Propagation Characteristics of Rectangular Hermite-Gaussian Array Beams Through Oceanic and Atmospheric Turbulence

This study investigates the propagation characteristics of rectangular Hermite-Gaussian array beams (RHGABs) through the oceanic and atmospheric turbulence considering ocean waves. Firstly, a theoretical model for the RHGABs in seawater-to-air propagation path is proposed. Secondly, we derive the expressions of the mean squared beam width and angular spread of RHGABs by using the extended Huygens-Fresnel principle and Rytov approximation. Finally, the effects of turbulent mediums on the RHGABs spreading are studied by analyzing the Rayleigh range. Our findings suggest that the Rayleigh range increases with the beam order of RHGABs. And the mean squared beam width of RHGABs in oceanic turbulence increases significantly higher than that in atmospheric turbulence. The results of this research can be applied to underwater optical communication systems.


I. INTRODUCTION
I N THE past decades, underwater laser communications have gained attention due to their advantages over traditional wireless transmission methods, including higher bandwidth, lower time latency, and enhanced security [1]. The high-order Gaussian beam and orbital angular momentum (OAM) in underwater laser communications has been widely studied [2], [3], [4], [5]. In practical maritime communication systems, as represented in Fig. 1, information can be transmitted from submarines to aircraft through the seawater-to-air link [6]. Consequently, the research on the propagation characteristics of laser beams through oceanic and atmospheric turbulence has become a key hotspot, particularly in cross-media scenarios.
Currently, using high-order beam and multi-beams are common solutions for resisting the effects of turbulence [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. Particularly, Hermite-Gaussian array beams (HGABs), as a kind of topical high-order beams, have been shown to be less sensitive to turbulence than Gaussian beams [12], [13]. Wang derived an analytical expression for the propagation factors of partially coherent elegant Hermite-cosh Gaussian beams in non-Kolmogorov [14]. Sayan examined the propagation distances and the zenith angles at the realistic propagation distances involved in uplink and downlink configurations [15]. Yuan studied the evolution properties of the M 2 factor of the Hermite-Gaussian beam in turbulent atmosphere [16]. Zhu derived the propagation formula of the radially polarized Hermite-cos-Gaussian correlated Schell-model beam in oceanic turbulence [17]. Li established the model of the received probability density of orbital angular momentum (OAM) modes of a new Hermite-Gaussian vortex beam propagating through the turbulent ocean of anisotropy [16]. Zhou derived the analytically general expressions of the centroid and the beam spot size of an elegant Hermite-Gaussian beam passing through an Airy transform optical system [19]. Wu studied the evolution behavior of Hermite-cosine-Gaussian beam propagating in oceanic turbulence [20]. Additionally, the rectangular Gaussian array beams are widely used in various fields, such as high-power systems and inertial confinement fusion [21]. Therefore, we aim to study the beam quality of rectangular Hermite-Gaussian beams propagating through the oceanic and atmospheric turbulence.
Our research employs the Rytov approximation and extended Huygens-Fresnel principle to model the light field propagation through the seawater-to-air path. Moreover, we derive expressions for parameters such as mean squared beam width, angular spread, and Rayleigh range, which reflect the characteristics of beam propagation.
The rest structure of this paper is organized as follows: in Section II, the analytical analyses of the beam transmission properties are given. In Section III, the numerical results are analyzed. And the conclusions are drawn in Section IV.

A. Light Field Propagating Though Double Turbulent Mediums
It can be illustrated from Fig. 1 that the light field in receptive plane is determined by the initial light field, the refractive index fluctuations caused by the turbulent mediums and rough sea surface. In our model, the rough sea surface is considered as a medium combined by the ocean, atmosphere, and interface, which is discussed separately. Noting the beam intensity attenuation at the rough sea surface. We consider the transmittance of RHGABs propagating through the seawater-air interface, which is denoted as Ξ.
First, the light field propagating in free space is obtained by the extended Huygens-Fresnel principle, which can be written as [28] where k is the wave number, U (free) (x, y, z) is the field intensity at (x, y, z), U 0 (x 0 , y 0 ,0) denotes the field intensity of the point (x 0 , y 0 , 0) in the field source plane (initial plane). Second, we can obtain the light field in individual turbulent medium according to Rytov approximation [28], i.e., where U (t) denotes the light field in an individual turbulent medium, U (free) is the light field in the free space, A denotes the ith order complex phase perturbation caused by turbulence.
Third, considering the light field transmitting through double turbulent mediums (A and B) continuously, and neglecting the higher-order terms, (2) is rewritten as where U (tt) denotes the light field propagating in double turbulent mediums, (i) B denotes the ith order complex phase perturbation caused by the second turbulent medium.
In practice, as mentioned in Fig. 1, the (3) is applied in the seawater-to-air propagation path. And since the complex phase B are determined by the spatial coordinates, the complex phase perturbations mentioned in (3) can be written as [28] (1) (1) where, r a = (x, y), r 1 = (r x1 , r y1 ), l denotes the propagation distance.
Then, upon substituting (1), (4), (5) to (3), we can obtain where z D is the distance from the initial plane to seawater-air interface, z is the distance from receiving plane to initial plane.
Here, RHGABs are composed of M × N equal Hermite-Gaussian beams. As shown in Fig. 1(a), the distance between two adjacent Hermite-Gaussian array beams in the x, y axis direction is x l , y l , respectively. The field distribution (z = 0) of the off-axis RHGABs centered at point (ix d , jy d ), reads as [29] where, H m (•) and H n (•) are the mth and nth order Hermite polynomials, respectively. And m and n are abbreviation for beam order in the following. Particularly, the RHGABs can be regard as Gaussian array beams (GABs) if the order m = n = 0. Thus, we can obtain the field intensity of the point (x 0 , y 0 , 0) in the field source plane, which can be written as We assume that the RHGABs are correlated, and the crossspectral density function (CSDF) of the RHGABs can be written as where the U (tt) * denotes the complex conjugate of U (tt) .

B. Intensity Calculation in Seawater to Air Propagation Path
Assuming r a1 =r a2 = r, the light intensity in the receiving plane (z > z D + d ε ) can be represented as (12). Noting that the expression of light intensity when z ≤ z D is similar to (12), which is not listed here.
Thus, we have where, I (out) (r, z) denotes the beam intensity after the reflection and absorption in the interface, I (in) (r, z) denotes the beam intensity before the reflection and absorption in the interface. Therefore, the average intensity of the RHGABs represented by (12) propagating in the whole propagation path reads as where, γ 1 is the probability of the laser propagation in the oceanic turbulence, γ 2 is the probability of the laser propagation in the atmospheric turbulence, d ε is the thickness of the wave layer. And we assume denotes the distribution function of the Gaussian distribution.

C. Mean Squared Beam Width and Angular Spread
The mean-squared beam width is written as [31] w 2 (z) = 4 ρ 2 I(r, z) dxdy I(r, z) dxdy Upon substitution (17) to (19). It is easy to find that (22) It means that, the mean squared beam width is not affect by the reflection and absorption in the seawater-air interface.
Upon substituting (18) to (19), and after straight calculation, we can obtain where Moreover, the angular spread can be represented as [17]

D. Rayleigh Range
The Rayleigh range z R characterizes the collimation interval of the beam, defined as the propagation distance hen the crosssectional area of the beam reaches twice at the field source [32], and z R can be expressed as: Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. Bring formula (23) into formula (30), and we can obtain and

III. NUMERICAL RESULT
In the following, the results are presented in Figs. 2 -6. Unless otherwise specified, the simulation parameters are selected as: M = N = 7, x d = y d = 3 cm, λ = 0.54 um, w 0 = 1 cm, m = n = 4, η = 0.01, α = 3.5, C n 2 = 10 −14 , d ε = 5 m, respectively. To compare the intensity of oceanic turbulence with that of atmospheric turbulence, we introduce parameter T (κ)dκ) as a factor that represents the relative contribution of oceanic and atmospheric turbulence. The surfaces in Fig. 2 represent the values of T corresponding to χ T is 10 −8 , 5×10 −8 and 10 −7 , from the bottom to the top, respectively. It illustrates that T increases with the increase of χ T and the decrease of ω and ε. That means the larger χ T or the smaller ω and ε are, the stronger the oceanic turbulence is. In the rest part, we will use different values of T to represent different intensities of the ocean turbulence. Fig. 3 presents the changes in the mean squared beam width of RHGABs at various propagation distances and turbulence intensities. Notably, in Fig. 3(a), the mean square beam width expands with the increase of propagation distance, whether in the atmosphere or ocean. At the distance of 50 m, an inflection point appears, which is exactly the distance from seawater-air interface to the initial plane. In general, oceanic turbulence has a greater impact on the variation of beam width corresponding to the atmospheric turbulence.  Additionally, in Fig. 3(b), the overall trend of the mean square beam width considering environmental noise is basically consistent with that ignoring environmental noise. At the seawater-air interface, however, the random fluctuations on the mean squared beam width occur when this model considers environmental noise. It means that the environmental noise is a noteworthy and important factor for seawater-to-air optical communication.   T in Fig. 4(a) and (b) are 1000 and 10000, respectively. These figures suggest that the larger value of k leads to faster beam width growth in the ocean, and greater contribution to overall propagation. The beam number plays a key role in the mean squared beam width, as a larger beam number results in a larger initial beam width, leading to a wider mean squared beam width in the propagation path. Fig. 5 illustrates the relationship between angular spread and RHGABs parameters (beam number, separated distance, initial beam width) under the conditions of atmospheric and oceanic turbulence, with a particular focus on the difference between single-beam and array beams. In Fig. 5(a), the impact of beam order on angular spread is depicted, the single-beam's angular spread increases as beam order increases. Fig. 5(b) shows that the angular spread remained constant for the single-beam as there is no distance variation between it and the adjacent beams. Fig. 5(c) reveals that the angular spread of the single-beam decreases with the increase of w 0 . Generally, the variation of the single beam is monotonic, while it is opposite for the array beams. The angular spread of array beams oscillates along with the single beam for different m, x d , and w 0 . Additionally, it is important to note that a larger turbulence intensity corresponds to a larger angular spread and a smaller amplitude of oscillation.

C. Rayleigh Range
To examine the correlation between the Rayleigh range and the beam order, beam number, we present Fig. 6. The gray dashed line represents the threshold. Here, the threshold is set as the Rayleigh range when m, n take the values of m = n = 5. As depicted, the Rayleigh range increases with the increase of beam order in the case of single beam. Conversely, in the case of multiple beams, the Rayleigh range oscillates with the increase of the beam order. Moreover, the peak values of Rayleigh range appear at the beam order of 3, 8, 15, and 23, but the overall trend is downward. In addition, the Rayleigh range expands with the beam number, at the same beam order. Fig. 7 shows that the Rayleigh range decreases with the increase of relative turbulence intensity T. It is of interesting that the Rayleigh range does not always decrease with the increase of the beam under the determined value of T. This phenomenon is consistent with the results in Fig. 6. It also can be reflected on the intersection of the curves in Fig. 7.

IV. CONCLUSION
In this paper, the propagation characteristics of RHGABs propagating through oceanic and atmospheric turbulence are studied. In the analytical model, we investigate the variation tendencies of Rayleigh range and angular spread. Our findings are as follows: the Rayleigh range increases with beam number under the same beam order. Moreover, the mean squared beam width increases more rapidly in oceanic turbulence than in the atmosphere. Furthermore, the angular spread of the RHGABs increases with the increase of the beam order. According to the above-mentioned results, one can increase the beam number to extend the effective communication distance. The conclusions in this paper are valuable for real optical communication system design.
Disclosures: The authors declare no conflicts of interest.