System assessment for design often involves averages, such as the standard deviation of the wavefront error, which are estimated by ray tracing through a sample of points within the entrance pupil. General-purpose sampling and weighting schemes are presented, and it is also shown that optical design can benefit from tailored versions of these schemes. It turns out that the type of Gaussian quadrature that has long been recognized for efficiency in this domain requires about 40% to 50% more ray tracing to attain comparable accuracy to generic versions of the new schemes. Even greater efficiency gains can be won, however, by tailoring such sampling schemes to the optical context where azimuthal variation in the wavefront is generally weaker than the radial variation. These new schemes are special cases of what is known in the mathematical world as cubature. Our initial results also led to the consideration of simpler sampling configurations that approximate the newfound cubature schemes. We report on the practical application of a selection of such schemes and make observations that aid in the discovery of cubature schemes relevant to optical design of systems with circular pupils.
System assessment for design often involves averages, such as rms wavefront error, that are estimated by ray tracing through a sample of points within the pupil. Novel general-purpose sampling and weighting schemes are presented and it is also shown that optical design can benefit from tailored versions of these schemes. It turns out that the type of Gaussian quadrature that has long been recognized for efficiency in this domain requires about 40-50% more ray tracing to attain comparable accuracy to generic versions of the new schemes. Even greater efficiency gains can be won, however, by tailoring such sampling schemes to the optical context where azimuthal variation in the wavefront is generally weaker than the radial variation. These new schemes are special cases of what is known in the mathematical world as cubature. Our initial results also led to the consideration of simpler sampling configurations that approximate the newfound cubature schemes. We report on the practical application of a selection of such schemes and make observations that aid in the discovery of novel cubature schemes relevant to optical design of systems with circular pupils.
Optical surfaces feature a wide range of length scales from “figure” down to “finish”, but the mid-spatial frequency structure (MSF) holds growing significance. Cost-effective production of systems demands answers to multiple layers of related questions, such as how best to quantify MSF, assess its optical impact, and employ existing production tools to meet MSF requirements. These answers evolve as new production technologies are introduced. I present general observations about a few of the associated challenges and attempt to clarify some essential aspects related to quantifying MSF as well as estimating its impact.
Gradient-orthogonal representations of aspheric shapes give a more effective and intuitive characterization that also copes with increasingly complex surfaces. Further, we have seen a range of applications where standard design codes (including CodeV® and Zemax®) can find systems with better optical performance when optimized in this representation. The examples presented here include a system with no global axis of symmetry and another with freeform surfaces. In all these particular cases, the end results can be retro-fitted in terms of conventional representations, but the optimizers fail to find the superior solutions unless an orthogonal basis is employed during the design process. Because the communication of shape is so much more effective in terms of a gradient-orthogonal description, our results give added motivation for the communities of design, fabrication, and testing to gain more experience with this new convention.
Aspheric surfaces provide significant benefits to an optical design. Unfortunately, aspheres are usually more difficult to fabricate than spherical surfaces, making the choice of whether and when to use aspheres in a design less obvious. Much of the difficulty comes from obtaining aspheric measurements with comparable quality and simplicity to spherical measurements. Subaperture stitching can provide a flexible and effective test for many aspheric shapes, enabling more cost-effective manufacture of high-precision aspheres. To take full advantage of this flexible testing capability, however, the designer must know what the limitations of the measurement are, so that the asphere designs can be optimized for both performance and manufacturability. In practice, this can be quite difficult, as instrument capabilities are difficult to quantify absolutely, and standard asphere polynomial coefficients are difficult to interpret. The slope-orthogonal “Q” polynomial representation for an aspheric surface is ideal for constraining the slope departure of aspheres. We present a method of estimating whether an asphere described by Q polynomials is measurable by QED Technologies’ SSI-A system. This estimation function quickly computes the testability from the asphere’s prescription (Q polynomial coefficients, radius of curvature, and aperture size), and is thus suitable for employing in lens design merit functions. We compare the estimates against actual SSI-A lattices. Finally, we explore the speed and utility of the method in a lens design study.
Aspheric surfaces can provide significant benefits to optical systems, but manufacturing high-precision
aspheric surfaces is often limited by the availability of surface metrology. Traditionally, aspheric measurements have
required dedicated null correction optics, but the cost, lead time, inflexibility, and calibration difficulty of null optics
make aspheres less attractive. In the past three years, we have developed the Subaperture Stitching Interferometer for
Aspheres (SSI-A®) to help address this limitation, providing flexible aspheric measurement capability up to 200 waves
of aspheric departure from best-fit sphere.
Some aspheres, however, have hundreds or even thousands of waves of departure. We have recently
developed Variable Optical Null (VON<sup>TM</sup>) technology that can null much of the aspheric departure in a subaperture. The
VON is automatically reconfigurable and is adjusted to nearly null each specific subaperture of an asphere. The VON
provides a significant boost in aspheric measurement capability, enabling aspheres with up to 1000 waves of departure
to be measured, without the use of null optics that are dedicated to each asphere prescription. We outline the basic
principles of subaperture stitching and the Variable Optical Null, demonstrate the extended capability provided by the
VON, and present measurement results from our new Aspheric Stitching Interferometer (ASI<sup>TM</sup>).
Two key criteria govern the characterization of nominal shapes for aspheric optical surfaces. An <i>efficient</i>
representation describes the spectrum of relevant shapes to the required accuracy by using the fewest decimal digits in
the associated coefficients. Also, a representation is more <i>effective</i> if it can, in some way, facilitate other processes - such as optical design, tolerancing, or direct human interpretation. With the development of better tools for their design,
metrology, and fabrication, aspheric optics are becoming ever more pervasive. As part of this trend, aspheric departures
of up to a thousand microns or more must be characterized at almost nanometre precision. For all but the simplest of
shapes, this is not as easy as it might sound. Efficiency is therefore increasingly important. Further, metrology tools
continue to be one of the weaker links in the cost-effective production of aspheric optics. Interferometry particularly
struggles to deal with steep slopes in aspheric departure. Such observations motivated the ideas described in what
follows for modifying the conventional description of rotationally symmetric aspheres to use orthogonal bases that boost
efficiency. The new representations can facilitate surface tolerancing as well as the design of aspheres with cost-effective
metrology options. These ideas enable the description of aspheric shapes in terms of decompositions that not
only deliver improved efficiency and effectiveness, but that are also shown to admit direct interpretations. While it's
neither poetry nor a cure-all, an old blight can be relieved.
The manufacturing of precision aspheres has traditionally been a long-lead-time, labor-intensive process that is made
even more expensive by the need for specific process expertise, dedicated tooling for polishing, and dedicated nulls for
metrology. These challenges have limited the widespread use of optical aspheres. New technology is currently being
developed to enable flexible and lower-cost manufacturing of precision aspheres, without the need for dedicated tools or
null optics. Subaperture Stitching Interferometry (SSI®) combined with Magnetorheological Finishing (MRF®) enable a
flexible and deterministic approach to finishing precision aspheres in a wide variety of materials and geometries. MRF
systems use highly stable, subaperture tools that perfectly conform to the changing curvature of aspheric optics during
the polishing process. This enables a single machine to process plano, spherical, and aspheric surfaces (both convex and
concave) without the delays and costs associated with maintaining and switching between sets of dedicated tooling. SSI
systems mathematically "stitch" together subaperture measurements to generate high-resolution, high-precision, fullaperture
aspheric surface measurements. By locally nulling and using maximum pixel resolution over a subaperture, the
SSI extends general-purpose, null-free interferometry to aspheres with departures from best-fit-sphere on the order of
100ë. When these technologies are combined with either the latest grinding and pre-polishing or diamond-turning
technology, fast, flexible prototyping, or small-batch production of precision aspheres is available at an attractive cost.
Interferometric tests of aspheres have traditionally relied on so-called "null correctors". These usually require significant time and expense to design and fabricate, and are specific to a particular asphere prescription. What's more, they are tedious to align and calibrate. Aspheres can also be tested without null correction (using a spherical wavefront), but such capability is extremely limited. A typical interferometer can acquire only a few micrometers of fourth-order aspheric departure due to high-density interference fringes. Furthermore, standard software packages do not compensate for the impact upon a non-null measurement of (i) the part's aspheric shape or (ii) the interferometer's optical aberrations. While fringe density and asphere compensation severely limit the practical utility of a non-null asphere measurement, subaperture stitching can directly address these issues. In 2004, QED Technologies introduced the Subaperture Stitching Interferometer (SSI<sup>(R)</sup>) to automatically stitch spherical surfaces (including hemispheres). The system also boosts accuracy with in-line calibration of systematic errors. We have recently added aspheric capability, extending non-null aspheric test capability by an order of magnitude or more. As demonstrated in the past on annular zones of nearly nulled data, subaperture stitching can extend the testable aspheric departure. We present a more generally applicable and robust method of stitching non-null aspheric phase measurements. By exploiting novel compensation schemes and in-line system error calibration, our subaperture stitching system can provide significantly better accuracy and increased testable aspheric departure over an unstitched non-null test. Examples of stitched non-null tests are analyzed in this paper, and cross-tested against corresponding null tests.
This paper summarizes some of latest developments by QED Technologies (QED) in the field of high-precision polishing and metrology. Magneto-Rheological Finishing (MRF) is a deterministic sub-aperture polishing process that overcomes many of the fundamental limitations of traditional finishing. The MR fluid forms a polishing tool that is perfectly conformal and therefore can polish a variety of shapes, including flats, spheres, aspheres, prisms, and cylinders, with round or non-round apertures. Over the past several years, QED's Q22 family of polishing platforms, based on the MRF process, have demonstrated the ability to produce optical surfaces with accuracies better than 30 nm peak-to-valley (PV) and surface micro-roughness less than 0.5 nm rms on an ever-widening variety of optical glass, single crystal, and glass-ceramic materials. The MRF process facilitates the correction of the transmitted wavefront of single elements and/or entire systems, as well as enabling the inducement of specific desired wavefront characteristics (i.e., other than making surfaces perfectly flat or spherical), which is beneficial for applications such as phase correction or other freeform applications. QED's Sub-aperture Stitching Interferometer (SSI) complements MRF by extending the effective aperture, accuracy, resolution, and dynamic range of a phase-shifting interferometer. This workstation performs automated sub-aperture stitching measurements of spheres, flats, and mild aspheres. It combines a six-axis precision stage system, a commercial
Fizeau interferometer, and specially developed software that automates measurement design, data acquisition, and the reconstruction of the full-aperture map of figure error. Aside from the correction of sub-aperture placement errors (such as tilts, optical power, and registration effects), the SSI software also accounts for reference-wave error, distortion, and other aberrations in the interferometer's imaging optics. By addressing these matters upfront, we avoid limitations encountered in earlier stitching work and significantly boost reproducibility beyond that of the integrated interferometer on its own.
Subaperture polishing technologies have radically changed the landscape of precision optics manufacturing and enabled the production of higher precision optics with increasingly difficult figure requirements. However, metrology is a critical piece of the optics fabrication process, and the dependence on interferometry is especially acute for computer-controlled, deterministic finishing. Without accurate full-aperture metrology, figure correction using subaperture polishing technologies would not be possible. QED Technologies has developed the Subaperture Stitching Interferometer (SSI) that extends the effective aperture and dynamic range of a phase measuring interferometer. The SSI's novel developments in software and hardware improve the capacity and accuracy of traditional interferometers, overcoming many of the limitations previously faced. The SSI performs high-accuracy automated measurements of spheres, flats, and mild aspheres up to 200 mm in diameter by stitching subaperture data. The system combines a six-axis precision workstation, a commercial Fizeau interferometer of 4" or 6" aperture, and dedicated software. QED's software automates the measurement design, data acquisition, and mathematical reconstruction of the full-aperture phase map. The stitching algorithm incorporates a general framework for compensating several types of errors introduced by the interferometer and stage mechanics. These include positioning errors, viewing system distortion, the system reference wave error, etc. The SSI has been proven to deliver the accurate and flexible metrology that is vital to precision optics fabrication. This paper will briefly review the capabilities of the SSI as a production-ready, metrology system that enables costeffective manufacturing of precision optical surfaces.
Interferometers are often used to measure optical surfaces and systems. The accuracy of such measurements is often limited by the ability to calibrate systematic errors such as reference wave and image distortion. Standard techniques for calibrating reference wave include the two-sphere and random-ball test. QED Technologies® (QED) recently introduced a Subaperture Stitching Interferometer (SSI®) that has the integrated ability to perform reference wave calibration. By measuring an optical surface in multiple locations, the stitching algorithm has the ability to compensate for reference wave and imaging distortion. Each of the three reference wave calibration methods has its own limitations that ultimately affect the accuracy of the measurement. The merits of each technique for reference wave calibration are reviewed and analyzed. By using the SSI-computed estimate and the random-ball test in tandem, a composite method for calibrating reference wave error is shown to combine the benefits of both individual techniques. The stitching process also calibrates for distortion, and plots are shown for different transmission optics. Measurements with and without distortion compensation are shown, and the residual difference is compared to theoretical predictions.
Many defense systems have a critical need for high-precision, complex optics. However, fabrication of high quality, advanced optics is often seriously hampered by the lack of accurate and affordable metrology. QED's Subaperture Stitching Interferometer (SSI<sup>®</sup>) provides a breakthrough technology, enabling the automatic capture of precise metrology data for large and/or strongly curved (concave and convex) parts.
QED’s SSI complements next-generation finishing technologies, such as Magnetorheological Finishing (MRF<sup>®</sup>), by extending the effective aperture, accuracy and dynamic range of a phase-shifting interferometer. This workstation performs automated sub-aperture stitching measurements of spheres, flats, and mild aspheres. It combines a six-axis precision stage system, a commercial Fizeau interferometer, and specially developed software that automates measurement design, data acquisition, and the reconstruction of the full-aperture figure error map. Aside from the correction of sub-aperture placement errors (such as tilts, optical power, and registration effects), our software also accounts for reference-wave error, distortion and other aberrations in the interferometer’s imaging optics. The SSI can automatically measure the full aperture of high numerical aperture surfaces (such as domes) to interferometric accuracy.
The SSI extends the usability of a phase measuring interferometer and allows users with minimal training to produce full-aperture measurements of otherwise untestable parts. Work continues to extend this technology to measure aspheric shapes without the use of dedicated null optics. This SSI technology will be described, sample measurement results shown, and various manufacturing applications discussed.
Subaperture stitching is a well-known technique for extending the effective aperture range of phase measuring interferometers. In the past, stitching has successfully been applied to improve the lateral coverage and/or resolution of plano interferometers (including interference microscopes). More recently, QED Technologies has developed a subaperture stitching interferometer (SSI®) for automatic stitching of spherical surfaces, including hemispheres. But stitching can also extend the amount of aspheric departure that can be measured in a non-null test.
Conventional interferometers have some capability to measure mild aspheric surfaces without null correction. The interference fringe resolution of the camera limits the asphericity that can be measured, while the difficulty in inferring the surface form from the measured phase degrades accuracy. Therefore, commercially available interferometers can only measure a few micrometers of fourth-order aspheric departure. Furthermore, standard measurement software does not compensate for the aspheric shape or for the interferometer imaging errors present in a non-null measurement. As a result, non-null aspheric measurements are more difficult, and less accurate, than a spherical null test. Examples are presented in this paper that illustrate these issues. Subaperture stitching can extend the testable aspheric departure of a non-null test. This has been demonstrated in the past on annular zones of near-null data. We present a more generally applicable and robust method of stitching non-null phase data, which can provide better accuracy and increased testable aspheric departure over an unstitched test.
Optical surfaces are routinely measured using phase-shifting interferometry. The fringe imaging and other interferometer optics introduce distortion into the measurements. Distortion causes a change in magnification as a function of field position, and is often not quantified and calibrated during measurements of optical surfaces. When calculating the figure of an optical surface, systematic errors such as distortion will ultimately limit the accuracy of the measurement. We present a method for improving the accuracy in interferometric measurements using subaperture stitching interferometry. QED's Subaperture Stitching Interferometer (SSI®) is a six-axis computer-controlled workstation that incorporates a standard Fizeau interferometer with our own stitching algorithms. The SSI is a commercially available product that automatically performs inline calibration of systematic errors such as reference wave and distortion. By measuring an optical surface in multiple orientations both on and off-axis, our stitching algorithms are shown to have the ability to measure the distortion (and other systematic errors) in an interferometer, and compensate for these errors automatically. Using the compensators obtained from stitched measurements, distortion values are calculated and plots are shown for several different transmission optics. Theoretical simulations displaying the effects of distortion on surface metrology are shown. Measurements are taken with and without distortion compensators, and the residual difference is analyzed.
This paper summarizes some of QED Technologies’ latest developments in the field of high-precision polishing and metrology. Magneto-Rheological Finishing (MRF) is a deterministic sub-aperture polishing process that overcomes many of the fundamental limitations of traditional finishing. MRF has demonstrated the ability to produce optical surfaces with accuracies better than 30 nm peak-to-valley (PV) and surface micro-roughness less than 0.5 nm rms on a wide variety of optical glasses, single crystals, and glass-ceramics. The MR fluid forms a polishing tool that is perfectly conformal and therefore can polish a variety of shapes, including flats, spheres, aspheres, prisms, and cylinders, with either round or rectangular apertures. QED’s Sub-aperture Stitching Interferometer (SSI) complements MRF by extending the effective aperture, accuracy, resolution, and dynamic range of a phase-shifting interferometer. This workstation performs automated sub-aperture stitching measurements of spheres, flats, and mild aspheres. It combines a six-axis precision stage system, a commercial Fizeau interferometer, and specially developed software that automates measurement design, data acquisition, and the reconstruction of the full-aperture map of figure error. Aside from the correction of sub-aperture placement errors (such as tilts, optical power, and registration effects), our software also accounts for reference-wave error, distortion, and other aberrations in the interferometer’s imaging optics. By addressing these matters up front, we avoid limitations encountered in earlier stitching work and significantly boost reproducibility beyond that of the integrated interferometer on its own.
We describe the application of both stitching interferometry and magneto-rheological finishing (MRF) to the surface metrology and final figure correction of large optics. These particular subaperture technologies help to address the need for flexible systems that improve both overall manufacturing time and cost effectiveness. MRF can achieve high volumetric removal rates with a small-footprint tool that is perfectly conformable and highly stable. This tool is therefore well suited to finishing large optics (including aspheres) and correcting mid-spatial frequency errors. The system does not need vacuum, reduces microroughness to below one nm rms on most materials, and is able to meet the figure tolerance specs for astronomical optics. Such a technology is ideally complemented by a system for the stitching of interferometric subaperture data. Stitching inherently enables the testing of larger apertures with higher resolution and, thanks to the inbuilt calibration, even to higher accuracy in many situations. Moreover, given the low-order character of the dominant residual uncertainties in the stitched full-aperture data, such an approach is well suited to adaptive mirrors because the actuators correct precisely these deformations. While this approach enables the non-null testing of parts with greater aspheric departure and can lead to a significantly reduced non-common air path in the testing of long-radius concave parts, it is especially effective for convex optics. That is, stitching is particularly well suited to the testing of secondary mirrors and, alongside the testing of the off-axis primary segments, these are clearly critical challenges for extremely large telescope (ELT) projects.
Stable superpositions of elemental ray contributions give new capabilities for the ray-based analysis of optical fields and systems.
These field estimates are constructed so that the results are effectively independent of the width of the field elements associated with each ray. The elements are thus free to be, more or less arbitrarily, localized or distributed at will. Since such elements thus need not disperse under diffraction, the new scheme is not a traditional beam summation method. In fact, it is this definitive property that leads to remarkable simplifications, and to significantly more accurate ray-based wave models. With this approach, error estimates become easily accessible, and a asymptotic treatment can finally be developed for fundamental processes such as refraction and diffraction.
Subaperture stitching is a well-known technique for extending the effective aperture and dynamic range of phase measuring interferometers. Several commercially available instruments can automatically stitch flat surfaces, but practical solutions for stitching spherical and aspherical surfaces are inherently more complex. We have developed an interferometer workstation that can perform high-accuracy automated subaperture stitching of spheres, flats, and mild aspheres up to 200 mm in diameter. The workstation combines a six-axis precision stage system, a commercial Fizeau interferometer of 4” or 6” aperture, and a specially developed software package that automates measurement design, subaperture data acquisition, and the mathematical reconstruction of a full-aperture phase map. The stitching algorithm incorporates a general constrained optimization framework for compensating for several types of errors introduced by the interferometer optics and stage mechanics. These include positioning errors, viewing system distortion, and the system reference wave. We present repeatability data, and compare stitched full-aperture measurements made with two different transmission spheres to a calibrated full-aperture measurement. We also demonstrate stitching’s ability to test larger aspheric departures on a 10 mm departure parabola, and compare the preliminary results with a full-aperture null test.
Our new method for optical modeling puts ray optics on a more solid foundation. This method not only delivers higher accuracy, but also offers estimates of its own errors. The conceptual framework is fully consistent with intuitive interpretations of rays and avoids the <i>ad hoc</i> leaps of standard ray-based modeling. These include problems in such areas as propagation, refraction, reflection, and diffraction. The model's higher accuracy also means that more applications now fall within the sope of ray-based system analysis. This is demonstrated via a simple example involving a waveguide with a smoothly varying refractive index. In particular, a low-order waveguide mode is modeled as it propagates to, and interacts with, a flat interface between the waveguide and a homogeneous medium.
Although rays form the basis of both our initiative thinking and our numerical modeling of optical systems, we must remember that they are no more than mathematical constructs with a relatively tenuous connection to the physical world. What's more, certain aspects of their link to wave-based models of optical systems have always been problematic. For example, the limit to what rays can tell us about the associated wave field remains unclear. The current state of the art is reviewed and a framework is then outlined that offers a more direct appreciation of these issues and promises extended capabilities for ray-based methods. The new approach involves modeling wave propagation by using--as an intermediate tool--a windowed Fourier transform of the field.
A generalization of Hamilton's formalism for geometrical optics is given here to provide more convenient descriptions of the optical properties of certain classes of systems. This generalization is made by replacing the usual points and planes that are used as references in the definition of characteristic functions by more general surfaces. In this context, the fractional Legendre transform is used in the semigeometrical estimation of field propagators to eliminate errors introduced by caustics.
The structure of the simulated annealing algorithm is presented and its rationale is discussed. A unifying heuristic is then introduced which serves as a guide in the design of all of the sub-components of the algorithm. Simply put this heuristic principle states that at every cycle in the algorithm the occupation density should be kept as close as possible to the equilibrium distribution. This heuristic has been used as a guide to develop novel step generation and temperature control methods intended to improve the efficiency of the simulated annealing algorithm. The resulting algorithm has been used in attempts to locate good solutions for one of the lens design problems associated with this conference viz. the " monochromatic quartet" and a sample of the results is presented. 1 Global optimization in the context oflens design Whatever the context optimization algorithms relate to problems that take the following form: Given some configuration space with coordinates r (x1 . . x) and a merit function written asffr) find the point r whereftr) takes it lowest value. That is find the global minimum. In many cases there is also a set of auxiliary constraints that must be met so the problem statement becomes: Find the global minimum of the merit function within the region defined by E. (r) 0 j 1 2 . . . p and 0 j 1 2 . . . q.
This course provides an overview of how aspheric surfaces are designed, manufactured, and measured. The primary goal of this course is to teach how to determine whether a particular aspheric surface design will be difficult to make and/or test. This will facilitate cost/performance trade off discussions between designers, fabricators, and metrologists.
We will begin with a discussion of what an asphere is and how they benefit optical designs. Next we will explain various asphere geometry characteristics, especially how to evaluate local curvature plots. We will also review flaws of the standard polynomial representation, and how the Forbes polynomials can simplify asphere analysis. Then we will discuss how various specifications (such as figure error and local slope) can influence the difficulty of manufacturing an asphere. Optical assembly tolerances, however, are beyond the scope of this course - we will focus on individual elements (lenses / mirrors).
The latter half of the course will focus on the more common technologies used to generate, polish, and/or measure aspheric surfaces (e.g. diamond turning, glass molding, pad polishing, interferometry). We'll give an overview of a few generic manufacturing processes (e.g. generate-polish-measure). Then we'll review the main strengths and weaknesses of each technology in the context of cost-effective asphere manufacturing.