Optical control of the Advanced Technology Solar Telescope

Robert Upton

Author Affiliations

Robert Upton^{1}

^{1}R. 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
$19\text{\hspace{0.17em} nm}$, subject to
$\text{10 \hspace{0.17em} nm}$ rms WFS noise. This result is obtained utilizing nine WFSs
distributed in the FOV with a
$\text{300 \hspace{0.17em} nm}$ 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
$\text{17 .8 \hspace{0.17em} nm}$. 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 Figures^{
a
}

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 ATST^{
a
}

Number

Expression

Name

4

ρ^{2} cos[2ϕ]

Third-order X astigmatism

5

2ρ^{2} − 1

Defocus

6

ρ^{2} sin[2ϕ]

Third-order 45° astigmatism

7

ρ^{3} cos[3ϕ]

X trefoil

8

[3ρ^{3} − 2ρ]cos[ϕ]

Third-order X coma

9

[3ρ^{3} − 2ρ]sin[ϕ]

Third-order Y coma

10

ρ^{3} sin[3ϕ]

Y trefoil

Zernike polynomials listed define the WFS modes for a given WFS.

Table 4

Order of ATST RBDOF Sensitivities^{
a
}

Iteration

RBDOFs Truncated

Sensitivity

Sensitivity Units

1

M1X

6.88 × 10^{−7}

mm^{−1}

2

M2Y

9.44 × 10^{−7}

mm^{−1}

3

M2X

6.72 × 10^{−6}

mm^{−1}

4

M2C

9.23 × 10^{−6}

deg^{−1}

5

M1Z

3.25 × 10^{−5}

mm^{−1}

6

M1Y

8.66 × 10^{−4}

mm^{−1}

7

M2B

1.23 × 10^{−3}

deg^{−1}

8

M2A

7.55 × 10^{−3}

deg^{−1}

9

M1C

1.22 × 10^{−2}

deg^{−1}

10

M2Z

0.8

mm^{−1}

11

M1B

1

deg^{−1}

12

M1A

1

deg^{−1}

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.

Table 5

Operational Sensitivities of the ATST RBDOFs^{
a
}

RBDOFs

Operational Sensitivity

M2C

9.23 × 10^{−9}

M1X

3.4 × 10^{−8}

M2Y

4.7 × 10^{−8}

M2X

3.4 × 10^{−7}

M2B

1.23 × 10^{−6}

M1Z

1.6 × 10^{−6}

M2A

7.55 × 10^{−6}

M1C

1.22 × 10^{−5}

M1Y

4.3 × 10^{−5}

M1B

1 × 10^{−3}

M1A

1 × 10^{−3}

M2Z

4 × 10^{−2}

Operational sensitivities quoted are unitless.

Table 6

Order of the ATST M1 LBMs Truncated Using the Mode Truncation Algorithm^{
a
}

Iteration

M1 LBMs Truncated

Sensitivity

Sensitivity Units

1

Z_{13}

0.91

λ^{−1}

2

Z_{10}

0.91

λ^{−1}

3

Z_{7}

0.91

λ^{−1}

4

Z_{8}

0.91

λ^{−1}

5

Z_{9}

1

λ^{−1}

6

Z_{4}

1

λ^{−1}

7

Z_{6}

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 Z_{6}^{
a
}

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 H_{SEC} and Eigenvalues for H_{SEC}^{T}H_{SEC}^{
a
}

Singular Values

Singular Value Units

Eigenvalues

Eigenvalue Units

472.24

λ.s^{−1}

2.23 × 10^{5}

λ^{2}.s^{−2}

471.61

λ.s^{−1}

2.22 × 10^{5}

λ^{2}.s^{−2}

42.61

λ.s^{−1}

1.82 × 10^{3}

λ^{2}.s^{−2}

2.74

λ.s^{−1}

7.50

λ^{2}.s^{−2}

0.13

λ.s^{−1}

1.57 × 10^{−2}

λ^{2}.s^{−2}

0.09

λ.s^{−1}

8.80 × 10^{−3}

λ^{2}.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 DOF^{2}.

Table 9

Order of the ATST M1 LBMs Truncated Using the Mode Truncation Algorithm^{
a
}

rms M1 LBM (nm)

Aberration before Correction (nm)

Aberration after Correction (nm)

Z_{4} 100

492

19

Z_{4} 200

514

19

Z_{4} 300

581

19

Z_{6} 100

489

20

Z_{6} 200

467

20

Z_{6} 300

543

20

Z_{13} 100

398

19

Z_{13} 200

457

19

Z_{13} 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 Figures^{
a
}

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 ATST^{
a
}

Number

Expression

Name

4

ρ^{2} cos[2ϕ]

Third-order X astigmatism

5

2ρ^{2} − 1

Defocus

6

ρ^{2} sin[2ϕ]

Third-order 45° astigmatism

7

ρ^{3} cos[3ϕ]

X trefoil

8

[3ρ^{3} − 2ρ]cos[ϕ]

Third-order X coma

9

[3ρ^{3} − 2ρ]sin[ϕ]

Third-order Y coma

10

ρ^{3} sin[3ϕ]

Y trefoil

Zernike polynomials listed define the WFS modes for a given WFS.

Table 4

Order of ATST RBDOF Sensitivities^{
a
}

Iteration

RBDOFs Truncated

Sensitivity

Sensitivity Units

1

M1X

6.88 × 10^{−7}

mm^{−1}

2

M2Y

9.44 × 10^{−7}

mm^{−1}

3

M2X

6.72 × 10^{−6}

mm^{−1}

4

M2C

9.23 × 10^{−6}

deg^{−1}

5

M1Z

3.25 × 10^{−5}

mm^{−1}

6

M1Y

8.66 × 10^{−4}

mm^{−1}

7

M2B

1.23 × 10^{−3}

deg^{−1}

8

M2A

7.55 × 10^{−3}

deg^{−1}

9

M1C

1.22 × 10^{−2}

deg^{−1}

10

M2Z

0.8

mm^{−1}

11

M1B

1

deg^{−1}

12

M1A

1

deg^{−1}

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.

Table 5

Operational Sensitivities of the ATST RBDOFs^{
a
}

RBDOFs

Operational Sensitivity

M2C

9.23 × 10^{−9}

M1X

3.4 × 10^{−8}

M2Y

4.7 × 10^{−8}

M2X

3.4 × 10^{−7}

M2B

1.23 × 10^{−6}

M1Z

1.6 × 10^{−6}

M2A

7.55 × 10^{−6}

M1C

1.22 × 10^{−5}

M1Y

4.3 × 10^{−5}

M1B

1 × 10^{−3}

M1A

1 × 10^{−3}

M2Z

4 × 10^{−2}

Operational sensitivities quoted are unitless.

Table 6

Order of the ATST M1 LBMs Truncated Using the Mode Truncation Algorithm^{
a
}

Iteration

M1 LBMs Truncated

Sensitivity

Sensitivity Units

1

Z_{13}

0.91

λ^{−1}

2

Z_{10}

0.91

λ^{−1}

3

Z_{7}

0.91

λ^{−1}

4

Z_{8}

0.91

λ^{−1}

5

Z_{9}

1

λ^{−1}

6

Z_{4}

1

λ^{−1}

7

Z_{6}

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 Z_{6}^{
a
}

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 H_{SEC} and Eigenvalues for H_{SEC}^{T}H_{SEC}^{
a
}

Singular Values

Singular Value Units

Eigenvalues

Eigenvalue Units

472.24

λ.s^{−1}

2.23 × 10^{5}

λ^{2}.s^{−2}

471.61

λ.s^{−1}

2.22 × 10^{5}

λ^{2}.s^{−2}

42.61

λ.s^{−1}

1.82 × 10^{3}

λ^{2}.s^{−2}

2.74

λ.s^{−1}

7.50

λ^{2}.s^{−2}

0.13

λ.s^{−1}

1.57 × 10^{−2}

λ^{2}.s^{−2}

0.09

λ.s^{−1}

8.80 × 10^{−3}

λ^{2}.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 DOF^{2}.

Table 9

Order of the ATST M1 LBMs Truncated Using the Mode Truncation Algorithm^{
a
}

rms M1 LBM (nm)

Aberration before Correction (nm)

Aberration after Correction (nm)

Z_{4} 100

492

19

Z_{4} 200

514

19

Z_{4} 300

581

19

Z_{6} 100

489

20

Z_{6} 200

467

20

Z_{6} 300

543

20

Z_{13} 100

398

19

Z_{13} 200

457

19

Z_{13} 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.