Optical design method of two-mirror astronomical telescopes with reduced misalignment sensitivities based on the nodal aberration theory

Hongcai Ma; Guohao Ju; Xiaobin Zhang

doi:10.1364/OSAC.1.000145

1. Introduction

The misalignment sensitivities of astronomical telescopes are important for tolerance budgeting and optical alignment [1–4]. On the one hand, an optical system with relaxed tolerance has stronger ability to resist surrounding environment perturbation and lower demand of complexity for support structure, especially for the space telescopes [5,6]. On the other hand, in the alignment stage, an optical system with low misalignment sensitivities is more convenient to align.

In traditional optical design process, the selected initial structure parameters (including radius of mirror, thickness between mirrors and so on) are optimized to a good image quality. Then the tolerance sensitivities of mirror are checked independently. To increase the efficiency of optical design, tolerances are more and more integrated into optimization process. Rogers described a desensitization method based on inclusion of the as-built performance directly into the optimization error function and examined the addition of a simple sensitivity parameter to the merit function [7,8]. This approach selected the weighting factor for the sensitivity parameter based on the numerical method, which required intensive computation.

Some methods for designing reflective mirror telescopes with reduced misalignment sensitivities have also been previously proposed [9,10]. Specifically, Scaduto et al proposed that by choosing proper conic constant of the secondary mirror (SM), a two-mirror telescope with low sensitivities of coma to corresponding decenter of the SM can be obtained [11]. However, the sensitivities of coma and astigmatic aberration to tilt misalignments still have not been explored. Besides, the sensitivities of astigmatic aberration to decenter misalignments have not been considered quantitatively.

Nodal aberration theory (NAT) can analytically describe the aberration field of the optical systems in the presence of misalignments [12–14], which can supply considerable insights for tolerance sensitivities. Recently, Bauman et al proposed an approach to optimize tolerances sensitivities in the guide of NAT [15]. By perturbing in a numerical optical model, the relations between the perturbations of decentered optics and double Zernike polynomials have been built. In the reference [15], the tolerance sensitivities are obtained by numerical samples and this method has the limitation that the analytic relations between misalignment sensitivities and optical structure parameters cannot be obtained. Consequently the factors influencing the misalignment sensitivities cannot be explained essentially, which is important for us to develop the optimization method of misalignment sensitivities.

Therefore, in the previous researches the systematic design method for astronomical telescopes with reduced misalignment sensitivities based on analytic theory has not been proposed. In this paper, we focus on building analytic relations between the misalignment sensitivities and the optical structure parameters for two-mirror telescopes based on the NAT. On this basis, an optical design method of two-mirror system with reduced misalignment sensitivities is further proposed.

This paper is organized as follows. In Section 2, we derive the sensitivities of coma and astigmatism aberration to lateral misalignments for the two-mirror telescopes based on NAT. In Section 3, a systematic optical design method for considering reduced misalignment sensitivities and good image quality simultaneously is proposed. Then the Mt. Hopkins Telescope is taken as an example to demonstrate the feasibility of the design method in Section 4. In Section 5, we summarize and conclude the paper.

2. Analytic expressions for misalignment sensitivities of two-mirror telescopes

In the two-mirror telescope, the location of the primary mirror (PM) is usually used as a reference and the misalignment sensitivities of SM are analyzed here. The lateral misalignments of SM such as two decenter and two tilt misalignments are considered. Four dominant low order terms (Z₅: Astig, axis at 0°/90°, Z₆: Astig, axis at ± 45°, Z₇: Coma,x-axis, Z₈: Coma, y-axis) of Fringe Zernike polynomials for exit-pupil wave-front are discussed here. The specific expressions of SM misalignment sensitivities for two-mirror telescopes are derived based on the NAT. In this section, the misalignment sensitivities will be expressed as the analytic function of field of view (FOV) and optical structure parameter.

The vector form of wave aberration expansion is the theoretic basis of NAT, which was discovered by Shack [16] and developed by Thompson [17–21]. Wave aberration function for the perturbed rotationally symmetric system is expressed as

\begin{matrix} W = \sum_{j} \sum_{p}^{\infty} \sum_{n}^{\infty} \sum_{m}^{\infty} {(W_{k l m})}_{j} {[(\vec{H} - {\vec{σ}}_{j}) \cdot (\vec{H} - {\vec{σ}}_{j})]}^{p} {[\vec{ρ} \cdot \vec{ρ}]}^{n} {[(\vec{H} - {\vec{σ}}_{j}) \cdot \vec{ρ}]}^{m}, \\ k = 2 p + m, l = 2 n + m, \end{matrix}

where

\vec{H}

is the normalized field vector,

\vec{ρ}

is the normalized pupil vector,

{\vec{σ}}_{j}

is the introduced aberration field decenter vector of surface j,

{(W_{k l m})}_{j}

is the corresponding wave aberration coefficient.

2.1 Sensitivities of astigmatism aberration to lateral misalignments

In the presence of misalignments, the third-order astigmatism aberration field based on NAT can be given by

W_{A S T} = \frac{1}{2} \sum_{j} W_{222 j} [{(\vec{H} - {\vec{σ}}_{j})}^{2} \cdot {\vec{ρ}}^{2}] .

Expanding the above Eq. (2), we can obtain

W_{A S T} = \frac{1}{2} [(\sum_{j} W_{222 j}) {\vec{H}}^{2} - 2 \vec{H} (\sum_{j} W_{222 j} {\vec{σ}}_{j}) + (\sum_{j} W_{222 j} {\vec{σ}}_{j}^{2})] \cdot {\vec{ρ}}^{2} .

Here, the aberration field decenter vector is also expressed as a linear combination of the misalignments [22,23] and only the linear term of the misalignments for the misalignment sensitivities is considered. Then we focus on the linear term of the aberration field decenter vector,

Δ W_{A S T} = \frac{1}{2} [- 2 \vec{H} {\vec{A}}_{222}] \cdot {\vec{ρ}}^{2},

where

{\begin{cases} {\vec{A}}_{222 x} = W_{222, S M}^{s p h} {\vec{σ}}_{S M, x}^{s p h} + W_{222, S M}^{a s p h} {\vec{σ}}_{S M, x}^{a s p h} \\ {\vec{A}}_{222 y} = W_{222, S M}^{s p h} {\vec{σ}}_{S M, y}^{s p h} + W_{222, S M}^{a s p h} {\vec{σ}}_{S M, y}^{a s p h} \end{cases},

where

{\vec{A}}_{222 x}

and

{\vec{A}}_{222 y}

are the x-component and y-component of

{\vec{A}}_{222}

,

W_{222, S M}^{s p h}

and

W_{222, S M}^{a s p h}

are the wave aberration coefficients of SM,

{\vec{σ}}_{S M, x}^{s p h}

,

{\vec{σ}}_{S M, x}^{a s p h}

,

{\vec{σ}}_{S M, y}^{s p h}

, and

{\vec{σ}}_{S M, y}^{a s p h}

are the aberration field decenter vectors of SM, sph and asph denote the spherical and aspherical contributions, respectively.

According to the vector multiplication, the Eq. (4) can be rewritten as

[\begin{matrix} - {\vec{H}}_{x} & {\vec{H}}_{y} \\ - {\vec{H}}_{y} & - {\vec{H}}_{x} \end{matrix}] [\begin{array}{l} {\vec{A}}_{222, x} \\ {\vec{A}}_{222, y} \end{array}] = [\begin{matrix} Δ C_{5}^{W_{222}} \\ Δ C_{6}^{W_{222}} \end{matrix}],

where

Δ C_{5}^{W_{222}}

and

Δ C_{6}^{W_{222}}

denote the 5th and 6th Fringe Zernike coefficients contributed from third-order astigmatism variation induced by misalignments. For two-mirror astronomical telescopes, we can neglect the fifth order aberrations. Therefore, the 5th and 6th Fringe Zernike coefficients variation of the two-mirror system exit-pupil wave-front (

Δ C_{5}^{s y s t e m}

and

Δ C_{6}^{s y s t e m}

) mainly come from third-order astigmatism variation.

Here, we assume that the stop aperture is located at the PM. Referred to Eq. (3-34)–Eq. (3-37) in [24], the aberration field decenter vectors are expressed with lateral misalignments of SM for two-mirror telescope as followed:

{\begin{cases} {\vec{σ}}_{S M, x}^{s p h} = \frac{B D E_{S M} - c_{S M} X D E_{S M}}{(1 + c_{S M} d) {\bar{u}}_{P M}} \\ {\vec{σ}}_{S M, y}^{s p h} = - \frac{A D E_{S M} + c_{S M} Y D E_{S M}}{(1 + c_{S M} d) {\bar{u}}_{P M}} \\ {\vec{σ}}_{S M, x}^{a s p h} = - \frac{X D E_{S M}}{d {\bar{u}}_{P M}} \\ {\vec{σ}}_{S M, y}^{a s p h} = - \frac{Y D E_{S M}}{d {\bar{u}}_{P M}} \end{cases},

where

{\bar{u}}_{P M}

is the paraxial chief ray incident angle at PM,

d

is the thickness between PM and SM which is negative,

c_{S M}

is the curvature of SM,

X D E_{S M}

and

Y D E_{S M}

are the decenter misalignments of SM, and

A D E_{S M}

and

B D E_{S M}

are the tilt misalignments of SM.

From Eq. (5) to Eq. (7), the misalignment sensitivities are expressed as the analytic function of FOV and optical structure parameter:

{\begin{cases} S_{X D E_{S M}}^{C_{5}} = \frac{\partial C_{5}^{s y s t e m}}{\partial X D E_{S M}} \approx \frac{Δ C_{5}^{s y s t e m}}{X D E_{S M}} = H_{x} \cdot (α_{A S T} + β_{A S T}) \\ S_{A D E_{S M}}^{C_{5}} = \frac{\partial C_{5}^{s y s t e m}}{\partial A D E_{S M}} \approx \frac{Δ C_{5}^{s y s t e m}}{A D E_{S M}} = H_{y} \cdot (- \frac{α_{A S T}}{c_{S M}}) \\ \frac{H_{y}}{H_{x}} \cdot S_{Y D E_{S M}}^{C_{6}} = S_{X D E_{S M}}^{C_{6}} = - S_{Y D E_{S M}}^{C_{5}} = \frac{H_{y}}{H_{x}} \cdot S_{X D E_{S M}}^{C_{5}} \\ \frac{H_{x}}{H_{y}} \cdot S_{B D E_{S M}}^{C_{6}} = - S_{A D E_{S M}}^{C_{6}} = S_{B D E_{S M}}^{C_{5}} = \frac{H_{x}}{H_{y}} \cdot S_{A D E_{S M}}^{C_{5}} \\ α_{A S T} = \frac{1}{{\bar{u}}_{P M}} W_{222, S M}^{s p h} \frac{c_{S M}}{(1 + c_{S M} d)} \\ β_{A S T} = \frac{1}{{\bar{u}}_{P M}} W_{222, S M}^{a s p h} (\frac{1}{d}) \end{cases},

where

S_{M i s a l i g n m e n t}^{C_{5} / C_{6}}

denote the sensitivities of C5/C6 with respect to certain misalignment, and

α_{A S T}

and

β_{A S T}

represent the intermediate factor.

The results of Eq. (8) show that the sensitivities of $C_{5}^{s y s t e m}$ and $C_{6}^{s y s t e m}$ to lateral misalignments are field-dependent. The relation among the misalignment sensitivities for different field points depends on the FOV functions. For a two-mirror system with a determined FOV, the misalignment sensitivity of one specific field point (except on-axis field) can characterize the misalignment sensitivities of all the field point. There also exists proportional relation among different misalignment sensitivities of astigmatism aberration. For example, the amplitude of misalignment sensitivity $S_{X D E_{S M}}^{C_{5}}$ equals to the misalignment sensitivity $S_{X D E_{S M}}^{C_{6}}$ and the amplitude of misalignment sensitivity $S_{Y D E_{S M}}^{C_{5}}$ is the negative of the misalignment sensitivity $S_{X D E_{S M}}^{C_{6}}$ .

Furthermore, astigmatic aberration coefficients $W_{222, S M}^{s p h}$ and $W_{222, S M}^{a s p h}$ of two-mirror telescope expressed by the structure parameters are derived in [25] as bellowed.

{\begin{cases} W_{222, S M}^{s p h} = \frac{1}{2} {(\frac{y_{1}}{f^{'}})}^{2} ξ^{0} L {[(\frac{- d}{L} - \frac{E P T}{f^{'}}) + \frac{2 f^{'}}{L (m_{2} - 1)}]}^{2} {\bar{u}}^{2}_{P M} \\ W_{222, S M}^{a s p h} = \frac{1}{2} {(\frac{y_{1}}{f^{'}})}^{2} ξ^{*} L {(\frac{- d}{L} - \frac{E P T}{f^{'}})}^{2} {\bar{u}}^{2}_{P M} \end{cases},

where

y_{1}

is the semi-diameter of PM,

f^{'}

is the focal length of the two-mirror telescope, EPT is the distance of entrance pupil compared to PM,

m_{2}

is the magnification of SM and

m_{2} = \frac{L - f^{'}}{- d}

,

ξ^{0} = \frac{{(m_{2} + 1)}^{3}}{4} {(\frac{m_{2} - 1}{m_{2} + 1})}^{2}

,

ξ^{*} = \frac{{(m_{2} + 1)}^{3}}{4} b_{s 2}

, and

b_{s 2}

is the conic constant of SM. The Gaussian optics of a two-mirror telescope is as shown in Fig. 1

Fig. 1 The Gaussian optics of a two-mirror telescope.

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.

Here, the EPT equals to zero. Substituting Eq. (9) for the $W_{222, S M}^{s p h}$ and $W_{222, S M}^{a s p h}$ in Eq. (8), the $S_{X D E_{S M}}^{C_{5}}$ and $S_{A D E_{S M}}^{C_{5}}$ are simplified as below:

{\begin{cases} S_{X D E_{S M}}^{C_{5}} = H_{x} \cdot \frac{1}{8} {(\frac{y_{1}}{f^{'}})}^{2} {\bar{u}}_{P M} (^{m_{2} + 1) 2} \frac{((L + d + f^{'}) + (- L + d + f^{'}) b_{s 2})}{L} \\ S_{A D E_{S M}}^{C_{5}} = - H_{y} \cdot \frac{1}{4} {(\frac{y_{1}}{f^{'}})}^{2} {\bar{u}}_{P M} (m_{2} + 1) L (L + d + f^{'}) \\ = - H_{y} \cdot \frac{1}{4} {(\frac{y_{1}}{f^{'}})}^{2} {\bar{u}}_{P M} \frac{{(d + f^{'})}^{2} - L^{2}}{d} \end{cases} .

From Eq. (10) we know the misalignment sensitivities of

C_{5}^{s y s t e m}

and

C_{6}^{s y s t e m}

is proportional to the square ratio of relative aperture

\frac{y_{1}}{f^{'}}

. Besides, the misalignment sensitivities of

C_{5}^{s y s t e m}

and

C_{6}^{s y s t e m}

is also proportional to the

{\bar{u}}_{P M}

.

The structure parameter conditions of $S_{X D E_{S M}}^{C_{5}} = 0$ or $S_{A D E_{S M}}^{C_{5}} = 0$ can easily be obtained. If the sensitivity of $C_{5}^{s y s t e m}$ to $X D E_{S M}$ equals to zero, we can obtain $L - d - f^{'} = 0$ or $b_{s 2} = \frac{L + d + f^{'}}{L - d - f^{'}}$ . If the sensitivity of $C_{5}^{s y s t e m}$ to $A D E_{S M}$ equals to zero, we can obtain $L - d - f^{'} = 0$ or $L + d + f^{'} = 0$ .

2.2 Sensitivities of coma aberration to lateral misalignments

Similar to the manner used in above section, we derive the misalignment sensitivities of coma aberration to lateral misalignments. According to NAT, the third-order coma aberration is expressed as

W = \sum_{j} W_{131 j} [(\vec{H} - {\vec{σ}}_{j}) \cdot \vec{ρ}] (\vec{ρ} \cdot \vec{ρ}) .

Expanding the above Eq. (11), we can obtain

W = {[(\sum_{j} W_{131 j}) \vec{H} - (\sum_{j} W_{131 j} {\vec{σ}}_{j})] \cdot \vec{ρ}} (\vec{ρ} \cdot \vec{ρ}) .

Here, we also focus the linear term of the aberration field decenter vector,

Δ W_{C O M A} = {- {\vec{A}}_{131} \cdot \vec{ρ}} (\vec{ρ} \cdot \vec{ρ}),

where

{\begin{cases} {\vec{A}}_{131 x} = W_{131, S M}^{s p h} {\vec{σ}}_{S M, x}^{s p h} + W_{131, S M}^{a s p h} {\vec{σ}}_{S M, x}^{a s p h} \\ {\vec{A}}_{131 y} = W_{131, S M}^{s p h} {\vec{σ}}_{S M, y}^{s p h} + W_{131, S M}^{a s p h} {\vec{σ}}_{S M, y}^{a s p h} \end{cases},

where

{\vec{A}}_{131 x}

and

{\vec{A}}_{131 y}

are the x-component and y-component of

{\vec{A}}_{131}

,

W_{131, S M}^{s p h}

and

W_{131, S M}^{a s p h}

are the wave aberration coefficients of SM.

According to the vector multiplication, Eq. (13) of aberration field can be expressed by a matrix, which is given by

\frac{1}{3} [\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}] [\begin{array}{l} {\vec{A}}_{131, x} \\ {\vec{A}}_{131, y} \end{array}] = [\begin{matrix} Δ C_{7}^{W_{131}} \\ Δ C_{8}^{W_{31}} \end{matrix}],

where

Δ C_{7}^{W_{131}}

and

Δ C_{8}^{W_{131}}

denote the 7th and 8th Fringe Zernike coefficients contributed from third-order coma variation. Here, the 7th and 8th Fringe Zernike coefficients variation of the two-mirror system exit-pupil wave-front (

Δ C_{7}^{s y s t e m}

and

Δ C_{8}^{s y s t e m}

) mainly come from third-order coma variation.

From Eqs. (14), (15), and (7), the misalignment sensitivities are derived as followed:

{\begin{cases} S_{X D E_{S M}}^{C_{7}} = \frac{\partial C_{7}^{s y s t e m}}{\partial X D E_{S M}} \approx \frac{Δ C_{7}^{s y s t e m}}{X D E_{S M}} = \frac{1}{3} (α_{c o m a} + β_{c o m a}) \\ S_{B D E_{S M}}^{C_{7}} = \frac{\partial C_{7}^{s y s t e m}}{\partial B D E_{S M}} \approx \frac{Δ C_{7}^{s y s t e m}}{B D E_{S M}} = - \frac{1}{3} \frac{α_{c o m a}}{c_{S M}} \\ S_{Y D E_{S M}}^{C_{8}} = S_{X D E_{S M}}^{C_{7}} \\ S_{A D E_{S M}}^{C_{8}} = - S_{B D E_{S M}}^{C_{7}} \\ S_{X D E_{S M}}^{C_{8}} = S_{B D E_{S M}}^{C_{8}} = S_{Y D E_{S M}}^{C_{7}} = S_{A D E_{S M}}^{C_{7}} = 0 \\ α_{c o m a} = \frac{1}{{\bar{u}}_{P M}} W_{131, S M}^{s p h} \frac{c_{S M}}{(1 + c_{S M} d)} \\ β_{c o m a} = \frac{1}{{\bar{u}}_{P M}} W_{131, S M}^{a s p h} (\frac{1}{d}) \end{cases},

where

S_{M i s a l i g n m e n t}^{C_{7} / C_{8}}

denote the sensitivities of C7/C8 to certain misalignment,

α_{c o m a}

and

β_{c o m a}

represent the intermediate factor.

The results of Eq. (16) show that the misalignment sensitivities of $C_{7}^{s y s t e m}$ and $C_{8}^{s y s t e m}$ is field-constant. There also exists proportional relation among different misalignment sensitivities of coma aberration. For example, the amplitude of misalignment sensitivity $S_{X D E_{S M}}^{C_{7}}$ equals to the misalignment sensitivity $S_{Y D E_{S M}}^{C_{8}}$ . The amplitude of misalignment sensitivity $S_{A D E_{S M}}^{C_{7}}$ equals the misalignment sensitivity $S_{B D E_{S M}}^{C_{8}}$ , which also equal to zero.

Furthermore, Coma aberration coefficients $W_{131, S M}^{s p h}$ and $W_{131, S M}^{a s p h}$ of the two-mirror telescope expressed by the structure parameters are derived in [25] as bellowed.

{\begin{cases} W_{131, S M}^{s p h} = \frac{1}{2} {(\frac{y_{1}}{f^{'}})}^{3} ξ^{0} [- d - \frac{L}{f^{'}} E P T + \frac{2 f^{'}}{(m_{2} - 1)}] {\bar{u}}_{P M} \\ W_{131, S M}^{a s p h} = \frac{1}{2} {(\frac{y_{1}}{f^{'}})}^{3} ξ^{*} [- d - \frac{L}{f^{'}} E P T] {\bar{u}}_{P M} \end{cases} .

Here, the EPT equals to zero. Substituting Eq. (17) for the $W_{131, S M}^{s p h}$ and $W_{131, S M}^{a s p h}$ in Eq. (16), the $S_{X D E_{S M}}^{C_{7}}$ and $S_{B D E_{S M}}^{C_{7}}$ are simplified as below:

{\begin{cases} S_{X D E_{S M}}^{C_{7}} = \frac{1}{24} {(\frac{y_{1}}{f^{'}})}^{3} (^{m_{2} + 1) 2} ((m_{2} - 1) - (m_{2} + 1) b_{s 2}) \\ S_{B D E_{S M}}^{C_{7}} = - \frac{1}{12} {(\frac{y_{1}}{f^{'}})}^{3} (m_{2} - 1) (m_{2} + 1) L \\ = - \frac{1}{12} {(\frac{y_{1}}{f^{'}})}^{3} L ({(\frac{f^{'} - L}{d})}^{2} - 1) \end{cases} .

From Eq. (18) we know, the misalignment sensitivities of $C_{7}^{s y s t e m}$ and $C_{8}^{s y s t e m}$ are proportional to the triple ratio of relative aperture $\frac{y_{1}}{f^{'}}$ .

The structure parameter conditions of $S_{X D E_{S M}}^{C_{7}} = 0$ or $S_{B D E_{S M}}^{C_{7}} = 0$ can easily be obtained. If the sensitivity of $C_{7}^{s y s t e m}$ to $X D E_{S M}$ equals to zero, we can get $L - d - f^{'} = 0$ or $b_{s 2} = \frac{m_{2} - 1}{m_{2} + 1}$ . The result of $b_{s 2} = \frac{m_{2} - 1}{m_{2} + 1}$ is consistent with the conclusion in Reference [11] and it equivalent to the coma-free-point located at infinite distance. If the sensitivity of $C_{7}^{s y s t e m}$ to $A D E_{S M}$ equals to zero, we can get $L - d - f^{'} = 0$ or $L + d + f^{'} = 0$ . It equivalent to the coma-free-point located at the vertex of secondary mirror.

3. Optical design method with reduced misalignment sensitivities for two-mirror telescope

According to the Eqs. (10) and (18), the conic constant of SM determines the misalignment sensitivities $S_{X D E_{S M}}^{C_{5}}$ and $S_{X D E_{S M}}^{C_{7}}$ . If $S_{X D E_{S M}}^{C_{5}} = 0$ or $S_{X D E_{S M}}^{C_{7}} = 0$ , specific conic constant of SM can be chosen to reduce the misalignment sensitivities. Then the conic constant cannot satisfy the third-order aberration correction and reduced misalignment sensitivities simultaneously. In this case the high-order even aspherics are required in the figure of SM to improve image quality.

To correct third-order spherical and third-order coma, the R-C telescope or Cassegrain telescope has a specific secondary conic constant, which is determined by the $d$ , $L$ and $f^{'}$ [26]. Based on the restrain of optical design, the structure parameters $d$ and L are always selected in a certain range. Therefore, the misalignment sensitivities can be reduced by adjusting the structure parameters $d$ and $L$ . This design method of optical system permits the optimization of both reduced misalignment sensitivities and good image quality.

In this paper, the design procedure for two-mirror telescope with reduced misalignment sensitivities based on the analytic theory is shown in the following steps:

1) Determine the initial $d_{0}$ , $L_{0}$ and ${f^{'}}_{0}$ and the upper limit and lower limit of the structure parameters $d$ and $L$ .
2) Derive analytic function of conic constant for corrections of certain aberrations.
3) Derive misalignment sensitivities functions with respect to variable $d$ and $L$ and its partial derivatives.
4) Minimize the root-mean-square (RMS) of misalignment sensitivities [specifically defined in Eq. (25)] by the monotonicity of the misalignment sensitivities functions.
5) Calculate the structure parameters $b_{s 1}$ , $b_{s 2}$ , $c_{P M}$ and $c_{S M}$ based on the corrections of certain aberrations.
6) Check the performance of the system in design software.

To furthermore demonstrate this design method, we take R-C telescope as an example. The R-C telescope is corrected for both third-order spherical aberration and third-order coma aberration, which result in the following conic constants [26]:

b_{s 1} = \frac{1}{{(c_{P M})}^{3}} \frac{[2 L d^{2} - {(L - f^{'})}^{2}]}{8 {(- d)}^{3} {(f^{'})}^{3}},

b_{s 2} = \frac{1}{{(c_{S M})}^{3}} \frac{[2 f^{'} {(L - f^{'})}^{2} + (f^{'} - (- d) - L) (f^{'} + (- d) - L) ((- d) - f^{'} - L)]}{8 {(- d)}^{3} L^{3}},

where

b_{s 1}

is the conic constant of PM, and

c_{P M}

is the curvature of the PM.

c_{P M}

and

c_{S M}

can be written as:

c_{P M} = \frac{L - f^{'}}{2 (- d) f^{'}} .

c_{S M} = \frac{L + (- d) - f^{'}}{2 (- d) L} .

Substituting Eqs. (20) and (22) for $b_{s 2}$ in Eq. (16), the $S_{X D E_{S M}}^{C_{5}}$ and $S_{X D E_{S M}}^{C_{7}}$ are rewritten furthermore as below:

{\begin{cases} S_{X D E_{S M}}^{C_{7}} = \frac{1}{12} {(\frac{y_{1}}{f^{'}})}^{3} (m_{2}^{3} + \frac{L}{d}) = \frac{1}{12} {(\frac{y_{1}}{f^{'}})}^{3} \frac{1}{{(- d)}^{3}} ({(L - f^{'})}^{3} - d^{2} L) \\ S_{X D E_{S M}}^{C_{5}} = H_{x} \cdot \frac{1}{4} {(\frac{y_{1}}{f^{'}})}^{2} {\bar{u}}_{P M} m_{2} (m_{2} + \frac{2 f^{'} + d}{L}) \\ = H_{x} \cdot \frac{1}{4} {(\frac{y_{1}}{f^{'}})}^{2} {\bar{u}}_{P M} \frac{(L - f^{'})}{d^{2} L} (L (L - f^{'}) - d (2 f^{'} + d)) \end{cases} .

All the misalignment sensitivities (including

S_{X D E_{S M}}^{C_{5}}

,

S_{A D E_{S M}}^{C_{5}}

,

S_{X D E_{S M}}^{C_{7}}

and

S_{B D E_{S M}}^{C_{7}}

) are expressed by the structure parameters

y_{1}

,

f^{'}

,

{\bar{u}}_{P M}

,

d

and

L

. Among them, the

d

and

L

are used as optimization variables.

System average misalignment sensitivities (SAMS) over the FOV is defined as below:

S_{s y s} = {(\frac{\iint_{F i e l d} \sum_{M i s a l i g n m e n t} \sum_{Z e r n i k e} {(S_{M i s a l i g n m e n t}^{Z e r n i k e})}^{2} d H_{x} d H_{y}}{\iint_{F i e l d} d H_{x} d H_{y}})}^{1 / 2},

where

S_{M i s a l i g n m e n t}^{Z e r n i k e}

represents the sensitivity of Zernike coefficient to the lateral misalignment. The SAMS means the RMS of misalignment sensitivity over the area FOV. It integrates the sum of misalignment sensitivity functions square over the FOV and then roots the average of that integration over the FOV.

According to Eqs. (8) and (16), the proportion relation of misalignment sensitivities show only four of them ( $S_{X D E_{S M}}^{C_{5}}$ , $S_{A D E_{S M}}^{C_{5}}$ , $S_{X D E_{S M}}^{C_{7}}$ and $S_{B D E_{S M}}^{C_{7}}$ ) are independent. So the misalignment sensitivities $S_{X D E_{S M}}^{C_{5}}$ , $S_{A D E_{S M}}^{C_{5}}$ , $S_{X D E_{S M}}^{C_{7}}$ and $S_{B D E_{S M}}^{C_{7}}$ for one certain FOV (except zero field point) can characterize that of all the FOV for two-mirror system. In this paper, the merit function is the SAMS for single FOV simplified as below:

S_{R M S} = \sqrt{{(S_{X D E_{S M}}^{C_{5}})}^{2} + {(S_{A D E_{S M}}^{C_{5}})}^{2} + {(S_{X D E_{S M}}^{C_{7}})}^{2} + {(S_{B D E_{S M}}^{C_{7}})}^{2}} .

To pursue the optimal $S_{R M S}$ , the minimum values of the dominant misalignment sensitivities are determined based on the monotonicity of the misalignment sensitivities functions with respect to the $d$ and $L$ . The monotonicity depends on the partial derivatives of misalignment sensitivities function.

Specifically, the partial derivatives of misalignment sensitivities $S_{X D E_{S M}}^{C_{5}}$ , $S_{A D E_{S M}}^{C_{5}}$ , $S_{X D E_{S M}}^{C_{7}}$ and $S_{B D E_{S M}}^{C_{7}}$ with respect to variable $d$ and $L$ are shown as followed, respectively:

{\begin{cases} \frac{\partial S_{X D E_{S M}}^{C_{5}}}{\partial d} = H_{x} \cdot \frac{1}{2} {(\frac{y_{1}}{f^{'}})}^{2} {\bar{u}}_{P M} \frac{(L - f)}{d^{3} L} (L (f^{'} - L) + f^{'} d) \\ \frac{\partial S_{X D E_{S M}}^{C_{5}}}{\partial L} = H_{x} \cdot \frac{1}{4} {(\frac{y_{1}}{f^{'}})}^{2} {\bar{u}}_{P M} \frac{1}{d} (\frac{2 (L - f^{'})}{d} - \frac{(2 f + d) f^{'}}{L^{2}}) \end{cases},

{\begin{cases} \frac{\partial S_{A D E_{S M}}^{C_{5}}}{\partial d} = - H_{y} \cdot \frac{1}{4} {(\frac{y_{1}}{f^{'}})}^{2} {\bar{u}}_{P M} \frac{d^{2} - {(f^{'})}^{2} + L^{2}}{d^{2}} \\ \frac{\partial S_{A D E_{S M}}^{C_{5}}}{\partial L} = H_{y} \cdot \frac{1}{4} {(\frac{y_{1}}{f^{'}})}^{2} {\bar{u}}_{P M} \frac{2 L}{d} \end{cases},

{\begin{cases} \frac{\partial S_{X D E_{S M}}^{C_{7}}}{\partial d} = \frac{1}{12} {(\frac{y_{1}}{f^{'}})}^{3} \frac{1}{d^{2}} (\frac{3 {(L - f)}^{3}}{d^{2}} - L) \\ \frac{\partial S_{X D E_{S M}}^{C_{7}}}{\partial L} = - \frac{1}{12} {(\frac{y_{1}}{f^{'}})}^{3} \frac{(3 {(L - f)}^{2} - d^{2})}{d^{3}} \end{cases},

{\begin{cases} \frac{\partial S_{B D E_{S M}}^{C_{7}}}{\partial d} = \frac{1}{6} {(\frac{y_{1}}{f^{'}})}^{3} \frac{L {(L - f^{'})}^{2}}{d^{3}} \\ \frac{\partial S_{B D E_{S M}}^{C_{7}}}{\partial L} = - \frac{1}{12} {(\frac{y_{1}}{f^{'}})}^{3} (\frac{(f^{'} - L) (f^{'} - 3 L)}{d^{2}} - 1) \end{cases} .

Based on the Eqs. (10), (18), (23), and (26)–(29), the monotonicity that satisfies reduced misalignment sensitivities is presented in Table 5 of the Appendix 1. The amplitude relation conditions of the parameters

d

,

L

and

f^{'}

are included. These conditions can guide the optical design of R-C telescope with reduced misalignment sensitivities. In addition, the influence on system misalignment sensitivities by of the mirror power distribution and other optical performance parameter, which can be written as the function of

d

and

L

, can be deduced in similar way.

4. An optical design example for R-C telescope with reduced misalignment sensitivities

In this section, the F/7.7, 1200 mm Mt. Hopkins telescope serves as the reference system. The optical structure parameters of the Mt. Hopkins are used as the initial design parameters, which is optimized for lower misalignment sensitivities. The thickness $d$ and $L$ are used as optimization variables. The variation range are $d \in [d_{0} - 10 % d_{0}, d_{0} + 10 % d_{0}]$ and $L \in [L_{0} - 10 % L_{0}, L_{0} + 10 % L_{0}]$ , respectively. In this range, we search for an optimal system with reduced misalignment sensitivities. The variation range of $d$ and $L$ are set approximately in this section, which mainly demonstrates the design method of the two-mirror telescopes with low misalignment sensitivities.

The optical parameters of the Mt. Hopkins telescope are given by McLeod [27] and are summarized in Table 6 in the Appendix 2. The optical system is an R-C telescope, with a maximum field of 17.67″. The Mt. Hopkins telescope is simulated in the optical analysis software CODEV [28]. A schematic layout of the telescope is given in Fig. 2

Fig. 2 Schematic layout of the 1.2m Mt. Hopkins telescope.

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. The Full-Field-Display showing the RMS spot diameter across the FOV is shown in Fig. 4(a). The average of RMS spot diameter over the FOV is 0.116λ.

The misalignment sensitivities of the initial system are calculated analytically according to Eqs. (10), (18), and (23) shown in Table 4. The selected field point is (0.3°, 0.3°) at the edge of FOV. The proportion relations of misalignment sensitivities show that the misalignment sensitivities of Z7/Z8 are dominant in the RMS of misalignment sensitivities.

Next, according to Table 5, the influence on the misalignment sensitivities of the initial system by the variables of $d$ and $L$ are listed in Table 1

Table 1. The Influence on Misalignment Sensitivities of the Initial System by the Variable of $d$ and $L$

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. One result from Table 1 is that reducing all the misalignment sensitivities at the same time leads to contradictory requirements on

d

and

L

. Therefore, we trade off these requirements with certain structure parameters for the goal of minimum

S_{R M S}

.According to Eqs. (26)–(29), the partial derivatives for the misalignment sensitivity functions with respect to the

d

and

L

are listed in Table 2

Table 2. The Partial Derivatives for the Misalignment Sensitivity Functions to the $d$ and $L$

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. In fact, the partial derivatives represent the rate of change for the misalignment sensitivity functions. Obviously, the rate of change for misalignment sensitivity to

d

is much larger than for misalignment sensitivity to

L

. The rates of change for Z7/Z8 sensitivities are larger than for Z5/Z6 sensitivities.To minimize the RMS of misalignment sensitivities, we optimize the value of

d

and

L

. From the Table 1 we know, with the

| d |

increasing, the misalignment sensitivities

S_{A D E_{S M}}^{C_{5}}

_,

S_{X D E_{S M}}^{C_{7}}

_,

S_{B D E_{S M}}^{C_{7}}

decrease and only the misalignment sensitivity

S_{X D E_{S M}}^{C_{5}}

increases. However, the rate of change for the misalignment sensitivity

S_{X D E_{S M}}^{C_{5}}

with respect to

d

is enough small to negligible. Here, we set the optimization weight of misalignment sensitivity

S_{X D E_{S M}}^{C_{5}}

as zero. Therefore, the maximum value of

d = 1.1 \times d_{0}

are selected as the optimal parameter.

With the $L$ increasing, the misalignment sensitivities $S_{X D E_{S M}}^{C_{5}}$ , $S_{A D E_{S M}}^{C_{5}}$ , $S_{X D E_{S M}}^{C_{7}}$ decrease and only the misalignment sensitivity $S_{B D E_{S M}}^{C_{7}}$ increases. Because of $\frac{\partial S_{B D E_{S M}}^{C_{7}}}{\partial L} : \frac{\partial S_{X D E_{S M}}^{C_{7}}}{\partial L} \approx 1 : 3$ , the increasing value of L is benefit for lower RMS of misalignment sensitivities. Therefore, the maximum value of $L = 1.1 \times L_{0}$ are selected.

The other structure parameters include mirror curvature and conic constant are calculated by Eqs. (19)–(22). In the optimization process, the variations of these parameters are shown in Fig. (3)

Fig. 3 In the optimization process, the variations of mirror curvature and conic constant when the $d$ and $L$ vary (a) and (b) the variations of mirror curvature and conic constant with the $d$ varies and (c) and (d) the variations of mirror curvature and conic constant with the $L$ varies.

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. Firstly, when the

d

increases from

d_{0}

to

1.1 \times d_{0}

and the

L

are fixed as

L_{0}

, the mirror curvature of PM and SM decrease while the conic constants increase as shown in Fig. 3(a) and 3(b). Secondly, when the

L

increases from

L_{0}

to

1.1 \times L_{0}

and the

d

are fixed as

1.1 \times d_{0}

, the mirror curvature of PM and SM decrease while the conic constants increase further as shown in Fig. 3(c) and 3(d). These results indicate that the decrease of the mirror curvature and the increase of conic constants for both PM and SM can soften the misalignment sensitivities of two-mirror system.

The optical parameters of the final optimized system are summarized in Table 3

Table 3. Optical Parameters for the Optimized System

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and the misalignment sensitivities of the optimized system are listed in Table 4

Table 4. The Misalignment Sensitivities of the Initial System and the Optimized System

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, respectively. The Full-Field-Display showing the RMS spot diameter across the FOV is shown in Fig. 4(b)

Fig. 4 Full-Field-Display showing the RMS spot diameter across the field for (a) the initial system and (b) the optimized system.

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.

According to the analytic relations between misalignment sensitivities and optical structure parameters, the misalignment sensitivities have no dependence on the image quality. Therefore, the misalignment sensitivities of optical systems with different image quality can be compared relatively. After optimization of initial system, the RMS of misalignment sensitivities reduces from 0.405λ to 0.312λ and the average of RMS spot diameter over the FOV decreases from 0.116λ to 0.065λ. In the variable range of $d$ and $L$ , by selecting the dominant rate of change for misalignment sensitivity functions with respect to $d$ and $L$ , the optimal structure parameter with reduced misalignment sensitivities are obtained. Besides, all the misalignment sensitivities decrease. The results also show that the proposed design method can optimize the misalignment sensitivities and image quality simultaneously.

Besides, a comparison of misalignment sensitivity functions and its partial derivatives between the analytic functions and numerical calculations is also made in the Appendix 3 in detail. The results show that the analytic misalignment sensitivity functions are consistent with the results obtained from the traditional numerical calculations.

5. Conclusion

In this paper, the misalignment sensitivities are expressed as the analytic function of FOV and optical structure parameters for two-mirror telescope based on NAT. The specific expressions of sensitivities of astigmatism and coma aberration to lateral misalignments are derived. The inherent relations among different misalignment sensitivities and optical structure parameters conditions for zero misalignment sensitivities have been summarized. The results show that the misalignment sensitivities of $C_{5}^{s y s t e m}$ and $C_{6}^{s y s t e m}$ are field-dependent and the misalignment sensitivities of $C_{7}^{s y s t e m}$ and $C_{8}^{s y s t e m}$ is field-constant. On this basis, a systematic optical design method for two-mirror telescope with reduced misalignment sensitivities is proposed. This analytic design method considers reduced misalignment sensitivities and good image quality simultaneously. The design steps for two-mirror telescope are introduced. Then the Mt. Hopkins Telescope is taken as an example to demonstrate the feasibility of the design method. After optimization, the RMS of misalignment sensitivities decreases from 0.405λ to 0.312λ and at the same time the image quality is improved. Besides, the analytical misalignment sensitivities function and its partial derivatives are consistent with the results obtained from the traditional numerical calculations.

The optimization result also shows that the misalignment sensitivities of two-mirror system will decrease when the mirror distance and optical system overall length increase. Though this result is identical to the known widely fact, this paper focuses the reason and the analytic expressions between misalignment sensitivities and optical structure parameters. These quantized relations are the basis for the analytic design method of the two-mirror telescopes with low misalignment sensitivities. The proposed design method in this paper can also be extended to the three-mirror anastigmat (TMA) telescopes.

Appendix 1

This Appendix and Table 5

Table 5. The Misalignment Sensitivities Influenced by the Variable of $d$ and $L$

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provides the monotonicity that satisfies reduced misalignment sensitivities used in Section 3.

Appendix 2

This Appendix and Table 6

Table 6. Optical Parameters for the Mt. Hopkins Telescope Based on McLeod

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provides the optical prescription for the Mt. Hopkins Telescope used in Section 4.

Appendix 3

This Appendix provides the comparisons of misalignment sensitivities and its rates of change between analytic functions and numerical calculations in Section 4. As a complementary account, the mathematic accuracies of misalignment sensitivity functions are demonstrated.

By decentering the SM 100um and tilting the SM 50″ in CODEV, respectively, the exit-pupil wave-front variation of fringe Zernike coefficient Z5~Z8 can be obtained. As a result, the numerically calculated misalignment sensitivities are listed in Table 7

Table 7. The Misalignment Sensitivities of the Initial System Based on the Numerical Calculations and the Deviation between the Analytic Functions and Numerical Calculations

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. Compared to the Table 4, the maximum deviation of misalignment sensitivities is less than 0.005λ between the analytic functions and numerical calculations and the deviation of RMS is only −0.007λ. The difference between the results of analytic method and numerical method is due mainly some additional aberration contributions to the low-order aberration fields that are not considered.

By perturbing the $d$ and $L$ with 100mm in CODEV, respectively, the rates of change for the misalignment sensitivities are also obtained by numerical calculations listed in Table 8

Table 8. The Rates of Change for the Misalignment Sensitivities to $d$ and $L$ based on the Numerical Calculations and the Deviation between the Analytic Functions and Numerical Calculations

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. Compared to Table 2, the maximum deviation of the rates of change for misalignment sensitivities is less than 0.005λ between the analytical functions and numerical calculations and the RMS deviation is only −0.006λ.

From Table 7 and Table 8, we can find that the misalignment sensitivity functions are very close to the numerical calculations. It is proven that the results of analytical functions and numerical calculations are consistent.

Funding

National Natural Science Foundation of China (NSFC) (61705223).

Acknowledgments

We thank Synopsys for the educational license of CODE V.

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Misalignment sensitivities	$S_{X D E_{S M}}^{C_{5}} ↓$	$S_{A D E_{S M}}^{C_{5}} ↓$	$S_{X D E_{S M}}^{C_{7}} ↓$	$S_{B D E_{S M}}^{C_{7}} ↓$
Variable monotonicity	$\| d \| ↓$	$\| d \| ↑$	$\| d \| ↑$	$\| d \| ↑$
Variable monotonicity	$L ↑$	$L ↑$	$L ↑$	$L ↓$
( $↓$ stands for decreasing, $↑$ stands for increasing)

	$S_{X D E_{S M}}^{C_{5}}$	$S_{A D E_{S M}}^{C_{5}}$	$S_{X D E_{S M}}^{C_{7}}$	$S_{B D E_{S M}}^{C_{7}}$	RMS
$\partial / \partial d$	−0.00006	0.00554	−0.04136	−0.03849	0.05677
$\partial / \partial L$	0.00099	−0.00056	0.00981	−0.00368	0.01054
The unit is 0.01λ/mm.

Surface	Type	Conic constant	Radius	Thickness
PM (stop)	Conic	−1.059113	−5227.582	−1887.793
SM	Conic	−3.686378	−2008.924	2618.7789
image	plane	0	infinit
Radius and Thickness are in mm.

	$S_{X D E_{S M}}^{C_{5}}$	$S_{A D E_{S M}}^{C_{5}}$	$S_{X D E_{S M}}^{C_{7}}$	$S_{B D E_{S M}}^{C_{7}}$	RMS
initial system	−0.01064	0.06367	−0.23975	−0.31072	0.39773
optimized system	−0.00868	0.05379	−0.16400	−0.25879	0.31118
The unit of decenter sensitivity is λ/100um and the unit of tilt sensitivity is λ/50″.

misalignment sensitivities	Conditions that satisfy reduced misalignment sensitivities
$\| S_{X D E_{S M}}^{C_{5}} \| ↓$	$S_{X D E_{S M}}^{C_{5}} \cdot \frac{\partial S_{X D E_{S M}}^{C_{5}}}{\partial d} \geq 0$	$\begin{array}{l} [(f^{'} (2 d + L) \geq L^{2} - d^{2}) \cap (f^{'} (d + L) \geq L^{2})] \\ \cup [(f^{'} (2 d + L) \leq L^{2} - d^{2}) \cap (f^{'} (d + L) \leq L^{2})] \end{array}$	$\| d \| ↑$
	$S_{X D E_{S M}}^{C_{5}} \cdot \frac{\partial S_{X D E_{S M}}^{C_{5}}}{\partial d} \leq 0$	$\begin{array}{l} [(f^{'} (2 d + L) \geq L^{2} - d^{2}) \cap (f^{'} (d + L) \leq L^{2})] \\ \cup [(f^{'} (2 d + L) \leq L^{2} - d^{2}) \cap (f^{'} (d + L) \geq L^{2})] \end{array}$	$\| d \| ↓$
	$S_{X D E_{S M}}^{C_{5}} \cdot \frac{\partial S_{X D E_{S M}}^{C_{5}}}{\partial L} \geq 0$	$\begin{array}{l} [(f^{'} (2 d + L) \geq L^{2} - d^{2}) \cap (f^{'} (\frac{2}{d} + \frac{2 f + d}{L^{2}}) \geq \frac{2 L}{d})] \\ \cup [(f^{'} (2 d + L) \leq L^{2} - d^{2}) \cap (f^{'} (\frac{2}{d} + \frac{2 f^{'} + d}{L^{2}}) \leq \frac{2 L}{d})] \end{array}$	$L ↓$
	$S_{X D E_{S M}}^{C_{5}} \cdot \frac{\partial S_{X D E_{S M}}^{C_{5}}}{\partial L} \leq 0$	$\begin{array}{l} [(f^{'} (2 d + L) \geq L^{2} - d^{2}) \cap (f^{'} (\frac{2}{d} + \frac{2 f^{'} + d}{L^{2}}) \leq \frac{2 L}{d})] \\ \cup [(f^{'} (2 d + L) \leq L^{2} - d^{2}) \cap (f^{'} (\frac{2}{d} + \frac{2 f^{'} + d}{L^{2}}) \geq \frac{2 L}{d})] \end{array}$	$L ↑$
$\| S_{A D E_{S M}}^{C_{5}} \| ↓$	$S_{A D E_{S M}}^{C_{5}} \cdot \frac{\partial S_{A D E_{S M}}^{C_{5}}}{\partial d} \geq 0$	$f^{'} \geq L - d \cup (- L - d \leq f^{'} \leq \sqrt{d^{2} + L^{2}})$	$\| d \| ↑$
	$S_{A D E_{S M}}^{C_{5}} \cdot \frac{\partial S_{A D E_{S M}}^{C_{5}}}{\partial d} \leq 0$	$f^{'} \leq - L - d \cup (\sqrt{d^{2} + L^{2}} \leq f^{'} \leq - L - d)$	$\| d \| ↓$
	$S_{A D E_{S M}}^{C_{5}} \cdot \frac{\partial S_{A D E_{S M}}^{C_{5}}}{\partial L} \geq 0$	$- L - d \leq f^{'} \leq L - d$	$L ↓$
	$S_{A D E_{S M}}^{C_{5}} \cdot \frac{\partial S_{A D E_{S M}}^{C_{5}}}{\partial L} \leq 0$	$f^{'} \leq - L - d \cup f^{'} \geq L - d$	$L ↑$
$\| S_{X D E_{S M}}^{C_{7}} \| ↓$	$S_{X D E_{S M}}^{C_{7}} \cdot \frac{\partial S_{X D E_{S M}}^{C_{7}}}{\partial d} \geq 0$	$f^{'} \leq L - \sqrt[3]{d^{2} L} \cup f^{'} \geq L - \sqrt[3]{\frac{d^{2} L}{3}}$	$\| d \| ↑$
	$S_{X D E_{S M}}^{C_{7}} \cdot \frac{\partial S_{X D E_{S M}}^{C_{7}}}{\partial d} \leq 0$	$L - \sqrt[3]{d^{2} L} \leq f \leq L - \sqrt[3]{\frac{d^{2} L}{3}}$	$\| d \| ↓$
	$S_{X D E_{S M}}^{C_{7}} \cdot \frac{\partial S_{X D E_{S M}}^{C_{7}}}{\partial L} \geq 0$	$\begin{array}{l} [(f^{'} \leq L - \sqrt[3]{d^{2} L}) \cap (f^{'} \leq L + \frac{d}{\sqrt{3}})] \\ \cup [(f^{'} \geq L - \sqrt[3]{d^{2} L}) \cap (L + \frac{d}{\sqrt{3}} \leq f^{'} \leq L - \frac{d}{\sqrt{3}})] \end{array}$	$L ↓$
	$S_{X D E_{S M}}^{C_{7}} \cdot \frac{\partial S_{X D E_{S M}}^{C_{7}}}{\partial L} \leq 0$	$\begin{array}{l} [(f^{'} \leq L - \sqrt[3]{d^{2} L}) \cap (L + \frac{d}{\sqrt{3}} \leq f^{'} \leq L - \frac{d}{\sqrt{3}})] \\ \cup [(f^{'} \geq L - \sqrt[3]{d^{2} L}) \cap (f^{'} \leq L + \frac{d}{\sqrt{3}} \cup f^{'} \geq L - \frac{d}{\sqrt{3}})] \end{array}$	$L ↑$
$\| S_{B D E_{S M}}^{C_{7}} \| ↓$	$S_{B D E_{S M}}^{C_{7}} \cdot \frac{\partial S_{B D E_{S M}}^{C_{7}}}{\partial d} \geq 0$	$f^{'} \leq L + d \cup f^{'} \geq L - d$	$\| d \| ↑$
	$S_{B D E_{S M}}^{C_{7}} \cdot \frac{\partial S_{B D E_{S M}}^{C_{7}}}{\partial d} \leq 0$	$L + d \leq f^{'} \leq L - d$	$\| d \| ↓$
	$S_{B D E_{S M}}^{C_{7}} \cdot \frac{\partial S_{B D E_{S M}}^{C_{7}}}{\partial L} \geq 0$	$\begin{array}{l} [L + d \leq f^{'} \leq L - d \cap (f^{'} - L) (f^{'} - 3 L) \leq d^{2}] \cup \\ [(f^{'} \leq L + d \cup f^{'} \geq L - d) \cap (f^{'} - L) (f^{'} - 3 L) \geq d^{2}] \end{array}$	$L ↓$
	$S_{B D E_{S M}}^{C_{7}} \cdot \frac{\partial S_{B D E_{S M}}^{C_{7}}}{\partial L} \leq 0$	$\begin{array}{l} [L + d \leq f^{'} \leq L - d \cap (f^{'} - L) (f^{'} - 3 L) \geq d^{2}] \cup \\ [(f^{'} \leq L + d \cup f^{'} \geq L - d) \cap (f^{'} - L) (f^{'} - 3 L) \leq d^{2}] \end{array}$	$L ↑$
( $↓$ stands for decreasing, $↑$ stands for increasing)

Optical design method of two-mirror astronomical telescopes with reduced misalignment sensitivities based on the nodal aberration theory

Abstract

1. Introduction

2. Analytic expressions for misalignment sensitivities of two-mirror telescopes

2.1 Sensitivities of astigmatism aberration to lateral misalignments

2.2 Sensitivities of coma aberration to lateral misalignments

3. Optical design method with reduced misalignment sensitivities for two-mirror telescope

4. An optical design example for R-C telescope with reduced misalignment sensitivities

5. Conclusion

Appendix 1

Appendix 2

Appendix 3

Funding

Acknowledgments

References

Cited By

Figures (4)

Tables (8)

Equations (29)

OSA Continuum

Surface	Type	Conic constant	Radius	Thickness	Y Semi-Aperture
PM (stop)	Conic	−1.040231	−4591.812	−1716.176	609.6
SM	Conic	−3.083152	−1532.687	2380.708	153.93
image	plane	0	infinit
Radius and Thickness are in mm.

	$S_{X D E_{S M}}^{C_{5}}$	$S_{A D E_{S M}}^{C_{5}}$	$S_{X D E_{S M}}^{C_{7}}$	$S_{B D E_{S M}}^{C_{7}}$	RMS
Numerical calculations	−0.00901	0.06860	−0.24411	−0.31544	0.40482
Deviation	−0.00163	−0.00493	0.00436	0.00472	−0.00708
The unit of decenter sensitivities and tilt sensitivities are λ/100um and λ/50″, respectively.

		$S_{X D E_{S M}}^{C_{5}}$	$S_{A D E_{S M}}^{C_{5}}$	$S_{X D E_{S M}}^{C_{7}}$	$S_{B D E_{S M}}^{C_{7}}$	RMS
$\partial / \partial d$	Numerical calculations	0.00069	0.00654	−0.04602	−0.04202	0.06266
$\partial / \partial d$	Deviation	0.00075	0.00101	−0.00466	−0.00353	−0.00589
$\partial / \partial L$	Numerical calculations	0.00090	−0.00071	0.01050	−0.00345	0.01111
$\partial / \partial L$	Deviation	−0.00009	−0.00015	0.00070	0.00023	−0.00057
The unit is λ/100mm.