# Second-Order Statistics of Self-Splitting Structured Beams in Oceanic Turbulence

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## Abstract

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## 1. Introduction

## 2. Cross-Spectral Density of SSS Beams in Oceanic Turbulence

## 3. Propagation Factors and Relative Radius of Curvature of SSS Beams in Oceanic Turbulence

## 4. Numerical Calculation and Analysis of Second-Order Statistics of SSS Beams

## 5. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Evolution of the SDOC modulus of an SSS beam when traversing through (

**a**) free space and (

**b**) oceanic turbulence.

**Figure 2.**Development of the normalized spectral intensity of an SSS beam traversing through (

**a**) free space and (

**b**) oceanic turbulence.

**Figure 3.**Cross line (${\rho}_{1y}-{\rho}_{2y}=0$) of the SDOC modulus at a propagation distance of z = 300 m for various oceanic turbulence parameters: (

**a**) the kinetic energy dissipation rate per unit mass of the fluid; (

**b**) the Kolmogorov inner scale; (

**c**) the dissipation rate of the mean-square temperature; (

**d**) the relative strength of the temperature to salinity undulations.

**Figure 4.**Cross line (${\rho}_{1y}-{\rho}_{2y}=0$) of the SDOC modulus at a propagation distance of z = 300 m for various (

**a**) beam orders and (

**b**) coherence lengths.

**Figure 5.**Ratio of the spectral intensity in the optical axis to the maximum intensity in the transverse plane of SSS beams for different beam orders and coherence lengths versus various oceanic turbulence parameters: (

**a**) the kinetic energy dissipation rate per unit mass of the fluid; (

**b**) the Kolmogorov inner scale; (

**c**) the dissipation rate of the mean-square temperature; (

**d**) the relative strength of the temperature to salinity undulations.

**Figure 6.**Ratio of the spectral intensity for the optical axis ($\rho $ = 0) against the maximum intensity in the transverse plane for the SSS beams traversing through oceanic turbulence for various (

**a**) beam orders and (

**b**) coherence lengths.

**Figure 7.**Normalized propagation factor for SSS beams traversing through oceanic turbulence for various (

**a**) beam orders and (

**b**) coherence lengths. The dark line on the graphs denotes the corresponding result for a GSM beam.

**Figure 8.**Relative radius of curvature of SSS beams traversing through oceanic turbulence for various (

**a**) beam orders and (

**b**) coherence lengths.

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**MDPI and ACS Style**

Liu, L.; Liu, Y.; Chang, H.; Huang, J.; Zhu, X.; Cai, Y.; Yu, J.
Second-Order Statistics of Self-Splitting Structured Beams in Oceanic Turbulence. *Photonics* **2023**, *10*, 339.
https://doi.org/10.3390/photonics10030339

**AMA Style**

Liu L, Liu Y, Chang H, Huang J, Zhu X, Cai Y, Yu J.
Second-Order Statistics of Self-Splitting Structured Beams in Oceanic Turbulence. *Photonics*. 2023; 10(3):339.
https://doi.org/10.3390/photonics10030339

**Chicago/Turabian Style**

Liu, Liming, Yulu Liu, Hao Chang, Jifei Huang, Xinlei Zhu, Yangjian Cai, and Jiayi Yu.
2023. "Second-Order Statistics of Self-Splitting Structured Beams in Oceanic Turbulence" *Photonics* 10, no. 3: 339.
https://doi.org/10.3390/photonics10030339