Compensation factors for 3rd order coma in three mirror anastigmatic (TMA) telescopes

S. Xu; Z. Cui; B. Qi

doi:10.1364/OE.26.000298

1. Introduction

Nodal aberration theory (NAT) is proposed by R. V. Shack and K. P. Thompson [1] in 1980 based on a shifted aberration field centers concept suggested by Buchroeder [2] and is developed up to fifth order aberrations, including 5th order spherical, coma, astigmatism and distortion, mainly attributed to Thompson in the last 30 years [3–5]. It plays an important role especially in facilitating the alignment of optical systems nowadays due to its capability of analyzing misalignment induced aberrations in optical systems with circular or near circular pupils [6]. By applying NAT and Full Field Display (FFD) techniques, Thompson et al. has received some notable successes in studying alignments of some TMA telescopes (one of them is the famous James Webb Space Telescope (JWST)) since 2006 [7]. Many authors also use NAT to analyze the behavior of freeform optical systems due to its power in analyzing the system behavior over the field of view [8–10]. NAT is also proved to be useful and less sensitive to measurement errors for applying active optics when compared to sensitivity table method [11].

Third order coma (just coma for short hereafter) is a well-known field-linear aberration that is induced by misalignments, including tilts and decenters, of mirrors of optical systems at the image plane. When mirrors are misaligned, filed-constant coma will be induced. It can either be positive or negative in terms of the sign of the tilt/decenter, which makes it possible to be completely compensated by imposing another tilt/decenter on the same mirror itself or, of course, other mirrors in the optical system. For example, NAT has been used to prove the coma-free point of the secondary mirror (SM) in the Cassegrain telescope by Ren [12], which, in fact, uses tilt induced coma of the SM to compensate the coma induced by decenter of the SM itself. Therefore, it is possible to find such completely compensating conditions in an optical system with multiple mirrors. In this paper, as an example, we will find out this possibility and how the compensation works in the TMA telescopes.

2. Coma compensation in misaligned coaxial TMA telescopes

A TMA system is designed to correct the residual aberrations of the two-mirror systems to achieve better performance, which is important especially in professional usage. The TMA telescope discussed in this paper (Fig. 1), with its aperture stop on the primary mirror (PM), has a real intermediate image plane between the SM and the tertiary mirror (TM).

Fig. 1 Gaussian optics of a coaxial TMA telescope

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Sign convention used here is the same as those used in [13], that is, $y_{1}, {s^{'}}_{2}, d_{2}, f^{'}$ and $u_{2}$ are numerally positive while $d_{1}, {s^{'}}_{3}, {u^{'}}_{2}$ and ${u^{'}}_{3}$ are numerally negative in Fig. 1.

The aberration field of a system is simply the sum of spherical and aspheric contributions of individual surface. Coma in an aligned rotational symmetry optical system can be expressed as

W_{131} = \sum_{j} W_{131 j} (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) = \sum_{j} W_{131 j}^{o} ({\vec{H}}^{o}_{j} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) + \sum_{j} W_{131 j}^{*} ({\vec{H}}^{*}_{j} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ})

Here, j represents the surface number, $\vec{H}$ and $\vec{ρ}$ represent positions at the image plane and entrance/exit pupil respectively. $W_{131}^{o}$ and $W_{131}^{*}$ represent the contribution factors of coma from the spherical part and the aspheric part respectively and equal to $\frac{1}{2} S_{I I}^{o}$ and $\frac{1}{2} S_{I I}^{*}$ , Seidel coefficients for coma, respectively.

While mirrors are misaligned in an optical system, according to NAT, the spherical and aspheric contribution centers of surfaces are changed and coefficients of aberrations, however, remain unchanged. That is, coma in a misaligned system can be expressed from Eq. (1) as

\begin{matrix} W_{131} = \sum_{j} W_{131 j}^{o} ((\vec{H} - {\vec{σ}}^{o}_{j}) \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) + \sum_{j} W_{131 j}^{*} ((\vec{H} - {\vec{σ}}^{*}_{j}) \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ = \sum_{j} W_{131 j}^{} (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) - \sum_{j} W_{131 j}^{o} ({\vec{σ}}^{o}_{j} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) - \sum_{j} W_{131 j}^{*} ({\vec{σ}}^{*}_{j} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \end{matrix}

Here, ${\vec{σ}}_{j}^{o}$ and ${\vec{σ}}_{j}^{*}$ represent the decentration vectors of the centers of the aberration fields with respect to spherical part and aspheric part of surface j respectively. From Eq. (1), Eq. (2) and the relationship between $W_{131}$ and $S_{I I}$ , the increased aberration $Δ W_{131}$ can be expressed as

Δ W_{131} = - \frac{1}{2} (\sum_{j} S_{I I, j}^{o} ({\vec{σ}}^{o}_{j} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) + \sum_{j} S_{I I, j}^{*} ({\vec{σ}}^{*}_{j} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}))

_{${\vec{σ}}_{j}^{o}$} and ${\vec{σ}}_{j}^{*}$ in Eq. (3) are usually different in most misaligned cases, which makes it necessary to derive the spherical and aspheric coefficients, $S_{I I, j}^{o}$ and $S_{I I, j}^{*}$ , of individual surface.

2.1 Coma coefficients of the SM and the TM

Seidel Coefficients of coma of surface j are defined as

{\begin{cases} S_{I I, j}^{o} = - A_{j} A_{p r j} y_{j} (\frac{{u^{'}}_{j}}{{n^{'}}_{j}} - \frac{u_{j}}{n_{j}}) \\ S_{I I, j}^{*} = (\frac{y_{p r j}}{y_{j}}) c_{j}^{3} ({n^{'}}_{j} - n_{j}) b_{s j} y_{j}^{4} \end{cases}

Where j means surface number, subscript pr means principle ray, $A = n i = n^{'} i^{'}$ represents the Snell Invariant, $c = 1 / r = 1 / 2 f^{'}$ represents curvature of a surface, b_s is the conic constant, y is the height of the incident ray on the surface, u is the angle of the incidence ray.

Coma coefficients of the SM are the same as those derived by R. N. Wilson [13] due to the structure of the first two mirrors in the TMA telescope illustrated in Fig. 1 is similar to the Cassegrain telescope that R. N. Wilson used. Note that, $s_{p r 1} = 0$ in [13] while the stop is on PM. The magnification of the TM is defined as $m_{3} = u_{3} / {u^{'}}_{3} = {u^{'}}_{2} / {u^{'}}_{3} = {s^{'}}_{3} / s_{3}$ . Then the relationship between ${f^{'}}_{C}$ , focal length of the Cassegrain telescope in [13], and $f^{'}$ , focal length of the TMA telescope in this paper, can be expressed as ${f^{'}}_{C} = f^{'} / m_{3}$ . Therefore, Seidel coefficients of the SM can be rewritten from R. N. Wilson as

{\begin{cases} S_{I I, 2}^{o} = - \frac{1}{4} (^{\frac{y_{1}_{}}{f^{'}}) 3} m_{3}^{3} (m_{2} + 1) (^{m_{2} - 1) 2} [d_{1} - \frac{2 f^{'}}{m_{3} (m_{2} - 1)}] u_{p r 1} \\ S_{I I, 2}^{*} = - \frac{1}{4} (^{\frac{y_{1}_{}}{f^{'}}) 3} m_{3}^{3} (^{m_{2} + 1) 3} d_{1} b_{s 2} u_{p r 1} \end{cases}

Here, m₂ is the magnification of the SM. b_s2 is the conic constant of the SM. u_pr1 represents the incident angle of the paraxial principle ray in the object space.

For the case in Fig. 1, $n_{1} = {n^{'}}_{2} = n_{3} = 1$ , ${n^{'}}_{1} = n_{2} = {n^{'}}_{3} = - 1$ and $i_{3} = - {i^{'}}_{3}$ . According to the reflection theorem, $2 i_{3} = {u^{'}}_{2} - {u^{'}}_{3}$ . ${u^{'}}_{3} = - y_{1} / f^{'}$ , which is shown in Fig. 1. Then, we have

A_{3} = n_{3} i_{3} = \frac{{u^{'}}_{2} - {u^{'}}_{3}}{2} = \frac{m_{3} - 1}{2} {u^{'}}_{3} = \frac{1 - m_{3}}{2} \frac{y_{1}}{f^{'}}

Note that, $u_{3} = m_{3} {u^{'}}_{3}$ . Then we have

\frac{{u^{'}}_{3}}{{n^{'}}_{3}} - \frac{u_{3}}{n_{3}} = - {u^{'}}_{3} - u_{3} = - (m_{3} + 1) {u^{'}}_{3} = (m_{3} + 1) \frac{y_{1}}{f^{'}}

y₃ can be calculated by object distance of the incident ray, s₃, and the angle of the incident ray, u₃. With $s_{3} = s^{'} / m_{3}$ , we have

y_{3} = s_{3} u_{3} = \frac{{s^{'}}_{3}}{m_{3}} \cdot m_{3} {u^{'}}_{3} = \frac{y_{1}}{f^{'}} {s^{'}}_{3}

From the reflection equation and $m_{3} = {s^{'}}_{3} / s_{3}$ , we have ${f^{'}}_{3} = {s^{'}}_{3} / (m_{3} + 1)$ , then

c_{3} = \frac{1}{2 {f^{'}}_{3}} = \frac{m_{3} + 1}{2 {s^{'}}_{3}}

The principle ray passes through the center of stop (vertex of PM) with an angular u_pr1, which gives $u_{p r 2} = {u^{'}}_{p r 1} = - u_{p r 1}$ . Relationship between ${u^{'}}_{p r 2}$ and $u_{p r 2}$ can be decided by using the reflection equation. Then, we have

y_{p r 3} = y_{p r 2} + d_{2} {u^{'}}_{p r 2} = - d_{1} u_{p r 1} - d_{2} (\frac{d_{1}}{{f^{'}}_{2}} + 1) u_{p r 2} = (\frac{d_{1} d_{2}}{{f^{'}}_{2}} + d_{2} - d_{1}) u_{p r 1}

Defining ${s^{'}}_{p r 2}$ as the image distance of the principle ray with respect to the SM. ${s^{'}}_{p r 2}$ and ${u^{'}}_{p r 3}$ can be calculated from the reflection equation. Method to calculate ${i^{'}}_{p r 3}$ is the same as the one to calculate $i_{3}$ in Eq. (6). Then we have

{\begin{cases} {s^{'}}_{p r 2} = \frac{d_{1} {f^{'}}_{2}}{d_{1} + {f^{'}}_{2}}, u_{p r 3} = {u^{'}}_{p r 2} = (\frac{d_{1}}{{f^{'}}_{2}} + 1) u_{p r 1} \\ {u^{'}}_{p r 3} = (\frac{{s^{'}}_{p r 2} - d_{2}}{{f^{'}}_{3}} - 1) u_{p r 3}, {i^{'}}_{p r 3} = \frac{{u^{'}}_{p r 3} - u_{p r 3}}{2} \end{cases}

From Eq. (11), the Snell Invariant of the TM is then given as

A_{p r 3} = {n^{'}}_{3} {i^{'}}_{p r 3} = \frac{(d_{2} + 2 {f^{'}}_{3}) (d_{1} + {f^{'}}_{2}) - d_{1} {f^{'}}_{2}}{2 {f^{'}}_{2} {f^{'}}_{3}} u_{p r 1}

By substituting Eqs. (6)-(10) and Eq. (12) in Eq. (4), Seidel coefficients of the TM can be expressed as

{\begin{cases} S_{I I, 3}^{o} = - \frac{1}{4} (^{\frac{y_{1}_{}}{f^{'}}) 3} (^{m_{3} + 1) 2} (1 - m_{3}) [\frac{d_{1} d_{2}}{{f^{'}}_{2}} + d_{2} - d_{1} + 2 {f^{'}}_{3} (\frac{d_{1}}{{f^{'}}_{2}} + 1)] u_{p r 1} \\ S_{I I, 3}^{*} = - \frac{1}{4} (^{\frac{y_{1}_{}}{f^{'}}) 3} (^{m_{3} + 1) 3} (\frac{d_{1} d_{2}}{{f^{'}}_{2}} + d_{2} - d_{1}) b_{s 3} u_{p r 1} \end{cases}

_{${f^{'}}_{2}$} and ${f^{'}}_{3}$ remain in Eq. (13), which is convenient when calculating the misalignment induced coma, $Δ W_{131}$ , in Eq. (3).

2.2 Contribution centers of the SM and the TM

In a rotationally symmetric TMA telescope, contribution center of spherical part of a mirror is along the line that passes through the center of curvature and the entrance/exit pupil while contribution center of aspheric is along the line that passes though the vertex of the surface and the entrance/exit pupil. This is the principle that is used to locate the contribution centers of the SM and the TM in a misaligned TMA telescope.

Considering the case that tilts and decenters are both imposed on the SM and the TM, which is shown in Fig. 2. Decenter described here excludes displacement along the axis of the surface. The center of the entrance, vertex and center of curvature of individual surface in a rotationally symmetric system are on a common line, which results that centers of symmetry of the aberration contributions still coincide when mirrors are displaced along the axis of the surface.

Fig. 2 The five lines in the object space of the TM in a misaligned TMA system

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As shown in Fig. 2, A₁ is the axis of the PM. O₂ and O₃ are the curvature centers of the SM and the TM respectively. V₂ and V₃ are the vertexes of the SM and the TM respectively. E'₂ is the image of E, center of the entrance, in the image space of the SM (or the object space of the TM). F₁ is the center of the primary image filed and F'₁ is the image of F₁ in the image space of the SM. A and B represent decenters of the SM and the TM respectively, α and β represent tilts of the SM and the TM respectively. A and β are positive while B and α are negative in Fig. 2. Lines L_O2, L_O3, L_V2, L_V3, and L_q all pass through E'₂ and pass through O₂, O₃, V₂, V₃ and F'₁ respectively. P is defined as the height of E'₂ from axis A₁ and s_E2 as the distance from V₂ to E'₂ along axis A₁. Right-handed coordinates are used in this paper.

Lines L_O2, L_O3, L_V2, L_V3, and L_q locate the contribution centers of spherical aberration of the SM and the TM, aspheric aberration of the SM and the TM and the field center respectively after the TM images them into the image space of the whole system. Slope angles of the five lines will be derived first in the image space of the SM, next in the image space of the whole system.

From the reflection equation, s_E2 can be given as

s_{E 2} = \frac{{f^{'}}_{2} d_{1}}{d_{1} + {f^{'}}_{2}}

O₂, E'₂ and E are on the common line L_O2. By applying similar triangles theorem, $P / (r_{2} α + A) = (d_{1} + s_{E 2}) / (d_{1} + r_{2})$ . From Eq. (14), P can be given as

P = \frac{2 d_{1} (2 {f^{'}}_{2} α + A)}{d_{1} + {f^{'}}_{2}}

Slop angles of the five lines with respect to axis A₁ can be directly given from Fig. 2 as

{\begin{cases} U_{V 2} = \frac{P - A}{s_{E 2}}, U_{O 2} = \frac{A + 2 {f^{'}}_{2} α - P}{2 {f^{'}}_{2} - s_{E 2}} \\ U_{V 3} = \frac{B - P}{d_{2} - s_{E 2}}, U_{O 3} = \frac{B + 2 {f^{'}}_{3} β - P}{2 {f^{'}}_{3} + d_{2} - s_{E 2}}, U_{Lq} = P / s_{E 2} \end{cases}

Here, $U_{V 2}, U_{O 2}, U_{V 3}, U_{O 3}$ and $U_{Lq}$ represent the slop angles of lines L_V2, L_O2, L_V3, L_O3, and L_q respectively. In Fig. 2, they are all numerically negative.

All slop angles in Eq. (16) should minus β when they are imaged into the image filed of the whole system by TM, giving

U_{x, 3} = U_{x} - β

As shown in Fig. 3, line L_V2 intersects the axis of the TM at a point W. s_V2,3 is defined as the length between W and V₃ along axis A₁, s_V is the length from point T to point W along A₁ and is positive in Fig. 3.

Fig. 3 L_V2 in the object space of the TM

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s_V2,3 is calculated as the object distance of line L_V2 due to β is small. Length from point T to point E'₂ equals to $P - B + (d_{2} - s_{E 2}) β$ . U_v2,3 and β is small that s_v approximately equals to $- [P - B + (d_{2} - s_{E 2}) β] / U_{V 2, 3}$ ( $U_{V 2, 3}$ is negative), which leads to

s_{V 2, 3} = s_{E 2} - d_{2} + s_{V} = s_{E 2} - d_{2} - \frac{P - B + (d_{2} - s_{E 2}) β}{U_{V 2, 3}}

Slope angle U'_V2,3 of line L_V2 in the image space of the whole system then can be derived by using parameters U_V2,3, s_V2,3 and f'₃. Slope angle U'_O2,3 and U'_Lq,3 can also be derived by the same method. U'_O3,3 is equal to U_O3,3 and U'_V3,3 is equal to −U_V3,3, then we have

{\begin{cases} {U^{'}}_{V 2, 3} = \frac{(A - P) (d_{2} +_{3} {f^{'}}_{3})}{s_{E 2} {f^{'}}_{3}} + \frac{B - A + β {f^{'}}_{3}}{{f^{'}}_{3}} \\ {U^{'}}_{O 2, 3} = \frac{B}{2 {f^{'}}_{3}} + β - \frac{A - P + 2 α {f^{'}}_{2}}{2 {f^{'}}_{2} - s_{E 2}} - \frac{(A + 2 α {f^{'}}_{2}) (d_{2} - s_{E 2}) + P (2 {f^{'}}_{3} - d_{2})}{{f^{'}}_{3} (2 {f^{'}}_{2} - s_{E 2})} \\ {U^{'}}_{V 3, 3} = β - \frac{B - P}{d_{2} - s_{E 2}} \\ {U^{'}}_{O 3, 3} = \frac{B - P - (d_{2} - s_{E 2}) β}{2 {f^{'}}_{3} + d_{2} - s_{E 2}} \\ {U^{'}}_{Lq, 3} = \frac{B + β {f^{'}}_{3}}{{f^{'}}_{3}} - \frac{P (d_{2} + {f^{'}}_{3})}{s_{E 2} {f^{'}}_{3}} \end{cases}

_{${U^{'}}_{V 2, 3}, {U^{'}}_{V 2, 3}, {U^{'}}_{V 3, 3}, {U^{'}}_{O 3, 3}$} and ${U^{'}}_{Lq, 3}$ are illustrated in Fig. 4. ${S^{'}}_{E^{'}}$ is the distance between E', center of the exit pupil, and the final image plane along axis A₁. Lines L_V2, L_O2, L_V3, L_O3, and L_q in the image field of the whole system all pass though E', center of exit pupil, and intersect the final image plane at points P_V2, P_O2, P_V3, P_O3, and P_Lq respectively. According to NAT, P_V2, P_V3 P_O2, P_O3 and P_Lq are the contribution centers of aspheric parts of the SM and TM, spherical parts of the SM and TM and the new center of the field. That is, vectors start from P_Lq to P_V2, P_O2, P_V3 and P_O3 are ${\vec{σ}}_{2}^{*}, {\vec{σ}}_{2}^{o}, {\vec{σ}}_{3}^{*}$ and ${\vec{σ}}_{3}^{o}$ in Eq. (3) respectively.

Fig. 4 Contribution centers of the SM and the TM

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_{${\vec{σ}}_{2}^{*}, {\vec{σ}}_{2}^{o}, {\vec{σ}}_{3}^{*}$} and ${\vec{σ}}_{3}^{o}$ have the same direction in Fig. 4. They can be expressed as

{\vec{σ}}_{j} = {s^{'}}_{E^{'}} ({U^{'}}_{x j, 3} - {U^{'}}_{Lq, 3}) \vec{e}

Where $\vec{e}$ is a normalized vector with a direction along $\vec{y}$ . From Eqs. (14), (15), (19) and (20), gives

{\begin{matrix} {\vec{σ}}_{2}^{*} = (\frac{m}{{f^{'}}_{2} {f^{'}}_{3}} + \frac{m + {f^{'}}_{2}}{d_{1} {f^{'}}_{3}}) A {s^{'}}_{E^{'}} \vec{e} = k_{A 2}^{*} A {s^{'}}_{E^{'}} \vec{e} \\ {\vec{σ}}_{2}^{o} = (\frac{m}{{f^{'}}_{2} {f^{'}}_{3}} - \frac{m - {f^{'}}_{2}}{{f^{'}}_{3} (d_{1} + 2 {f^{'}}_{2})}) (A + 2 {f^{'}}_{2} α) {s^{'}}_{E^{'}} \vec{e} = (k_{A 2}^{o} A + k_{α 2}^{o} α) {s^{'}}_{E^{'}} \vec{e} \\ {\vec{σ}}_{3}^{*} = \frac{n + t}{n {f^{'}}_{2} {f^{'}}_{3}} (d_{2} A - {f^{'}}_{2} B + 2 d_{2} {f^{'}}_{2} α) {s^{'}}_{E^{'}} \vec{e} = (k_{A 3}^{*} A + k_{B 3}^{*} B + k_{α 3}^{*} α) {s^{'}}_{E^{'}} \vec{e} \\ {\vec{σ}}_{3}^{o} = \frac{n + t}{{f^{'}}_{2} {f^{'}}_{3} (n + 2 t)} [(d_{2} + 2 {f^{'}}_{3}) A - {f^{'}}_{2} B + 2 (d_{2} {f^{'}}_{2} + 2 {f^{'}}_{2} {f^{'}}_{3}) α - 2 {f^{'}}_{2} {f^{'}}_{3} β] {s^{'}}_{E^{'}} \vec{e} \\ = (k_{A 3}^{o} A + k_{B 3}^{o} B + k_{α 3}^{o} α + k_{β 3}^{o} β) {s^{'}}_{E^{'}} \vec{e} \end{matrix}

Where

{\begin{cases} m = d_{2} - {f^{'}}_{2} + {f^{'}}_{3} \\ n = d_{1} d_{2} - d_{1} {f^{'}}_{2} + d_{2} {f^{'}}_{2} \\ t = d_{1} {f^{'}}_{3} + {f^{'}}_{2} {f^{'}}_{3} \end{cases}

From Eqs. (3), (5), (13), and (21), we have

\begin{matrix} Δ W_{131} = - \frac{1}{2} [(S_{I I, 2}^{o} {\vec{σ}}_{2}^{o} + S_{I I, 2}^{*} {\vec{σ}}_{2}^{*} + S_{I I, 3}^{o} {\vec{σ}}_{3}^{o} + S_{I I, 3}^{*} {\vec{σ}}_{3}^{*}) \cdot \vec{ρ}] (\vec{ρ} \cdot \vec{ρ}) \\ = - \frac{{s^{'}}_{E^{'}}}{2} [S_{I I, 2}^{o} (k_{A 2}^{o} A + k_{α 2}^{o} α) + S_{I I, 2}^{*} k_{A 2}^{*} A + S_{I I, 3}^{o} (k_{A 3}^{o} A + k_{B 3}^{o} B + k_{α 3}^{o} α + k_{β 3}^{o} β) + \\ S_{I I, 3}^{*} (k_{A 3}^{*} A + k_{B 3}^{*} B + k_{α 3}^{*} α)] (\vec{e} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ = - \frac{{s^{'}}_{E^{'}}}{2} (k_{A} A + k_{B} B + k_{α} α + k_{β} β) (\vec{e} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \end{matrix}

Where

{\begin{cases} k_{A} = S_{I I, 2}^{o} k_{A 2}^{o} + S_{I I, 2}^{*} k_{A 2}^{*} + S_{I I, 3}^{o} k_{A 3}^{o} + S_{I I, 3}^{*} k_{A 3}^{*} \\ k_{B} = S_{I I, 3}^{o} k_{B 3}^{o} + S_{I I, 3}^{*} k_{B 3}^{*} \\ k_{α} = S_{I I, 2}^{o} k_{α 2}^{o} + S_{I I, 3}^{o} k_{α 3}^{o} + S_{I I, 3}^{*} k_{α 3}^{*} \\ k_{β} = S_{I I, 3}^{o} k_{β 3}^{o} \end{cases}

If vectors ${\vec{σ}}_{2}^{*}, {\vec{σ}}_{2}^{o}, {\vec{σ}}_{3}^{*}$ and ${\vec{σ}}_{3}^{o}$ have different directions in the image space, then the compensation factors along $\vec{x}$ and $\vec{y}$ can be given as

{\begin{cases} f_{m x} = k_{A} A_{x} + k_{B} B_{x} + k_{α} α_{x} + k_{β} β_{x} \\ f_{m y} = k_{A} A_{y} + k_{B} B_{y} + k_{α} α_{y} + k_{β} β_{y} \end{cases}

Here, A_x(B_x) and A_y(B_y) represent the components of misaligned parameter A(B) along $\vec{x}$ and $\vec{y}$ directions respectively, α_x(β_x) and α_y(β_y) represent the components of misaligned parameter α(β) in x = 0 plane and y = 0 plane respectively. α_x(β_x) is negative in Fig. 5. f_mx and f_my in Eq. (23) are the compensation factors for 3rd coma in TMAs.

Fig. 5 α_x(β_x) of a surface

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2.3 Coma compensation

Coma compensation in TMAs is accomplished just by simply setting the compensation factors in Eq. (23) to zero. Factors of misaligned parameters in Eq. (23) are constants in a specific TMA system and are independent from each other with respect to misalignment induced coma. That means coma is linearly affected by individual misaligned parameter. If one of the misaligned parameters in Eq. (23) becomes non-zero in a system, the other three parameters then form an equation to eliminate the coma. In practice, the best way to compensate the aberrations induced by tilts/decenters of a mirror is to apply the inverse tilts/decenters of the same mirror, which leads to an aligned system. However, Eq. (23) is very useful for applying active optics to correct the residual coma to achieve better performance when some mirrors in the system are fixed. Even though every change of a misalignment parameter can cause other aberrations, for example, astigmatism, only one of the four parameters in Eq. (23) is needed to eliminate the induced coma and leaves the other three to balance the other aberrations.

There are six cases when applying single-parameter compensation (use a single parameter to compensate the coma induced by another misaligned parameter). Only three of them are independent and can be calculated by Eqs. (5), (13), (21), and (22) as

{\begin{cases} k_{β} = - (m_{3} - 1) (^{m_{3} + 1) 2} [2 d_{2} - 2 d_{1} + r_{3} + \frac{2 d_{1} (2 d_{2} + r_{3})}{r_{2}}] \\ \frac{B}{β} = - \frac{k_{β}}{k_{B}} = - T, \frac{B}{α} = - \frac{k_{α}}{k_{B}} = G, \frac{A}{α} = - \frac{k_{α}}{k_{A}} = - 1 / (\frac{1}{r_{2}} + \frac{H}{G M}) \end{cases}

where

{\begin{matrix} M = b_{s 3} (m_{3} + 1) + (1 - m_{3}), T = r_{3} (1 - m_{3}) / M, H = (^{m_{2} + 1) 3} m_{3}^{3} b_{s 2} / {(m_{3} + 1)}^{2} \\ N = \frac{m_{3}^{3} (1 - m_{2}^{2})}{{(m_{3} + 1)}^{2} M} (\frac{d_{1} - d_{1} m_{2} + m_{2} r_{1}}{d_{1} + r_{2}}), G = N r_{2} + 2 d_{2} + 2 T \end{matrix}

The compensation factors in Eq. (23) and the other three cases of single-parameter compensation can be established by Eq. (24). The simplest case is that using tilt induced coma of the TM to compensate the coma induced by decenter of the TM itself and leads to the coma free point, Z_CFP3, for the TM. Note that, ${s^{'}}_{3} = r_{3} / [2 (m_{3} + 1)]$ , then we have

Z_{C F P 3} = \frac{B}{β} = - \frac{k_{β}}{k_{B}} = - {s^{'}}_{3} / [(\frac{m_{3} + 1}{2}) {1 - (\frac{m_{3} + 1}{m_{3} - 1})} b_{s 3}]

Equation (25) is exactly the negative form of the CFP of the SM (replace s₂ with s'₂ in [13]) in 2-mirror systems. Z_CFP3 is numerically positive in the system in Fig. 1 (bs3 is negative), which means that the CFP locates on axis A₁ between the center of curvature and the vertex of the TM. The last formula in Eq. (24) is Z_CFP2, the CFP of the SM.

A rotational symmetric coaxial TMA telescope is given in Table 1. Diameter of the stop is 500mm.

Table 1. Optical parameters of a coaxial TMA telescope. Units are mm and degrees.

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The TMA telescope in Table 1 is shown in Fig. 6(a). FFDs with respect to coma in the aligned and misaligned system are illustrated in Fig. 6(b) and Fig. 6(c) respectively.

Fig. 6 (a) Plot of the coaxial TMA in Table 1. (b) FFD in terms of coma at the image plane in the aligned system. (c) FFD when α_y = 0.02° is imposed on the SM.

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Figure 6(b) shows that coma near the edge of the field is corrected to improve the imaging quality. Field-constant coma is induced in Fig. 6(c) when system is misaligned.

Considering the case that B_y = 2mm with a direction along $\vec{y}$ , values of α_y and β_y are arbitrarily chosen to 0.01° and 0.02° respectively. By setting the compensation factors in Eq. (23) to zero, we have A_y = –0.0271mm. FFD after compensating of this case is shown in Fig. 7(a). Another general case is that while B_y = 2mm, α_y = 0.01°, β_y = 0.02°, and A_y = –0.0271mm already exist in the system, misalignment B_x = 0.5mm exists on the TM at the same time. With the same method, β_x and A_x are arbitrarily chosen to –0.03° and 0.02mm respectively, then we have α_x = 0.006663°. FFD after compensating of this case is shown in Fig. 7(b). Note that, signs of α_x and β_x in Code V are both inverse when compared to the sign convention in this paper.

Fig. 7 (a) B_y = 2mm, α_y = 0.01°, β_y = 0.02° and A_y = –0.0271mm are imposed on the surfaces. (b) On the basis of (a), B_x = 0.5mm, β_x = –0.03°, A_x = 0.02mm and α_x = 0.006663° are imposed on the surfaces at the same time.

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Coma in Figs. 7(a) and 7(b) are corrected to the aligned case in Fig. 6(b). The cases in Fig. 7 show that, only one misaligned parameter is needed to eliminate the coma while the other three parameters can be used to correct the other aberrations, for example, astigmatism.

3. Coma compensation in misaligned off-axis TMA telescopes

Off-axis TMA systems are designed to solve the obscuration problem that exists in reflecting coaxial systems with multiple mirrors. They are usually designed from the coaxial TMAs with the same structure parameters that include radii, spacings and conic constants [14]. These parameters are all the variables that the compensation factors need. Therefore, the compensation factors can be used to design the off-axis TMAs.

An off-axis TMA is shown in Fig. 8. The system tilts the SM and the TM at the same time and only uses parts of the mirrors to solve the obscuration problem that usually caused by the SM in coaxial TMAs. It actually can be treated as a misaligned coaxial TMA telescope.

Fig. 8 An off-axis TMA system

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From section 2, coma compensation in coaxial TMA telescope can be accomplished by simply setting the compensation factors, f_mx and f_my, to zero. Actually, parameters A, B, α and β in the aligned off-axis TMA shown in Fig. 8 must satisfy the zero condition of the compensation factors to eliminate the coma, which is illustrated below. That means $k_{A} A + k_{B} B + k_{α} α + k_{β} β$ equals to zero in Fig. 8. Misalignment parameters, ΔA, ΔB, Δα and Δβ, in a misaligned off-axis TMA system then form an equation for coma compensation as

Δ W_{131} = - \frac{{s^{'}}_{E^{'}}}{2} (k_{A} Δ A + k_{B} Δ B + k_{α} Δ α + k_{β} Δ β) (\vec{e} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ})

Equation (26) is the same form as the one in Eq. (22) and leads to the same compensation factors as those in coaxial TMAs.

An off-axial TMA telescope is given in Table 2. Diameter of the stop is 1000mm. YDE of the center of the stop is 771.605mm. YDE means decenter along $\vec{y}$ with respect to the axis of the primary mirror.

Table 2. Optical parameters for an off-axial TMA telescope. Units are mm and degrees.

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The TMA in Table 2 and FFDs for the aligned system and the case with Δα_y = 0.01° imposed on the SM are shown in Figs. 9(a)-9(c) respectively.

Fig. 9 (a) Plot of the off-axis TMA in Table 2. (b) FFD when system is aligned. (c) FFD when Δα_y = 0.01° is imposed on the SM

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Coma in an aligned off-axis TMA is almost clear as shown in Fig. 9(b) and misalignment induced coma is also constant over the field of view as shown in Fig. 9(c).

When Δα_y = 0.01° are imposed on SM, compensating parameters ΔA_y, ΔB_y and Δβ_y are separately calculated from Eq. (23) to 0.114mm, 2.611mm and –0.2483°. FFDs of these three cases are illustrated in Figs. 10(a)-10(c) respectively.

Fig. 10 FFDs in terms of coma at the image field when use (a) ΔA_y, (b) ΔB_y and (c) Δβ_y to accomplish the compensation while Δα_y = 0.01° are imposed on SM.

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The three cases in Fig. 10 shows that the compensation factors can eliminate the misalignment induced coma all over the field when compared to the aligned case in Fig. 9(b). The slight differences between Figs. 10(a)-10(c) and the aligned case in Fig. 9(b) are caused by the approximations that applied in the derivation and the changes of the parameters of surfaces. Note that, a misaligned system can be treated as a new system and will have a different field behavior.

Actually, compensation factors are applicable for tilt/decenter parameters of the SM and the TM in Table 2. Assuming decenter of the TM, tilts of the SM and the TM have already been set, by using the compensation factors, decenter of the SM is then calculated to be 13.5459mm, which is very close to the value, 13.4922mm, in Table 2. This case is shown in Fig. 11.

Fig. 11 Residual coma when use the compensation factors to redesign the TMA in Table 2.

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From Fig. 11 we know that tilts and decenters of SM and TM in an aligned off-axis TMA telescope satisfy the zero condition of the compensation factors, which means the compensation factors can be used to design off-axis TMAs. In fact, tilt and decenter parameters in an aligned off-axis TMA must satisfy the zero condition of the compensation factors due to the reason that coma can only be eliminated while the compensation factors equal to zero.

From the NAT and derivation of the compensation factors, we know that, the compensation factors exist not only in the TMAs but also in any reflecting telescope with multiple mirrors and circular pupils. They just have different forms in telescopes with different number of mirrors.

4. Conclusion

The compensation factors show that misalignment induced coma is linearly affected by misaligned parameters. All the four misaligned parameters, A, B, α, and β, can be used to accomplish coma compensation in misaligned TMAs. By setting the compensation factors to zero, existence of CFP for the SM in TMA systems is proved and Z_CFP of the TM is proved to be the negative form of the Z_CFP of the SM in the Cassegrain telescope. Compensation factors can also be used to design off-axis TMAs due to its capability of eliminating coma in the system. In this application, only one tilt/decenter parameter is needed to eliminate coma and the other three parameters can be used to correct the other aberrations. Compensation factors for 3rd order coma are not limited to TMAs. They exist in any reflecting telescope with multiple mirrors and circular/near circular pupils and can be calculated by the same method in this paper.

References and links

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

2. R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1976).

3. 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]

4. 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]

5. 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]

6. 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]

7. K. P. Thompson, T. Schmid, and J. P. Rolland, “Recent Discoveries from Nodal Aberration Theory,” Proc. SPIE 7652, 76522Q (2010). [CrossRef]

8. H. Shi, H. Jiang, X. Zhang, C. Wang, and T. Liu, “Analysis of nodal aberration properties in off-axis freeform system design,” Appl. Opt. 55(24), 6782–6790 (2016). [CrossRef] [PubMed]

9. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Extending Nodal Aberration Theory to include mount-induced aberrations with application to freeform surfaces,” Opt. Express 20(18), 20139–20155 (2012). [CrossRef] [PubMed]

10. A. Bauer, J. P. Rolland, and K. P. Thompson, “Ray-based optical design tool for freeform optics: coma full-field display,” Opt. Express 24(1), 459–472 (2016). [CrossRef] [PubMed]

11. X. Zhang, S. Xu, H. Ma, and N. Liu, “Optical compensation for the perturbed three mirror anastigmatic telescope based on nodal aberration theory,” Opt. Express 25(11), 12867–12883 (2017). [CrossRef] [PubMed]

12. B. Ren, G. Jin, and X. Zhong, “Third-order coma-free point in two-mirror telescopes by a vector approach,” Appl. Opt. 50(21), 3918–3923 (2011). [CrossRef] [PubMed]

13. R. N. Wilson, Reflecting Telescope Optics I (Springer, 1996).

14. K. P. Thompson, K. Fuerschbach, T. Schmid, and J. P. Rolland, “Using Nodal Aberration Theory to understand the aberrations of multiple unobscured Three Mirror Anastigmatic (TMA) telescopes,” Proc. SPIE 7433, 74330B (2009). [CrossRef]

Surface	Type	Conic constant	Radius(mm)	Spacing(mm)
Primary(STOP)	Conic	–0.88187	–1711.0871	–683.0066
Secondary	Conic	–1.8928	–480.2491	1348.499
Tertiary	Conic	–0.37946	–662.7358	–603.065
Image	Flat	–	–	–

Surface	Type	Conic constant	Radius(mm)	Spacing(mm)	Axis Tilt (degree)	Axis Offset (YDE, mm)
Primary(STOP)	Conic	–0.9366	–4449.008	–1773.631	–	–
Secondary	Conic	–2.7952	–1304.154	3042.86	1.5884	13.4922
Tertiary	Conic	–0.3877	–1872.508	–2287.68	3.9224	145.961
Image	Flat	–	–	–	6.4097	–10.895

Surface	Type	Conic constant	Radius(mm)	Spacing(mm)
Primary(STOP)	Conic	–0.88187	–1711.0871	–683.0066
Secondary	Conic	–1.8928	–480.2491	1348.499
Tertiary	Conic	–0.37946	–662.7358	–603.065
Image	Flat	–	–	–

Surface	Type	Conic constant	Radius(mm)	Spacing(mm)	Axis Tilt (degree)	Axis Offset (YDE, mm)
Primary(STOP)	Conic	–0.9366	–4449.008	–1773.631	–	–
Secondary	Conic	–2.7952	–1304.154	3042.86	1.5884	13.4922
Tertiary	Conic	–0.3877	–1872.508	–2287.68	3.9224	145.961
Image	Flat	–	–	–	6.4097	–10.895

Compensation factors for 3rd order coma in three mirror anastigmatic (TMA) telescopes

Abstract

1. Introduction

2. Coma compensation in misaligned coaxial TMA telescopes

2.1 Coma coefficients of the SM and the TM

2.2 Contribution centers of the SM and the TM

2.3 Coma compensation

3. Coma compensation in misaligned off-axis TMA telescopes

4. Conclusion

References and links

Cited By

Figures (11)

Tables (2)

Equations (29)

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