Scintillation mitigation via the cross phase of the Gaussian Schell-model beam in a turbulent atmosphere

Hui Zhang; Hui Zhang; Hui Zhang; Lingling Zhao; Lingling Zhao; Lingling Zhao; Yaru Gao; Yaru Gao; Yaru Gao; Yaru Gao; Yangjian Cai; Yangjian Cai; Yangjian Cai; Yangjian Cai; Yangsheng Yuan; Yangsheng Yuan; Yangsheng Yuan

doi:10.1364/OE.501006

1. Introduction

Scintillation is a fluctuation in intensity that occurs when light propagates in a random medium. This phenomenon significantly affects the image and signal quality [1–13]. Scintillation affects the signal carried by a laser beam through the atmosphere, and the signal quality can be improved by reducing the scintillation of the beams in FSO communications. The theoretical results showed that the scintillation of a partially coherent beam was lower than that of a fully coherent beam [14–16]. Partially coherent Laguerre Gaussian beams with a large radial mode number were applied to reduce the scintillations induced by atmospheric turbulence [17]. The nonuniform polarization of the incoherent beam array with multiple beamlets was modulated to reduce scintillation [18]. Partially coherent beams with special correlations have small scintillations when they propagate through a turbulent atmosphere, such as symmetric and asymmetrical multi-Gaussian Schell-model beams [19,20], pseudo-Bessel-Gaussian Schell-mode beams [21], and the intensity-intensity cross-correlation beams [22], et al. Partially coherent beams carrying the twisted and vortex phases were used to optimize the scintillation perturbed by atmospheric turbulence [23,24].

The cross phase is one of the special phases of beams with a quadratic phase structure. The cross phase of the beams was applied in optical vortex manipulation, coherent mode conversion, beam rotation, and self-reconstruction. A twisting phase was added to the vortex beam to detect the signs and values of the topological charge [25]. Subsequently, the cross phase was combined with a partially coherent vortex beam to apply orbital angular momentum measurements at extremely low coherence [26]. A high-order cross phase was proposed, and this new type of phase structure was used to modulate the optical vortices to obtain polygonal shaping and multi-singularity manipulation [27]. The cross phase was used to design a novel optical vortex beam generator based on a metasurface, and the intensity and singularity distributions were easily modulated [28]. By imposing the cross phase on the Hermite-Gaussian and Laguerre-Gaussian modes, their interconversion can be achieved by changing the parameters of the cross phase [29]. Gaussian Schell-model beams with a cross phase have twist and modulation effects, which provide a new degree of freedom that can be used to rotate the intensity distribution of the beam exceeding 90${\circ }$ and enhance the z-coherence [30,31]. The self-reconstruction properties of a beam disturbed by obstacles can be enhanced by introducing a cross phase structure into the coherence function [32]. The second-order spatial coherence structure of the cross phase measurement can be applied to recover image information carried by partially coherent beams disturbed by an obstacle or turbulent atmosphere [33,34].

In this study, we theoretically and experimentally investigated the scintillation index of a Gaussian Schell-model (GSM) beam with cross phase propagation in a turbulent atmosphere. The results show that the scintillation index of this beam can be controlled by modulating its quadratic phase structure. A hot plate was used to emulate atmospheric turbulence, and the scintillation index was measured using a charge-coupled device (CCD) camera. Our results can be applied to image and FSO communications.

2. Theory

The cross-spectral density function (CSD) can be used to calculate the evolution parameters of beam propagation through a turbulent atmosphere, which is characterized by second-order statistical properties [35]. The CSD of a GSM beam with a cross phase at the source plane is expressed as

(1)$${\Gamma _2}\left( {\widetilde {\mathbf{r}},0} \right) = \exp \left( { - {{\widetilde {\mathbf{r}}}^T}{{\mathbf{M}}^{ - 1}}\widetilde {\mathbf{r}}} \right),$$

where ${\widetilde {\mathbf{r}}^T} = \left ( {{{\mathbf{r}}_1},{{\mathbf{r}}_2}} \right )$ describes two arbitrary points at $z$=0, and the parameter ${{\mathbf{M}}^{ - 1}}$, which notes the tensor of the partially coherent complex curvature, is obtained by

(2)$${{\mathbf{M}}^{ - 1}} = \left( {\begin{array}{cc} {\left( {\frac{1}{{4{\sigma ^2}}} + \frac{1}{{2{\delta ^2}}}} \right){\mathbf{{ I}}} + \frac{1}{2}iku{\mathbf{J}}} & { - \frac{1}{{2{\delta ^2}}}{\mathbf{{ I}}}}\\ { - \frac{1}{{2{\delta ^2}}}{\mathbf{{ I}}}} & {\left( {\frac{1}{{4{\sigma ^2}}} + \frac{1}{{2{\delta ^2}}}} \right){\mathbf{{ I}}} - \frac{1}{2}iku{\mathbf{J}}} \end{array}} \right),$$

where ${\mathbf{{I}}}$ is a 2$\times$2 unit matrix and $k = 2\pi /\lambda$ is the wavenumber with wavelength $\lambda$. Here, $\delta$ and $\sigma$ are the coherence length and beam width, respectively. The factor $u$ denotes the strength factor of the cross phase. ${\mathbf{J}}$ is an antisymmetric matrix, as follows:

(3)$${\mathbf{J}} = \left( {\begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}} \right)$$

Using the extended Huygens-Fresnel integral, the average intensity at the receiver plane can be expressed as [2]

(4)$$\begin{aligned} \left\langle {I\left( {{\mathbf{p}},z} \right)} \right\rangle &= \frac{1}{{{{\left( {\lambda z} \right)}^2}}}\int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {{d^2}{{\mathbf{r}}_1}\int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {{d^2}{{\mathbf{r}}_2}} } } } {\Gamma _2}\left( {{{\mathbf{r}}_1},{{\mathbf{r}}_2},0} \right)\\ &\times \exp \left( {\frac{{ik}}{{2z}}{{\left| {{{\mathbf{r}}_1} - {\mathbf{p}}} \right|}^2} - \frac{{ik}}{{2z}}{{\left| {{{\mathbf{r}}_2} - {\mathbf{p}}} \right|}^2}} \right)\left\langle {\exp \left[ {\psi \left( {{{\mathbf{r}}_1},{\mathbf{p}}} \right)} \right]\exp \left[ {{\psi ^*}\left( {{{\mathbf{r}}_2},{\mathbf{p}}} \right)} \right]} \right\rangle \end{aligned},$$

where $\left \langle {} \right \rangle$ denotes the ensemble averaging over the medium statistics, $z$ is the propagation distance, and [19]

(5)$$\left\langle {\exp \left[ {\psi \left( {{{\mathbf{r}}_1},{\mathbf{p}}} \right)} \right]\exp \left[ {{\psi ^*}\left( {{{\mathbf{r}}_2},{\mathbf{p}}} \right)} \right]} \right\rangle = \exp \left( { - \frac{{{{\left| {{{\mathbf{r}}_1} - {{\mathbf{r}}_2}} \right|}^2}}}{{\rho _0^2}}} \right)$$

Here, where ${\mathbf{p}} = \left ( {{p_x},{p_y}} \right )$ is the position vector at the receiver plane and ${\rho _0} = {\left ( {0.545C_n^2{k^2}z} \right )^{ - 3/5}}$ is the coherence length of a spherical wave propagating through atmospheric turbulence with the structure constant $C_n^2$.

Substituting Eq. (1) and Eq. (5) into Eq. (4), the average intensity of a GSM beam with a cross phase in a turbulent atmosphere can be obtained as follows:

(6)$$\left\langle {I\left( {{\mathbf{p}},z} \right)} \right\rangle = \frac{{{k^2}}}{{4{z^2}}}\frac{{\exp \left( {{{\widetilde {\mathbf{p}}}^T}{\mathbf{M}}_t^{ - 1}{{\mathbf{M}}_r}^{ - 1}{\mathbf{M}}_t^{ - 1}\widetilde {\mathbf{p}}} \right)}}{{{{\left( {\det \left| {{{\mathbf{M}}_r}} \right|} \right)}^{1/2}}}},$$

where ${\widetilde {\mathbf{p}}^T} = \left ( {{\mathbf{p}},{\mathbf{p}}} \right )$, and

(7)$${\mathbf{M}}_s^{ - 1} = \left( {\begin{array}{cc} {\left( { - \frac{{ik}}{{2z}} + \frac{1}{{\rho _0^2}}} \right){\mathbf{{ I}}}} & { - \frac{1}{{\rho _0^2}}{\mathbf{{ I}}}}\\ { - \frac{1}{{\rho _0^2}}{\mathbf{{ I}}}} & {\left( {\frac{{ik}}{{2z}} + \frac{1}{{\rho _0^2}}} \right){\mathbf{{ I}}}} \end{array}} \right)$$

(8)$${\mathbf{M}}_t^{ - 1} = \left( {\begin{array}{cc} { - \frac{{ik}}{{2z}}{\mathbf{{ I}}}} & {0{\mathbf{{ I}}}}\\ {0{\mathbf{{ I}}}} & {\frac{{ik}}{{2z}}{\mathbf{{ I}}}} \end{array}} \right)$$

(9)$${{\mathbf{M}}_r}^{ - 1} = {\left( {{{\mathbf{M}}^{ - 1}} + {\mathbf{M}}_s^{ - 1}} \right)^{ - 1}}$$

The average squared intensity of the GSM beam with a cross phase at the receiver plane is expressed as [16]

(10)$$\begin{aligned} \left\langle {{I^2}\left( {{\mathbf{p}},z} \right)} \right\rangle &= \frac{1}{{{{\left( {\lambda z} \right)}^4}}}\int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {{d^2}{{\mathbf{r}}_1}\int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {{d^2}{{\mathbf{r}}_2}} } } } \int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {{d^2}{{\mathbf{r}}_3}} } \int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {{d^2}{{\mathbf{r}}_4}} } \\ &\times {\Gamma _4}\left( {{{\mathbf{r}}_1},{{\mathbf{r}}_2},{{\mathbf{r}}_3},{{\mathbf{r}}_4},0} \right)\Gamma _4^m\left( {{{\mathbf{r}}_1},{{\mathbf{r}}_2},{{\mathbf{r}}_3},{{\mathbf{r}}_4},{\mathbf{p}}} \right)\\ &\times \exp \left( {\frac{{ik}}{{2z}}{{\left| {{{\mathbf{r}}_1} - {\mathbf{p}}} \right|}^2} - \frac{{ik}}{{2z}}{{\left| {{{\mathbf{r}}_2} - {\mathbf{p}}} \right|}^2} + \frac{{ik}}{{2z}}{{\left| {{{\mathbf{r}}_3} - {\mathbf{p}}} \right|}^2} - \frac{{ik}}{{2z}}{{\left| {{{\mathbf{r}}_4} - {\mathbf{p}}} \right|}^2}} \right) \end{aligned},$$

where ${\Gamma _4}\left ( {{{\mathbf{r}}_1},{{\mathbf{r}}_2},{{\mathbf{r}}_3},{{\mathbf{r}}_4},0} \right ) = \left \langle {E\left ( {{{\mathbf{r}}_1}} \right ){E^*}\left ( {{{\mathbf{r}}_2}} \right )E\left ( {{{\mathbf{r}}_3}} \right ){E^*}\left ( {{{\mathbf{r}}_4}} \right )} \right \rangle$ is a fourth-order correlation function of the beam in the source plane. We assume that the detection time in the receiver plane is larger than the characteristic time of the source field, that is, the detector is considered a "slow" detector [36]. The fourth-order correlation function at $z$=0 is approximated as

(11)$${\Gamma _4}\left( {{{\mathbf{r}}_1},{{\mathbf{r}}_2},{{\mathbf{r}}_3},{{\mathbf{r}}_4},0} \right) \approx \left\langle {E\left( {{{\mathbf{r}}_1}} \right){E^*}\left( {{{\mathbf{r}}_2}} \right)} \right\rangle \left\langle {E\left( {{{\mathbf{r}}_3}} \right){E^*}\left( {{{\mathbf{r}}_4}} \right)} \right\rangle$$

$\Gamma _4^m\left ( {{{\mathbf{r}}_1},{{\mathbf{r}}_2},{{\mathbf{r}}_3},{{\mathbf{r}}_4},{\mathbf{p}}} \right )$ is the fourth-order correlation function for the turbulent medium at the receiver plane, and is expressed as follows [36]:

(12)$$\begin{aligned} &\Gamma _4^m\left( {{{\mathbf{r}}_1},{{\mathbf{r}}_2},{{\mathbf{r}}_3},{{\mathbf{r}}_4},{\mathbf{p}}} \right) = \left[ {1 + 2{B_\chi }({{\mathbf{r}}_1} - {{\mathbf{r}}_3}) + 2{B_\chi }({{\mathbf{r}}_2} - {{\mathbf{r}}_4})} \right]\\ &{\rm{ }} \times \exp \left[ { - 0.5{D_\psi }\left( {{{\mathbf{r}}_1} - {{\mathbf{r}}_2}} \right) - 0.5{D_\psi }\left( {{{\mathbf{r}}_3} - {{\mathbf{r}}_4}} \right) - 0.5{D_\psi }\left( {{{\mathbf{r}}_2} - {{\mathbf{r}}_3}} \right) - 0.5{D_\psi }\left( {{{\mathbf{r}}_1} - {{\mathbf{r}}_4}} \right)} \right] \\ &{\rm{ }} \times \exp \left[ {0.5{D_\psi }\left( {{{\mathbf{r}}_1} - {{\mathbf{r}}_3}} \right) + 0.5{D_\psi }\left( {{{\mathbf{r}}_2} - {{\mathbf{r}}_4}} \right) + i{D_{\chi s}}\left( {{{\mathbf{r}}_2} - {{\mathbf{r}}_4}} \right) - i{D_{\chi s}}\left( {{{\mathbf{r}}_1} - {{\mathbf{r}}_3}} \right)} \right] \end{aligned}$$

Here, ${D_\psi }({{\mathbf{r}}_s} - {{\mathbf{r}}_d}) = 2|{{\mathbf{r}}_s} - {{\mathbf{r}}_d}{|^2}/\rho _0^2$ is the wave structure function and ${B_\chi }({{\mathbf{r}}_s} - {{\mathbf{r}}_d}) = \sigma _\chi ^2|{{\mathbf{r}}_s} - {{\mathbf{r}}_d}{|^2}/\rho _0^2$ is the log-amplitude phase structure function with $\sigma _\chi ^2 = 0.124C_n^2{k^{7/6}}{z^{11/6}}$. ${D_{\chi s}}({{\mathbf{r}}_s} - {{\mathbf{r}}_d}) = 2|{{\mathbf{r}}_s} - {{\mathbf{r}}_d}{|^2}/\rho _{\chi s}^2$ is a log-amplitude correlation function of $\rho _{\chi s}^2 = {(0.114C_n^2{k^{13/6}}{z^{5/6}})^{ - 1/2}}$ ($s$=1,2,3,4, $d$=1,2,3,4).

Substituting Eq. (11) and Eq. (12) into Eq. (10), the average squared intensity of a GSM beam with a cross phase at the receiver plane can be obtained as

(13)$$\begin{aligned} \left\langle {{I^2}\left( {\overline {\mathbf{p}} ,z} \right)} \right\rangle &= \frac{{{k^4}}}{{16{z^4}}}\left[ {\frac{{\exp \left( {{{\overline {\mathbf{p}} }^T}{\mathbf{M}}_d^{ - 1}{{\mathbf{M}}_{rh1}}^{ - 1}{\mathbf{M}}_d^{ - 1}\overline {\mathbf{p}} } \right)}}{{{{\left( {\det \left| {{{\mathbf{M}}_{rh1}}} \right|} \right)}^{1/2}}}}} \right.\\ &\left. { + 2\sigma _\chi ^2\frac{{\exp \left( {{{\overline {\mathbf{p}} }^T}{\mathbf{M}}_d^{ - 1}{{\mathbf{M}}_{rh2}}^{ - 1}{\mathbf{M}}_d^{ - 1}\overline {\mathbf{p}} } \right)}}{{{{\left( {\det \left| {{{\mathbf{M}}_{rh2}}} \right|} \right)}^{1/2}}}} + 2\sigma _\chi ^2\frac{{\exp \left( {{{\overline {\mathbf{p}} }^T}{\mathbf{M}}_d^{ - 1}{{\mathbf{M}}_{rh3}}^{ - 1}{\mathbf{M}}_d^{ - 1}\overline {\mathbf{p}} } \right)}}{{{{\left( {\det \left| {{{\mathbf{M}}_{rh3}}} \right|} \right)}^{1/2}}}}} \right] \end{aligned}$$

Here,

(14)$$\begin{aligned} &{\overline {\mathbf{p}} ^T} = \left( {{\mathbf{p}},{\mathbf{p}},{\mathbf{p}},{\mathbf{p}}} \right),{{\mathbf{M}}_{rhi}}^{ - 1} = {\left( {{\mathbf{M}}_m^{ - 1} + {\mathbf{M}}_{hi}^{ - 1}} \right)^{ - 1}},{\mathbf{M}}_{hi}^{ - 1} = \left( {\begin{array}{cc} {{{\mathbf{A}}_{hi}}} & {{{\mathbf{B}}_{hi}}}\\ {{{\mathbf{B}}_{hi}}} & {{{\mathbf{A}}_{hi}}} \end{array}} \right),\left( {i = 1,2,3} \right),\\ &{{\mathbf{A}}_{h1}} = {{\mathbf{B}}_{h1}} = \left( {\begin{array}{cc} {{Q_1}{\mathbf{I}}} & {{Q_2}{\mathbf{I}}}\\ {{Q_2}{\mathbf{I}}} & {Q_1^*{\mathbf{I}}} \end{array}} \right),{{\mathbf{A}}_{h2}} = \left( {\begin{array}{cc} {{Q_3}{\mathbf{I}}} & {{Q_2}{\mathbf{I}}}\\ {{Q_2}{\mathbf{I}}} & {Q_1^*{\mathbf{I}}} \end{array}} \right),{{\mathbf{B}}_{h2}} = \left( {\begin{array}{cc} {{Q_4}{\mathbf{I}}} & {{Q_2}{\mathbf{I}}}\\ {{Q_2}{\mathbf{I}}} & {{Q_5}{\mathbf{I}}} \end{array}} \right),\\ &{{\mathbf{A}}_{h3}} = \left( {\begin{array}{cc} {{Q_1}{\mathbf{I}}} & {{Q_2}{\mathbf{I}}}\\ {{Q_2}{\mathbf{I}}} & {Q_3^*{\mathbf{I}}} \end{array}} \right),{{\mathbf{B}}_{h3}} = \left( {\begin{array}{cc} {Q_5^*{\mathbf{I}}} & {{Q_2}{\mathbf{I}}}\\ {{Q_2}{\mathbf{I}}} & {Q_4^*{\mathbf{I}}} \end{array}} \right),\\ &{\mathbf{M}}_m^{ - 1} = \left( {\begin{array}{cc} {{{\mathbf{M}}^{ - 1}}} & {0{\mathbf{I}}}\\ {0{\mathbf{I}}} & {{{\mathbf{M}}^{ - 1}}} \end{array}} \right),{\mathbf{M}}_d^{ - 1} = \left( {\begin{array}{cc} {{\mathbf{M}}_t^{ - 1}} & {0{\mathbf{{ I}}}}\\ {0{\mathbf{{ I}}}} & {{\mathbf{M}}_t^{ - 1}} \end{array}} \right),\\ &{Q_1} ={-} \frac{{ik}}{{2z}} + \frac{1}{{\rho _0^2}} + \frac{i}{{\rho _{\chi s}^2}},{Q_2} ={-} \frac{1}{{\rho _0^2}},{Q_3} ={-} \frac{{ik}}{{2z}} + \frac{2}{{\rho _0^2}} + \frac{i}{{\rho _{\chi s}^2}},{Q_4} ={-} \frac{i}{{\rho _{\chi s}^2}},{Q_5} = \frac{1}{{\rho _0^2}} + \frac{i}{{\rho _{\chi s}^2}}. \end{aligned}$$

The scintillation index of a GSM beam with a cross phase in a turbulent atmosphere is defined as [2]

(15)$${m^2}\left( {{\mathbf{p}},z} \right) = \frac{{\left\langle {{I^2}\left( {{\mathbf{p}},z} \right)} \right\rangle }}{{{{\left\langle {I\left( {{\mathbf{p}},z} \right)} \right\rangle }^2}}} - 1$$

Equation (15) is the main analytical expression used to investigate the scintillation index. Our results agree well with those in [23] for a twist factor ${\mu _0}$. Theoretical and experimental results show that a GSM beams with the cross phase can be used to reduce scintillation in a turbulent atmosphere.

3. Numerical results

In this section, we study numerically the scintillation index of a GSM with the cross phase in a turbulent atmosphere, set the wavelength $\lambda$=1550 nm, $\sigma$=20 mm, $\delta$=20 mm, and $C_n^2{\rm {\ =\ }}5 \times {10^{{\rm {\ }\hbox{-}{\rm \ }}16}}{{\rm {m}}^{ - 2/3}}$, unless otherwise specified.

Figure 1 shows the scintillation index of the GSM beams with a cross phase of different strength factors u in atmospheric turbulence at different propagation distances. As shown, the scintillation index is slightly influenced by the cross phase of the GSM beams at different turbulent strengths in the near field, but the opposite is true for the far field. The scintillation indices of GSM beams with and without a strength factor increased as the propagation distance increased. Moreover, the scintillation index of the GSM beam with the cross phase is lower than that of the GSM beam ($u$=0). As the strength factor increases, the scintillation index decreases. This result indicates that a GSM beam carrying the cross phase with a large strength factor can be used to mitigate the induced turbulent atmosphere. Furthermore, the cross phase maintains its role of resisting turbulence as the structure constant $C_n^2$ increases.

Fig. 1. Scintillation index of the GSM beams with the cross phase of the different strength factor $u$ in the atmospheric turbulence for the different propagation distances at (a) $C_n^2{\rm {\ =\ }}1 \times {10^{{\rm {\ }\hbox{-}{\rm \ }}16}}{{\rm {m}}^{ - 2/3}}$, (b) $C_n^2{\rm {\ =\ }}5 \times {10^{{\rm {\ }\hbox{-}{\rm \ }}16}}{{\rm {m}}^{ - 2/3}}$, (c) $C_n^2{\rm {\ =\ }}1 \times {10^{{\rm {\ }\hbox{-}{\rm \ }}15}}{{\rm {m}}^{ - 2/3}}$, and (d) $C_n^2{\rm {\ =\ }}5 \times {10^{{\rm {\ }\hbox{-}{\rm \ }}15}}{{\rm {m}}^{ - 2/3}}$.

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Figure 2 shows that the scintillation indices of GSM beams with cross phases of different strength factors are affected by the coherence length. As shown in Fig. 2, under the same conditions, the scintillation index is small when the GSM beam carries a cross phase with low coherence length propagation in a turbulent atmosphere. The scintillation indices with different strength factors changed significantly when the beams had a high coherence length (Figs. 2(a) and 2(d)). This phenomenon indicates that the scintillation index can be reduced by a beam with a small coherence length and large strength factor.

Fig. 2. Scintillation index of the GSM beams with the cross phase of the different strength factor u in the atmospheric turbulence for the different propagation distance, the coherence length is (a) $\delta$=11 mm, (b) $\delta$=14 mm, (c) $\delta$=17 mm, and (d) $\delta$=20 mm.

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Figure 3 illustrates the scintillation index of the GSM beams with a cross phase of different strength factors in atmospheric turbulence for different propagation distances. As shown in Fig. 3, the GSM beam with the cross phase has the same scintillation index for an arbitrary wavelength in the near field. A beam with long wavelength has a small scintillation index when the propagation distance is increased further. The scintillation index of the GSM beam carrying the cross phase was smaller than that of the GSM beam in the far field. Under the same conditions, the scintillation index is small for a GSM beam carrying a cross phase with a large strength factor and long wavelength when it propagates through a turbulent atmosphere.

Fig. 3. Scintillation index of the GSM beams with the cross phase of the different strength factor u in the atmospheric turbulence for the different propagation distance, the wavelength is (a) $\lambda$=980 nm, (b) $\lambda$=1064 nm, (c) $\lambda$=1310 nm, and (d) $\lambda$=1550 nm.

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Figure 4 shows that the scintillation indices of the GSM beams with a cross phase are affected by the coherence length, strength factor, and structure constant. Our numerical results show that the scintillation index of partially coherent beams with a long coherence length decreases rapidly when the strength factor increases, whereas it decreases slowly for low coherence lengths. As shown in Fig. 4, the scintillation index decreases when the coherence length increases, and increases in a strongly turbulent atmosphere. The values of the scintillation index tended to be the same at large strength factors; this phenomenon can be interpreted that the second term of the tensor element of Eq. (1) is further larger than the first term for a larger strength factor. The diagonal element of the tensor of the partially coherent complex curvature is then determined by the second term.

Fig. 4. Scintillation index of the GSM beams with the cross phase for the different coherence length and strength factor at $z$=2 $km$. (a) $C_n^2{\rm {\ =\ }}1 \times {10^{{\rm {\ }\hbox{-}{\rm \ }}16}}{{\rm {m}}^{ - 2/3}}$, (b) $C_n^2{\rm {\ =\ }}5 \times {10^{{\rm {\ }\hbox{-}{\rm \ }}16}}{{\rm {m}}^{ - 2/3}}$, (c) $C_n^2{\rm {\ =\ }}1 \times {10^{{\rm {\ }\hbox{-}{\rm \ }}15}}{{\rm {m}}^{ - 2/3}}$, and (d) $C_n^2{\rm {\ =\ }}5 \times {10^{{\rm {\ }\hbox{-}{\rm \ }}15}}{{\rm {m}}^{ - 2/3}}$.

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Figures 5 and 6 show that the scintillation indices of the GSM beams with a cross phase are affected by the values of the coherence length, beam width, and strength factor. The scintillation index increases as the beam width increases at the small strength factor $u = 0.2{\rm {k}}{{\rm {m}}^{ - 1}}$ (Fig. 5); the contrary results are obtained at the large strength factor (Fig. 6). These results indicate that the scintillation indices various trends based on the beam width were determined using the strength factor values. From Figs. 1–6, the scintillation index of GSM beams with a cross phase is smaller than that of GSM beams when they propagate in a turbulent atmosphere. Therefore, GSM beams carrying a quadratic phase structure can be used to mitigate atmospheric turbulence disturbance when these beams are applied in imaging or FSO communications.

Fig. 5. Scintillation index of the GSM beam with the cross phase against the coherence length and the beam width.

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Fig. 6. Scintillation index of the GSM beam with the cross phase against the strength factor and the beam width.

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4. Experiment

Figure 7 shows the experimental setup for generating the GSM beam with a cross phase and measuring the scintillation index of the generated beam through thermally induced turbulence. A continuous laser beam ($\lambda$=1550 nm) amplified using an erbium-doped fiber amplifier (EDFA) is collimated using a collimator (Col.). The collimated beam passed through a beam expansion system comprising thin lenses L1 (focal length $f$=150 mm) and L2 ($f$= 300 mm). The expanded Gaussian beam focused by L3 was sent to a rotating ground-glass disk (RGGD) to generate partially coherent beams with Gaussian statistics, which were then collimated by L4. The collimated beam is shaped by a Gaussian amplitude filter (GAF) to produce a GSM beam (beam width $\sigma$=20 mm is determined by the size of the GAF). A multiple-diffraction-order GSM beam with a cross phase is generated by the GSM beam through a spatial light modulator (SLM) loaded with a holograph. The desired GSM beam with a cross phase was filtered using a 4$f$ system with a circular aperture (CA) located at the confocal plane ($f$=100 mm). This beam is reflected by a reflecting mirror (RM) through a hot plate. A turbulent atmosphere was simulated using a graphene electric hot plate ($40 \times 30$ cm). The turbulence strength was determined based on the temperature of the hot plate. The GSM beam with a cross phase was focused by L7 ($f$=150 mm), and its intensity distribution was recorded using a charge-coupled device camera (CCD). The scintillation index was calculated using normalized irradiance variance.

Fig. 7. Experimental setup for measuring the scintillation index of GSM beam with the cross phase through a thermally induced turbulence. EDFA, erbium-doped fiber amplifier; Col., Collimator; L1-L7, lens; RGGD, rotating ground-glass disk; GAF, Gaussian amplitude filter; SLM, spatial light modulator; CA, circular aperture; RM, reflecting mirror; CCD, charge coupled device camera; PC, personal computer.

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In this experiment, we capture 3000 frames to calculate the scintillation index of the GSM beam with the cross phase propagating in a turbulent atmosphere. These frames have different beam cross-sections. The beam cross section of each frame can be represented by the matrix ${I_n}\left ( {{x_k},{y_j}} \right )$, where ${{x_k}}$ and ${{y_j}}$ are pixel spatial coordinates, and $n$ ($n=1, 2,\;{\ldots }\;3000$) denotes the cross section of each beam.

The scintillation index of the beam at the centroid in the receiver plane can be analyzed as follows [24].

(16)$$m_c^2 = \frac{{\sum\limits_N {{I_n}^2\left( {{{\overline x }_n},{{\overline y }_n}} \right)} }}{{N{{\overline I }^2}}} - 1,$$

where N denotes the total number of frames, and $\left ( {{{\overline x }_n},{{\overline y }_n}} \right )$ is the coordinate of the centroid for each frame.

(17)$${\overline x _n} = \frac{{\sum\limits_k {\sum\limits_j {{x_k}{I_n}\left( {{x_k},{y_j}} \right)} } }}{{\sum\limits_k {\sum\limits_j {{I_n}\left( {{x_k},{y_j}} \right)} } }},{\overline y _n} = \frac{{\sum\limits_k {\sum\limits_j {{y_i}{I_n}\left( {{x_k},{y_j}} \right)} } }}{{\sum\limits_k {\sum\limits_j {{I_n}\left( {{x_k},{y_j}} \right)} } }},$$

where $\overline I$ describes the average intensity at the centroid for the total frames,

(18)$$\overline I = \frac{{\sum\limits_N {{I_n}\left( {{{\overline x }_n},{{\overline y }_n}} \right)} }}{N}$$

Figure 8 shows the experimental results for the intensity distribution of the GSM beam with the cross phase at the focal plane. From Fig. 8, the shapes of the intensity distribution of the GSM beam with the cross phase at the focal plane are almost the same for the different coherence length, because the coherence length only effected on the beam spreading. The values of the strength factor effect on the intensity distribution are obvious. The shapes of them are elongated and rotated according to the cross phase’s changes. For the case of the cross phase at $u$ = 0, the shape of the intensity distribution of this beam will be degenerated into a GSM beam (Figs. 8(a1) and 8(a2)).

Fig. 8. Intensity distribution of the GSM beam with the cross phase at the focal plane with the different strength factor and coherence length for temperatures different from room temperature $\Delta$T = 75 K.

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Figure 9 shows the experimental results of the scintillation index of the GSM beam with the cross phase for two different temperature values. The scintillation indices were calculated from the statistical average of the intensity of 3000 frames at the centroid of the beam through a hot plate. From Figs. 4 and 9, the scintillation index of the GSM beam with the cross phase from the theoretical calculations and experimental results show the same trend. The turbulent atmosphere was approximately emulated by the hot plate because it is a random process. Figure 9(a) shows the scintillation index of the GSM beam with a cross phase for the different coherence lengths at $\delta$=20 mm and 25 mm. The GSM beam with the cross phase exhibited smaller scintillation index with a lower coherence length and larger strength factor. The same trend can be obtained for a larger temperature difference, that is, strong turbulence, and a larger scintillation index for larger for the same coherence length and strength factor (Fig. 9(b)).

Fig. 9. Scintillation index of the GSM beam with the cross phase from experimental results at two temperatures different from room temperature (a) $\Delta$T = 75 K and (b) $\Delta$T = 105 K.

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Fig. 10. Scintillation index of the GSM beam with the cross phase from experimental results for the different coherence length.

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Figure 10 shows the scintillation index of the GSM beam with the cross phase varying according to temperature difference $\Delta$T. From Fig. 10, the scintillation index increases when the temperature difference increases. Because the large temperature difference denotes the strong turbulent atmosphere. In the same condition, the scintillation index shows relative smaller values with smaller coherence length. This result is consistent with our theoretical results for the scintillation index of the GSM beam with the cross phase propagates through turbulent atmosphere (Figs. 2 and 4). From Figs. 9 and 10, the strength factor and coherence length can be chosen to reduce scintillation of the GSM beam with the cross phase induced by the atmospheric turbulence.

5. Conclusion

We theoretically and experimentally investigated the scintillation index of GSM beams with a cross phase in atmospheric turbulence. A cross phase was used to mitigate the scintillation of the turbulent disturbance. The numerical results show that the scintillation index can be reduced by appropriately choosing the strength factor of the cross phase, coherence length, beam width, and wavelength. The experimental results are in agreement with this theory. Our results provide a new method for mitigating scintillation caused by a turbulent atmosphere.

Funding

National Key Research and Development Program of China (2022YFA1404800, 2019YFA0705000); National Natural Science Foundation of China (NSFC) (12192254, 11974219, 11974218, 12174227, 11904211, 92250304); Natural Science Foundation of Shandong Province (ZR2019MA028).

Disclosures

The authors declare no conflicts of interest.

Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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Scintillation mitigation via the cross phase of the Gaussian Schell-model beam in a turbulent atmosphere

Abstract

1. Introduction

2. Theory

3. Numerical results

4. Experiment

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (10)

Equations (18)

Optics Express