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Abstract
We have previously presented broadband variable transmission sphere (BVTS) systems based on the Alvarez lens for use in Fizeau interferometry. A BVTS system may replace multiple conventional transmission spheres with one system composed of two plano-freeform optical elements. In this paper, we discuss retrace error as a source of aberrations in Fizeau interferometry measurements when using BVTS systems. We explore a design optimization method to decrease retrace error by optimizing freeform surfaces to reduce targeted Fringe Zernike wavefront aberrations. Significant reduction in peak-to-valley retrace error is demonstrated.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
1.1 Broadband variable transmission spheres
We have previously reported on the use of Broadband Variable Transmission Spheres as potential alternatives to conventional transmission spheres [1,2]. In Fizeau interferometry, transmission spheres transform incident wavefronts into high-quality spherical wavefronts to match the shapes of nominally spherical test surfaces for non-contact surface characterization [2–10]. In this approach, a transmission sphere with appropriate f-number must be chosen such that the center of curvature of the surface under test and the focal plane of the transmission sphere are aligned, so that the curvature of the test wavefront matches the curvature of the test surface [11–13]. Traditional transmission spheres are designed for specific f-numbers and source wavelengths which limits the ranges of surface curvatures that an individual transmission sphere can measure. BVTS systems leverage the concept of Alvarez lenses to create a variable f-number transmission sphere functioning for a broad range of source wavelengths. BVTS systems have the potential to reduce the need for multiple transmission spheres, reduce the time needed for changing and calibrating transmission spheres to test different surface curvatures, and to increase the range of surface curvatures measurable with a single system. In Ref. [1], we investigated two BVTS designs and simulated surface measurements using these BVTS systems and concluded that they faced limitations in surface measurement accuracy due to residual system aberrations. In this paper we explore the nature of retrace errors and their significance in BVTS system performance limitations. We also present an optimization method to reduce retrace errors in BVTS systems and present results for several examples that show significant reductions in retrace errors.
1.2 Retrace error
In Fizeau interferometry it is assumed that light reflected from the reference and test surfaces, respectively, follow the same optical path back through the optical elements of the system, especially the transmission sphere [14–16]. This assumption requires that the test surface shape be perfectly matched to the incident wavefront shape; retrace error results when this assumption is not met. The simplest example of retrace error, ignoring additional surfaces or material impurities, is illustrated in Fig. 1. In Fig. 1(a) the rays enter an ideal system from the left to the reference and test surfaces and retrace back through the transmission sphere along the same path. In contrast, rays in Fig. 1(b) deviate before or at the test surface. Accordingly, the rays incident on the reference surface retrace through the system along the same path but the rays incident on the test surface do not, resulting in a path difference which we identify as retrace error. It is beneficial to separate retrace error into two types: alignment errors and transmitted wavefront errors. For spherical transmission spheres the primary measurement errors due to misalignment with the test surface are defocus and coma [14]. Similarly, aberrations in the transmitted wavefronts will result in retrace error corresponding to the imperfect spherical wave incident on the test surface. For spherical transmission spheres, transmitted wavefront errors are primarily due to surface deviations and machining errors or material impurities [14].
In our earlier work we noted the presence of significant aberrations in simulated surface measurements using BVTS [1]. In our idealized simulation, the test surfaces were not aberrated and alignment was not a contributing factor. This suggests that the aberrations present in the simulations are caused by retrace errors resulting from transmitted wavefront errors from the BVTS systems themselves.
2. Initial designs and simulation methods
2.1 Initial BVTS system designs
We previously presented two example BVTS system designs using Zemax OpticStudio [1]. Zygo’s Transmission Sphere Selection Guide provided a guideline for systems with 101.6 mm entrance pupil diameter (EPD) [4]. F-numbers of the BVTS systems range from ± f/15 to ± f/80. Source wavelengths were chosen from visible to long-wave infrared laser wavelengths (0.6328 µm, 1.55 µm, 3.39 µm, and 10.6 µm). These BVTS systems used Zinc sulfide (ZnS) as the optical material because of its transmissivity and relatively low dispersion over the entire band of interest, but this concept can also be applied to other materials depending on manufacturability and transmissivity.
We now briefly summarize the designs and outcomes of these systems from Ref. [1]. In the first system, we added selected low-order polynomials to the basic Alvarez freeform surface and optimized the surface coefficients across the full range of f-numbers and design wavelengths with optimization targets of less than λ/10 peak to valley (PV) wavefront error (centroid) and diffraction limited spot sizes. A second system was also designed using techniques developed by Grewe et al. [17,18]; additional polynomial terms were added to the freeform surfaces to address wavefront aberrations and the system was optimized using the same approach as the first system. The surface equations for both example BVTS designs are shown in Table 5 of Appendix A. Both example BVTS systems had diffraction-limited spot sizes and PV wavefront error of less than λ/10 for the IR source wavelengths. However, as discussed above, neither BVTS system resulted in the desired performance in the simulated measurements of test surfaces, showing that optimization metrics beyond spot size and PV wavefront error are required.
2.2 Fizeau interferometer simulation
A model for a 101.6 mm EPD Fizeau interferometer was built in VirtualLab Fusion to simulate our BVTS systems, as illustrated in Fig. 2 [4,11]. Collimated sources were used with an aperture added after the BVTS to block unwanted light from interacting with the edges of the BVTS when configured to the maximum lateral shift. Light reflected from the reference and test surfaces were reflected to the detector by the beamsplitter, and the resulting interferogram was used for analysis.
To isolate the retrace error for the BVTS transmission sphere in the interferometer, only reflections from the reference surface and the test surface were considered; no reflections from intermediate surfaces of the BVTS system contributed to the interferogram, thereby ensuring that only transmitted wavefront retrace error is present in our simulated surface measurements. Alignment retrace error is not present because the simulated components were simulated with perfect alignment. All components except for the transmission sphere and test surface were ‘ideal’ components as defined in VirtualLab Fusion, limiting them to basic optical functions rather than objects with simulated material properties. For example, the ‘ideal’ beamsplitter is not a cube or plate but rather a simple field re-direction function. In contrast to the other components, the transmission sphere and tests surface have ideal shapes with no aberrations or deviations from their design function, and include simulated material properties and real thicknesses. The reference and test wavefronts interfere with each other at the detector, resulting in an interferogram corresponding to variations in the test wavefront with respect to the reference wavefront. The corresponding BVTS configuration was simulated for each ideal wavefront and phase data from each resulting interferogram was exported as an ASCII file for analysis.
A MATLAB script was written to import and evaluate phase data from the simulated interferograms obtained in VirtualLab Fusion using two algorithms to ensure proper phase unwrapping [19,20]. Because the surface parameters are known in simulation, the ideal interferogram and the deviation in the surface measurement from the wavefront shape can be calculated. Calculating the difference between the simulated and ideal surface maps provides a surface measurement height map of retrace errors induced by the BVTS transmission sphere; ideally, the resulting height map would be zero everywhere. Surface measurement PV and root mean square (RMS) magnitudes were calculated and the ZernikeCalc MATLAB function was used to fit Fringe Zernike polynomials to the retrace-error map for aberration analysis [21].
3. Retrace error simulation methods and results
3.1 Retrace errors of BVTS examples 1 and 2
We previously reported on surface measurement data for BVTS Examples 1 and 2 obtained from the simulated Fizeau interferometer, as described in Section 2 of Ref. [1]. The PV errors of surface measurements reported for BVTS Examples 1 and 2 (in Tables 6 through 8 of Ref. [1]) were filtered by removal of the first four Fringe Zernike terms, a common practice for surface measurements. However, evaluation of the total retrace error requires inclusion of all the terms. To this end, we analyzed simulated wavefront data for test surfaces chosen to match a range of commercially available transmission spheres [4]. Specifically, r-numbers (test surface radius of curvature divided by test area diameter) from -81.5 to 80 were evaluated for 100-mm-diameter test surfaces. The resulting PV retrace errors are presented in Table 1.
Table 1. PV Retrace Errors for BVTS Examples 1 and 2
Since the first four Fringe Zernike terms are not removed in the retrace error evaluation, the magnitudes of the PV retrace errors are higher than the PV surface measurement errors previously reported [1]. From this, only the LWIR systems in either of these two BVTS cases can be expected to perform with less than λ/10 PV retrace error. This does not preclude the other source wavelengths from potential use, but drastically limits their expected performance in these BVTS systems. For these reasons, it is important to identify the primary contributors to retrace error as well as to determine if there is the potential to address them.
The largest differences between the two initial BVTS systems were in x-tilt, y-tilt, oblique astigmatism, and vertical coma, as demonstrated in Tables 7 and 8 in Ref. [1]. If we compare tilt terms between the systems, it is obvious that BVTS Example 1 possesses unwanted aberrations in both tilt terms while BVTS Example 2 has a negligible y-tilt with a significantly larger x-tilt. While the combination of x- and y-tilts is undesired, the overall magnitude is less in the first system. For the second system, x-tilt terms and fringes across the source wavelengths proved challenging for the phase unwrapping procedure, which dominated the PV retrace error as can be seen comparing results from Table 1 above and Table 5 of Ref. [1]. The tilt terms present must be due to transmitted wavefront error, implying the freeform surfaces are introducing a tilt term to the wavefront and exacerbating it upon the second pass through the BVTS.
As to why these aberrations arise, the analytical design approach behind the Alvarez lens concept (and therefore the BVTS systems) assumes the freeform surfaces are in the same plane [17]. However, there must be some space between the freeform surfaces to avoid collisions. As a consequence, the wavefront modified by the first freeform surface has propagated some distance to the second freeform surface and therefore is deviated from an ideal shape [17]. While symmetry breaking of the freeform surfaces can be used to reduce aberrations for imaging systems, surface symmetry is more desirable in the current case where the test wavefront returns through the BVTS.
3.2 Optimization method for reduced retrace error: BVTS example 3
To reduce retrace error, optimization metrics must consider the wavefront aberrations generated by the freeform surfaces in the BVTS [17,22–24]. Optimizing the freeform surfaces with the target of uniform interferograms (corresponding to ideal measurements of ideal surfaces) from the simulated Fizeau system for the full measurement range of the BVTS is computationally challenging. Instead, we optimize the BVTS system across the full range as an imaging system to minimize PV wavefront error and to reduce the aberrations with the highest contribution in the focal plane; reducing transmitted wavefront error should reduce retrace error [14]. From the Zemax OpticStudio User Manual, wavefront error with respect to centroid is calculated by removing piston and tilt [25]. Since tilt terms were present in our previous results, the optimization should target minimized PV wavefront error referenced to the chief ray instead, which does not remove the tilt terms.
For a new optimized design addressing the measurement aberrations of the initial systems, we started from the BVTS Example 2 system and created a merit function in Zemax OpticStudio to minimize PV wavefront error (chief ray) and specific Fringe Zernike polynomial wavefront aberrations in the image plane. The Fringe Zernike terms with the highest variation across all configurations for the BVTS Example 2 system were assigned targets of 0 and equal weighting for 40 configurations in the multi-configuration editor (10 different f-numbers each for four wavelengths). The terms with the highest initial contribution were piston (Z1), x-tilt (Z2), defocus (Z4), horizontal coma (Z7), spherical (Z9), horizontal trefoil (Z10), secondary horizontal coma, (Z14), horizontal tetrafoil (Z17), secondary horizontal trefoil (Z19), and horizontal pentafoil (Z26). We previously reported that significant contributions from x-tilt, defocus, horizontal coma, and spherical were present in the surface measurement aberrations for this system (Table 8 of Ref. [1]). Therefore, by optimizing the surface coefficients to minimize the image plane wavefront aberrations, we intend to reduce the surface measurement aberrations accordingly.
The original 4th order polynomial system of BVTS Example 2 showed improvement up to the Z9 (Spherical) aberration term after optimization. Since higher order surface polynomials were needed to address higher order aberrations, we used a more general form for the freeform surfaces for the new BVTS design:
Using the surface form of Eq. (1), we incremented one order (defined by the sum of m and n in Eq. (1)) at a time, optimizing with each iteration. The merit function had diminishing improvements (on the order of 10−5) as 10th order surface polynomials were added. While optimizing, it was observed that the y term offered no advantage and the x2 term contributed a significant amount of x-tilt (Z2) aberration regardless of optimization. The surface coefficients for the resulting example BVTS design are shown in Table 6 of Appendix A, and the resulting PV wavefront error (chief ray) in the focal plane is shown in Table 2.Table 2. PV Wavefront Errors (chief ray) for BVTS Example 3
The PV wavefront error (chief ray) in the focal plane did not significantly improve (and in some configurations slightly worsened) when compared to corresponding results for BVTS Examples 1 and 2 (presented in Tables 3 and 5 of Ref. [1]). This strongly suggests, again, that diffraction limited spot size and the total PV wavefront error of the BVTS systems in the focal plane are not the best optimization metrics for the BVTS systems. However, as will be discussed below, the retrace error was significantly reduced for the third system compared to the initial systems, likely due to reduction in magnitude of the selected Fringe Zernike wavefront aberration terms.
Table 3. PV Retrace Errors for Ideal Surface Measurements
3.3 Retrace errors of BVTS example 3
As just discussed, BVTS Example 3 was optimized with the goal of reduced retrace errors. We simulated the use of this new design for the same ideal surfaces as BVTS Examples 1 and 2. The resulting PV retrace errors for BVTS Example 3 are presented in Table 3.
4. Discussion
The PV retrace errors for BVTS Example 3 are drastically reduced for the shorter source wavelengths in comparison to the PV retrace errors reported in Table 1 for the two initial designs. These errors also remain below λ/10 for the 10.6 µm and much of 3.39 µm, which was not achieved in either of the previous BVTS system examples. While perhaps still insufficient for precise surface measurements, across the full ranges tested, the PV error for BVTS Example 3 is less than λ/2 for all IR source wavelengths (except the r/16 concave configuration for 1.55 µm) and less than λ for the visible source wavelength, indicating significantly improved performance over the previous systems. To visually illustrate this improvement, Figs. 3 and 4 compare representative results of wrapped interferogram phase for the three BVTS systems for a range of r-numbers. Constant phase across the interferogram corresponds to zero retrace error (no path difference for an ideal surface). Notably, x-tilt is visibly present in BVTS Example 2, and the phase is much more constant for BVTS Example 3. The improved PV error metric of the retrace error maps and the more constant phase for BVTS Example 3 indicate that our optimization method is successful in reducing retrace error. It is feasible that further reductions may be possible with refined optimization procedures.
Fig. 3. Interferogram phase (radians from –π to + π) for 1.55 µm source with each BVTS Example for convex surfaces, 100 mm diameter.
Fig. 4. Interferogram phase (radians from –π to + π) for 1.55 µm source with each BVTS Example for concave surfaces, 100 mm diameter.
While wrapped phase is useful for visually demonstrating the aberration effects, unwrapped phase is used to determine the height maps and is more conducive to quantification of the constancy of the interferogram and surface measurements. To this end, the root-mean-square (RMS) error of the unwrapped interferogram phase maps was calculated for each case to provide a quantitative metric for comparison. RMS is useful for determining the variation of values with respect to the arithmetic mean; a smaller RMS value corresponds to a greater degree of constancy. RMS values and the standard deviation σ of the RMS values for the unwrapped interferograms are reported in Table 4. As can be seen, the non-symmetrical aberrations of Example 1 introduce a degree of inconsistency and the tilt terms in Example 2 introduce significant deviation for the faster r-numbers. Example 3 demonstrates a significant improvement over the initial two systems, along with a more consistent performance across the range of positive and negative r-number surfaces.
Table 4. RMS of Example Unwrapped Interferogram Phase Profiles
5. Conclusions
We have demonstrated that BVTS systems can replace several conventional transmission spheres with one Alvarez lens-based system, but that the systems can fall short of target performance (except in the LWIR) due to retrace error. We have demonstrated an improved optimization method targeting specific wavefront aberrations to significantly reduce retrace error. The improved BVTS system brings the shorter source wavelengths closer to viability for surface measurements. It is likely that the initial BVTS systems are ambitious in their source wavelength range; reduced ranges and changes in optical materials may result in further improvements for shorter source wavelengths when coupled with updated optimization metrics. Tolerancing studies and experimental characterization of BVTS systems are desirable to evaluate the effects of manufacturing and alignment errors on system performance. Future work to develop optimization methods for the design of BVTS systems related directly to constant interferogram phase along with the demonstrated transmitted wavefront aberration optimization may further enable improved surface measurements.
Appendix A
The form of the freeform surfaces for the BVTS examples are specified by Eq. (1) with the following surface coefficients amn:
Table 5. Surface Coefficients for BVTS Examples 1 and 2
Table 6. Surface Coefficients for BVTS Example 3
Acknowledgments
The authors would like to acknowledge valuable discussions with Dr. Chris Evans from the University of North Carolina at Charlotte. J.K. would like to acknowledge the support of the DoD SMART Scholarship Program. This work is synergistic with the NSF I/UCRC Center for Freeform Optics (IIP-1822049 and IIP-1822026).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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