Holography

Holography records a light wave in a way that preserves both its amplitude and its phase. A photograph stores only intensity, so the depth and direction information carried by the phase is lost. In holography, light scattered from an object (the object beam) is overlapped on a recording medium with a coherent reference beam from the same laser, and the interference pattern they form is recorded. Illuminating the developed hologram with the reference beam again diffracts light into a copy of the original object wave, so a viewer sees the object in three dimensions with full parallax. The recorded fringes are fine: for a 632.8 nm helium-neon laser and beams crossing at 30°, they are about 1.22 µm apart.

Recording and reconstruction

If the object wave at the plate is $O$ and the reference wave is $R$, the recorded intensity is

$$I = |O|^2 + |R|^2 + O R^* + O^* R.$$

The cross terms carry the phase of $O$ relative to $R$. Reilluminating the hologram with $R$ multiplies the transmittance by $R$, and the term $|R|^2 O$ is a copy of the object wave. The other terms give an undiffracted beam and a conjugate (twin) image. Dennis Gabor proposed the method in 1948 to improve electron microscopy, with object and reference on the same axis, where the twin image overlaps the reconstructed image. Leith and Upatnieks introduced the off-axis reference beam in 1962, which separates the images by angle and made laser holography practical. Gabor received the 1971 Nobel Prize in Physics.

Fringe spacing

Two plane waves crossing at a full angle $\theta$ produce fringes with period

$$\Lambda = \frac{\lambda}{2\sin(\theta/2)}.$$

For $\lambda = 632.8$ nm and $\theta = 30°$, $\Lambda = 632.8/(2 \sin 15°) \approx 1.22$ µm, or about 820 lines/mm. A hologram is therefore a complex diffraction grating, and the recording material must resolve these fringes: silver halide holographic emulsions, dichromated gelatin and photopolymers resolve several thousand lines per millimeter.

Coherence and stability

The object and reference paths must match to within the coherence length of the laser, or the fringes lose contrast. A single-frequency laser allows path differences of meters; a multimode laser limits the usable depth of the scene to a fraction of its cavity length. The setup must also be mechanically still during the exposure: a mirror that moves by $\lambda/4$, about 160 nm at 632.8 nm, changes the reflected path by half a wavelength and shifts the fringes by half a period, washing them out. Holography tables are therefore vibration-isolated; pulsed lasers record moving subjects. Diffuse objects add speckle to the reconstructed image.

Transmission and reflection holograms

In a transmission hologram the object and reference beams reach the plate from the same side, and the image is viewed by light passing through it. In a reflection hologram (Denisyuk), the beams arrive from opposite sides, so the fringes lie almost parallel to the surface with spacing near $\lambda/2n$: about 211 nm in a material of index 1.5 at 632.8 nm. These stacked fringes act as a Bragg mirror that reflects only a narrow band, satisfying the Bragg condition, so a reflection hologram can be viewed in white light and appears in a single color.

Thin and thick (volume) holograms also differ in diffraction efficiency. A thin sinusoidal amplitude hologram diffracts at most 6.25% of the light into the image, a thin sinusoidal phase hologram 33.9%, and a thick phase hologram approaching 100% at the Bragg angle.

Digital holography and holographic optical elements

In digital holography a camera sensor records the interference pattern and a computer reconstructs the wave by numerical diffraction, using the methods of Fourier optics. The pixel pitch $p$ limits the angle between object and reference beams to about $\lambda/2p$; for 3.45 µm pixels at 632.8 nm this is about 5.3°. Digital holographic microscopy uses the recovered phase to measure cell thickness and surface height. The reverse process, computer-generated holography, calculates a fringe pattern and displays it on a spatial light modulator to shape beams, form optical traps or project images.

Holographic optical elements are holograms recorded to act as lenses, gratings, beam splitters or combiners. They are used in head-up and augmented-reality displays and as volume Bragg gratings that stabilize the wavelength of laser diodes. Holographic interferometry compares two exposures of an object to map sub-micrometer deformations.

Common questions

Why does holography need a laser?

The object and reference beams must interfere stably over the whole recording area and depth of the scene, which requires spatial and temporal coherence. Gabor's 1948 experiments used a filtered mercury lamp; holograms of extended three-dimensional scenes became practical only with lasers in the early 1960s.

What is the difference between a hologram and a 3D display?

A hologram reconstructs the actual light wave from the object, so the eye focuses at the correct depth and sees parallax as it moves. Stereoscopic displays show one flat image to each eye.

Can a hologram be cut in half?

Yes. Each part of a transmission hologram of a diffuse object receives light from the whole object, so a fragment still reconstructs the full scene through a smaller window.

References: D. Gabor, Nature 161, 777 (1948); E. N. Leith, J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962); P. Hariharan, Optical Holography, 2nd ed. (Cambridge University Press, 1996); J. W. Goodman, Introduction to Fourier Optics, 4th ed. (W. H. Freeman, 2017).