Abstract
Mid-spatial frequency (MSF) structures on optical surfaces degrade system performance and a perturbation model is typically used to simplify the assessment of their effects. In this simple model, MSF phase structures are dragged along the nominal rays of a system to yield estimates of wavefronts in the exit pupil that may be used for further analysis. However, the validity of the perturbation model remains an open area of study. We extend our previous assessment of the validity of this model [K. Liang, Opt. Express 27, 3390 (2019) [CrossRef] that was focused on the analysis of single-frequency MSF structures in two dimensions to now include error estimates for broad-spectra MSF structures in three dimensions.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Mid-spatial frequency (MSF) structures are inevitable in most aspheric and freeform optical systems due to the subaperture tools that are used during the manufacturing process. Their characteristic frequencies lie between those of the common low-order aberrations and high-order scattering, and their detrimental effects on optical performance remain an active area of research. For example, there have been many efforts towards simplifying the tolerancing of optical parts afflicted with MSF [1–6]. To this end, the perturbation model is often used to cut down on the computation time needed to understand the propagation of MSF structures. This model, in which the MSF phase structure (which can vary significantly from part to part) is simply dragged along rays of the nominal system, is often used in order to avoid the need for new ray tracing for each MSF realization [7]. However, the validity of this perturbation model requires further treatment; its analysis in two dimensions was presented in Ref. 8 and those results are extended in Appendix A in a manner that we now generalize to three dimensions.
The mathematical framework used in this manuscript to estimate the error incurred by the perturbation model is based on an asymptotic analysis of the Helmholtz wave equation for the propagation of a monochromatic field in free space. A key step is the placement of the MSF structure at one asymptotic order beneath the nominal wavefront. This framework is similar to that in Ref. 8, but the solution now includes contributions from every asymptotic order (under appropriate approximations). This extension permits the analysis of MSF structures with broad spatial-frequency spectra.
2. Asymptotic propagation estimate based on nominal rays
The propagation of a monochromatic scalar field, $\textrm {Re}\left [ U(\boldsymbol{r}) e^{-{\textrm {i}} \omega t} \right ]$, in a homogeneous medium is governed by the Helmholtz equation
where $k = \omega /c = 2\pi /\lambda$ is the wavenumber in the medium. The field is taken to be propagating towards larger $z$ and, at some reference plane $z = z_{\textrm {M}}$, we take the initial value of the field to be given nominally by $U(\boldsymbol{r}_{\perp},z_{\textrm {M}}) = U_0 A(\boldsymbol{r}_{\perp}) \exp [ {\textrm {i}}k W(\boldsymbol{r}_{\perp})]$, where $U_0$ is a constant with field units and $\boldsymbol{r}_{\perp} = (x,y)$ are the transverse coordinates. At $z = z_{\textrm {M}}$, we also superpose an MSF phase factor of the form $\exp [{\textrm {i}} \phi (\boldsymbol{r}_{\perp})]$, where $\phi (\boldsymbol{r}_{\perp})$ is taken to have zero mean and the magnitude of its variation is less than $\pi$. Moreover, given its characterization as an MSF structure, $\phi (\boldsymbol{r}_{\perp})$ is assumed to vary more rapidly than either $W(\boldsymbol{r}_{\perp})$ or $A(\boldsymbol{r}_{\perp})$. The goal now is to derive an estimate, in a manner that is similar to that in Ref. 8, of how this MSF phase structure affects the field under propagation.We begin by writing $U(\boldsymbol{r}) = U_0 \exp [ {\textrm {i}}k \Phi (\boldsymbol{r})]$, where $\Phi (\boldsymbol{r})$ is a complex quantity that accounts for spatial variations in both the phase and amplitude. With this, Eq. (1) becomes
Eq. (2) can be solved upon expressing $\Phi$ as an asymptotic series in the parameter $({\textrm {i}}k)^{-1}$: By using Eq. (3) with Eq. (2) and separating terms of equal powers of $k$, we arrive atWe begin with Eq. (4), the well-known Hamilton-Jacobi or Eikonal equation, which can be solved in terms of nominal rays by using the following parametrization involving $\boldsymbol{\xi} = (\xi ,\eta ,s)$:
It is furthermore shown in Appendix B that Eq. (5) can also be solved in terms of the parametrization in Eq. (6). For $N=1$, the parametrized solution is
In what follows, the method used for proceeding to larger values of $N$ differs from that presented in Ref. 8. In order to appreciate the three-dimensional results, however, it is helpful to revisit the two-dimensional case and present the mathematical framework upon which the full three-dimensional treatment will follow by analogy, see Appendix A. As a reminder of the derivation in Ref. 8, recall that only the first correction to the perturbation model was analyzed. That is, the series in Eq. (3) is truncated at $N=2$, hence the field is taken to be approximated by
It is convenient, and necessary with regards to the derivations in Appendices A and B, to now work in image space in cases where, to a good approximation, a wavefront propagating from a point object source converges onto a point on the image plane. As in Ref. 8, we consider for simplicity only the on-axis object point, whose ideal image is located at the origin. It should be noted, however, that the analysis that follows can be used for off-axis object points as well since the choice of the origin was made out of convenience and similar methods can be applied to off-axis field points. Furthermore, we assume that the MSF content on each optical surface is adequately resolved in its corresponding conjugate plane in image space. Under these assumptions, the dominant error of the perturbation model is associated with the process of simply dragging these MSF structures along the nominally converging rays from their conjugate planes to the exit pupil. Henceforth, we will work with a single optical surface (with MSF structures) whose conjugate plane is located at $z= z_{\textrm {M}}$. Furthermore, the locations of the exit pupil plane and the image plane are taken to be $z = z_{\textrm {P}}$ and $z = 0$, respectively, see Fig. 1. With this framework, the nominal (converging) wavefront and obliquity factor are given by
To assess the error incurred by the perturbation model, one must go beyond the $N = 1$ term in Eqs. (3) and (5). It is shown in Appendix B, under the approximations that $\phi$ is small and that it varies more rapidly than the nominal quantities, that
The perturbation model is given by
3. Simple field error estimates in a homogeneous medium
The root-mean-squared error (RMSE) of the perturbation model, $\epsilon$, can be estimated as a function of propagation distance by integrating over the transverse plane the squared modulus of the difference between $U_\textrm {P}$ and the corrected field estimate in Eq. (15). This is achieved by changing the variable of integration from $(x,y)$ to $(\xi ,\eta )$ by using the differential area transformation
In Section 4, we use both Eqs. (21) and (22) to obtain rules of thumb regarding the validity of the perturbation model. Although Eqs. (21) and (22) were formally derived for systems with arbitrary NA, the remainder of this work will focus on systems with low to moderate NA. This is because the appropriate analysis for high-NA systems should involve a vector field treatment and the scalar formalism described here is sometimes insufficient. Furthermore, the results for systems with low to moderate NA may be of more interest for manufacturers due to their ease of interpretation and utility. However, for those interested in the behavior of $\epsilon$ in the high-NA regime, a discussion of those results from this scalar treatment is included in Appendix C.
4. Rules of thumb for low to moderate NA
In this section, we provide rules of thumb for the error incurred by the perturbation model for an imaging system with low to moderate NA. In this case the expressions for $\epsilon$ in Eqs. (21) and (22) can be simplified by using $\chi \approx 1$ and $A \approx 1$. Furthermore, $\hat {\mathcal {W}}$ can be simplified to $\nabla _{\perp} ^{2}$, the transverse Laplacian operator. With this, Eqs. (22) and (21) become
4.1 Rules of thumb for sinusoidal MSF structures in the milled and turned geometries
For MSF structures whose spectra are well-localized, it turns out that $\epsilon \approx \epsilon _{\textrm {F}}$ in Eq. (23) is sufficient. Note that Eq. (23) can be re-expressed as a normalized RMSE (NRMSE):
For completeness and in anticipation of the discussion regarding MSF phases with broad spectra, Fig. 2(b) shows how NRMSE values from a turned MSF surface calculated numerically compare with both $\epsilon _{\textrm {F}}$ and $\epsilon _{\textrm {C}}$ of Eqs. (23) and (24), respectively, as functions of $z_{\textrm {M}}/z_{\textrm {P}}$. Figure 2(b) is an alternative way to view the same data as Fig. 2(a) without having to introduce the notion of $r_1$, which may obscure the effects of what happens when, for example, the MSF is placed near the exit pupil or the focus. Near $z_{\textrm {M}}/z_{\textrm {P}} = 1$, $\epsilon _{\textrm {F}}$ is accurate and, as is indicated by the previous discussion regarding Fig. 2(a), the perturbation model is valid there since $\epsilon /\varphi < 1/3$. Furthermore, we point out that it is possible for the perturbation model to be valid [for instance, the case of $C = 5$ in Fig. 2(b)] even for large values of $z_{\textrm {M}}/z_{\textrm {P}}$, such as those beyond the image plane. This is in keeping with the fact that $\epsilon /\varphi$ is a closed curve if one were to join the $z_{\textrm {M}}/z_{\textrm {P}} = \pm \infty$ edges of Fig. 2(b), as discussed in Ref. 8. For larger values of $C$, however, the NRMSE begins to oscillate, due to the Talbot effect, for values of $z_{\textrm {M}}/z_{\textrm {P}}$ that are sufficiently far from unity (so that $r_1^{2}/\mathcal {R}^{2}$ is large) and this behavior is captured only by the complete error estimate. A notable feature of Fig. 2(b) is the region near $z_{\textrm {M}}/z_{\textrm {P}} = 0$, where the complete error estimate oscillates rapidly; it is evident that the perturbation model is not valid in this region and this fact is represented by the translucency of the plot. Recall that $\lambda = 632$ nm in these simulations and note that the effect of varying $\lambda$ on the plots of Fig. 2(b) is to change the value of $C$ to which they correspond, proportionally to $\lambda ^{-1/2}$. For example, if $\lambda$ were increased by a factor of 4, the plot for $C=20$ would correspond instead to $C = 10$. Figure 2(a), on the other hand, is explicitly independent of $\lambda$.
The simple error estimate in Eq. (29) is the analogous rule of thumb, specifically for the milled and turned sinusoidal MSF groove geometries, to the one-dimensional version in Ref. 8. Although it works well for MSF structures in the form of Eqs. (27) and (28), it was observed in Ref. 8 that such an estimate appears to overestimate the error incurred by MSF structures that possess a broad spectrum; this was demonstrated with synthetic MSF structures with spectra that obeyed a power-decay law. For these specific examples, the simple analog of Eq. (24) proved to be sufficient since it accurately predicted the NRMSE behavior in the region $0 < \epsilon /\varphi <1/3$, which is where the perturbation model was considered acceptable. However, it turns out that the consideration of MSF data requires an extension of the rule of thumb predicted by Eq. (25). For such MSF structures, it is necessary to consider the more complete NRMSE expression of Eq. (24).
4.2 Rules of thumb for MSF structures with broad spatial spectra
To begin, we present Fig. 3(a) so that it can be used as a reference for further discussion regarding the ineffectiveness of Eq. (25) when applied to MSF structures that are more complicated than those given by Eqs. (27) and (28). There, it is evident that such a simple estimate for $\epsilon /\varphi$ is useful only for small values of $r_1^{2}/\mathcal {R}^{2}$. The behavior of the numerically calculated NRMSE departs from the simple estimate very quickly in some examples. Although it is fortunate that Eq. (25) overestimates the true NRMSE, it fails as a rule of thumb for MSF structures with broad spectra (such as those seen in Fig. 3). For instance, someone interested in using the perturbation model with an optical system with MSF structures similar to that color-coded purple in Fig. 3(a), operating at $r_1^{2}/\mathcal {R}^{2} \approx 0.8$ with an NRMSE threshold of $20\%$, would erroneously believe based on Eq. (25) that this model should not be used. Therefore, there is a need for a more complete estimate that is accurate for MSF structures with broad spectra; this is provided by Eq. (24), which can be rewritten in the Fourier domain, after normalization by $\varphi ^{2}$, as
For the numerical simulations performed with the MSF structures shown in Fig. 3, the MSF data were pre-processed to have zero mean and normalized to have RMS of $\pi /8$. As was done with the sinusoidal examples in Sec. 4.1, field propagation was modeled by using the angular spectrum. Specifically, the nominal field is initially generated over an array of points at $z_{\textrm {P}}$ and propagated to the location $z_{\textrm {M}}$, where it is then multiplied by an exponential containing $\phi$, the MSF structure. Depending on the value of the ratio $|z_{\textrm {M}}/z_{\textrm {P}}|$, the dimensions of the MSF array must be scaled in order to ensure that the converging beam sees the same MSF phase over its diameter. Such a consideration was technically less cumbersome for the analysis in Sec. 4.1 because the MSF structures considered there are periodic and new arrays for $\phi$ at any $z_{\textrm {M}}$ can be sampled directly from an analytic sinusoidal function with the appropriate number of cycles. Although the calculated NRMSE does not oscillate quickly near $z_{\textrm {M}}/z_{\textrm {P}} = 0$ for MSF phases with broad spectra [compare the complete estimates in Fig. 2(b) and Fig. 3(b)], this region is similarly marked with translucency (to indicate the invalidity of the perturbation model regardless of numerically calculated results) because the simulation procedure described earlier in this process becomes invalid as the MSF phase is placed near the vicinity of the caustic at $z = 0$.
Figure 3 shows that, in the paraxial regime, the more complete estimate in Eq. (30) accurately predicts the NRMSE due to the perturbation model. As mentioned earlier, the simple estimate in Eq. (25) always overestimates the NRMSE predicted by Eq. (30). That is, the perturbation model appears to be more valid for MSF structures with broad spectra, as compared to those that are approximated well with a single frequency. One can understand this by noting that the real MSF data used in the analysis is dominated by low-frequency contributions. As a result, the plots shown in Fig. 3(b) are more akin to the plots in Fig. 2(b) with small values of $C$ (such as $C = 5$ and $C = 10$) that display a less oscillatory behavior. In other words, the calculated NRMSE for MSF data with broad spectra do not display the oscillatory Talbot re-imaging behavior seen in Fig. 2(b) for the larger $C$ values because the MSF data contain multiple frequencies that wash out the Talbot effect (in particular, validity is mainly governed by the low spatial frequencies).
A further point should be made regarding the relationship between Eqs. (25) and (30), in which the former is a first-order approximation in $r_1^{2} |\boldsymbol{\kappa}|^{2}$ of the latter. Equation (25) is sufficiently accurate for the MSF examples of Eqs. (27) and (28) in Fig. 2(b) within a region that shrinks with larger $C$ (or $\boldsymbol{\kappa}$). However, despite this shrinking region of accuracy, Eq. (25) is sufficiently accurate for $0 \le \epsilon _{\textrm F}/\phi <1$, which is the region that is relevant for assessing the validity of the perturbation model. The same cannot be said for the MSF examples in Fig. 3(b), whose spectra include both small and large values of $\boldsymbol{\kappa}$. Although the presence of large spatial frequencies consistently explains the small region of accuracy of Eq. (25), it is evident that this simple error estimate fails far before $\epsilon /\varphi \approx 1$.
5. Concluding remarks
We investigated the validity of the perturbation model in three dimensions by using an asymptotic framework like that in Ref. 8. However, we expanded upon the findings in Ref. 8 not only in the consideration of one extra spatial dimension, but also in the completeness of estimates for the error incurred by the perturbation model. That is, through further consistent approximations within the asymptotic framework, it is possible to solve for the complete correctional term [not just the first correction seen in Eq. (22)], which proves to be necessary when considering realistic MSF structures. This more complete approach gives accurate rules of thumb for the validity of the perturbation model. In particular, for the case of imaging systems with low to moderate NA, a general rule of thumb for the validity of the perturbation model for small-amplitude MSF structures is provided in Eq. (30). This more complete error estimate replaces the one in Ref. 8, which involved a troublesome fourth-order spectral moment of the MSF phase structure. These possibly-divergent moments are evidently replaced by the well-behaved integral of Eq. (30). As observed in Ref. 8, when an optical system has more than a single instance of MSF, the total (mean squared) error incurred upon using the perturbation model is simply given by the sum of the (mean squared) error for each individual MSF structure, provided that the MSF structures are statistically uncorrelated with each other. Furthermore, we mention that our results, which are derived by imaging the MSF structure to its conjugate location in image space and then propagating to the exit pupil, would be the same if we had instead imaged the MSF to the neighborhood of the aperture stop and performed the propagation steps there before imaging to the pupil. This invariance is discussed in Appendix B of Ref. 8.
The general framework in Sec. 2 was specialized to that of imaging systems (in fact, the complete error estimates are accurate only for a spherical nominal wavefront in image space), where only three locations in image space are relevant: the image plane, the exit pupil plane, and the plane that is conjugate to the MSF interface itself. The simplest error estimate, $\epsilon _{\textrm {F}}$, is suitable for imaging systems with low to moderate numerical apertures and MSF structures that are well described with a single spatial frequency with $\epsilon /\varphi < 1$. This error estimate is subsumed by the more general one described in Eq. (30), which is nicely represented in the Fourier domain; this estimate was shown to be sufficient in estimating the validity of the perturbation model for MSF structures with broad spectra. The real MSF data used in the analysis contained low spatial frequencies, which dominated the general behavior of the validity of the perturbation model. This, along with the washing-out of the Talbot effect, accounts for why the NRMSE, for many of the real MSF data used in this analysis, is low and not oscillatory when compared with the plots in Fig. 2. Moreover, the simple error estimate of Eq. (25) fails for both the MSF examples in Figs. 2 and 3 outside a domain that shrinks with the presence of higher spatial frequencies. However, the estimate is always sufficiently accurate within the range $\epsilon _{\textrm {F}}/\varphi < 1$ for MSF structures whose spectra are well-localized. For MSF structures with broad spatial spectra, one must instead use the complete error estimate, $\epsilon _{\textrm {C}}$, given by Eq. (30). Even though it was derived in the context of low to moderate NA, we remark that the results in Appendix C lead us to expect that Eq. (30) is, to within a factor of 2 or so, useful for numerical apertures up to 0.8. Therefore, even in the analysis of systems with appreciable NA, it is possible to use Eq. (30) to obtain a rough error estimate of the perturbation model rather than involving the more complicated, and incomplete, discussion regarding systems with high NA in Appendix C.
A. Extended derivation for two spatial dimensions
In this appendix, an alternative approximation for the field, $U(x,z)$, in two spatial dimensions is presented; the definitions of symbols correspond to those given in Appendix A of Ref. 8. One begins by considering the general differential equation for $\overline {\Phi _N}$:
An approximation is now made with regards to Eq. (34): the only term retained on the right-hand side is the one that involves $\partial _{\xi }^{2}$. The reason for making this simplification, which is discussed more in Appendix B, is ultimately due to the fact that the derivatives of $\phi$ (which varies quickly under its characterization as MSF) are large when compared to the other terms. These other terms contain nominal quantities and their transverse derivatives. By doing this, and explicitly using Eq. (35) in the definition of $\mathbb{J}^{-1}$, Eq. (34) is approximated as
With Eq. (41), it is now possible to calculate
The method presented here differs from that shown in Appendix A of Ref. 8 mainly in two ways. First, the new derivation includes every term in the summation seen in Eq. (43); in Ref. 8, this summation was truncated at $N = 2$. Second, the specification of $W$ to be a converging spherical wavefront was used. These two differences, along with approximations that $\phi$ is small and varies more quickly than the nominal quantities, $W$ and $A$, allows for a field estimate that gives rise to a complete error estimate upon using the perturbation model that includes contributions from all $N$. Although the method in Ref. 8 included higher-order corrections in $\overline {\Phi _2}$ that accounted for larger $\phi$, these terms were ultimately discarded in the development of a simple rule of thumb.
B. MSF-independent rays derivation in three spatial dimensions
Equation (4) can be solved with the method of characteristics, which leads to solutions given in the parametrization of Eq. (6). When making this change of variables, it is convenient to use the transpose of the Jacobian matrix
To begin, we show that Eq. (7) is the solution to the Eikonal equation in Eq. (4). By using Eq. (45), and with some simplification, we find that
For $N=1$, Eq. (5) reduces to
By parameterizing in terms of $\boldsymbol {\xi }$ and using Eq. (47), the left-hand side of Eq. (49) simply becomes $\partial _s \overline {\Phi _1}$. For the right-hand side, the parametrization leads toIt is useful to restate the remaining equations for $N\ge 2$. Once again, the left-hand side of Eq. (5) can be simplified to $\partial _s \overline {\Phi _N}$. That is,
Equation (55) is a simplified recursive differential equation for $\overline {\Phi _N}$. At this point, we assume the following form of a converging spherical wavefront for $W$:
Having obtained $\overline {\Phi _2}$, it is now possible to consider the equation for $\overline {\Phi _3}$:
The processes between Eqs. (61) and (63) can be iterated to give [using an approximation akin to that leading to Eq. (41) in Appendix A]
C. Discussion of the high NA regime
In this section, we discuss the NRMSE expressions given by Eqs. (21) and (22) for systems with high NA. In particular, we examine how going into the high NA regime complicates the simple rules of thumb seen in Sec. 4. The discussion in this section highlights a non-trivial aspect of the generalization to three dimensions; the rule-of-thumb expressions in Eqs. (21) and (22). We restate here that the following results may be inappropriate for a rigorous treatment of high NA systems, where polarization effects can no longer be ignored for field calculations.
As observed earlier, the main distinction between the low/moderate NA and high NA error estimates is comprised of the appearance of the obliquity factor $\chi$ and the differential operator $\hat {\mathcal {W}}$ (in place of $\nabla _\perp ^{2}$) for the latter. For convenience, we re-state the approximate NRMSE formulas, taken from Eqs. (22) and (23), as
To make a connection with the rules of thumb found in Ref. 8 regarding the high NA regime, it is prudent to ask which part of $\epsilon _{\textrm {h}}/\varphi$ is responsible for most the behavior of $\mathcal {Q}$ as the numerical aperture of the system is increased. From Eq. (69), it is clear that the two possible sources are the factor of $\chi ^{-6}$ (in the angled brackets of the numerator) and the differential operator $\hat {\mathcal {W}}$. For MSF structures $\phi$ with more than a few cycles across the aperture, we can further approximate $\epsilon _{\textrm {h}}/\varphi$ by writing the average of a product [$\chi ^{-6}$ and $(\nabla _\perp ^{2} \phi )^{2}$] as a product of averages. This approximation is justified because $\chi ^{-6}$ varies much more slowly across the aperture, even for large numerical apertures, when compared with $\phi$. Furthermore, as can be seen from Eq. (13), the differential operator $\hat {\mathcal {W}}$ can be approximated by $\nabla _\perp ^{2}$ so long as the dominant variation of $\phi$ is in $r$ rather than $\theta$. However, if the variation in $\theta$ dominates, then $\hat {\mathcal {W}} \approx \chi ^{2} \nabla _\perp ^{2}$. The NRMSE of the perturbation model, for $\phi$ that vary dominantly in $r$ or $\theta$, is given by
As a final comment on the comparison of the expressions in Eq. (69), it should be noted that, although Fig. 4 illustrates an intriguing geometrical dependence of $\mathcal {Q}$, its actual value is very close to unity for systems with a moderately large numerical aperature. For an imaging system with a numerical aperture of 0.6, for example, it can be seen that $\mathcal {Q} \lesssim 1.2$. Therefore, for the purposes of obtaining a rule-of-thumb error estimate for using the perturbation model in systems with moderate numerical apertures, it may be sufficient to use $\epsilon _{\textrm {lm}}/\varphi$ in Eq. (69).
Funding
National Science Foundation (1338877); Excellence Initiative of Aix-Marseille Université-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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