Absolute surface metrology with a phase-shifting interferometer for incommensurate transverse spatial shifts E. E. Bloemhof National Science Foundation, Arlington, Virginia 22230, USA ([email protected]) Received 26 August 2013; revised 24 November 2013; accepted 23 December 2013; posted 3 January 2014 (Doc. ID 196439); published 4 February 2014
We consider the detailed implementation and practical utility of a novel absolute optical metrology scheme recently proposed for use with a phase-shifting interferometer (PSI). This scheme extracts absolute phase differences between points on the surface of the optic under test by differencing phase maps made with slightly different transverse spatial shifts of that optic. These absolute phase (or height) differences, which for single-pixel shifts are automatically obtained in the well-known Hudgin geometry, yield the underlying absolute surface map by standard wavefront reconstruction techniques. The PSI by itself maps surface height only relative to that of a separate reference optic known or assumed to be flat. In practice, even relatively high-quality (and expensive) transmission flats or spheres used to reference a PSI are flat or spherical only to a few dozen nanometers peak to valley (P-V) over typical 4 in. apertures. The new technique for removing the effects of the reference surface is in principle accurate as well as simple, and may represent a significant advance in optical metrology. Here it is shown that transverse shifts need not match the pixel size; somewhat counterintuitively, the single-pixel spatial resolution of the PSI is retained even when transverse shifts are much coarser. Practical considerations for shifts not necessarily commensurate with pixel size, and broader applications, are discussed. OCIS codes: (120.3180) Interferometry; (120.6650) Surface measurements, figure. http://dx.doi.org/10.1364/AO.53.000792
1. Absolute Surface Metrology from Phase-Shifting Interferometry
Measurement of the nanometer-scale absolute surface figure of an optic under test with a phase-shifting interferometer (PSI) has been a subject of considerable study because a PSI normally measures that surface only relative to the unknown underlying shape of its reference surface. That is, the measurement of a test optic is really the measurement of the two-surface cavity it forms with the reference surface, called the transmission flat or transmission sphere. Many solutions have been proposed to extract the surface of the test optic alone [1–5]. A number of these involve introduction of one or more additional surfaces, then comparing the resulting pairwise combinations of surface shapes. Recently, a novel approach was proposed [6] that may give improved simplicity and accuracy with only two auxiliary measurements (small-amplitude lateral shifts of the test optic in each of the transverse 792
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coordinate directions) and no auxiliary optic. The new approach may also generalize to other applications. For simplicity, the original presentation of the technique assumed transverse shifts in x- and y-directions equal to the pixel size (really, to the center-to-center spacing of pixels). While intuitively it is clear that the new technique still yields absolute metrology when this restriction is relaxed, it is useful to examine how the computations should be generalized for the more realistic case of transverse shifts not equal to pixel dimensions. The results may be of considerable practical importance, and they include the surprising conclusion that measurements with shifts substantially larger than the pixel size effectively retain the spatial resolution set by the PSI pixel scale. 2. Commensurate Case: Shifts Equal to Pixel Size
A brief review of the new transverse-shifting technique for absolute metrology [6] is now given; the
experimental situation is shown in Fig. 1. A PSI returns a map of the test optic surface shape by combining data from a few phase-shifted exposures obtained in parallel or in rapid succession, by using the four-bin or similar algorithm. Each exposure is made at a distinct piston phase value of the reference surface through, e.g., piezoelectric actuation. Overall piston of a PSI map is unknown, and tip/tilt is generally not of interest, so tip/tilt of the test optic need not be maintained to any degree of precision during transverse shifts. What is generally desired is the higher-order shape of the test optic, showing local surface structures, and it suffices to reference every map (cavity or absolute) to the mean piston and overall tip/tilt of a flat under test. Because the PSI measures a cavity phase, the starting maps it produces inherit the imperfect flatness of the reference surface. This dependence on reference surface shape can be expressed mathematically as follows. Any line of sight at transverse position x; y through test and PSI optics to a given pixel of the PSI camera can be written, for one choice of sign convention, as Φ0 x; y φmirror x; y − ψ ref x; y;
(1)
where mirror and reference flat phase maps are φmirror x; y and ψ ref x; y, respectively. The PSI returns a phase map Φ0 x; y that is the difference of these two unknown maps. The new approach for removing dependence on the reference flat map, and thus obtaining the absolute shape of the test optic, is to shift the test optic laterally and form differences with the original map. This effectively results in double differences of phase (height) from which the reference shape is accurately cancelled. Explicitly, the auxiliary maps obtained with small shifts in x and y are ΦΔx x; y φmirror x Δx; y − ψ ref x; y ΦΔy x; y φmirror x; y Δy − ψ ref x; y;
(2)
Fig. 1. Measurement schematic (from [6]): the surface height of the mirror under test is mapped at subnanometer precision, but only relative to the surface shape of the transmission flat (reference surface). The reference surface is typically known to be flat to perhaps tens of nanometers over transverse scales of interest, but will have unknown spatial structure inherited by the PSI measurement. Phases are taken as proportional to surface displacements in a common direction, since the PSI measures distance between the two surfaces; an alternative sign convention, with phases proportional to surface heights, would give a PSI measurement equal to the sum of the test and reference phases.
because the reference surface shape ψ ref x; y is fixed with respect to the camera whose pixels record the PSI images (the line of sight of a given pixel intercepts the same portion of the reference surface, as nothing has moved transversely inside the PSI, but views a new position on the test optic). This elimination of the reference surface by differencing the two transversely shifted PSI maps with the original map is seen mathematically by subtracting Eq. (1) from Eq. (2): ΦΔx − Φ0 φmirror x Δx; y − φmirror x; y ΦΔy − Φ0 φmirror x; y Δy − φmirror x; y:
(3)
These are absolute measurements of x- and y- height differences at an array of points on the test optic; these may or may not form an interlinked grid or lattice. In an alternative sign convention, phases at reference and test-optic surfaces are taken to be proportional to surface heights, with height axes pointing in opposite directions; then the reference phases should enter Eqs. (1) and (2) additively. Those reference phases of course still cancel out on differencing Eqs. (1) and (2) to obtain Eq. (3). The convention used here is perhaps more physical, measuring phase in a common spatial direction, and mathematically emphasizing that the PSI measures the test surface relative to the reference surface by returning the gap, or position difference, between the two. When transverse shifts are equal to one pixel, the data lie in the Hudgin geometry, as shown in Fig. 2. The recovery of the absolute wavefront (test optic surface height) from this grid of absolute differences is a standard exercise in wavefront reconstruction, discussed in [7–9]. The more general case, in which shifts are incommensurate with pixel size (spanning multiple pixels and/or fractional pixels) is considered in more detail later in the current paper. Reviewing the commensurate case in which an auxiliary PSI map is obtained at a transverse shift of one pixel in each of the x- and y-directions (Fig. 2), the absolute phase differences of Eq. (3) form an overconstrained system of linear equations in the absolute phase values:
Fig. 2. Schematic of absolute test surface reconstruction in the Hudgin geometry for the commensurate case where transverse shifts are equal to PSI pixel dimensions. Circles at pixel center denote nodes at which the phase (surface height) is desired; arrows denote phase steps (differences) measured in an absolute way by the transverse shifting technique, from Eq. (3). Reconstruction of these measurements yields the full absolute surface map. 10 February 2014 / Vol. 53, No. 5 / APPLIED OPTICS
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⃗ ⃗n; S⃗ AW
(4)
where wavefront slopes or differences S are related to the underlying wavefront or surface height map W, including a noise term represented by n. The wavefront values W sought are arranged here in a vector of length N 2 , where each phase map measures N × N. Absolute wavefront differences S found by differencing shifted and nominal PSI maps are arranged in a vector of length 2NN − 1, and the sparse matrix A, with dimension N 2 × 2NN − 1, has nonzero elements given by Aq p − 1N − 1; q p − 1N −1 Aq p − 1N − 1; 1 q p − 1N 1 for p 1; …; N;
q 1; …; N − 1
Ar NN − 1; r −1 Ar NN − 1; r N 1 for r 1; …; NN − 1:
(5)
The solution to Eq. (4) that minimizes the errors in a least-squares solution for this overconstrained problem is well known; neglecting the noise term it is ⃗ ⃗ AT A−1 AT S: W
(6)
There is one technicality in the current situation: the matrix A is not of full row rank, and so AT A−1 does not exist. This reflects the fact that the overall piston term of the phase map W is not defined. The situation is easily remedied [8] by extending A with a row of 1’s and extending the vector S with a single 0, equivalent to setting overall piston to zero. This procedure yields the underlying absolute phase map, free of contamination from the additive phase map introduced by the reference surface (transmission flat). Just two additional auxiliary maps, with the test optic shifted transversely in x and y, are required in addition to a map at the original position. An example of the very high intrinsic fidelity of the technique is shown in the simulation of Fig. 3. In the next sections it will be seen that the new technique is experimentally robust and does not depend on precise single-pixel transverse shifts, i.e., the incommensurate transverse shifts likely to be obtained in practice do not significantly degrade the utility of this approach.
Fig. 3. Simulation of absolute phase extraction despite phase corruption by the reference surface. (a) Underlying phase or surface height of test optic to be extracted, (b) phase map of corrupting PSI reference surface, (c) difference of these, the resulting distorted cavity phase map measured by the PSI, (d),(e) absolute surface differences evaluated along x- and y-axes from transverse shifting technique, and (f) reconstructed surface of test optic, in good agreement with panel (a).
themselves is straightforward. As will be seen, reconstruction requires shift knowledge, rather than control of the shifts, and tolerances on that knowledge are relatively loose for practical purposes. In this section transverse shifts that are even multiples of pixel size are considered. So if the pixel dimensions are x0 ; y0 , the transverse measurement shifts have the form Δx; Δy Mx0 ; Ny0 , with M and N integers. A concrete illustration is given in Fig. 4, with M 3 and N 2. The differencing of transversely shifted phase maps outlined in Section 2 now produces absolute phase/height differences connecting a grid of nodes separated by intervening pixels not included in the grid. Each camera pixel in the difference maps, Eq. (3), measures the absolute phase difference between two surface points separated by the transverse shift, even when that shift is large, but application of the Hudgin geometry machinery now takes some care.
3. Absolute Maps from Shifts that are Multiples of the Pixel Size
In the more general case, transverse shifts are not equal to the pixel dimension, so the precise Hudgin geometry will not obtain. Pixels are typically small, so shifts may be several times larger. However, the optics of the PSI generally provide clear transverse registration of the test optic, by sighting on a surface feature or on the edges of the optic. So accurate determination of spatial shifts from the PSI data 794
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Fig. 4. Schematic of absolute test surface reconstruction when transverse shifts are multiples of pixel dimensions. Phase values are sought on the nodes represented by circles at pixel center. Transverse shifts, by Eq. (3), provide absolute phase steps (differences) between pixels connected by arrows. Though noncontiguous, pixels in this subgrid may be treated by standard Hudgin geometry reconstruction.
The simple approach presented here is to separately analyze each subgrid for which shifted pixel positions correspond to the position of another pixel in the unshifted map (Fig. 4). Each such subgrid may be mapped into a familiar Hudgin geometry, and the standard reconstruction machinery of Section 2 may be applied without difficulty. The absolute reconstructed wavefront will be obtained for points at the subgrid positions of Fig. 4, and the optic has been mapped at some modest loss of spatial resolution. Note that the reconstruction procedure of Section 2 is not sensitive to the spacing between nodes on the phase wavefront. It is possible to retain in a straightforward way the spatial frequency content of the input phase maps by incorporating reconstructed maps for pixels on all possible subgrids. For the 3 × 2 example of Fig. 4, there are M × N 3 × 2 6 distinct interleaved systems (subgrids) of nodes, each of which can be mapped to or from the canonical contiguous-pixel case of Fig. 2. The standard Hudgin-geometry reconstruction algebra therefore yields absolute wavefront solutions for each of the subgrids, M × N in number, with the obvious interpretation that each wavefront is rigorously known only on a disjoint set of pixels such as those shaded in Fig. 4. The parallel processing of M × N 3 × 2 6 subgrids is suggested in Fig. 5. The subgrid maps, M × N in number, may easily be combined into a solution at all pixel locations of the optic under test. In this way, despite the lowerspatial-frequency content of each individual subgrid, the spatial resolution of the map as a whole is preserved. The explicit combined wavefront map W as a function of submaps W mn, M × N in number, is Wm i − 1 · Δx; n j − 1 · Δy W mn i; j i 1; …; imax ; j 1; …jmax m 1; …; M; n 1; …; N where M Δx∕x0 ; N Δy∕y0 :
(7)
Equation (7) provides the mapping, in both directions, between subgrids on the full optic and contiguous N × N maps for Hudgin-geometry reconstruction. In practice, simple combination of raw reconstructed phase maps from the subgrids tends to work
Fig. 5. Schematic of absolute test surface reconstruction when transverse shifts are multiples of pixel dimensions. Absolute phase differences are known between pixels (shaded) separated by x- and y-shifts, as drawn in Fig. 4. Similar absolute connections are known for other subgrids indicated by the other five rectangles in this 3 × 2 example.
well for maps of reasonably large size because the subgrid maps tend to have very similar means. (The parent PSI maps were set to zero mean, and the departure of subgrid maps from this same zero mean is a measure of the severity of artifacts due to partitioning the full map.) Single surface features of moderate size, real or spurious, tend to affect all grid maps in common unless they have spatial scales finer than Δx or Δy, so will not bias the mean height of one or a few grid maps. Distributions of fine features, comparable to the pixel size, will by chance tend to have comparable average impact on each subgrid map, for maps of reasonably large size, while the effect of a single defect in one pixel is diluted by imax jmax ∼ several thousand, for situations of practical interest. The mean values of subgrid maps may be adjusted after reconstruction to match, though the variation in numerical experiments is so small that combined full maps are not much affected. The effect of combining subgrid maps is potentially to introduce a jitter on scales of a few pixels, and PSI maps are rarely relied upon at spatial frequencies that high. Other techniques may be applied to deal with this jitter, most simply a two-dimensional box-car averaging of the phase map, with some loss of spatial resolution, or detection and correction of high-spatial-frequency jitter with Fourier techniques. The effectiveness of combining subsampled subgrid maps recovered by Hudgin reconstruction is shown in a numerical simulation, in Fig. 6, of the example that has been presented in Fig. 5. An underlying test surface phase map is shown in the first panel; it is the surface to be extracted. The second panel shows the unknown additively corrupting reference map, and the third panel shows the difference of these two, which is the map measured by the PSI. Pixels on the first subgrid are shown in the fourth panel; these may be mapped into a contiguous Hudgin geometry, shown next. The reconstruction from absolute differences is shown in the next panel, stretched 3∶2 to match the original sampling; this is then mapped back onto the full grid. Applying this procedure to the remaining five subgrids and combining the outputs gives the final panel of Fig. 6, which agrees well with the input test surface of the first panel. For this example, the spread (standard deviation) in means over the subgrids was only roughly 2% of those mean map levels, which from a practical point of view is negligible compared to other noise sources in a PSI measurement. The final panel of Fig. 6 is presented with no adjustment of level, and no artifacts from the subgrid procedure are noticeable. Though the use of subgrids is driven by the structure of the measurements, they provide a significant computational advantage as well. The size of matrix A appearing in Eq. (6) is 2n3 n − 1, where n × n is the size of the phase map that A will process, and this grows very rapidly with size of the underlying phase map. When the total map is split into subgrids M × N in number, the evaluation of A and of the product of 10 February 2014 / Vol. 53, No. 5 / APPLIED OPTICS
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Fig. 7. Schematic of absolute test surface reconstruction when transverse shifts contain fractional pixels. Phase values are sought on a pseudogrid of nodes represented by circles. Transverse shifts provide absolute phase differences between positions connected by arrows. The geometry for bilinear interpolation [Eq. (8)] is shown at lower right. Fig. 6. Simulation of reconstruction when transverse shifts are Δx 3, Δy 2 pixels. (a) Underlying surface of test optic, to be extracted, (b) unknown corrupting reference surface, (c) PSI measurement, the difference of test and reference surfaces, (d) sampling of PSI measurement at every (3,2) pixels in x; y, to create the first subgrid with Hudgin geometry, (e) 40 × 40 remapping of the data points in previous panel, omitting points on other subgrids, (f) reconstruction of slopes evaluated from previous panel and shifted maps, stretched to actual 3∶2 aspect ratio on test optic, (g) mapping of this reconstructed absolute surface onto test optic, and (h) combination of reconstructions for all six subgrids.
matrices in Eq. (6) still need only be done once, but now for maps smaller by factors of M and N along the two axes. One could construct a large matrix, a generalization of A from Eq. (5), expressing difference connections now jumping several pixels, whose inversion would solve the entire problem with all subgrids included. However, the system would still naturally divide into distinct subgrids that would not be linked in any fundamental way. This approach would end up being similar to the approach already discussed, but with a much larger computational load. 4. Absolute Maps from Fractional-Pixel Shifts
The most general case of transverse shifting, and the case most readily achieved experimentally, involves shifts that may be larger than a single pixel but are not precise multiples of the pixel dimensions. As shown in Fig. 7, nodes (defined by shifts Δx, Δy) at which a single pixel samples the wavefront may commonly be multiple pixels plus a fractional part. Accurate knowledge of transverse shifts is relatively easy to obtain, either from the optical displays provided by the PSI itself or from mechanical readouts on the stage on which the test optic is mounted. Accurate control of the shifts is not difficult, but generally not required. If the standard Hudgin-geometry machinery is to be used, the procedure is conceptually similar to that of the previous section. Subgrids of camera pixels that are separated by the actual transverse shifts in x and y are isolated, the Hudgin-geometry wavefront 796
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reconstruction is performed for each subgrid, and the solutions defined over subgrids are combined into a single, contiguous map revealing the fully sampled absolute underlying phase of the test surface. Prior to reconstruction, the case with fractionalpixel shifts generally requires some form of regularization of the data. Some rather simple approaches will give good results, in terms both of sensitivity and spatial resolution. Simplest of all is to construct the network or grid of Hudgin nodes as a lattice with spacing Δx; Δy Mx0 ; Ny0 , then to approximate the phase value at the position of a Hudgin node by the value measured by the PSI camera pixel nearest in position. The impact on spatial resolution of the reconstructed absolute phase maps is small, of the order of the camera pixel dimension; if all frameworks shifted from the nominal by integer pixel values up to the rounded shift values in each direction are employed, essentially all of the information in the unreconstructed absolute phase map is retained. The combination of different, relatively offset frames by assuming common (say, zero) means works very well, as in the previous section. A more sophisticated approach can be taken, making a better approximation of noninteger-pixel Hudgin node phase values by bilinear interpolation among the values registered to PSI camera pixels: φtest − ψ ref effective fφtest − ψ ref x1 ;y1 x2 − xy2 − y φtest − ψ ref x2 ;y1 x − x1 y2 − y φtest − ψ ref x1 ;y2 x2 − xy − y1 φtest − ψ ref x2 ;y2 x − x1 y − y1 g
1 : x0 y0
(8)
The geometry is shown in Fig. 7. If possible, somewhat computationally intensive, is to Fourier transform the grid of camera pixel absolute phase values, pad the higher frequencies with zeros, and inverse Fourier transform. The result, as is well known, is
a function on a finer grid, although no actual higherfrequency information is generated. This denser array of pseudopixels may permit more accurate interpolation to match the Hudgin subgrid, defined by the transverse mechanical shifts achieved, to the pixel grid. Though this is elegant, and perhaps useful in situations demanding higher precision, the simpler methods above will generally suffice since any high-spatial-frequency noise they generate, on the order of the inverse of the pixel spacing, is usually not of interest with PSI measurements. Another drawback with the padded Fourier transform technique is that a large padded fraction is needed to allow a final grid size on which the Hudgin nodes may be accurately placed. 5. Conclusions
This study has considered the general, practical case in which absolute maps are obtained from as few as three interferometric measurements with relative spatial shifts that are not equal to the pixel dimension, and may have slight clocking errors. The benefits of the technique are obtained in a simple and natural way, with no loss of spatial resolution. The technique presented here is a double difference that takes advantage of the fact that the unknown reference surface is rigidly fixed with respect to the camera of the PSI. As is evident from the algebra [e.g., Eq. (1)], the absolute surface map of the optic under test, once found, may be combined with the raw observations to recover the absolute surface of the reference surface, or transmission flat. Error sources in this technique are of two kinds: imaging fidelity, and surface height fidelity through the reconstruction process. The first refers to registering pairs of PSI frames with small transverse shifts. Mechanical stability of optical elements in a PSI will be excellent over the time scales necessary to obtain two image frames. The test optic itself will have fiducial edges to which the PSI pixels can easily be referred to a precision of one pixel, with subpixel precision possible. Hence, transverse distortions of the final phase map can be kept smaller than a pixel, which is entirely acceptable because PSI maps are rarely relied on to pixel resolution. Surface height or wavefront reconstruction in the Hudgin geometry has been shown to have a relatively benign dependence on the number of elements in the measurement grid. Typical input measurement errors σ in give reconstructed wavefront errors σ 2wf Eσ 2in a b ln N 2 σ 2in , where a, b are constants and N is the linear grid size; the factor E is typically of order unity [7–9]. Individual PSI wavefront measurements limited by instrumental noise sources can be much better than 1 nm peak to valley (P-V), substantially better than knowledge of the absolute flatness of the reference surface. If vibration and atmospheric turbulence are well controlled, the transverse shifting technique presented here may provide absolute surface measurements with accuracy substantially better than λ∕100P-V.
Also apparent in the algebra is that the differencing of Eq. (3) will remove reference phase surfaces of any number that lie in series with the optic under test, whether in reflection or transmission. It will also remove phase surfaces that are additive with the test surface, e.g., series of transmission windows or precision mirrors in an interferometric beam train, just as well as the subtractive phase of the reference flat with which the test optic forms a cavity. With series of many optics, some care must be taken in controlling tilt as the optic under test is translated, to avoid beam walk on other optics in the train. The technique can be applied to the study of noncommon paths in adaptive optics systems. Here steering mirrors typically available in the wavefront sensor beam train can effectively provide transverse shifts on the entire optical train upstream of the dichroic, isolating wavefront errors in the upstream and downstream beam trains. Interestingly, in this application the dichroic is typically in a slow converging beam of spherical waves, so the pupil tracking by two steering mirrors actually involves an offset and a tilt in each axis. More generally, the absolute metrology method discussed here will work with spherical optics. In this case the transverse motion traced out on the test optic should be rotations about the common center of curvature of the transmission sphere and the spherical test optic, rather than linear displacements transverse to the measurement beam. This novel approach to absolute surface measurement through transverse shifting appears useful in a variety of applications. It is simple, requiring no external optics, a very simple mechanical setup, and a minimal number of exposures on the optic under test. The simplicity of the data processing that results helps to assure high accuracy. Support from the National Science Foundation is acknowledged. Any opinion, finding, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. References 1. G. Schulz and J. Schwider, “Interferometric testing of smooth surfaces,” Progress in Optics, E. Wolf, ed. (Elsevier, 1976), Vol. XIII, pp. 93–167. 2. C. Ai and J. C. Wyant, “Absolute testing of flats by using even and odd functions,” Appl. Opt. 32, 4698–4705 (1993). 3. J. Grzanna, “Absolute testing of optical flats at points on a square grid: error propagation,” Appl. Opt. 33, 6654–6661 (1994). 4. K. R. Freischlad, “Absolute interferometric testing based on a reconstruction of rotational shear,” Appl. Opt. 40, 1637–1648 (2001). 5. U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. 45, 5856–5865 (2006). 6. E. E. Bloemhof, “Absolute surface metrology by differencing spatially shifted maps from a phase-shifting interferometer,” Opt. Lett. 35, 2346–2348 (2010). 7. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. A 67, 375–378 (1977). 8. J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980). 9. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998).
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We consider the detailed implementation and practical utility of a novel absolute optical metrology scheme recently proposed for use with a phase-shifting interferometer (PSI). This scheme extracts absolute phase differences between points on the surface of the optic under test by differencing phase maps made with slightly different transverse spatial shifts of that optic. These absolute phase (or height) differences, which for single-pixel shifts are automatically obtained in the well-known Hudgin geometry, yield the underlying absolute surface map by standard wavefront reconstruction techniques. The PSI by itself maps surface height only relative to that of a separate reference optic known or assumed to be flat. In practice, even relatively high-quality (and expensive) transmission flats or spheres used to reference a PSI are flat or spherical only to a few dozen nanometers peak to valley (P-V) over typical 4 in. apertures. The new technique for removing the effects of the reference surface is in principle accurate as well as simple, and may represent a significant advance in optical metrology. Here it is shown that transverse shifts need not match the pixel size; somewhat counterintuitively, the single-pixel spatial resolution of the PSI is retained even when transverse shifts are much coarser. Practical considerations for shifts not necessarily commensurate with pixel size, and broader applications, are discussed. OCIS codes: (120.3180) Interferometry; (120.6650) Surface measurements, figure. http://dx.doi.org/10.1364/AO.53.000792
1. Absolute Surface Metrology from Phase-Shifting Interferometry
Measurement of the nanometer-scale absolute surface figure of an optic under test with a phase-shifting interferometer (PSI) has been a subject of considerable study because a PSI normally measures that surface only relative to the unknown underlying shape of its reference surface. That is, the measurement of a test optic is really the measurement of the two-surface cavity it forms with the reference surface, called the transmission flat or transmission sphere. Many solutions have been proposed to extract the surface of the test optic alone [1–5]. A number of these involve introduction of one or more additional surfaces, then comparing the resulting pairwise combinations of surface shapes. Recently, a novel approach was proposed [6] that may give improved simplicity and accuracy with only two auxiliary measurements (small-amplitude lateral shifts of the test optic in each of the transverse 792
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coordinate directions) and no auxiliary optic. The new approach may also generalize to other applications. For simplicity, the original presentation of the technique assumed transverse shifts in x- and y-directions equal to the pixel size (really, to the center-to-center spacing of pixels). While intuitively it is clear that the new technique still yields absolute metrology when this restriction is relaxed, it is useful to examine how the computations should be generalized for the more realistic case of transverse shifts not equal to pixel dimensions. The results may be of considerable practical importance, and they include the surprising conclusion that measurements with shifts substantially larger than the pixel size effectively retain the spatial resolution set by the PSI pixel scale. 2. Commensurate Case: Shifts Equal to Pixel Size
A brief review of the new transverse-shifting technique for absolute metrology [6] is now given; the
experimental situation is shown in Fig. 1. A PSI returns a map of the test optic surface shape by combining data from a few phase-shifted exposures obtained in parallel or in rapid succession, by using the four-bin or similar algorithm. Each exposure is made at a distinct piston phase value of the reference surface through, e.g., piezoelectric actuation. Overall piston of a PSI map is unknown, and tip/tilt is generally not of interest, so tip/tilt of the test optic need not be maintained to any degree of precision during transverse shifts. What is generally desired is the higher-order shape of the test optic, showing local surface structures, and it suffices to reference every map (cavity or absolute) to the mean piston and overall tip/tilt of a flat under test. Because the PSI measures a cavity phase, the starting maps it produces inherit the imperfect flatness of the reference surface. This dependence on reference surface shape can be expressed mathematically as follows. Any line of sight at transverse position x; y through test and PSI optics to a given pixel of the PSI camera can be written, for one choice of sign convention, as Φ0 x; y φmirror x; y − ψ ref x; y;
(1)
where mirror and reference flat phase maps are φmirror x; y and ψ ref x; y, respectively. The PSI returns a phase map Φ0 x; y that is the difference of these two unknown maps. The new approach for removing dependence on the reference flat map, and thus obtaining the absolute shape of the test optic, is to shift the test optic laterally and form differences with the original map. This effectively results in double differences of phase (height) from which the reference shape is accurately cancelled. Explicitly, the auxiliary maps obtained with small shifts in x and y are ΦΔx x; y φmirror x Δx; y − ψ ref x; y ΦΔy x; y φmirror x; y Δy − ψ ref x; y;
(2)
Fig. 1. Measurement schematic (from [6]): the surface height of the mirror under test is mapped at subnanometer precision, but only relative to the surface shape of the transmission flat (reference surface). The reference surface is typically known to be flat to perhaps tens of nanometers over transverse scales of interest, but will have unknown spatial structure inherited by the PSI measurement. Phases are taken as proportional to surface displacements in a common direction, since the PSI measures distance between the two surfaces; an alternative sign convention, with phases proportional to surface heights, would give a PSI measurement equal to the sum of the test and reference phases.
because the reference surface shape ψ ref x; y is fixed with respect to the camera whose pixels record the PSI images (the line of sight of a given pixel intercepts the same portion of the reference surface, as nothing has moved transversely inside the PSI, but views a new position on the test optic). This elimination of the reference surface by differencing the two transversely shifted PSI maps with the original map is seen mathematically by subtracting Eq. (1) from Eq. (2): ΦΔx − Φ0 φmirror x Δx; y − φmirror x; y ΦΔy − Φ0 φmirror x; y Δy − φmirror x; y:
(3)
These are absolute measurements of x- and y- height differences at an array of points on the test optic; these may or may not form an interlinked grid or lattice. In an alternative sign convention, phases at reference and test-optic surfaces are taken to be proportional to surface heights, with height axes pointing in opposite directions; then the reference phases should enter Eqs. (1) and (2) additively. Those reference phases of course still cancel out on differencing Eqs. (1) and (2) to obtain Eq. (3). The convention used here is perhaps more physical, measuring phase in a common spatial direction, and mathematically emphasizing that the PSI measures the test surface relative to the reference surface by returning the gap, or position difference, between the two. When transverse shifts are equal to one pixel, the data lie in the Hudgin geometry, as shown in Fig. 2. The recovery of the absolute wavefront (test optic surface height) from this grid of absolute differences is a standard exercise in wavefront reconstruction, discussed in [7–9]. The more general case, in which shifts are incommensurate with pixel size (spanning multiple pixels and/or fractional pixels) is considered in more detail later in the current paper. Reviewing the commensurate case in which an auxiliary PSI map is obtained at a transverse shift of one pixel in each of the x- and y-directions (Fig. 2), the absolute phase differences of Eq. (3) form an overconstrained system of linear equations in the absolute phase values:
Fig. 2. Schematic of absolute test surface reconstruction in the Hudgin geometry for the commensurate case where transverse shifts are equal to PSI pixel dimensions. Circles at pixel center denote nodes at which the phase (surface height) is desired; arrows denote phase steps (differences) measured in an absolute way by the transverse shifting technique, from Eq. (3). Reconstruction of these measurements yields the full absolute surface map. 10 February 2014 / Vol. 53, No. 5 / APPLIED OPTICS
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⃗ ⃗n; S⃗ AW
(4)
where wavefront slopes or differences S are related to the underlying wavefront or surface height map W, including a noise term represented by n. The wavefront values W sought are arranged here in a vector of length N 2 , where each phase map measures N × N. Absolute wavefront differences S found by differencing shifted and nominal PSI maps are arranged in a vector of length 2NN − 1, and the sparse matrix A, with dimension N 2 × 2NN − 1, has nonzero elements given by Aq p − 1N − 1; q p − 1N −1 Aq p − 1N − 1; 1 q p − 1N 1 for p 1; …; N;
q 1; …; N − 1
Ar NN − 1; r −1 Ar NN − 1; r N 1 for r 1; …; NN − 1:
(5)
The solution to Eq. (4) that minimizes the errors in a least-squares solution for this overconstrained problem is well known; neglecting the noise term it is ⃗ ⃗ AT A−1 AT S: W
(6)
There is one technicality in the current situation: the matrix A is not of full row rank, and so AT A−1 does not exist. This reflects the fact that the overall piston term of the phase map W is not defined. The situation is easily remedied [8] by extending A with a row of 1’s and extending the vector S with a single 0, equivalent to setting overall piston to zero. This procedure yields the underlying absolute phase map, free of contamination from the additive phase map introduced by the reference surface (transmission flat). Just two additional auxiliary maps, with the test optic shifted transversely in x and y, are required in addition to a map at the original position. An example of the very high intrinsic fidelity of the technique is shown in the simulation of Fig. 3. In the next sections it will be seen that the new technique is experimentally robust and does not depend on precise single-pixel transverse shifts, i.e., the incommensurate transverse shifts likely to be obtained in practice do not significantly degrade the utility of this approach.
Fig. 3. Simulation of absolute phase extraction despite phase corruption by the reference surface. (a) Underlying phase or surface height of test optic to be extracted, (b) phase map of corrupting PSI reference surface, (c) difference of these, the resulting distorted cavity phase map measured by the PSI, (d),(e) absolute surface differences evaluated along x- and y-axes from transverse shifting technique, and (f) reconstructed surface of test optic, in good agreement with panel (a).
themselves is straightforward. As will be seen, reconstruction requires shift knowledge, rather than control of the shifts, and tolerances on that knowledge are relatively loose for practical purposes. In this section transverse shifts that are even multiples of pixel size are considered. So if the pixel dimensions are x0 ; y0 , the transverse measurement shifts have the form Δx; Δy Mx0 ; Ny0 , with M and N integers. A concrete illustration is given in Fig. 4, with M 3 and N 2. The differencing of transversely shifted phase maps outlined in Section 2 now produces absolute phase/height differences connecting a grid of nodes separated by intervening pixels not included in the grid. Each camera pixel in the difference maps, Eq. (3), measures the absolute phase difference between two surface points separated by the transverse shift, even when that shift is large, but application of the Hudgin geometry machinery now takes some care.
3. Absolute Maps from Shifts that are Multiples of the Pixel Size
In the more general case, transverse shifts are not equal to the pixel dimension, so the precise Hudgin geometry will not obtain. Pixels are typically small, so shifts may be several times larger. However, the optics of the PSI generally provide clear transverse registration of the test optic, by sighting on a surface feature or on the edges of the optic. So accurate determination of spatial shifts from the PSI data 794
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Fig. 4. Schematic of absolute test surface reconstruction when transverse shifts are multiples of pixel dimensions. Phase values are sought on the nodes represented by circles at pixel center. Transverse shifts, by Eq. (3), provide absolute phase steps (differences) between pixels connected by arrows. Though noncontiguous, pixels in this subgrid may be treated by standard Hudgin geometry reconstruction.
The simple approach presented here is to separately analyze each subgrid for which shifted pixel positions correspond to the position of another pixel in the unshifted map (Fig. 4). Each such subgrid may be mapped into a familiar Hudgin geometry, and the standard reconstruction machinery of Section 2 may be applied without difficulty. The absolute reconstructed wavefront will be obtained for points at the subgrid positions of Fig. 4, and the optic has been mapped at some modest loss of spatial resolution. Note that the reconstruction procedure of Section 2 is not sensitive to the spacing between nodes on the phase wavefront. It is possible to retain in a straightforward way the spatial frequency content of the input phase maps by incorporating reconstructed maps for pixels on all possible subgrids. For the 3 × 2 example of Fig. 4, there are M × N 3 × 2 6 distinct interleaved systems (subgrids) of nodes, each of which can be mapped to or from the canonical contiguous-pixel case of Fig. 2. The standard Hudgin-geometry reconstruction algebra therefore yields absolute wavefront solutions for each of the subgrids, M × N in number, with the obvious interpretation that each wavefront is rigorously known only on a disjoint set of pixels such as those shaded in Fig. 4. The parallel processing of M × N 3 × 2 6 subgrids is suggested in Fig. 5. The subgrid maps, M × N in number, may easily be combined into a solution at all pixel locations of the optic under test. In this way, despite the lowerspatial-frequency content of each individual subgrid, the spatial resolution of the map as a whole is preserved. The explicit combined wavefront map W as a function of submaps W mn, M × N in number, is Wm i − 1 · Δx; n j − 1 · Δy W mn i; j i 1; …; imax ; j 1; …jmax m 1; …; M; n 1; …; N where M Δx∕x0 ; N Δy∕y0 :
(7)
Equation (7) provides the mapping, in both directions, between subgrids on the full optic and contiguous N × N maps for Hudgin-geometry reconstruction. In practice, simple combination of raw reconstructed phase maps from the subgrids tends to work
Fig. 5. Schematic of absolute test surface reconstruction when transverse shifts are multiples of pixel dimensions. Absolute phase differences are known between pixels (shaded) separated by x- and y-shifts, as drawn in Fig. 4. Similar absolute connections are known for other subgrids indicated by the other five rectangles in this 3 × 2 example.
well for maps of reasonably large size because the subgrid maps tend to have very similar means. (The parent PSI maps were set to zero mean, and the departure of subgrid maps from this same zero mean is a measure of the severity of artifacts due to partitioning the full map.) Single surface features of moderate size, real or spurious, tend to affect all grid maps in common unless they have spatial scales finer than Δx or Δy, so will not bias the mean height of one or a few grid maps. Distributions of fine features, comparable to the pixel size, will by chance tend to have comparable average impact on each subgrid map, for maps of reasonably large size, while the effect of a single defect in one pixel is diluted by imax jmax ∼ several thousand, for situations of practical interest. The mean values of subgrid maps may be adjusted after reconstruction to match, though the variation in numerical experiments is so small that combined full maps are not much affected. The effect of combining subgrid maps is potentially to introduce a jitter on scales of a few pixels, and PSI maps are rarely relied upon at spatial frequencies that high. Other techniques may be applied to deal with this jitter, most simply a two-dimensional box-car averaging of the phase map, with some loss of spatial resolution, or detection and correction of high-spatial-frequency jitter with Fourier techniques. The effectiveness of combining subsampled subgrid maps recovered by Hudgin reconstruction is shown in a numerical simulation, in Fig. 6, of the example that has been presented in Fig. 5. An underlying test surface phase map is shown in the first panel; it is the surface to be extracted. The second panel shows the unknown additively corrupting reference map, and the third panel shows the difference of these two, which is the map measured by the PSI. Pixels on the first subgrid are shown in the fourth panel; these may be mapped into a contiguous Hudgin geometry, shown next. The reconstruction from absolute differences is shown in the next panel, stretched 3∶2 to match the original sampling; this is then mapped back onto the full grid. Applying this procedure to the remaining five subgrids and combining the outputs gives the final panel of Fig. 6, which agrees well with the input test surface of the first panel. For this example, the spread (standard deviation) in means over the subgrids was only roughly 2% of those mean map levels, which from a practical point of view is negligible compared to other noise sources in a PSI measurement. The final panel of Fig. 6 is presented with no adjustment of level, and no artifacts from the subgrid procedure are noticeable. Though the use of subgrids is driven by the structure of the measurements, they provide a significant computational advantage as well. The size of matrix A appearing in Eq. (6) is 2n3 n − 1, where n × n is the size of the phase map that A will process, and this grows very rapidly with size of the underlying phase map. When the total map is split into subgrids M × N in number, the evaluation of A and of the product of 10 February 2014 / Vol. 53, No. 5 / APPLIED OPTICS
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Fig. 7. Schematic of absolute test surface reconstruction when transverse shifts contain fractional pixels. Phase values are sought on a pseudogrid of nodes represented by circles. Transverse shifts provide absolute phase differences between positions connected by arrows. The geometry for bilinear interpolation [Eq. (8)] is shown at lower right. Fig. 6. Simulation of reconstruction when transverse shifts are Δx 3, Δy 2 pixels. (a) Underlying surface of test optic, to be extracted, (b) unknown corrupting reference surface, (c) PSI measurement, the difference of test and reference surfaces, (d) sampling of PSI measurement at every (3,2) pixels in x; y, to create the first subgrid with Hudgin geometry, (e) 40 × 40 remapping of the data points in previous panel, omitting points on other subgrids, (f) reconstruction of slopes evaluated from previous panel and shifted maps, stretched to actual 3∶2 aspect ratio on test optic, (g) mapping of this reconstructed absolute surface onto test optic, and (h) combination of reconstructions for all six subgrids.
matrices in Eq. (6) still need only be done once, but now for maps smaller by factors of M and N along the two axes. One could construct a large matrix, a generalization of A from Eq. (5), expressing difference connections now jumping several pixels, whose inversion would solve the entire problem with all subgrids included. However, the system would still naturally divide into distinct subgrids that would not be linked in any fundamental way. This approach would end up being similar to the approach already discussed, but with a much larger computational load. 4. Absolute Maps from Fractional-Pixel Shifts
The most general case of transverse shifting, and the case most readily achieved experimentally, involves shifts that may be larger than a single pixel but are not precise multiples of the pixel dimensions. As shown in Fig. 7, nodes (defined by shifts Δx, Δy) at which a single pixel samples the wavefront may commonly be multiple pixels plus a fractional part. Accurate knowledge of transverse shifts is relatively easy to obtain, either from the optical displays provided by the PSI itself or from mechanical readouts on the stage on which the test optic is mounted. Accurate control of the shifts is not difficult, but generally not required. If the standard Hudgin-geometry machinery is to be used, the procedure is conceptually similar to that of the previous section. Subgrids of camera pixels that are separated by the actual transverse shifts in x and y are isolated, the Hudgin-geometry wavefront 796
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reconstruction is performed for each subgrid, and the solutions defined over subgrids are combined into a single, contiguous map revealing the fully sampled absolute underlying phase of the test surface. Prior to reconstruction, the case with fractionalpixel shifts generally requires some form of regularization of the data. Some rather simple approaches will give good results, in terms both of sensitivity and spatial resolution. Simplest of all is to construct the network or grid of Hudgin nodes as a lattice with spacing Δx; Δy Mx0 ; Ny0 , then to approximate the phase value at the position of a Hudgin node by the value measured by the PSI camera pixel nearest in position. The impact on spatial resolution of the reconstructed absolute phase maps is small, of the order of the camera pixel dimension; if all frameworks shifted from the nominal by integer pixel values up to the rounded shift values in each direction are employed, essentially all of the information in the unreconstructed absolute phase map is retained. The combination of different, relatively offset frames by assuming common (say, zero) means works very well, as in the previous section. A more sophisticated approach can be taken, making a better approximation of noninteger-pixel Hudgin node phase values by bilinear interpolation among the values registered to PSI camera pixels: φtest − ψ ref effective fφtest − ψ ref x1 ;y1 x2 − xy2 − y φtest − ψ ref x2 ;y1 x − x1 y2 − y φtest − ψ ref x1 ;y2 x2 − xy − y1 φtest − ψ ref x2 ;y2 x − x1 y − y1 g
1 : x0 y0
(8)
The geometry is shown in Fig. 7. If possible, somewhat computationally intensive, is to Fourier transform the grid of camera pixel absolute phase values, pad the higher frequencies with zeros, and inverse Fourier transform. The result, as is well known, is
a function on a finer grid, although no actual higherfrequency information is generated. This denser array of pseudopixels may permit more accurate interpolation to match the Hudgin subgrid, defined by the transverse mechanical shifts achieved, to the pixel grid. Though this is elegant, and perhaps useful in situations demanding higher precision, the simpler methods above will generally suffice since any high-spatial-frequency noise they generate, on the order of the inverse of the pixel spacing, is usually not of interest with PSI measurements. Another drawback with the padded Fourier transform technique is that a large padded fraction is needed to allow a final grid size on which the Hudgin nodes may be accurately placed. 5. Conclusions
This study has considered the general, practical case in which absolute maps are obtained from as few as three interferometric measurements with relative spatial shifts that are not equal to the pixel dimension, and may have slight clocking errors. The benefits of the technique are obtained in a simple and natural way, with no loss of spatial resolution. The technique presented here is a double difference that takes advantage of the fact that the unknown reference surface is rigidly fixed with respect to the camera of the PSI. As is evident from the algebra [e.g., Eq. (1)], the absolute surface map of the optic under test, once found, may be combined with the raw observations to recover the absolute surface of the reference surface, or transmission flat. Error sources in this technique are of two kinds: imaging fidelity, and surface height fidelity through the reconstruction process. The first refers to registering pairs of PSI frames with small transverse shifts. Mechanical stability of optical elements in a PSI will be excellent over the time scales necessary to obtain two image frames. The test optic itself will have fiducial edges to which the PSI pixels can easily be referred to a precision of one pixel, with subpixel precision possible. Hence, transverse distortions of the final phase map can be kept smaller than a pixel, which is entirely acceptable because PSI maps are rarely relied on to pixel resolution. Surface height or wavefront reconstruction in the Hudgin geometry has been shown to have a relatively benign dependence on the number of elements in the measurement grid. Typical input measurement errors σ in give reconstructed wavefront errors σ 2wf Eσ 2in a b ln N 2 σ 2in , where a, b are constants and N is the linear grid size; the factor E is typically of order unity [7–9]. Individual PSI wavefront measurements limited by instrumental noise sources can be much better than 1 nm peak to valley (P-V), substantially better than knowledge of the absolute flatness of the reference surface. If vibration and atmospheric turbulence are well controlled, the transverse shifting technique presented here may provide absolute surface measurements with accuracy substantially better than λ∕100P-V.
Also apparent in the algebra is that the differencing of Eq. (3) will remove reference phase surfaces of any number that lie in series with the optic under test, whether in reflection or transmission. It will also remove phase surfaces that are additive with the test surface, e.g., series of transmission windows or precision mirrors in an interferometric beam train, just as well as the subtractive phase of the reference flat with which the test optic forms a cavity. With series of many optics, some care must be taken in controlling tilt as the optic under test is translated, to avoid beam walk on other optics in the train. The technique can be applied to the study of noncommon paths in adaptive optics systems. Here steering mirrors typically available in the wavefront sensor beam train can effectively provide transverse shifts on the entire optical train upstream of the dichroic, isolating wavefront errors in the upstream and downstream beam trains. Interestingly, in this application the dichroic is typically in a slow converging beam of spherical waves, so the pupil tracking by two steering mirrors actually involves an offset and a tilt in each axis. More generally, the absolute metrology method discussed here will work with spherical optics. In this case the transverse motion traced out on the test optic should be rotations about the common center of curvature of the transmission sphere and the spherical test optic, rather than linear displacements transverse to the measurement beam. This novel approach to absolute surface measurement through transverse shifting appears useful in a variety of applications. It is simple, requiring no external optics, a very simple mechanical setup, and a minimal number of exposures on the optic under test. The simplicity of the data processing that results helps to assure high accuracy. Support from the National Science Foundation is acknowledged. Any opinion, finding, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. References 1. G. Schulz and J. Schwider, “Interferometric testing of smooth surfaces,” Progress in Optics, E. Wolf, ed. (Elsevier, 1976), Vol. XIII, pp. 93–167. 2. C. Ai and J. C. Wyant, “Absolute testing of flats by using even and odd functions,” Appl. Opt. 32, 4698–4705 (1993). 3. J. Grzanna, “Absolute testing of optical flats at points on a square grid: error propagation,” Appl. Opt. 33, 6654–6661 (1994). 4. K. R. Freischlad, “Absolute interferometric testing based on a reconstruction of rotational shear,” Appl. Opt. 40, 1637–1648 (2001). 5. U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. 45, 5856–5865 (2006). 6. E. E. Bloemhof, “Absolute surface metrology by differencing spatially shifted maps from a phase-shifting interferometer,” Opt. Lett. 35, 2346–2348 (2010). 7. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. A 67, 375–378 (1977). 8. J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980). 9. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998).
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