Ghost imaging (GI) is an imaging technique based on second-order intensity correlation^[1–10]. 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^[12–16]. 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^[17–20].

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^[23–28]. 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.

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 $\lambda$. A light beam from the source propagates to the object through an optical system with a PSF: $$h(x,\alpha )=\frac{e^{-ikz}}{i\lambda z}\exp \left[\frac{-i\pi }{\lambda z}{(x-\alpha )}^2\right],$$(1)where $k=\frac{2\pi }{\lambda }$, and $z$ is the distance between the source and the object. $x$ and $\alpha$ are the transverse coordinates on the source and the object plane, respectively. The light field at the object plane is $$E(\alpha )=\int E(x)h(x,\alpha )\mathrm{d}x,$$(2)where $E(x)$ denotes the light field of source plane at $x$. The light intensity at the object plane is $$I(\alpha )=E^{*}(\alpha )E(\alpha ).$$(3)

If $T(\alpha )$ represents the light field transmission function of the object, the light intensity at the bucket detector is given by $$B_t=\int I(\alpha ){|T(\alpha )|}^2\mathrm{d}\alpha .$$(4)

The intensity fluctuations correlation between $I(\alpha )$ and $B_t$ is $$\begin{gathered} \mathrm{\Delta }{G}^{(2)}(\alpha )=\langle \mathrm{\Delta }I(\alpha )\mathrm{\Delta }{B}_t\rangle =\langle [I(\alpha )-\langle I(\alpha )\rangle ][B_t-\langle B_t\rangle ]\rangle =\langle [I(\alpha )-\langle I(\alpha )\rangle ][\int I({\alpha }^{\prime }){|T({\alpha }^{\prime })|}^2\mathrm{d}{\alpha }^{\prime }-\langle \int I({\alpha }^{\prime }){|T({\alpha }^{\prime })|}^2\mathrm{d}{\alpha }^{\prime }\rangle ]\rangle \\ =\int {|\int G^{(1)}(x,x^{\prime })T({\alpha }^{\prime })h^{*}(x,\alpha )h(x^{\prime },{\alpha }^{\prime })\mathrm{d}x\mathrm{d}{x}^{\prime }|}^2\mathrm{d}{\alpha }^{\prime }, \end{gathered}$$(5)where $\langle \cdots \rangle$ represents the ensemble average, and $G^{(1)}(x,x^{\prime })=\langle E^{*}(x)E(x^{\prime })\rangle$ 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 $x_0$, we have $G^{(1)}(x,x_0)=I_0\delta (x-x_0)$, where $I_0$ is the intensity of the source. Substituting $G^{(1)}(x,x^{\prime })$ and Eq. (1) into Eq. (5), after some calculations, we arrive at $$\mathrm{\Delta }{G}^{(2)}(\alpha )=I_0^2\int {|T({\alpha }^{\prime })|}^2{\mathrm{sinc}}^2[\frac{2\pi R}{\lambda z}(\alpha -{\alpha }^{\prime })]\mathrm{d}{\alpha }^{\prime },$$(6)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 ${\mathrm{sinc}}^2$ function in Eq. (6).

$I_0$ 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, $\mathrm{\Delta }{G}^{(2)}(\alpha )$ in Eq. (6) should be substituted with $\mathrm{\Delta }{G}^{(2)}(I_0,\alpha )$. Fluctuations of $I_0$ lead to fluctuations of $\mathrm{\Delta }{G}^{(2)}(I_0,\alpha )$. We use the concept of cumulants to describe the fluctuations of $\mathrm{\Delta }{G}^{(2)}(I_0,\alpha )$ since they contain more information than $\mathrm{\Delta }{G}^{(2)}(I_0,\alpha )$ itself. The cumulant-generating function $K(s,\alpha )$ is defined as $$\begin{gathered} K(s,\alpha )=\ln \{\langle \exp [s\mathrm{\Delta }{G}^{(2)}(I_0,\alpha )]\rangle \}=\sum _{n=1}^{\infty }{\kappa }_n(\alpha )\frac{s^n}{n!} \\ =\mu (\alpha )\times s+{\sigma }^2(\alpha )\times \frac{s^2}{2}+\cdots , \end{gathered}$$(7)where ${\kappa }_n(\alpha )$ is the $n$th-order cumulants, $\mu (\alpha )=\langle \mathrm{\Delta }{G}^{(2)}(I_0,\alpha )\rangle$, and $\sigma (\beta )=\langle \mathrm{\Delta }{G}^{(2)}(I_0,\alpha )-\langle \mathrm{\Delta }{G}^{(2)}(I_0,\alpha )\rangle \rangle$. The $n$th-order cumulant is given by $${\kappa }_n(\alpha )={\frac{{\mathrm{d}}^{(n)}K(s,\alpha )}{\mathrm{d}{s}^{(n)}}|}_{s=0}.$$(8)

In order to minimize the imaging time, we consider the second-order cumulants, which can be written as $$\begin{gathered} {\kappa }_2(\alpha )=\langle {[\mathrm{\Delta }{G}^{(2)}(I_0,\alpha )-\langle \mathrm{\Delta }{G}^{(2)}(I_0,\alpha )\rangle ]}^2\rangle \\ =\int {\kappa }_2(\alpha ,{\alpha }^{\prime })\mathrm{d}{\alpha }^{\prime }+L(\alpha ), \end{gathered}$$(9)where $$\begin{gathered} {\kappa }_2(\alpha ,{\alpha }^{\prime })=\langle {[\mathrm{\Delta }{G}^{(2)}(I_0,\alpha ,{\alpha }^{\prime })-\langle \mathrm{\Delta }{G}^{(2)}(I_0,\alpha ,{\alpha }^{\prime })\rangle ]}^2\rangle \\ =\langle {(I_0^2-\langle I_0^2\rangle )}^2\rangle \times {|T({\alpha }^{\prime })|}^4\times {\mathrm{sinc}}^4[\frac{2\pi R}{\lambda z}(\alpha -{\alpha }^{\prime })], \end{gathered}$$(10)and $$\begin{gathered} L(\alpha )={\int }_{{\alpha }^{\prime }}{\int }_{{\alpha }^{\prime \prime }\ne {\alpha }^{\prime }}\langle [\mathrm{\Delta }{G}^{(2)}(I_0,\alpha ,{\alpha }^{\prime }) \\ -\langle \mathrm{\Delta }{G}^{(2)}(I_0,\alpha ,{\alpha }^{\prime })\rangle ][\mathrm{\Delta }{G}^{(2)}(I_0,\alpha ,{\alpha }^{\prime \prime }) \\ -\langle \mathrm{\Delta }{G}^{(2)}(I_0,\alpha ,{\alpha }^{\prime \prime })\rangle ]\rangle \mathrm{d}{\alpha }^{\prime }\mathrm{d}{\alpha }^{\prime \prime }=\langle {(I_0^2-\langle I_0^2\rangle )}^2\rangle \times {\int }_{{\alpha }^{\prime }}{\int }_{{\alpha }^{\prime \prime }\ne {\alpha }^{\prime }}{|T({\alpha }^{\prime })|}^2{|T({\alpha }^{\prime \prime })|}^2\times {\mathrm{sinc}}^2[\frac{2\pi R}{\lambda z}(\alpha -{\alpha }^{\prime })]{\mathrm{sinc}}^2[\frac{2\pi R}{\lambda z}(\alpha -{\alpha }^{\prime \prime })]\mathrm{d}{\alpha }^{\prime }\mathrm{d}{\alpha }^{\prime \prime }, \end{gathered}$$(11)where $\mathrm{\Delta }{G}^{(2)}(I_0,\alpha ,{\alpha }^{\prime })=I_0^2{|T({\alpha }^{\prime })|}^2{\mathrm{sinc}}^2[\frac{2\pi R}{\lambda z}(\alpha -{\alpha }^{\prime })]$ is the correlation between the intensity fluctuations at $\alpha$ and ${\alpha }^{\prime }$. As a matter of fact, ${\kappa }_2(\alpha )$ contains the information about fluctuations of $\mathrm{\Delta }{G}^{(2)}(I_0,\alpha )$ and ${\kappa }_2(\alpha ,{\alpha }^{\prime })$ about those of $\mathrm{\Delta }{G}^{(2)}(I_0,\alpha ,{\alpha }^{\prime })$. $L(\alpha )$ is the cross-information generated correlating $\mathrm{\Delta }{G}^{(2)}(I_0,\alpha ,{\alpha }^{\prime })-\langle \mathrm{\Delta }{G}^{(2)}(I_0,\alpha ,{\alpha }^{\prime })\rangle$ and $\mathrm{\Delta }{G}^{(2)}(I_0,\alpha ,{\alpha }^{\prime \prime })-\langle \mathrm{\Delta }{G}^{(2)}(I_0,\alpha ,{\alpha }^{\prime \prime })\rangle$ for all different ${\alpha }^{\prime }$ and ${\alpha }^{\prime \prime }$ (${\alpha }^{\prime }\ne {\alpha }^{\prime \prime }$). From Eq. (9), we see that ${\kappa }_2(\alpha )$ is written in terms of ${\kappa }_2(\alpha ,{\alpha }^{\prime })$ and $L(\alpha )$. According to Eqs. (10) and (11), the FWHMs of ${\kappa }_2(\alpha ,{\alpha }^{\prime })$ and $L(\alpha )$ are influenced by $z$ and $\lambda$. In particular, they increase with $z$ or $\lambda$. One has $L(\alpha )=0$ if there is no cross interference between any two points on the object plane. In the following, we make use of ${\kappa }_2(\alpha )$ instead of $\mathrm{\Delta }{G}^{(2)}(I_0,\alpha )$ 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 ${\mathrm{sinc}}^4$ function, whereas in traditional GI the form of PSF scales as ${\mathrm{sinc}}^2$. 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 $\lambda =550\mathrm{nm}$, $z=0.8\mathrm{m}$, and $R=1\mathrm{mm}$. For a pinhole-like object at ${\alpha }^{\prime }=0$, the imaging results are shown in Fig. 2.

From Eq. (9), we see that $L(\alpha )$ affects the resolution of SCGI. In order to understand how, we consider a situation where the object is made of two pinholes placed at ${\alpha }^{\prime }$ and ${\alpha }^{\prime \prime }=-{\alpha }^{\prime }$, 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 $L(\alpha )=0$ is clearer than for the case $L(\alpha )\ne 0$. This is because cross information, which cannot distinguish between ${\kappa }_2(\alpha ,{\alpha }^{\prime })$ and ${\kappa }_2(\alpha ,{\alpha }^{\prime \prime })$, is present when $L(\alpha )\ne 0$.

Compared to traditional GI, the resolution of SCGI improves even when $L(\alpha )\ne 0$. 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 (${\alpha }_0$ and ${\alpha }_0^{\prime }$, we set ${\alpha }_0=-{\alpha }_0^{\prime }$ for the sake of simplicity) that may be resolved into distinct objects^[39]. For traditional GI, since the intensity PSF is a ${\mathrm{sinc}}^2$ function, the Rayleigh distance is $d_1=|{\alpha }_0-{\alpha }_0^{\prime }|$ when $\mathrm{\Delta }{G}^{(2)}(I_0,0)/\mathrm{\Delta }{G}^{(2)}(I_0,{\alpha }_0)\approx 0.81$ [we assume ${|T({\alpha }_0)|}^2={|T(-{\alpha }_0)|}^2=1$]. On the other hand, from Eq. (9), we have that ${\kappa }_2(0)/{\kappa }_2({\alpha }_0)=0.6561<0.81$ for $d_1=|{\alpha }_0-{\alpha }_0^{\prime }|$ 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 $\mathrm{\Delta }{G}^{(2)}(I_0,\alpha )$. ${\kappa }_2(\alpha )$ is the fluctuation information of $\mathrm{\Delta }{G}^{(2)}(I_0,\alpha )$ and contains more information than $\mathrm{\Delta }{G}^{(2)}(I_0,\alpha )$, such that it can be used in CSGI to further enhance the spatial resolution just as in traditional GI.

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 $a=0.8\mathrm{mm}$, slits center distance $b=1.2\mathrm{mm}$, and slit height $g=8\mathrm{mm}$. The distance between the source and the object is $z=0.8\mathrm{m}$.

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 ${\kappa }_2(\alpha )$ in the same conditions. Results are obtained by averaging over 50,000 exposure frames. For ${\kappa }_2(\alpha )$, we get $\mathrm{\Delta }{G}^{(2)}(I_0,\alpha )$ every 5000 steps. The experimental results are shown in Figs. 4(a) and 4(b). According to our theoretical analysis, the PSF of $\mathrm{\Delta }{G}^{(2)}(I_0,\alpha )$ can be narrowed by ${\kappa }_2(\alpha )$. 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 ${\kappa }_2(\alpha )$. That means the PSF of $\mathrm{\Delta }{G}^{(2)}(I_0,\alpha )$ is wider than the PSF of ${\kappa }_2(\alpha )$. The resolution obtained by ${\kappa }_2(\alpha )$ is improved compared to $\mathrm{\Delta }{G}^{(2)}(I_0,\alpha )$. The experimental results are consistent with our theoretical analysis.

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 ${\kappa }_2(\alpha )$ can be used in the CSGI scheme. Here, for CSGI, we obtain $\mathrm{\Delta }{G}^{(2)}(I_0,\alpha )$ with 3000 steps. Experimental results obtained by CSGI and CSGI with ${\kappa }_2(\alpha )$ are shown in Figs. 4(c) and 4(d), respectively. We see that the resolution obtained by CSGI with ${\kappa }_2(\alpha )$ is improved compared to CSGI.

In conclusion, we have used ${\kappa }_2(\alpha )$ instead of $\mathrm{\Delta }{G}^{(2)}(I_0,\alpha )$ 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 ${\kappa }_2(\alpha )$ to the CSGI scheme and obtained images with higher resolution than those obtained by CSGI. Similarly, ${\kappa }_2(\alpha )$ 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.