Abstract

A formalism is presented for optical waveguiding by means of Bragg reflection. The theory provides analytic expressions for field modal profiles, dispersion, and attenuation. Waveguiding in line defects of photonic bandgap crystals, a special case of the theory, is used as an example. Quantized allowed widths of the guiding channel are predicted.

© 2002 Optical Society of America

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References

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  1. P. Yeh and A. Yariv, Opt. Commun. 19, 427 (1976).
    [CrossRef]
  2. A. Y. Cho, A. Yariv, and P. Yeh, Appl. Phys. Lett. 30, 471 (1977).
    [CrossRef]
  3. P. Yeh, A. Yariv, and E. Marom, J. Opt. Soc. Am 68, 1196 (1978).
    [CrossRef]
  4. S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joanopoulos, Opt. Lett. 25, 1297 (2000).
    [CrossRef]
  5. M. Loncar, D. Dedeljkovic, T. Doll, A. Scherer, J. Vuckovic, and T. Pearsall, Appl. Phys. Lett. 25, 1937 (2000).
    [CrossRef]
  6. K. Sakoda, Phys. Rev. B 51, 4672 (1995).
    [CrossRef]
  7. See, for example, A. Yariv, Optical Electronics in Modern Communication, 5th ed. (Oxford U. Press, Oxford, 1997).
  8. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), p. 478.
  9. See, for example, N. Ashcroft and D. Mermin, Solid State Physics (Holt, Rinehart & Winston, New York, 1976), p. 133.
  10. M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, Electron. Lett. 37, 293 (2001).
    [CrossRef]

2001 (1)

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, Electron. Lett. 37, 293 (2001).
[CrossRef]

2000 (2)

S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joanopoulos, Opt. Lett. 25, 1297 (2000).
[CrossRef]

M. Loncar, D. Dedeljkovic, T. Doll, A. Scherer, J. Vuckovic, and T. Pearsall, Appl. Phys. Lett. 25, 1937 (2000).
[CrossRef]

1995 (1)

K. Sakoda, Phys. Rev. B 51, 4672 (1995).
[CrossRef]

1978 (1)

P. Yeh, A. Yariv, and E. Marom, J. Opt. Soc. Am 68, 1196 (1978).
[CrossRef]

1977 (1)

A. Y. Cho, A. Yariv, and P. Yeh, Appl. Phys. Lett. 30, 471 (1977).
[CrossRef]

1976 (1)

P. Yeh and A. Yariv, Opt. Commun. 19, 427 (1976).
[CrossRef]

Ashcroft, N.

See, for example, N. Ashcroft and D. Mermin, Solid State Physics (Holt, Rinehart & Winston, New York, 1976), p. 133.

Cho, A. Y.

A. Y. Cho, A. Yariv, and P. Yeh, Appl. Phys. Lett. 30, 471 (1977).
[CrossRef]

Chow, E.

Dedeljkovic, D.

M. Loncar, D. Dedeljkovic, T. Doll, A. Scherer, J. Vuckovic, and T. Pearsall, Appl. Phys. Lett. 25, 1937 (2000).
[CrossRef]

Doll, T.

M. Loncar, D. Dedeljkovic, T. Doll, A. Scherer, J. Vuckovic, and T. Pearsall, Appl. Phys. Lett. 25, 1937 (2000).
[CrossRef]

Joanopoulos, J. D.

Johnson, S. G.

Lin, S. Y.

Loncar, M.

M. Loncar, D. Dedeljkovic, T. Doll, A. Scherer, J. Vuckovic, and T. Pearsall, Appl. Phys. Lett. 25, 1937 (2000).
[CrossRef]

Marom, E.

P. Yeh, A. Yariv, and E. Marom, J. Opt. Soc. Am 68, 1196 (1978).
[CrossRef]

Mermin, D.

See, for example, N. Ashcroft and D. Mermin, Solid State Physics (Holt, Rinehart & Winston, New York, 1976), p. 133.

Notomi, M.

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, Electron. Lett. 37, 293 (2001).
[CrossRef]

Pearsall, T.

M. Loncar, D. Dedeljkovic, T. Doll, A. Scherer, J. Vuckovic, and T. Pearsall, Appl. Phys. Lett. 25, 1937 (2000).
[CrossRef]

Sakoda, K.

K. Sakoda, Phys. Rev. B 51, 4672 (1995).
[CrossRef]

Scherer, A.

M. Loncar, D. Dedeljkovic, T. Doll, A. Scherer, J. Vuckovic, and T. Pearsall, Appl. Phys. Lett. 25, 1937 (2000).
[CrossRef]

Shinya, A.

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, Electron. Lett. 37, 293 (2001).
[CrossRef]

Takahashi, C.

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, Electron. Lett. 37, 293 (2001).
[CrossRef]

Takahashi, J.

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, Electron. Lett. 37, 293 (2001).
[CrossRef]

Vuckovic, J.

M. Loncar, D. Dedeljkovic, T. Doll, A. Scherer, J. Vuckovic, and T. Pearsall, Appl. Phys. Lett. 25, 1937 (2000).
[CrossRef]

Yamada, K.

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, Electron. Lett. 37, 293 (2001).
[CrossRef]

Yariv, A.

P. Yeh, A. Yariv, and E. Marom, J. Opt. Soc. Am 68, 1196 (1978).
[CrossRef]

A. Y. Cho, A. Yariv, and P. Yeh, Appl. Phys. Lett. 30, 471 (1977).
[CrossRef]

P. Yeh and A. Yariv, Opt. Commun. 19, 427 (1976).
[CrossRef]

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), p. 478.

See, for example, A. Yariv, Optical Electronics in Modern Communication, 5th ed. (Oxford U. Press, Oxford, 1997).

Yeh, P.

P. Yeh, A. Yariv, and E. Marom, J. Opt. Soc. Am 68, 1196 (1978).
[CrossRef]

A. Y. Cho, A. Yariv, and P. Yeh, Appl. Phys. Lett. 30, 471 (1977).
[CrossRef]

P. Yeh and A. Yariv, Opt. Commun. 19, 427 (1976).
[CrossRef]

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), p. 478.

Yokohama, I.

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, Electron. Lett. 37, 293 (2001).
[CrossRef]

Appl. Phys. Lett. (2)

M. Loncar, D. Dedeljkovic, T. Doll, A. Scherer, J. Vuckovic, and T. Pearsall, Appl. Phys. Lett. 25, 1937 (2000).
[CrossRef]

A. Y. Cho, A. Yariv, and P. Yeh, Appl. Phys. Lett. 30, 471 (1977).
[CrossRef]

Electron. Lett. (1)

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, Electron. Lett. 37, 293 (2001).
[CrossRef]

J. Opt. Soc. Am (1)

P. Yeh, A. Yariv, and E. Marom, J. Opt. Soc. Am 68, 1196 (1978).
[CrossRef]

Opt. Commun. (1)

P. Yeh and A. Yariv, Opt. Commun. 19, 427 (1976).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (1)

K. Sakoda, Phys. Rev. B 51, 4672 (1995).
[CrossRef]

Other (3)

See, for example, A. Yariv, Optical Electronics in Modern Communication, 5th ed. (Oxford U. Press, Oxford, 1997).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), p. 478.

See, for example, N. Ashcroft and D. Mermin, Solid State Physics (Holt, Rinehart & Winston, New York, 1976), p. 133.

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Figures (2)

Fig. 1
Fig. 1

Triangular array of holes bounding a clear GC as the model used in the present analysis.

Fig. 2
Fig. 2

Transverse modal field distributions for the lowest-order low-loss modes of the structure of Fig. 1. Note that these are two physically distinct waveguides with different GC widths.

Equations (23)

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2E+ω2μϵ0n2rE=0.
ExW/2=Ei exp-ikx+βz+Er exp-i-kx+βz,
n2x,z=l,mn2l,m expiKl,m·r,
Kl,m=l2πbxˆ+m2πazˆ,
n2l,m=1ab-b/2b/2-a/2a/2n2x,zexp-iKl,m·rdxdz,
kikxˆ+βzˆ,    kr=-kxˆ+βzˆ
Ex,z=Eixexp-iki·r+Erxexp-ikr·r.
-2ikEixexp-iki·r+2ikErxexp-ikr·r=-ω2μϵ0l,m0n2l,m expiKl,m·r×Ei exp-iki·r+Er exp-ikr·r.
kr-ki=±Kl,m,    ki=k,β,    kr=-k,β
Eix=κl,0Er,    Erx=κl,0*Ei,
β2=ω/c2n20,0-lπb21/2,  l=1,2,.
rupper=ErxEixx=+W/2=-κl,0*|κl,0uppertanhκl,0upperL, rlower=EixErxx=-W/2=-κl,0*|κl,0lowertanhκl,0lowerL,
n2l,0=1ab-b/2b/2-a/2a/2-n02-1Aholeδxδz+δx-b2δz-a2exp-il2πbxdxdz=-n02-1Aholeab1+exp-ilπ,
n21,0=0,    n22,0=-2(n02-1Aholeab,
κ2,0=i2πn02-1Aholeaλ02,
r=i tanhκ2,0L.
Er expikW/2Ei exp-ikW/2=r,    Ei expikW/2Er exp-ikW/2=r,
r2 exp-i2kW=1.
2kW-2θr=s2π,  s=1,2,,  rrexpiθr.
W=b4,3b4,5b4...
W=b2, Ex,z=exp-iβz×E0 cos2πbxxb8E0 cos2πbxexp-κ2,0x-b8xb8E0 cos2πbxexpκ2,0x+b8x-b8
W=3b4, Ex,z=exp-iβz×E0 sin2πbxx3b8E0 sin2πbxexp-κ2,0x-3b8x3b8E0 sin2πbxexpκ2,0x+3b8x-3b8.
α=16π exp-2κLβb2

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