwer Scientific, Dordrecht, The Netherlands, 1996), pp. 133–151.
[CrossRef]

Garcia-Vidal, F.

F. Wijnands, J. Pendry, P. Roberts, P. Bell, L. Moreno, F. Garcia-Vidal, “Numerical method for calculating spontaneous emission rate near a surface using Green’s functions,” in Microcavities and Photonic Bandgaps, J. Rarity, C. Weisbuch, eds. (Kluwer, Scientific, Dordrecht, The Netherlands, 1996), pp. 299–308.
[CrossRef]

Gaylord, T. K.

Ho, K.

K. Ho, C. Chan, C. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Hoose, J.

Joannopoulos, J.

R. Meade, K. Brommer, A. Rappe, J. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10,961–10,964 (1991).
[CrossRef]

P. Villeneuve, Sh. Fan, J. Joannopoulos, “Microcavities in photonic crystals,” in Microcavities and Photonic Bandgaps, J. Rarity, C. Weisbuch, eds. (Kluwer Scientific, Dordrecht, The Netherlands, 1996), pp. 133–151.
[CrossRef]

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Li, L.

Loewen, E.

E. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997).

Maystre, D.

E. Popov, B. Bozhkov, D. Maystre, J. Hoose, “Integral method for echelles covered with lossless or absorbing thin dielectric layers,” Appl. Opt. 38, 47–55 (1999).
[CrossRef]

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]

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

Meade, R.

R. Meade, K. Brommer, A. Rappe, J. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10,961–10,964 (1991).
[CrossRef]

Moharam, M. G.

Montiel, F.

Moreno, L.

F. Wijnands, J. Pendry, P. Roberts, P. Bell, L. Moreno, F. Garcia-Vidal, “Numerical method for calculating spontaneous emission rate near a surface using Green’s functions,” in Microcavities and Photonic Bandgaps, J. Rarity, C. Weisbuch, eds. (Kluwer, Scientific, Dordrecht, The Netherlands, 1996), pp. 299–308.
[CrossRef]

Neviere, M.

Paraire, N.

Pendry, J.

F. Wijnands, J. Pendry, P. Roberts, P. Bell, L. Moreno, F. Garcia-Vidal, “Numerical method for calculating spontaneous emission rate near a surface using Green’s functions,” in Microcavities and Photonic Bandgaps, J. Rarity, C. Weisbuch, eds. (Kluwer, Scientific, Dordrecht, The Netherlands, 1996), pp. 299–308.
[CrossRef]

Popov, E.

Rappe, A.

R. Meade, K. Brommer, A. Rappe, J. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10,961–10,964 (1991).
[CrossRef]

Roberts, P.

F. Wijnands, J. Pendry, P. Roberts, P. Bell, L. Moreno, F. Garcia-Vidal, “Numerical method for calculating spontaneous emission rate near a surface using Green’s functions,” in Microcavities and Photonic Bandgaps, J. Rarity, C. Weisbuch, eds. (Kluwer, Scientific, Dordrecht, The Netherlands, 1996), pp. 299–308.
[CrossRef]

Satpathy, S.

Z. Zhang, S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[CrossRef] [PubMed]

Soukoulis, C.

K. Ho, C. Chan, C. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Tayeb, G.

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]

Villeneuve, P.

P. Villeneuve, Sh. Fan, J. Joannopoulos, “Microcavities in photonic crystals,” in Microcavities and Photonic Bandgaps, J. Rarity, C. Weisbuch, eds. (Kluwer Scientific, Dordrecht, The Netherlands, 1996), pp. 133–151.
[CrossRef]

Wijnands, F.

F. Wijnands, J. Pendry, P. Roberts, P. Bell, L. Moreno, F. Garcia-Vidal, “Numerical method for calculating spontaneous emission rate near a surface using Green’s functions,” in Microcavities and Photonic Bandgaps, J. Rarity, C. Weisbuch, eds. (Kluwer, Scientific, Dordrecht, The Netherlands, 1996), pp. 299–308.
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Zhang, Z.

Z. Zhang, S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[CrossRef] [PubMed]

Appl. Opt. (1)

J. Opt. Soc. Am A (1)

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]

J. Opt. Soc. Am. (1)

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

Phys. Rev. B (1)

R. Meade, K. Brommer, A. Rappe, J. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10,961–10,964 (1991).
[CrossRef]

Phys. Rev. Lett. (4)

Z. Zhang, S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

K. Ho, C. Chan, C. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Pure Appl. Opt. (1)

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

Other (4)

F. Wijnands, J. Pendry, P. Roberts, P. Bell, L. Moreno, F. Garcia-Vidal, “Numerical method for calculating spontaneous emission rate near a surface using Green’s functions,” in Microcavities and Photonic Bandgaps, J. Rarity, C. Weisbuch, eds. (Kluwer, Scientific, Dordrecht, The Netherlands, 1996), pp. 299–308.
[CrossRef]

E. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997).

P. Villeneuve, Sh. Fan, J. Joannopoulos, “Microcavities in photonic crystals,” in Microcavities and Photonic Bandgaps, J. Rarity, C. Weisbuch, eds. (Kluwer Scientific, Dordrecht, The Netherlands, 1996), pp. 133–151.
[CrossRef]

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Schematic representation of three scattering systems consisting of cylindrical objects.

Fig. 2
Fig. 2

Reflectivity of a system without penetration and various numbers of layers. Identical cylinders with radius 0.2 µm; horizontal distance, d = 1 µm; c = 0.5 µm; vertical distance, h = 1.73205 µm (i.e., cylinders lie in the corners of equilateral triangles), n 1 = 1, n 2 = n 3 = 2, normal incidence, TE polarization. (a) N = 7 (i.e., the total number of rod layers is 15), (b) N = 15, (c) N = 100.

Fig. 3
Fig. 3

Reflectivity of a system of circular dielectric rods for 4 rod radii r. d = h = 1 µm, c = 0.5 µm. n 1 = 1, n 2 = n 3 = 1.5, normal incidence, TE polarization. Solid curves, N = 5; dotted curves, N = 25: (a) no interpenetration, (b) the penetration limit. (f) r = 2/4; the cylinders in consecutive layers shifted horizontally by c touch one another.

Fig. 4
Fig. 4

Reflectivity in TM polarization of a system of dielectric rods. Parameters other than that of r are listed in Fig. 3.

Tables (2)

Tables Icon

Table 1 Eigenvalues of the Transmission Matrix for Two Values of Truncation Parameter Ma

Tables Icon

Table 2 Comparison of Computation Times (s)

Equations (33)

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

Fzjx, y=nbnj+expiβnjy+bnj-exp-iβnjyexpiαnx, Fxjx, y=n iqjβnjbnj+expiβnjy-bnj-exp-iβnjyexpiαnx,
qj=1TE1/k2nj2TM,
αm=α0+mK, βmj2=k2nj2-αm2.
K=2π/d,
α0=knNsin θ0N
Ezx, y=m Emyexpiαmx,
Hxx, y=1ωμ0m Hmyexpiαmx.
Hzx, y=1ωμ0m Hmyexpiαmx,
Exx, y=-m Emyexpiαmx.
dEnydy=iHny, dHnydy=-iαn2Eny+i m n-myEmy,
dHnydy=i m n-myEmy, dEnydy=iHny-iαnm n-myαmHmy,
x, y=m myexpim2πd x1k2nj2x, y.
E˜np±yj=δnp, H˜np±yj±i dE˜np±dyy=yj=±βn1δnp,
H˜np±yj=δnp, E˜np±yj=±1k2n12 βn1δnp,
bnj+1,+bnj+1,-=T11jT12jT21jT22jbnj,+bnj,-.
Tj=T11jT12jT21jT22j
T11jT12jT21jT22j=12E˜nm+yj+1+H˜nm+yj+1/q1βm1E˜nm-yj+1+H˜nm-yj+1/q1βm1E˜nm+yj+1-H˜nm+yj+1/q1βm1E˜nm-yj+1-H˜nm-yj+1/q1βm1.
T=TN-1TjN-3T1.
Tj=VΦW,
Tjp=VΦpW,
V=V11V12V21V22,  W=W11W12W21W22.
T=T11N-1T12N-1T21N-1T22N-1V11+V12QV11+V12QW11T121+W12T221,
Qmn=Φ2m/Φ1nN-3W21T121+W22T221×W11T121+W12T221-1mn.
bnj,+bn0,-=S11jS12jS21jS22jbn0,+bnj,-.
S22j=S22j-1Z22j,
S12j=T12j+T11jS12j-1Z22j,
S11j=T11j-T12jS21j-1S11j-1,
S11j=S21j-1-S22jT21jS11j-1,
Z22j=T22j+T21jS12j-1-1,
bnj,-=S22j-1bn1,-,
bnj,+=S12jbnj,-.
bn1,-=S22NbnN,-.
bnj,-=Z22j+1-1Z22j+2-1 Z22N-1-1bnN,-.

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