OSA / IODC/OF&T 2010
OTuA6.pdf
The Forbes Polynomial: A more predictable surface for fabricators Kevin P. Thompson,1,* Florian Fournier,2 Jannick P. Rolland,2 G.W. Forbes3 1
Optical Research Associates, 3 Graywood Lane, Suite 100, Pittsford, NY, 14534, USA 2 EvoOpticks, 3 Graywood Lane, Suite 201, Pittsford, NY, 14534, USA 3 QED Technologies, Inc., 1040 University Ave., Rochester, NY, 14607, USA *
[email protected]
Abstract: The Forbes polynomial is an important new surface type that allows coefficient tolerances to be meaningful and allows the difficulty of fabrication to be quantified. Implications and guidance to fabricators will be presented. ©2010 Optical Society of America OCIS codes: (220.1250) Aspherics; (220.4610) Optical fabrication
Greg Forbes, in his paper in this conference [1], has discussed the mathematical motivations for changing the standard description of aspheric surfaces and proposed a new description that is motivated by new capabilities for finishing and in conjunction measuring aspheric parts. This talk is meant to represent a strong support for this new standard, from the perspective of optical design as a context for optical fabricators to prioritize recognizing the significance and importance of this new, more effective formulation to replace power series aspheres. There are a number of motivations for changing from the historical power series based definition of a rotationally symmetric, nonspherical transmitting optical surface for imaging applications to the Forbes polynomial definition; many of these motivations are compelling for the fabricator. One key motivation is that of enabling a reduction in the price of an aspheric surface by substantially improving the ability to predict the cost of fabricating the surface. By establishing metrics that clearly and predictably quantify whether an optical surface falls into the manufacturing envelope for an emerging class of polishing and in-process measurement equipment based on MRF (Magneto-Rheological Finishing) and other emerging methods, a fabricator will be empowered to confidently predict the difficultly to manufacture a particular surface description and as a result its cost. As described in the earlier paper, the coefficients that specify the power-series based description of a rotationally symmetric optical surface are failing on a number of fronts. Some of the causes can be attributed to the optical design process and others to intrinsic limitations in the traditional power series description of the aspheric departure. To understand why the power series description is failing, and why it is failing now, it is interesting to look at the history of the power series aspheric surface description. There are two perspectives here, the first is the use of power series aspheres at all in the context of the design of lens versus mirror systems and the second is their usage in specifically computer-aided optical design. In the context of their introduction into the process of optical design, almost all references point to Ernst Abbe. In 1899, Abbe filed a US Patent, 697,959, titled simply “Lens System”. The figures from the filing, which is readily accessible, are shown in Figure 1.
R1=17.3 R2=24.86 d1=2.9 d2=1.2 d3=3.0 n1=1.52034 n2=1.53322
Figure 1. Figures from Abbe’s patent 697,959, filed 1899 (granted 1902) These figures support a discussion wherein Abbe presents a new option for reducing the field aberrations in the lens shown in the third panel. Figure 2 provides Abbe’s explanation of the reasoning behind the introduction of a power series expansion of the departure of the surface shape from a sphere into the optical design process.
OSA / IODC/OF&T 2010
OTuA6.pdf
Figure 2. Text from Abbe’s patent 697,959 Abbe introduces what so far is proving to be the first appearance of a power series polynomial approximation of an optical surface in the lens design literature, with the specific text from the patent illustrated in Figure 3.
Figure 3. The first appearance of a power series description of an aspheric surface in the context of optical design, from Abbe’s patent 697,959. Certainly, it would be beyond conception, even to Abbe, that the “….,” in Figure 3 would come to pass, even if it did take 100 years. Looking at computer-aided optical design, it appears the aspheric surfaces were introduced into the computer aided design hardware (in the earliest days there was no such thing as software, yet) by a number of groups who were working the problem simultaneously around 1949-1951, each affiliated with a specific early computer at Harvard, NBS, and Los Alamos. James G. Baker, one of the earliest adopters of aspheres in lens design, who also mastered their fabrication, was leading one of the teams, working at Harvard. In Chapter III “ASPHERIC RAY-TRACING” of the then classified report under the direction of Baker; written in the case by G.H. Conant, Jr. one finds the following statement, “The series expansion method for the aspheric components, the method of which we are making use, was mentioned some 53 years ago by Abbe in German and English patents, as referred to in the von Rohr book (The Formation of Optical Images). One of the most recent formulations for aspheric calculations is that by Feder (JOSA 41, 630 (1951), where an iterative procedure suitable for automatic computation is described” For those who have never seen the early days of computers, the methods used in the earliest days involved the layout of “plug boards”. Figure 4 shows the new plug board layout as modified to incorporate aspheric raytracing. This perspective illustrates that the power series asphere was introduced at a time in history that could not anticipate the challenges it is applied to today. Therefore, it should be no surprise that it needs to be replaced, and within optical design, CODE V® has implemented the Forbes polynomial within lens optimization and analysis. There are two reasons for the failure of the power series asphere at this point in time. First, the field of lithography, in particular, is reaching new thresholds in operational numerical aperture and the naturally occurring
OSA / IODC/OF&T 2010
OTuA6.pdf
Figure 4.Aspheric ray-trace plug board, from the earliest days of computer-aided optical design. limiting spherical aberration requires aspheres of orders that exceed available number of significant digits, as clearly demonstrated in the earlier paper by Forbes [1]. This is the scientifically compelling need. Separately, in the last decade, the commercial optical design programs have reformatted their input and output to make the opportunity to use more coefficients essentially too convenient. Somewhat akin to the concept of designing a telescope with empty magnification, it is all too easy for a novice optical designer to invoke considerably more aspheric coefficients than are called for by the problem at hand. This second cause is seen recently in patents filed within the cell phone industry, as discussed in a paper by McGuire at this conference, [2]. For the fabricator, especially one without an in-house optical design group, which is not uncommon, it is difficult to know if the asphere they are being asked to fabricate has the proper number of terms. With the emergence of optical finishing equipment that is particularly suited to the fabrication of aspheres, it has become an industry challenge to establish a price point that encourages optical designers to introduce them throughout their optical designs. While there is a “rule of thumb” that says that if an asphere is added an element can be removed, a recent comprehensive study by Cakmakci [3] of 3188 lenses from the LensView™ database, indicates that in fact it requires the introduction of two aspheric surfaces to effectively eliminate a lens in an optical system. What are potentially the most powerful features of the Forbes polynomial for aspheric surfaces are first that they have units of length, and therefore their value is their contribution to the surface departure in, for example, microns. This will be a descending series and once it falls below a threshold, say 0.05 microns, the remaining terms become irrelevant. This fact will be clear to any fabricator and, in addition, it can be used in the optical design process to automatically control in real-time the set of “live” coefficients. It is this feature that is expected to lead shortly to a quantitative metric for the cost of an asphere, as to a large extent the cost is directly related to the number of aspheric terms and the slope. The second powerful feature is that unlike the fact that power series coefficients are statistically meaningless if one tries to establish a tolerance on the coefficient, as shown by Rogers [4], Forbes polynomial coefficients can be given tolerances that are meaningful. It is these two features that are anticipated, and in fact expected, to accelerate the industry acceptance to this new standard for the specification of rotationally symmetric nonspherical surfaces. REFERENCES [1]Forbes, G.W., “Better ways to specify aspheric shapes can facilitate design, fab., and testing alike”, IODC, (2010). [2] McGuire, J.P. “Manufacturable mobile phone optics: higher order aspheres are not always better”, IODC, (2010). [3] Cakmakci, O., et al., “Design efficiency of 3188 optical designs”, SPIE Proc. 7060, 7060S-1, (2008). [4] Rogers, J.R., “Slope error tolerances for optical surfaces “, Presented at OSA OptiFab, paper TD04-4 (2007).
