Abstract

We present a theoretical analysis of wave propagation in rectangular- and square-lattice photonic crystals by approximating the arbitrary refractive-index function with a staircase profile. This profile has a number of tunable parameters that allow one to fit the band structures of the staircase and the desired structure over a large frequency range. The staircase profile is such that its corresponding two-dimensional wave equation decomposes into two one-dimensional equations by means of a constant decoupling parameter. We show that basic features of the band structure and Bloch waves in photonic crystals can be analyzed theoretically through this simple approximation.

© 2003 Optical Society of America

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References

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  1. J. D. Joannopolous, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).
  2. P. Tran, Phys. Rev. B 52, 10673 (1995).
    [CrossRef]
  3. B. Garlak, S. Enoch, and G. Tayeb, J. Opt. Soc. Am. A 17, 1012 (2000).
    [CrossRef]
  4. D. Felbaque, G. Tayeb, and D. Maystre, J. Opt. Soc. Am. A 11, 2526 (1994).
    [CrossRef]
  5. B. Garlak, S. Enoch, and G. Tayeb, J. Opt. Soc. Am. A 19, 1547 (2002).
    [CrossRef]
  6. D. C. Dobson, J. Comput. Phys. 149, 363 (1999).
    [CrossRef]
  7. S. Chaowei, C. H. Chan, and K. F. Chan, in Proceedings of IEEE Antennas and Propagation Society International Symposium, 2002 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), Vol. 4, pp. 356–359.
  8. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  9. S. Khorasani and A. Adibi, Opt. Commun. 216, 439 (2003).
    [CrossRef]
  10. A. Krokhin, P. Halevi, and J. Arriaga, Phys. Rev. B 65, 115220 (2002).
    [CrossRef]

2003 (1)

S. Khorasani and A. Adibi, Opt. Commun. 216, 439 (2003).
[CrossRef]

2002 (3)

A. Krokhin, P. Halevi, and J. Arriaga, Phys. Rev. B 65, 115220 (2002).
[CrossRef]

B. Garlak, S. Enoch, and G. Tayeb, J. Opt. Soc. Am. A 19, 1547 (2002).
[CrossRef]

S. Chaowei, C. H. Chan, and K. F. Chan, in Proceedings of IEEE Antennas and Propagation Society International Symposium, 2002 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), Vol. 4, pp. 356–359.

2000 (1)

1999 (1)

D. C. Dobson, J. Comput. Phys. 149, 363 (1999).
[CrossRef]

1995 (2)

J. D. Joannopolous, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

P. Tran, Phys. Rev. B 52, 10673 (1995).
[CrossRef]

1994 (1)

D. Felbaque, G. Tayeb, and D. Maystre, J. Opt. Soc. Am. A 11, 2526 (1994).
[CrossRef]

1984 (1)

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Adibi, A.

S. Khorasani and A. Adibi, Opt. Commun. 216, 439 (2003).
[CrossRef]

Arriaga, J.

A. Krokhin, P. Halevi, and J. Arriaga, Phys. Rev. B 65, 115220 (2002).
[CrossRef]

Chan, C. H.

S. Chaowei, C. H. Chan, and K. F. Chan, in Proceedings of IEEE Antennas and Propagation Society International Symposium, 2002 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), Vol. 4, pp. 356–359.

Chan, K. F.

S. Chaowei, C. H. Chan, and K. F. Chan, in Proceedings of IEEE Antennas and Propagation Society International Symposium, 2002 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), Vol. 4, pp. 356–359.

Chaowei, S.

S. Chaowei, C. H. Chan, and K. F. Chan, in Proceedings of IEEE Antennas and Propagation Society International Symposium, 2002 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), Vol. 4, pp. 356–359.

Dobson, D. C.

D. C. Dobson, J. Comput. Phys. 149, 363 (1999).
[CrossRef]

Enoch, S.

Felbaque, D.

D. Felbaque, G. Tayeb, and D. Maystre, J. Opt. Soc. Am. A 11, 2526 (1994).
[CrossRef]

Garlak, B.

Halevi, P.

A. Krokhin, P. Halevi, and J. Arriaga, Phys. Rev. B 65, 115220 (2002).
[CrossRef]

Joannopolous, J. D.

J. D. Joannopolous, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

Khorasani, S.

S. Khorasani and A. Adibi, Opt. Commun. 216, 439 (2003).
[CrossRef]

Krokhin, A.

A. Krokhin, P. Halevi, and J. Arriaga, Phys. Rev. B 65, 115220 (2002).
[CrossRef]

Maystre, D.

D. Felbaque, G. Tayeb, and D. Maystre, J. Opt. Soc. Am. A 11, 2526 (1994).
[CrossRef]

Meade, R. D.

J. D. Joannopolous, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

Tayeb, G.

Tran, P.

P. Tran, Phys. Rev. B 52, 10673 (1995).
[CrossRef]

Winn, J. N.

J. D. Joannopolous, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

J. Comput. Phys. (1)

D. C. Dobson, J. Comput. Phys. 149, 363 (1999).
[CrossRef]

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

D. Felbaque, G. Tayeb, and D. Maystre, J. Opt. Soc. Am. A 11, 2526 (1994).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

S. Khorasani and A. Adibi, Opt. Commun. 216, 439 (2003).
[CrossRef]

Phys. Rev. B (2)

A. Krokhin, P. Halevi, and J. Arriaga, Phys. Rev. B 65, 115220 (2002).
[CrossRef]

P. Tran, Phys. Rev. B 52, 10673 (1995).
[CrossRef]

Other (3)

J. D. Joannopolous, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

S. Chaowei, C. H. Chan, and K. F. Chan, in Proceedings of IEEE Antennas and Propagation Society International Symposium, 2002 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2002), Vol. 4, pp. 356–359.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

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Figures (2)

Fig. 1
Fig. 1

Staircase approximation of Fig. 1: profiles of χx and ψy functions as given by Eq. (2) and the corresponding 2D unit cell.

Fig. 2
Fig. 2

Structure for TE modes (electric field normal to the plane of periodicity) of allowed bands of the square-lattice PC of Si rods (r=0.4L, where L is the lattice constant). Solid curves, original profile of Fig. 1 computed by the PWE; dotted curves, staircase profile of Fig. 2 computed from analytical expressions given by Eqs. (6a) and (6b). (a) Band structure along high-symmetry directions, (b) constant-frequency contour map of the first band, (c) constant-frequency contour map of the second band.

Equations (14)

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

x,y=χx+ψy,
χx=a+ux+½t-ux-½tb-ax½X,  χx=χx+mX,
ψy=c+uy+½s-uy-½sd-cy½Y,  ψy=ψy+nY,
2x2+2y2Ax,y+ω2c2x,yAx,y=0,
2x2Λx+ω2c2χx+βΛx=0,
2y2Ψy+ω2c2ψy-βΨy=0,
cosκX=q112exp-jωXcχ-½X+β1/2+q222exp+jωXcχ-½X+β1/2,
cosηY=p112exp-jωYcψ-½Y-β1/2+p222exp+jωYcψ-½Y-β1/2.
cosκX=coskbtcoskaX-t-ka2+kb22kakbsinkbtsinkaX-t,
cosηY=coskdscoskcY-s-kc2+kd22kckdsinkdssinkcY-s,
fx,y=mnfmn expj2πmxX+nyY,
fmn=1XYunit cellfx,y×exp-j2πmxX+nyYdxdy,
χx+ψy=1-tXa+tXb+1-sYc+sYd+2πb-am>0sinπmt/Xm×cos2πmxX+2πd-cn>0sinπns/Yn×cos2πnyY.
t=Xπcos-1Rf20Rf10,  a=f002-πtRf10X sinπt/X, b=a+πRf10sinπt/X.

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