Diffraction

Aperture

drag on it to paint it open (with Close, to paint it shut)

On the screen at distance z

From the side

the middle row of the screen, as z goes from 0 (top) to twice the screen's distance (bottom)

Experiments

Start here

Light comes through the aperture (left) and lands on a screen a distance z behind it (middle). The picture underneath is the same light seen from the side, on its way from the aperture at the top to twice the screen's distance at the bottom; the dashed line is the screen.

Drag the z slider: the screen moves away, and the slit's sharp shadow with ripples at its edges turns into a wide pattern of fixed shape. Drag on the aperture to paint it open (Close paints it shut), or pick another shape from the buttons.

Look for this: close to the aperture the light nearly keeps the aperture's shape; far away it spreads into the aperture's Fourier transform, wide where the aperture is narrow. The readout under the slider says which of the two you are seeing.

1. One slit, near and far

Slide $z$ from short to long. At first the slit's shadow is sharp, with ripples inside its edges (Fresnel diffraction: the near field); then the ripples grow and merge; far away the pattern settles into the $\mathrm{sinc}^2$ of a slit (Fraunhofer diffraction: the far field), wider the narrower the slit, with zeros at angles $\theta=\pm\lambda/w,\pm2\lambda/w\ldots$ for a slit of width $w$. The readout under the slider keeps the Fresnel number $N_F=a^2/\lambda z$, with $a=w/2$ the half width: the near field is $N_F\gg1$ and the far field $N_F\ll1$; at $N_F=0.3$ the pattern is still 5% off its $\mathrm{sinc}^2$, at $N_F=0.1$ within 1%. Past 60 mm the screen zooms out with the light, so the pattern keeps its size, and the scale bar gives angles. The slit's 1.4 mm length takes longer: its far field begins near a metre. Far field (10 m) gives the Fraunhofer pattern to within 1%.

2. The edge of a shadow

A straight edge: light reaches into the geometrical shadow a little, and outside it the brightness oscillates. Everything scales with $\sqrt{\lambda z}$ (105 µm here): the first bright fringe is $0.86\sqrt{\lambda z}$ = 90 µm outside the edge and 1.37 times as bright as the open beam, the next fringes come closer and closer together, and right at the edge the light is a quarter of the open beam's (half the amplitude). Move the screen back and the fringes spread out as $\sqrt z$.

3. The spot of Arago and Poisson

An opaque disk. At the centre of its shadow is a bright spot, as bright as the light would be there with no disk at all (to 1% here), in a shadow that is dark around it (6% of the open beam at its darkest). Poisson, judging Fresnel's wave theory in 1818, derived this spot to show the theory absurd; Arago looked, and it was there. Every point of the disk's rim is the same distance from the centre, so the waves from the rim all arrive in step. Slide $z$ out: once the disk covers less than one Fresnel zone ($a^2/\lambda z<1$) the shadow fills in.

4. Young's two slits

Two slits a distance $d$ apart ($d$ = 250 µm, each slit $d/4$ wide): fringes spaced by $\lambda/d$ in angle (2.2 mrad; 0.18 mm on the screen at 80 mm), dark at $\theta=\pm\lambda/2d,\pm3\lambda/2d\ldots$, inside the single slit's envelope. The envelope's first zero, at $\theta=4\lambda/d$, falls on the fourth fringe, which is missing. The side view shows the fringes form: two diverging beams, overlapping and interfering.

5. The Talbot carpet

A grating of period $d$ makes a perfect image of itself at the Talbot length $z_T=2d^2/\lambda$ (8.4 mm for this grating, $d$ = 48 µm), and an image moved over by $d/2$ at half that: the screen is set there, and the side view spans one $z_T$, from the grating at the top to its image at the bottom. A quarter of the way the pattern has half the period. This grating is a sine, with only the orders 0 and ±1, so its carpet is that simple; a sharp-edged grating weaves a finer one. The "Talbot carpet", found by Henry Fox Talbot in 1836, is the same pattern as the quantum revival of a particle in a box, with $z$ for time.

6. A Fresnel zone plate

Rings, open and closed alternately, with radii $r_n=\sqrt{n\lambda f}$: every open ring sends light to the point $z=f$ on the axis in step, so the plate focuses like a lens, to a spot about $0.61\lambda f/R$ in radius for a plate of radius $R$ (25 µm here); and also at $f/3$, $f/5$..., from the harmonics of the rings, which send as much light the other way too, diverging as if from $-f$, $-f/3$... behind the plate. The screen is set at $z=f$, zoomed in 4 times, on the linear scale ($I$), which shows the true contrast: with $N$ open rings each adding its share in step, the focus is $(2N)^2$ times as bright as the light falling on the plate (324 for $N=9$; 305 here, the pixels softening the finest rings a little), and everything else is dark. Switch to $\sqrt I$ to see the faint rings around it, and move the screen to $f/3$ and $f/5$; the side view shows the light converging. The plate's rings stop before they get narrower than 2.5 pixels: drawn on a grid, finer rings would alias, sending light where a real plate doesn't.

7. A lens is a phase

A thin lens changes only the phase of the light, by $-kr^2/2f$: the aperture is drawn in colour, the colour being the phase. At $z=f$ the light gathers into the focus, the Airy pattern of the lens's round opening (switch the scale to $\sqrt I$ to see its rings), of radius $0.61\lambda f/R$ = 28 µm for this 1.2 mm lens and 1600 times the incident intensity. The phase is in proportion to $k$, so in white light the lens keeps its focus for every colour, as glass does; the zone plate, an amplitude mask, has $f\propto1/\lambda$ and focuses red at 42 mm and blue at 61 mm. Switch on White light with each and compare the side views.

8. White light

Three wavelengths (red, green, blue) at once, through the grating, a metre away. Each colour is bent by its own angle: the first orders are at $\theta=\lambda/d$, 9.4 mrad for blue to 13.5 for red, so each order is a spectrum with red on the outside, and the zero order stays white. Slide the screen in to 60 mm: close to the grating each colour weaves its own Talbot carpet ($z_T=2d^2/\lambda$: 7.1 mm red, 8.4 green, 10.2 blue), and the three fall in and out of step as coloured fringes.

9. Speckle

A random aperture, like rough glass or a laser-lit wall: the light arriving at each point is a sum of many waves with random phases, and the screen is a random pattern of bright and dark grains, as many dark as bright. The grains have the size of the far-field pattern of the whole lit patch, about $\lambda z/D$: 0.11 mm here, for the 1 mm patch at 200 mm. Move the screen back and they grow in proportion; the pattern changes, but never smooths out.

How it works

Light of wavelength $\lambda$ passes through an aperture you can paint, and lands on a screen a distance $z$ behind it. The light just past the aperture is a sum of plane waves (its Fourier transform: the angular spectrum, the light sorted by direction), and each travels on with its own phase $e^{ik_zz}$, $k_z=\sqrt{k^2-k_x^2-k_y^2}$. Close to the aperture the screen shows a sharp shadow with fringes at its edges (Fresnel diffraction); far away, the Fourier transform of the aperture itself (Fraunhofer).

Which you see depends on the Fresnel number $N_F=a^2/\lambda z$ for an aperture of radius $a$. Seen from a point on the screen, paths through the middle and the rim of the aperture differ by about $a^2/2z$, and $N_F$ counts the half wavelengths in that difference: the Fresnel zones the aperture covers. Many zones ($N_F\gg1$): each edge makes its own fringes, on the scale of $\sqrt{\lambda z}$, and the shadow is sharp. Less than one ($N_F\ll1$): what is left of the path difference is linear across the aperture, $-xx'/z$, so the screen adds up $e^{-ikxx'/z}$ over it: the Fourier transform of the aperture, a pattern of fixed shape in angle $\theta=x/z$ that grows in proportion to $z$. The handover is around the Fraunhofer distance $a^2/\lambda$, where $N_F=1$; the far field proper needs $z\gg a^2/\lambda$ (within 1% of the Fourier transform from $N_F=0.1$ on). The slider goes to 10 m, many Fraunhofer distances for the slits, holes and grating here; Far field moves the screen there.

How it is computed

The aperture is 512 by 512 pixels of $\Delta x=8$ µm, each the average of the aperture over its area, so that edges and rings fall between pixels smoothly. Such a grid carries angles up to $\lambda/2\Delta x$ (34 mrad in green light): light bent further, by detail finer than a pixel, is lost. Up to $z_c=N\Delta x^2/\lambda$ (60 mm in green) the plane waves travel on that grid, and the screen shows its middle 2 mm. By $z_c$ the steepest of them have crossed half the grid, and on a fixed grid they would come back in at the other side; so from there on the demo uses the Fresnel transform, whose pixels grow in proportion to $z$, as $\lambda z/N\Delta x$. The screen then spans a fixed range of angles, ±17 mrad in green, and zooms out as the light spreads. At $z_c$ the two methods are the same calculation. Past $z_c$ the phase (Field) is shown relative to a spherical wave from the aperture, whose rings would be finer than the pixels, and the side view brightens its rows in proportion to $z^2$, to make up for the spreading.

The screen's brightness is in proportion to the amplitude, $\sqrt I$, with the brightest 0.5% of it white: that shows faint fringes. The linear scale, $I$, is the intensity itself with the brightest point white: the true contrast. Zoom magnifies the middle of the screen, reading the field between its pixels as the band-limited function it is (a sum of plane waves), not as squares. In white light a phase painted on the aperture (the lens, the rough glass) is taken as a thickness of glass: red light sees 550/650 of the phase the slider's green light sees, blue 550/450 of it.

Light and quantum mechanics

For waves near the axis, $k_z\approx k-(k_x^2+k_y^2)/2k$, and the field obeys $\partial_zU=\frac{i}{2k}\nabla_\perp^2U$: the free Schrödinger equation, with $z$ for time and $1/k$ for $\hbar/m$. Diffraction is the spreading of a wave packet. (Up to $z_c$ this demo uses the exact $k_z$; the Fresnel transform beyond it is paraxial, which at the angles shown changes the intensity by a fraction of a percent.)

The idea of the BrainFlux LightPropagation applet, finished in 2026. Source: github.com/gallicchio/brainflux, MIT licence.