1 Introduction

The study of spectral properties of pulsed laser beams during their propagation through various media and optical systems has been extensively developed due to its significance in applications such as spectroscopy, optical communication, coding, and information transmission (Yadav et al. 2007a, 2007b, 2008; Han 2009). Spectral changes can arise from correlated sources (Wolf 1987), focusing of light waves (Gbur et al. 2001), scattering in spatially random media (Wolf 1989; Wang et al. 2017) and even turbulence (Liu et al. 2016; Benzehoua and Belafhal 2023a). Moreover, such changes can also occur when the light pulse propagates through an aperture (Pu et al. 1999). There are two main spectral changes: spectral shifts and spectral switches. During the propagation of a laser beam, its spectrum can frequently shift to lower or higher frequencies, which is known as a red or blue shift. Over the years, various studies have explored the spectral changes of laser beams as they propagate through different media and optical systems. In 2016, Zou et al. investigated the spectral anomalies of a pulsed chirped super-Gaussian beam diffracted (Zou and Hu 2016). Similarly, Liu et al. investigated the effect of oceanic turbulence on the spectral properties of a chirped Gaussian pulsed beam (Liu et al. 2016). In another study, Liu et al. explored the spectral and coherence properties of pulsed Gaussian Schell model beams as they propagated through a turbulent atmosphere (Liu et al. 2017). In 2018, Ding et al. examined the impact of oceanic turbulence on the spectral changes of partially coherent pulsed beams (Ding et al. 2018). The authors found that factors such as turbulence kinetic energy dissipation rate, relative intensity decrease for temperature and salinity fluctuations, and the mean quadratic temperature dissipation rate can all affect the spectral switches induced by turbulence. Furthermore, Duan et al. studied the impact of biological tissues and spatial correlation on the spectral changes of a vortex beam modeled by Schell Gaussian (Duan et al. 2020). In 2021, Jo et al. investigated the spectral properties of a chirped Gaussian pulsed beam propagating through a turbulent atmospheric path (Jo et al. 2021). Most recently, Jo et al. conducted a study on the impact of oceanic turbulence on the spectral changes of a diffracted chirped Gaussian pulsed beam (Jo et al. 2022). In this context, Benzehoua and Belafhal have significantly contributed to the field of spectral beam propagation through noteworthy studies. Their initial investigation delved into the spectral characteristics of pulsed Laguerre higher-order cosh-Gaussian beams when traversing turbulent atmospheres (Benzehoua and Belafhal 2023b). In a subsequent study, their focus shifted to the analysis of pulsed vortex beams within oceanic turbulence conditions (Benzehoua and Belafhal 2023c). Their research extended to the examination of how atmospheric turbulence influences the spectral changes in diffracted pulsed hollow higher-order cosh-Gaussian beams (Benzehoua and Belafhal 2023d).

Benzehoua et al. have also explored spectrum changes in pulsed chirped Generalized Hermite cosh-Gaussian beams through turbulent biological tissues (Benzehoua et al. 2023).

On the other hand, the study of marine turbulence and turbulent atmosphere has laid the foundation for the field of physics (Eckart and Ferris 1956). Since the introduction of the first functional laser beam in 1960 (Maiman 1960), the propagation properties of laser beams through turbulent atmosphere and turbulent maritime environments have intrigued researchers due to their significance in communication and laser defense. While research on laser beam propagation through turbulent atmosphere has been extensively explored (Ez-Zariy et al. 2016; Boufalah et al. 2018; Hricha et al. 2022; Nossir et al. 2021; Chib et al. 2022; Bayraktar 2021a, 2021b, 2023), there has been relatively less research on laser beam propagation in turbulent maritime conditions. A study by Bayraktar (Bayraktar 2021a, b, c) showed that an astigmatic hyperbolic sinusoidal Gaussian beam transforms into a ring-shaped pattern during its propagation in oceanic turbulence.

Humidity variation emerges as another influential factor affecting optical turbulence in the marine medium, which complicates the beam propagation process (Friehe et al. 1975). Understanding how optical beams interact with the maritime atmosphere has become crucial, given the wide range of applications, including free-space optical communication and remote sensing. Several models have been developed in the literature to describe the marine atmospheric spectrum. Compared to atmospheric conditions above land, this medium has distinct effects on the propagation of optical radiation. One such contribution is the analytical marine atmospheric spectrum proposed by Grayshan et al. (2008), which approximates Hill's Salton Sea numerical spectrum (Grayshan et al. 2008). Additionally, Khannous and Belafhal presented a novel atmospheric spectrum model for marine atmospheric turbulence in various spaces, offering broad applicability (Khannous and Belafhal 2018).

The study focuses on examining the spectral characteristics of a pulsed LHChG beam when it travels through a turbulent maritime environment. The pulsed LHChG beam is a versatile form of laser beam, which can also include pulsed Laguerre-Gaussian (LG) and Gaussian beams as special cases. Its intensity in the central region can be manipulated by changing the beam parameters, making it a useful tool for various applications. The cosh function used in the pulsed LHChG beam combines Gaussian functions. Additionally, the Laguerre-Gaussian beam, which is a Laguerre polynomial modulated by a Gaussian envelope, offers distinct advantages. The present article aims to further investigate the analytical formulations of spatial coherence length and spectral density in marine atmospheric turbulence, using a novel spectrum proposed by (Khannous and Belafhal 2018). These sections will present the electric field distribution of the pulsed LHChG beam in the Cartesian coordinate system at the source plane. Additionally, in Sect. 2 we will present the theoretical model describing the spectral behaviors of the pulsed beam as it propagates through turbulent maritime environments. A graphical analysis of the impact of beam and turbulent oceanic parameters on the evolution behavior of the pulsed LHChG beam will be presented in Sect. 3. Finally, we will summarize our findings to gain a clear understanding of laser beam properties in maritime environments.

2 Theoretical model on the propagation of pulsed LHChG beam through maritime atmospheric turbulence

In the space–time domain, the distribution of the electric field of the pulsed LHChG beam at the source plane (z = 0) is defined in the cylindrical coordinate system as follows (Benzehoua and Belafhal 2023a)

$$ E_{l,m}^{n} \left( {\rho ,\phi ,z = 0,t} \right) = A_{0} \left( {\frac{\sqrt 2 }{{w_{0} }}\rho } \right)^{l} L_{m}^{l} \left( {\frac{{2\rho^{2} }}{{w_{0}^{2} }}} \right)cosh^{n} \left( {\Omega \rho^{2} } \right)\exp \left( { - \frac{{\rho^{2} }}{{w_{0}^{2} }}} \right){\text{e}}^{il\phi } f\left( t \right), $$
(1)

where \(\rho\) and \(\phi\) are the radial and azimuthal coordinates, \(A_{0}\) is a constant (we'll take it equal to 1 for simplicity), \(L_{m}^{l} (.)\) denotes the Laguerre polynomial with m and l being the radial and angular mode orders, \(w_{0}\) is the width of the Gaussian part size, \(e^{il\phi }\) is the beam phase term, and \(\Omega\) is the displacement parameter associated with the cosh part. In the previous definition (1), \(f(t)\) is the time envelope of the initial pulsed beam.

By applying the following formula (Abramowitz and Stegun 1970)

$$ \begin{array}{*{20}c} {cosh^{n} \left( {r^{2} } \right) = \frac{1}{{2^{n} }}\sum\limits_{s = 0}^{n} {\frac{n!}{{\left( {n - s} \right)!s!}}e^{{a_{sn} r^{2} }} } ,\;\;\;{\text{with}}} & {a_{sn} = 2s - n} \\ \end{array} $$
(2)

and by using the relationship between Laguerre–Gauss and Hermite-Gauss modes

$$ e^{il\phi } \rho^{l} L_{m}^{l} \left( {\rho^{2} } \right) = \frac{{\left( { - 1} \right)^{m} }}{{2^{2m + l} m!}}\sum\limits_{p = 0}^{m} {\sum\limits_{q = 0}^{l} {i^{q} \left( {\begin{array}{*{20}c} m \\ p \\ \end{array} } \right)\left( {\begin{array}{*{20}c} l \\ q \\ \end{array} } \right)\,H_{2p + l - q} \left( x \right)} } H_{2m - 2p + q} \left( y \right), $$
(3)

where \(H_{N} \left( . \right)\) is the Hermite polynomial of order N, and \(\left( {\begin{array}{*{20}c} m \\ p \\ \end{array} } \right)\) is the binomial coefficient,

Equation (1) can be rewritten in Cartesian coordinates as follows

$$ \begin{aligned} E_{l,m}^{n} \left( {x,y,z = 0,t} \right) &= \exp \left( { - \frac{{x^{2} }}{{w_{0}^{2} }} - \frac{{y^{2} }}{{w_{0}^{2} }}} \right)\,\sum\limits_{p = 0}^{m} {\sum\limits_{q = 0}^{l} {\,\frac{{\left( { - 1} \right)^{m} }}{{2^{2m + l + 1} m!}}i^{q} \left( {\begin{array}{*{20}c} m \\ p \\ \end{array} } \right)\,\left( {\begin{array}{*{20}c} l \\ q \\ \end{array} } \right)H_{2p + l - q} \left( {\frac{\sqrt 2 x}{{w_{0} }}} \right)} H_{2m - 2p + q} \left( {\frac{\sqrt 2 y}{{w_{0} }}} \right)} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, &\quad \times \frac{1}{{2^{n} }}\sum\limits_{s = 0}^{n} {\frac{n!}{{\left( {n - s} \right)!s!}}\exp \left( {\Omega a_{sn} x^{2} } \right)\exp \left( {\Omega a_{sn} y^{2} } \right)f\left( t \right)} . \hfill \\ \end{aligned} $$
(4)

Let's assume that the incident pulsed beam has a Gaussian shape (Agrawal 1999)

$$ f\left( t \right) = e^{{ - t^{2} /T^{2} }} e^{{ - i\omega_{0} t}} $$
(5)

where T represents the pulse duration, and \(\omega_{0}\) denotes the central frequency.

By performing the Fourier transform of Eq. (4), we can obtain the following result, which represents the field in the space-frequency domain

$$ E_{l,m}^{n} (x,y,z = 0,\omega ) = \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^{ + \infty } {E(x,y,z = 0,t)e^{ - i\omega t} } dt\, = E(x,y,z = 0)f(\omega ) $$
(6)

where \(f(\omega )\) is the initial Fourier spectrum of the pulsed beam and is given by

$$ f(\omega ) = \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^{ + \infty } {f(t){\text{e}}^{ - i\omega t} } dt $$
(7)

By substituting Eq. (5) into Eq. (7), the calculation of the integral yields

$$ S^{\left( 0 \right)} \left( \omega \right) = \left| {f\left( \omega \right)} \right|^{2} = \left| {\frac{T}{\sqrt 2 }\exp \left( {\frac{{ - T^{2} \left( {\omega - \omega_{0} } \right)^{2} }}{4}} \right)} \right|^{2} $$
(8)

where \(S^{\left( 0 \right)} \left( \omega \right)\) is the original power spectrum of the incident beam.

The cross-spectral density function of the pulsed LHChG beam at the input plane is given by

$$ W_{l,m}^{n} \left( {r_{1} ,r_{2} ,0,\omega } \right) = E_{l,m}^{n} \left( {x_{1} ,y_{1} ,0,\omega } \right)E_{l,m}^{n*} \left( {x_{2} ,y_{2} ,0,\omega } \right), $$
(9)

where \({\mathbf{r}}_{1} = \left( {x_{1} ,y_{1} } \right)\) and \({\mathbf{r}}_{{\mathbf{2}}} = \left( {x_{2} ,y_{2} } \right)\) are two arbitrary position vectors in the source plane.

In the following, we focus on studying the propagation characteristics of the pulsed LHChG beam through a turbulent medium, with particular emphasis on its behavior in the presence of maritime turbulence.

Based on the extended Huygens-Fresnel integral formula and Rytov’s theory, the cross-spectral density function of the pulsed LHChG beam propagating through a turbulent medium can be expressed as follows (Andrews and Phillips 2005)

$$ \begin{aligned} W_{l,m}^{n} \left( {{{\varvec{\uprho}}}_{{\mathbf{1}}} ,{{\varvec{\uprho}}}_{{\mathbf{2}}} ,z,\omega } \right) &= \left( {\frac{k}{2\pi z}} \right)^{2} S^{\left( 0 \right)} \left( \omega \right)\int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {W_{l,m}^{n} \left( {{\mathbf{r}}_{{\mathbf{1}}} ,{\mathbf{r}}_{{\mathbf{2}}} ,0} \right)\exp \left[ { - \frac{ik}{{2z}}\left( {{{\varvec{\uprho}}}_{{\mathbf{1}}} - {\mathbf{r}}_{{\mathbf{1}}} } \right)^{2} + \frac{ik}{{2z}}\left( {{{\varvec{\uprho}}}_{{\mathbf{2}}} - {\mathbf{r}}_{{\mathbf{2}}} } \right)^{2} } \right]} } \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, &\quad \times \left\langle {\exp \left[ {\psi^{ * } \left( {{\mathbf{r}}_{{\mathbf{1}}} ,{{\varvec{\uprho}}}_{{\mathbf{1}}} } \right) + \psi \left( {{\mathbf{r}}_{{\mathbf{2}}} ,{{\varvec{\uprho}}}_{{\mathbf{2}}} } \right)} \right]} \right\rangle_{m} d^{2} {\mathbf{r}}_{{\mathbf{1}}} d^{2} {\mathbf{r}}_{{\mathbf{2}}} , \hfill \\ \end{aligned} $$
(10)

where \({{\varvec{\uprho}}}_{{\mathbf{1}}}\) and \({{\varvec{\uprho}}}_{{\mathbf{2}}}\) are two position vectors in the reception plane, \(k = \frac{\omega }{c}\) is the wavenumber with c representing the speed of radiation source in vacuum, \(\left\langle . \right\rangle_{m}\) denotes the ensemble average over the statistics of the medium covering log-amplitude and phase fluctuations due to turbulent atmosphere, \(\psi \left( {{\mathbf{r}},{{\varvec{\uprho}}}} \right)\) represents the random part of the complex phase, and the asterisk denotes the complex conjugate. The expression for the ensemble average is obtained from the Rytov quadratic approximation, given by (Andrews and Phillips 2005)

$$ \left\langle {\exp \left[ {\psi^{ * } \left( {{\mathbf{r}}_{{\mathbf{1}}} ,{{\varvec{\uprho}}}_{{\mathbf{1}}} } \right) + \psi \left( {{\mathbf{r}}_{{\mathbf{2}}} ,{{\varvec{\uprho}}}_{{\mathbf{2}}} } \right)} \right]} \right\rangle_{m} \approx \exp \left\{ { - \frac{1}{{\rho_{0}^{2} }}\left[ {\left( {{\mathbf{r}}_{{\mathbf{1}}} - {\mathbf{r}}_{{\mathbf{2}}} } \right)^{2} + \left( {{\mathbf{r}}_{{\mathbf{1}}} - {\mathbf{r}}_{{\mathbf{2}}} } \right)\left( {{{\varvec{\uprho}}}_{{\mathbf{1}}} - {{\varvec{\uprho}}}_{{\mathbf{2}}} } \right) + \left( {{{\varvec{\uprho}}}_{{\mathbf{1}}} - {{\varvec{\uprho}}}_{{\mathbf{2}}} } \right)^{2} } \right]} \right\}. $$
(11)

Thus, the spatial coherence length of spherical wave propagation can be presented in the integral form (Belafhal et al. 2021)

$$ \rho_{0} = \left[ {\frac{{\pi^{2} }}{3}k^{2} z\int\limits_{0}^{ + \infty } {\kappa^{3} \Phi_{n} \left( \kappa \right)d\kappa } } \right]^{ - 1/2} , $$
(12)

where \(\Phi_{n} \left( \kappa \right)\) is the power spectrum of marine atmospheric turbulence with \(\kappa\) as the spatial frequency. In the present study, we will adopt the model developed by (Khannous and Belafhal 2018)

$$ \Phi_{n} \left( \kappa \right) = 0.033C_{n}^{2} e^{{ - \frac{{\kappa^{2} }}{{\kappa_{Hn}^{2} }}}} \left( {1 + b\frac{\kappa }{{\kappa_{Hn} }}e^{{ - \alpha ^{\prime}\frac{{\kappa^{2} }}{{\kappa_{Hn}^{2} }}}} + c\left( {\frac{\kappa }{{\kappa_{Hn} }}} \right)^{\frac{7}{6}} e^{{ - \beta \frac{{\kappa^{2} }}{{\kappa_{Hn}^{2} }}}} } \right)\frac{1}{{\left( {\kappa^{2} + \kappa_{0}^{2} } \right)^{\frac{11}{6}} }}, $$
(13)

where \(b = - 3.930,\) \(\,c = 6.109,\) \(\alpha ^{\prime} = 204.278,\) \(\beta = 4.623,\) \(\kappa_{Hn} = {{6.262} \mathord{\left/ {\vphantom {{6.262} {l_{0} }}} \right. \kern-0pt} {l_{0} }}\) and \(\kappa_{0} = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } {L_{0} }}} \right. \kern-0pt} {L_{0} }},\) with \(l_{0}\) and \(L_{0}\) as the inner scale size and outer scale size of the turbulence and \(C_{n}^{2}\), which describes the turbulence level, is the constant of the refraction index structure.

In the upcoming analysis, we will determine the analytical formula of the spatial coherence length. Substituting Eq. (13) into Eq. (12), the integral part takes the following form

$$ \begin{aligned} \int\limits_{0}^{ + \infty } {\kappa^{3} \Phi_{n} \left( \kappa \right)} d\kappa & = 0.033C_{n}^{2} \left[ {\int\limits_{0}^{ + \infty } {\kappa^{3} \frac{{e^{{ - {{\kappa^{2} } \mathord{\left/ {\vphantom {{\kappa^{2} } {\kappa_{Hn}^{2} }}} \right. \kern-0pt} {\kappa_{Hn}^{2} }}}} }}{{\left( {\kappa^{2} + \kappa_{0}^{2} } \right)^{{{{11} \mathord{\left/ {\vphantom {{11} 6}} \right. \kern-0pt} 6}}} }}d\kappa + \frac{b}{{\kappa_{Hn} }}\int\limits_{0}^{ + \infty } {\kappa^{4} \frac{{e^{{ - {{\kappa^{2} \left( {1 + \alpha ^{\prime}} \right)} \mathord{\left/ {\vphantom {{\kappa^{2} \left( {1 + \alpha ^{\prime}} \right)} {\kappa_{Hn}^{2} }}} \right. \kern-0pt} {\kappa_{Hn}^{2} }}}} }}{{\left( {\kappa^{2} + \kappa_{0}^{2} } \right)^{{{{11} \mathord{\left/ {\vphantom {{11} 6}} \right. \kern-0pt} 6}}} }}d\kappa } } } \right. \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, & \quad + \left. {\frac{c}{{\kappa_{Hn}^{{{7 \mathord{\left/ {\vphantom {7 6}} \right. \kern-0pt} 6}}} }}\int\limits_{0}^{ + \infty } {\kappa^{{{{25} \mathord{\left/ {\vphantom {{25} 6}} \right. \kern-0pt} 6}}} \frac{{e^{{ - {{\kappa^{2} \left( {1 + \beta } \right)} \mathord{\left/ {\vphantom {{\kappa^{2} \left( {1 + \beta } \right)} {\kappa_{Hn}^{2} }}} \right. \kern-0pt} {\kappa_{Hn}^{2} }}}} }}{{\left( {\kappa^{2} + \kappa_{0}^{2} } \right)^{{{{11} \mathord{\left/ {\vphantom {{11} 6}} \right. \kern-0pt} 6}}} }}d\kappa } } \right]. \hfill \\ \end{aligned} $$
(14)

By invoking the integral formula (Belafhal et al. 2021)

$$ \int\limits_{0}^{ + \infty } {\frac{{t^{2\mu - 1} e^{{ - \varepsilon t^{2} }} }}{{\left( {1 + ct^{2} } \right)^{\nu } }}} dt = \frac{\Gamma \left( \mu \right)}{{2c^{\mu } }}U\left( {\mu ;\mu + 1 - \nu ;\frac{\varepsilon }{c}} \right), $$
(15)

with \(c \ne 0\) and \(U\left( {\mu ;\mu + 1 - \nu ;\frac{\varepsilon }{c}} \right)\) denoting the confluent hypergeometric of the second kind and after tedious but straightforward integral calculations, the expression of the spatial coherence length can be arranged as

$$ \begin{gathered} \rho_{0} = \left\{ {0.011\frac{{\pi^{2} k^{2} z}}{2}C_{n}^{2} \kappa_{0}^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}}} \left[ {U\left( {2;\frac{7}{6};\frac{{\kappa_{0}^{2} }}{{\kappa_{Hn}^{2} }}} \right) + \Gamma \left( \frac{5}{2} \right)b\left( {\frac{{\kappa_{0} }}{{\kappa_{Hn} }}} \right)U\left( {\frac{5}{2};\frac{5}{3};\frac{{\left( {1 + \alpha^{\prime}} \right)\kappa_{0}^{2} }}{{\kappa_{Hn}^{2} }}} \right)} \right.} \right. \hfill \\ \,\,\,\,\,\,\,\,\left. {\left. { + \Gamma \left( {\frac{31}{{12}}} \right)c\left( {\frac{{\kappa_{0} }}{{\kappa_{Hn} }}} \right)^{{{7 \mathord{\left/ {\vphantom {7 6}} \right. \kern-0pt} 6}}} U\left( {\frac{31}{{12}};\frac{7}{4};\frac{{\left( {1 + \beta } \right)\kappa_{0}^{2} }}{{\kappa_{Hn}^{2} }}} \right)} \right]} \right\}^{ - 1/2} . \hfill \\ \end{gathered} $$
(16)

By taking \({{\varvec{\uprho}}}_{{\mathbf{1}}} = {{\varvec{\uprho}}}_{2} = {{\varvec{\uprho}}}\) and substituting Eqs. (9, 16) into Eq. (10), we obtain the following result

$$ \begin{gathered} W_{l,m}^{n} \left( {{{\varvec{\uprho}}},{{\varvec{\uprho}}},z,\omega } \right) = S^{\left( 0 \right)} \left( \omega \right)\left( {\frac{k\,}{{2\pi \,z}}} \right)^{2} \frac{1}{{2^{4m + 2l + 2n} \left( {m!} \right)^{2} }}\sum\limits_{p = 0}^{m} {\sum\limits_{q = 0}^{l} {\sum\limits_{{r_{1} = 0}}^{m} {\sum\limits_{{r_{2} = 0}}^{l} {\sum\limits_{{s_{1} = 0}}^{n} {\sum\limits_{{s_{2} = 0}}^{n} {\left( {\begin{array}{*{20}c} m \\ p \\ \end{array} } \right)} } \,\left( {\begin{array}{*{20}c} l \\ q \\ \end{array} } \right)\left( {\begin{array}{*{20}c} m \\ {r_{1} } \\ \end{array} } \right)\,\left( {\begin{array}{*{20}c} l \\ {r_{2} } \\ \end{array} } \right)} } \,\left( {\begin{array}{*{20}c} n \\ {s_{1} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} n \\ {s_{2} } \\ \end{array} } \right)\,i^{q} \left( { - i} \right)^{{r_{2} }} } } \hfill \\ \times \int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {H_{2p + l - q} \left( {{{\sqrt 2 x_{1} } \mathord{\left/ {\vphantom {{\sqrt 2 x_{1} } {w_{0} }}} \right. \kern-0pt} {w_{0} }}} \right)\exp \left[ { - \beta_{1} x_{1}^{2} + 2\left( {v_{x} + \frac{{x_{2} }}{{\rho_{0}^{2} }}} \right)x_{1} } \right]H_{2m - 2p + q} \left( {{{\sqrt 2 y_{1} } \mathord{\left/ {\vphantom {{\sqrt 2 y_{1} } {w_{0} }}} \right. \kern-0pt} {w_{0} }}} \right)\exp \left[ { - \beta_{1} y_{1}^{2} + 2\left( {v_{y} + \frac{{y_{2} }}{{\rho_{0}^{2} }}} \right)y_{1} } \right]} } dx_{1} dy_{1} \hfill \\ \times \int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {H_{{2r_{1} + l - r_{2} }} \left( {{{\sqrt 2 x_{2} } \mathord{\left/ {\vphantom {{\sqrt 2 x_{2} } {w_{0} }}} \right. \kern-0pt} {w_{0} }}} \right)\exp \left[ { - \beta_{2}^{{}} x_{2}^{2} - \frac{ik}{z}\rho_{x} x_{2} } \right]H_{{2m - 2r_{1} + r_{2} }} \left( {{{\sqrt 2 y_{2} } \mathord{\left/ {\vphantom {{\sqrt 2 y_{2} } {w_{0} }}} \right. \kern-0pt} {w_{0} }}} \right)\exp \left[ { - \beta_{2}^{{}} y_{2}^{2} - \frac{ik}{z}\rho_{y} y_{2} } \right]dx_{2} dy_{2} } } , \hfill \\ \end{gathered} $$
(17)

where

$$ \beta_{1} = \frac{1}{{w_{0}^{2} }} + \frac{1}{{\rho_{0}^{2} }} + \frac{ik}{{2z}} - \Omega a_{{s_{1} n}} , $$
(18)
$$ \beta_{2} = \frac{1}{{w_{0}^{2} }} + \frac{1}{{\rho_{0}^{2} }} - \frac{ik}{{2z}} - \Omega a_{{s_{2} n}} , $$
(19)

and

$$ v_{\alpha } = \frac{{ik\rho_{\alpha } }}{z}, $$
(20)

with (α = x or y).

By employing the formulas for the development of the Hermite function (Abramowitz and Stegun 1970; Belafhal et al. 2020)

$$ H_{n} \left( x \right) = \sum\limits_{k = 0}^{{\left[ {{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-0pt} 2}} \right]}} {\left( { - 1} \right)^{k} \frac{n!}{{k!\left( {n - 2k} \right)!}}\left( {2x} \right)^{n - 2k} } , $$
(21)

and

$$ H_{n} \left( {x + y} \right) = \frac{1}{{2^{{{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-0pt} 2}}} }}\sum\limits_{k = 0}^{n} {\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)H_{k} \left( {\sqrt 2 x} \right)} H_{n - k} \left( {\sqrt 2 y} \right), $$
(22)

as well as the following integrals

$$ \int\limits_{ - \infty }^{ + \infty } {\exp \left( { - px^{2} + 2qx} \right)H_{m} \left( {\alpha x} \right)\,dx} = \sqrt {\frac{\pi }{p}} \exp \left( {\frac{{q^{2} }}{p}} \right)\left[ {1 - \frac{{\alpha^{2} }}{p}} \right]^{{{m \mathord{\left/ {\vphantom {m 2}} \right. \kern-0pt} 2}}} H_{m} \left[ {\frac{\alpha q}{{p\sqrt {1 - \left( {{{\alpha^{2} } \mathord{\left/ {\vphantom {{\alpha^{2} } p}} \right. \kern-0pt} p}} \right)} }}} \right], $$
(23)

and

$$ \int\limits_{ - \infty }^{ + \infty } {x^{l} \exp \left( { - px^{2} + 2qx} \right)H_{m} \left( {\alpha x} \right)\,dx} = \sqrt {\frac{\pi }{p}} \frac{{\exp \left( {{{q^{2} } \mathord{\left/ {\vphantom {{q^{2} } p}} \right. \kern-0pt} p}} \right)}}{{2^{l} }}\sum\limits_{k = 0}^{{\left[ {{m \mathord{\left/ {\vphantom {m 2}} \right. \kern-0pt} 2}} \right]}} {\frac{{\left( { - 1} \right)^{k} m!}}{{k!\left( {m - 2k} \right)!}}} H_{m + l - 2k} \left( {\frac{iq}{{\sqrt p }}} \right)\left( {\frac{\alpha }{i\sqrt p }} \right)^{m + l - 2k} , $$
(24)

with \(\Re e\,\{ p > 0\}\), and after laborious algebraic calculations, the spectral intensity of the pulsed LHChG beam propagating through a turbulent maritime environment is expressed as follows

$$ S_{l,m}^{n} \left( {\rho_{x} ,\rho_{y} ,z,\omega } \right) = W_{l,m}^{n} \left( {{{\varvec{\uprho}}},{{\varvec{\uprho}}},z,\omega } \right) = S^{\left( 0 \right)} \left( \omega \right)M_{l,m}^{n} \left( {\rho_{x} ,\rho_{y} ,z,\omega } \right) $$
(25)

where

$$ \begin{aligned} M_{l,m}^{n} \left( {\rho_{x} ,\rho_{y} ,z,\omega } \right) &= \frac{{k^{2} \,}}{{4\,z^{2} }}\frac{1}{{2^{5(m + 0.5l) + 2n} \left( {m!} \right)^{2} }}\sum\limits_{p = 0}^{m} {\sum\limits_{q = 0}^{l} {\sum\limits_{{r_{1} = 0}}^{m} {\sum\limits_{{r_{2} = 0}}^{l} {\sum\limits_{{s_{1} = 0}}^{n} {\sum\limits_{{s_{2} = 0}}^{n} {\left( {\begin{array}{*{20}c} m \\ p \\ \end{array} } \right)} } \,\left( {\begin{array}{*{20}c} l \\ q \\ \end{array} } \right)\left( {\begin{array}{*{20}c} m \\ {r_{1} } \\ \end{array} } \right)\,\left( {\begin{array}{*{20}c} l \\ {r_{2} } \\ \end{array} } \right)} } \,} } \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, &\quad \times \left( {\begin{array}{*{20}c} n \\ {s_{1} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} n \\ {s_{2} } \\ \end{array} } \right)\,i^{q} \left( { - i} \right)^{{r_{2} }} S_{x} S_{y} , \hfill \\ \end{aligned} $$
(26)

with

$$ \begin{gathered} S_{x} = \frac{1}{{\sqrt {\beta_{1} b_{x} } }}\exp \left( {\frac{{v_{x}^{2} }}{{\beta_{1} }} + \frac{{\gamma_{x}^{2} }}{{b_{x} }}} \right)\left( {1 - \frac{2}{{w_{0}^{2} \beta_{1} }}} \right)^{{\frac{2p + l - q}{2}}} \sum\limits_{\mu = 0}^{2p + l - q} {\sum\limits_{{f_{1} = 0}}^{{\left[ {{\mu \mathord{\left/ {\vphantom {\mu 2}} \right. \kern-0pt} 2}} \right]}} {\sum\limits_{{h_{1} = 0}}^{{\left[ {{{(2r_{1} + l - r_{2} )} \mathord{\left/ {\vphantom {{(2r_{1} + l - r_{2} )} 2}} \right. \kern-0pt} 2}} \right]}} {\left( {\begin{array}{*{20}c} {2p + l - q} \\ \mu \\ \end{array} } \right)} } } \hfill \\ \,\,\,\,\,\,\, \times \frac{{\left( { - 1} \right)^{{f_{1} + h_{1} }} }}{{2^{{\mu - 2f_{1} }} }}\frac{\mu !}{{f_{1} !\left( {\mu - 2f_{1} } \right)!}}\frac{{\left( {2q + l - r_{1} } \right)!}}{{h_{1} !\left( {2r_{1} + l - r_{2} - 2h_{1} } \right)!}}\left( {\frac{{2\sqrt 2 Q_{x} }}{{\rho_{0}^{2} }}} \right)^{{\mu - 2f_{1} }} \hfill \\ \,\,\,\,\,\,\,\, \times \left( {\frac{\sqrt 2 }{{iw_{0} \sqrt {b_{x} } }}} \right)^{{2r_{1} + l - r_{2} + \mu - 2f_{1} - 2h_{1} }} H_{2p + l - q - \mu } \left( {\sqrt 2 \alpha_{x} v_{x} } \right)H_{{2r_{1} + l - r_{2} + \mu - 2f_{1} - 2h_{1} }} \left( {\frac{{i\gamma_{x} }}{{\sqrt {b_{x} } }}} \right), \hfill \\ \end{gathered} $$
(27)
$$ \begin{aligned} S_{y} & = \frac{1}{{\sqrt {\beta_{1} b_{y} } }}\exp \left( {\frac{{v_{y}^{2} }}{{\beta_{1} }} + \frac{{\gamma_{y}^{2} }}{{b_{y} }}} \right)\left( {1 - \frac{2}{{w_{0}^{2} \beta_{1} }}} \right)^{{\frac{2m - 2p + q}{2}}} \sum\limits_{\mu = 0}^{2p + l - q} {\sum\limits_{{f_{2} = 0}}^{{\left[ {{\sigma \mathord{\left/ {\vphantom {\sigma 2}} \right. \kern-0pt} 2}} \right]}} {\sum\limits_{{h_{2} = 0}}^{{\left[ {{{(2m - 2r_{1} + r_{2} )} \mathord{\left/ {\vphantom {{(2m - 2r_{1} + r_{2} )} 2}} \right. \kern-0pt} 2}} \right]}} {\left( {\begin{array}{*{20}c} {2p + l - q} \\ \sigma \\ \end{array} } \right)} } } \hfill \\ \,\,\,\,\,\,\,&\quad \times \frac{{\left( { - 1} \right)^{{f_{2} + h_{2} }} }}{{2^{{\sigma - 2f_{2} }} }}\frac{\sigma !}{{f_{2} !\left( {\sigma - 2f_{2} } \right)!}}\frac{{\left( {2m - 2r_{1} + r_{2} } \right)!}}{{h_{2} !\left( {2m - 2r_{1} + r_{2} - 2h_{2} } \right)!}}\left( {\frac{{2\sqrt 2 Q_{y} }}{{\rho_{0}^{2} }}} \right)^{{\sigma - 2f_{2} }} \hfill \\ \,\,\,\,\,\,\,\, &\quad \times \left( {\frac{\sqrt 2 }{{iw_{0} \sqrt {b_{y} } }}} \right)^{{2m - 2r_{1} + r_{2} + \mu - 2f_{2} - 2h_{2} }} H_{2m - 2p + q - \sigma } \left( {\sqrt 2 \alpha_{y} v_{y} } \right)H_{{2m - 2p + q + \sigma - 2f_{2} - 2h_{2} }} \left( {\frac{{i\gamma_{y} }}{{\sqrt {b_{y} } }}} \right), \hfill \\ \end{aligned} $$
(28)
$$ b_{\alpha } = \beta_{2\alpha }^{{}} - \frac{1}{{\beta_{1\alpha } \rho_{0}^{4} }}, $$
(29)
$$ \gamma_{\alpha } = v_{\alpha } \left( {\frac{1}{{\beta_{1\alpha } \rho_{0}^{2} }} - 1} \right), $$
(30)

and

$$ \begin{array}{*{20}c} {Q_{\alpha } = \frac{\sqrt 2 }{{w_{0} \beta_{1} \sqrt {1 - \left( {{2 \mathord{\left/ {\vphantom {2 {w_{0}^{2} \beta_{1} }}} \right. \kern-0pt} {w_{0}^{2} \beta_{1} }}} \right)} }},} & {\left( {\alpha = x{\text{or}}y} \right)} \\ \end{array} $$
(31)

From Eq. (25), we observe that the spectral intensity of the pulsed LHChG beam propagating through a turbulent medium is the product of the original spectrum \(S^{\left( 0 \right)} \left( \omega \right)\) and the spectral modifier \(M_{l,m}^{n}\). The first term depends on the pulse duration T and the central frequency \(\omega_{0}\). The second term depends on the radial coordinate, beam parameters, and parameters of the turbulent maritime environment.

In the next Section, based on our previous results, we will numerically examine the change in behavior of a pulsed LHChG beam as it travels through the turbulence in a maritime environment.

3 Graphical simulation results and discussion

Numerical calculations are conducted using Eqs. (25, 26) to investigate the impact of turbulence on the spectral modifier and spectral intensity of pulsed LHChG beams. The relative spectral shift is represented by \({{\delta \omega } \mathord{\left/ {\vphantom {{\delta \omega } {\omega_{0} }}} \right. \kern-0pt} {\omega_{0} }} = {{\left( {\omega_{\max } - \omega_{0} } \right)} \mathord{\left/ {\vphantom {{\left( {\omega_{\max } - \omega_{0} } \right)} {\omega_{0} }}} \right. \kern-0pt} {\omega_{0} }}\), where \(\omega_{\max }\) corresponds to the frequency related to the maximum spectral intensity. Specifically, for \({{\delta \omega } \mathord{\left/ {\vphantom {{\delta \omega } {\omega_{0} }}} \right. \kern-0pt} {\omega_{0} }} > 0\), the spectral intensity experiences a blue shift, while for \({{\delta \omega } \mathord{\left/ {\vphantom {{\delta \omega } {\omega_{0} }}} \right. \kern-0pt} {\omega_{0} }} < 0\), it undergoes a red shift.

To simplify the analysis, the normalized original spectrum is \(S^{00} (\omega ) = S^{\left( 0 \right)} \left( \omega \right)/S^{\left( 0 \right)} \left( {\omega_{0} } \right)\), with the normalized spectral intensity denoted as \(S_{l,m}^{n} (\omega ) = S_{l,m}^{n} (\rho_{x} ,\rho_{y} ,z,\omega )/S_{l,m}^{n} (\rho_{x} ,\rho_{y} ,z,\omega_{\max } )\), and the normalized spectral modifier given by \(M_{l,m}^{n} (\omega_{0} ) = M_{l,m}^{n} (\rho_{x} ,\rho_{y} ,z,\omega_{0} )/M_{(l,m)\max }^{n} (\rho_{x} ,\rho_{y} ,z,\omega_{0} )\), where \(M_{(l,m)\max }^{n} (\rho_{x} ,\rho_{y} ,z,\omega_{0} )\) represents the maximum value of at the observation point \((\rho_{x} ,\rho_{y} ,z)\). In the numerical simulations, unless otherwise specified in the figures, the parameters of the pulsed beam and the turbulence are defined: \(\Omega = 0.1\,m^{ - 1}\), \(T = 5fs,\) \(\lambda = 632.8nm\), \(w_{0x} = w_{0y} = 2cm\), \(L_{0} = 10m\), \(l_{0} = 1mm\), \(m = l = 1\) and n = 3. Figure 1 shows the graphical representation of the spectral modifier of the pulsed LHChG beam in free space and in maritime turbulence at different propagation distances.

Fig. 1
figure 1

Normalized spectral modifier of the pulsed LHChG beam propagating in maritime turbulence at different propagation distances with a \(C_{n}^{2} = 0\), b \(C_{n}^{2} = 1 \times 10^{ - 16} m^{ - 2/3}\) and c \(C_{n}^{2} = 2 \times 10^{ - 15} m^{ - 2/3}\)

Full size image

According to Fig. 1, the profile of the pulsed LHChG beam undergoes various evolutions during its propagation in free space and in a turbulent maritime medium. In free space, we observe that the positions of the spectral minima depend on the propagation distance z.

Furthermore, it is evident that the lobes become broader as the propagation distance increases. However, we also notice that the profile of the beam gradually changes from a hollow shape to a Gaussian shape in the far field as the propagation distance increases. Comparing Fig. 1b, c allows us to deduce that the transformation of the LHChG beam into a Gaussian-type beam occurs more rapidly in the presence of intense turbulence.

Then we present the results of the influence of initial beam parameters, maritime turbulence, and transverse positions on the spectral intensity, as well as the relative spectral shifts of the pulsed LHChG beam. The following calculation parameters were used: \(\Omega = 0.1\,m^{ - 1}\), \(\lambda = 632.8nm,\)\(C_{n}^{2} = 1 \times 10^{ - 16} m^{ - 2/3}\), \(w_{0x} = w_{0y} = 2cm\), \({\text{c = 3 }} \times 10^{8} \,m/s,\)\(z = 5\,km\), \(T = 5fs,\)\(\rho_{y} = 0,\)\(L_{0} = 10m\), \(l_{0} = 1mm\), \(m = l = 1\) et n = 3.

Figure 2 illustrates the normalized spectral intensity of the pulsed LHChG beam, both on-axis and off axis, for different values of the refractive index structure constant. The circles represent the normalized initial spectrum. Based on the results in Fig. 2a, the axial spectral intensity of the pulsed LHChG beam is shifted towards the blue. It appears that the blue shift slightly decreases as the value of the refractive index structure constant increases. However, as shown in Fig. 2b, a red shift in the off axis spectral intensity can be observed for all values of the refractive index structure constant, and this red shift decreases as the refractive index structure constant increases. This indicates that both the on-axis and off-axis spectra approach the original spectrum when the maritime environment becomes less turbulent (\(C_{n}^{2} = 1 \times 10^{ - 16} m^{ - 2/3}\)).

Fig. 2
figure 2

Normalized spectral intensity of the pulsed LHChG beam propagating in maritime turbulence for different values of the refractive index structure constant: a on-axis (0) and b off axis \((\rho_{x} = 0.3m)\)

Full size image

Figure 3 presents the normalized on-axis and off axis spectral intensity of the pulsed LHChG beam for different pulse durations T. The circles indicate the normalized initial spectrum. As shown in Fig. 3a, the axial spectral intensity of the pulsed LHChG beam is clearly blue-shifted. This blue shift decreases as the pulse duration increases. Concrete examples are given by these values: for T = 5 fs, we have \(\delta \omega /\omega_{0} = 0.0212\), for T = 8 fs, it is \(\delta \omega /\omega_{0} = 0.0091,\) and for T = 12 fs, it is \(\delta \omega /\omega_{0} = 0.0030 \). But the results in Fig. 4b show that the off axis spectral intensity is red-shifted for all pulse durations. The amount of red shift increases with decreasing pulse duration. For example, for T = 5, 8, and 12 fs, the corresponding values are \(\delta \omega /\omega_{0}\): = −0.2253, −0.0960, and −0.0475, respectively. This suggests that both the on-axis and off axis spectra approach the initial spectrum when the pulse duration T is set to 12 fs.

Fig. 3
figure 3

Normalized spectral intensity of a pulsed LHChG beam propagating in maritime turbulence, with different values of T, for both cases: a on-axis (0) and b off axis \((\rho_{x} = 0.3m)\)

Full size image
Fig. 4
figure 4

Relative spectral shift for different values of: a the refractive index structure constant, and b the pulse duration

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Figure 4 shows the off axis relative spectral shifts of the pulsed LHChG beam in maritime turbulence as a function of the \(\rho_{x}\) transverse coordinates for different values of the refractive index structure constant and pulse duration T as a function of the \(\rho_{x}\) position parameter. It is observed that the axial spectral shift is oriented towards the blue, and this shift becomes more pronounced with increasing refractive index structure constant and pulse duration T. Additionally, the blue spectral shift gradually transitions towards a red spectral shift as the position parameter \(\rho_{x}\) increases. These observations underscore a rapid spectral transition phenomenon in pulsed LHChG beams. As this parameter increases, the non-axial spectral shift becomes more similar for various turbulent maritime environments.

4 Conclusion

We investigated the spreading characteristics of the propagation characteristics of a pulsed LHChG beam in maritime atmospheric turbulence. The study establishes a formula for spectral changes using the Huygens-Fresnel principle and the Fourier Transform method. Additionally, the study explores the relative spectral shift of the pulsed LHChG beam at different radial coordinates through numerical calculations. The findings reveal that the spectral intensity depends on key factors such as the structure constant of the refractive index, pulse duration, and beam order. Remarkably, the on-axis spectral intensity experiences a blue shift, while the off-axis spectral intensity undergoes a red shift as the radial coordinate increases. The results hold significant potential for applications in information coding and transmission, making this study a crucial contribution to the field of laser beam propagation in turbulent maritime environments.