Because the intensity in two-beam interference fringes varies sinusoidally with the phase difference, it is difficult to locate the fringe maxima or minima, in a photograph of the interference pattern, to better than a tenth of the fringe spacing. In addition, when the number of fringes is small and they are unequally spaced, errors are introduced by the need for nonlinear interpolation to determine the fractional fringe order at any point.

Computer-aided evaluation

One way to obtain higher accuracy is by using a CCD camera interfaced with a computer to sample and store the values of the intensity in the interference pattern at an array of points. These values can then be digitized and processed, using a number of techniques, to obtain the fractional fringe order at these points [Robinson & Reid, 1993]. Preprocessing is usually necessary to reduce speckle noise as well as to correct for local variations in image brightness.

Fourier-transform techniques

An additional tilt introduced in one of the beams (say, along the x direction) generates background fringes corresponding to a spatial carrier frequency. These fringes are modulated by the additional phase difference between the beams due to the changes in the object [Takeda, Ina & Kobayashi, 1982]. If the spatial carrier frequency is sufficiently high, the Fourier transform of the intensity distribution in the interference pattern can be processed to obtain the phase difference.

Computer-generated holograms can produce wavefronts with any prescribed amplitude and phase distribution and have, therefore, found many applications. The production of such holograms has been discussed by Lee [1978], Yaroslavskii and Merzlyakov [1980] and Dallas [1980], and essentially involves two steps.

The first step is to calculate the complex amplitude of the object wave at the hologram plane; for simplicity, this is usually taken to be the discrete Fourier transform (see Appendix B) of the complex amplitude at an N×N set of points in the object plane. The second step involves using the N×N computed values of the discrete Fourier transform to produce a transparency (the hologram) which reconstructs the object wave when it is suitably illuminated.

Two approaches have been followed for this purpose. In the first, which is analogous to off-axis holography, the complex amplitudes of a plane reference wave and the object wave, at each point in the hologram plane, are added, and the squared modulus of their sum is evaluated. These values are used to produce a transparency whose amplitude transmittance is real and positive everywhere.

An alternative is to produce a transparency that records both the amplitude and the phase of the object wave in the hologram plane. This transparency can be thought of as the superposition of two transparencies, one of constant thickness having a transmittance at each point proportional to the amplitude of the object wave, and the other with uniform transmittance but having thickness variations proportional to the phase of the object wave.

Holography is now used widely as a display medium. In addition, it is firmly established as a tool for scientific and engineering studies, and has found a remarkably wide range of applications for which it is uniquely suited.

This book is intended as an introduction to the subject for science and engineering students, as well as people with a scientific background who would like to learn more about holography and its applications. Key topics are presented at a level that is accessible to anyone with a basic knowledge of physics. A comprehensive bibliography and references to original papers identify sources of additional information. Numerical problems (and solutions) are provided at the end of each chapter, to clarify the principles discussed and give the reader a feel for numbers.

After a brief historical retrospect, the first three chapters review image formation by a hologram, the characteristics of the reconstructed image, and the basic types of holograms, while the next three chapters discuss available light sources, the characteristics of hologram recording media and practical recording materials.

These six chapters are followed by three chapters describing methods for the production of different types of holograms for displays, including multicolor holograms, and methods for making copies of holograms, as well as a chapter describing the production of computer-generated holograms. Following these, the next two chapters review some of the most important technical applications of holography, such as high-resolution imaging, holographic optical elements, and holographic information storage and processing.

In principle, a multicolor image can be produced by a hologram recorded with three suitably chosen wavelengths, when it is illuminated once again with these wavelengths. However, a problem is that each hologram diffracts, in addition to the wavelength used to record it, the other two wavelengths as well. The cross-talk images produced in this fashion overlap with, and degrade, the desired multicolored image. This problem has been overcome, and several methods are now available to produce multicolor images [Hariharan, 1983].

Multicolor reflection holograms

The first technique employed to eliminate cross-talk made use of the high wavelength selectivity of volume reflection holograms. If such a hologram is recorded with three wavelengths, one set of fringe planes is produced for each wavelength. When the hologram is illuminated with white light, each set of fringe planes diffracts a narrow band of wavelengths centered on the original wavelength used to record it, giving a multicolor image free from cross-talk [Upatnieks, Marks & Federowicz, 1966].

Higher diffraction efficiency can be obtained by superimposing three bleached volume reflection holograms recorded on two plates, one with optimum characteristics for the red, and the other with optimum characteristics for the green and blue. Brighter images can also be obtained if the final holograms are produced using real images of the object projected by primary holograms whose aperture is limited by a suitably shaped stop [Hariharan, 1980a].

So far, we have treated a hologram recorded on a photographic film as equivalent, to a first approximation, to a grating of negligible thickness with a spatially varying transmittance. However, if the thickness of the recording medium is larger than the average spacing of the fringes, volume effects cannot be neglected. It is even possible, as mentioned in Section 1.7, to produce holograms in which the interference pattern that is recorded consists of planes running almost parallel to the surface of the recording material; such holograms reconstruct an image in reflected light.

In addition, with modified processing techniques, or with other recording materials, it is possible to reproduce the variations in the intensity in the interference pattern produced by the object and reference beams as variations in the refractive index, or the thickness, of the hologram. Accordingly, holograms recorded in a medium whose thickness is much less than the spacing of the interference fringes (thin holograms) can be classified as amplitude holograms and phase holograms.

Similarly, holograms recorded in thick media (volume holograms) can be subdivided into transmission amplitude holograms, transmission phase holograms, reflection amplitude holograms and reflection phase holograms.

In the next few sections we review the characteristics of these six types of holograms. For simplicity, we consider only gratings produced by the interference of two plane wavefronts.

Holography makes it possible to store a wavefront and reconstruct it at a later time. As a result, interferometric techniques can be used to compare two wavefronts which were originally separated in time or space, or even wavefronts of different wavelengths. In addition, since a hologram reconstructs the shape of an object with a rough surface faithfully, down to its smallest details, large scale changes in the shape of almost any object can be measured with interferometric precision [Brooks, Heflinger & Wuerker, 1965; Burch, 1965; Collier, Doherty & Pennington, 1965; Haines & Hildebrand, 1965; Stetson & Powell, 1965]. Holographic interferometry is now used extensively in nondestructive testing, aerodynamics, heat transfer and plasma diagnostics [Vest, 1979; Rastogi, 1994] as well as in studies of the behavior of anatomical structures and prostheses under stress [Greguss, 1975; von Bally, 1979; Podbielska, 1991, 1992].

Real-time interferometry

Equations (1.6)–(1.9) show that if a hologram is replaced in its original position in the same optical system used to record it, and illuminated with the original reference wave, it reconstructs the original object wave. If, then, the shape of the object changes slightly, the directly transmitted object wave will interfere with the reconstructed object wave to produce, as shown in fig. 13.1, a fringe pattern that maps the changes in the shape of the object.