Qualitative theory of bright solitons: The soliton sketch

Journal of the Optical Society of America B: Optical Physics

Volumn 13, Issue 6, 1996, Pages 1146-1150

  Snyder, A W ^a   Mitchell, D J ^a   Buryak, A ^a  

^a AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030541116     PISSN: 07403224     EISSN: None     Source Type: Journal    
DOI: 10.1364/JOSAB.13.001146     Document Type: Article

References (19)

1. J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Voge, and P. W. E. Smith, Opt. Lett. 15, 471 (1990).
2. J. S. Aitchison, K. Al-Hemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, Electron. Lett. 28, 1879 (1992).
3. R. de la Fuente, A. Barthelemy, and C. Froehly, Opt. Lett. 16, 783 (1991).
4. M. Segev, G. C. Valley, B. Crosignani, P. di Porto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
5. R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 15, 1005 (1964).
6. A. W. Snyder, D. J. Mitchell, L. Poladian, and F. Ladouceur, Opt. Lett. 16, 21 (1991). These multistable solitons of circular symmetry are analogous to those in one dimension, first discussed by A. E. Kaplan, Phys. Rev. Lett. 55, 1291 (1985).
7. A. W. Snyder, D. J. Mitchell, L. Poladian, and F. Ladouceur, Opt. Lett. 16, 21 (1991). These multistable solitons of circular symmetry are analogous to those in one dimension, first discussed by A. E. Kaplan, Phys. Rev. Lett. 55, 1291 (1985).
8. 2. If they did, their induced waveguide δn[I(x)] would have a negative value at the waveguide center (x = 0). But a mode of such a (linear) waveguide must itself be double peaked. This is inconsistent with the assumed, and, in fact, the known, intensity profile I(x) of a fundamental bright soliton.
9. Nonlinear couplers have previously been shown to lend themselves to physically meaningful bifurcation diagrams, in which the power of one core is displayed versus total power. A. W. Snyder, D. J. Mitchell, L. Poladian, D. Rowland, and Y. Chen, J. Opt. Soc. Am. B 8, 2102 (1991).
10. Why does the qualitative theory work so well when, as we show in Section 5, it assumes that all solitons have the same shape? This would appear contradictory, except for the power-law nonlinearity, because the shapes of the various soliton-induced waveguides n[I(x)] can differ significantly from one another. But we know that the fundamental modes of waveguides are relatively insensitive to the shape of the refractive-index profile, depending instead on the integrated profile area (Section 14-10 of Ref. 10).
11. F. N. Sears, M. W. Zemansky, and H. D. Young, University Physics, 6th ed. (Addison-Wesley, Reading, Mass., 1982), pp. 721 and 789. This has been a standard text in high schools over the years.
12. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983). See Chapter 1. Note Eq. (2) holds for plane-wave reflection at a plane interface as well as for rays of an arbitrary graded-index medium.
13. A. W. Snyder and D. J. Mitchell, Opt. Lett. 18, 101 (1993). Exact analytical solutions are given for 1-D solitons of the power-law nonlinearity; i.e., the constants of proportionality are given.
14. This is a trivial consequence of the fact that the wave equation can be written in a nondimensionalized form for a power-law nonlinearity.
15. m increase as P increases.
16. m is a monotonic function of β for solitons of circular symmetry (see also Ref. 6).
17. D. J. Mitchell and A. W. Snyder, J. Opt. Soc. Am. B 10, 1572 (1993).
18. B. Gidas, W. Ni, and L. Nirenberg, in Math Analysis and Applications Part A, L. Nachbin and L. Scwartz, eds., Vol. 7A of Advances in Mathematics Supplementary Studies (Academic, New York, 1981), p. 369.
19. A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, Mod. Phys. Lett. B 9, 1479 (1995). It is here demonstrated that optical spatial solitons can be approached rigorously by use of only linear waveguide physics. This linear perspective generates the fundamental equations, motivates possible soliton classes, and shows how analytical solutions can be lifted directly from linear physics.