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 parabola. 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

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. (b) Same data as in Fig. 4(a) but with both tilt data but with both tilt and focus removed. The and focus removed. (c) Repeatability of the surface height mea- surface profile data for l eqs are shown in Fig. 2(c). For surement by the MWLPSI. comparison, Fig. 2(d) shows the result of single-wavelength PSI for l = 6328 Å; one can see how the 2p ambiguity problem limits the phase measurement range both tilt and focus removed. After temporal averaging of the conventional single-wavelength PSI. The result of five sets of data to get rid of air turbulance problems, shown in Fig. 2(a) shows the correct wave front under the repeatability of the surface height measurement by

test because l eql is long enough, but that shown in Fig. the MWLPSI is better than 25-Å rms (l = 6328 Å), as

2(c) contains 2p ambiguities since l eqs is too short The shown in Fig. 4(c). first step is to use the phase data of l as a reference eql III. Conclusion to correct the 2p ambiguities in the phase data of l eqs by the method mentioned earlier. Figure 3(a) shows A method to enhance the capability of TWLPSI has

the corrected phase data for l eqs, and Fig. 3(b) shows been developed. By introducing the phase data of a the same data but with both tilt and focus removed. third wavelength, one can measure the phase of a very

This corrected phase data for l eqs can be used as in- steep wave front. Since all necessary phase calculations termediate reference data so that one can go ahead and have been done inside a computer, no hologram is re- perform the second step 2p ambiguity correction work, quired as in two-wavelength holography. A sample test i.e., correct the 2p ambiguities in any single-wavelength measuring the surface height of a defocused off-axis phase data. Figure 4(a) shows the final result for l = parabola (wave front sag = 200 µm) has been performed 6328 Å. Figure 4(b) shows the same result but with to verify the capability of the MWLPSI. After tem-

806 APPLIED OPTICS / VoI. 24, No. 6 1 15 March 1985 poral averaging of five sets of data, the repeatability of the measurement is better than 25-Å rms (l = 6328 3. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna. R. Spolaczyk. Å). and K. Merkel. Digital Wave-Front Measuring Interferometry: Some Systematic Error Sources,” Appl. Opt. 22, 3421(1983). References 4. J. C. Wyant, “Testing Aspherics Using Two-Wavelength Holog- 1. J. H. Bruning, Fringe Scanning Interferometers,” in Optical Shop raphy,” Appl. Opt. 10, 2113 (1971). Testing, D. Malacera, Ed. (Wiley, New York, 1978). 5. C. Polhemus, “Two-Wavelength Interferometry,” Appl. Opt. 12, 2. C. Koliopoulos, “Interferometric Optical Phase Measurement 2071 (1973). Techniques,” Ph.D. Dissertation, Optical Sciences Center, U. 6. Y.-Y. Cheng and J. C. Wyant, “Two-Wavelength Phase Shifting Arizona (1981). Interferometry, Appl Opt. 23, 4639 (1984).