Abstract
The propagation properties of the Hermite-cosine-hyperbolic-Gaussian beam (HChGB) in a chiral medium are investigated. The analytical formula for a HChGB beam propagating through a chiral medium is derived theoretically based on the Huygens-Fresnel integral, and the propagation properties are illustrated numerically and discussed. Results show that the evolution properties of the HChGB in a chiral medium are closely related to the beam order, the chirality factor, and the decentered parameter b.
Similar content being viewed by others
Availability of data and materials
No datasets is used in the present study.
References
Baimuratov, A.S., Gun’ko, Y.K., Shalkovskiy, A.G., Baranov, A.V., Fedorov, A.V., Rukhlenko, I.D.: Optical activity of chiral nanoscrolls. Adv. Opt. Mater. 5, 1600982–1600987 (2017)
Bayraktar, M.: Propagation of cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis. Indian J. Phys. (2021). https://doi.org/10.1007/s12648-021-02189-9
Beaulieu, S., Comby, A., Descamps, D., Fabre, B., Garcia, G.A., Géneaux, R., Harvey, A.G., Légaré, F., Mašín, Z., Nahon, L., Nahon, L., Ordonez, A.F., Petit, S., Pons, B., Mairesse, Y., Smirnova, O., Blanchet, V.: Photoexcitation circular dichroism in chiral molecules. Nat. Phys. 14, 484–489 (2018)
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(3), 313–319 (2020)
Casperson, L.W., Tovar, A.A.: Hermite-Sinusoidal-Gaussian Beams in complex optical systems. J. Opt. Am. A 15, 954–961 (1998)
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)
Chen, R.P., Dai, C.Q.: The effect of Kerr nonlinearity on a Cosh-Gaussian beam. Lasers Eng. 19(3), 255–264 (2010)
Chern, R.L., Chang, P.H.: Negative refractive and backward wave in chiral mediums: illustrations of Gaussian beams. J. Appl. Phys. 113(15), 153504–153509 (2013)
Collins, S.A.: Lens-system diffraction integral written in terms of matrix optics. J. Opt. Soc. Am. 60, 1168–1177 (1970)
Deng, D.: Propagation properties of off-axis cosh-Gaussian beam combination through first–order optical systems. Phys. Lett. A 333, 485–495 (2004)
Deng, D.: Propagation of elegant cosine Gaussian laser beams. Opt. Commun. 259, 409–414 (2006)
Deng, F., Yu, W., Huang, J., Zhao, R., Lin, J., Deng, D.: Propagation of Airy-Gaussian beams in a chiral medium. Eur. Phys. J. D, 70–87 (2016)
Du, X., Zhao, D.: Fractional Fourier transform of off-axial elliptical cosh-Gaussian beams. Optik 119, 379–382 (2008)
Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: URSS Academy of Sciences, "Integrals and Series". Overseas Publishers Association (1986)
Gill, J.S.: Self-focusing of Cosh-Gaussian laser beam in a Kerr medium with linear absorption. Advanced Photonics & Renewable Energy, OSA Technical Digest (CD) (2010)
Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals Series, and Products, 5th edn. Academic Press, New York (1994)
Hricha, Z., Belafhal, A.: Propagation properties of elegant Hermite-cosh-Gaussian laser beams. Phys. Chem. News 16, 37–40 (2004)
Hricha, Z., Belafhal, A.: Focusing properties of focal shift of hyperbolic-cosine-Gaussian beams. Opt. Commun. 253, 242–249 (2005a)
Hricha, Z., Belafhal, A.: A comparative parametric characterization of elegant and standard Hermite-cosh-Gaussian beams. Opt. Commun. 253, 231–241 (2005b)
Hricha, Z., Belafhal, A.: Focusing properties of focal Hermite-cosh-Gaussian beams. Opt. Commun. 253, 242–249 (2005c)
Hricha, Z., Lazrek, M., Yaalou, M., Belafhal, A.: Propagation of vortex Hermite-cosine-hyperbolic-Gaussian beams in turbulent atmosphere. Opt. Quant. Elect. 53(11), 624–638 (2021a)
Hricha, Z., Yaalou, M., Belafhal, A.: Introduction of the vortex Hermite-Cosh-Gaussian beam and the analysis of its intensity pattern upon propagation. Opt. Quant. Elect. 53, 80–87 (2021b)
Hricha, Z., El Halba, E.M., Belafhal, A.: Circular cosine-hyperbolic-Gaussian beam and its paraxial propagation properties. Opt. Commun. 502(20), 127400 (2021c)
Hricha, Z., El Halba, E.M., Belafhal, A.: Focusing properties and focal shift of vortex Hermite-cosh- Gaussian beams. Opt. Quant. Elect. 54, 551–570 (2022a)
Hricha, Z., Yaalou, M., Belafhal, A.: Effects of turbulent atmosphere on the propagation properties of vortex Hermite-cosine-hyperbolic-Gaussian beams. Opt. Quant. Elect. 53, 624–635 (2022b)
Hricha, Z., El Halba, E.M., Belafhal, A.: Focusing properties and focal shift of vortex Hermite-cosh-Gaussian beams. Opt. Quant. Elect. 54, 551–570 (2022c)
Hricha, Z., Lazrek, M., Belafhal, A.: Paraxial propagation of cosine-hyperbolic-Gaussian beams in a chiral medium. J. Mod. Opt. 69(12), 1–11 (2022d)
Hricha, Z., El Halba, E.M., Belafhal, A.: Propagation of circular cosine-hyperbolic Gaussian beams in strongly nonlocal nonlinear media. Opt. Quant. Elect. 55, 473–486 (2023)
Hricha, Z., Yaalou, M., Belafhal, A.: Propagation properties of vortex cosine-hyperbolic-Gaussian beams in strongly nonlocal nonlinear media. J.Q.S.R.T.; 265, 107554–107560 (2020)
Hua, S., Liu, Y.W., Zhang, H.J., Tang, L.Z., Feng, Y.C.: Propagation of an Airy-Gaussian-vortex beam in a chiral medium. Opt. Commun. 388, 29–37 (2017)
Hui, Y., Cui, Z., Li, Y., Zhao, W., Han, Y.: Propagation and dynamical characteristics of a Bessel-Gaussian beam in a chiral medium. J. Opt. Soc. Am. A 35(8), 1299–1305 (2018)
Ibnchaikh, M., Dalil-Essakali, L., Hricha, Z., Belafhal, A.: Parametric characterization of truncated Hermite-cosh-Gaussian beams. Opt. Commun. 190, 29–36 (2001)
Jaggard, D.L., Michelson, A.R., Papas, C.H.: On electromagnetic waves in a chiral media. Appl. Phys. 18, 211–216 (1979)
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(1), 100–107 (2017)
Konar, S., Mishra, M., Jana, S.: Nonlieanr 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)
Kwon, D.H., Werner, P.L., Werner, D.H.: Optical planar chiral metamaterial designs for strong circular dichroism and polarization rotation. Opt. Exp. 16, 11802–11807 (2008)
Lazrek, M., Hricha, Z., Belafhal, A.: Propagation of vortex Hermite-cosh-Gaussian beams in a gradient-index medium. Opt. Quant. Elect. 54, 67–78 (2022)
Lazrek, M., Hricha, Z., Belafhal, A.: Properties of circular cosh-Gaussian beams focused by a thin lens system under a turbulent atmosphere. Opt. Quant. Elect. 55, 554-1–18 (2023)
Lee, S., Wang, Z.B., Feng, C., Jiao, J., Khan, A., Li, L.: Circular dichroism in planar extrinsic chirality metamaterial at oblique incident beam. Opt. Commun. 309, 201–204 (2013)
Li, J., Zhuang, S., Xie, Y., Huang, C.: Gradient force pattern of truncated hyperbolic-cosine Gaussian beam through phase plate focusing system. Opt. Eng. 46(12), 124201–124208 (2007)
Liu, X.Y., Zhao, D.M.: Propagation of vortex Airy beam in chiral medium. Opt. Commun. 321, 6–10 (2014)
Lü, B., Ma, H., Zhang, B.: Propagation properties of cosh-Gaussian beams. Opt. Commun. 164, 165–170 (1999)
Mahajan, R., Undefined, R., Gill, T.S., Kaur, R.: Stability and dynamics of a cosh-Gaussian laser beam in relativistic thermal quantum plasma. Laser Part. Beams 26(3), 314–352 (2018)
Patil, S.D., Navare, S.T., Takale, M.V., Dongare, M.B.: Self-focusing of cosh-Gaussian laser beams in a parabolic medium with linear absorption. Opt. Lasers Eng. 47, 604–606 (2009)
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 Tech. 44, 314–317 (2012)
Pendry, J.B.: A chiral route to negative refraction. Science 306, 1353–1355 (2004)
Sersic, I., van de Haar, M.A., Arango, F.B., Koenderink, A.F.: Ubiquity of optical activity in planar metamaterial scatterers. Phys. Rev. Lett. 108, 428–437 (2012)
Siegman, A.E.: Lasers. University Science Books, Mill Valley, Sausalito (1986)
Tovar, A.A., Casperson, L.W.: Production and propagation of Hermite-sinusoidal-Gaussian laser beams. J. Opt. Am. A 15, 2425–2432 (1998)
Wani, M.A., Kant, N.: Self-Focusing of Hermite-cosh-Gaussian laser beams in Plasma under density transition. Adv. in Opt. 154, 497–502 (2018)
Wen, W., Chu, X.: Propagation of symmetric tunable dual airy beam through ABCD optical system. Opt. Commun. 333, 38–44 (2014)
Xie, J.T., Zhang, J.B., Ye, J.R., Liu, H., Liang, Z., Long, S., Zhou, K., Deng, D.: Paraxial propagation of the first order chirped Airy-vortex beams in the chiral medium. Opt. Exp. 26(5), 5845–5856 (2018)
Yaalou, M., Hricha, Z., Lazrek, M., Belafhal, A.: Diffraction characteristics of Hermite-Cosh-Gaussian beams through an Airy transform optical system. J. Mod. Opt. 67(9), 771–781 (2020)
Yang, X., Wu, Z., Qu, T.: Paraxial propagation of the cosh-Airy vortex beams in a chiral medium. Chin. Phys. B 29(3), 034201–034217 (2020)
Zeng, Z., Deng, D.: Paraxial propagation of Pearcey Gaussian beams with the astigmatic phase in the chiral medium. J. Opt. Soc. Am. B 37(1), 30–37 (2020)
Zhang, Y.Y., Wang, L., Zhang, Z.Y.: Circular dichroism in planar achiral plasmonic L-shaped nanostructure arrays. IEEE Photon. J. 9, 4800407–4800413 (2017)
Zhuang, F., Du, X., Zhao, D.: Polarization modulation for a stochastic electromagnetic beam passing through a chiral medium. Opt. Lett. 36(14), 2683–2685 (2011)
Zhuang, F., Du, X., Ye, Y., Zhao, D.: Evolution of Airy beams in a chiral medium. Opt. Lett. 37(11), 1871–1873 (2012)
Funding
No funding is received from any organization for this work.
Author information
Authors and Affiliations
Contributions
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.
Corresponding author
Ethics declarations
Conflict of interests
The authors have no financial or proprietary interests in any material discussed in this article.
Ethical approval
This article does not contain any studies involving animals or human participants performed by any of the authors. We declare this manuscript is original, and is not currently considered for publication elsewhere. We further confirm that the order of authors listed in the manuscript has been approved by all of us.
Consent to participate
Informed consent was obtained from all authors.
Consent for publication
The authors confirm that there is informed consent to the publication of the data contained in the article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Yaalou, M., Hricha, Z. & Belafhal, A. Paraxial propagation of Hermite cosine-hyperbolic-Gaussian beams in a chiral medium. Opt Quant Electron 55, 1281 (2023). https://doi.org/10.1007/s11082-023-05585-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11082-023-05585-z