• Chinese Optics Letters
  • Vol. 20, Issue 11, 112602 (2022)
Huan Zhao1, Xiaoqian Wang1、**, Chao Gao1, Zhuo Yu1、2, Shuang Wang1, Lidan Gou1, and Zhihai Yao1、*
Author Affiliations
  • 1Department of Physics, Changchun University of Science and Technology, Changchun 130022, China
  • 2School of Physics and Electronics, Baicheng Normal University, Baicheng 137000, China
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    DOI: 10.3788/COL202220.112602 Cite this Article Set citation alerts
    Huan Zhao, Xiaoqian Wang, Chao Gao, Zhuo Yu, Shuang Wang, Lidan Gou, Zhihai Yao. Second-order cumulants ghost imaging[J]. Chinese Optics Letters, 2022, 20(11): 112602 Copy Citation Text show less

    Abstract

    Ghost imaging (GI) is a technique to retrieve images by correlating intensity fluctuations. In this Letter, we present a novel scheme for GI referred to as second-order cumulants GI (SCGI). The image is retrieved from fluctuation information, and resolution may be enhanced compared to traditional GI. We experimentally performed SCGI image reconstruction, and the results are in agreement with theoretical predictions.

    1. Introduction

    Ghost imaging (GI) is an imaging technique based on second-order intensity correlation[110]. GI has many advantages compared to traditional imaging techniques. For example, thermal lens-less GI may be performed[11] even in the presence of atmospheric and instrumental fluctuations[1216]. The resolution is an important factor in evaluating image quality[17,18], and how to improve the resolution of GI is a key factor in the development of GI[1720].

    The spatial resolution of a GI system is limited by the point spread function (PSF) of the system, just as in traditional imaging[18]. In general, the resolution of GI is taken to be the full width at half-maximum (FWHM) of the PSF, which itself is approximately equal to the average size of the speckles[18]. In traditional imaging systems, many schemes have been proposed to improve resolution via reducing the impact of PSF[21,22]. Some schemes to improve the resolution of GI have also been proposed. Compressed sensing GI (CSGI) reduces the effect of PSF on the imaging quality using sparsity constraints to improve resolution[2328]. Han’s group reported a proof-of-principle experiment, where the resolution of a thermal light two-arm microscope scheme is improved by employing second-order intensity correlation imaging to narrow PSF[29]. The scheme has been implemented based on high-pass spatial-frequency filtering of the correlated intensity fluctuations[30]. The narrowing of PSF by higher-order correlation of non-Rayleigh speckle fields has been reported[31]. Other schemes to enhance the resolution of GI have also been suggested, such as spatial low-pass filtering[18,20], localizing and thresholding[32], preconditioned deconvolution methods[33], optical random speckle encoding based on hybrid wavelength and phase modulation scheme[34], deep neural network constraints[35], and speeded up robust features new sum of modified Laplacian (SURF-NSML)[36].

    In this Letter, we put forward a novel scheme to enhance the resolution of GI by narrowing the PSF, referred to as second-order cumulants GI (SCGI). In our scheme, the fluctuation information of GI is exploited, which contains more information than traditional GI data and allows one to improve the resolution. We theoretically analyze the feasibility of the scheme and its performance in improving the resolution by second-order cumulants and experimentally verify results using a double-slit object. We also show that second-order cumulants can be used together with other modified GI schemes, such as CSGI, to obtain images with higher resolution.

    2. Methods

    A typical GI experimental setup is shown in Fig. 1, where a double-slit object is employed to assess the resolution[29,37,38]. In this system, there is a monochromatic source of light at wavelength λ. A light beam from the source propagates to the object through an optical system with a PSF: h(x,α)=eikziλzexp[iπλz(xα)2],where k=2πλ, and z is the distance between the source and the object. x and α are the transverse coordinates on the source and the object plane, respectively. The light field at the object plane is E(α)=E(x)h(x,α)dx,where E(x) denotes the light field of source plane at x. The light intensity at the object plane is I(α)=E*(α)E(α).

    Experimental setup to assess the resolution of GI. There is a digital micro-mirror device (DMD) in the source. The light illuminates the object and then is collected by a bucket detector (Dt) with no spatial resolution.

    Figure 1.Experimental setup to assess the resolution of GI. There is a digital micro-mirror device (DMD) in the source. The light illuminates the object and then is collected by a bucket detector (Dt) with no spatial resolution.

    If T(α) represents the light field transmission function of the object, the light intensity at the bucket detector is given by Bt=I(α)|T(α)|2dα.

    The intensity fluctuations correlation between I(α) and Bt is ΔG(2)(α)=ΔI(α)ΔBt=[I(α)I(α)][BtBt]=[I(α)I(α)][I(α)|T(α)|2dαI(α)|T(α)|2dα]=|G(1)(x,x)T(α)h*(x,α)h(x,α)dxdx|2dα,where represents the ensemble average, and G(1)(x,x)=E*(x)E(x) is the first-order correlation function at the source. We consider a situation where light comes as a point-like source and is randomly and uniformly distributed on the source plane. If the light spot is located at x0, we have G(1)(x,x0)=I0δ(xx0), where I0 is the intensity of the source. Substituting G(1)(x,x) and Eq. (1) into Eq. (5), after some calculations, we arrive at ΔG(2)(α)=I02|T(α)|2sinc2[2πRλz(αα)]dα,where R is the radius of the light source. Obviously, the image resolution is constrained by this PSF, and the resolution is determined by the first zero of the sinc2 function in Eq. (6).

    I0 is usually assumed constant in traditional GI, i.e., one assumes that the emitting power of the light source is perfectly stable. In fact, the emitting power of the light source cannot be kept stable. Thus, ΔG(2)(α) in Eq. (6) should be substituted with ΔG(2)(I0,α). Fluctuations of I0 lead to fluctuations of ΔG(2)(I0,α). We use the concept of cumulants to describe the fluctuations of ΔG(2)(I0,α) since they contain more information than ΔG(2)(I0,α) itself. The cumulant-generating function K(s,α) is defined as K(s,α)=ln{exp[sΔG(2)(I0,α)]}=n=1κn(α)snn!=μ(α)×s+σ2(α)×s22+,where κn(α) is the nth-order cumulants, μ(α)=ΔG(2)(I0,α), and σ(β)=ΔG(2)(I0,α)ΔG(2)(I0,α). The nth-order cumulant is given by κn(α)=d(n)K(s,α)ds(n)|s=0.

    In order to minimize the imaging time, we consider the second-order cumulants, which can be written as κ2(α)=[ΔG(2)(I0,α)ΔG(2)(I0,α)]2=κ2(α,α)dα+L(α),where κ2(α,α)=[ΔG(2)(I0,α,α)ΔG(2)(I0,α,α)]2=(I02I02)2×|T(α)|4×sinc4[2πRλz(αα)],and L(α)=ααα[ΔG(2)(I0,α,α)ΔG(2)(I0,α,α)][ΔG(2)(I0,α,α)ΔG(2)(I0,α,α)]dαdα=(I02I02)2×ααα|T(α)|2|T(α)|2×sinc2[2πRλz(αα)]sinc2[2πRλz(αα)]dαdα,where ΔG(2)(I0,α,α)=I02|T(α)|2sinc2[2πRλz(αα)] is the correlation between the intensity fluctuations at α and α. As a matter of fact, κ2(α) contains the information about fluctuations of ΔG(2)(I0,α) and κ2(α,α) about those of ΔG(2)(I0,α,α). L(α) is the cross-information generated correlating ΔG(2)(I0,α,α)ΔG(2)(I0,α,α) and ΔG(2)(I0,α,α)ΔG(2)(I0,α,α) for all different α and α (αα). From Eq. (9), we see that κ2(α) is written in terms of κ2(α,α) and L(α). According to Eqs. (10) and (11), the FWHMs of κ2(α,α) and L(α) are influenced by z and λ. In particular, they increase with z or λ. One has L(α)=0 if there is no cross interference between any two points on the object plane. In the following, we make use of κ2(α) instead of ΔG(2)(I0,α) to reconstruct the image of the object, and, for this reason, we refer to our scheme as SCGI.

    From Eqs. (9)–(11), we see that the intensity PSF of SCGI corresponds to a sinc4 function, whereas in traditional GI the form of PSF scales as sinc2. In turn, the FWHM of the PSF in Eq. (6) is larger than that in Eq. (9). In order to address a concrete example, we set λ=550nm, z=0.8m, and R=1mm. For a pinhole-like object at α=0, the imaging results are shown in Fig. 2.

    PSF of a pinhole object at α′ = 0 for λ = 550 nm, z = 0.8 m, and R = 0.001 m by traditional GI (blue solid curve) and SCGI (red dash curve).

    Figure 2.PSF of a pinhole object at α′ = 0 for λ = 550 nm, z = 0.8 m, and R = 0.001 m by traditional GI (blue solid curve) and SCGI (red dash curve).

    From Eq. (9), we see that L(α) affects the resolution of SCGI. In order to understand how, we consider a situation where the object is made of two pinholes placed at α and α=α, respectively. The imaging results are shown in Fig. 3. Looking at Figs. 3(a) and 3(b), we see that the image of the object for L(α)=0 is clearer than for the case L(α)0. This is because cross information, which cannot distinguish between κ2(α,α) and κ2(α,α), is present when L(α)0.

    Image of the two-pinhole object by SCGI when α′ = −0.1 mm, α′′ = 0.1 mm, λ = 550 nm, R = 0.001 m, and z = 0.8 m: (a) L(α) = 0; (b) L(α) ≠ 0.

    Figure 3.Image of the two-pinhole object by SCGI when α′ = −0.1 mm, α′′ = 0.1 mm, λ = 550 nm, R = 0.001 m, and z = 0.8 m: (a) L(α) = 0; (b) L(α) ≠ 0.

    Compared to traditional GI, the resolution of SCGI improves even when L(α)0. This may be seen as follows, using the Rayleigh criterion to assess the resolution of the GI[39], i.e., looking at the minimum separation between two incoherent point sources (α0 and α0, we set α0=α0 for the sake of simplicity) that may be resolved into distinct objects[39]. For traditional GI, since the intensity PSF is a sinc2 function, the Rayleigh distance is d1=|α0α0| when ΔG(2)(I0,0)/ΔG(2)(I0,α0)0.81 [we assume |T(α0)|2=|T(α0)|2=1]. On the other hand, from Eq. (9), we have that κ2(0)/κ2(α0)=0.6561<0.81 for d1=|α0α0| by Eq. (9), i.e., SCGI shows enhanced resolution compared to traditional GI.

    Second-order cumulants can also be used in other modified GI schemes, such as CSGI, which itself aims at improving the resolution of GI by reducing the effect of PSF on the information carried by ΔG(2)(I0,α). κ2(α) is the fluctuation information of ΔG(2)(I0,α) and contains more information than ΔG(2)(I0,α), such that it can be used in CSGI to further enhance the spatial resolution just as in traditional GI.

    3. Results

    We experimentally verify our theoretical predictions by using a computational GI setup. The light source is a projector (XE11F), and there is a digital mirror device (DMD) in the source. An optical spatial filter with a central wavelength of 550 nm is inserted in the light beam behind the projector. A bucket detector is composed of a lens and an optical detection circuit (LSSPD-2.5-3 P-08.26). The object is a double slit with width a=0.8mm, slits center distance b=1.2mm, and slit height g=8mm. The distance between the source and the object is z=0.8m.

    First, we demonstrate that second-order cumulants can be used in traditional GI to enhance the resolution. In particular, we measure the resolution of traditional GI and traditional GI with κ2(α) in the same conditions. Results are obtained by averaging over 50,000 exposure frames. For κ2(α), we get ΔG(2)(I0,α) every 5000 steps. The experimental results are shown in Figs. 4(a) and 4(b). According to our theoretical analysis, the PSF of ΔG(2)(I0,α) can be narrowed by κ2(α). In Fig. 4(a), we find that the double slit cannot be distinguished. However, in Fig. 4(b), the double slit can be distinguished using κ2(α). That means the PSF of ΔG(2)(I0,α) is wider than the PSF of κ2(α). The resolution obtained by κ2(α) is improved compared to ΔG(2)(I0,α). The experimental results are consistent with our theoretical analysis.

    Experimental results for a double slit: (a)–(d) show results by traditional GI, SCGI [κ2(α) of traditional GI], CSGI, and SCGI [κ2(α) of CSGI], respectively.

    Figure 4.Experimental results for a double slit: (a)–(d) show results by traditional GI, SCGI [κ2(α) of traditional GI], CSGI, and SCGI [κ2(α) of CSGI], respectively.

    We then verify that second-order cumulants can also be used in other modified GI schemes. In particular, with the same setup, we prove experimentally that κ2(α) can be used in the CSGI scheme. Here, for CSGI, we obtain ΔG(2)(I0,α) with 3000 steps. Experimental results obtained by CSGI and CSGI with κ2(α) are shown in Figs. 4(c) and 4(d), respectively. We see that the resolution obtained by CSGI with κ2(α) is improved compared to CSGI.

    4. Discussion

    In conclusion, we have used κ2(α) instead of ΔG(2)(I0,α) to reconstruct the image of the object in the GI system. We have termed this protocol SCGI. Our theoretical analysis and experimental results show that the resolution of GI can be enhanced by SCGI without changing the experimental setup of GI. In order to verify the performance of the protocol, we applied κ2(α) to the CSGI scheme and obtained images with higher resolution than those obtained by CSGI. Similarly, κ2(α) can also be used in other modified GI to enhance the resolution, such as spatial low-pass filter, localizing, and thresholding schemes. We will discuss them elsewhere.

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    Huan Zhao, Xiaoqian Wang, Chao Gao, Zhuo Yu, Shuang Wang, Lidan Gou, Zhihai Yao. Second-order cumulants ghost imaging[J]. Chinese Optics Letters, 2022, 20(11): 112602
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