Second-Order Sensitivity Matrix Method for Aligning Off-Axis Telescopes

Zhu Liu; Qi Peng; Ge Ren; Chunsheng Xiang; Yufeng Tan

doi:10.3788/LOP202259.0811002

Journals >Laser & Optoelectronics Progress >Volume 59 >Issue 8 >Page 0811002 > Article

Laser & Optoelectronics Progress
Vol. 59, Issue 8, 0811002 (2022)

Second-Order Sensitivity Matrix Method for Aligning Off-Axis Telescopes

Zhu Liu^1,2, Qi Peng^1,2,*, Ge Ren^1,2, Chunsheng Xiang^1,2, and Yufeng Tan^1,2

Author Affiliations

¹Key Laboratory of Optical Engineering, Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu , Sichuan 610209, China

²University of Chinese Academy of Sciences, Beijing 100049, China

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DOI: 10.3788/LOP202259.0811002 Cite this Article Set citation alerts

Zhu Liu, Qi Peng, Ge Ren, Chunsheng Xiang, Yufeng Tan. Second-Order Sensitivity Matrix Method for Aligning Off-Axis Telescopes[J]. Laser & Optoelectronics Progress, 2022, 59(8): 0811002 Copy Citation Text

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Abstract

Detecting a decenter or tilt is a key problem in aligning an off-axis telescope system. Therefore, this paper proposes a second-order sensitivity matrix method. When the ranges of decenter and tilt are from -3 mm to 3 mm and from -3 mrad to 3 mrad, respectively, the relation curves between the Zernike coefficient and misalignment are drawn. All their goodness of fit is larger than 0.99. Based on these curves, two sensitivity matrixes are solved and verified using simulation experiments. The results are as follows: The root mean squares (RMSs) of the x-axis decenter and y-axis decenter are 0.0023 mm and 0.0043 mm, respectively, and the RMSs of x-axis tilt and y-axis tilt are 0.0177 mrad and 0.0031 mrad, respectively. The simulation results show that this method is more precise than the traditional sensitivity matrix method.

Keywords

imaging system misalignment error off-axis telescope second-order sensitivity matrix Zernike coefficient

f (x, y) = \sum_{p = 0}^{\infty} \sum_{q = 0}^{\infty} Z_{p q} V_{p q} (ρ, θ)

，（1）

\begin{array}{l} f (ρ, θ) = Z_{1} + Z_{2} ρ c o s θ + Z_{3} ρ s i n θ + Z_{4} (2 ρ^{2} - 1) + Z_{5} ρ^{2} c o s 2 θ + Z_{6} ρ^{2} s i n 2 θ + \\ Z_{7} (3 ρ^{2} - 2) ρ c o s θ + Z_{8} (3 ρ^{2} - 2) ρ s i n θ + Z_{9} (6 ρ^{4} - 6 ρ^{2} + 1) + \dots, \end{array}

（2）

Δ Z = A \times Δ B

，（3）

\begin{array}{l} Δ Z = {[\begin{matrix} Δ Z_{1} & \dots & Δ Z_{m} \end{matrix}]}^{T} = \\ {[\begin{matrix} Z_{1} & \dots & Z_{m} \end{matrix}]}^{T} - {[\begin{matrix} Z_{1}^{i} & \dots & Z_{m}^{i} \end{matrix}]}^{T} \end{array}

，（4）

A = [\begin{matrix} \frac{\partial Z_{1}}{\partial x_{1}} & \dots & \frac{\partial Z_{1}}{\partial x_{n}} \\ ⋮ & ⋮ \\ \frac{\partial Z_{m}}{\partial x_{1}} & \dots & \frac{\partial Z_{m}}{\partial x_{n}} \end{matrix}]

，（5）

\begin{array}{l} Δ B = {[\begin{matrix} Δ x_{1} & \dots & Δ x_{n} \end{matrix}]}^{T} = \\ {[\begin{matrix} x_{1} & \dots & x_{n} \end{matrix}]}^{T} - {[\begin{matrix} x_{1}^{i} & \dots & x_{n}^{i} \end{matrix}]}^{T} \end{array}

，（6）

Φ (Δ B) = {(A \times Δ B - Δ Z)}^{T} \times (A \times Δ B - Δ Z)

。（7）

\nabla Φ (Δ X) = \nabla [{(A \times Δ B - Δ Z)}^{T} (A \times Δ B - Δ Z)] = 0

，（8）

Δ B = {(A^{T} \times A)}^{- 1} \times A^{T} \times Δ Z

。（9）

Δ Z = A_{s}^{'} \times Δ B + \frac{1}{2} A^{″} \times Δ B^{\{2\}} + δ (Δ Z)

，（10）

Δ Z = [Z_{1}^{} \dots Z_{m}^{}]^{T} - [Z_{1}^{i} \dots Z_{m}^{i}]^{T}

，（11）

A_{s}^{'} = [\begin{matrix} {(\frac{δ Z_{1}}{δ x_{1}})}_{c o n s t a n t} & \dots & {(\frac{δ Z_{1}}{δ x_{n}})}_{c o n s t a n t} \\ ⋮ & ⋮ \\ {(\frac{δ Z_{m}}{δ x_{1}})}_{c o n s t a n t} & \dots & {(\frac{δ Z_{m}}{δ x_{n}})}_{c o n s t a n t} \end{matrix}]

，（12）

Δ B = [x_{1}^{} \dots x_{n}^{}]^{T} - [x_{1}^{i} \dots x_{n}^{i}]^{T}

，（13）

A^{″} = [\begin{matrix} \frac{δ^{2} Z_{1}}{δ {x_{1}}^{2}} & \dots & \frac{δ^{2} Z_{1}}{δ {x_{n}}^{2}} \\ ⋮ & ⋮ \\ \frac{δ^{2} Z_{m}}{δ {x_{1}}^{2}} & \dots & \frac{δ^{2} Z_{m}}{δ {x_{n}}^{2}} \end{matrix}]

，（14）

Δ B^{{2}} = (Δ B \times Δ B^{T}) \times I_{(n)} \times {[\begin{matrix} 1 \\ 0 \\ ⋮ \\ 0 \end{matrix}]}_{(n \times 1)} + (Δ B \times Δ B^{T}) \times I_{(n)} \times {[\begin{matrix} 0 \\ 1 \\ ⋮ \\ 0 \end{matrix}]}_{(n \times 1)} + \dots + (Δ B \times Δ B^{T}) \times I_{(n)} \times {[\begin{matrix} 0 \\ 0 \\ ⋮ \\ 1 \end{matrix}]}_{(n \times 1)}

，（15）

δ (Δ Z) = \frac{A^{(n)}}{(n + 1)!} \times Δ B^{{n}}

，（16）

Zhu Liu, Qi Peng, Ge Ren, Chunsheng Xiang, Yufeng Tan. Second-Order Sensitivity Matrix Method for Aligning Off-Axis Telescopes[J]. Laser & Optoelectronics Progress, 2022, 59(8): 0811002

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