Interference and Diffraction: How Boundaries Form Fringes

01From the material source through the aperture to the screen

Why does matter passing through an aperture produce an ordered pattern of bright and dark regions? ASD treats the localized material core, its surrounding accompanying field and the apparatus boundaries together. Whether the core passes through an opening depends on its effective size and its interaction with the material; the outer envelope responds to the whole aperture. Once the prepared source state is specified, one can first calculate the field reaching the screen and then examine how local interactions become records.

Let the main propagation direction be $x$, with a barrier at $x=0$ and a detector screen at $x=L$. The transverse coordinates at the aperture and screen are $y$ and $Y$, respectively, and $L$ is the propagation distance. We use the low-energy envelope developed in the effective wave equation, taking its propagation coefficients to be $B^{ij}=\delta^{ij}$: the selected propagation region is isotropic. Here $m_0$ is the mass parameter obtained by energy matching for this envelope channel, and $\hbar$ retains the mode-action matching specified on the matter-wave page.

When the incident state has been prepared, propagation is approximately linear and the barrier does not appreciably excite internal degrees of freedom, a transmission function $T(y)$ describes the aperture. The boundary condition is $\psi(0,y)=T(y)\psi_{\rm in}(y)$, where $\psi_{\rm in}$ is the incident transverse amplitude. The function $T$ need not take only the values zero and one: surface potentials and absorption may enter its complex amplitude. Apparatus physics thus enters the boundary condition before propagation determines the downstream field.

02From the envelope equation to the aperture integral

For a nearly monoenergetic incident state, factor out the common time oscillation and the rapid phase along $x$, as in Chapter 5, (5.3.1):

Here $\Psi$ is the full effective envelope, $k$ is the principal longitudinal wave number and $E_{\rm kin}$ is the kinetic energy of this free propagation state. The function $\psi$ describes the remaining spatial variation. Substitution into the source-free, zero-potential effective equation cancels the time-phase and principal propagation terms, leaving

The paraxial approximation means that propagation is predominantly along $x$ and that the remaining amplitude varies slowly along it. Its explicit condition is $|\partial_x^2\psi|\ll|2k\partial_x\psi|$; the equation then becomes $i\partial_x\psi_{\rm par}=-(2k)^{-1}\partial_y^2\psi_{\rm par}$. Fourier transformation in the transverse coordinate gives each component of transverse wave number $k_y$, at propagation distance $L$, the factor $\exp[-ik_y^2L/(2k)]$. Inverting that transform yields

The wavelength of the propagation state is $\lambda=2\pi/k$. The integral adds complex amplitudes across the aperture with their propagation phases, retaining both their strengths and their relative phases at a screen position. The common factor $e^{ikL-iE_{\rm kin}t/\hbar}$ does not change the relative intensity pattern on that screen.

This Fresnel integral requires a thin aperture, near-axis propagation and weak multiple scattering. If thickness, internal excitation or surface interactions cannot be represented by a transmission condition at $x=0$, the material and field must be included in a fuller boundary-value problem; changing a geometric slit width alone is insufficient.

03One boundary solution gives the single-slit envelope and double-slit fringes

Consider two rectangular slits of width $a$ and center separation $d$. With $\operatorname{rect}(u)$ equal to one for $|u|<1/2$ and zero outside, their aperture is

Assume approximately uniform, coherent illumination across both slits. Let $D$ denote the transverse size of the whole aperture. The Fraunhofer far-field approximation additionally requires $D^2/(\lambda L)\ll1$. The quadratic aperture-position phase in the propagation kernel can then be neglected, while the linear phase associated with screen direction must be retained. The screen amplitude is consequently given by the aperture transform:

The $\operatorname{sinc}$ factor comes from the finite width of each slit; the cosine comes from the relative phase of their center positions. At an observation direction relative to the $x$ axis with angle $\phi$, $k_y=k\sin\phi$, and the inter-slit phase difference is $kd\sin\phi$. Taking the squared modulus gives

The common illumination and propagation normalization are included in $I_0$. At small angles, $Y\simeq L\phi$: the fringe spacing is $\Delta Y\simeq\lambda L/d$, and the first zero of the single-slit envelope is at $|Y|\simeq\lambda L/a$. Narrowing the slits broadens the envelope; increasing their separation reduces the fringe spacing. Closing one slit removes the double-slit cosine factor while leaving single-slit diffraction. Both effects therefore follow from the same propagation equation with different aperture conditions.

The interactive figure displays the relative far-field intensity and its parameter dependence. It does not simulate individual core trajectories or detection positions, and it is not a complete numerical solution for a particular electron apparatus.

04How much of the fringe pattern survives coherence averaging?

Real sources have finite bandwidth, and materials and environments modify both path amplitudes and phases. Write the two responses as $A_1d_1e^{i\theta_1}$ and $A_2d_2e^{i\theta_2}$, where $A_1,A_2$ set amplitude scales, $d_1,d_2$ are complex responses varying with preparation or apparatus microstate, and $\Delta\theta=\theta_2-\theta_1$. Angle brackets denote the corresponding average. Chapter 5, (5.3.7)–(5.3.8), defines

Here $I_j=|A_j|^2\langle|d_j|^2\rangle$ is the mean single-path intensity, and $\mathcal V$ is the visibility obtained by scanning phase while keeping these response parameters fixed. The normalized mutual-coherence coefficient $\Gamma_{12}$ measures the phase correlation of the two complex responses; its upper bound follows directly from Cauchy–Schwarz. Intensity imbalance reduces visibility even when the paths remain coherent, while spectral averaging, phase drift and material response change $\Gamma_{12}$. The coefficient does not replace analysis of all readable apparatus variables and their conditional distributions.

For the Gaussian narrowband packet used in the final manuscript, define $\Delta\omega$ as angular-frequency standard deviation and use the minimum root-mean-square time width. Then $\tau_{\rm coh}=1/(2\Delta\omega)$ and $\ell_{\rm coh}=v_g/(2\Delta\omega)$, where $v_g$ is group velocity. The two path group delays must differ by less than this scale to retain high contrast. Other spectral profiles and width conventions require their own coefficients; the general order-of-magnitude statement $\tau_{\rm coh}\sim1/\Delta\omega$ is not the Gaussian equality. The observation angle is written $\phi$ on this page (manuscript $\theta$ in 5.3.6); optical phases $\theta_j$ and $\Delta\theta$ retain their separate meaning.

05How a potential enters the phase comparison

In addition to aperture geometry, potentials accumulate phase along a path. For two paths with equal leading speeds and a small potential difference, the effective envelope phase evolution gives

The quantities $V_1(t),V_2(t)$ are the potential energies encountered along the paths. This equation defines the potential phase difference as path one minus path two, $\Delta\theta_V=\theta_1-\theta_2$. To insert it into the preceding coherence expression, whose convention is $\Delta\theta=\theta_2-\theta_1$, reverse its sign. The gravitational expression gives a magnitude and is unaffected by this convention. The gravitational example uses a weak uniform field, $V=M_Pgz$, with passive gravitational mass $M_P$, local gravitational acceleration $g$ and height $z$; the passive gravitational mass symbol does not denote the proton mass. If two horizontal path segments have height separation $h_0$, length $l$ and speed $v$, their potential difference is $M_Pgh_0$ and their traversal time is $l/v$. This gives the effective area $\mathcal A=h_0l$ in the registered equation. With inertial mass $M_I$ and the further matching $P=M_Iv$ and $\lambda=h/P$, the phase magnitude is also $M_PM_Ig\mathcal A\lambda/(2\pi\hbar^2)$, where $h=2\pi\hbar$.

The COW neutron-interference experiment provides a comparison for this gravitational phase. A real crystal interferometer also requires geometry, elastic deformation and rotation corrections. ASD must obtain the potential, mass matching and phase response from the same longitudinal spatial background.

LVE likewise requires comparison of complete paths. For a constant velocity vector, Chapter 5, (5.1.16), writes the direction-odd term as an endpoint contribution. It cancels between paths with common endpoints. A directional term on a single arm does not by itself establish an additional interference signal.

06From field intensity to measured counts

For nearly monochromatic, similarly directed propagation with an approximately constant normalization coefficient, $I$ may represent the relative distribution of envelope action weight or mean energy flux. This answers how a boundary redistributes a continuous field. Screen records additionally involve absorption, efficiency and time integration. Calculating a fringe profile and calculating individual-particle event statistics are thus successive tasks.

Electron crystal diffraction tests the wave-number–momentum relation. Tonomura and colleagues recorded individual electron events accumulating into fringes, using an electron biprism and a position counter rather than the two rectangular slits modeled here. Their shared two-path superposition structure supports comparison of physical relations, but quantitative data require the actual boundaries and detector response.

A field solution alone also does not determine the material core's landing position. Chapter 5 distinguishes local overlap coupling of a finite core to the field from the phase flow obtained by amplitude–phase decomposition of the envelope; they are not automatically the same core motion law. Even if phase flow is separately adopted as an effective guidance law and the initial core distribution is matched to the field weight, the continuity equation preserves that distribution rather than proving convergence from arbitrary initial states. Event rates and conditional position distributions require the joint apparatus model on the detection and statistics page.

07Field establishment and a testable response

Field establishment in the apparatus is constrained by retarded propagation in the underlying second-order equation. When the core speed is below $c$, some disturbances from preparation can reach the aperture first, but this does not determine their amplitude, transverse extent or coherence there; those follow from source history and boundary response. A rapid aperture change affects only the part of the system in its future causal domain.

Finite propagation is not unique to ASD. An apparatus-level discriminator requires calculation of response amplitude, establishment time and detector waveform for a specified switching process, followed by subtraction of the comparison-model result under the same conditions. OP-QM-006 accordingly requires an observable difference, uncertainty budget and falsification criterion. The present fringe formulas provide an effective wave-phenomenology requirement.