Ming Wen, Hongcai Ma, and Chengshan Han, "Active alignment of complex perturbed pupil-offset off-axis telescopes using the extension of nodal aberration theory," Appl. Opt. 60, 3874-3887 (2021)
This paper presents an optical alignment strategy for complex perturbed pupil-offset off-axis reflective telescopes, based on the extension of nodal aberration theory (NAT). First, the direct expansion of the wave aberration function in the vector form for perturbed off-axis systems is given, which is especially convenient for the expansion of the corresponding higher-order terms. The inherent vector relationships between the contributions generated by the aberrations of the on-axis parent systems through pupil transformation are disclosed in detail, which is helpful to understand the aberration behavior of off-axis systems. Then, according to the inherent vector relationships, an analytical alignment model based on NAT for complex cases of perturbed off-axis telescopes is established. It can quantitatively separate the effects of misalignments and surface figure errors from the total aberration fields. The alignment model is solved by using particle swarm optimization algorithm. Then, an optical alignment example of the off-axis three-mirror anastigmatic telescope with misalignments and complex surface figure errors based on the proposed method is demonstrated. After correction, the perturbed telescope can be nearly restored to the nominal states. Finally, Monte Carlo simulations are carried out to show the effectiveness and accuracy of the proposed method.
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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Contributions Generated by the Aberrations of the On-Axis Parent Systems through Pupil Transformation to Zernike Polynomial Vectors
Aberration Type
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Table 2.
Inherent Vector Relationships between Derived Aberrations for Perturbed Off-Axis Systems Based on Fifth-Order Expansion
Aberration Type
Derived Aberration Contributions Generated through Pupil Transformation
Inherent Relationships between the Derived Aberration Contributions
None
None
Table 3.
Inherent Vector Relationships between Derived Aberrations for Perturbed Off-Axis Systems Based on Partial Seventh-Order Expansion
Aberration Type
Derived Aberration Contributions Generated through Pupil Transformation
Inherent Relationships between the Derived Aberration Contributions
None
Table 4.
Introduced Figure Errors (I-V) and Computed Figure Errors (C-V)a
I-V
0.0600
−0.0500
−0.0600
0.0400
−0.0400
0.0500
C-V
0.0595
−0.0496
−0.0596
0.0403
−0.0398
0.0502
${}_FC_i^{{\rm PM}}$ denotes the surface figure errors. The Fringe Zernike coefficients are in $\lambda$.
Table 5.
Introduced Misalignments (I-M) and Computed Misalignments (C-M) of SMa
I-M
0.1500
−0.1500
−0.0150
0.0150
C-M
0.1489
−0.1513
−0.0151
0.0149
XDE and YDE are in mm. ADE and BDE are in degrees.
Table 6.
Introduced Misalignments (I-M) and Computed Misalignments (C-M) of TMa
I-M
−0.2500
0.2500
0.0200
−0.0200
C-M
−0.2478
0.2473
0.0197
−0.0202
XDE and YDE are in mm. ADE and BDE are in degrees.
Table 7.
Four Different Cases Considered in Monte Carlo Simulationsa
XDE, YDE
ADE, BDE
M-E
Case 1
[−0.05, 0.05]
[−0.005, 0.005]
[−0.02, 0.02]
\
Case 2
[−0.5, 0.5]
[−0.02, 0.02]
[−0.05, 0.05]
\
Case 3
[−1.5, 1.5]
[−0.05, 0.05]
[−0.1, 0.1]
\
Case 4
[−0.5, 0.5]
[−0.02, 0.02]
[−0.05, 0.05]
3%
XDE and YDE are in mm, ADE and BDE are in degrees, the Fringe Zernike coefficients are in $\lambda$, and M-E denotes the measurement errors, $i = 5,6,10,11,17,18$.
Tables (7)
Table 1.
Contributions Generated by the Aberrations of the On-Axis Parent Systems through Pupil Transformation to Zernike Polynomial Vectors
Aberration Type
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Table 2.
Inherent Vector Relationships between Derived Aberrations for Perturbed Off-Axis Systems Based on Fifth-Order Expansion
Aberration Type
Derived Aberration Contributions Generated through Pupil Transformation
Inherent Relationships between the Derived Aberration Contributions
None
None
Table 3.
Inherent Vector Relationships between Derived Aberrations for Perturbed Off-Axis Systems Based on Partial Seventh-Order Expansion
Aberration Type
Derived Aberration Contributions Generated through Pupil Transformation
Inherent Relationships between the Derived Aberration Contributions
None
Table 4.
Introduced Figure Errors (I-V) and Computed Figure Errors (C-V)a
I-V
0.0600
−0.0500
−0.0600
0.0400
−0.0400
0.0500
C-V
0.0595
−0.0496
−0.0596
0.0403
−0.0398
0.0502
${}_FC_i^{{\rm PM}}$ denotes the surface figure errors. The Fringe Zernike coefficients are in $\lambda$.
Table 5.
Introduced Misalignments (I-M) and Computed Misalignments (C-M) of SMa
I-M
0.1500
−0.1500
−0.0150
0.0150
C-M
0.1489
−0.1513
−0.0151
0.0149
XDE and YDE are in mm. ADE and BDE are in degrees.
Table 6.
Introduced Misalignments (I-M) and Computed Misalignments (C-M) of TMa
I-M
−0.2500
0.2500
0.0200
−0.0200
C-M
−0.2478
0.2473
0.0197
−0.0202
XDE and YDE are in mm. ADE and BDE are in degrees.
Table 7.
Four Different Cases Considered in Monte Carlo Simulationsa
XDE, YDE
ADE, BDE
M-E
Case 1
[−0.05, 0.05]
[−0.005, 0.005]
[−0.02, 0.02]
\
Case 2
[−0.5, 0.5]
[−0.02, 0.02]
[−0.05, 0.05]
\
Case 3
[−1.5, 1.5]
[−0.05, 0.05]
[−0.1, 0.1]
\
Case 4
[−0.5, 0.5]
[−0.02, 0.02]
[−0.05, 0.05]
3%
XDE and YDE are in mm, ADE and BDE are in degrees, the Fringe Zernike coefficients are in $\lambda$, and M-E denotes the measurement errors, $i = 5,6,10,11,17,18$.