rotary stage, positioned so the gap is over the center of rotation. The laser beam used to interrogate the sample originates from a frequency stabilized laser of ...
MEASUREMENT OF THE ABSOLUTE DISTANCE BETWEEN TWO PARTIALLY REFLECTIVE SURFACES Vivek G. Badami, Leslie L. Deck, and Lars A. Selberg Zygo Corporation Middlefield, CT, USA
This paper describes an interferometric technique for measuring the separation between two plane-parallel, uncoated, dielectric surfaces with separations of a few mm. This technique is similar to the one described by Coppola et al [3], but differs from it in some significant respects. A description of the method is provided along with an uncertainty analysis that supports the techniques’ potential to perform measurements with sub-nm uncertainty. Preliminary experimental results are also included. THE MEASURAND The objective of the measurement is to determine the absolute separation between two plane surfaces. An assembly of the type used in the measurements described here is shown in Figure 1, and is comprised of two wedged optical flats assembled with an intervening annular spacer to form a rigid assembly. The measurand is the separation between the two inner surfaces denoted by d in the figure and is hereafter referred to as the gap. MEASUREMENT PRINCIPLE The method is based on the change in the optical path difference (OPD) between the beams reflected from the two surfaces of the sample due to physical rotation (in the direction
shown in Figure 1), which for a given rotation angle is a function of d. WEDGED OPTICAL FLATS
SPACER
2 1 θi
d
θt
SAMPLE ROTATION
INTRODUCTION The absolute distance between two surfaces is a parameter of interest in many applications. Examples include wavelength standards, etalons, gage block calibrations, the measurement of sphere diameters, etc. Many interferometric techniques exist for making such a measurement. Of these, the highest precision methods that do not involve modifications to the surfaces are based on the method of exact fractions [1] and wavelength tuning [2]. The former method is limited due to the phase resolution attainable from analysis of a static fringe pattern, while the latter technique requires a wide tuning range to measure relatively small separations (of the order of a few mm).
FIGURE 1. Interference between reflections from the two surfaces of the sample. This dependence of the OPD between the beams (beams 1 & 2) reflected from the two surfaces of the plane-parallel transparent region (bounded by the wedged optical flats and depicted in Figure 1) on the transmitted angle θt is given by 2ndcosθt, where n is the refractive index of the medium and θt is in turn a function of the angle of incidence θi. The change in OPD causes an intensity variation due to interference between the reflected beams. Therefore, the gap may be extracted from knowledge of the rotation angle and the corresponding intensity variation. EXPERIMENTAL SETUP & PROCEDURE A schematic of the experimental setup is shown in Figure 2. All adjustments and motions are relative to the coordinate system shown. The axis of rotation is assumed to be coincident with the z axis, while the y axis is coincident with the input beam from the laser. The artifact or sample to be measured is mounted via a six degree-of-freedom (6DOF) adjustment (not shown) to a high-accuracy rotary stage, positioned so the gap is over the center of rotation. The laser beam used to interrogate the sample originates from a frequency stabilized laser of known wavelength. The beam passes through an uncoated wedged plate beamsplitter (B/S) before entering the
artifact. The surface of the B/S closest to the sample serves as the splitting surface. The AXIS OF ROTATION ROTARY y TABLE
WEDGED PLATE B/S FREQUENCY STABILIZED LASER
BAFFLE
ARTIFACT
z
MOTION CONTROLLER
TRIGGER
LENS
x
APERTURE
LARGE AREA PHOTODETECTOR
A/D CONTROL COMPUTER
FIGURE 2. Schematic of experimental setup. wedge in the B/S and the wedged optical flats of the sample prevent spurious reflections. The reflected beams from the two surfaces of interest of the artifact are reflected by the B/S and enter the detector assembly. The beam passes through a baffle and is then incident on a lens, the focus of which is located in the plane of a variable aperture (iris diaphragm). The beam passes through the aperture and strikes a largearea photodetector. The aperture and the baffle work to minimize stray light. Note that the signal from the artifact is monitored in reflection, rather than in transmission as in the technique described by Coppola et al [3]. The arrangement described here provides a much higher S/N ratio as there is no DC light intensity in the measured signal. x
y
φ z
αsample
αBS
ENTRANCE FLAT
AXIS OF ROTATION
FIGURE 3. Wedge and relative orientation angles. Alignments Several critical alignments & measurements are performed during setup. These include aligning the plane of the wedge of the entrance flat of the sample and B/S to the plane of rotation, positioning the center of the spacer over the
center of rotation and establishing the nominal ‘zero’ position of the rotary stage. The latter corresponds to the situation where the input beam is normal to the surface(s) of interest. Several angles are required as inputs to the mathematical model of the system in addition to the rotary stage angle. The wedge angles of the B/S and the entrance flat of the sample (αBS and αsample in Figure 3) are measured along with the angle that establishes the relationship between the B/S and the entrance flat of the sample (φ in Figure 3). These angles are used to compute the angle of incidence at the gap as a function of stage rotation. The sample is rotated about the z axis by the rotary stage under computer control. The output of the detector is digitized and the data acquisition is triggered by pulses from the rotary stage encoder, thus rendering data collection immune to variations in stage velocity. This also permits registration of successive runs. A typical measurement commences by driving the stage to one end of the desired travel. The stage is then rotated through the desired travel while the voltage from the detector is recorded simultaneously and synchronously with the stage position. This is designated as one scan. The stage then returns to its start position before starting the next scan. Any given measurement is comprised of five such scans which are then averaged. Typical rotations are ±5° about the zero position at a scan speed of 1°/sec. UNCERTAINTY ANALYSIS This uncertainty analysis is based on a Monte Carlo simulation and is performed in accordance with the methodology outlined in Supplement 1 [4] to the ISO Guide to the Expression of Uncertainty in Measurement (GUM) [5]. The influence quantities, their respective uncertainties and their contributions are listed in Table 1. All contributing distributions are assumed to be uniform distributions. The analysis generates a simulated intensity trace as a function of angle. This simulated data is then subjected to the same fitting procedure used to process the real data while the input parameters to the fitting function are varied in accordance with the distributions assigned to each parameter. The significant contributors are the uncertainties in index of air in the gap, the measurement of the rotation angle, measurement of αBS and the
wavelength. The uncertainty in the index of air in the gap is derived from uncertainties in measuring temperature and pressure of 0.1°C and 1 mm Hg respectively. The uncertainty in the measurement of rotation angle is derived from the stage manufacturer’s specification. The wavelength is measured by beating the laser against an iodine stabilized laser (measurement uncertainty of 2.5x10-11) and the uncertainty is derived from the specification on the tolerances on the components of the control loop used to stabilize the frequency. Uncertainties in the index of the various fused silica optical flats are based on the variability in the index as reported by Malitson [6]. Uncertainty estimates for angle measurements are based on what is easily achievable with the equipment at hand. TABLE 1. Influence quantities and contributions for a 1.5 mm gap. Uncertainties values listed are the half-widths of uniform distributions. Contribution (pm) Index of air in gap 0.18 ppm 155 Rotation angle 3.6 arcsec 60.4 10 arc sec 52.4 αBS Wavelength 0.05 ppm 43.3 10 arcsec 31.4 αsample Measured intensity 1% 11 10 arcsec 2.65 φ Index of B/S 10 ppm 0.45 Ambient air index 5 ppm 0.24 Index of entrance flat 10 ppm 0.03 Expanded uncertainty (k=2) 353.9 The contribution from the uncertainty associated with the index of air in the gap dominates the resulting expanded uncertainty (k=2) of ~354 pm. Influence Quantity
Uncertainty
DATA PROCESSING The data reduction process consists of performing a nonlinear least-squares fit to the measured intensity, with the rotation angle as an independent parameter and the gap as a free parameter. The model function for the fit requires the critical angles shown in Figure 3, the vacuum wavelength, the refractive indices of the entrance flat of the sample and B/S, and the polarization direction. Snell's law and Fresnel’s equations are then used to calculate the optical paths lengths and reflection coefficients, respectively, of the two reflected beams to compute the expected interference pattern as a function of beam launch angle at the B/S and
the gap. The square of the residual to the fit is used as a merit function for the fitting routine. Excellent agreement was observed between the data and the fit (Figure 4).
FIGURE 4. Comparison of measured and fit intensities. The fringe order is calculated from an a priori estimate of the gap (uncertainty ~50 nm) based on measurements of the gage blocks used as spacers in the artifact (see next section), and the method is used to determine only the fringe fraction. The merit function has a well defined minimum which is used to compute the fringe fraction. This fringe fraction and fringe order values are combined to calculate the gap. The technique has the potential to also determine the fringe order. This will be validated in future work. TABLE 2. Measurement results. Gap
Cal. Value
R1 R2 RN
(μm) 1270.0121 1778.0254 1500.0000
Meas. Value. (μm) 1270.0216 1778.0461 1500.1058
Δ
σ
(nm) 9.55 20.65 105.82
(nm) 0.22 0.36 0.10
EXPERIMENTAL RESULTS The method was tested on three samples, two of which were constructed with Starrett® square croblox® gage blocks (0.050” & 0.070”) as spacers, while the third one was constructed with two closely matched rectangular gage blocks of the same nominal size (1.5 mm) as spacers. The three gaps are designated R1, R2 and RN respectively. In the case of the first two samples, the gap is formed by the space enclosed by the hole in the block and the optical flats and is nominally sealed. In the third case, the two blocks are placed parallel to one another with a space between them. The gap of interest is the separation between the flats in the region between the blocks. The samples are
assembled by a procedure similar to optical contacting and then mounted in spring loaded cells for ease of handling. The results of the measurements of the physical gap are shown in Table 2. The table lists the calibrated value of the gage block used as the spacer (the average value of the two blocks is listed for RN), the mean measured value of the gap, the deviation of the measured value from the calibrated value Δ, and the standard deviation σ of the five measurements used to compute the mean measured value. The measurements must be compared to the reference values in the context of the uncertainties of the measured values and the calibrated values. The expanded uncertainty (k=2) of the measured value is estimated in an earlier section to be ~0.35 nm. The uncertainty in the reference value of the gap is derived from the uncertainty in the calibrated value of the gage block (~30 nm), and uncertainties associated with the ‘film’ between the two contacting faces of the block and the flats, and the correction of the calibrated length to the measurement temperature (~2 nm). While the films are assumed to have zero thickness, the uncertainty in this value is assumed to be 25 nm [7] each, which when added in quadrature to the uncertainty in block length results in an expanded uncertainty (k=2) of ~46.5 nm. When viewed in the context of these uncertainties, it can be seen that the measurements for R1 and R2 agree to within this uncertainty. It is also evident that the deviation in the case of RN exceeds the uncertainty in the calibrated value. The discrepancy is believed to arise as consequence of the method of construction of the artifact. This artifact is constructed by sandwiching two rectangular gage blocks placed parallel to one another. Due to the slight difference in the dimensions of the blocks and lack of coplanarity of the contacting faces, it is quite likely that the contact between the flats and the blocks is not as intimate as in the case of the other cavities. It is also possible that the flats are distorted due to the same reasons, with the result that the gap produced does not agree with the dimensions of the spacers. For these reasons, it is probably inappropriate to use the calibrated values of the gage blocks as reference values for RN. The measurements have also proved to be stable over time and a sense for the reproducibility of the measurement can be derived from measurements of the gaps almost six months apart under similar (but not
identical) conditions. R1 and R2 show values that agree to within 2 nm over this period. RN, however shows a difference of 12 nm. This instability is probably an indication of lack of stability and marginal quality of the assembly. CONCLUSION A method for making absolute distance measurements between two partially reflective, uncoated, plane-parallel surfaces separated by a few mm has been described. The supporting uncertainty analysis suggests that the technique is capable of sub-nm uncertainties. Results of preliminary experiments with well constructed samples show that the results agree to within the uncertainty of the calibrated references. It is also clear that the uncertainties of the reference values used in these measurements is large compared to the estimated uncertainty of the technique and another technique with commensurate uncertainty is required to completely test this technique. ACKNOWLEDGEMENTS The authors would like to acknowledge the input and assistance of Dr. Xavier Colonna de Lega, Andrew Stein and Andrew Carlson. REFERENCES [1] Tilford CR. Analytical procedure for determining lengths from fractional fringes. Applied Optics. 1977; 16(7): 1857-1860. [2] Deck L. Absolute distance measurements using FTPSI with a widely tunable IR laser. Proc SPIE. 2002; 4778: 218-226. [3] Coppola G, Ferraro P, Iodice M, and De Nicola S, inventors; Consiglio Nazionale Delle Ricerche, assignee; Interferometric system for the simultaneous measurement of the index of refraction and of the thickness of transparent materials, and related procedure. United States patent US 7046373, 2006 May 16. [4] BIPM. Evaluation of measurement data – Supplement 1 to the "Guide to the expression of uncertainty in measurement" – Propagation of distributions using a Monte Carlo method. JCGM 101:2008. [5] ISO. Guide to the expression of uncertainty in measurement. 1st ed. Geneva: ISO, 1995. [6] Malitson IH. Interspecimen comparison of the refractive index of fused silica. JOSA. 1965; 55(10): 1205-1209. [7] Doiron T and Beers JS. Gage Block Handbook. NIST Monograph 180, 2005.
