Use of pupil-difference moments for predicting optical performance impacts of generalized mid-spatial frequency surface errors

Luke A. DeMars; Thomas J. Suleski

doi:10.1364/OE.503735

1. Introduction

Sub-aperture manufacturing methods are key enablers for freeform optics [1]. However, one drawback to sub-aperture processes is the potential to introduce mid-spatial frequency (MSF) surface errors [2–6], which can take considerable effort to mitigate [7–16]. MSF surface errors fall between ‘low-frequency’ form errors and ‘high-frequency’ roughness errors, with ambiguous lower and upper bounds [17]. Understanding the impacts of MSF errors on optical performance is important to the larger optics community as such errors can cause small angle scatter [18,19], degrade image performance [20–23], and result in self-imaging [24,25] and distortion [26,27].

MSF surface errors are commonly observed with a broad range of structured distributions (e.g., raster, radial, azimuthal) that are related to the manufacturing method [17,28]. Previous methods for predicting the performance impacts of MSF surface errors on image quality have relied on assumptions of random error distributions [18,29,30]. A common approach is to estimate the Strehl ratio from the surface RMS error [30]. This approach will provide a baseline performance estimation for random surfaces in this work. However, different MSF distributions with similar simple surface metrics (i.e., root mean square (RMS) surface error (σ), slope, Peak-Valley (PV)) can have drastically different optical performance [1,31,32]. Therefore, specification methods that connect general MSF surface errors to optical performance are desirable.

To this end, additional metrics such as the minimum modulation curve (MMC) [33] and Q′ [31] have been recently proposed. The MMC metric quantifies the lowest values in the 2D modulation transfer function (MTF) as a function of radial image space spatial frequency and azimuthal angle, while Q′ quantifies the normalized area under the MMC and is analogous to the Strehl ratio. These methods are inspired by the 1D modulation transfer function (MTF) and its relationship to the Strehl ratio [34], but better quantify and distinguish between the optical performance impacts of deterministic MSF surface errors and also agree with predictions from earlier statistical methods [30] for cases with uniform or random MSF surface errors [31]. However, we note that previous connections of the MMC and Q′ to surface specifications were largely empirical [31]. Pupil-Difference Probability Distributions (PDPD) were recently introduced by Alonso and Liang as a tool to show the impact of MSF groove structures and their random variations on the optical transfer function (OTF) [35,36]. Building on that work, we have recently shown that the PDPD moments have desirable properties for specification of general MSF surface errors and for connecting those specifications to relative optical modulation [32].

In this paper, we build on the work from [32] to demonstrate use of the PDPD moments to connect both random and structured MSF error distributions to the MMC, Strehl ratio, and Q′. We first summarize the measures of optical performance used in this work. We then investigate the relationship of the 2^nd PDPD moments to σ to quantitatively explore the convergence of predicted optical performance from random and deterministic MSF error distributions. For cases that do not converge, we then provide a procedure relating the 2^nd and 4^th PDPD moments to the minimum modulation curve [33] and Q′ [31]. Finally, we demonstrate the application of these methods to several examples with different MSF distributions and levels of randomness and compare how estimated optical performance values compare to predictions based on earlier methods assuming random error distributions.

2. Measures of optical performance for MSF errors

2.1 Strehl ratio

The Strehl ratio S quantifies the ratio of the peak value of the point spread function (PSF) of an aberrated optical system to the peak value for an ideal diffraction-limited system [34]. While there are multiple methods to calculate the Strehl ratio [37–43], we use two forms in this work. The first definition is calculated from the area under the aberrated (‘AB’) MTF normalized by the area under the diffraction-limited (‘DL’) MTF [34], as shown in Eq. (1):

(1)$$S = \frac{{\int\!\!\!\int {MT{F_{AB}}({f_x},{f_y})} d{f_x}d{f_y}}}{{\int\!\!\!\int {MT{F_{DL}}({f_x},{f_y})} d{f_x}d{f_y}}}. $$

The second definition is an exponentially decaying approximation proposed for use with MSF errors by Youngworth and Stone [30], as shown in Eq. (2):

(2)$$S \approx \exp [ - {k^2}{(\Delta n)^2}{\sigma ^2}], $$

where Δn is the index contrast and k is the wavenumber. With either definition of the Strehl ratio, the aberrated MTF can be estimated by multiplying S by the diffraction-limited MTF, as shown in Eq. (3).

(3)$$MT{F_{AB}}({f_x},{f_y}) \approx (S)MT{F_{DL}}({f_x},{f_y})$$

One challenge with the Strehl ratio is that it can result in misleading predictions of optical performance with deterministic MSF errors. To illustrate this point, we calculate the Strehl ratio for both a Gaussian error distribution, Fig. 1(a), and for a structured raster MSF signature, Fig. 1(b). The Gaussian surface has a PV = 598 nm and an σ = 70 nm. The structured raster MSF signature used in this example has a period of 0.2 mm, PV = 200 nm, and σ = 70 nm. Note that we scaled the Gaussian surface to have nearly the same σ and did not control the PV. In order to achieve a specific PV for the Gaussian surface, the relative phase angles of the spectra would need to be controlled [44,45]. However, for the following examples, having an equivalent σ is sufficient. Both surfaces are superimposed onto separate but identical plano-convex lenses at the aperture stop with 100 mm focal length, diameter D = 2 mm, and Δn = 0.49 at a source wavelength of 532 nm, as shown in Fig. 1(c).

Fig. 1. (a) Gaussian surface error; (b) structured raster MSF error, (c) optical system used in example; (d) MTF and Strehl estimate for Gaussian surface case; (e) MTF and Strehl estimate for Structured MSF case.

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For both cases, we then calculate the PSF from the absolute magnitude squared of the propagated field via a Fourier Transform to the image plane under far-field assumptions, and then calculate the 2D MTF by taking the magnitude of the Fourier transform of the PSF. A cross-section of the 2D MTF along the x-axis for the lens with Gaussian surface is shown in Fig. 1(d) and for the lens with the structured MSF in Fig. 1(e). In addition, the MTF is estimated from Eq. (3), with S = 0.84 from both Eqs. (1) and (2) in each case. From Fig. 1(d), we see that random errors are well estimated by the Strehl ratio but Fig. 1(e) shows that the Strehl ratio underestimates the impacts of the deterministic MSF error on MTF.

2.2 MMC and Q’

As discussed in Section 1, the minimum modulation curve (MMC) [33] and Q′ [31] were recently proposed to better estimate optical performance impacts of more general MSF errors. The MMC is a useful metric because it accounts for the directionality that MSF errors may have but also reduces to estimates given by Strehl ratio for random surface error distributions [31]. Analogous to the Strehl ratio, Q′ is quantified by calculating the area under the MMC normalized by the area under the diffraction-limited MTF [31], as shown in Eq. (4). The MMC can be estimated from Q′ using Eq. (5).

(4)$${Q^{\prime}} = \frac{{\int {MMC(\rho )d\rho } }}{{\int {MT{F_{DL}}(\rho )d\rho } }}. $$

(5)$$MMC(\rho ) \approx (Q^{\prime})MT{F_{DL}}(\rho ). $$

In Fig. 2 below, we calculate the MMC and Q′ from the MSF example in Fig. 1. Using Eq. (4), we calculate Q′ = 0.69 and then use Eq. (5) to estimate the MMC. We see that Q′ better estimates the performance impacts of the MSF error on this lens because it captures the oscillations in the MTF.

Fig. 2. MMC and Q′ predictions of optical performance impacts of MSF errors from Fig. 1.

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3. Convergence of performance estimates between random and deterministic MSF error distributions

The examples in Section 2 illustrate differences in optical performance predictions for random and deterministic MSF error distributions. We now consider use of the relative optical modulation M to investigate the convergence of performance estimates between random and deterministic cases. As described in [32], M is a multiplicative factor to the MTF that captures information on the MSF error distribution. Under the assumptions of a symmetric PDPD, small drops in relative modulation, and an MSF error surface at the exit pupil, the 2^nd and 4^th PDPD moments can be used to estimate the relative modulation as shown in Eq. (6):

(6)$$M({\tau _x},{\tau _y}) \approx 1 - {k^2}{(\Delta n)^2}\frac{{\left\langle {H_C^2({\tau_x},{\tau_y})} \right\rangle }}{{2!}} + {k^4}{(\Delta n)^4}\frac{{\left\langle {H_C^4({\tau_x},{\tau_y})} \right\rangle }}{{4!}}. $$

In Eq. (6) the pupil shift vector τ_i is a generalization of image space spatial frequency and is used to describe the location of the PDPD moment value in the PDPD moment map [32]. For PDPD moment maps shown in this work, τ_x ranges from –D to D, and τ_yranges from 0 to D. Note that the negative axis is not shown for τ_y to avoid redundancy in the PDPD moment maps [32,46].

We now use Eq. (6) to compare performance estimates between surfaces with random and structured MSF error distributions. We start by assuming the PDPD has a zero-mean pupil-difference for all pupil shifts, as was done for the area structure function [46]. We then expand out the 2^nd PDPD moment terms, as shown in Eq. (7):

(7)$$\left\langle {H_c^2({\tau_x},{\tau_y})} \right\rangle = \left\langle {{h^2}(x,y)} \right\rangle + \left\langle {{h^2}(x - {\tau_x},y - {\tau_y})} \right\rangle - 2\left\langle {h(x,y)h(x - {\tau_x},y - {\tau_y})} \right\rangle. $$

When the height distribution h(x,y) is stationary [46,47], the first and second terms are equivalent and the 2^nd PDPD moment can be approximated as:

(8)$$\left\langle {H_c^2({\tau_x},{\tau_y})} \right\rangle \approx 2\left\langle {{h^2}(x,y)} \right\rangle - 2\left\langle {h(x,y)h(x - {\tau_x},y - {\tau_y})} \right\rangle = 2{\sigma ^2} - 2\left\langle {h(x,y)h(x - {\tau_x},y - {\tau_y})} \right\rangle. $$

This shows that the 2^nd PDPD moment is proportional to σ² and an autocovariance term. The autocovariance term can oscillate from -σ² (perfectly anti-correlated) to +σ² (perfectly correlated) and will tend towards zero at small pupil-shifts for random surfaces [29].

To further illustrate this point, we calculate the 2^nd PDPD moments for the raster sinusoidal MSF error and random Gaussian surface used in Section 2.1. The 2^nd PDPD moments of the Gaussian and structured MSF are calculated using the procedure defined in [32] and are shown in Fig. 3(a) and 3(b), respectively. We then take the cross-sections of the 2^nd PDPD moments along the +τ_x direction from zero to D at τ_y = 0 for both cases, as shown in Figs. 3(c) and 3(d). From Fig. 3(c), we see that the 2^nd PDPD moment of the Gaussian surface lies close to the mean of 2σ². From Fig. 3(d), we see that the 2^nd PDPD moment oscillates around the mean of 2σ² and peaks at twice the mean due to the deterministic structure of the raster MSF error. This description holds well until we reach the edges of the 2^nd PDPD moment due to a decreasing number of data points in the PDPD. It is clear that the 2^nd PDPD moment amplitude for random surface errors with little correlation will lie close to 2σ², and for structured raster MSF errors the 2^nd PDPD moment amplitude will oscillate around the mean of 2σ² up to a maximum of twice the mean. We also note that the asymptote reached for the 2^nd PDPD amplitude of a Gaussian surface is the same as the structure function of a Gaussian random phase screen [48].

Fig. 3. (a) 2^nd PDPD moment of Gaussian surface error, (b) 2^nd PDPD moment of Structured MSF, (c) cross-section along the +τ_x_, direction from zero to D at τ_y = 0 of 2^nd PDPD moment from Gaussian surface, and (d) cross-section of 2^nd PDPD moment along +τ_x_, from zero to D at τ_y = 0 from the structured MSF example.

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To see how Eq. (6) relates to the Strehl ratio associated with random surfaces, we consider only up to the second term of Eq. (6) and substitute 2σ²in the second term, which is the 2^nd PDPD moment result for random surfaces. In this case, the estimated relative modulation, shown in Eq. (9), is approximately equal to the first two terms in a Taylor series expansion of the Strehl ratio from Eq. (2), which is valid under a weak aberration scenario [37].

(9)$${M_{uncorrelated}}({f_x},{f_y}) \approx 1 - {k^2}{(\Delta n)^2}{\sigma ^2} \approx S. $$

If the same k, Δn, and system from Section 2.1 are used with Eq. (9), then M_uncorrelated ≈ 0.84, which is the same result that was reached using Eqs. (1) and (2).

This implies that the Strehl ratio is a good measure of optical performance impact when surface errors are random. This result is supported by previous works that assume Gaussian statistics of surface errors to estimate the impact on the transfer function [18,29]. However, we note here that (unlike previous works) the use of the PDPD moments makes no assumption of Gaussian statistics to relate the surface specification to optical performance.

Applying a similar process for the structured MSF case, we have an upper bound estimate of the 2^nd PDPD moments impact on the relative modulation, shown in Eq. (10). Once again, using the same k and Δn, we estimate M_correlated ≈ 0.67 which is closer to the Q′ estimate in Section 2.2.

(10)$${M_{correlated}}({f_x},{f_y}) \approx 1 - 2{k^2}{(\Delta n)^2}{\sigma ^2} \approx Q^{\prime}. $$

Note that the slight deviation between the Q′ from Section 2.2 and M_correlated is attributed to the truncation of the series used to derive Eq. (10). We see clearly from this analysis that Strehl ratio can be a poor estimate of optical performance for structured MSF errors and that the use of PDPD moments can facilitate better performance predictions in such cases.

Up to this point, we have only shown how the 2^nd PDPD moment relates to σ and how the relative modulation (up to the 2^nd term) converges to the Strehl ratio under a random surface assumption. A logical next step would be to consider a similar process including the 4^th PDPD moment. To do so we would need to assume that the Gaussian moment theorem [47] could be applied to the higher-order terms in the 4^th PDPD moment expansion. However, we emphasize that MSF error distributions may not be Gaussian. Rather than making assumptions on the surface statistics, we propose a procedure that utilizes the generality of the PDPD moments to predict the impacts of MSF errors on optical performance regardless of their correlation.

4. Optical performance predictions from PDPD moments

4.1 Standard error in PDPD moments

Before estimating the optical performance using the PDPD moments, the influence on regions in the PDPD moment maps with few data points must be considered. Figure 4 shows that the number of data points changes from many to few as the pupil shift varies in the PDPD calculation, and each instance of the PDPD has a finite amount of data points from which to calculate the moments, as shown in Fig. 4(c).

Fig. 4. (a) PDPD using many points, (b) PDPD using few points, (c) map of number of data points used in each PDPD pupil shift instance.

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To assess the amount of error due to the discreteness of calculating PDPD moment data, we propose use of the standard error [49]. The standard error can be thought of as the variance in calculating a moment from a limited data set. Not all moments have easy-to-express relationships to standard error. However, the 2^nd moment has a tractable form and does not make statistical assumptions in its formulation. As defined by Rao, the standard error of the 2^nd moment (the variance) is shown in Eq. (11) [49] where m is the number of points in the distribution, σ² is the variance, and κ is the 4^th moment. We must rewrite Eq. (11) as a function of pupil shift to apply this method to the 2^nd PDPD moments. The standard error of the 2^nd PDPD moment is shown in Eq. (12). The normalized standard error (NSE) can then be calculated by dividing through by the 2^nd PDPD moments, as shown in Eq. (13). Note that the NSE is undefined for a 2^nd PDPD moment of zero value (which occurs at zero pupil-shift).

(11)$$SE[{\sigma ^2}] = \sqrt {\frac{{{{(m - 1)}^2}}}{{{m^3}}}(\kappa + \frac{{(m - 3)}}{{(m - 1)}}{\sigma ^4}} )$$

(12)$$SE[\left\langle {H_C^2({\tau_x},{\tau_y})} \right\rangle ] = \sqrt {\frac{{{{(m({\tau _x},{\tau _y}) - 1)}^2}}}{{m{{({\tau _x},{\tau _y})}^3}}}(\left\langle {H_C^4({\tau_x},{\tau_y})} \right\rangle + \frac{{(m({\tau _x},{\tau _y}) - 3)}}{{(m({\tau _x},{\tau _y}) - 1)}}{{\left\langle {H_C^2({\tau_x},{\tau_y})} \right\rangle }^2}} )$$

(13)$$NSE = \frac{{SE[\left\langle {H_C^2({\tau_x},{\tau_y})} \right\rangle ]}}{{\left\langle {H_C^2({\tau_x},{\tau_y})} \right\rangle }}. $$

A similar process can be performed for the 4^th PDPD moment, but the expression is less tractable. The standard errors for specific 4^th moment values can be calculated by bootstrapping or jackknifing approaches [50], but this is outside the scope of the present work.

With the normalized standard error map, a particular threshold can be applied (for example, 1%), and then a mask of all the 2^nd PDPD moments with a normalized standard error less than 1% can be multiplied by the 2^nd PDPD moment map to exclude those values from assessment, as shown in Figs. 5 and 6, respectively. The same mask should also be applied to the 4^th PDPD moment map since those pupil shift locations won’t be considered in the 2^nd PDPD moment map.

Fig. 5. Normalized standard error map for structured MSF example from Fig. 1(b) with a 1% threshold.

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Fig. 6. Example masking out points in the 2^nd PDPD moment with normalized standard errors greater than 1%.

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4.2 PDPD moments relationship with Q′ and estimating the MMC

To connect the PDPD moments to Q′, we start at Eq. (6). We note that Eq. (6) captures the full range of relative modulation. From the form of Eq. (6) it is clear that the minimum relative modulation will be found for the condition in which the 2^nd PDPD moment is a maximum. The 4^th PDPD maxima also follows this trend. As a result, we assert that Q′ can be estimated by using the PDPD moment maxima in Eq. (6) to obtain:

(14)$${Q^{\prime}} \approx 1 - \frac{{{k^2}{{(\Delta n)}^2}{{\left\langle {H_c^2({\tau_x},{\tau_y})} \right\rangle }_{\max }}}}{{2!}} + \frac{{{k^4}{{(\Delta n)}^4}{{\left\langle {H_c^4({\tau_x},{\tau_y})} \right\rangle }_{\max }}}}{{4!}}. $$

The MMC can then be estimated from Eq. (5). It should be noted that the maxima in the 2^nd and 4^th PDPD moments may not always occur at the same pupil shift values, in which case Eq. (14) may lose some accuracy. In this case, we suggest looking for the relative modulation’s lowest value, estimated from Eq. (6), and then reporting 2^nd and 4^th PDPD moment values at that particular pupil shift to better estimate the MMC. To account for this point and the standard error of the PDPD moments, we propose the following process:

1) Calculate the 2^nd and 4^th PDPD moments from the MSF surface error.
2) Calculate the standard error of the 2^nd PDPD moments.
3) Apply a chosen threshold to the normalized standard error and crop out 2^nd and 4^th PDPD moments for all pupil shift values above that threshold.
4) Apply Eq. (6) to estimate the relative modulation.
5) Find the lowest relative modulation from Eq. (6) and calculate the 2^nd and 4^th PDPD moments for those pupil shift values.
6) Estimate Q′ using Eq. (14).
7) Apply Eq. (5) to estimate the MMC.

We use the MSF example from Section 2.2 with the same Δn and k values as before to illustrate the workflow, as shown in Fig. 7. The estimate of Q′≈ 0.68 is within 1% of the estimate from Eq. (4).

Fig. 7. Workflow for example MSF surface from Section 2.2. (a) Cropped 2^nd and 4^th PDPD moments based on NSE; (b) estimates of relative modulation and Q′ from PDPD moments; (c) estimated MMC using Eq. (5).

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2. Examples: estimating optical performance of different MSF distributions

We now compare specification and performance predictions for several different MSF surface error distributions (Fig. 8) using the general procedures outlined in Section 4.2 and the procedures for random surface errors.

Fig. 8. (a) Radial sinusoidal MSF error; (b) Azimuthal sinusoidal MSF error; (c) Experimental raster MSF error.

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We also include the results from the earlier raster and Gaussian surfaces in this analysis for additional comparison. The σ for all examples differ slightly due to their periodicity and distribution but are all ∼70 nm.

To compare the procedure in Section 4.2 to the procedures for random surfaces, we do the following: (1) Tabulate the maxima of the 2^nd and 4^th PDPD moments following Section 4.2; (2) Calculate 2σ² for each surface. These values are included to show that performance estimates from Section 4.2 and random surface methods converge when the 2^nd PDPD moment peak value is close to 2σ²; (3) Calculate Q′ values from the MMC (Eq. (4)) and PDPD moments (Eq. (14)); and (4) Calculate the Strehl ratio from the MTF (Eq. (1)) and from σ (Eq. (2)), all using the same Δn and k values from Section 2. A summary of these results is shown in Table 1.

Table 1. Surface statistics and optical performance metrics for example MSF surfaces

View Table

From Table 1, we note that cases with a 2^nd PDPD maximum close to 2σ² tend to be appropriately specified by σ, and its performance will be closely estimated by the Strehl ratio. The 2^nd PDPD maximum of the example Gaussian surface is closest to 2σ², and we observe that performance in this case is well predicted by the Strehl ratio. A general trend can be observed that larger deviations between the 2^nd PDPD maxima and 2σ², correspond to larger deviations between the Strehl ratio performance estimates and Q′. Results are also shown in Fig. 9 for each case through plots of the diffraction-limited performance, the MTF estimated from Eq. (2) with Eq. (3), and the MMC estimated using Eq. (14) with Eq. (5).

Fig. 9. MMC comparisons with diffraction-limited case and Strehl estimate of MTF for (a) Raster MSF surface example, (b) Azimuthal MSF surface example, (c) Experimental MSF surface example, (d) Radial MSF surface example, and (e) Gaussian surface example.

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An additional observation from Table 1 is that the MMC calculation of Q′ does not always closely match the Q′ estimate from Section 4.2. This is because the PDPD moment maxima are calculated from the lowest relative modulation value, while the MMC calculation of Q′ comes from the average of the lowest 2D MTF values. This nuance is best visualized in Fig. 9(b) and 9(c), where we see the MMC estimated by Section 4.2 methods only touches one point on the MMC, at the point of lowest relative optical modulation.

3. Summary and discussion

We have demonstrated a procedure using the PDPD moments of MSF surface errors to predict reductions in optical performance and shown that the 2^nd and 4^th PDPD moments can be utilized to estimate the optical performance impacts of both random and deterministic MSF surface errors. For random surfaces, the estimates using PDPD moments converge to estimates using the Strehl ratio with random surface errors. We showed that optical performance is well predicted using methods based on the Strehl ratio for surfaces for which the 2^nd PDPD moment maximum is close to the mean. However, we also saw a large variation in optical performance predictions for the range of MSF surface error distributions presented. As a result, we saw it valuable and necessary to utilize the 2^nd and 4^th PDPD moment procedures for most cases. Use of PDPD moments as proposed herein enable specification, improved predictions of the impacts of generalized MSF errors on optical performance in imaging systems, and better understanding of the sources of performance degradation at the cost of increased calculation complexity and time in comparison to the Strehl ratio or direct calculation of MTF_AB. Ultimately a decision must be made on how precisely the optical performance must be predicted to determine if it is necessary to utilize the PDPD moments.

We note a limitation in that the moment maps can have large standard errors when used with low-resolution height maps. It is recommended to use surface height maps that are nominally of the size of (at least) 1000 × 1000 pixels if possible. Using smaller height maps may not properly resolve the MSF errors and may also produce large errors in optical performance predictions. A separate challenge is that the PDPD moment maps can be challenging to interpret due to the multi-dimensional nature of the statistics. We utilize the maxima of the moment maps in the current work but note that additional information regarding the surface errors is also contained, such as period [46] and slope [51].

With the above limitations in mind, we provide this work to help readers gauge if a given MSF error distribution can be adequately specified using σ or if the additional information provided by the PDPD moments is needed for their specific case. We also note that the maxima of the PDPD moments provide single-value metrics that can be reported with little interpretation of the moment maps as a whole. This result suggests that the calculation of the maxima could be automated, and the values used as straightforward pass/fail criteria for specification and acceptance testing.

Lastly, we note opportunities for future expansion of this work. As discussed in [32], in cases where the 3^rd PDPD moment is non-zero, the complex component of the relative modulation will also be non-zero, and the estimate from Section 4.2 may not be valid. We plan to extend this work to include the analysis of the 3^rd PDPD moment and provide additional procedures that consider a complex component to the relative modulation. In addition, we note that the aberrated 2D MTF can be estimated by multiplying Eq. (6) by the 2D diffraction-limited MTF. This approach could help to facilitate defining optical performance over specific band limits in image space and enable quantifying the impacts of MSF surface errors at targeted image space spatial frequencies.

Funding

National Science Foundation I/UCRC Center for Freeform Optics (IIP-1822026, IIP-1822049).

Acknowledgments

The authors would like to acknowledge helpful discussions with Dr. Angela Allen and Dr. Chris Evans from UNC Charlotte, Dr. Jannick Rolland from the University of Rochester, and Dr. Miguel Alonso of Aix Marseille Université and the University of Rochester. This work was previously included in the first author’s dissertation [52].

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. J. P. Rolland, M. A. Davies, T. J. Suleski, et al., “Freeform optics for imaging,” Optica 8(2), 161–176 (2021). [CrossRef]

2. A. Sohn, L. Lamonds, and K. Garrard, “Modeling of vibration in single-point diamond turning,” in Proc. ASPE 21st Annual Meeting (2006), pp. 15–20.

3. S. Rakuff and P. Beaudet, “Thermally induced errors in diamond turning of optical structured surfaces,” Opt. Eng. 46(10), 103401 (2007). [CrossRef]

4. M. Pohl and R. Borret, “Simulation of mid-spatials from the grinding process,” Eur. Opt. Soc.-Rapid 11, 16010 (2016). [CrossRef]

5. B. Zhong, H. Huang, X. Chen, et al., “Modelling and simulation of mid-spatial-frequency error generation in CCOS,” J. Eur. Opt. Soc.-Rapid Publ. 14(1), 4 (2018). [CrossRef]

6. S. Wan, C. Wei, Z. Hong, et al., “Modeling and analysis of the mid-spatial- frequency error characteristics and generation mechanism in sub-aperture optical polishing,” Opt. Express 28(6), 8959–8973 (2020). [CrossRef]

7. D. D. Walker and C. Dunn, “Pseudo-random tool paths for CNC sub-aperture polishing and other applications,” Opt. Express 16(23), 18942–18949 (2008). [CrossRef]

8. D. Nelson, J. D. Nelson, A. Gould, et al., “Incorporating VIVE into the precision optics manufacturing process,” Proc. SPIE 81261, 812613 (2011). [CrossRef]

9. T. Wang, H. Cheng, W. Zhang, et al., “Restraint of path effect on optical surface in magnetorheological jet polishing,” Appl. Opt. 55(4), 935–942 (2016). [CrossRef]

10. H. Shahinian, M. Hassan, H. Cherukuri, et al., “Fiber-based tools: material removal and mid-spatial frequency error reduction,” Appl. Opt. 56(29), 8266–8274 (2017). [CrossRef]

11. A. Beaucamp, K. Takizawa, Y. Han, et al., “Reduction of mid-spatial frequency errors on aspheric and freeform optics by circular-random path polishing,” Opt. Express 29(19), 29802–29812 (2021). [CrossRef]

12. S.-W. Liu, H.-X. Wang, Q.-H. Zhang, et al., “Smoothing process of conformal vibration polishing for mid-spatial frequency errors: characteristics research and guiding prediction,” Appl. Opt. 60(13), 3925–3935 (2021). [CrossRef]

13. K. Wan, S. Wan, C. Jiang, et al., “Sparse bi-step raster path for suppressing the mid-spatial-frequency error by fluid jet polishing,” Opt. Express 30(5), 6603–6616 (2022). [CrossRef]

14. C. Wang, Y. Han, H. Zhang, et al., “Suppression of mid-spatial-frequency waviness by a universal random tree-shaped path in robotic bonnet polishing,” Opt. Express 30(16), 29216–29233 (2022). [CrossRef]

15. G. Chen and J. Qiao, “Femtosecond-laser-enabled simultaneous figuring and finishing of glass with a subnanometer optical surface,” Opt. Lett. 47(15), 3860–3863 (2022). [CrossRef]

16. J. Coniglio, J. Beck, J. Odle, et al., “Force controlled mid-spatial frequency error smoothing using machine learning,” Proc. SPIE 12188, 121880P (2022). [CrossRef]

17. D. M. Aikens, J. E. DeGroote, and R. N. Youngworth, “Specification and control of mid-spatial frequency wavefront errors in optical systems,” in Frontiers in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA1.

18. J. E. Harvey and A. K. Thompson, “Scattering effects from residual optical fabrication errors,” Proc. SPIE 2576, 155–174 (1995). [CrossRef]

19. X. Lu and H. Gross, “Wigner function-based modeling and propagation of partially coherent light in optical systems with scattering surfaces,” Opt. Express 29(10), 14985–15000 (2021). [CrossRef]

20. J. P. Marioge and S. Slansky, “Effect of figure and waviness on image quality,” J. Opt. 14(4), 189–198 (1983). [CrossRef]

21. J. M. Tamkin, W. J. Dallas, and T. D. Milster, “Theory of point-spread function artifacts due to structured mid-spatial frequency surface errors,” Appl. Opt. 49(25), 4814–4824 (2010). [CrossRef]

22. J. M. Tamkin, T. D. Milster, and W. Dallas, “Theory of modulation transfer function artifacts due to mid-spatial-frequency errors and its application to optical tolerancing,” Appl. Opt. 49(25), 4825–4835 (2010). [CrossRef]

23. J. M. Tamkin and T. D. Milster, “Effects of structured mid-spatial frequency surface errors on image performance,” Appl. Opt. 49(33), 6522–6536 (2010). [CrossRef]

24. V. P. Sivokon and M. D. Thorpe, “Theory of bokeh image structure in camera lenses with an aspheric surface,” Opt. Eng. 53(6), 065103 (2014). [CrossRef]

25. K. Liang, G. W. Forbes, and M. A. Alonso, “Validity of the perturbation model for the propagation of MSF structures in 3D,” Opt. Express 28(14), 20277–20295 (2020). [CrossRef]

26. G. Erdei, G. Szarvas, and E. Lorincz, “Tolerancing surface accuracy of aspheric lenses used for imaging purposes,” Proc. SPIE 5249, 718 (2004). [CrossRef]

27. K. Achilles, K. Uhlendorf, and D. Ochse, “Tolerancing the impact of mid-spatial frequency surface errors of lenses on distortion and image homogeneity,” Proc. SPIE 9626, 96260A (2015). [CrossRef]

28. L. A. DeMars, S. Rueda, and T. J. Suleski, “MSFLib: A Data Library of Mid-Spatial Frequency Surface Errors for Optical Modeling and Specification,” in Optica Design and Fabrication Congress 2023 (IODC,OFT), Technical Digest Series (Optica Publishing Group, 2023), paper OW4B.2.

29. R. J. Noll, “Effect of Mid- and High-Spatial Frequencies on Optical Performance,” Opt. Eng. 18(2), 137–142 (1979). [CrossRef]

30. R. N. Youngworth and B. D. Stone, “Simple estimates for the effects of mid-spatial-frequency surface errors on image quality,” Appl. Opt. 39(13), 2198–2209 (2000). [CrossRef]

31. H. Aryan, G. D. Boreman, and T. J. Suleski, “Simple methods for estimating the performance and specification of optical components with anisotropic mid-spatial frequency surface errors,” Opt. Express 27(22), 32709–32721 (2019). [CrossRef]

32. L. A. DeMars and T. J. Suleski, “Pupil-difference moments for estimating relative modulation from general mid-spatial frequency surface errors,” Opt. Lett. 48(9), 2492–2495 (2023). [CrossRef]

33. H. Aryan, G. D. Boreman, and T. J. Suleski, “The Minimum Modulation Curve as a tool for specifying optical performance: application to surfaces with mid-spatial frequency errors,” Opt. Express 27(18), 25551–25559 (2019). [CrossRef]

34. G. D. Boreman, Modulation Transfer Function in Optical and Electro-Optical Systems, 2nd ed. (SPIE PRESS, 2021), Chap. 1.

35. K. Liang and M. A. Alonso, “Understanding the effects of groove structures on the MTF,” Opt. Express 25(16), 18827–18841 (2017). [CrossRef]

36. K. Liang and M. A. Alonso, “Effects on the OTF of MSF structures with random variations,” Opt. Express 27(24), 34665 (2019). [CrossRef]

37. V. N. Mahajan, “Strehl Ratio for Primary Aberrations in Terms of Their Aberration Variance,” J. Opt. Soc. Am. 73(6), 860–861 (1983). [CrossRef]

38. J. Ojeda-Castaneda, E. Andres, and P. Montes, “Phase-space representation of the Strehl ratio: ambiguity function,” J. Opt. Soc. Am. A 4(2), 313–317 (1987). [CrossRef]

39. A. van den Bos, “Aberration and the Strehl ratio,” J. Opt. Soc. Am. A 17(2), 356–358 (2000). [CrossRef]

40. C. J. R. Sheppard, “Maréchal condition and the effect of aberrations on Strehl intensity,” Opt. Lett. 39(8), 2354–2357 (2014). [CrossRef]

41. M. A. Alonso and G. W. Forbes, “Strehl ratio as the Fourier transform of a probability density of error differences,” Opt. Lett. 41(16), 3735–3738 (2016). [CrossRef]

42. H. Aryan, K. Liang, M. A. Alonso, et al., “Predictive models for the Strehl ratio of diamond-machined optics,” Appl. Opt. 58(12), 3272–3276 (2019). [CrossRef]

43. B. D. Strycker, “The Strehl ratio as a phase histogram,” Appl. Opt. 62(18), 5035 (2023). [CrossRef]

44. A. van den Bos, “A New Method for Synthesis of Low-Peak-Factor Signals,” IEEE Trans. Acoust., Speech, Signal Process. 35(1), 120–122 (1987). [CrossRef]

45. A. Van Den Bos, “Rayleigh wave-front criterion: comment,” J. Opt. Soc. Am. A 16(9), 2307–2309 (1999). [CrossRef]

46. L. He, C. J. Evans, and A. Davies, “Two-quadrant area structure function analysis for optical surface characterization,” Opt. Express 20(21), 23275–23280 (2012). [CrossRef]

47. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995), Chap. 2.

48. J. W. Goodman, Statistical Optics, 2nd ed. (John Wiley & Sons, 2015), Chap. 8.

49. C. R. Rao, Linear Statistical Inference and Its Applications, 2nd ed. (John Wiley & Sons, 1973), Chap. 6.

50. B. Efron, The Jackknife, the Bootstrap and Other Resampling Plans (Society of Industrial and Applied Mathematics, 1982).

51. R. Zhelem, “Specification of optical surface accuracy using the structure function,” Proc. SPIE 8083, 808310 (2011). [CrossRef]

52. L. DeMars, “Specification and modeling of mid-spatial frequency errors in optical systems,” Ph.D. dissertation (University of North Carolina at Charlotte, 2023).

Surface	2^nd PDPD_max (nm²)	2σ² (nm²)	4^th PDPD_max (nm⁴)	Strehl (Eq. (1))	Strehl_σ (Eq. (2))	Q′ (Eq. (4))	Q′_PDPD (Eq. (14)
Raster	2.07x $10^{4}$	1.01x $10^{4}$	6.48x $10^{8}$	0.84	0.84	0.69	0.68
Azimuthal	1.99x $10^{4}$	1.00x $10^{4}$	5.87x $10^{8}$	0.84	0.85	0.84	0.69
Experimental	1.51x $10^{4}$	1.00x $10^{4}$	7.15x $10^{8}$	0.85	0.85	0.82	0.78
Radial	1.40x $10^{4}$	1.01x $10^{4}$	3.64x $10^{8}$	0.84	0.84	0.84	0.78
Gaussian	1.07x $10^{4}$	1.00x $10^{4}$	3.58x $10^{8}$	0.85	0.85	0.84	0.84

Use of pupil-difference moments for predicting optical performance impacts of generalized mid-spatial frequency surface errors

Abstract

1. Introduction

2. Measures of optical performance for MSF errors

2.1 Strehl ratio

2.2 MMC and Q’

3. Convergence of performance estimates between random and deterministic MSF error distributions

4. Optical performance predictions from PDPD moments

4.1 Standard error in PDPD moments

4.2 PDPD moments relationship with Q′ and estimating the MMC

2. Examples: estimating optical performance of different MSF distributions

3. Summary and discussion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Tables (1)

Equations (14)

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