## Abstract

In diamond-machined freeform manufacturing processes, a tool-tip often leaves behind characteristic mid-spatial frequency (MSF) structures on the optical surface. Unwanted movement between the tool-tip and the part results in MSF structures with random variations. Here, we analyze the effects of these MSF structures on the system’s optical performance and derive simple analytic estimates for the optical transfer function in terms of the parameters of these structures. These expressions are expected to aid in MSF tolerancing.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Precision freeform manufacturing allows the fabrication of surfaces with large numbers of degrees of freedom. However, these processes typically involve the use of subaperture tools, which leave behind surface grooves whose characteristic frequencies are between those of figure and roughness. These errors, known as mid-spatial frequency (MSF) errors, introduce unwanted effects that are difficult to characterize and complicate tolerancing on manufactured parts [1,2].

In prior work, we explored the effects on the optical transfer function (OTF) of wavefront errors, including quadratic aberrations [3] as well as MSF groove structures due to subaperture tools [4]. In the latter case, two types of MSF groove errors common in freeform manufacturing processes were analyzed: parallel straight grooves resulting from diamond milling, and concentric circular (or spiral) grooves resulting from diamond turning. The key theoretical tool proposed there to obtain analytic expressions for the OTF was referred to as the pupil-difference probability density (PDPD), which is defined as the probability density that two random points in the pupil with given vector separation have a given difference in wavefront error. This prior study assumed regular grooves of equal height and width [4]. However, real processes present tool-tip or spindle vibrations, causing the MSF structures to have profiles are not perfectly periodic. Diamond machining usually involves a fixed tool-tip that makes contact with the optical part as the part is moving (rotating or translating). A local coordinate system is used in which the three directions are called the thrust, *cutting*, and *feed* directions of the machine, as shown in Fig. 1(a). It is well established that there can be tool-tip vibration in all three of these directions [5–10], and so incorporating such effects into the models for the OTF is of central importance. Vibrations in the thrust and cutting directions cause random variations $t_i$ in the depth of each groove, following a probability density $K_{\textrm T}(t_i)$ with standard deviation $\mu$, as shown in Fig. 1(c). Vibrations in the feed direction, on the other hand, introduce random lateral groove displacements $f_i$ following a probability density $K_{\textrm F}(f_i)$ with standard deviation $\sigma$. Our goal is to provide simple approximate formulas for the OTF of milled and turned surfaces that incorporate these effects.

The outline of this work is as follows: A review of the PDPD is provided in Sec. 2. The analysis of MSF structures including the effects of tool-tip vibrations is given in Sec. 3 for milled surfaces and in Sec. 4 for turned surfaces. For readers more interested in the formulas than the formal derivations, Sec. 5 provides a self-contained summary of the main results.

## 2. Review of general PDPD theory

The PDPD, denoted as $P(\eta ,\;{\boldsymbol \rho })$, is defined as the probability density that for two random points in the system’s pupil whose vector separation is ${\boldsymbol \rho }$, the difference in aberration function $W$ is equal to some prescribed quantity $\eta$. Mathematically, this can be written in the following form:

## 3. Effects of tool-tip vibrations in MSF structures for milled surfaces

The PDPD due to parallel MSF grooves (e.g. resulting from diamond milling) can be approximately expressed in terms of the one-dimensional PDPD of its cross-section [4] by using the scalar pupil displacement variables $q_{\textrm n}= {\boldsymbol q} \cdot \hat {\textrm n}$ and $\rho _{\textrm n}= {\boldsymbol \rho } \cdot \hat {\textrm n}$, where $\hat {\textrm n}$ is the unit vector normal to the grooves. The grooves of the nominal MSF structure (without tool-tip vibrations) can be modeled as a piecewise-parabolic function, where $h$ and $T$ are the grooves’ nominal peak-to-valley (PV) and width (or period), respectively, as illustrated in Fig. 1(b). It was shown in [4] that such a MSF structure leads to a very simple nominal PDPD (and Fourier transform):

At this point, it is useful to define the transition function $\mathcal {T}_{\rho _{\textrm n}}$ as

#### 3.1 Pistons model for thrust vibrations

Let us start by considering fluctuations in the height of the ridges resulting from variations in thrust (or cutting). We model these fluctuations by letting each parabolic segment be shifted vertically by an amount $t_i$, where each shift is assumed to be independent from all others, following a zero-mean probability density $K_{\textrm {T}}(t_i)$ whose standard deviation $\mu$ is assumed to be small compared to $h$ [see Fig. 1(c)]. To begin, we try an overly simplistic model that we call the *pistons model*, where each complete parabolic segment is rigidly displaced, causing unphysical discontinuities in $W$ as illustrated in Fig. 2(a). The difference of the two oppositely-shifted copies of the error, $\Delta W=W(q_{\textrm n}-\rho _{\textrm n}/2)-W(q_{\textrm n}+\rho _{\textrm n}/2)$, is then also discontinuous.

One can see from Fig. 2 that the rectangular contribution for each straight segment of $\Delta W$ is shifted in $\eta$ by the difference $t_i - t_j$. Hence, the mean effect of thrust vibrations is simply given by

#### 3.2 Realistic model for thrust vibrations

It turns out that an expression can be found based on a more realistic model with continuous grooves, whose form is almost as simple as that of the pistons model. The key is to study the shift $s$ of the intersections of each segment of $\Delta W$. As can be seen from Fig. 4(b), $s$ depends not only on $\rho _{\textrm n}$ but also on the shifts $t_{i}$ and $t_{i+1}$ of two contiguous grooves, as well as on the shift $t_{j}$ of the groove in the displaced replica that overlaps with this junction. The value of $s$ can be found from simple geometric considerations to be

For $K_{\textrm {T}}$ given by Eq. (8), we have the following simple expression for $\tilde {P}_{1,\textrm {T}}(k,\rho _{\textrm n})$:

Although the form of $K_{\textrm {T}}$ in Eq. (8) is a realistic model for thrust vibrations, it is worth noting that a zero-mean Gaussian form for $K_{\textrm {T}}$ of the same standard deviation, namely

#### 3.3 Feed vibrations model

We now model the effects of feed vibrations by letting each parabolic segment of the groove structure be displaced laterally by an amount $f_i$, as shown in Fig. 5(a). The displacements $f_i$ follow a zero-mean probability density $K_{\textrm {F}}$ whose standard deviation $\sigma$ is assumed to be small compared to the nominal groove width $T$ [see Fig. 1(c)]. These displacements modify the saw-tooth structure of $\Delta W$ such that (unlike for thrust vibrations) the slopes of the different line segments in $\Delta W$ change, as shown in Fig. 5(b). Therefore, not only do the limits of the contributions to the PDPD change with the fluctuations, but also their relative weights.

The lateral displacements of the grooves cause each straight segment of $\Delta W$ to provide a different contribution to the PDPD (illustrated by the green rectangle with width $w$ and height $\alpha$) when compared to the nominal case (gray, with width $w_0$ and height $w_0^{-1}$). Due to the statistical symmetry of the intersections, we can make the contributions symmetric around the origin in $\eta$ to facilitate the calculation. The expressions for $w$ and $\alpha$ are then

#### 3.4 Combination of thrust and feed vibrations

The combination of the two effects described so far, without the sinusoidal fitting technique discussed in Sec. 3.3, is given by the product of Eqs. (16) and (22):

#### 3.5 Standard deviation estimate

Tolerancing requires not only an estimate for the mean of the OTF but also for its spread. Based on numerical simulations we propose the following expression for the standard deviation of $\tilde {P}_{1,\textrm {C}}$ in terms of the dimensionless combinations $kh$ (nominal PV in waves), $2R/T$ (number of cycles), $\sigma /T$ (relative feed vibration strength), and $\mu /h$ (relative thrust vibration strength):

## 4. Effects of tool-tip vibrations in MSF structures for turned surfaces

We now extend the model to surfaces with concentric grooves such as those approximating the MSF geometry on diamond-turned surfaces, where the description of the groove shapes given earlier applies to the radial cross section, so that the relevant one-dimensional parameter is $\rho = |{\boldsymbol \rho }|$. As shown in [4], the PDPD for such groove structures can be decomposed in two contributions: one due to the parts of the overlap region where the grooves from two shifted copies of the error align roughly, and one where they do not (as shown in Appendix C.). The latter usually dominates; this gives rise to a strong attenuation of the oscillations in the OTF and makes the effect of variations along the grooves (which break the rotational symmetry) negligible.

#### 4.1 Effects of thrust and feed vibrations

As shown in [4], the (Fourier-transformed) PDPD of a nominal groove structure with turned geometry, $\tilde {P}_2$, can be calculated from the corresponding quantity for the radial cross-section, $\tilde {P}_1$, by expressing the latter as a Fourier series in $\rho$ and multiplying each term by an appropriate factor. Since $\tilde {P}_{1,\textrm {C}}$ in Eq. (28) is already a truncated Fourier series, this transformation is straightforward, and (as shown in Appendix B. with $a_0 \rightarrow 2G_0$ and $a_1 \rightarrow G_1$) it leads to

#### 4.2 Standard deviation estimate

Based on numerical simulations we propose an estimate for the standard deviation of $\tilde {P}_{2,\textrm {C}}$:

## 5. Summary of results

The OTF estimates derived in this work can be summarized by the following simple expression:

The corresponding estimates for the standard deviations of the OTFs are given by

## 6. Concluding remarks

Diamond-machined freeform optical surfaces are inevitably affected by the presence of MSF structures, which cause degradation in optical quality and performance. In this work, we extend the analysis of these errors initiated in [4] by incorporating the effects of vibrations in the thrust, cutting and feed directions, resulting from unwanted relative movement between the diamond tool-tip and the optical element under fabrication. The study was based on a mathematical representation referred to as the PDPD, which allows a geometric/visual interpretation of the effects of these errors.

By working in the PDPD domain, we show how thrust and feed vibrations modify the nominal PDPD through several probability integrals. Although most of the examples presented assumed that the vibrations follow a normal distribution, the theory is general for any probability distribution. Parallel and concentric MSF groove patterns were considered, and expressions for the corresponding mean OTF and spread were given. It should be noted, though, that several approximations were assumed when modeling MSF structures with random variations in both milled and turned geometries. For example, in the consideration of milled surfaces, Eq. (28) does not account for variations along the grooves’ direction; these effects can be incorporated by multiplying the result by the Fourier transform of a PDPD describing those variations. Furthermore, for turned surfaces, Eq. (31) does not account for variations that break the rotational symmetry of turned MSF structures; the effect of these variations are negligible.

The results serve to further complement previous tolerancing tools regarding MSF structures found on freeform optical surfaces. Since all realistic diamond-machining processes involve tool-tip vibrations MSF structures, the expressions presented here provide a means to further predict the degradation of the OTF beyond the nominally periodic MSF structures.

## Appendix A: Milled feed vibration derivation

We begin by defining the characteristic functions $\tilde {K}_{\textrm {F}}$ as

The exact forms of the coefficients $C_0$ and $C_1$ in Eq. (24) are given by

## Appendix B: Review of PDPD theory for diamond-turned surfaces

This appendix summarizes the results in [4] for finding the Fourier transform of the PDPD for turned surfaces ($\tilde {P}_2$) from that for milled surfaces ($\tilde {P}_1$) with the same cross-section, and provides corrections to two typographic errors in [4]. This relation was found to take the form

The action of $\hat {\mathcal {Q}}_x$ is simplest to calculate if $\tilde {P}_1$ is expressed as a Fourier series:

Note that, for $\mu = 0$ and $\sigma = 0$, $a_m$ were found in [4] to be given by

## Appendix C: Definitions of overlap area functions

The area functions $A(\rho )$ and $B(\rho )$ are given for $\rho\;<\;R$ by

## Funding

National Science Foundation (NSF) I/UCRC Center for Freeform Optics (IIP-1338877); Aix-Marseille Université.

## Acknowledgments

MAA received funding from the Excellence Initiative of Aix-Marseille University – A*MIDEX, a French “Investissements d’Avenir” programme.

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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