† Corresponding author. E-mail:
In ghost imaging, an illumination light is split into test and reference beams which pass through two different optical systems respectively and an image is constructed with the second-order correlation between the two light beams. Since both light beams are diffracted when passing through the optical systems, the spatial resolution of ghost imaging is in general lower than that of a corresponding conventional imaging system. When Gaussian-shaped light spots are used to illuminate an object, randomly scanning across the object plane, in the ghost imaging scheme, we show that by localizing central positions of the spots of the reference light beam, the resolution can be increased by a factor of
Unlike conventional imaging that is based on the first-order correlation of light, correlated imaging concepts including auto-correlation imaging schemes[1–4] and ghost imaging schemes[5–13] employ the second-order correlation of light. It results in some unique features and has attracted a lot of attention in last decades. In fact, correlated imaging has been applied in various fields including optical lithography,[14, 15] remote imaging,[16] microscopy imaging,[17] x-ray imaging,[18] and Terahertz imaging.[19]
In these applications, the spatial resolution of image is a major issue. In conventional imaging, the resolution is given by Rayleigh criterion[20] which comes from the finite-size of the entrance pupil of the imaging system. In auto-correlation imaging, the resolution is determined by the product of the point spread functions (PSFs) of the two light beams.[1–4] In ghost imaging, the resolution is in general quantified by the full width at half-maximum (FWHM) of the peak in a Hanbury-Brown and Twiss (HBT) intensity correlation measurement, or by use of the transverse coherence length of the illumination speckles, namely the averaged size of the speckles.[1, 21–23] In the last decade, a great deal of resolution enhancement proposals have been suggested, such as compressive sensing technique[24–26] and non-Rayleigh speckle fields,[27, 28] low-pass spatial filter scheme,[23] and high-pass spatial-frequency filter scheme.[29] In a recent publication, Yang et al.[30] reported a new correlated imaging scheme using the orbital angular momentum correlations of light. This scheme may be used to image an object with a very high azimuthal resolution. Although those investigations to the resolution of ghost imaging have been done, to our knowledge, how the optical systems in the test and the reference path of ghost imaging individually affect the imaging resolution is not clear. Is the resolution of ghost imaging also given by the product of the two light beamʼs PSFs as in auto-correlation imaging?
In this work, a Gaussian-shaped spot illumination scheme is employed to analyze the resolution issue of ghost imaging. We demonstrate that the resolution of ghost imaging is determined by the convolution of the PSFs of the test and reference light beams. Then, we propose localization and threshold methods to diminish the effect of the PSFs and enhance the resolution of ghost imaging. In photo-activated localization microscopy (PALM) scheme,[31] a sub-diffraction image is constructed by localizing the positions of the fluorophores.[31, 32] In our scheme, we show that by localizing the central positions of all the illuminated spots of the reference light beam, the resolution of ghost imaging can be enhanced by a factor of
A typical ghost imaging schematic is shown in Fig.
Suppose that the light beam in the test path propagates to the object through an optical system with a PSF
![]() |
![](cpb_28_4_044202/cpb_28_4_044202_ieqn7.gif)
![](cpb_28_4_044202/cpb_28_4_044202_ieqn8.gif)
![]() |
Suppose that the PSF of the reference path is
![]() |
![]() |
If the light source is in a Gaussian-type quantum state, the second-order correlation of the bucket signals and the reference signals can be written as[13]
![]() |
![](cpb_28_4_044202/cpb_28_4_044202_ieqn10.gif)
In order to explicitly analyze spatial resolution of ghost imaging, we consider the case where the illumination light is a point-like source which randomly and uniformly distributed on the source plane. If the light spot is located at
![]() |
As is well known, the image in a conventional imaging system can be expressed in the form[36]
![]() |
![](cpb_28_4_044202/cpb_28_4_044202_ieqn14.gif)
![](cpb_28_4_044202/cpb_28_4_044202_ieqn15.gif)
![](cpb_28_4_044202/cpb_28_4_044202_ieqn16.gif)
![](cpb_28_4_044202/cpb_28_4_044202_ieqn17.gif)
In order to work out an explicit expression of the resolution of ghost imaging, we assume that all the PSFs in Eqs. (
![]() |
![]() |
![](cpb_28_4_044202/cpb_28_4_044202_ieqn19.gif)
![](cpb_28_4_044202/cpb_28_4_044202_ieqn20.gif)
![](cpb_28_4_044202/cpb_28_4_044202_ieqn21.gif)
![](cpb_28_4_044202/cpb_28_4_044202_ieqn22.gif)
![](cpb_28_4_044202/cpb_28_4_044202_ieqn23.gif)
![](cpb_28_4_044202/cpb_28_4_044202_ieqn24.gif)
![](cpb_28_4_044202/cpb_28_4_044202_ieqn25.gif)
![](cpb_28_4_044202/cpb_28_4_044202_ieqn26.gif)
As shown in Eqs. (
For this end, as similar as in the localization process,[31] we precisely label central positions of light spots of each illumination pattern in the reference light beam. In mathematics, it means that the PSF
![]() |
Here, in order to understand how the PSFs affect the resolution, we study the case where the object is a pinhole at
Then we choose the post-selection method[35] to diminish the effect of the PSF
![]() |
Next, to increase the signal strength of the bucket detector and improve the anti-noise ability of the imaging system, the multi-spot illumination scheme is the most commonly used in ghost imaging.[37–43] If each illumination pattern contains M light spots, based on the theory developed in the preceding section, we can drive out the imaging formula
![]() |
![](cpb_28_4_044202/cpb_28_4_044202_ieqn36.gif)
![]() |
The first term is the desired image and the second one is the background noise resulting from cross talks between different illumination spots. It is clear that the localizing and thresholding methods proposed in the preceding section for resolution enhancement can be used to the multi-spot illumination case if central positions of all spots in an illumination pattern are recorded prior to the illumination.
The experimental setup is shown in Fig.
In experiment, the FWHM of the Gaussian-shaped spot shinning the object plane varies from
To further raise the resolution, we set a threshold
In Fig.
Finally, we would like to emphasize two points. At first, the threshold method may also be suitable to a gray-scale object although the binary object is used in the present demonstration. For a gray-scaled object, light spots shinning on the low gray-scale regions would be abandoned if the threshold is set too high, and the contrast resolution would be decreased and the image get distorted. For a gray-scaled object, one should appropriately divide the object into several zones according to its gray distribution and set different threshold values for different zones. Then, the ghost imaging process is performed zone-by-zone and the whole image of the object can be constructed by combining the images of the different zones. It means that for a gray-scale object one should dynamically set a threshold value to the bucket measurement. Secondly, the localization and threshold methods also can be implemented when the light source of ghost imaging is not point-like and not Gaussian shape. In that case, Gaussian-distributed fit allows the reference pattern transform to be the overlaps of the Gaussian spots and the central positions of the spots also can be tracked.[32, 46] Therefore, the two methods to enhance the resolution are applicable to any other ghost imaging protocols.
In this work, we sequentially analyzed the diffraction effects of the test and reference paths, and individual contributions to the spatial resolution of ghost imaging are quantified. Next, we showed that the PSF of the reference path can be removed by localizing central positions of the reference patterns and the image resolution can be enhanced by a factor of
[1] | |
[2] | |
[3] | |
[4] | |
[5] | |
[6] | |
[7] | |
[8] | |
[9] | |
[10] | |
[11] | |
[12] | |
[13] | |
[14] | |
[15] | |
[16] | |
[17] | |
[18] | |
[19] | |
[20] | |
[21] | |
[22] | |
[23] | |
[24] | |
[25] | |
[26] | |
[27] | |
[28] | |
[29] | |
[30] | |
[31] | |
[32] | |
[33] | |
[34] | |
[35] | |
[36] | |
[37] | |
[38] | |
[39] | |
[40] | |
[41] | |
[42] | |
[43] | |
[44] | |
[45] | |
[46] |