The surface of the object which in nature is rough relative to the wavelength of light beams is a rough target. When the light is incident on the rough target, the beam will be scattered, and the receiving end will receive the light intensity pattern of alternating light and dark waves, which is called speckle pattern. The echo scattering characteristic of the light field is one of the key technologies for the integration of underwater laser communication and detection in the future. In ocean detection technology, underwater scattering technology, and underwater wireless optical communication technology, the transmission characteristics of beams through oceanic turbulence and echo characteristics of beams through the rough surface play a crucial role in turbulent conditions. The physical characteristics of the rough target surface (surface roughness, coherence length, deformation degree, motion velocity, and rotation velocity) play an important role in influencing echo characteristics. The current studies mainly adopt the fractal method, wavelet transform, deep learning, and other methods to process the laser speckle pattern collected at the receiving end. The surface roughness, deformation degree, translation velocity, and rotation velocity of the rough target can be identified. The current underwater detection technology is mainly laser, but the laser loss during detection is serious, thus resulting in limited rough target information reflected by the received speckle. Since the vortex beam features hollow intensity distribution, phase helix, and orthogonality of orbital angular momentum (OAM), it can carry more information than the Gaussian beam. Laguerre-Gaussian (LG) beams are typical vortex beams with significant advantages in light scattering and target recognition. At present, there are two aspects to study the scattering characteristics of light beams through rough targets. The first is the speckle characteristics of light beams through the rough surface in free space, and the second is the speckle characteristics of light beams through the rough surface in atmospheric turbulence. The propagation theory of vortex beams through oceanic turbulence is very mature, but the scattering characteristics of vortex beams through Gaussian random rough surface in weak oceanic turbulence are rarely studied.

Spatial coherence property is a part of the echo scattering property, which can reflect the coherence of the echo field between two points in space. The spatial complex coherence degree of the beam is utilized to represent the spatial coherence property of the echo field. The spatial distribution of the speckle is related to the surface coherence length of the rough target. We build a double-path transmission model of LG beam through Gaussian random rough surface in weak oceanic turbulence by referring to the scattering characteristics of vortex beams through the rough surface in atmospheric turbulence. Based on the generalized Huygens-Fresnel diffraction principle, the intensity of the echo speckle field of LG beams reflected by a rough surface with Gaussian distribution in oceanic turbulence is derived. The influences of the LG beam's light source parameters, oceanic turbulence intensity, and rough surface roughness on speckle field complex coherence degree are investigated.

The effects of light source parameters, oceanic turbulence, and rough target surface parameters on the complex coherence of the echo speckle field are analyzed numerically. Figs. 4-10 show that complex coherence decreases with the increasing topological charge, waist radius, and wavelength of LG beams, decreases with the increase in oceanic turbulence intensity, and rises with the increasing coherence length of the rough surface. Additionally, when the coherent length of the rough surface is larger than that of spherical wave propagating in oceanic turbulence, the complex coherence degree does not change significantly. This shows that the influence of rough surfaces on complex coherence is much less than that of oceanic turbulence.

This study is based on the generalized Huygens-Fresnel diffraction principle and the relative advantages of vortex beams in suppressing turbulence effect due to the special helical phase characteristics of the beams. Then the analytical expression of the scattering intensity of LG beams through the rough surface in oceanic turbulence is innovated, and the theoretical expression of the complex coherence of the scattering field at the receiving end is obtained. The results indicate that the complex coherence decreases with the increasing topology charge, waist radius, and wavelength of the LG beam, decreases with the rising oceanic turbulence intensity, and rises with the increasing dry length of the rough surface. However, when the coherent length of the rough surface is larger than that of the spherical wave propagating in oceanic turbulence, the complex coherence degree does not change significantly. This shows that the influence of rough surfaces on complex coherence is much less than that of oceanic turbulence. The analytical expression of the light field scattered by LG beams through the rough surface and the complex coherence of the light field of the back wave derived in this paper provides a theoretical basis for underwater target detection.