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Freeform optics have emerged as promising components in diverse applications due to the potential for superior optical performance. There are many research fields in the area ranging from fabrication to measurement, with metrology being one of the most challenging tasks. In this paper, we describe a new variant of lateral shearing interferometer with a tunable laser source that enables 3D surface profile measurements of freeform optics with high speed, high vertical resolution, large departure, and large field-of-view. We have verified the proposed technique by comparing our measurement result with that of an existing technique and measuring a representative freeform optic.
© 2014 Optical Society of America
1. Introduction
One of the recent growth areas in the optical industry is systems based on freeform optics, which can be defined as a component with a non-rotationally symmetric optical surface about any axis. Freeform optics have shown rapid growth because of the creation of new paradigms in the areas from optical design and fabrication to measurement. Freeform optical surfaces lead to excellent system performance in terms of wavefront aberration, system size, and design degrees of freedom as compared to conventional optical surfaces. Freeform optics are becoming widely used in many applications such as space optics, astronomy, medical instrumentation, and instrumentation for the semiconductor industry. This growth has been possible because of rapid advances in manufacturing technology. However, there are still many challenges that must be overcome to realize widespread application. The measurement of freeform surfaces is very challenging, in particular, and considered an urgent area for research and development. Fabrication capabilities cannot advance without parallel advances in measurement technology.
There are many approaches under developement for 3D surface measurements of freeform optics [1, 2]. A point contact method with a stylus is an intuitive approach but it is time-consuming for complete 3D surface mapping. Deflectometry is a structured light based technique that measures the slope profile of the freeform surface. This is done by quantifying the distortion of a sinusoidal pattern reflected from the freeform surface under test [3, 4]. The method is relatively simple, repeatable, and easy to implement, but sensitive to calibration errors. In general, optical interferometric methods offer superior measurement resolution. Interferometry extracts the surface profile by interpreting the interference pattern between a wave reflected from the surface under test and a reference wave. The test and reference waves cannot be too different, however, else fringe densities become too high. In most configurations, this limits measurable surfaces to those that are nearly flat or nearly spherical, or it requires a complicated test configuration and the use, for example, of a custom CGH (computer generated hologram). It is not applicable to general freeform surface metrology. Lateral shearing interferometry is a variation of standard optical interferometry and has emerged as a promising candidate for freeform surface metrology [5–14]. The lateral shearing interferometer offers the advantage that the reference wave is generated by duplicating the measurement wave and displacing it laterally by a small amount, thus keeping fringe densities low and measurable. The interference directly reveals the slope profile of the surface under test, and the height profile is recovered then through integration.
In this paper, we discuss a new variant of lateral shearing interferometer for 3D surface mapping of freeform optics. The underlying basis is to incorporate the principle of wavelength scanning interferometry into the Michelson-based lateral shearing interferometry for high-speed, high-resolution, large departure, and large-area measurements. Our proposed technique, based on the use of a swept-wavelength laser for phase shifting, allows for high speed measurements without any mechanical scanning process, making it suitable for vibration-desensitized measurements for harsh environments. Also the wavelength scanning technique with the Fourier-based analysis is capable of profiling smooth or rough and discontinuous surfaces without 2π-ambiguity [15]. The wavelength scanning range, at least required for phase shifting, can be optimized according to the average OPD (optical path difference) between the original and sheared wavefronts, and this OPD can be varied. As an example, a small tuning range is sufficient to interpret the interference patterns with large OPD using Fourier technique. Therefore, a wide-range tunable source is not necessarily required for 3D inspection of optical freeform surfaces in this investigation.
2. Principle
Figure 1 shows a schematic diagram of our proposed method. The light source is a tunable laser and its maximum tuning range is from 765 nm to 781 nm. A CMOS 4 megapixel camera captures a series of digitized intensity images of the interference fringe pattern as the wavelength is tuned. A collimated beam from the tunable laser can be adjusted to p-polarization or s-polarization after passing through a rotatable half-wave plate. The two polarization states are used to realize two different modes for the instrument, a measurement mode and a calibration mode. The p-polarization state is used for the measurement mode, in which case the beam is transmitted through the polarizing beam splitter. Double pass through a quarter-wave plate after reflection from the surface under test converts the beam to s-polarization which is then reflected at the polarizing beam splitter. This beam is then divided by a beam splitter into two duplicated waves, one of which is laterally displaced (or sheared) along the x- or y-direction.
Fig. 1 A schematic diagram of lateral shearing interferometer for measurement of freeform optics; s-pol: s-polarized light, p-pol: p-polarized light, CL: collimating lens, HWP: half wave plate, PBS: polarizing beam splitter, QWP: quarter wave plate, BS: beam splitter, RP: right-angle prism, IL: imaging lens; (a) optical layout and beam path for measurement mode and (b) optical layout and beam path for calibration mode.
The x-sheared wave is obtained by translating the right-angle prism I along the z-axis and the y-sheared wave is obtained by rotating the right-angle prism I about the z-axis, as illustrated in Fig. 2. In this way, the laterally sheared interferograms are generated and then imaged onto the 2D detector array by the imaging optics. The imaging optics are adjusted so the surface of the part under test and/or the calibration grid are in focus at the camera.
Fig. 2 The details of lateral shearing part illustrating how to obtain the lateral sheared interferograms in two orthogonal directions; (a) the x-sheared interferogram and its corresponding grid shifted images from measurement and calibration modes when translating the right-angle prism I along the z-axis (b) the y-sheared interferogram and its corresponding grid shifted images from measurement and calibration modes when rotating the right-angle prism I about the z-axis. The y-sheared wave front exiting the beam splitter is tilted at an angle along the y-axis (out of the plane of the page).
The s-polarization is used for the calibration mode, in which case the beam illuminates a grid and is then transmitted by the polarizing beam splitter after double pass through a quarter-wave plate. The amount of lateral shear, S, can be measured by counting the number of the pixels on the grid laterally shifted in the x – or y – direction. The grid spacing is calibrated by measuring the distance using a micrometer when one pixel moves and overlaps the adjacent pixel. So the shear amount is controlled precisely by a micrometer in the micro-level precision and the interval of shear is calibrated by measuring the amount of the grid movement. Here, the amount of shear determines the lateral resolution, because the wavefront is reconstructed only at intervals of shear. In this set-up, 1 pixel corresponds to 250 μm. The polarization state of the beam is controlled for two reasons. One is to allow expedient switching between measurement and calibration modes, enabling convenient and frequent estimates of the amount of lateral shear in the two orthogonal directions. The second reason is to increase the signal-to-noise ratio of the sheared interferogram by collecting only the beam reflected from the target and blocking the unintended background beams reflected from other optics in the system.
The interference signal according to the wavelength of the source used can be described as
Where is the slow-varying spectral density distributions of the original and the displaced beams, is , is the wave-number , is the initial wave-number,is the additional spectral carrier component, and represents the difference between the original and the sheared waves, or , where are the coordinates of an arbitrary point on the target and is the amount of shear in the x- or y-direction. Here, is set by moving the right-angle prism II from position a1 to a2 on the order of several hundred micrometers, where position a1 corresponds to no average optical path difference between the original and sheared wave fronts. The Fourier-transformed signal of Eq. (1) can be mathematically expressed asWhere and are the Fourier transforms of and , respectively. Here, the induced spectral component causes a linear frequency shift to and its complex conjugate in the frequency domain. The interference frequency , shifted in the amount of in the positive domain, contains the spatial phase distribution given by , which is the quantity of interest. The term can be extracted with two different approaches. The first is to recover by use of the double-Fourier transform technique. The inverse Fourier transform of the isolated signal, (using a suitable window) can be applied, leading toNext, we calculate the complex logarithm and take the imaginary part of Eq. (3), giving. The value of can be calculated from the derivative of , following references from 15 to 17,Another approach to recover is to obtain the phase information directly from the complex amplitude of the Fourier transformed signal as follows [18]:Hence can be calculated using either Eq. (4) or Eq. (5). In this investigation, we used Eq. (5) in order to obtain with high resolution. Once is obtained in the x- and y-directions, the surface height map can then be obtained from slope integration with the Southwell’s zonal method [19, 20], for example.3. Experimental results
We have tested our proposed technique by measuring two representative samples, a 1 inch silver coated flat mirror and a freeform shaped optic. Table 1 summarizes the experimental parameters used for the measurements. We modulate the laterally sheared interferogram by setting to several hundred micrometers and using a small tuning range of 5 nm, allowing for good signal-to-noise analysis with the Fourier technique. The additional OPD causes an unequal-path configuration between the original and sheared wavefronts, which results in defocus issues especially for objects with high slopes. This defocus leads to the system performance degradation due to poor fringe visibility and serious measurement errors such as distortions when imaging high-slope optical surfaces. However, the depth-of-field imaging capability of our system is designed to provide the focusing distance in the range of several millimeters, so the defocus issues can be negligible when measuring the object with a height within several millimeters. We ignore this effect in this study.
Table 1. Details of the apparatus used for experiments.
Figure 3 shows the comparison of the measurement results of the flat mirror with our suggested method and a Zygo Fizeau interferometer. A point-by-point intercomparison is not practical, but the overall trend of the surface variation can be compared. Figure 3(a) and (b) show the measured x-, y-sheared interferograms and their corresponding surface slopes, respectively. Faint nearly-horizontal and nearly-vertical stripes in Fig. 3(a) and (b), respectively, can be seen in the interferograms and come from unwanted reflections from optical components. These stripes do not modulate and therefore do not directly contribute to the measurement. The surface map is reconstructed from the integration of the x-and y-slope maps and is shown in Fig. 3 (c) with piston and tilt removed. The same area measured with the Fizeau interferometer is shown in Fig. 3(d). Our measurement shows a PV of 50.3 nm and the Fizeau measurement shows a PV of 53.7 nm. The shear amount in lateral shearing interferometry sets a boundary on the measurement resolution that can be achieved. The shear amount was 250 μm for our measurement, thus spatial frequency structure in the surface on the order of several thousand m−1 and higher cannot be resolved. This accounts for the difference in the high spatial frequency information that can be seen in the figures.
Fig. 3 Comparisons of measurement results: (a) the x-sheared interferogram viewed in monochromatic light and its corresponding 3D surface slope map, (b) the y-sheared interferogram viewed in monochromatic light and its corresponding 3D surface slope map, (c) the reconstructed 3D surface map with x-and y-slope integrations after the piston and tilt being subtracted, and (d) 3D surface map measured with the Zygo Fizeau interferometer.
Figure 4 shows a measurement result of a freeform optic using our technique. Such a freeform surface is very challenging to measure with any other method, so an independent full 3D surface profile measurement is not available for comparison. We measured the part with a contact stylus to provide point-by-point data for an approximate comparison. The stylus was calibrated by measuring a step-height standard. For the profilometry data, we used a standard a low-pass Gaussian filter with a short-wavelength cutoff of 8 μm for smoothing [21]. An exact comparison at the same line of A-A′ is impossible in practice and it may cause a difference between two measurement results owing to inconsistency in measuring points. However, the results of Fig. 4 (a) and (b) show plausible 3D surface slope maps that match the interferogram structure and Fig. 4 (d) compares line profiles from our measurement result and the profilometer scan that we estimate to be separated by no more than 0.05 mm.
Fig. 4 A freeform optical surface measurement result: (a) the x-sheared interferogram viewed in monochromatic light and its corresponding 3D surface slope map, (b) the y-sheared interferogram viewed in monochromatic light and its corresponding 3D surface slope map, (c) the reconstructed 3D surface map with x-and y-slope integrations with piston and tilt removed, and (d) comparison to a stylus measurement of the line profile A-A′ of (c)
4. Conclusion
Freeform optics are becoming widely used in diverse applications, yet many research challenges remain. Metrology is one of the most urgent and challenging aspects. We have proposed a new variant of lateral shearing interferometry with a tunable laser source that enables 3D surface profile measurements of freeform optics with high speed, high vertical resolution, large departure, and large field-of-view. Our method was verified by comparing the measurement result of a flat mirror with a Fizeau interferometer and measuring a representative freeform optic surface. We anticipate that the proposed technique will be a promising metrological 3D tool for freeform optics measurements even under challenging environmental conditions.
Acknowledgements
This material is partially based upon work supported by Korea Research Council of Fundamental Science & Technology – Grant funded by the Korean Government (KRCF-2013-CAP-1345194477) and by the industrial strategic technology development program, ‘Development of micro machining technology for 700 nm-patterning on DOE (diffractive optical element)’ funded by the Ministry of Trade, Industry & Energy (MOITE, Korea).
References and links
1. E. Savio, L. D. Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” Annals of the CIRP 56(2), 810–835 (2007). [CrossRef]
2. R. Henselmans, L. A. Cacace, G. F. Y. Kramer, P. C. J. N. Rosielle, and M. Steinbuch, “The NANOMEFOS non-contact measurement machine for freeform optics,” Precis. Eng. 35(4), 607–624 (2011). [CrossRef]
3. C. Faber, E. Olesch, R. Krobot, and G. Häusler, “Deflectometry challenges interferometry – the competition gets tougher!,” Proc. SPIE 8493, 0R-1-0R-15(2013).
4. G. Häusler, C. Faber, E. Olesch, and S. Ettl, “Deflectometry vs. Interferometry,” Proc. SPIE 8788, 1C–1-1C–11(2013).
5. M. V. R. K. Murty, “The use of a single plane parallel plate as a lateral shearing interferometer with a visible gas laser source,” Appl. Opt. 3(4), 531–534 (1964). [CrossRef]
6. J. C. Wyant, “Double frequency grating lateral shear interferometer,” Appl. Opt. 12(9), 2057–2060 (1973). [CrossRef] [PubMed]
7. M. P. Rimmer, “Method for evaluating lateral shearing interferograms,” Appl. Opt. 13(3), 623–629 (1974). [CrossRef] [PubMed]
8. M. P. Rimmer and J. C. Wyant, “Evaluation of large aberrations using a lateral-shear interferometer having variable shear,” Appl. Opt. 14(1), 142–150 (1975). [CrossRef] [PubMed]
9. S. Okuda, T. Nomura, K. Kamiya, H. Miyashiro, K. Yoshikawa, and H. Tashiro, “High-precision analysis of a lateral shearing interferogram by use of the integration method and polynomials,” Appl. Opt. 39(28), 5179–5186 (2000). [CrossRef] [PubMed]
10. H.-H. Lee, J.-H. You, and S.-H. Park, “Phase-shifting lateral shearing interferometer with two pairs of wedge plates,” Opt. Lett. 28(22), 2243–2245 (2003). [CrossRef] [PubMed]
11. P. Liang, J. Ding, Z. Jin, C.-S. Guo, and H.-T. Wang, “Two-dimensional wave-front reconstruction from lateral shearing interferograms,” Opt. Express 14(2), 625–634 (2006). [CrossRef] [PubMed]
12. S. Ettl, J. Kaminski, M. C. Knauer, and G. Häusler, “Shape reconstruction from gradient data,” Appl. Opt. 47(12), 2091–2097 (2008). [CrossRef] [PubMed]
13. F. Dai, F. Tang, X. Wang, and O. Sasaki, “Generalized zonal wavefront reconstruction for high spatial resolution in lateral shearing interferometry,” J. Opt. Soc. Am. A 29(9), 2038–2047 (2012). [CrossRef] [PubMed]
14. H.-G. Rhee, Y.-S. Ghim, J. Lee, H.-S. Yang, and Y.-W. Lee, “Correction of rotational inaccuracy in lateral shearing interferometry for freeform measurement,” Opt. Express 21(21), 24799–24808 (2013). [CrossRef] [PubMed]
15. M. Takeda and H. Yamamoto, “Fourier-transform speckle profilometry: three-dimensional shape measurements of diffuse objects with large height steps and/or spatially isolated surfaces,” Appl. Opt. 33(34), 7829–7837 (1994). [CrossRef] [PubMed]
16. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]
17. Y.-S. Ghim and S.-W. Kim, “Thin-film thickness profile and its refractive index measurements by dispersive white-light interferometry,” Opt. Express 14(24), 11885–11891 (2006). [CrossRef] [PubMed]
18. L. L. Deck, “Fourier-transform phase-shifting interferometry,” Appl. Opt. 42(13), 2354–2365 (2003). [CrossRef] [PubMed]
19. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70(8), 998–1006 (1980). [CrossRef]
20. D. Malacara-Doblado, I. Ghozeil, “Hartmann, Hartmann-Shack, and Other Screen Tests,” in optical shop testing 3rd ed., Wiley Series in Pure and Applied Optics (Wiley, 2007), 361–397.
21. ASME B46.1–2002, “Terminology and measurement procedures for profiling, contact, skidless instruments,”in Surface texture (Surface roughness, waviness, and lay). (Amer. Soc. of Mech. Engrs., 2003), Section 3.
