Optics 057 · Diffraction, Fourier optics, and computational imaging

Aperture Diffraction Studio

How is the diffraction from a hole related to that from an obstacle of the same shape? They are complements: the opening's diffracted field plus the obstacle's adds up to the unobstructed wave, which is Babinet's principle. Away from the straight-through beam, their far-field patterns are therefore the same. A round hole gives Airy rings in the far field, and a round opaque disk can show a bright spot in the middle of its shadow, the Arago spot.

Draw a binary aperture and propagate its complex field onto a 3D receiving plane. Compare slits, Airy rings, polygons and letters, swap the mask for its complement, and watch the Arago spot come out of the same calculation.

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 residual / mask cell pitch

\epsilon_B,\Delta\xi

$0.00\pi$

Model regime $\text{valid model regime}$

Model checked against analytic results

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 (data unchanged)

\log_{10}G

4

Probe position on the centre line (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. A smaller opening spreads the pattern, and rotating the mask rotates the whole field. Swap opening and obstacle to check Babinet's principle at the amplitude level: the total intensities are not the same. In the Arago spot scene, a bright center appears inside the shadow of the opaque disk. Shift-drag the receiving face in a bench view to change the distance. The mask and detector detail views show the objects themselves. Exposure changes brightness only; Play animates markers on the incident wave and does not make a new measurement.

The model is scalar, monochromatic Fresnel diffraction of an ideal, infinite incident plane wave. The aperture is finite, and its complement is an infinite screen with that shape blocked. Edges are rasterized by area coverage and integrated cell by cell, and the detector takes point samples. An ideal far-field reference is shown separately, and display gain is explicit. The axial layout is logarithmic, the mask and detector have different transverse magnifications, and support hardware takes no part in the optics. This is not a Maxwell solution for real material edges or a noisy experiment with a finite beam.

Physics tutorial

Audit the physics of Aperture Diffraction Studio

BackgroundA hole does not cast a picture of its outline on a distant screen. Every open patch of the mask adds a complex field at the screen, and complementary masks add as amplitudes: $U_A+U_{1-A}=U_0$ . The Lab propagates both masks with the same calculation, so the detector pattern always belongs to the mask on display.

Why it mattersHow is the diffraction from a hole related to that from an obstacle of the same shape?

Start with the essentials

Focus question: How is the diffraction from a hole related to that from an obstacle of the same shape?
One-sentence intuition: Add the amplitudes first, then take the intensity: $I=|U|^2$ . A circular opening gives Airy rings in the far-field limit, while an opaque disk can bring back a bright on-axis point inside its near-field shadow. The two patterns come from different fields.

Core mathematical model

Fresnel propagation from the drawn mask to the screen

\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}

The mask sits at zero axial distance, light travels forward in vacuum, and every coordinate in the integral is in metres. The incident amplitude is one. Only the common longitudinal phase is dropped. The editor produces a binary design, which is rasterized into cells whose amplitude is the area average, and the propagation kernel is integrated over each cell. The field does not come from a periodic FFT or a stored pattern.

Babinet's principle holds for amplitudes

\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}

Outside the finite design area, the opening transmits nothing and its complement transmits fully. The unblocked, infinite plane wave has unit field everywhere on the screen. So the total intensity behind the obstacle generally differs from the intensity behind the opening. Only the scattered part, left after the background is subtracted, equals the negative of the aperture field. The two total images are not equal.

Ideal far-field references for a circle and a slit

\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}

For a circle, size means the diameter, never a radius of curvature. For a slit, size is the full width; the slit is one millimetre long, which adds a second sinc factor. The gold curve assumes an ideal smooth shape in the far-field limit, and the cyan curve uses the rasterized Fresnel field. For a polygon, size is the diameter of the circle through its vertices; for letters and drawings, it is the side of the design square. In the near field the computed curve need not match the gold one.

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 on-axis result is derived independently: integrate radially over an ideal circular opening, then apply Babinet's principle. It assumes an infinite uniform plane wave and scalar Fresnel diffraction. The rasterized calculation should approach this value on its own; it is not set to exactly one, and no dot is drawn on top. In a real experiment, a finite beam, rough edges, misalignment and partial coherence can change the result.

Sampling and where the Fresnel approximation holds

\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 Fresnel number treats the size setting as a diameter, so it is not the full extent of every polygon, slit or drawing. The phase-error bound instead uses the corners of the full design area and of the detector window. The detector window adjusts to wavelength, distance and size. The window power counts only light from the compact opening, not the total energy of an infinite plane wave. The simulated detector samples points; it does not integrate over pixels. Convergence of the raster and the independent limits are tested separately.

How detector brightness is displayed

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

Both detector maps use this same brightness mapping. Gain does not change the raw readouts, the field values or the log-intensity profile. The scene enlarges the mask and the receiver separately and draws the optical rail on a logarithmic scale. The computed dots are field samples on transverse windows that adapt in size; they are not photons and do not show light in flight. The mechanical supports are not part of the ideal optical model.

Common difficulties

Complementary masks do not give the same total intensity

Typical misconceptionAn opaque disk should give the same rings as a circular hole, or else a centre that stays dark.

Better mental modelAdd the complex amplitudes before squaring. The unblocked background wave interferes with the light scattered by the disk and can bring back a bright spot on the axis, inside the geometric shadow.

Run the experiment

01
Scene 1: Draw an aperture
Start with the circle and shrink its diameter. Then rotate a slit or a letter and compare the mask with the whole detector map.
What to observe: A smaller opening spreads into a wider far-field pattern instead of making a smaller picture of the mask.
02
Scene 2: Babinet complement
Draw an opening and release to propagate it. Then switch to the complementary obstacle without changing the wavelength or the distance.
What to observe: The total intensity behind the complementary obstacle changes, even though its scattered intensity, once the background is subtracted, matches the opening's.
03
Scene 3: Poisson–Arago spot
Select Arago spot. Look at the opaque disk in the mask view, then at the bright central spot on the receiving screen.
What to observe: Under the ideal plane-wave model, the computed intensity at the centre of the disk's shadow approaches one.

Aperture Diffraction Studio

Audit the physics of Aperture Diffraction Studio

Core mathematical model

Fresnel propagation from the drawn mask to the screen

Babinet's principle holds for amplitudes

Ideal far-field references for a circle and a slit

The bright point behind an opaque disk

Sampling and where the Fresnel approximation holds

How detector brightness is displayed

Common difficulties

Complementary masks do not give the same total intensity

Run the experiment

Scene 1: Draw an aperture

Scene 2: Babinet complement

Scene 3: Poisson–Arago spot

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Aperture, propagation, measurement