## Abstract

Due to the absence of rotational symmetry, off-axis astronomical telescopes with off-set pupil become subject to rotational misalignments. Rotational misalignments of large off-axis mirrors with reference to their geometric center can greatly degrade the imaging quality. This paper presents an in-depth discussion on the net aberration fields of off-axis astronomical telescopes induced by rotational misalignments. Aberration function of off-axis telescopes with rotational misalignments is derived based on the framework of nodal aberration theory. Expressions of several important aberrations are obtained under some approximations. Then the specific field characteristics of these aberrations are presented and explicated. Meanwhile, we demonstrate that rotational misalignments can be converted to a kind of surface decenters; on the other hand, the effects of rotational misalignments have their special features which are different from the effects of general surface decenters. Besides, some other insightful discussions are further presented. This work is also applicable to the rotational misalignments of the off-axis segments of primary mirror in segmented mirror astronomical telescopes.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Compared to rotationally symmetric on-axis astronomical telescopes with pupil obstruction, off-axis telescopes with offset pupil have many advantages in astronomical observation, such as lower scattering property, larger emissivity throughput, and higher dynamic range [1]. Besides, unobscured off-axis telescopes also have simpler and sharper diffraction pattern, meaning not only higher resolution but also some advantages in ellipticity performance [2], which is of cardinal importance for weak gravitational lensing measurement [3]. While off-axis telescopes have been widely applied in the area of remote sensing, to date only few large off-axis astronomical telescopes have been constructed [4,5]. Apart from the fabrication of large off-axis surfaces, another main challenge that hinders the construction of large off-axis astronomical telescopes lies in the fine alignment as well as active optical compensation and alignment of this class of telescopes [6,7]. Due to the absence of rotational symmetry, the effects of misalignments in off-axis telescopes become more complicated. It is of great significance to illuminate the effects of different kinds of misalignments on the net aberration fields of off-axis astronomical telescopes.

Segmented or synthetic-aperture primary mirrors are efficient solutions to the problems with monolithic primary mirror manufacture and testing, transportation and launch [8,9]. In fact, segmented mirror telescopes and synthetic-aperture telescopes can be seen as an array of off-axis telescopes. Each off-axis segment (or each sub-aperture) of the primary mirror and secondary mirrors compose an off-axis telescope. While the aperture size of each off-axis segment is comparatively small, the magnitude of aperture offset with respect to optical axis can be very large, which can greatly increase the sensitivity of segments to misalignments. In the alignment of off-axis segments, the insights of the aberrational effects of segment misalignments can help diagnose the misalignment states of the segments and thus facilitate the whole alignment process.

While numerical methods have been used to calculate the misalignments of off-axis systems [10–12], these methods can hardly contribute to an in-depth understanding of effects of misalignments on net aberration fields; on the other hand, these methods still cannot provide any valuable insights or theoretical guidance for the alignment or active alignment of off-axis systems. Nodal aberration theory (NAT), discovered by Shack [13] and developed by Thompson [14–18], is an analytic tool in studying the aberration behavior of on-axis optical systems which contain misaligned, or intentionally tilted and/or decentered components [19,20]. This theory was first extended to study optical systems with plane symmetry [21]. Then, this theory was extended to analytically express the effects of freeform surfaces on the net aberration fields several years ago [22–24]. Recently, the correctness of nodal aberration theory was verified through experiment with a Ritchey-Chrétien telescope [25]. However, when discussing the effects of misalignments, the traditional nodal aberration theory is only applicable to those optical systems comprised of rotationally symmetric surfaces [22]. In other words, traditional nodal aberration theory is not directly applicable to off-axis systems.

In our previous work, we systematically discussed the effects of lateral misalignments (decenter and tip-tilt of optical surfaces in the lateral direction) and axial misalignments (dislocation of optical surfaces along the axial direction) in off-axis two-mirror astronomical telescopes by extending nodal aberration theory to off-axis telescopes [26,27]. While these works can contribute to a better understanding of the effects of misalignments in off-axis telescopes, they are not enough. There exists another kind of misalignments in off-axis systems, i.e., rotational misalignments. In practice, it is very possible that the location of off-axis mirror is correct while it has some rotational error with respect to its geometric center. On-axis telescopes are immune to this kind of misalignments due to the presence of rotational symmetry. However, rotational symmetry is broken due to pupil offset in off-axis telescopes and only a bilateral symmetry is preserved. Rotational misalignments of large off-axis mirrors can greatly degrade the imaging quality while the effects of them are still not well known, which require further study for better understand.

In this paper, we continue to present an in-depth and systematic discussion on the net aberration fields of off-axis astronomical telescopes induced by rotationally misalignments. This work concentrates on three problems: (1) how to analytically express the effects of rotationally misalignments on the net aberration fields; (2) what special characteristics of the aberration fields that will be induced by rotationally misalignments; (3) what valuable insights or theoretical guidance can be provided by the knowledge of the effects of rotational misalignments in off-axis systems. This work will contribute to a deep understanding of the effects of rotational misalignments on the net aberration fields of off-axis telescopes.

This paper is organized as follows. In Section 2, we derive the aberration function for off-axis astronomical telescopes with rotational misalignments. In Section 3, an off-axis telescope suitable for space application is taken as an example to illustrate the special aberration field characteristics that will be introduced by rotational misalignments. Then we continue to some further discussions in Section 4. In Section 5, we summarize and conclude the paper.

## 2. Aberration function of off-axis astronomical telescopes with rotational misalignments

In this section, we derive the aberration function of off-axis astronomical telescopes with rotational misalignments. To this end, we first convert this problem to an equivalent case which can be handled by nodal aberration theory. We point out and demonstrate that rotational misalignments can be converted to a kind of decenters which have equivalent effects with rotational misalignments.

#### 2.1 Derivation procedure

Net image plane aberration of rotationally symmetric optical systems consists of the sum of all individual surface contributions, which can be descried by the wave aberration expansion in vector form, given by [15]

*j*.

Before deriving the aberration function for off-axis systems with rotational misalignments, we here first introduce the aberration function of a nominal off-axis system, which can be seen as an off-axis portion of a larger rotationally symmetric on-axis system. In the following we call it on-axis parent system. The transformation between the pupil coordinate of the on-axis parent system and the off-axis portion is shown in Fig. 1(a), which can be expressed as

where $\overrightarrow{\rho}$ denotes the on-axis pupil vector normalized by the half aperture size of the on-axis parent pupil, $\overrightarrow{\rho}\text{'}$ is the off-axis pupil vector normalized by the half aperture size of the off-axis pupil, $\overrightarrow{s}$ represents the location of the off-axis pupil relative to the on-axis parent pupil which is normalized by the half aperture size of the on-axis parent pupil, and $b$ represents the scale factor of the aperture size of the off-axis portion relative to the parent on-axis pupil.Then by substituting Eq. (2) into Eq. (1), we can obtain the aberration function of the nominal off-axis system as

Now we begin to derive the aberration function of off-axis systems with rotational misalignments. The coordinate transformation between the real off-axis surface with rotational misalignments and its parent surface (which is rotationally symmetric) is shown in Fig. 1(b). We suppose the off-axis surface *j* is rotated by an angle ${\delta}_{j}$ with reference to its geometric center, ${P}_{j}$. In this process, the center of its parent surface is translated from $O$ to ${O}_{j}\text{'}$, and the position of ${O}_{j}\text{'}$ with reference to $O$ is represented by ${\overrightarrow{v}}_{j}$ which is normalized by the half aperture size of the parent surface, ${R}_{j}$. The position of ${P}_{j}$ with reference to ${O}_{j}\text{'}$ is represented by ${\overrightarrow{t}}_{j}$ which is also normalized by half aperture size of the parent surface. ${}_{m}\overrightarrow{\rho}{}_{j}$ is a position vector defined with reference to the center of the on-axis parent surface *j* normalized by the half aperture size of this surface, ${}_{m}\overrightarrow{\rho}{\text{'}}_{j}$ is a position vector defined with reference to the center of the off-axis surface *j* normalized by the half aperture size of the off-axis surface, and ${}_{m}b{}_{j}$ is the corresponding scale factor between them. This coordinate transformation describes the relation between the relative positions of the rotated off-axis surface and its parent surface.

We here first reveal some inherent relations between Fig. 1(a) and Fig. 1(b). While Fig. 1(a) illustrates pupil coordinate transformation which is in system level and Fig. 1(b) describes the coordinate transformation in surface level, they become equivalent for the optical surface which serves as the aperture stop. In effect, for the optical surface serving as the aperture stop, the surface level coordinate transformation also determines the system level pupil coordinate transformation. In this case, the magnitude of ${\overrightarrow{t}}_{j}$ equals to the magnitude of $\overrightarrow{s}$and ${}_{m}b{}_{j}=b$.

Then we convert the derivation of the aberration function for the off-axis system with rotational misalignments into two separated procedures:

- (1) Derive the aberration function for the on-axis parent system with decentered surface.
We can see from Fig. 1(b) that each on-axis parent surface is decentered from $O$ to ${O}_{j}\text{'}$. The decenter vector of each surface can be expressed as

$${\overrightarrow{V}}_{j}=\left[\begin{array}{c}{S}_{j}\mathrm{sin}{\delta}_{j}\\ {S}_{j}\left(1-\mathrm{cos}{\delta}_{j}\right)\end{array}\right],$$where ${S}_{j}$ is the distance from the geometric center of the off-axis surface to the center of the parent surface for surface

*j*. The normalized decenter vector ${\overrightarrow{v}}_{j}={\overrightarrow{V}}_{j}/{R}_{j},$ where ${R}_{j}$ is the half aperture size of the parent surface.Since the surfaces in the on-axis parent system are rotationally symmetric, this process can be analytically discussed with nodal aberration theory. Supposing the position of the shifted aberration field center associated with

*j*th decentered surface can be expressed as ${\overrightarrow{\sigma}}_{j}$, the aberration function of the on-axis parent system can be given by$$W={\displaystyle \sum _{j}{\displaystyle \sum _{p}^{\infty}{\displaystyle \sum _{n}^{\infty}{\displaystyle \sum _{m}^{\infty}{W}_{klm}{}_{j}{\left[\left(\overrightarrow{H}-{\overrightarrow{\sigma}}_{j}\right)\cdot \left(\overrightarrow{H}-{\overrightarrow{\sigma}}_{j}\right)\right]}^{p}{\left(\overrightarrow{\rho}\cdot \overrightarrow{\rho}\right)}^{n}{\left[\left(\overrightarrow{H}-{\overrightarrow{\sigma}}_{j}\right)\cdot \overrightarrow{\rho}\right]}^{m}}}}}.$$The position of the aberration field center associated with each surface, ${\overrightarrow{\sigma}}_{j}$, can be analytically calculated with ${\overrightarrow{V}}_{j}$ and other system parameters [28].

- (2) Apply a system level pupil coordinate transformation to the aberration function of the decentered on-axis parent system.
This system level coordinate transformation is determined by off-axis surface which serves as the aperture stop. It may be seen from Fig. 1(b) that the positions of the each off-axis surface with reference to its parent surface are different. However, we should keep in mind that a system has only one exit pupil. The position of this off-axis pupil relative to the on-axis parent pupil is determined by the stop surface. The coordinate transformation between the off-axis pupil and the on-axis parent pupil in this case is expressed as

where ${\overrightarrow{t}}_{0}$ specially represents the position of the off-axis surface which serves as the aperture stop with reference to its parent surface and ${\overrightarrow{t}}_{0}$ has the same magnitude with $\overrightarrow{s}$.

On the other hand, the relation between $\overrightarrow{s}$ and ${\overrightarrow{t}}_{0}$ can be expressed as

and

where ${\overrightarrow{v}}_{0}$ specially represents the ${\overrightarrow{v}}_{j}$ in Fig. 1(b) for the surface which serves as the aperture stop, ${\delta}_{0}$ is the rotational error of the off-axis stop surface, and $|\cdot |$represents the magnitude of a vector or the absolute value of a constant. For misalignment level perturbations, ${\delta}_{0}$ is very small (usually $\left|{\delta}_{0}\right|$<0.001). Therefore, we can consider that ${\overrightarrow{t}}_{0}=\overrightarrow{s}$. In other words, while the parent pupil is decentered, the magnitude of this decenter is so small compared to the magnitude of pupil offset that it nearly does not affect the relative position between the off-axis pupil and the parent pupil.

Then by substituting Eq. (6) into Eq. (5) and considering ${\overrightarrow{t}}_{0}=\overrightarrow{s}$, we can obtain the aberration function for off-axis systems with rotational misalignments as

$$W={\displaystyle \sum _{j}{\displaystyle \sum _{p}^{\infty}{\displaystyle \sum _{n}^{\infty}{\displaystyle \sum _{m}^{\infty}{W}_{klm}{}_{j}{\left[\left(\overrightarrow{H}-{\overrightarrow{\sigma}}_{j}\right)\cdot \left(\overrightarrow{H}-{\overrightarrow{\sigma}}_{j}\right)\right]}^{p}{\left[\left(\overrightarrow{\rho}+\overrightarrow{s}\right)\cdot \left(\overrightarrow{\rho}+\overrightarrow{s}\right)\right]}^{n}{\left[\left(\overrightarrow{H}-{\overrightarrow{\sigma}}_{j}\right)\cdot \left(\overrightarrow{\rho}+\overrightarrow{s}\right)\right]}^{m}}}}},$$where for simplicity we assume $b=1$ (i.e. the off-axis surface has the same aperture size as the on-axis parent surface) and neglect the primes.

Through the above derivation, we can see that the effects of rotational misalignments in off-axis telescopes can be converted to a kind of lateral misalignments which are given by Eq. (4). We can use similar method with reference [26] to expand Eq. (9), convert the pupil dependence of each term into existing aberration types and group the aberration coefficients according to their pupil dependence. While the field dependence of each aberration type is generally an infinite sum and it is very hard to derive the exact expression for the field dependence of each aberration type, we can obtain some concise expressions that may be not accurate but can provide a direct expression on the aberration field characteristics. When only the third-order aberrations of the on-axis parent system are considered, we can obtain the aberration function of the off-axis system with rotational misalignments as

$$W={C}_{20}\cdot \left(\overrightarrow{\rho}\cdot \overrightarrow{\rho}\right)+{\overrightarrow{C}}_{22}\cdot {\overrightarrow{\rho}}^{2}+{\overrightarrow{C}}_{31}\cdot \overrightarrow{\rho}\left(\overrightarrow{\rho}\cdot \overrightarrow{\rho}\right)+{C}_{40}\cdot {\left(\overrightarrow{\rho}\cdot \overrightarrow{\rho}\right)}^{2},$$where

$${C}_{20}={W}_{020}+{\displaystyle \sum _{j}{W}_{220Mj}\left(\overrightarrow{H}-{\overrightarrow{\sigma}}_{j}\right)\cdot \left(\overrightarrow{H}-{\overrightarrow{\sigma}}_{j}\right)}+2{\displaystyle \sum _{j}{W}_{131j}\overrightarrow{s}}\cdot \left(\overrightarrow{H}-{\overrightarrow{\sigma}}_{j}\right)+4{W}_{040}\left(\overrightarrow{s}\cdot \overrightarrow{s}\right),$$$${\overrightarrow{C}}_{22}=\frac{1}{2}{\displaystyle \sum _{j}{W}_{222j}\left(\overrightarrow{H}-{\overrightarrow{\sigma}}_{j}\right)\left(\overrightarrow{H}-{\overrightarrow{\sigma}}_{j}\right)}+{\displaystyle \sum _{j}{W}_{131j}\overrightarrow{s}\left(\overrightarrow{H}-{\overrightarrow{\sigma}}_{j}\right)}\text{+}2{W}_{040}{\overrightarrow{s}}^{2},$$

#### 2.1 Validation for the deviation procedure

In the above subsection, we propose that an off-axis system with rotational misalignments (of each surface) can be seen as a result of selecting a proper off-axis portion of a parent on-axis system in which each surface is decentered. This process can further be interpreted as a decenter of each off-axis surface. The equivalent decenter of each surface is related to the rotational misalignments of each surface, as shown in Eq. (4).

In this subsection, we will perform some suitable simulations to validate this statement. On one hand, we introduce rotational misalignments to an off-axis three-mirror anastigmatic (TMA) telescope. On the other hand, we introduce the corresponding decenter misalignments to each surface which are calculated with Eq. (4). Here an off-axis TMA telescope suitable for space application designed by Gilberto Moretto [29] is used, which was once a preliminary off-axis optical configuration for the Supernova/Acceleration Probe (SNAP) Mission. The aperture size of the primary mirror is 2m and the F number is 10. Optical layout of this telescope is presented in Fig. 2 and the optical design parameters are presented in Appendix A.

The introduced rotational misalignments of each off-axis surface are shown in Table 1 and the corresponding decenter parameters obtained with Eq. (4) are shown in Table 2, where *XDE* and *YDE* are the vertex decenters of a mirror in the $x-z$and $y-z$ plane, respectively. The definition for the sign of their directions is in accordance with CODE V. The aberration coefficients of two certain field points, $({0}^{\text{o}},{0}^{\text{o}})$ and $({0.3}^{\text{o}},{0.3}^{\text{o}}),$ for the two cases are shown in Table 3. We can see that the aberration coefficients for the two cases are the same with each other. This result can validate the statement that the effects of rotational misalignments are equivalent to the effects of certain surface decenters obtained with Eq. (4).

Therefore, since we have demonstrated the accuracy of the aberration expressions, i.e., Eqs. (10-14), in [26] for off-axis telescopes with lateral misalignments and analyze the deviation between them and the real ray trace data, in this paper we will no longer repeat it and directly use them to analyze the special aberration field characteristics of off-axis telescopes with rotational misalignments.

## 3. Aberration field characteristics of off-axis astronomical telescopes with rotational misalignments

While in the Section 2 we state that rotational misalignments can be converted to a kind of surface decenters in off-axis telescopes, the effects of rotational misalignments have their special features and there are surely differences between the effects of rotational misalignments and general decenter misalignments. In this section, we will present and explicate the characteristic field dependencies of different aberrations of off-axis astronomical telescopes induced by rotational misalignments. Meanwhile, we mainly focus on the net aberration contribution of rotational misalignments and the interactions between rotational misalignments and nominal aberration can partly be referred to [26] (since rotational misalignments can be converted to special kind of lateral misalignments). We try to provide an intuitive view of the aberration field characteristics induced by rotational misalignments and help grasp the main points.

#### 3.1 Astigmatism

Astigmatism of off-axis astronomical telescopes in the presence of rotational misalignments shown in Eq. (10) and Eq. (12) can be rewritten as

Now we begin to reveal some special features for the astigmatic aberration field induced by rotational misalignments. Considering that the rotational error ${\delta}_{j}$ is usually very small, the equivalent decenter obtained with Eq. (4) can be rewritten as

We can see that the direction of this equivalent decenter is in $x$ direction, which is perpendicular to the direction of pupil offset. Therefore, the position of the shifted aberration field center associated with each surface, ${\delta}_{j}$, is also in the $x$ direction ($+x$ or $-x$direction). According to Eq. (19), we can know that the direction of ${\overrightarrow{A}}_{131}$ is in $x$ direction. Therefore, Eq. (20) can be rewritten asThen the direction of the astigmatism induced by rotational misalignments can be approximated as

For the case of the off-axis SNAP telescope shown in Fig. 2, $\xi \left(\overrightarrow{s}\right)=3\pi /2$. According to Eq. (23), we can obtain that the direction of the astigmatism induced by rotational misalignments is roughly $3\pi /4$ or $\pi /4$. Full filed displays (FFDs) provided by CODE V for astigmatism in the off-axis SNAP telescope without and with the rotational misalignments presented in Table 1 are shown in Fig. 3. The length of the line in Fig. 3 represents the magnitude of the astigmatism in certain scale, and the direction of the line represents the direction of the astigmatism. We can see that a large astigmatism which is nearly constant in the field is induced by rotational misalignments. Importantly, we can easily recognize that the direction of this astigmatism is about $\pi /4$.

We can also know that if $\xi \left(\overrightarrow{s}\right)=\pi /2$ (in $+y$direction), the direction of the astigmatism induced by rotational misalignments is still $3\pi /4$ or $\pi /4$. Besides, if $\overrightarrow{s}$ is in $x$direction, i.e. $\xi \left(\overrightarrow{s}\right)=0\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{or}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\pi $, the direction of ${\overrightarrow{V}}_{j}$is in the $y$ direction in this case, and therefore the direction of astigmatism is still $3\pi /4$ or $\pi /4$. On the other hand, if the direction of $\overrightarrow{s}$is $3\pi /4$ or $\pi /4$, the direction of the astigmatism induced by rotational misalignments will be 0 or $\pi /2$.

Besides, in on-axis astronomical telescopes, field-constant astigmatism can generally be attributed to astigmatic figure errors. However, in off-axis telescopes, this is not the case. Apart from lateral misalignments, rotational misalignments can also introduce a large field-constant astigmatism. Therefore, it is very hard to separate the effects of astigmatic figure errors between misalignments purely by analyzing astigmatic aberration field.

#### 3.2 Coma

Coma aberration field of off-axis astronomical telescopes in the presence of rotational misalignments shown in Eq. (10) and Eq. (13) can be rewritten as

_{${K}_{131}=-\left|{\overrightarrow{A}}_{131}\right|$}if the vector ${\overrightarrow{A}}_{131}$ points to $+x$ direction and ${K}_{131}=\left|{\overrightarrow{A}}_{131}\right|$ if this vector points to $-x$ direction. Therefore, the direction of the coma induced by rotational misalignments can be expressed asConsidering $\xi \left({\overrightarrow{e}}_{x}\right)=0$, the direction of coma is determined by the sign of ${K}_{131}$(i.e. the direction of coma is 0 or $\pi $).

For the case of the off-axis SNAP telescope shown in Fig. 2, the FFD for coma in the nominal off-axis SNAP telescope is shown in Fig. 4(a), and the FFD for coma in the presence of the rotational misalignments presented in Table 1 are shown in Fig. 4(b). The length of the comet in Fig. 4 indicates the magnitude of coma in certain scale and the head of the comet points to the direction of coma. We can see that a field-constant coma is induced by rotational misalignments. Importantly, we can easily recognize that the direction of this coma is about $\pi $.

Direction of the field-constant coma induced by the rotational misalignments is also related to the direction of pupil offset. Specifically, the direction of this coma lies perpendicular to the direction of pupil offset. In other words, if the direction of pupil offset is in $x$ direction, the direction of the coma induced by rotational misalignments will lie in $y$ direction.

#### 3.3 Medial focal surface

In reference [26] we show that lateral misalignments (decenters and tip-tilts) in off-axis astronomical telescopes can generally introduce a field-constant focal shift. In this paper, we will show that, while rotational misalignments can be converted to a kind of surface decenters, they do not introduce any field-constant focal shift. The reason is that the equivalent surface decenters corresponding to rotational misalignments obtained with Eq. (4) have a special direction, i.e. they are always perpendicular to the direction of pupil offset.

Medial focal surface of off-axis astronomical telescopes in the presence of rotational misalignments shown in Eq. (10) and Eq. (11) can be rewritten as

Then we continue to discuss the field-constant term. As presented before, the direction of the equivalent decenters of rotational misalignments obtained with Eq. (4) are perpendicular to the direction of pupil offset (which is represented by the direction of $\overrightarrow{s}$), and therefore the direction of ${\overrightarrow{A}}_{131}$ is perpendicular to direction of $\overrightarrow{s}$, i.e.

We can deduce that there is no field-constant focal shift will be induced by rotational misalignments in off-axis telescopes.FFDs for medial focal surface of the off-axis SNAP telescope in the nominal state and in the presence of rotational misalignments are shown in Fig. 5(a) and Fig. 5(b), respectively. We can hardly recognize the differences between them. On the other hand, FFD for medial focal surface of the off-axis SNAP telescope with some lateral misalignments (not in the direction perpendicular to the direction of pupil offset) is shown in Fig. 5(c), where we can easily recognize a large focal shift. Therefore, while rotational misalignments can be converted to a kind of lateral misalignments, its effects are different from general lateral misalignments in off-axis systems.

## 4. Other discussions

In this section, some other brief but insightful discussions concerning the effects of rotational misalignments in off-axis astronomical telescopes will be presented.

#### 4.1 Compensating effects of lateral misalignments to rotational misalignments

In the previous section, we point out that the effects of rotational error of a surface in off-axis telescopes can be seen as a decenter of this surface in the direction perpendicular to pupil offset. In this perspective, we can deduce that while the large off-axis astronomical telescopes are sensitive to rotational misalignments, in practice we do not need to provide adjustable ability to directly correct this kind of misalignments after initial alignment. The effects of rotational misalignments can well be compensated by introducing some proper lateral misalignments (decenters and tip-tilts) to the system.

Here we use a simple example to illustrate the compensating effects of lateral misalignments to rotational misalignments. In the presence of the rotational misalignments presented in Table 1, we can introduce a lateral misalignment of the secondary mirror to compensate for the aberrations induced by rotational misalignments. In practice the secondary mirror is usually smaller in size and weight and it is more convenient to adjust this mirror. The lateral misalignments of the secondary mirror needed for compensation can be computed analytically. Here we take the tertiary mirror as the coordinate reference, i.e., we consider that tertiary mirror is well aligned and the misalignments of the primary mirror and the secondary mirror are defined with reference to the tertiary mirror. In this case, the relationships between the shifted aberration field center for each individual surface and the lateral misalignment parameters can be given by [28] [30]

Both the field-constant astigmatism and coma in the off-axis telescope with lateral misalignments are mainly generated from misalignment-induced filed-constant coma of the on-axis parent system through pupil coordinate transformation. Besides, the field-linear astigmatism in off-axis telescope is nearly the same as that in the on-axis parent system. Therefore, the compensating conditions between the equivalent lateral misalignments of rotational misalignments and the lateral misalignments introduced deliberately can be expressed as

Besides, in the alignment or active alignment of large off-axis telescopes, it is not proper to calculate the lateral misalignments and rotational misalignments at the same time, especially when using sensitivity table method. The sensitivity matrix in this case is ill-conditioned due to the compensating effects between these two kinds of misalignments.

We should also point out that it is not always reasonable to compensate for the effects of rotational misalignments with lateral misalignments. The lateral misalignments needed to compensate for the effects of rotational misalignments may be very large that exceed the adjustable ability of the adjustment mechanism. Therefore, in the initial alignment it is important to guarantee the accuracy of this degree of freedom.

#### 4.2 Magnitude of pupil offset and the sensitivity of astigmatism to rotational misalignments

As shown in Section 3, astigmatism is the dominant aberration in the presence of rotational misalignments. It is of importance to understand the relationship between the sensitivity of astigmatism to rotational misalignments and the magnitude of pupil offset. For example, in the optical design of large off-axis telescopes, we hope that we can select a proper magnitude of pupil offset with which the system is less sensitive to rotational misalignments. For large segmented mirror astronomical telescopes, different off-axis segments of the primary mirror have different magnitude of pupil offset, and the tolerances of different segments for the rotational errors are different. This subsection discusses the influence of the magnitude of pupil offset on the sensitivity of the astigmatic aberration field to rotational misalignments.

Equation (21) can be rewritten as

*j*(if this surface is the aperture stop, then we have that ${R}_{j}\text{'}={R}_{j}$). We can see that the magnitude of ${\overrightarrow{V}}_{j}$is proportional to the magnitude of pupil offset. In other words, the equivalent lateral misalignments corresponding to rotational misalignments are proportional to the magnitude of pupil offset. Considering that the magnitude of ${\overrightarrow{\sigma}}_{j}$ is linearly correlated with the lateral misalignments of the system, as shown in Eqs. (36-41), the magnitude of ${\overrightarrow{A}}_{131}$ is also proportional to the magnitude of pupil offset. Therefore, the magnitude of ${\overrightarrow{A}}_{131}\overrightarrow{s}$, $\left|{\overrightarrow{A}}_{131}\overrightarrow{s}\right|$, is proportion to the square of the magnitude of pupil offset. In other word, the sensitivity of astigmatism to rotational misalignments is proportional to the square of the magnitude of pupil offset.

FFD for the astigmatism of the off-axis SNAP telescope with the same rotational misalignments presented in Table 1 with a half magnitude of pupil offset is shown in Fig. 7(a). Comparing the scales shown in Fig. 7(a) and Fig. 3(b), we can see that the magnitude of astigmatism in Fig. 7(a) is only about a quarter of the magnitude of astigmatism in Fig. 3(b).

On the other hand, while rotational misalignments in off-axis systems can be converted to equivalent lateral misalignments, the relationships between the sensitivity of astigmatism to these two kinds of misalignments and the magnitude of pupil offset are different. The sensitivity of astigmatism to lateral misalignments is linearly related to the magnitude of pupil offset [26]. FFD for astigmatism of the off-axis SNAP telescope with the same lateral misalignments presented in Table 2 while the magnitude of pupil offset is reduced by half is shown in Fig. 7(b), where we can see that the magnitude is half that presented in Fig. 3(b). This is another difference between the effects of rotational misalignments and lateral misalignments.

We can also deduce that the sensitivity of astigmatism to rotational misalignments increases rapidly with the increase in the magnitude of pupil offset. Therefore, in the optical design of off-axis astronomical telescopes, we can reduce the magnitude of pupil offset to reduce the sensitivity of wavefront aberration to rotational misalignments (under the premise that an unobstructed configuration can be obtained). We can also deduce that for large segmented mirror astronomical telescopes, the segments far away from the optical axis are much harder to align than those near the optical axis.

## 5. Conclusions

This paper extends nodal aberration theory to include the effects of rotational misalignments in off-axis astronomical telescopes and presents a systematic discussion on it. The aberration function of off-axis telescopes with rotational misalignments is derived. The expressions of several important aberrations are obtained under some approximations. Then the specific field characteristics of these aberrations are presented and explicated. Some other insightful discussions concerning the effects of rotational misalignments are also presented.

Some conclusions are presented as follows:

- (1) Rotational misalignments can be converted to a kind of surface decenters in off-axis astronomical telescopes. The underlying reasons for this are that each surface in off-axis systems can generally be seen as an off-axis portion of a rotationally symmetric parent surface and rotational errors are usually comparatively small for misalignment level perturbations.
- (2) While rotational misalignments can be converted to a kind of surface decenters in off-axis telescopes, the effects of rotational misalignments have their special features and there are differences between the effects of rotational misalignments and general decenter misalignments. Specifically, the directions of astigmatism and coma induced by rotationally misalignment are nearly certain, which are determined by the direction of pupil offset, while directions of astigmatism and coma induced by lateral misalignments can be arbitrary. Besides, rotational misalignments nearly do not affect medial focal surface while lateral misalignments in off-axis telescopes can generally introduce a field-constant focal shift.
- (3) Since rotational error of a surface in off-axis telescopes can be seen as a decenter of this surface in the direction perpendicular to pupil offset, the effects of rotational misalignments can well be compensated by introducing some proper lateral misalignments (decenters and tip-tilts) to the system. While large off-axis astronomical telescopes are sensitive to rotational misalignments, in practice we generally do not need to provide adjustable ability to directly correct this kind of misalignments after initial alignment. On the other hand, in general it is not appropriate to compensate for the effects of lateral misalignments by introducing rotational misalignments.
- (4) Sensitivity of astigmatism (the dominant aberration induced by rotational misalignments) to rotational misalignments is proportional to the square of the magnitude of pupil offset while of astigmatism to lateral misalignments is linearly related to the magnitude of pupil offset. This fact indicates that as the magnitude of pupil offset increases, the sensitivity of astigmatism to rotational misalignments increases more rapidly. Therefore, for those off-axis segments (of the segmented primary mirror) far away from the optical axis, we should particularly guarantee the alignment accuracy for this degree of freedom.

Note that not all off-axis astronomical telescopes can be seen as an off-axis portion of an on-axis parent telescope with rotational symmetry. There can be freeform overlays in the nominal design of the off-axis telescope. However, the rotation errors of freeform overlays can easily be handled with a rotating coordinate transformation. We do not further discuss them in this paper.

This work can contribute to a more complete understanding of the effects of misalignments in off-axis astronomical telescopes.

## 6 Appendix A

Basic optics parameters of the off-axis SNAP telescope are shown in Table 4. Its high-order aspheric parameters are shown in Table 5. The aberration coefficients for the spherical base curve and the aspheric departure from the spherical base curve of each individual surface are shown in Table 6, which are directly obtained from the optical simulation software.

## Funding

National Natural Science Foundation of China (NSFC) (61705223); National Key Research and Development Program (2016YFE0205000).

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