## Simple physics of quadratic spatial solitons

Optics Express, Vol. 10, Issue 9, pp. 388-396 (2002)

http://dx.doi.org/10.1364/OE.10.000388

Acrobat PDF (387 KB)

### 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

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

_{0}+n

_{2}I, with n

_{2}a constant and I the local intensity. [6

6. G. I. Stegeman and M. Segev, “Optical Solitons and Their Interactions: Universality and Diversity,” Sci. **286**, 1518–1523 (1999). [CrossRef]

^{(2)}-active media. All of the afore-mentioned media exhibit a saturable nonlinear response, i. e. the variation in optical path-length upon propagation depends in a sub-linear way on the field intensity at each point in the transverse beam profile. [7

7. A. D. Boardman and K. Xie, “Theory of spatial solitons,” Radio Science **28**, 891–899 (1993). [CrossRef]

^{(2)}-active media for multi-frequency solitons: there is no change in the refractive index for the quadratic solitons, because they consist of multiple frequency waves coupled together by the second-order nonlinearity χ

^{(2)}. Because of this complexity, they are invariably discussed in terms of the detailed numerical solutions to the pertinent nonlinear wave equations and not in terms of the underlying physics.[8–12] Fortunately, it is possible to explain in a simple way how self-focusing processes occur during parametric interactions. In fact, one can obtain a great deal of insight and construct quite simple pictures for some of the properties of these quadratic solitons and their interaction “forces” without relying on the detailed solutions. It is our goal in this Article to show this. However, it is important to note at the outset that there are properties such as stability, soliton fusion etc. which cannot be handled in a simple way and require detailed mathematical attention.

*a*

_{FF}) and a single harmonic field (of envelope

*a*

_{SH}). Diffraction can occur in only one dimension (y), i. e. in the plane (y, z) of a slab waveguide, and the nonlinear wave evolution along z is described by the well-known equations for SHG:

*ω*

_{FF}

*n*

^{3}

*c*

^{3}

*ε*

_{0})

^{-1/2}the nonlinear coupling strength and

_{FF}- k

_{SH}is the wavevector mismatch, and near the phase-matching condition n = n

_{FF}∼ n

_{SH}. The other symbols have their usual definitions. When only the fundamental (FF) is input, in the plane-wave case (the second harmonic (SH) wave initially grows π/2 out of phase with the fundamental. This relative phase changes with propagation distance when Δk ≠ 0 because, in the general case of a wavevector (and phase-velocity) mismatch, v

_{ω}≠v

_{2ω}. Up-(FF→SH) and down-(SH→FF) conversion occur successively due to this velocity mismatch and the FF develops a nonlinear phase-shift through “cascading”, as pictured in Figure 1. When k

_{SH}≠2k

_{FF}, the up-converted phase-fronts acquire a time lag/gap with respect to the un-converted FF and, once down-conversion takes place and the energy flows back into the FF, this results in a nonlinear phase-shift in the FF, with size and sign depending on intensity and mismatch, for a given nonlinearity. [13

13. G. I. Stegeman, M. Sheik-Bahae, E. VanStryland, and G. Assanto, “Large nonlinear phase shifts in second-order nonlinear optical processes,” Opt. Lett. **18**, 13–15 (1993). [CrossRef] [PubMed]

14. G. Leo, G. Assanto, and W. E. Torruellas, “Bidimensional spatial solitary waves in quadratically nonlinear bulk media,” J. Opt. Soc. Am. B **14**, 3134–3142 (1997). [CrossRef]

*e*

^{-y2/wo2}), undergoing SHG, forms a narrower SH beam (∝

*e*

^{-y2/wo2}). The two waves interact parametrically where both intensities are higher, in such a way that the down-converted photons at FF (∝

*e*

^{-3y2/wo2}) occur preferentially on axis leading also to compress the otherwise diffracting FF beam. When the diffraction (L

_{D}=π

_{pg}=1/

*γ*√

*I*) are comparable to each other, i. e., when this narrowing mechanism is balanced by linear diffraction, a solution with an invariant transverse profile can propagate. The latter is a bright spatial soliton and, by its own nature, contains both frequency components. For this very reason, quadratic solitons are also referred to as

*simultons*.[16

16. M. J. Werner and P. D. Drummond, “Simulton solutions for the parametric amplifier,” J. Opt. Soc. Am. B **10**, 2390–2393 (1993). [CrossRef]

10. A. V. Buryak and Y. S. Kivshar, “Spatial optical solitons governed by quadratic nonlinearity,” Opt. Lett. **19**, 1612–1614 (1994). [CrossRef] [PubMed]

^{2}

*a*(y)/∂y

^{2}= 0) and stationary (∂|

*a*(y)|/∂z = 0) solutions, |

*a*

_{FF}(y)| and |

*a*

_{SH}(y)| must be independent of z. This implies that the field profiles in the plane (y, z) must be constant. More information can be gained by examining the structure of the coupled mode equations, in the phase-matched case. After propagation for a short distance Δz, the evolution of the fields is given by

*a*

_{FF}and Δ

*a*

_{SH}must correspond to a pure phase rotation of the fields, i. e. they must be orthogonal to

*a*

_{FF}and

*a*

_{SH}, respectively. Given the “

*i*” pre-factor in the equations (3), the two fields must therefore be parallel to each other. Furthermore, since they must remain parallel to one another in the soliton, they must also rotate together. As indicated in Figure 4, this is in contrast to the standard phase-matched SHG case, for which the fields are orthogonal to one another. Note that this is a nonlinear phase rotation since it is proportional to the product of two fields, and occurs for all solitons by virtue of their nonlinear propagation. Now, equations (1) and (2) apparently predict the amount of rotation to be different for different transverse positions, i. e. a function of y. However, it is again a well-known property of solitons that any changes occur uniformly across the envelope, including phase. Thus the soliton experiences an “averaged” nonlinear phase rotation. Furthermore, the higher the intensity of the soliton, the faster the nonlinear rotation.

*a*still must remain parallel and rotate together. Thus there is an additional nonlinear contribution to the nonlinear phase rotation for the field which lags in phase due to Δk ≠ 0, so that the envelopes remain in phase.

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). [CrossRef] [PubMed]

*effective forces*between quadratic spatial solitons. To this extent, two simultons are assumed whose fields overlap in space while propagating in essentially the same direction. In the first approximation, this allows the role of relative transverse velocities and additional terms deriving from tilt-induced wavevector-mismatch to be neglected. Identifying by

*a*

_{FF}and

*a*

_{SH}the field amplitudes in one of the solitons, and by

*b*

_{FF}and

*b*

_{SH}those in the other one, the nonlinear polarization terms driving the FF will take the form:

*a*and

*b*are responsible for the soliton-soliton interaction. In Figure 5 below the contributions of each term across the soliton transverse field distributions are identified.

*a*

_{FF}

*, a*

_{SH}] and [

*b*

_{FF}

*, b*

_{SH}], i. e.,

*a*

_{FF}= ∠

*b*

_{FF}(= ∠

*a*

_{SH}= ∠

*b*

_{SH}):

*b*

_{SH}

*a*

_{SH}

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). [CrossRef]

*eigen*-nature of each simulton, when considering the actual spatial overlap of the pertinent field distributions with the nonlinear perturbing polarization (4) at FF or (5) at SH, respectively, the weights of the individual quadratic terms in (10) thru (13) are comparable, i. e., for simultons of similar power and size (|

*f*|x indicates the peak amplitude):

*a*

_{FF}= ∠

*b*

_{FF}-

*π*(= ∠

*a*

_{SH}= ∠

*b*

_{SH}-

*π*) and :

*a*

_{FF}= ∠

*b*

_{FF}-

*π/2*(= ∠

*a*

_{SH}= ∠

*b*

_{SH}-

*π/2*):

*a*

_{SH}, whereas (21) in (9) shows a reduction in

*b*

_{SH}, despite the rotation slow-down due to the self-terms from the neighboring soliton. Thus

*a*

_{SH}grows while

*b*

_{SH}decreases or, equivalently, the [

*a*

_{FF}

*, a*

_{SH}] simulton takes energy from the i[

*b*

_{FF}

*, b*

_{SH}] soliton, undergoing amplification through the interaction.

*b*

_{FF}

*, b*

_{SH}] simulton will be amplified at the expense of [

*a*

_{FF}

*, a*

_{SH}].

21. B. Costantini, C. De Angelis, A. Barthelemy, A. Laureti Palma, and G. Assanto, “Polarization multiplexed χ^{(2)} solitary waves interactions,” Opt. Lett. **22**, 1376–1378 (1997). [CrossRef]

## Acknowledgements

*Solitonic Gateless Computing*.

## References and links

1. | A. D. Boardman and A. P. Sukhorukov, |

2. | S. Trillo and W. E. Torruellas, |

3. | M. Segev and G. Stegeman, “Self-Trapping of Optical Beams: Spatial Solitons,” Phys. Today |

4. | M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically Assisted Self-Confinement and Waveguiding in planar Nematic Liquid Crystal cells”, Appl. Phys. Lett. |

5. | A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, “Unification of Linear and Nonlinear Wave Optics,” Mod. Phys. Lett. B |

6. | G. I. Stegeman and M. Segev, “Optical Solitons and Their Interactions: Universality and Diversity,” Sci. |

7. | A. D. Boardman and K. Xie, “Theory of spatial solitons,” Radio Science |

8. | Y. N. Karamzin and A. P. Sukhorukov, “Mutual focusing of high-power light beams in media with quadratic nonlinearity”, Sov. Phys.-JETP , |

9. | K. Hayata and M. Koshiba, “Multidimensional solitons in quadratic nonlinear media,” Phys. Rev. Lett. |

10. | A. V. Buryak and Y. S. Kivshar, “Spatial optical solitons governed by quadratic nonlinearity,” Opt. Lett. |

11. | C.R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ |

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 |

13. | G. I. Stegeman, M. Sheik-Bahae, E. VanStryland, and G. Assanto, “Large nonlinear phase shifts in second-order nonlinear optical processes,” Opt. Lett. |

14. | G. Leo, G. Assanto, and W. E. Torruellas, “Bidimensional spatial solitary waves in quadratically nonlinear bulk media,” J. Opt. Soc. Am. B |

15. | G. Assanto, “Diffraction with Second-Harmonic Generation for the formation of self-guided or ‘solitary’ waves,” in |

16. | M. J. Werner and P. D. Drummond, “Simulton solutions for the parametric amplifier,” J. Opt. Soc. Am. B |

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. |

18. | A. V. Buryak, Y. S. Kivshar, and S. Trillo, “Stability of tree-wave parametric solitons in diffractive quadratic media,” Phys. Rev. Lett. |

19. | D. M. Baboiu and G. I. Stegeman, “Solitary-wave interactions in quadratic media near type I phase-matching conditions,” J. Opt. Soc. Am. |

20. | A. V. Buryak and V. V. Steblina, “Soliton collisions in bulk quadratic media: comprehensive analytical and numerical study,” J. Opt. Soc. Am. B |

21. | B. Costantini, C. De Angelis, A. Barthelemy, A. Laureti Palma, and G. Assanto, “Polarization multiplexed χ |

22. | S. K. Johansen, O. Bang, and M. P. Soerensen, “Escape velocities in bulk χ |

23. | V. V. Steblina, Y. S. Kivshar, and A. V. Buryak, “Scattering and spiraling of solitons in a bulk quadratic medium,” Opt. Lett. |

**OCIS Codes**

(190.4180) Nonlinear optics : Multiphoton processes

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

(190.5940) Nonlinear optics : Self-action effects

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 12, 2002

Revised Manuscript: April 18, 2002

Published: May 6, 2002

**Citation**

Gaetano Assanto and George Stegeman, "Simple physics of quadratic spatial solitons," Opt. Express **10**, 388-396 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-9-388

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### References

- 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).
- M. Segev and G. Stegeman, �??Self-Trapping of Optical Beams: Spatial Solitons,�?? Phys. Today 51, 43-48 (1998). [CrossRef]
- 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]
- 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]
- G. I. Stegeman and M. Segev, �??Optical Solitons and Their Interactions: Universality and Diversity,�?? Sci. 286, 1518-1523 (1999). [CrossRef]
- A. D. Boardman and K. Xie, �??Theory of spatial solitons,�?? Radio Science 28, 891-899 (1993). [CrossRef]
- 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).
- K. Hayata and M. Koshiba, "Multidimensional solitons in quadratic nonlinear media," Phys. Rev. Lett. 71, 3275-3278 (1993). [CrossRef] [PubMed]
- A. V. Buryak and Y. S. Kivshar, "Spatial optical solitons governed by quadratic nonlinearity," Opt. Lett. 19, 1612-1614 (1994). [CrossRef] [PubMed]
- C.R. Menyuk, R. Schiek, and L. Torner, "Solitary waves due to �?(2) :�?(2) cascading," J. Opt. Soc. Am. B 11, 2434-2443 (1994). [CrossRef]
- 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]
- G. I. Stegeman, M. Sheik-Bahae, E. VanStryland, and G. Assanto, "Large nonlinear phase shifts in second-order nonlinear optical processes," Opt. Lett. 18, 13-15 (1993). [CrossRef] [PubMed]
- 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]
- 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).
- M. J. Werner and P. D. Drummond, �??Simulton solutions for the parametric amplifier,�?? J. Opt. Soc. Am. B 10, 2390-2393 (1993). [CrossRef]
- 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]
- 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]
- 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]
- A. V. Buryak and V. V. Steblina, "Soliton collisions in bulk quadratic media: comprehensive analytical and numerical study," J. Opt. Soc. Am. B 16, 245-255 (1999). [CrossRef]
- B. Costantini, C. De Angelis, A. Barthelemy, A. Laureti Palma, and G. Assanto, "Polarization multiplexed �?(2)solitary waves interactions," Opt. Lett. 22, 1376-1378 (1997). [CrossRef]
- S. K. Johansen, O. Bang, and M. P. Soerensen, "Escape velocities in bulk �?(2) soliton interactions," Phys. Rev. E 65, 026601-026604 (2002). [CrossRef]
- V. V. Steblina, Y. S. Kivshar, and A. V. Buryak, "Scattering and spiraling of solitons in a bulk quadratic medium," Opt. Lett. 23, 156-158 (1998). [CrossRef]

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