Interferometrically

Interferometry is a technique which uses the ''interference'' of superimposed waves to extract information. Interferometry typically uses electromagnetic waves and is an important investigative technique in the fields of astronomy, fiber optics, engineering metrology, optical metrology, oceanography, seismology, spectroscopy (and its applications to chemistry), quantum mechanics, nuclear and particle physics, plasma physics, remote sensing, biomolecular interactions, surface profiling, microfluidics, mechanical stress/strain measurement, velocimetry, optometry, and making holograms.

Interferometers are devices that extract information from interference. They are widely used in science and industry for the measurement of microscopic displacements, refractive index changes and surface irregularities. In the case with most interferometers, light from a single source is split into two beams that travel in different optical paths, which are then combined again to produce interference; two incoherent sources can also be made to interfere under some circumstances though. The resulting interference fringes give information about the difference in optical path lengths. In analytical science, interferometers are used to measure lengths and the shape of optical components with nanometer precision; they are the highest precision length measuring instruments in existence. In Fourier transform spectroscopy they are used to analyze light containing features of absorption or emission associated with a substance or mixture. An astronomical interferometer consists of two or more separate telescopes that combine their signals, offering a resolution equivalent to that of a telescope of diameter equal to the largest separation between its individual elements.

Basic principles

Interferometry makes use of the principle of superposition to combine waves in a way that will cause the result of their combination to have some meaningful property that is diagnostic of the original state of the waves. This works because when two waves with the same frequency combine, the resulting intensity pattern is determined by the phase difference between the two waves—waves that are in phase will undergo constructive interference while waves that are out of phase will undergo destructive interference. Waves which are not completely in phase nor completely out of phase will have an intermediate intensity pattern, which can be used to determine their relative phase difference. Most interferometers use light or some other form of electromagnetic wave.

Typically (see Fig. 1, the well-known Michelson configuration) a single incoming beam of coherent light will be split into two identical beams by a beam splitter (a partially reflecting mirror). Each of these beams travels a different route, called a path, and they are recombined before arriving at a detector. The path difference, the difference in the distance traveled by each beam, creates a phase difference between them. It is this introduced phase difference that creates the interference pattern between the initially identical waves. If a single beam has been split along two paths, then the phase difference is diagnostic of anything that changes the phase along the paths. This could be a physical change in the path length itself or a change in the refractive index along the path.

As seen in Fig. 2a and 2b, the observer has a direct view of mirror ''M''₁ seen through the beam splitter, and sees a reflected image ''M''′₂ of mirror ''M''₂. The fringes can be interpreted as the result of interference between light coming from the two virtual images ''S''′₁ and ''S''′₂ of the original source ''S''. The characteristics of the interference pattern depend on the nature of the light source and the precise orientation of the mirrors and beam splitter. In Fig. 2a, the optical elements are oriented so that ''S''′₁ and ''S''′₂ are in line with the observer, and the resulting interference pattern consists of circles centered on the normal to ''M''₁ and ''M'₂''. If, as in Fig. 2b, ''M''₁ and ''M''′₂ are tilted with respect to each other, the interference fringes will generally take the shape of conic sections (hyperbolas), but if ''M''′₁ and ''M''′₂ overlap, the fringes near the axis will be straight, parallel, and equally spaced. If S is an extended source rather than a point source as illustrated, the fringes of Fig. 2a must be observed with a telescope set at infinity, while the fringes of Fig. 2b will be localized on the mirrors.

Use of white light will result in a pattern of colored fringes (see Fig. 3). The central fringe representing equal path length may be light or dark depending on the number of phase inversions experienced by the two beams as they traverse the optical system. (See Michelson interferometer for a discussion of this.)

Categories

Interferometers and interferometric techniques may be categorized by a variety of criteria:

Homodyne versus heterodyne detection

In homodyne detection, the interference occurs between two beams at the same wavelength (or carrier frequency). The phase difference between the two beams results in a change in the intensity of the light on the detector. The resulting intensity of the light after mixing of these two beams is measured, or the pattern of interference fringes is viewed or recorded. Most of the interferometers discussed in this article fall into this category.

The heterodyne technique is used for (1) shifting an input signal into a new frequency range as well as (2) amplifying a weak input signal (assuming use of an active mixer). A weak input signal of frequency f₁ is mixed with a strong reference frequency f₂ from a local oscillator (LO). The nonlinear combination of the input signals creates two new signals, one at the sum f₁ + f₂ of the two frequencies, and the other at the difference f₁ − f₂. These new frequencies are called heterodynes. Typically only one of the new frequencies is desired, and the other signal is filtered out of the output of the mixer. The output signal will have an intensity proportional to the product of the amplitudes of the input signals.

The most important and widely used application of the heterodyne technique is in the superheterodyne receiver (superhet), invented in 1917-18 by U.S. engineer Edwin Howard Armstrong and French engineer Lucien Lévy. In this circuit, the incoming radio frequency signal from the antenna is mixed with a signal from a local oscillator (LO) and converted by the heterodyne technique to a lower fixed frequency signal called the intermediate frequency (IF). This IF is amplified and filtered, before being applied to a detector which extracts the audio signal, which is sent to the loudspeaker.
: Optical heterodyne detection is an extension of the heterodyne technique to higher (visible) frequencies.

While optical heterodyne interferometry is usually done at a single point it is also possible to perform this widefield.

Double path versus common path

A double-path interferometer is one in which the reference beam and sample beam travel along divergent paths. Examples include the Michelson interferometer, the Twyman–Green interferometer, and the Mach–Zehnder interferometer. After being perturbed by interaction with the sample under test, the sample beam is recombined with the reference beam to create an interference pattern which can then be interpreted.

A '' common-path interferometer'' is a class of interferometer in which the reference beam and sample beam travel along the same path. Fig. 4 illustrates the Sagnac interferometer, the fibre optic gyroscope, the point diffraction interferometer, and the lateral shearing interferometer. Other examples of common path interferometer include the Zernike phase-contrast microscope, Fresnel's biprism, the zero-area Sagnac, and the scatterplate interferometer.

Wavefront splitting versus amplitude splitting

Wavefront splitting inferometers

A wavefront splitting interferometer divides a light wavefront emerging from a point or a narrow slit (''i.e.'' spatially coherent light) and, after allowing the two parts of the wavefront to travel through different paths, allows them to recombine. Fig. 5 illustrates Young's interference experiment and Lloyd's mirror. Other examples of wavefront splitting interferometer include the Fresnel biprism, the Billet Bi-Lens, and the Rayleigh interferometer.

In 1803, Young's interference experiment played a major role in the general acceptance of the wave theory of light. If white light is used in Young's experiment, the result is a white central band of constructive interference corresponding to equal path length from the two slits, surrounded by a symmetrical pattern of colored fringes of diminishing intensity. In addition to continuous electromagnetic radiation, Young's experiment has been performed with individual photons, with electrons, and with buckyball molecules large enough to be seen under an electron microscope.

Lloyd's mirror generates interference fringes by combining direct light from a source (blue lines) and light from the source's reflected image (red lines) from a mirror held at grazing incidence. The result is an asymmetrical pattern of fringes. The band of equal path length, nearest the mirror, is dark rather than bright. In 1834, Humphrey Lloyd interpreted this effect as proof that the phase of a front-surface reflected beam is inverted.

Amplitude-splitting inferometers

An amplitude splitting interferometer uses a partial reflector to divide the amplitude of the incident wave into separate beams which are separated and recombined.

The Fizeau interferometer is shown as it might be set up to test an optical flat. A precisely figured reference flat is placed on top of the flat being tested, separated by narrow spacers. The reference flat is slightly beveled (only a fraction of a degree of beveling is necessary) to prevent the rear surface of the flat from producing interference fringes. Separating the test and reference flats allows the two flats to be tilted with respect to each other. By adjusting the tilt, which adds a controlled phase gradient to the fringe pattern, one can control the spacing and direction of the fringes, so that one may obtain an easily interpreted series of nearly parallel fringes rather than a complex swirl of contour lines. Separating the plates, however, necessitates that the illuminating light be collimated. Fig 6 shows a collimated beam of monochromatic light illuminating the two flats and a beam splitter allowing the fringes to be viewed on-axis.

The Mach–Zehnder interferometer is a more versatile instrument than the Michelson interferometer. Each of the well separated light paths is traversed only once, and the fringes can be adjusted so that they are localized in any desired plane. Typically, the fringes would be adjusted to lie in the same plane as the test object, so that fringes and test object can be photographed together. If it is decided to produce fringes in white light, then, since white light has a limited coherence length, on the order of micrometers, great care must be taken to equalize the optical paths or no fringes will be visible. As illustrated in Fig. 6, a compensating cell would be placed in the path of the reference beam to match the test cell. Note also the precise orientation of the beam splitters. The reflecting surfaces of the beam splitters would be oriented so that the test and reference beams pass through an equal amount of glass. In this orientation, the test and reference beams each experience two front-surface reflections, resulting in the same number of phase inversions. The result is that light traveling an equal optical path length in the test and reference beams produces a white light fringe of constructive interference.

The heart of the Fabry–Pérot interferometer is a pair of partially silvered glass optical flats spaced several millimeters to centimeters apart with the silvered surfaces facing each other. (Alternatively, a Fabry–Pérot ''etalon'' uses a transparent plate with two parallel reflecting surfaces.) As with the Fizeau interferometer, the flats are slightly beveled. In a typical system, illumination is provided by a diffuse source set at the focal plane of a collimating lens. A focusing lens produces what would be an inverted image of the source if the paired flats were not present, i.e., in the absence of the paired flats, all light emitted from point A passing through the optical system would be focused at point A'. In Fig. 6, only one ray emitted from point A on the source is traced. As the ray passes through the paired flats, it is multiply reflected to produce multiple transmitted rays which are collected by the focusing lens and brought to point A' on the screen. The complete interference pattern takes the appearance of a set of concentric rings. The sharpness of the rings depends on the reflectivity of the flats. If the reflectivity is high, resulting in a high Q factor (i.e., high finesse), monochromatic light produces a set of narrow bright rings against a dark background. In Fig. 6, the low-finesse image corresponds to a reflectivity of 0.04 (i.e., unsilvered surfaces) ''versus'' a reflectivity of 0.95 for the high-finesse image.

Fig. 6 illustrates the Fizeau, Mach–Zehnder, and Fabry–Pérot interferometers. Other examples of amplitude splitting interferometer include the Michelson, Twyman–Green, Laser Unequal Path, and Linnik interferometer.

Michelson-Morley

Michelson and Morley (1887) and other early experimentalists using interferometric techniques in an attempt to measure the properties of the luminiferous aether, used monochromatic light only for initially setting up their equipment, always switching to white light for the actual measurements. The reason is that measurements were recorded visually. Monochromatic light would result in a uniform fringe pattern. Lacking modern means of environmental temperature control, experimentalists struggled with continual fringe drift even though the interferometer might be set up in a basement. Since the fringes would occasionally disappear due to vibrations by passing horse traffic, distant thunderstorms and the like, it would be easy for an observer to "get lost" when the fringes returned to visibility. The advantages of white light, which produced a distinctive colored fringe pattern, far outweighed the difficulties of aligning the apparatus due to its low coherence length