The point-spread function (PSF) is fundamental importance in estimating the imaging resolution in optical imaging systems. By using the Collins formula, a analytical imaging formula for ghost imaging system is obtained. The intensity fluctuation correlation function can be viewed as the convolution of the original object and a PSF. The imaging resolution is determined by the width of PSF. Based on the optical transfer matrix theory, we present the analytical formula describing the width of the PSF, by which one can estimate imaging resolution of a new-designed imaging scheme when compared with that of the existing imaging system. Several typical ghost imaging systems are chosen to verify experimentally our theoretical results.
© 2016 Optical Society of America
Ghost imaging is a new technique that forms an object image or its diffraction pattern through measuring two correlated optical fields. The first demonstration of ghost imaging utilized a biphoton source, and the result was interpreted as a quantum phenomenon owing to the entanglement of the source photons . The theoretical [2,3] and experimental [4,5] results later demonstrated that ghost imaging could be achieved with the pseudo-thermal light. However, the spatial correlation properties of quantum entangled source and a thermal light source are different. Two-photon geometric optics pointed out that the imaging in coincidence counts of an aperture placed in one of the down-conversion beams is found to be the analog of a simple spherical mirror system . In the classical case, the classical thermal source looks like a phase-conjugate mirror in correlated imaging with classical thermal source . Moreover, some results have proved that the two light beams in the object and reference arms can have different wavelengths for vacuum propagation [8,9].
Due to the novel physical peculiarities of ghost imaging[10–13], more and more attention has been paid to its potential applications in practice[14–20]. For ghost imaging in an optically harsh environment, the effects from atmosphere turbulence and scattering media on imaging quality may be decreased by using the second-order correlation of light[21–26]. To achieve ghost imaging with high resolution, high-order ghost imaging [27, 28], computational ghost imaging [29, 30], compressive sensing [31, 32] and the shaped light  could be the effective methods. Many studies have presented a analytical imaging formula for a particular ghost-imaging system which can be used to estimate imaging resolution of ghost imaging [8, 21, 26]. In this paper, by using the Collins formula, our analytical results show that the intensity fluctuation correlation function of thermal ghost imaging can be represented as a convolution of the object and a Gaussian function which can be viewed as PSF of ghost imaging system, and we give a general analytical formula of the width of the PSF can be used to different ghost imaging systems. Here we use this theory to analyze several typical examples of ghost imaging schemes. The experimental results agree well with the theoretical analysis. Our results are quite helpful for estimating the difference of imaging resolution of different ghost imaging systems.
A standard ghost imaging system is depicted in Fig. 1. An incoherent light is split into two beams by a beam splitter (BS), then the two light beams are sent through two different optical systems. In the test arm, an unknown object is imbedded between the light source and the detector CCD-1. The other path is the reference arm, the light beam propagates freely, and is measured by a reference detector CCD-2. The intensity distributions recorded on CCD-1 and CCD-2 are correlated by a correlator to obtain the correlation function of the intensity fluctuations.
For the test arm, the field Et (xt) at the test detector CCD-1 can be given by34], Ht1(u, ξ) and Ht2(ξ, xt) have the forms
The information of the object imaged can be extracted by computing the correlation of the intensity fluctuations 
Suppose the intensity distribution of the light source is the Gaussian type, the first-order correlation function for a completely incoherent source can be expressed as35] and with the help of Eqs. (1)–(3) and (5), the intensity fluctuation correlation function in Eq. (4) can be rewritten as
For the object imaged, suppose 〈t(ξ)t*(ξ′)〉 = T (ξ)δ(ξ − ξ′), and a large bucket detector is used in the test arm. Ghost-image is proportional to
Equation (11) is the key formula in this paper, and it determines the imaging quality in ghost imaging system. Some papers are discussed about the formula for ghost imaging [8, 21, 26, 35]. The optical transfer matrices of the optical systems from the source to the object and from the source to the reference detector CCD-2 can be given when a new ghost imaging scheme is designed. Then one can estimate imaging resolution of this ghost imaging system by Eq. (11). Note that the optical system Ht2(ξ, xt) in the test path does not affect the width of the PSF. From Eq. (11), thermal ghost imaging always gives the best imaging resolution when ΔPSF attains its minimum value. Here we can obtain the optimal resolution condition for different ghost imaging systems Ar Bt1 = Br At1. The difference of imaging resolution for different ghost imaging systems relies on the optical transfer matrix element Bt1 in the optical system Ht1(u, ξ) when the optimal condition is satisfied.
As we all know, the light source is the main parameter that determines the width of the PSF. From Eq. (11), it is obvious that the transverse size of the source ρs is included in the ΔPSF, from which an increase of ρs can decrease the width of the PSF, which results in the improvement of imaging resolution. The result is in agreement with those in previous works [16,21,36,37].
3. Experiment results
In this section, several types of ghost imaging schemes are chosen to verify our analytical results. Firstly, we analyze a lensless ghost imaging system, as shown in Fig. 2(a). In the test arm, an object with the transmission function t(ξ) is placed between the light source and a bucket detector CCD-1. In the reference arm, the beam propagates freely, and is measured by a multi pixel detector CCD-2. In this case, the optical transfer matrix elements for the optical systems Hr (u, xr) and Ht1(u, ξ) in Fig. 2(a) can be represented as
The width of the PSF can be written as
From Eq. (13), ΔPSF attains its minimum value when Lt1 = Lr, under which ghost-image always gives the best resolution. It is noted that ΔPSF depends on the value of Lt1 when Lt1 = Lr is satisfied.
During of the experimental implement, the wavelength of the source is λ = 532nm and ρs = 4mm. The object is a simple double-slit with the slit width 0.2mm and center-to-center separation 0.4mm. The effects of the object-to-source distance Lt1 on ghost imaging are shown in Figs. 2(b) and 2(c). When the propagation distance Lt1 is small, high quality ghost-image can be obtained [see Fig. 2(b)]. From Fig. 2(c), ghost-image becomes blurred with an increase of the distance Lt1. It is clear that the experimental results (open circles) agree well with the theoretical predictions (solid curves). We plot the normalized PSF which corresponds to the images in Figs. 2(b) and 2(c) as the function of xr in Fig. 2(d), the PSF becomes broader when the propagation distance Lt1 is increased, which indicates that the imaging resolution becomes worse. The results are in good agreement with the experiment results in Figs. 2(b) and 2(c).
Next, let us analyze the case with a lens which is located in the test arm , as shown in Fig. 3(a). Unlike a lensless ghost imaging, a lens with focal length ft is located at a distance Lt1 from the source and Lt2 from the object in the test arm. In this case, the optical transfer matrices for Hr (u, xr) keeps unchanged, and we do not show it. Ht1(u, ξ) can be represented as
The width of the PSF can be expressed asEq. (15), ΔPSF attains its minimum value when Eq. (16) is satisfied. We also expect to observe an inverted image magnified by a factor of M = (Lt1 − Lr1)/Lt2.
To estimate imaging quality of different ghost imaging systems, the experimental results for lensless ghost imaging are also given when the experimental scheme in Fig. 3(a) is implemented. Lt1 + Lt2 = Lr1 = 500mm is chosen when the test arm does not include the lens, and the corresponding result is plotted in Fig. 3(b). Here we see a blurred ghost-image due to the large distance from the source to the object, as discussed in the above. When a lens (the focal length ft = 100mm) is inserted into the test arm, the demagnification and the magnification images are obtained in Figs. 3(c) Lt1 = 200mm, Lt2 = 300mm and (d) Lt1 = 350mm, Lt2 = 150mm, respectively. During of this process, the value of Lr1 satisfies Eq. (16). Here a ghost-image magnified by a factor M = (Lt1 − Lr1)/Lt2 = 1/2 is obtained in Fig. 3(c) when the object distance Lt2 > 2ft, and the imaging resolution is enhanced greatly when compared with the result in lensless system. Under ft < Lt2 < 2ft, ghost-image with M = (Lt1 − Lr1)/Lt2 = 2 is observed in Fig. 3(d). It is clearly seen that imaging resolution increases when a lens inserted into the test arm. The cause for this phenomenon lies in the nature of the lens, which shortens the diffraction length |Lt1 + Lt2 − Lt1Lt2/ft | when the distance of the object-to-source Lt1 + Lt2 remains unchanged. The PSF for the images in Figs. 3(b)–3(d) is plotted in Fig. 3(e), and the width of the PSF obviously becomes narrow when the lens is inserted into the test arm, which means that the imaging quality can be enhanced. The results are very helpful, one can insert a lens into the test arm to obtain better ghost-image when the distance of object-to-source remains unchanged.
Then, we consider the case when a lens is located in the reference arm [see Fig. 4(a)]. Here there exists a lens with focal length fr between the source and CCD-2 in the reference arm, and the distance from the source to the lens and from the lens to CCD-2 are Lr1 and Lr2, respectively. Similarly, the optical transfer matrices for Ht1(u, ξ) is the same as Eq. (12). Hr (u, xr) has
The width of the PSF can be rewritten as
When the optimal resolution condition is satisfied, one can obtain the Gaussian thin-lens equationEq. (18) is that the distance of the test arm Lt1 affects the width of the PSF if the optimal resolution condition is satisfied. The lens in the reference arm has no effect on imaging resolution, and it only changes the magnification factor of ghost-image.
When the reference arm does not include the lens, Lt1 = Lr1 = 200mm is chosen, we get a image with good resolution [see Fig. 4(b)]. By using the experimental schematic depicted in Fig. 4(a), the results are shown in Figs. 4(c) Lr1 = 350mm, Lr2 = 300mm and (d) Lr1 = 500mm, Lr2 = 150mm. During of the experimental implement, Lt1 = 200mm is fixed and the lens in the reference arm has the focal length fr = 100mm. It is shown that, by using the reference lens, ghost-images with M = Lr2/(Lr1 − Lt1) are obtained, while imaging resolution has no changes. The corresponding PSF is depicted in Fig. 4(e). Note that the PSF has no changes when a lens is inserted into the reference arm. That is to say, imaging resolution is not affected when a reference lens is considered.
Finally, we discuss the combined effects of a test lens and a reference lens, and the experimental setup is shown in Fig. 5(a). In the test arm, the light beam goes through a test thin lens (the focal length ft) and the object imaged, then is detected by the detector CCD-1. In the reference arm, the light propagates through a reference thin lens (the focal length fr) and then to CCD-2. Here we choose Lt1 = ft and Lr1 = fr. This imaging system is actually a new-designed ghost telescope scheme [38, 39]. The optical transfer matrices for Ht1(u, ξ) and Hr (u, xr) are the same as Eqs. (14) and (17), respectively. The width of the PSF can be expressed as
It is shown that ΔPSF, i.e., imaging resolution depends on the value of ft when the optimal resolution condition is satisfied. Here the image of the object magnified by a factor M = ft/fr is obtained.
By setting Lt1 = ft = Lr1 = fr = 200mm, the experimental results are depicted in Figs. 5(b) and (c). For comparison, the parameters Lt2 = 200mm and 500mm are chosen in Figs. 5(b) and 5(c), respectively. The parameter Lr2 satisfies Eq. (21). For a small propagation distance, a high resolution ghost-image can be reconstructed [see Fig. 5(b)]. It is shown that, from Fig. 5(c), imaging resolution does not change when the propagation distance is increased. In other words, one can enhance the imaging distance of ghost imaging with unchanged imaging resolution in a ghost telescope system. The imaging scheme may be helpful for high-resolution far-distance ghost imaging . Figure 5(d) displays the PSF corresponding to Figs. 5(b) and 5(c), here two curves obviously overlap for different distances Lt2, which means that the imaging resolution keeps unchanged.
Note that a simple double-slit is chosen as the object imaged in the above discussion. To make our result more general, a complex object, the letter “GI” is used to implement the corresponding experiment in ghost telescope system. A lensless ghost imaging system which has been investigated extensively [17, 18, 41–43] is treated as the comparison object. The PSF in the two imaging systems is presented firstly in Fig. 6(a). It is shown that the PSF in lensless system is wider than that in ghost telescope system. So we can infer that, the resolution of the newly-designed ghost telescope system is better than that in a lensless system, which is verified by the experimental results in Figs. 6(b) and 6(c).
It is noted the transfer functions in a lensless ghost imaging system is discussed by Cheng. The results show that with the help of the transfer functions, one can quantitatively analyze imaging quality . By comparing our results with those in Ref. , the PSF is the spatial domain version of the transfer function of the imaging system, and the imaging resolution can be improved by decreasing the width of the PSF. Formally, the optical transfer function(OTF) is defined as the Fourier transform of the PSF. One can enhance imaging quality by increasing the bandwidth of the OTF when compared with the spectrum of the object. During ghost imaging process, the OTF provides a comprehensive and well-defined characterization of optical systems , we will attempt to present a general analytical formula of the OTF which can be used to different ghost imaging systems in our future work.
In conclusion, we have given a analytical imaging formula to estimate the imaging resolution of different ghost imaging systems. Based on the Collins formula and the optical transfer matrix theory, the analytical formula for ghost-images is derived. It is shown that the intensity fluctuation correlation function can be viewed as the convolution of the original object and the PSF, and the imaging resolution is determined by the width of the PSF. To verify our theory, four kinds of typical ghost imaging schemes are analyzed experimentally and theoretically. The results show that, based on our analytical formula, the imaging resolution of a new ghost imaging system can be evaluated when compared with that of the existed imaging schemes. Our results are quite helpful for designing new ghost imaging schemes.
National Natural Science Foundation of China (61372102, 61571183).
We thank L. Guo for helpful discussions.
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