## Abstract

The relationship between transmission area of an object imaged and the visibility of correlated imaging is investigated in a lensless system. We show that they are not in simple inverse proportion, as usually depicted. The changes of the visibility will be quite different when the transmission area is varied by different manners, which may motivate people to seek a new understanding about the influence factors of the visibility.

© 2007 Optical Society of America

## 1. Introduction

Correlated imaging has attracted much attention in recent years [1–6]. The initial works about correlated imaging rely on the the quantum entanglement of the photon pairs created by spontaneous parametric down-conversion (SPDC) [2, 3]. In order to resolve double-slit experiments with SPDC light, a four-order correlation function in the {**r**} space was developed [7]. Abouraddy *et al*. discussed the role of entanglement in two-photon imaging, and showed that entanglement is a prerequisite for achieving distributed quantum imaging [8]. While their work leads to some debate. Bennink *et al*. showed that coincidence imaging does not require entanglement, and provided an experimental demonstration using a classical source [9]. Using classical statistical optics, Cheng and Han studied a particular aspect of coincidence imaging with incoherent sources [10]. The first experimental demonstration of two-photon correlated imaging with true thermal light from a hollow cathode lamp was also reported [11].

Visibility is one of the most important factors to characterize an imaging system. The visibility, usually defined as the ratio of the intensity fluctuation correlation to the spatial intensity correlation in a correlated imaging system, has been investigated in some papers. Cai and Zhu showed that the visibility of the fringes increases with a decrease of the source’s transverse size[12, 13]. Gatti *et al*. investigated experimentally fundamental properties of coherent ghost imaging using spatially incoherent beams generated from a pseudo-thermal source, in particular, a laconic description about the visibility was given: it roughly scales as the ratio of the coherence area to the object transmissive area[14].

This paper discusses the effects from the transmission area of an object on correlated imaging with incoherent light source. We show that the effects are quite different when transmission area is varied by different manners, and the visibility is not simple inversely proportional to the object transmissive area (as depicted in Ref. [14]), which may yield a new understanding about the factors that influence the visibility.

## 2. The model and analytical results

The imaging system appropriate for correlated imaging is shown in Fig. 1. The light beam from the source *S* is incoherent, and divided into two beams by a beam splitter, they travel through a test arm and a reference arm, which are described by their impulse response functions *h*
_{1}(*x*
_{1},*u*
_{1}) and *h*
_{2}(*x*
_{2},*u*
_{2}), respectively. The test arm usually includes an object to be imaged. Detector *D _{t}* and

*D*are used to record the intensity distribution at

_{r}*u*

_{1}and

*u*

_{2}, respectively.

The optical field in the source can be represented by *E*(*x*). After propagating through two different optical systems, the field has

The coincident counting rate between two detectors is proportional to the second-order correlation function *G*
^{(2)}(*u*
_{1},*u*
_{2})[4, 15]

where *E*(*u _{i}*) (

*i*= 1,2) is the optical field in the test (reference) detector. Substituting Eq. (1) into Eq. (2), we have

$$\phantom{\rule{4.9em}{0ex}}\times {h}_{1}({x}_{1},{u}_{1}){h}_{2}({x}_{2},{u}_{2}){h}_{2}^{*}({x}_{2}^{\prime},{u}_{2})$$

$$\phantom{\rule{4.9em}{0ex}}\times {h}_{1}^{*}({x}_{1}^{\prime},{u}_{1}){\mathrm{dx}}_{1}{\mathrm{dx}}_{2}{\mathrm{dx}}_{2}^{\prime}{\mathrm{dx}}_{1}^{\prime},$$

where 〈*E*(*x*
_{1})*E*(*x*
_{2})*E**(*x*́_{2})*E**(*x*́_{1})〉is the second-order correlation function at the light source, We represent it by *G*
^{(2)}(*x*
_{1}, *x*
_{2}, *x*́_{2}, *x*́_{1}) in the following in order to outline the parallelism with the formalism in Eq. (2).

In many cases, the fluctuation of a classical light field can be characterized by a Gaussian field statistics with zero mean [16], for which one obtains

$$\phantom{\rule{5.2em}{0ex}}+{G}^{\left(1\right)}({x}_{1},{x}_{2}^{\prime}){G}^{\left(1\right)}({x}_{2},{x}_{1}^{\prime}),$$

where *G*
^{(1)}(*x _{i}*,

*x*) is the first-order correlation function of the fluctuating source field, and arbitrary order correlation function is thus expressed via the first-order correlation function

_{j}along with the relation *G*
^{(1)}(*x _{i}*,

*x*) = [

_{j}*G*

^{(1)}(

*x*,

_{j}*x*)]*. By using Eqs. (4) and (5), we can simplify Eq. (3) as

_{i}$$\phantom{\rule{4.9em}{0ex}}\times \int \int \u3008E\left({x}_{2}\right){E}^{*}\left({x}_{2}^{\prime}\right)\u3009{h}_{2}({x}_{2},{u}_{2}){h}_{2}^{*}({x}_{2}^{\prime},{u}_{2}){\mathrm{dx}}_{2}{\mathrm{dx}}_{2}^{\prime}$$

$$\phantom{\rule{4.9em}{0ex}}+\int \int \u3008E\left({x}_{1}\right){E}^{*}\left({x}_{2}^{\prime}\right)\u3009{h}_{1}({x}_{1},{u}_{1}){h}_{2}^{*}({x}_{2}^{\prime},{u}_{2}){\mathrm{dx}}_{1}{\mathrm{dx}}_{2}^{\prime}$$

$$\phantom{\rule{4.9em}{0ex}}\times \int \int \u3008E\left({x}_{2}\right){E}^{*}\left({x}_{1}^{\prime}\right)\u3009{h}_{2}({x}_{2},{u}_{2}){h}_{1}^{*}({x}_{1}^{\prime},{u}_{1}){\mathrm{dx}}_{2}{\mathrm{dx}}_{1}^{\prime}$$

$$\phantom{\rule{4.9em}{0ex}}=\u3008{I}_{1}\left({u}_{1}\right)\u3009\u3008{I}_{2}\left({u}_{2}\right)\u3009+{\mid \int \int \u3008E\left({x}_{1}\right){E}^{*}\left({x}_{2}^{\prime}\right)\u3009{h}_{1}({x}_{1},{u}_{1}){h}_{2}^{*}({x}_{2}^{\prime},{u}_{2}){\mathrm{dx}}_{1}{\mathrm{dx}}_{2}^{\prime}\mid}^{2}$$

$$\phantom{\rule{4.9em}{0ex}}=\u3008{I}_{1}\left({u}_{1}\right)\u3009\u3008{I}_{2}\left({u}_{2}\right)\u3009+{\Delta G}^{\left(2\right)}({u}_{1},{u}_{2}).$$

The second-order correlation of radiation, firstly demonstrated by the Hanbury-Brown and Twiss (HBT) experiment[17], is a quantitative description of the intensity correlation or two-photon correlation. In a lensless imaging system, for the thermal sources only the Fourier trans-form of the modulus squared of the amplitude transmittance of the object can be obtained via HBT interferometry, i.e., the phase information about the object loses and only transmitted intensity information is remained[18] Therefore a complete image cannot be formed without further image processing. Indeed, the best one can do is to recover an autocorrelation of the image from the HBT data. While, the Fourier transform of the amplitude transmittance of the object is obtined in our imaging system[10], so it retrieves the amplitude transmittance rather than the intensity transmittance of the object, as done in HBT experiment.

The information of the object imaged is extracted by measuring the spatial correlation function of the intensities 〈*I*
_{1}
*I*
_{2}〉. By subtracting the background term 〈*I*
_{1}〉〈*I*
_{2}〉, we can obtain the correlation function of the intensity fluctuations, all information about the object is contained in it

We now investigate the visibility, which is defined as Eq. (8)

It has been known from the discussion in Ref. [19] that Δ*G*
^{(2)} ≤ 〈*I*{*u*
_{1})〉〈*I*(*u*
_{2})〉 in the thermal case, so the visibility defined in Eq. (8) is never above 0.5. As we know that the visibility is determined by both the object imaged and the imaging system. According to our definition, the effects from the variety of the object can be ignored. So the changes of the visibility discussed in the following are only induced by the correlated imaging system.

## 3. Numerical results

For incoherent light, we assume that its intensity distribution is of the Gaussian type. Then the second-order correlation function for completely incoherent light source can be written as

where *G*
_{0} is a normalized constant, a is the transverse size of the source.

In the test arm, an object (transmission function *t*(*x*́)) is located at a distance *z*
_{1} from the source *S* and a distance *z*
_{2} from the detector *D _{t}*. Thus the impulse response function can be expressed in the Fresnel-diffraction approximation as

$$\phantom{\rule{4.2em}{0ex}}\times \frac{{e}^{{-ikz}_{2}}}{{i\lambda z}_{2}}\mathrm{exp}\left[\frac{-i\pi}{{\lambda z}_{2}}{\left({u}_{1}-{x}^{\prime}\right)}^{2}\right]{\mathrm{dx}}^{\prime}.$$

The reference arm contains nothing but free-space propagation from *S* to *D _{r}*. Thus the corresponding impulse response function under the paraxial approximation is

we know that, from the conclusion in Ref. [10], such a coincidence imaging system realizes the function of Fourier transform imaging under the condition of a large, uniform, fully incoherent light source. Here, we take a double slit with the slit width ω = 0.075mm and the distance between two slits *d* = 0.15mm as the object imaged. The transverse size of the source *a* = 1mm, other parameters are chosen as λ = 532nm, *z* = 175mm, *z*
_{1} = 75mm, and *z*
_{2} = *z*-*z*
_{1}.

By using a pointlike test detector located at *u*
_{1} = 0 and substituting Eqs. (9)–(11) into Eq. (6), we can get the normalized conditional intensity correlation *G*
^{(2)}(*u*
_{1} = 0,*u*
_{2}). For simplicity, we assume that the object simply transmits the light over a region of area *S _{obj}* and stops it elsewhere. Here, we consider two methods to change the transmission area. Firstly, we increase the number of slits with the same period parameters. The interference-diffraction pattern is given in Fig. 2(a). From our simulations it clearly emerges that, under the given parameters, the visibility decreases slightly with an increase of the number of slits, i.e., the transmission area.

To make our results more general, in Fig. 2(b), we give the visibility under different number of slits *n*. Here, we only depict the dots for finite slits because the interference-diffraction pattern will be deformed when *n* is much larger. It is clear that the visibility decreases with the increasing *n*.

Secondly, we increase the slit width of the double-slit under a given slit distance *d* = 0.15mm, the results are depicted in Fig. 3(a). An increase of the slit width leads to an increase of the visibility. It should be noticed, by comparing with the curves in Fig. 2(a), the change of the visibility by widening the slits is much bigger than that by increasing the number of slits. That can be simply explained as follows: the high frequency component almost has no changes when the number of slits is increased with the same period parameters, while the increasing slit width of the double-slit will make the high frequency component decrease rapidly, therefore the changes of visibility in Fig. 3 are much greater than those in Fig. 2.

In Fig. 3(b), we show the dependence of the visibility on the normalized slit width ω/*d*. Here we obtain a curve about the visibility different from that in Fig. 2(b). The visibility gets worse with a decrease of the slit width. Interestingly, Gatti *et al*. changed the transmissive area by reducing the slit length, and showed an increase of the visibility in their recent work [14]. It is obvious that the change of visibility is completely different though the two methods both reduce the transmission area. While we believe that two results are both helpful for estimating the visibility of correlated imaging with complex objects.

We also notice that a diaphragm is used at a distance *z*
_{0} from the thermal source in Ref. [14], which means that the spatial intensity distribution of the source used in their experiment is uniform, not Gaussian type, as chosen in our work. Then there exists a problem whether the spatial intensity distribution of the incoherent source leads to the notable difference between our conclusion and that in Ref. [14]. By using a finite, uniform, incoherent source, we reconsider the relation between the visibility and the slit width of the double-slit. The same conclusion is drawn that the visibility increases with the increase of the slit width. So the difference mentioned above is independent of the intensity distribution of the source, and it only derives from the variety manners of the object transmissive area.

From the results in Figs. 2(b) and 3(b), the variation ranges of the visibility we obtain coincide with the prediction value in Ref. [19]. Notice that a fundamental difficulty in experimentally observing the interference-diffraction pattern is the small relative magnitude of the second term on the right-hand side of Eq. (6), i.e., the poor visibility, when the coherence time of the detected field is much smaller than the detection time interval. Therefore the term becomes difficult to observe in the presence of an undiminished first term in Eq. (6)[15]. During the practical experiment implementation, we usually retrieve the desired information,i.e., the intensity fluctuation correlation by subtracting the background, as shown in Fig. 4. Comparing the curves which are normalized by 〈*I*
_{1})〉〈*I*
_{2}〉 in our experiment, we can draw the same conclusion that different variety manners of the transmission area will produce different effects on the visibility.

## 4. Conclusion

A simple equation used to depict the relationship between the coherence area *A _{coh}*, the object transmissive area

*A*, and the visibility had been given in Ref. [14], in which they indicated that the visibility roughly scales as the ratio of

_{obj}*A*to

_{coh}*A*. Here we show that the influence factors of the visibility are much more complex than the above depiction in a correlated imaging system, even only for the same factor (the object transmissive area), different variety manners can produce completely different effects. This may lead to a new understanding about the factors that influence the visibility. On the other hand, with the technology development of correlated imaging, one is not satisfied with the correlated imaging only for simple objects, and more complex objects are being considered. Here there are still many questions unresolved, such as what limits the visibility for complex objects. Based on the above conclusions, we can estimate the visibility for many complex objects, and thus supply the corresponding theoretic basis for the realistic implementation of correlated imaging with complex objects.

_{obj}## Acknowledgements

One of the author (Y.B.) is sincerely grateful to the anonymous referees for their helpful comments. The work is partly supported by the National Natural Science Foundation of China, Project No. 60477007, the Shanghai Optical-Tech Special Project, Project No. 034119815, and Shanghai Fundamental Research Project (06JC14069).

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