Abstract

Using intuitive physics, we show how to read the stability and the qualitative characteristics of all self-guided (bounded) beams directly from the graph of refractive index versus intensity that characterizes any nonlinear medium. Our approach predicts new soliton classes. It reveals important differences between solitons of one and two transverse dimensions. It links the physical characteristics of solitons and their possible multistabilities directly with topological features of the material nonlinearity n(I). The soliton sketch is introduced as a physical alternative to the usual bifurcation diagrams.

© 1996 Optical Society of America

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  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).
    [CrossRef] [PubMed]
  2. J. S. Aitchison, K. Al-Hemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, Electron. Lett. 28, 1879 (1992).
    [CrossRef]
  3. R. de la Fuente, A. Barthelemy, and C. Froehly, Opt. Lett. 16, 783 (1991).
    [CrossRef]
  4. M. Segev, G. C. Valley, B. Crosignani, P. di Porto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
    [CrossRef] [PubMed]
  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).
    [CrossRef] [PubMed]
  7. It is possible to anticipate why bright solitons, whose spatial intensity profile I (x) is monotonically decreasing, cannot have a maximum value of intensity within the gap I1 < Im < I2. 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.
  8. 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).
    [CrossRef]
  9. 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).
  10. 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.
  11. 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.
  12. 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.
    [CrossRef] [PubMed]
  13. This is a trivial consequence of the fact that the wave equation can be written in a nondimensionalized form for a power-law nonlinearity.
  14. An identical stability criterion results for perturbations that preserve P and soliton shape but allow for a change in Im. An increase in Im means that soliton width must be decreased or equivalently that diffraction must be increased. This is read from Fig. 1 (upper) by movement along the straight line (Im/P). Furthermore, an increase in Im changes δn and hence refraction. The direction of this change is read from the bold curve of Fig. 1 (upper). Only when the direction of δn is opposite that of diffraction (negative feedback) can the soliton be stable. From Fig. 1 (upper) we observe that this requires that Im increase as P increases.
  15. It has been shown previously that a fundamental self-guided beam propagating in a homogeneous medium of arbitrary nonlinearity is stable provided (dβ/dP) > 0, where β is the soliton propagation constant (see Ref. 16 for details and literature). This result holds for arbitrary perturbations in both one and two transverse dimensions. Replacing β in the stability criterion (dβ/dP) > 0, by the maximum soliton intensity Im necessitates that Im be a monotonic function of β. Then stability is given by (dIm/dP) > 0, as derived in this paper from elementary physics. It is straightforward to provide an analytic proof that β is a monotonic function of Im for 1-D solitons. Exhaustive numerical investigations, including that for the example of Fig. 1, confirms that Im is a monotonic function of β for solitons of circular symmetry (see also Ref. 6).
  16. D. J. Mitchell and A. W. Snyder, J. Opt. Soc. Am. B 10, 1572 (1993).
    [CrossRef]
  17. 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.
  18. 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.
    [CrossRef]

1995 (1)

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.
[CrossRef]

1994 (1)

M. Segev, G. C. Valley, B. Crosignani, P. di Porto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

1993 (2)

1992 (1)

J. S. Aitchison, K. Al-Hemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, Electron. Lett. 28, 1879 (1992).
[CrossRef]

1991 (3)

1990 (1)

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 15, 1005 (1964).

Aitchison, J. S.

Al-Hemyari, K.

J. S. Aitchison, K. Al-Hemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, Electron. Lett. 28, 1879 (1992).
[CrossRef]

Barthelemy, A.

R. de la Fuente, A. Barthelemy, and C. Froehly, Opt. Lett. 16, 783 (1991).
[CrossRef]

Chen, Y.

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 15, 1005 (1964).

Crosignani, B.

M. Segev, G. C. Valley, B. Crosignani, P. di Porto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

de la Fuente, R.

R. de la Fuente, A. Barthelemy, and C. Froehly, Opt. Lett. 16, 783 (1991).
[CrossRef]

di Porto, P.

M. Segev, G. C. Valley, B. Crosignani, P. di Porto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

Froehly, C.

R. de la Fuente, A. Barthelemy, and C. Froehly, Opt. Lett. 16, 783 (1991).
[CrossRef]

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 15, 1005 (1964).

Gidas, B.

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.

Grant, R. S.

J. S. Aitchison, K. Al-Hemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, Electron. Lett. 28, 1879 (1992).
[CrossRef]

Ironside, C. N.

J. S. Aitchison, K. Al-Hemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, Electron. Lett. 28, 1879 (1992).
[CrossRef]

Jackel, J. L.

Kivshar, Y. S.

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.
[CrossRef]

Ladouceur, F.

Leaird, D. E.

Love, J. D.

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.

Mitchell, D. J.

Ni, W.

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.

Nirenberg, L.

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.

Oliver, M. K.

Poladian, L.

Rowland, D.

Sears, F. N.

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.

Segev, M.

M. Segev, G. C. Valley, B. Crosignani, P. di Porto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

Sibbett, W.

J. S. Aitchison, K. Al-Hemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, Electron. Lett. 28, 1879 (1992).
[CrossRef]

Silberberg, Y.

Smith, P. W. E.

Snyder, A. W.

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.
[CrossRef]

D. J. Mitchell and A. W. Snyder, J. Opt. Soc. Am. B 10, 1572 (1993).
[CrossRef]

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.
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

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.

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 15, 1005 (1964).

Valley, G. C.

M. Segev, G. C. Valley, B. Crosignani, P. di Porto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

Voge, E. M.

Weiner, A. M.

Yariv, A.

M. Segev, G. C. Valley, B. Crosignani, P. di Porto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

Young, H. D.

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.

Zemansky, M. W.

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.

Electron. Lett. (1)

J. S. Aitchison, K. Al-Hemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, Electron. Lett. 28, 1879 (1992).
[CrossRef]

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

Mod. Phys. Lett. B (1)

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.
[CrossRef]

Opt. Lett. (4)

Phys. Rev. Lett. (2)

M. Segev, G. C. Valley, B. Crosignani, P. di Porto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 15, 1005 (1964).

Other (8)

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).

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.

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.

This is a trivial consequence of the fact that the wave equation can be written in a nondimensionalized form for a power-law nonlinearity.

An identical stability criterion results for perturbations that preserve P and soliton shape but allow for a change in Im. An increase in Im means that soliton width must be decreased or equivalently that diffraction must be increased. This is read from Fig. 1 (upper) by movement along the straight line (Im/P). Furthermore, an increase in Im changes δn and hence refraction. The direction of this change is read from the bold curve of Fig. 1 (upper). Only when the direction of δn is opposite that of diffraction (negative feedback) can the soliton be stable. From Fig. 1 (upper) we observe that this requires that Im increase as P increases.

It has been shown previously that a fundamental self-guided beam propagating in a homogeneous medium of arbitrary nonlinearity is stable provided (dβ/dP) > 0, where β is the soliton propagation constant (see Ref. 16 for details and literature). This result holds for arbitrary perturbations in both one and two transverse dimensions. Replacing β in the stability criterion (dβ/dP) > 0, by the maximum soliton intensity Im necessitates that Im be a monotonic function of β. Then stability is given by (dIm/dP) > 0, as derived in this paper from elementary physics. It is straightforward to provide an analytic proof that β is a monotonic function of Im for 1-D solitons. Exhaustive numerical investigations, including that for the example of Fig. 1, confirms that Im is a monotonic function of β for solitons of circular symmetry (see also Ref. 6).

It is possible to anticipate why bright solitons, whose spatial intensity profile I (x) is monotonically decreasing, cannot have a maximum value of intensity within the gap I1 < Im < I2. 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.

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.

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Figures (3)

Fig. 1
Fig. 1

Upper, Material nonlinearity in which the refractive index is n = n ¯ + δ n ( I m ) with | δ n | n ¯. The solid curve is a hypothetical nonlinearity. Lower left, Qualitative soliton sketch for solitons of circular symmetry, displaying the locus of intersections of straight lines (Im/P) in the upper graph. Lower right, Soliton sketch found by numerical solution of the wave equation.

Fig. 2
Fig. 2

Upper, Same as Fig. 1. Lower left and right, Qualitative and exact sketches for 1-D solitons, respectively.

Fig. 3
Fig. 3

Soliton (open circle) of power P is perturbed to P + δP, keeping its maximum intensity Im constant but making it wider. Refraction is then greater than diffraction (see text). The soliton will thus flow initially in a manner to increase Im. If this flow is toward the intensity maximum Im necessary for a soliton (open circle) to exist at P + δP, then the soliton is stable (left); otherwise it is unstable (right). Hence stability requires (dIm/dP) > 0.

Equations (4)

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θ d 1 / ρ ,
θ c δ n .
δ n ( I m ) ( I m / P )
δ n ( I m ) ( I m / P ) 2

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