Ghost imaging through inhomogeneous turbulent atmosphere along an uplink path and a downlink path

Wei Tan; Xianwei Huang; Suqin Nan; Yanfeng Bai; Xiquan Fu

doi:10.1364/OSAC.387075

1. Introduction

Ghost imaging (GI), which is also called correlated imaging or “two-photon” imaging, is a novel nonlocal imaging method that exploits the correlation between the test beam and the reference beam to reconstruct the information of an unknown object. Since it was first achieved in an experiment based on a second-order spatial correlation of entangled photon pairs generated by spontaneous parametric down-conversion (SPDC) in 1995 [1,2], GI has shown potential applications in remote sensing [3], lidar [4,5], military detection [6] and medical imaging [7,8] because of its excellent characteristic of strong anti-interference. In the last few years, many works have been reported to improve the quality of GI with thermal light sources [9–18]. For example, high-order GI [9,10], differential GI [11,12], pseudo-inverse GI [13,14], fluorescence GI [15], compressive sensing GI [16,17], and fuzzy c-means clustering GI [18]. In addition, machine learning algorithms [19–23] were also introduced into GI to quickly and accurately acquire ghost-image under low sampling condition [24–26].

The disturbance of channels, including scattering [27], fading [28] and turbulence [29], is unavoidable in real environment. In particular, the factors such as light fluctuation, scintillation and beam wander because of the existence of turbulent atmosphere can result in the quality degradation of GI [29]. In 2009, Cheng first introduced turbulent atmosphere into GI and theoretically analyzed the influence of turbulent atmosphere on the imaging quality [30]. Since then, GI through turbulent atmosphere has increasingly attracted attention and many relevant works have been implemented in theoretically and experimentally [31–44]. Almost all of the above works of GI through turbulent atmosphere were focused on the horizontal propagation path where the turbulent atmosphere is thought to be homogeneous. In fact, the beam may propagate along a slant path (including uplink and downlink paths) in turbulent atmosphere for some real scenes, such as satellite communication and remote sensing. During of this process, the turbulent atmosphere should be inhomogeneous. Note that the group of Zhang theoretically gave the formulas of lensless [36] and lens [37] GI through non-Kolmogorov turbulent atmosphere along a slant path. Actually, the imaging formulas only involved uplink path and the numerical results were also missing on detail. In this paper, we present a numerical demonstration of GI through turbulent atmosphere along an uplink path and a downlink path. The numerical imaging formula of GI is derived to investigate the influence of the zenith angle on the imaging quality based on optical coherent theory and multi-phase screens method. Specifically, we explain the difference of the imaging quality in GI under uplink and downlink paths by the phase modulation effect. Our results may provide new ideas for the application of GI on the satellite communication and the remote sensing.

The rest of this paper is organized as follows. In Section 2, we present a model of thermal light lensless GI through turbulent atmosphere along an uplink path and a downlink path and the relative theory. In Section 3, we present the numerical results of the GI setup and the relative discussion. Finally, we summarize the findings in Section 4.

2. Model and theory

The setup of a lensless GI system with turbulent atmosphere along an uplink path and a downlink path is shown in Fig. 1. Here the pseduo-thermal light is split into two beams by the non-polarizing beam splitter (BS), and then the two beams propagate through a reference arm and a test arm, respectively. In the reference arm, there is a high resolution reference detector D_r. The test arm contains an unknown object t(x,y) and a bucket detector D_t without spatial resolution which is used to collect the photons transmitted from the object. The propagation distance from the source to the reference detector, from the source to the object and from the object to the bucket detector are set as Z₀, Z₁ and Z₂, respectively. The notation $\vec{n}$ is the normal vector to the ground, h₀ is the distance between the source (uplink path) or bucket detector (downlink path) and the ground, and θ is the zenith angle. The intensity distributions detected by the two detectors are correlated by a correlator to obtain the correlation function of the intensity fluctuations that can be used to reconstruct the information of the imaging object. Here we assume that turbulent atmosphere only exists in the test arm because the reference arm can be ignored in computational GI [3,6,45].

Fig. 1. Setup of a lensless GI system with turbulent atmosphere along an uplink path (a) and a downlink path (b).

Download Full Size | PDF

Based on the classical optical coherent theory, the correlation function of the intensity fluctuations between the two detectors is given by [16,46]

(1)$$G(x,y)\textrm{ = }\frac{1}{N}\sum\limits_{n\textrm{ = }1}^N {I_r^n(x,y)I_t^n} - \frac{1}{N}\sum\limits_{n\textrm{ = }1}^N {I_r^n(x,y)} \frac{1}{N}\sum\limits_{n\textrm{ = }1}^N {I_t^n}, $$

where $I_r^n(x,y)$, $I_t^n$ and N are the intensity recorded in D_r in the nth measurement, the total intensity collected by the D_t in the nth measurement and the total number of measurements, respectively.

According to Eq. (1), $I_r^n(x,y)$ and $I_t^n$ should be obtained to realize GI. Assuming the optical wave propagates along the z-axis and considering the beam propagation model with turbulent atmosphere, the scalar wave equation in parabolic approximation can be written as [47]

(2)$$2ik\frac{{\partial A}}{{\partial z}} + \nabla _ \bot ^2A + 2{k^2}\Delta nA = 0, $$

where $\nabla _ \bot ^2 = {{{\partial ^2}} \mathord{\left/ {\vphantom {{{\partial^2}} {\partial {x^2} + }}} \right.} {\partial {x^2} + }}{{{\partial ^2}} \mathord{\left/ {\vphantom {{{\partial^2}} {\partial {y^2}}}} \right.} {\partial {y^2}}}$ is Laplace operator, k = 2π/λ is the wave number in free space related to the wavelength λ, A is a slowly varying field amplitude in z, and Δn << 1 is the refractive index disturbance induced by the turbulence. The effect of turbulence can be described by random phase approximations since the refractive index disturbance Δn is small enough. Therefore, the vacuum diffraction propagation and the phase modulation induced by turbulent atmosphere are thought to be independent. The effect of turbulence is simulated by using the multi-phase screens method [48], which can be argued that the entire turbulence medium can be well modeled by a finite number of phase screens, which is placed on the rear surface of the media segment. We suppose that the phase screens are independent of each other.

In the multi-phase screens method, the propagation distance z is divided to sub-distances Δz_j = z_j - z_j_-1, z₀ = 0, j = 1, 2, 3…P. Let A(x,y,z_j_-1) be the complete solution to Eq. (2) at z_j_-1 plane, the solution to Eq. (2) at z_j = z_j_-1 + Δz_j plane satisfies the approximation relation

(3)$$A(x,y,{z_j}) = \exp \left( {\frac{i}{{2k}}\int_{{z_{j - 1}}}^{{z_j}} {\nabla_ \bot^2} dz} \right) \times \exp [iS(x,y,{z_j})]A(x,y,{z_{j - 1}}), $$

where $S({x,y,{z_j}} )\textrm{ = }k\int_{{z_{j - 1}}}^{{z_j}} {\Delta n(x,y,{z_j})dz}$ is the random phase modulation due to the turbulence corresponding to the jth phase screen. The wave field of a beam after propagating through a distance of z can be expressed as

(4)$$A({x,y,z} )\textrm{ = }\prod\limits_{j = 1}^P {\exp \left( {\frac{i}{{2k}}\int_{{z_{j - 1}}}^{{z_j}} {\nabla_ \bot^2dz} } \right)\exp [{iS({x,y,{z_j}} )} ]A({x,y,{z_0}} )}, $$

where A(x,y,z₀) is the initial wave field. Using Fourier transform algorithm, Eq. (3) can be rewritten as

(5)$$A({x,y,{z_j}} )= {{{\cal F}}^{\textrm{ - }1}}\left\{ {{{\cal F}}\{{\exp [{iS({x,y,{z_j}} )} ]A({x,y,{z_{j - 1}}} )} \}\exp \left( { - i\frac{{{K_x}^2 + {K_y}^2}}{{2k}}\Delta {z_j}} \right)} \right\}, $$

where ${{\cal F}}$ and ${{{\cal F}}^{\textrm{ - }1}}$ are the Fourier transform and inverse Fourier transform, respectively, and K_x(K_y) is the spatial wave number in the x(y) direction.

According to Eq. (5), S(x,y,z_j) is the key to calculation. Due to the random property of S(x,y,z_j), the analytical solution cannot be obtained. Therefore, the numerical method is employed to construct the phase screen. The spectral inversion method [47] is employed with high calculation efficiency in the following work.

Basd on the (modified) von Kármán spectrum, for a turbulent medium with a thickness of Δz, the spectrum ${\Phi _\phi }(\kappa )$ of the phase screen can be expressed as [49,50]

(6)$${\Phi _\phi }(\kappa ) = 2\pi {k^2}\Delta z \times 0.033C_n^2(h)\exp [ - {(\kappa /{\kappa _m})^2}]{({\kappa ^2} + {\kappa _0}^2)^{ - 11/6}}, $$

where κ is the spatial wave number, κ_m = 5.92/l₀, l₀ is the inner scale of turbulence, and κ₀ = 2π/L₀, L₀ is the outer scale of turbulence, $C_n^2(h)$ is the index structure parameter, h is the altitude from the ground. For a beam propagation along a horizontal path, $C_n^2$ can be regard as a constant. However, in a vertical or slant path, $C_n^2(h)$ is the function of altitude h. Several $C_n^2(h)$ profile models are employed for ground-to-space or space-to-ground applications. Here, the Hufnagel-Valley_5/7 (H-V_5/7) turbulent atmosphere profile model is used, which can be described as [50]

(7)$$\begin{array}{c} C_n^2(h) = 5.94 \times {10^{ - 53}}{(v/27)^2}{h^{10}}\exp ( - h/1000)\\ + 2.7 \times {10^{ - 16}}\exp ( - h/1500)\\ + {C_0}\exp ( - h/100), \end{array}$$

where v = 21 m/s is the rms windspeed, and C₀ = 1.7 × 10⁻¹⁴ m^−2/3 is the nominal value of $C_n^2(0)$ at the ground. For the slant path with zenith angle θ, Eq. (7) can be rewritten as

(8)$$\begin{array}{c} C_n^2(z,\theta ) = 5.94 \times {10^{ - 53}}{(v/27)^2}{(z^{\prime}\cos \theta + {h_0})^{10}}\exp ({ - (z^{\prime}\cos \theta + {h_0})/1000} )\\ + 2.7 \times {10^{ - 16}}\exp ({ - (z^{\prime}\cos \theta + {h_0})/1500} )\\ + {C_0}\exp ({ - (z^{\prime}\cos \theta + {h_0})/100} ), \end{array}$$

where h = z′cosθ + h₀. For the uplink path, z′ = z, and z′ = L-z for the downlink path withe L being the distance between the source and the bucket detector. Δh_j = Δz_jcosθ is the vertical distance between two phase screens (j-1 and j). It is clear that $\theta \textrm{ = }{0^ \circ }$ and ${90^ \circ }$ correspond to the vertical and horizontal propagation cases, respectively. Selection of $C_n^2(\Delta {z_j})$ between two-phase screens and the distribution of random phase screens on the path are vital for the slanted path case. Here $C_n^2(\Delta {z_j})$ is set to the average refractive index structure parameter of the turbulence between phase screens, which can be written as

(9)$$C_n^2(\Delta {z_j})\textrm{ = }\frac{1}{{\Delta {z_j}}}\int_{{z_{j - 1}}}^{{z_j}} {C_n^2(z,\theta )} dz. $$

The equivalent Rytov index-interval phase screen is employed, which has shown its stability and reliability [51]. Here, we assume that plane wave propagates between phase screens. The Rytov index can be written as

(10)$$\beta _\textrm{0}^\textrm{2}\textrm{ = 1}\textrm{.23}C_n^2({{z_{j - 1}}} ){k^{^{7/6}}}{({\Delta {z_j}} )^{11/6}}. $$

We can deduce the interval between adjacent phase screens by setting $\beta _\textrm{0}^\textrm{2}$ = C, where C is a constant. Then the number of phase screens can be easily obtained under the condition that the transmission distance is fixed. The random phase screen can be generated by [52]

(11)$$S(x,y) = {{{\cal F}}^{\textrm{ - }1}}\left( {M \cdot \frac{{2\pi }}{{{L_S}}}\sqrt {{\Phi _\phi }(\kappa )} } \right), $$

where M is a complex gaussian random matrix with a mean of 0 and a variance of 1, and L_S is the width of the phase screen. Then the phase screen can be simulated by Eq. (11).

According to the GI system shown in Fig. 1 and Eq. (4), the wave field on the object plane, the bucket detector plane and the reference detector plane have the forms

(12)$$A({x,y,{Z_1}} )\textrm{ = }\prod\limits_{j = 1}^{{P_o}} {\exp \left( {\frac{i}{{2k}}\int_{{z_{j - 1}}}^{{z_j}} {\nabla_ \bot^2dz} } \right)\exp [{iS({x,y,{z_j}} )} ]A({x,y,{z_0}} )}, $$

(13)$$A({x,y,{Z_1} + {Z_2}} )\textrm{ = }\prod\limits_{j = 1}^{{P_t}} {\exp \left( {\frac{i}{{2k}}\int_{{z_{j - 1}}}^{{z_j}} {\nabla_ \bot^2dz} } \right)} \exp [{iS({x,y,{z_j}} )} ]A({x,y,{Z_1}} )t(x,y), $$

(14)$$A({x,y,{Z_0}} )= \prod\limits_{j = 1}^{{P_r}} {\exp \left( {\frac{i}{{2k}}\int_{{z_{j - 1}}}^{{z_j}} {\nabla_ \bot^2dz} } \right)\exp [{iS({x,y,{z_j}} )} ]A({x,y,{z_0}} )}, $$

where P_o, P_t, and P_r are the number of phase screens corresponding to the wave field of a beam after propagating through a distance of Z₁, Z₁+Z₂ and Z₀, respectively. t(x,y) denotes the transmission function of the imaging object. The intensity distribution of the source is I_s(x,y) = |A(x,y,z₀)|². It is noted that $C_n^2$ = 0 inside S(x,y,z) in Eq. (14) because no turbulence is considered in the reference arm. Thus, Eq. (1) can be rewritten as

(15)$$\begin{aligned} G(x,y) &= \frac{1}{N}\sum\limits_{n\textrm{ = }1}^N {{{|{{A^n}(x,y,{Z_0})} |}^\textrm{2}}\left( {\sum\limits_{({x,y} )} {{{|{{A^n}(x,y,{Z_1} + {Z_2})} |}^2}} } \right)} \\ & - \frac{1}{N}\sum\limits_{n\textrm{ = }1}^N {{{|{{A^n}(x,y,{Z_0})} |}^2}} \frac{1}{N}\sum\limits_{n = 1}^N {\left( {\sum\limits_{({x,y} )} {{{|{{A^n}(x,y,{Z_1} + {Z_2})} |}^2}} } \right)} . \end{aligned}$$

The unknown object imaged can be retrieved by Eq. (15), and the effect from the slant path on GI through turbulent atmosphere can be discussed.

3. Results and discussion

Based on the model and the theory in section 2, we can compare the influence of beam propagation along a slant path (uplink and downlink paths) on the quality of GI through turbulent atmosphere by using numerical simulation. In the following cases, we set $\beta _\textrm{0}^\textrm{2}$= C = 0.004, l₀ = 8.5 mm, L₀ = 10 m, h₀ = 1 m, λ = 632.8 nm, Z₀ = Z₁= 1 km, Z₂ = 0.05 km, and N = 4000. Nx = Ny = 256, and Nx(Ny) is the number of sampling points on the x(y) axis. A double-slit, which has a slit width w = 22.5 mm and a center-to-center separation d = 67.5 mm, is chosen as the imaging object. The size of source is chosen as ρ_s = 10 mm.

Firstly, we consider the change of the beam distribution after propagating along a vertical path (θ = 0). Figures 2(b)–(d) show the optical field distribution of a collimating Gaussian beam (beam waist radius w₀ = 10 mm) shown in Fig. 2(a) after propagating through 1 km at free space, uplink path, and downlink path, respectively. One can see that the light spot experiences wander and deformation in the case of vertical propagation path. Moreover, the wander distance and the deformation degree of the light spot in an uplink path are more serious than those in a downlink path by comparing Figs. 2(c) with (d), which means that the influence of turbulent atmosphere along an uplink path is bigger than that along a downlink path for the beam propagation. At the same time, as can be seen from Figs. 2(b) and (d), the wander distance and the deformation degree of the light spot in downlink path are very small. In other words, the influence of turbulent atmosphere along a downlink path on the beam propagation is not obvious.

Fig. 2. (a) is the initial collimating Gaussian beam. (b)–(d) are the optical field distributions after propagating through 1 km under free space, vertical uplink path, and vertical downlink path, respectively.

Download Full Size | PDF

In Fig. 2, we simply compare the influence of turbulent atmosphere on the beam propagation along the uplink and downlink paths. Here the number of phase screens are set as 6, which can be deduced according to Eq. (10). The phase screen distribution is shown in Fig. 3(a). To further figure out this phenomenon, the relationships between quality index (beam wander and scintillation index) and the altitude from the ground h ∈ [1,1001 m] are investigated. The results are shown in Figs. 3(b) and (c). It can be seen that with h increasing, the beam wander and scintillation index both significantly increase in uplink path, reaching 8.499 mm and 0.0294 at h = 1001 m, respectively. For the downlink path, the beam wander and scintillation index both slowly increase, reaching 2.097 mm and 0.012 at h = 1 m, respectively. Obviously, the beam quality is more sensitive to the propagation distance in an uplink path than that along a downlink path.

Fig. 3. (a)–(d) are the phase screen distribution, the beam wander, scintillation index, and mean phase fluctuation, respectively.

Download Full Size | PDF

This result can also be explained by the phase modulation effect. According to Eq. (5), the turbulence effect on the beam propagation is determined by the random phase S(x,y,z). Figure 3(d) shows the relationship between the mean phase fluctuation S_avr and h, where S_avr = 〈S_max(x,y,z)−S_min(x,y,z)〉. For a small h value, the mean phase fluctuation is quite large with a strong phase modulation effect. With h increasing, the modulation effect gradually becomes weak. For the uplink path, the phase modulation effect varies from strong to weak, while the case is quite different for the downlink path, where the phase modulation effect varies from weak to strong. As is known to all, strong phase modulation has a great influence on the beam during the propagation progress, while weak phase modulation has litter or no effect. This opposite phase modulation process results in significant difference shown in Figs. 2(b) and (c). The difference will lead to the difference of the imaging quality of the ghost-image because the information of an object is retrieved by measuring the spatial intensity correlation between two beams, as shown in Fig. 1.

Under the same condition, the reconstructed ghost images are shown in Figs. 4(b)–(d). Here a high-quality ghost-image is obtained for both free space and a downlink path, whereas the ghost-image has poorer quality in the case of an uplink path, which is in agreement with the conclusion in Fig. 2. Therefore, imaging quality of GI through turbulent atmosphere along a downlink path is better than that along an uplink path.

Fig. 4. (a) is the imaging object. (b)–(d) are the corresponding ghost-images under free space, vertical uplink path, and vertical downlink path, respectively.

Download Full Size | PDF

Next, we study the influence of the zenith angle on GI along a slant path. Figures 5(a)–(c) and 5(d)–(f) show the ghost images at uplink and downlink paths under different zenith angles, respectively. When the zenith angle is increased from ${15^ \circ }$ to ${75^ \circ }$, the quality of the reconstructed ghost-image is gradually degraded, which can be explained by Eq. (8): a large zenith angle corresponds to a large turbulence parameter at the same propagation distance. In addition, the imaging quality of the ghost images at the downlink path are better than that at the uplink path under the same parameters, which is similar with the conclusion in Fig. 2. Moreover, it is noteworthy that a high-quality ghost-image can still be maintained for the downlink path when the zenith angle $\theta \le \textrm{4}{5^ \circ }$ according to Figs. 5(d) and (e) though the imaging quality under an uplink path is quite bad.

Fig. 5. Retrieved ghost images at an uplink path (a)–(c) and a downlink (d)–(f) path under zenith angles θ =${15^ \circ }$, ${45^ \circ }$ and ${75^ \circ }$, respectively.

Download Full Size | PDF

In order to evaluate the imaging quality of the ghost images quantitatively, we introduce the contrast-to-noise ratio (CNR) and visibility (V) [53], which have the forms

(16)$$\textrm{CNR} = \frac{{\left\langle {{I_S}} \right\rangle - \left\langle {{I_N}} \right\rangle }}{{\sqrt {{V_S} + {V_N}} }}, $$

and

(17)$$V = \frac{{\left\langle {{I_S}} \right\rangle - \left\langle {{I_N}} \right\rangle }}{{\left\langle {{I_S}} \right\rangle \textrm{ + }\left\langle {{I_N}} \right\rangle }}, $$

where $\left\langle { \cdot{\cdot} \cdot } \right\rangle$ is the ensemble average. I_S(I_N) is the signal(noise) point intensity values of the ghost images, which correspond to the location where the binary object is 1(0). Both I_S and I_N are integer values from 0 to 255. V_S and V_N are the variance of the intensity values of the signal and noise points, respectively. Generally, the higher CNR or V is, the better quality of the ghost-image has. Given zenith angle $\theta \in [0,{90^ \circ }]$, Fig. 6 shows how CNR and V vary with θ under uplink and downlink paths. It is shown that both CNR and V decrease slowly for small angles and declines rapidly under large angles. While, the CNR and V values of the ghost images at an uplink path are always worse than those at a downlink path. It is noted that the slant path degenerates into the horizontal path when $\theta \textrm{ = }{90^ \circ }$, which has been discussed in many works [30–44].

Fig. 6. (a) CNR and (b) V of the retrieved ghost images at different zenith angles.

Download Full Size | PDF

4. Conclusions

In conclusion, we have numerically investigated the effects of turbulent atmosphere on the quality of GI along an uplink path and a downlink path. According to the optical coherent theory and multi-phase screens method, we present a numerical imaging formula of GI through turbulent atmosphere along a slant path, which can be used to discuss the effect from the slant path on the quality of GI. The results demonstrate that the quality of the ghost-image is gradually degraded when increasing the zenith angle for both uplink and downlink paths, while the imaging quality of GI along a downlink path is much better than that along an uplink path. These results may be valuable for GI applications in satellite communication and remote sensing.

Funding

National Natural Science Foundation of China (61871431, 61971184); Natural Science Foundation of Hunan Province (2017JJ1014).

Disclosures

The authors declare no conflicts of interest.

References

1. D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ‘ghost’ interference and diffraction,” Phys. Rev. Lett. 74(18), 3600–3603 (1995). [CrossRef]

2. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52(5), R3429–R3432 (1995). [CrossRef]

3. B. I. Erkmen, “Computational ghost imaging for remote sensing,” J. Opt. Soc. Am. A 29(5), 782–789 (2012). [CrossRef]

4. C. Q. Zhao, W. L. Gong, M. L. Chen, E. R. Li, H. Wang, W. D. Xu, and S. S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101(14), 141123 (2012). [CrossRef]

5. M. Chen, E. Li, W. Gong, Z. Bo, X. Xu, C. Zhao, X. Shen, W. Xu, and S. S. Han, “Ghost imaging lidar via sparsity constraints in real atmosphere,” Opt. Photonics J. 03(02), 83–85 (2013). [CrossRef]

6. H. C. Liu and S. Zhang, “Computational ghost imaging of hot objects in long-wave infrared range,” Appl. Phys. Lett. 111(3), 031110 (2017). [CrossRef]

7. D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganin, “Experimental x-ray ghost imaging,” Phys. Rev. Lett. 117(11), 113902 (2016). [CrossRef]

8. S. Chen, “X-ray ghost images could cut radiation doses,” Science 359(6383), 1452 (2018). [CrossRef]

9. Y. F. Bai and S. S. Han, “Ghost imaging with thermal light by third-order correlation,” Phys. Rev. A 76(4), 043828 (2007). [CrossRef]

10. K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “High-order thermal ghost imaging,” Opt. Lett. 34(21), 3343–3345 (2009). [CrossRef]

11. F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential Ghost Imaging,” Phys. Rev. Lett. 104(25), 253603 (2010). [CrossRef]

12. M. F. Li, Y. R. Zhang, K. H. Luo, L. A. Wu, and H. Fan, “Time-correspondence differential ghost imaging,” Phys. Rev. A 87(3), 033813 (2013). [CrossRef]

13. C. Zhang, S. X. Guo, J. S. Cao, J. Guan, and F. L. Gao, “Object reconstitution using pseudo-inverse for ghost imaging,” Opt. Express 22(24), 30063–30073 (2014). [CrossRef]

14. W. L. Gong, “High-resolution pseudo-inverse ghost imaging,” Photonics Res. 3(5), 234–237 (2015). [CrossRef]

15. H. Ghanbari-Ghalehjoughi, S. Ahmadi-Kandjani, and M. Eslami, “High quality computational ghost imaging using multi-fluorescent screen,” J. Opt. Soc. Am. A 32(2), 323–328 (2015). [CrossRef]

16. X. H. Shi, X. W. Huang, S. Q. Nan, H. X. Li, Y. F. Bai, and X. Q. Fu, “Image quality enhancement in low-light-level ghost imaging using modified compressive sensing method,” Laser Phys. Lett. 15(4), 045204 (2018). [CrossRef]

17. H. Y. Huang, C. Zhou, T. Tian, D. Q. Liu, and L. J. Song, “High-quality compressive ghost imaging,” Opt. Commun. 412(1), 60–65 (2018). [CrossRef]

18. Y. Zhou, T. Zhang, F. Zhong, and S. X. Guo, “Enhancing image quality of ghost imaging by fuzzy c-means clustering method,” AIP Adv. 9(7), 075006 (2019). [CrossRef]

19. J. Cai, J. W. Luo, S. L. Wang, and S. Yang, “Feature selection in machine learning: A new perspective,” Neurocomputing 300, 70–79 (2018). [CrossRef]

20. W. H. Chen, J. Y. An, R. F. Li, L. Fu, G. Q. Xie, M. Z. A. Bhuiyan, and K. Q. Li, “A novel fuzzy deep-learning approach to traffic flow prediction with uncertain spatial–temporal data features,” Futur. Gener. Comp. Syst. 89, 78–88 (2018). [CrossRef]

21. J. Y. An, L. Fu, M. Hu, W. H. Chen, and J. W. Zhan, “A novel fuzzy-based convolutional neural network method to traffic flow prediction with uncertain traffic accident information,” IEEE Access 7, 20708–20722 (2019). [CrossRef]

22. K. Wang, K. L. Li, L. Q. Zhou, Y. K. Hu, Z. Y. Chen, J. Liu, and C. Chen, “Multiple convolutional neural networks for multivariate time series prediction,” Neurocomputing 360, 107–119 (2019). [CrossRef]

23. J. G. Chen, K. L. Li, K. Bilal, X. Zhou, K. Q. Li, and P. S. Yu, “A bi-layered parallel training architecture for large-scale convolutional neural networks,” IEEE Trans. Parallel Distrib. Syst. 30(5), 965–976 (2019). [CrossRef]

24. M. Lyu, W. Wang, H. Wang, H. C. Wang, G. W. Li, N. Chen, and G. H. Situ, “Deep-learning-based ghost imaging,” Sci. Rep. 7(1), 17865 (2017). [CrossRef]

25. Y. C. He, G. Wang, G. X. Dong, S. T. Zhu, H. Chen, A. X. Zhang, and Z. Xu, “Ghost imaging based on deep learning,” Sci. Rep. 8(1), 6469 (2018). [CrossRef]

26. F. Wang, H. Wang, H. C. Wang, G. W. Li, and G. H. Situ, “Learning from simulation: An end-to-end deep-learning approach for computational ghost imaging,” Opt. Express 27(18), 25560–25572 (2019). [CrossRef]

27. Y. K. Xu, W. T. Liu, E. F. Zhang, Q. Li, H. Y. Dai, and P. X. Chen, “Is ghost imaging intrinsically more powerful against scattering,” Opt. Express 23(26), 32993–33000 (2015). [CrossRef]

28. T. Aulin, “A modified model for the fading signal at a mobile radio channel,” IEEE Trans. Veh. Technol. 28(3), 182–203 (1979). [CrossRef]

29. X. L. Liu, F. Wang, C. Wei, and Y. J. Cai, “Experimental study of turbulence-induced beam wander and deformation of a partially coherent beam,” Opt. Lett. 39(11), 3336–3339 (2014). [CrossRef]

30. J. Cheng, “Ghost imaging through turbulent atmosphere,” Opt. Express 17(10), 7916–7921 (2009). [CrossRef]

31. P. L. Zhang, W. L. Gong, X. Shen, and S. S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82(3), 033817 (2010). [CrossRef]

32. N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A 84(6), 063824 (2011). [CrossRef]

33. K. W. C. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shapiro, P. B. Dixon, G. A. Howland, J. C. Howell, J. H. Eberly, M. N. O’Sullivan, B. Rodenburg, and R. W. Boyd, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84(4), 043807 (2011). [CrossRef]

34. P. B. Dixon, G. A. Howland, K. W. C. Chan, C. O’Sullivan-Hale, B. Rodenburg, N. D. Hardy, J. H. Shapiro, D. S. Simon, A. V. Sergienko, R. W. Boyd, and J. C. Howell, “Quantum ghost imaging through turbulence,” Phys. Rev. A 83(5), 051803 (2011). [CrossRef]

35. R. E. Meyers, K. S. Deacon, and Y. H. Shih, “Positive-negative turbulence-free ghost imaging,” Appl. Phys. Lett. 100(13), 131114 (2012). [CrossRef]

36. Y. X. Zhang and Y. G. Wang, “Computational lensless ghost imaging in a slant path non-Kolmogorov turbulent atmosphere,” Optik 123(15), 1360–1363 (2012). [CrossRef]

37. X. Wang and Y. X. Zhang, “Lens ghost imaging in a non-Kolmogorov slant turbulence atmosphere,” Optik 124(20), 4378–4382 (2013). [CrossRef]

38. J. Chen and J. Lin, “Unified theory of thermal ghost imaging and ghost diffraction through turbulent atmosphere,” Phys. Rev. A 87(4), 043810 (2013). [CrossRef]

39. S. M. Zhao, B. Wang, L. Y. Gong, Y. B. Sheng, W. W. Cheng, X. L. Dong, and B. Y. Zheng, “Improving the atmosphere turbulence tolerance in holographic ghost imaging system by channel coding,” J. Lightwave Technol. 31(17), 2823–2828 (2013). [CrossRef]

40. X. Yang, Y. Zhang, L. Xu, C. H. Yang, Q. Wang, Y. H. Liu, and Y. Zhao, “Increasing the range accuracy of three-dimensional ghost imaging ladar using optimum slicing number method,” Chin. Phys. B 24(12), 124202 (2015). [CrossRef]

41. L. L. Tang, Y. F. Bai, C. Duan, S. Q. Nan, Q. Shen, and X. Q. Fu, “Effects of incident angles on reﬂective ghost imaging through atmospheric turbulence,” Laser Phys. 28(1), 015201 (2018). [CrossRef]

42. C. L. Luo, P. Lei, Z. L. Li, J. Q. Qi, X. X. Jia, F. Dong, and Z. M. Liu, “Long-distance ghost imaging with an almost non-diffracting Lorentz source in atmospheric turbulence,” Laser Phys. Lett. 15(8), 085201 (2018). [CrossRef]

43. X. L. Liu, F. Wang, M. H. Zhang, and Y. J. Cai, “Effects of atmospheric turbulence on lensless ghost imaging with partially coherent light,” Appl. Sci. 8(9), 1479 (2018). [CrossRef]

44. W. Tan, X. W. Huang, S. Q. Nan, Y. F. Bai, and X. Q. Fu, “Effect of the collection range of a bucket detector on ghost imaging through turbulent atmosphere,” J. Opt. Soc. Am. A 36(7), 1261–1266 (2019). [CrossRef]

45. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), 061802 (2008). [CrossRef]

46. E. Gatti, M. Brambilla, L. A. Bache, and Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93(9), 093602 (2004). [CrossRef]

47. J. M. Martin and S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27(11), 2111–2126 (1988). [CrossRef]

48. S. M. Zhao, J. Leach, L. Y. Gong, J. Ding, and B. Y. Zheng, “Aberration corrections for free-space optical communications in atmosphere turbulence using orbital angular momentum states,” Opt. Express 20(1), 452–461 (2012). [CrossRef]

49. G. Gbur, “Partially coherent beam propagation in atmospheric turbulence,” J. Opt. Soc. Am. A 31(9), 2038–2045 (2014). [CrossRef]

50. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

51. H. Chen, X. L. Ji, X. Q. Li, T. Wang, Q. Zhao, and H. Zhang, “Energy focus ability of annular beams propagating through atmospheric turbulence along a slanted path,” Opt. Laser Technol. 71, 22–28 (2015). [CrossRef]

52. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).

53. X. H. Shi, H. X. Li, Y. F. Bai, and X. Q. Fu, “Negative influence of detector noise on ghost imaging based on the photon counting technique at low light levels,” Appl. Opt. 56(26), 7320–7326 (2017). [CrossRef]

Ghost imaging through inhomogeneous turbulent atmosphere along an uplink path and a downlink path

Abstract

1. Introduction

2. Model and theory

3. Results and discussion

4. Conclusions

Funding

Disclosures

References

Cited By

Figures (6)

Equations (17)

OSA Continuum