The principle of phase screen method [7]is considering the beam propagate through vacuum and phase screens along the transmission path alternately as shown inFig.1. The propagation distance L is divided into many parts, turbulence effects of each part is considered to be a phase screen, the light field between two adjacent phase screen can be expressed as
Here,
$\Delta {z_{j + 1}} = {z_{j + 1}} - {z_j}$
is the distance between two phase screen located at
${z_{j + 1}}$
and
${z_j}$
;
${\kappa _x}$
and
${\kappa _y}$
denotes the spatial frequency, the grids are divided to be
$N \times N$
and the resolution is
$\Delta x$
,
$F$
and
${F^{ - 1}}$
represents Fourier transform and Fourier inverse transform, respectively;
$\varphi \left( {x,y} \right)$
reveals the phase fluctuation caused by turbulence which can be expressed as [7]
Here,
$x = m\Delta x$
,
$y = m\Delta y$
represents the spatial domain,
$\Delta x$
,
$\Delta y$
is the sample interval, m, n are integers;
${\kappa _x} = {m^{'}}\Delta {\kappa _x}$
,
${\kappa _y} = {n^{'}}\Delta {\kappa _y}$
represents the frequency domain,
$\Delta {\kappa _x}$
,
$\Delta {\kappa _y}$
is the sample interval,
$m'$
,
$n'$
are integers;
$a\left( {{\kappa _x},{\kappa _y}} \right)$
is the Fourier transform of the Gaussian random matrix with the mean value 0 and variance 1;
${\varPhi _\theta }\left( {{\kappa _x}{\rm{,}}\;{\kappa _y}} \right)$
represents the phase power spectrum, which is related to the power spectrum as
The parameter of phase screen mainly include the grid size
$\Delta x$
, screen size x and the number of phase screen along the propagation path NPS. Since FFT method is usually used to calculate the beam propagation for program, the sample rule request is derived by Ref.[9]
Here,
$\lambda $
is wavelength;L is the propagation distance. Considering that
$x = \sqrt {\lambda LN} $
,
$\Delta x = \sqrt {\lambda L/N} $
, the relationships between grid size
$\Delta x$
, screen size x and grids number N is decided. Usually the grids number N is decided firstly, and with higher N, the resolution of sample is more approximate while the computation complexity is higher. In this paper, the influence of different N is shown and compared.
Meanwhile, the description of beam should also be considered. For Gaussian beam, the beam width
${w_0}$
should be much bigger than
$\Delta x$
. Also, consider about the characteristics of turbulence, the sampling should be less than half of the coherence length
$\Delta x \leqslant {r_0}/2$
. Here the oceanic turbulence coherence length
${r_0} \approx 2.1{\rho _0}$
[10], spatial coherence scale
${\rho _0}$
for oceanic turbulence is derived by Ref.[11].
The number of phase screen along the path is decided by the turbulence strength. According to Ref.[12], the fluctuation between two phase screens should be small enough and meet the requirements of Rytov variance
$\sigma _R^2\left( {\Delta z} \right) < 0.1$
, then the requirement of phase screen number could be expressed as
Rytov variance
$\sigma _R^2$
can be approximately expressed as[13]:
The example of parameter settings is as follows:
(1) Consider about the strongest turbulence condition as an example, L=50 m,
$\varepsilon = {10^{ - 6}}{{\rm{m}}^2}{{\rm{s}}^{ - 3}}$
,
${\chi _T} = {10^{ - 6}}{K^2}{s^{ - 1}}$
,
$\omega = - 2$
. Substituting into Eq.(14),
${r_0} \approx 0.7\;{\rm{mm}}$
is obtained. Consider that
$\Delta x \leqslant {r_0}/2$
,
$\Delta x < < {w_0}$
, the grid size should fulfill
$\Delta x \leqslant 0.35\;{\rm{mm}}$
, grids number should fulfill
$N \geqslant 238$
.
(2) Use Eq.(15) to calculate the phase screen number, and the distance between adjacent phase screens are should meet
$\Delta z \leqslant 2.5$
. During the simulation in this paper, the distance between screens is 2.5 m, which is much lower than the atmospheric turbulence simulation conditions.