## Abstract

Some simple measures of the difficulty of a variety of steps in asphere fabrication are defined by reference to fundamental geometric considerations. It is shown that effective approximations can then be exploited when an asphere’s shape is characterized by using a particular orthogonal basis. The efficiency of the results allows them to be used not only as quick manufacturability estimates at the production end, but more importantly as part of an efficient design process that can boost the resulting optical systems’ cost-effectiveness.

©2011 Optical Society of America

## 1. Introduction

Aspheric optical surfaces deliver higher performing, more compact, and lighter systems in a range of applications. Aspheres become of even greater value as their manufacturability is integrated more tightly into the design phase. Depending on the production processes and volume, cost-effectiveness can be boosted by accounting for the difficulty of steps like polishing, measuring, molding, and/or assembling the aspheric components. Explicit recipes for estimating cost/difficulty are currently not easy to come by, however. For such estimates to be embedded usefully within the design process, computational efficiency is critical. A particular class of highly efficient estimates of manufacturability is considered in this work.

Brute-force analysis of “as-built performance” is conceptually straightforward. Such Monte-Carlo-style processes enable designers either to determine or account for tolerances on parameters such as alignment, surface figure, mid-spatial frequency errors (MSF), homogeneity, etc. A variety of related aspects that are specific to aspheres was discussed recently by Epple and Wang [1], and by McGuire [2]. In the former, the key idea is to exploit aspheres to loosen system assembly tolerances, and the latter treats a particular example where the removal of high-order terms from the aspheres is also used to loosen tolerances. By using perturbation methods like those applied by Youngworth and Stone [3], these processes are accelerated significantly because no additional ray tracing is needed to evaluate the perturbed system. (These same methods are invoked below in Sections 3 and 7.)

Many of the asphere-related efforts in the area of “design for manufacturability” are based on the vague notion that *the less deviation from a sphere, the better*. Foreman [4,5] was more explicit when he hypothesized that it is the radial rate of change of an asphere’s meridional radius of curvature that offers a useful relative measure of manufacturability. Absolute reference values and justifications for such measures of difficulty are hard to find. Kumler [6] presented a variety of practical considerations related to polishing and testing, and treated an example where adding higher order terms to aspheres can reduce their difficulty for the manufacturer. He takes a significant step towards coupling measures of manufacturability directly to the capabilities of current fabrication and metrology tools. This allows him to go beyond relative statements, such as his guideline that “the greater the slope of the aspheric departure from a best-fit sphere, the more difficult the asphere,” to state absolute values as reference points for this slope, namely 2 microns per mm. His reference value is driven by both polishing and metrology considerations, and is in keeping with his other machine-specific absolute numbers for edge thicknesses, margins for diameters of lens blanks, etc. Such work offers important empirical quantification of the fact that an asphere’s cost and achievable figure and MSF tolerances depend strongly on its shape. It is part of an essentially unending project, of course, because innovations mean that such reference values are constantly evolving, and they vary between production technologies.

The slope discussed by Kumler directly yields the difficulty of full-aperture interferometric tests because that slope is proportional to fringe density. In that context, the allowed maximal fraction of the Nyquist sampling rate is the natural measure. In relation to sub-aperture pad polishing, du Jeu [7] uses a prescribed level of maximal tool misfit to select pad size, and the relative pad size then serves as an effective measure of difficulty. He gives compact equations for the case of the standard rotated conic sections. The same ideas are applicable for more complex aspheres, but the equations become burdensome. Further, the surface must then be analyzed at all radial zones since the outer edge need no longer be the most challenging zone. The main objective of what follows is to derive computationally efficient estimates of geometrically based measures of difficulty like the two just mentioned. Such estimates can serve as the foundation for determining whether an asphere can plausibly be made within spec and, if so, identifying the types of processes that may be required and hence its approximate cost. The method of specifying an asphere’s nominal shape turns out to hold the key to efficiently evaluating the associated measures of difficulty. Although initial steps have been made in this direction [8], they did not exploit some powerful algorithms that were reported recently [9].

The specification of aspheric shape is reviewed in Section 2 where a generalization is also introduced to extend applicability. Even for the cases of conic null tests and simple CGH nulls, the estimation of whether such full-aperture interferometric tests are workable is shown in Section 3 to follow incredibly simply with this particular characterization of shape. As discussed in Section 4, more sophisticated measures of manufacturability are coupled to variation in the surface’s local principal curvatures. Stitched interferometric metrology is used as one example, and it happens to be closely related to the analysis of difficulty that was discussed by du Jeu. The computation of the elements that are required for efficiently estimating such measures of manufacturability is treated in Sections 5 and 6.

## 2. A tailored characterization of shape

The standard characterization of a rotationally symmetric asphere’s shape is to express its sag in cylindrical polar coordinates as $z=f(\rho )$. For a conventional conic section of axial curvature *c* and conic constant *κ*, this takes the form

where “:=” denotes a definition and

The cosine of the angle between the surface normal and the part’s axis is then equal to

More generally, the sag can be expressed as a conic section with some added departure that is expressed as a linear combination of the elements of some basis. If ${\rho}_{\mathrm{max}}$ denotes one half of the part’s clear aperture and $u:=\rho /{\rho}_{\mathrm{max}}$, an effective option is to write

where ${Q}_{m}(x)$ is a polynomial of order *m*. Notice that the added deviation from the conic vanishes at both the aperture’s center and its edge (where $u=0$ and $u=1$). Also note that Eq. (2.1) of [9] is just Eq. (2.4) above, but with $\kappa =0$, hence $\sigma (\rho )\text{\hspace{0.17em}}\equiv \text{\hspace{0.17em}}\varphi (\rho )$ and *c* is then the curvature of the best-fit sphere.

When measured along the normal to the conic instead of along the optical axis, the departure from the conic in Eq. (2.4) is —to first order in this departure— equal to

This follows upon multiplication by the cosine factor that converts displacement along *z* to displacement along the normal. The particular family of polynomials treated in [9] is chosen to be orthogonalized so that the mean square slope of this normal departure is given simply by

where the angle brackets denote a weighted average over the aperture:

The particular weight chosen in Eq. (2.7) is not critical, but this one delivers a minimax-like truncation error in slope, see [10].

For an asphere characterized with respect to the best-fit sphere, i.e. when $\kappa =0$, the Nyquist slope for the normal departure is $(\lambda /4)/(2{\rho}_{\mathrm{max}}/N)$ in a conventional full-aperture interferometric test at wavelength *λ* on an $N\times N$ pixel grid. While it is the peak slope that ultimately drives testability, the rms slope is a useful and more readily accessible measure. It follows from Eq. (2.6) that the rms fringe density in Nyquist units, say $\overline{\gamma}$, is given by

This result provides a valuable estimate of non-null testability: it involves little more than summing the squares of the aspheric coefficients; there is no need to evaluate even a single sag value let alone determine the best-fit sphere or any aspheric departure samples and their rates of change. The representation in Eq. (2.4) was constructed so that this analysis can all be done up front and in closed form. Alternatively, if the largest acceptable value for this rms fringe density is written as ${\overline{\gamma}}_{\mathrm{max}}$, Eq. (2.8) gives the powerful design constraint of [10]:

It turns out that a useful constraint to control the line spacing of a CGH null takes a similar form since that spacing is also coupled directly to ${\scriptscriptstyle \frac{\text{d}}{\text{d}\rho}}\delta ({u}^{2})$.

Through the familiar Gram-Schmidt process, Eqs. (2.6) and (2.7) completely determine the polynomials that appear in Eq. (2.4). For ray tracing etc., it is worth noting that Eq. (2.4) and its first two derivatives can be expressed simply as

Here, the arguments to $\varphi (\rho )$, $\sigma (\rho )$, and $\delta ({u}^{2})$ have been suppressed for brevity. Highly robust and efficient algorithms are given in Section 3 of [9] for evaluating the sum in Eq. (2.5) as well as its derivatives of any order. These algorithms use compact recurrence relations to determine the results without evaluating a single member of the polynomial basis. This means that $\delta ({u}^{2})$ and its derivatives can be evaluated readily for Eqs. (2.10)-(2.12). Note that when working with respect to a best-fit sphere, $\kappa =0$ and $\sigma \text{\hspace{0.17em}}\equiv \text{\hspace{0.17em}}\varphi $, so Eqs. (2.11) and (2.12) can be simplified slightly.

## 3. Benefits of the option for a non-zero conic constant

One of the motivations for the generalization offered in Section 2 is that the best-fit sphere used in [9,10] becomes an unworkable hyper-hemisphere for extremely “fast” parts, viz. when $|f({\rho}_{\mathrm{max}})|\text{\hspace{0.17em} \hspace{0.17em}}\ge \text{\hspace{0.17em} \hspace{0.17em}}{\rho}_{\mathrm{max}}$. In contrast, the conic component of Eq. (2.4) typically allows such surfaces to be handled without complication. In my opinion, the polynomials written as ${Q}^{\text{con}}$ in [10] can now therefore be forgotten; just as in [9], the superscript on ${Q}^{\text{bfs}}$ has thus been dropped.

As another example of a new capability that Eq. (2.4) offers, consider a case where the asphere of interest is close to some conic section for which an interferometric null test, see Fig. 1 , is already in hand. In this case, it is straightforward to show that the associated fringe density is again approximately proportional to ${\scriptscriptstyle \frac{\text{d}}{\text{d}u}}\delta ({u}^{2})$. There are only two minor modifications to the results presented in Section 2. First, rather than carrying the usual factor of two for reflection at nominally normal incidence, the additive departure from the conic impacts the raw phase map with a multiplicative factor of four times the cosine of the angle of incidence of the test rays at the asphere. (Remember that, as sketched in Fig. 1, the rays are no longer normal to the asphere in such tests.) The factor of two has become four because the test wave now bounces off the asphere twice, and the cosine factor follows from the perturbation methods used in [3] —see their Eq. (1) and the associated references. Second, the measured fringe density can sometimes carry an additional factor associated with the mapping from each pixel to its associated point on the asphere. The details depend on the configuration, but in many cases the effect is negligible. For example, no such additional factor is required either for the test at left in Fig. 1 or for the conventional test of [10], which uses the best-fit sphere. Similarly, the weak variation in the cosine factor can also be ignored when the asphere is not fast, such as for a typical (nearly parabolic) telescope primary. For many purposes, it is therefore sufficient to note that, in place of the earlier factor of two, there is roughly a factor of four for the fringe density in these double-pass conic null tests. Consequently Eqs. (2.8) and (2.9) carry over, but with the factor of 8 replaced by 16. That is, the rms fringe density can again be estimated with remarkable ease for these cases. If it is ever needed, however, the key to a more accurate result is the combination of the cosine factor together with the fact that position in the interferogram is closely proportional to the sine of the angle between the interferometer’s axis and the test rays as they return to the transmission sphere.

Because Eqs. (2.5)-(2.7) are entirely independent of the conic shape, the polynomial basis used here is precisely that of [10]. Just as in [9], these same polynomials are also ideally suited to working with obstructed systems, i.e. annular apertures like those in Fig. 1 where the aperture corresponds to say $\epsilon <u<1$ for some $\epsilon \ge 0$. Much as described in [9], the end result involves expressing the sag over the reduced region of interest as

where the normal departure is now given by

That is, the additive departure now vanishes at the inside and outside edges of the annulus, and the orthogonal polynomials are stretched to span that range. By using the appropriately weighted average given in Eq. (5.1) of [9], the mean square slope over the annulus is again found to be given by the expression on the right-hand side of Eq. (2.6). This leads to remarkably simple feasibility estimates for interferometric tests of annular apertures with either a conic null test or —when $\kappa =0$— a conventional non-null test. Also, it follows from the similarity of Eqs. (2.10) and (3.1), that Eqs. (2.11) and (2.12) carry over without change. Notice that setting $\epsilon =0$ reduces Eqs. (3.1) and (3.2) precisely to Eqs. (2.4) and (2.5).

## 4. Significance of the principal curvatures

The two principal radii of curvature at each point on an asphere can be found by intersecting neighboring surface normals. As can be seen in Fig. 2 , for points that lie in a plane containing the part’s axis, the standard expression for curvature in two dimensions is applicable, namely

I refer to this as the in-plane curvature. On the other hand, by considering surface normals at points on the cone in Fig. 1 (which meets the shaded plane of symmetry at ${90}^{\circ}$) the other principal radius of curvature is seen to be the displacement from the surface down to the normal’s intersection with the optical axis. The inverse of this radius is referred to here as the out-of-plane curvature and can be seen to satisfy

These two curvatures are sometimes distinguished by using labels such as radial, tangential, sagittal, meridional, azimuthal, etc.; the notation IP and OOP is an attempt to avoid the confusion that can follow in making these distinctions. Since these curvatures characterize the local shape, it is intuitive that sub-aperture tools (either for polishing or metrology) will primarily be sensitive to these values and, in particular, to the difference between them.

It follows from Eqs. (4.1) and (4.2) that ${c}_{\text{IP}}(\rho )$ can be derived simply from ${c}_{\text{OOP}}(\rho )$ since

To evaluate ${c}_{\text{IP}}({\rho}_{0})$ graphically, therefore, just draw the tangent to ${c}_{\text{OOP}}(\rho )$ at $\rho ={\rho}_{0}$ and determine this tangent’s height at $\rho =2{\rho}_{0}$. Further, consider expanding ${c}_{\text{OOP}}(\rho )$ about the axis —where ${c}_{\text{OOP}}(0)={c}_{\text{IP}}(0)={f}^{\u2033}(0)={c}_{0}$, say— to find

It follows immediately from Eq. (4.3) that the *j*th coefficient from the analogous expansion for ${c}_{\text{IP}}(\rho )$ is precisely $(2j+1)\text{\hspace{0.17em}}{c}_{j}$ in place of ${c}_{j}$. Near the axis, ${c}_{\text{IP}}(\rho )$ therefore varies three times more rapidly, i.e. as ${c}_{0}+3{c}_{1}{\rho}^{2}+O(4)$ in place of ${c}_{0}+{c}_{1}{\rho}^{2}+O(4)$. The relative coefficient strength is even more pronounced at higher orders. Any measures of curvature variations therefore tend to be dominated by ${c}_{\text{IP}}(\rho )$. Notice, for example, that if ${c}_{\text{OOP}}(\rho )$ achieves an extreme value at say $\rho ={\rho}_{0}$, where${c}_{\text{OOP}}^{\prime}({\rho}_{0})=0$, ${c}_{\text{IP}}(\rho )$ has already achieved a more extreme value at some $\rho <{\rho}_{0}$ because Eq. (4.3) and its derivative reveal that ${c}_{\text{IP}}({\rho}_{0})={c}_{\text{OOP}}({\rho}_{0})$ and ${c}_{\text{IP}}{}^{\prime}({\rho}_{0})=\rho \text{\hspace{0.17em}}{c}_{\text{OOP}}^{\u2033}({\rho}_{0})$, hence $\mathrm{sgn}[{c}_{\text{IP}}{}^{\prime}({\rho}_{0})]=\mathrm{sgn}[{c}_{\text{OOP}}^{\u2033}({\rho}_{0})]$. On the other hand, if ${c}_{\text{OOP}}(\rho )$ takes its most extreme value at the aperture’s edge, it follows from Eq. (4.3) that ${c}_{\text{IP}}(\rho )$ takes an even more extreme value there. An intuitive consequence therefore is that ${c}_{\text{IP}}(\rho )$ is always the first to change sign.

#### 4.1 Pad polishing

Much as in [7], consider sub-aperture polishing of an asphere with a circular pad of diameter *T* and a spherical face of curvature ${c}_{\text{pad}}$. The difficulty of this step is driven largely by the misfit between the pad and the asphere. Empirical studies can determine a maximal acceptable level of tool misfit, where the answer can be expected to depend on both dynamic and mechanical properties such as the conformability of the pad. The polishing time and difficulty is then driven by the largest tool size that is compatible with this bound. Typically $\left|{c}_{\text{pad}}\right|(T/2)\ll 1$ and the pad’s sag at its edge is then approximately ${\scriptscriptstyle \frac{1}{2}}{c}_{\text{pad}}{(T/2)}^{2}={c}_{\text{pad}}{T}^{2}/8$. Of course, the optimal value of ${c}_{\text{pad}}$ generally falls between the extreme values of the principal curvatures over the aperture. When $\kappa =0$, it is reasonable to adopt the best-fit sphere’s curvature —i.e. choose ${c}_{\text{pad}}=c$ of Section 2— and this is the case considered now. It follows from the discussion in the previous paragraph that, in the most challenging radial zones of the part, the misfit is then typically $|{c}_{\text{IP}}(\rho )-c|\text{\hspace{0.17em}}{T}^{2}/8$ and, for numerical efficiency, a useful estimate of rms tool misfit can be defined by

A process for efficiently evaluating entities such as the one inside the square root of Eq. (4.5) is the chief subject of what follows. With that in hand, it is straightforward to estimate difficulty by solving Eq. (4.5) for *T* in order to obtain the size of the largest tool that respects some maximal acceptable level of $\overline{\mu}$, say ${\overline{\mu}}_{\mathrm{max}}$.

#### 4.2 Subaperture stitching

The metrology is oftentimes one of the most critical challenges in regular “grind-and-polish asphere production”. Subaperture stitching [11] offers greater capability than the full-aperture testing considered in Section 2, and it is more flexible than the null test discussed in Section 3. As indicated in Fig. 3
, this process involves a lattice of subapertures that are ultimately fused into a full-aperture metrology map. In this case, testability tends to be driven by the number of subapertures that are required. Of course, the size of any one illuminated patch grows with the numerical aperture (NA) of the transmission sphere (TS), but the individual subapertures must be small enough for the fringes to be resolved. It is now shown that the TS’s optimal NA is determined by the strength of the aspheric departure. When $\kappa =0$, the asphere’s nominal NA is just $\eta =c\text{\hspace{0.17em}}{\rho}_{\mathrm{max}}$, and its ratio to the NA of the TS, say $\chi \text{\hspace{0.17em}}:=\text{\hspace{0.17em} \hspace{0.17em}}\eta /{\eta}_{\text{TS}}$, gives a simple measure of what is referred to here as “stitchability”. As *χ* gets larger, the test becomes slower and the end result’s measurement uncertainty grows.

Consider the configuration where the interferometer’s optical axis passes through the asphere at the center of the illuminated subaperture in Fig. 3 and is normal to the surface at that point. Its transverse coordinate axes are denoted by $(x,y)$, where the *x* axis is taken to lie in the asphere’s associated plane of symmetry. If the interferometer is focused at the part and the center of curvature of the TS is displaced by $1/{c}_{\text{test}}$ from the surface, the unwrapped measured phase can be approximated by

where $({\ell}_{x},{\ell}_{y})$ are the transverse direction cosines of the test ray as it enters the interferometer. To a good approximation, as indicated in Section 3, position on an interferometer’s detector is proportional to these direction cosines. So, when an $N\times N$ detector is filled with the light through a TS of numerical aperture ${\eta}_{\text{TS}}$,

where $-N/2<\text{\hspace{0.17em} \hspace{0.17em}}(j,k)\text{\hspace{0.17em} \hspace{0.17em}}<N/2$ are just the interferogram’s pixel indices.

When, as suggested in Fig. 3, the test is configured so that

it follows from Eqs. (4.6) and (4.7) that the interferogram’s phase is described by

Along the *j* axis, the phase change between adjacent pixels therefore reaches a prescribed fraction, say *γ*, of the Nyquist sampling rate when

and this is readily solved for *j*:

The same expression emerges from the *k* axis. Notice that, while the size of the illuminated region on the part grows linearly with ${\eta}_{\text{TS}}$, Eq. (4.11) states that the fraction of that region in which the fringes are resolvable falls like $1/{\eta}_{\text{TS}}{}^{\text{\hspace{0.17em}}2}$. It follows that the largest area of resolvable fringes at the part is found when the expression in Eq. (4.11) is equal to $N/2$, and this condition yields the numerical aperture of the locally optimal TS:

The expression on the right-hand side of Eq. (4.12) varies across the aperture. To ensure that the detector is never underfilled, its minimum value should be used to select${\eta}_{\text{TS}}$. This occurs near the maximum of $|{c}_{\text{IP}}-{c}_{\text{OOP}}|$. A more effective solution allows some underfilled subapertures in order to gain larger subapertures everywhere else. One natural option is to adopt an rms value rather than the extreme value, e.g. choose ${\eta}_{\text{TS}}$ to be given by

Of course, ${\eta}_{\text{TS}}$ need never exceed the part’s NA, i.e. keep ${\eta}_{\text{TS}}<\eta =c\text{\hspace{0.17em}}{\rho}_{\mathrm{max}}$. So it is in fact the minimum of *η* and the value given by Eq. (4.13) that is desired. Furthermore, since only a handful of TS’s are typically available in practice, the preferred one most closely matches this resulting value while also having a transmission surface of sufficiently large radius of curvature to accommodate the part. The key is that Eq. (4.13) is a core component of stitchability estimation. As remarked at Eq. (4.5), the chief subject of what follows is the determination of an efficient process for evaluating averages like those that appear inside these square roots. The end result cannot be as simple as the final one in Eq. (2.8), but that expression sets the standard for what is sought next.

## 5. First-order approximations for the principal curvatures

The particular manufacturability estimates discussed in the previous section are simplest when $\kappa =0$ and they are, in fact, most relevant to cases where there is no need for a non-zero conic constant. It is useful in such cases to simplify these estimates by using a series expansion to low-orders in the aspheric departure coefficients. This turns out to be one of the unique strengths of Eq. (2.4).

It turns out that, with Eq. (2.10), Eq. (4.2) can be expressed as

where

Notice that ${\epsilon}_{\text{OOP}}({u}^{2})$ is considered “small” here because, on account of Eq. (2.5), it is homogeneous of degree one in the aspheric coefficients, i.e. in $\left\{\text{\hspace{0.17em}}{a}_{m}\right\}$. Accordingly, Eq. (5.1) is now expanded as a series:

The first-order component of the relative change in ${c}_{\text{OOP}}$ is precisely ${\epsilon}_{\text{OOP}}({u}^{2})$. It is important to appreciate, however, that this dimensionless entity need not be smaller than unity for this series expansion to be accurate. The singularity in the denominator of Eq. (5.1) can be used to show that the series in Eq. (5.3) converges provided

This must hold for all $0<u<1$, of course. Up to third-order terms are included in Eq. (5.3) to give an idea of the impact of truncation at lower orders. A complete analysis reveals that just the first-order estimate is adequate for most purposes provided Eq. (5.4) is respected with an additional safety margin of a factor of three or four reduction on its right-hand side. In particular, the error in the first-order correction when expressed as a fraction of the correction itself, i.e. $|c\text{\hspace{0.17em}}[1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\epsilon}_{\text{OOP}}({u}^{2})]\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{c}_{\text{OOP}}(u\text{\hspace{0.17em}}{\rho}_{\mathrm{max}})|/[c\text{\hspace{0.17em}}{\epsilon}_{\text{OOP}}({u}^{2})]$, is then found to remain under 20% on all but the outer radial zones of extremely “fast” parts, meaning $\left|\eta \right|$ greater than about $0.8$.

Similarly, Eq. (4.1) can be expressed as

where

In this case, Eq. (5.5) can be expanded to find

[These expressions can also be generated directly by using Eq. (4.3) with Eqs. (5.2) and (5.3): the first step is to observe that ${\epsilon}_{\text{IP}}({u}^{2})\text{\hspace{0.17em} \hspace{0.17em}}=\text{\hspace{0.17em} \hspace{0.17em}}{\epsilon}_{\text{OOP}}({u}^{2})+2{u}^{2}{\epsilon}_{\text{OOP}}^{\prime}({u}^{2})$.] The location of the singularity in Eq. (5.5) is the same as that of Eq. (5.1), so the condition for convergence is again just Eq. (5.4). That is, there is no analogous independent constraint on the magnitude of ${\epsilon}_{\text{IP}}({u}^{2})$. It is readily seen that, even with the factor mentioned above as a safety margin, Eq. (5.4) allows $|{\epsilon}_{\text{OOP}}({u}^{2})|$ to exceed unity over all but the outer radial zones on fast parts. Keep in mind too that the variation in $|{\epsilon}_{\text{IP}}({u}^{2})|$ is even stronger.

In short, the first-order estimates given by

*Eqs. (5.8)*

*are exact equalities on axis*, i.e. at $u=0$. More generally, the impressive accuracy of Eqs. (5.8) across the aperture can be appreciated by confirming that, even though $|{\epsilon}_{\text{IP}}({u}^{2})|$ repeatedly approaches 10 and occasionally exceeds it, $|{\epsilon}_{\text{OOP}}({u}^{2})|$ remains about a factor of two or more below the convergence limit given in Eq. (5.4) (with $\eta \approx 0.39$ in this case). Do not forget, however, that the sole purpose of these linear approximations is to enable significant simplifications of manufacturability estimates. This is demonstrated in the next section for the case of stitched metrology.

## 6. Stitchability matrices

The stitchability measure introduced between Eqs. (4.5) and (4.6), namely $\chi \text{\hspace{0.17em}}:=\text{\hspace{0.17em} \hspace{0.17em}}\eta /{\eta}_{\text{TS}}$, can now be approximated by slightly rearranging Eq. (4.13) and using Eqs. (5.8) to find

The last line in Eq. (6.1) follows from Eqs. (5.2) and (5.6). Since $\delta ({u}^{2})$ of Eq. (2.5) is linear in the aspheric coefficients, i.e. in $a=({a}_{0},{a}_{1},\mathrm{...},{a}_{M})$, Eq. (6.1) can be re-arranged and expressed in terms of three numerical matrices:

As shown in Appendix A, these matrices involve averages that can be evaluated in closed form. In place of the $a\cdot a$, i.e. ${a}^{2}$, of Eq. (2.8), Eq. (6.2) involves sums of products of the coefficients rather than just the sum of their squares.

The operation count in evaluating the right-hand side of Eq. (6.2) can be almost halved by using Cholesky decomposition so that only upper-triangular matrices are involved, i.e.

For further verification of code based on the results derived in Appendix A, the initial sub-block of these triangular matrices has the form

In this way, stitchability can be evaluated with only about ${\scriptscriptstyle \frac{3}{2}}(M+2)(M+1)$ arithmetic operations. There are options for further approximations to accelerate this for “slow” parts (i.e. ${\eta}^{2}\ll 1$), but even in this form, such constraints can be built directly into the design process itself.

Equation (6.2) can be used in two different ways. The first is to solve that equation for *χ* and use the result for any specific asphere to evaluate its associated extension factor, i.e. the ratio of the NA of the surface to the NA of the optimal TS. This not only determines the number of subapertures required for total cover of the surface but gives some idea of the uncertainty in the final metrology map for any particular stitching platform. Alternatively, a maximal acceptable value, say ${\chi}_{\mathrm{max}}\approx 5$ (or whatever is appropriate for the current optical tolerances and metrology platform), can be adopted for *χ* in Eq. (6.2). A constraint is then applied during design to keep the expression on the right-hand side of Eq. (6.2) bounded accordingly.

To give a graphical idea of this constraint, a comparison of the stitchable domain with the domain for full-aperture testability is presented in Fig. 5
for the case of just two aspheric coefficients, i.e. $M=1$ in Eq. (2.4). Since the maximum value of ${Q}_{0}(x)$ is precisely $1/4$, the peak of the normal departure is generally about ${a}_{0}/4$. Notice that the full-aperture testability extends only out to ${a}_{0}\approx 20\mu m$, i.e. an aspheric departure of about $5\mu m$. (The plots of ${Q}_{m}(x)$ in [9] reveal that about $8\mu m$ or so is testable with ${a}_{1}\approx 20\mu m$.) Even for an extension bound of just ${\chi}_{\mathrm{max}}=4$, the option to stitch the metrology evidently extends the strength of testable aspheres by up to a factor of ten for slow parts, i.e. those with small *η*; for faster parts, there is then an additional factor of up to two or three. Notice also how the orientation of these “ellipses of stitchability” depends on the NA of the part. That is, the strongest aspheres that can be stitched have two coefficients of the same sign for slow parts, but their signs are different on fast parts (say $\eta >0.85$). For any value of *M*, eigen-reduction of the $M\times M$ matrix in Eq. (6.2) readily yields the principal axes of the associated ellipsoids.

As alluded to after Eq. (4.13), the constraint that follows from Eq. (6.2) is not sufficient. When that constraint is satisfied, Eq. (6.2) can be used to solve for *χ* (now less than ${\chi}_{\mathrm{max}}$ by design) and the optimal NA of the TS is then ${\eta}_{\text{TS}}\text{\hspace{0.17em}}\approx \text{\hspace{0.17em} \hspace{0.17em}}\eta /\chi =\text{\hspace{0.17em} \hspace{0.17em}}c\text{\hspace{0.17em}}{\rho}_{\mathrm{max}}/\chi $. The test therefore relies on a TS being available that not only matches this NA sufficiently but also has a reference surface of large enough radius of curvature to avoid collision with the part. Further, the part’s CA must fit within the metrology system’s mechanical envelope and the hardware must have sufficient travel on its mechanical stages to achieve the configuration for each subaperture in keeping with Eq. (4.8) (where $1/{c}_{\text{test}}$ is the displacement from the part to the center of curvature of the reference surface). These are among the primary considerations, but there is also a secondary condition: the part cannot be arbitrarily small. In particular, the validity of Eq. (6.1) requires both

The former ensures that the relative variation in ${c}_{\text{test}}=({c}_{\text{IP}}+{c}_{\text{OOP}})/2$ across the aperture does not approach unity. (This is oftentimes nearly synonymous with the constraint related to the travel of the mechanical stages.) The latter was discussed after Eq. (5.4).

It follows from Eqs. (5.2) and (5.6) that, for fixed values of the aspheric coefficients as well as of *η*, each of Eqs. (6.7) leads trivially to a minimal value for the part size, i.e. ${\rho}_{\mathrm{max}}$. If the larger of these two aperture sizes is written as ${\text{CA}}_{\mathrm{min}}$, then Eqs. (6.7) can be reduced to the condition ${\rho}_{\mathrm{max}}>\text{\hspace{0.17em} \hspace{0.17em}}{\scriptscriptstyle \frac{1}{2}}{\text{CA}}_{\mathrm{min}}$. Plots of ${\text{CA}}_{\mathrm{min}}$are presented in Fig. 6
, where it can be seen that this secondary condition is typically met for clear apertures of more than 7mm or so; ${\text{CA}}_{\mathrm{min}}$ is larger than this only for parts that have both low-NA as well as asphericity near the limits associated with Eq. (6.2). In these atypical cases, brute-force evaluation of Eq. (4.11) is required in place of Eqs. (6.1) and (6.2). A key observation, however, is that this secondary validity condition is automatically satisfied for a wide and important class of aspheres.

Keep in mind that the primary constraint for stitchability follows from Eq. (6.2) —which limits the local astigmatism across the asphere— and that there are other practical and analytical matters that can also play a role. Analogous pad-related measures of manufacturability follow in the same way. As discussed briefly at the end of Appendix A, Eqs. (5.6) and (5.8a) can be used with Eq. (4.5) to get the corresponding matrices for that particular pad-related measure. Unlike Eq. (6.2), which is intended for the sorts of more moderate aspheres that fall near or within the capture range of stitched metrology, the pad-related results are valid for a wider class of aspheres with more strongly varying curvatures. In particular, Eq. (6.7a) can be dropped for this case. Again, however, there are secondary practical conditions related to envelopes of the polishing machine, etc.

## 7. Concluding Remarks

Because full-aperture interferometric tests are a workhorse in the deterministic fabrication of mild aspheres, constraints such as Eq. (2.9) provide valuable support for the process of design for manufacturability. The generalization of that constraint for conic null tests and even for the testing of annular apertures was presented in Section 3. Fundamental geometric entities were used in Section 4 to derive and define the additional basic measures of manufacturability given in Eqs. (4.5) and (4.13). Useful approximations of the underlying entities were presented in Eqs. (5.2), (5.6), and (5.8). These linear approximations enabled the introduction of the numerical matrices of Section 6 that can be computed robustly and efficiently to arbitrary orders by using the methods developed in Appendix A. For greater efficiency during design, these matrices need only be computed once and stored. As discussed after Eq. (6.6), results like those derived here can be used either up front as design constraints or as aides to support quick manufacturability estimates at the production end of the process.

A number of other options are created by the results presented here. For example, a common design challenge for a system that has multiple spherical elements is to determine which of them can be most effective when converted to aspheres. When coupled with the sort of perturbation methods used in [3], any of the constraints discussed above can now be used to efficiently identify those surfaces that deliver the best optical performance (say rms wavefront error) for a given level of difficulty of manufacturability. In such work, the spheres all have known values for $\eta =c\text{\hspace{0.17em}}{\rho}_{\mathrm{max}}$, so the composite matrix in Eq. (6.2) —or its analogue for any other critical process— can be fully evaluated for each surface to facilitate this step that can proceed via standard matrix methods.

Finally, note that the diversity of processes and steps involved in the fabrication of optical aspheres means that an equal variety of manufacturability estimates is required. For example, other metrology processes such as various zonal interferometric tests and CGH nulls can be analyzed in similar fashions. The developments presented above establish that the characterization of shape used in Eq. (2.4) is well matched to supporting such estimates by exploiting the efficient approximations of Section 5 and the analytical methods presented in Appendix A. It is also worth re-emphasising that the introduction of a conic constant to Eq. (2.4) has given the associated polynomials even greater generality for these applications.

## Appendix A

As shown in [9], for some purposes it is convenient to adopt an auxiliary basis so that Eq. (2.5) can also be expressed as

where ${P}_{m}(x)$ is a scaled Jacobi polynomial of order *m* that satisfies

see Eqs. (A.4) and (A.6) of [9]. As shown in Section 3 of [9], the coefficients in either of the sums in Eq. (A.1) can be readily interchanged, i.e. $\left\{\text{\hspace{0.17em}}{b}_{m}\right\}$ found from $\left\{\text{\hspace{0.17em}}{a}_{m}\right\}$ and vice versa. Eq. (A.2) suggests that a change of variables, namely $u=\mathrm{cos}\theta $, be used in Eq. (2.7) to see that

In these terms, each prime in Eq. (6.1) represents the operator

When expressed in terms of *θ*, the first piece inside the braces of Eq. (6.1) is found to be

where

Similarly, the second piece inside the braces becomes

where

Both ${E}_{m}(\theta )$ and ${F}_{m}(\theta )$ are symmetric about $\theta =\pi /2$, which in terms of Eq. (A.3) means that $g(-u)\equiv g(u)$, so this average can be modified for such functions to become

When multiplied out, the average inside the square root of Eq. (6.1) now evidently involves three basic types of integrals, namely

Each of these can be evaluated in closed form.

The first of the three integrals of interest can be expressed as

where

The second and third can be re-expressed as

where ${\delta}_{mn}$ is Kronecker’s delta function. Notice that, because negative subscripts appear in Eq. (A.8) when $m=0$, absolute values have been introduced in the final part of Eq. (A.17); this result is necessarily an even function of both *m* and *n*. By changing variables to $z=\mathrm{exp}[i\text{\hspace{0.17em}}\tau ]$ in Eq. (A.15), that integral can be evaluated as a contour integral around the unit circle in the complex plane:

This can first be evaluated for $m,n\ge 0$, and then absolute values again inserted at the end. After factoring the numerator, the residue of the singularity at the origin involves just the coefficient in ${z}^{m+n-2}$ from the expansion of $(1+{z}^{2}+{z}^{4}+\mathrm{...}+{z}^{2m-2})(1+{z}^{2}+{z}^{4}+\mathrm{...}+{z}^{2n-2})$. It follows that the result is zero unless $m+n$ is even and, more generally, that

With this, ${I}_{mn}$, ${J}_{mn}$, and ${K}_{mn}$ are now all known in closed form. Further, the expressions for ${J}_{mn}$ and ${K}_{mn}$ in Eqs. (A.16) and (A.17) can be simplified to

In vector notation, the argument of the square root in Eq. (6.1) can be written as

where the elements of the matrices ${\text{B}}_{k}$ are written as ${b}_{mn}^{k}$ and are given by

where, from Eqs.(A.6) and (A.8), we have

All of these matrix elements can now be evaluated by using Eqs.(A.14), (A.19), and (A.20). With

and ${v}_{mn}:=\text{Min}(m,n)$, it turns out that Eq.(A.22) can be expressed as

Similarly, with

it is found that

Finally, with

the elements of ${\text{B}}_{2}$ are given by

The initial sub-block of these matrices is presented for verification of any implementation of Eqs. (A.26)-(A32):

To express the constraint in terms of the original coefficients, namely **a** of Eq. (A.1) rather than **b**, the change-of-basis matrix of [9] (a lower-triangular band matrix) is used, hence $a={\text{L}}^{T}b$ and

where

In this way, the constraint can be constructed efficiently for any number of terms. Of course these numerical matrices need be computed and stored only once. Also, note that a different weighting was used in [8] to that used in Eq. (A.3), so the resulting matrices are not precisely the same.

Finally, note that Eqs. (5.6) and (5.8a) can be used to derive analogous matrices for working with Eq. (4.5). The first step is to work from those equations to see that

In this case, the term inside the braces of Eq. (A.38) is found to be

where

Equation (A.39) is a composite analog of Eqs. (A.6) and (A.8). With this result, the integrals required to evaluate the average in Eq. (A.38) all follow once again from Eqs. (A.15) and (A.19). The associated matrices that are analogous to those introduced in Eq. (A.21) can therefore now be determined in the same fashion.

## Acknowledgments

I thank Dr. John R. Rogers and Dr. Kevin P. Thompson for recommending that a conic constant be retained in the expression for characterizing an asphere’s shape in terms of the polynomials formerly written as ${Q}_{m}^{\text{bfs}}(x)$. That provided the clear motivation for Eq. (2.4). I also grateful to the anonymous reviewers for their careful reading of this work and their suggested improvements.

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