Abstract

An extension of the recently introduced nonlinear finite-difference time-domain technique [Opt. Lett. 21, 1138 (1996)] for the study of electromagnetic wave propagation in a non-linear Kerr medium to include absorption is presented. The optical limiting and switching of short pulses by use of a nonlinear quarter-wave reflector (a one-dimensional photonic bandgap structure) with a defect is studied. Comparison with an optical limiter and with an optical switch with a perfect nonlinear quarter-wave reflector shows that introducing a defect can improve the performance of these devices.

© 1997 Optical Society of America

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References

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  1. P. Tran, “Optical switching with a nonlinear photonic crystal: a numerical study,” Opt. Lett. 21, 1138–1140 (1996).
    [CrossRef] [PubMed]
  2. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell'sequations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).
  3. M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonicband gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
    [CrossRef] [PubMed]
  4. For a review of photonic switches see Y. Silberberg, “Photonicswitching devices,” Opt. News 5(2), 7–12(1989).
  5. For a review of optical bistability see H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, Orlando, Fla., 1985).
  6. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
    [CrossRef] [PubMed]
  7. D. L. Mills and S. E. Trullinger, “Gap solitons in nonlinear periodic structure,” Phys. Rev. B 36, 947–952 (1987).
    [CrossRef]
  8. K. Hayata, A. Misawa, and M. Koshiba, “Spatial polarization instabilities due to transverse effects in nonlinearguided-wave systems,” J. Opt. Soc. Am. B 7, 1268–1280 (1990), and references therein.
    [CrossRef]
  9. C. De Angelis and G. F. Nalesso, “Spatial soliton switching modes of nonlinear optical slab waveguides,” J. Opt. Soc. Am. B 10, 55–59 (1993), and references therein.
    [CrossRef]
  10. J. U. Kang, G. I. Stegeman, and J. S. Aitchison, “One-dimensional spatial soliton dragging, trapping, and all-opticalswitching in AlGaAs waveguides,” Opt. Lett. 21, 189–191 (1996).
    [CrossRef] [PubMed]
  11. R. M. Joseph and A. Taflove, “FDTD Maxwell's equations models for nonlinear electrodynamics and optics,” IEEE Trans. Antennas Propag. 45, 364–374 (1997).
    [CrossRef]
  12. R. W. Ziolkowski, “The incorporation of microscopic material models into the FDTD approachfor ultrafast optical pulse simulations,” IEEE Trans. Antennas Propag. 45, 375–391 (1997).
    [CrossRef]
  13. P. Yang and K. N. Liou, “Finite-difference time domain method for light scattering by smallice crystals in three-dimensional space,” J. Opt. Soc. Am. A 13, 2072–2085 (1996).
    [CrossRef]
  14. G. Mur, “Absorbing boundary conditions for finite-difference approximation ofthe time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 1073–1077 (1981).
    [CrossRef]

1997 (2)

R. M. Joseph and A. Taflove, “FDTD Maxwell's equations models for nonlinear electrodynamics and optics,” IEEE Trans. Antennas Propag. 45, 364–374 (1997).
[CrossRef]

R. W. Ziolkowski, “The incorporation of microscopic material models into the FDTD approachfor ultrafast optical pulse simulations,” IEEE Trans. Antennas Propag. 45, 375–391 (1997).
[CrossRef]

1996 (3)

1994 (1)

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonicband gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

1993 (1)

1990 (1)

1987 (2)

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

D. L. Mills and S. E. Trullinger, “Gap solitons in nonlinear periodic structure,” Phys. Rev. B 36, 947–952 (1987).
[CrossRef]

1981 (1)

G. Mur, “Absorbing boundary conditions for finite-difference approximation ofthe time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 1073–1077 (1981).
[CrossRef]

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell'sequations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

Aitchison, J. S.

Bloemer, M. J.

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonicband gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

Bowden, C. M.

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonicband gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

Chen, W.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

De Angelis, C.

Dowling, J. P.

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonicband gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

Hayata, K.

Joseph, R. M.

R. M. Joseph and A. Taflove, “FDTD Maxwell's equations models for nonlinear electrodynamics and optics,” IEEE Trans. Antennas Propag. 45, 364–374 (1997).
[CrossRef]

Kang, J. U.

Koshiba, M.

Liou, K. N.

Mills, D. L.

D. L. Mills and S. E. Trullinger, “Gap solitons in nonlinear periodic structure,” Phys. Rev. B 36, 947–952 (1987).
[CrossRef]

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Misawa, A.

Mur, G.

G. Mur, “Absorbing boundary conditions for finite-difference approximation ofthe time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 1073–1077 (1981).
[CrossRef]

Nalesso, G. F.

Scalora, M.

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonicband gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

Stegeman, G. I.

Taflove, A.

R. M. Joseph and A. Taflove, “FDTD Maxwell's equations models for nonlinear electrodynamics and optics,” IEEE Trans. Antennas Propag. 45, 364–374 (1997).
[CrossRef]

Tran, P.

Trullinger, S. E.

D. L. Mills and S. E. Trullinger, “Gap solitons in nonlinear periodic structure,” Phys. Rev. B 36, 947–952 (1987).
[CrossRef]

Yang, P.

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell'sequations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

Ziolkowski, R. W.

R. W. Ziolkowski, “The incorporation of microscopic material models into the FDTD approachfor ultrafast optical pulse simulations,” IEEE Trans. Antennas Propag. 45, 375–391 (1997).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell'sequations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

R. M. Joseph and A. Taflove, “FDTD Maxwell's equations models for nonlinear electrodynamics and optics,” IEEE Trans. Antennas Propag. 45, 364–374 (1997).
[CrossRef]

R. W. Ziolkowski, “The incorporation of microscopic material models into the FDTD approachfor ultrafast optical pulse simulations,” IEEE Trans. Antennas Propag. 45, 375–391 (1997).
[CrossRef]

IEEE Trans. Electromagn. Compat. (1)

G. Mur, “Absorbing boundary conditions for finite-difference approximation ofthe time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 1073–1077 (1981).
[CrossRef]

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

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

Opt. Lett. (2)

Phys. Rev. B (1)

D. L. Mills and S. E. Trullinger, “Gap solitons in nonlinear periodic structure,” Phys. Rev. B 36, 947–952 (1987).
[CrossRef]

Phys. Rev. Lett. (2)

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonicband gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

Other (2)

For a review of photonic switches see Y. Silberberg, “Photonicswitching devices,” Opt. News 5(2), 7–12(1989).

For a review of optical bistability see H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, Orlando, Fla., 1985).

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Figures (12)

Fig. 1
Fig. 1

Transmission as a function of the frequency (the band structure) for the quarter-wave reflector (shown at the bottom). The arrow indicates the frequency of operation of the band-edge optical limiter.

Fig. 2
Fig. 2

Field intensity inside the quarter-wave reflector when a plane wave with frequency ω/c0=0.865 (2π/λ0) is incident upon it.

Fig. 3
Fig. 3

Band structure of the quarter-wave reflector with a defect (shown at the bottom). The arrow indicates the frequency of operation of the defect-mode optical limiter.

Fig. 4
Fig. 4

Field intensity inside the structure shown at the bottom of Fig. 3 when a plane wave with frequency ω/c0=0.973 (2π/λ0) is incident upon it.

Fig. 5
Fig. 5

Band structure of a structure corresponding to placing the structure shown in Fig. 3 a distance λ0/4 in front of the structure shown in Fig. 1.

Fig. 6
Fig. 6

Electric field intensity inside the structure of Fig. 5 when a plane wave with frequency ω/c0=0.973 (2π/λ0) is incident upon it.

Fig. 7
Fig. 7

Transmitted intensity as a function of the incident field strength for the band-edge and the defect-mode optical limiters.

Fig. 8
Fig. 8

Fractional transmitted intensity as a function of the incident field strength for the band-edge and the defect-mode optical limiters.

Fig. 9
Fig. 9

Fractional transmitted intensity, normalized to the fractional transmitted intensity in the linear regime, as a function of the incident field strength for the band-edge and the defect-mode optical limiters.

Fig. 10
Fig. 10

Same as Fig. 7 but with absorption included.

Fig. 11
Fig. 11

Transmission as a function of the frequency for (a) a perfect quarter-wave reflector and (b) one with a defect. The arrows indicate the frequencies of the pump and probe beams used.

Fig. 12
Fig. 12

Spectrum of the transmitted energy with (solid curves) and without (dashed curves) the pump beam for (a) a perfect quarter-wave reflector and (b) one with a defect.

Equations (13)

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1c0D(r, t)t=×H(r, t),
1c0H(r, t)t=-×E(r, t),
D(r, t)=E(r, t)+χ|E(r, t)|2E(r, t).
|D(r, t)|2=2A+2χA2+χ2A3,
1c0D(r, t)t+4πc0J(r, t)=×H(r, t),
1c0D(r, t)t+τc0[r+χ|E(r, t)|2]E(r, t)
-τc0χ|E(r, t)|2E(r, t)=×H(r, t),
1c0[eτ tD(r, t)]t
=eτ t×H(r, t)+τc0χeτ t|E(r, t)|2E(r, t).
D(r, Δt)=exp(-τΔt)D(r, 0)+(c0Δt)×exp(-τΔt/2)×H(r, Δt/2)+(τΔt)χ2[|E(r, Δt)|2E(r, Δt)+exp(-τΔt)|E(r, 0)|2E(r, 0)].
rE(r, Δt)+χ1-τΔt2|E(r, Δt)|2E(r, Δt)
=exp (-τΔt)D(r, 0)+(c0Δt)×exp(-τΔt/2)×H(r, Δt/2)+(τΔt)χ2exp(-τΔt)|E(r, 0)|2E(r, 0).
Ez(x, t)=E0 exp{-iω0/c0[c0t-0(x-x0)]-[c0t-0(x-x0)]2(Δω/c0)2/2},

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