1R. Upton (rupton@noao.edu) is with the New Initiatives Office, National Optical Astronomy Observatory, Association of Universities for Research in Astronomy, 950 North Cherry Avenue, Tucson, Arizona 85719.
The Advanced Technology Solar Telescope (ATST) is an off-axis Gregorian astronomical
telescope design. The ATST is expected to be subject to thermal
and gravitational effects that result in misalignments of its mirrors and warping
of its primary mirror. These effects require active, closed-loop correction
to maintain its as-designed diffraction-limited optical performance.
The simulation and modeling of the ATST with a closed-loop correction
strategy are presented. The correction strategy is derived from the linear mathematical
properties of two Jacobian, or influence, matrices that map the ATST
rigid-body (RB) misalignments and primary mirror figure errors to wavefront sensor (WFS)
measurements. The two Jacobian matrices also quantify the sensitivities of the
ATST to RB and primary mirror figure perturbations.
The modeled active correction strategy results in a decrease of the rms
wavefront error averaged over the field of view (FOV) from 500 to
, subject to
rms WFS noise. This result is obtained utilizing nine WFSs
distributed in the FOV with a
rms astigmatism
figure error on the primary mirror.
Correction of the ATST RB perturbations is demonstrated for an
optimum subset of three WFSs with corrections improving the
ATST rms wavefront error from 340 to
. In addition to the
active correction of the ATST, an analytically robust sensitivity analysis that
can be generally extended to a wider class of optical systems is presented.
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Optical Prescription of the ATST Given to Two Significant Figuresa
Element
ROC (mm)
Thickness (mm)
Diameter (mm)
Conic
M1 (Primary)
−16,000
−9200
4400
−1
M2 (Secondary)
2081.26
8158.93
800
−0.54
M3 (Boresight 1)
Infinity
−3400
85.46
0
M4 (Pupil relay 1)
5979.24
4281.98
400
−0.37
M5 (Deformable mirror)
Infinity
−375
213.18
0
M6 (Boresight 2)
Infinity
16,082.13
237.26
0
M7 (Pupil relay 2)
−12,701.62
−5300
600
−0.27
M8 (Corrector)
120,000
5508.88
323.36
0
M9 (Fold)
Infinity
−5000
124
0
Camera
−6000
311.94
The ROC is the radius of curvature. The camera is modeled as an ideal optical element with zero coma, distortion, and spherical aberration contributions. The final focal ratio is 50.6.
Table 2
RBDOFs for M1 and M2
RBDOFs
Descriptor
M1X
M1 X decenter
M1Y
M1 Y decenter
M1Z
M1 Z decenter
M1A
M1 X tilt
M1B
M1 Y tilt
M1C
M1 Z tilt
M2X
M2 X decenter
M2Y
M2 Y decenter
M2Z
M2 Z decenter
M2A
M2 X tilt
M2B
M2 Y tilt
M2C
M2 Z tilt
Table 3
Zernike Polynomials Used to Define the Optical Performance of the ATSTa
Number
Expression
Name
4
ρ2 cos[2ϕ]
Third-order X astigmatism
5
2ρ2 − 1
Defocus
6
ρ2 sin[2ϕ]
Third-order 45° astigmatism
7
ρ3 cos[3ϕ]
X trefoil
8
[3ρ3 − 2ρ]cos[ϕ]
Third-order X coma
9
[3ρ3 − 2ρ]sin[ϕ]
Third-order Y coma
10
ρ3 sin[3ϕ]
Y trefoil
Zernike polynomials listed define the WFS modes for a given WFS.
Sensitivities are formed using the mode truncation algorithm. The inverse condition number quoted quantifies the sensitivity of each mode. Hence the RBDOFs listed are arranged in order from least-to-greatest sensitivity.
Order of the ATST M1 LBMs Truncated Using the Mode Truncation Algorithma
Iteration
M1 LBMs Truncated
Sensitivity
Sensitivity Units
1
Z13
0.91
λ−1
2
Z10
0.91
λ−1
3
Z7
0.91
λ−1
4
Z8
0.91
λ−1
5
Z9
1
λ−1
6
Z4
1
λ−1
7
Z6
1
λ−1
Inverse condition number quoted quantifies the sensitivity of each mode. The M1 LBMs are arranged in order from least-to-greatest sensitivity.
Table 7
Torsional Motions of M1 and M2 that Result in the Dominant Aberration 45° Astigmatism Z6a
Torsional Mode
Operational Sensitivity
Units
M2C
9.23 × 10−9
deg−1
M1X
3.4 × 10−8
mm−1
M2X
3.4 × 10−7
mm−1
M2B
1.23 × 10−6
deg−1
M1C
1.22 × 10−5
deg−1
M1B
1 × 10−3
deg−1
Torsional motions are listed in order of least-to-greatest sensitivity.
Table 8
Singular Values of HSEC and Eigenvalues for HSECTHSECa
Singular Values
Singular Value Units
Eigenvalues
Eigenvalue Units
472.24
λ.s−1
2.23 × 105
λ2.s−2
471.61
λ.s−1
2.22 × 105
λ2.s−2
42.61
λ.s−1
1.82 × 103
λ2.s−2
2.74
λ.s−1
7.50
λ2.s−2
0.13
λ.s−1
1.57 × 10−2
λ2.s−2
0.09
λ.s−1
8.80 × 10−3
λ2.s−2
Units quoted for the singular values are nominally waves per differential change in the DOF (s). The units quoted for the eigenvalues are nominally waves per differential change in the DOF2.
Table 9
Order of the ATST M1 LBMs Truncated Using the Mode Truncation Algorithma
rms M1 LBM (nm)
Aberration before Correction (nm)
Aberration after Correction (nm)
Z4 100
492
19
Z4 200
514
19
Z4 300
581
19
Z6 100
489
20
Z6 200
467
20
Z6 300
543
20
Z13 100
398
19
Z13 200
457
19
Z13 300
538
19
Inverse condition number quoted quantifies the sensitivity of each mode. The M1 LBMs are arranged in order from least-to-greatest sensitivity.
Tables (9)
Table 1
Optical Prescription of the ATST Given to Two Significant Figuresa
Element
ROC (mm)
Thickness (mm)
Diameter (mm)
Conic
M1 (Primary)
−16,000
−9200
4400
−1
M2 (Secondary)
2081.26
8158.93
800
−0.54
M3 (Boresight 1)
Infinity
−3400
85.46
0
M4 (Pupil relay 1)
5979.24
4281.98
400
−0.37
M5 (Deformable mirror)
Infinity
−375
213.18
0
M6 (Boresight 2)
Infinity
16,082.13
237.26
0
M7 (Pupil relay 2)
−12,701.62
−5300
600
−0.27
M8 (Corrector)
120,000
5508.88
323.36
0
M9 (Fold)
Infinity
−5000
124
0
Camera
−6000
311.94
The ROC is the radius of curvature. The camera is modeled as an ideal optical element with zero coma, distortion, and spherical aberration contributions. The final focal ratio is 50.6.
Table 2
RBDOFs for M1 and M2
RBDOFs
Descriptor
M1X
M1 X decenter
M1Y
M1 Y decenter
M1Z
M1 Z decenter
M1A
M1 X tilt
M1B
M1 Y tilt
M1C
M1 Z tilt
M2X
M2 X decenter
M2Y
M2 Y decenter
M2Z
M2 Z decenter
M2A
M2 X tilt
M2B
M2 Y tilt
M2C
M2 Z tilt
Table 3
Zernike Polynomials Used to Define the Optical Performance of the ATSTa
Number
Expression
Name
4
ρ2 cos[2ϕ]
Third-order X astigmatism
5
2ρ2 − 1
Defocus
6
ρ2 sin[2ϕ]
Third-order 45° astigmatism
7
ρ3 cos[3ϕ]
X trefoil
8
[3ρ3 − 2ρ]cos[ϕ]
Third-order X coma
9
[3ρ3 − 2ρ]sin[ϕ]
Third-order Y coma
10
ρ3 sin[3ϕ]
Y trefoil
Zernike polynomials listed define the WFS modes for a given WFS.
Sensitivities are formed using the mode truncation algorithm. The inverse condition number quoted quantifies the sensitivity of each mode. Hence the RBDOFs listed are arranged in order from least-to-greatest sensitivity.
Order of the ATST M1 LBMs Truncated Using the Mode Truncation Algorithma
Iteration
M1 LBMs Truncated
Sensitivity
Sensitivity Units
1
Z13
0.91
λ−1
2
Z10
0.91
λ−1
3
Z7
0.91
λ−1
4
Z8
0.91
λ−1
5
Z9
1
λ−1
6
Z4
1
λ−1
7
Z6
1
λ−1
Inverse condition number quoted quantifies the sensitivity of each mode. The M1 LBMs are arranged in order from least-to-greatest sensitivity.
Table 7
Torsional Motions of M1 and M2 that Result in the Dominant Aberration 45° Astigmatism Z6a
Torsional Mode
Operational Sensitivity
Units
M2C
9.23 × 10−9
deg−1
M1X
3.4 × 10−8
mm−1
M2X
3.4 × 10−7
mm−1
M2B
1.23 × 10−6
deg−1
M1C
1.22 × 10−5
deg−1
M1B
1 × 10−3
deg−1
Torsional motions are listed in order of least-to-greatest sensitivity.
Table 8
Singular Values of HSEC and Eigenvalues for HSECTHSECa
Singular Values
Singular Value Units
Eigenvalues
Eigenvalue Units
472.24
λ.s−1
2.23 × 105
λ2.s−2
471.61
λ.s−1
2.22 × 105
λ2.s−2
42.61
λ.s−1
1.82 × 103
λ2.s−2
2.74
λ.s−1
7.50
λ2.s−2
0.13
λ.s−1
1.57 × 10−2
λ2.s−2
0.09
λ.s−1
8.80 × 10−3
λ2.s−2
Units quoted for the singular values are nominally waves per differential change in the DOF (s). The units quoted for the eigenvalues are nominally waves per differential change in the DOF2.
Table 9
Order of the ATST M1 LBMs Truncated Using the Mode Truncation Algorithma
rms M1 LBM (nm)
Aberration before Correction (nm)
Aberration after Correction (nm)
Z4 100
492
19
Z4 200
514
19
Z4 300
581
19
Z6 100
489
20
Z6 200
467
20
Z6 300
543
20
Z13 100
398
19
Z13 200
457
19
Z13 300
538
19
Inverse condition number quoted quantifies the sensitivity of each mode. The M1 LBMs are arranged in order from least-to-greatest sensitivity.