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\title{\centering Propagation intensity and phase distribution of  Beam }

%%%  Authors
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\author[1 \cauthfn{1}]{\href{https://su.edu.ye/coss/en/our_team/mohammed-saleh-saleh-qusailah/}{M S. Q}~\orcidlink{0000-0001-7412-061X}}
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\author[2]{\href{https://su.edu.ye/coss/en/our_team/abdu-ahmed-h-alkelly/}{A A. Al}}
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\author[1]{W A. Al}
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\affil[1]{Department of Physics, Faculty of Science, University of Sana'a, Sana'a, Yemen}
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\affil[2]{Department of Physics, Faculty of Sci, University of S S , Country}
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\corrauth{\cauthfn{1} {m.qusailah@su.edu.ye}}

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\runningtitle{\usefont{T1}{qhv}{b}{it}\selectfont{Propagation intensity  and phase distribution of Beam}}
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\runningauthor{\usefont{T1}{qhv}{b}{it}\selectfont{M S.Q et al.}}

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\keywords{%
	Propagation%
	, Intensity% 
	, Phase%
	, Beam%
	, Distribution of Beam%
}



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%%%%    frontmatter  on the First Page  titel, Authors , abstract , article info 
	\begin{frontmatter}
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		\begin{abstract}
			\lipsum[1] % Dummy text
		\end{abstract}		
	\end{frontmatter}
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\section{Introduction}
\lipsum[2] % Dummy text
\cite{zhao2010paraxial},  \cite{zhou2010average}, \lipsum[3] % Dummy text
\cite{alkelly2021coherence}
\section{Model of a  beam}\label{sec:TheoryModel}
At the source plane, the electric fields of the LGVB beam can be
expressed as\cite{ni2013propagation}:
%\normalsize{%}
\begin{gather}
	\begin{split}
		E\left( r_0,0\right) &=\frac{w_{0x} w_{0y}}{\left(w^{2}_{0x}+x^{2}_0 \right)\left(w^{2}_{0y}+y^{2}_0 \right)} \\ 
		&  \times\;\exp \left(-\frac{x^{2}_0+y^{2}_0}{w^{2}_0}\right)
		\times{\left(\frac{x_0+iy_0}{w_0}\right)}^{M},
	\end{split}
	\label{eq:Er0z0} 
\end{gather}

where $r_0 =(x_0, y_0)$ is the position vector at the source plane $z_0,
w_{0x}$  and $w_{0y}$ are the parameters related to the beam widths in the $x$ and $y$ directions, respectively, $w_0$ is the waist width of the Gaussian part of the Lorentz–Gauss vortex beam, and $M$ is the topological charge of Lorentz–Gauss vortex beam\cite{liu2018average}:
\begin{gather}
	\label{eq:wrro*}
	\begin{split}
		&W(r_{01};r_{02};0)=\left\langle E\left( r_{01},0\right)E^{*}\left( r_{02},0\right)\right\rangle \\
		&\quad=\sqrt{I(r_{01}, 0)I(r_{02},0)}\times\mu(x_{01}-x_{02},  y_{01}-y_{02}) ,
	\end{split}
\end{gather}

where $\mu {(}x_{{01}}-x_{{02}},y_{{01}}-y_{{02}}{)}$ represents the spectral degree of coherence and is represented as follows: 
%
\begin{gather}		\label{eq:mu}
	\hskip-20pt\begin{split}
		&\mu{(x_{01}-x_{02},y_{01}-y_{02})}=\\ &\qquad\exp\left[-\frac{{(}x_{{01}}-x_{{02}}{{)}}^{{2}}}{{2}{\sigma}^{{2}}_x}-\frac{{(}y_{{01}}-y_{{02}}{{)}}^{{2}}}{{2}{\sigma }^{{2}}_y}\right],  
	\end{split}
\end{gather} 	

when $\sigma$ turns into infinity, $\mu(x_{01}-x_{02},  y_{01}-y_{02}) = 1$, thus becoming  
\begin{gather}
	\label{eq:mu*}
	\begin{split}
		W(r_{01};r_{02};0)&=\sqrt{I(r_{01}, 0)I(r_{02},0)},
	\end{split}
\end{gather} 

and when $r_{01}=r_{02}$ then 
\begin{gather}
	\label{eq:mu**}
	\begin{split}
		W(r_{01};r_{02};0)&=I(r_{01}, 0)
	\end{split}
\end{gather} 

Substituting Eq. (\ref{eq:Er0z0}) into Eq.(\ref{eq:mu}), partially coherent Lorentz-Gauss  vortex beam at the source plane can be expressed as\cite{liu2017propagation}:	
\begin{gather}\label{eq:2*} 
	\begin{split}
		W(r_{01};r_{02};0)&=\frac{1}{w_{0}^{2M}}\frac{w_{0x} w_{0y}}{\left(w^{2}_{0x}+x^{2}_{01} \right)\left(w^{2}_{0y}+y^{2}_{01}\right)}\\
		&\times\exp \left(-\frac{x^{2}_{01}+y^{2}_{01}}{w^{2}_0}\right) 	\times\left(x_{01}+i y_{01} \right)^{M}\\
		&\times \frac{w_{0x} w_{0y}}{\left(w^{2}_{0x}+x^{2}_{02} \right)\left(w^{2}_{0y}+y^{2}_{02}\right)}\\ 
		&\times\exp \left(-\frac{x^{2}_{02}+y^{2}_{02}}{w^{2}_0}\right) 	\times\left(x_{02}-i y_{02} \right)^{M}\\
		&\times\exp\left[- \frac{(x_{01}- x_{02})^{2}}{2 \sigma_{x}^{2}}- \frac{(y_{01}- y_{02})^{2}}{2 \sigma_{y}^{2}}\right] ,
	\end{split}
\end{gather}

Where $ \sigma_{x} $ and  $ \sigma_{y} $ are the spatial coherence length in the $ x $ and $ y $ directions respectively . By considering in Eq.(\ref{eq:2*}), the relation between the Hermite-Gauss function and the Lorentz distribution function \cite{liu2017propagation}:
\begin{gather}
	\hskip-00pt\begin{split} 
		&\frac{1}{\left(w^{2}_{0x}+x^{2} \right)\left(w^{2}_{0y}+y^{2} \right)}=\\
		&\quad\;\frac{\pi}{2 w^{2}_{0x} w^{2}_{0y}}\sum_{m=0}^{N}\sum_{n=0}^{N}a_{2m}a_{2n}\\
		&\quad\;H_{2m}\left(\frac{x}{w_{0x}}\right) H_{2n}\left( \frac{y}{w_{0y}}\right)\\
		&\quad\times\;\exp\left(-\frac{x^{2}}{2w^{2}_{0x}} -\frac{y^{2}}{2w^{2}_{0y}}\right).
	\end{split}\label{eq:21}
\end{gather}

Where $H_{2m} $ and $H_{2n} $ are the $2m$ and $2n$ order Hermite polynomial, and $H_{2m}$ can be expressed as\cite{jeffrey2008handbook} 
\begin{equation}
	\label{eq:3}
	H_{2m}(x)=\sum_{L=0}^{m}\frac{(-1)^{L}(2m)!}{L!(2m-2L)!}(2x)^{2m-2L}.
\end{equation}

Similarly, $H_{2n} $ can be expressed as in Eq.(\ref{eq:3}), only m replaced by n.
And the expanded coefficients  $a_{2m} $ can be rewritten  as: \cite{ni2013propagation} 
\begin{gather} \label{eq:a2m}
	\hskip-20pt\begin{split} 
		a_{2m}&=\frac{(-1)^{m}}{2^{2m-1}\sqrt{\pi}}
		\bigg\{
		\frac{1}{m!}\sqrt{\frac{\pi}{2}}\;\textbf{e}^{\left( {1}/{4}\right) ^{2}}\text{erfc}\left[\frac{1}{\sqrt{2}}\right]\\
		&+\sum_{s=1}^{m}\frac{2^{2s}}{(2s)!(m-s)!}
		\bigg[\sqrt{\frac{\pi}{2}}\;\textbf{e}^{\left( {1}/{4}\right)^{2}}\times\\  
		&\text{erfc}\left[\dfrac{1}{\sqrt{2}}\right] +\sum_{t=1}^{s}(-1)^{t}(2t-3)!!\bigg]
		\bigg\}.
	\end{split}
\end{gather}

where $\text{erfc}\left[x\right] =1- \text{erf}\left[x\right]$, and  
with an increasing in numbers $2m$ the values of $a_{2m}$  will dramatically decrease.
Now,  recalling the following equation \cite{jeffrey2008handbook}: 
\begin{equation}\label{eq:4}
	\left(x+i y \right)^{M}=\sum_{L=0}^{M} \frac{M!  i^{L}}{L! (M-L)!}x^{M-L}y^{L},
\end{equation}

\lipsum[4] % Dummy text

\subsection{Simulation and discussion}
In this section, the average intensity, phase, and vortex properties of the beam propagating. Figures \ref{fig:f1} and \ref{fig:f2} ...

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\begin{figure}[H]
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	\caption{Normalized intensity and the phase distribution with  different values of  $w _{0x} =w_{0y}$.}
	\label{fig:f1}
\end{figure}

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				\includegraphics[width=\linew,height=\lineh,frame]{example-image} 
				\includegraphics[width=\linew,height=\lineh,frame]{example-image} 
				\caption{Normalized intensity and the phase distribution with  different values of  $w _{0x} =w_{0y}$.}
				\label{fig:f2}
			\end{figure}
}
	\lipsum[5] % Dummy text%
	
	\section{Conclusion}
	\lipsum[6] % Dummy text%.\\

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