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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 9 — May. 1, 2006
  • pp: 4005–4012
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New all-optical switch based on the spatial soliton repulsion

Yaw-Dong Wu  »View Author Affiliations


Optics Express, Vol. 14, Issue 9, pp. 4005-4012 (2006)
http://dx.doi.org/10.1364/OE.14.004005


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Abstract

We propose an all-optical switching device based on the interaction property between optical spatial solitons. By launching the nonlinear symmetric modes for the relative phase relation Π into the uniform nonlinear medium, the repulsive property between spatial solitons will be observed. Based on the repulsive property, a new all-optical switching device will be proposed.

© 2006 Optical Society of America

1. Introduction

All-optical switching devices based on the optical Kerr effect in a nonlinear waveguide have been object of great interest for high-bit rate optical communication systems and ultra-fast information processing systems. In the past, a number of all-optical switching devices have been proposed by using a nonlinear interferometer [1–2

1. K. J. Blow, N. J. Doran, and B. K. Nayar, “Experimental demonstration of optical soliton switching in an all-fiber nonlinear Sagnac Interferometer,” Opt. Lett. 14, 754 (1989). [CrossRef] [PubMed]

], a nonlinear directional coupler [3

3. S. R. Friberg, A. M. Weiner, Y. Silberberg, B. G. Sfez, and P. S. Smith, “Femotosecond switching in dual-core-fiber nonlinear coupler,” Opt. Lett. 13, 904 (1988). [CrossRef] [PubMed]

], and a nonlinear waveguide junction [4–6

4. Y. Silberberg and B. G. Sfez, “All-optical phase- and power-controlled switching in nonlinear waveguide junctions,” Opt. Lett. 13, 1132 (1988). [CrossRef] [PubMed]

]. Further, a number of theoretical studies about wave propagating along the guided-wave systems made from linear and nonlinear material have also been reported [7–18

7. Y. D. Wu, M. H. Chen, and H. J. Tasi, “Analyzing multiplayer optical waveguides with nonlinear cladding and substrates,” J. Opt. Soc. Am. B. 19, 1737 (2002). [CrossRef]

].

Recently, the application of the spatial soliton in the all-optical device [19–28

19. Y. D. Wu, M. H. Chen, and C. H. Chu, “All-optical logic device using bent nonlinear tapered Y-junction waveguide structure,” Fiber Integr. Opt. 20, 517 (2001).

] has been discussed ardently. Spatial solitons are the result of balance between diffraction effect and self-focusing effect. The light beams propagating through a bulk medium tend to broaden due to diffraction. However, the spatial broadening due to diffraction can be compensated by using the features of nonlinear optics which make the evolutions of light beams sharpen due to self-focusing. Once the diffraction effect and the self-focusing effect balance for each other, the spatial solitons can be formed [29–30

29. P. A. Belanger and P. Mathieu, “Dark soliton in a Kerr defocusing medium,” Appl. Opt. 26, 111 (1987). [CrossRef] [PubMed]

]. This is a similar situation that exists in the time domain for the optical pulses. The optical pulses propagating along a fiber tend to broaden due to group velocity dispersion. Once the group velocity dispersion effect and the self-phase modulation effect balance for each other, the temporal optical solitons can be formed.

The interaction between spatial solitons has attracted much attention because they resemble real particles in the interaction properties [31–32

31. S. Blair and K. Wagner, “Spatialcsoliton angular deflection logic gates,” Appl. Opt. 38, 6749 (1999). [CrossRef]

]. When one fundamental soliton is launched parallel to another, they will attract or repel each other, depending on the relative phase relation between them. The interaction is attractive with the relative phase of 0 in phase, whereas the interaction is repulsive with the relative phase of π out of phase. The strength of interaction also depends upon the initial separation distance between spatial solitons.

Based on the repulsion between spatial solitons, a new all-optical switch will be proposed. By launching the nonlinear symmetric modes with the relative phase of π into the uniform nonlinear medium, the repulsive property between spatial solitons is observed. These characteristics are investigated by using the beam propagation method (BPM) [33

33. A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface,” Phy. Rev. A. 39, 1809 (1989). [CrossRef]

]. By fixing the input signal power and changing the control power, the numerical results show that this device could really function as an all-optical switch.

2. Analysis

The structure of the proposed all-optical switching device is shown in Fig. 1. It is divided into three sections: the input section, the uniform nonlinear medium section and the output section. The lengths of the three sections are L 1, L 2, and L 3, respectively. In the input section, the straight signal guide is in the center and the two outward guides are control guides. The two control guides are parallel to the signal guide and thus the control beam and the signal beam can repel each other in the nonlinear medium. The separated distance between the control guide and the signal guide is denoted S, and the width of each guiding film is denoted W. In the nonlinear medium section, the spatial solitons are repelling each other. In the output section, the nonlinear waveguides are used to couple out the spatial solitons excited by the input signal beams. The separation between each output guide is sufficiently large to prevent the mutual coupling. Since the control beam is useless in the following process, we may use a lossy medium to attenuate it. For simplicity, we consider the case of TE waves propagating along the structure as

εxzt=Exzexp[j(βk0zwt)]
(1)
Fig. 1. The proposed 1×N all-optical switching device.

where k 0 is the wave number in the free space and β is the effective refractive index, and we have taken the field to be homogeneous in the y direction. Taking into account the slowly varying envelope approximation, we obtain the following equation for E(x, z):

2k0Ez+2Ex2+k02[ni2xzE2β2]E=0i=f,c,u
(2)

where the subscripts f, c, and u are used to denote the guiding film, the cladding, and the uniform nonlinear medium, respectively. For a Kerr-type nonlinear medium, the square of the refractive index ni2 can be expressed as:

ni2=nio2+αE2
(3)

where nio is the linear refractive index of the nonlinear medium and α is the nonlinear coefficient (α=0, for the linear medium). All the refractive indices in the proposed structure are ni = nco in the cladding of the input and output sections, ni=nfo2+αE2 in the guiding film of the input section, ni=nfo2+αE2 in the guiding film of the output section, and ni=nuo2+αE2 in the uniform nonlinear medium section.

3. Results and discussions

Fig. 2. The position shift Δd is plotted as a function of the left normalized control power Pc /Po when the width of each guiding film W=1.5μm and the separated distance S=2.5μm.

We use the BPM to simulate TE waves propagating along the structure. For the calculations, we choose the following numerical data: the transverse sampling points N=1280, a longitudinal step length Δz = 0.05 μm, the width of each guiding film W=1.5μm, the free space wavelength λ =1.55μm, L1=50μm, L2=220μm, L3=50μm, n f0 = 1.53, n c0 = n u0 = 1.5, α =6.3786μm 2 /V 2, S=2.5μm, the optimum input signal power is fixed at P 0 =0.07W/mm, and the input control power is varied from 0.82P 0 to 1.16P 0 The symbol Δd is used to denote the position shift of the output signal beam propagating throughout the uniform nonlinear medium and the symbol Pc/P0 is used to denote the normalized control power. The position shift Δd as a function of the normalized control power Pc/P0 with the left control beam is shown in Fig. 2. Symmetry guarantees similar results when the right control beam is on. The results shown above can be used to design an all-optical 1×N switching device by using the position shift of the output signal beam. For example, we proposed a 1×9 all-optical switching device, as show in Fig. 3.

When there is no control beam, the output signal beam will propagate straight through the output waveguide E, as shown in Fig. 4. When the left control beam is on, the output signal beam will swing in the uniform nonlinear medium by using the repulsive property between optical spatial solitons. When the control power reaches Pc = 0.832P0, the output signal beam will be switched to the right output waveguide A with the position shift Δd = 2μm, as shown in Fig. 5, when the control power reaches Pc = 0.864P0, the output signal beam will be switched to the right output waveguide B with the position shift Δd = 4μm, as shown in Fig. 6, when the control power reaches Pc = 0.91P0, the output signal beam will be switched to the right output waveguide C with the position shift Δd = 6μm, as shown in Fig. 7, and, last, when the control power reaches Pc = 1.02P0, the output signal beam will be switched to the right output waveguide D with the position shift Δd = μm, as shown in Fig. 8. Symmetry guarantees similar results when the right control beam is on. The output signal beams will be switched to the left output guides from one waveguide to another. These results shown above can be used to design an all-optical switching device by using the position shift of the output signal beam.

Fig. 3. The proposed 1×9 all-optical switching device.
Fig. 4. The evolutions of the signal beam propagating along the structure without the control beam.
Fig. 5. The evolutions of the signal beam propagating along the structure with the left control beam at Pc=0.832Po, Δd=2μm.
Fig. 6. The evolutions of the signal beam propagating along the structure with the left control beam at Pc=0.864Po, Δd=4μm
Fig. 7. The evolutions of the signal beam propagating along the structure with the left control beam at Pc=0.91Po, Δd=6μm
Fig. 8. The evolutions of the signal beam propagating along the structure with the left control beam at Pc=1.02 Po, Δd=8μm
Fig. 9. The position shift Δd is plotted as a function of the left normalized control power Pc /Po when the signal power Po=0.07W/mm and the width of each guiding film W= 1.5 μm.
Fig. 10. The position shift Δd is plotted as a function of the left normalized control power Pc /Po when the signal power Po=0.07W/mm and and the separated distance S=2.5 μm.

In order to further understand the influence of the separated distance S and the width of each guiding film W of the position shift, more detail results are shown in Figs. 9–10. Figure 9 shows that when we fix the width of each guiding film W=1.5μm and Po=0.07W/mm, and increase the separated distance S, the position shift will also decrease consequently. Because when S increases, the repulsive force between the spatial solitons will decrease. Figure 10 shows that when we fix S=2.5 μm and Po=0.07W/mm, and increase the width of each guiding film W, the position shift will increase at the beginning and then decrease consequently. Because when the normalized control power Pc/Po is less than 0.85, as W increases, the repulsive force between the spatial solitons will also increase; but when Pc/Po is large than 0.85, as W increases, the repulsive force between the spatial solitons will decrease.

4. Conclusions

In the paper, we have designed a new 1×N all-optical switch by using the spatial soliton repulsion. The proposed structure is an all-optical switch controlled by two control beams. If there is no control beam, the signal beam will be coupled out in the central output guide. When the control beam is on, the output signal beam will swing in the uniform nonlinear medium. The numerical results show that the proposed waveguide structure could really function as a all-optical switching device. It is a potential key component in the application of optical signal processing and optical computing systems.

Acknowledgments

This work was partly supported by National Science Council R. O. C. under Grant No. 94-2215-E-151-001.

References and links

1.

K. J. Blow, N. J. Doran, and B. K. Nayar, “Experimental demonstration of optical soliton switching in an all-fiber nonlinear Sagnac Interferometer,” Opt. Lett. 14, 754 (1989). [CrossRef] [PubMed]

2.

L. Thylen, N. Finalayson, C. T. Seaton, and G. I. Stegeman, “All-optical guided-wave Mach-Zender switching device,” Appl. Phys. Lett. 51, 1304 (1987). [CrossRef]

3.

S. R. Friberg, A. M. Weiner, Y. Silberberg, B. G. Sfez, and P. S. Smith, “Femotosecond switching in dual-core-fiber nonlinear coupler,” Opt. Lett. 13, 904 (1988). [CrossRef] [PubMed]

4.

Y. Silberberg and B. G. Sfez, “All-optical phase- and power-controlled switching in nonlinear waveguide junctions,” Opt. Lett. 13, 1132 (1988). [CrossRef] [PubMed]

5.

J. P. Sabini, N. Finalyson, and G. I. Stegeman, “All-optical switching in nonlinear X junctions.” Appl. Phys. Lett. 55, 1176 (1989). [CrossRef]

6.

H. Fouckhardt and Y. Silberberg, “All-optical switching in waveguide X junctions,” J. Opt. Soc. Am. B. 7, 803 (1990). [CrossRef]

7.

Y. D. Wu, M. H. Chen, and H. J. Tasi, “Analyzing multiplayer optical waveguides with nonlinear cladding and substrates,” J. Opt. Soc. Am. B. 19, 1737 (2002). [CrossRef]

8.

S. She and S. Zhang, “Analysis of nonlinear TE waves in a periodic refractive index waveguide with nonlinear cladding,” Opt. Commun. 161, 141 (1999). [CrossRef]

9.

U. Trutschel, F. Lederer, and M. Golz, “Nonlinear guided waves in multiplayer systems,” IEEE J. Quantum Electron 25, 194 (1989). [CrossRef]

10.

S. She and S. Zhang, “Analysis of nonlinear TE waves in a periodic refractive index waveguide with nonlinear cladding,” Opt. Commun. 161, 141 (1999). [CrossRef]

11.

Y. D. Wu, M. H. Chen, and H. J. Tasi, “A general method for analyzing the multilayer optical waveguide with nonlinear cladding and substrate,” SPIE Design, Fabrication, and Characterization of Photonic Device II , 4594, 323 (2001).

12.

Y. D. Wu and M. H. Chen, “The fundamental theory of the symmetric three layer nonlinear optical waveguide structure and the numerical simulation,” J. Nat. Kao. Uni. of App. Sci. 32, 133 (2002).

13.

M. H. Chen, Y. D. Wu, and R. Z. Tasy, “Analyses of antisymmetric modes of three-layer nonlinear optical waveguide,” J. Nat. Kao. Uni. of App. Sci. , 34, 1 (2005).

14.

H. Murata, M. Izutsu, and T. Sueta. “Optical bistability and all-optical switching in novel waveguide junctions with localized optical nonlinearity,” J. Lightwave Technol. 16, 833 (1998). [CrossRef]

15.

Y. D. Wu, “Analyzing multilayer optical waveguide with a localized arbitrary nonlinear guiding film,” IEEE J. Quantum Electron. 40, 529 (2004). [CrossRef]

16.

Y. D. Wu and D. H. Cai, “Analytical and numerical analyses of TE-polarized waves in the planner optical waveguides with the nonlinear guiding film,” J. Eng. Technol. and Edu. 1, 19 (2004).

17.

Yi-Fan Li and K. Iizuka, “Unified nonlinear waveguide dispersion equations with our spurious roots,” IEEE J. Quantum. Electron. 31, 791 (1995). [CrossRef]

18.

Y. D. Wu and M.-H. Chen, “Method for analyzing multilayer nonlinear optical waveguide,” Opt. Express. 137982 (2005). [CrossRef] [PubMed]

19.

Y. D. Wu, M. H. Chen, and C. H. Chu, “All-optical logic device using bent nonlinear tapered Y-junction waveguide structure,” Fiber Integr. Opt. 20, 517 (2001).

20.

Y. D. Wu, “Nonlinear all-optical switching device by using the spatial soliton collision,” Fiber Integr. Opt. 23, 387 (2004). [CrossRef]

21.

Y. D. Wu, “New all-optical wavelength auto-router based on spatial solitons,” Opt. Express. 12, 4172 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-18-4172 [CrossRef] [PubMed]

22.

Y. D Wu, Y. F. Laio, M. H. Chen, and K. H. Chiang, “Nonlinear all-optical phase and power-controlled switch by using the spatial solitons interaction,” in Nonlinear Optical Phenomena and Applications; Q. Gong, Y. Cui, and R. A. Lessard; Eds. Proc SPIE. 5646, 334–344 (2005). [CrossRef]

23.

Y. D. Wu, Y. F. Laio, and M. H. Chiang, “A new all-optical phase-controlled routing switch,” in Nonlinear Optical Phenomena and Applications; Q. Gong, Y. Cui, and R. A. Lessard; Eds. Proc. SPIE. 5646, 345–357 (2005). [CrossRef]

24.

Y. D. Wu, “1×N all-optical switching device by using the phase modulation of spatial solitons,” Appl. Opt. 44, 4114 (2005). [CrossRef]

25.

Y. D. Wu, “All-optical logic gates by using multibranch waveguide structure with localized optical nonlinearity,” IEEE J. Sel. Top. Quantum. Electron. 11, 307 (2005). [CrossRef]

26.

T. Yabu, M. Geshiro, T. Kitamura, K. Nishida, and S. Sawa, “All-optical logic gates containing a two-mode nonlinear waveguide,” IEEE J. Qunatum. Electron. 38, 37 (2002). [CrossRef]

27.

F. Garzia and M. Bertolotti, “All-optical security coded key,” Opt. Quantum. Electron. 33, 527 (2001). [CrossRef]

28.

Y. H. Pramono and Endarko, “Nonlinear waveguides for optical logic and computation,” J. Nonlinear Opt. Phys. Mater. 10, 209 (2001). [CrossRef]

29.

P. A. Belanger and P. Mathieu, “Dark soliton in a Kerr defocusing medium,” Appl. Opt. 26, 111 (1987). [CrossRef] [PubMed]

30.

J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, and P. W. E. Smith, “Observation of spatial optical solitons in a nonlinear glass waveguide,” Opt. Lett. 15, 471 (1990). [CrossRef] [PubMed]

31.

S. Blair and K. Wagner, “Spatialcsoliton angular deflection logic gates,” Appl. Opt. 38, 6749 (1999). [CrossRef]

32.

H. F. Chou, C. F. Lin, and G. C. Wang, “An interative finite difference beam propagation method for modeling second-order nonlinear effects in optical waveguides,” J. J. Lightwave Technol. 16, 1686 (1998). [CrossRef]

33.

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface,” Phy. Rev. A. 39, 1809 (1989). [CrossRef]

OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(190.3270) Nonlinear optics : Kerr effect
(230.1150) Optical devices : All-optical devices

ToC Category:
Nonlinear Optics

History
Original Manuscript: January 23, 2006
Revised Manuscript: April 7, 2006
Manuscript Accepted: April 8, 2006
Published: May 1, 2006

Citation
Yaw-Dong Wu, "New all-optical switch based on the spatial soliton repulsion," Opt. Express 14, 4005-4012 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-9-4005


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References

  1. K. J. Blow, N. J. Doran, and B. K. Nayar, "Experimental demonstration of optical soliton switching in an all-fiber nonlinear Sagnac Interferometer," Opt. Lett. 14, 754 (1989). [CrossRef] [PubMed]
  2. L. Thylen, N. Finalayson, C. T. Seaton, and G. I. Stegeman, "All-optical guided-wave Mach-Zender switching device," Appl. Phys. Lett. 51, 1304 (1987). [CrossRef]
  3. S. R. Friberg, A. M. Weiner, Y. Silberberg, B. G. Sfez, and P. S. Smith, "Femotosecond switching in dual-core-fiber nonlinear coupler," Opt. Lett. 13, 904 (1988). [CrossRef] [PubMed]
  4. Y. Silberberg and B. G. Sfez, "All-optical phase- and power-controlled switching in nonlinear waveguide junctions," Opt. Lett. 13, 1132 (1988). [CrossRef] [PubMed]
  5. J. P. Sabini, N. Finalyson, and G. I. Stegeman, "All-optical switching in nonlinear X junctions." Appl. Phys. Lett. 55, 1176 (1989). [CrossRef]
  6. H. Fouckhardt and Y. Silberberg, "All-optical switching in waveguide X junctions," J. Opt. Soc. Am. B. 7, 803 (1990). [CrossRef]
  7. Y. D. Wu, M. H. Chen, and H. J. Tasi, "Analyzing multiplayer optical waveguides with nonlinear cladding and substrates," J. Opt. Soc. Am. B. 19, 1737 (2002). [CrossRef]
  8. S. She and S. Zhang, "Analysis of nonlinear TE waves in a periodic refractive index waveguide with nonlinear cladding," Opt. Commun. 161, 141 (1999). [CrossRef]
  9. U. Trutschel, F. Lederer, and M. Golz, "Nonlinear guided waves in multiplayer systems," IEEE J. Quantum Electron 25, 194 (1989). [CrossRef]
  10. S. She and S. Zhang, "Analysis of nonlinear TE waves in a periodic refractive index waveguide with nonlinear cladding," Opt. Commun. 161, 141 (1999). [CrossRef]
  11. Y. D. Wu, M. H. Chen, and H. J. Tasi, "A general method for analyzing the multilayer optical waveguide with nonlinear cladding and substrate," SPIE Design, Fabrication, and Characterization of Photonic Device II,  4594, 323 (2001).
  12. Y. D. Wu and M. H. Chen, "The fundamental theory of the symmetric three layer nonlinear optical waveguide structure and the numerical simulation," J. Nat. Kao. Uni. of App. Sci. 32,133 (2002).
  13. M. H. Chen, Y. D. Wu, and R. Z. Tasy, "Analyses of antisymmetric modes of three-layer nonlinear optical waveguide," J. Nat. Kao. Uni. of App. Sci.,  34, 1 (2005).
  14. H. Murata, M. Izutsu, and T. Sueta. "Optical bistability and all-optical switching in novel waveguide junctions with localized optical nonlinearity," J. Lightwave Technol. 16, 833 (1998). [CrossRef]
  15. Y. D. Wu, "Analyzing multilayer optical waveguide with a localized arbitrary nonlinear guiding film," IEEE J. Quantum Electron. 40, 529 (2004). [CrossRef]
  16. Y. D. Wu and D. H. Cai, "Analytical and numerical analyses of TE-polarized waves in the planner optical waveguides with the nonlinear guiding film," J. Eng. Technol. and Edu. 1, 19 (2004).
  17. Yi-Fan Li and K. Iizuka, "Unified nonlinear waveguide dispersion equations with our spurious roots," IEEE J. Quantum. Electron. 31, 791 (1995). [CrossRef]
  18. Y. D. Wu and M.-H. Chen, "Method for analyzing multilayer nonlinear optical waveguide," Opt. Express. 137982 (2005). [CrossRef] [PubMed]
  19. Y. D. Wu, M. H. Chen, and C. H. Chu, "All-optical logic device using bent nonlinear tapered Y-junction waveguide structure," Fiber Integr. Opt. 20, 517 (2001).
  20. Y. D. Wu, "Nonlinear all-optical switching device by using the spatial soliton collision," Fiber Integr. Opt. 23, 387 (2004). [CrossRef]
  21. Y. D. Wu, "New all-optical wavelength auto-router based on spatial solitons," Opt. Express. 12, 4172 (2004). [CrossRef] [PubMed]
  22. Y. D Wu, Y. F. Laio, M. H. Chen, and K. H. Chiang, "Nonlinear all-optical phase and power-controlled switch by using the spatial solitons interaction," in Nonlinear Optical Phenomena and Applications; Q. Gong, Y. Cui, R. A. Lessard; Eds.Proc SPIE. 5646, 334-344 (2005). [CrossRef]
  23. Y. D. Wu, Y. F. Laio, and M. H. Chiang, "A new all-optical phase-controlled routing switch," in Nonlinear Optical Phenomena and Applications; Q. Gong, Y. Cui, R. A. Lessard; Eds. Proc. SPIE. 5646, 345-357 (2005). [CrossRef]
  24. Y. D. Wu, "1xN all-optical switching device by using the phase modulation of spatial solitons," Appl. Opt. 44, 4114 (2005). [CrossRef]
  25. Y. D. Wu, "All-optical logic gates by using multibranch waveguide structure with localized optical nonlinearity," IEEE J. Sel. Top. Quantum. Electron. 11, 307 (2005). [CrossRef]
  26. T. Yabu, M. Geshiro, T. Kitamura, K. Nishida, and S. Sawa, "All-optical logic gates containing a two-mode nonlinear waveguide," IEEE J. Qunatum. Electron. 38, 37 (2002). [CrossRef]
  27. F. Garzia, and M. Bertolotti, "All-optical security coded key," Opt. Quantum. Electron. 33, 527 (2001). [CrossRef]
  28. Y. H. Pramono, and Endarko, "Nonlinear waveguides for optical logic and computation," J. Nonlinear Opt. Phys. Mater. 10, 209 (2001). [CrossRef]
  29. P. A. Belanger and P. Mathieu, "Dark soliton in a Kerr defocusing medium," Appl. Opt. 26, 111 (1987). [CrossRef] [PubMed]
  30. J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, and P. W. E. Smith, "Observation of spatial optical solitons in a nonlinear glass waveguide," Opt. Lett. 15, 471 (1990). [CrossRef] [PubMed]
  31. S. Blair and K. Wagner, "Spatialcsoliton angular deflection logic gates," Appl. Opt. 38, 6749 (1999). [CrossRef]
  32. H. F. Chou, C. F. Lin, and G. C. Wang, "An interative finite difference beam propagation method for modeling second-order nonlinear effects in optical waveguides," J. Lightwave Technol. 16, 1686 (1998). [CrossRef]
  33. A. B. Aceves, J. V. Moloney, and A. C. Newell, "Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface," Phy. Rev. A. 39, 1809 (1989). [CrossRef]

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