## Abstract

Statistical aspects of Young's double-slit diffraction experiment are analysed quantitatively. It is shown that the signal-to-noise ratio and the spatial resolution in the detected diffraction pattern satisfy a duality relationship which implies that both of them cannot be improved simultaneously beyond a certain limit if the total number of particles forming the image is fixed. As a consequence of this duality, it is possible to estimate the minimal number of particles that have to be detected in order for two slits separated by a given distance to be resolved with a confidence level corresponding to a pre-defined signal-to-noise ratio, e.g. according to the Rose criterion. These results are related to the recently introduced imaging system quality characteristic which combines the spatial resolution and the noise sensitivity, and allows one to estimate the efficiency with which imaging quanta are utilised in a system to deliver maximal amount of information about the imaged object. The presented results can be useful for applications where the imaging quanta are at a premium or where minimization of the radiation dose is important.

© 2015 Optical Society of America

## 1. Introduction

It is well known that, in accordance with the laws of quantum physics, very small particles (such as electrons, ions, molecules etc.) can exhibit detectable wave properties. The latter manifest themselves e.g. in the process of diffraction. Young’s double-slit diffraction is a famous experiment [1], which is conceptually simple and informative, and is often used to demonstrate the wave properties of particles. On the other hand, corpuscular properties of the particles manifest themselves through the appearance of distinct spots in the image that correspond to detection events of individual particles during the diffraction pattern formation in the Young's double-slit experiments performed at low flux levels. The combination of wave and corpuscular properties of matter observed in these experiments is considered a manifestation of Bohr's Complementarity Principle [2].

Recently, several groups have published results of diffraction experiments in which formation of Young’s double-slit diffraction pattern has been carried out “explicitly” by individual particles [3–6]. Specifically, the experimental conditions were such that the particles were emitted one by one and the time required to emit, propagate and detect a single particle was small compared to an average time between the emissions of sequential particles. It has been shown, in particular, that while it is impossible to predict in advance the exact detection point for each individual particle, after detecting a large number of particles the image starts to resemble the expected interference pattern.

In this paper we take a different view on the image formation in the Young’s experiment. Namely, we consider the following question: given the statistical nature of the image formation, how can one quantify the status of the image formed by a finite number of photons and extract information about the slits from the image? In order to answer this question, in this paper we employ our recently introduced imaging system quality index [7–9] and the detectability/distinguishability figure of merit from statistical decision making theory [10,11].

Although this paper is primarily devoted to the analysis of statistical properties of image formation in the classical Young's double-slit experiment, the results obtained here have relevance to other problems of imaging science. In particular, as the issue of distinguishability of two closely located features lies at the heart of the definition of the spatial resolution of imaging systems, the quantitative dependence of such distinguishability on the number of image forming particles becomes very important under low-light imaging conditions or when the dose delivered to the imaged specimen must be minimized, as e.g. in some biological and medical imaging techniques. A link to such problems can be established for the central results of the present paper via the notion of the imaging quality characteristic [7–9]. Details of such connections and further related results will be the subject of our forthcoming paper [12].

The present paper is organized as follows. In Section 2 an imaging task corresponding to Young's experiment is formulated and the relevant figure of merit is introduced. In Sections 3 and 4, the duality between noise and spatial resolution in contact images and in far-field images of two identical slits is investigated. Discussion of the results and conclusions are given in Section 5.

## 2. Formulation of the problem

Consider two identical slits located in the same plane, having the width *b* and transversely separated by distance *d*. The slits are illuminated uniformly by a source located at distance *R*_{1} upstream from the slits. The image of the slits is acquired in the image plane located at distance *R*_{2} downstream from the slits. A schematic diagram of this imaging setup is shown in Fig. 1(a).

In the present paper, we consider a basic problem where one is interested in deciding whether it is possible to reliably distinguish (resolve) two slits by analysing their image, in the presence of noise. We aim at establishing a quantitative relationship between the relevant geometrical parameters, including the slits’ width, the distance between two slits (which corresponds to the spatial resolution of the setup), some figure of merit (FoM) that objectively quantifies our ability to resolve the features of interest in the presence of noise and a corresponding “quality” characteristic of the setup that simultaneously characterizes its spatial resolution and noise sensitivity [7–9].

In this study we shall use, as the FoM, a signal-to-noise ratio (SNR) similar to that of the ideal observer from Bayesian decision making theory [10,11]. In terms of this theory, we are interested in discriminating the following two signals known exactly. The first signal, ${S}_{1}(x;b,d)$, corresponds to the imaging scheme presented in Fig. 1(a) and consisting of two identical slits with width *b* separated by the distance *d*. The second signal, ${S}_{2}(x;b)$, corresponds to the imaging scheme presented in Fig. 1(b) and consisting of a single slit with the same width *b* but illuminated with twice the incident fluence compared to the case of two slits. Hence, formally, ${S}_{2}(x;b)={S}_{1}(x;b,0)$. Note that these signals correspond to one-dimensional (1D) photon fluence distributions in the image plane integrated over the length of the slits (i.e. integrated along the *y* axis).

In general, our ability to discriminate the above two signals strongly depends on the nature and parameters of the noise that affects the signals. In the following we restrict our consideration to the case of spatially uncorrelated multiplicative noise for which the variance of noise in each signal depends on the signal. In particular, in the case of the Poisson (photon counting) noise, the variance of noise (per unit length) in the signals is equal to the signals (assuming that the latter are measured in the units of photon fluence), $st{d}_{n,i}^{2}={S}_{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}(i=1,2)$. We assume for simplicity that a photon counting detector is used in this experiment. Note, however, that the nature of the particles (photons, electrons, neutrons, etc.) used to form the image is generally irrelevant to the results presented below. The square of the SNR is written as follows:

In the next two sections, two limits of the above imaging scheme will be considered separately: 1) contact images of the slits (with a negligibly small distance *R*_{2}) and 2) Fraunhofer diffraction by the slits. In both cases, perfect slits are considered, with the transmittance function of individual slits, $T(x;b)$, given by the following expression:

## 3. Noise-resolution duality in contact images of the slits

Let *F* and 2*F* be the uniform photon fluence in the beam incident on the screen containing two slits and one slit, respectively. Note that here we consider (for simplicity) the one-dimensional case, so the fluence is integrated along the *y* axis, i.e. *F* is expressed in photons per unit of length. Then, in the case of a negligibly small distance *R*_{2}, the images produced by two slits and one slit, respectively, can be expressed as follows:

Substituting Eq. (3) into Eq. (1), one obtains:

An example dependence of the *SNR _{I}* on the ratio $d/b$ is shown in Fig. 2(a) (black line). Equation (5) also allows one to express the minimum total number of photons,

*N*, required to reliably distinguish (with the signal-to-noise ratio

_{tot}*SNR*) the considered two images,

_{I}*S*

_{1}and

*S*

_{2}, as a function of the ratio $d/b$. An example dependence of

*N*on the ratio $d/b$, for

_{tot}*SNR*= 5, is shown in Fig. 2(a) (blue line).

_{I}Figure 2(b) shows the dependence of the ratio $SN{R}_{I}^{2}/({N}_{tot}\text{\hspace{0.17em}}d/b)$ on $d/b$. Analysis of Eq. (5) and Fig. 2(b) shows that

*S*

_{1}and

*S*

_{2}, as characterized by$SN{R}_{I}$, cannot exceed the square root of twice the total number of photons used to form the image. For example, in order to achieve

*SNR*= 5 (which is related to the Rose criterion [13]), one needs at least 12.5(

_{I}*b*/

*d*) photons to form the corresponding image. This result is not very useful at the limit of

*d*>>

*b*(where it only tells us that the required number of photons is non-negative), but tells us that at least 13 photons are required to achieve

*SNR*= 5 at the limit

_{I}*d = b*.

According to Eq. (6), at a fixed incident fluence, the spatial resolution and the SNR can be traded for each other, but it is impossible to improve both of them simultaneously beyond a certain well-defined limit. Therefore, this relationship expresses a form of duality existing between the image noise and the spatial resolution in the double-slit experiment in the contact imaging regime. Note that Eq. (6) is also quite similar in essence to the (1D case of) “noise-resolution duality principle” introduced in [7]: ${Q}_{S}^{2}=SN{R}_{1}^{2}/(F\Delta X)\le const$. The latter quantity, ${Q}_{S}$, termed the “intrinsic quality” characteristic of the imaging system in [7], is a function of the spatial resolution $\Delta X$, defined as the width of the system's point-spread function (PSF), and the signal-to-noise ratio, $SN{R}_{1}$, which corresponds to the output signal and noise produced by the system in the case of uniform illumination with incident fluence *F*. By analogy with this definition, the expression estimated in Eq. (6) can be viewed as the square of the “imaging quality” in the double-slit experiment in the contact imaging geometry.

## 4. Noise-resolution duality in far-field images of two identical slits

Consider the Fraunhofer diffraction pattern formed at the effective distance $z\equiv {R}_{1}{R}_{2}/({R}_{1}+{R}_{2})$ from two identical slits with the width *b* and separated by the distance *d* (in this section we assume that *d* >> *b*). In order to satisfy the far-field (Fraunhofer) diffraction conditions, the following inequality should be fulfilled:

*λ*is the wavelength of the incident illumination.

In the case of partially coherent quasi-monochromatic incident beam and/or detector with finite spatial resolution, such that the visibility $V\equiv ({I}_{\mathrm{max}}-{I}_{\mathrm{min}})/({I}_{\mathrm{max}}+{I}_{\mathrm{min}})$ of the interference fringes is in the range (0, 1) and the fringes are shifted by a distance *x*_{0} with respect to those in the case of a perfect imaging system (with coherent incident beam and perfect detector), the photon fluence in the image created by two identical slits is written as follows [1]:

*F*is the photon fluence produced by any one slit, when the second one is closed, at the centre of the image, $\Delta x=\lambda z/d$ is the distance between the fringes in the interference pattern formed by the slits and $\mathrm{sinc}(x)=\mathrm{sin}(\pi x)/(\pi x)$. Equation (8) corresponds to the case of the slits’ width

*b*much smaller than the effective spatial coherence length of the imaging system (see below). Note that for brevity we omitted the dependence of the signals on parameters other than the spatial coordinate

*x*in the left-hand side of Eq. (8) and below. Also, it should be noted that, for convenience, the coordinate

*x*refers to the slits plane, i.e. the images are considered back-projected onto this plane.

Our aim is to distinguish the above interference pattern formed by two identical slits from the intensity profile formed with the same total number of photons by a single slit:

The intensity profile described by Eq. (9) formally corresponds to the case of incoherent illumination of the slits, so that*V*= 0 in Eq. (8).

Figure 3(a) shows examples of the signal *S*_{1}, described by Eq. (8), for two values of the visibility, as well as the signal *S*_{2} described by Eq. (9). Figure 3(b) shows the corresponding difference, *S*_{1} – *S*_{2}, for the same values of the visibility as in Fig. 3(a).

Using Eqs. (8) and (9), the SNR of the ideal observer corresponding to one central period of the interference pattern is (as long as *d* >> *b*, one can neglect the effect of the finite width of the slits on the central part of the interference pattern):

It is of practical importance to estimate the minimum number of photons that form the interference pattern from two identical slits which is distinguishable from the pattern formed by a single slit (with the same total number of photons). This question obviously has a close relationship to definitions of the spatial resolution of an imaging system [see e.g. Chap.7 in 10]. The total number of photons can be calculated by integrating either one of the signals, *S*_{1} or *S*_{2}, e.g.:

*S*

_{1}and

*S*

_{2}, on the interference pattern visibility, as described by Eq. (15), is shown in Fig. 4(a). For this SNR value, the minimum required number of photons,

*N*= 40.4, corresponds to a fully coherent incident wave,

_{tot,min}*V*= 1. An example of the dependence of the square of the SNR of the ideal observer on the interference pattern visibility, as described by Eq. (14), is shown in Fig. 4(b).

Comparing Eq. (14) with Eq. (5), one can see that (a) the $SN{R}_{I}^{2}$ corresponding to both the near- and far-field “images” of the slits is linearly proportional to the total number of photons ${N}_{tot}$ used to form the corresponding image, and (b) the ratio $Q=SN{R}_{I}^{2}/{N}_{tot}$ in the near-field case depends only on the ratio *d / b* of the distance between the slits to their width, while in the far-field case, $Q$ depends only on the visibility *V*. It should be noted, however, that the visibility *V* of the interference pattern usually strongly depends on the distance *d* between the slits and on the resolution of the imaging system. The latter, obviously, depends on the size of the primary source, on the detector’s resolution and on the imaging geometry (in particular, on the magnification). For instance, in the case of a spatially incoherent source with the Gaussian distribution of intensity and a detector with the Gaussian PSF, it can be easily shown that there is no shift of the interference fringes, *x*_{0} = 0, and the visibility is given by the following expression:

Analysis of Eq. (16) indicates that by increasing the distance *d* between the slits, and hence decreasing the period $\Delta x=\lambda z/d$ of the diffraction pattern, the visibility is decreased. Then, according to Eq. (14), the corresponding $SN{R}_{I}^{2}$ in the diffraction pattern will also decrease. Equivalently, according to Eq. (15), the minimal number of photons required to achieve a certain level of SNR, will increase monotonically as a function of *d*.

These results agree with the naturally expected behaviour, according to which a larger transverse coherence length of the illuminating beam in the slits plane, and/or more photons, and/or better resolving detector will be required in order to distinguish two slits separated by a larger distance. Indeed, the ratio ${\sigma}_{sys}/\Delta x$ in Eq. (16) can be alternatively presented as follows:

where ${l}_{sys}^{-2}={l}_{c}^{-2}+{l}_{det}^{-2}$, ${l}_{c}\equiv \lambda {R}_{1}/(2\pi {\sigma}_{src})$ is the transverse (spatial) coherence length of the incident wave in the slits plane and ${l}_{det}\equiv \lambda {R}_{2}/(2\pi {\sigma}_{det})$ is a parameter closely resembling the spatial coherence length*l*, but reflecting the spatial resolution of the detector rather than properties of the incident illumination. Figure 4(c) shows the actual dependences of the visibility of the interference pattern, as well as the minimum number of photons, on the ratio $d/{l}_{sys}$.

_{c}Using Eqs. (14), (16) and (17) one can express the ratio $SN{R}_{I}^{2}\text{\hspace{0.17em}}(d/{l}_{sys})/{N}_{tot}$ as a function of $d/{l}_{sys}$. This function is shown in Fig. 4(d) and allows one to write the following inequality, which closely resembles the “noise-resolution duality” Eq. (6) obtained in the near-field regime:

The ratio ${l}_{sys}/d=\Delta x/(2\pi {\sigma}_{sys})$ plays the role of a normalised spatial resolution in the interference pattern formed by two slits, in the far field. By analogy with the previous results, Eq. (18) can be viewed as an estimation of the imaging quality in the double-slit diffraction experiment in the far field. As in the contact imaging case, the SNR and the spatial resolution in the imaging plane can be traded for each other, but it is impossible to improve them both simultaneously beyond a certain limit. Note however, that in the far-field case the dependence of the imaging quality on the distance between the slits (i.e. on the spatial resolution in the object plane) is opposite to that in the near-field case.

## 5. Conclusions

Summarising the above results, we can state that the distinguishability between the images corresponding to two slits and one slit, as described by the SNR of the ideal observer, has a very simple and physically transparent behaviour, both in the near and far fields. Namely, in both cases $SN{R}_{I}^{2}$ depends linearly on the total number of photons (or other particles) forming each image. In the case of near-field imaging, $SN{R}_{I}^{2}$ is a non-decreasing function of the ratio $d/b$ of the distance between the slits and their width. In the case of the far-field imaging, $SN{R}_{I}^{2}$ is a monotonically increasing function of the ratio ${l}_{sys}/d$ of the effective spatial coherence length to the distance between the slits (assuming that *d >> b*). In both cases, the ratio of $SN{R}_{I}^{2}$ and the (normalised) spatial resolution in the image plane cannot exceed the product of the total number of photons forming the image and an absolute dimensionless constant. This result is a manifestation of a duality existing between the SNR and the spatial resolution in the image plane, which is similar to the recently demonstrated noise-resolution duality in linear shift-invariant systems [7,8].

Our results clarify the statistical aspects of the famous Young's double-slits experiment, which has been used over the years for demonstration of the dual wave and corpuscular nature of light, electrons and even whole molecules [1–6,16]. Using this simple classical experiment as a model, the present paper gives an explicit expression for the minimal number of the relevant events (e.g. photon detections) that need to be registered in order to resolve objectively (with a pre-defined level of SNR) a certain property of the illuminated object (e.g. to distinguish the images of two slits and one slit). We have demonstrated that a new characteristic, termed “imaging system quality”, can be used as a quantitative measure of the efficiency with which an imaging system utilises photons or other particles to form the images which simultaneously have high signal-to-noise ratio and fine spatial resolution. We believe that these results are relevant to standard definitions of SNR and resolution in imaging systems. They may also find practical application in areas where either the observations have to be performed with low number of imaging quanta, as in some astronomical studies, or where the radiation dose delivered to the imaged object needs to be carefully controlled and limited, as in some biological and medical imaging techniques [9].

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