For hundreds of years, spherical surfaces have been the norm for optics simply because they are easy to make: randomly rubbing two rocks together while allowing free rotation abrades high points and reduces their mated common surface to a spherical section: only spheres have the same local geometry at all points. It is little surprise, then, that most optical surfaces are sections of spheres. However, non-spherical optics, or ‘aspheres’ can deliver better optical performance.

Aspheres are far more challenging to manufacture than spherical optics for several reasons. Polishing tools must then cope with varying curvatures. In order to meet optical tolerances, manufacturers must be able to measure the surface topography to accuracies well below 1μm: an essential operation that is complex and expensive for aspheres. High-precision computer-controlled machines are increasingly allowing optics makers to meet these challenges, and thereby enabling aspheres to play a greater role in cost-effective optical systems.

Surprisingly, a simple aspect of this technology has fallen behind the game: the specification of desired aspheric shapes. Optical designers, manufacturers, and metrologists are painfully aware that the traditional method for characterizing the nominal shape of aspheric optics is inadequate.^{1,2}

The standard characterization of an optical surface quantifies its deviation from a conic section, because conic sections have especially useful optical properties. In particular, a rotationally symmetric part's nominal shape is conventionally expressed in Cartesian coordinates as:

where *r*^{2}=*x*^{2} + *y*^{2}.^{3} The first part of Equation 1 describes a conic of axial curvature *c* with conic parameter ε. The additive polynomial pieces are plotted in Figure 1. Their clear similarity in shape leads to difficulties that rapidly become critical as more terms are used. This representation is sorely ill-conditioned. All of the difficulties can be avoided, however, by orthogonalizing these additive pieces.^{4} Loosely speaking, this means that the elemental shapes of Figure 1 are linearly combined to form oscillatory polynomials whose pairwise products integrate to zero, much like sinusoidal harmonics. This simple step has a striking impact.

**Figure 1.** The first six members in the sum in Equation

1 with

*u*=

*r*/

*r*_{max} where

*r*_{max} is the aperture radius. The similarity in shapes becomes a problem as more terms are added.

Designers like to explicitly control a surface's axial intercept and axial curvature while also being able to represent a conic exactly. Without sacrificing these strengths of Equation 1, its problems can be solved by normalizing the transverse variable to *u*=*r*/*r*_{max} and writing:

where the new basis members, namely *Q*^{con}_{m}, are plotted in Figure 2. When fitting given shapes with an orthogonal basis, the optimal value of any one coefficient is independent of *M*. Further, the mean square value of the additive piece in Equation 2 is just a weighted sum of squares, namely Σ _{m}*a*_{m}^{2} /(2*m* + 5). This method avoids cancellation problems and allows us to interpret the coefficients intuitively.

**Figure 2.** The first six orthogonal polynomials in the sum in Equation

2. The color coding is as for Figure

1.

**Figure 3.** The first six members in the sum in Equation

3. The color coding is again as for Figure

1.

For those working in manufacturing and testing, it is helpful to have ready access to the slope of the deviation between the part and its best-fitting sphere. Aspheres are currently far more cost effective if this slope is constrained. Accordingly, a second basis has been introduced specifically to support the design of mild aspheres by replacing the additive polynomial in Equation 2 with:

These new polynomials are chosen so that the mean of the square of the derivative (i.e. d/d*u*) of the expression in Equation 3 is precisely Σ _{m}*a*_{m}^{2}. This non-standard orthogonalization leads to the basis members plotted in Figure 3. These elements change the surface's shape without changing its best-fit sphere. The designer can optimize cost-effective solutions by simply constraining Σ_{m}*a*_{m}^{2} to ensure that the asphere can be finished and tested without prohibitive complexity.

**Table 1.** Coefficients (in nm) from Equations 1 and 2 for describing the curve in Figure 4.

Some benefits of using orthogonal bases can be appreciated by examining Table 1. These values fit the departure map of Figure 4. For Equation 1, the coefficients are normalized by re-expressing the deviation in terms of *u* so that all entries in the table can be given in nanometer units. The coefficients for Equation 1 with *M*=7 and 8 are given in Columns 2 and 4. It turns out that although the overall fit differs by only a few nanometers, the individual coefficients change by up to about a million times this (e.g. see *a*_{6}). Cancellation issues are also demonstrated in Column 3, which holds changes for the values in Column 2 that surprisingly combine to modify the overall fit by less than 1nm.

**Figure 4.** A sample aspheric surface deviates from a conic section by 60μm from peak to valley over a clear aperture of arbitrary size.

By contrast, the corresponding coefficients for Equation 2 with *M*=8 are given in the last column. Notice that far fewer digits are needed. Also, the coefficients for *M*=7 follow simply upon dropping the last entry. This 5nm entry alone accounts identically for the ungainly difference between Columns 2 and 4. It also follows upon inspection that the truncation error with *M*=7 for Equation 2 will be about 5nm with the form of *Q*_{8}^{con}. Given that the mean square departure is Σ _{m}*a*_{m}^{2} /(2*m* + 5), the rough magnitude of the low-order terms follows intuitively from the amplitude of the departure plotted in Figure 4.

There are other benefits associated with these orthogonal representations.^{4} Remarkably, these minor changes to Equation 1 deliver numerical robustness, allow aspheres to be described with fewer digits, and enable intuitive understanding of a part's shape upon inspection of the coefficient values. What's more, constraining aspheres by using Equation 3 facilitates the design of cost-effective solutions that should further boost the prevalence of aspheric optics. We are currently working with optical design software companies to demonstrate the effectiveness of these ideas in real systems, and to incorporate them as new tools for optical designers.

Greg Forbes

QED Technologies Inc.

Sydney, Australia

Greg Forbes is senior scientist at QED Technologies where he has developed innovations and software for optical polishing and metrology systems. He is an OSA Fellow and former faculty member of The Institute of Optics, University of Rochester.