## Multiple-Beam Interferometry

The technique of multiple-beam interferometry is based upon situating two surfaces of high reflectivity in close proximity and using a lens to converge beams which have undergone multiple-reflection between the surfaces.

An arrangement in which the two opposed surfaces are parallel is utilized in the Fabry-Perot interferometer. If the two planes are not parallel, then interference fringes appear localized in the wedge space. The situation is essentially the same as that encountered in the wedge space in two-beam interferometry.

### Characteristics of Multiple-Beam Interference Patterns

In multiple-beam interferometry, the breadth of the interference fringes becomes extremely narrow. That is, the contour lines on the "map" become narrow and precision of measurement of surface topography improves accordingly. When the optimal conditions are satisfied, the breadth of the fringes in multiple-beam interferometry is on the order of 1/50 of the corresponding breadth in the two-beam method, and hence the precision is improved by a factor of approximately 50. Because, as discussed in the article on two-beam interferometry, the limit of measurement of level differences utilizing that technique is about 25 nanometers, the limit of measurement in multiple-beam interferometry is therefore on the order of 0.5 nanometer (5 Angstroms).

As shown in Figure 1(a), when the incident light enters the wedge space, multiple-reflection between the opposing surfaces occurs. The beams arriving at point x in the figure include the non-reflected beam (1), the twice-reflected beam (2), the fourfold-reflected beam (3), etc. At each reflection, the intensity drops in accordance with the reflectivity of the surface, and therefore the quantity of light of multiple-beams finally collected by the lens is governed by the reflectivity. The greater the number of multiple reflections of the beams contributing to the interference, the finer and sharper the fringes. Figure 1(b) illustrates the relation between the breadth of the interference fringes and the reflectivity for the case in which the reflectivities of the specimen surface and the reference plate (reference mirror) are identical. It can be seen from the figure that the reflectivity determines the breadth of the interference fringes.

A comparison between the intensity distributions of two-beam interference fringes and multiple-beam interference fringes corresponding to 90 percent reflectivity is shown in Figure 1(c). The two-beam interference fringes are of the form (cosθ)E2, and the breadth of the ridges and valleys is nearly the same. On the other hand, as shown in the lower curve of Figure 1(c), the multiple-beam interference fringes display a sharply peaked intensity distribution. As in the case of two-beam interferometry, successive interference fringes appear localized at every level change of half a wavelength. However, because the fringes are narrow, information concerning the regions between the fringes is likely to be unavailable.

### Conditions for High Precision Measurement

Proceeding from the Airy formula for multiple reflection between parallel plane surfaces, Samuel Tolansky performed a detailed analysis of the intensity distribution of the multiple-interference fringes appearing in a wedge space, and from the results of this study deduced the conditions for obtaining the highest measurement precision, as summarized by the following five points:

• Coating the surface of the reference plate with a film of high reflectivity and low absorption.
• Coating the specimen with a uniform film of high reflectivity, faithfully conforming with the original topography of the specimen.
• Making the distance between the two surfaces (t in Figure 1(a)) as small as possible, at most 10 millimeters, and preferably of the order of the wavelength of the light.
• Collimating the incident light beam to a parallelism within 3 degrees.
• Making the incident light as closely perpendicular to the reference plate as possible.

The last two conditions are comparatively easy to achieve provided that the previous one is satisfied. In summary, the surfaces of the reference plate and the specimen should be brought into close proximity, reducing as far as possible the distance t in Figure 1(a), and the surface of the reference mirror should be coated by silver vapor deposition or with a multi-layer film of low absorption.

### Optical Systems for Multiple-Beam Interferometry

The basic function of optical systems for multiple-beam interferometry is to allow observation of the interference pattern appearing in the wedge space formed by two surfaces. Observation of the fringes produced by reflection can be performed with a microscope in two possible ways. Figure 2 illustrates an optical system appropriate for the measurement of film thickness, etc., utilizing a low power objective lens. Because the working distance of the low power objective is relatively long, a half mirror can be inserted between the objective and the specimen. Additionally, the low power objective possesses the advantage of a large depth of field, which permits observation over a wide field of view.

Although the low power lens has certain benefits, a high magnification objective lens is necessary for measurement of the minute topography and relief features of the specimen. With such an objective, because the working distance is reduced to less than 1 millimeter, an optical system of the type depicted in Figure 3 must be utilized. This is the optical system of an ordinary reflecting microscope. It is ideal to converge the light at the rear focal point of the objective, so that the light beam is directed perpendicularly to the specimen surface. Commercially available objective lenses are tolerably suitable for this purpose.

Low-pressure mercury lamps of the type used for spectroscopy were formerly employed as light sources in such interferometric systems. Using this type of lamp, sharp spectral lines can be obtained, and additionally, the spectral arrangement of colors can be used to identify the sequential order of the interference fringes. The disadvantage of these lamps is low illumination intensity.

Recently, however, bright halogen lamps combined with interference filters have come into general use for interferometry. With achromatic objectives, green (546-nanometer) monochromatic light is the least expensive type of illumination, and provides excellent results.

### Specimen Preparation

Adequate preparation of the specimen is essential in order to obtain the highest precision in multiple-beam interferometry. Because level differences of atomic dimensions are to be measured, contaminants must be removed from the surface. Therefore, the surface should be thoroughly cleaned, except for cases where the surface is mechanically weak or the specimen is chemically unstable. The same requirements apply to cleaning of the reference flat. Commercially marketed multiple-beam interferometric sets provide a selection of multilayer optical flat with various reflectivities matched to the reflectivity of specimens (cleaning of these plates may erode the coating and therefore requires great care). Use of flats matched to the specimen reflectivity permits the formation of multiple-beam interference fringes, but, as illustrated in Figure 1(b), the sharpness of the fringes depends upon the reflectivities of the specimen and the reference plate. Consequently, imparting a high reflectivity to both surfaces is the most effective means of ensuring the formation of a distinct pattern.

Especially small specimens should be cleaned after first being mounted on a glass slide with Canada balsam or another mounting medium. Chemically and mechanically robust materials such as glass and quartz should be cleaned by the following procedure:

• Apply a small quantity of a suitable surfactant (such as a detergent of the type used for kitchen utensils) to a wad of absorbent cotton, and remove the dirt from the specimen by vigorous scrubbing. Ultrasonic cleaning is also suitable for this purpose. However, grease is not easily removed without scrubbing.
• Wet a wad of absorbent cotton with hydrogen peroxide and clean in the same manner as described in the step above.
• Clean the specimen in the same manner with distilled water.
• Wipe the surface thoroughly with dry absorbent cotton until the fog disappears instantly after breathing onto the surface.

Specimens with low reflectivity should be coated by vacuum deposition with a material such as aluminum, silver, or gold. Silver is particularly easy to apply, and is of high reflectivity.

A vacuum-deposited silver film having a thickness of 50-100 nanometers is appropriate. Coating should be performed by rapid vapor deposition so as to be completed in 20 to 30 seconds. In order to accomplish this, a sufficient quantity of silver is placed in a vapor deposition boat, and a shutter is interposed between the specimen and the boat. After the silver is white hot, the shutter is opened and then closed, providing a simple means of controlling the thickness of the coating by varying the time that the shutter is open. In this setup, the distance between the specimen and the boat should be at least 20-30 centimeters.

If the thickness of the deposited silver film is roughly 50-100 nanometers, then the film will appear bluish violet when viewed against a bright light source. An excessively thick film will have a surface that behaves like a mirror, and will not transmit light.

The quality of the silver film on the reference plate can be evaluated as follows. If the silvered surfaces of two such plates are opposed and subtend a wedge as illustrated in Figure 4, and a bright light source is viewed through the wedge, then a sequence of similar images of the light source will appear. The number of these images should be counted; the presence of 25 or more indicates a satisfactory vapor-deposited film, having a reflectance of at least 90 percent. As a rough criterion, the more slowly the color of the images shifts from blue to red, the better the quality of the film.

### Practical Applications of Multiple-Beam Interferometry

When surface topography is measured by multiple-beam interferometry, the specimen and the reference plate make contact at some point. Hence, this method is not appropriate for specimens that are sensitive to contamination or for very soft specimens. Moreover, specimens which for some reason cannot be coated with silver are not suitable.

As previously explained in the discussion of conditions for sharpness of interference fringes, the closest possible proximity of the specimen and the reference plate is essential. In order to realize this, the specimen and reference flat are clamped in a jig having three screws that must be adjusted while observing the interference fringes through the microscope. As in the case of two-beam interferometry, the pattern must be adjusted so that the fringes appear perpendicular to the step to be measured, and the dispersion of the fringes is also adjusted. Measurement can be performed if at least three fringes appear in the view field.

The continuity of the interference fringes can be ascertained from the color arrangement of the fringes formed under illumination by white light, as in two-beam interferometry. Examples are presented in Figure 5 that illustrate the reasoning involved. Suppose that the interference pattern is observed using a 546-nanometer filter with a low-pressure mercury lamp of the type employed for spectroscopy, and that a shift of the fringes appears as shown in Figure 5(a). Obviously, the upper and lower rows of fringes have been mutually displaced by a step on the surface, but the original manner of continuity of the fringes is not evident. If the monochromatic filter is removed, interference fringes with yellow and orange spectral fringes appear, and the proper connection of the fringes can be ascertained by comparing the arrangement of the colored fringes. In Figure 5(b), B-B' represents a sequence of interference fringes of the same order, while in Figure 5(c), the fact that B-C' represents interference fringes of the same order is equally obvious by inspection of the arrangement of the colored fringes.

The method of measuring the level differences from the amount of shift displayed by the fringes is similar to that described in the article on two-beam interferometry and illustrated in that section in Figure 7(c). However, in the case of multiple-beam interferometry, the spacing of the interference fringes is not necessarily equidistant, and therefore height is calculated by the following method. In Figure 5(b), for example, the height, h, of a step can be determined by the following formula:

Likewise, for the example illustrated in Figure 5(c), the appropriate formula is:

A multiple-beam interferometric photomicrograph of the pits, termed trigons (triangular depressions), in a natural diamond is presented in Figure 6(a). The larger depressions are 10-20 nanometers in depth, while the depths of the smaller ones are on the order of 2-4 nanometers.

Figure 6(b) illustrates a multiple-beam interference pattern, formed by trigons, utilizing transmitted light, which can be observed in this case since the diamond itself transmits light. Transmitted multiple-beam interference displays bright fringes. This contrasts with patterns formed by opaque specimens, which show dark fringes. Light transmitted through specimens which possess double refractive indices forms exceedingly complicated interference patterns and therefore such specimens are not suitable for measurement by means of transmitted light. In such cases, reliable measurements of topographic irregularities can still be obtained by interference of reflected light.

As discussed in the related article on two-beam interferometry, high dispersion interference permits the utilization of intensity variations due to extremely minute differences in height. This is illustrated by the example of the trigons of a natural diamond shown in Figure 7(a). The surface shown in this micrograph is nearly the same as that shown in Figure 6(a), but in that instance, the surface of the specimen and the reference plate subtend a wedge, so that a small number of interference fringes are formed. However, in the formation of the pattern of Figure 7(a), the two surfaces are positioned almost in parallel, thereby spreading a single fringe over a large area and facilitating detailed observation. The contrast displayed by minute level changes can be enhanced to an even greater extent than is possible when utilizing two-beam interferometry. As shown in Figure 7(b), an elevation difference dt gives rise to an intensity difference dI. The gradient of the intensity distribution of a multiple-beam interference fringe is far more abrupt than that of a two-beam interference fringe, and the contrast due to minute level variations displays correspondingly greater sensitivity.

A hexagonal ferrite crystal (magnetoplumbite) is presented in Figure 6 ((c) and (d)). This is the same crystal as that of Figure 8(a) in the article on two-beam interferometry. Here, Figure 6(c) is a portion of the multiple-beam interference pattern photographed at high magnification. Figure 6(d) is a phase contrast photomicrograph of the same pattern portion, superimposed upon an interference pattern. In this manner, level differences can be displayed quantitatively on a single photograph. Moreover, one can compensate for the loss of information in the intermediate regions between fringes that results from excessive spacing between the fringes in the multiple-beam interference pattern.

### Conclusion

The examples given constitute a mere fragment of the vast field of applications of interferometry. However, it is hoped that these examples have served to illustrate the fact that the application of interferometry to various material phenomena such as syntheses, dissolution, fracture, deformation, and film formation permits the acquisition of information that would be difficult to obtain by other means.

To summarize, interferometry is an extremely simple and high precision method, and therefore can be used routinely with the same convenience as an ordinary ruler on one's desk. In fact, a conventional microscope can immediately be converted into an interferometer at any time by merely mounting the appropriate attachment.

### 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.