Abstract

High-performance objectives pose strict limitations on residual errors present in the system. External mechanical influences can induce structural vibrations in the optical system which causes a displacement of the lenses present in the system. This will influence the imaging performance, causing degraded images or broadened structures in a lithographic processes. In this paper an adaptive state observer for the detection of structural vibrations of the optical elements of an imaging system based on a series of wavefront tilt measurements is introduced. The observer output is used as an input for a closed-loop PD control to mitigate the lens displacements directly.

© 2015 Optical Society of America

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References

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  1. K. F. Beckstette, “Ultrapräzise Oberflächenbearbeitung am Beispiel von Lithografieoptiken (Ultraprecise Surface Figuring for Lithography Optics),” Tech. Mess. 69(12), 526 (2002).
    [Crossref]
  2. J. Holterman, T. J. de Vries, and F. Auer, “Decoupling of collocated actuator-sensor-pairs for active vibration control,” in 21st Benelux Meeting on Systems and Control Book of Abstracts (2002), p. 66.
  3. K. Washisu, “Control Apparatus For Image Blur Correction,” US 6757488 B2 (2004).
  4. S. Kakiuchi, “Camera Provided With Camera-Shake Compensation Functionality,” US 2006/0008263 A1 (2006).
  5. S.-J. Ko, S.-H. Lee, and K.-H. Lee, “Digital image stabilizing algorithms based on bit-plane matching,” IEEE Trans. Consumer Electronics 44(3), 617–622 (1998).
    [Crossref]
  6. J. H. Burge, “An easy way to relate optical element motion to system pointing stability,” Proc. SPIE 6288, 62880I (2006).
    [Crossref]
  7. R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61(5), 656 (1971).
  8. B. D. Anderson and J. B. Moore, Optimal Filtering (Courier Dover Publications, 2012).
  9. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207 (1976).
    [Crossref]
  10. D. R. Neal, J. Copland, and D. A. Neal, “Shack-Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002).
    [Crossref]
  11. M. M. Morrison, “Inertial Measurement Unit,” US 4711125 (1987).
  12. M. Araki, “PID control,” Control Systems, Robotics and Automation 2, 1–23, (2002).
  13. H. Gilbergs, N. Wengert, K. Frenner, P. Eberhard, and W. Osten, “Reconstruction of dynamical perturbations in optical systems by opto-mechanical simulation methods,” Proc. SPIE 8326, 83262N (2012).
    [Crossref]
  14. J. Watson, “Tip-Tilt Correction for Astronomical Telescopes using Adaptive Control,” Wescon - Integrated Circuit Expo (1997), pp. 490–494.

2012 (1)

H. Gilbergs, N. Wengert, K. Frenner, P. Eberhard, and W. Osten, “Reconstruction of dynamical perturbations in optical systems by opto-mechanical simulation methods,” Proc. SPIE 8326, 83262N (2012).
[Crossref]

2006 (1)

J. H. Burge, “An easy way to relate optical element motion to system pointing stability,” Proc. SPIE 6288, 62880I (2006).
[Crossref]

2002 (3)

K. F. Beckstette, “Ultrapräzise Oberflächenbearbeitung am Beispiel von Lithografieoptiken (Ultraprecise Surface Figuring for Lithography Optics),” Tech. Mess. 69(12), 526 (2002).
[Crossref]

D. R. Neal, J. Copland, and D. A. Neal, “Shack-Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002).
[Crossref]

M. Araki, “PID control,” Control Systems, Robotics and Automation 2, 1–23, (2002).

1998 (1)

S.-J. Ko, S.-H. Lee, and K.-H. Lee, “Digital image stabilizing algorithms based on bit-plane matching,” IEEE Trans. Consumer Electronics 44(3), 617–622 (1998).
[Crossref]

1976 (1)

1971 (1)

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61(5), 656 (1971).

Anderson, B. D.

B. D. Anderson and J. B. Moore, Optimal Filtering (Courier Dover Publications, 2012).

Araki, M.

M. Araki, “PID control,” Control Systems, Robotics and Automation 2, 1–23, (2002).

Auer, F.

J. Holterman, T. J. de Vries, and F. Auer, “Decoupling of collocated actuator-sensor-pairs for active vibration control,” in 21st Benelux Meeting on Systems and Control Book of Abstracts (2002), p. 66.

Beckstette, K. F.

K. F. Beckstette, “Ultrapräzise Oberflächenbearbeitung am Beispiel von Lithografieoptiken (Ultraprecise Surface Figuring for Lithography Optics),” Tech. Mess. 69(12), 526 (2002).
[Crossref]

Burge, J. H.

J. H. Burge, “An easy way to relate optical element motion to system pointing stability,” Proc. SPIE 6288, 62880I (2006).
[Crossref]

Copland, J.

D. R. Neal, J. Copland, and D. A. Neal, “Shack-Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002).
[Crossref]

de Vries, T. J.

J. Holterman, T. J. de Vries, and F. Auer, “Decoupling of collocated actuator-sensor-pairs for active vibration control,” in 21st Benelux Meeting on Systems and Control Book of Abstracts (2002), p. 66.

Eberhard, P.

H. Gilbergs, N. Wengert, K. Frenner, P. Eberhard, and W. Osten, “Reconstruction of dynamical perturbations in optical systems by opto-mechanical simulation methods,” Proc. SPIE 8326, 83262N (2012).
[Crossref]

Frenner, K.

H. Gilbergs, N. Wengert, K. Frenner, P. Eberhard, and W. Osten, “Reconstruction of dynamical perturbations in optical systems by opto-mechanical simulation methods,” Proc. SPIE 8326, 83262N (2012).
[Crossref]

Gilbergs, H.

H. Gilbergs, N. Wengert, K. Frenner, P. Eberhard, and W. Osten, “Reconstruction of dynamical perturbations in optical systems by opto-mechanical simulation methods,” Proc. SPIE 8326, 83262N (2012).
[Crossref]

Holterman, J.

J. Holterman, T. J. de Vries, and F. Auer, “Decoupling of collocated actuator-sensor-pairs for active vibration control,” in 21st Benelux Meeting on Systems and Control Book of Abstracts (2002), p. 66.

Kakiuchi, S.

S. Kakiuchi, “Camera Provided With Camera-Shake Compensation Functionality,” US 2006/0008263 A1 (2006).

Ko, S.-J.

S.-J. Ko, S.-H. Lee, and K.-H. Lee, “Digital image stabilizing algorithms based on bit-plane matching,” IEEE Trans. Consumer Electronics 44(3), 617–622 (1998).
[Crossref]

Lee, K.-H.

S.-J. Ko, S.-H. Lee, and K.-H. Lee, “Digital image stabilizing algorithms based on bit-plane matching,” IEEE Trans. Consumer Electronics 44(3), 617–622 (1998).
[Crossref]

Lee, S.-H.

S.-J. Ko, S.-H. Lee, and K.-H. Lee, “Digital image stabilizing algorithms based on bit-plane matching,” IEEE Trans. Consumer Electronics 44(3), 617–622 (1998).
[Crossref]

Moore, J. B.

B. D. Anderson and J. B. Moore, Optimal Filtering (Courier Dover Publications, 2012).

Morrison, M. M.

M. M. Morrison, “Inertial Measurement Unit,” US 4711125 (1987).

Neal, D. A.

D. R. Neal, J. Copland, and D. A. Neal, “Shack-Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002).
[Crossref]

Neal, D. R.

D. R. Neal, J. Copland, and D. A. Neal, “Shack-Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002).
[Crossref]

Noll, R. J.

Osten, W.

H. Gilbergs, N. Wengert, K. Frenner, P. Eberhard, and W. Osten, “Reconstruction of dynamical perturbations in optical systems by opto-mechanical simulation methods,” Proc. SPIE 8326, 83262N (2012).
[Crossref]

Platt, B. C.

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61(5), 656 (1971).

Shack, R. V.

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61(5), 656 (1971).

Washisu, K.

K. Washisu, “Control Apparatus For Image Blur Correction,” US 6757488 B2 (2004).

Watson, J.

J. Watson, “Tip-Tilt Correction for Astronomical Telescopes using Adaptive Control,” Wescon - Integrated Circuit Expo (1997), pp. 490–494.

Wengert, N.

H. Gilbergs, N. Wengert, K. Frenner, P. Eberhard, and W. Osten, “Reconstruction of dynamical perturbations in optical systems by opto-mechanical simulation methods,” Proc. SPIE 8326, 83262N (2012).
[Crossref]

Control Systems, Robotics and Automation (1)

M. Araki, “PID control,” Control Systems, Robotics and Automation 2, 1–23, (2002).

IEEE Trans. Consumer Electronics (1)

S.-J. Ko, S.-H. Lee, and K.-H. Lee, “Digital image stabilizing algorithms based on bit-plane matching,” IEEE Trans. Consumer Electronics 44(3), 617–622 (1998).
[Crossref]

J. Opt. Soc. Am. (2)

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61(5), 656 (1971).

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207 (1976).
[Crossref]

Proc. SPIE (3)

D. R. Neal, J. Copland, and D. A. Neal, “Shack-Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002).
[Crossref]

H. Gilbergs, N. Wengert, K. Frenner, P. Eberhard, and W. Osten, “Reconstruction of dynamical perturbations in optical systems by opto-mechanical simulation methods,” Proc. SPIE 8326, 83262N (2012).
[Crossref]

J. H. Burge, “An easy way to relate optical element motion to system pointing stability,” Proc. SPIE 6288, 62880I (2006).
[Crossref]

Tech. Mess. (1)

K. F. Beckstette, “Ultrapräzise Oberflächenbearbeitung am Beispiel von Lithografieoptiken (Ultraprecise Surface Figuring for Lithography Optics),” Tech. Mess. 69(12), 526 (2002).
[Crossref]

Other (6)

J. Holterman, T. J. de Vries, and F. Auer, “Decoupling of collocated actuator-sensor-pairs for active vibration control,” in 21st Benelux Meeting on Systems and Control Book of Abstracts (2002), p. 66.

K. Washisu, “Control Apparatus For Image Blur Correction,” US 6757488 B2 (2004).

S. Kakiuchi, “Camera Provided With Camera-Shake Compensation Functionality,” US 2006/0008263 A1 (2006).

B. D. Anderson and J. B. Moore, Optimal Filtering (Courier Dover Publications, 2012).

J. Watson, “Tip-Tilt Correction for Astronomical Telescopes using Adaptive Control,” Wescon - Integrated Circuit Expo (1997), pp. 490–494.

M. M. Morrison, “Inertial Measurement Unit,” US 4711125 (1987).

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Figures (7)

Fig. 1
Fig. 1

(a) A lenslet array is illuminated by a plane wave and generates spot images on the CCD detector. (b) A tilt in the incident wave leads to an evenly distributed shift in the spot positions. The spots positions of a) are marked with red dots.

Fig. 2
Fig. 2

(a) The setup consists of three lenses, each mounted on a physical pendulum. A solid state laser (λ = 532nm) is focused on a pinhole to act as a point light source. A Shack-Hartmann sensor is used for the detection. For reference measurements each lens pendulum is equipped with an IMU (not depicted). (b) The pendulums hold the lenses at a distance L = 160mm from the rotation axis.

Fig. 3
Fig. 3

(a) Simulation of the wavefront x-tilt in the aperture of the wavefront sensor as a function of the angular displacement of each lens. For small angles a linear dependence can be assumed. (b) Measured angular data from the gyroscopes (green dots) and a damped harmonic oscillator fit (black line). The data is reproduced accurately, with a slight degradation at lower amplitudes, where holding friction becomes noticeable.

Fig. 4
Fig. 4

Results of the time varying state observer applied to experimental data. The reference measurement from the IMUs (small markers) is reproduced with slight deviations due to motion blur in the wavefront measurement.

Fig. 5
Fig. 5

Optical setup used for the PD control experiment. The object plane (left side) is projected to the image plane with a 4x reduction. The three single lenses as well as the lens group in the center can be controlled using a linear actuation system. For the detection of the wavefront tilt a Shack-Hartmann sensor located after the image plane is used (not depicted). The distance from the object plane to the image is 857.7mm

Fig. 6
Fig. 6

Schematic of the full PD control loop. A seperate PD controller is implemented for each of the n optical elements in the system.

Fig. 7
Fig. 7

Comparison of the resulting lens oscillations for the same initial conditions with and without PD control. (a) The exact values of the decentrations (small markers) are reproduced accurately after 1s. (b) The controller dampens the vibrations to zero in 5s. The measurements are conducted on a quasi-static setup for high repeatability of the oscillations for identical initial conditions.

Equations (19)

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α ¨ + δ α ˙ + ω 2 α = 0
α = α 0 e δ t sin ( ω t + ϕ ) .
T = i = 1 n S i α i = i = 1 n S i α 0 , i e δ i t sin ( ω i t + ϕ i )
x k + 1 = A k x k + B k u k ,
y k = C k x k ,
x = ( α 0 , 1 α 0 , n ϕ 1 ϕ n )
T ( t ) = i = 1 n S i α 0 , i e δ i t [ sin ( ω i t ) cos ( ϕ i ) + cos ( ω i t ) sin ( ϕ i ) ]
T ( t ) = i = 1 n S i α 0 , i cos ( ϕ i ) e δ i t sin ( ω i t ) + i = 1 n S i α 0 , i sin ( ϕ i ) e δ i t cos ( ω i t )
= i = 1 n ( c i e δ i t sin ( ω i t ) + s i e δ i t cos ( ω i t ) )
x = ( c 1 c n s 1 s n ) C ( t ) = ( e δ 1 t sin ( ω 1 t ) e δ n t sin ( ω n t ) e δ 1 t cos ( ω 1 t ) e δ n t cos ( ω n t ) ) ,
x ^ k | k 1 = A x ^ k 1 + Bu = x ^ k 1 + Bu
P ^ k | k 1 = A P ^ k 1 A = P ^ k 1 ,
x ^ k = x ^ k | k 1 + K ^ k y ˜ k
P ^ k = P ^ k | k 1 K ^ k S k K ^ k
y ˜ k = y k C k x ^ k | k 1
S k = C k P k | k 1 C k + R k
K ^ k = P k | k 1 C k S k 1 .
u i , k = K p α i , k + K d α i , k α i , k 1 Δ t
B k = ( S 1 cos ( ϕ 1 , k ) S n cos ( ϕ n , k ) S 1 sin ( ϕ 1 , k ) S n sin ( ϕ n , k ) ) u k = ( u 1 , k u n , k u 1 , k u n , k ) ,

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