Optics 057 · Diffraction, Fourier optics, and computational imaging

Aperture Diffraction Studio

Draw a binary aperture and propagate its complex field onto a real 3D receiving plane. Compare slits, Airy rings, polygons and letters; complement the same mask and reveal the Arago spot without replacing the computation.

Size-based Fresnel scale / receiver distance

N_d,z

$0.50$

Probe / central TOTAL intensity

I_p/I_0,I_c/I_0

$50\%$

Babinet amplitude closure / mask cell pitch

\epsilon_B,\Delta\xi

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Aperture Diffraction Studio

Size (circle diameter / slit width / design side)

d

0.30\,\mathrm{mm}

Receiver distance (logarithmic rail)

z

2.50\,\mathrm m

Vacuum wavelength

\lambda

532\,\mathrm{nm}

Shape index (or use shape buttons below)

s

1

Mask rotation

\alpha

0^{\circ}

Complement (off: opening, on: obstacle)

c

0

Display gain decimal exponent (not data)

\log_{10}G

4

Centre-line probe (fraction of window)

x_p/L

0

Pick a slit, circle, polygon or letter, or draw your own opening below the bench and release to calculate. Smaller openings spread the pattern; rotation rotates the mask and the entire field. Swap opening and obstacle to inspect Babinet at the amplitude level: total intensities are not identical. In Arago spot, a bright centre reappears inside the opaque disk shadow. Shift-drag the receiving face in a bench view to change distance. Mask and detector detail views show the actual objects. Exposure affects brightness only; Play animates incident-wave markers, not a new measurement.

Scalar monochromatic Fresnel model with an ideal infinite incident plane wave; finite compact aperture and infinite-plane complement. Raster edge coverage and cell quadrature, point-sample detector. Separate ideal far-field reference, explicit display gain. Logarithmic axial layout and different mask / detector transverse magnifications; support hardware excluded from optics. Not a material-edge Maxwell solution or a noisy finite-beam experiment.

CONTINUE EXPLORING

Aperture, propagation, measurement

Change one physical assumption at a time: slit count, coherence, aperture shape or propagation distance.

Physics tutorial

Audit the physics of Aperture Diffraction Studio

BackgroundA hole does not make a photograph of its outline on a distant screen. Each transmitted element contributes a complex field; complementarity acts on amplitudes: $U_A+U_{1-A}=U_0$ . Here the two masks share one propagation calculation, so the illuminated mask and displayed detector cannot drift into unrelated illustrations.

Why it mattersHow does the same drawn shape change diffraction when it is an opening or an opaque obstacle?

Start with the essentials

Focus question: How does the same drawn shape change diffraction when it is an opening or an opaque obstacle?
One-sentence intuition: Intensity is measured only after summing amplitudes: $I=|U|^2$ . A circle opening makes Airy rings in its far-field limit; an opaque disk can restore a bright on-axis point in its near-field shadow. These are different fields, not the same ring texture.

Core mathematical model

From the edited mask to the receiving plane

\begin{aligned}U_A(x,y;z)&=\frac{1}{i\lambda z}\iint A(\xi,\eta)\\&\quad e^{i\pi[(x-\xi)^2+(y-\eta)^2]/(\lambda z)}\,\mathrm d\xi\,\mathrm d\eta\\I_A/I_0&=|U_A|^2\end{aligned}

Mask plane is at zero axial distance, propagation is forward in vacuum, and all coordinates in the integral are metres. Incident amplitude is one. Only the common longitudinal phase is removed. The editor supplies a binary design, rasterized into area-averaged amplitude cells; each cell kernel is integrated. No FFT period or stored pattern is substituted.

Babinet is an amplitude identity

\begin{aligned}U_{1-A}&=1-U_A\\|U_{1-A}|^2&=1+|U_A|^2-2\operatorname{Re}U_A\\|U_{1-A}-1|^2&=|U_A|^2\end{aligned}

The opening vanishes outside the finite design domain; the complementary transmission is one outside it. The unblocked infinite plane wave has unit field everywhere on the receiving plane. Thus the obstacle total intensity generally differs from the aperture intensity. Only its background-subtracted scattered contribution is the negative aperture field. No equality of the two TOTAL images is claimed.

Airy and slit curves are separate limiting references

\begin{aligned}I_{\rm circle}/I_0&=\left(\frac{\pi d^2}{4\lambda z}\right)^2\left[\frac{2J_1(q)}q\right]^2\\q&=\frac{\pi d r}{\lambda z}\\r_1&\simeq1.21967\,\frac{\lambda z}{d}\\x_{1,\rm slit}&=\frac{\lambda z}{d}\end{aligned}

Circle size is diameter, never curvature radius. Slit size is full width; its finite length is one millimetre and supplies a second sinc factor. The gold curve assumes an ideal continuous shape and the far-field limit, while cyan uses the rasterized Fresnel field. Polygon size is vertex-circle diameter; letter and drawing size is design-square side. Near field need not agree with gold.

The bright point behind an opaque disk

\begin{aligned}U_A(0;z)&=1-e^{i\pi a^2/(\lambda z)}\\U_{\rm disk}(0;z)&=e^{i\pi a^2/(\lambda z)}\\I_{\rm disk}(0;z)/I_0&=1,\qquad a=d/2\end{aligned}

This independent on-axis result follows by radial integration for an ideal circular opening, followed by Babinet. It requires the infinite uniform plane-wave and scalar Fresnel assumptions. The numerical raster should approach it; it is not forced to exactly one or painted as a bright dot. Finite-beam illumination, rough edges, misalignment and partial coherence can change a real experiment.

Sampling and approximation boundaries

\begin{aligned}\Delta \xi&=\frac{1.6\,\mathrm{mm}}{256}\\N_d&=\frac{(d/2)^2}{\lambda z}\\\delta\phi_{\max}&\le\frac{\pi r_{\max}^4}{4\lambda z^3}\end{aligned}

The size-based Fresnel number is a diameter convention, not the full support radius of every polygon, slit or drawing. The phase bound uses the full design-domain and detector corners instead. Detector window adapts to wavelength, distance and size; window power is for the compact opening only, not total energy of an infinite plane wave. The simulated detector point-samples rather than integrating pixels. Raster convergence and independent limits are tested separately.

Display gain is not a physical measurement

\begin{aligned}B&=\min\!\left(1,\frac{\ln(1+G I/I_0)}{\ln5}\right)\\G&=10^g\end{aligned}

The same declared transfer applies to both detector maps. Raw readouts, field values and the log-intensity profile do not change with gain. The scene enlarges the mask and receiver separately and uses a logarithmic optical rail. Computed dots are field samples on adaptive transverse windows, never photons or a physical time-of-flight animation. The supports are excluded from the ideal optical model.

Common difficulties

Complementarity does not copy total intensity

Typical misconceptionAn opaque disk must show the same total rings as a circular opening, or a permanently black centre.

Better mental modelAdd complex amplitudes before squaring. The unblocked background interferes with the complementary contribution and can restore a bright on-axis spot inside the geometrical shadow.

Run the experiment

01
Scene 1: Draw an aperture
Start with the circle, reduce its diameter, then rotate a slit or letter and compare the mask with the whole detector map.
What to observe: A smaller opening produces a wider far-field pattern, not a smaller photograph of the mask.
02
Scene 2: Babinet complement
Draw an opening, release to propagate it, and swap to the complementary obstacle without changing wavelength or distance.
What to observe: The complementary TOTAL intensity changes even though its background-subtracted field has the same scattered intensity as the opening.
03
Scene 3: Poisson–Arago spot
Select Arago spot. Inspect the opaque disk in the mask view and the central bright spot on the actual receiving face.
What to observe: The circular disk centre approaches unit intensity under the ideal plane-wave model; no centre pixel is overwritten.