Abstract

Non-degenerate wavelength computational ghost imaging with thermal light source is studied theoretically and experimentally. The acquired computational ghost images are of high quality when the wavelength of computed light is different from the light detected by bucket detector. Compared to the necessary light of short wavelength in previous ghost imaging, the use of longer wavelength light is demonstrated to bring about ghost images with higher spatial resolution, in strong atmospheric turbulence.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Ghost imaging is a transverse active-imaging technique that exploits the correlation between two light beams to image an object without spatially resolving measurements of the light beam that has undergone object interaction [1]. Over the past two decades, GI has been found to have some unique advantages over conventional optical imaging technologies, such as super-resolution [2,3], good imaging quality in harsh optical environment [4,5]. Recently, the experiments of GI with X-ray and electronics sources [68] have been reported, indicating that GI has become a powerful comprehensive tool in exploring and analyzing the internal characteristics of complex material, e.g. biomolecular structures and nanomaterials.

GI is considered to have broad applied prospects [913], especially in remote sensing and laser radar. One of the key reasons that many applications have not yet realized in practice is the complexity of the optical path structure of conventional GI. Fortunately, in 2008, Shapiro proposed a novel GI scheme-computational ghost imaging (CGI) [14], in which the image is reconstructed by correlating a calculated pattern with the signal of a bucket detector located behind the object illuminated by modulated light by spatial light modulator (SLM). Deterministic modulation of a laser beam with an SLM can provide the signal field used for target interrogation, while the on-target intensity pattern needed for the reference field can then be calculated via diffraction theory [15,16].

CGI is currently the most promising imaging solution for remote sensing and laser radar because it has only one optical path, which is different from the conventional GI essentially. Although GI can bring about good imaging quality in harsh optical environment, it is not negligible that the ghost image will be significantly degraded with strong turbulence and large propagation distance. 2011, Meyers et al proposed turbulence-free GI, which played a crucial role in applications [4]. Previous works have shown that the resolution of non-degenerate wavelength GI is higher than that of degenerate wavelength GI [17,18]. In addition, the source with shorter wavelength illuminating the object is effective for ghost images with higher resolution in atmospheric turbulence [19,20].

Non-degenerate wavelength CGI with thermal light source relies on a modification of the conventional CGI set-up (Fig. 1). A binocular charge-coupled device (CCD) camera is employed to replace the bucket detector, compared to the conventional CGI scheme. The binocular CCD camera can not only output the bucket signal but also be used as a tool to measure the distance between the SLM and the object [21,22].

 

Fig. 1. Setup of the non-degenerate-wavelength computational ghost imaging with thermal light source. SLM: spatial light modulator, BBD: binocular bucket detector.

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2. Theory

In the following, we theoretically illustrate the concept of nondegenerate wavelength CGI with thermal light source. The collected light by the detector is named as signal light, while the calculated patterns by the SLM is named as reference light. To explore the properties of nondegenerate wavelength CGI, a continuous laser beam $E_{s}$ is projected to an SLM and the modulated light illuminates the object after propagating a distance of $z$ in free space. The light field detected by the binocular CCD camera can be expressed as [5,23]

$$E_{s}(x_{s})=\frac{-1}{\lambda_{1}\sqrt{z_{1}z_{2}}}\int dudyE_{s}^{{\prime} }\left( \lambda_{s},u\right) e^{\frac{i\pi}{\lambda_{s}z_{1}}\left( y-u\right) }\,T(y)e^{\frac{i\pi}{\lambda_{s}z_{2}}\left( x_{s}-y\right) },$$
where $T(y)$ represents the information of object. The quantities $u$, $y$, $x$ represent the transverse coordinates of the SLM, the object plane and the bucket detector plane, respectively. $z_{1}$ is the distance from the SLM to the object. $z_{2}$ represents the distance from object to the binocular CCD camera.

In practice, $z_{1}$ and $z_{2}$ need to be measured in real time to get the calculated light field according to the diffraction principle. Fortunately, the distance from the object to detector can be accurately measured by the binocular parallax principle without adding other equipments [21,22]. For simplicity and practicality, we assume $z_{1}=z_{2}=L_{1}$. In order to obtain a nondegenerate-wavelength computational ghost image, the wavelength of the calculated light $E_{2}$ should be different from that of the detected light $E_{s}$, i.e., $\lambda _{s}\neq \lambda _{r}$. Furthermore, the propagation distance $L_{2}=z_{3}$ of the calculated light is determined by the optimal imaging conditions of the non-degenerate GI ($\lambda _{s} L_{1}=\lambda _{r}L_{2}$) [17]. Thus, the distribution of the computed light is given by

$$E_{r}(x_{r})=\left( \frac{-i}{\lambda_{r}\sqrt{z_{3}}}\right) ^{1/2}\int duE_{r}^{{\prime}}\left( \lambda_{r},u\right) e^{\frac{i\pi}{\lambda _{r}z_{3}}\left( u-x_{r}\right) }\,.$$
In order to reconstruct the image, the calculated patterns at the object plane are cross-correlated with the signal output by the detector, i.e.,
$$\begin{aligned}& G(x_{s},x_{r})=\left\langle I_{s}\left( x_{s}\right) I_{r}\left( x_{r}\right) \right\rangle -\left\langle I_{s}\left( x_{s}\right) \right\rangle \left\langle I_{r}\left( x_{r}\right) \right\rangle \\ & =\left\langle E_{s}^{{\ast}}\left( x_{s}\right) E_{s}\left( x_{s}\right) E_{r}^{{\ast}}\left( x_{r}\right) E_{r}\left( x_{r}\right) \right\rangle -\left\langle E_{s}^{{\ast}}\left( x_{s}\right) E_{s}\left( x_{s}\right) \right\rangle \left\langle E_{r}^{{\ast}}\left( x_{r}\right) E_{r}\left( x_{r}\right) \right\rangle \\ & =\frac{1}{\lambda_{s}^{2}\lambda_{r}z_{1}z_{2}z_{3}}\int du_{1} du_{1}^{{\prime}}du_{2}du_{2}^{{\prime}}dydy^{{\prime}}C\left( \lambda _{s},\lambda_{r};u_{1},u_{1}^{{\prime}},u_{2},u_{2}^{{\prime}}\right) T(y)T^{{\ast}}(y)\\ & \times e^{\frac{i\pi}{\lambda_{s}z_{1}}\left[ \left( y-u_{1}\right) ^{2}-\left( y^{{\prime}}-u_{1}^{{\prime}}\right) ^{2}\right] } e^{\frac{i\pi}{\lambda_{s}z_{2}}\left[ \left( x_{s}-y\right) ^{2}-\left( x_{s}^{{\prime}}-y^{{\prime\prime}}\right) ^{2}\right] }e^{\frac{i\pi }{\lambda_{r}z_{3}}\left[ \left( u_{2}-x_{r}\right) ^{2}-\left( u_{2}^{{\prime}}-x_{r}^{{\prime}}\right) ^{2}\right] }, \end{aligned}$$
where
$$\begin{aligned}& C\left( \lambda_{s},\lambda_{r};u_{1},u_{1}^{{\prime}},u_{2} ,u_{2}^{{\prime}}\right) \\ & =\left\langle E_{s}^{{\prime}\ast}\left( \lambda_{s},u_{1}\right) E_{s}^{{\prime}}\left( \lambda_{s},u_{1}^{{\prime}}\right) E_{r} ^{{\prime}\ast}\left( \lambda_{r},u_{2}\right) E_{r}^{{\prime}}\left( \lambda_{r},u_{2}^{{\prime}}\right) \right\rangle \\ & =\left\langle E_{s}^{{\prime}\ast}\left( \lambda_{s}\right) E_{s}^{{\prime}}\left( \lambda_{s}\right) \right\rangle \left\langle E_{r}^{{\prime}\ast}\left( \lambda_{r}\right) E_{r}^{{\prime}}\left( \lambda_{r}\right) \right\rangle \left\langle V\left( u_{1}\right) V^{{\ast} }\left( u_{2}^{{\prime}}\right) \right\rangle \left\langle V\left( u_{2}\right) V^{{\ast}}\left( u_{1}^{{\prime}}\right) \right\rangle \end{aligned}$$
is the intensity cross-correlation function of the light beams in the spatial and temporal frequency domain evaluated at the output plane of the SLM. $E_{a}\left ( \lambda _{a},u_{a}\right ) =E_{a}^{{\prime }}\left ( \lambda _{a}\right ) V\left ( u_{a}\right )$, $a=s,r$. $E_{s}^{{\prime }}$ and $E_{r}^{{\prime }}$ are the independent light fields at the input plane of the SLM. In order to simplify the calculation, we assume that $\left \langle E^{\ast }\left ( \lambda \right ) E\left ( \lambda \right ) \right \rangle =I_{0}=1$. In addition, the SLM mask function $V\left ( u\right )$ possesses spatial correlations that follow Gaussian statistics.
$$\left\langle V\left( u_{1}\right) V^{{\ast}}\left( u_{2}^{{\prime}}\right) \right\rangle =e^{-\frac{u_{1}^{2}+u_{1}^{{\prime}2}+u_{2}^{2}+u_{2} ^{{\prime}2}}{4\omega}}e^{-\frac{\left( u_{1}-u_{2}^{{\prime}}\right) ^{2}+\left( u_{2}-u_{1}^{{\prime}}\right) ^{2}}{2l_{c}^{2}}},$$
where $\omega$ is the transverse size of the laser beam and $l_{c}$ is the correlation parameter of the random amplitude caused by the SLM. Substituting Eq. (5) into Eq. (3), we can thus write the ghost image of Eq. (3) as
$$G(x_{r})=\frac{\pi^{2}}{\lambda_{s}\lambda_{r}z_{1}z_{3}\sqrt{ABCD}}\int dyO\left( y\right) \exp\left( \frac{S^{2}}{4A}+\frac{P^{2}}{4B}+\frac {Q^{2}}{4C}+\frac{R^{2}}{4D}\right) ,$$
where,
$$\begin{aligned}A & =\frac{1}{4\omega^{2}}+\frac{1}{2l_{c}^{2}}-\frac{i\pi}{\lambda_{s}z_{1} },B=\frac{1}{4\omega^{2}}+\frac{1}{2l_{c}^{2}}+\frac{i\pi}{\lambda_{s}z_{1} },\\ C & =\frac{1}{4\omega^{2}}+\frac{1}{2l_{c}^{2}}-\frac{i\pi}{\lambda_{r}z_{3} }-\frac{1}{4Bl_{c}^{4}},D=\frac{1}{4\omega^{2}}+\frac{1}{2l_{c}^{2}} +\frac{i\pi}{\lambda_{r}z_{3}}-\frac{1}{4Al_{c}^{4}},\\ S & =\frac{2i\pi y}{\lambda_{s}z_{1}},P=\frac{2i\pi y}{\lambda_{s}z_{1}},Q=\frac{i\pi y}{Bl_{c}^{2}\lambda_{s}z_{1}}-\frac{2i\pi x_{2}}{\lambda_{r}z_{3}},R=\frac{2i\pi x_{2}}{\lambda_{r}z_{3}}-\frac{i\pi y}{Al_{c} ^{2}\lambda_{s}z_{1}},\end{aligned}$$
$\left \langle T(y)T^{\ast }(y)\right \rangle =\lambda _{s}O\left ( y\right ) \delta \left ( y-y^{{\prime }}\right ) .$ From Eq. (6), we can see that the correlation function $G(x_{r})$ has the same form as that of two-color GI [5,17]. Therefore, the non-degenerate wavelength thermal CGI feasible in practice even when the difference of the wavelengths is very large. Different from the two-color GI, as the reference light field is completely calculated, the wavelength can be set arbitrarily,

We calculate the exponential function $\exp \left ( \frac {S^{2}}{4A} +\frac {P^{2}}{4B}+\frac {Q^{2}}{4C}+\frac {R^{2}}{4D}\right )$ on the right of Eq. (6), and then give the standard form of GI. This calculation process is simple but cumbersome, so the result is directly given by

$$G(x_{r})=\frac{\pi^{2}}{\lambda_{s}\lambda_{r}z_{1}z_{3}\sqrt{ABCD}} \exp\left( -\frac{x_{s}^{2}}{2W_{fov}^{2}}\right) \int dyO\left( y\right) \exp\left( -\frac{\left( y-mx_{s}^{2}\right) }{2W_{psf}^{2}}\right) ,$$
where $W_{fov}$ and $m$ represent the view of field and magnification factor [5], respectively.
$$W_{psf}=Re\sqrt{\frac{1}{2\left( K_{1}^{2}+K_{2}^{2}+K_{3}^{2}+K_{4} ^{2}\right) }}$$
is the width of point spread function (PSF).
$$\begin{aligned}K_{1} & =\frac{\pi}{\sqrt{A}\lambda_{s}z_{1}},K_{2}=\frac{\pi}{\sqrt {B}\lambda_{s}z_{1}},K_{3}=\frac{\pi}{2B\sqrt{C}\lambda_{s}z_{1}l_{c}^{2}},\\ K_{4} & =\frac{\pi}{2\sqrt{D}\lambda_{s}z_{1}}\left( \frac{1}{Bl_{c}^{2}}-\frac{1}{Al_{c}^{2}}\right).\end{aligned}$$
Equation (8) is plotted in Fig. 2 as a function of variable $\lambda _{r}$, $\lambda _{s}$ and $L_{1}=z_{1}$. It is seen that the width of the PSF attains its smallest value when $\lambda _{s}L_{1}=\lambda _{r}L_{2}$, in previous work [15]. Figure 2(a) compares the resolution of conventional CGI with single wavelength light source (blue dot line) and nondegenerate wavelength CGI (other kinds of lines), from which we found that there is no significant difference between them (minimum value of PSF). Compared to this one, the spatial resolution changes significantly as the wavelength of signal light changes. This result has illustrated the results that the spatial resolution of non-degenerate wavelength CGI depends more strongly on signal light $\lambda _{s}$ than reference light $\lambda _{r}$ (Fig. 2(b)).

 

Fig. 2. (a) Width of the point-spread function as a function of $\lambda _{s}=800nm$, $\lambda _{r}=400,600,800,1000,1200,1400nm$. Parameters used are $\omega =5cm$, $l_{c}=1mm$, $z_{1}=1km$. (b)The corresponding plot of PSF as a function of $\lambda _{s}=800nm$, $\lambda _{r}=400,600,800,1000,1200,1400nm$.

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In the case of atmospheric turbulence, based on the extended Huygens-Fresnel integral, the propagation function from the source plane to the detector plane is given by [5,23]

$$\begin{aligned}E_{s}(x_{s}) & =\frac{-1}{\lambda_{s}\sqrt{z_{1}z_{2}}}\int dudyE_{s} ^{{\prime}}\left( \lambda_{s},u_{1}\right) e^{\frac{i\pi}{\lambda_{s}z_{1} }\left( y-u_{a}\right) }\, \\ & \times\exp\left[ \phi_{1}(y,u)\right] T(y)\exp\left[ \phi_{2} (x,y)\right] T(y)e^{\frac{i\pi}{\lambda_{s}z_{2}}\left( x_{s}-y\right) }, \end{aligned}$$
where $\phi _{1}$ and $\phi _{2}$ characterize the atmospheric turbulence effects in the SLM-to-object path and the object-to-bucket-detector path, respectively. Because the reference arm can be computed by the simulation program according to the diffraction theory, the computed light field is not affected by atmospheric turbulence. Thus, the imaging function of nondegenerate-wavelength CGI with atmospheric turbulence can be expressed as
$$\begin{aligned}G(x_{r}) & =\frac{\pi^{2}}{\lambda_{s}\lambda_{r}z_{1}z_{3}\sqrt{A^{{\prime }}B^{{\prime}}C^{{\prime}}D^{{\prime}}}}\exp\left( -\frac{x_{s}^{2} }{2W_{fov}^{2}}\right) \\ & \times\int dyO\left( y\right) \exp\left( -\frac{\left( y-mx_{s} ^{2}\right) }{2W_{psf}^{2}}\right) . \end{aligned}$$
The PSF has the same form as Eq. (8), i.e.,
$$W_{psf}=Re\sqrt{\frac{1}{2\left( K_{1}^{2}+K_{2}^{2}+K_{3}^{2}+K_{4} ^{2}\right) }},$$
with
$$\begin{aligned}A^{{\prime}} & =\frac{1}{4\omega^{2}}+\frac{1}{2l_{c}^{2}}+\frac{1} {2\rho^{2}}-\frac{i\pi}{\lambda_{s}z_{1}},B^{{\prime}}=\frac{1}{4\omega^{2} }+\frac{1}{2l_{c}^{2}}+\frac{1}{2\rho^{2}}+\frac{i\pi}{\lambda_{s}z_{1}} -\frac{1}{4A^{{\prime}}\rho^{4}}, \\ C^{{\prime}} & =\frac{1}{4\omega^{2}}+\frac{1}{2l_{c}^{2}}+\frac{1} {2\rho^{2}}-\frac{i\pi}{\lambda_{r}z_{3}}-\frac{1}{4B^{{\prime}}l_{c}^{4}}\\ D^{{\prime}} & =\frac{1}{4\omega^{2}}+\frac{1}{2l_{c}^{2}}+\frac{i\pi }{\lambda_{r}z_{3}}-\frac{1}{4A^{{\prime}}l_{c}^{2}}-\frac{1}{16A^{{\prime }2}B^{{\prime}}l_{c}^{4}\rho^{4}}-\frac{1}{64A^{{\prime}2}B^{{\prime} 2}C^{{\prime}}l_{c}^{8}\rho^{4}}, \end{aligned}$$
$$\begin{aligned}K_{1} & =\frac{\pi}{\sqrt{A^{{\prime}}}\lambda_{s}z_{1}},K_{2}=\frac{\pi }{\sqrt{B^{{\prime}}}\lambda_{s}z_{1}}\left( 1-\frac{1}{2A^{{\prime}}\rho^{2}}\right) ,\\ K_{3} & =\frac{\pi}{2B^{{\prime}}\sqrt{C^{{\prime}}}\lambda_{s}z_{1} l_{c}^{2}}\left( 1-\frac{1}{2A^{{\prime}}\rho^{2}}\right) ,\\ K_{4} & =\frac{\pi}{2\sqrt{D^{{\prime}}}\lambda_{s}z_{1}}\left[ -\frac {1}{A^{{\prime}}l_{c}^{2}}+\frac{1}{B^{{\prime}}l_{c}^{2}}\left( 1-\frac {1}{2A^{{\prime}}\rho^{2}}\right) \right. \\ & \left. \times\left( \frac{1}{2A^{{\prime}}\rho^{2}}+\frac{1} {8A^{{\prime}}B^{{\prime}}C^{{\prime}}l_{c}^{4}\rho^{2}}\right) \right] ,\end{aligned}$$
where $\rho =\left ( 0.55C_{n,s}^{2}k_{s}^{2}z_{0}\right ) ^{-\frac {3}{5}}$, $C_{n,s}^{2}$ is the turbulence strength.

Equation (11) is plotted in Fig. 3 as a function of $\lambda _{r}$ and $\lambda _{s}$. We test the resolution of nondegenerate wavelength thermal CGI with the wavelength $\lambda _{s}$ fixed and the wavelength $\lambda _{r}$ changed. The results are shown in Figs. 3(a)-3(d). The results show that the spatial resolution of non-degenerate wavelength CGI with thermal light is equal to that of conventional CGI in tuebulence. The spatial resolution will be reduced with the increase of turbulence. We fix the $\lambda _{r}$ and change $\lambda _{s}$, the results are shown in Figs. 3(e)-3(h). In weak turbulence ($C_{n,s}^{2}=1\times 10^{-15}$), ghost images with higher resolution can be obtained when the light of signal arm carries shorter wavelength, which is the same as the previous works [5,16,19]. However, in strong turbulence ($C_{n,s}^{2}=1\times 10^{-12}$), ghost images with higher resolution can be obtained when the light of signal arm carries longer wavelength, which is different from previous works [5,16,19].

 

Fig. 3. The PSF of the nondegenerate wavelength CGI with thermal light source. Parameters used are $\omega =5cm$, $l_{c}=1mm$, $z_{1}=1km$. (a-d) are the curve lines for $C_{n,s}^{2}=10^{-15}$, $10^{-14}$, $10^{-13}$, $10^{-12}$, $\lambda _{s}=800nm$, $\lambda _{r}=400,600,800,1000,1200,1400nm$. (e-h) are the curve lines for $C_{n,s}^{2}=10^{-15}$, $10^{-14}$, $10^{-13}$, $10^{-12}$, $\lambda _{r}=800nm$, $\lambda _{s}=400,600,800,1000,1200,1400nm$.

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3. Experiments

To experimentally demonstrate the non-degenerate wavelength CGI, we used the imaging setup described in the previous section to image a Rubik’s Cube. The setup is based on a two-dimensional amplitude-only ferroelectric liquid crystal spatial light modulator (FLC-SLM, Meadowlark Optics A512-450-850), with $512\times 512$ addressable $15\mu m\times 15\mu m$ pixels. The light sources are CW solid-state lasers with wavelength 532nm and 635 nm respectively. The bucket detector is replaced by a binocular camera (CCD) with $640\times 480$ addressable $3.0\mu m\times 3.0\mu m$. A $5\times 5\times 5cm^{3}$ object (Rubik’s Cube) located at a distance $L_{1}=100cm$ from the SLM, and a lens collects the reflected light onto the detector. In each realization a grayscale with $512\times 512$ is sent to the SLM. The distance between the two detectors of the binocular camera is 12 cm. The measurement error of binocular ranging is about 1%. The accuracy of the measurement can be higher by optimizing the algorithm and calibration.

Figure 4 compares the nondegenerate-wavelength CGI with different wavelengths of signal light and reference light. The experiment results show that a high-quality ghost image can be obtained even when the wavelengths of light used in the signal and reference arms are very different. Furthermore, changing the wavelength of the reference light does not significantly improve the resolution of ghost image. The color of non-degenerate wavelength ghost image depends on the light detected by the bucket detector [24,25]. It is noteworthy that the red color in Figs. 4(a)-4(d) is produced by laser-activated coatings, which does not affect the experimental results.

 

Fig. 4. The nondegenerate-wavelength computational ghost image with 500000 realizations. (a,e) object,(b-d) Parameters used are $\lambda _{s}=532nm$, $\lambda _{r}=400nm,532nm,635nm$. (f-h)The corresponding results with $\lambda _{s}=635nm$, $\lambda _{r}=400nm,532nm,635nm$.

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To verify the effect of strong atmospheric turbulence on non-degenerate wavelength CGI, turbulence is introduced by adding heating elements at ${550}^{\circ }\mathrm {C}$ underneath optical path [4]. Heating of the air causes temporal and spatial fluctuations on its index of refraction that makes the classical image of the object jitter about randomly on the image plane causing a “blurred” picture. Figure 5 shows that the spatial resolution of ghost image obtained by 635 nm (Fig. 5(d)) is better than that of 532 nm (Fig. 5(c)). The reason for this phenomenon is that strong turbulence has less effect on long wavelength laser than that on short wavelength laser [26]. Further studies have shown that this conclusion applies to all types of ghost imaging.

 

Fig. 5. The nondegenerate-wavelength computational ghost image with 500000 realizations. (a,b) object, (c) Parameters used are $\lambda _{r}=\lambda _{s}=532nm$. (d)The corresponding results with $\lambda _{r}=\lambda _{s} =635nm$.

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4. Discussions and conclusion

Shapiro creatively proposed the concept of CGI based on the modulation of light field by SLM. Compared with conventional GI, the CGI is more suitable for laser radar, remote sensing and other fields. Chan et al. proposed two-color GI based on the modulation effect of spatial light modulator on different wavelength light, and proved that the resolution of ghost image depends on each of these wavelengths in non-degenerate wavelength GI with thermal light [17]. Actually, two-color GI is still a conventional GI rather than CGI. Here, our results show that the resolution of non-degenerate wavelength CGI with thermal light depends primarily on the wavelength used to illuminate the object, but not on the calculated light. It is not difficult to understand that the correlation function of non-degenerate wavelength CGI is the same as that of two-color GI, because they are all based on the modulation of light field by spatial light modulator. Thus, the spatial intensity pattern of the light on the signal arm and computed (reference) arm is the same, which are similar to that of conventional GI [27].

In conclusion, we have shown that nondegenerate wavelength CGI with thermal light source is not only feasible but can also give a ghost image with high quality. The key to nondegenerate wavelength CGI is to use the binocular ranging principle to measure the distance of object in real time to obtain the calculated light field. Moreover, there is no fundamental limit in the extent for the wavelength in the reference arm as the corresponding field is completely calculated. It is out of the restriction of the spatial light modulator, which is better than the conventional two-color GI. More important, the results show that the longer wavelength laser is sent to illuminate the object in strong atmospheric turbulence, the higher resolution of ghost image can be obtained. In addition, this scheme may have potential imaging applications for moving objects, which will be discussed in detail in our future works.

Funding

National Natural Science Foundation of China (11704221, 11574178, 61675115); Natural Science Foundation of Shandong Province (ZR2016AP09, ZR2016JL005); Taishan Scholar Foundation of Shandong Province (tsqn201812059).

Acknowledgments

The authors wish to thank Dongfeng Shi for his contribution to resolution calculation.

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  1. T. B. Pittman, Y. 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]
  2. H. Wang, S. Han, and M. I. Kolobov, “Quantum limits of super-resolution via Sparsity constraint,” Opt. Express 20(21), 23235 (2012).
    [Crossref]
  3. C. Zhang, W. L. Gong, and S. S. Han, “Improving imaging resolution of shaking targets by Fourier-transform ghost diffraction,” Appl. Phys. Lett. 102(2), 021111 (2013).
    [Crossref]
  4. R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett. 98(11), 111115 (2011).
    [Crossref]
  5. D. F. Shi, C. Y. Fan, P. F. Zhang, H. Shen, J. H. Zhang, C. H. Qiao, and Y. J. Wang, “Two-wavelength ghost imaging through atmospheric turbulence,” Opt. Express 21(2), 2050–2064 (2013).
    [Crossref]
  6. 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]
  7. H. Yu, R. Lu, S. Han, H. Xie. G. Du. T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard X rays,” Phys. Rev. Lett. 117(11), 113901 (2016).
    [Crossref]
  8. A. Zhang, Y. He, L. Wu, L. Chen, and B. Wang, “Tabletop x-ray ghost imaging with ultra-low radiation,” Optica 5(4), 374–377 (2018).
    [Crossref]
  9. M. Bina, D. Magatti, M. Molteni, A. Gatti, L. A. Lugiato, and F. Ferri, “Backscattering Differential Ghost Imaging in Turbid Media,” Phys. Rev. Lett. 110(8), 083901 (2013).
    [Crossref]
  10. S. Li, F. Cropp, K. Kabra, T. J. Lane, G. Wetzstein, P. Musumeci, and D. Ratner, “Electron Ghost Imaging,” Phys. Rev. Lett. 121(11), 114801 (2018).
    [Crossref]
  11. D. Ratner, J. P. Cryan, T. J. Lane, S. Li, and G. Stupakov, “Pump-Probe Ghost Imaging with SASE FELs,” Phys. Rev. X 9(1), 011045 (2019).
    [Crossref]
  12. X. D. Qiu, D. K. Zhang, W. H. Zhang, and L. X. Chen, “Structured-Pump-Enabled Quantum Pattern Recognition,” Phys. Rev. Lett. 122(12), 123901 (2019).
    [Crossref]
  13. R. I. Khakimov, B. M. Henson, D. K. Shin, S. S. Hodgman, R. D. Dall, K. G . H. Baldwin, and A. G. Truscott, “Ghost imaging with atoms,” Nature 540(7631), 100–103 (2016).
    [Crossref]
  14. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), 061802 (2008).
    [Crossref]
  15. Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79(5), 053840 (2009).
    [Crossref]
  16. B. I. Erkmen, “Computational ghost imaging for remote sensing,” J. Opt. Soc. Am. A 29(5), 782–789 (2012).
    [Crossref]
  17. K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A 79(3), 033808 (2009).
    [Crossref]
  18. S. Karmakar and Y. Shih, “Two-color ghost imaging with enhanced angular resolving power,” Phys. Rev. A 81(3), 033845 (2010).
    [Crossref]
  19. N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A 84(6), 063824 (2011).
    [Crossref]
  20. P. Ben Dixon, G. A. Howland, K. W. Clifford 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]
  21. X. J. Zhao and C. J. Li, “Research of Key-tech in a Binocular Real-time Ranging System,” Laser & Infrared 36(9), 874–877 (2006).
  22. C. W. Wen, W. J. Chen, and Y. Zhang, “Embedded binocular stereo vision ranging system,” Computer Systems & Applications 21(10), 143–147 (2012).
  23. J. Cheng, “Ghost imaging through turbulent atmosphere,” Opt. Express 17(10), 7916–7921 (2009).
    [Crossref]
  24. D. Y. Duan, S. J. Du, and Y. J. Xia, “Multiwavelength ghost imaging,” Phys. Rev. A 88(5), 053842 (2013).
    [Crossref]
  25. X. L. Yin, Y. J. Xia, and D. Y. Duan, “Theoretical and experimental study of the color of ghost imaging,” Opt. Express 26(15), 18944–18949 (2018).
    [Crossref]
  26. M. W. Fitzmaurice, J. L. Bufton, and P. O. Minott, “Wavelength dependence of laser-beam scintillation,” J. Opt. Soc. Am. 59(1), 7–10 (1969).
    [Crossref]
  27. B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77(4), 043809 (2008).
    [Crossref]

2019 (2)

D. Ratner, J. P. Cryan, T. J. Lane, S. Li, and G. Stupakov, “Pump-Probe Ghost Imaging with SASE FELs,” Phys. Rev. X 9(1), 011045 (2019).
[Crossref]

X. D. Qiu, D. K. Zhang, W. H. Zhang, and L. X. Chen, “Structured-Pump-Enabled Quantum Pattern Recognition,” Phys. Rev. Lett. 122(12), 123901 (2019).
[Crossref]

2018 (3)

2016 (3)

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]

H. Yu, R. Lu, S. Han, H. Xie. G. Du. T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard X rays,” Phys. Rev. Lett. 117(11), 113901 (2016).
[Crossref]

R. I. Khakimov, B. M. Henson, D. K. Shin, S. S. Hodgman, R. D. Dall, K. G . H. Baldwin, and A. G. Truscott, “Ghost imaging with atoms,” Nature 540(7631), 100–103 (2016).
[Crossref]

2013 (4)

D. F. Shi, C. Y. Fan, P. F. Zhang, H. Shen, J. H. Zhang, C. H. Qiao, and Y. J. Wang, “Two-wavelength ghost imaging through atmospheric turbulence,” Opt. Express 21(2), 2050–2064 (2013).
[Crossref]

M. Bina, D. Magatti, M. Molteni, A. Gatti, L. A. Lugiato, and F. Ferri, “Backscattering Differential Ghost Imaging in Turbid Media,” Phys. Rev. Lett. 110(8), 083901 (2013).
[Crossref]

C. Zhang, W. L. Gong, and S. S. Han, “Improving imaging resolution of shaking targets by Fourier-transform ghost diffraction,” Appl. Phys. Lett. 102(2), 021111 (2013).
[Crossref]

D. Y. Duan, S. J. Du, and Y. J. Xia, “Multiwavelength ghost imaging,” Phys. Rev. A 88(5), 053842 (2013).
[Crossref]

2012 (3)

2011 (3)

R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett. 98(11), 111115 (2011).
[Crossref]

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

P. Ben Dixon, G. A. Howland, K. W. Clifford 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]

2010 (1)

S. Karmakar and Y. Shih, “Two-color ghost imaging with enhanced angular resolving power,” Phys. Rev. A 81(3), 033845 (2010).
[Crossref]

2009 (3)

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79(5), 053840 (2009).
[Crossref]

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

K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A 79(3), 033808 (2009).
[Crossref]

2008 (2)

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

B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77(4), 043809 (2008).
[Crossref]

2006 (1)

X. J. Zhao and C. J. Li, “Research of Key-tech in a Binocular Real-time Ranging System,” Laser & Infrared 36(9), 874–877 (2006).

1995 (1)

T. B. Pittman, Y. 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]

1969 (1)

. H. Baldwin, K. G

R. I. Khakimov, B. M. Henson, D. K. Shin, S. S. Hodgman, R. D. Dall, K. G . H. Baldwin, and A. G. Truscott, “Ghost imaging with atoms,” Nature 540(7631), 100–103 (2016).
[Crossref]

Ben Dixon, P.

P. Ben Dixon, G. A. Howland, K. W. Clifford 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]

Bina, M.

M. Bina, D. Magatti, M. Molteni, A. Gatti, L. A. Lugiato, and F. Ferri, “Backscattering Differential Ghost Imaging in Turbid Media,” Phys. Rev. Lett. 110(8), 083901 (2013).
[Crossref]

Boyd, R. W.

P. Ben Dixon, G. A. Howland, K. W. Clifford 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]

K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A 79(3), 033808 (2009).
[Crossref]

Bromberg, Y.

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79(5), 053840 (2009).
[Crossref]

Bufton, J. L.

Cantelli, V.

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]

Chan, K. W. C.

K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A 79(3), 033808 (2009).
[Crossref]

Chen, L.

Chen, L. X.

X. D. Qiu, D. K. Zhang, W. H. Zhang, and L. X. Chen, “Structured-Pump-Enabled Quantum Pattern Recognition,” Phys. Rev. Lett. 122(12), 123901 (2019).
[Crossref]

Chen, W. J.

C. W. Wen, W. J. Chen, and Y. Zhang, “Embedded binocular stereo vision ranging system,” Computer Systems & Applications 21(10), 143–147 (2012).

Cheng, J.

Clifford Chan, K. W.

P. Ben Dixon, G. A. Howland, K. W. Clifford 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]

Cropp, F.

S. Li, F. Cropp, K. Kabra, T. J. Lane, G. Wetzstein, P. Musumeci, and D. Ratner, “Electron Ghost Imaging,” Phys. Rev. Lett. 121(11), 114801 (2018).
[Crossref]

Cryan, J. P.

D. Ratner, J. P. Cryan, T. J. Lane, S. Li, and G. Stupakov, “Pump-Probe Ghost Imaging with SASE FELs,” Phys. Rev. X 9(1), 011045 (2019).
[Crossref]

Dall, R. D.

R. I. Khakimov, B. M. Henson, D. K. Shin, S. S. Hodgman, R. D. Dall, K. G . H. Baldwin, and A. G. Truscott, “Ghost imaging with atoms,” Nature 540(7631), 100–103 (2016).
[Crossref]

Deacon, K. S.

R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett. 98(11), 111115 (2011).
[Crossref]

Du, S. J.

D. Y. Duan, S. J. Du, and Y. J. Xia, “Multiwavelength ghost imaging,” Phys. Rev. A 88(5), 053842 (2013).
[Crossref]

Duan, D. Y.

Erkmen, B. I.

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

B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77(4), 043809 (2008).
[Crossref]

Fan, C. Y.

Ferri, F.

M. Bina, D. Magatti, M. Molteni, A. Gatti, L. A. Lugiato, and F. Ferri, “Backscattering Differential Ghost Imaging in Turbid Media,” Phys. Rev. Lett. 110(8), 083901 (2013).
[Crossref]

Fitzmaurice, M. W.

Gatti, A.

M. Bina, D. Magatti, M. Molteni, A. Gatti, L. A. Lugiato, and F. Ferri, “Backscattering Differential Ghost Imaging in Turbid Media,” Phys. Rev. Lett. 110(8), 083901 (2013).
[Crossref]

Gong, W. L.

C. Zhang, W. L. Gong, and S. S. Han, “Improving imaging resolution of shaking targets by Fourier-transform ghost diffraction,” Appl. Phys. Lett. 102(2), 021111 (2013).
[Crossref]

Han, S.

H. Yu, R. Lu, S. Han, H. Xie. G. Du. T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard X rays,” Phys. Rev. Lett. 117(11), 113901 (2016).
[Crossref]

H. Wang, S. Han, and M. I. Kolobov, “Quantum limits of super-resolution via Sparsity constraint,” Opt. Express 20(21), 23235 (2012).
[Crossref]

Han, S. S.

C. Zhang, W. L. Gong, and S. S. Han, “Improving imaging resolution of shaking targets by Fourier-transform ghost diffraction,” Appl. Phys. Lett. 102(2), 021111 (2013).
[Crossref]

Hardy, N. D.

P. Ben Dixon, G. A. Howland, K. W. Clifford 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]

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

He, Y.

Henson, B. M.

R. I. Khakimov, B. M. Henson, D. K. Shin, S. S. Hodgman, R. D. Dall, K. G . H. Baldwin, and A. G. Truscott, “Ghost imaging with atoms,” Nature 540(7631), 100–103 (2016).
[Crossref]

Hodgman, S. S.

R. I. Khakimov, B. M. Henson, D. K. Shin, S. S. Hodgman, R. D. Dall, K. G . H. Baldwin, and A. G. Truscott, “Ghost imaging with atoms,” Nature 540(7631), 100–103 (2016).
[Crossref]

Howell, J. C.

P. Ben Dixon, G. A. Howland, K. W. Clifford 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]

Howland, G. A.

P. Ben Dixon, G. A. Howland, K. W. Clifford 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]

Kabra, K.

S. Li, F. Cropp, K. Kabra, T. J. Lane, G. Wetzstein, P. Musumeci, and D. Ratner, “Electron Ghost Imaging,” Phys. Rev. Lett. 121(11), 114801 (2018).
[Crossref]

Karmakar, S.

S. Karmakar and Y. Shih, “Two-color ghost imaging with enhanced angular resolving power,” Phys. Rev. A 81(3), 033845 (2010).
[Crossref]

Katz, O.

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79(5), 053840 (2009).
[Crossref]

Khakimov, R. I.

R. I. Khakimov, B. M. Henson, D. K. Shin, S. S. Hodgman, R. D. Dall, K. G . H. Baldwin, and A. G. Truscott, “Ghost imaging with atoms,” Nature 540(7631), 100–103 (2016).
[Crossref]

Kolobov, M. I.

Lane, T. J.

D. Ratner, J. P. Cryan, T. J. Lane, S. Li, and G. Stupakov, “Pump-Probe Ghost Imaging with SASE FELs,” Phys. Rev. X 9(1), 011045 (2019).
[Crossref]

S. Li, F. Cropp, K. Kabra, T. J. Lane, G. Wetzstein, P. Musumeci, and D. Ratner, “Electron Ghost Imaging,” Phys. Rev. Lett. 121(11), 114801 (2018).
[Crossref]

Li, C. J.

X. J. Zhao and C. J. Li, “Research of Key-tech in a Binocular Real-time Ranging System,” Laser & Infrared 36(9), 874–877 (2006).

Li, S.

D. Ratner, J. P. Cryan, T. J. Lane, S. Li, and G. Stupakov, “Pump-Probe Ghost Imaging with SASE FELs,” Phys. Rev. X 9(1), 011045 (2019).
[Crossref]

S. Li, F. Cropp, K. Kabra, T. J. Lane, G. Wetzstein, P. Musumeci, and D. Ratner, “Electron Ghost Imaging,” Phys. Rev. Lett. 121(11), 114801 (2018).
[Crossref]

Lu, R.

H. Yu, R. Lu, S. Han, H. Xie. G. Du. T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard X rays,” Phys. Rev. Lett. 117(11), 113901 (2016).
[Crossref]

Lugiato, L. A.

M. Bina, D. Magatti, M. Molteni, A. Gatti, L. A. Lugiato, and F. Ferri, “Backscattering Differential Ghost Imaging in Turbid Media,” Phys. Rev. Lett. 110(8), 083901 (2013).
[Crossref]

Magatti, D.

M. Bina, D. Magatti, M. Molteni, A. Gatti, L. A. Lugiato, and F. Ferri, “Backscattering Differential Ghost Imaging in Turbid Media,” Phys. Rev. Lett. 110(8), 083901 (2013).
[Crossref]

Meyers, R. E.

R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett. 98(11), 111115 (2011).
[Crossref]

Minott, P. O.

Molteni, M.

M. Bina, D. Magatti, M. Molteni, A. Gatti, L. A. Lugiato, and F. Ferri, “Backscattering Differential Ghost Imaging in Turbid Media,” Phys. Rev. Lett. 110(8), 083901 (2013).
[Crossref]

Musumeci, P.

S. Li, F. Cropp, K. Kabra, T. J. Lane, G. Wetzstein, P. Musumeci, and D. Ratner, “Electron Ghost Imaging,” Phys. Rev. Lett. 121(11), 114801 (2018).
[Crossref]

O’Sullivan, M. N.

K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A 79(3), 033808 (2009).
[Crossref]

O’Sullivan-Hale, C.

P. Ben Dixon, G. A. Howland, K. W. Clifford 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]

Paganin, D. M.

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]

Pelliccia, D.

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]

Pittman, T. B.

T. B. Pittman, Y. 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]

Qiao, C. H.

Qiu, X. D.

X. D. Qiu, D. K. Zhang, W. H. Zhang, and L. X. Chen, “Structured-Pump-Enabled Quantum Pattern Recognition,” Phys. Rev. Lett. 122(12), 123901 (2019).
[Crossref]

Rack, A.

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]

Ratner, D.

D. Ratner, J. P. Cryan, T. J. Lane, S. Li, and G. Stupakov, “Pump-Probe Ghost Imaging with SASE FELs,” Phys. Rev. X 9(1), 011045 (2019).
[Crossref]

S. Li, F. Cropp, K. Kabra, T. J. Lane, G. Wetzstein, P. Musumeci, and D. Ratner, “Electron Ghost Imaging,” Phys. Rev. Lett. 121(11), 114801 (2018).
[Crossref]

Rodenburg, B.

P. Ben Dixon, G. A. Howland, K. W. Clifford 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]

Scheel, M.

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]

Sergienko, A. V.

P. Ben Dixon, G. A. Howland, K. W. Clifford 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]

T. B. Pittman, Y. 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]

Shapiro, J. H.

P. Ben Dixon, G. A. Howland, K. W. Clifford 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]

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

B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77(4), 043809 (2008).
[Crossref]

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

Shen, H.

Shi, D. F.

Shih, Y.

R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett. 98(11), 111115 (2011).
[Crossref]

S. Karmakar and Y. Shih, “Two-color ghost imaging with enhanced angular resolving power,” Phys. Rev. A 81(3), 033845 (2010).
[Crossref]

T. B. Pittman, Y. 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]

Shin, D. K.

R. I. Khakimov, B. M. Henson, D. K. Shin, S. S. Hodgman, R. D. Dall, K. G . H. Baldwin, and A. G. Truscott, “Ghost imaging with atoms,” Nature 540(7631), 100–103 (2016).
[Crossref]

Silberberg, Y.

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79(5), 053840 (2009).
[Crossref]

Simon, D. S.

P. Ben Dixon, G. A. Howland, K. W. Clifford 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]

Strekalov, D. V.

T. B. Pittman, Y. 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).
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Stupakov, G.

D. Ratner, J. P. Cryan, T. J. Lane, S. Li, and G. Stupakov, “Pump-Probe Ghost Imaging with SASE FELs,” Phys. Rev. X 9(1), 011045 (2019).
[Crossref]

Truscott, A. G.

R. I. Khakimov, B. M. Henson, D. K. Shin, S. S. Hodgman, R. D. Dall, K. G . H. Baldwin, and A. G. Truscott, “Ghost imaging with atoms,” Nature 540(7631), 100–103 (2016).
[Crossref]

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Wang, H.

Wang, Y. J.

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C. W. Wen, W. J. Chen, and Y. Zhang, “Embedded binocular stereo vision ranging system,” Computer Systems & Applications 21(10), 143–147 (2012).

Wetzstein, G.

S. Li, F. Cropp, K. Kabra, T. J. Lane, G. Wetzstein, P. Musumeci, and D. Ratner, “Electron Ghost Imaging,” Phys. Rev. Lett. 121(11), 114801 (2018).
[Crossref]

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Xia, Y. J.

Xie. G. Du. T. Xiao, H.

H. Yu, R. Lu, S. Han, H. Xie. G. Du. T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard X rays,” Phys. Rev. Lett. 117(11), 113901 (2016).
[Crossref]

Yin, X. L.

Yu, H.

H. Yu, R. Lu, S. Han, H. Xie. G. Du. T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard X rays,” Phys. Rev. Lett. 117(11), 113901 (2016).
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Zhang, C.

C. Zhang, W. L. Gong, and S. S. Han, “Improving imaging resolution of shaking targets by Fourier-transform ghost diffraction,” Appl. Phys. Lett. 102(2), 021111 (2013).
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X. D. Qiu, D. K. Zhang, W. H. Zhang, and L. X. Chen, “Structured-Pump-Enabled Quantum Pattern Recognition,” Phys. Rev. Lett. 122(12), 123901 (2019).
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Zhang, J. H.

Zhang, P. F.

Zhang, W. H.

X. D. Qiu, D. K. Zhang, W. H. Zhang, and L. X. Chen, “Structured-Pump-Enabled Quantum Pattern Recognition,” Phys. Rev. Lett. 122(12), 123901 (2019).
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C. W. Wen, W. J. Chen, and Y. Zhang, “Embedded binocular stereo vision ranging system,” Computer Systems & Applications 21(10), 143–147 (2012).

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X. J. Zhao and C. J. Li, “Research of Key-tech in a Binocular Real-time Ranging System,” Laser & Infrared 36(9), 874–877 (2006).

Zhu, D.

H. Yu, R. Lu, S. Han, H. Xie. G. Du. T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard X rays,” Phys. Rev. Lett. 117(11), 113901 (2016).
[Crossref]

Appl. Phys. Lett. (2)

C. Zhang, W. L. Gong, and S. S. Han, “Improving imaging resolution of shaking targets by Fourier-transform ghost diffraction,” Appl. Phys. Lett. 102(2), 021111 (2013).
[Crossref]

R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett. 98(11), 111115 (2011).
[Crossref]

Computer Systems & Applications (1)

C. W. Wen, W. J. Chen, and Y. Zhang, “Embedded binocular stereo vision ranging system,” Computer Systems & Applications 21(10), 143–147 (2012).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Laser & Infrared (1)

X. J. Zhao and C. J. Li, “Research of Key-tech in a Binocular Real-time Ranging System,” Laser & Infrared 36(9), 874–877 (2006).

Nature (1)

R. I. Khakimov, B. M. Henson, D. K. Shin, S. S. Hodgman, R. D. Dall, K. G . H. Baldwin, and A. G. Truscott, “Ghost imaging with atoms,” Nature 540(7631), 100–103 (2016).
[Crossref]

Opt. Express (4)

Optica (1)

Phys. Rev. A (9)

T. B. Pittman, Y. 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]

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

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79(5), 053840 (2009).
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Phys. Rev. Lett. (5)

X. D. Qiu, D. K. Zhang, W. H. Zhang, and L. X. Chen, “Structured-Pump-Enabled Quantum Pattern Recognition,” Phys. Rev. Lett. 122(12), 123901 (2019).
[Crossref]

M. Bina, D. Magatti, M. Molteni, A. Gatti, L. A. Lugiato, and F. Ferri, “Backscattering Differential Ghost Imaging in Turbid Media,” Phys. Rev. Lett. 110(8), 083901 (2013).
[Crossref]

S. Li, F. Cropp, K. Kabra, T. J. Lane, G. Wetzstein, P. Musumeci, and D. Ratner, “Electron Ghost Imaging,” Phys. Rev. Lett. 121(11), 114801 (2018).
[Crossref]

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]

H. Yu, R. Lu, S. Han, H. Xie. G. Du. T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard X rays,” Phys. Rev. Lett. 117(11), 113901 (2016).
[Crossref]

Phys. Rev. X (1)

D. Ratner, J. P. Cryan, T. J. Lane, S. Li, and G. Stupakov, “Pump-Probe Ghost Imaging with SASE FELs,” Phys. Rev. X 9(1), 011045 (2019).
[Crossref]

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Figures (5)

Fig. 1.
Fig. 1. Setup of the non-degenerate-wavelength computational ghost imaging with thermal light source. SLM: spatial light modulator, BBD: binocular bucket detector.
Fig. 2.
Fig. 2. (a) Width of the point-spread function as a function of $\lambda _{s}=800nm$, $\lambda _{r}=400,600,800,1000,1200,1400nm$. Parameters used are $\omega =5cm$, $l_{c}=1mm$, $z_{1}=1km$. (b)The corresponding plot of PSF as a function of $\lambda _{s}=800nm$, $\lambda _{r}=400,600,800,1000,1200,1400nm$.
Fig. 3.
Fig. 3. The PSF of the nondegenerate wavelength CGI with thermal light source. Parameters used are $\omega =5cm$, $l_{c}=1mm$, $z_{1}=1km$. (a-d) are the curve lines for $C_{n,s}^{2}=10^{-15}$, $10^{-14}$, $10^{-13}$, $10^{-12}$, $\lambda _{s}=800nm$, $\lambda _{r}=400,600,800,1000,1200,1400nm$. (e-h) are the curve lines for $C_{n,s}^{2}=10^{-15}$, $10^{-14}$, $10^{-13}$, $10^{-12}$, $\lambda _{r}=800nm$, $\lambda _{s}=400,600,800,1000,1200,1400nm$.
Fig. 4.
Fig. 4. The nondegenerate-wavelength computational ghost image with 500000 realizations. (a,e) object,(b-d) Parameters used are $\lambda _{s}=532nm$, $\lambda _{r}=400nm,532nm,635nm$. (f-h)The corresponding results with $\lambda _{s}=635nm$, $\lambda _{r}=400nm,532nm,635nm$.
Fig. 5.
Fig. 5. The nondegenerate-wavelength computational ghost image with 500000 realizations. (a,b) object, (c) Parameters used are $\lambda _{r}=\lambda _{s}=532nm$. (d)The corresponding results with $\lambda _{r}=\lambda _{s} =635nm$.

Equations (15)

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E s ( x s ) = 1 λ 1 z 1 z 2 d u d y E s ( λ s , u ) e i π λ s z 1 ( y u ) T ( y ) e i π λ s z 2 ( x s y ) ,
E r ( x r ) = ( i λ r z 3 ) 1 / 2 d u E r ( λ r , u ) e i π λ r z 3 ( u x r ) .
G ( x s , x r ) = I s ( x s ) I r ( x r ) I s ( x s ) I r ( x r ) = E s ( x s ) E s ( x s ) E r ( x r ) E r ( x r ) E s ( x s ) E s ( x s ) E r ( x r ) E r ( x r ) = 1 λ s 2 λ r z 1 z 2 z 3 d u 1 d u 1 d u 2 d u 2 d y d y C ( λ s , λ r ; u 1 , u 1 , u 2 , u 2 ) T ( y ) T ( y ) × e i π λ s z 1 [ ( y u 1 ) 2 ( y u 1 ) 2 ] e i π λ s z 2 [ ( x s y ) 2 ( x s y ) 2 ] e i π λ r z 3 [ ( u 2 x r ) 2 ( u 2 x r ) 2 ] ,
C ( λ s , λ r ; u 1 , u 1 , u 2 , u 2 ) = E s ( λ s , u 1 ) E s ( λ s , u 1 ) E r ( λ r , u 2 ) E r ( λ r , u 2 ) = E s ( λ s ) E s ( λ s ) E r ( λ r ) E r ( λ r ) V ( u 1 ) V ( u 2 ) V ( u 2 ) V ( u 1 )
V ( u 1 ) V ( u 2 ) = e u 1 2 + u 1 2 + u 2 2 + u 2 2 4 ω e ( u 1 u 2 ) 2 + ( u 2 u 1 ) 2 2 l c 2 ,
G ( x r ) = π 2 λ s λ r z 1 z 3 A B C D d y O ( y ) exp ( S 2 4 A + P 2 4 B + Q 2 4 C + R 2 4 D ) ,
A = 1 4 ω 2 + 1 2 l c 2 i π λ s z 1 , B = 1 4 ω 2 + 1 2 l c 2 + i π λ s z 1 , C = 1 4 ω 2 + 1 2 l c 2 i π λ r z 3 1 4 B l c 4 , D = 1 4 ω 2 + 1 2 l c 2 + i π λ r z 3 1 4 A l c 4 , S = 2 i π y λ s z 1 , P = 2 i π y λ s z 1 , Q = i π y B l c 2 λ s z 1 2 i π x 2 λ r z 3 , R = 2 i π x 2 λ r z 3 i π y A l c 2 λ s z 1 ,
G ( x r ) = π 2 λ s λ r z 1 z 3 A B C D exp ( x s 2 2 W f o v 2 ) d y O ( y ) exp ( ( y m x s 2 ) 2 W p s f 2 ) ,
W p s f = R e 1 2 ( K 1 2 + K 2 2 + K 3 2 + K 4 2 )
K 1 = π A λ s z 1 , K 2 = π B λ s z 1 , K 3 = π 2 B C λ s z 1 l c 2 , K 4 = π 2 D λ s z 1 ( 1 B l c 2 1 A l c 2 ) .
E s ( x s ) = 1 λ s z 1 z 2 d u d y E s ( λ s , u 1 ) e i π λ s z 1 ( y u a ) × exp [ ϕ 1 ( y , u ) ] T ( y ) exp [ ϕ 2 ( x , y ) ] T ( y ) e i π λ s z 2 ( x s y ) ,
G ( x r ) = π 2 λ s λ r z 1 z 3 A B C D exp ( x s 2 2 W f o v 2 ) × d y O ( y ) exp ( ( y m x s 2 ) 2 W p s f 2 ) .
W p s f = R e 1 2 ( K 1 2 + K 2 2 + K 3 2 + K 4 2 ) ,
A = 1 4 ω 2 + 1 2 l c 2 + 1 2 ρ 2 i π λ s z 1 , B = 1 4 ω 2 + 1 2 l c 2 + 1 2 ρ 2 + i π λ s z 1 1 4 A ρ 4 , C = 1 4 ω 2 + 1 2 l c 2 + 1 2 ρ 2 i π λ r z 3 1 4 B l c 4 D = 1 4 ω 2 + 1 2 l c 2 + i π λ r z 3 1 4 A l c 2 1 16 A 2 B l c 4 ρ 4 1 64 A 2 B 2 C l c 8 ρ 4 ,
K 1 = π A λ s z 1 , K 2 = π B λ s z 1 ( 1 1 2 A ρ 2 ) , K 3 = π 2 B C λ s z 1 l c 2 ( 1 1 2 A ρ 2 ) , K 4 = π 2 D λ s z 1 [ 1 A l c 2 + 1 B l c 2 ( 1 1 2 A ρ 2 ) × ( 1 2 A ρ 2 + 1 8 A B C l c 4 ρ 2 ) ] ,

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