This paper introduces the path forward for the integration of freeform optical surfaces, particularly those related to φ-polynomial surfaces, including Zernike polynomial surfaces, with nodal aberration theory. With this formalism, the performance of an optical system throughout the field of view can be anticipated analytically accounting for figure error, mount-induced errors, and misalignment. Previously, only misalignments had been described by nodal aberration theory, with the exception of one special case for figure error. As an example of these new results, three point mounting error that results in a Zernike trefoil deformation is studied for the secondary mirror of a two mirror and three mirror telescope. It is demonstrated that for the case of trefoil deformation applied to a surface not at the stop, there is the anticipated field constant contribution to elliptical coma (also called trefoil) as well as a newly identified field dependent contribution to astigmatism: field linear, field conjugate astigmatism. The magnitude of this astigmatic contribution varies linearly with the field of view; however, it has a unique variation in orientation with field that is described mathematically by a concept that is unique to nodal aberration theory known as the field conjugate vector.
©2012 Optical Society of America
Nodal aberration theory (NAT) describes the aberration fields of optical systems when the constraint of rotational symmetry is not imposed. Until recently, the theory, discovered by Shack  and developed by Thompson , has been limited to optical imaging systems made of rotationally symmetric components, or offset aperture portions thereof, that are tilted and/or decentered. Recently, the special case of an astigmatic optical surface located at the aperture stop (or pupil) was introduced into NAT by Schmid et al.  and analyzed for the case of a primary mirror in a two mirror telescope. At the stop surface, the beam footprint is the same for all field points, so all field angles receive the same contribution from the astigmatic surface. The net astigmatic field dependence, as predicted by NAT and as validated by real ray tracing, takes on characteristic nodal features that allow the presence and magnitude/orientation of astigmatic figure error to be readily distinguished from the presence and magnitude/orientation of any misalignment of the secondary mirror.
Previously, Fuerschbach et al.  described how NAT and the full field display (FFD) could be used for the optical design of fully nonsymmetric optical systems utilizing freeform, φ-polynomial based optical surfaces. These displays plot the magnitude and orientation of a FRINGE Zernike decomposition of the wavefront optical path difference (OPD) over a grid of points throughout the field of view (FOV) on a term by term basis and provide a visual aid for observing the nodal behavior when the system symmetry is broken. Fuerschbach’s work was guided by concepts of NAT, but the design approach itself was empirical.
In this paper, we present for the first time a path based in NAT for developing an analytic theory for the aberration fields of nonsymmetric optical systems with freeform surfaces. With this theory, the zeroes (nodes) of the aberration contributions, which are distributed throughout the FOV, can be anticipated analytically and targeted directly for the correction or control of the aberrations in an optical system with freeform surfaces. We consider an optical surface defined by a conic plus a φ-polynomial (Zernike polynomial) overlay, where significantly, the freeform overlay can be placed anywhere within the optical imaging system. Under these more general conditions, the aberration contributions of the freeform surface contribute both field constant and field dependent terms to the net aberration fields of the optical system.
Unexpectedly, we find that the impact of integrating φ-polynomial freeform surfaces into NAT does not introduce new forms of field dependence; rather, the freeform parameters link directly with the terms presented for the generally multinodal field dependence of the sixth order wavefront aberrations derived for tilted and decentered rotationally symmetric surfaces, which have not been studied in detail. An important example, the impact of three point mount-induced error (trefoil) on the field dependence of astigmatism, is presented here. With this extension to NAT, it is now possible to describe the impact of alignment, fabrication figure errors, and mount-induced errors considering all the surfaces in the optical system. This path will lead to a general, unrestricted aberration theory for optical systems that involve φ-polynomial freeform surfaces.
2. Formulating Nodal Aberration Theory for freeform φ-polynomial surfaces away from the aperture stop
To analytically characterize the impact of a φ-polynomial optical surface away from the stop on the net aberration fields, first consider a classical Schmidt telescope configuration. The telescope is composed of a rotationally symmetric third order (fourth order in wavefront) aspheric corrector plate in coincidence with a mechanical aperture that is the stop of the optical system, located at the center of curvature of a spherical mirror. In such a configuration, the net aberration contribution of the aspheric corrector plate, , is described by the overall third order spherical aberration it induces, given by
Nominally, the Schmidt telescope is corrected for third order spherical aberration by the corrector plate and for third order coma and astigmatism by locating the stop at the center of curvature of the spherical mirror, leaving only field curvature as the limiting third order aberration. The case where an aspheric corrector plate located in the stop or pupil of an optical system is decentered from the optical axis was previously treated in the context of NAT by Thompson  and was more recently revisited by Wang et al. . If the aspheric plate is instead shifted axially (i.e. longitudinally along the optical axis) relative to a physical aperture stop, as shown in Fig. 1 , the beam for an off-axis field point will begin to displace across the aspheric plate. The amount of relative beam displacement,, is given by
Conceptually, the beam displacement on the corrector plate when it is shifted away from the stop can be thought of as a field dependent decenter of the aspheric corrector when it is located at the aperture stop. Therefore, the net aberration contribution of the aspheric corrector described by Eq. (1) must be modified to account for this effect. The modified aberration contribution, , taking into account the displacement parameter is then given byEq. (3), the original spherical aberration contribution from the aspheric plate generates lower order field dependent aberration components as the plate is shifted away from the stop. Note that the operation of vector multiplication, introduced in , is being used in this expansion. The aberration terms that are generated by this expansion are the conventional third order field aberration terms summarized in Table 1 , which could be anticipated since the field aberrations are the product of spherical aberration in the presence of a stop shift from the center of curvature.
Figure 2(a) -2(d) demonstrates the generation of astigmatism and coma for an example F/1.4 Schmidt telescope analyzed using a FFD over a ± 4° FOV. The aberration components of the displays are calculated based on real ray optical data using either a generalized Coddington close skew ray trace for astigmatism  or a FRINGE Zernike polynomial fit to the wavefront OPD data in the exit pupil for coma and any higher order aberration terms. In Fig. 2(e), the magnitude of the generated coma and astigmatism is evaluated at two specific field points for several longitudinal positions of the fourth order aspheric corrector plate. From the figure it can be seen that as the plate moves longitudinally away from the aperture stop along the optical axis, third order field linear coma is generated linearly with the distance from the aperture stop. In addition, third order field quadratic astigmatism is generated quadratically with distance from the aperture stop, matching the predictions described in Table 1. These observed dependencies parallel observations made by Burch  when he introduced his “see-saw diagram” concept and by Rakich  when he used the “see-saw diagram” to simplify the third order analysis of optical systems.
What has been recognized for the first time in the context of NAT is that this method for generating the aberration terms displayed in Eq. (3) is not restricted to rotationally symmetric corrector plates and it can be applied, with interpretation, to the general class of φ-polynomial surfaces. This approach is a pathway for melding freeform optical surfaces into NAT. More significantly, the outcome is that freeform surfaces in the φ-polynomial family fit directly into the existing discoveries for the field dependent nodal properties of the characteristic aberrations in the traditional wave aberration expansion through sixth order that are developed in [2, 10–12].
3. The astigmatic aberration field induced by three point mount-induced trefoil
In optical testing, φ-polynomials, e.g. Zernike polynomials, are often used to fit deformations to optical surfaces. The deformation of particular interest is the self weight deflection of an optic located away from the aperture stop being held at three points, a kinematically stable condition. An error of this nature is usually measured interferometrically by measuring the optic in its on-axis, null configuration while in its in-use mounting configuration; or, the error can be simulated by the use of finite element methods. In either the measured or simulated case, the deformation is quantified based on the values of its FRINGE Zernike coefficients, a commonly used φ-polynomial set . A discussion of the FRINGE Zernike set is found in Gray et al. . The predominant surface error that arises with this mount configuration is trefoil, in optical testing terminology, (FRINGE polynomial terms and) displayed in Fig. 3 and given by
Three point mount-induced error can be introduced in the vector multiplication environment of NAT with the following observation, which is the basis for nodal aberration theory,Eq. (6), which defines a new orientation, , displayed in Fig. 3 and given by
In the case of the three point deformation described above, the aberration contribution is field constant when the mount-induced deformed surface is located at the aperture stop and develops a field dependent contribution as the surface is shifted longitudinally away from the aperture stop. In Section 2, when describing the aspheric corrector plate of the Schmidt telescope, the field constant aberration that results is third order spherical aberration. By analogy, if a surface placed at the stop is deformed by a three point mount-induced error that causes predominately a FRINGE Zernike trefoil deformation, it will introduce a field constant aberration. From the vector pupil dependence in Eq. (7), it can be deduced that the trefoil deformation will induce field constant, elliptical coma that is predicted by NAT (see Eq. (19) of ). Based on this observation, it can be added to the total aberration field as3].
If a surface with three point mount-induced trefoil error is now placed away from the stop as it would be for a typical telescope secondary mirror, the beam footprint for an off-axis field angle will begin to displace across the surface resulting in the emergence of a number of field dependent terms. Using Eq. (9) and replacing with leads to a specific set of additive terms for the wavefront expansion when a surface with a mount-induced trefoil error is located away from the stop,2], is used,Eq. (12), Eq. (11) takes on the formEq. (14), three additional field dependent aberration terms are generated in addition to the anticipated field constant elliptical coma (trefoil) term. The third and fourth terms are distortion and piston that do not affect the image quality but affect the mapping and phase. Here we are focusing on the image quality; therefore, these terms will not be directly addressed.Equation (15) is a form of field linear astigmatism that was first seen in the derivation for the nodal structure of field quartic fifth order (sixth order in wavefront) astigmatism by Thompson . This linear astigmatism term has not previously been isolated as an observable field dependence and it represents the first time any aberration with conjugate field dependence has been linked to an observable quantity. The magnitude and, more significantly, the orientation of the astigmatic line images are illustrated in Fig. 4 . This form of astigmatic field dependence was reported in the literature by Stacy , but its analytical origin has remained undiscovered until now. The fact that a trefoil mount error generates an aberration besides that of elliptical coma is non-intuitive and represents a paradigm shift in the understanding of the aberration behavior of Zernike based, freeform optical surfaces. In fact, the discovery of a direct link between the aberrational influence of freeform surfaces and NAT is quite unexpected. It opens a path to developing an analytic understanding of the aberrations of freeform surfaces in optical systems directly within the context of the traditional aberration theory of Seidel, and those that followed, including H.H. Hopkins.
4. Astigmatic field dependence of a reflective telescope in the presence of a three point mount-induced surface deformation on the secondary mirror
Depending on the telescope optical configuration, the third order aberrations, i.e. spherical aberration, coma, and astigmatism, may or may not be corrected. For the case of a two mirror telescope, the system is corrected for third order spherical aberration and may be corrected for third order coma depending on the conic distribution of the mirrors. Whether or not coma is corrected, third order astigmatism remains uncorrected. If a third mirror is added, the telescope system may also be corrected for third order astigmatism. In either the two or three mirror case, when the secondary mirror is deformed by a three point mount, it will generate a field dependent astigmatic contribution, assuming the secondary mirror is not the stop surface. Under these conditions, the astigmatic response of the telescope is of interest because it reveals information into the as-built state of the telescope. In the case described above, the astigmatic response, , of the telescope takes the nodal form
To emphasize, Eq. (16) presents the magnitude and orientation of the astigmatic FRINGE Zernike coefficients (Z5/6) that would be measured if an interferogram was collected at the field point in the FOV of the perturbed telescope. The perturbation, in this case, is a three point kinematic mount deformation on the secondary mirror, characterized by , and is directly related to the measured values of the FRINGE Zernike trefoil (Z10/11) following Eq. (10).
To exploit the strength of NAT for developing insight into the relationships between alignment, fabrication, uncorrected aberration fields, and now mount-induced errors, the next step is to understand the nodal response of the astigmatism to these deviations from a nominal design depending on whether the system is corrected for third order astigmatism.
4.1 Astigmatic reflective telescope configuration () in the presence of a three point mount-induced surface deformation on the secondary mirror
In order to determine the possible nodal geometry for the case where residual third order astigmatism exists, the term inside the brackets of Eq. (16) is set equal to zero, as represented in Eq. (17),Eq. (17) is to establish a path for arrangingandin a form that can be solved, ideally using previously developed techniques. This step is accomplished by multiplying both sides of Eq. (17) by unity in the form of2] has been applied. Since Eq. (18) is a unit, scalar formulation, it does not affect the magnitude or orientation of either vector in Eq. (17). Multiplying the identity in Eq. (18) through Eq. (17) yieldsEq. (18), Eq. (19) takes the formEq. (20) exhibiting equilateral trinodal behavior with a fourth zero located on-axis at. The method for finding the three equilateral node locations is described in Appendix A where the solutions are described in terms of a reduced field coordinate, , and given by11, 12] for characterizing the cubic nodal behavior of elliptical coma and fifth order astigmatism. In this case, the vectors are proportional to, which is directly computed from a measurement or simulation of the mount-induced trefoil deformation on the secondary mirror, as visualized in Fig. 5(b) . For the special case of an astigmatic telescope in its nominal state, other than a mount-induced deformation on the secondary mirror, the four field points at which astigmatism is found to be zero are illustrated in Fig. 5(a).
4.2 Anastigmatic reflective telescope configuration () in the presence of a three point mount-induced surface deformation on the secondary mirror
For the case where the telescope configuration is corrected for third order astigmatism, the first term inside the brackets of Eq. (16) is set to zero yieldingEq. (22) it can be seen that the only astigmatic contribution is now from the mount-induced perturbation on the secondary mirror. In this case, the nodal solution is trivial where if the term inside the brackets of Eq. (22) is set to zero, the only solution is located on-axis at.
For both the astigmatic and anastigmatic cases presented above, the astigmatism takes on a unique distribution throughout the FOV when there is a mount-induced error on the secondary mirror. These unique distributions are significant because by measuring only the FRINGE Zernike pair (Z5/6) and reconstructing the nodal geometry from these measurements, it can be determined whether the as-built telescope is dominated by mount error versus other errors like alignment or residual figure error.
5. Validation of the nodal properties of a reflective telescope with three point mount-induced figure error on the secondary mirror
5.1 Astigmatic reflective telescope configuration () in the presence of a three point mount-induced surface deformation on the secondary mirror
As a validation of the predicted nodal behavior summarized in Fig. 5(a) for the case of a two mirror telescope with a mount-induced perturbation on the secondary mirror, an F/8, 300 mm Ritchey-Chrétien telescope, displayed in Fig. 6(a) , has been simulated in commercially available lens design software, in this case, CODE V®. The aberration performance throughout the FOV in terms of a total measure of image quality, the RMS wavefront error (RMS WFE), is displayed in Fig. 6(b). The RMS WFE increases as a function of FOV because of the uncorrected field quadratic astigmatism.
When it comes to assembling and aligning an optical system of this type, it is becoming increasingly common to measure the system interferometrically and use information that is available about significant characteristic aberrations through a polynomial fit to the wavefront OPD. Figure 7 displays separately the FRINGE Zernike astigmatism (Z5/6) and FRINGE Zernike trefoil (Z10/11) that would be measured at selected, discrete points in the FOV. As can be seen from Fig. 7(a), the system suffers from third order astigmatism. The higher order aberrations, like elliptical coma, are near zero, which is expected for a system with a modest f/number and FOV. When a 0.5λ, 0° orientation, trefoil mount error is added to the secondary mirror, the aberration displays are modified as shown in Fig. 7(b). The astigmatic contribution has developed a quadranodal behavior and there is now a field constant contribution to the elliptical coma. The astigmatic behavior matches the general case shown in Fig. 5(a) where the orientation angle,, has been set to zero. A quantitative evaluation of the zeroes in the display for astigmatism from Fig. 7(b) confirms the predictions made by NAT described in Section 4.1. The displays are based on real ray data and the zero locations for the astigmatic contribution are independent of NAT so they are an excellent validation of the theoretical developments presented in this paper.
5.2 Anastigmatic reflective telescope configuration () in the presence of a three point mount-induced surface deformation on the secondary mirror
In the case of an anastigmatic telescope with a mount-induced perturbation on the secondary mirror, the nodal behavior is simplified as discussed in Section 4.2 where the node is on-axis at . As a validation for this prediction, a relevant three mirror anastigmat geometry based on the James Webb Space Telescope (JWST)  has been simulated and analyzed for a trefoil perturbation on the secondary mirror. The optical system operates at F/20 with a 6.6 m entrance pupil diameter and is shown in Fig. 8(a) . In order to yield an accessible focal plane, the FOV is biased so that an off-axis portion of the tertiary mirror is utilized. The RMS WFE of the system is displayed in Fig. 8(b) over a ± 0.2° FOV and the portion of the field that is utilized for the biased system is bounded by the red rectangle. In the center of the on-axis FOV, the RMS WFE is well behaved because the third order aberrations are well corrected. The performance does increase at the edge of the FOV due to higher order aberration contributions.
Following a similar approach to that outlined in Section 5.1, the individual aberration contributions that make up the total RMS WFE can be evaluated over the FOV. Figure 9 displays separately the FRINGE Zernike astigmatism (Z5/6) and FRINGE Zernike trefoil (Z10/11) that would be measured at selected, discrete points in the FOV for the JWST-like system. As can be seen from Fig. 9(a), the system is anastigmatic and the elliptical coma is near zero throughout the FOV. If a 0.5λ, 0° orientation, trefoil error is added to the secondary mirror, the aberration displays are modified as shown in Fig. 9(b). The astigmatic contribution has developed field linear, field conjugate astigmatism with a single node centered on-axis. The node lies outside the usable FOV for the field biased telescope. As with the previous case, there is also a field constant contribution to the elliptical coma. Both contributions match the theoretical developments presented in this paper.
6. Extending Nodal Aberration Theory to include decentered freeform φ-polynomial surfaces away from the aperture stop
In the case of the JWST-like geometry in Fig. 8(a), the tertiary mirror is an off-axis section of a larger rotationally symmetric surface. If a trefoil deformation is to be applied to the tertiary mirror, the error must be centered with respect to the off-axis portion of the surface, not the larger parent surface. Therefore, an additional parameter must be defined that accounts for a shift of the nonsymmetric deformation from the reference axis that is defined to be the optical axis ray (OAR) . Following the method used in  for the decenter of an aspheric cap of an optical surface, the nonsymmetric deformation is treated as a zero-power thin plate. When the nonsymmetric deformation is shifted, there is a freeform sigma vector that can be expressed as2], asEq. (16) with the effective field heightand generalizing the perturbation to be on the jth optical surface, takes the formEq. (25) is best found numerically and may be quadranodal but degenerates to special cases where only three or two nodes exist. For the anastigmatic case where the third order astigmatism is zero, Eq. (25) simplifies to
As a validation of these predictions, the JWST-like system evaluated in Section 5.2 is reevaluated where the 0.5λ, 0° orientation, trefoil error is now added to the off-axis section of the tertiary mirror. In this case, the aberration displays are modified as shown in Fig. 10 . The astigmatic contribution has developed field linear, field conjugate astigmatism with a single node now centered off-axis. The node has moved off-axis because the trefoil deformation is no longer located along the OAR and now lies in the center of the field biased FOV. It is also interesting to note that for this configuration, the induced astigmatic contribution is larger than the induced field constant contribution to the elliptical coma. At the tertiary mirror, the beam footprints for each field are widely spread about the optical surface; as a result, the field dependent contribution has a larger net effect than the field constant contribution.
We have shown in this paper a method for integrating freeform optical surfaces, particularly those related to φ-polynomial surfaces, including Zernike polynomial surfaces, with NAT. When a freeform optical surface is placed at a surface away from the aperture stop, there is the anticipated field constant contribution as well as a field dependent contribution to the net aberration fields. This behavior has been studied for both the case of a two mirror and three mirror telescope with a three point mount-induced trefoil deformation on the secondary or tertiary mirror. The deformation induces a new type of astigmatic field dependence, field conjugate, field linear astigmatism, which in the presence of conventional third order field quadratic astigmatism yields quadranodal behavior. With this outcome as a basis, an aberration theory that fully supports developing optical design strategies for fully nonsymmetric imaging optical systems with freeform surfaces is under development.
The new development in NAT is also being extended to include a methodology for separating the effects of mount-induced error, misalignment induced astigmatism, and astigmatic figure error. This work is particularly relevant to the current generation of European Southern Observatory (ESO) ground based telescopes where the alignment technology on-site is based on a thin substrate active primary mirror combined with a secondary mirror that can be adjusted in tilt and decenter around external pivot points that includes the center of curvature (to maintain boresight) and the coma free pivot point (for final alignment).
8. Appendix A
This appendix provides a method for solving the three equilateral node locations in the special case where the secondary mirror of an otherwise ideal two mirror telescope has been deformed by a three point mounting error characterized by . In order to find the nodal response, the term inside the bracket of Eq. (20) is rearranged, and set to zero, taking the formEq. (27) is substituted with a new reduced field vector written in complex notation asEq. (28) has the same orientation, θ, as but with a magnitude equal to the cube root of . In this new form, Eq. (27) takes the form11] for solving the nodes of a cubic vector equation, that has been applied to the case of elliptical coma and fifth order astigmatism [11, 12] in tilted and decentered systems, the node locations for a trinodal form are governed by two vectors, and, which, in this case are equal, and given byFig. 11(a) where the solutions are plotted in the reduced field coordinate. In Fig. 11(b), the four nodal solutions have been re-mapped into the conventional field coordinate.
We thank the Frank J. Horton Research Fellowship, the II-VI Foundation, the National Science Foundation (EECS-1002179), and the NYSTAR Foundation (C050070) for supporting this research. We also thank Synopsys Inc. for the student license of CODE V®.
References and links
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