Verification of a polarization-insensitive optical interferometer system with subnanometric capability

B Verification of a polarization-insensitive optical interferometer system with subnanometric capability VTTERWQRTH E I N E M A N N

M. J. Downs, K. P. Birch, M. G. Cox, and J. W. Nunn National Physical Laboratory, Teddington, Middlesex, UK The performance of a length-measuring interferometer system designed to be insensitive to stray reflections and polarization effects resulting in a subnanometric measurement capability is described. Results from the mathematical analysis of the interferometer signals, which provided accurate fringe subdivision and a/lowed a l(x of O. 15 nm to be realized from this system, are also described. The motion of a piezoelectric transducer (PZT) was characterized over a l-t~m range using the system, and the results were used to confirm this subnanometric measurement capability of the interferometer.

Introduction The development of lasers with their intense collimated beams and potential coherence lengths of tens of meters has made optical interferometry the predominant technique for the high-precision measurement of displacement. These systems reversibly count the fringes produced by the interferometer, and in short-distance applications, they must resolve to fractions of a fringe to achieve the required measurement precision. In most applications, these instruments are used in the free atmosphere, and to realize their potential accuracy-approaching 1 part in 10s---it is essential to correct the wavelength of the illuminating radiation for changes in the refractive index of air that otherwise might limit this performance by orders of magnitude. Over short distances (less than 100 i~m), the achievable accuracy becomes proportionally less sensitive to changes in the refractive index of the atmosphere, except the variations that cause anomalous path length differences between the two arms of the interferometer. When subnanometric resolution is required, the lengths of the airpaths in the two arms of the interferometer must be minimized; a sufficiently accurate method for fringe subdivision must be used; and the fidelity of the signals generated from the interferometer must be maintained. Unfortunately, unwanted reflections and polarization effects cause fringe distortion and impose stringent

Address reprint requests to Michael J. Downs, National Physical Laboratory, Queen's Road, Teddington, Middlesex TWIt OLW UK. Precision Engineering 17:84-88, 1995 © Elsevier Science Inc., 1995 655 Avenue of the Americas, N e w York, NY 10010

limitations on the instrument when resolving to fractions of a fringe. The performance of an interferometer system 1 designed to overcome these problems and the results produced by a mathematical technique employed to both analyze the electronic signals produced by the interferometer and realize accurate fringe subdivision are described. Piezoelectric transducers (PZT) are employed in a wide range of applications where precise motion is required over a short distance. These devices require the application of a high voltage to induce a length change, and it is widely known that the relationship between these two parameters is not a linear function. The characterization of a PZT at the subnanometric level is demonstrated using the interferometer.

Optical configuration The interferometer system, 1 shown in Figure 1, has been described elsewhere to demonstrate the use of polarization techniques for signal derivation. 2 This interferometer will operate with any suitable visible radiation source with adequate coherence for the application. For the process described later, a 6-mW unstabilized polarised helium-neon (HeNe) laser was chosen. The interferometer includes a beamsplitter, which is wedged to reduce the effects of unwanted reflections to an acceptable level, and a suitably orientated compensator plate, with identical wedge angle and thickness, used to facilitate the alignment of the system by correcting for the deviation and displacement caused by the beamsplitter. As seen in the figure, the electronic signals are generated by photodetectors monitoring the two 0141-6359/95/$10.00 SSDI 0141-6359(94)00004-J

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interferograms produced by the optical system. The phase difference between these signals is approximately 90°, as required for bidirectional fringe counting, and subdivision is generated by employing thin metal films for the beamsplitter design. For the three-layer stack chosen, the phase difference is close to 90° for both the perpendicular and parallel polarization components. This makes the system insensitive to polarization effects and easier to align, avoiding the cyclic error problems caused by polarization leakage that affects heterodyne interferometers 3-5 when they are used in applications that require subnanometric resolution. A precise horizontal movement of the retroreflector in the measurement arm is provided by attaching the component to a parallel strip hinge motion, which incorporates a location arrangement for mounting precision length transducers. This is configured so that the direction of the motion of any transducer will be coaxial with the interferometer measurement axis, satisfying the Abbe measurement principle, thus providing a calibration facility for those devices. To assess the performance of the interferometer, as shown in Figure 1, a PZT is chosen to modulate the optical path of the interferometer and to demonstrate the subnanometric measurement capability of the instrument.

employed, significant errors may arise in highaccuracy applications. The magnitudes of these errors are shown in Figure 2 where the arctangent function has been applied to calculate the path differences from simulated interferometer signals with individual assumed DC offsets and gain differences of 10% and a deviation from phase quadrature of 10°. In all cases, the optical path difference is assumed to be varying uniformly, and the error magnitudes are calculated for a wavelength 633 G -4

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Electronic processing The two electrical signals produced by conventional two-beam interferometers vary sinusoidally as the optical path difference is changed linearly. Accurate bidirectional fringe counting and subdivision to nanometric precision requires these two electronic signals to have zero mean DC levels, equal amplitudes, and to be in phase quadrature. When these criteria are achieved, precision fringe subdivision can be obtained through the use of an arctangent function. In practice, these ideal criteria cannot be achieved, and, unless correction techniques are

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Downs et aL: Optical interferometer system nm. The figure shows that errors of several nanometers will occur under these circumstances, which typically represent the practical condition, and therefore, a correction technique will be required to obtain a subnanometric measurement accuracy. Further possible sources of error can be generated in the electronic conversion of the optical signals; for example, nonlinearity in the photodetector amplifiers. In addition, noise is produced by rapid fluctuations in the laser light level and in the electronics. These produced a total noise level and nonlinearity of 0.05% (for a bandwidth of 1 MHz), which corresponded to an insignificant optical path difference of 0.05 nm.

Interferometer signal correction The technique employed to correct for errors in practical signals is that developed by Heydemann s and improved by Birch. 7 The magnitudes of the two electrical signals produced by the interferometer are recorded by an analog to digital convertor at approximately equal intervals as the optical path length is changed over a few fringes. These data are analyzed to produce a best fit ellipse from which correction values for the DC offsets, the difference in the amplitudes, and the deviation from phase quadrature are derived. It should be noted that this technique does not require a uniform rate of change in the optical path and that the system is relatively insensitive to the magnitudes of the signal corrections required. Once derived, these data are subsequently used in a mathematical algorithm to transform instantaneous interferometer signals into "ideal" values to which an arctangent function can be applied and accurate fringe fractions determined. The synchronization of fringe counting and subdivision is achieved by employing appropriate software algorithms. The speed of the correction process is dependent upon the rate at which data are collected and on the system noise. For the interferometer employed, eight datapoints per fringe were found to be adequate, and the complete correction process took about 5 seconds with the processor employed. Once the process has been completed, the time to transform the practical interferometer signals into fringe fractions, typically a few milliseconds, is limited by the analog-to-digital conversion of these signals and the use of the algorithm. The signal correction technique was verified using idealized data to ensure that the developed software was not limiting the achievable precision of the interferometer. The results from the idealized data showed variations of significantly less than 0.1 nm, which adequately confirmed the accuracy of the technique. 8 This process also confirmed that the manner in which the data were collected and analyzed did not contribute to the overall noise of the system.

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Results The limiting precision of the interferometer was determined initially without the signal-correction techniques discussed in the previous section. The two electrical signals produced by the interferometer were monitored on an oscilloscope, and the photodetector amplifier electronics associated with each signal were adjusted visually to give the same amplitude and no DC offset. The oscilloscope was set up to allow these signals to describe a Lissajous figure as the optical path was varied. The phase difference between the signals of approximately 7/2 was adjusted by using electronic signal mixing to produce a circular figure on the oscilloscope display. Following the above adjustment, the path was slowly scanned over three fringes by the application of a ramp voltage to the PZT, and 100 instantaneous interferometric signal levels were recorded using an analog-to-digital convertor. The instantaneous phases obtained from the visually optimized signals were compared with those derived after employing the mathematical signal correction technique. The differences between the phases, therefore, represented the residual length measuring error after visual optimisation. This difference is shown plotted as curve (a) in Figure 3 and is seen to vary from about -0.5 to 2.2 nm. The magnitude of this error will, of course, vary depending upon the precision of the visual adjustment obtained from the oscilloscope. In practice, the limit of this adjustment was found to introduce an error of a few nanometers. The residual differences between the datapoints obtained from the instantaneous electrical signals and the best fit ellipse were also used to estimate the error in determining the induced change in optical path. 7 These errors are shown plotted as curve (b) in Figure 3. The sources of error may be attributed to air turbulence, mechanical vi-

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Downs et aL : Optical interferometer system

bration, electronic noise, and intensity variations in the laser. The curve shows there is no indication of any significant systematic errors at the subnanometric level, and the standard deviation for these data was calculated to be only 0.15 nm. (Strict identification of the level of any systematic errors requires further mathematical analysis, and this would need to be pursued if an accuracy of better than 0.1 nm is required from the interferometer.) The procedure described above was repeated five times over a 30-min period, with negligible variations being obtained in both the calculated correction values derived from the fitted ellipse and the standard deviations of the error. These results and the standard deviation of 0.15 nm underline the excellent performance of the interferometer system. These measurements were performed in a laboratory with fairly modest thermal and mechanical isolation from the surroundings. This technique has also been applied to a common-path interferometer system employing a Jamin beamsplitter and, as might be expected, preliminary experimental results suggest a further improvement in overall performance resulting from the insensitivity of that system to mechanical effects.

Characterization of the piezoelectric transducer motion The motion of the piezoelectric transducer used to vary the optical path may also be characterized in the following manner. A slowly varying voltage was applied to the PZT, which produced a corresponding change in the optical path in the interferometer of about 1 p,m (or three fringes). The applied voltage and interferometer signal levels were recorded during this process. The interferometer correction techniques previously discussed were used to derive accurately the induced change in optical path difference produced by the PZT. The optical path difference was recorded at 100 uniformly spaced values of applied voltage, and Figure 4(a) shows a plot of the departure from a linear fit. The underlying function is evidently nonlinear: the approximating least-squares straight line differs from the data by amounts up to 50 nm. Because the underlying curve possesses a single turning point and exhibits no rapid change in curvature, it was thought appropriate to model it by a polynomial. Accordingly, the class of polynomials was considered as a suitable family of models for the data. Polynomials of successive degrees 0, 1, 2. . . . were fitted to the data by the method of least-squares. All data points were accorded the same weight, because there was no reason to expect that values in some parts of the range were any more accurate than elsewhere. To determine polynomials of high degree, should it be necessary, the library routine T1FE from DASL--the NPL Data Approximation Subrou-

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zation tine Library--was used. 9't° For reasons of numerical stability, this routine normalizes the range of the independent variable, applied voltage to ( - 1 , 1) and represents the fitted polynomials in terms of the Chebyshev polynomials of the first kind in this variable. 1°'~ Routine T1FE provides least-squares polynomials and the corresponding rms residuals for all degrees up to a specified maximum. The rms residual stabilized to 0.17 nm at degree-22, a value that compares favorably with the known optimum performance of the current system of 0.15 nm. Note that a degree less than 22; i.e., a degree for which the rms residual has not saturated, would not fully describe the systematic components in the PZT motion. Models other than polynomials are, of course, capable of representing data such as those considered here. Because polynomial splines, especially cubic splines, have the ability to model broad classes of data, splines from the class of cubic splines with increasing number of uniformly spaced knots (the points where neighboring spline segments join) were considered. Uniform knots are appropriate because of the lack of violent changes in behavior of the underlying function. Library rou-

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Downs et aL: Optical interferometer system tine B1FE from DASL was used to provide cubic spline fits with 0, 1, 2 . . . . uniformly spaced knots. B1FE represents splines in terms of B-splines 11'12 for numerical stability and also provides values of the corresponding rms residuals. It was found that the rms residual generally decreased as the number of uniform knots increased until, once 14 or more knots were used, the rms residual stabilized at 0.17 n m. This value matches that for polynomials to two significant figures. To this degree of accuracy, the 14-knot cubic spline is indistinguishable from the degree-22 polynomial. It seems reasonable to conclude that, at least for the dataset analyzed, the data can be represented by a smooth characterization function to an accuracy limited by an estimated noise level having a standard deviation of 0.17 nm. Although the number of parameters, 18, of a 14 uniform-knot cubic spline is less than that, 23, of a 22nd degree polynomial, on the evidence of these data, it is recommended that, because of their simplicity, polynomials be used to provide characterization curves for optical path difference as a function of applied voltage for a PZT. These mathematical techniques applied to a single scan of the PZT motion, which are readily available as software packages, have demonstrated a subnanometric calibration capability. The above data collection and analysis procedure was repeated for five further measurement runs, and in each case, the standard deviation from the polynomial and spline fits agreed to within 0.02 nm and had a range of 0.17 to 0.30 nm. This range is attributable to variations in the motion of the PZT and mechanical noise. An examination of the data obtained for the six measurement runs indicated that the reproducibility of the displacement characteristics obtained from the PZT was typically of the order of 5 nm. This together with the nonlinearity demonstrates the limitations in the use of these devices in the absence of additional control systems.

Conclusion The performance of a relatively inexpensive interferometer system designed to be capable of subnanometric resolution with any source adequately coherent for the application and a mathematical technique employed for correcting the associated

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electrical signals has been evaluated. The results of the evaluation show that subnanometric resolution has been achieved. Path difference accuracies of about 3 and 0.15 nm are obtained if the interferometer signals are visually and mathematically optimized, respectively. The measurement capability of the instrument in determining the displacement at the subnanometric level has also been demonstrated using a piezoelectric transducer.

Acknowledgments The authors gratefully acknowledge the helpful advice of their colleagues at the National Physical Laboratory.

References 1 Downs, M. J. and Rowley, W. R. C. "A proposed design for a polarization-insensitive optical interferometer system with subnanometric capability," Prec Eng, 1993, 15, 281286 2 Downs, M. J. "A proposed design for an optical interferometer with subnanometric resolution," Nanotechno/ogy, 1990, 1, 27-30 3 Rosenbluth, A. E. and Bobroff, N. "Optical sources of nonlinearity in heterodyne interferometers," Prec Eng, 1990, 12, 7-11 4 0 l d h a m , N. M., Kramar, J. A., Hetrik, P. S. and Teague, E. C. "Electronic limitation in phase meters for heterodyne interferometry," NIST, Gaithersburg, MD, U.S. Dept. of Commerce, Technology Administration 5 Tanaka, M., Yamagami, T. and Nakayama, K. "Linear interpolation of periodic error in a heterodyne laser interferometer at subnanometre levels," IEEE Trans on Instru and Meas, 1989, 38, 552-554 6 Heydemann, P. L. M. "Determination and correction of quadrature fringe measurement errors in interferometers," App/Opt, 1981, 20, 3,382-3,384 7 Birch, K. P. "Optical fringe subdivision with nanometric accuracy," Prec Eng, 1990, 12, 195-198 8 Birch, K. P. "The precise determination of refractometric parameters for atmospheric gases," Ph.D. Thesis, Southampton University 1988 9 Anthony, G. T. and Cox, M. G. "The National Physical Laboratory's Data Approximation Subroutine Library," in Algorithms for Approximation, J. C. Mason and M. G. Cox, eds. Oxford: Clarendon Press, 1987, pp. 669-687 10 Cox, M.G. "The NPL Data Approximation Subroutine Library: Current and planned facilities," NAG Newsletter, 1987, 2, 3-16 11 Clenshaw, C. W. and Hayes, J. G. "Curve and surface fitting," J Inst Math Appl 1963, 1, 164-183 12 Cox, M. G. "Algorithms for spline curves and surfaces," in

Fundamental developments of computer-aided geometric modelling, L. Piegl, ed. London: Academic Press, 1993, pp. 51-76

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