Skip to main content
SpringerLink
Log in

Focusing properties and focal shift of a vortex cosine-hyperbolic Gaussian beam

  • Published:
Optical and Quantum Electronics Aims and scope Submit manuscript

Abstract

In this paper, we investigate the focusing properties of a vortex-cosh-Gaussian (vChG) beam passing through a converging thin lens. Based on the Huygens–Fresnel diffraction integral, we derived the analytical propagation equation as well as the beam width expression of a focused vChGB. It is shown that the focusing properties including the focal shift of the focused vChGB are crucially dependent on the incident beam parameters namely the decentered parameter and the topological charge in addition to the Gaussian Fresnel number \(N_{F}\). From typical numerical examples, it is found that the focused vChGB is transformed into a multi-lobes structure shape at the real focus plane, and the principal maximum intensity of the beam is located away from the axis. The amount of focal shift, which is determined from the minimum spot size criterion, is strongly dependent on the Fresnel number, the decentered parameter and the vortex charge m. The obtained results may be useful for the applications of the vChGBs in beam shaping and beam focusing.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Belafhal, A., Hricha, Z., Dalil-Essakali, L., Usman, T.: A note on some integrals involving Hermite polynomials and their applications. Adv. Math. Mod. Appl. 5, 313–319 (2020)

    Google Scholar 

  • Borghi, R., Santasiero, M., Vicalvi, S.: Focal shift in focused flat-topped beams. Opt. Commun. 154, 243–248 (1998)

    Article  ADS  Google Scholar 

  • Cai, Y., Ge, D.: Propagation of various dark hollow beams through an aperture paraxial ABCD optical system. Phys. Lett. A 357, 72–80 (2006)

    Article  ADS  Google Scholar 

  • Cai, Y., Lu, X., Lin, Q.: Hollow Gaussian beam and its propagation. Opt. Lett. 28, 1084–1086 (2003)

    Article  ADS  Google Scholar 

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

    Article  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 

  • Green, P.L., Hall, D.G.: Focal shift in vector beams. Opt. Exp. 4(10), 411–419 (1999)

    Article  ADS  Google Scholar 

  • Hricha, Z., Belafhal, A.: Focusing properties and focal shift hyperbolic-cosine-Gaussian beams. Opt. Commun. 35, 242–249 (2005a)

    Article  ADS  Google Scholar 

  • Hricha, Z., Belafhal, A.: Focal shift in the axisymmetric Bessel-modulated Gaussian beam. Opt. Commun. 255, 235–240 (2005b)

    Article  ADS  Google Scholar 

  • Hricha, Z., Dalil-Essakali, L., Belafhal, A.: Axial Intensity and focal shifts in focused partially coherent conical Bessel-Gauss beams. Opt. Quant. Elec. 35, 101–107 (2003)

    Article  Google Scholar 

  • Hricha, Z., Yaalou, M., Belafhal, A.: Introduction of a new vortex cosine-hyperbolic-Gaussian beam and the study of its propagation properties in fractional fourier transform optical system. Opt. Quant. Electron. 52, 296–302 (2020)

    Article  Google Scholar 

  • Kuga, T., Torii, Y., Shiokawa, N., Hirano, T., Shimizu, Y., Sasada, H.: Novel optical trap of atoms with a doughnut beam. Phys. Rev. Lett. 78, 4713–4716 (1997)

    Article  ADS  Google Scholar 

  • Lan, J., Li, B.: Focusing hollow Gaussian beam with variation of radial polarization. Optik 207, 163774–163785 (2020)

    Article  ADS  Google Scholar 

  • Li, Y., Wolf, E.: Focal shift in diffracted converging spherical waves. Opt. Commun. 39, 211–215 (1981)

    Article  ADS  Google Scholar 

  • Li, Y., Wolf, E.: Focal shift in focused truncated Gaussian beam. Opt. Commun. 42, 151–156 (1982)

    Article  ADS  Google Scholar 

  • Li, J., Sun, P., Ma, H., Zhou, S.: Focus properties of cosh-Gaussian beams with the power-exponent phase vortex. J. Opt. Soc. Am. A 37, 483–490 (2020)

    Article  ADS  Google Scholar 

  • Liu, X., Pu, J.: Focal shift of the partially coherent vortex beams focused by an aperture lens. Optik 120, 574–578 (2009a)

    Article  ADS  Google Scholar 

  • Liu, X., Pu, J.: Focal shift of the partially vortex beams focused by an aperture lens. Optik 120(12), 574–578 (2009b)

    Article  ADS  Google Scholar 

  • Liu, H., Dong, Y., Zhang, J., Li, S., Lü, Y.: The diffraction propagation properties of double-half inverse Gaussian hollow beams. Opt. Laser Technol. 56, 404–408 (2014)

    Article  ADS  Google Scholar 

  • Liu, D., Wang, Y., Wang, G., Yin, H.: Propagation properties of flat-topped vortex hollow beam in uniaxial crystals orthogonal to the optical axis. Optik 127, 7842–7851 (2016)

    Article  ADS  Google Scholar 

  • Liu, Z., Chen, J., Zhao, D.: Experimental study of propagation properties of vortex beams in oceanic turbulence. Appl. Opt. 56(12), 3577–3582 (2017)

    Article  ADS  Google Scholar 

  • Lü, B., Huang, W.: Focal shift in unapertured Bessel-gauss beams. Opt. Commun. 109, 43–46 (1994)

    Article  ADS  Google Scholar 

  • Luo, M., Chen, Q., Huan, L., Zhao, D.: Propagation of stochastic electromagnetic vortex through the turbulent biological tissues. Phys. Lett. A 378, 308–314 (2014)

    Article  ADS  Google Scholar 

  • Mei, Z., Zhao, D.: Controllable dark-hollow beams and their propagation characteristics. J. Opt. Soc. Am. A 22, 1898–1902 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  • Mei, Z., Zhao, D.: Controllable elliptical dark-hollow beams. J. Opt. Soc. Am. A 23, 919–925 (2006a)

    Article  ADS  Google Scholar 

  • Mei, Z., Zhao, D.: Generalized beam propagation factor of hard-edged diffracted controllable dark-hollow beams. Opt. Commun. 263, 261–266 (2006b)

    Article  ADS  Google Scholar 

  • Mei, Z., Zhao, D.: Non-paraxial propagation of controllable dark-hollow beams. J. Opt. Soc. Am. A 25, 537–542 (2008)

    Article  ADS  Google Scholar 

  • Siegman, A.E.: Lasers, University Science Books, Mill Valley, California (1986)

  • Wang, Z., Lin, Q., Wang, Y.: Control of atomic rotation by elliptical hollow beam carrying zero angular momentum. Opt. Commun. 240, 357–362 (2004)

    Article  ADS  Google Scholar 

  • Wang, J., Ni, X.-W., Gu, B., Wang, H.-T.: Focal shift of flat-topped beams through a lens system with and without aperture. Optik 123, 1440–1443 (2012)

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  • Yaalou, M., El Halba, E.M., Hricha, Z., Belafhal, A.: Generation of spiraling Bessel beams from Dark/Antidark Gaussian beams diffracted by a curved fork-shaped hologram. Opt. Quant. Electron. 51, 242–254 (2019a)

    Article  Google Scholar 

  • Yaalou, M., El Halba, E.M., Hricha, Z., Belafhal, A.: Propagation characteristics of Dark and Antidark Gaussian beams in a turbulent atmosphere. Opt. Quant. Electron. 51, 255–266 (2019b)

    Article  Google Scholar 

  • Yin, J., Gao, W., Zhu, Y.: Generation of dark hollow beams and their applications. Prog. in Opt. 44, 119–204 (2003)

    Article  ADS  Google Scholar 

  • Zeng, X., Hui, Z., Zhang, M.: M2 factor of controllable dark-hollow beams through a multi-apertured ABCD system. App. Opt. 57(27), 7667–7672 (2018a)

    Article  ADS  Google Scholar 

  • Zeng, J., Liu, X., Wang, F., Zhao, C., Cai, Y.: Partially coherent fractional vortex beam. Opt. Exp. 26(21), 26830–26844 (2018b)

    Article  ADS  Google Scholar 

  • Zhao, X., Pang, X., Zhang, J., Wan, G.: Transverse focal shift in Vortex beams. IEEEE Photon. J. 10(1), 1–17 (2018)

    Google Scholar 

  • Zhou, G.Q., Cai, Y., Dai, C.Q.: Hollow vortex Gaussian beams. Sci. Chin. 56(5), 896–903 (2013)

    Google Scholar 

Download references

Author information

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, M. Lazrek & A. Belafhal

Authors
  1. Z. Hricha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. M. El Halba
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Lazrek
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. Belafhal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Z. Hricha or A. Belafhal.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hricha, Z., Halba, E.M.E., Lazrek, M. et al. Focusing properties and focal shift of a vortex cosine-hyperbolic Gaussian beam. Opt Quant Electron 53, 449 (2021). https://doi.org/10.1007/s11082-021-03105-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-021-03105-5

Keyword

Search

Navigation