Instantaneous measurement Fizeau interferometer with high spatial resolution

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1 Copyright 2011 Society of Photo-Optical Instrumentation Engineers. This paper was published in Proceedings of SPIE and is made available as an electronic reprint with permission of SPIE. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited. Instantaneous measurement Fizeau interferometer with high spatial resolution Daniel M. Sykora and Peter de Groot Zygo Corporation, 21 Laurel Brook Road, Middlefield, CT, USA ABSTRACT Modern precision optical manufacturing places high demands on instrument design, both for flexible response to challenging environments and high lateral resolution for measuring both surface form and mid spatial frequency waviness. Here we report on a Fizeau-type interferometer optimized for light-efficient, single-frame carrier fringe acquisition for instantaneous metrology at high lateral resolution. Fully coherent optics and a pixel camera provide high slope acceptance and an instrument transfer function (ITF) above 50% at 250 cycles/aperture for all zoom settings, as demonstrated for an etched phase step and custom periodic artifact. The instrument further provides an ITF of 500 cycles/aperture using optional temporal phase shifting interferometry on the same platform. Keywords: Instantaneous interferometry, spatial resolution, lateral resolution, instrument transfer function 1. INTRODUCTION Fabrication of precision optical surfaces increasingly requires the use of computer controlled sub-aperture polishing techniques to achieve specified figure and surface finish tolerances. Accurate characterization of figure errors that define correction feed maps and qualification of any mid spatial frequency signatures left behind by the tool path are fundamental metrology requirements for cost-effective fabrication. These fabrication requirements impose high demands on a metrology system, including large slope acceptance with minimized rigid-body retrace 1,2 and excellent imaging fidelity versus frequency using an instrument transfer function (ITF) to quantify performance. 3,4,5,6 The metrology space for much of the optical test community covers plano, spherical, and mild aspheric optical surfaces ranging in diameters from 25-mm up to several meters. Preferred metrology for optical testing in controlled environments is phase-shifting interferometry (PSI), while more challenging environments require dynamic or instantaneous interferometers with single-frame acquisitions at integration times as short as 100 sec. Figure 1: ZYGO DynaFiz dynamic interferometer. A system-level evaluation of retrace and ITF performance is provided for an interferometer capable of both conventional PSI and instantaneous interferometry using carrier fringes for linear spatial phase shifting (Figure 1). 7 Instrument design optimizes the balance between high-speed measurements and lateral resolution for both data acquisition methods. 8 The paper will outline how slope acceptance and lateral resolution are related, how carrier fringe methods for instantaneous measurements may be optimized for lateral resolution, and finally, how a well-designed, optically coherent zoom system supports single-frame metrology of spatial frequencies beyond 9.0 cycles/mm.

2 2. SLOPE ACCEPTANCE Modern interferometers use digital electronic cameras to leverage advanced computing power for processing interferograms. The discrete sampling by these sensors establishes the fundamental imaging resolution limits (i.e., the Nyquist frequency) at the detector plane, and provides design and performance targets for slope acceptance and lateral resolution as a function of imaging magnification. Instrument slope acceptance relates to the tilt angle between interfering reference and test wavefronts that yields unvignetted fringes at the detector plane with period equal to the targeted maximum spatial frequency (i.e. in line pairs or cycles/aperture). Minimizing noise may drive the targeted resolution to some fraction of Nyquist (e.g., 0.9). These considerations comprise preventing the detection of higher frequency aliased fringes, suppressing diffracted reflections from the sensor array, and limiting the energy passed from unwanted reflections throughout the system. With a target of lateral resolution, R (in fractional Nyquist), a collimated cavity illumination aperture, A, and the instrument wavelength,, one may calculate the necessary slope acceptance, S x,y, as R S ( m, p ) m p 2 A x, y x, y x, y, (1) where m is the instrument magnification, and p x,y equals the number of acquired pixels in either the x or y direction. With instrument parameters set to = 632.8nm, R = 0.9, and A = 101.6mm, the required slope acceptance at three different magnifications is summarized in Table 1, along with the corresponding surface slope of a test surface and tilt fringes. A surface under test is sampled in reflection, and so the measureable slope on the surface of the part is equal to ½ the instrument slope acceptance, while the equivalent tilt fringes or cycles/aperture, C x,y, is simply p Cx, y ( px, y ) R 2 Table 1: Targeted instrument limits for slope acceptance to support 0.9 Nyquist lateral resolution at three separate magnifications for temporal PSI. Equivalent surface slope and cycles/aperture are also displayed. Pixels Mag Slope Acceptance Surface Slope cycles/aperture X 3.2mrad 1.6mrad X 5.8mrad 2.9mrad X 10.1mrad 5.0mrad 540 Maximized slope acceptance allows for measurement of surfaces with large figure error. A more significant benefit to the optical test community results when large slopes are measured with minimal instrument retrace errors (e.g., < 0.25 waves peak-to-valley) leading to an expanded measurement space and improved convergence rates for deterministic corrections. 3. HIGH SLOPES AND RETRACE ERROR Designing a system for high slopes requires not only careful matching of camera resolution and system aperture as described above, but also lens design to minimize retrace error. This is particularly important in spatial phase shifting methods that rely on carrier fringes generated by introducing a tilt offset between the object and reference surfaces, resulting in a nominal fringe density as high as ½ Nyquist. Good instrument design minimizes calibration requirements and maximizes resolution for such a system. Qualification of a system over the full slope acceptance range is performed by measuring a plano cavity in an on-axis null alignment and differencing this map from subsequent measurements taken with roughly 100, 250, 400, and 500 tilt fringes. The results in Figure 2 demonstrate < 0.1-wave of Zernike-type retrace errors with as much as 250 rigid-body tilt fringes for all three magnifications (1.00X, 1.73X, and 3.00X), and < 0.25-wave of retrace errors for nearly all points sampled out to 500 fringes of tilt. Figure 2 also recasts the same data versus slope on the test surface, and shows clearly xy, (2)

3 how increasing magnification expands the slope acceptance (over functionally smaller apertures) while maintaining retrace performance. These empirical slope acceptance limits correlate well with those outlined in Table 1. A straightforward in situ calibration compensates any residual retrace bias caused by the introduction of carrier fringes to < 5nm, using the optic under test. In this way, a synthetic cavity null alignment is established and the curves of Figure 2 effectively pass through zero at the carrier frequency, thereby bounding rigid-body retrace for slope deviation away from an instantaneous synthetic null to values typically < 0.15-waves.8 Figure 2: Uncalibrated retrace measured as a function of rigid-body tilt with respect to on-axis null for the interferometer of Figure 1. Retrace is reported as a 36-term Zernike PV versus tilt fringes (top) and versus test surface slope (bottom) for all three magnifications (1.00X, 1.73X, and 3.00X). 4. LATERAL RESOLUTION WITH INSTANTANEOUS INTERFEROMETRY The basic limits of the carrier fringe method may be understood by considering what happens to the spatial phase shift with increasing tilt. Assume a nominal fringe density of 4 pixels/fringe (PPF) or ½ Nyquist, as illustrated in Figure 3. If we tilt one way, the fringes can be up to Nyquist before we reach the limit implied by aliasing. If we tilt the other way, we can (in principle) go as far as null. These two directions are ± the number of fringes over the aperture, and consequently the cutoff frequency in cycles/aperture is equal to the fringe density in fringes/aperture. Thus for a camera, we have along the x and y axes a resolution cutoff frequency of 300 cycles/aperture. Along a diagonal, it is larger by the square root of 2, or 420 cycles/aperture. Where an instrument attenuates frequencies beyond 0.9 Nyquist, these resolution cutoffs are practically limited to 270 and 380 cycles/aperture, respectively. Servin and Estrada present a similar argument mathematically; noting that the fundamental requirement is that the phase gradient for the imposed spatial phase shift must be greater than local wavefront slope, regardless of the spatial phase shifting method. 9

4 Figure 3: Acquired interferogram sub-regions illustrating carrier fringes aligned for maximum lateral resolution in the Y direction (left), the X direction (middle), and along the diagonal (right). The resolution cutoff frequency is increased by the square root of 2 for the diagonal orientation corresponding to 420 cycles/aperture over a pixel camera. Practically, it is neither advised nor necessary to have a sharp decline in instrument response at the cutoff frequency, as this can generate ripple artifacts (Gibbs phenomenon) in the final 3D profile. It is preferable to engineer a gradual decline or attenuation of the instrument response as a function of frequency, with the result that the cutoff frequency represents a reduced response rather than a hard limit. A compromise is ~50% ITF at the ½ Nyquist cutoff, declining to zero at Nyquist, with approximately 4 pixels/fringes. Design compromises are required of all single-frame algorithms in order to balance lateral resolution limits and potential errors in the phase calculation, leading to a convergence in limiting ITF behavior between carrier fringes and alternative methods that are based on spatial phase shifting. Considering a specific example, when comparing our carrier fringe technique with the pixelated camera described by Millerd et al, 10 the only difference is the pixel pattern for the spatial phase shift. Kimbrough and Millerd show that the lateral resolution limits with a pixelated camera are comparable to the limits of the carrier fringe method, assuming a carrier fringe density as shown in Figure For the pixelated camera, lateral resolution may be extended beyond ½ Nyquist by means of compact, 2 2 pixel convolution kernels; however, small kernels lead to increasing phase calculation errors with large slope. 12 Balancing these considerations leads to ITF performance in pixelated-camera phase shifting comparable to that reported below for carrier fringes assuming that the carrier fringe density is at ½ Nyquist as in Figure 3. Even though both carrier fringe and pixelated camera interferometers may report non-zero ITF beyond ½ Nyquist, there are very reasonable questions about the usefulness of any spatial frequency information at these high frequencies. The caution is that the frequencies detected close to Nyquist may very well relate to aliased information caused by coupling with the spatial phase shift frequency. Ultimately, to gain full access to the lateral resolution afforded by a given camera format and matching coherent optical design, a PSI measurement is the best option. 5. MEASURED ITF FOR A STEP ARTIFACT There have been a number of reported techniques for qualifying the ITF of interferometric metrology systems. 3,4,5,6 Fundamentally, coherent interferometers are non-linear systems where spatial frequencies may couple together in the imaged wavefront. This is at odds with the desire to treat an interferometer as a linear system and qualify its ITF by using well-known techniques for calculating the modulation transfer function (MTF) of an optical imaging system. De Groot and Colonna de Lega have shown that restricting ourselves to surface height variations << /4 limits diffracted energy to the -1, 0, and +1 orders, placing us in the regime where an interferometer functionally responds as a linear system. 5 But is this a valid characterization of a Fizeau interferometer? If the intended application is metrology of optically polished surfaces, then one only need look at the power spectral density (PSD) of such surfaces to understand that the answer is YES the amplitude of mid and high spatial frequency content should be < 30nm (i.e., < /20). If the intent is to qualify the instrument for metrology of diffractive structures (i.e. gratings) with arbitrary depth, then the answer is NO; but thankfully this is not the designed intent, nor is it a common application, of a Fizeau interferometer. With the above limitations in mind, a step-height artifact was fabricated by etching a step of height ~35nm onto a polished glass substrate. Using lithographic fabrication techniques this allows for a nearly ideal edge. Shown in Figure 4 is the etched substrate measured at 1.73X magnification using the interferometer of Figure 1. The orange peel rippling in the surface finish is not ideal and will pollute the qualification of ITF at lower frequencies, but that is not our primary interest.

5 Figure 4: Step-height artifact measured at 1.73X magnification. A slice profile is inset and shows a step height of ~35nm, satisfying the criterion of surface height variations << /4, required for ITF characterization using linear systems theory. Qualification of a one-dimensional (1D) ITF is performed by carefully setting imaging focus for the object plane of the step artifact, measuring the step in a vertical or horizontal configuration, and finally by taking the square-root of the ratio of the PSD from the measured step to the PSD of an ideal step, PSD( x, y ) measured ITF( xy, ). (3) PSD( ) x, y ideal A program was written to calculate the ideal step height, and average ITF across many 1D slices. This reduces the noise in the resulting curve and reports the mean 1D ITF that describes a selected field of view. Figure 5 reports on the measured ITF using a temporal PSI algorithm at three magnifications (1.00X, 1.73X, and 3.00X) to qualify the interferometer of Figure 1. This provides a baseline describing the full lateral resolution capabilities of the hardware. ITF curves are often plotted versus fractional Nyquist or cycles/aperture on a Log 10 scale. While the curves look better, graphing on a logarithmic scale has at least two drawbacks: 1) more than half of the plot describes a regime where even a poorly designed instrument would perform well; and 2) it fails to highlight with clarity increasing resolution with image magnification. When plotted on a linear frequency scale, Figure 5 clearly illustrates the significant gains in resolution with magnification due to carefully designed coherent imaging. At all three magnifications, frequencies at ½ Nyquist result in an ITF of ~ 80%, only 10% lower than the detector sampling limit. Figure 5: Measured ITF using a PSI algorithm at 1.00X, 1.73X, and 3.00X magnification. ITF data is displayed in cycles/mm with a linear scale (left) and recast in cycles/aperture using a Log 10 scale (right). The sampling limit of the detector is additionally displayed on the Log 10 plot.

6 Figure 6: Measured ITF on a using a diagonal carrier fringe algorithm set to 4 pixels/fringe at 1.00X, 1.73X, and 3.00X magnification. ITF data is displayed in cycles/mm with a linear scale (left) and recast in cycles/aperture using a Log 10 scale (right). The sampling limit of the detector is additionally displayed on the Log 10 plot. Figure 7: Measured ITF at 1X magnification for: 1) coherent imaging using PSI and carrier fringe algorithms; and 2) a conventional laser Fizeau interferometer with incoherent imaging. The ITF performance of the same instrument was qualified using an instantaneous single-frame carrier fringe immediately following each PSI measurement, without making any adjustments to the cavity or instrument. This facilitates a rigorous comparison of lateral resolution differences between PSI and a carrier fringe method. Figure 6 summarizes the ITF results when using carrier fringe acquisition in the same manner and over the same limits as that of Figure 5. We see the expected gradual decline in instrument response owing to the designed attenuation of the carrier fringe algorithm with a more rapid falloff after ½ Nyquist. Even so, significant lateral resolution is retained and inspection of the data shows ITF still > 60% at 300 cycles/aperture for 1.00X and 1.73X magnifications, and an instrument response for 3.00X magnification of > 40% out to 9.0 cycles/mm. Finally, Figure 7 offers a comparison between the ITF of this instrument and a system with custom fixed magnification incoherent imaging. Notice that the instrument response for this instantaneous carrier fringe method utilizing coherent imaging is equivalent out to 3 cycles/mm to a temporal PSI system incorporating incoherent imaging. 6. VALIDATING MEASURED ITF WITH A CUSTOM PERIODIC ARTIFACT An alternative approach to evaluate a systems frequency response and lateral resolution across the field of view has concentrated on the fabrication, testing, and modeling of an artifact with periodic patterns that are linearly chirped in frequency. 13 Building on this concept we specified a similar custom artifact at a diameter of 100mm and added additional patterns with constant frequency as well as sampling in the azimuthal direction that contained periods as large as 0.25 cycles/mm and as small as 9.0 cycles/mm; Figure 8 and Table 2 provide a detailed description.

7 (a) Figure 8: (a) Photo of the Ø 100mm custom periodic artifact, after photolithography process and wet etch, but before final chromium layer was applied for uniform reflectivity. Details on the 12 different segments are summarized in Table 2. (b) Phase measurement at 3.00X magnification of final pattern height showing surface relief of 36.25nm. Segment Table 2: Detailed description of the pattern profiles in each of the twelve segments pictured in Figure 8. Description 1 Linear chirp in frequency, 9.0 cycles/mm 0.25 cycles/mm, center to edge 2 Linear chirp in frequency, 9.0 cycles/mm at mid-radius, expanding period towards center/edge 3 Linear chirp in frequency, 0.25 cycles/mm 9.0 cycles/mm, center to edge cycles/mm along azimuth at 2/3 radius, growing towards edge and wrapping to the center cycles/mm along azimuth at 1/3 radius, growing towards edge and wrapping to the center cycles/mm along azimuth at center, azimuthal period growing with radius to edge Segment Description Segment Description cycles/mm cycles/mm cycles/mm cycles/mm cycles/mm cycles/mm The custom periodic artifact of Figure 8 facilitates validation of the stated lateral resolution across the full field of view of an instrument. A rotation and measurement of the artifact at 30 degree increments enables a synthesized full-field measurement of each segment s pattern through data parsing and recombination. The periodic artifact also provides a nice way of graphically validating the ITF results of Section 5 for a few selected segments. Segment 1 is a chirped grating that ranges from 0.25 cycles/mm to 9.0 cycles/mm edge to center, respectively. Figure 9 displays a phase map and corresponding slice at 1.00X magnification through segment 1 for both PSI and instantaneous carrier fringe acquisition. For the PSI profile we see roughly 67% amplitude at 4.0 cycles/mm, which correlates well with the ITF results in Figure 5. Inspection of the carrier fringe profile shows similar agreement to Figure 6 with roughly 32% amplitude falling at 3.5 cycles/mm. (b)

8 Figure 9: Acquired and processed phase maps of chirped grating segments acquired with PSI and instantaneous carrier fringe algorithms at 1.00X magnification. A slice profile is sampled in segment 1 for each acquisition and plotted below along with an attenuated height envelope defined by the ITF results of Figure 5 and Figure 6. Inspection of segments with constant radial period allow for a direct point-to-point comparison between the ITF curves and the profile amplitude. Segment 9 (4.5 cycles/mm) was measured with instantaneous carrier fringe at 1.73X magnification and the resulting phase map and profile are displayed in Figure 10. Locating 4.5 cycles/mm on the 1.73X curve of Figure 6 shows an ITF of 66%. Multiplying the pattern depth (36.25nm) by 0.66 yields 23.95nm. The actual profile height matches this prediction. Figure 10 also shows a phase map and profile from segment 12 (9.0 cycles/mm) as measured with PSI at 3.00X magnification. Locating 9.0 cycles/mm on the 3.00X curve of Figure 5 shows an ITF of 76%, predicting an amplitude of 27.55nm.

9 Figure 10: Phase maps of the six constant period radial grating segments. 1.73X magnification for instantaneous interferometry is shown for 1.5, 3.0, and 4.5 cycles/mm, and 3.0X magnification by PSI is shown for 6.0, 7.5, and 9.0 cycles/mm. Slice profiles sampled from the 4.5 cycles/mm and 9.0 cycles/mm segments are shown below with ITF adjusted reference lines for the expected pattern height. 7. SUMMARY AND CONCLUSION Metrological integrity of both low-order figure and mid spatial frequency content is a requirement for today s precision optical surfaces where sub-aperture polishing is increasingly adopted to support tighter customer demands. Many of these surfaces require testing within environments or configurations where the only practical metrology solution is single-frame instantaneous interferometry. This paper introduced and qualified a unique Fizeau interferometer designed and optimized at three magnifications for lateral resolution and slope acceptance in both instantaneous carrier fringe and temporal phase shifting (PSI) configurations thereby enabling, with one instrument, an increased measurement space for the optical test community. Slope acceptance was demonstrated for 1X, 1.73X, and 3X magnifications up to 0.9 Nyquist with residual rigid-body retrace < 0.25 waves PV, supporting accurate measurement of figure error even for mild

10 aspheres. In addition, ITF was demonstrated as > 60% and > 80% out to ~ ½ Nyquist for single-frame carrier fringe acquisition and temporal PSI acquisition, respectively. At 3X magnification (~ 35mm field-of-view) and using singleframe acquisition, periodic content out to 9.0 cycles/mm may still be imaged with a system response > 40%. Lastly, correlation of the predicted attenuation by measurement of a custom periodic artifact, employing both chirped radial gratings and constant radial grating patterns, validates the ITF results. The author gratefully acknowledges contributions to an ITF processing utility from Mike Holmes and Mike Woynar (ZYGO), and also thanks Ulf Griesmann (NIST), for assistance in creating the mask pattern used in fabrication of the custom periodic artifact. REFERENCES [1] Evans, C. J., Compensation for errors introduced by nonzero fringe densities in phase measuring interferometers, CIRP Ann. Int. Inst. Prod. Eng. Res., 42/l, (1993). [2] Huang, C., Propagation errors in precision Fizeau interferometry, Appl. Opt. 32, (1993). [3] Wolfe, C. R., Downie, J. D., and Lawson, J. K., Measuring the spatial frequency transfer function of phasemeasuring interferometers for laser optics, Proc. SPIE 2870, (1996). [4] Novak, E., Ai, C., and Wyant, J. C., Transfer function characterization of laser Fizeau interferometer for high spatial frequency phase measurements Proc. SPIE 3134, (1997). [5] de Groot, P., and Colonna de Lega, X., Interpreting interferometric height measurements using the instrument transfer function, in Proc. FRINGE 2005, W. Osten, Ed., (Springer Verlag, Berlin Heidelberg, 2006). [6] Yashchuk, V. V., McKinney, W. R., and Takacs, P. Z., Binary pseudorandom grating standard for calibration of surface profilometers, Opt. Eng. 47, (2008). [7] Sykora, D. M., and de Groot, P., Instantaneous interferometry: another view, OF&T 2010 Technical Digest, OMA1 (2010). [8] Sykora, D. M., and Holmes, M. L., Dynamic measurements using a Fizeau interferometer, Proc. SPIE 8082, 80821R (2011). [9] Servin, M., and Estrada, J. C., "Error-free demodulation of pixelated carrier frequency interferograms," Opt. Express 18, (2010). [10] Millerd, J., Brock, N., Hayes, J., North-Morris, M., Novak, M., and Wyant, J., "Pixelated phase-mask dynamic interferometer," Proc. SPIE 5531, (2004). [11] Kimbrough, B., and Millerd, J., The spatial frequency response and resolution limitations of pixilated mask spatial carrier based phase shifting interferometry, Proc. SPIE 7790, 77900K (2010). [12] Kimbrough, B. T., Pixelated mask spatial carrier phase shifting Interferometry algorithms and associated errors, Appl. Opt. 45, (2006). [13] Chu, J., Wang, Q., Lehan, J. P., Guangjun, G., and Griesmann, U., Spatially resolved height response of phaseshifting interferometers measured using a patterned mirror with varying spatial frequency, Opt. Eng. 49, (2010).