**Expressions used in section 4.1.- 4.3.**

When the filter function is equal to

$$F\_{n,m}(\kappa, z) = 2N\_n^2 \langle \gamma\_z \kappa \rangle [1 - J\_0(s\_z \kappa)]$$

The results are determined by the sizes of *sz* and *γzD*. If |*sz*|≥*γzD*, then

$$I\_n(z) = \binom{n+1}{n+1} \Gamma\left(n - \frac{5}{6}\right) \binom{\frac{n}{\prime} \Gamma\left(\frac{n}{\prime} \right) \Gamma\left(\frac{n}{\prime} \right) \Gamma\left(\frac{n}{\prime} \right) \Gamma\left(\frac{n}{\prime} \right)}}{2^n \pi^{\prime\prime} \Gamma\left(\frac{n}{\prime} \right) \Gamma\left(\frac{n}{\prime} \right) \Gamma\left(\frac{n}{\prime} \right) \Gamma\left(\frac{n}{\prime} \right)} - 1}$$
 
$$= \frac{\frac{\pi}{3} \frac{\Gamma\_2\left(n \cdot 5 \mid 6, \ n \cdot 5 \mid 6, \ n \cdot 3 \mid 2; n + 3 \mid 2; n + 2, 2n + 3; \left\lfloor \gamma\_\pm D \mid \beta\_\pm \right\rfloor^2\right)}{2^{2n + 5/3} \Gamma\left(\cdot n + 11 \mid 6\right) \Gamma\left(\frac{n}{\prime} \right) \Gamma\left(\frac{n}{\prime} \right) \Gamma\left(\frac{n}{\prime} \right) \Gamma\left(\frac{n}{\prime} \right)}}$$

Further, if |*sz*|≫*γD*, an asymptotic series of small parameter |*γzD* /*sz*| can be found. When expanding to second-order, the results are

$$I\_n(\mathbf{z}) = (1+n)\Gamma\{n \cdot \frac{5}{6}\} (2D\gamma\_z)^{\frac{5}{6}} \Big[ \frac{\Gamma\left(\frac{7}{3}\right) / \Gamma\left(n + \frac{23}{6}\right)}{2^{43} \sqrt{\pi} \, \Gamma\left(\frac{17}{6}\right)} - \frac{2^{\frac{1}{2}} \gamma\_z D \, / s\_z \, ^{[2n+13]}}{\Gamma\left(\frac{11}{6} \, \cdot \, n \right) \mathbb{Z}^n \Gamma(2+n)^2} \left[ \frac{\left(n \cdot \frac{5}{6}\right)^2}{n+2} + \frac{2^{-33}}{\left(\gamma\_z D \, / s\_z\right)^2} \right] \Big] \tag{79}$$

if |*sz*|<*γzD*, then

*In*(*z*)=21/3(*<sup>n</sup>* <sup>+</sup> 1)(*γzD*)5/3{ *<sup>Γ</sup>*( <sup>7</sup> <sup>3</sup> )*Γ*(*<sup>n</sup>* - <sup>5</sup> 6 ) *<sup>π</sup> <sup>Γ</sup>*( <sup>17</sup> <sup>6</sup> )*Γ*(*<sup>n</sup>* <sup>+</sup> <sup>23</sup> 6 ) - - *<sup>Γ</sup>*( <sup>7</sup> <sup>3</sup> )*Γ*(*<sup>n</sup>* - <sup>5</sup> 6 ) *<sup>π</sup> <sup>Γ</sup>*( <sup>17</sup> <sup>6</sup> )*Γ*(*<sup>n</sup>* <sup>+</sup> <sup>23</sup> <sup>6</sup> ) <sup>3</sup> *F*2 (- 11 <sup>6</sup> , - *<sup>n</sup>* - <sup>17</sup> <sup>6</sup> , *<sup>n</sup>* - <sup>5</sup> <sup>6</sup> ; - <sup>4</sup> <sup>3</sup> , 1;| *sz <sup>γ</sup>z<sup>D</sup>* <sup>|</sup><sup>2</sup> )- - *<sup>Γ</sup>*(- 7 3 ) *<sup>π</sup> <sup>Γ</sup>*( <sup>10</sup> <sup>3</sup> ) <sup>|</sup> *sz <sup>γ</sup>z<sup>D</sup>* <sup>|</sup>14/3 3 *F*2 ( 1 <sup>2</sup> , - *<sup>n</sup>* - <sup>1</sup> <sup>2</sup> , *n* + 3 2 ; 10 <sup>3</sup> , <sup>10</sup> <sup>3</sup> ;| *sz <sup>γ</sup>z<sup>D</sup>* <sup>|</sup><sup>2</sup> )}

Further if |*sz*|≪*γzD*, the second-order asymptotic expansions of parameter | *sz <sup>γ</sup>z<sup>D</sup>* | are

$$I\_n(\boldsymbol{z}) = \frac{\binom{\boldsymbol{\Gamma}\_{\boldsymbol{n}} + \boldsymbol{n}}{\sqrt{\boldsymbol{n}}} \frac{\Gamma\left(\frac{\boldsymbol{\gamma}}{\boldsymbol{\delta}}\right)}{\Gamma\left(\frac{\boldsymbol{n}}{\boldsymbol{\delta}}\right)} \left(\frac{\boldsymbol{\gamma}\_{\boldsymbol{z}} \boldsymbol{D}}{2}\right)^{\frac{\boldsymbol{\gamma}}{\boldsymbol{\delta}}} \Big| \frac{\boldsymbol{s}\_{\boldsymbol{z}}}{\boldsymbol{\gamma}\_{\boldsymbol{z}} \boldsymbol{D}} \Big| \frac{\boldsymbol{2} \left[\boldsymbol{1} \boldsymbol{1} \left(\boldsymbol{u} + \frac{\boldsymbol{\gamma}}{\boldsymbol{\delta}}\right) - \boldsymbol{5} \boldsymbol{N}\left(\boldsymbol{u} + \frac{\boldsymbol{\gamma}}{\boldsymbol{\delta}}\right)\right]}{2 \boldsymbol{\Gamma}\left(\boldsymbol{u} + \frac{\boldsymbol{n}}{\boldsymbol{\delta}}\right)} \Big| \frac{\boldsymbol{s}\_{\boldsymbol{z}}}{\boldsymbol{\gamma}\_{\boldsymbol{z}} \boldsymbol{D}} \Big| \frac{\boldsymbol{s}\_{\boldsymbol{z}}}{\boldsymbol{\gamma}\_{\boldsymbol{z}} \boldsymbol{D}} \Big| \tag{80}$$

**Author details**

**References**

Jingyuan Chen and Xiang Chang

Yunnan Astronomical Observatory, Chinese Academy of Science, China

tional Society for Optical Engineering, Bellingham

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#### **Expressions used in section 4.4.**

When the filter function is equal to

$$F\_{n,m}(\kappa, z) = [N\_n(\kappa) \cdot N\_n(\alpha\_z \kappa)]^2$$

The integrations can be expressed by

$$\begin{split} \boldsymbol{I}\_{n}(\boldsymbol{\omega}) &= \frac{(n+1)\boldsymbol{I}\left(n - \frac{5}{6}\right)\boldsymbol{\Omega}^{-\frac{5}{3}}}{2^{2n+\frac{11}{3}}\boldsymbol{I}\left(\frac{17}{6}\right)\boldsymbol{I}\left(n+2\right)} \left| \frac{8\boldsymbol{I}\left(\frac{4}{3}\right)\boldsymbol{I}\left(2n+3\right)\boldsymbol{\alpha}\_{z}^{\frac{2}{3}} + 1}{3\boldsymbol{I}\left(n+\frac{3}{2}\right)\boldsymbol{I}\left(n+\frac{23}{6}\right)} - 1 \right| \\\\ &\sim \frac{2^{2n+1}\boldsymbol{\alpha}\_{z}^{\frac{n}{3}}}{6n+17} \left| 5\left(\boldsymbol{\alpha}\_{z}^{2} - 1\right) \right|\_{2} \, \_{1}F\_{1}\{1/\boldsymbol{6}, \ n-5\left/\left\{6; n+2; \boldsymbol{\alpha}\_{z}^{2}\right\} - 1\right\} \right| \\\\ &\sim \left(6\boldsymbol{\alpha}\_{z}^{2}n - 6n - 22\right) \, \_{2}F\_{1}\{-5\left/\left\{6, \left\{6; n+2; \boldsymbol{\alpha}\_{z}^{2}\right\} \right\} \} \right| \end{split}$$

When the beacon is high enough (*H* ≫*z*), we can obtain the following asymptotic solution expanding to second-order turbulence moments:

$$I\_n(z) = \frac{8(n+1)\prod 108n(n+2)\cdot 55}{623\cdot \pi \cdot \Gamma(\cdot 10/3)\Gamma(n+23/6)} \Gamma^2(-\frac{8}{3})\Gamma\left(n - \frac{5}{6}\right)\left(\frac{z}{H}\right)^2 D^{5/3} \tag{81}$$

#### **Expressions used in section 4.5.**

When the filter function is equal to

$$F\_{n,m}(\kappa, z) = N\_n^{-2}(\kappa) \| 1 - G\_s(\kappa, z) \|^2$$

For the Gaussian source Eq. (12), the results can be expressed as:

*<sup>I</sup> <sup>n</sup>*(*z*)= <sup>1</sup> <sup>2</sup> (1 <sup>+</sup> *<sup>n</sup>*)*Γ*(*<sup>n</sup>* - <sup>5</sup> <sup>6</sup> ){ <sup>2</sup> 1 <sup>3</sup> *<sup>Γ</sup>*( <sup>7</sup> <sup>3</sup> )*<sup>D</sup>* 5 3 *<sup>Γ</sup>*( <sup>17</sup> <sup>6</sup> )*Γ*(*<sup>n</sup>* <sup>+</sup> <sup>23</sup> 6 ) + + (*θσ <sup>z</sup>*)-2*<sup>n</sup>*+5/3*<sup>D</sup>* <sup>2</sup>*<sup>n</sup>* 24*<sup>n</sup> Γ*(*n* + 2) <sup>2</sup> 2 *<sup>F</sup>*2(*<sup>n</sup>* - <sup>5</sup> / 6, *<sup>n</sup>* <sup>+</sup> <sup>3</sup> / 2;*<sup>n</sup>* <sup>+</sup> 2, <sup>2</sup>*<sup>n</sup>* <sup>+</sup> 3; - *<sup>D</sup>* <sup>2</sup> 4(*θ<sup>r</sup> <sup>z</sup>*)2 )- -2*n*+1/6 2 *<sup>F</sup>*2(*<sup>n</sup>* - <sup>5</sup> / 6, *<sup>n</sup>* <sup>+</sup> <sup>3</sup> / 2;*<sup>n</sup>* <sup>+</sup> 2, <sup>2</sup>*<sup>n</sup>* <sup>+</sup> 3; - *<sup>D</sup>* <sup>2</sup> 2(*θ<sup>r</sup> <sup>z</sup>*)2 )}

If the widths of the distributed source are very large (*θrz* ≫*D*), expanding the solutions to second-order terms of small parameter *D* /(*θrz*), the results are obtained as follows:

$$I\_{n}(\mathbf{z}) = \frac{\binom{\mathbf{I}\_{1} + \mathbf{n}}{\mathbf{I}\_{2} + \mathbf{n}} \Gamma\left(\mathbf{n} \cdot \frac{\mathbf{z}}{\sigma}\right)}{2^{2\beta} \mathbf{D}^{-5\beta}} \left[\frac{\Gamma\left(\frac{\gamma}{\delta}\right) / \Gamma\left(\frac{\gamma}{\delta}\right)}{\sqrt{\pi} \, \Gamma\left(\mathbf{n} + \frac{2\gamma}{\delta}\right)} - \frac{(2\beta, z / D)^{5/8 - 2\alpha}}{\mathbf{Z}^{\pi \times 1} \Gamma(\mathbf{n} + 2) \mathbf{Z}^{\dagger}} \left[\left(\mathbf{Z}^{\pi \times 1/6} \cdot \mathbf{1}\right) + \frac{\left(\mathbf{z} \cdot \frac{\mathbf{z}}{\sigma}\right) (2^{\pi \times 7/8} \cdot \mathbf{1})}{2(2 + \pi) (2\alpha \beta \cdot \mathbf{z} / D)^{2}}\right]\right] \tag{82}$$
