To introduce our approach, let us consider the Schrödinger and Maxwell equations for moving particles and light propagation in field free space, respectively. Taking into account only the stationary solutions

, with energy E, and h the Planck constant, the time dependence can be eliminated so that either the Schrödinger or the Maxwell equations are written in a unified form corresponding to the Helmholtz equation

where is the electromagnetic field with associated photons of momentum p, or the complex amplitude probability function of particles of momentum p. The two-point correlation, with, and asterisk denoting ensemble average and complex conjugated respectively, is a well-established method to describe optical interference [1] . This mathematical approach is extended here to account for the propagation of single particles through an interferometer. Moreover, an original physical interpretation of the model is given.

Let us consider a space volume delimited by two parallel planes at a distance named, respectively, input (I_P) and output plane (O_P), Figure 1. The two-point correlation at the O_P can be expressed as a modal expansion, whose coefficients are given by the two-point correlation at the I_P. In turn, the non-paraxial modes, obtained by solving Equation (1) by the Green’s function method [2] , define the modal kernel, i.e.

The reduced coordinates and univocally refer to pairs of points, with separation vectors suffixed B, to the midpoint between them at the positions suffixed A, so that on I_P and on O_P. The region around each position suffixed A includes pairs of points for which the two-point correlation takes on non-null values. It is called the structured support of spatial coherence. Outside the structured support, the two-point correlation nullifies or is negligible. The modal kernel

is a scalar, dimensionless, complex valued and deterministic function, essentially determined by the boundary conditions of the arrangement. is the complex transmission function of I_P.

By assuming the spatial symmetry of Equation (3) for each point in I_P, the modal kernel defines a cone with vertex at the input plane and base at the output plane, Figure 1. The electromagnetic irradiance or the particle

density distribution at I_P and O_P are the physical observables given by and

respectively. From Equation (2) follows

where

is the contribution of the structured support centred at a given to the physical observable at any. To characterize physically the two point correlation function, it is useful to introduce the dimensionless function in the integral of Equation (4b), with the Dirac delta function, thus obtaining

,

with

and

given, respectively, by the first and second term of the dimensionless function. From Equation (3) it follows that, in Equation (5a), is real and positive definite, so that. Accordingly, Equation (5a) denotes the transport of the physical observable from a fixed to any. For this reason, the vertex of the cone defined by the kernel of Equation (5a), which acts as source of, is named real point emitter. is independent of the two-point correlation for. The hermitic symmetry of and for the change leads to the real part of the kernel in Equation (5b), denoted Re. Indeed, it results from the association of the integrand values for the two degrees of freedom in orientation of. Consequently, oscillates between positive and negative values with average zero without affecting the total number of particles (Appendix 1). For this reason, the vertex of the cone defined by the kernel of Equation (5b), which acts as source of, is named virtual point emitter. The physical effect of is to distribute particles, arising from the real point emitters, along well defined regions of the interferometer. From Equations (4) and (5) the physical observable is then given by

with

and

In summary, a model for particle propagation from I_P to O_P is obtained by considering, 1) two sets of point emitters, with different physical attributes, distributed on I_P and, 2) the corresponding cone geometries confined within the volume delimited by I_P and O_P. The real point emitters represent the photon irradiance or the particle density at a given point, independently of the two-point correlation conditions. The cones associated to photon or particle propagation have different scales but similar geometries given by

.

The virtual point emitters are determined by the corresponding two-point correlation over the structured support centred at the emitter position. The correlation cones depend on the shape and size of the experimental setup and the particle momentum according with the modal kernel of Equation (5b). The virtual cones modulate the space between I_P and O_P, thus contributing to the formation of the final interference pattern. In conclusion, the propagation of particles between two planes is reduced to define two types of cones, i.e. the real and the virtual ones. The mathematical procedure to define the minimum number and distribution of real and virtual point emitters to describe the final interference pattern is analogous to what reported in optics. (see [2] and references therein).

Although particle interference has also been described in terms of trajectories (see [3] -[7] and references therein), our approach does not define single paths but rather it delineates preferential regions for particles propagation through an interferometer. Indeed, Equation (1) does not define the dynamic of a discrete ensemble of single particles but it describes the particle spatial probability density.

In this section we give a pictorial physical description of the coherence and real particle distribution through the interferometer as derived from the present model. The setup is divided into two stages in which, the output plane of the first stage is the input plane of the second one. With reference to Figure 2, the SM-stage is the volume delimited by the source (S_P) and the mask (M_P) planes placed at a distance z' to each other. The second stage MD is confined between M_P and the detector plane (D_P) placed at a distance z. The effective particle source, of size a, is at S_P, while a mask with two holes at a distance b, is at M_P. A conventional squared modulus detector is at D_P.

The behaviour of photons or particles in both stages is described by inserting, in Equations (2)-(6), the corresponding experimental parameters (the integration domains, the distance between planes, etc.). Figure 3 shows two cases of high and zero correlation, respectively. The experimental parameters of the SM-stage (i.e. the source diameter a, the distance z', etc.) determine the correlation cone that provides the structured support of spatial coherence at M_P as in graphs, (a) for high correlation and, (c) for zero correlation. In Figure 3(a), the lateral coherence (the central maximum diameter of the correlation cone) is larger than the two-pinhole distance b while in Figure 3(c) the lateral coherence is shorter than b.

Under the high correlation conditions of Figure 3(a), a virtual point emitter, indicated by the red dot, is activated between the two holes. It generates, in the MD-stage, a cone whose internal structure is schematically represented by the bright and dark regions shown in Figure 3(a). Particles propagate preferentially along the bright regions so that a system of interference fringes is formed at the detector plane. Figure 3(b) shows an overall view of particle propagation from the source to the detector.

In case of zero correlation conditions in the SM-stage, the virtual source is not activated at M_P leaving the MD-stage completely uncorrelated, Figure 3(c). Particles passing through the two holes are randomly distributed at the detector plane, Figure 3(d).

Let us proof the validity of the model by considering a two-slit or alternatively a double-hole, single electron experiment realized with the Young setup of Figure 2 (see Appendix 2 for details). Thermionic emission of single electrons is modelled as a uniform, random and spatially incoherent effective source at S_P. The term “uniform” means that the emission probability is the same for all the source points and any of them emits the same average number of electrons per unit time with the same probability. For a spatial incoherent source , which means that virtual point emitters are not activated at S_P so that, in the SM-stage, Equation (2) becomes

,

(in optics, the paraxial approach of this expression is given by the Van Cittert-Zernike theorem [1] ). For an ideal point source at, it turns out that and. Therefore only one correlation cone, given by the modal kernel, specifies the structured support at M_P. Accordingly a finite lateral coherence length is defined at a plane, at any distance from the source. This is a substantial different result with respect to the one obtained with a wave approach according to which an infinite transversal coherence is considered. By enlarging the effective source size the number of correlation cones increases. Their superposition in the SM-stage reduces the resulting structured support dimensions (lateral coherence) at M_P, as discussed previously. An important consequence is that the effective source size determines the coherence conditions for interference, independently of the electron emission statistics.

The two holes of the Young mask are modelled with mathematical points at, it means that diffraction effects are neglected. When the electron beam strikes on the mask, then a real point emitter is activated in the hole crossed by a single electron. At the same time, if the correlation cone “illuminates” both holes, one virtual point emitter is activated at the midpoint between them, Figure 3(a). The final interference fringe system is given by the contribution of real and virtual point emitters according to Equation (6a),

,

where

with

By taking into account only the fraction S₀ of electrons which go through the mask, we define P the probability of an electron passing the pinhole at the position, and the probability of electrons passing at. Therefore, and represent the correlation functions of the real point emitters at the pinholes. By applying the Schwartz inequality to the two-point correlation

, we obtain, where is the modulus

of the complex degree of spatial coherence [1] at M_P and the condition assures that electrons go through both pinholes.

Simulations of the build-up of single electron interference patterns with high, partial and very low correlation conditions are shown in Figure 4. The entries are in accordance with reported experimental setups [8] -[10] , (i.e. two-slit spacing 330 nm, detector resolution 128 × 32 pixels, p = 1.59 × 10⁻²² kg∙m∙s⁻¹). The two-point correlation at M_P is adjusted by changing the distance z' in the SM-stage to obtain three different coherence conditions while, the distance z in the MD-stage is fixed. The resulting detection events of 70,000 electrons are calculated. Detector sensitivity is modeled by adjusting the minimum number of “hits” on a specific pixel, during the integration time, in order to produce the record of electron arrivals. The background and the shot noises of conventional physical detectors are also taken into account. Zero electron loss is assumed so that the emission statistics determines the random electron arrivals to the detector.

Images of Figure 4 have two identical frames. The upper ones show the random electron hits on D_P at the pixel indicated at top right. The number of arrivals is reported at top left. The lower frames show the interference figures resulting from the build-up of 70,000 single electrons. The cumulative, single electron detection process is accurately simulated with the present model in accordance with observations [8] [9] (for a historical review of single electron interference experiments see [10] ). The movies reporting the build-up of interference fringes with high, partial and no coherent conditions are available respectively at

https://youtu.be/gcKUWLjXvBQ

https://youtu.be/R4zBLL1Wv10

https://youtu.be/wgCb7O9eUqE

The potential of our model is presented to fit the interference pattern obtained with single electrons striking on a line grating (see Appendix 3 for details). Figure 5 shows the left lobe of the observed pattern. It consists of an ensemble of bright spots with decreasing intensity as evidenced also by the calculated intensity profile.

Simulations were carried out by considering a uniform effective source whose size is controlled with the condenser lens system of the microscope [8] [9] . Under these conditions, a resulting sinc-function correlation envelope is obtained at M_P. Due to the particular construction features of the grating, to obtain a careful fit of the interference peak modulation a Gaussian transmission function was used. Because images were recorded with non-standard electron optical ray path, microscope lens aberrations and non-linear response of the photographic recording were not considered. The excellent agreement with the experimental pattern of Figure 5 was obtained with an effective source diameter 130 nm.

Starting from these promising results, we simulate the single molecule interference patterns of Phthalocyanine (PcH₂) and its derivative (F₂₄PcH₂) [11] . The molecular beam, generated by a Knudsen cell in a horizontal interferometer configuration, is limited by a collimating slit which determines the effective source size. Subsequently, molecules go through a linear transmission grating and interference patterns are revealed by a dedicated detector. In Figure 6, our simulations (red solid lines) are contrasted with the experimental curves reported in [11] . The simulation results are obtained with the experimental setup parameters which, for the small PcH₂ and large F₂₄PcH₂ molecules are, respectively, 1) collimating slit 1 μm wide, collimating slit-grating distance 0.702 m, 2) collimating slit 3 μm wide, collimating slit-grating distance 0.566 m [11] . The transmission function for both collimating and grating slits is assumed constant in Figure 6(a) and Figure 6(b), and Gaussian in Figure 6(c) and Figure 6(d). The molecule velocity distribution, centred at values corresponding to the de Broglie wavelengths 5.2 pm for PcH₂ and 2.1 pm for F₂₄PcH₂, is corrected for vertical dispersion due to gravity (for a more direct comparison of our simulations with those of [11] , the de Broglie wavelength is used instead of particle momentum). Moreover, an effective width reduction of the grating slits was considered, due to van der Waals interaction between grating bars and passing molecules [11] - [14] . Only the curves in Figure 6(c) and Figure 6(d) are in excellent agreement with the experimental results thus confirming that the assumption of a Gaussian transmission function is crucial. The maximum molecule transmission assumed centred on the slit optical axis is a further confirmation that the interaction between molecules and grating walls is essential to describe the final intensity distribution. Our simulation in Figure 6(d) is obtained only with a rigorous evaluation of the two-point correlation conditions (Appendix 4), without resorting to the ad hoc Gaussian correction to smear the interference peaks used in [11] .

It turns out that the lateral coherence distance at the grating plane is larger for PcH₂ molecules than for F₂₄PcH₂ molecules. Because the grating period is the same in both experiments, the number of correlated slits is higher for the small molecules with respect to the bigger ones. It must be pointed out that the numerical fit of the experimental data of Figure 6(d) was obtained, by the Vienna group, using diffraction integrals in the paraxial approximation convoluted with an ad hoc Gaussian with a standard deviation of 3 μm [11] . To explain the observed smearing of the interference fringes, loss of contrast of Figure 6(d) with respect to Figure 6(c), molecular fragmentation within the beam and/or surface diffusion were assumed. Alternatively, our model, based on a

rigorous evaluation of the two-point correlation conditions (Appendix 4), gives directly an excellent fit of the observed results advancing no further hypothesis.

Talbot carpets are near field interference figures which are repeated at specific distances away from the grating plane. The transversal dimension of the maxima and their periods, which depends on the grating distance, can be consistently reduced with respect to the grating slit size. In principle, this peculiarity is worth considering to realize molecular nanostructures on a large scale. Figure 7 illustrates a Talbot carpet of a thought experiment, based on realistic set up parameters, which could be realized with Tetracene molecules moving with a velocity of 2000 m/s produced by a SuMBD source [15] . A collimating slit, width 0.2 μm, is placed in front of the source at 1 m from a linear grating. The grating period is 500 nm and individual slit width 50 nm. As it can be seen in Figure 7, the period of the principal maxima is about 10 nm, a value much smaller that the slit width (50 nm). This result demonstrates clearly how our model can be adopted for nanolithography applied to organic electronics.