Scintillation of Computational Ghost Imaging with a Finite Bucket Detector through Atmospheric Turbulence

Abstract

Based on the extended Huygens–Fresnel principle and infinitely long phase screen, the scintillation index and the aperture averaging effect of partially coherent beams in computational ghost imaging (CGI) with a finite bucket detector through atmospheric turbulence is investigated analytically and numerically. The signal–to–noise ratio (SNR) is used to evaluate the image quality of computational ghost imaging. It is found that a strong phase modulation effect due to increasing turbulence intensity, leads to a degradation in image quality, as well as an increase in the scintillation index. In addition, the scintillation–saturation phenomenon occurs for strong turbulence. On the other hand, reducing the propagation distance, and the degree of source coherence results in a decrease in the scintillation index and an improvement of image quality. However, deteriorating the degree of beam source coherence could weaken the aperture averaging effect. Thus, the optimal beam and bucket detector aperture size require a trade–off between the scintillation index, the aperture averaging effects, and the image quality in CGI.

1. Introduction

Ghost imaging (GI), also known as correlated imaging, is a new type of active illumination–based imaging technique that can restore an object’s image by intensity correlation measurements. GI was first achieved in 1995 using entangled photons generated in a spontaneous parametric down–conversion (SPDC source) by Pittman et al. [1]. Several years later, it was proved that GI could be realized by using a classical source [2,3], such as an incoherent thermal or pseudo–thermal source, which broadened the application of ghost imaging. In 2008, Shapiro et al. proposed a computational ghost imaging scheme [4], and its experimental demonstration was performed in 2009 [5], which reduced the complexity of the experimental setup and developed more advanced computational image reconstruction techniques.

In practice, imaging through turbulence is unavoidable in many optical applications. The properties of GI in atmospheric propagation have been studied by some experiments and theoretical analysis [6,7,8,9]. For example, GI through turbulent atmosphere is investigated analytically in Refs. [10,11]. In addition, ghost imaging through inhomogeneous turbulent atmosphere along an uplink path and a downlink path is studied in detail [12]. The visibility and quality of the ghost image in two different atmospheric turbulences are discussed in [13]. It has been found that image quality is influenced by the phase modulation effect due to turbulence strength and propagation distance [14]. In order to improve the imaging quality, Shi et al. proposed an adaptive optical ghost imaging (AOGI) system, which presents a better performance of GI in the atmosphere [15]. In addition, a great deal of work has been focused on sophisticated algorithms over these years to improve the imaging quality for a GI system, such as differential ghost imaging (DGI) [16], normalized ghost imaging (NGI) [17], compressive sensing computational ghost imaging (CSCGI) [18], three–dimensional CGI [19] and so on.

The random phase modulation caused by atmospheric turbulence results in scintillations which seriously restrict the reception of signals. In recent years, some studies have been carried out concerning the scintillation effect in atmospheric turbulence [20,21,22,23,24,25,26]. For example, the scintillation properties of a pseudo–partially coherent beam (PPCB) propagating through turbulent atmosphere by using numerical simulation is investigated [27]. The scintillation index of various beam models in the atmosphere is also studied, such as elliptical Gaussian beam, flat–topped beam and so on [28,29,30,31,32,33]. Until now, the scintillation effect of CGI has not been investigated concerning turbulent atmosphere. In addition, in all these studies about CGI cited above, the single pixel bucket detector is assumed to be large enough to receive all the light field information. In practice, a finite size bucket detector is often encountered, which results in the scintillation effect. Thus, it is necessary to study the effect of the collection range of a bucket detector on the scintillation index of CGI, especially for long–distance CGI through turbulent atmosphere. Furthermore, as the receiver aperture size increases, intensity fluctuation at the receiver decreases. This phenomenon is known as the aperture averaging effect, which has a mitigating impact on the scintillation index. Thus, some interesting questions arise. What is the relation between the scintillations and the imaging quality of CGI with a finite bucket detector? How does the partial coherence affect the aperture averaging factor in the CGI system through atmospheric turbulence? Based on the extended Huygens integral, the analytical imaging formula has been derived. In addition, we investigate the scintillation effect, the aperture averaging effect, and the imaging quality of CGI with partially coherent beams (PCBs) through atmospheric turbulence, concerning a finite bucket detector using a simulation method.

2. Theory

The CGI scheme [4,34] is shown in Figure 1. A partially coherent light source E(x), which is produced by a programmable spatial light modulator (SLM), propagates through atmospheric turbulence, illuminates the object, and then is collected by the Single Pixel Bucket Detector in the test arm. On the other hand, the reference arm in ghost imaging (GI) is replaced by the precomputed light field in CGI, which can be calculated using the transmitter–plane beam profile and paraxial free–space beam propagation theory. Therefore, the object image can be obtained by using the correlator to compute the spatial correlation functions of intensity fluctuations from the collected light field and precalculated light field, respectively. In addition, the propagation distances from the source to the object and from the object to the bucket detector are represented as z₀ and z₁, respectively. The precalculated light field is assumed to propagate in free space over a distance z₂.

Based on the ghost imaging theory [35,36], the second–order correlation function containing the imaging information of the object is described by

$$g_2(u_1,u_2)=〈I(u_1)I(u_2)〉-〈I(u_1)〉〈I(u_2)〉,$$

(1)

where $I(u_1)$ and $I(u_2)$ represent the intensities collected by the bucket detector plane and are precalculated by the computer, respectively. $\langle \rangle$ denotes the ensemble average on the source field. According to the extended Huygens–Fresnel integral [37], the response function $h_1(u_1,x_1)$ of the turbulent test arm at the bucket detector can be expressed as

$$\begin{array}{ll}{h}_1(u_1,x_1)=& {(-\frac{1}{{\lambda }^2{z}_0{z}_1})}^{1/2}\\ & \times {\displaystyle {\int }_{-\infty }^{\infty }\exp \left[-\frac{i\pi }{\lambda z_0}{(x_1-u_0)}^2-\frac{i\pi }{\lambda z_1}{(u_0-u_1)}^2\right]}\\ & \times \exp \left[{\psi }_0(x_1,u_0)\right]\exp \left[{\psi }_1(u_0,u_1)\right]H\left(u_0\right)d{u}_0,\end{array}$$

(2)

where ψ₀ and ψ₁ represent the random part of the complex phase perturbation caused by atmospheric turbulence from the source to the object and from the object to the bucket detector, respectively. x₁ and x₂ represent two points in the source field, respectively. A simple double slit is employed as an object $H(u_0)$ in this simulation which satisfies

$$H(u_0)=\left\{\begin{array}{l}1,-\frac{d}{2}-\frac{a}{2}<u_0<-\frac{d}{2}+\frac{a}{2}\\ 1,\frac{d}{2}-\frac{a}{2}<u_0<\frac{d}{2}+\frac{a}{2}\\ 0\end{array}\right.,$$

(3)

where a = 2 cm, b = 8 cm, d = 4 cm denotes the width, the height, and the separation of the double slit, respectively.

Similarly, the precomputed response function of the turbulence–free path is given by

$$h_2(u_2,x_2)={(-\frac{i}{\lambda z_2})}^{1/2}{\displaystyle {\int }_{-\infty }^{\infty }\exp \left[-\frac{i\pi }{\lambda z_0}{(x_2-u_2)}^2\right]}.$$

(4)

It is assumed that the two–point correlation function of a Gaussian Schell–model beam (GSM) with partial spatial coherence can be written as [27,38]

$$〈E(x_1)E^{*}(x_2)〉=\exp [-\frac{x_1^2+x_2^2}{4{\sigma }_l^2}]\exp [-\frac{{(x_1-x_2)}^2}{2{\sigma }_{\mu }^2}],$$

(5)

where ${\sigma }_l$ is the transverse size of the light source, while ${\sigma }_{\mu }$ represents the lateral correlation radius which determines the spectral degree of source coherence. When ${\sigma }_{\mu }\to \infty$, it is corresponding to a fully coherent beam (Gaussian beam).

Then, substituting Equations (2) and (4) into Equation (1), the fourth–order correlation function is obtained as

$$\begin{array}{ll}{G}^{(2)}(u_1,u_2)=& \langle I(u_1)I(u_2)\rangle \\ & =\frac{1}{{\lambda }^3{z}_0{z}_1{z}_2}{\displaystyle \int {\displaystyle \int {\displaystyle \int {\displaystyle \int {\displaystyle \int {\displaystyle \int d{x}_{1}d{x^{\prime }}_{1}d{x}_{2}d{x^{\prime }}_{2}d{u}_{0}d{u^{\prime }}_0}}}}}}\\ & \times H(u_0)H^{*}({u^{\prime }}_0)\langle E(x_1)E^{*}({x^{\prime }}_2)E(x_2)E^{*}({x^{\prime }}_1)\rangle \\ & \times \langle \exp \left[{\psi }_0(x_1,u_0)+{\psi }_0^{*}({x^{\prime }}_1,{u^{\prime }}_0)\right]\rangle \\ & \times \langle \exp \left[{\psi }_1(u_1,u_0)+{\psi }_1^{*}({u^{\prime }}_1,{u^{\prime }}_0)\right]\rangle \\ & \times \exp \{\frac{i\pi }{\lambda z_0}\left[{\left(u_0-x_1\right)}^2-{\left({u^{\prime }}_0-{x^{\prime }}_1\right)}^2\right]\}\\ & \times \exp \{\frac{i\pi }{\lambda z_1}\left[{\left(u_1-u_0\right)}^2-{\left({u^{\prime }}_1-{u^{\prime }}_0\right)}^2\right]\}\\ & \times \exp \{\frac{i\pi }{\lambda z_2}\left[{\left(u_2-x_2\right)}^2-{\left({u^{\prime }}_2-{x^{\prime }}_2\right)}^2\right]\}.\end{array}$$

(6)

Additionally, the last term of (1) can be described as

$$\begin{array}{ll}\langle I(u_1)\rangle \langle I(u_1)\rangle =& \frac{1}{{\lambda }^3{z}_0{z}_1{z}_2}{\displaystyle \int {\displaystyle \int {\displaystyle \int {\displaystyle \int {\displaystyle \int {\displaystyle \int d{x}_{1}d{x^{\prime }}_{1}d{x}_{2}d{x^{\prime }}_{2}d{u}_{0}d{u^{\prime }}_0}}}}}}\\ & \times H(u_0)H^{*}({u^{\prime }}_0)\langle E(x_1)E^{*}({x^{\prime }}_1)\rangle \langle E(x_2)E^{*}({x^{\prime }}_2)\rangle \\ & \times \langle \exp \left[{\psi }_0(x_1,u_0)+{\psi }_0^{*}({x^{\prime }}_1,{u^{\prime }}_0)\right]\rangle \\ & \times \langle \exp \left[{\psi }_1(u_1,u_0)+{\psi }_1^{*}({u^{\prime }}_1,{u^{\prime }}_0)\right]\rangle \\ & \times \exp \{\frac{i\pi }{\lambda z_0}\left[{\left(u_0-x_1\right)}^2-{\left({u^{\prime }}_0-{x^{\prime }}_1\right)}^2\right]\}\\ & \times \exp \{\frac{i\pi }{\lambda z_1}\left[{\left(u_1-u_0\right)}^2-{\left({u^{\prime }}_1-{u^{\prime }}_0\right)}^2\right]\}\\ & \times \exp \{\frac{i\pi }{\lambda z_2}\left[{\left(u_2-x_2\right)}^2-{\left({u^{\prime }}_2-{x^{\prime }}_2\right)}^2\right]\}.\end{array}$$

(7)

Supposing the light source is a Gaussian Schell–model which obeys Gaussian variate of zero mean and the fourth–order correlation of a partially coherent source is also expressed as

$$\begin{array}{l}〈E(x_1)E^{\ast }({x^{\prime }}_1)E(x_2)E^{\ast }({x^{\prime }}_2)〉\\ =〈E(x_1)E^{\ast }({x^{\prime }}_1)〉〈E(x_2)E^{\ast }({x^{\prime }}_2)〉+〈E(x_1)E^{\ast }({x^{\prime }}_2)〉〈E(x_2)E^{\ast }({x^{\prime }}_1)〉.\end{array}$$

(8)

On substituting Equations (6)–(8) into Equation (1), we obtain the expression of second–order correlation functions

$$\begin{array}{ll}{g}_2(u_1,u_2)=& \langle I(u_1)I(u_2)\rangle -\langle I(u_1)\rangle \langle I(u_2)\rangle \\ & =\frac{1}{{\lambda }^3{z}_0{z}_1{z}_2}{\displaystyle \int {\displaystyle \int {\displaystyle \int {\displaystyle \int {\displaystyle \int {\displaystyle \int d{x}_{1}d{x^{\prime }}_{1}d{x}_{2}d{x^{\prime }}_{2}d{u}_{0}d{u^{\prime }}_0}}}}}}\\ & \times H(u_0)H^{*}({u^{\prime }}_0)\langle E(x_1)E^{*}({x^{\prime }}_2)\rangle \langle E(x_2)E^{*}({x^{\prime }}_1)\rangle \\ & \times \langle \exp \left[{\psi }_0(x_1,u_0)+{\psi }_0^{*}({x^{\prime }}_1,{u^{\prime }}_0)\right]\rangle \\ & \times \langle \exp \left[{\psi }_1(u_1,u_0)+{\psi }_1^{*}({u^{\prime }}_1,{u^{\prime }}_0)\right]\rangle \\ & \times \exp \{\frac{i\pi }{\lambda z_0}\left[{\left(u_0-x_1\right)}^2-{\left({u^{\prime }}_0-{x^{\prime }}_1\right)}^2\right]\}\\ & \times \exp \{\frac{i\pi }{\lambda z_1}\left[{\left(u_1-u_0\right)}^2-{\left({u^{\prime }}_1-{u^{\prime }}_0\right)}^2\right]\}\\ & \times \exp \{\frac{i\pi }{\lambda z_2}\left[{\left(u_2-x_2\right)}^2-{\left({u^{\prime }}_2-{x^{\prime }}_2\right)}^2\right]\}.\end{array}$$

(9)

The statistical averages due to the atmospheric turbulence can be written approximately as [20,39]

$$\begin{array}{l}〈\exp \left[{\psi }_i(x,u_0)+{\psi }_i^{*}(x^{\prime },{u^{\prime }}_0)\right]〉\\ =\exp [\frac{{(x-x^{\prime })}^2+(x-x^{\prime })(u_0-{u^{\prime }}_0)+{(u_0-{u^{\prime }}_0)}^2}{-{\rho }_i^2}],\end{array}$$

(10)

where $\langle \rangle$ denotes the statistics averages caused by the turbulent atmosphere, ${\rho }_i={(0.55C_n^2{k}^2{z}_i)}^{-3/5}$ denotes the coherence length of a spherical wave propagating through a turbulent medium, and $C_n^2$ represents the refractive index structure parameter describing the strength of atmospheric turbulence in the path z_i.

By using the integral formula

$${\displaystyle {\int }_{-\infty }^{\infty }\exp (-p^2{x}^2\pm qx)}dx=\exp (\frac{q^2}{4p^2})\frac{\sqrt{\pi }}{p},$$

(11)

According to the result of [11], Equation (9) turns out to be

$$\begin{array}{l}g(u_1,u_2)=\frac{\pi }{{\lambda }^2{z}_0{z}_2\sqrt{A_1{A}_2{B}_1{B}_2}}{\displaystyle \int d{u}_0{\left|H\left(u_0\right)\right|}^2}\\ \times \exp \left\{-\frac{{\pi }^2{u}_2^2}{A_1{\lambda }^2{z}_2^2}+\frac{\left(\frac{2i\pi u_2}{\lambda z_2}-\frac{2i\pi u_2}{A_1{\rho }_2^2{z}_1}\right)}{4A_2}+\frac{(\frac{i\pi u_2}{\lambda z_2{A}_2{\sigma }_u^2}-\frac{2i\pi u_0}{\lambda z_0}-\frac{i\pi u_2}{A_1{A}_2{\rho }_2^2\lambda z_2{\sigma }_u^2})}{4B_1}\right\}\\ \times \exp \{\frac{[\frac{2i\pi u_0}{\lambda z_0}-\frac{i\pi u_2}{A_1{\sigma }_{\mu }^2\lambda z_2}+\frac{\frac{i\pi u_2}{\lambda z_2}-\frac{i\pi u_2}{A_1{\rho }_2^2\lambda z_2}}{A_1{A}_2{\sigma }_{\mu }^2{\rho }_2^2}}{4B_2}\\ +\frac{{\left(\frac{1}{2B_1{\rho }_0^2}+\frac{1}{8A_1{A}_2{B}_1{\rho }_2^2{\sigma }_u^2}\right)\left(\frac{i\pi u_2}{\lambda z_2{A}_2{\sigma }_u^2}-\frac{2i\pi u_0}{\lambda z_0}-\frac{i\pi u_2}{A_1{A}_2{\rho }_2^2\lambda z_2{\sigma }_u^2}\right)]}^2}{4B_2}\},\end{array}$$

(12)

where

$$A_1=\frac{1}{4{\sigma }_l^2}+\frac{1}{2{\sigma }_{\mu }^2}+\frac{1}{{\rho }_0^2}-\frac{i\pi }{\lambda z_0},$$

(13)

$$A_2=\frac{1}{4{\sigma }_l^2}+\frac{1}{2{\sigma }_{\mu }^2}+\frac{1}{{\rho }_0^2}+\frac{i\pi }{\lambda z_2}-\frac{1}{A_1{\rho }_2^4},$$

(14)

$$B_1=\frac{1}{4{\sigma }_l^2}+\frac{1}{2{\sigma }_{\mu }^2}+\frac{1}{{\rho }_0^2}-\frac{i\pi }{\lambda z_0}-\frac{1}{4A_2{\sigma }_u^4},$$

(15)

$$B_2=\frac{1}{4{\sigma }_l^2}+\frac{1}{2{\sigma }_{\mu }^2}+\frac{1}{{\rho }_0^2}+\frac{i\pi }{\lambda z_0}-\frac{1}{4A_1{\sigma }_{\mu }^4}-\frac{1}{4A_1{}^2{A}_2{\rho }_2^4{\sigma }_{\mu }^4}-\frac{{\left(\frac{2}{{\rho }_0^2}+\frac{1}{2A_1{A}_2{\rho }_2^2{\sigma }_{\mu }^4}\right)}^2}{4B_1}.$$

(16)

Signal–to–noise ratio (SNR) is defined to evaluate the CGI imaging quality in the turbulent atmosphere as follows

$$\mathrm{SNR}=\frac{\mathrm{S}\left(\mathrm{g}\right(u_1,u_2\left)\right)}{\mathrm{N}\left(\mathrm{g}\right(u_1,u_2\left)\right)},$$

(17)

where $S\left(g(u_1,u_2)\right)$ represents the signal of CGI, which can be expressed as the mean value of the second–order correlation function of the object’s transmitting part

$$S\left(g(u_1,u_2)\right)=〈g(u_1,u_2)\times H\left(u_0\right)〉.$$

(18)

The noise of CGI can be defined as the standard deviation of the second–order correlation function of the object background part

$$N\left(g(u_1,u_2)\right)=\mathrm{std}\left(g{(u_1,u_2)}_{back}\right).$$

(19)

Thus, the SNR of CGI is expressed as

$$\mathrm{SNR}=\frac{\mathrm{S}\left(\mathrm{g}\right(u_1,u_2\left)\right)}{\mathrm{N}\left(\mathrm{g}\right(u_1,u_2\left)\right)}=\frac{〈g(u_1,u_2)\times H(u_0)〉}{\mathrm{std}(g{(u_1,u_2)}_{back})}.$$

(20)

In terms of the test arm, the light field propagates through atmospheric turbulence, illuminates the object, and is collected by the Single Pixel Bucket Detector. Thus, the total intensity at the bucket detector plane can be written as

$$\begin{array}{l}〈I_u{}_1〉=\frac{1}{2\lambda z_0\sqrt{C_1{C}_2}}{\displaystyle \int d{u}_0{\left|H(u_0)\right|}^2}\\ \times \exp \left\{\frac{{\left[\frac{2\mathrm{i}\pi u_0}{\lambda z_0}-\left(\frac{1}{2C_1{\sigma }_{\mu }^2}+\frac{2}{{\rho }_0^2}\right)\times \frac{2i\pi u_0}{\lambda z_0}\right]}^2}{4C_2}+{\left[\frac{i\pi u_0}{\sqrt{C_1}\lambda z_0}\right]}^2\right\}\end{array}$$

(21)

where

$$C_1=\frac{1}{4{\sigma }_l^2}+\frac{1}{2{\sigma }_{\mu }^2}+\frac{1}{{\rho }_0^2}+\frac{i\pi }{\lambda z_0},$$

(22)

$$C_2=\frac{1}{4{\sigma }_l^2}+\frac{1}{2{\sigma }_{\mu }^2}+\frac{1}{{\rho }_0^2}-\frac{i\pi }{\lambda z_0}-{\left(\frac{1}{2\sqrt{C_1}{\sigma }_{\mu }^2}+\frac{1}{\sqrt{C_1}{\rho }_0^2}\right)}^2.$$

(23)

However, the atmospheric turbulence causes the phase modulation effect on the test arm of CGI, resulting in random fluctuations in the light intensity at the bucket receiver, which is called the scintillation effect. In addition, considering a finite aperture (i.e., the Single Pixel Bucket Detector) with a diameter of D, the aperture averaging scintillation index is defined as

$${\sigma }_I^2(D)=\frac{〈I_{u1}^2(D)〉-{〈I_{u1}(D)〉}^2}{{〈I_{u1}(D)〉}^2}.$$

(24)

where $I_{u1}(D)$ is the total intensity within a receiving aperture. Equation (24) measures the intensity variance of fluctuation in the light field at the plane of the bucket receiver.

The aperture averaging factor is defined as the ratio of the aperture averaging scintillation index ${\sigma }_I^2(D)$ to the point scintillation index ${\sigma }_I^2(0)$ at receiving the aperture center point

$$A(D)=\frac{{\sigma }_I^2(D)}{{\sigma }_I^2(0)}.$$

(25)

When A(D) decreases, the aperture averaging effect of the scintillation index can be strengthened.

3. Numerical Simulation Model

Concerning the partially coherent light source in CGI, Gaussian Schell–model beam is an analytically tractable model. However, a practical PCB may not follow the GSM theory and is better examined in detail through a numerical simulation approach, such as a wave optics simulation. Gu et al. proposed that the random Gaussian rough surface can simulate the optical–rough surface [40]. In this simulation, the Monte Carlo method is utilized to generate a dynamic Gaussian–correlated random surface to simulate the partially coherent light source model developed by SLM [40,41]. Thus, the initial partially coherent light field can be expressed as

$$u(x_0,y_0,0)=u_0(x_0,y_0,0)\exp \left[\frac{-i4\pi h^{\prime }(x_0,y_0)}{\lambda }\right],$$

(26)

where $u_0(x_0,y_0,0)$ is the initial optical field of the corresponding fully coherent beam, $\exp \left[-i4\pi h^{\prime }(x_0,y_0)/\lambda \right]$ is the random phase function, and the two–dimensional rough surface $h^{\prime }(x_0,y_0)$ can be described as follows

$$\begin{array}{ll}{h}^{\prime }(x_0,y_0)& =\frac{1}{G_x{G}_y}{\displaystyle \sum _{m_k=-M_0/2+1}^{M_0/2}{\displaystyle \sum _{n_k=-N_0/2+1}^{N_0/2}F(k_{m_k},k_{n_k})}}\\ & \exp \left[i(k_{m_k}{x}_0,k_{n_k}{y}_0)\right],\end{array}$$

(27)

where G_x and G_y are the length of a generated two–dimensional random rough surface in x and y directions, respectively, and can be expressed as $G_x=M_0\cdot \Delta x_0,$ $G_y=N_0\cdot \Delta y_0$. M₀ and N₀ are discrete points in each direction with the same spacing $\Delta x_0$ and $\Delta y_0$ between the points, respectively. (x₀, y₀) is the sampling point of the two–dimensional rough surface and satisfies $x_0=m_0\Delta x_0, y_0=n_0\Delta y_0$, $m_0=-M_0/2+1,\dots ,M_0/2$, $n_0=-N_0/2+1,\dots ,N_0/2.$ $F(k_{m_k},k_{n_k})$ is expressed as

$$\begin{array}{l}F(k_{m_k},k_{n_k})=2\pi {\left[G_x{G}_{y}S(k_{m_k},k_{n_k})\right]}^{1/2}\\ \times \left\{\begin{array}{ll}\frac{\left[N^{(j)}(0,1)+i{N}^{(j)}(0,1)\right]}{\sqrt{2}},& m_k\ne 0,M_0/2 \mathrm{and} n_k\ne 0,N_0/2,j=0,1,\dots ,j\\ N^{(j)}(0,1), & m_k=0,M_0/2 \mathrm{and} n_k=0,N_0/2,j=0,1,\dots ,j\end{array}\right.,\end{array}$$

(28)

where j is jth realization of random face, and N^(j) (0,1) denotes a sequence of normally distributed numbers in [0, 1] with zero mean and unity standard deviation. $k_{m_k}=2\pi m_k/G_x,k_{n_k}=2\pi n_k/G_y$. $S(k_x,k_y)$ is the power spectral density of Gaussian rough surface, i.e.,

$$S(k_x,k_y)={\delta }_h{}^2\frac{l_x{l}_y}{4\pi }\exp \left(-\frac{k_x^2{l}_x^2+k_y^2{l}_y^2}{4}\right),$$

(29)

where δ_h is the root–mean–square roughness, and l_x, l_y is the coherence length satisfying l_x = l_y. Their variation significantly impacts the fluctuation in height and frequency of rough surfaces, which affects the coherence of the light source [37]. When l_x, l_y → ∞ and δ_h → 0, it is corresponding to a fully coherent beam (Gaussian beam).

4. Results

4.1. Influence of the Atmospheric Coherence Length r₀

Based on the infinitely long phase screen [42,43], we design a computer code of CGI through atmospheric turbulence to investigate the scintillation effect, the aperture averaging effect, and the imaging quality. In this model, the light source (wavelength λ = 632 nm and size ρ_s = 0.01 m) generated by a CW laser propagates through paths that are all horizontal links with the distance z₀ = z₁ = z₂ = 2000 m. The atmospheric coherence length r₀ is known as an ensemble quantity that describes the average strength of the atmospheric turbulence over the propagation path z_i and can be written as: $r_0={\left(0.423C_n^2{\left(2\pi /\lambda \right)}^2{z}_i\right)}^{-\frac{3}{5}}$.

Figure 2 shows the aperture averaging scintillation index ${\sigma }_I^2(D)$ and the aperture averaging factor A(D) versus the aperture diameter D for different values of the atmospheric coherence length r₀, respectively, where each point is the averaging result of 6000 simulation realizations. From Figure 2a, it is clear that ${\sigma }_I^2(D)$ decreases as D increases due to the aperture averaging effect. Additionally, at the same aperture diameter size, the smaller the atmospheric coherence length r₀ (i.e., the stronger the atmospheric turbulence) results in the larger the scintillation index. As to the atmospheric coherence length r₀ = 0.06 m, the point scintillation index ${\sigma }_I^2(D=0)$ is 0.93. However, when r₀ = 0.02 m, the point scintillation index ${\sigma }_I^2(D=0)$ increases to 1.249, because the strong phase modulation effect due to atmospheric turbulence, causes a decrease in the receiver intensity correlation, and successfully increasing the receiver aperture averages all the intensity fluctuations. In addition, it is noted from Figure 2b that A(D) decreases when D increases or r₀ decreases, which implies that increasing aperture diameter size and turbulence level results in more aperture averaging.

The images of the double slit in different strengths of turbulence are reconstructed in Figure 3 after statistics from over 12,000 samples. It is clear from Figure 3a–d that the strong turbulence fluctuation has a significant influence on the imaging quality. On the one hand, CGI in free space has a significantly concentrated intensity distribution in its double–slit transmission region, and its reconstructed image can be distinguished easily. However, the image blurred and changed beyond recognition with strong turbulence (r₀ = 0.02 m) in the test arm. On the other hand, the corresponding values of SNR are equal to 8.46, 11.58, 14.87, and 16.89 for atmospheric coherence length r₀ = 0.02 m, 0.03 m, 0.06 m, ∞ (i.e., in free space), respectively. Namely, strong turbulence (small r₀) causes an evident scintillation effect and further degrades the image quality, nevertheless, this can be weakened by increasing the aperture diameter size.

4.2. Influence of the Propagation Distance z₀

Figure 4a–c shows the aperture scintillation index ${\sigma }_I^2(D)$ of partially coherent beams propagating through the test arm of CGI versus the distance z₀ for different values of the aperture diameter in all turbulence conditions, such as weak ($C_n^2=5\times {10}^{-16}{\mathrm{m}}^{-2/3}$), moderate ($C_n^2=2.5\times {10}^{-15}{\mathrm{m}}^{-2/3}$), and strong ($C_n^2=5\times {10}^{-13}{\mathrm{m}}^{-2/3}$) turbulences. It is seen from Figure 4a that the scintillation index increases slowly as z₀ increases in weak turbulence and increases rapidly as z₀ increases in moderate turbulence (Figure 4b) due to a strong phase modulation effect [12], which is caused by the accumulation of an atmospheric turbulence effect. Under the condition of strong turbulence (Figure 4c), the scintillation index fluctuates around 1.25 and no longer increases as z₀ increases, owing to the scintillation–saturation phenomenon. Among them, the point scintillation index at the center of the receiver detector is the largest. Expectedly, increasing the receiver aperture diameter from 0 m to 0.0625 m, results in strong aperture averaging effects, especially in a long propagation distance, because increasing distance z₀ causes a loss in intensity correlation at receiver aperture, and increasing the receiver aperture mitigates the scintillation effect. Furthermore, at the same propagation distance, as turbulence strength increases, strong phase modulation has a great influence on the beam, leading to an increasing scintillation index of ghost imaging until the scintillation–saturation effect occurs for strong turbulence. The influence of the distance z₀ on the SNR of CGI has also been investigated in Figure 4d. It can be seen clearly that both are in free space and in the atmosphere; therefore, increasing z₀ can decrease SNR, leading to the degradation of the imaging quality, which indicates that CGI works well within a distance less than 2800 m. Thus, long distance results in a high–scintillation index which can be suppressed with a large bucket detector, and a low SNR, corresponds to bad imaging quality, respectively.

4.3. Influence of Coherence Parameters

Figure 5a,c shows the changes of scintillation index ${\sigma }_I^2(D)$ versus different coherence parameters (i.e., coherence length (l_x = l_y) and root–mean–square roughness δ_h) for different values of the aperture diameter in the atmosphere, respectively. It can be noticed from Figure 5a that in atmospheric turbulence, the point scintillation index (D = 0 m) increases rapidly as l_x, l_y increase. When the aperture diameter increases, ${\sigma }_I^2(D)$ increases slowly as l_x, l_y increases. Similarly, for small apertures (D = 0 m, 0.0078 m), ${\sigma }_I^2(D)$ decreases rapidly as δ_h increases (see Figure 5c), and when the aperture diameter increases, ${\sigma }_I^2(D)$ decreases slowly as δ_h increases. Furthermore, when D increases to 0.0625 m, the change of ${\sigma }_I^2(D)$ decreases to about 0.2 and is independent of l_x, l_y, and δ_h due to the aperture averaging effect. The evolution of SNR for CGI with different coherence parameters (i.e., coherence length (l_x = l_y) and root–mean–square roughness δ_h) of PCBs both in free space and in the atmosphere is studied in Figure 5b,d, respectively. It is found that in free space, coherence parameters (coherence length (l_x = l_y) and root–mean–square roughness δ_h) have little influence on the SNR of CGI. However, in the atmosphere, SNR decreases as l_x, l_y increases (see Figure 5b), and SNR increases as δ_h increases (see Figure 5d). Thus, it implies that when PCBs are utilized in CGI, better coherence can cause a stronger scintillation effect and worse imaging quality for a small size bucket detector.

The aperture averaging scintillation index ${\sigma }_I^2(D)$, as functions of the aperture diameter D and different coherence parameters (i.e., coherence length (l_x = l_y) and root–mean–square roughness δ_h) of PCBs in CGI, are shown in Figure 6a,c, respectively. It can be seen that ${\sigma }_I^2(D)$ decreases rapidly as D becomes large. Moreover, the scintillation index increases significantly when l_x, l_y increases or δ_h decreases, respectively. Namely, CGI is greatly affected by the turbulent atmosphere for PCB light source with better coherence.

The curves of the aperture averaging factor A(D) versus aperture diameter D for different coherence parameters (i.e., coherence length (l_x = l_y) and root–mean–square roughness δ_h) are shown in Figure 6b,d, respectively. It is known from Figure 6b,d that the aperture averaging factor A(D) decreases as D increases. In addition, when l_x, l_y decreases or δ_h increases, A(D) increases at the same aperture diameter (see Figure 6b,d). This means that the scintillation index can be significantly mitigated by deteriorating the coherence of the light source, but meanwhile, the aperture averaging effect is obviously weakened. The turbulence has a greater impact on the scintillation effect of the PCB source with better coherence, resulting in stronger intensity fluctuation at receiver aperture. Thus, for CGI with a PCB source, those with better coherence show more aperture averaging than those with worse coherence.

From Figure 5 to Figure 6, it can be seen that reducing the light source coherence and increasing the bucket detector size can decrease the scintillation index, which contributes to improving the image quality. However, reducing the coherence of the light source can also weaken the aperture averaging effect, which indicates that spatial coherence degradation of the CGI light source brings no significant benefits to the aperture averaging effect. Furthermore, a large aperture is favorable to the aperture averaging effect.

5. Conclusions

We investigated the effects of CGI with a finite bucket detector through atmospheric turbulence in detail. Based on the extended Huygens–Fresnel integral, we successfully defined the SNR, the scintillation index, and the aperture averaging factor of CGI, and derived the analytical formula of computational ghost imaging with PCB. By using the infinitely long–turbulence phase screen method to simulate the CGI in the atmosphere, it is found that the strong phase modulation effect is due to the increasing turbulence intensity, which leads to a degradation in image quality, as well as an increase in the scintillation index. In addition, the scintillation–saturation phenomenon occurs for strong turbulence. On the other hand, increasing the propagation distance, increasing a bucket detector size, and reducing the degree of source coherence, can cause a decrease of the scintillation index and an improvement of image quality, which indicates that utilizing the PCB source and a large bucket detector has a great advantage in CGI in the atmosphere. Furthermore, a large aperture strengthens the aperture averaging effect; however, this may be weakened as the coherence degree of beam source deteriorates. Therefore, the optimal beam and the Single Pixel Bucket Detector aperture size require a trade–off between the scintillation index, aperture averaging effects and the image quality in CGI. These results may be valuable for computational ghost imaging applications in remote imaging and astronomical observation.