Abstract

Nonlinear evolution of coupled forward and backward fields in a multi-layered film is numerically investigated. We examine the role of longitudinal and transverse modulation instabilities in media of finite length with a homogeneous nonlinear susceptibility χ(3). The numerical solution of the nonlinear equations by a beam-propagation method that handles backward waves is described.

© 2002 Optical Society of America

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References

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  1. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
    [Crossref] [PubMed]
  2. C. M. de Sterke and J. E. Sipe, “Gap Solitons,” Progress in Optics 33, 203–260 (1994).
  3. M. Scalora, J.P. Dowling, C.M. Bowden, and M. J. Bloemer,“ Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
    [Crossref] [PubMed]
  4. M. Scalora, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
    [Crossref]
  5. J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
    [Crossref]
  6. M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–1081 (1996).
    [Crossref]
  7. G. G. Luther and C. J. McKinstrie, “ Transverse modulational instability of collinear waves,” J. Opt. Soc. Am. B 7, 1125–1141 (1990).
    [Crossref]
  8. L.W. Liou, X.D. Cao, C.J. McKinstrie, and G.P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
    [Crossref] [PubMed]
  9. Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. 15, 1282–1284 (1990).
    [Crossref] [PubMed]
  10. B. J. Eggleton, C. M. deSterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
    [Crossref]
  11. B. J. Eggleton, C. M. deSterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
    [Crossref]
  12. N. M. Lichinitzer, C. J. McKinstrie, C. M. de Sterke, and G. P. Agrawal, “Spatiotemporal instabilities in nonlinear bulk media with Bragg gratings,” J. Opt. Soc. Am. B 18, 45–54 (2001).
    [Crossref]
  13. M. Scalora and M. Crenshaw, “Beam propagation method that handles reflections,” Optics Commun. 108, 191–196 (1994).
    [Crossref]

2001 (1)

1997 (1)

1996 (2)

B. J. Eggleton, C. M. deSterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
[Crossref]

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–1081 (1996).
[Crossref]

1994 (5)

C. M. de Sterke and J. E. Sipe, “Gap Solitons,” Progress in Optics 33, 203–260 (1994).

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

M. Scalora, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[Crossref]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[Crossref]

M. Scalora and M. Crenshaw, “Beam propagation method that handles reflections,” Optics Commun. 108, 191–196 (1994).
[Crossref]

1992 (1)

L.W. Liou, X.D. Cao, C.J. McKinstrie, and G.P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[Crossref] [PubMed]

1990 (2)

1987 (1)

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

Agrawal, G. P.

Agrawal, G.P.

L.W. Liou, X.D. Cao, C.J. McKinstrie, and G.P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[Crossref] [PubMed]

Bendickson, J. M.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–1081 (1996).
[Crossref]

Bloemer, M. J.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–1081 (1996).
[Crossref]

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

M. Scalora, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[Crossref]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[Crossref]

Bowden, C. M.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–1081 (1996).
[Crossref]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[Crossref]

M. Scalora, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[Crossref]

Bowden, C.M.

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

Cao, X.D.

L.W. Liou, X.D. Cao, C.J. McKinstrie, and G.P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[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]

Crenshaw, M.

M. Scalora and M. Crenshaw, “Beam propagation method that handles reflections,” Optics Commun. 108, 191–196 (1994).
[Crossref]

de Sterke, C. M.

deSterke, C. M.

B. J. Eggleton, C. M. deSterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[Crossref]

B. J. Eggleton, C. M. deSterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
[Crossref]

Dowling, J. P.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–1081 (1996).
[Crossref]

M. Scalora, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[Crossref]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[Crossref]

Dowling, J.P.

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

Eggleton, B. J.

B. J. Eggleton, C. M. deSterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[Crossref]

B. J. Eggleton, C. M. deSterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
[Crossref]

Flynn, R. J.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–1081 (1996).
[Crossref]

Fork, R. L.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–1081 (1996).
[Crossref]

Leavitt, R. P.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–1081 (1996).
[Crossref]

Ledbetter, H. S.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–1081 (1996).
[Crossref]

Lichinitzer, N. M.

Liou, L.W.

L.W. Liou, X.D. Cao, C.J. McKinstrie, and G.P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[Crossref] [PubMed]

Luther, G. G.

McKinstrie, C. J.

McKinstrie, C.J.

L.W. Liou, X.D. Cao, C.J. McKinstrie, and G.P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[Crossref] [PubMed]

Mills, D. L.

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

Reinhardt, S. B.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–1081 (1996).
[Crossref]

Scalora, M.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–1081 (1996).
[Crossref]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[Crossref]

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

M. Scalora, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[Crossref]

M. Scalora and M. Crenshaw, “Beam propagation method that handles reflections,” Optics Commun. 108, 191–196 (1994).
[Crossref]

Silberberg, Y.

Sipe, J. E.

B. J. Eggleton, C. M. deSterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
[Crossref]

C. M. de Sterke and J. E. Sipe, “Gap Solitons,” Progress in Optics 33, 203–260 (1994).

Slusher, R. E.

B. J. Eggleton, C. M. deSterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[Crossref]

B. J. Eggleton, C. M. deSterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
[Crossref]

Tocci, M. D.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–1081 (1996).
[Crossref]

Electron. Lett. (1)

B. J. Eggleton, C. M. deSterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
[Crossref]

J. Appl. Phys. (2)

M. Scalora, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[Crossref]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[Crossref]

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

Opt. Lett. (1)

Optics Commun. (1)

M. Scalora and M. Crenshaw, “Beam propagation method that handles reflections,” Optics Commun. 108, 191–196 (1994).
[Crossref]

Phys. Rev. A (1)

L.W. Liou, X.D. Cao, C.J. McKinstrie, and G.P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[Crossref] [PubMed]

Phys. Rev. E (1)

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–1081 (1996).
[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 photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[Crossref] [PubMed]

Progress in Optics (1)

C. M. de Sterke and J. E. Sipe, “Gap Solitons,” Progress in Optics 33, 203–260 (1994).

Supplementary Material (3)

» Media 1: GIF (197 KB)     
» Media 2: GIF (216 KB)     
» Media 3: GIF (156 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

Transmission curve and input pulse spectrum. The parameters have been scaled as discussed in the text. The frequency is shifted relative to the laser line, which is placed at the first transmission maximum on the high frequency side.

Fig. 2.
Fig. 2.

Animation of the pulse evolution through the nonlinear medium with initial detuning δ = 1.12, i.e. the central frequency of the laser is tuned to the first transmission maximum on the high frequency branch. The initial amplitude is A=0.2. Top panel is the pulse amplitude launched outside the medium. The center panel is the forward-propagating pulse amplitude and the bottom panel is the backward propagating pulse amplitude. The numbers in brackets denote the maximum value of the intensity in that plot (196 KB gif animation).

Figure 3:
Figure 3:

The pulse amplitude launched in the forward direction is shown in the top panel. The bottom panel is the backward propagating pulse amplitude. See Figure 2 for a discussion of parameters. For this case A=0.6 (216 KB gif animation).

Figure 4:
Figure 4:

A three-dimensional representation of the final frame in Fig. 3. The pulse amplitude in the forward direction is shown in the top panel. The bottom panel is the backward propagating pulse amplitude.

Fig. 5:
Fig. 5:

The spatial spectrum, H(q), of the pulse after traversing the PBG, as defined in Eq. (11) and the peak is normalized to unity. The three curves correspond to the amplitudes A=0.2 (solid blue line), 0.3 (dash-dotted red line), and 0.4 (dashed black line).

Fig. 6.
Fig. 6.

Top panel is the pulse amplitude in the forward-propagating direction and the bottom panel is the backward propagating pulse amplitude. The parameters are the same as in discussed earlier, except δ = -1.12 and A= 0.7 (156 KB gif animation).

Equations (16)

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

1 v E f t = E f z + i F 2 E f + E f + i κ E b + ( E f 2 + 2 E b 2 ) E f
1 v E b t = + E b z + i F 2 E b + E b + i κ E f + ( E b 2 + 2 E f 2 ) E b
E f ( x , y , 0 , t ) = S ( x , y , t ) ,
E b ( x , y , L , t ) = 0 .
E f ( q , z , ω ) = S ( q , ω ) ( cos ( Δ z ) + i Ω Δ sin ( Δ z ) + κ 2 Δ sin ( Δ L ) sin ( Δ z ) [ Δ cos ( Δ L ) + i Ω sin ( Δ L ) ] )
E b ( q , z , ω ) = S ( q , ω ) i κ sin ( Δ ( z L ) ) [ Δ cos ( Δ L ) + i Ω sin ( Δ L ) ]
L ff = v ( z + i F 2 + ) ; ,
L bb = v ( z + i F 2 + ) ; .
N ff = ivη ( E f 2 + 2 E b 2 ) ; N bb = ivη ( E b 2 + 2 E f 2 ) ;
K fb = ivκ ; K bf = ivκ .
U t = ( L + V ) U .
U ( t + Δ t ) = exp ( Δ tL 2 ) exp ( Δ tV ) exp ( Δ tL 2 ) U ( t ) .
exp ( Δ tV ) = exp ( Δ tK 2 ) exp ( Δ tN ) exp ( Δ tK 2 ) .
K 2 = δ 2 κ 2 .
F ( x , z , 0 ) = A exp ( ( z z 0 ) 2 σ z 2 ) exp ( x 2 σ x 2 ) .
H ( q ) = ∫∫ e iqx E f ( x , z , t ) dzdx ,

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