1. Introduction

It is often important to characterize surface and transmitted wavefront errors in terms of the spatial content. The errors are typically analyzed in three spatial domains [1–3]: figure, ripple (or mid-spatial frequency) and roughness, often with arbitrary cut-offs. These errors can affect optical system performance. For example, mid-spatial frequency errors can lead to self-focusing and power loss in a high-power laser [1].

This paper introduces the area SF to analyze the spatial content of optical surfaces. Section 1 discusses the linear SF which is used for specification of some astronomical optics. The linear SF is well suited to the description of isotropic surfaces. Section 2 introduces the area SF, its calculation and visualization and Section 3 shows examples both for simulated data and for optical surfaces. Section 4 compares the SF with other area analyses of spatial content. Section 5 indicates further work needed to apply the area SF in specification and acceptance testing over the full spatial domain for anisotropic surfaces of arbitrary aperture shape.

Currently, PSD is commonly used for describing the spatial frequency characteristics [4]. It is plotted in units of length cubed versus spatial frequency in inverse length units. The definition and calculation of the PSD varies among implementations, and differences in the details of the calculation can lead to remarkably different results. The PSD typically uses a small fraction of the surface data, as it is usually calculated over profiles. In addition, a typical calculation of the linear PSD conveys no information on surface anisotropy. The area PSD is typically calculated over square or rectangular areas after removal of low order “figure”. The linear SF complements the PSD and can be calculated for profile or areal data. It is the expectation of the squared height difference as a function of separation. The SF was first used by astronomers to describe astronomical seeing (atmospheric turbulence). It started with a model developed by Tatarski [5] and Fried [6] whose work was based on Kolmogorov statistics. For Kolmogorov turbulence, the structure function, S(r), can be expressed as

The SF was introduced for use in surface metrology in the 1970’s [9,10], and the initial work was applied to surface finish characterization and fractal analysis [11,12].

Figure 1 shows the calculation of the linear SF for area data. Conceptually all possible point pairs of separation τ₁ are selected, their squared height differences calculated, and then averaged to obtain SF(τ₁). This process is repeated for all separations τ within the measurement area, generating the SF (Fig. 1(b)). The structure function can be computed over any chosen dynamic range for any aperture shape, albeit the number of points contributing to the average decreases as τ tends toward the aperture size. Separation into figure, roughness and mid-spatial frequencies can still be done but it is not necessary. The linear SF compresses 2D height information to 1D and consequently is insensitive to anisotropic structure. Figure 2 shows calculated linear SFs for two quite different surfaces generated with unit coefficients for two different Zernike terms. The two surfaces in Fig. 2(a) and 2(c) have markedly different azimuthal frequencies, but the two linear SFs are almost indistinguishable.

 

Fig. 1 Calculation of the linear SF. (a) Input map. (b) Linear SF as a function of separation.

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Fig. 2 Comparison of linear SFs. (a) Map 1. (b) Linear SF of map 1. (c) Map 2. (d) Linear SF of map 2.

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The area structure function retains information on surface anisotropy. It was introduced by Sayles and Thomas in 1977 in polar terms [10]. Two more recent publications show area SF in Cartesian coordinates for rough surfaces such as cylinder liners [13, 14]. Modern optics manufacturing is often “small tool” to enable manufacture of aspheric and freeform surfaces. Suppression of information on surface isotropy will lead to an optimistic prediction of optical performance when using a linear SF compared to an area SF specification.

This paper introduces a 2-quadrant area SF for characterization of optical surfaces over the full spatial bandwidth of an arbitrary aperture in terms of both amplitude and orientation. This new calculation of the area SF in two quadrants is necessary to represent surface anisotropy. Area SF provides a quantitative means of surface specification but further work is required to translate optical system performance requirements into area SF specifications.

2. Calculation of the area structure function

The area SF can be expressed as

Based on Eq. (2), the area SF can be calculated as follows: (1) duplicate the height map, (2) offset the duplicate map by ${\tau }_x$ and ${\tau }_y$ and overlay on the original map, (3) calculate the squared height differences in the overlap region, and (4) average to obtain $S({\tau }_x,{\tau }_y)$.

Because ${\tau }_x$ and ${\tau }_y$ are positive integers, $S({\tau }_x,{\tau }_y)$ is in the first quadrant of the Cartesian coordinate system. For rotationally invariant surfaces, positive values of ${\tau }_x$ and ${\tau }_y$are sufficient. However, for asymmetric surfaces, Thomas’ equation (Eq. (2)) is insufficient, because $S({\tau }_x,{\tau }_y)$ is not equal to $S(-{\tau }_x,{\tau }_y)$ in general. The area SF, including$S(-{\tau }_x,{\tau }_y)$, represents offset and overlap in the second quadrant. For example, in Fig. 3(b) , we calculate the area SF in two directions (move the duplicated map in the upper-right and upper-left directions respectively), and the result is shown in Fig. 3(c) and 3(d). The area SF in the first quadrant is not the same as that in the second quadrant ($S({\tau }_x,{\tau }_y)\ne S(-{\tau }_x,{\tau }_y)$) for some surfaces. Because the SF is the squared value, $S({\tau }_x,{\tau }_y)=S(-{\tau }_x,-{\tau }_y)$and $S(-{\tau }_x,{\tau }_y)=S({\tau }_x,-{\tau }_y)$, ie the first and third quadrant contain the same information. Rotationally varying surfaces that are asymmetric about both the x- and y-axes require the two-quadrant area SF for a complete description of the spatial content.

 

Fig. 3 Calculation of the area SF. (a) Input map. (b) Move the duplicated map in 2 directions. (c and d) area SF in 2 quadrants (the value is between 0 and 8 μm²).

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3. Physical interpretation of the area structure function

3.1 Simulation data

The SF analysis of a surface with figure error (for example low-order Zernike components) is useful for building physical understanding, as shown in Fig. 3. Consider the 2-quadrant SF graphs in Fig. 3(c) and 3(d). At the origin, ${\tau }_x$ and ${\tau }_y$equal zero, and the SF is always identically zero – that is the expectation of the squared height difference between the surface map and itself with no separation is zero. Moving to small ${\tau }_x$and ${\tau }_y$ (small separation) near the origin, SF values are low. This is to be expected with only low spatial frequency errors. In the first quadrant (of Fig. 3(c)), the SF value is high at the edge (large separation), capturing a cubic right-left feature of the surface – that is a significant height difference between the upper-right and lower-left edges of the surface. For the same separation in the second quadrant, the SF values are relatively low on the edge, conveying the similarity in the surface heights at the upper-left and lower-right. The long spatial wavelength components of this surface have a strong directional component. The area SF shows this for all spatial wavelengths. Also, the SF values are directly connected to height errors, an intuitive property for the optics manufacturer. The values are the surface variance as a function of separation, and the square root of the SF is the surface RMS as a function of separation.

3.2 Diamond turned surface

Figure 4(a) shows the analysis for a Fizeau interferometer measurement of a diamond-turned aluminum flat. The area SF of the input map (Fig. 4(c)) is dominated by 2 features: an orientation-independent peak at a separation of approximately 25 mm, and an asymmetry due to astigmatism in the part. The 25 mm peak is due to the conical shape to the part, a common form error when there is a squareness error in the machine. The astigmatism is not likely caused by a machine error, and more likely the result of mount-induced deformations. If mid-spatial frequency error is the primary interest, the SF analysis can be applied after removal of the form error, as is done for a PSD analysis. Figure 4(b) shows the residual error in this part after removal of 36 Zernike terms. The corresponding area SF is shown in Fig. 4(d). The annuli in the area SF at 10 mm and 22 mm separation are primarily due to the central high in the residual error (Zernikes do not fit a cone well), dropping to the low zone at a radius of about 10 mm, and then the edge that is again low.

 

Fig. 4 Interpretation of the 2-quadrant SF calculation for a diamond-turned aluminum flat. (a) Input map. (b) Residual error after removing the first 36 Zernike terms. (c) Area SF for part (a) of the input map in 2 quadrants. (d) Area SF of the residual error, part (c) in 2 quadrants.

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Note the highest frequency “ripple” in the area structure function with a separation repeat length of about 4 mm. This ripple is rotationally invariant, that is the structure function values are constant for values of τ_x and τ_y for which $\tau =\sqrt{{\tau }_x{}^2+{\tau }_y{}^2}$is constant.

4. Area structure function compared to other area analyses

In this section, the area SF is compared to the power spectral density (PSD) and the autocorrelation function (ACF). All three analyses are mathematically related, but only for a stationary surface. This means, all three provide the same general information in this limit, but the units and physical interpretation vary. A surface with form error is not stationary. The ACF and SF analyses can be sensibly applied to such surface maps, whereas the PSD analysis should not be applied to form errors. Low spatial frequency content in circular apertures can be described with values of Zernike coefficients, where as the ACF and SF represent this in a map that highlights the surface anisotropy and key length scales of interest.

For mid-to-high spatial frequency analysis, form error is usually removed, and the residual may or may not be stationary. Regardless, all three analyses can be used. The ACF and PSD can be related to the modulation transfer function (MTF). The SF can also be related to MTF if the surface is stationary. Work is underway to explore the connection between SF and MTF.

4.1 Stationary surface

The SF can be described as

For a stationary surface [10],

Thus, the SF is closely related to the ACF for a stationary surface. The overall scales of the two functions are different, and the locations of the maxima and minima are inverted.

The PSD is the Fourier transform of the autocovariance function [15,16]

The residual surface error of a conventionally polished surface is, typically, stationary and characterized by a speckled or “orange peel” texture. Figure 5(a) and 5(b) are the SF and ACF for the residual error in a large polished flat. Figure 5(c) is the difference between $s({\tau }_x,{\tau }_y)$ and $2{\sigma }^2\left\{1-R({\tau }_x,{\tau }_y)\right\}.$ As expected, most of the differences are zero except at the edges where there are only a few calculated points. Figure 5(d) is the area PSD of the residual error.

 

Fig. 5 Comparison between SF, ACF and PSD for a polished surface after removing 36 Zernikes. (a) Area SF of the residual error. (b) Area ACF of the residual error. (c) Deviation = SF-2σ²(1-ACF). (d) Area PSD of the residual error.

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4.2 Non-stationary surfaces

Optical surfaces, particular when figure is considered, are typically not stationary. In this case, Eq. (4) is not correct and hence the simple relationship between SF and ACF does not apply.

Figure 6 shows the analysis of the diamond-turned aluminum flat. Comparison of Fig. 6(b) and 6(c) suggests that the area SF provides a clear visualization of the spatial content of the surface as a function of separation and orientation compared to the ACF, perhaps a result of a product vs. a difference as a function of separation. As Sayles and Thomas [10] point out, because the area ACF is the sum of terms each of which is the product of two amplitudes, it is less intuitive. Figure 6(d) shows a PSD analysis of the data. The values between different spatial frequencies are not as clear as the area SF. Further, the PSD analysis strongly depends on filtering, the choice of windows and zero padding.

 

Fig. 6 Analysis of a diamond-turned aluminum flat. (a) Residual error after removing the first 36 Zernike terms. (b) Area SF. (c) Area ACF. (d) Area PSD.

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5. Conclusions and future work

This paper introduces the area structure function as a complementary method for specifying and representing the spatial content of optical surface errors. Surface anisotropy is not represented in a linear SF, nor in a single quadrant area SF. The two quadrant area SF introduced here allows for a complete representation.

Future work will include an investigation of the impact of different measurement coordinate systems, the combination of data obtained with multiple instruments into a single SF analysis, and the connection between SF and the modulation transfer function.