Precision interferometer for measuring photoelastic constants Roy M. Waxier, Deane Horowitz, and Albert Feldman

U.S. National Bureau of Standards, Inorganic Materials Division, Washington, D.C. 20234. Received 20 September 1976. Twyman-Green and Fizeau interferometers have been

used to measure the peizooptic constants of glasses and crystals, and the coefficients have been calculated from the observed shift of optical interference fringes.1-3 Evaluation of the piezooptic constants to three significant figures requires measurement of the stress induced change in optical path-length to a precision of at least one part in 103. This corre­sponds to fringe shift measurements within λ/10 to λ/100 for materials capable of withstanding stresses as high as 108 N/m2. For optical materials such as KC1 which can withstand stresses on the order of 106 N/m2, it becomes necessary to measure fringe shifts within λ/100 to λ/1000 in order to attain the same precision.

It is the purpose of this Letter to describe an interferometer, based on a design by Dyson,4 that is capable of precise mea­surements of photoelastic constants at low stress levels. In this arrangement, the beams in the two arms of the interfer­ometer, which are polarized at 90° to each other, are recom-bined, and the state of polarization of the combined beam is analyzed with a Soleil-Babinet compensator. Thus, the measurement of fringe shifts is converted to a measurement of relative optical path difference with its attendant greater precision.5-7 Moreover, this interferometer has great stability because the two beams are in close proximity and traverse the same optics.' Other workers have employed this type of in­terferometer.8 However, we are the first to use it, with ap­propriate modifications, for measuring photoelastic constants. Green,8 using two versions of this interferometer for mea­suring thermal expansion and change of refractive index with temperature, reports a precision of measurement of λ/500.

In this Letter we describe the interferometer and present data obtained on fused silica at 0.6328 μm. This glass was selected because it can be obtained as a pure material and because the coefficients that we measured with the application of small stresses may be compared with previously reported coefficients obtained with high stresses.1

A schematic of the interferometer is shown in Fig. 1. Lin-

Fig. 1. Interferometer for measuring stress-induced change in optical pathlength.

20 APPLIED OPTICS / Vol. 16, No. 1 / January 1977

early polarized light from a He-Ne laser operating at 0.6328 μm is incident upon a Wollaston prism, which separates the light into two linearly polarized beams, one beam with vertical polarization and the other with horizontal polarization. We can balance the intensity of the two beams by changing the polarization angle of the incident radiation. The Wollaston prism is placed at the focus of a lens that serves to collimate the two beams and make them parallel, and the specimen is positioned in one of these beams. The beams then pass through a quarter-wave plate preceded by a second lens which brings them together at the focal point where they are incident upon a plane mirror. The two reflected beams then return through the system; one beam retraces its path while the other is reflected to the other side as shown. Double passage through the quarter-wave plate means that it acts effectively as a half-wave plate. Because the quarter-wave plate is ori­ented with principal axes at 45° from the horizontal, the state of polarization of each beam is rotated through 90°. Thus, the states of polarization having been interchanged, when the beams are combined at the Wollaston prism, exit as a single beam at an angle with respect to the incident beam. This arrangement is similar to the one used by Green,8 but a no­table difference is that, here, the quarter-wave plate is placed after the specimen rather than before it.

Figure 1 was drawn to bring out the key elements of the interferometer; however, there are other attendant parts of the whole measuring setup that are important. These include two Glan-Thompson polarizing prisms in graduated amounts, a spatial filter, a beam expander, and a Soleil-Babinet com­pensator. One Glan-Thompson prism serves to polarize the incident beam. The spacial filter and the beam expander are placed in the exit beam to eliminate spurious laser radiation. The Soleil-Babinet compensator and the second prism are then used to analyze the state or polarization of the exit beam. A silicon matrix vidicon camera and a TV monitor is employed to provide a convenient method for observation of the output fringe pattern.

The specimen was prepared from high quality fused silica that was free of striae and birefringence. A plate of this ma­terial, about 1 cm thick, was ground and polished to better than one-tenth of an interference fringe when examined in a Twyman-Green interferometer. From this plate a specimen was cut to the dimensions of 1.067 × 1.261 × 5.5 cm. The four remaining unpolished faces were precision ground on a surface grinder. Reexamination of the specimen in a polariscope showed that no optical strain had been introduced by the machining.

The specimen was then mounted in a stressing apparatus. The apparatus and the method of mounting the specimen have been described in detail elsewhere.9 The force applied to the specimen was measured in pounds by a load cell and voltmeter calibrated together by the method of deadweight loading by the Engineering Mechanics Section of NBS.

The specimen, held in the stressing apparatus, was placed in one arm of the interferometer, and loads up to 180 kg (400 lb) were applied in 22.5-kg (50-lb) increments. Calculations show that the highest pressure was 1.34 × 107 N/m2, and the incremental pressures were 1.68 × 106 N/m2. The effect of stress was to change the optic path in one arm of the inter­ferometer and, hence, to alter the state of polarization of the output beam. When no load was applied to the specimen, the Soleil-Babinet compensator was set to produce a null on the TV screen. For each incremental stress load, the compensator was reset to produce a null and the compensator drum reading recorded. The drum was calibrated in terms of the fringe shift. A zero stress reading was taken before and after a reading of the effect of each incremental load, and the zero stress readings were averaged. This procedure was followed

in order to compensate for a slight drift which was attributed to small temperature changes. All the elements shown in Fig. 1 were covered by a wooden box to lessen the effects of air currents and temperature gradients.

For a solid in the glassy state there are only two piezooptic constants, q11 and q12, which relate the change in applied stress to the induced change in refractive index. When a brick shaped specimen of glass is placed in uniaxial compression, there is a change in refractive index Δn1 for light polarized along the direction of stress according to the equation

where ΔP is the change in stress, and n is the zero stress re­fractive index. There is a second change in refractive index Δn2 for light polarized perpendicular to the direction of stress, and this is

Figure 1 shows the specimen intercepting a horizontally po­larized beam so that the data obtained for this position are used to obtain q12· By placing the specimen in the return path of this beam, where the radiation is vertically polarized, we obtain data for q11.

The changes in optical pathlength vs applied load for the two directions of polarization are shown in Fig. 2. It can be seen from the figure that the data conform very closely to a straight line. Both sets of points were fitted to a straight line by the method of least squares, and the standard deviation of the points from the fit corresponded to a fringe deviation of about λ/500. The piezooptic constants were then determined according to the equations

and

where ANi and AN2 are the fringe shifts for radiation that is polarized vertically and horizontally, respectively, t is the thickness of the specimen, and s i 2 is an elastic compliance component whose value for fused silica is -2.16 X 10~12

Fig. 2. The effect of uniaxial stress on the optical pathlength in a specimen of fused silica.

January 1977 / Voï. 16, No. 1 / APPLIED OPTICS 21

Table I. Piezooptic Constants of Fused Silica in Units of 1 0 - 1 2 m 2 / N

mVN.1 In Table I we present the values of q11 and q12 ob­tained from the linear least square fit to the change of optic path as a function of stress. The errors shown in the table are the standard deviation of the data points from the line and hence represent the precision of measurement. The accuracy, however, is not as good because of possible systematic errors, but these we believe to be less than 2%. Such systematic er­rors would include stress gradients in the specimen. For comparison, values of q11 and q12 derived from data reported by Primak and Post1 are shown in Table I, and it can be seen that there is good agreement between the two sets of values.

This work was supported in part by the Department of Defense Advanced Research Projects Agency.

References 1. W. Primak and D. J. Post, J. Appl. Phys. 30, No. 5, 779 (1959). 2. A. Feldman, R. M. Waxier, and D. Horowitz, in Optical Properties

of Highly Transparent Solids, S. S. Mitra and B. Bendow, Eds. (Plenum, New York, 1975), pp. 517-525.

3. A. Feldman, D. Horowitz, and R. M. Waxier, in Laser Induced Damage in Optical Materials: 1975, A. J. Glass and A. H. Guen-ther, Eds. (NBS Special Publication 435, April 1976), pp. 164-169.

4. J. Dyson, Interferometry as a Measuring Tool (The Machinery Publishing Co., Brighton, 1970).

5. M. Francon, Optical Interferometry (Academic, New York, 1966).

6. W. Krug, J. Rienitz, and G. Schultz, Contributions to Interference Microscopy (Hilger & Watts, London, 1964).

7. J. Dyson, Appl. Opt. 7, 569 (1968). 8. F. Green, in Optical Instruments and Techniques, 1969, J. H.

Dickson, Ed. (Oriel Press, New-castle upon Tyne, England, 1970), pp. 189-198.

9. A. Feldman and W. J. McKean, Rev. Sci. Instrum. 46, 1588 (1975).

22 APPLIED OPTICS / Vol. 16, No. 1 / January 1977