# Experimental Study on Phase Scintillation of Optical Transmission in Atmospheric Turbulence
## 2 Theoretical Analysis  
The propagation of a beam through atmospheric turbulence involves complex physical processes, including optical phenomena such as intensity attenuation, intensity scintillation, and beam wander. The received optical field intensity can be expressed as [10]. \[u(r)=A(r)\exp(\mathbf{i}\phi(r)) \tag{1}\]  
where \(r\) is the propagation path vector of light and \(u(r)\) is the received optical field intensity, which can be expressed as a combination of the amplitude \(A(r)\) and phase \(\phi(r)\) of the field intensity. As a light beam propagates through atmospheric turbulence, its optical field is affected by the turbulence, leading to changes in both the phase and amplitude of the optical field. These changes are random and vary with time and space. To study the phase changes, a wavefront sensor can be placed at the receiving end to observe the wavefrontphase distribution and variations on the receiving plane after the light beam has passed through atmospheric turbulence. The receiving end uses a centrally obscured Maksutov antenna, resulting in the wavefront sensor detecting an annular spot, as shown in Figure 1. Therefore, this paper employs annular Zernike polynomials to decompose the wavefront phase. Annular Zernike polynomials are a type of polynomial used to describe wavefront aberrations in optical systems. They are an extension of traditional Zernike polynomials, which are suitable for optical systems with annular apertures, such as telescopes with central obstructions. These polynomials are orthogonal within the annular region and can effectively represent wavefront aberrations for annular apertures. The wavefront sensor used in this study detects a spot array of 32 \(\times\) 32, with the central dark spot array being 8 \(\times\) 8. The ratio of the inner to outer diameter is 0.25, and the obscuration ratio \(\varepsilon\) is 0.25. According to the working principle of the Shack-Hartmann wavefront sensor, the spot array formed on the \(CCD\) target surface can be regarded as the focal point of each microlens, and the coordinates of each spot at this time will be (\(x_{c0}\), \(y_{c0}\)). In this paper, the calibration accuracy of the Shack-Hartmann wavefront sensor is very important and will directly affect the measurement accuracy of the beam quality. The actual laser beam will introduce more or less wavefront aberrations after passing through a series of optical systems and the atmosphere during the ejection process. This phase distortion of the beam is incident on the _Shack-Hartmann_ wavefront sensor and the formation of spot arrays on the \(CCD\) target surface. The center-of-mass position of these spot arrays will deviate from the center-of-mass position of the spot arrays in the ideal plane wavefront. The offset of the spot center-of-mass coordinates (\(x_{c}\), \(y_{c}\)), with respect to the calibrated origin (\(x_{c0}\), \(y_{c0}\)), is calculated. \[\Delta x=x_{c}-x_{c0} \tag{2}\]  
\[\Delta y=y_{c}-y_{c0} \tag{3}\]  
Then, the average slopes \(G_{x}\) and \(G_{y}\) of the wavefront phase of the incident beam in the \(x\) and \(y\) directions within the sub-aperture are  
\[G_{x}=\frac{1}{s}\iint\limits_{s}\frac{\partial\phi(x,y)}{\partial x}dxdy= \frac{\Delta x}{f} \tag{4}\]  
\[G_{y}=\frac{1}{s}\iint\limits_{s}\frac{\partial\phi(x,y)}{\partial y}dxdy= \frac{\Delta y}{f} \tag{5}\]  
In Equations (4) and (5), \(f\) denotes the focal length of the microlens array, \(S\) is the subaperture area, and \(\Delta x\) and \(\Delta y\) are the offsets from the center reference point in the \(x\) and \(y\) directions, respectively. Based on the above equation, the slope of the wavefront is calculated, and then, the phase of the wavefront is calculated by reconstruction. Figure 1: Received spot.