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Multiple-Wavelength Phase-Shifting Interferometry

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Multiple-Wavelength Phase-Shifting Interferometry Multiple-wavelength phase-shifting interferometry Yeou-Yen Cheng and James C. Wyant This paper describes a method to enhance the capability of two-wavelength phase-shifting interferometry. By introducing the phase data of a third wavelength, one can measure the phase of a very steep wave front. Experiments have been performed using a linear detector array to measure surface height of an off-axis pa- rabola. For the wave front being measured the optical path difference between adjacent detector pixels was as large as 3.3 waves. After temporal averaging of five sets of data, the repeatability of the measurement is better than 25-Å rms (l = 6328 Å). I. Introduction alent wavelength l eq according to Eq, (1) with the as- Conventional single-wavelength phase-shifting in- sumption that the difference of OPD between any ad- 1-3 terferometry (PSI) is a technique which can perform jacent pixels is <l eq/2: direct phase measurement by first obtaining three or more intensity readings of fringe patterns with some known amounts of phase shifts and then using these intensity readings to calculate the phase value at each data point. Since the phase distribution across the interferogram is a measured modulo 2p one needs an Once these D OPDs between pixels were obtained, the assumption to remove the 2p discontinuities in the relative wave-front (OPD) plot or relative surface height measured phase data. The fundamental assumption plot could be obtained by integrating all these D OPD for single-wavelength PSI is that the difference of op- values. The problem for the first method of the two- tical path difference (OPD) between any adjacent pixels wavelength phase-shifting interferometer (TWLPSI) is £l/2. This assumption sets a limit to the phase is that the calculated phase data for l eq are quite noisy measurement range of single-wavelength PSI. When due to the error amplification effect,6 which makes the the slope of the test surface is steep enough that the measurement precision poorer than that of single- phase change between any adjacent pixels is larger than wavelength PSI. In the second more accurate method, p, the 2p ambiguity problem will ruin the result of the the phase data for l eq, obtained by the first method are phase measurement. By using either a higher resolu- used as a reference only, so that one can go ahead and tion detector array or a longer wavelength light source, use it to correct the 2x ambiguities in the single-wave- it is possible to overcome this problem. One can use length phase data and maintain its measurement pre- either an IR light source or two shorter visible wave- cision. lengths to synthesize a longer equivalent wavelength as There are several ways to make the 2p ambiguity is used in many two-wavelength techniques.4-6 corrections. A simple way is by finding the difference In Ref. 6 two methods were presented to solve the 2p between the reference data (noisier but without 2p ambiguity problem. For the first method two sets of ambiguities) and the data with 2p ambiguities. In phase data (with 2p ambiguities) for l a and l b, are general, if the noise in the reference data is not too large, stored in the microcomputer, which will then calculate one can get a kind of step function and let one know the phase difference between pixels for a longer equiv- where an integer number of 2p values should be added or subtracted. A problem using the first method of the TWLPSI is that as the wave front under test becomes steeper and steeper a longer l eq is needed to solve the 2p ambiguity problem. Due to the error amplification effect, the The authors are with University of Arizona, Optical Sciences amplitude of the high-frequency structure on the cal- Center, Tucson, Arizona 85721. culated phase data for l eq has a larger amplitude for Received 5 October 1984. longer l eq and makes it harder to correct the 2p am- 0003-6935/85/060804-04$0200/0. biguities in single-wavelength phase data. In this paper © 1985 Optical Society of America. we propose a method that can overcome this problem 804 APPLIED OPTICS / Vol. 24, No. 6 / 15 March 1985 ambiguities in l a, l b, or l c. Actually, more correction steps could be applied if one had more phase data from other wavelengths. II. Experimental Setup and Results The experimental setup is shown in Fig. 1. An argon-ion laser and a He-Ne laser were used as the light sources so that the equivalent wavelength could be changed from 1.93 to 28.5 µm. Note that the diverging lens in the setup was also used as an imaging lens. Since the mirror under test (an off-axis parabola with ap- parent f/5) is not a very steep aspheric surface, one can defocus it to get a very fine fringe pattern. The fringe patterns of different wavelengths are sampled by a Reticon RC1728H linear array (only 1024 pixels being used); the analog signal is then converted into a l0-bit Fig. 1. Experimental setup for the MWLPSI. digitized signal which is fed into a HP9836 microcom- by including the phase data of a third wavelength. To puter for processing. Due to the finite width of each select the right wavelengths, one has to make sure that pixel’s response curve, the largest measurable wave the OPD difference between adjacent pixels is less than front slope is -0.8 l /pixel for this particular detector half of the longest equivalent wavelength (l eql/2). The array. To test the capability of the multiple-wave- idea is to use the input phase data for l a, l b, and l c length phase-shifting interferometer (MWLPSI), one (assume l a < l b < l c) to calculate the phase data for needs a wave front much steeper than that mentioned the longest equivalent wavelength (l eql) and that for the above and requires a nearly ideal point detector array shortest equivalent wavelength (l eqs). A good combi- to sample it. Although this kind of ideal point detector nation of wavelengths is for the ratio of l eql/ l eqs and array is not yet available, one can simulate it by using l eqs/ l b to be 3 or 4. Now there are two steps to make the whole array to take intensity readings and then use the corrections: (1) use the phase data of l eql to correct every four or five pixels to do phase calculations. In our 2p ambiguities in the phase data of l eqs; (2) use the 2p case, 1024 pixel elements were used to take intensity ambiguity corrected phase data of l eqs to correct the 2p data for fringe patterns, and only 240 phase values were SURFACE PROFILE SURFACE PROFILE TWLBG RMS-29.1 µM P-V-95.7 µM TWLBG RMS- 8.4 µM P-V- 2.2 µM Distance on surface In mm Distance on surface in mm (a) (b) SURFACE PROFILE SURFACE PROFILE TWLBR RMS- 3.3µM P-V-30.5 µM 6328 RMS- 1.1 µM P-V- 4.1 µM 0.0 13.2 26.4 39.6 52.8 66.1 Distance on surface In mm Distance on surface In mm (c) (d) Fig. 2. (a) Two-wavelength surface height plot for l eql, where l eql = 6.45 µm. (b) Some data as in Fig. 2(a) but with both tilt and focus removed. (c) Two-wavelength surface height plot for l eqs, where l eqs = 1.93 µm. (d) Single wavelength surface height plot for l = 6328 Å obtained by single-wavelength PSI. 15 March 1985 / Vol. 24, No. 8 / APPLIED OPTICS 805 SURFACE PROFILE SURFACE PROFILE TWLBR RMS-30.2 µM P-V-99.9 µM 6328 RMS=30.6 µM P-V-101.1 µM t I 6.6 13.2 26.4 36.6 52.6 66.1 Distance on surface in mm Distance on surface in mm (a) (a) SURFACE PROFILE SURFACE PROFILE 6328 RMS- 0.4 µM P-V- 1.9 µM TWLBR RMS- 0.4 µM P-V- 2.0 µM 0.0 13.2 26.4 39.6 52.8 66. 1 0.8 13.2 26.4 38.6 52.6 66.1 Distance on surface in mm Distance on surface in mm (b) (b) SURFACE PROFILE Fig. 3. (a) 2p ambiguity corrected surface height plot for l eqs. (b) DIFF. RMS- 21 A. P-V- 104 A. Same data as in Fig. 3(a) but with both tilt and focus removed. calculated so that the largest wave front slope becomes 3.3 l /pixel, and the TWLPSI stops working when l eq, is shorter than 3.3 µm. Three typical wavelenths used in this experiment are l a = 4765 Å, l b = 5145 Å , and l c = 6326 Å [where l eql = l a l b/(l b - l a) = 6.45 µm, l eqs = l a l b /(l b - l a) = 1.93 µm]. The computer did the 0.0 13.2 26.4 39.6 52.8 66.l calculations for the phase data for l a, l b, and l c, then Distance on surface in mm calculates the phase data for l eql and l eqs, according to (c) Eq. (1). After integration, the surface profile data for l Fig. 4. (a) 2x ambiguity corrected single-wavelength surface height eql are shown in Fig. 2(a). Figure 2(b) shows the same plot for l = 6328 Å.
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