Ghost imaging (GI), as a nonlocal imaging technique that reconstructs an unknown object by calculating the intensity fluctuation correlation between two spatially independent beams, was first implemented by using entangled photon pairs in 1995 ^[1,2]. Since 2002, the theoretical and the experimental results have demonstrated that GI could also be implemented with thermal light ^[3–6]. In recent years, most works have focused on GI with thermal light because it can provide more practical applications when compared with the entangled source, and some new methods have been proposed to improve imaging quality, such as pseudo-inverse GI ^[7], compressive sensing GI ^[8,9], three-dimensional GI ^[10], differential GI ^[11], high-order GI ^[12], normalized GI ^[13], and binary ghost imaging (BGI) ^[14,15].

When considering the practical applications of GI over a long distance, the influences from the path disturbance, such as atmospheric turbulence ^[16,17] and scattering ^[18–22], have to be considered. Here, the scattering medium is usually classified into Mie scattering ^[18,22] and Rayleigh scattering ^[23] according to the size of the scattering particles and the light wavelength. In most cases, Mie scattering is more common. Gong and Han first demonstrated that the image of a transmission object with high quality can still be obtained in GI with Mie scattering media ^[18]. Xu et al. proved that GI with a transmission target has intrinsic robustness in the scattering medium when compared with traditional imaging ^[22]. The robustness is only for imaging resolution. In fact, signal-to-noise ratio (SNR) or contrast-to-noise ratio (CNR) will be affected with the occurrence of the scattering medium. In addition, reflective GI is closer to the demands of practical applications ^[24,25] than GI with the transmission target. In this paper, the effect of the scattering medium on reflective GI under different incident angles is investigated experimentally. We are surprised to find that the existence of the scattering medium can make the imaging quality of reflective GI improve when BGI is applied by comparing it with the result with no scattering medium. In other words, the scattering medium can play an active role in improving imaging quality. The corresponding numerical results also verify our conclusion.

Figure 1 presents the experimental setup of reflective GI in the scattering medium. The pseudo-thermal light source which is obtained by modulating a CW laser (λ=532nm) from a diode pump solid-state laser into a slowly rotating ground glass is divided by a non-polarizing beam splitter (BS) into two beams, and then the two beams propagate through two paths (a test path and a reference path), which are described by their impulse response functions, ht(u2,xt) and hr(u1,xr), respectively. Here, ui (i=1, 2) represents the transverse coordinate of the source plane. The test beam propagates through a transparent cube glass with a side length of d=40mm, and the container is full of the suspension liquid which is composed of the NaCl solution with a density of ρ=1.19g/cm3 and a certain concentration of latex spherical calibration particles (GBW(E)120002a with a particle diameter of a=3.5μm). The solution satisfies the condition of the Mie scattering (2πa/λ=41.31<50) ^[23] and is employed to simulate the scattering medium. A reflective object is immersed in the scattering medium. Then, all the randomly scattered and reflected photons from the surface of the object are collected by a bucket detector Dt. The reference path contains nothing, and the beam propagates freely from the source to a CCD camera Dr with high spatial resolution, which is used to record the intensity distribution at xr.

According to the optical coherence theory, the information about the object imaged can be constructed by the correlation function of intensity fluctuations between Dt and Dr ^[4] as follows: where ⟨E*(u1)E(u2)⟩ is the first-order correlation function of the source and has where a is the transverse size of the source, and δ(u) is the delta function.

In the test arm, a reflective object with a smooth surface is placed at a distance z1 from the source, and the bucket detector is fixed at the distance z2 from the object. The corresponding impulse response function can be written as where x1 and x2 signify the transverse coordinates of the incident and the output planes of the scattering medium, respectively. t(ξ) represents the reflectivity of the object with ξ being the object plane, and θi and θo, respectively, signify the incident angle and the reflective angle. Here, the length of the scattering medium is ignored because its length is much shorter than that of the test arm, and the random phase of the scattering medium is considered as the phase screen ^[21,26]. ϕ(x1) and ϕ(x2) represent the phase shifts of the random phase screens before and after the reflective object, respectively. Their spatial autocorrelation function is obtained according to the statistical property of the random phase screen ^[26] as follows: where σ is the variance of the random phase which reflects the strength of the scattering medium, and s is a constant which is associated with the autocorrelation function of the phase.

As for the reference path, the light travels freely. Thus, the impulse response function has the form Substituting Eqs. (2)–(5) into Eq. (1), and following the related calculation methods in our previous work ^[27], Eq. (1) can be simplified as where sinc(x)=sin(πx)/πx. If there are N shot measurements, then the SNR can be defined as the ratio of the imaging signal and its fluctuation according to Ref. ^[28] as follows: where M=∫dξdx1dx1′[t(ξ)]2exp{-σ2[1-e-(x1-x1′)2s2]}. From Eq. (7) and considering Eq. (6), the SNR decreases with the increase of the scattering intensity or the decrease of the incident angle. It is obvious that the existence of the scattering medium has a negative effect on the SNR.

In our experiment, the imaging object is chosen in the form of two very thin reflecting lines with a slit width of 0.4 mm and a center-to-center separation of 0.8 mm on the black stainless steel, which is immersed in the middle of the scattering medium. A lens with a focal length of 50 mm is used to focus all the reflected and scattered photons onto the active area of the bucket detector. The transverse size of the source a=1mm, the distances satisfy z0=z1=28cm, and z2=50mm. The length of the scattering medium before and after the object Δz=d/2=20mm. Obviously, Δz≪z0.

First, we experimentally investigate the effects from the scattering medium on reflective GI under different incident angles. Here, we add 0 mL, 1.2 mL, and 2.4 mL latex spherical calibration particles to NaCl solution, which correspond to no scattering, moderate scattering, and strong scattering, respectively. The corresponding results are plotted in Figs. 2(a)–2(d), and all patterns are obtained with 3000 realizations. The relationships between the SNR and the CNR ^[29] of the ghost-images are shown in Figs. 2(b)–2(d), and the concentration of the scattering medium under different incident angles is plotted in Figs. 2(e) and 2(f), respectively. It is shown that the background is not good, though no scattering medium is considered in Fig. 2(a), which mainly comes from the noise of the bucket detector and can be decreased greatly by a modified compressive sensing (MCS) method we presented in Ref. ^[29], as shown in Fig. 2(b). Here, the resolution is defined as the product of the number of pixels of center-center separation and the pixel size of the detector 3.45 μm. By comparing the results when moderate scattering is used in Fig. 2(c) with those under no scattering in Fig. 2(b), the imaging resolution almost stays unchanged [the resolution is 220×3.45μm=0.76mm in Figs. 2(a1)–2(c1) and 163×3.45μm=0.56mm in Figs. 2(a2)–2(c2)], which confirms the robustness of reflective GI ^[22]. The information of the object can be hardly extracted under the strong scattering, which means that the robustness is destroyed [see Figs. 2(d)]. During this process, it is also noted that an increase of the scattering strength or a decrease of the incident angle results in the decrease of the SNR and the CNR, which are in accordance with the analytical results in Eq. (7).

We are now in a position to discuss the role that the scattering medium plays in reflective GI. As we all know, the scattering is generally believed to have a negative influence on the imaging process. In fact, the CNR of reflective GI does get worse with the existence of the scattering medium, although GI has robustness for the imaging resolution when compared with the traditional imaging, as shown in Fig. 2. Next, we focus on improving the poor CNR of reflective GI with the scattering medium and re-analyzing the role of the scattering medium.

We noted that Guo et al. proposed a scheme to improve the CNR of the ghost-image by adding the random noises with the probability distribution functions being rectangular and triangular to the row data before quantization in binary ghost imaging ^[14]. In our reflective GI scheme with the scattering medium, we calculate the probability distribution function of the speckle patterns from the scattering medium, and it is approximately triangular [see Fig. 1(b)], which is the same as that depicted in Fig. 4(c) in Ref. ^[14]. It is natural to ask if the effect from the scattered medium behaves as the random noise which can be used to improve imaging quality when the data of the test detector are recorded with less bits, i.e., experiencing quantization. To answer this question, contrast experiments of GI and BGI under different conditions are implemented, and the corresponding results are shown in Fig. 3. Here, the value of the CNR of every figure is marked at bottom right. During BGI, 1-bit quantization is used on the data of the test detector, which turns the data into two levels (0 and 1). According to the usual analysis ^[15], the less bits recorded will result in the decline of the imaging quality, as shown in Figs. 3(a) and 3(c). It is quite interesting to find that the CNR is improved under different concentrations of the scattering medium and incident angles when the scattering medium is used, as shown in Figs. 3(b), 3(d), and 3(e). The reason for this phenomenon is the existence of the scattering medium by comparing Figs. 3. From Ref. ^[14], adding random noise with the triangular or rectangular distribution to the input signal at the bucket detector can randomize the output, which leads to the independent and homogeneous quantization error, thus less distortion between the input and the output. The probability distribution function of the speckle patterns from the scattering medium shown in Fig. 1(b) is triangular, which is similar with that of the random noise used in Ref. ^[14]. So, the randomly distributed speckle from the scattering medium behaves as the random noise, and can be used to compensate for imaging quality decline after quantization. In other words, the scattering medium can have a positive effect on imaging quality when BGI is applied.

Next, we further confirm our conclusion by the numerical results from Eq. (6). During this process, σ=0.2μm, s=20μm, and other parameters are the same as those in Fig. 2(a2). Figure 4(a) shows the retrieved images (128×128pixels) of GI and BGI with no scattering, and the corresponding CNRs are 3.96 and 3.83, respectively. Ghost-images of GI and BGI with the scattering medium are presented in Fig. 4(b), and the CNRs are 1.59 and 1.64, respectively. From the simulation results, one can easily see that BGI will deteriorate imaging quality when no scattering is considered. The case is quite different with the occurrence of the scattering medium, and BGI makes an active effect on improving imaging quality, which is in accordance with the experimental results. In addition, the changes of the theoretical values of the CNR (see Fig. 4) are basically the same as those in the experiment (see Fig. 3).

We also implement the corresponding experiments by using the carbon powder from a selenium drum as the scattering particles, and the results show that BGI still works to improve imaging quality. Whether this method is universally applicable needs further discussion in future work when other kinds of scattering media are chosen.

In conclusion, we have experimentally investigated reflective GI with a scattering medium. It is shown that the SNR and the CNR of the ghost-image decrease when the strength of the scattering medium is increased, although GI has the intrinsic robustness for imaging resolution. During this process, the scattering medium has a negative influence. The case becomes quite different when BGI is used. Our results show that the CNR is improved effectively, i.e., the scattering medium can have a positive effect on imaging quality. By the experimental and the numerical results, it is shown that the existence of the scattering medium is just the cause of this phenomenon.