  ## Two-Beam Interferometry ### Principles and Applications of Two-Beam Interferometry

Among the various forms of interferometry, two-beam interferometry is especially simple and straightforward in principle, as well as practice, and is therefore utilized for a broad range of applications. This technique will be described in detail below, with principal reference to applications devised for the topographic measurements of materials surfaces.

The interference phenomenon first encountered in high school physics is that of Newton's rings, which constitute the localized concentric interference fringes observed in the neighborhood of the point of contact when a plano-convex lens with a large radius of curvature is placed against a flat glass plate. These interference rings were first studied by Isaac Newton. Interference phenomena cannot be explained by merely regarding light as composed of rays that propagate along straight lines, as is assumed in geometrical optics. Subsequent to Newton, English physicist Thomas Young, proceeding from the viewpoint of wave optics, explained Newton's rings as a light interference phenomenon. That is, light is a wave motion possessing crests and valleys; if crests are superimposed upon crests and valleys upon valleys, then the waves will be mutually reinforced, whereas if crests are superimposed upon valleys, then the two will mutually cancel. Consequently, a succession of alternately bright and dark interference fringes will arise.

The distance corresponding to one period of the light wave, for example, between one crest and the next one, is called one wavelength, and is conventionally denoted by λ. Newton's rings successively appear as the air gap between the glass plate and the lens surface increases by half a wavelength. Therefore, if incident light of different wavelengths is utilized, corresponding to red and green, for example, then the interval between successive rings will be wider for red light than for green.

Figure 1(a) illustrates the Newton's rings formed in an apparatus that allows a convex lens to be clamped, under the tension of screws, into contact with a flat glass plate. The order of the interference fringes, counted from the dark disk at the center of the pattern, is denoted by N(where N = 0, 1, 2, ...). When dealing with optical interference, it is important to note that the light waves are superimposed in the medium through which the light propagates (considering interference in terms of optical distance). If the gap between the lens and the glass plate is in a vacuum or in air, then the refractive index (n) of the medium is 1. On the other hand, if this gap is filled with water, then n = 1.333, and as a consequence, when traversing this medium, the velocity of the light is diminished. Thus, the wavelength is effectively shortened, and the distances between successive rings are reduced. In a converse manner, this relationship can be utilized for the purpose of measuring the refractive index of the medium.

If the angle of incidence of the light waves is denoted by Φ and thickness, or distance between the glass plate and the lens, by t, then interference fringes will appear when the following condition is satisfied:

In the case illustrated by Figure 1(a), the angle of incidence equals zero, cosΦ = 1 (as would always be the case for perpendicular incidence), and therefore the above relation becomes:

which rearranges to

Thus, in a vacuum or in air, since n = 1, the relation reduces to:

where N = 1, 2, 3, .... (any positive integer). Thus, each time the thickness of the gap changes by λ/2 (by half a wavelength), an interference fringe appears (Figure 1(b)).

Referring again to Figure 1(a), the interference fringes appear like contour lines on the top surface of the apparatus at intervals corresponding to height differences of half a wavelength. This is equivalent to the contour lines formed by Newton's interference fringes at intervals of half a wavelength on the surface of an object. In the case of a map, sea level constitutes the reference plane, while in the case of the interference fringes the glass plate serves as the reference. In optical interferometry, this reference plane is known as the reference plate (or reference mirror). Alternatively, this type of interference pattern is known as "fringes of equal thickness".

Figure 1(c) illustrates the relationship between the spacing of the interference fringes and the angle of the air wedge formed by a reference plate (reference mirror) and the surface of the specimen. In this case, from the above discussion, the interference fringes are obviously not circular like Newton's rings, but form straight lines. As the angle increases, the spacing between the interference fringes diminishes.

The red trace in Figure 1(d) plots the intensity distribution of the interference fringes of the Newton's rings in Figure 1(a), measured with a microphotometer. In this example, the distribution displays broad contour lines. In a topographic map, the finer the contour line interval, the more delicate the details that can be represented. Similarly, the breadth of the interference fringes governs the precision obtainable in the interferometric measurement of the irregularities of a surface. The illustrated method has the disadvantage that the specimen must be placed in contact with the reference plate (reference mirror).

### Applications of Contact Method

Figure 2(a) is a photomicrograph of a natural diamond with edges of 0.8 millimeter in length, examined by the Newton's rings method, which was illustrated above for a lens in contact with a glass plate. The specimen was illuminated with light having a wavelength (λ) of 546 nanometers (green), obtained by means of a multilayer filter. Using this simple method, half-wavelength (273-nanometer) contour lines can be visualized on the specimen. Since the horizontal distances are known immediately from the magnification or the scale, the angle of inclination of the surface can be calculated. Furthermore, it is possible to obtain the cross-sections that will appear when the diamond is cut in various directions. Since light of accurately known wavelength is used, height can be measured with extremely high precision. The protuberance of this diamond surface is 12 λ high, or approximately 6.5 micrometers, and the inclination of the surface is measured to be approximately 1 degree and 16 minutes.

Figure 2(b) shows the microstructure of the surface, ignoring quantitative measurement. This sort of image can be obtained if the angle of the wedge indicated in Figure 1(c) is close to zero. Such a small wedge angle has the effect of broadening the interference fringe of the zeroth-order, which then covers the entire field of view. If the wedge angle is reduced to zero, the surfaces of the crystal and the reference plate become parallel. This method of broadening a single interference to attain high contrast is known as the high dispersion method, and permits the visualization of height differences as small as 2 nanometers.

The high dispersion method utilizes the fact that a single interference fringe actually possesses a continuous intensity distribution in terms of the air gap, as shown in Figure 2(c). Corresponding to an infinitesimal height difference dt, the intensity within a single interference fringe changes by dI, as depicted in Figure 2(c). By utilizing the portion of the curve with a relatively large value of dI/dt, one can observe detailed surface structure with high sensitivity. Moreover, with this method, halos of the kind seen in phase contrast microscopy do not occur, and the entire surfaces can be visualized with high contrast from any direction. The disadvantage of this method is extreme sensitivity to mechanical vibration and consequent difficulty in maintaining the same contrast over an extended period of time.

Etch pits on a (111) surface of a diamond are shown in Figure 3(a). After etching by an oxidant, a high dispersion fringe was visualized by two-beam interference. The image clearly shows two types of etch pits - flat bottomed and point bottomed. The flat-bottomed pits are extremely shallow, with depths of about one-fourth of a wavelength (137 nanometers) or less. In contrast, calculating from the number of interference fringes, the deepest point-bottomed pits are about three wavelengths (1.64 micrometers) deep. Additionally, the interference fringes on the rim of the crystal (in the upper right-hand corner of the image) show that the edge of the crystal has been rounded by the action of dissolution. Moreover, since the length of the sides of the pits can be measured, the gradient of the pits can be calculated. The inclination of the deepest pit, in the direction from a vertex to the center, is approximately 8 degrees and 50 minutes. Thus, the slopes of the pits, as determined quantitatively, are in fact considerably gentler than would be inferred from casual inspection. Figure 3(a) illustrates an example in which both observation and measurement can be accomplished with a single photomicrograph.

The previous examples have illustrated surface microstructure resulting from crystal growth and dissolution visualized by two-beam interferometry. The method is also applicable in the study of fracture surfaces produced by physical destruction. Figure 3(b) is a two-beam interferometric photomicrograph of the cleavage plane produced in a diamond by a giant pulse laser, with the two sides of the plane matched for comparison. The fact that the two sides are not identical indicates that some fragments, albeit minute, were lost when the crystal was cleaved. A triangular depression appears at the center of the left-hand margin of the photograph; this is the point where the laser beam was focused by the lens. The traces on the surface show that the light instantaneously traversed the diamond and was partially reflected at the opposite wall; the depression in the center of the photograph was produced when this reflection occurred. Furthermore, it can be determined that the fracture was initiated within the crystal at a point about 100 micrometers from the reflecting surface.

So-called fractography is the practice of examining fracture surfaces by light or scanning electron microscopy and analyzing the fracture mechanism on the basis of the observed patterns. In the illustrated analysis, two-beam interferometry can be applied to obtain quantitative information concerning the topographic irregularities of the fracture surface and the stream-like configuration known as the river pattern.

### Michelson Interferometer

All of the interference patterns described above are created by methods in which the surface of the specimen is placed nearly in contact with the reference disk. However, in the case of specimens such as semiconductors, which are extremely sensitive to contamination, or soft and easily deformed specimens, it is desirable to form an interference pattern without contact. One such method relies upon use of a Michelson type interferometer, which has several variants.

The principle of the Michelson interferometer, as illustrated in Figure 4(a), is quite simple. The essential elements of the design are as follows. A beam emitted by the light source is split into two beams of nearly equal intensity by a half mirror (beam splitter), one of these beams being directed onto a flat reference mirror and the other onto the specimen surface. The light produced by reflection of these two beams is then made to interfere. When observed from the viewing port, interference occurs between the image of the reference mirror and the image of the specimen surface. Since the light waves reflected by the specimen and the reference mirror originate from the splitting of a beam emitted by the same light source, these waves are mutually coherent, and consequently a two-beam interference pattern is obtained. The object inserted into the optical path between the beam splitter and the reference mirror, is a glass plate of the same composition and thickness as the beam splitter. Because of the presence of this plate, the two divided light beams arrive at the viewing port after propagating through the equivalent optical distance (the product of refractive index and thickness of the optical components). Note that in this type of interferometer, the beam splitter and specimen surface are separated by an appreciable distance; thus, an interference pattern is obtained without contact.

### Non-Contact Interferometric Instruments

A two-beam interferometer functions by dividing originally coherent light into two beams of equal intensity, directing one beam onto the reference mirror and the other onto the specimen, and measuring the optical path difference (the difference in optical distances) between the resulting two reflected light waves. In order to implement this method, various instrument types have been devised, employing several devices to split the light wave and to provide the appropriate optical paths.

The Watson interference objective, manufactured by the Watson Company (Great Britain), is a compact variant of the Michelson interferometer, designed to be installed on a microscope. The construction of the instrument is shown in Figure 4(b). In comparing this illustration with the schematic representation of the Michelson principle (Figure 4(a)), it can be seen that an objective lens is interposed close to the beam splitter, permitting the measurement of minute specimens.

In this interferometer, the inclination of the reference mirror with respect to the optical axis is equivalent to using a reference plate that is not parallel to the surface of the specimen, thus creating an air wedge, as illustrated in Figure 1(c). The greater the inclination of the reference mirror relative to the specimen surface, the narrower the spacing between the interference fringes. The direction of the individual interference fringes is the same as the direction of the line of intersection of the planes of the specimen surface and the reference mirror image. Shifting the reference mirror in the left or right direction has the effect of varying the distance between the specimen surface and the image of the mirror. If these two surfaces are parallel, then a high-dispersion pattern is formed, and if the reference mirror image plane is shifted away from the specimen plane by a distance of half a wavelength, the respective orders of the interference fringes are changed by one. A shift through one-fourth of a wavelength inverts the pattern by transforming bright fringes to dark fringes, and vice versa.

Another instrument variation is provided by the Linnik interferometer, which utilizes a high magnification objective lens in the application of the interference technique to observation of minute details. The principle employed is that of the Michelson interferometer. Figure 4(c) presents the basic arrangement, comprising a light source, a collimator, a beam-splitting prism, an eyepiece, uniform objective lenses with completely identical optical distances, a specimen surface, which gives rise to an image, and a reference mirror, which gives rise to a reflection image. Because uniform objective lenses are difficult to manufacture, only a small number of such instruments have been marketed.

The Mirau interference objective is an interference objective of comparatively high magnification (10x, 20x or 40x) used in instruments produced by the Nikon Corporation. The principle of the device, as illustrated in Figure 5(a), relies on placing a reflection reference mirror in the center of the objective lens, and interposing a half mirror between the objective lens and the specimen. These components are so arranged that an interference pattern will appear if the system is focused upon the specimen. If the specimen is inclined, localized interference fringes will appear as previously explained in the description of two-beam interferometry. Characteristic of the design category, non-contact measurements can be performed with this instrument.

Figure 5(b) illustrates the design of the Nikon low magnification interference objective, a new type of objective (for TI instruments and having magnifications of 2.5x and 5x) recently developed by the company. The use of a low magnification objective has two advantages. First, the working distance is comparatively long (11.1 millimeters for the 2.5x objective and 9 millimeters for the 5x objective). Secondly, a wide field of view can be observed with the benefit of a large depth of field. The outstanding feature of this interference objective is the fact that the center of the reference mirror lies on the optical axis of the objective lens. In order to realize this feature, the reference mirror is located on the plane surface of a hemisphere. The angle of the mirror is adjusted by means of two screws behind the hemisphere. An immediate change to conventional bright field observation can be performed by inserting a light-blocking shield in front of the mirror.

In use, because the center of the reference mirror coincides with the optical axis, the interference pattern immediately appears at the center of the field of view. When using the previously mentioned Watson interferometer or Mirau objective, displaying the interference pattern at the center of the field of view often requires troublesome manipulation, and this tends to discourage more widespread use of interferometry. Additionally, since the present lens is suitable for a field-of-view number up to 20, and the maximum diameter of the actual field is equal to the field-of-view number divided by the magnification, specimens of diameters up to about 8 millimeters (for magnification 2.5x), or 4 millimeters (for magnification 5x) can be interferometrically measured with a single observation.

If graduations are inscribed on the reference mirror, then the scale can be conveniently recorded on the photomicrograph or digital image. In the Watson interferometer, special illumination is required for the interference objective, but this is not necessary for the Nikon TI or Mirau instruments; the same light path is used for bright field observation and interferometric measurement.

### Points Requiring Attention in Applications of Interferometry

In the preceding discussion, several examples of practical two-beam interferometry have been described referring to a conventional classification into contact and non-contact methods. However, there is actually no essential difference in the nature of the interference fringes between these two methods, and obviously both low and high spread patterns are obtainable by either method. The non-contact method clearly possesses a wide range of applicability. In the contact method, the order of the interference fringes increases from the zeroth, first and second, etc., in a unidirectional manner, since the reference surface and the specimen are separated by a wedge-like air gap. On the other hand, in the non-contact method, the intersections between the images of the reference mirror and the specimen form the fringes of zeroth-order, on both sides of which, first order, second order, and higher order fringes appear. This sort of pattern is shown in Figure 6(b), which was deliberately taken without a filter, using white light, and therefore the order of the fringes are readily distinguished. A zeroth-order dark fringe is visible in the center of the field, flanked on both sides by higher order fringes. The orders of the fringes are easily recognized if one observes the red fringes (a monochrome version of the original color image is presented here). Up to six orders of red fringes appear on the right side and up to five orders of red fringes on the left side of the zeroth-order fringe.

If this pattern were photographed with monochromatic light, the interference pattern would be displayed with almost the same contrast, but the continuity of the fringes would become obscure at the sites of abrupt changes in level. Consequently, in any type of interferometry, the continuity of the fringes should first be ascertained with white light in order to avoid substantial errors. Identification of the orders of interference fringes under monochromatic light is particularly difficult when using high grade interference filters with narrow half-band width.

Precision of measurement is improved by creating finer interference fringes. Formerly, light sources such as low-pressure sodium lamps were used for this purpose. Currently, however, because high-grade interference filters are available, halogen lamps are often used in combination with these improved filters. In such cases, two points should be noted. First, unlike the use of sodium light, the maximum transmitted wavelength usually varies to some extent according to the particular interference filter being used. Therefore, it is most important to precisely ascertain the characteristic wavelength of the filter. Secondly, the filter must be inserted perpendicular to the optical axis, since even a slight obliquity will cause a shift toward shorter wavelengths.

### Measuring Differences of Elevation

The measurement of vertical surface irregularities and topography using contour lines has already been discussed and illustrated in Figures 2 and 3. Here, an example is presented wherein such irregularities are measured using three interference fringes. Figure 7 illustrates differential interference and two-beam interference photomicrographs (Figure 7(a) and 7(b), respectively) of the edge of a razor blade, which permit the determination of the roughness of the mechanically finished surface.

In this case, the relationship between the reference mirror and the specimen is similar in principle to that of Figure 1(c), in which the interference fringes successively appear as the height changes by half a wavelength. Suppose that, corresponding to an abrasion groove of depth t in Figure 7, interference fringes appear as shown in Figure 7(c). If the photomicrograph is enlarged, then both D and L can be measured. Using these values, the depth t can be calculated from the formula:

The spacing of interference fringes shown in Figure 7 is somewhat narrow, which could make measurement difficult. In the study of some specimens, the dispersion of the interference fringes can be altered to enhance the visualization of surface features. Figures 8(a) through 8(d) show the spiral growth of magnetoplumbite (a hexagonal ferrite), with the angles of the hexagonal spiral displaying a zigzag form. The dispersion in Figure 8(b) is small, and the shifts in the interference fringes are not evident except at the thickest steps, but nevertheless, the orders of the fringes can be enumerated.

If the dispersion is increased to the extent of Figure 8(c), then the characteristics of the irregularities at the thin as well as the thick steps become visible. The fringes at the center of the image are particularly interesting. Because the pattern is due to spiral growth, the fringes should uniformly shift in the same direction, but here, the individual fringes are not only rectilinear but display a minute transverse oscillatory pattern. This can be interpreted as indicating a cross section of the form illustrated in Figure 8(e). The levels indicated by the broken lines represent the spiral steps formed immediately after the growth of the crystal, the successive level differences being of the order of 1.2 nanometers. This is due to the crystal undergoing quasi-two dimensional dissolution, resulting in the present cross-section represented by the solid gray region in Figure 8(e), analogous to the ridges now remaining in the Colorado plateau after a period of erosion. In the present case, the ridges are of the order of 60 nanometers in height. When the dispersion is increased still further, as in Figure 8(d), the image becomes more suitable for qualitative observation than for measurement. Steps are well visualized, although only at the sites where the interference fringes have expanded. On the other hand, when this specimen is viewed by bright field microscopy, only the thickest steps are visualized, as illustrated in the phase contrast photomicrograph presented in Figure 8(a).

A point to be noted when measuring such steps is that the interference fringes should be adjusted so that they run perpendicular to the steps that are to be measured, which facilitates the subsequent procedure. Additionally, if color photographs cannot be taken, the specimen should first be observed with white light, and the continuity of the interference fringes confirmed, in order to facilitate subsequent measurement.

When using a high dispersion, a single interference fringe becomes relatively broad and therefore steps can be measured by comparing the widths of the fringes themselves. In such cases, the widths of the light and dark fringes become nearly equal, as shown, for example, in Figure 8(c). In other words, a single broad interference fringe in itself represents a level difference of λ/4. The shift in a single interference fringe is diagrammed in Figure 8(f), in which the level difference is located at the center.

### Absolute Level Differences and Limits of Measurement

In both Figures 2(a) and 3(a), triangular contour lines appeared, but were interpreted from the beginning as representing elevations in one case and depressions in the other. However, this distinction is by no means self-evident, and was decided only on the basis of the assumption that crystal growth results in elevations while etching results in depressions. Thus, the microscopist is faced with the problem of how to perceive the absolute distinction between depression and elevation. If the level differences are relatively large, then these can be distinguished by adjusting the focus of the objective, but if the level differences are of the order of the wavelength of the light used in the observation, then this cannot be done. One possible method, however, is the use of a phase-contrast microscope. By interchanging positive and negative phase contrast, one can reverse the contrast of the halo appearing at the high side of a step and thus definitely distinguish between high and low. However, this can, in fact, be accomplished by two-beam interferometry even without a phase contrast microscope. Using white light to examine the surface of a convex object, the interference fringes are arrayed in increasing order from the vicinity of the reference plate (reference mirror) toward the periphery of the object. As explained in reference to Figure 6(b), the order of the fringes can be determined from their color arrangement. Conversely, in the case of a concave object, the order of the fringes increases from the periphery toward the center. That is, it suffices to perform the measurements while bearing in mind the sequence of colors in the interference fringes and the size of the air gap.

The limit of measurement of level differences depends upon the extent to which the breadth of the interference fringes (the skirts of the intensity peak) can be defined. If measurement down to 1/10 of a fringe width can be performed, then the limit of measurement should be of the order of λ/10, or roughly 25 nanometers.

### Contributing Authors

Hiroshi Komatsu - Institute for Materials Research, Tohoku University, Sendai, Japan.

Thomas J. Fellers and Michael W. Davidson - National High Magnetic Field Laboratory, 1800 East Paul Dirac Dr., The Florida State University, Tallahassee, Florida, 32310.

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