1.

Introduction

Momentum exchange theory (MET) is presented as an alternative picture for photon diffraction based upon quantized momentum transfer with the scattering lattice or aperture.^2–6 MET draws from the early ideas of Duane⁷ and later formalized by Landé.^8–11 MET starts from a momentum representation for scattered particles and postulates the probability distribution for diffraction scattering is determined by at least two important factors: (1) the momentum transfer states of the scattering lattice and (2) the distance over which momentum is transferred between the lattice and scattered particle. The distinct feature of MET is this dependence on the specific path of the scattered particle and location of momentum transfer. Classical optical wave (COW) theory leaves us with an erroneous understanding of diffraction in suggesting that the probability for detecting a diffracted photon is determined on observation at the point of detection.^2,5 This picture is conceptually flawed as the probability must be determined at the location of scattering to conform to conservation of momentum. The current paper compares and contrasts MET with optical wave theories to elucidate these underlying momentum exchange selection rules. This picture is important as it illuminates our debates over the wave-particle nature of light and quantum uncertainty, framing the contrasts between the standard (Copenhagen) interpretation and several alternative interpretations.^12–16

For clarity, it is important to emphasize here the distinction between diffraction, which relates to the scattering and probability distribution for individual photons, and interference, which is a cooperative phenomenon that relates to the probabilities of detection (absorption?) when there is a multiplicity of photons (e.g., two electromagnetic beams) that can influence each other at the point of detection. An individual particle cannot interfere with itself. Detection of a photon is dependent on its polarization and phase (orientation?) and the polarizability of the electronic states of a detector (antenna or molecule) it interacts with. Thus, based upon the characteristics of detection, multiple photons, concurrently at a detector, can affect the potential for each to be absorbed (can interfere). Observed optical phenomena can involve diffraction, or interference, or both (e.g., laser holography).

In the 1920s, Duane⁷ and later Ehrenfest and Epstein¹⁷ showed diffraction could be explained in terms of a momentum change by interaction with a scattering lattice dependent on the geometry and momentum states of the lattice that can be analyzed according to Fourier’s theorem. These early ideas seemed to be lost with the initial formulations of wave mechanics, but were later revived by Landé, who used them in his formulation of quantum theory.^8–11 The primary learning from these formulations is that the probability distribution for the scattering of a particle is determined by the geometry of the scattering lattice, much as the angular distribution of light diffracted in classical optical theory is the Fourier transform of the aperture or obstacle.^18,19 These ideas languished because of the success of classical electrodynamics and wave mechanics to explain both optical diffraction and interference, despite their conceptual challenges. More recently, Van Vliet²⁰ refreshed these early ideas of Duane and Landé with an updated treatment of diffraction by linear momentum quantization through periodic structures, again stressing this approach avoids a dual nature to quantum particles. Momentum conservation is preserved without retardation. An additional contribution of this work was the examination of systems with reduced symmetries. Such descriptions are more consistent with later formulations of quantum theory, such as the statistical interpretation of Ballentine²¹ or path integral approach of Feynman.²²

Storey et al.²³ joined this debate on interpretations examining a familiar “welcher Weg” (which way) experiment associated with double-slit diffraction, describing the photon scattering in terms of a momentum representation and quantized momentum transfer. The observation was again put forward that the interference pattern was determined by the uncertainty principle. Here, it was described that a microwave radiation exchange within the diffracting aperture is responsible for a transverse momentum “kick” to the scattered photon. Wiseman et al²⁴ provide further perspective describing a formalism to analyze momentum transfer in welcher Weg double-slit diffraction experiments using the Wigner function. This question was also analyzed using Bohm’s formulation of quantum mechanics in which scattered particles have a definite position and momentum at all times.²⁵ This approach conceptualizes the interaction and momentum exchange at the diffracting slits and suggests the potential application of Bohmian trajectories to single-slit and other diffraction configurations. This treatment demonstrated the utility of defined particle paths and opens the possibility that this formalism can accommodate geometrically defined transfer functions that are dependent on the separation of the path from the aperture boundary proposed by MET.

The argument for the MET picture will be built in succeeding sections. Section 2 provides a contrast and comparison of optical wave theories with MET noting a common connection in the Fourier transform. Section 3 examines a momentum representation of the equations of classical optical theory to point out their dependence on momentum exchange at the scattering aperture. Section 4 reviews single-slit diffraction, demonstrating that scattering probabilities are dependent on momentum state functions of the slit and suggesting that the scattering will be dependent on the path of the photons. Section 5 provides a similar analysis of straight-edge diffraction connecting momentum exchange probabilities to the analysis of Fresnel zones. Section 6 provides an analysis, working backward from the straight-edge diffraction pattern observed, to define the probability distribution for momentum exchange with photons in the plane of the straight edge. This analysis demonstrates the dependence of the scattering on the path of the photon. Section 7 extends the analysis to multiple slits and the Talbot effect. Section 8 reviews diffraction by a circular disc to examine negative (attractive) momentum dispersion of photons in the plane of the disc. Section 9 generalizes the MET formalism for broader application. Section 10 looks at the implications and predictions for other experimental configurations.

2.

Contrasting Pictures of Diffraction

To aid our understanding of this more integrated picture, we contrast MET with the predominant historical pictures that have emerged for diffraction phenomena.

2.4.

Geometric Correlations of Theories

The distinctions and similarities between these three different pictures of diffraction are illustrated in Fig. 1, which was detailed in an earlier paper.⁵ The diagrams depict the diffraction of monochromatic light with a detection point at P and a reference point for diffraction at O. We assume the $x,y$ plane is normal to the propagation vector of the light that aligns with the segment $\overline{\mathrm{AP}}$. Our plane of interest contains the propagation vector (light ray) from the source to point P and the reference point, O. Figure 1(a) illustrates the Huygens–Fresnel principle of COW, where Huygens wavelets are generated in the $x,y$ plane. The segment $\overline{\mathrm{BP}}$ depicts the path of one wavelet that can be compared with the path and phase of $\overrightarrow{\mathrm{AP}}$. In COW, the diffraction field intensity is calculated via the weighted sum (integral) of all the Huygens wavelet paths that converge on the detection point, P. The intensity calculation is derived from the relative phase contribution of each wavelet, dependent on the relative lengths of the segments, e.g., $\overline{\mathrm{AP}}$ and $\overline{\mathrm{BP}}$. As the lengths of two sides and one angle of a triangle always determine the length of the third side, the lengths of $\overline{\mathrm{AP}}$ and $\overline{\mathrm{BP}}$ uniquely determine $\overline{\mathrm{AB}}$. Similarly, $\overline{\mathrm{OP}}$ and $\overline{\mathrm{AP}}$ will determine $\overline{\mathrm{OA}}$ and thus, $\overline{\mathrm{OB}}$ is also defined. To clarify, this phase difference, $\mathrm{\Delta }\text{phase}=-i2\pi \mathrm{\Lambda }/\lambda$, between any unperturbed (reference) wavelet of length, $\overline{\mathrm{AP}}$, and any dispersed wavelet of length $\overrightarrow{\mathrm{BP}}$ is a function of the difference in the lengths, $\mathrm{\Lambda }=\overline{\mathrm{BP}}-\overline{\mathrm{AP}}$. As $\overline{\mathrm{BP}}=\overline{\mathrm{BA}}/\sin \theta$ and $\overline{\mathrm{AP}}=\overline{\mathrm{BA}}/\tan \theta$, the phase variation for a Huygens wavelet at a detection point will always depend on the angle of deflection and a reference distance along the $x$ or normal axis

Eq. (2)

$$\mathrm{\Lambda }=\frac{\overline{\mathrm{BA}}(1-\cos \theta )}{\sin \theta }\mathrm{.}$$

Assuming the Fraunhofer region of diffraction and using the small angle approximation

Eq. (3)

$$\mathrm{\Lambda }\cong \overline{\mathrm{BA}}\sin (\theta /2)\cong \frac{\overline{\mathrm{BA}}}{2}\sin \theta .$$

Fig. 1

Three different conceptual pictures for optical diffraction related to a reference point at O with light intensity measured at P. (a) COW picture based upon the Huygens–Fresnel principle summing the interference between wavelets emanating in the $x,y$ plane; (b) the BDW theory picture with $\overline{\mathrm{OP}}$ representing the diffraction wave; and (c) the MET picture with the scattering of photons by quantized momentum exchange determined by the proximity of the scattering boundary point.

This parallels the Fourier transform dependence on the transverse dimensions in this formalism; recognizing the Fourier transform directly connects an aperture function to a far-field function.³⁷ Two wavelet trajectories can be related through the relation of each to the phase of a reference wavelet. I emphasize that these geometric relations are axiomatic and not the result of alternative theoretical considerations. There is a fundamental connection between an angle of deflection and the “phase separation.”

Figure 1(b) illustrates BDW theory, where $\overrightarrow{\mathrm{OP}}$ represents the ray associated with the diffraction wave scattered from our reference point, O, that might be associated with the edge of an aperture. $\overrightarrow{\mathrm{AP}}$ is the path of the unperturbed ray and $\overrightarrow{\mathrm{BP}}$ is the path of a perturbed ray. This approach sums the phases of all the rays reaching the detection point, P. From the geometry of this depiction, we can assert that the phases at the detection point will be dependent on the distances, $\overline{\mathrm{OA}}$ and $\overline{\mathrm{OB}}$, and the deflection angle.

As the relation of Eq. (3) is axiomatic, the scattering probability distribution derived via the Huygens–Fresnel principle can always be reframed in terms of momentum exchange probabilities determined by geometries of the scattering lattice. I will demonstrate that by adopting a momentum representation for the mathematical formalism used in COW and BDW, we can develop a connection to these probabilities and intensity profiles that our geometric analysis demonstrates must exist. Familiar tools to make this connection will be the Fresnel zones and the Fresnel number that can be related to momentum exchange. Figure 1(c) illustrates MET, where photons passing the reference point O near B are scattered by the exchange of momentum and deflected to P, with the probability being dependent on the selection rules for momentum exchange. To clarify, in the momentum representation, where the photon momentum is $p=h/\lambda =\hslash k$, the rotation of the photon trajectory in the $x,z$ plane, $\theta$, is connected to a change in momentum along the perpendicular ($x$) axis, $\mathrm{\Delta }{p}_x=p\sin \theta$, and the momentum change (reduction) along the $z$-axis, $\mathrm{\Delta }{p}_z=p(1-\cos \theta )$. Thus, the relative phase variation will depend on as follows

Eq. (4)

$$\mathrm{\Delta }\text{phase}=\frac{-2\pi \mathrm{\Lambda }}{\lambda }\cong \frac{-\pi \overline{\mathrm{BA}}\mathrm{\Delta }{p}_x}{h}\cong \frac{-\pi \overline{\mathrm{BA}}}{{\lambda }_e}\cong \frac{-\overline{\mathrm{BA}}}{2}{\mathrm{k}}_e,$$

where ${\lambda }_e$ used here is associated with an effective wavelength defining momentum exchange and the phase change is related to the ratio, $\overline{\mathrm{BA}}/{\lambda }_e$. With this conversion, our probabilities for momentum exchange must involve a geometric summation (integral) at the aperture that parallels the summation of phases we determine in the far field. This will be the approach developed further in the next sections of this paper. We will note an advantage to this approach is that probabilities can be associated with individual light paths that simplify the analysis of a broadened set of aperture configurations and experimental conditions. The approach also provides insight to the phase problem or the Fresnel phase-shift that is associated with the classical or Fourier representation of diffraction.

3.

Mathematical Foundations for Photon Diffraction

An excellent visualization of slit diffraction of photons was provided by Harris et al.,²⁶ who compared the experimental results from high-resolution diffraction patterns with the theoretical predictions of optical wave theory in the Fraunhofer and Fresnel regions and found an excellent agreement between the calculated and observed patterns. This work produced plots of the scattering patterns at different distances from the slit that provide an indication of how the light scattering pattern near the slit progressively fans out as a detector screen moves farther from the diffracting slit. From this, we can visualize the distribution of paths followed by scattered photons. We can use the COW formalism to relate to a description of photon diffraction in terms of quantized momentum transfer. Adopting this approach does not endorse the wave picture that classical optical theory presents, but rather it is used to demonstrate the axiomatic compatibility with our MET formulation for particles. I will start with a generalized description of light diffraction based on the Rayleigh–Sommerfeld and the Fresnel–Kirchoff theories of diffraction. I consider diffraction of light by an aperture in the $x,y$ plane at $z=0$. I assume a light source treated as a monochromatic plane wave initially propagating along the $z$-axis. This can be experimentally modeled with laser light. An additional assumption is that the source of light is sufficiently dilute that the response of any detector used is linear to the intensity of light—to the number of photons reaching the detector. This also assumes that there is no co-operative phenomenon (interference) observed at the detector, where photons that are “out-of-phase” might be observed to interfere due to the response characteristic of the detector or amplifier.

Starting with the first Rayleigh–Sommerfeld diffraction formula, the electromagnetic field amplitude at a screen or detector at $X,Y,Z$ is $U(X,Y,Z)$, where the square of the amplitude is proportional to the energy flux or flux of photon particles. This formulation is reviewed in multiple texts on optical diffraction.^19,30,38 If this flux is normalized across the distribution, it can be used to provide probabilities for detection of photons. The first integral formula is

The Kirchoff approximation to the integral integrates only over the area of the aperture. With the dispersion of light at $(x,y,0)$, the propagation vector has a length

This approximation thus assumes $Z\cong r$. Equation (5) becomes

As pointed out by many authors, the form of Eq. (8) provides a Fourier transformation of the field flux and is a two-dimensional convolution with respect to $x$ and $y$. The exponentials correspond to the transfer function.^38–41 As the momentum of a photon can be expressed in terms of the wavenumber or wavelength, I will make this relation explicit in our analysis by the substitution, $p=h/\lambda$. I also note that the momentum deflections along the $x$ and $y$ axes can be represented by

These substitutions provide a momentum representation to our scattering equations (Huygen’s wavelets) and are extremely useful to illustrate the connection between particle diffraction and momentum transfer. I have assumed the photon scattering to be elastic—the energy and wavelength of light is not changed. Making these substitutions and separating the exponentials

Assuming the illumination, $U(x,y,0)$, is uniform over the aperture under consideration allows us to separate the integral

4.

Single-Slit Diffraction

There are several aperture configurations for which solutions to the Rayleigh–Sommerfeld formula have been examined in detail. Analytical solutions to various aperture problems have been examined in detail by Steane and Rutt.⁴¹ One familiar example is a long narrow slit, $d$ wide, running along the $y$-axis, and centered at $x=0$. The separate integrals of Eq. (12) are solvable in the traditional fashion using the Fresnel integrals,⁴² integrating over $x$ from $-d/2$ to $d/2$ and assuming the $y$ integral from $-\infty$ to $+\infty$ gives a constant. (Note: this assumption that the $y$ integral provides a constant, uniform dispersion is acceptable for uniform illumination but would not be accurate for a narrow laser beam; see Fig. 6.) Noting Eqs. (9) and (10), integration can be done with the substitution in the argument of $v=\sqrt{2Z/\lambda }\sin \theta$ and $v^{\prime }=\sqrt{2Z/\lambda }\sin \varphi$ and using the Cornu spiral, integral tables, or computer algorithms to generate numerical solutions.

In the long distance, Fraunhofer limit for slit diffraction, the angular distribution for the intensity, $I_X$, exhibits the relation

Fig. 2

Single-slit diffraction in the Fraunhofer limit. The effective wavelength for momentum transfer, ${\lambda }_e$, is shown for four different values of $n$ corresponding to maxima in the diffraction pattern. These are related to geometrically harmonic solutions to Schrödinger’s wave equation for a one-dimensional square well. The significance of the numbered inflection points is described later in Table 2.

Figure 3 is a picture of a green laser pointer diffracted by a single, narrow slit, where the paths for the scattered photons are illuminated by a water vapor fog. This type of experimental set up helps to illustrate how the photon paths are determined at the slit aperture, though the scattering pattern can be predicted by the formalism of a wave.

Fig. 3

Green (520 nm) laser pointer diffracted by a narrow slit with beam illuminated by a fog generator. Picture illustrates the probabilities of the paths for scattered photons. (Credit: Quantum Optics Lab Olomouc 2012.)

In single-slit experiments examining the near-field, there appears a minima in the center of the diffraction pattern (see Harris et al.²⁶). Fresnel’s criteria for such minima assume that the integration is over four Fresnel zones. I have pointed out previously that to obtain such a pattern, the magnitude of momentum transfer must be dependent on the position within the slit as well as specific momentum eigenvalues of the slit.⁶ This relation will be clarified through our examination of straight-edge diffraction.

5.

Straight-Edge Diffraction

The classical derivation of the scattering intensity at a distant screen for edge diffraction of monochromatic light can be found in most optics textbooks.^19,42 The derivation usually employs the Rayleigh–Sommerfeld and Fresnel–Kirchoff theories of diffraction. The experimental setup is illustrated in Fig. 4, with a long, straight edge located at $x=0$.

Fig. 4

(a) Diagram of Fresnel diffraction of monochromatic light by a long, straight edge at $x=0$, $z=0$, and distant detector screen at $z=Z$. The six different photon scattering trajectories are shown, where $\mathrm{\Delta }{p}_x=\mathbf{g}h/x$, $\mathbf{g}=1$ from Eq. (27), three converging at the detection point, $X$, $Y$, $Z$. (b) Typical Fresnel intensity pattern produced by edge on distant screen.

From the first Rayleigh–Sommerfeld diffraction formula, Eq. (5), we have derived the illumination field, $U(x,y,0)$, in Eq. (12). For edge diffraction, integration over the $y$-dimension will generate a constant that can be combined in the coefficient, thus, we can focus on the $x$ dependence. Using a familiar substitution, we obtain as follows

Having used a standard derivation from COW, we can further deconstruct the components to these equations. In the illuminated region, where $X^{\prime }$ is positive, we can separate this irradiance approximation into the sum of two functions representing attractive and repulsive deflections. These component functions are plotted in Fig. 5

Fig. 5

Contributions to the observed diffraction pattern from a straight edge, ${\mathrm{A}}_{X^{\prime }}$. ${\mathrm{\Gamma }}_{X^{\prime }}$ is the relative contribution from repulsive, positive deflections, Eq. (24), and ${\mathrm{B}}_{X^{\prime }}$ is the relative contribution from attractive, negative deflections, Eq. (26). ${\mathrm{\Omega }}_{X^{\prime }}$ is a derived component of ${\mathrm{\Gamma }}_{X^{\prime }}$ defined in Eq. (35).

The approximate sum of the attractive, negative deflections is calculated as the difference as follows

Fresnel provided us with two useful conceptual tools to describe the summation and interference of Huygens wavelets. One is the Fresnel number associated with the scattering to $X$, $Z$, where $N_{\mathrm{F}}=X^2/\lambda r\cong X^{\prime 2}/2$. We note the integral, ${\mathrm{\Gamma }}_{X^{\prime }}$, spans the phase interval, $N_{\mathrm{F}}$. A second, related tool is the Fresnel half-period zones analysis. The half period zones define the dimensional spacing at the scattering object or aperture that correspond to half-period phase differences in dispersed wavelets reaching a specific detection point. Insight to our edge diffraction is gained referencing these tools. MET relates these to momentum exchange criteria.

Table 1 provides the relevant dimensions for a Fresnel zones analysis of our edge diffraction experiment. The dimension from $x=0$ to the edge of each Fresnel half-period zone is calculated such that there is a half-period difference at $Z$, column B. We note that these zone edges approximate the $X^{\prime }$ inflection points in our analysis of the diffraction pattern of Fig. 5, column E. Also calculated, related to Eq. (4), is the effective wavelength that would be associated with a momentum exchange at the aperture that leads to a photon deflection equal to the length to the zone edge, column C. Our definition of the momentum exchange coefficient that can be related to the Fresnel number, column D, will be developed in the next section that clarifies how these are connected to the calculated coefficient at the profile inflection points, column F.

Table 1

Parameters associated with a Fresnel analysis of edge diffraction in Fresnel units. Description of each column is provided in the text.

6.

Phenomenological Description and Experimental Curve Fitting

Converting to a momentum representation gives us an alternative interpretation for optical diffraction formulas in terms of momentum exchange corresponding to microwave energies. We demonstrated the irradiance predications of COW can be constructed as the sum of attractive and repulsive momentum exchange terms. In this next section, we will “reverse engineer” these terms, assuming that they represent the experimentally observed diffraction patterns and develop the momentum exchange descriptions that will generate these patterns. This analysis makes a few simple assumptions: (a) photons behave as particles with specific paths and quantized momentum; (b) photons are scattered in the vicinity of an aperture or lattice by momentum transfer with that lattice; (c) the momentum transfer probabilities are defined by two factors: (1) the momentum exchange states associated with the lattice and (2) the distance over which momentum is transferred.

The probability for scattering in MET must result from a distinct ensemble of exchange potential functions within the aperture and their cross-section at the path point of photon scattering. Such an assumption is necessary to generate the sharp definition of optical diffraction patterns. We note that BDW theory draws upon the geometry and symmetry of the scattering boundary or aperture to define the interference with the incident wave.³³ For example, Schwinger et al.⁴³ use the “more physical” approach of BDW and describe the scattered wave by a long, straight edge as a cylindrical field, concentric to the edge. We make the distinction between MET and BDW in that the field in the proximity of the edge or aperture determines the scattering and not interference at the point of detection as in BDW.

The similar symmetry requirements may be clarified by our observation of the scattering of a laser beam by a long, narrow slit. Note, in Fig. 6, the laser light is scattered perpendicular to the length of the slit. The scattering field contributions parallel to the edge must cancel or not contributed—the parallel scattering is no wider than the original beam.

Fig. 6

Photo of observed diffraction pattern for a He–Ne laser through a 0.1-mm vertical slit. Photon scattering is perpendicular to the slit. (Credit: Jordgette 2010.)

In determining how these fields/potentials interact, we will make an analogy to Thomson scattering by an electromagnetic field in x-ray crystallography, where the structure factor for a crystal’s geometry is expressed in a reciprocal phase space that defines x-ray reflections.⁴⁴ This reciprocal space representation can be correlated to momentum eigenfunctions of the crystal lattice that determine the scattering profile. We should be able to define optical scattering from an aperture in terms of a reciprocal geometry that is correlated with momentum transfer states. A difference from crystallography is that we do not immediately recognize a periodic pattern of the scattering electromagnetic field in reciprocal space. Our analysis will help us converge on the scattering probabilities for this momentum transfer in the plane of an aperture.

Our analysis of edge diffraction will assume that the momentum transfer is a reciprocal function of the distance, $x$, between the photon path and edge of the aperture. As with our analysis for Eq. (21), we assume a summation of deflected photon paths that converge upon the screen point $(X,0,Z)$ scattered from points in the plane of the aperture, $(x,\mathrm{0,0})$ and thus, $\mathrm{\Delta }x=(X-x)$. As with BDW analysis and experimental observations, we assume the $y$-contributions to deflections will cancel. Making the substitution, $\mathrm{\Delta }{p}_x=gh/x$, into Eq. (9), the net potential momentum transfer with an exchange coefficient, $g$, is expressed by

The diffraction pattern will be determined by the probabilities for the different positive and negative values of $g$. Expressing distances in terms of the Fresnel coefficient, $u$, we solve Eq. (27)

For each absolute value of $g$, there can be up to three different values of $x^{\prime }$ that deflect to each value of $X^{\prime }$. We can determine the contribution to the intensity from each deflection cross-section. For $g$ positive, a positive deflection (repulsion) from the edge, we obtain

Here, I have used the convention in physics to take the absolute value of the differential cross-section, so that the flux contributions will sum positively. This provides a generic equation allowing us to calculate the intensities when the illumination is not uniform. The intensity contribution at $X^{\prime }$ from positive deflections will be the sum (integral) of the contributions from the different values of $x^{\prime }$ from 0 to $X^{\prime }$

If we assume the illumination beyond the edge is uniform, expressed by the unobstructed intensity, $\mathrm{\varphi }(x_0^{\prime })$, Eq. (32) reduces to

This provides our basis to calculate Pr(+g) by fitting to our experimental observations. Equation (33) correlates with Eq. (24) that relates to our positive deflections. Assuming Eq. (24) reflects an observed experimental profile, we connect these relations

We note the integral will only be real for $g\le X^{\prime 2}/8$, so our integration is limited to $g=0\to X^{\prime 2}/8$. As $g$ approaches $X^{\prime 2}/8$, our denominator approaches infinity. We can approximate the square root term by a series expansion and adding a delta function, $\mathrm{\Delta }$, as $g$ approaches $X^{\prime 2}/8$

Fig. 7

Relative probability distribution for the different values of $g$ (dashed). The distribution for positive $g$ is obtained by fitting the momentum exchanges of Eqs. (33) and (35) to the curve, ${\mathrm{\Gamma }}_{X^{\prime }}$, of Fig. 5. The shape of this profile can be confirmed experimentally. (a) Solid curve is the harmonic function in Eq. (36). (b) Solid curve is the sum of Gaussian distributions at $+g$ maxima in Eq. (39).

Though this curve in Fig. 7 can be derived from experimental observations, we have correlated it to calculated Fresnel pattern of Fig. 4 and the resulting inflection points are tabulated in Table 1. By connecting these relations, we observe the harmonic probability profile in Fig. 7(a) can be modeled by a diminishing function with a frequency approximated by the form

The local maxima are at increments of $\mathrm{\Delta }p=h/2x$. Alternatively, these probability maxima for $g$ might be associated with momentum transition states for the aperture with behavior resembling spectral lines with broadening defined by Gaussian or Lorentzian statistical distributions. In this case, we might model the probability of $+g$ by a summation resembling

Though our curve fitting set $b\cong 0.8$, for insight to the geometric selection rules for momentum exchange and the Fresnel zone analysis of optical theory, we will examine setting $b=1$, such that the inflection points for $g$ correspond to $g=m/4$. These track the values for $g_m$ in Table 1, column D. The probability of Eq. (36) then relates to the sine function

Local maxima in momentum exchange from Eq. (27) are thus approximated where

From Eq. (41), the higher probabilities for momentum exchange are found when the effective wavelength is an odd fraction of $4x$. This relation illuminates the connection between the momentum exchange selection rules and the Fresnel phase-shift of $-\pi /2$ adopted in Eq. (5). With Eq. (40), we come full-circle on our predictions of Sec. 2.4, demonstrating diffraction as a harmonic function of momentum transfer in the plane of the aperture.

The relationship between the effective wavelength for momentum exchange and the distance to the edge is graphically illustrated in Fig. 8, where a sine function for different solutions for ${\lambda }_e$ in Eq. (41) is superimposed on the $x$-axis of our edge experiment with the value set to 0 at $x=0$. We note in Fig. 8 how the selection rules for momentum exchange identify an effective wavelength (momentum) that relates to the exchange distance to the photon path. This suggests a geometric relationship between the properties of the momentum exchange particles or electromagnetic potential near the aperture and the path of the scattered photons.

Fig. 8

Straight-edge diffraction. Graphical representation of four solutions to Eq. (41), where momenta exchange are maxima. Superimposed on the $x$-axis is $\sin 2\pi x/{\lambda }_e$ for the different values of the effective wavelength for momentum exchange, ${\lambda }_e$. The photon path is at $x_0$ from the edge.

When we bring two long, straight edges toward each other, we create a single-slit diffraction configuration. We can directly relate the predicted exchange maxima from Fig. 8 with the single-slit scattering predictions shown in Fig. 2. We assume a slit separation, $d$. We can equate the relations for the maxima in the effective wavelength probabilities for both experimental configurations

Thus, the maxima in the probability for exchange are observed when the values for $x$, the distance from the photon path to the edge, are given by

These positions correspond to the inflection points for the sinusoidal functions plotted in Figs. 2 and 8. The calculations for the different values for $n$ and $j$ are shown in Table 2. This table shows the values for $x$ associated with these maxima and provides a number label for these inflection points in the figures. The values for $n$ define the momentum eigenvalues for the maxima in exchange and the values for $j$ define the distance of the photon paths from the edge that will exchange momentum with these values. The square of the harmonic functions plotted in Figs. 2 and 8 is related to the probability of momentum exchange at each point. We note the amplification of specific values of ${\lambda }_e$ from multiple paths. Scattering solutions are shown for half of the single slit as each half is symmetric. The total dispersion (diffraction pattern) would be the sum across all points in the slit.

Table 2

Equating the maxima in the probabilities for the effective wavelengths for single-slit and straight-edge diffraction allows us to calculate the distance of the photon path from the edge for these momentum transfers. Position numbers correspond to the location of the inflections at these values of x in Figs. 2 and 8.

This diffraction by a narrow slit is sketched in Fig. 9 for a configuration in the near-field, where the half width of the slit, $d/2$, is equal to two Fresnel zones (total of four over slit), $Z\cong d^2/8\lambda$. The figure depicts the higher probability scattering trajectories only from positions 1 through 6 labeled in Fig. 2 and Table 2. The magnitudes of the deflections are defined by the effective wavelengths for momentum exchange, column one of Table 2. Figure 9 helps us visualize how this momentum exchange scattering can replace the concept of Huygens wavelets produced at the aperture. At $d/2$, there is the highest probability for scattering and the most probable six deflections are shown. Figure 9 illustrates how we would see an intensity minima at the center of the profile for this screen distance, as predicted by Fresnel and experimentally documented (see Harris et al.²⁶). In the far-field, Fraunhofer limit where the intensity profile can be described by Eq. (13), the peak in the center of the profile corresponds to $n=0$, $j=0$, and $g=1/4$. In this case, the momentum exchange determining this center is $\mathrm{\Delta }{p}_S=\pm h/4x$, for each half-slit width from $x=0$ to $d/2$.

Fig. 9

(a) Deflection of parallel light paths from different positions within a single slit of width $d$. Higher probability deflections correspond to positions 1 through 6 in Fig. 2 and Table 2. (b) Fresnel intensity pattern at a distance $Z\cong d^2/8\lambda$ from the aperture corresponding to $\sim 4$ Fresnel zones. (c) Intensity pattern at $Z\cong 2d^2/\lambda$, one Fresnel zone, first minima at $2d$ from center where ${\lambda }_e=d$ (see intensities in Harris et al.²⁶).

7.

Multiple Slits and Talbot Effect

We may examine the COW treatment of multiple-slit diffraction provided by Born and Wolf¹ that merges the intensity profile for a single-slit with that for multiple slits [see Eqs. (13) and (14)]

Harris et al.²⁶ provide useful experimental results for double-slit optical diffraction in the far-field, where the slit separation (12.68 mm) is large compared to the slits’ width (0.65 mm). The experimental intensity pattern is consistent with the theoretical predictions with deflection increments at $\mathrm{\Delta }p=jh/L$. The pattern also exhibited the contribution of each of the narrow slits to the dispersion. We have seen from our analysis [Fig. 9(c)] that in the far-field, the most probable contribution to the scattering from a single slit corresponds to $\mathrm{\Delta }{p}_S=\pm h/2\mathrm{d}$, with these two dispersion profiles summing to a single peak distribution at the distant screen and additional peaks at $\mathrm{\Delta }{p}_S=(2n+1)h/2d$. With the existence of a second or multiple slits in the experiment, the dispersion with momentum exchange increments, $\mathrm{\Delta }p=jh/L$, can be added to the single-slit profiles if an additional functional dependence is imposed as with Eq. (44). Equation (44) was suitable for the Fraunhofer region, but we would like to consider deflections in the nearer field, Fresnel region by integrating the phase shift of $\pi /2$ into our analysis. In this case, the most probable momentum deflections for the double slit would correspond to a form

Similar to our analysis of a single slit in Fig. 2, we note that the most probable values for momentum exchange due to the slit separations also resemble the solutions to a one-dimensional square well, where the width of the well is $2L$ and the origin for the $x$ dimension is centered at one slit

7.1.

Talbot Effect

A comprehensive review of the Talbot effect and its many applications in modern optics has been presented by Wen et al.⁴⁵ The Talbot effect is associated with Fresnel diffraction of coherent, monochromatic light by an extended, multiple-slit grating, periodic in the transverse direction. The effect is observed in the formation of a perfect image of the grating (Talbot image or Fourier image) at a distance, $2Z_T=2L^2/\lambda$ and a contrast reversed image (also a Talbot image) at $Z_T=L^2/\lambda$, where $Z_T$ is referred to as the Talbot distance. These Talbot images have also been termed “Fresnel images” or “self images.”

An optical experiment to observe the Talbot effect is sketched in Fig. 10. We may use the Fresnel–Kirchoff diffraction formula, Eq. (8), to derive a solution

Eq. (48)

$$U(X)\propto {\int }_{-\infty }^{\infty }U(x)\exp \left[\frac{i\pi }{\lambda Z}{(X-x)}^2\right]\mathrm{d}x={\int }_{-\infty }^{\infty }U(x)\exp \left[i\pi \lambda Z{\left(\frac{\mathrm{\Delta }{p}_x}{h}\right)}^2\right]\mathrm{d}x.$$

Fig. 10

The optical Talbot effect. Diagram of the dispersion of monochromatic light through an extended grating on left with narrow, parallel slits at increments of $L$. The six most probable momentum exchanges at each slit are shown as rays connecting to the primary Talbot image on the right. This illustrates the “Talbot carpet” with double-frequency and triple-frequency images.

Following Goodman’s Fourier analysis,¹⁹ the grating, with slits perpendicular to $x$, can be modeled as a transmitting structure with amplitude transmittance

Eq. (49)

$$U(x)=\frac{1}{2}[1+q\cos (\frac{2\pi x}{L}\left)\right].$$

I would note that if $q$ is 1 [Eq. (49)], this transmission function has the same form as the square of our momentum eigenfunctions in Eq. (47)—these are Fourier frequency components of our grating geometry. This points to the clear connection between MET and the Fourier analysis to describe multiple-slit diffraction. Using a transfer function approach to develop a solution, the intensity distribution was found to be as follows

Eq. (50)

$$I(X)\propto 1+2q\cos \left(\frac{\pi \lambda Z}{L^2}\right)\cos \left(\frac{2\pi x}{L}\right)+q^2{\cos }^2\left(\frac{2\pi x}{L}\right).$$

We observe when

Eq. (51)

$$\frac{\pi \lambda Z}{L^2}=2\pi n\text{or}Z=\frac{2n{L}^2}{\lambda }=2n{Z}_T,$$

there is a perfect image of the grating reformed at the screen (see Fig. 10). At odd multiples $Z_T$, we see a contrast reversal with an intense spot whenever $x=(2j+1)L/2$. This is the secondary Talbot image. In Fig. 10, light rays representing the six most probable dispersions from each slit are drawn from each grating slit to these spots at $Z_T$. Dispersions between these rays must have low probability to retain the dark areas between these spots. We can calculate the higher probability momentum exchanges associated with the deflections of these rays. The effective wavelength, ${\lambda }_e$, for these transfers is

Eq. (52)

$${\lambda }_e\cong \frac{\lambda Z}{\mathrm{\Delta }x}=\frac{2\lambda Z_T}{(2j+1)L}=\frac{2L}{(2j+1)},\mathrm{\Delta }{p}_M=\frac{h}{{\lambda }_e}\cong \frac{(2j+1)h}{2L},$$

consistent with Eq. (46) and confirming the Talbot effect is due to quantized momentum exchange at the grating.

8.

Diffraction by an Opaque Circular Disc: Poisson’s Spot

We can learn more about the probabilities for momentum exchange by examining a very similar experiment, the diffraction by an opaque circular disc, which can give our experimental predictions for negative $g$. A typical experiment can be configured as our Fig. 4, replacing our long straight edge with a circular disc. This configuration is assumed in Fig. 11. This experiment has an interesting history. In 1818, Poisson used Fresnel’s wave theory to predict a spot in the middle of a circular shadow, thereby suggesting that Fresnel’s theory was absurd. This spot was later observed by Arago giving strong endorsement to Fresnel’s wave theory of light in preference over Newton’s corpuscular theory.⁴⁶ This observation has subsequently been referred to as Poisson’s spot or the spot of Arago. It is useful to demonstrate that this spot can be explained in terms of photon dispersion through momentum exchange.

Fig. 11

Diffraction by an opaque circular disc with similar experimental conditions to Fig. 4. Different deflected photon paths are shown. Poisson’s spot or spot of Arago is observed at the center of the shadow. Depicted is an experimental configuration, where the opaque disc has a radius of three Fresnel half zones (shown). Dimensions shown are in Fresnel units.

Lucke⁴⁷ has noted the problems of accurately predicting the intensity profile within the shadow of an opaque disc using the familiar wave theory diffraction integrals. He has demonstrated that the Rayleigh–Sommerfeld formulas can lead to accurate descriptions near the central axis, though the approximations become less accurate toward the edge. Lucke has also shown that Fourier propagation, an alternative approach to analyzing a diffraction problem, will reproduce the more accurate Rayleigh–Sommerfeld equation predictions. As previously noted, the Fourier components to the intensity can always be related to light path rotation and momentum exchange at the aperture.