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Propagation of circular cosine-hyperbolic Gaussian beams in strongly nonlocal nonlinear media

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Abstract

The propagation properties of a circular cosine-hyperbolic Gaussian beam (CiChGB) in a strongly nonlocal nonlinear media (SNNM) are theoretically investigated. Based on the Snyder-Michell model and Collins formula, the closed-form expressions of the complex amplitude, beam width, and curvature radius for an on-axis incident CiChGB in SNNM are derived, and their evolution behaviors are illustrated with numerical examples. It is demonstrated that a CiChGB evolves periodically in SNNM, similarly to a spatial breather, due to the interplay between the nonlinearity process and diffraction. The period of evolution and beam intensity pattern is closely dependent on the input power, initial beam parameters, and SNNM factor. Under the critical input power, the intensity distribution, width and curvature radius of the beam can keep invariant upon propagation, so the CiChGB behaves like a soliton. The research results could have potential applications in beam shaping, optical switch, and micromanipulation.

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References

  • Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover (1970)

    MATH  Google Scholar 

  • Bang, O., Krolikowski, W., Wyller, J., Rusmussen, J.J.: Collapse arrest and soliton stabilization in nonlocal nonlinear media. Phys. Rev. E 66, 046619–046622 (2002)

    ADS  MathSciNet  Google Scholar 

  • Belanger, P.A.: Beam propagation and ABCD ray matrices. Opt. Lett. 16(4), 196–198 (1991)

    ADS  Google Scholar 

  • Bian, L., Tang, B.: Evolution properties of hypergeometric-Gaussian type-II beams in strongly nonlocal nonlinear media. J. Opt. Soc. Am. B 35, 1362–1367 (2018)

    ADS  Google Scholar 

  • Buccoliero, D., Desyanitkev, A.S., Krolikowski, W., Kivshar, Y.S.: Spiraling multi-vortex solitons in nonlocal nonlinear media. Opt. Lett. 32, 198–200 (2008a)

    ADS  Google Scholar 

  • Buccoliero, D., Desyanitkev, A.S., Krolikowski, W., Kivshar, Y.S.: Laguerre and Hermite soliton clusters in nonlocal nonlinear media. Phys. Rev. Lett. 98, 053901–053904 (2008b)

    ADS  Google Scholar 

  • Casperson, L.W., Hall, D.G., Tovar, A.A.: Sinusoidal-Gaussian beams in complex optical systems. J. Opt. Soc. Am. A 14, 3341–3348 (1997)

    ADS  MathSciNet  Google Scholar 

  • Chen, S., Qi, Z., Xie, J., Lin, G., Zhan, Y., Ma, S., Deng, D.: Propagation properties of Airy Gaussian vortex beams in strongly nonlocal nonlinear media. Opt. Commun. 429, 72–79 (2018)

    ADS  Google Scholar 

  • Chu, X.: Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere. Opt. Express 15(26), 17613–17618 (2007)

    ADS  Google Scholar 

  • Collins, S.A.: Lens-system diffraction integral written in terms of matrix optics. J. Opt. Soc. Am 60, 1168–1177 (1970)

    ADS  Google Scholar 

  • Conti, C., Peccianti, M., Assanto, G.: Observation of optical spatial solitons in a highly nonlocal medium. Phys. Rev. Lett. 92, 113902–113905 (2004)

    ADS  Google Scholar 

  • Dai, Z., Yang, Z.J., Zhang, S., Pang, Z.G., You, K.: Propagation dynamics of modified hollow Gaussian beams in strongly nonlocal nonlinear media. Laser. Phys. 25, 025401–025409 (2015)

    ADS  Google Scholar 

  • Dai, Z., Yang, Z., Tang, S., Pang, Z.: Cosh-squared-Gaussian breathers induced by off-waist incidence in highly nonlocal media. Optik 217, 164766–1647780 (2020)

    ADS  Google Scholar 

  • Dai, Z., Yang, Z., Ling, X., Zhang, S., Panget, Z.: Interaction trajectory of solitons in nonlinear media with an arbitrary degree of nonlocality. Ann. Phys. 366, 13–21 (2021)

    ADS  MathSciNet  Google Scholar 

  • Dai, Z., Wang, Y., Wang, Y.C., Yang, Z., Pang, Z.: Propagation properties of controllable anomalous hollow beams in strongly nonlocal nonlinear media. Res. Phys. 42, 105951 (2022)

    Google Scholar 

  • Deng, D.M., Guo, Q.: Propagation of Laguerre–Gaussian beams in strongly nonlocal nonlinear media. J. Opt. A Pure Appl. Opt. 10, 35101–35108 (2008a)

    Google Scholar 

  • Deng, D.M., Guo, Q.: Ince–Gaussian beams in strongly nonlocal nonlinear media. J. Phys. B At. Mol. Opt. Phys. 41, 145401–145408 (2008b)

    ADS  Google Scholar 

  • El Halba, E.M., Hricha, Z., Belafhal, A.: Fractional Fourier transforms of circular cosine-hyperbolic-Gaussian beams. J. Mod. Opt. 69(19), 1086–1093 (2022)

    ADS  Google Scholar 

  • Eyyuboglu, H.T., Baykal, Y.: Analysis of reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere. Opt. Express 12(20), 4659–4674 (2004)

    ADS  Google Scholar 

  • Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals Series, and Products, 5th edn. Academic Press, New York (1994)

    MATH  Google Scholar 

  • Guo, Q., Lu, B., Yi, F., Chi, S., Xie, Y.: Large shift of nonlocal optical spatial solitons. Phys. Rev. E 69, 016602–016609 (2004)

    ADS  Google Scholar 

  • Hricha, Z., Belafhal, A.: Propagation properties of elegant and standard Hermite-cosh-Gaussian laser beams. Phys. Chem. News 16, 37–40 (2004)

    Google Scholar 

  • Hricha, Z., Belafhal, A.: Focusing properties and focal shift of hyperbolic-cosine-Gaussian beams. Opt. Commun. 253, 242–249 (2005a)

    ADS  Google Scholar 

  • Hricha, Z., Belafhal, A.: Comparative parametric characterization of elegant and standard Hermite-cosh-Gaussian beams. Opt. Commun. 253, 231–241 (2005b)

    ADS  Google Scholar 

  • Hricha, Z., Dalil-Essakali, L., Ibnchaikh, M., Belafhal, A.: Kurtosis factor of some truncated and nontruncated laser beams. Phys. Chem. News. 3, 11–16 (2001)

    Google Scholar 

  • Hricha, Z., Yaalou, M., Belafhal, A.: Propagation of hollow sinh-Gaussian beams in strongly nonlocal nonlinear media. Opt. Commun. 478, 126400 (2021a)

    Google Scholar 

  • Hricha, Z., Yaalou, M., Belafhal, A.: Propagation properties of vortex cosine-hyperbolic-Gaussian beams in strongly nonlocal nonlinear media. J. Quant. Spectrosc. Radiat. Transf. 5, 107554–107561 (2021b)

    Google Scholar 

  • Hricha, Z., El Halba, E.M., Lazrek, M., Belafhal, A.: Focusing properties and focal shift of a vortex cosine-hyperbolic Gaussian beam. Opt. Quant. Elect. 53(8), 449–465 (2021c)

    Google Scholar 

  • Hricha, Z., Lazrek, M., El Halba, E., Belafhal, A.: Parametric characterization of vortex cosine-hyperbolic-Gaussian beams. Res. Opt. 5, 100120–100127 (2021d)

    Google Scholar 

  • Hricha, Z., Yaalou, M., Belafhal, A.: Evolution properties of vortex beams through strongly nonlocal nonlinear media. Chin. J. Phys. 77, 1419–1430 (2022a)

    MathSciNet  Google Scholar 

  • Hricha, Z., Lazrek, M., Belafhal, A.: Paraxial propagation of cosine-hyperbolic-Gaussian beams in a chiral medium. J. Mod. Opt. 60(20), 1159–1169 (2022b)

    ADS  Google Scholar 

  • Hricha, Z., El Halba, E.M., Belafhal, A.: Circular cosine-hyperbolic-Gaussian beam, and its paraxial propagation properties. Opt. Commun. 502, 127400 (2022c)

    Google Scholar 

  • Izdebkaya, Y.V., Desyatnikove, A.S., Kivsha, Y.S.: Self-induced mode transformation in nonlocal nonlinear media. Phys. Rev. Lett. 111, 123902 (2013)

    ADS  Google Scholar 

  • Kaur, S., Kaur, M., Kaur, R., Gill, T.S.: Propagation characteristics of Hermite-cosh-Gaussian laser beams in a rippled density plasma. Laser Part. Beams 35, 100–107 (2017)

    ADS  Google Scholar 

  • Konar, S., Mishra, M., Jana, S.: Nonlinear evolution of cosh-Gaussian laser beams and generation of flat-top spatial solitons in cubic quantic nonlinear media. Phys. Lett. A 362, 505–510 (2007)

    ADS  Google Scholar 

  • Lazrek, M., Hricha, Z., Belafhal, A.: Partially coherent vortex cosh-Gaussian beam, and its paraxial propagation. Opt. Quant. Electron. 53(12), 694–710 (2021)

    Google Scholar 

  • Lazrek, M., Hricha, Z., Belafhal, A.: Propagation properties of vortex cosine-hyperbolic-Gaussian beams through oceanic turbulence. Opt. Quant. Electron. 54(3), 172–185 (2022)

    Google Scholar 

  • Lü, B., Ma, H., Zhang, B.: Propagation properties of cosh-Gaussian beams. Opt. Commun. 164, 165–170 (1999)

    ADS  Google Scholar 

  • Lu, D.Q., Hu, W., Zheng, Y.J., Liang, Y.B., Cao, L.G., Guo, Q.: Self-induced fractional Fourier transform and revivable higher-order spatial solitons in strongly nonlocal nonlinear media. Phys. Rev. A 78, 043815 (2008)

    ADS  Google Scholar 

  • Lu, D.Q., Hu, W., Guo, Q.: The relation between optical beam propagation in free space and strongly nonlocal nonlinear media. Europhys. Lett. 86, 4404–4410 (2009)

    Google Scholar 

  • Mahajan, R., Richa, T.S.G., Kaur, R., Aggarwal, M.: Stability and dynamics of a cosh-Gaussian laser beam in relativistic thermal quantum plasma. Laser Part. Beams 26(3), 314–352 (2018)

    Google Scholar 

  • Patil, S.D., Takale, M.V., Navare, S.T., Dongare, M.B.: Focusing of Hermite cosh-Gaussian laser beams in collisionless magnetoplasma. Laser Part. Beams 28, 343–349 (2010)

    ADS  Google Scholar 

  • Patil, S.D., Takale, M.V., Navare, S.T., Fulari, V.J.N., Dongare, M.B.: Relativistic self-focusing of cosh-Gaussian laser beams in a plasma. Opt. Laser Technol. 44, 314–317 (2012)

    ADS  Google Scholar 

  • Peccianti, M., Conti, C., Assanto, G.: Interplay between nonlocality and nonlinearity in nematic liquid crystals. Opt. Lett. 30, 415–417 (2005)

    ADS  Google Scholar 

  • Pedri, P., Santos, L.: Two-dimensional bright solitons in dipolar Bose–Einstein condensates. Phys. Rev. Lett. 95, 200404–200407 (2005)

    ADS  Google Scholar 

  • Rotschild, C., Cohen, O., Manela, O., Sejev, M.: Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and vortex-ring solitons. Phys. Rev. Lett. 95, 213904 (2005)

    ADS  Google Scholar 

  • Rusmussen, P.D., Bang, O., Krolikowski, W.: Theory of nonlocal soliton interaction in nematic liquid crystals. Phys. Rev. E 72, 066611–066617 (2005)

    ADS  MathSciNet  Google Scholar 

  • Shen, S., Yang, Z.J., Li, X.L., Zhang, S.M.: Periodic propagation of complex-valued hyperbolic-cosine-Gaussian solitons and breathers with complicated light field structure in strongly nonlocal nonlinear media. Commun. Nonlinear Sci. Numer. Simul. 103, 106005 (2021)

    MathSciNet  MATH  Google Scholar 

  • Shen, S., Yang, Z.J., Pang, Z.G., Ge, Y.R.: The complex-valued astigmatic cosine-Gaussian soliton solution of the nonlocal nonlinear Schrödinger equation and its transmission characteristics. Appl. Math. Lett. 125, 107755 (2022)

    MATH  Google Scholar 

  • Siegman, A.E.: New developments in laser resonator. In: Proceeding of SPIE, pp. 12242–12256 (1990)

  • Skupin, S., Bang, O., Edmundson, D., Krolikowski, W.: Stability of two-dimensional spatial solitons in nonlocal nonlinear media. Phys. Rev. E 73, 066603 (2006)

    ADS  Google Scholar 

  • Skupin, S., Saffman, M.P., Krolikowski, W.: Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor. Phys. Rev. Lett. 98, 263902–263905 (2007)

    ADS  Google Scholar 

  • Snyder, A.W., Mitchell, D.J.: Accessible solitons. Science 276, 1538–1541 (1997)

    Google Scholar 

  • Song, L., Jiang, X., Yang, Z., Li, X., Zhang, S.: Propagation of Lorentz beam in strongly nonlocal media. J. Nonlinear Opt. Phys. Mater. 27(4), 1850047 (2018)

    Google Scholar 

  • Song, L., Yang, Z., Li, X.: Spiraling anomalous vortex beam arrays in strongly nonlocal nonlinear media. Phys. Rev. A 99, 063817 (2019)

    ADS  Google Scholar 

  • Terhalle, B., Trager, D., Tang, L., Imbrock, J., Denz, C.: Structure analysis of two-dimensional multipole solitons in local nonlinear media. Phys. Rev. E 74, 057601–057609 (2006)

    ADS  Google Scholar 

  • Wang, Y.B., Yang, Z.J., Pan, Z.G., Li, X.L., Zhang, S.M.: Shape-variable four-petal Gaussian vortex breathers in strongly nonlocal nonlinear media. Res. Phys. 15, 102583–102588 (2019)

    Google Scholar 

  • Wani, M.A., Kant, N.: Self-Focusing of Hermite-cosh-Gaussian laser beams in Plasma under density transition. Adv. Opt. 2, 1–5 (2014)

    Google Scholar 

  • Weber, H.: Propagation of higher-order intensity moments in quadratic-index media. Opt. Quant. Electron. 24, S1027–S1049 (1992)

    Google Scholar 

  • Yang, Z.J., Lu, D., Hu, W., Zheng, Y., Gao, X., Guo, Q.: Propagation of optical beams in SNNM. Phys. Lett. A 374, 4007–4013 (2010a)

    ADS  MATH  Google Scholar 

  • Yang, Z.J., Lu, D.Q., Hu, W., Zheng, Y.Z., Gao, X.H.: Hollow Gaussian beams in strongly nonlocal nonlinear media. Chin. Phys. B 19, 124212–124220 (2010b)

    ADS  Google Scholar 

  • Zhou, G., Chen, R., Ru, G.: Propagation of an Airy beam in strongly nonlocal nonlinear media. Laser. Phys. Lett. 11, 105001–105011 (2014)

    ADS  Google Scholar 

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Authors and Affiliations

  1. Laboratory LPNAMME, Laser Physics Group, Department of Physics, Faculty of Sciences, Chouaïb Doukkali University, P. B 20, 24000, El Jadida, Morocco

    Z. Hricha, E. M. El Halba & A. Belafhal

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  1. Z. Hricha
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  2. E. M. El Halba
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  3. A. Belafhal
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All authors contributed to the study conception and design. All authors performed simulations, data collection and analysis and commented the present version of the manuscript. All authors read and approved the final manuscript.

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Hricha, Z., El Halba, E.M. & Belafhal, A. Propagation of circular cosine-hyperbolic Gaussian beams in strongly nonlocal nonlinear media. Opt Quant Electron 55, 473 (2023). https://doi.org/10.1007/s11082-023-04758-0

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