Abstract

Rotating patterns can be produced by two coaxial spiraling elliptic beams in nonlocal nonlinear media. The two constituent beams carry the orbital angular momentum (OAM), which has the same or opposite signs. The resultant patterns exhibit revolving and spinning rotations similar to the Sun-Earth system. The revolving-typed rotation comes of the overall OAM, while the spinning-typed rotation results from the respective OAM of the constituent beam. A kind of soliton patterns can stably exist, for which a vortex is nested centrally in the rotating square-shaped optical envelope and four peaks appear at four corners. The centrally nested vortex splits into two single vortex for r1, with r being the power ratio between the two constituent beams. The two single vortexes align vertically for r<1 and horizontally for r>1, and their spacing increases with |r1|. The theoretical results give a possibility of the transforming from cross-phase-typed OAM into the helical-phase-typed OAM, which may find potential applications in beam shaping and controlling. In addition, the square optical envelope may exhibit advantages in the integration of optical components.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
  16. G. Liang, Y. Wang, Q. Guo, and H. Zhang, “Anisotropic diffraction induced by orbital angular momentum during propagations of optical beams,” Opt. Express. 26(7), 8084–8094 (2018).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  23. D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27(6), 3135 (1983).
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  24. S. Hwang, T. Kim, J. Lee, and T. J. Yu, “Design of square-shaped beam homogenizer for petawatt-class Ti:sapphire amplifier,” Opt. Express. 25(9), 9511–9520 (2017).
    [Crossref] [PubMed]
  25. Z. W. Cao, S. L. Gong, and Y. Gao, “Characterization of TC17 titanium alloy treated by square-spot laser shock peening,” Adv. Mat. Res. 652–654, 2378–2383 (2013).
  26. C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91(7), 073901 (2003).
    [Crossref] [PubMed]
  27. C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
    [Crossref] [PubMed]
  28. G. Liang and Q. Wang, “Controllable conversion between Hermite Gaussian and Laguerre Gaussian modes due to cross phase,” Opt. Express 27(8), 10684–10691 (2019).
    [Crossref]

2019 (1)

2018 (4)

Y. Shen, Z. Wan, Y. Meng, X. Fu, and M. Gong, “Polygonal vortex beams,” IEEE Photonics J. 10(4), 1503016 (2018).
[Crossref]

Y. Shen, X. Fu, and M. Gong, “Truncated triangular diffraction lattices and orbital-angular-momentum detection of vortex SU(2) geometric modes,” Opt. Express. 26(20), 25545–25557 (2018).
[Crossref] [PubMed]

P. H. Tuan, Y. H. Hsieh, Y. H. Lai, K. F. Huang, and Y. F. Chen, “Characterization and generation of high-power multi-axis vortex beams by using off-axis pumped degenerate cavities with external astigmatic mode converter,” Opt. Express. 26(16), 20481–20491 (2018).
[Crossref] [PubMed]

G. Liang, Y. Wang, Q. Guo, and H. Zhang, “Anisotropic diffraction induced by orbital angular momentum during propagations of optical beams,” Opt. Express. 26(7), 8084–8094 (2018).
[Crossref] [PubMed]

2017 (1)

S. Hwang, T. Kim, J. Lee, and T. J. Yu, “Design of square-shaped beam homogenizer for petawatt-class Ti:sapphire amplifier,” Opt. Express. 25(9), 9511–9520 (2017).
[Crossref] [PubMed]

2013 (2)

Z. W. Cao, S. L. Gong, and Y. Gao, “Characterization of TC17 titanium alloy treated by square-spot laser shock peening,” Adv. Mat. Res. 652–654, 2378–2383 (2013).

G. Liang and Q. Guo, “Spiraling elliptic solitons in nonlocal nonlinear media without anisotropy,” Phys. Rev. A 88(4), 043825 (2013).
[Crossref]

2010 (2)

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010).
[Crossref] [PubMed]

F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, “Twin-vortex solitons in nonlocal nonlinear media,” Opt. Lett. 35(5), 628–630 (2010).
[Crossref] [PubMed]

2005 (4)

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt. 47, 291–391 (2005).
[Crossref]

A. I. Yakimenko, Y. A. Zaliznyak, and Y. S. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71(6), 065603 (2005).
[Crossref]

D. Briedis, D. E. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, “Ring vortex solitons in nonlocal nonlinear media,” Opt. Express 13(2), 435–443 (2005).
[Crossref] [PubMed]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
[Crossref] [PubMed]

2004 (1)

2003 (2)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref] [PubMed]

C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91(7), 073901 (2003).
[Crossref] [PubMed]

2001 (2)

P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. 78(2), 249–251 (2001).
[Crossref]

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[Crossref]

2000 (1)

I. Freund, “Optical vortex trajectories,” Opt. Commun. 181(1–3), 19–33 (2000).
[Crossref]

1999 (1)

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[Crossref]

1997 (3)

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79(13), 2450 (1997).
[Crossref]

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144(4–6), 210–213 (1997).
[Crossref]

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
[Crossref]

1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. R. Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826 (1995).
[Crossref] [PubMed]

1993 (1)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40(1), 73–87 (1993).
[Crossref]

1983 (1)

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27(6), 3135 (1983).
[Crossref]

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[Crossref]

Allen, L.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[Crossref]

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144(4–6), 210–213 (1997).
[Crossref]

Anderson, D.

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27(6), 3135 (1983).
[Crossref]

Assanto, G.

C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91(7), 073901 (2003).
[Crossref] [PubMed]

Babiker, M.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[Crossref]

Bang, O.

Berry, M. V.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[Crossref]

Briedis, D.

Buccoliero, D.

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010).
[Crossref] [PubMed]

Cao, Z. W.

Z. W. Cao, S. L. Gong, and Y. Gao, “Characterization of TC17 titanium alloy treated by square-spot laser shock peening,” Adv. Mat. Res. 652–654, 2378–2383 (2013).

Carmon, T.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
[Crossref] [PubMed]

Chen, Y. F.

P. H. Tuan, Y. H. Hsieh, Y. H. Lai, K. F. Huang, and Y. F. Chen, “Characterization and generation of high-power multi-axis vortex beams by using off-axis pumped degenerate cavities with external astigmatic mode converter,” Opt. Express. 26(16), 20481–20491 (2018).
[Crossref] [PubMed]

Cohen, O.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
[Crossref] [PubMed]

Conti, C.

C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91(7), 073901 (2003).
[Crossref] [PubMed]

Courtial, J.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144(4–6), 210–213 (1997).
[Crossref]

Dennis, M. R.

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010).
[Crossref] [PubMed]

Desyatnikov, A. S.

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010).
[Crossref] [PubMed]

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt. 47, 291–391 (2005).
[Crossref]

Dholakia, K.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144(4–6), 210–213 (1997).
[Crossref]

Dunlop, H. R.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. R. Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826 (1995).
[Crossref] [PubMed]

Edmundson, D.

Firth, W. J.

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79(13), 2450 (1997).
[Crossref]

Freund, I.

I. Freund, “Optical vortex trajectories,” Opt. Commun. 181(1–3), 19–33 (2000).
[Crossref]

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. R. Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826 (1995).
[Crossref] [PubMed]

Fu, X.

Y. Shen, Z. Wan, Y. Meng, X. Fu, and M. Gong, “Polygonal vortex beams,” IEEE Photonics J. 10(4), 1503016 (2018).
[Crossref]

Y. Shen, X. Fu, and M. Gong, “Truncated triangular diffraction lattices and orbital-angular-momentum detection of vortex SU(2) geometric modes,” Opt. Express. 26(20), 25545–25557 (2018).
[Crossref] [PubMed]

Galajda, P.

P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. 78(2), 249–251 (2001).
[Crossref]

Gao, Y.

Z. W. Cao, S. L. Gong, and Y. Gao, “Characterization of TC17 titanium alloy treated by square-spot laser shock peening,” Adv. Mat. Res. 652–654, 2378–2383 (2013).

Gong, M.

Y. Shen, X. Fu, and M. Gong, “Truncated triangular diffraction lattices and orbital-angular-momentum detection of vortex SU(2) geometric modes,” Opt. Express. 26(20), 25545–25557 (2018).
[Crossref] [PubMed]

Y. Shen, Z. Wan, Y. Meng, X. Fu, and M. Gong, “Polygonal vortex beams,” IEEE Photonics J. 10(4), 1503016 (2018).
[Crossref]

Gong, S. L.

Z. W. Cao, S. L. Gong, and Y. Gao, “Characterization of TC17 titanium alloy treated by square-spot laser shock peening,” Adv. Mat. Res. 652–654, 2378–2383 (2013).

Grier, D. G.

Guo, Q.

G. Liang, Y. Wang, Q. Guo, and H. Zhang, “Anisotropic diffraction induced by orbital angular momentum during propagations of optical beams,” Opt. Express. 26(7), 8084–8094 (2018).
[Crossref] [PubMed]

G. Liang and Q. Guo, “Spiraling elliptic solitons in nonlocal nonlinear media without anisotropy,” Phys. Rev. A 88(4), 043825 (2013).
[Crossref]

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. R. Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826 (1995).
[Crossref] [PubMed]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. R. Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826 (1995).
[Crossref] [PubMed]

Hsieh, Y. H.

P. H. Tuan, Y. H. Hsieh, Y. H. Lai, K. F. Huang, and Y. F. Chen, “Characterization and generation of high-power multi-axis vortex beams by using off-axis pumped degenerate cavities with external astigmatic mode converter,” Opt. Express. 26(16), 20481–20491 (2018).
[Crossref] [PubMed]

Hu, B.

Huang, K. F.

P. H. Tuan, Y. H. Hsieh, Y. H. Lai, K. F. Huang, and Y. F. Chen, “Characterization and generation of high-power multi-axis vortex beams by using off-axis pumped degenerate cavities with external astigmatic mode converter,” Opt. Express. 26(16), 20481–20491 (2018).
[Crossref] [PubMed]

Hwang, S.

S. Hwang, T. Kim, J. Lee, and T. J. Yu, “Design of square-shaped beam homogenizer for petawatt-class Ti:sapphire amplifier,” Opt. Express. 25(9), 9511–9520 (2017).
[Crossref] [PubMed]

Indebetouw, G.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40(1), 73–87 (1993).
[Crossref]

Kartashov, Y. V.

Kim, T.

S. Hwang, T. Kim, J. Lee, and T. J. Yu, “Design of square-shaped beam homogenizer for petawatt-class Ti:sapphire amplifier,” Opt. Express. 25(9), 9511–9520 (2017).
[Crossref] [PubMed]

Kivshar, Y. S.

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010).
[Crossref] [PubMed]

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt. 47, 291–391 (2005).
[Crossref]

A. I. Yakimenko, Y. A. Zaliznyak, and Y. S. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71(6), 065603 (2005).
[Crossref]

Krolikowski, W.

Ladavac, K.

Lai, Y. H.

P. H. Tuan, Y. H. Hsieh, Y. H. Lai, K. F. Huang, and Y. F. Chen, “Characterization and generation of high-power multi-axis vortex beams by using off-axis pumped degenerate cavities with external astigmatic mode converter,” Opt. Express. 26(16), 20481–20491 (2018).
[Crossref] [PubMed]

Lee, J.

S. Hwang, T. Kim, J. Lee, and T. J. Yu, “Design of square-shaped beam homogenizer for petawatt-class Ti:sapphire amplifier,” Opt. Express. 25(9), 9511–9520 (2017).
[Crossref] [PubMed]

Liang, G.

G. Liang and Q. Wang, “Controllable conversion between Hermite Gaussian and Laguerre Gaussian modes due to cross phase,” Opt. Express 27(8), 10684–10691 (2019).
[Crossref]

G. Liang, Y. Wang, Q. Guo, and H. Zhang, “Anisotropic diffraction induced by orbital angular momentum during propagations of optical beams,” Opt. Express. 26(7), 8084–8094 (2018).
[Crossref] [PubMed]

G. Liang and Q. Guo, “Spiraling elliptic solitons in nonlocal nonlinear media without anisotropy,” Phys. Rev. A 88(4), 043825 (2013).
[Crossref]

Manela, O.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
[Crossref] [PubMed]

Meng, Y.

Y. Shen, Z. Wan, Y. Meng, X. Fu, and M. Gong, “Polygonal vortex beams,” IEEE Photonics J. 10(4), 1503016 (2018).
[Crossref]

Mitchell, D. J.

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
[Crossref]

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[Crossref]

Ormos, P.

P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. 78(2), 249–251 (2001).
[Crossref]

Padgett, M. J.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[Crossref]

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144(4–6), 210–213 (1997).
[Crossref]

Peccianti, M.

C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91(7), 073901 (2003).
[Crossref] [PubMed]

Petersen, D. E.

Rotschild, C.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
[Crossref] [PubMed]

Segev, M.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
[Crossref] [PubMed]

Shen, Y.

Y. Shen, Z. Wan, Y. Meng, X. Fu, and M. Gong, “Polygonal vortex beams,” IEEE Photonics J. 10(4), 1503016 (2018).
[Crossref]

Y. Shen, X. Fu, and M. Gong, “Truncated triangular diffraction lattices and orbital-angular-momentum detection of vortex SU(2) geometric modes,” Opt. Express. 26(20), 25545–25557 (2018).
[Crossref] [PubMed]

Skryabin, D. V.

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79(13), 2450 (1997).
[Crossref]

Snyder, A. W.

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
[Crossref]

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[Crossref]

Torner, L.

F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, “Twin-vortex solitons in nonlocal nonlinear media,” Opt. Lett. 35(5), 628–630 (2010).
[Crossref] [PubMed]

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt. 47, 291–391 (2005).
[Crossref]

Tuan, P. H.

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G. Liang, Y. Wang, Q. Guo, and H. Zhang, “Anisotropic diffraction induced by orbital angular momentum during propagations of optical beams,” Opt. Express. 26(7), 8084–8094 (2018).
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S. Hwang, T. Kim, J. Lee, and T. J. Yu, “Design of square-shaped beam homogenizer for petawatt-class Ti:sapphire amplifier,” Opt. Express. 25(9), 9511–9520 (2017).
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A. I. Yakimenko, Y. A. Zaliznyak, and Y. S. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71(6), 065603 (2005).
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G. Liang, Y. Wang, Q. Guo, and H. Zhang, “Anisotropic diffraction induced by orbital angular momentum during propagations of optical beams,” Opt. Express. 26(7), 8084–8094 (2018).
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S. Hwang, T. Kim, J. Lee, and T. J. Yu, “Design of square-shaped beam homogenizer for petawatt-class Ti:sapphire amplifier,” Opt. Express. 25(9), 9511–9520 (2017).
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Figures (8)

Fig. 1
Fig. 1 Evolutions of soliton intensity (the first row) and phase (the second row) when r = 1, P0 = Pc and Θ = Θc. The numbers labeled above every plots are the propagation distances in half periods.
Fig. 2
Fig. 2 Intensity (the first row) and phase (the second row) for different values of r labeled above every figure.
Fig. 3
Fig. 3 Intensity for different values of r labeled above every figure.
Fig. 4
Fig. 4 Same as Fig. 1 but plots are for r = 1.3.
Fig. 5
Fig. 5 Evolutions of breathers intensity (the first row) and phase (the second row) in half period of T = 2 π / P 0 γ when P0 = 1.5 Pc and Θ = Θc. The distances labeled above every plots are the same as that in Fig. 1
Fig. 6
Fig. 6 Same as Fig. 5 but plots are for Θ = 0.5Θc (the first row) and Θ = 1.5Θc (the second row), but keeping P0 = Pc.
Fig. 7
Fig. 7 Evolutions of the beam φ = A(GbcrGcb) exp(iΘxy) with b = 2, c = 1 and Θ = 3/8. White dashed and green dashed lines denote the profiles of Gbc exp(iΘ1 xy) and Gcb exp(iΘ2 xy), respectively.
Fig. 8
Fig. 8 Evolutions of the beam φ = A[Gbc exp (iΘ1 xy) − Gcb exp (iΘ2 xy)] with b = 2, c = 1, Θ1 = 3/4 and Θ2 = 1/8. White dashed and green dashed lines denote the profiles of Gbc exp (iΘ1 xy) and Gcb exp (iΘ2 xy), respectively.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

i φ z + 1 2 2 φ 1 2 γ 2 P 0 ( x 2 + y 2 ) φ = 0 ,
φ ( X , Y , Z ) = A [ G b c exp  ( i Θ X Y + i λ Z ) r G c b exp  ( i Θ X Y + i λ Z ) ] ,
P 0 = A 2 π b c [ r 2 + 1 α ( r , Θ ) ] ,
σ M P 0 = Θ ( b 2 c 2 ) ( r 2 + 1 + α 3 / 4 r 2 ) 2 ( r 2 + 1 α ) ,
P c = ( b 2 + c 2 ) 2 4 b 4 c 4 γ 2 , Θ c = b 2 c 2 2 b 2 c 2 ,
φ ( X , Y , Z ) A ( b 2 c 2 ) 2 b 2 c 2 ( X 2 + i Θ 4 b 2 c 2 b 2 c 2 X Y Y 2 ) .
I = | φ | 2 r 2 d r , ω = M / I .

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