# Extending Nodal Aberration Theory to include mount-induced aberrations with application to freeform surfaces Open Access

## Abstract

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.

## 1. Introduction

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, $WCorrector,Stop$, is described by the overall third order spherical aberration it induces, given by

$WCorrector,Stop=W040(ASPH)(ρ→·ρ→)2,$
where $W040(ASPH)$ denotes the spherical aberration wave aberration contribution from the aspheric corrector plate and$ρ→$ is a normalized two-dimensional pupil vector that denotes a location in the pupil of the Schmidt telescope.

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,$Δh→$, is given by

$Δh→≡(y¯y)H→=(u¯ty)H→,$
where$y$is the paraxial marginal ray height on the aspheric plate, $y¯$is the paraxial chief ray height on the aspheric plate, $u¯$is the paraxial chief ray angle, $t$is the distance between the aspheric corrector plate and the mechanical aperture that is the optical system stop, and$H→$is the normalized two-dimensional field vector that locates the field point of interest in the image plane (i.e. 0≤|$H→$|≤1). Fig. 1 When the aspheric corrector plate of a Schmidt telescope is displaced longitudinally from the aperture stop along the optical axis, the beam for an off-axis field point will displace along the corrector plate. The amount of relative beam displacement defined by Eq. (2) depends on the paraxial quantities for the marginal ray height,$y$, the chief ray height, $y¯$, the chief ray angle, $u¯$, and the distance between the stop and the corrector plate, $t$.

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, $WCorrector,Not Stop$, taking into account the displacement parameter $Δh→$ is then given by

$WCorrector,Not Stop=W040(ASPH)[(ρ→+Δh→)·(ρ→+Δh→)]2 =W040(ASPH)[(ρ→ ·ρ→)2+4(Δh→ ·ρ→)(ρ→ ·ρ→)+4(Δh→ ·Δh→)(ρ→ ·ρ→)+2(Δh→2 ·ρ→2)+4(Δh→ ·Δh→)(Δh→ ·ρ→)+(Δh→ ·Δh→)2].$
As can be seen from Eq. (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. Table 1. Field aberration terms that are generated from the longitudinal shift of an aspheric plate from the stop surface in a Schmidt telescope.

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. Fig. 2 Generation of coma and astigmatism as the aspheric corrector plate in a Schmidt telescope is moved longitudinally (along the optical axis) from the physical aperture stop located at the center of curvature of the spherical primary mirror for various positions (a-d). For each field point in the FFD, the plot symbol conveys the magnitude and orientation of the aberration. (e) Plots of the magnitude of coma and astigmatism generated as the aspheric plate is moved longitudinally for two field points, (0°, 2°) (blue square) and (0°, 4°) (red triangle).

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, 1012].

## 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 $Z10$and$Z11$) displayed in Fig. 3 and given by

$(Z10Z11)=(z10 ρ3cos(3ϕ)z11 ρ3sin(3ϕ)),$
where $z10$and $z11$ represent the FRINGE Zernike coefficient values for the trefoil term, $ρ$ is the normalized radial coordinate, and $ϕ$ represents the azimuthal angle in the exit pupil (for an on-axis test point). In optical testing, the FRINGE Zernike set is described in a right-handed coordinate system with $ϕ$ measured counter-clockwise from the$x^−$axis. The magnitude,$|z10/11|MNTERR$, and orientation, $ξMNTERR10/11Test$, of the trefoil mount-induced error (MNTERR) is then calculated from the coefficients by
$|z10/11|MNTERR =z102+z112$
$ξMNTERR10/11Test=13tan−1(z11z10),$
where the superscript Test denotes the optical testing coordinate system. Fig. 3 Surface map describing the characteristic error induced in some cases by a kinematic three point mount on an optical surface as measured interferometrically, over the full aperture of the part, on-axis. The error is quantified by its magnitude $|z10/11|MNTERR$ and its orientation $ξMNTERR10/11$that is measured clockwise with respect to the $y^−$axis. The coordinate system assumes that the part is evaluated looking from the interferometer towards the part. P and V denote where the surface error is a peak rather than a valley.

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,

$if ρ→=ρ(sin(ϕ)cos(ϕ)), then ρ→3=ρ3(sin(3ϕ)cos(3ϕ)),$
where consistent with commercial optical design raytrace programs, a right-handed coordinate system is employed with $ϕ$ measured clockwise from the $y^−$axis. To implement a coordinate system for the mount-induced deformation that is consistent with its generated aberration field within the context of the real ray based environment of NAT, there is a reversal of the terms in Eq. (6), which defines a new orientation, $ξMNTERR10/11$, displayed in Fig. 3 and given by

$ξMNTERR10/11=13tan−1(z10z11).$

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 as

$W333,Stop=−14(C→MNTERR3333·ρ→3),$
where $C→MNTERR3333$is a two-dimensional vector that describes the magnitude and orientation of the mount-induced error, which is related to the overall Zernike trefoil by
$C→MNTERR3333≡4|z10/11|MNTERRexp(i3ξMNTERR10/11).$
The method of introducing new aberration contributions into NAT parallels the successful introduction of astigmatic figure error, $B→FIGERR2222$, for a mirror surface at an aperture stop in .

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 $ρ→$ with $ρ→+Δh→$ 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,

$W333,Not Stop=−14[C→MNTERR3333· (ρ→+Δh→)3] =−14[C→MNTERR3333·ρ→3+3C→MNTERR3333·Δh→ρ→2+3C→MNTERR3333·Δh→2ρ→+C→MNTERR3333·Δh→3].$
To map the impact of these additive terms on the overall field dependent wave aberration expansion of an optical system, the pupil dependence needs to be converted into existing aberration types. To this end, an additional vector operation, introduced in , is used,
$A→·B→C→=A→B→*·C→,$
where $B→*$is a conjugate vector with the standard properties of a conjugate variable in the mathematics of complex numbers
$B→*=|B→|exp(−iβ)=−Bxx^+Byy^.$
By applying the vector identity of Eq. (12), Eq. (11) takes on the form
$W333,Not Stop=−14[C→MNTERR3333·ρ→3+3C→MNTERR3333Δh→*·ρ→2+3C→MNTERR3333Δh→*2·ρ→+C→MNTERR3333·Δh→3].$
As can be seen from Eq. (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.

The second term is seen to be an astigmatic term based on the $ρ→2$aperture dependence. When Eq. (1) is used to replace$Δh→$ in the astigmatic term of Eq. (14), it becomes

$−34C→MNTERR3333Δh→*·ρ→2=−34(y¯jyj)C→MNTERR333,j3H→*·ρ→2,$
where the index j has been introduced to generalize the result to include a multi-element optical system where the mount-induced deformation is on the jth optical surface. 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. Fig. 4 The characteristic field dependence of field linear, field conjugate astigmatism that is generated by a mount-induced trefoil error on an optical surface placed away from the aperture stop of an optical system.

## 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, $WAST$, of the telescope takes the nodal form

$WAST=[12W222H→2−34(y¯SMySM)C→MNTERR333,SM3H→*] ·ρ→2,$
where the subscript SM signifies that the mount-induced trefoil deformation is on the secondary mirror surface and depending on whether the telescope is anastigmatic, $W222$may or may not be equal to zero.

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 $H→$ 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 $C→MNTERR333,SM3$, 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 ($W222≠0$) 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),

$12W222H→2−34(y¯SMySM) C→MNTERR333,SM3H→*= 0→.$
The first step in solving the vector formulation represented in Eq. (17) is to establish a path for arranging$H→2$and$H→*$in 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 of
$1=H→H→*|H→|2=H→*H→|H→|2,$
where a vector multiplication relation presented in  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) yields
$1|H→|2[12W222H→3−34(y¯SMySM) C→MNTERR333,SM3H→*H→]H→*=0→.$
Again, making use of the identity in Eq. (18), Eq. (19) takes the form
$[12W222H→3|H→|2−34(y¯SMySM) C→MNTERR333,SM3]H→*=0→.$
It can now be seen based on the powers of $H→$that there is a quadranodal astigmatic response in the FOV to a mount-induced trefoil deformation on the secondary mirror with the term in the brackets of Eq. (20) exhibiting equilateral trinodal behavior with a fourth zero located on-axis at$H→=0→$. 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, $Π→$, and given by
$2(x¯→MNTERR222) , −(x¯→MNTERR222)+i3(x˜→MNTERR222) , −(x¯→MNTERR222)−i3(x˜→MNTERR222),$
where the definition of the governing vectors $x¯→MNTERR222$and$x˜→MNTERR222$ is described in Appendix A. The solution vectors follow a notation introduced in [11, 12] for characterizing the cubic nodal behavior of elliptical coma and fifth order astigmatism. In this case, the vectors are proportional to$C→MNTERR333,SM$, 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). Fig. 5 (a) In the presence of conventional third order field quadratic astigmatism and Zernike trefoil at a surface away from the stop, e.g., a two mirror telescope with three point mount-induced error on the secondary mirror, the astigmatic field dependence displays four nodes, i.e., quadranodal behavior. The nodal behavior is displayed in a reduced field coordinate, $Π→$. The node located by $2(x¯→MNTERR222)$ has an orientation angle of $ξMNTERR10/11$ and a magnitude that is proportional to $|C→MNTERR333,SM|$. The two related nodes on the equilateral triangle are then advanced by 120° and 240° for this special case. (b) A measurement or simulation of the mount-induced error on the secondary mirror yields the magnitude and orientation of$C→MNTERR333,SM$.

#### 4.2 Anastigmatic reflective telescope configuration ($W222=0$) 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 yielding

$WAST=[−34(y¯SMySM)C→MNTERR333,SM3H→*] ·ρ→2.$
In Eq. (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$H→=0→$.

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 ($W222≠0$) 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. Fig. 6 (a) Layout for a F/8, 300 mm Ritchey-Chrétien telescope and (b) a Full Field Display (FFD) of the RMS wavefront error (RMS WFE) of the optical system at 0.633µm over a ± 0.2° FOV. Each circle represents the magnitude of the RMS WFE at a particular location in the FOV.

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,$ξMNTERR10,11$, 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. Fig. 7 Displays of the magnitude and orientation of FRINGE Zernike astigmatism (Z5/6) and FRINGE Zernike trefoil, elliptical coma, (Z10/11) throughout the FOV for (a) a Ritchey-Chrétien telescope in its nominal state and (b) the telescope when 0.5λ of three point mount-induced error oriented at 0° has been added to the secondary mirror. It is important to recognize that these displays of data are full field displays that are based on a Zernike polynomial fit to real ray trace optical path difference data evaluated on a grid of points in the FOV. For each field point, the plot symbol conveys the magnitude and orientation of the Zernike coefficients pairs, Z5/6 on the left and Z10/11 on the right.

#### 5.2 Anastigmatic reflective telescope configuration ($W222=0$) 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 $H→=0→$. 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. Fig. 8 (a) Layout for a JWST-like telescope geometry and (b) a Full Field Display (FFD) of the RMS WFE of the optical system at 1.00 µm over a ± 0.2° FOV. The system utilizes a field bias (outlined in red) to create an accessible focal plane.

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. Fig. 9 Displays of the magnitude and orientation of FRINGE Zernike astigmatism (Z5/6) and FRINGE Zernike trefoil, elliptical coma, (Z10/11) throughout the FOV for (a) a JWST-like telescope in its nominal state and (b) the telescope when 0.5λ of three point mount-induced error oriented at 0° has been added to the secondary mirror.

## 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 $(σ→FF)j$ that can be expressed as

$(σ→FF)j=(δv→FF*)jy¯j,$
where $(δv→FF*)j$is the distance between the optical axis ray and the freeform departure vertex. For the case of a freeform, φ-polynomial surface, the freeform vertex corresponds to the origin of the unit circle that bounds the polynomial set. To compute the overall aberration field from the shifted freeform deformation, a new effective aberration field height$(H→FF)j$ is defined, following the notation of , as
$(H→FF)j=H→− (σ→FF)j.$
The astigmatic response of a telescope with a mount-induced trefoil deformation can now be modified to account for the new effective aberration field height. Updating Eq. (16) with the effective field height$(H→FF)j$and generalizing the perturbation to be on the jth optical surface, $WAST$ takes the form
$WAST=[12W222H→2−34(y¯jyj)C→MNTERR333,j3(H→FF*)j] ·ρ→2.$
The nodal solution for the astigmatic response represented in Eq. (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
$WAST=[−34(y¯jyj)C→MNTERR333,j3(H→FF*)j] ·ρ→2,$
where it can be seen there is a single node located at$H→= (σ→FF)j$.

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. Fig. 10 Displays of the magnitude and orientation of FRINGE Zernike astigmatism (Z5/6) and FRINGE Zernike trefoil, elliptical coma, (Z10/11) throughout the FOV for a JWST-like telescope with 0.5λ of three point mount-induced error oriented at 0° on the off-axis tertiary mirror.

## 7. Conclusion

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 $C→MNTERR333,SM3$. In order to find the nodal response, the term inside the bracket of Eq. (20) is rearranged, and set to zero, taking the form

$H→3|H→|2−32W222(y¯SMySM) C→MNTERR333,SM3=0→.$
The first term of Eq. (27) is substituted with a new reduced field vector $Π→3$written in complex notation as
$Π→3≡H→3|H→|2=|H→|3ei3θ|H→|2=|H→|ei3θ=(|H→|13)3ei3θ,$
where the new vector represented in Eq. (28) has the same orientation, θ, as $H→$ but with a magnitude equal to the cube root of $|H→|$. In this new form, Eq. (27) takes the form
$Π→3−32W222(y¯SMySM) C→MNTERR333,SM3=0→.$
Following the method proposed by Thompson and detailed in  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, $x¯→$and$x˜→$, which, in this case are equal, and given by
$x¯→MNTERR222=x˜→MNTERR222=[32W222(y¯SMySM)]13(C→MNTERR333,SM3)13.$
In terms of these cubic equation solution vectors $x¯→MNTERR222$and$x˜→MNTERR222$, which are best kept independent for later generalizations, the three node locations referenced to the intersection of the OAR with the image plane are, for this case, equidistant from the on-axis node with $WRC−TEL,MNTERR = 0$at
$2(x¯→MNTERR222) , −(x¯→MNTERR222)+i3(x˜→MNTERR222) , −(x¯→MNTERR222)−i3(x˜→MNTERR222).$
The four field points at which astigmatism is found to be zero are illustrated in Fig. 11(a) where the solutions are plotted in the $Π→$reduced field coordinate. In Fig. 11(b), the four nodal solutions have been re-mapped into the conventional $H→$field coordinate. Fig. 11 In the presence of conventional third order field quadratic astigmatism and Zernike trefoil at a surface away from the stop, e.g., a two mirror telescope with a three point mount-induced error on the secondary mirror, the astigmatic field dependence displays four nodes, i.e., quadranodal behavior. (a) The nodal behavior is displayed in a reduced field coordinate,$Π→$, where the node located by $2(x¯→MNTERR222)$ has an orientation angle of $ξMNTERR10/11$ and a magnitude that is proportional to $|C→MNTERR333,SM|$. The two related nodes on the equilateral triangle are then advanced by 120° and 240° for this special case. (b) When the nodal solutions are re-mapped to the conventional field coordinate,$H→$, the node located by $2(x¯→MNTERR222)$ has an orientation angle of $ξMNTERR10/11$ and a magnitude that is proportional to $|C→MNTERR333,SM3|$. The two related nodes on the equilateral triangle are then advanced by 120° and 240°.

## Acknowledgments

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®.

1. R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).

2. K. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22(7), 1389–1401 (2005). [CrossRef]   [PubMed]

3. T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express 18(16), 17433–17447 (2010). [CrossRef]   [PubMed]

4. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing phi-polynomial surfaces,” Opt. Express 19(22), 21919–21928 (2011). [CrossRef]   [PubMed]

5. K. P. Thompson, “Aberration fields in unobscured mirror systems,” J. Opt. Soc. Am. 103, 159–165 (1980).

6. J. Wang, B. Guo, Q. Sun, and Z. Lu, “Third-order aberration fields of pupil decentered optical systems,” Opt. Express 20(11), 11652–11658 (2012). [CrossRef]   [PubMed]

7. K. P. Thompson, “Reinterpreting Coddington: correcting 150 years of confusion,” in Robert Shannon and Roland Shack, Legends in Applied Optics, J. E. Harvey, and R. B. Hooker, eds. (SPIE Press, 2005), 41–49.

8. C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc. 103, 159–165 (1942).

9. A. Rakich, “Calculation of third-order misalignment aberrations with the optical plate diagram,” Proc. SPIE 7652, 765230, 765230-11 (2010). [CrossRef]

10. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26(5), 1090–1100 (2009). [CrossRef]   [PubMed]

11. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the comatic aberrations,” J. Opt. Soc. Am. A 27(6), 1490–1504 (2010). [CrossRef]   [PubMed]

12. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the astigmatic aberrations,” J. Opt. Soc. Am. A 28(5), 821–836 (2011). [CrossRef]   [PubMed]

13. Synopsys Inc, “Zernike Polynomials,” in CODE V Reference Manual (2012), Volume IV, Appendix C.

14. R. W. Gray, C. Dunn, K. P. Thompson, and J. P. Rolland, “An analytic expression for the field dependence of Zernike polynomials in rotationally symmetric optical systems,” Opt. Express 20(15), 16436–16449 (2012). [CrossRef]

15. J. E. Stacy and S. A. Macenka, “Optimization of an unobscured optical system using vector aberration theory,” Proc. SPIE 679, 21–24 (1986).

16. J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004). [CrossRef]

17. K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A 26(6), 1503–1517 (2009). [CrossRef]   [PubMed]

### References

• View by:

1. R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).
2. K. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22(7), 1389–1401 (2005).
[Crossref] [PubMed]
3. T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express 18(16), 17433–17447 (2010).
[Crossref] [PubMed]
4. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing phi-polynomial surfaces,” Opt. Express 19(22), 21919–21928 (2011).
[Crossref] [PubMed]
5. K. P. Thompson, “Aberration fields in unobscured mirror systems,” J. Opt. Soc. Am. 103, 159–165 (1980).
6. J. Wang, B. Guo, Q. Sun, and Z. Lu, “Third-order aberration fields of pupil decentered optical systems,” Opt. Express 20(11), 11652–11658 (2012).
[Crossref] [PubMed]
7. K. P. Thompson, “Reinterpreting Coddington: correcting 150 years of confusion,” in Robert Shannon and Roland Shack, Legends in Applied Optics, J. E. Harvey, and R. B. Hooker, eds. (SPIE Press, 2005), 41–49.
8. C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc. 103, 159–165 (1942).
9. A. Rakich, “Calculation of third-order misalignment aberrations with the optical plate diagram,” Proc. SPIE 7652, 765230, 765230-11 (2010).
[Crossref]
10. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26(5), 1090–1100 (2009).
[Crossref] [PubMed]
11. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the comatic aberrations,” J. Opt. Soc. Am. A 27(6), 1490–1504 (2010).
[Crossref] [PubMed]
12. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the astigmatic aberrations,” J. Opt. Soc. Am. A 28(5), 821–836 (2011).
[Crossref] [PubMed]
13. Synopsys Inc, “Zernike Polynomials,” in CODE V Reference Manual (2012), Volume IV, Appendix C.
14. R. W. Gray, C. Dunn, K. P. Thompson, and J. P. Rolland, “An analytic expression for the field dependence of Zernike polynomials in rotationally symmetric optical systems,” Opt. Express 20(15), 16436–16449 (2012).
[Crossref]
15. J. E. Stacy and S. A. Macenka, “Optimization of an unobscured optical system using vector aberration theory,” Proc. SPIE 679, 21–24 (1986).
16. J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]
17. K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A 26(6), 1503–1517 (2009).
[Crossref] [PubMed]

#### 2010 (3)

A. Rakich, “Calculation of third-order misalignment aberrations with the optical plate diagram,” Proc. SPIE 7652, 765230, 765230-11 (2010).
[Crossref]

#### 2004 (1)

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### 1986 (1)

J. E. Stacy and S. A. Macenka, “Optimization of an unobscured optical system using vector aberration theory,” Proc. SPIE 679, 21–24 (1986).

#### 1980 (2)

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).

K. P. Thompson, “Aberration fields in unobscured mirror systems,” J. Opt. Soc. Am. 103, 159–165 (1980).

#### 1942 (1)

C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc. 103, 159–165 (1942).

#### Atcheson, P. D.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### Atkinson, C. B.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### Au, D.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### Bronowicki, A. J.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### Bujanda, E.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### Burch, C. R.

C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc. 103, 159–165 (1942).

#### Cohen, A.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### Davies, D.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### Lightsey, P. A.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### Lundquist, R.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### Lynch, R.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### Macenka, S. A.

J. E. Stacy and S. A. Macenka, “Optimization of an unobscured optical system using vector aberration theory,” Proc. SPIE 679, 21–24 (1986).

#### Menzel, M. T.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### Mohan, M.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### Nella, J.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### Pohner, J.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### Rakich, A.

A. Rakich, “Calculation of third-order misalignment aberrations with the optical plate diagram,” Proc. SPIE 7652, 765230, 765230-11 (2010).
[Crossref]

#### Reynolds, P.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### Rivera, H.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### Shack, R. V.

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).

#### Shuckstes, D. V.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### Simmons, D. D. F.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### Smith, R. C.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### Stacy, J. E.

J. E. Stacy and S. A. Macenka, “Optimization of an unobscured optical system using vector aberration theory,” Proc. SPIE 679, 21–24 (1986).

#### Sullivan, P. C.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### Texter, S. C.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### Waldie, D. D.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### Woods, R.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

#### J. Opt. Soc. Am. (1)

K. P. Thompson, “Aberration fields in unobscured mirror systems,” J. Opt. Soc. Am. 103, 159–165 (1980).

#### Mon. Not. R. Astron. Soc. (1)

C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc. 103, 159–165 (1942).

#### Proc. SPIE (4)

J. E. Stacy and S. A. Macenka, “Optimization of an unobscured optical system using vector aberration theory,” Proc. SPIE 679, 21–24 (1986).

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE 5487, 576–587 (2004).
[Crossref]

A. Rakich, “Calculation of third-order misalignment aberrations with the optical plate diagram,” Proc. SPIE 7652, 765230, 765230-11 (2010).
[Crossref]

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).

#### Other (2)

K. P. Thompson, “Reinterpreting Coddington: correcting 150 years of confusion,” in Robert Shannon and Roland Shack, Legends in Applied Optics, J. E. Harvey, and R. B. Hooker, eds. (SPIE Press, 2005), 41–49.

Synopsys Inc, “Zernike Polynomials,” in CODE V Reference Manual (2012), Volume IV, Appendix C.

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### Figures (11)

Fig. 1 When the aspheric corrector plate of a Schmidt telescope is displaced longitudinally from the aperture stop along the optical axis, the beam for an off-axis field point will displace along the corrector plate. The amount of relative beam displacement defined by Eq. (2) depends on the paraxial quantities for the marginal ray height,$y$, the chief ray height, $y ¯$, the chief ray angle, $u ¯$, and the distance between the stop and the corrector plate, $t$.
Fig. 2 Generation of coma and astigmatism as the aspheric corrector plate in a Schmidt telescope is moved longitudinally (along the optical axis) from the physical aperture stop located at the center of curvature of the spherical primary mirror for various positions (a-d). For each field point in the FFD, the plot symbol conveys the magnitude and orientation of the aberration. (e) Plots of the magnitude of coma and astigmatism generated as the aspheric plate is moved longitudinally for two field points, (0°, 2°) (blue square) and (0°, 4°) (red triangle).
Fig. 3 Surface map describing the characteristic error induced in some cases by a kinematic three point mount on an optical surface as measured interferometrically, over the full aperture of the part, on-axis. The error is quantified by its magnitude $| z 10/11 | MNTERR$ and its orientation $ξ MNTERR 10/11$that is measured clockwise with respect to the $y ^ −$axis. The coordinate system assumes that the part is evaluated looking from the interferometer towards the part. P and V denote where the surface error is a peak rather than a valley.
Fig. 4 The characteristic field dependence of field linear, field conjugate astigmatism that is generated by a mount-induced trefoil error on an optical surface placed away from the aperture stop of an optical system.
Fig. 5 (a) In the presence of conventional third order field quadratic astigmatism and Zernike trefoil at a surface away from the stop, e.g., a two mirror telescope with three point mount-induced error on the secondary mirror, the astigmatic field dependence displays four nodes, i.e., quadranodal behavior. The nodal behavior is displayed in a reduced field coordinate, $Π →$. The node located by $2( x ¯ → MNTERR 222 )$ has an orientation angle of $ξ MNTERR 10/11$ and a magnitude that is proportional to $| C → MNTERR 333,SM |$. The two related nodes on the equilateral triangle are then advanced by 120° and 240° for this special case. (b) A measurement or simulation of the mount-induced error on the secondary mirror yields the magnitude and orientation of$C → MNTERR 333,SM$.
Fig. 6 (a) Layout for a F/8, 300 mm Ritchey-Chrétien telescope and (b) a Full Field Display (FFD) of the RMS wavefront error (RMS WFE) of the optical system at 0.633µm over a ± 0.2° FOV. Each circle represents the magnitude of the RMS WFE at a particular location in the FOV.
Fig. 7 Displays of the magnitude and orientation of FRINGE Zernike astigmatism (Z5/6) and FRINGE Zernike trefoil, elliptical coma, (Z10/11) throughout the FOV for (a) a Ritchey-Chrétien telescope in its nominal state and (b) the telescope when 0.5λ of three point mount-induced error oriented at 0° has been added to the secondary mirror. It is important to recognize that these displays of data are full field displays that are based on a Zernike polynomial fit to real ray trace optical path difference data evaluated on a grid of points in the FOV. For each field point, the plot symbol conveys the magnitude and orientation of the Zernike coefficients pairs, Z5/6 on the left and Z10/11 on the right.
Fig. 8 (a) Layout for a JWST-like telescope geometry and (b) a Full Field Display (FFD) of the RMS WFE of the optical system at 1.00 µm over a ± 0.2° FOV. The system utilizes a field bias (outlined in red) to create an accessible focal plane.
Fig. 9 Displays of the magnitude and orientation of FRINGE Zernike astigmatism (Z5/6) and FRINGE Zernike trefoil, elliptical coma, (Z10/11) throughout the FOV for (a) a JWST-like telescope in its nominal state and (b) the telescope when 0.5λ of three point mount-induced error oriented at 0° has been added to the secondary mirror.
Fig. 10 Displays of the magnitude and orientation of FRINGE Zernike astigmatism (Z5/6) and FRINGE Zernike trefoil, elliptical coma, (Z10/11) throughout the FOV for a JWST-like telescope with 0.5λ of three point mount-induced error oriented at 0° on the off-axis tertiary mirror.
Fig. 11 In the presence of conventional third order field quadratic astigmatism and Zernike trefoil at a surface away from the stop, e.g., a two mirror telescope with a three point mount-induced error on the secondary mirror, the astigmatic field dependence displays four nodes, i.e., quadranodal behavior. (a) The nodal behavior is displayed in a reduced field coordinate,$Π →$, where the node located by $2( x ¯ → MNTERR 222 )$ has an orientation angle of $ξ MNTERR 10/11$ and a magnitude that is proportional to $| C → MNTERR 333,SM |$. The two related nodes on the equilateral triangle are then advanced by 120° and 240° for this special case. (b) When the nodal solutions are re-mapped to the conventional field coordinate,$H →$, the node located by $2( x ¯ → MNTERR 222 )$ has an orientation angle of $ξ MNTERR 10/11$ and a magnitude that is proportional to $| C → MNTERR 333,SM 3 |$. The two related nodes on the equilateral triangle are then advanced by 120° and 240°.

### Tables (1) Table 1 Field aberration terms that are generated from the longitudinal shift of an aspheric plate from the stop surface in a Schmidt telescope.

### Equations (31)

$W Corrector,Stop = W 040 ( ASPH ) ( ρ → · ρ → ) 2 ,$
$Δ h → ≡( y ¯ y ) H → =( u ¯ t y ) H → ,$
$W Corrector,Not Stop = W 040 ( ASPH ) [ ( ρ → +Δ h → )·( ρ → +Δ h → ) ] 2 = W 040 ( ASPH ) [ ( ρ → · ρ → ) 2 +4( Δ h → · ρ → )( ρ → · ρ → )+4( Δ h → ·Δ h → )( ρ → · ρ → ) +2( Δ h → 2 · ρ → 2 )+4( Δ h → ·Δ h → )( Δ h → · ρ → )+ ( Δ h → ·Δ h → ) 2 ].$
$( Z 10 Z 11 )=( z 10 ρ 3 cos( 3ϕ ) z 11 ρ 3 sin( 3ϕ ) ),$
$| z 10/11 | MNTERR = z 10 2 + z 11 2$
$ξ MNTERR 10/11 Test = 1 3 tan −1 ( z 11 z 10 ),$
$if ρ → =ρ( sin( ϕ ) cos( ϕ ) ), then ρ → 3 = ρ 3 ( sin( 3ϕ ) cos( 3ϕ ) ),$
$ξ MNTERR 10/11 = 1 3 tan −1 ( z 10 z 11 ).$
$W 333,Stop =− 1 4 ( C → MNTERR 333 3 · ρ → 3 ),$
$C → MNTERR 333 3 ≡4 | z 10/11 | MNTERR exp( i3 ξ MNTERR 10/11 ).$
$W 333,Not Stop =− 1 4 [ C → MNTERR 333 3 · ( ρ → +Δ h → ) 3 ] =− 1 4 [ C → MNTERR 333 3 · ρ → 3 +3 C → MNTERR 333 3 ·Δ h → ρ → 2 +3 C → MNTERR 333 3 ·Δ h → 2 ρ → + C → MNTERR 333 3 ·Δ h → 3 ].$
$A → · B → C → = A → B → * · C → ,$
$B → * =| B → |exp( −iβ )=− B x x ^ + B y y ^ .$
$W 333,Not Stop =− 1 4 [ C → MNTERR 333 3 · ρ → 3 +3 C → MNTERR 333 3 Δ h → * · ρ → 2 +3 C → MNTERR 333 3 Δ h → *2 · ρ → + C → MNTERR 333 3 ·Δ h → 3 ].$
$− 3 4 C → MNTERR 333 3 Δ h → * · ρ → 2 =− 3 4 ( y ¯ j y j ) C → MNTERR 333,j 3 H → * · ρ → 2 ,$
$W AST =[ 1 2 W 222 H → 2 − 3 4 ( y ¯ SM y SM ) C → MNTERR 333,SM 3 H → * ] · ρ → 2 ,$
$1 2 W 222 H → 2 − 3 4 ( y ¯ SM y SM ) C → MNTERR 333,SM 3 H → * = 0 → .$
$1= H → H → * | H → | 2 = H → * H → | H → | 2 ,$
$1 | H → | 2 [ 1 2 W 222 H → 3 − 3 4 ( y ¯ SM y SM ) C → MNTERR 333,SM 3 H → * H → ] H → * = 0 → .$
$[ 1 2 W 222 H → 3 | H → | 2 − 3 4 ( y ¯ SM y SM ) C → MNTERR 333,SM 3 ] H → * = 0 → .$
$2( x ¯ → MNTERR 222 ) , −( x ¯ → MNTERR 222 )+i 3 ( x ˜ → MNTERR 222 ) , −( x ¯ → MNTERR 222 )−i 3 ( x ˜ → MNTERR 222 ),$
$W AST =[ − 3 4 ( y ¯ SM y SM ) C → MNTERR 333,SM 3 H → * ] · ρ → 2 .$
$( σ → FF ) j = ( δ v → FF * ) j y ¯ j ,$
$( H → FF ) j = H → − ( σ → FF ) j .$
$W AST =[ 1 2 W 222 H → 2 − 3 4 ( y ¯ j y j ) C → MNTERR 333,j 3 ( H → FF * ) j ] · ρ → 2 .$
$W AST =[ − 3 4 ( y ¯ j y j ) C → MNTERR 333,j 3 ( H → FF * ) j ] · ρ → 2 ,$
$H → 3 | H → | 2 − 3 2 W 222 ( y ¯ SM y SM ) C → MNTERR 333,SM 3 = 0 → .$
$Π → 3 ≡ H → 3 | H → | 2 = | H → | 3 e i3θ | H → | 2 =| H → | e i3θ = ( | H → | 1 3 ) 3 e i3θ ,$
$Π → 3 − 3 2 W 222 ( y ¯ SM y SM ) C → MNTERR 333,SM 3 = 0 → .$
$x ¯ → MNTERR 222 = x ˜ → MNTERR 222 = [ 3 2 W 222 ( y ¯ SM y SM ) ] 1 3 ( C → MNTERR 333,SM 3 ) 1 3 .$
$2( x ¯ → MNTERR 222 ) , −( x ¯ → MNTERR 222 )+i 3 ( x ˜ → MNTERR 222 ) , −( x ¯ → MNTERR 222 )−i 3 ( x ˜ → MNTERR 222 ).$

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