The image quality in ghost imaging is vital in practical applications. Through theoretical analysis, we find that for thermal light the average intensity as well as the fluctuations of an arbitrary incident field can greatly influence the image quality. Based on this, we suggest an easily realizable scheme to improve the visibility by generating speckles of non-Gaussian intensity distributions with a spatial light modulator. Numerical simulation demonstrates that this method can significantly improve the visibility, and the effect on the imaging resolution is also discussed. This method may thus be helpful in promoting the implementation of ghost imaging in real applications.

In ghost imaging (GI), a light beam is divided into two arms, one of which contains an object and a “bucket” detector which only measures the total light intensity, while the other contains a “reference” detector which never interacts with the object. Contrary to intuition, an image may be constructed from the second-order correlation of the intensities at the two detectors. The first experiment was demonstrated with entangled light obtained from spontaneous parametric down conversion,^{1} and the phenomenon was once considered as a characteristic of the nonlocality of entanglement,^{2} hence the name “ghost” imaging was coined. Later, it was found that thermal light can also be used to realize GI,^{3–11} which has many advantages as the source is readily available and has higher intensity so less data acquisition time is needed. However, a fundamental disadvantage is that there is always a large background, which restricts the maximum visibility to 1/3, compared to 1 for entangled light. Many methods have been introduced to increase this visibility, such as high-order GI,^{12–16} or improved algorithms for image reconstruction.^{17–20} It was also later realized that the essential role of the reference detector is merely to provide a means to measure the spatial distribution of the field at the object;^{21} if this distribution could be generated artificially then the reference detector would be unnecessary. This computational GI was first realized experimentally with a spatial light modulator (SLM),^{22} which can vary the intensity and/or phase at each point in the plane of an incident beam, and has also been used in other variations of GI.^{23,24} In this paper we show theoretically that the visibility of pseudothermal GI is related to the intensity as well as the fluctuations and speckle size of an arbitrary field incident on the object; to our knowledge this has never been noted before. Based on this, we demonstrate a scheme in which image visibilities can be increased by using a non-Gaussian light distribution with artificially generated random fluctuations.

The experimental setup of thermal GI is shown in Fig. 1(a). A pseudothermal light source is formed by a laser beam illuminating a rotating ground glass disk to produce a randomly changing speckle pattern, which is then divided by a beamsplitter into the reference and object arms. The spatially resolving reference detector D_{a} is customarily a charge coupled device (CCD). In the other arm, the beam is incident on an object and then detected by the bucket detector D_{b}. Both object and D_{a} are at the same distance from the source, so the speckles in the two planes are identical.

The image is retrieved from *N* exposure frames, with the speckle intensity *I*_{a}(*x*_{i}) on each pixel *x*_{i} of the reference detector being multiplied by the total intensity *I*_{b} at the bucket detector in the same measurement. The results are averaged, and the image is then obtained from the normalized second-order correlation function

Here

*t*(

*x*

_{i}) is the transmission function of the object, and ⟨…⟩ denotes averaging over the

*N*measurement times. The covariance between

*I*

_{b}and

*I*

_{a}(

*x*

_{i}) is defined as cov(

*I*

_{b},

*I*

_{a}(

*x*

_{i})) = ⟨

*I*

_{b}·

*I*

_{a}(

*x*

_{i})⟩ − ⟨

*I*

_{b}⟩⟨

*I*

_{a}(

*x*

_{i})⟩.

For good resolution, the spatial coherence area *S*_{c}, i.e., the average speckle size of the field, must be smaller than the required details of the image. In this case the gray scale can be treated as constant throughout the coherence area. Then the second-order correlation function can be expressed as

where *N*_{c} = *S*_{c}/*S*_{0} is the number of pixels in the coherence area, and *S*_{0} is the area of one pixel; *D*(*I*_{a}(*x*_{i})) is the variance of the intensities of *N* measurements at point *x*_{i}, and is defined as

where *j* is the measurement number. For a uniform incident field, we have ⟨*I*_{a}(*x*_{i})⟩ = ⟨*I*_{a}⟩ and *D*(*I*_{a}(*x*_{i})) = *D*(*I*_{a}). We note that the average intensity of *I*_{b} is

where *N*_{m} = *S*_{m}/*S*_{0} and

Without losing generality, we consider a binary object without any gray scale, which means that the transmission function *t*(*x*) equals 1 in the transparent area, and 0 otherwise. For a point *x*_{i} corresponding to the area within the transparent region, the second-order correlation function is given by

For a point *x*_{i} in the blocked background area, the second-order correlation function is

From the definition of the visibility

*P*(

*I*

_{a}) =

*D*(

*I*

_{a})/⟨

*I*

_{a}⟩

^{2}. If this is large, then the visibility is increased.

It is remarkable that, to our knowledge, there has not been any previous discussion about the effect of the total incident intensity on GI. The reason may be that, for natural thermal sources, *D*(*I*_{a}) = ⟨*I*_{a}⟩^{2}, so Eq. (8b) contains no dependence on the average intensity or variance. However, with the recent availability of SLMs, we can now control the speckles to obey any distribution, and not be restricted to a Gaussian state when using pseudothermal light. Therefore, a different relation between the intensity and variance may be synthesized, and the imaging visibility increased accordingly.

Figure 1(b) is a schematic of GI with speckles generated by an SLM, which is quite simple and similar to that in computational GI.^{22} A uniform light source illuminates the SLM, which is programmed to produce a field with the desired intensity distribution. The speckles are then focused onto the object plane, and a bucket detector records the total light intensity passing through the object. The image is retrieved from the second-order correlation between the synthesized speckle field *I*_{a}(*x*_{i}) and the bucket signals *I*_{b}. Below we describe our numerical simulation experiment to investigate the influence of speckle intensity and distribution on GI.

The SLM in the simulation is assumed to consist of 100 pixels. The intensity from each pixel is programmed to fluctuate independently while maintaining a uniform average distribution, thus the generated speckles have a spatial coherence area *S*_{c} of one pixel. We first take a simple double-slit as object, with different ratios of the object to coherence area of *S*_{m}/*S*_{c} = 10, 20 and 40. After multiplying the randomly generated speckle patterns by the transmission function of the double-slit and integrating, the value of the total intensity on the bucket detector can be obtained. This is then used to retrieve the image through correlation with the original intensity distribution of the speckles, and averaging over a large number of frames.

We first simulate Gaussian state speckles (as found in the commonly used pseudothermal light source made from a laser beam illuminating a rotating ground glass), so *P*(*I*_{a}) = 1. The object is then imaged and the visibility is calculated with the definition in Eq. (8a). This corresponds to the situation of conventional GI which has very low visibility. From Eq. (8b) we can see that if the ratio of variance to average intensity squared *P*(*I*_{a}) is higher, the image visibility will improve. We program the speckle intensity as:

where *I*_{a0}(*x*_{i}) is the intensity of the original Gaussian speckles, and *n* ⩾ 1 is a control parameter. As *n* increases, larger intensities in the speckles increase faster than smaller intensities, and so the programmed speckles will fluctuate more strongly. Thus we have

When *n* is an integer, for a Gaussian field we have

so we can see that the larger the value of *n*, the larger *P*(*I*_{a}) will be, hence we can adjust the degree of fluctuations of speckles on the object by choosing different *n*. For *n* > 1 , the speckles will have stronger fluctuations than with a Gaussian state.

Figure 2 shows the visibility *V* as a function of *P*(*I*_{a}) for *N* = 5000 and *S*_{m}/*S*_{c} = 10, 20 and 40. We can see that the numerical results (data points) agree with the theory (dashed curves) very well, and that the visibility increases steadily with increasing *P*(*I*_{a}). The circles, squares and crosses are the numerical simulation results derived from Eq. (8a), while the dashed curves are plotted from Eq. (8b). It is remarkable that the visibility can exceed 1/3, the limit of second-order thermal GI with Gaussian statistics. Theoretically, *V* could even reach 1, if the fluctuations were large enough relative to the average intensity.

Figure 3 shows the images of a simulated double-slit of slit width 10 pixels separated by 30 pixels, so that *S*_{m}/*S*_{c} = 20. A typical realization of a speckle pattern with a Gaussian distribution (*P*(*I*_{a}) = 1) is shown in Fig. 3(a1). The image after averaging over 5000 frames is shown in Fig. 3(b1); the visibility is 0.0241, calculated from the definition of Eq. (8a). The values of

*n*= 2 (

*P*(

*I*

_{a}) = 5) and

*n*= 2.5 (

*P*(

*I*

_{a}) = 9.8) are plotted, and it is obvious that the fluctuations are much larger. The corresponding ghost images are shown in Figs. 3(b2) and (b3), where the visibilities are, respectively,

*V*= 0.1078 and

*V*= 0.1877. It is clear that the visibility is greatly improved as the speckles fluctuate more strongly.

From Eq. (5), the influences of *P*(*I*_{a}) on the imaging results are similar for any values of *t*(*x*_{i}). Therefore, for gray-scale objects our method of increasing imaging visibility should also work. We confirm this by imaging the cartoon face of a panda with gray tones, using real speckles instead of simulated ones. Figure 4(a) shows the result of usual GI, which has a visibility of 7.9 × 10^{−4}. By programming the real speckles with Eq. (9), we obtain speckles with stronger fluctuations. In Figs. 4(b) and 4(c), values of *P*(*I*_{a}) = 5 and *P*(*I*_{a}) = 9.8 are used, giving visibilities of 3.6 × 10^{−3} and 6.3 × 10^{−3}, respectively. Once again, the image visibility is significantly improved.

The imaging resolution of GI is determined by the spatial coherence area of the field, i.e., the size of the speckles.^{10} To obtain the best resolution we program the computer to control each SLM pixel independently, in which case the coherence area will be just one pixel, as in the simulation of Fig. 3. In traditional GI, a speckle usually covers several pixels, with the diameter of a speckle defined as the full-width-half-max (FWHM) of its intensity. If we use nonlinear programming, it is possible to change the FWHM. To examine the effect of this on the speckle size and consequently the resolution, we calculate the normalized second-order self-correlation function *g*^{(2)}(*x*) and compare the results using real speckles and programmed ones. In Fig. 5(a) is the plot with original speckles (*P*(*I*_{a}) = 1, blue line), the FWHM of which reflects the transverse coherence length of the field and is 66.6 μm here. For (b) and (c), speckles of *P*(*I*_{a}) = 5 and *P*(*I*_{a}) = 9.8 are used, giving FWHM values of 62.1 and 59.5 μm (green and red lines), respectively. The curves are scaled to give a maximum value of 2 for the case of *P*(*I*_{a}) = 1. We can see that the program for Eq. (9) makes the speckles sharper and narrower, and the coherence area is reduced so the resolution is improved. It should be emphasized that Eq. (9) is just one means to obtain speckles with stronger fluctuations, and the coherence area is decided by the particular algorithm used to generate the non-Gaussian speckles. However, the results in Fig. 5 show that we can indeed design small speckles with non-Gaussian intensity fluctuations in such a way that the visibility and resolution are both improved simultaneously.

In conclusion, we have analyzed the dependence of the visibility of thermal GI on the average intensity as well as on the intensity fluctuations of the light source. If the variance is increased relative to the average intensity, the visibility can be greatly increased. Based on this, we have simulated experiments in which speckles with large, non-Gaussian fluctuations generated by an SLM are synthesized and used to illuminate a simple double-slit and an object with gray tones, resulting in greatly improved visibility compared with traditional GI. The ratio of object size to spatial coherence area is also a factor to be considered, but in real situations the former cannot be changed, nor is the latter easily adjustable. Moreover, although a large coherence area can increase the visibility it will degrade the resolution at the same time. Using an SLM to create various artificial field distributions to increase the visibility of GI is feasible and easily realizable. This method may thus be helpful in practical applications of GI, especially in biomedical laboratory experiments.

This work was supported by the National Basic Research Program of China under Grant No. 2010CB922904, the National Natural Science Foundation of China under Grant No. 60978002, and the Hi-Tech Research and Development Program of China under Grant Project No. 2011AA120102.