Abstract

Enhancement of optical nonlinearity in one-dimensional photonic-crystal structures with a defect is considered theoretically. It is shown that a large enhancement can be obtained for absorption saturation and degenerate four-wave mixing efficiency as a result of large optical field amplitude of the localized photonic-defect mode at the defect layer. The figure of merit of the use of the photonic-crystal structure is derived especially for systems in which the concentration of the nonlinear optical material can be arbitrarily adjusted. Optical bistability is also predicted for optically dense samples. They can be applied in real photonic devices because of their simple structure and the large enhancement obtained.

© 1997 Optical Society of America

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References

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  1. C. M. Soukoulis, ed., Photonic Band Gaps and Localization (Plenum, New York, 1993).
  2. S. Radic, N. George, and G. P. Agrawal, “Theory of low-threshold optical switching in nonlinear phase-shifted periodic structures,” J. Opt. Soc. Am. B 12, 671–680 (1995).
    [Crossref]
  3. M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
    [Crossref]
  4. R. Shimano, S. Inouye, M. Kuwata-Gonokami, T. Nakamura, M. Yamanishi, and I. Ogura “Efficient phase conjugation wave generation from a GaAs single quantum well in a microcavity,” Jpn. J. Appl. Phys. 34, L817–L820 (1995).
    [Crossref]
  5. T. Hattori, N. Tsurumachi, N. Muroi, H. Nakatsuka, and E. Ogino, “Enhancement of optical nonlinearity in one-dimensional photonic crystals,” Prog. Crystal Growth Charact. 33, 183–186 (1996).
  6. For example, M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  7. H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry–Perot interferometer,” Phys. Rev. Lett. 36, 1135–1138 (1976).
    [Crossref]
  8. H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann “Optical bistability in semiconductors,” Appl. Phys. Lett. 35, 451–453 (1979).
    [Crossref]
  9. D. Pellat, R. Azoulay, G. Leroux, L. Dugrand, Y. Rafflé, R. Kuszelwicz, and J. L. Oudar, “Optical bistability at 980 nm in a strained InGaAs/GaAs multiple quantum well microcavity with resonant periodic nonlinearity,” Appl. Phys. Lett. 62, 2489–2491 (1993).
    [Crossref]
  10. A. Szöke, V. Daneu, J. Goldhar, and N. A. Kurnit, “Bistable optical element and its applications,” Appl. Phys. Lett. 15, 376–379 (1969).
    [Crossref]

1996 (1)

T. Hattori, N. Tsurumachi, N. Muroi, H. Nakatsuka, and E. Ogino, “Enhancement of optical nonlinearity in one-dimensional photonic crystals,” Prog. Crystal Growth Charact. 33, 183–186 (1996).

1995 (3)

S. Radic, N. George, and G. P. Agrawal, “Theory of low-threshold optical switching in nonlinear phase-shifted periodic structures,” J. Opt. Soc. Am. B 12, 671–680 (1995).
[Crossref]

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[Crossref]

R. Shimano, S. Inouye, M. Kuwata-Gonokami, T. Nakamura, M. Yamanishi, and I. Ogura “Efficient phase conjugation wave generation from a GaAs single quantum well in a microcavity,” Jpn. J. Appl. Phys. 34, L817–L820 (1995).
[Crossref]

1993 (1)

D. Pellat, R. Azoulay, G. Leroux, L. Dugrand, Y. Rafflé, R. Kuszelwicz, and J. L. Oudar, “Optical bistability at 980 nm in a strained InGaAs/GaAs multiple quantum well microcavity with resonant periodic nonlinearity,” Appl. Phys. Lett. 62, 2489–2491 (1993).
[Crossref]

1979 (1)

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann “Optical bistability in semiconductors,” Appl. Phys. Lett. 35, 451–453 (1979).
[Crossref]

1976 (1)

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry–Perot interferometer,” Phys. Rev. Lett. 36, 1135–1138 (1976).
[Crossref]

1969 (1)

A. Szöke, V. Daneu, J. Goldhar, and N. A. Kurnit, “Bistable optical element and its applications,” Appl. Phys. Lett. 15, 376–379 (1969).
[Crossref]

Agrawal, G. P.

Azoulay, R.

D. Pellat, R. Azoulay, G. Leroux, L. Dugrand, Y. Rafflé, R. Kuszelwicz, and J. L. Oudar, “Optical bistability at 980 nm in a strained InGaAs/GaAs multiple quantum well microcavity with resonant periodic nonlinearity,” Appl. Phys. Lett. 62, 2489–2491 (1993).
[Crossref]

Bloemer, M. J.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[Crossref]

Born, M.

For example, M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Bowden, C. M.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[Crossref]

Daneu, V.

A. Szöke, V. Daneu, J. Goldhar, and N. A. Kurnit, “Bistable optical element and its applications,” Appl. Phys. Lett. 15, 376–379 (1969).
[Crossref]

Dowling, J. P.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[Crossref]

Dugrand, L.

D. Pellat, R. Azoulay, G. Leroux, L. Dugrand, Y. Rafflé, R. Kuszelwicz, and J. L. Oudar, “Optical bistability at 980 nm in a strained InGaAs/GaAs multiple quantum well microcavity with resonant periodic nonlinearity,” Appl. Phys. Lett. 62, 2489–2491 (1993).
[Crossref]

George, N.

Gibbs, H. M.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann “Optical bistability in semiconductors,” Appl. Phys. Lett. 35, 451–453 (1979).
[Crossref]

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry–Perot interferometer,” Phys. Rev. Lett. 36, 1135–1138 (1976).
[Crossref]

Goldhar, J.

A. Szöke, V. Daneu, J. Goldhar, and N. A. Kurnit, “Bistable optical element and its applications,” Appl. Phys. Lett. 15, 376–379 (1969).
[Crossref]

Gossard, A. C.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann “Optical bistability in semiconductors,” Appl. Phys. Lett. 35, 451–453 (1979).
[Crossref]

Hattori, T.

T. Hattori, N. Tsurumachi, N. Muroi, H. Nakatsuka, and E. Ogino, “Enhancement of optical nonlinearity in one-dimensional photonic crystals,” Prog. Crystal Growth Charact. 33, 183–186 (1996).

Inouye, S.

R. Shimano, S. Inouye, M. Kuwata-Gonokami, T. Nakamura, M. Yamanishi, and I. Ogura “Efficient phase conjugation wave generation from a GaAs single quantum well in a microcavity,” Jpn. J. Appl. Phys. 34, L817–L820 (1995).
[Crossref]

Kurnit, N. A.

A. Szöke, V. Daneu, J. Goldhar, and N. A. Kurnit, “Bistable optical element and its applications,” Appl. Phys. Lett. 15, 376–379 (1969).
[Crossref]

Kuszelwicz, R.

D. Pellat, R. Azoulay, G. Leroux, L. Dugrand, Y. Rafflé, R. Kuszelwicz, and J. L. Oudar, “Optical bistability at 980 nm in a strained InGaAs/GaAs multiple quantum well microcavity with resonant periodic nonlinearity,” Appl. Phys. Lett. 62, 2489–2491 (1993).
[Crossref]

Kuwata-Gonokami, M.

R. Shimano, S. Inouye, M. Kuwata-Gonokami, T. Nakamura, M. Yamanishi, and I. Ogura “Efficient phase conjugation wave generation from a GaAs single quantum well in a microcavity,” Jpn. J. Appl. Phys. 34, L817–L820 (1995).
[Crossref]

Leroux, G.

D. Pellat, R. Azoulay, G. Leroux, L. Dugrand, Y. Rafflé, R. Kuszelwicz, and J. L. Oudar, “Optical bistability at 980 nm in a strained InGaAs/GaAs multiple quantum well microcavity with resonant periodic nonlinearity,” Appl. Phys. Lett. 62, 2489–2491 (1993).
[Crossref]

McCall, S. L.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann “Optical bistability in semiconductors,” Appl. Phys. Lett. 35, 451–453 (1979).
[Crossref]

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry–Perot interferometer,” Phys. Rev. Lett. 36, 1135–1138 (1976).
[Crossref]

Muroi, N.

T. Hattori, N. Tsurumachi, N. Muroi, H. Nakatsuka, and E. Ogino, “Enhancement of optical nonlinearity in one-dimensional photonic crystals,” Prog. Crystal Growth Charact. 33, 183–186 (1996).

Nakamura, T.

R. Shimano, S. Inouye, M. Kuwata-Gonokami, T. Nakamura, M. Yamanishi, and I. Ogura “Efficient phase conjugation wave generation from a GaAs single quantum well in a microcavity,” Jpn. J. Appl. Phys. 34, L817–L820 (1995).
[Crossref]

Nakatsuka, H.

T. Hattori, N. Tsurumachi, N. Muroi, H. Nakatsuka, and E. Ogino, “Enhancement of optical nonlinearity in one-dimensional photonic crystals,” Prog. Crystal Growth Charact. 33, 183–186 (1996).

Ogino, E.

T. Hattori, N. Tsurumachi, N. Muroi, H. Nakatsuka, and E. Ogino, “Enhancement of optical nonlinearity in one-dimensional photonic crystals,” Prog. Crystal Growth Charact. 33, 183–186 (1996).

Ogura, I.

R. Shimano, S. Inouye, M. Kuwata-Gonokami, T. Nakamura, M. Yamanishi, and I. Ogura “Efficient phase conjugation wave generation from a GaAs single quantum well in a microcavity,” Jpn. J. Appl. Phys. 34, L817–L820 (1995).
[Crossref]

Oudar, J. L.

D. Pellat, R. Azoulay, G. Leroux, L. Dugrand, Y. Rafflé, R. Kuszelwicz, and J. L. Oudar, “Optical bistability at 980 nm in a strained InGaAs/GaAs multiple quantum well microcavity with resonant periodic nonlinearity,” Appl. Phys. Lett. 62, 2489–2491 (1993).
[Crossref]

Passner, A.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann “Optical bistability in semiconductors,” Appl. Phys. Lett. 35, 451–453 (1979).
[Crossref]

Pellat, D.

D. Pellat, R. Azoulay, G. Leroux, L. Dugrand, Y. Rafflé, R. Kuszelwicz, and J. L. Oudar, “Optical bistability at 980 nm in a strained InGaAs/GaAs multiple quantum well microcavity with resonant periodic nonlinearity,” Appl. Phys. Lett. 62, 2489–2491 (1993).
[Crossref]

Radic, S.

Rafflé, Y.

D. Pellat, R. Azoulay, G. Leroux, L. Dugrand, Y. Rafflé, R. Kuszelwicz, and J. L. Oudar, “Optical bistability at 980 nm in a strained InGaAs/GaAs multiple quantum well microcavity with resonant periodic nonlinearity,” Appl. Phys. Lett. 62, 2489–2491 (1993).
[Crossref]

Scalora, M.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[Crossref]

Shimano, R.

R. Shimano, S. Inouye, M. Kuwata-Gonokami, T. Nakamura, M. Yamanishi, and I. Ogura “Efficient phase conjugation wave generation from a GaAs single quantum well in a microcavity,” Jpn. J. Appl. Phys. 34, L817–L820 (1995).
[Crossref]

Szöke, A.

A. Szöke, V. Daneu, J. Goldhar, and N. A. Kurnit, “Bistable optical element and its applications,” Appl. Phys. Lett. 15, 376–379 (1969).
[Crossref]

Tocci, M. D.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[Crossref]

Tsurumachi, N.

T. Hattori, N. Tsurumachi, N. Muroi, H. Nakatsuka, and E. Ogino, “Enhancement of optical nonlinearity in one-dimensional photonic crystals,” Prog. Crystal Growth Charact. 33, 183–186 (1996).

Venkatesan, T. N. C.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann “Optical bistability in semiconductors,” Appl. Phys. Lett. 35, 451–453 (1979).
[Crossref]

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry–Perot interferometer,” Phys. Rev. Lett. 36, 1135–1138 (1976).
[Crossref]

Wiegmann, W.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann “Optical bistability in semiconductors,” Appl. Phys. Lett. 35, 451–453 (1979).
[Crossref]

Wolf, E.

For example, M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Yamanishi, M.

R. Shimano, S. Inouye, M. Kuwata-Gonokami, T. Nakamura, M. Yamanishi, and I. Ogura “Efficient phase conjugation wave generation from a GaAs single quantum well in a microcavity,” Jpn. J. Appl. Phys. 34, L817–L820 (1995).
[Crossref]

Appl. Phys. Lett. (4)

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[Crossref]

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann “Optical bistability in semiconductors,” Appl. Phys. Lett. 35, 451–453 (1979).
[Crossref]

D. Pellat, R. Azoulay, G. Leroux, L. Dugrand, Y. Rafflé, R. Kuszelwicz, and J. L. Oudar, “Optical bistability at 980 nm in a strained InGaAs/GaAs multiple quantum well microcavity with resonant periodic nonlinearity,” Appl. Phys. Lett. 62, 2489–2491 (1993).
[Crossref]

A. Szöke, V. Daneu, J. Goldhar, and N. A. Kurnit, “Bistable optical element and its applications,” Appl. Phys. Lett. 15, 376–379 (1969).
[Crossref]

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

Jpn. J. Appl. Phys. (1)

R. Shimano, S. Inouye, M. Kuwata-Gonokami, T. Nakamura, M. Yamanishi, and I. Ogura “Efficient phase conjugation wave generation from a GaAs single quantum well in a microcavity,” Jpn. J. Appl. Phys. 34, L817–L820 (1995).
[Crossref]

Phys. Rev. Lett. (1)

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry–Perot interferometer,” Phys. Rev. Lett. 36, 1135–1138 (1976).
[Crossref]

Prog. Crystal Growth Charact. (1)

T. Hattori, N. Tsurumachi, N. Muroi, H. Nakatsuka, and E. Ogino, “Enhancement of optical nonlinearity in one-dimensional photonic crystals,” Prog. Crystal Growth Charact. 33, 183–186 (1996).

Other (2)

For example, M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

C. M. Soukoulis, ed., Photonic Band Gaps and Localization (Plenum, New York, 1993).

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Figures (11)

Fig. 1
Fig. 1

Model of 1-D PC structures.

Fig. 2
Fig. 2

Calculated transmission spectrum of a 1-D PC with a defect with N=5, κ=0.003, and nAdA=640 nm/4.

Fig. 3
Fig. 3

Calculated field pattern of light at the frequency of the midgap transmission peak that is incident normally upon the PC structure with N=5 and with no absorbance of the defect layer.

Fig. 4
Fig. 4

Normalized transmittance change of the PC sample with N=5 plotted as a function of the extinction coefficient as obtained in Eq. (23).

Fig. 5
Fig. 5

Dependence of the intensity-enhancement factor G on the extinction coefficient κ for the PC structure of N=5.

Fig. 6
Fig. 6

Normalized transmittance change for the PC sample with N=5 and for a naked sample plotted as a function of the linear transmittance.

Fig. 7
Fig. 7

Ratio of the normalized transmittance change for the PC sample with N=5 and that for a naked sample, plotted as a function of the linear transmittance.

Fig. 8
Fig. 8

Relation between the input intensity and the transmittance for several values of y0, which characterizes the magnitude of the linear transmittance.

Fig. 9
Fig. 9

Range of the normalized input intensity, x, where bistability is observed, as a function of the linear transmittance, T0.

Fig. 10
Fig. 10

Normalized DFWM generation efficiency for the PC sample with N=5 and for a naked sample plotted as a function of the linear transmittance.

Fig. 11
Fig. 11

Ratio of the normalized DFWM generation efficiency for the PC sample with N=5 and that for a naked sample, plotted as a function of the linear transmittance.

Equations (58)

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t=-22 coshπκnX+sinhπκnX(nB/nA)2N1+iκ/nX+1+iκ/nX(nB/nA)2N,
T=|t|2.
G=T nX2πκsinhπκnXcoshπκnXnBnA2N nX2nX2+κ2+nAnB2N+sinhπκnX 2nX2nX2+κ2.
(nB/nA)N1,
t=-22+πκnXnBnA2N,
T=42+πκnXnBnA2N2,
G=T2nBnA2N,
nXκ.
κ=κ01+I/Is.
I=GIin,
(y0-y)(y+2)2-xy=0,
x2nBnA2N IinIS,
yπnXnBnA2Nκ,
y0πnXnBnA2Nκ0,
T=4(y+2)2.
I(z)=2GIin cos2(πz/dX)(-dX/2<z<dX/2).
κ(z)=κ01+I(z)/IS,
κ¯=-dX/2dX/2κ(z)I(z)dz-dX/2dX/2I(z)dz=κ0 2(1+p-1)p1+p,
p2GIin/IS.
dκ¯dIin=-32κG/IS.
dTdIin=dTdκ¯dκ¯dIin=1IS24πκnXnBnA4N2+πκnXnBnA2N-5.
ΔdTdIinIS.
ΔPC=24πκnXnBnA4N2+πκnXnBnA2N-5.
Δnaked=exp(-αL)[1-exp(-αL)].
α=4πκλ,
κ(nX/π)(nA/nB)2N
αL1,
ΔPC=3πκ4nXnBnA4N
Δnaked=4πκLλ
FΔPC/Δnaked=38nBnA4N.
κ=nX2πnAnB2N,
T=0.64,
ΔPCmax=3843125nBnA2N.
Δnakedmax=14
FΔΔPCmax/Δnakedmax=15363125nBnA2N.
κ=nXπnAnB2N,
T=4/9,
ΔTPCmax=29nBnA2N.
ΔTnakedmax=1
FΔ/TΔTPCmaxΔTnakedmax=29nBnA2N.
ΔPC(T)=32nBnA2NT2(1-T),
Δnaked(T)=T(1-T).
F(T)ΔPC(T)/Δnaked(T)=32nBnA2N T1+T,
y0-14(y+y2+8y0y)(y+2)2-xy=0,
x2nBnA2N IinIS,
yπnXnBnA2Nκ¯,
y0πnXnBnA2Nκ0,
T=4(y+2)2.
Eout=Eout(1)+Eout(3).
Iout=c|Eout|2=c{|Eout(1)|2+2|Eout(1)Eout(3)|}
T=IoutIin=T0+2T0Eout(3)Ein,
Eout(1)=T0Ein.
dTdIin=2T0 ddIinEout(3)Ein.
|Eout(3)|=Iin2T0dTdIin|Ein|.
IDFWM=14TdTdIin2Iin3.
WIDFWMIS2/Iin3
WPC=916nBnA4NT3(1-T)2
Wnaked=14T(1-T)2

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