Abstract

We present a detailed time-domain analysis of a promising nonlinear optical device consisting of alternating layers of nonlinear materials with oppositely signed Kerr coefficients. We study propagation of nonsolitonic (Gaussian) pulses through the device, whose transmittance characteristics point to potential uses in all-optical switches and limiters. If the optical structure has no linear built-in grating, the pulse experiences a nonsolitonic (amplitude-decaying) propagation in the structure, which exhibits limiting properties depending on the bandwidth of the pulse. We elucidate the conditions under which double imaging occurs within the dynamically formed grating under the pulse propagation. In the presence of the linear out-of-phase grating, we observe strong envelope compression and reshaping of a Gaussian pulse, resulting in stable high-amplitude, multiple-peak oscillations as it propagates through the nonlinear optical structure.

© 2003 Optical Society of America

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References

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  1. C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203-259 (1994).
    [CrossRef]
  2. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627-1630 (1996).
    [CrossRef] [PubMed]
  3. N. G. R. Broderick, D. Taverner, and D. J. Richardson, “Nonlinear switching in fiber Bragg gratings,” Opt. Express 3, 447-453 (1998).
    [CrossRef] [PubMed]
  4. N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, “Optical pulse compression in fiber Bragg gratings,” Phys. Rev. Lett. 79, 4566-4569 (1997).
    [CrossRef]
  5. N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, “Experimental observation of nonlinear pulse compression in nonuniform Bragg gratings,” Opt. Lett. 22, 1837-1839 (1997).
    [CrossRef]
  6. N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427-1429 (1992).
    [CrossRef]
  7. C. J. Herbert, W. S. Capinsky, and M. S. Malcuit, “Optical power limiting with nonlinear periodic structures,” Opt. Lett. 17, 1037-1039 (1992).
    [CrossRef] [PubMed]
  8. L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550-555 (2000).
    [CrossRef]
  9. J. He and M. Cada, “Optical bistability in semiconductor periodic structures,” IEEE J. Quantum Electron. 27, 1182-1188 (1991).
    [CrossRef]
  10. D. Pelinovsky, L. Brzozowski, J. Sears, and E. H. Sargent, “Stable all-optical limiting in nonlinear periodic structures. I. Analysis,” J. Opt. Soc. Am. B 19, 43-53 (2002).
    [CrossRef]
  11. D. Pelinovsky and E. H. Sargent, “Stable all-optical limiting in nonlinear periodic structures. II. Computations,” J. Opt. Soc. Am. B 19, 1873-1889 (2002).
    [CrossRef]
  12. W. N. Ye, L. Brzozowski, E. H. Sargent, and D. Pelinovsky, “Nonlinear propagation of ultrashort pulses in nonlinear periodic materials with oppositely-signed Kerr coefficients,” in IEEE LEOS Annual General Meeting Proceedings (Institute of Electrical and Electronics Engineers, San Diego, 2001), pp. 441-442.
  13. E. Johnson and E. H. Sargent, “Function and sensitivity of signal processing systems using addition followed by limiting,” J. Lightwave Technol. (to be published).
  14. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160-163 (1987).
    [CrossRef] [PubMed]
  15. D. L. Mills and S. E. Trullinger, “Gap solitons in nonlinear periodic structures,” Phys. Rev. B 36, 947-952 (1987).
    [CrossRef]
  16. J. E. Sipe and H. G. Winful, “Nonlinear Schro¨dinger solitons in a periodic structure,” Opt. Lett. 13, 132-133 (1988).
    [CrossRef]
  17. C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149-5165 (1988).
    [CrossRef] [PubMed]
  18. D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746-1749 (1989).
    [CrossRef] [PubMed]
  19. A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37-42 (1989).
    [CrossRef]
  20. R. Rangel-Rojo, S. Yamada, H. Matsuda, and D. Yankelevich, “Large near-resonance third-order nonlinearity in an azobenzene functionalized polymer film,” Appl. Phys. Lett. 72, 1021-1023 (1998).
    [CrossRef]
  21. H. S. Loka, S. D. Benjamin, and P. W. E. Smith, “Optical characterization of GaAs for ultrafast switching devices,” IEEE J. Quantum Electron. 34, 1426-1437 (1998).
    [CrossRef]
  22. L. Qian, S. D. Benjamin, P. W. E. Smith, B. J. Robinson, and D. A. Thompson, “Picosecond carrier lifetime and large optical nonlinearities in InGaAsP grown by helium-plasma-assisted molecular beam epitaxy,” Opt. Lett. 22, 108-110 (1997).
    [CrossRef] [PubMed]
  23. E. Garmire, “Resonant optical nonlinearities in semiconductors,” IEEE J. Sel. Top. Quantum Electron. 6, 1094-1110 (2000).
    [CrossRef]
  24. J. D. Begin and M. Cada, “Exact analytic solutions to the non-linear wave equation for a saturable Kerr-like medium: modes of non-linear optical waveguides and couplers,” IEEE J. Quantum Electron. 30, 3006-3016 (1994).
    [CrossRef]
  25. A. Underhill, C. Hill, A. Charlton, S. Oliver, and S. Kreshaw, “Third-order NLO properties of PMMA films co-dispersed with metal dithiolene oligomers,” Synth. Met. 71, 1703–1704 (1995).
    [CrossRef]

2002 (2)

2000 (2)

L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550-555 (2000).
[CrossRef]

E. Garmire, “Resonant optical nonlinearities in semiconductors,” IEEE J. Sel. Top. Quantum Electron. 6, 1094-1110 (2000).
[CrossRef]

1998 (3)

R. Rangel-Rojo, S. Yamada, H. Matsuda, and D. Yankelevich, “Large near-resonance third-order nonlinearity in an azobenzene functionalized polymer film,” Appl. Phys. Lett. 72, 1021-1023 (1998).
[CrossRef]

H. S. Loka, S. D. Benjamin, and P. W. E. Smith, “Optical characterization of GaAs for ultrafast switching devices,” IEEE J. Quantum Electron. 34, 1426-1437 (1998).
[CrossRef]

N. G. R. Broderick, D. Taverner, and D. J. Richardson, “Nonlinear switching in fiber Bragg gratings,” Opt. Express 3, 447-453 (1998).
[CrossRef] [PubMed]

1997 (3)

1996 (1)

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

1995 (1)

A. Underhill, C. Hill, A. Charlton, S. Oliver, and S. Kreshaw, “Third-order NLO properties of PMMA films co-dispersed with metal dithiolene oligomers,” Synth. Met. 71, 1703–1704 (1995).
[CrossRef]

1994 (2)

J. D. Begin and M. Cada, “Exact analytic solutions to the non-linear wave equation for a saturable Kerr-like medium: modes of non-linear optical waveguides and couplers,” IEEE J. Quantum Electron. 30, 3006-3016 (1994).
[CrossRef]

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203-259 (1994).
[CrossRef]

1992 (2)

N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427-1429 (1992).
[CrossRef]

C. J. Herbert, W. S. Capinsky, and M. S. Malcuit, “Optical power limiting with nonlinear periodic structures,” Opt. Lett. 17, 1037-1039 (1992).
[CrossRef] [PubMed]

1991 (1)

J. He and M. Cada, “Optical bistability in semiconductor periodic structures,” IEEE J. Quantum Electron. 27, 1182-1188 (1991).
[CrossRef]

1989 (2)

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746-1749 (1989).
[CrossRef] [PubMed]

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37-42 (1989).
[CrossRef]

1988 (2)

J. E. Sipe and H. G. Winful, “Nonlinear Schro¨dinger solitons in a periodic structure,” Opt. Lett. 13, 132-133 (1988).
[CrossRef]

C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149-5165 (1988).
[CrossRef] [PubMed]

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 structures,” Phys. Rev. B 36, 947-952 (1987).
[CrossRef]

Aceves, A. B.

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37-42 (1989).
[CrossRef]

Begin, J. D.

J. D. Begin and M. Cada, “Exact analytic solutions to the non-linear wave equation for a saturable Kerr-like medium: modes of non-linear optical waveguides and couplers,” IEEE J. Quantum Electron. 30, 3006-3016 (1994).
[CrossRef]

Benjamin, S. D.

Broderick, N. G. R.

Brown, T. G.

N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427-1429 (1992).
[CrossRef]

Brzozowski, L.

D. Pelinovsky, L. Brzozowski, J. Sears, and E. H. Sargent, “Stable all-optical limiting in nonlinear periodic structures. I. Analysis,” J. Opt. Soc. Am. B 19, 43-53 (2002).
[CrossRef]

L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550-555 (2000).
[CrossRef]

Cada, M.

J. D. Begin and M. Cada, “Exact analytic solutions to the non-linear wave equation for a saturable Kerr-like medium: modes of non-linear optical waveguides and couplers,” IEEE J. Quantum Electron. 30, 3006-3016 (1994).
[CrossRef]

J. He and M. Cada, “Optical bistability in semiconductor periodic structures,” IEEE J. Quantum Electron. 27, 1182-1188 (1991).
[CrossRef]

Capinsky, W. S.

Charlton, A.

A. Underhill, C. Hill, A. Charlton, S. Oliver, and S. Kreshaw, “Third-order NLO properties of PMMA films co-dispersed with metal dithiolene oligomers,” Synth. Met. 71, 1703–1704 (1995).
[CrossRef]

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]

Christodoulides, D. N.

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746-1749 (1989).
[CrossRef] [PubMed]

de Sterke, C. M.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203-259 (1994).
[CrossRef]

C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149-5165 (1988).
[CrossRef] [PubMed]

Eggleton, B. J.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

Garmire, E.

E. Garmire, “Resonant optical nonlinearities in semiconductors,” IEEE J. Sel. Top. Quantum Electron. 6, 1094-1110 (2000).
[CrossRef]

He, J.

J. He and M. Cada, “Optical bistability in semiconductor periodic structures,” IEEE J. Quantum Electron. 27, 1182-1188 (1991).
[CrossRef]

Herbert, C. J.

Hill, C.

A. Underhill, C. Hill, A. Charlton, S. Oliver, and S. Kreshaw, “Third-order NLO properties of PMMA films co-dispersed with metal dithiolene oligomers,” Synth. Met. 71, 1703–1704 (1995).
[CrossRef]

Ibsen, M.

N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, “Optical pulse compression in fiber Bragg gratings,” Phys. Rev. Lett. 79, 4566-4569 (1997).
[CrossRef]

N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, “Experimental observation of nonlinear pulse compression in nonuniform Bragg gratings,” Opt. Lett. 22, 1837-1839 (1997).
[CrossRef]

Joseph, R. I.

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746-1749 (1989).
[CrossRef] [PubMed]

Kreshaw, S.

A. Underhill, C. Hill, A. Charlton, S. Oliver, and S. Kreshaw, “Third-order NLO properties of PMMA films co-dispersed with metal dithiolene oligomers,” Synth. Met. 71, 1703–1704 (1995).
[CrossRef]

Krug, P. A.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

Laming, R. I.

N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, “Experimental observation of nonlinear pulse compression in nonuniform Bragg gratings,” Opt. Lett. 22, 1837-1839 (1997).
[CrossRef]

N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, “Optical pulse compression in fiber Bragg gratings,” Phys. Rev. Lett. 79, 4566-4569 (1997).
[CrossRef]

Loka, H. S.

H. S. Loka, S. D. Benjamin, and P. W. E. Smith, “Optical characterization of GaAs for ultrafast switching devices,” IEEE J. Quantum Electron. 34, 1426-1437 (1998).
[CrossRef]

Malcuit, M. S.

Matsuda, H.

R. Rangel-Rojo, S. Yamada, H. Matsuda, and D. Yankelevich, “Large near-resonance third-order nonlinearity in an azobenzene functionalized polymer film,” Appl. Phys. Lett. 72, 1021-1023 (1998).
[CrossRef]

Mills, D. L.

D. L. Mills and S. E. Trullinger, “Gap solitons in nonlinear periodic structures,” 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]

Oliver, S.

A. Underhill, C. Hill, A. Charlton, S. Oliver, and S. Kreshaw, “Third-order NLO properties of PMMA films co-dispersed with metal dithiolene oligomers,” Synth. Met. 71, 1703–1704 (1995).
[CrossRef]

Pelinovsky, D.

Prelewitz, D. F.

N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427-1429 (1992).
[CrossRef]

Qian, L.

Rangel-Rojo, R.

R. Rangel-Rojo, S. Yamada, H. Matsuda, and D. Yankelevich, “Large near-resonance third-order nonlinearity in an azobenzene functionalized polymer film,” Appl. Phys. Lett. 72, 1021-1023 (1998).
[CrossRef]

Richardson, D. J.

Robinson, B. J.

Sankey, N. D.

N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427-1429 (1992).
[CrossRef]

Sargent, E. H.

Sears, J.

Sipe, J. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203-259 (1994).
[CrossRef]

J. E. Sipe and H. G. Winful, “Nonlinear Schro¨dinger solitons in a periodic structure,” Opt. Lett. 13, 132-133 (1988).
[CrossRef]

C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149-5165 (1988).
[CrossRef] [PubMed]

Slusher, R. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

Smith, P. W. E.

Taverner, D.

Thompson, D. A.

Trullinger, S. E.

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

Underhill, A.

A. Underhill, C. Hill, A. Charlton, S. Oliver, and S. Kreshaw, “Third-order NLO properties of PMMA films co-dispersed with metal dithiolene oligomers,” Synth. Met. 71, 1703–1704 (1995).
[CrossRef]

Wabnitz, S.

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37-42 (1989).
[CrossRef]

Winful, H. G.

Yamada, S.

R. Rangel-Rojo, S. Yamada, H. Matsuda, and D. Yankelevich, “Large near-resonance third-order nonlinearity in an azobenzene functionalized polymer film,” Appl. Phys. Lett. 72, 1021-1023 (1998).
[CrossRef]

Yankelevich, D.

R. Rangel-Rojo, S. Yamada, H. Matsuda, and D. Yankelevich, “Large near-resonance third-order nonlinearity in an azobenzene functionalized polymer film,” Appl. Phys. Lett. 72, 1021-1023 (1998).
[CrossRef]

Appl. Phys. Lett. (2)

R. Rangel-Rojo, S. Yamada, H. Matsuda, and D. Yankelevich, “Large near-resonance third-order nonlinearity in an azobenzene functionalized polymer film,” Appl. Phys. Lett. 72, 1021-1023 (1998).
[CrossRef]

N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427-1429 (1992).
[CrossRef]

IEEE J. Quantum Electron. (4)

H. S. Loka, S. D. Benjamin, and P. W. E. Smith, “Optical characterization of GaAs for ultrafast switching devices,” IEEE J. Quantum Electron. 34, 1426-1437 (1998).
[CrossRef]

L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550-555 (2000).
[CrossRef]

J. He and M. Cada, “Optical bistability in semiconductor periodic structures,” IEEE J. Quantum Electron. 27, 1182-1188 (1991).
[CrossRef]

J. D. Begin and M. Cada, “Exact analytic solutions to the non-linear wave equation for a saturable Kerr-like medium: modes of non-linear optical waveguides and couplers,” IEEE J. Quantum Electron. 30, 3006-3016 (1994).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

E. Garmire, “Resonant optical nonlinearities in semiconductors,” IEEE J. Sel. Top. Quantum Electron. 6, 1094-1110 (2000).
[CrossRef]

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

Opt. Express (1)

Opt. Lett. (4)

Phys. Lett. A (1)

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37-42 (1989).
[CrossRef]

Phys. Rev. A (1)

C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149-5165 (1988).
[CrossRef] [PubMed]

Phys. Rev. B (1)

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

Phys. Rev. Lett. (4)

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746-1749 (1989).
[CrossRef] [PubMed]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, “Optical pulse compression in fiber Bragg gratings,” Phys. Rev. Lett. 79, 4566-4569 (1997).
[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]

Prog. Opt. (1)

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203-259 (1994).
[CrossRef]

Synth. Met. (1)

A. Underhill, C. Hill, A. Charlton, S. Oliver, and S. Kreshaw, “Third-order NLO properties of PMMA films co-dispersed with metal dithiolene oligomers,” Synth. Met. 71, 1703–1704 (1995).
[CrossRef]

Other (2)

W. N. Ye, L. Brzozowski, E. H. Sargent, and D. Pelinovsky, “Nonlinear propagation of ultrashort pulses in nonlinear periodic materials with oppositely-signed Kerr coefficients,” in IEEE LEOS Annual General Meeting Proceedings (Institute of Electrical and Electronics Engineers, San Diego, 2001), pp. 441-442.

E. Johnson and E. H. Sargent, “Function and sensitivity of signal processing systems using addition followed by limiting,” J. Lightwave Technol. (to be published).

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Figures (12)

Fig. 1
Fig. 1

Bragg soliton propagation in the system (1)–(2) with nnl=0, n0k=-0.1, n2k=1/2π×10-12 cm2/W. Shown are (a) intensity of the forward wave and (b) intensity of the backward wave. Parameters of the Bragg soliton are V=0.5, Ω=0.01.

Fig. 2
Fig. 2

Decaying Gaussian pulse propagation in the system (1)–(2) without built-in linear grating: nnl=0, n0k=0, n2k=1/2π×10-12 cm2/W. Shown are (a) intensity of the forward wave and (b) intensity of the backward wave. Parameters of the Gaussian pulse are Ipeak=55 GW/cm2, FWHM=27 fs.

Fig. 3
Fig. 3

Propagation of a launched Gaussian pulse in the system with out-of-phase, built-in linear grating n0k=-0.1; compression–decompression cycling is observed. The other parameters are the same as in Fig. 2. Shown are (a) intensity of the forward wave; (b) intensity of the backward wave; (c) top view of the three-dimensional graph of panel (a); (d) top view of the graph of panel (b).

Fig. 4
Fig. 4

Nonlinear periodic device consists of alternating layers of nonlinear materials with oppositely-signed Kerr coefficients. The refractive indices of two adjacent layers are n01+nnl1I and n02+nnl2I. We study three cases: (a) no linear grating; (b) in-phase, linear, built-in grating; (c) out-of-phase, linear, built-in grating.

Fig. 5
Fig. 5

Steady state analysis: transmittance as a function of incident intensity level for various device lengths: L=70 μm, 180 μm and 290 μm. Inset: transmitted intensity level versus incident intensity for the same device, demonstrating characteristic limiting behavior.

Fig. 6
Fig. 6

Pulse transmittance as a function of pulse width for a fixed peak pulse intensity of Ipeak=4 GW/cm2. The transmittance of the device with length L=70 μm, 180 μm, and 290 μm drops to a limiting value.

Fig. 7
Fig. 7

Input and output intensities of pulse through a 180-μm-long device for an input pulse width of (a) 605 fs or characteristic length of 180 μm, and (b) 1440 fs or characteristic length of 435 μm.

Fig. 8
Fig. 8

Heuristic analysis of pulse shaping in a 180-μm-long nonlinear grating. The time-dependent, instantaneous transmittance is attributable to contributions from the forward- and backward-propagating pulse for an input pulse width of (a) 605 fs or characteristic length of 180 μm, and (b) 1440 fs or characteristic length of 435 μm.

Fig. 9
Fig. 9

(a) Total pulse transmitted intensity versus total pulse incident intensity; (b) corresponding energy transmittance as a function of pulse energy incident for linear in- and out-of-phase built-in gratings. Pulse width of 605 fs and device length of 180 μm were fixed for all cases.

Fig. 10
Fig. 10

Output temporal response of the device with length L=70 μm, 180 μm, 290 μm, 360 μm, 720 μm, and 1080 μm for a fixed input pulse with Ipeak=4 GW/cm2 and FWHM=605 fs. Pulse compression, reshaping, and double-peak oscillations are observed.

Fig. 11
Fig. 11

(a) Rate of change in amplitude of the forward-propagating wave; (b) top view of (a); (c) a simplified two-dimensional diagram of a Gaussian incident pulse and a compressed pulse; (d) top view of the intensity profile with respect to time and space. An incident pulse with Ipeak=4 GW/cm2, FWHM=605 fs is launched at the input of a 180-μm-long device.

Fig. 12
Fig. 12

Output transmitted pulse shapes when the intensity of an incident Gaussian pulse is set to (a) Ipeak=2 GW/cm2 and (b) Ipeak=6 GW/cm2. The width of the pulse is FWHM=605 fs and the device is fixed to L=180 μm.

Equations (32)

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iA+Z+A+T+n0kA-+nnl(|A+|2+2|A-|2)A+
+n2k[(2|A+|2+|A-|2)A-+A+2A¯-]=0,
-iA-Z-A-T+n0kA++nnl(2|A+|2+|A-|2)A-
+n2k[(|A+|2+2|A-|2)A++A-2A¯+]=0.
nln=n01+n022,nnl=nnl1+nnl22,
n0k=n01-n02π,n2k=nnl1-nnl2π.
Iin(T)=|A+(Z=0, T)|2,Iref(T)=|A-(Z=0, T)|2,
Iout(T)=|A+(Z=L, T)|2,|A-(Z=L, T)|2=0.
Iin(T)=Ipeakexp-(T-μ)22σ2,
FWHM=2σ(2 ln 2)1/2.
T (|A+|2+|A-|2)+Z (|A+|2-|A-|2)=0.
ddT0L(|A+|2+|A-|2)dZ=Iin(T)-Iref(T)-Iout(T).
Iin=Iref+Iout,
ζ=Z-VT(1-V2)1/2,τ=T-VZ(1-V2)1/2,A±(Z, T)=(1±V)1/2a±(ζ, τ),
ia+ζ+a+τ+n0ka-+n2k[(2(1+V)|a+|2
+(1-V)|a-|2)a-+(1+V)a+2a¯-]=0,
-ia-ζ-a-τ+n0ka++n2k[((1+V)|a+|2
+2(1-V)|a-|2)a++(1-V)a-2a¯+]=0,
a+=[Q(ζ)]1/2exp{i[ϕ(ζ)-ψ(ζ)]+iΩτ},
a-=[Q(ζ)]1/2exp[iϕ(ζ)+iΩτ].
Qζ=Hψ=-2Q sin ψ(n0k+2n2kQ),
ψζ=-HQ=2Ω-2 cos ψ(n0k+4n2kQ),
H=2Q(n0k+2n2kQ)cos ψ-2ΩQ.
cos ψ=Ωn0k+2n2kQ.
Qζ2+U(Q)=0,
U(Q)=-4Q2[(n0k+2n2kQ)2-Ω2].
Qsol=|n0k|-|Ω|2n2k.
Q(ζ)=n0k2-Ω22n2k(|Ω|cosh γζ+|n0k|),
A+=u(Z, T),A-=iy(Z, T),
uZ+uT=-[n0k+n2k(u2+y2)]y,
yZ-yT=-[n0k+n2k(u2+y2)]u.
ωfilter=4(nnl1-nnl2)Ipeakπ(n01-n02) ω0.

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