Phase-measuring interferometry (PMI) is the metrology tool of choice for the optical industry, measuring key parameters such as surface figure, transmitted wavefront, homogeneity, and radius of curvature for process control and quality assurance. Previously confined to the quality-control laboratory, PMI has become necessary technology for monitoring surface figure and transmitted wavefront on the production floor. Manufacturing tolerances for optical components as diverse as cell phone lenses and large telescope mirrors require interferometric-level metrology in environments in which low-noise testing facilities are impractical. New instantaneous PMI (IPMI) techniques can operate in these production environments with success, but care must be taken regarding systematic errors to assure good metrology. The Basics
An interferometer measures the phase difference between two wavefronts. To measure an optical component, a reference surface is compared to the test part, where each component produces a separate wavefront. Modern interferometers use computerized data acquisition techniquesPMIto measure these phase differences. The classic PMI technique, phase-shifting interferometry (PSI), uses multiple frames of data acquired in sequence while shifting the phase. Phase shifting is performed through either mechanically changing the length of the interferometer cavity or by changing the wavelength during data acquisition such as wavelength specific PSI (WSPSI). Both PSI approaches provide very low measurement uncertainty.
Recently, new techniques have been invented for enhanced performance: Fourier-transform PSI, for measuring multiple surfaces, and IPMI, for data acquisition in production environments.1 We are going to focus on the latter.
In all PSI techniques, each pixel independently extracts the wavefront phase using the measured intensity variation as a function of phase shift. The measured intensity I depends on phase φ via the equation
I = A + B cos φ (1)
where A and B are the offset and amplitude of the observed intensity variation. Since A, B, and φ are unknowns, we require at least three independent intensity measurements to recover the phase; typically, more are taken. In all PSI techniques, each pixel independently extracts the phase information.
For optimal performance, it is important that the optical rays trace back over the same paths in the reference surface and test parts. This is true in the most common Fizeau PSI interferometers. The higher the quality of the test part and the better the fringe alignment, the lower the measurement uncertainty. Optical rays tracing back over the same paths negate imperfections in the optical system, even if the wavefront within the system has measurable errors. This is not true with all PMI techniques or interferometer configurations. This natural alignment reduces measurement uncertainties in high-end commercial Fizeau interferometer systems to much better than λ/300. With proper temperature control, reference surface calibration, and careful handling of the test partthe kinds of conditions afforded in test laboratoriessuch commercial systems can achieve measurement uncertainties of better than λ/1000.
IPMI systems sacrifice this natural alignment to obtain rapid phase data. The key innovation in IPMI is capturing the interferometer phase data in a single camera frame to freeze out vibration and turbulence effects. To get around the multiple measurement requirements inherent in equation 1, IPMI encodes the intensity variation spatially rather than temporally, using either hardware or software techniques.
To spatially encode the intensity variation, hardware techniques apply specialized optical elements to replicate phase-shifted images, and simultaneously display the images on the same or multiple cameras. Software techniques leverage a constant phase shift across a single image with a tilted interferometer wavefront, and mathematically extract the phase variation across the field.2 An advantage of the latter technique is an easy upgrade for existing PSI interferometer systems. Note that the optical rays trace back over different paths so that imperfections in the optical system are not negated but measured; we call these ray-mapping errors. Calibration
All IPMI systems have separated optical paths, which increases measurement uncertainty. Uncalibrated system specifications of commercial IPMI instruments range from λ/10 to λ/20, 15 to 30 times worse than a PSI system. Some applications require system calibration to minimize these large systematic errors.
The most common calibration technique is measuring a high-quality calibration part. We assume the calibration part has very small wavefront errors compared to the interferometer. The measured result should therefore reveal only the systematic errors in the interferometer. This calibration measurement is saved and subtracted from future test part measurements.
National standards institutes can measure calibration parts to very low uncertainties. Once the calibration part leaves the calibration lab, however, numerous factors can degrade its usefulness. If a part was calibrated under tight laboratory conditions (typically 20°C ± 0.1K), the measurement only applies in an equal environment. This is especially true for parts mounted in metal housingsdifferential thermal expansion warps the part when the measurement temperature deviates from the test temperature, and thermal cycling will cause thermal gradients through the test part, also warping the calibration part.
The actual gradients will depend on the interferometer temperature, the air temperature, the amount of space between the test part and the reference, and the temperature cycling period of the production environment. It is not unusual for production or even large laboratories to exhibit thermal cycles of ±0.5°C over various periods. This magnitude of variation could cause uncertainties of 100 nm in a 300-mm flat, or λ/6 uncertainty. Because the calibration part is affected by the production-environment temperature instabilities and measurement uncertainties during calibration, this calibration process, though simple, has limitations.
The hardware IPMI approaches may be limited to the calibration part method, but the software-based carrier-fringe IPMI technique has several other calibration options. Carrier-fringe IPMI requires the introduction of significant tilt in the measurement cavity. The interferometer contributes an increasing systematic error with increasing cavity tilt because of ray-mapping errors, as noted above. Tilt-induced ray-mapping errors are, however, well understood, mathematically well behaved, and therefore easily calibrated and removed from the results.3,4 Before calibration, these errors are typically less than λ/20, small by IPMI standards but significant enough to require removal from the data via calibration.
Extensive calibration processes that were previously considered impractical are now straightforward with today's computers. Ray-mapping errors introduce non-axially-symmetric aberrations, typically coma and astigmatism. The first calibration technique leverages the mathematical smoothness of the aberrations. After acquisition of a series of IPMI measurements with increasing tilt, we can calculate and remove ray-mapping errors from the series. This technique is good to λ/50 and requires no special hardware or outside calibration.
The second technique uses the gold-standard PSI data acquisition to measure the non-axially-symmetric aberrations induced in the measurement cavity. This data is stored, just like in the calibration part method and removed from future carrier fringe data sets. The system is internally calibrated off the gold-standard PMI system without reliance on external measurement artifacts. Both techniques are good to λ/50 uncertainties, sufficient for most production requirements. After Calibration
With the interferometer calibrated, the remaining systematic errors are introduced by the reference surface, environment, and test part. The interferometer reference surface and test part experience the same influences, primarily due to temperature effects and mounting stresses. The smaller the temperature variation, the better the measurement. Your measurement uncertainty is never better than the total range of the measured variations. Quantifying the magnitude of these variations establishes the best-case systematic errors in the test setup. The best method for accomplishing this involves acquiring data over a long period, typically over several days at different times. The variations are primarily due to temperature changes.
The remaining environmental influences of turbulence and vibration are primarily random in nature. Averaging results minimizes these effects. In general, the random noise decreases as the inverse square root of the number of averages; thus, a 10X reduction in random environmental noise requires averaging 100 measurements together. This is a very simple operation in all systems and very fast due to rapid data processing.
New IPMI technologies allow measurement in production environments. The uncertainty of those measurements is determined by calibration quality and environmental control, of which temperature stability is the most important parameter. Now that data acquisition is possible in harsh environments, assuring meaningful results requires more attention to metrology detail. The choice of technology approach greatly affects the calibration methods available. oe
1. L. Deck, Appl. Opt. 42, p. 2354 (2003).
2. L. Deck, Proc. SPIE 5532, p. 159 (2004).
3. C. Evans, Annals of the CIRP 42, p. 577 (1993).
4. C. Evans, Optical Fabrication and Test Workshop, p. 259 (1994).
Robert Smythe is corporate vice president and director of interferometer products at Zygo Corp., Middlefield, CT.