Abstract

Spatial solitons in quadratically nonlinear media result from the interplay of parametric gain, diffraction and cascading phase shift. Their main features are well understood in mathematical terms, and several experiments have been successfully carried out which demonstrate their observability and most important properties. Here we provide an intuitive interpretation of some of the underlying physics, outlining the processes that govern their excitation, propagation and interaction forces.

© 2002 Optical Society of America

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References

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    [CrossRef]
  2. S. Trillo and W. E. Torruellas, Spatial Solitons (Springer-Verlag, Berlin, 2001).
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    [CrossRef]
  4. M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, "Electrically Assisted Self-Confinement andWaveguiding in planar Nematic Liquid Crystal cells", Appl. Phys. Lett. 77, 7-9 (2000).
    [CrossRef]
  5. A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, �??Unification of Linear and Nonlinear Wave Optics,�?? Mod. Phys. Lett. B 9, 1479-1506 (1995).
    [CrossRef]
  6. G. I. Stegeman and M. Segev, �??Optical Solitons and Their Interactions: Universality and Diversity,�?? Sci. 286, 1518-1523 (1999).
    [CrossRef]
  7. A. D. Boardman and K. Xie, �??Theory of spatial solitons,�?? Radio Science 28, 891-899 (1993).
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  8. Y. N. Karamzin and A. P. Sukhorukov, "Mutual focusing of high-power light beams in media with quadratic nonlinearity," Sov. Phys.-JETP 41, 414-416 (1976).
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  12. A. V. Buryak, Y. S. Kivshar, and V. V. Steblina, "Self-trapping of light beams and parametric solitons in diffractive quadratic media," Phys. Rev. A 52, 1670-1674 (1995).
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  14. G. Leo, G. Assanto, andW. E. Torruellas, "Bidimensional spatial solitary waves in quadratically nonlinear bulk media," J. Opt. Soc. Am. B 14, 3134-3142 (1997).
    [CrossRef]
  15. G. Assanto, "Diffraction with Second-Harmonic Generation for the formation of self-guided or 'solitary' waves," in Diffractive optics and Optical Microsystems, A. N. Chester and S. Martellucci eds., 65-74 (Plenum Press, New York, 1997).
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  17. R. A. Fuerst, M. T. G. Canva, G. I. Stegeman, G. Leo, and G. Assanto, "Robust generation, properties, and potential applications of quadratic spatial solitons generated by optical parametric amplification," Opt. & Quantum Electron. 30, 907-921 (1998).
    [CrossRef]
  18. A. V. Buryak, Y. S. Kivshar, and S. Trillo, �??Stability of tree-wave parametric solitons in diffractive quadratic media,�?? Phys. Rev. Lett. 77, 5210-5213 (1996).
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  19. D. M. Baboiu and G. I. Stegeman, "Solitary-wave interactions in quadratic media near type I phase-matching conditions," J. Opt. Soc. Am. 14, 3143-3150 (1997).
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    [CrossRef]
  22. S. K. Johansen, O. Bang, and M. P. Soerensen, "Escape velocities in bulk �?(2) soliton interactions," Phys. Rev. E 65, 026601-026604 (2002).
    [CrossRef]
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    [CrossRef]

Appl. Phys. Lett.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, "Electrically Assisted Self-Confinement andWaveguiding in planar Nematic Liquid Crystal cells", Appl. Phys. Lett. 77, 7-9 (2000).
[CrossRef]

Diffractive optics and Optical Microsyst

G. Assanto, "Diffraction with Second-Harmonic Generation for the formation of self-guided or 'solitary' waves," in Diffractive optics and Optical Microsystems, A. N. Chester and S. Martellucci eds., 65-74 (Plenum Press, New York, 1997).

J. Opt. Soc. Am.

D. M. Baboiu and G. I. Stegeman, "Solitary-wave interactions in quadratic media near type I phase-matching conditions," J. Opt. Soc. Am. 14, 3143-3150 (1997).
[CrossRef]

J. Opt. Soc. Am. B

Mod. Phys. Lett. B

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, �??Unification of Linear and Nonlinear Wave Optics,�?? Mod. Phys. Lett. B 9, 1479-1506 (1995).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

R. A. Fuerst, M. T. G. Canva, G. I. Stegeman, G. Leo, and G. Assanto, "Robust generation, properties, and potential applications of quadratic spatial solitons generated by optical parametric amplification," Opt. & Quantum Electron. 30, 907-921 (1998).
[CrossRef]

Phys. Rev. A

A. V. Buryak, Y. S. Kivshar, and V. V. Steblina, "Self-trapping of light beams and parametric solitons in diffractive quadratic media," Phys. Rev. A 52, 1670-1674 (1995).
[CrossRef] [PubMed]

Phys. Rev. E

S. K. Johansen, O. Bang, and M. P. Soerensen, "Escape velocities in bulk �?(2) soliton interactions," Phys. Rev. E 65, 026601-026604 (2002).
[CrossRef]

Phys. Rev. Lett.

A. V. Buryak, Y. S. Kivshar, and S. Trillo, �??Stability of tree-wave parametric solitons in diffractive quadratic media,�?? Phys. Rev. Lett. 77, 5210-5213 (1996).
[CrossRef] [PubMed]

K. Hayata and M. Koshiba, "Multidimensional solitons in quadratic nonlinear media," Phys. Rev. Lett. 71, 3275-3278 (1993).
[CrossRef] [PubMed]

Phys. Today

M. Segev and G. Stegeman, �??Self-Trapping of Optical Beams: Spatial Solitons,�?? Phys. Today 51, 43-48 (1998).
[CrossRef]

Radio Sci.

A. D. Boardman and K. Xie, �??Theory of spatial solitons,�?? Radio Science 28, 891-899 (1993).
[CrossRef]

Sci.

G. I. Stegeman and M. Segev, �??Optical Solitons and Their Interactions: Universality and Diversity,�?? Sci. 286, 1518-1523 (1999).
[CrossRef]

Sov. Phys.-JETP

Y. N. Karamzin and A. P. Sukhorukov, "Mutual focusing of high-power light beams in media with quadratic nonlinearity," Sov. Phys.-JETP 41, 414-416 (1976).

Other

A. D. Boardman and A. P. Sukhorukov, Soliton Driven Photonics (Kluwer Acad. Publ., Dordrecht, 2001).
[CrossRef]

S. Trillo and W. E. Torruellas, Spatial Solitons (Springer-Verlag, Berlin, 2001).

Supplementary Material (2)

» Media 1: MPG (206 KB)     
» Media 2: MPG (205 KB)     

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Figures (7)

Fig. 1
Fig. 1

Intuitive sketch on the origin of the “cascading” phase shift in Type I SHG with plane waves.

Fig. 2
Fig. 2

Simulated evolution of the transverse intensity of (a) a linearly diffracting 1D beam and (b) a quadratic spatial soliton excited by a gaussian FF beam. (c) The FF phase-front evolution for the latter case. Propagation distances are in units of the diffraction length LD, whereas the transverse coordinate is in units of the input waist w0. Here ΔkLD=2.

Fig. 3
Fig. 3

Beam narrowing through SHG. The input is an FF beam. From top to bottom: Linear diffraction at low powers, weak SHG below threshold for soliton generation, and spatial soliton formation.

Fig. 4
Fig. 4

(205 + 205KB) Animations showing the phase rotation of FF and SH field vectors in standard SHG (left) and quadratic soliton propagation (right).

Fig. 5
Fig. 5

Soliton collisions: schematic illustration of the interaction terms (in green) due to overlapping envelopes at FF and SH for solitons “a” and “b”.

Fig. 6
Fig. 6

Soliton collisions: as in Figure 5, but taking into account the spatial overlap with the eigen-distributions.

Fig. 7
Fig. 7

BPM propagation (z versus y) 3D-graphs for the various cases of interactions mentioned above. Resulting FF intensity for equi-power Gaussian beams launched parallel to z. Units are real (distances and intensity) but should be viewed as arbitrary.

Equations (23)

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2 i k FF z a FF ( y ) 2 y 2 a FF ( y ) = 2 k F F Γ [ a S H ( y ) a F F * ( y ) ] e i Δ k z
2 i k SH z a SH ( y ) 2 y 2 a SH ( y ) = 2 k SH Γ [ a FF 2 ( y ) ] e i Δ k z
Δ a FF ( y ) = i Γ a SH ( y ) a FF * ( y ) Δ z Δ a SH ( y ) = i Γ a FF 2 ( y ) Δ z
P FF = ε o χ ( 2 ) ( a SH a FF * + b SH b FF * + a SH b FF * + b SH a FF * )
P SH = ε o χ ( 2 ) ( a FF a FF + b FF b FF + 2 a FF b FF )
2 i k FF z a FF 2 y 2 a FF = 2 k F F Γ [ a S H a F F * + δ ( a F F ) ]
2 i k FF z b FF 2 y 2 b FF = 2 k F F Γ [ b S H b F F * + δ ( b F F ) ]
2 i k SH z a SH 2 y 2 a SH = 2 k S H Γ [ a F F 2 + δ ( a S H ) ]
2 i k SH z b SH 2 y 2 b SH = 2 k S H Γ [ b F F 2 + δ ( b S H ) ]
δ ( a F F ) = a SH b FF * + b SH a FF * + b SH b FF *
δ ( b F F ) = a SH b FF * + b SH a FF * + a SH a FF *
δ ( a SH ) = b FF 2 + 2 a FF b FF
δ ( b SH ) = a FF 2 + 2 a FF b FF
y d y ( a FF * / a FF x ) a SH b FF * y d y ( a FF * / a FF x ) b SH a FF * y d y ( b FF * / b FF x ) a SH b FF * y d y ( b FF * / b FF x ) b SH a FF *
y d y ( a SH * / a SH x ) a FF b F F y d y ( a SH * / a SH x ) b F F 2 y d y ( b SH * / b SH x ) a FF b F F y d y ( b SH * / b SH x ) a F F 2
δ ( a F F ) = a SH b FF * b SH a FF * + b SH b FF *
δ ( b F F ) = a SH b FF * b SH a FF * + a SH a FF *
δ ( a SH ) = b FF 2 2 a F F b F F
δ ( b SH ) = a FF 2 2 a F F b F F
δ ( a FF ) = i a SH b FF * + i b SH a FF * + b SH b FF *
δ ( b FF ) = i a SH b FF * + i b SH a FF * + a SH a FF *
δ ( a SH ) = b FF 2 + 2 i a FF b FF
δ ( b SH ) = a FF 2 2 i a FF b FF

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