Abstract

We show that, in the high wave-vector-mismatch (cascading) limit, the well-known paraxial description of parametric frequency conversion in quadratic media entails effective lensing effects, which can have a self-focusing or a self-defocusing nature, critically depending on the mismatch sign, the selected wave, and the launching condition (second-harmonic generation or downconversion). Numerical and experimental evidence of this behavior is reported.

© 2002 Optical Society of America

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  30. W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450–2453 (1997).
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    [Crossref]
  40. P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, S. Trillo, C. Conti, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities in self-trapping due to parametric frequency conversion,” Phys. Rev. Lett. 87, 183902 (2001).
    [Crossref]
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    [Crossref]
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2001 (2)

R. Schiek, H. Fang, R. Malendevich, and G. I. Stegeman, “Measurement of modulational instability gain of second-order nonlinear optical eigenmodes in a one-dimensional system,” Phys. Rev. Lett. 86, 4528–4531 (2001).
[Crossref] [PubMed]

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, S. Trillo, C. Conti, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities in self-trapping due to parametric frequency conversion,” Phys. Rev. Lett. 87, 183902 (2001).
[Crossref]

2000 (2)

X. Liu, K. Beckwitt, and F. W. Wise, “Noncollinear generation of optical spatiotemporal solitons and application to ultrafast logic,” Phys. Rev. E 61, R4722–R4725 (2000).
[Crossref]

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[Crossref] [PubMed]

1999 (3)

M. Vaupel, A. Maitre, and C. Fabre, “Observation of pattern formation in optical parametric oscillators,” Phys. Rev. Lett. 83, 5278–5281 (1999).
[Crossref]

B. Bourliaguet, V. Couderc, A. Barthelemy, G. W. Ross, P. G. R. Smith, D. C. Hanna, and C. De Angelis, “Observation of quadratic spatial solitons in periodically poled lithium niobate,” Opt. Lett. 24, 1410–1412 (1999).
[Crossref]

X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. 82, 4631–4634 (1999).
[Crossref]

1998 (6)

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett. 81, 570–573 (1998).
[Crossref]

P. Di Trapani, G. Valiulis, W. Chinaglia, and A. Andreoni, “Two-dimensional spatial solitary waves from traveling-wave parametric amplification of the quantum noise,” Phys. Rev. Lett. 80, 265–268 (1998).
[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]

L. Bergé, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303, 259–372 (1998).
[Crossref]

P. Di Trapani, A. Berzanskis, S. Minardi, S. Sapone, and W. Chinaglia, “Observation of optical vortices and J0 Bessel-like beams in quantum-noise parametric amplification,” Phys. Rev. Lett. 81, 5133–5136 (1998).
[Crossref]

D. V. Petrov and L. Torner, “Observation of topological charge pair nucleation in parametric wave mixing,” Phys. Rev. E 84, 7903–7907 (1998).
[Crossref]

1997 (7)

A. V. Buryak, Y. S. Kivshar, and S. Trillo, “Parametric spatial solitary waves due to type II second harmonic generation,” J. Opt. Soc. Am. B 14, 3110–3118 (1997).
[Crossref]

A. V. Buryak and Yu. S. Kivshar, “Multistability of three-wave parametric self-trapping,” Phys. Rev. Lett. 78, 3286–3289 (1997).
[Crossref]

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450–2453 (1997).
[Crossref]

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisoliton-like generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[Crossref]

M. T. G. Canva, R. A. Fuerst, D. Baboiu, G. I. Stegeman, and G. Assanto, “Quadratic spatial soliton generation by seeded downconversion of a strong harmonic pump beam,” Opt. Lett. 22, 1683–1685 (1997); errata,  26, 105 (2001).
[Crossref]

R. A. Fuerst, M. T. G. Canva, D. M. Baboiu, and G. I. Stegeman, “Properties of type II quadratic solitons excited by imbalanced fundamental waves,” Opt. Lett. 22, 1748–1750 (1997).
[Crossref]

C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial solitons and induced Kerr effects in quasi-phase-matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997).
[Crossref]

1996 (5)

T. A. B. Kennedy and S. Trillo, “Squeezing of cw light in a single-mode dispersive χ(2) medium,” Phys. Rev. A 54, 4396–4407 (1996).
[Crossref] [PubMed]

A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, “Self-trapping of light beams and parametric solitons in diffractive quadratic media,” Phys. Rev. A 52, 1670–1674 (1996).
[Crossref]

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[Crossref]

R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138–1141 (1996).
[Crossref]

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, “Stability of three-wave parametric solitons in diffractive media,” Phys. Rev. Lett. 77, 5210–5213 (1996).
[Crossref] [PubMed]

1995 (5)

A. G. Kalocsai and J. W. Haus, “Asymptotic wave-wave processes beyond cascading in quadratic nonlinear optical materials,” Phys. Rev. E 52, 3166–3183 (1995).
[Crossref]

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, “Instability of solitons governed by quadratic nonlinearity,” Phys. Rev. Lett. 75, 591–594 (1995).
[Crossref] [PubMed]

Ch. Bosshard, R. Speiter, M. Zgonick, and P. Günter, “Kerr nonlinearity via cascaded optical rectification and the linear electro-optics effect,” Phys. Rev. Lett. 74, 2816–2819 (1995).
[Crossref] [PubMed]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[Crossref] [PubMed]

L. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruellas, and G. I. Stegeman, “Stationary trapping of light beams in bulk second-order nonlinear media,” Opt. Commun. 121, 149–155 (1995).
[Crossref]

1994 (4)

M. Segev, G. C. Valley, B. Crosignani, P. Di Porto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied fields,” Phys. Rev. Lett. 73, 3211–3214 (1994).
[Crossref] [PubMed]

M. J. Werner and P. D. Drummond, “Strongly coupled nonlinear parametric solitary waves,” Opt. Lett. 19, 613–615 (1994).
[Crossref] [PubMed]

A. G. Kalocsai and J. W. Haus, “Nonlinear Schrödinger equation for optical media with quadratic nonlinearity,” Phys. Rev. A 49, 574–585 (1994).
[Crossref] [PubMed]

C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2):χ(2) cascading,” J. Opt. Soc. Am. B 11, 2434–2443 (1994).
[Crossref]

1993 (2)

1992 (2)

1974 (1)

Yu. N. Karamzin and A. P. Sukhorukov, “Nonlinear interaction of diffracted light beams in a medium with quadratic nonlinearity: mutual focusing of beams and limitation on the efficiency of optical frequency converters,” JETP Lett. 20, 339–342 (1974).

1967 (1)

L. A. Ostroskii, “Self-action of light in crystals,” Pis'ma Zh. Eksp. Teor. Fiz. 5, 331–334 (1967).

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

Aitchison, J. S.

Andreoni, A.

P. Di Trapani, G. Valiulis, W. Chinaglia, and A. Andreoni, “Two-dimensional spatial solitary waves from traveling-wave parametric amplification of the quantum noise,” Phys. Rev. Lett. 80, 265–268 (1998).
[Crossref]

Assanto, G.

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]

M. T. G. Canva, R. A. Fuerst, D. Baboiu, G. I. Stegeman, and G. Assanto, “Quadratic spatial soliton generation by seeded downconversion of a strong harmonic pump beam,” Opt. Lett. 22, 1683–1685 (1997); errata,  26, 105 (2001).
[Crossref]

Baboiu, D.

Baboiu, D. M.

R. A. Fuerst, M. T. G. Canva, D. M. Baboiu, and G. I. Stegeman, “Properties of type II quadratic solitons excited by imbalanced fundamental waves,” Opt. Lett. 22, 1748–1750 (1997).
[Crossref]

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisoliton-like generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[Crossref]

Baek, Y.

R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138–1141 (1996).
[Crossref]

Bang, O.

C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial solitons and induced Kerr effects in quasi-phase-matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997).
[Crossref]

Barthelemy, A.

Beckwitt, K.

X. Liu, K. Beckwitt, and F. W. Wise, “Noncollinear generation of optical spatiotemporal solitons and application to ultrafast logic,” Phys. Rev. E 61, R4722–R4725 (2000).
[Crossref]

Bergé, L.

L. Bergé, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303, 259–372 (1998).
[Crossref]

Berzanskis, A.

P. Di Trapani, A. Berzanskis, S. Minardi, S. Sapone, and W. Chinaglia, “Observation of optical vortices and J0 Bessel-like beams in quantum-noise parametric amplification,” Phys. Rev. Lett. 81, 5133–5136 (1998).
[Crossref]

Bosshard, Ch.

Ch. Bosshard, R. Speiter, M. Zgonick, and P. Günter, “Kerr nonlinearity via cascaded optical rectification and the linear electro-optics effect,” Phys. Rev. Lett. 74, 2816–2819 (1995).
[Crossref] [PubMed]

Bourliaguet, B.

Bramati, A.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, S. Trillo, C. Conti, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities in self-trapping due to parametric frequency conversion,” Phys. Rev. Lett. 87, 183902 (2001).
[Crossref]

Buryak, A. V.

A. V. Buryak and Yu. S. Kivshar, “Multistability of three-wave parametric self-trapping,” Phys. Rev. Lett. 78, 3286–3289 (1997).
[Crossref]

A. V. Buryak, Y. S. Kivshar, and S. Trillo, “Parametric spatial solitary waves due to type II second harmonic generation,” J. Opt. Soc. Am. B 14, 3110–3118 (1997).
[Crossref]

A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, “Self-trapping of light beams and parametric solitons in diffractive quadratic media,” Phys. Rev. A 52, 1670–1674 (1996).
[Crossref]

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, “Stability of three-wave parametric solitons in diffractive media,” Phys. Rev. Lett. 77, 5210–5213 (1996).
[Crossref] [PubMed]

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, “Instability of solitons governed by quadratic nonlinearity,” Phys. Rev. Lett. 75, 591–594 (1995).
[Crossref] [PubMed]

Caironi, D.

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett. 81, 570–573 (1998).
[Crossref]

Canva, M. T. G.

Cappellini, G.

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

Chinaglia, W.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, S. Trillo, C. Conti, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities in self-trapping due to parametric frequency conversion,” Phys. Rev. Lett. 87, 183902 (2001).
[Crossref]

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[Crossref] [PubMed]

P. Di Trapani, A. Berzanskis, S. Minardi, S. Sapone, and W. Chinaglia, “Observation of optical vortices and J0 Bessel-like beams in quantum-noise parametric amplification,” Phys. Rev. Lett. 81, 5133–5136 (1998).
[Crossref]

P. Di Trapani, G. Valiulis, W. Chinaglia, and A. Andreoni, “Two-dimensional spatial solitary waves from traveling-wave parametric amplification of the quantum noise,” Phys. Rev. Lett. 80, 265–268 (1998).
[Crossref]

Chisari, R.

Clausen, C. B.

C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial solitons and induced Kerr effects in quasi-phase-matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997).
[Crossref]

Conti, C.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, S. Trillo, C. Conti, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities in self-trapping due to parametric frequency conversion,” Phys. Rev. Lett. 87, 183902 (2001).
[Crossref]

Couderc, V.

Crosignani, B.

M. Segev, G. C. Valley, B. Crosignani, P. Di Porto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied fields,” Phys. Rev. Lett. 73, 3211–3214 (1994).
[Crossref] [PubMed]

M. Segev, B. Crosignani, A. Yariv, and B. Fisher, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
[Crossref] [PubMed]

Danielius, R.

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett. 81, 570–573 (1998).
[Crossref]

De Angelis, C.

Di Porto, P.

M. Segev, G. C. Valley, B. Crosignani, P. Di Porto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied fields,” Phys. Rev. Lett. 73, 3211–3214 (1994).
[Crossref] [PubMed]

Di Trapani, P.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, S. Trillo, C. Conti, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities in self-trapping due to parametric frequency conversion,” Phys. Rev. Lett. 87, 183902 (2001).
[Crossref]

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[Crossref] [PubMed]

P. Di Trapani, A. Berzanskis, S. Minardi, S. Sapone, and W. Chinaglia, “Observation of optical vortices and J0 Bessel-like beams in quantum-noise parametric amplification,” Phys. Rev. Lett. 81, 5133–5136 (1998).
[Crossref]

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett. 81, 570–573 (1998).
[Crossref]

P. Di Trapani, G. Valiulis, W. Chinaglia, and A. Andreoni, “Two-dimensional spatial solitary waves from traveling-wave parametric amplification of the quantum noise,” Phys. Rev. Lett. 80, 265–268 (1998).
[Crossref]

Drummond, P. D.

Dubietis, A.

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett. 81, 570–573 (1998).
[Crossref]

Etrich, C.

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” in Progress in Optics, E. Wolf, ed. (Pergamon, London, 2000).

Fabre, C.

M. Vaupel, A. Maitre, and C. Fabre, “Observation of pattern formation in optical parametric oscillators,” Phys. Rev. Lett. 83, 5278–5281 (1999).
[Crossref]

Fang, H.

R. Schiek, H. Fang, R. Malendevich, and G. I. Stegeman, “Measurement of modulational instability gain of second-order nonlinear optical eigenmodes in a one-dimensional system,” Phys. Rev. Lett. 86, 4528–4531 (2001).
[Crossref] [PubMed]

Firth, W. J.

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450–2453 (1997).
[Crossref]

Fisher, B.

M. Segev, B. Crosignani, A. Yariv, and B. Fisher, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
[Crossref] [PubMed]

Fuerst, R. A.

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]

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisoliton-like generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[Crossref]

M. T. G. Canva, R. A. Fuerst, D. Baboiu, G. I. Stegeman, and G. Assanto, “Quadratic spatial soliton generation by seeded downconversion of a strong harmonic pump beam,” Opt. Lett. 22, 1683–1685 (1997); errata,  26, 105 (2001).
[Crossref]

R. A. Fuerst, M. T. G. Canva, D. M. Baboiu, and G. I. Stegeman, “Properties of type II quadratic solitons excited by imbalanced fundamental waves,” Opt. Lett. 22, 1748–1750 (1997).
[Crossref]

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

Günter, P.

Ch. Bosshard, R. Speiter, M. Zgonick, and P. Günter, “Kerr nonlinearity via cascaded optical rectification and the linear electro-optics effect,” Phys. Rev. Lett. 74, 2816–2819 (1995).
[Crossref] [PubMed]

Hagan, D. J.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[Crossref]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[Crossref] [PubMed]

Hanna, D. C.

Haus, J. W.

A. G. Kalocsai and J. W. Haus, “Asymptotic wave-wave processes beyond cascading in quadratic nonlinear optical materials,” Phys. Rev. E 52, 3166–3183 (1995).
[Crossref]

A. G. Kalocsai and J. W. Haus, “Nonlinear Schrödinger equation for optical media with quadratic nonlinearity,” Phys. Rev. A 49, 574–585 (1994).
[Crossref] [PubMed]

Herminghaus, S.

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993).
[Crossref] [PubMed]

Hutchings, D. C.

Ironside, C. N.

Kalocsai, A. G.

A. G. Kalocsai and J. W. Haus, “Asymptotic wave-wave processes beyond cascading in quadratic nonlinear optical materials,” Phys. Rev. E 52, 3166–3183 (1995).
[Crossref]

A. G. Kalocsai and J. W. Haus, “Nonlinear Schrödinger equation for optical media with quadratic nonlinearity,” Phys. Rev. A 49, 574–585 (1994).
[Crossref] [PubMed]

Karamzin, Yu. N.

Yu. N. Karamzin and A. P. Sukhorukov, “Nonlinear interaction of diffracted light beams in a medium with quadratic nonlinearity: mutual focusing of beams and limitation on the efficiency of optical frequency converters,” JETP Lett. 20, 339–342 (1974).

Kennedy, T. A. B.

T. A. B. Kennedy and S. Trillo, “Squeezing of cw light in a single-mode dispersive χ(2) medium,” Phys. Rev. A 54, 4396–4407 (1996).
[Crossref] [PubMed]

Kilius, J.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, S. Trillo, C. Conti, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities in self-trapping due to parametric frequency conversion,” Phys. Rev. Lett. 87, 183902 (2001).
[Crossref]

Kivshar, Y. S.

C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial solitons and induced Kerr effects in quasi-phase-matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997).
[Crossref]

A. V. Buryak, Y. S. Kivshar, and S. Trillo, “Parametric spatial solitary waves due to type II second harmonic generation,” J. Opt. Soc. Am. B 14, 3110–3118 (1997).
[Crossref]

Kivshar, Yu. S.

A. V. Buryak and Yu. S. Kivshar, “Multistability of three-wave parametric self-trapping,” Phys. Rev. Lett. 78, 3286–3289 (1997).
[Crossref]

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, “Stability of three-wave parametric solitons in diffractive media,” Phys. Rev. Lett. 77, 5210–5213 (1996).
[Crossref] [PubMed]

A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, “Self-trapping of light beams and parametric solitons in diffractive quadratic media,” Phys. Rev. A 52, 1670–1674 (1996).
[Crossref]

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, “Instability of solitons governed by quadratic nonlinearity,” Phys. Rev. Lett. 75, 591–594 (1995).
[Crossref] [PubMed]

Lawrence, B.

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisoliton-like generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[Crossref]

Lederer, F.

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” in Progress in Optics, E. Wolf, ed. (Pergamon, London, 2000).

Leo, G.

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]

Liu, X.

X. Liu, K. Beckwitt, and F. W. Wise, “Noncollinear generation of optical spatiotemporal solitons and application to ultrafast logic,” Phys. Rev. E 61, R4722–R4725 (2000).
[Crossref]

X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. 82, 4631–4634 (1999).
[Crossref]

Maitre, A.

M. Vaupel, A. Maitre, and C. Fabre, “Observation of pattern formation in optical parametric oscillators,” Phys. Rev. Lett. 83, 5278–5281 (1999).
[Crossref]

Malendevich, R.

R. Schiek, H. Fang, R. Malendevich, and G. I. Stegeman, “Measurement of modulational instability gain of second-order nonlinear optical eigenmodes in a one-dimensional system,” Phys. Rev. Lett. 86, 4528–4531 (2001).
[Crossref] [PubMed]

Malomed, B. A.

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” in Progress in Optics, E. Wolf, ed. (Pergamon, London, 2000).

Mazilu, D.

L. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruellas, and G. I. Stegeman, “Stationary trapping of light beams in bulk second-order nonlinear media,” Opt. Commun. 121, 149–155 (1995).
[Crossref]

Menyuk, C. R.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[Crossref] [PubMed]

C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2):χ(2) cascading,” J. Opt. Soc. Am. B 11, 2434–2443 (1994).
[Crossref]

Mihalache, D.

L. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruellas, and G. I. Stegeman, “Stationary trapping of light beams in bulk second-order nonlinear media,” Opt. Commun. 121, 149–155 (1995).
[Crossref]

Minardi, S.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, S. Trillo, C. Conti, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities in self-trapping due to parametric frequency conversion,” Phys. Rev. Lett. 87, 183902 (2001).
[Crossref]

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[Crossref] [PubMed]

P. Di Trapani, A. Berzanskis, S. Minardi, S. Sapone, and W. Chinaglia, “Observation of optical vortices and J0 Bessel-like beams in quantum-noise parametric amplification,” Phys. Rev. Lett. 81, 5133–5136 (1998).
[Crossref]

Moloney, J. V.

A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1992).

Nayfeh, A. H.

A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1993).

Newell, A. C.

A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1992).

Ostroskii, L. A.

L. A. Ostroskii, “Self-action of light in crystals,” Pis'ma Zh. Eksp. Teor. Fiz. 5, 331–334 (1967).

Pelinovsky, D. E.

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, “Instability of solitons governed by quadratic nonlinearity,” Phys. Rev. Lett. 75, 591–594 (1995).
[Crossref] [PubMed]

Peschel, T.

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” in Progress in Optics, E. Wolf, ed. (Pergamon, London, 2000).

Peschel, U.

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” in Progress in Optics, E. Wolf, ed. (Pergamon, London, 2000).

Petrov, D. V.

D. V. Petrov and L. Torner, “Observation of topological charge pair nucleation in parametric wave mixing,” Phys. Rev. E 84, 7903–7907 (1998).
[Crossref]

Piskarskas, A.

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[Crossref] [PubMed]

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett. 81, 570–573 (1998).
[Crossref]

Qian, L. J.

X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. 82, 4631–4634 (1999).
[Crossref]

Ross, G. W.

Sapone, S.

P. Di Trapani, A. Berzanskis, S. Minardi, S. Sapone, and W. Chinaglia, “Observation of optical vortices and J0 Bessel-like beams in quantum-noise parametric amplification,” Phys. Rev. Lett. 81, 5133–5136 (1998).
[Crossref]

Schiek, R.

R. Schiek, H. Fang, R. Malendevich, and G. I. Stegeman, “Measurement of modulational instability gain of second-order nonlinear optical eigenmodes in a one-dimensional system,” Phys. Rev. Lett. 86, 4528–4531 (2001).
[Crossref] [PubMed]

R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138–1141 (1996).
[Crossref]

C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2):χ(2) cascading,” J. Opt. Soc. Am. B 11, 2434–2443 (1994).
[Crossref]

Segev, M.

M. Segev, G. C. Valley, B. Crosignani, P. Di Porto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied fields,” Phys. Rev. Lett. 73, 3211–3214 (1994).
[Crossref] [PubMed]

M. Segev, B. Crosignani, A. Yariv, and B. Fisher, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
[Crossref] [PubMed]

Skryabin, D. V.

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450–2453 (1997).
[Crossref]

Smith, P. G. R.

Speiter, R.

Ch. Bosshard, R. Speiter, M. Zgonick, and P. Günter, “Kerr nonlinearity via cascaded optical rectification and the linear electro-optics effect,” Phys. Rev. Lett. 74, 2816–2819 (1995).
[Crossref] [PubMed]

Steblina, V. V.

A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, “Self-trapping of light beams and parametric solitons in diffractive quadratic media,” Phys. Rev. A 52, 1670–1674 (1996).
[Crossref]

Stegeman, G. I.

R. Schiek, H. Fang, R. Malendevich, and G. I. Stegeman, “Measurement of modulational instability gain of second-order nonlinear optical eigenmodes in a one-dimensional system,” Phys. Rev. Lett. 86, 4528–4531 (2001).
[Crossref] [PubMed]

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]

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisoliton-like generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[Crossref]

R. A. Fuerst, M. T. G. Canva, D. M. Baboiu, and G. I. Stegeman, “Properties of type II quadratic solitons excited by imbalanced fundamental waves,” Opt. Lett. 22, 1748–1750 (1997).
[Crossref]

M. T. G. Canva, R. A. Fuerst, D. Baboiu, G. I. Stegeman, and G. Assanto, “Quadratic spatial soliton generation by seeded downconversion of a strong harmonic pump beam,” Opt. Lett. 22, 1683–1685 (1997); errata,  26, 105 (2001).
[Crossref]

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[Crossref]

R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138–1141 (1996).
[Crossref]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[Crossref] [PubMed]

L. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruellas, and G. I. Stegeman, “Stationary trapping of light beams in bulk second-order nonlinear media,” Opt. Commun. 121, 149–155 (1995).
[Crossref]

Sukhorukov, A. P.

Yu. N. Karamzin and A. P. Sukhorukov, “Nonlinear interaction of diffracted light beams in a medium with quadratic nonlinearity: mutual focusing of beams and limitation on the efficiency of optical frequency converters,” JETP Lett. 20, 339–342 (1974).

Torner, L.

D. V. Petrov and L. Torner, “Observation of topological charge pair nucleation in parametric wave mixing,” Phys. Rev. E 84, 7903–7907 (1998).
[Crossref]

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[Crossref]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[Crossref] [PubMed]

L. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruellas, and G. I. Stegeman, “Stationary trapping of light beams in bulk second-order nonlinear media,” Opt. Commun. 121, 149–155 (1995).
[Crossref]

C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2):χ(2) cascading,” J. Opt. Soc. Am. B 11, 2434–2443 (1994).
[Crossref]

Torruellas, W. E.

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisoliton-like generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[Crossref]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[Crossref] [PubMed]

L. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruellas, and G. I. Stegeman, “Stationary trapping of light beams in bulk second-order nonlinear media,” Opt. Commun. 121, 149–155 (1995).
[Crossref]

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

Trillo, S.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, S. Trillo, C. Conti, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities in self-trapping due to parametric frequency conversion,” Phys. Rev. Lett. 87, 183902 (2001).
[Crossref]

A. V. Buryak, Y. S. Kivshar, and S. Trillo, “Parametric spatial solitary waves due to type II second harmonic generation,” J. Opt. Soc. Am. B 14, 3110–3118 (1997).
[Crossref]

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisoliton-like generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[Crossref]

T. A. B. Kennedy and S. Trillo, “Squeezing of cw light in a single-mode dispersive χ(2) medium,” Phys. Rev. A 54, 4396–4407 (1996).
[Crossref] [PubMed]

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, “Stability of three-wave parametric solitons in diffractive media,” Phys. Rev. Lett. 77, 5210–5213 (1996).
[Crossref] [PubMed]

G. Cappellini, S. Trillo, S. Wabnitz, and R. Chisari, “Two wave mixing in a quadratic nonlinear medium: bifurcations, spatial instabilities, and chaos,” Opt. Lett. 17, 637–639 (1992).
[Crossref]

Valiulis, G.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, S. Trillo, C. Conti, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities in self-trapping due to parametric frequency conversion,” Phys. Rev. Lett. 87, 183902 (2001).
[Crossref]

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[Crossref] [PubMed]

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett. 81, 570–573 (1998).
[Crossref]

P. Di Trapani, G. Valiulis, W. Chinaglia, and A. Andreoni, “Two-dimensional spatial solitary waves from traveling-wave parametric amplification of the quantum noise,” Phys. Rev. Lett. 80, 265–268 (1998).
[Crossref]

Valley, G. C.

M. Segev, G. C. Valley, B. Crosignani, P. Di Porto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied fields,” Phys. Rev. Lett. 73, 3211–3214 (1994).
[Crossref] [PubMed]

VanStryland, E. W.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[Crossref] [PubMed]

Vaupel, M.

M. Vaupel, A. Maitre, and C. Fabre, “Observation of pattern formation in optical parametric oscillators,” Phys. Rev. Lett. 83, 5278–5281 (1999).
[Crossref]

Wabnitz, S.

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisoliton-like generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[Crossref]

G. Cappellini, S. Trillo, S. Wabnitz, and R. Chisari, “Two wave mixing in a quadratic nonlinear medium: bifurcations, spatial instabilities, and chaos,” Opt. Lett. 17, 637–639 (1992).
[Crossref]

Wang, Z.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[Crossref] [PubMed]

Werner, M. J.

Wise, F. W.

X. Liu, K. Beckwitt, and F. W. Wise, “Noncollinear generation of optical spatiotemporal solitons and application to ultrafast logic,” Phys. Rev. E 61, R4722–R4725 (2000).
[Crossref]

X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. 82, 4631–4634 (1999).
[Crossref]

Wright, E. M.

L. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruellas, and G. I. Stegeman, “Stationary trapping of light beams in bulk second-order nonlinear media,” Opt. Commun. 121, 149–155 (1995).
[Crossref]

Wulle, T.

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993).
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Figures (5)

Fig. 1
Fig. 1

Schematic of cascading by two-wave mixing of photons at frequencies ω and 2ω. DFG and SFG, (degenerate) downconversion and upconversion, respectively.

Fig. 2
Fig. 2

Beam widths versus propagation distance as obtained from numerical integration of Eqs. (1), with y=0, in the limit of large absolute wave-vector mismatch (δk=±100) in four qualitatively distinct cases: (a) SHG, positive mismatch; (b) SHG, negative mismatch; (c) OPA, positive mismatch; (d) OPA, negative mismatch. Dashed lines, initial width x0=8 of both FF and SH beams.

Fig. 3
Fig. 3

Output beam FWHM diameters d of the FF and SH beams measured as a function of wave-vector mismatch Δk in a SHG experiment with fixed input energy and input beam diameter. Dashed line, linear limit of FF diffractive spreading.

Fig. 4
Fig. 4

SH to FF power imbalance Q2/Q1 of the parametric solitons versus mismatch δk for several values of total soliton power Q=Q1+Q2. Unstable soliton branches are plotted as bold curves.

Fig. 5
Fig. 5

Soliton intensity versus x (profile sections at y=0: SH field |u20(x, y=0)|2, thinner curves; FF field |u10(x, y=0)|2, thicker curves;) for δk=-10 and total power Q=378: (a) unstable case, β=5.01; (b) stable case, β=13.83. Note the different horizontal and vertical scales.

Equations (34)

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i u1z+σ122u1+u2u1* exp(-iδkz)=0,
i u2z+σ222u2+u122 exp(iδkz)=0.
u1=A+εA1+ε2A2+,
u2=B+εB1+ε2B2+,
Lm=iε ζ+Lm(0)+εL(1)+ε2L(2)+,
Lm(0)=i z0+σm2 2x2+2y2,L(n)=i zn.
ζA=0,ζB=0;
iζA1+L1(0)A+A*B exp(-iζ)=0,
iζB1+L2(0)B+12A2 exp(iζ)=0.
L1(0)A=L2(0)B=0.
A1=-A*B exp(-iζ)+a1(x, y, zn|n0),
B1=12A2 exp(iζ)+b1(x, y, zn|n0).
-iζA2+L1(0)(A*B)exp(-iζ)-A*b1 exp(-iζ)
-Ba1* exp(-iζ)
=L1(0)a1+L(1)A+½ |A|2A-|B|2A,
-iζB2-½ L2(0)(A2)exp(iζ)-Aa1 exp(iζ)
=L2(0)b1+L(1)B-|A|2B.
A2=L1(0)(A*B)exp(-iζ)-A*b1 exp(-iζ)-a1*B exp(-iζ)+a2,
B2=L2(0)A22exp(iζ)+Aa1 exp(iζ)+b2,
L(1)A+½ |A|2A-|B|2A=-L1(0)a1,
L(1)B-|A|2B=-L2(0)b1.
L(1)A+½ |A|2A-|B|2A=0,
L(1)B-|A|2B=0,
L1(0)a1=0,L2(0)b1=0.
i Az+σ122A+12δk|A|2A-1δk|B|2A=0,
i Bz+σ222B-1δk|A|2B=0.
i a1z+σ122a1=0,
i b1z+σ222b1=0.
u1=A+1δk[-BA* exp(-iδkz)+a1],
u2=B+1δk A22 exp(iδkz)+b1.
u1(x, y, z=0)=F(x, y),u2(x, y, z=0)=0,
i Az+122A+12δk|A|2A=0.
u1(x, z=0)=A0 exp[-(x/x0)2],
u2(x, z=0)=B0 exp[-(x/x0)2],

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