Overview article
Optical holography is a technique which enables an optical wavefront to be recorded and later re-constructed. Holography is best known as a method of generating three-dimensional images but it also has a wide range of other applications.
A hologram is made by superimposing a second wavefront (normally called the reference beam) on the wavefront of interest, thereby generating an interference pattern which is recorded on a physical medium. When only the second wavefront illuminates the interference pattern, it is diffracted to recreate the original wavefront. Holograms can also be computer-generated by modelling the two wavefronts and adding them together digitally. The resulting digital image is then printed it onto a suitable mask or film and illuminated by a suitable source to reconstruct the wavefront of interest.
Basic physics.
To understand the process, it is helpful to understand interference and diffraction. Interference occurs when one or more wavefronts are superimposed. Diffraction occurs when a wavefront encounters an object. The process of producing a holographic reconstruction is explained below purely in terms of interference and diffraction. It is somewhat simplified but is accurate enough to give an understanding of how the holographic process works.
For those unfamiliar with these concepts, it is worthwhile to read those articles before reading further in this article.
A simple hologram can be made by superimposing two plane waves from the same light source on a light recording medium such as a photographic emulsion. The two waves interfere, giving a straight-line fringe pattern whose intensity varies sinusoidally across the medium. The spacing of the fringe pattern is determined by the angle between the two waves, and by the wavelength of the light.
The recorded light pattern is a diffraction grating, which is a structure with a repeating pattern. A simple example is a metal plate with slits cut at regular intervals. A light wave that is incident on a grating is split into several waves; the direction of these diffracted waves is determined by the grating spacing and the wavelength of the light.
When the recorded light pattern is illuminated by only one of the plane waves used to create it, it can be shown that one of the diffracted waves is a re-construction of the other plane wave.
 When a plane wave is added to a point source and the resulting interference pattern recorded, a point source hologram is produced. This is effectively a Fresnel zone plate which acts as a lens. If the plane wave is normally incident on the recording plate, three waves are diffracted by the plate
 the original plane wave
 a wave which appears to diverge from the point source - this is a reconstruction of the original point source wave
 a wave which is focused to a point on the other side of the plate at the same distance as the original point source
This is known as an in-line hologram. Its usefulness is limited by the fact that all three waves are superimposed.
If the plane wave illuminates the recording plate at non-normal incidence, then the three diffracted waves are now as follows:
 the original plane wave
 a wave which appears to diverge from the original point source - this is the re-constructed wave
 a wave which converges to a point which is deflected from the normal by twice the angle of incidence of the plane wave - this is known as the conjugate wave.
The three waves are now separated in space. This is known as an off-axis hologram. It was first developed by Leith and Upatnieks and was a vital step in enabling 3-d images to be produced with holography.
Theory underlying the holographic process.
General form.
The complex amplitude of a monochromatic electromagnetic wave can be represented by
formula_1
where A represents the amplitude of the vector, and formula_2 its phase.
To make a hologram, two waves are added together to give a total complex amplitude which can be represented as 
formula_3
where R refers to the recording wavefront, known as the reference wavefront, and O refer to the wavefront being recorded. The dependence on r has been omitted for clarity.
The intensity of the combined beams is the average value of the complex amplitude times its complex conjugate:
formula_4
A recording medium is exposed to the two beams and then developed. Assume that the amplitude transmittance, formula_5 of the developed photographic plate is linearly related to the intensity of the interference pattern.
formula_6
where formula_7 is a constant background transmittance and formula_8 is constant,
When the developed photographic plate is illuminated only by the reference beam, formula_9, the amplitude of the light transmitted through the plate, UH, is given by
formula_10
This can be split into three terms: 
formula_11
formula_12 is a modified version of the reference wave. The first term is a reduced amplitude version, the second is also a reduced amplitude version if the reference wave amplitude is uniform. The third term produces a halo round the transmitted reference wave which is negligible when the amplitude of the object wave is much less than that of the reference wave
formula_13 is the reconstructed object wave which is identical to the original wave except that its amplitude is reduced. When the object wave is generated by light scattered from an object or objects, a virtual image of the object(s) is formed when a lens is placed in the reconstructed wave.
formula_14 is known as the conjugate wave. It is similar to the object wave but has the opposite curvature. When the object wave is generated by light scattered from an object or a series of objects, a real image is formed on the opposite side of the hologram plate to where the object was located and is deflected from the normal axis by twice the angle between the reference wave and the normal direction.
Two plane waves.
The reference and object wave are monochromatic plane waves of uniform amplitude
formula_15
where the vectors formula_16 are related to the unit vectors formula_17 which define the direction of travel and the wavelength formula_18 of the two waves by
formula_19
The expression for formula_20 in the previous section can be written as
formula_21
from which it can be seen that the transmission of the holographic recording varies sinusoidally, i.e.it is a diffraction grating, where spacing of the fringes depends on the angle between the two waves and the orientation of the holographic plate.
The reconstructed wave is:
formula_22
which is a plane wave travelling in the direction formula_23, i.e. it is the reconstructed object wave with a modified amplitude.
The conjugate wave is:
formula_24
which is a plane wave travelling in the direction formula_25
This shows that the transmission of the photographic plate varies sinusoidally so it acts as a diffraction grating. If one of the waves is normally incident to the hologram plate, and the other is incident at an angle formula_26, the spacing of the fringes is formula_27 and the first order diffracted waves will be at angles given by formula_28. One of these is the reconstructed object plane wave, the other is the conjugate wave.
Plane wave and point source.
Consider a point source located at the origin which illuminates a photographic plate which is located at a distance formula_29 normal to the z axis. The phase difference between the source and a point (x, y, z) is given approximately by
formula_30
where formula_31
The complex amplitude of the object beam is then given by
formula_32.
The form of the interference pattern for various directions of incidence for the reference wave are shown on the right.
The reference plane wave is
formula_33
The reconstructed wave is described by the second term:
formula_34
The conjugate wave is described by the third term
formula_35
It emerges at twice the angle between the object and references wave and converges to a point which is at the same distance from the plate as the original point source. 
General object and a point source.
When light is scattered off a general object, the amplitude of the scattered light at any point formula_36 can be represented by
 formula_37
where both the phase and the amplitude vary with formula_38.
The distance travelled by a wave emitted from a point source located at a point formula_39 to a point formula_40 travels a distance formula_41, so that the phase change is formula_42 so that its amplitude is given by
 formula_43
where formula_44 is constant.
The reconstructed object wave is
formula_45
The conjugate wave is given by
formula_46
Recording a hologram.
Items required.
To make a hologram, the following are required: 
These requirements are inter-related, and it is essential to understand the nature of optical interference to see this. Interference is the variation in intensity which can occur when two light waves are superimposed. The intensity of the maxima exceeds the sum of the individual intensities of the two beams, and the intensity at the minima is less than this and may be zero. The interference pattern maps the relative phase between the two waves, and any change in the relative phases causes the interference pattern to move across the field of view. If the relative phase of the two waves changes by one cycle, then the pattern drifts by one whole fringe. One phase cycle corresponds to a change in the relative distances travelled by the two beams of one wavelength. Since the wavelength of light is of the order of 0.5 μm, it can be seen that very small changes in the optical paths travelled by either of the beams in the holographic recording system lead to movement of the interference pattern which is the holographic recording. Such changes can be caused by relative movements of any of the optical components or the object itself, and also by local changes in air-temperature. It is essential that any such changes are significantly less than the wavelength of light if a clear well-defined recording of the interference is to be created.
The exposure time required to record the hologram depends on the laser power available, on the particular medium used and on the size and nature of the object(s) to be recorded, just as in conventional photography. This determines the stability requirements. Exposure times of several minutes are typical when using quite powerful gas lasers and silver halide emulsions. All the elements within the optical system have to be stable to fractions of a μm over that period. It is possible to make holograms of much less stable objects by using a pulsed laser which produces a large amount of energy in a very short time (μs or less). These systems have been used to produce holograms of live people. A holographic portrait of Dennis Gabor was produced in 1971 using a pulsed ruby laser. 
Thus, the laser power, recording medium sensitivity, recording time and mechanical and thermal stability requirements are all interlinked. Generally, the smaller the object, the more compact the optical layout, so that the stability requirements are significantly less than when making holograms of large objects.
Another very important laser parameter is its coherence. This can be envisaged by considering a laser producing a sine wave whose frequency drifts over time; the coherence length can then be considered to be the distance over which it maintains a single frequency. This is important because two waves of different frequencies do not produce a stable interference pattern. The coherence length of the laser determines the depth of field which can be recorded in the scene. A good holography laser will typically have a coherence length of several meters, ample for a deep hologram.
The objects that form the scene must, in general, have optically rough surfaces so that they scatter light over a wide range of angles. A specularly reflecting (or shiny) surface reflects the light in only one direction at each point on its surface, so in general, most of the light will not be incident on the recording medium. A hologram of a shiny object can be made by locating it very close to the recording plate.
Hologram classifications.
There are three important properties of a hologram which are defined in this section. A given hologram will have one or other of each of these three properties, e.g. an amplitude modulated, thin, transmission hologram, or a phase modulated, volume, reflection hologram.
Amplitude and phase modulation holograms.
An amplitude modulation hologram is one where the amplitude of light diffracted by the hologram is proportional to the intensity of the recorded light. A straightforward example of this is photographic emulsion on a transparent substrate. The emulsion is exposed to the interference pattern, and is subsequently developed giving a transmittance which varies with the intensity of the pattern – the more light that fell on the plate at