Holography

1Record

the object's light $O$ and the reference $R$ meet on the plate

Drag the point (the dot) nearer or farther, up or down; sideways in the front view. The hologram and the view through it follow.

The object from the front

2Relight

the developed plate, lit by $R$ alone, sends on three beams

Click anywhere here to focus the camera (the eye) at that depth; click an image to focus on it.

The hologram

Cut it, or look through a window

Its Fourier transform

Looking through it

Experiments

Start here

A hologram is a photograph of interference. The plate records the fringes that light from an object makes with a second beam from the same laser, the reference; lit again by that beam, the plate sends the object's light on as it was, and the object is there behind it, in depth. The page shows the two steps, Record above and Relight below, for one bright point 60 mm behind the plate.

Drag the point in the Record view (nearer or farther, up or down) or in the front view (sideways): the hologram's rings, their Fourier transform and the view through the plate follow. Click in the Relight view to focus the camera at that depth, or click the twin to focus on it.

Look for the three beams the lit plate sends on, each in its own colour: the zero order straight on, the image where the point was, and the twin converging in front of the plate; and in the transform, one island of fringes for each.

1. One point: the hologram is a zone plate

One point 60 mm behind the plate, and the reference straight along the axis ($\theta$ = 0). The hologram is a set of rings. Why: light from the point reaches the plate a distance $r$ from its middle after travelling $\sqrt{z^2+r^2}\approx z+r^2/2z$, farther than at the middle by $r^2/2z$, and each half wavelength farther turns bright into dark. So the rings are brightest and darkest at $r_n=\sqrt{n\lambda z}$, bright for even $n$ and dark for odd: 10 of them out to the edge of the plate, ever closer together, round a bright middle. This is a Fresnel zone plate.

Lit again, a zone plate is a lens with two focal lengths, $+z$ and $-z$. It spreads part of the light as if from a point 60 mm behind (the image), it gathers part to a point 60 mm in front (the twin), and the rest goes straight on (the zero order). The camera, focused at 60 mm, sees the image as a sharp dot. Round it is the zero order, a square $L/d$ = 21 mrad wide, which is the plate itself out of focus, and the twin, twice as wide (it is out of focus by $1/z+1/d$). Click the twin in the Relight view: now the twin is the sharp dot. In line, all three beams leave along the axis, and no focus gets rid of the other two. This is the in-line arrangement of Dennis Gabor's first holograms (1948).

2. Tilt the reference: the three beams part

The same point, with the reference tilted up by 1°: the page as it opens. Drag the angle down to 0 and back up. The rings' centre is where the point's light arrives parallel to the reference, so as the reference tilts up it slides up, to $z\tan\theta$ above the axis, and at 0.61°, where $\tan\theta=L/2z$, it leaves the plate. Then the plate holds arcs of a zone plate centred above it: a lens and a grating in one.

Now the three beams leave in three directions (the Relight view): the zero order straight on, at $\theta$; the image along the object's own light; and the twin, bent twice as far, at $2\theta$, converging to a real point 60 mm in front and $2z\sin\theta$ = 2.1 mm up. The camera sees them side by side: the sharp image, the zero order 17 mrad below it, and the twin 35 mrad below, twice as wide. (Light leaving the plate upwards comes into the camera from below.) In the transform, the image and twin islands part at the same 0.61°: the tilt must be more than the spread of directions in which the object's light reaches the plate, here $\pm L/2z$ = ±10.7 mrad. This is the off-axis hologram of Emmett Leith and Juris Upatnieks (1962). Past about 1.2° the twin reaches the bottom of the view and comes back at the top: the pixels cannot bend light more than $\lambda/2p$ = 63 mrad (experiment 12).

3. A whole object: how far to tilt

A letter F, $W$ = 2 mm tall and 120 mm behind the plate, lit through ground glass so that each of its points lights the whole plate, with the reference at 0.4°. Raise the angle slowly and watch first the twin, then a haze, slide off the F in the view. Why there: the F's light reaches the plate from directions spread over $\pm B$, with $B=(W+L)/2z$ = 14 mrad (half the angle the F subtends, plus half the angle the plate subtends). So the image island of the transform is $2B$ wide, and so is the twin's, and they clear each other once $\sin\theta>B$, at 0.78°. The middle island holds $|O|^2$, the F's light interfering with itself. At any one point of the plate that light arrives from directions up to $W/z$ = 17 mrad apart, so this island reaches $\pm W/z$, and it clears the image island once $\sin\theta>B+W/z$, at 1.74°. (For an object much bigger than the plate this is the textbook rule, $\sin\theta>3B$.) Past 2.85° the image island reaches the edge of the transform: fringes finer than two pixels.

4. Focus through depth

Three points, 50, 80 and 130 mm behind the plate, and the camera's focus sweeping back and forth: each point sharpens at its own depth. The hologram recorded the object's whole wave, depth and all. Out of focus, a point is a small square $L\,|1/d-1/z|$ wide: the plate's own outline, as the blur of an out-of-focus camera has the shape of its aperture. Focused at 80 mm, the point at 50 mm is a square 9.6 mrad wide. Stop the sweep and click a point's image in the Relight view, or press Focus on the image for the selected point.

5. Parallax through a window

Three points straight behind one another, 40, 60 and 100 mm behind the plate, seen through a window 0.3 mm wide that sweeps up and down the plate (or drag it on the hologram). The camera is focused at 60 mm. The point at 60 mm stays put; the nearer one slides one way, the farther one the other: parallax, as when you move your head in front of a window. The plate recorded light arriving from every direction it can see, so each window shows the scene from its own position. Moving the window by $y_w$ moves a point at depth $z$ by $y_w(1/d-1/z)$ in this view: 0.9 mm moves the nearer point 15 pixels one way and the farther one 12 the other. A photograph cannot do this.

6. Cut the hologram

Through ground glass, every point of the F lights the whole plate, so every part of the plate holds the whole F. Press Cover half (under the hologram): the whole letter is still there, speckled and blurred from side to side, because the plate is now half as wide that way. Then Uncover all and drag out a small square on the hologram. The F survives in any piece, coarser and grainier as the piece shrinks, until the grains are as big as its strokes. A piece of width $D$ resolves directions $\lambda/D$ apart, like a lens of that size, and shows the scene only from where it is (experiment 5). The grains are speckle, from the random phases of the ground glass. (Try the bars too: with half the plate covered, the upright bars blur and the lying ones do not.)

7. Light it with the conjugate: a real image

Two points, 50 and 90 mm behind the plate, and the plate lit with the reference tilted the other way, down instead of up. At the plate that wave is $R^*$, so the plate passes $$HR^*=(|R|^2+|O|^2)\,R^*+R^{*2}\,O+|R|^2\,O^*,$$ and now the last term is the one to look at: $O^*$, the object's wave with its curvature reversed, converges to a real image of each point straight in front of where the point was. The camera is focused 50 mm in front, on the nearer point; focus at −90 mm for the other (or click it in the Relight view). A screen placed there would catch them. The real image is inside out (pseudoscopic): the point that was nearer the plate is still nearer it, so to someone looking from in front, the depths are reversed. (The reference sent backwards through the plate, from the front, puts the same $R^*$ on the plate, but its light runs backwards, retracing the object's light: then the real image forms behind the plate, where the object was.)

8. Relight in another colour

The F was recorded in red (633 nm) and is lit with green (532 nm), and the camera is still focused where the F was, so it is out of focus. The fringes bend light by $\sin\theta=\lambda/\Lambda$, so they bend green light less: a zone plate's focal length goes as $1/\lambda$, and the image forms farther away, at $150\times633/532$ = 178 mm. Press Focus on the image. The F is as big as before but farther away, so it looks smaller, by $532/633$ = 0.84. It has also moved down towards the zero order, by $(1-532/633)\sin\theta$ = 5.6 mrad, because the reference's own fringes bend green light less too. Slide the colour and watch it move. In white light every colour would make its own image, which is why these holograms need a laser to be seen. (Rainbow and reflection holograms get round this.)

9. A binary hologram

The hologram has been rounded to one bit: each pixel black or white, split at the average brightness (zoom in: magnified 4 times). The F survives, a little noisier: the fringes are still in the right places, and that is what matters. Computer-generated holograms are often binary: printed, etched, or shown on micromirror displays. Try more bits under Laser and camera: with 4 bits (16 grey levels) the image is all but perfect.

Try it on one point too (experiment 1, then 1 bit): the rings turn black and white, like an ordinary zone plate cut from opaque and clear rings. A square wave of the phase has harmonics, $e^{\pm3i\phi}$, $e^{\pm5i\phi}$, ..., so this zone plate has more focal lengths: focus at $z/3$ = 20 mm and there is another sharp dot.

10. A digital hologram of Fourier

Joseph Fourier's portrait, 5 mm across and 150 mm behind the plate, lit through ground glass. Its hologram looks like noise and the transform has the three islands, but through the plate there is Fourier. The engraving has more detail than the hologram can hold. The plate tells apart directions only $\lambda/L$ = 0.5 mrad apart, so the portrait is 67 directions across, and an off-axis hologram can give only part of its directions to the image: the rest go to the zero order and the twin. The grain is speckle again.

11. A lensless Fourier hologram

Here the reference is a point source in the object's own plane, 5 mm below the F, both 150 mm behind the plate. The waves from each point of the F and from the reference curve alike, so together they make straight fringes, and the hologram's transform is a picture of the object: the F turned half round in the image island, and upright in the twin island. One FFT makes the image. Through the plate, the image and the twin are both in focus at 150 mm, side by side, and the zero order is the reference's own point source, a bright dot 33 mrad below the F. X-ray holograms of nanostructures are made this way, with a pinhole beside the sample for the reference.

12. Fringes two pixels apart

The reference is as steep as the pixels allow: fringes $\lambda/\sin\theta=2p$ = 10 µm apart, at $\theta$ = 3.63°. A camera with pixels $p$ apart cannot record fringes finer than $2p$, so it cannot record light arriving at more than $\lambda/2p$ = 63 mrad to the reference. The image island reaches the edge of the transform and comes back on the other side, and the twin, which leaves at twice the angle, comes back right on top of the image. With two pixels per fringe, the camera cannot tell a fringe tilted one way from one tilted the other. Lower the angle until the islands come clear of the edges. The best angle depends on the object, but it is always well short of the limit.

The same limit holds for the object's own light. A point $z$ behind the plate makes rings $\lambda z/r$ apart at radius $r$, so at the plate's edge they are finer than two pixels if the point is nearer than $Np^2/\lambda$ = 10 mm. Bring the point of experiment 1 closer and watch the outer rings turn into a moiré.

How it works

1. Record. Light from the object, $O$, and a reference beam from the same laser, $R$, fall together on a plate. (Here the plate is a camera sensor, 256 × 256 pixels of 5 µm, so $L$ = 1.28 mm across, and the laser is red, $\lambda$ = 633 nm.) The plate records only the brightness, $$H=|O+R|^2=|O|^2+|R|^2+OR^*+O^*R.$$ The last two terms add up to $2|O||R|\cos(\phi_O-\phi_R)$: fringes, bright where the two waves arrive in step and dark where they arrive out of step. Where the fringes fall records the phase of $O$, which an ordinary photograph throws away.

2. Relight. Develop the plate so that it lets light through in proportion to $H$, take the object away, and light the plate with $R$ alone. It passes on $$HR=(|R|^2+|O|^2)\,R+|R|^2\,O+R^2\,O^*.$$ The first term goes straight on: the zero order. The second is the object's own wave, $O$, as it was: look through the plate and the object is there, behind it, in depth. That is a virtual image: the light only seems to come from there. The third is the complex conjugate, $O^*$, whose curvature is reversed: where $O$ spread out from the object, $O^*$ converges, to a real image in front of the plate, the twin. Tilt the reference and the three beams leave in three directions.

What is on the page. The side views stretch the heights (by the factor in their corner), or the plate would be a speck. Beside the Relight view are the three terms and where each one's light goes. The hologram is the recorded brightness; zoom in to see its fringes. Its Fourier transform sorts the fringes by spacing and direction: fringes of period $\Lambda$ bend light by $\sin\theta=\lambda/\Lambda$, so each island of the transform is one of the three beams. Looking through it is what a camera held against the lit plate sees, focused at a distance $d$: behind the plate ($d>0$), where a virtual image is, or in front ($d<0$), where a screen would catch a real one. It is drawn in angles, $x/d$, so the picture keeps its size as you focus. Click in the Relight view to focus at that depth; click an image or a twin to focus on it.

How it is computed

The points' light is exact spherical waves, $e^{ikr}/r$, with $r$ the distance to each pixel. A picture's light reaches the plate by the Fresnel diffraction integral (paraxial, which is accurate at these few degrees), computed by one FFT on a grid in which each sample of the picture is one direction of the view. With the ground glass, every sample has a random phase, so every point of the picture lights the whole plate.
The camera records the brightness at the centre of each pixel, without noise; rounding to a few grey levels is optional.
The view: the light leaving the plate goes through a lens that makes light from a point at distance $d$ parallel (its phase is exact for a point on the axis), and one FFT sorts the light by direction: together, a camera focused at $d$. The directions are $\lambda/L$ apart, out to $\pm\lambda/2p$. For $d<0$ this is also the Fresnel transform: the light on a screen $|d|$ in front of the plate. A plate of pixels is a grating of period $p$, which sends a copy of everything $\lambda/p$ away, so light that would leave at more than $\lambda/2p$ shows up on the other side. The brightest spots are drawn white, so that fainter light can be seen.
The side views, the three terms and the islands' extents (the note under the transform) come from the paraxial holography equations: each wave at the plate has a curvature and a tilt, and the plate adds and subtracts them. The tests check them against the computed images.

New in 2026 for BrainFlux. Source: github.com/gallicchio/brainflux, MIT licence.