Abstract
The propagation properties of the Whittaker–Gaussian (WG) beam propagating in turbulent atmosphere are investigated in detail based on the extended Huygens–Fresnel diffraction integral and the Rytov method. An analytical expression for the on-axis average intensity distribution of the WG beam in the turbulent atmosphere is derived. In particular, some numerical examples are illustrated and analyzed with various parameter settings to show the influence of the atmospheric turbulent and the source beam parameters on the behavior of the studied beam. However, the evolution characteristics of the WG beam spreading in the atmospheric turbulent are impacted by the atmospheric turbulence strength \(C_{n}^{2}\) and the initial beam parameters including the parameter \(\mu\), the beam order \(m\), the beam waist width \(\omega_{0}\) and the wavelength \(\lambda\). According to the explored results here, our study can be beneficial in some practical applications for both the remote sensing domain and optical communication systems.
Similar content being viewed by others
Availability of data and materials
No datasets is used in the present study.
References
Andrews, L.C., Phillips, R.L.: Laser Beam Propagation through Random Medium Bellingham. SPIE Press, Bellingham (1998)
Arul Teen, Y.P., Nathiyaa, T., Rajesh, K.B., Karthick, S.: Bessel Gaussian beam propagation through turbulence in free space optical communication. Opt. Mem. Neural Netw. 27, 81–88 (2018)
Bandres, M.A., Gutiérrez-Vega, J.C.: Circular beams. Opt. Lett. 33, 177–179 (2008)
Belafhal, A., Hennani, S., Ez-zariy, L., Chafq, A., Khouilid, M.: Propagation of truncated Bessel-modulated Gaussian beams in turbulent atmosphere. Phys. Chem. News 62, 36–43 (2011)
Benzehoua, H., Belafhal, A.: The Effects of Atmospheric Turbulence on the Spectral Changes of Diffracted Pulsed Hollow Higher-Order cosh-Gaussian Beam. Opt. Commun. 541, 1–9 (2023)
Born, M., Wolf, E.: Principles of Optics, 7th (expanded) edn. Cambridge University Press, Cambridge (1999)
Boufalah, F., Dalil-Essakali, L., Ez-zariy, L., Belafhal, A.: Introduction of generalized Bessel–Laguerre–Gaussian beams and its central intensity travelling a turbulent atmosphere. Opt. Quant. Electron. 50, 305–324 (2018)
Cai, Y., He, S.: Propagation of various dark hollow beams in a turbulent atmosphere. Opt. Express 14, 1353–1367 (2006)
Chib, S., Dalil-Essakali, L., Belafhal, A.: Comparative analysis of some Schell-model beams propagating through turbulent atmosphere. Opt. Quant. Electron. 54, 1–17 (2022)
Deng, Y., Wang, H., Ji, X., Li, X., Yu, H., Chen, L.: Characteristics of high-power partially coherent laser beams propagating upwards in the turbulent atmosphere. Opt. Express 28, 27927–27939 (2020)
Ebrahim, A.A.A., Swillam, M.A., Belafhal, A.: Atmospheric turbulent effects on the propagation properties of a general model vortex higher-order cosh-Gaussian beam. Opt. Quant. Electron. 55, 1–13 (2023)
Elmabruk, K., Eyyuboğlu, H.T.: Analysis of flat-topped Gaussian vortex beam scintillation properties in atmospheric turbulence. Opt. Eng. 58, 066115–066115 (2019)
Friehe, C.A., La Rue, J.C., Champagne, F.H., Gibson, C.H., Dreyer, G.F.: Effects of temperature and humidity fluctuations on the optical refractive index in the marine boundary layer. J. Opt. Soc. Am. 65, 1502–1511 (1975)
Gbur, G.: Partially coherent beam propagation in atmospheric turbulence. J. Opt. Soc. Am. A 31, 2038–2045 (2014)
Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series and Products, 5th edn. Academic Press, New York (1994)
Hajjarian, Z., Kavehrad, M., Fadlullah, J.: Spatially multiplexed multi-input-multi-output optical imaging system in a turbid, turbulent atmosphere. Appl. Opt. 49, 1528–1538 (2010)
Jabczyński, J.K., Gontar, P.: Impact of atmospheric turbulence on coherent beam combining for laser weapon systems. Def. Technol. 17, 1160–1167 (2021)
Khannous, F., Boustimi, M., Nebdi, H., Belafhal, A.: Theoretical investigation on the Hollow Gaussian beams propagating in atmospheric turbulent. Chin. J. Phys. 54, 194–204 (2016)
Lopezmago, D., Bandres, M.A., Gutiérrezvega, J.C.: Propagation of Whittaker Gaussian beams. Proc. SPIE 7430, 286–293 (2009)
Ma, H., Li, J., Chen, Y.: Research on the propagation of partially coherent cosh-Gaussian beams through an ABCD optical system in non-Kolmogorov turbulence by effective tensor approach. Opt. Appl. 51, 147–158 (2021)
Nabil, H., Balhamri, A., Belafhal, A.: Partially coherent laser beams propagating in jet engine exhaust induced turbulence. Opt. Quant. Electron. 54, 404–427 (2022)
Navidpour, S.M., Uysal, M., Kavehrad, M.: BER performance of free-space optical transmission with spatial diversity. IEEE Trans. Wirel. Commun. 6, 2813–2819 (2007)
Nossir, N., Dalil-Essakali, L., Belafhal, A.: Behavior of the central intensity of generalized Humbert–Gaussian beams against the atmospheric turbulence. Opt. Quant. Electron. 53, 1–3 (2021)
Nossir, N., Dalil-Essakali, L., Belafhal, A.: Propagation analysis of Whittaker–Gaussian laser beam in a gradient-index medium. Opt. Quant. Electron. 55, 1–10 (2023)
Saad, F., El Halba, E.M., Belafhal, A.: A theoretical study of the on-axis average intensity of generalized spiraling Bessel beams in a turbulent atmosphere. Opt. Quant. Electron. 49, 1–12 (2017)
Srivastava, H.M., Manocha, H.L.: A Treatise on Generating Functions. Halsted Press, Wiley, New York (1984)
Wang, C., Liu, L., Liu, L., Yu, J., Wang, F., Cai, Y., Peng, X.: Second-order statistics of a Hermite–Gaussian correlated Schell-model beam carrying twisted phase propagation in turbulent atmosphere. Opt. Express 31, 13255–13268 (2023)
Wen, J.J., Breazeal, M.A.: A diffraction beam field expressed as the superposition of Gaussian beams. J. Acoust. Soc. Am. 83, 1752–1756 (1988)
Xu, Z., Liu, X., Cai, Y., Ponomarenko, S.A., Liang, C.: Structurally stable beams in the turbulent atmosphere: Dark and antidark beams on incoherent background. J. Opt. Soc. Am. A 39, C51–C57 (2022)
Yaalou, M., El Halba, E.M., Hricha, Z., Belafhal, A.: Propagation characteristics of dark and antidark Gaussian beams in turbulent atmosphere. Opt. Quant. Electron. 51, 1–10 (2019)
Zhu, J., Li, X., Tang, H., Zhu, K.: Propagation of multi-cosine-Laguerre–Gaussian correlated Schell-model beams in free space and atmospheric turbulence. Opt. Express 25, 20071–20086 (2017)
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 interest
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 for publication
The authors confirm that there is informed consent to the publication of the data contained in the article.
Consent to participate
Informed consent was obtained from all authors.
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
Nossir, N., Dalil-Essakali, L. & Belafhal, A. Effects of moderate to weak atmospheric turbulence on the propagation properties of the Whittaker–Gaussian laser beam. Opt Quant Electron 56, 189 (2024). https://doi.org/10.1007/s11082-023-05830-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11082-023-05830-5