The design and use of optical interferometers. Optical interferometers based on both two-beam interference and multiple-beam interference of light are extremely powerful tools for metrology and spectroscopy. A wide variety of measurements can be performed, ranging from determining the shape of a surface to an accuracy of less than a millionth of an inch (25 nanometers) to determining the separation, by millions of miles, of binary stars. In spectroscopy, interferometry can be used to determine the hyperfine structure of spectrum lines. By using lasers in classical interferometers as well as holographic interferometers and speckle interferometers, it is possible to perform deformation, vibration, and contour measurements of diffuse objects that could not previously be performed.
Basic classes of interferometers
There are two basic classes of interferometers: division of wavefront and division of amplitude. Figure 1 shows two arrangements for obtaining division of wavefront. For the Young's double-pinhole interferometer (Fig. 1a), the light from a point source illuminates two pinholes. The light diffracted by these pinholes gives the interference of two point sources. For the Lloyd's mirror experiment (Fig. 1b), a mirror is used to provide a second image S2 of the point source S1, and in the region of overlap of the two beams the interference of two spherical beams can be observed. There are many other ways of obtaining division of wavefront; however, in each case the light leaving the source is spatially split, and then, by use of diffraction, mirrors, prisms, or lenses, the two spatially separated beams are superimposed.
Figure 2 shows one technique for obtaining division of amplitude. For division-of-amplitude interferometers, a beam splitter of some type is used to pick off a portion of the amplitude of the radiation, which is then combined with a second portion of the amplitude. The visibility of the resulting interference fringes is a maximum when the amplitudes of the two interfering beams are equal. See also: Interference of waves
The Michelson interferometer (Fig. 3) is based on division of amplitude. Light from an extended source S is incident on a partially reflecting plate (beam splitter) P1. The light transmitted through P1 reflects off mirror M1 back to plate P1. The light that is reflected proceeds to M2, which reflects it back to P1. At P1, the two waves are again partially reflected and partially transmitted, and a portion of each wave proceeds to the receiver R, which may be a screen, a photocell, or a human eye. Depending on the difference between the distances from the beam splitter to the mirrors M1 and M2, the two beams will interfere constructively or destructively. Plate P2 compensates for the thickness of P1. Often when a quasi-monochromatic light source is used with the interferometer, compensating plate P2 is omitted.
The function of the beam splitter is to superimpose (image) one mirror onto the other. When the mirrors' images are completely parallel, the interference fringes appear circular. If the mirrors are slightly inclined about a vertical axis, vertical fringes are formed across the field of view. These fringes can be formed in white light if the path difference in part of the field of view is made zero. Just as in other interference experiments, only a few fringes will appear in white light, because the difference in path will be different for wavelengths of different colors. Accordingly, the fringes will appear colored close to zero path difference and will disappear at larger path differences, where the fringe maxima and minima for the different wavelengths overlap. If light reflected off the beam splitter experiences a one-half-cycle relative phase shift, the fringe of zero path difference is black and can be easily distinguished from the neighboring fringes. This makes use of the instrument relatively easy.
The Michelson interferometer can be used as a spectroscope. Consider first the case of two close spectrum lines as a light source for the instrument. As the mirror M1 is shifted, fringes from each spectral line will cross the field. At certain path differences between M1 and M2, the fringes for the two spectral lines will be out of phase and will essentially disappear; at other points they will be in phase and will be reinforced. By measuring the distance between successive maxima in fringe contrast, it is possible to determine the wavelength difference between the lines.
This is a simple illustration of a very broad use for any two-beam interferometer. As the path length L is changed, the variation in intensity I(L) of the light coming from an interferometer gives information on the basis of which the spectrum of the input light can be derived. The equation for the intensity of the emergent energy can
be written as Eq. (1), where β is a constant and I(λ) is the intensity of the incident light at different wavelengths λ. This equation applies when the mirror M1 is moved linearly with time from the position where the path difference with M2 is zero to a position that depends on the longest wavelength in the spectrum to be examined. From Eq. (1), it is possible mathematically to recover the spectrum I(λ). In certain situations, such as in the infrared beyond the wavelength region of 1.5 μm, this technique offers a large advantage over conventional spectroscopy in that its utilization of light is extremely efficient. See also: Infrared spectroscopy
If the Michelson interferometer is used with a point source instead of an extended source, it is called a Twyman-Green interferometer. The use of the laser as the light source for the Twyman-Green interferometer has made it an extremely useful instrument for testing optical components. The great advantage of a laser source is that it makes it possible to obtain bright, good-contrast interference fringes, even if the path lengths for the two arms of the interferometer are quite different. See also: Laser
Figure 4 shows a Twyman-Green interferometer for testing a flat mirror. The laser beam is expanded to match the size of the sample being tested. Part of the laser light is transmitted to the reference surface, and part is reflected by the beam splitter to the flat surface being tested. Both beams are reflected back to the beam splitter, where they are combined to form interference fringes. An imaging lens projects the surface under test onto the observation plane.
Fringes (Fig. 5) show defects in the surface being tested. If the surface is perfectly flat, then straight, equally spaced fringes are obtained. Departure from the straight, equally spaced condition shows directly how the surface differs from being perfectly flat. For a given fringe, the difference in optical path between light going from the laser to the reference surface to the observation plane and light going from the laser to the test surface to the observation plane is a constant. (The optical path is equal to the product of the geometric path times the refractive index.) Between adjacent fringes (Fig. 5), the optical path difference changes by one wavelength, which for a helium–neon laser corresponds to 633 nm. The number of straight, equally spaced fringes and their orientation depend on the tip–tilt of the reference mirror. That is, by tipping or tilting the reference mirror, the difference in optical path can be made to vary linearly with distance across the laser beam.
Deviations from flatness of the test mirror also cause optical path variations. A height change of half a wavelength will cause an optical path change of one wavelength and a deviation from fringe straightness of one fringe. Thus, the fringes give surface height information, just as a topographic map gives height or contour information.
The existence of the essentially straight fringes provides a means of measuring surface contours relative to a tilted plane. This tilt is generally introduced to indicate the sign of the surface error, that is, whether the errors correspond to a hill or a valley. One way to get this sign information is to push in on the piece being tested when it is in the interferometer. If the fringes move toward the right when the test piece is pushed toward the beam splitter, then fringe deviations from straightness toward the right correspond to high points (hills) on the test surface and deviations to the left correspond to low points (valleys).
The basic Twyman-Green interferometer (Fig. 4) can be modified (Fig. 6) to test concave-spherical mirrors. In the interferometer, the center of curvature of the surface under test is placed at the focus of a high-quality diverger lens so that the wavefront is reflected back onto itself. After this retroflected wavefront passes through the diverger lens, it will be essentially a plane wave, which, when it interferes with the plane reference wave, will give interference fringes similar to those shown in Fig. 5 for testing flat surfaces. In this case it indicates how the concave-spherical mirror differs from the desired shape. Likewise, a convex-spherical mirror can be tested. Also, if a high-quality spherical mirror is used, the high-quality diverger lens can be replaced with the lens to be tested.
One of the most commonly used interferometers in optical metrology is the Fizeau interferometer, which can be thought of as a folded Twyman-Green interferometer. In the Fizeau interferometer, the two surfaces being compared, which can be flat, spherical, or aspherical, are placed in close contact. The light reflected off these two surfaces produces interference fringes. For each fringe, the separation between the two surfaces is a constant. If the two surfaces match, straight, equally spaced fringes result. Surface height variations between the two surfaces cause the fringes to deviate from straightness or equal separations, where one fringe deviation from straightness corresponds to a variation in separation between the two surfaces by an amount equal to one-half of the wavelength of the light source used in the interferometer. The wavelength of a helium source, which is often used in a Fizeau interferometer, is 587.56 nm; hence one fringe corresponds to a height variation of approximately 0.3 μm.
The Mach-Zehnder interferometer (Fig. 7) is a variation of the Michelson interferometer and, like the Michelson interferometer, depends on amplitude splitting of the wavefront. Light enters the instrument and is reflected and transmitted by the semitransparent mirror M1. The reflected portion proceeds to M3, where it is reflected through the cell C2 to the semitransparent mirror M4. Here it combines with the light transmitted by M1 to produce interference. The light transmitted by M1 passes through a cell C1, which is similar to C2 and is used to compensate for the windows of C1.
The major application of this instrument is in studying airflow around models of aircraft, missiles, or projectiles. The object and associated airstream are placed in one arm of the interferometer. Because the air pressure varies as it flows over the model, the index of refraction varies, and thus the effective path length of the light in this beam is a function of position. When the variation is an odd number of half-waves, the light will interfere destructively and a dark fringe will appear in the field of view. From a photograph of the fringes, the flow pattern can be derived mathematically.
A major difference between the Mach-Zehnder and the Michelson interferometers is that in the Mach-Zehnder the light goes through each path in the instrument only once, whereas in the Michelson the light traverses each path twice. This double traversal makes the Michelson interferometer extremely difficult to use in applications where spatial location of index variations is desired. The incoming and outgoing beams tend to travel over slightly different paths, and this lowers the resolution because of the index gradient across the field.
In a lateral-shear interferometer, an example of which is shown in Fig. 8, a wavefront is interfered with a shifted version of itself. A bright fringe is obtained at the points where the slope of the wavefront times the shift between the two wavefronts is equal to an integer number of wavelengths. That is, for a given fringe, the slope or derivative of the wavefront is a constant. For this reason, a lateral-shear interferometer is often called a differential interferometer.
Another type of shearing interferometer is a radial-shear interferometer. Here, a wavefront is interfered with an expanded version of itself. This interferometer is sensitive to radial slopes.
The advantages of shearing interferometers are that they are relatively simple and inexpensive, and because the reference wavefront is self-generated, an external wavefront is not needed. Because an external reference beam is not required, the source requirements are reduced from those of an interferometer such as a Twyman-Green. For this reason, shearing interferometers, in particular lateral-shear interferometers, are finding much use in applications such as adaptive optics systems for correction of atmospheric turbulence, where the light source has to be a star, or planet, or perhaps just reflected sunlight. See also: Adaptive optics
Michelson stellar interferometer
A Michelson stellar interferometer can be used to measure the diameter of stars that are as small as 0.01″ of arc. This task is impossible with a ground-based optical telescope because the atmosphere limits the resolution of the largest telescope to not much better than 1″ of arc.
The Michelson stellar interferometer is a simple adaptation of Young's two-slit experiment. In its first form, two slits are placed over the aperture of a telescope. If the object being observed is a true point source, the image will be crossed with a set of interference bands. A second point source separated by a small angle from the first will produce a second set of fringes. At certain values of this angle, the bright fringes in one set will coincide with the dark fringes in the second set. The smallest angle α at which the coincidence occurs will be that angle subtended at the slits by the separation of the peak of the central bright fringe from the nearest dark fringe. This angle is given by Eq. (2),
where d is the separation of the slits, λ is the dominant wavelength of the two sources, and α is their angular separation. The measurement of the separation of the sources is performed by adjusting the separation d between the slits until the fringes vanish.
Consider now a single source in the shape of a slit of finite width. If the slit subtends an angle at the telescope aperture that is larger than α, the interference fringes will be reduced in contrast. For various line elements at one side of the slit, there will be elements of angle α that will cancel the fringes from the first element. By induction, it is clear that for a separation d′ such that the slit source subtends an angle as given by Eq. (3),
the fringes from a single slit will vanish completely. For additional information on the Michelson stellar interferometer, See also: Diffraction .
All the interferometers discussed above are two-beam interferometers. The Fabry-Perot interferometer (Fig. 9) is a multiple-beam interferometer in that the two glass plates are partially silvered on the inner surfaces and the incoming wave is multiply reflected between the two surfaces. The position of the fringe maxima is the same for multiple-beam interference as two-beam interference; however, as the reflectivity of the two surfaces increases and the number of interfering beams increases, the fringes become sharper.
A quantity of particular interest in a Fabry-Perot interferometer is the ratio of the separation of adjacent maxima to the half-width of the fringes. It can be shown that this ratio, known as the finesse, is given by Eq. (3), where R is the reflectivity of the silvered surfaces.
The multiple-beam Fabry-Perot interferometer is of considerable importance in modern optics for spectroscopy. All the light rays incident on the Fabry-Perot at a given angle will result in a single circular fringe of uniform irradiance. With a broad diffuse source, the interference fringes will be narrow concentric rings, corresponding to the multiple-beam transmission pattern. The position of the fringes depends on the wavelength. That is, each wavelength gives a separate fringe pattern. The minimum resolvable wavelength difference is determined by the ability to resolve close fringes. The ratio of the wavelength λ to the least resolvable wavelength difference Δλ is known as the chromatic resolving power R. At nearly normal incidence it is given by Eq. (5),
where n is the refractive index between the two mirrors separated by a distance d. For a wavelength of 500 nm, nd = 10 mm, and R = 90%, the resolving power is well over 106. See also: Reflection of electromagnetic radiation; Resolving power (optics)
When Fabry-Perot interferometers are used with lasers, they are generally used in the central spot scanning mode. The interferometer is illuminated with a collimated laser beam, and all the light transmitted through the Fabry-Perot is focused onto a detector, whose output is displayed on an oscilloscope. Often one of the mirrors is on a piezoelectric mirror mount. As the voltage to the piezoelectric crystal is varied, the mirror separation is varied. The light output as a function of mirror separation gives the spectral frequency content of the laser source.
A wave recorded in a hologram is effectively stored for future reconstruction and use. Holographic interferometry is concerned with the formation and interpretation of the fringe pattern that appears when a wave, generated at some earlier time and stored in a hologram, is later reconstructed and caused to interfere with a comparison wave. It is the storage or time-delay aspect that gives the holographic method a unique advantage over conventional optical interferometry.
A hologram can be made of an arbitrarily shaped, rough scattering surface, and after suitable processing, if the hologram is illuminated with the same reference wavefront used in recording the hologram, the hologram will produce the original object wavefront. If the hologram is placed back into its original position, a person looking through the hologram will see both the original object and the image of the object stored in the hologram. If the object is now slightly deformed, interference fringes will be produced that tell how much the surface is deformed. Between adjacent fringes the optical path between the source and viewer has changed by one wavelength. Although the actual shape of the object is not determined, the change in the shape of the object is measured to within a small fraction of a wavelength, even though the object's surface is rough compared to the wavelength of light.
Double-exposure holographic interferometry (Fig. 10) is similar to real-time holographic interferometry described above, except now two exposures are made before processing: one exposure with the object in the undeformed state and a second exposure after deformation. When the hologram reconstruction is viewed, interference fringes will be seen that show how much the object was deformed between exposures.
The advantage of the double-exposure technique over the real-time technique is that there is no critical replacement of the hologram after processing. The disadvantage is that continuous comparison of surface displacement relative to an initial state cannot be made, but rather only the difference between two states is determined.
In time-average holographic interferometry (Fig. 11), a time-averaged hologram of a vibrating surface is recorded. If the maximum amplitude of the vibration is limited to some tens of light wavelengths, illumination of the hologram yields an image of the surface on which is superimposed several interference fringes that are contour lines of equal displacement of the surface. Time-average holography enables the vibrational amplitudes of diffusely reflecting surfaces to be measured with interferometric precision. See also: Holography
A random intensity distribution, called a speckle pattern, is generated when light from a highly coherent source, such as a laser, is scattered by a rough surface. The use of speckle patterns in the study of object displacements, vibration, and distortion is becoming of more importance in the nondestructive testing of mechanical components. For example, time-averaged speckle photographs can be used to analyze the vibrations of an object in its plane. In physical terms, the speckles in the image are drawn out into a line as the surface vibrates, instead of being double as in the double-exposure technique. The diffraction pattern of this smeared-out speckle-pattern recording is related to the relative time spent by the speckle at each point of its trajectory (Fig. 12). See also: Nondestructive evaluation
Speckle interferometry can be used to perform astronomical measurements similar to those performed by the Michelson stellar interferometer. Stellar speckle interferometry is a technique for obtaining diffraction-limited resolution of stellar objects despite the presence of the turbulent atmosphere that limits the resolution of ground-based telescopes to approximately 1″ of arc. For example, the diffraction limit of the 200-in.-diameter (5-m) Palomar Mountain telescope is approximately 0.02″ of arc, 1/50 the resolution limit set by the atmosphere.
The first step of the process is to take a large number, perhaps 100, of short exposures of the object, where each photo is taken for a different realization of the atmosphere. Next, the optical diffraction pattern, that is, the squared modulus of the Fourier transform of all the short-exposure photographs, is added. By taking a further Fourier transform of each ensemble-averaged diffraction pattern, the ensemble average of the spatial autocorrelation of the diffraction-limited images of each object is obtained. See also: Speckle
Electronic phase-measurement techniques can be used in interferometers such as the Twyman-Green, where the phase distribution across the interferogram is being measured. Phase-shifting interferometry is often used for these measurements because it provides for rapid, precise measurement of the phase distribution. In phase-shifting interferometry, the phase of the reference beam in the interferometer is made to vary in a known manner. This can be achieved, for example, by mounting the reference mirror on a piezoelectric transducer. By varying the voltage on the transducer, the reference mirror is moved a known amount to change the phase of the reference beam a known amount. A solid-state detector array is used to detect the intensity distribution across the interference pattern. This intensity distribution is read into computer memory three or more times, and between each intensity measurement the phase of the reference beam is changed by a known amount. From these three or more intensity measurements, the phase across the interference pattern can be determined to within a fraction of a degree.