Abstract

Precision optical systems that utilize laser beams as working media usually suffer from thermal aberrations caused by absorbed energy. Based on a specially designed three-lens system, the causes and contributions of mechanical structures to the system’s thermal aberrations are studied. The contribution of three thermal effects, surface deformation, change of refractive index, and stress birefringence on the system’s thermal aberrations, is analyzed respectively through an integrated optomechanical simulation method. The impact of the structure’s thermal dissipating capability and structure configuration on the system’s thermal aberrations is analyzed, too. Experiments have been carried out to validate the correctness and accuracy of the simulation method. Both the simulated and tested results can provide a reference for structure design and thermal aberration analysis of the similar optical systems.

© 2013 Optical Society of America

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References

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  1. K. Mann, A. Bayer, U. Leinhos, M. Schöneck, and B. Schäfer, “Measurement of wavefront distortions in DUV optics due to lens heating,” Proc. SPIE 7973, 79732B (2011).
    [CrossRef]
  2. J. Zhou, Y. Zhang, P. Engblom, M. Hyatt, E. Wu, M. Snajdr, A. Devilliers, Y. He, C. Hickman, P. Liu, D. de Lang, B. Geh, E. Byers, and S. Light, “Improving aberration control with application specific optimization using computational lithography,” Proc. SPIE 7640, 76400K (2010).
    [CrossRef]
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    [CrossRef]
  5. Y. Uehara, T. Matsuyama, T. Nakashima, Y. Ohmura, T. Ogata, K. Suzuki, and N. Tokuda, “Thermal aberration control for low k1 lithography,” Proc. SPIE 6520, 65202V (2007).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. K. B. Doyle, V. L. Genberg, and G. J. Michels, Integrated Optomechanical Analysis (SPIE, 2002).
  13. SigFit Reference Manual, version 2012R1d, Sigmadyne Inc.
  14. HPFS Fused silica standard grade semiconductor optics, Corning Inc..
  15. K. B. Doyle, “Thermo-elastic wavefront and polarization error analysis of a telecommunication optical circulator,” Proc. SPIE 4093, 18–27 (2000).
    [CrossRef]
  16. K. B. Doyle, “Numerical methods to compute optical errors due to stress birefringence,” Proc. SPIE 476934–42 (2002).
    [CrossRef]
  17. J. Schroeder, “Brillouin scattering and pockels coefficients in silicate glasses,” J. Non-Cryst. Solids 40, 549–566 (1980).
    [CrossRef]
  18. D. Su, E. Miao, Y. Sui, and H. Yang, “Absolute surface figure testing by shift-rotation method using Zernike polynomials,” Opt. Lett. 37, 3198–3200 (2012).
    [CrossRef]

2012 (1)

2011 (2)

H.-S. Yang, H. Kihm, I. K. Moon, G.-J. Jung, S.-C. Choi, K.-J. Lee, H.-Y. Hwang, S.-W. Kim, and Y.-W. Lee, “Three-shell based lens barrel for the effective athermalization of an IR optical system,” Appl. Opt. 50, 6206–6213 (2011).
[CrossRef]

K. Mann, A. Bayer, U. Leinhos, M. Schöneck, and B. Schäfer, “Measurement of wavefront distortions in DUV optics due to lens heating,” Proc. SPIE 7973, 79732B (2011).
[CrossRef]

2010 (2)

J. Zhou, Y. Zhang, P. Engblom, M. Hyatt, E. Wu, M. Snajdr, A. Devilliers, Y. He, C. Hickman, P. Liu, D. de Lang, B. Geh, E. Byers, and S. Light, “Improving aberration control with application specific optimization using computational lithography,” Proc. SPIE 7640, 76400K (2010).
[CrossRef]

T. Legero, T. Kessler, and U. Sterr, “Tuning the thermal expansion properties of optical reference cavities with fused silica mirrors,” J. Opt. Soc. Am. B 27, 914–919 (2010).
[CrossRef]

2008 (1)

T. Nakashima, Y. Ohmura, T. Ogata, Y. Uehara, H. Nishinaga, and T. Matsuyama, “Thermal aberration control in projection lens,” Proc. SPIE 6924, 69241V (2008).
[CrossRef]

2007 (2)

J. de Klerk, C. Wagner, R. Droste, L. Levasier, L. Jorritsma, E. V. Setten, H. Kattouw, J. Jacobs, and T. Heil, “Performance of a 1.35NA ArF immersion lithography system for 40 nm applications,” Proc. SPIE 6520, 65201Y (2007).
[CrossRef]

Y. Uehara, T. Matsuyama, T. Nakashima, Y. Ohmura, T. Ogata, K. Suzuki, and N. Tokuda, “Thermal aberration control for low k1 lithography,” Proc. SPIE 6520, 65202V (2007).
[CrossRef]

2006 (1)

2005 (1)

2002 (1)

K. B. Doyle, “Numerical methods to compute optical errors due to stress birefringence,” Proc. SPIE 476934–42 (2002).
[CrossRef]

2000 (1)

K. B. Doyle, “Thermo-elastic wavefront and polarization error analysis of a telecommunication optical circulator,” Proc. SPIE 4093, 18–27 (2000).
[CrossRef]

1980 (1)

J. Schroeder, “Brillouin scattering and pockels coefficients in silicate glasses,” J. Non-Cryst. Solids 40, 549–566 (1980).
[CrossRef]

Abdusamatov, K. I.

Bayer, A.

K. Mann, A. Bayer, U. Leinhos, M. Schöneck, and B. Schäfer, “Measurement of wavefront distortions in DUV optics due to lens heating,” Proc. SPIE 7973, 79732B (2011).
[CrossRef]

Bigelow, M.

M. Bigelow and N. Harned, “Taking optical precision to the extreme,” oemagazine, 32–33 (2004).
[CrossRef]

Byers, E.

J. Zhou, Y. Zhang, P. Engblom, M. Hyatt, E. Wu, M. Snajdr, A. Devilliers, Y. He, C. Hickman, P. Liu, D. de Lang, B. Geh, E. Byers, and S. Light, “Improving aberration control with application specific optimization using computational lithography,” Proc. SPIE 7640, 76400K (2010).
[CrossRef]

Choi, S.-C.

de Klerk, J.

J. de Klerk, C. Wagner, R. Droste, L. Levasier, L. Jorritsma, E. V. Setten, H. Kattouw, J. Jacobs, and T. Heil, “Performance of a 1.35NA ArF immersion lithography system for 40 nm applications,” Proc. SPIE 6520, 65201Y (2007).
[CrossRef]

de Lang, D.

J. Zhou, Y. Zhang, P. Engblom, M. Hyatt, E. Wu, M. Snajdr, A. Devilliers, Y. He, C. Hickman, P. Liu, D. de Lang, B. Geh, E. Byers, and S. Light, “Improving aberration control with application specific optimization using computational lithography,” Proc. SPIE 7640, 76400K (2010).
[CrossRef]

Devilliers, A.

J. Zhou, Y. Zhang, P. Engblom, M. Hyatt, E. Wu, M. Snajdr, A. Devilliers, Y. He, C. Hickman, P. Liu, D. de Lang, B. Geh, E. Byers, and S. Light, “Improving aberration control with application specific optimization using computational lithography,” Proc. SPIE 7640, 76400K (2010).
[CrossRef]

Doyle, K. B.

K. B. Doyle, “Numerical methods to compute optical errors due to stress birefringence,” Proc. SPIE 476934–42 (2002).
[CrossRef]

K. B. Doyle, “Thermo-elastic wavefront and polarization error analysis of a telecommunication optical circulator,” Proc. SPIE 4093, 18–27 (2000).
[CrossRef]

K. B. Doyle, V. L. Genberg, and G. J. Michels, Integrated Optomechanical Analysis (SPIE, 2002).

Droste, R.

J. de Klerk, C. Wagner, R. Droste, L. Levasier, L. Jorritsma, E. V. Setten, H. Kattouw, J. Jacobs, and T. Heil, “Performance of a 1.35NA ArF immersion lithography system for 40 nm applications,” Proc. SPIE 6520, 65201Y (2007).
[CrossRef]

Engblom, P.

J. Zhou, Y. Zhang, P. Engblom, M. Hyatt, E. Wu, M. Snajdr, A. Devilliers, Y. He, C. Hickman, P. Liu, D. de Lang, B. Geh, E. Byers, and S. Light, “Improving aberration control with application specific optimization using computational lithography,” Proc. SPIE 7640, 76400K (2010).
[CrossRef]

Geh, B.

J. Zhou, Y. Zhang, P. Engblom, M. Hyatt, E. Wu, M. Snajdr, A. Devilliers, Y. He, C. Hickman, P. Liu, D. de Lang, B. Geh, E. Byers, and S. Light, “Improving aberration control with application specific optimization using computational lithography,” Proc. SPIE 7640, 76400K (2010).
[CrossRef]

Genberg, V. L.

K. B. Doyle, V. L. Genberg, and G. J. Michels, Integrated Optomechanical Analysis (SPIE, 2002).

Harned, N.

M. Bigelow and N. Harned, “Taking optical precision to the extreme,” oemagazine, 32–33 (2004).
[CrossRef]

He, Y.

J. Zhou, Y. Zhang, P. Engblom, M. Hyatt, E. Wu, M. Snajdr, A. Devilliers, Y. He, C. Hickman, P. Liu, D. de Lang, B. Geh, E. Byers, and S. Light, “Improving aberration control with application specific optimization using computational lithography,” Proc. SPIE 7640, 76400K (2010).
[CrossRef]

Heil, T.

J. de Klerk, C. Wagner, R. Droste, L. Levasier, L. Jorritsma, E. V. Setten, H. Kattouw, J. Jacobs, and T. Heil, “Performance of a 1.35NA ArF immersion lithography system for 40 nm applications,” Proc. SPIE 6520, 65201Y (2007).
[CrossRef]

Hickman, C.

J. Zhou, Y. Zhang, P. Engblom, M. Hyatt, E. Wu, M. Snajdr, A. Devilliers, Y. He, C. Hickman, P. Liu, D. de Lang, B. Geh, E. Byers, and S. Light, “Improving aberration control with application specific optimization using computational lithography,” Proc. SPIE 7640, 76400K (2010).
[CrossRef]

Hwang, H.-Y.

Hyatt, M.

J. Zhou, Y. Zhang, P. Engblom, M. Hyatt, E. Wu, M. Snajdr, A. Devilliers, Y. He, C. Hickman, P. Liu, D. de Lang, B. Geh, E. Byers, and S. Light, “Improving aberration control with application specific optimization using computational lithography,” Proc. SPIE 7640, 76400K (2010).
[CrossRef]

Jacobs, J.

J. de Klerk, C. Wagner, R. Droste, L. Levasier, L. Jorritsma, E. V. Setten, H. Kattouw, J. Jacobs, and T. Heil, “Performance of a 1.35NA ArF immersion lithography system for 40 nm applications,” Proc. SPIE 6520, 65201Y (2007).
[CrossRef]

Jorritsma, L.

J. de Klerk, C. Wagner, R. Droste, L. Levasier, L. Jorritsma, E. V. Setten, H. Kattouw, J. Jacobs, and T. Heil, “Performance of a 1.35NA ArF immersion lithography system for 40 nm applications,” Proc. SPIE 6520, 65201Y (2007).
[CrossRef]

Jung, G.-J.

Kattouw, H.

J. de Klerk, C. Wagner, R. Droste, L. Levasier, L. Jorritsma, E. V. Setten, H. Kattouw, J. Jacobs, and T. Heil, “Performance of a 1.35NA ArF immersion lithography system for 40 nm applications,” Proc. SPIE 6520, 65201Y (2007).
[CrossRef]

Kessler, T.

Khankov, S. I.

Kihm, H.

Kim, S.-W.

Lee, K.-J.

Lee, Y.-W.

Legero, T.

Leinhos, U.

K. Mann, A. Bayer, U. Leinhos, M. Schöneck, and B. Schäfer, “Measurement of wavefront distortions in DUV optics due to lens heating,” Proc. SPIE 7973, 79732B (2011).
[CrossRef]

Levasier, L.

J. de Klerk, C. Wagner, R. Droste, L. Levasier, L. Jorritsma, E. V. Setten, H. Kattouw, J. Jacobs, and T. Heil, “Performance of a 1.35NA ArF immersion lithography system for 40 nm applications,” Proc. SPIE 6520, 65201Y (2007).
[CrossRef]

Light, S.

J. Zhou, Y. Zhang, P. Engblom, M. Hyatt, E. Wu, M. Snajdr, A. Devilliers, Y. He, C. Hickman, P. Liu, D. de Lang, B. Geh, E. Byers, and S. Light, “Improving aberration control with application specific optimization using computational lithography,” Proc. SPIE 7640, 76400K (2010).
[CrossRef]

Liu, C.

Liu, P.

J. Zhou, Y. Zhang, P. Engblom, M. Hyatt, E. Wu, M. Snajdr, A. Devilliers, Y. He, C. Hickman, P. Liu, D. de Lang, B. Geh, E. Byers, and S. Light, “Improving aberration control with application specific optimization using computational lithography,” Proc. SPIE 7640, 76400K (2010).
[CrossRef]

Lü, B.

Mann, K.

K. Mann, A. Bayer, U. Leinhos, M. Schöneck, and B. Schäfer, “Measurement of wavefront distortions in DUV optics due to lens heating,” Proc. SPIE 7973, 79732B (2011).
[CrossRef]

Matsuyama, T.

T. Nakashima, Y. Ohmura, T. Ogata, Y. Uehara, H. Nishinaga, and T. Matsuyama, “Thermal aberration control in projection lens,” Proc. SPIE 6924, 69241V (2008).
[CrossRef]

Y. Uehara, T. Matsuyama, T. Nakashima, Y. Ohmura, T. Ogata, K. Suzuki, and N. Tokuda, “Thermal aberration control for low k1 lithography,” Proc. SPIE 6520, 65202V (2007).
[CrossRef]

Miao, E.

Michels, G. J.

K. B. Doyle, V. L. Genberg, and G. J. Michels, Integrated Optomechanical Analysis (SPIE, 2002).

Moon, I. K.

Nakashima, T.

T. Nakashima, Y. Ohmura, T. Ogata, Y. Uehara, H. Nishinaga, and T. Matsuyama, “Thermal aberration control in projection lens,” Proc. SPIE 6924, 69241V (2008).
[CrossRef]

Y. Uehara, T. Matsuyama, T. Nakashima, Y. Ohmura, T. Ogata, K. Suzuki, and N. Tokuda, “Thermal aberration control for low k1 lithography,” Proc. SPIE 6520, 65202V (2007).
[CrossRef]

Nishinaga, H.

T. Nakashima, Y. Ohmura, T. Ogata, Y. Uehara, H. Nishinaga, and T. Matsuyama, “Thermal aberration control in projection lens,” Proc. SPIE 6924, 69241V (2008).
[CrossRef]

Ogata, T.

T. Nakashima, Y. Ohmura, T. Ogata, Y. Uehara, H. Nishinaga, and T. Matsuyama, “Thermal aberration control in projection lens,” Proc. SPIE 6924, 69241V (2008).
[CrossRef]

Y. Uehara, T. Matsuyama, T. Nakashima, Y. Ohmura, T. Ogata, K. Suzuki, and N. Tokuda, “Thermal aberration control for low k1 lithography,” Proc. SPIE 6520, 65202V (2007).
[CrossRef]

Ohmura, Y.

T. Nakashima, Y. Ohmura, T. Ogata, Y. Uehara, H. Nishinaga, and T. Matsuyama, “Thermal aberration control in projection lens,” Proc. SPIE 6924, 69241V (2008).
[CrossRef]

Y. Uehara, T. Matsuyama, T. Nakashima, Y. Ohmura, T. Ogata, K. Suzuki, and N. Tokuda, “Thermal aberration control for low k1 lithography,” Proc. SPIE 6520, 65202V (2007).
[CrossRef]

Schäfer, B.

K. Mann, A. Bayer, U. Leinhos, M. Schöneck, and B. Schäfer, “Measurement of wavefront distortions in DUV optics due to lens heating,” Proc. SPIE 7973, 79732B (2011).
[CrossRef]

Schöneck, M.

K. Mann, A. Bayer, U. Leinhos, M. Schöneck, and B. Schäfer, “Measurement of wavefront distortions in DUV optics due to lens heating,” Proc. SPIE 7973, 79732B (2011).
[CrossRef]

Schroeder, J.

J. Schroeder, “Brillouin scattering and pockels coefficients in silicate glasses,” J. Non-Cryst. Solids 40, 549–566 (1980).
[CrossRef]

Setten, E. V.

J. de Klerk, C. Wagner, R. Droste, L. Levasier, L. Jorritsma, E. V. Setten, H. Kattouw, J. Jacobs, and T. Heil, “Performance of a 1.35NA ArF immersion lithography system for 40 nm applications,” Proc. SPIE 6520, 65201Y (2007).
[CrossRef]

Snajdr, M.

J. Zhou, Y. Zhang, P. Engblom, M. Hyatt, E. Wu, M. Snajdr, A. Devilliers, Y. He, C. Hickman, P. Liu, D. de Lang, B. Geh, E. Byers, and S. Light, “Improving aberration control with application specific optimization using computational lithography,” Proc. SPIE 7640, 76400K (2010).
[CrossRef]

Sterr, U.

Su, D.

Sui, Y.

Suzuki, K.

Y. Uehara, T. Matsuyama, T. Nakashima, Y. Ohmura, T. Ogata, K. Suzuki, and N. Tokuda, “Thermal aberration control for low k1 lithography,” Proc. SPIE 6520, 65202V (2007).
[CrossRef]

Tan, F.

Tokuda, N.

Y. Uehara, T. Matsuyama, T. Nakashima, Y. Ohmura, T. Ogata, K. Suzuki, and N. Tokuda, “Thermal aberration control for low k1 lithography,” Proc. SPIE 6520, 65202V (2007).
[CrossRef]

Uehara, Y.

T. Nakashima, Y. Ohmura, T. Ogata, Y. Uehara, H. Nishinaga, and T. Matsuyama, “Thermal aberration control in projection lens,” Proc. SPIE 6924, 69241V (2008).
[CrossRef]

Y. Uehara, T. Matsuyama, T. Nakashima, Y. Ohmura, T. Ogata, K. Suzuki, and N. Tokuda, “Thermal aberration control for low k1 lithography,” Proc. SPIE 6520, 65202V (2007).
[CrossRef]

Wagner, C.

J. de Klerk, C. Wagner, R. Droste, L. Levasier, L. Jorritsma, E. V. Setten, H. Kattouw, J. Jacobs, and T. Heil, “Performance of a 1.35NA ArF immersion lithography system for 40 nm applications,” Proc. SPIE 6520, 65201Y (2007).
[CrossRef]

Wang, W.

Wu, E.

J. Zhou, Y. Zhang, P. Engblom, M. Hyatt, E. Wu, M. Snajdr, A. Devilliers, Y. He, C. Hickman, P. Liu, D. de Lang, B. Geh, E. Byers, and S. Light, “Improving aberration control with application specific optimization using computational lithography,” Proc. SPIE 7640, 76400K (2010).
[CrossRef]

Yang, H.

Yang, H.-S.

Yoder, P. R.

P. R. Yoder, Mounting Optics in Optical Instruments, 2nd ed. (SPIE, 2008).

Zhang, Y.

J. Zhou, Y. Zhang, P. Engblom, M. Hyatt, E. Wu, M. Snajdr, A. Devilliers, Y. He, C. Hickman, P. Liu, D. de Lang, B. Geh, E. Byers, and S. Light, “Improving aberration control with application specific optimization using computational lithography,” Proc. SPIE 7640, 76400K (2010).
[CrossRef]

Zhou, J.

J. Zhou, Y. Zhang, P. Engblom, M. Hyatt, E. Wu, M. Snajdr, A. Devilliers, Y. He, C. Hickman, P. Liu, D. de Lang, B. Geh, E. Byers, and S. Light, “Improving aberration control with application specific optimization using computational lithography,” Proc. SPIE 7640, 76400K (2010).
[CrossRef]

Appl. Opt. (2)

J. Non-Cryst. Solids (1)

J. Schroeder, “Brillouin scattering and pockels coefficients in silicate glasses,” J. Non-Cryst. Solids 40, 549–566 (1980).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Opt. Technol. (1)

Opt. Lett. (1)

Proc. SPIE (7)

K. B. Doyle, “Thermo-elastic wavefront and polarization error analysis of a telecommunication optical circulator,” Proc. SPIE 4093, 18–27 (2000).
[CrossRef]

K. B. Doyle, “Numerical methods to compute optical errors due to stress birefringence,” Proc. SPIE 476934–42 (2002).
[CrossRef]

J. de Klerk, C. Wagner, R. Droste, L. Levasier, L. Jorritsma, E. V. Setten, H. Kattouw, J. Jacobs, and T. Heil, “Performance of a 1.35NA ArF immersion lithography system for 40 nm applications,” Proc. SPIE 6520, 65201Y (2007).
[CrossRef]

Y. Uehara, T. Matsuyama, T. Nakashima, Y. Ohmura, T. Ogata, K. Suzuki, and N. Tokuda, “Thermal aberration control for low k1 lithography,” Proc. SPIE 6520, 65202V (2007).
[CrossRef]

K. Mann, A. Bayer, U. Leinhos, M. Schöneck, and B. Schäfer, “Measurement of wavefront distortions in DUV optics due to lens heating,” Proc. SPIE 7973, 79732B (2011).
[CrossRef]

J. Zhou, Y. Zhang, P. Engblom, M. Hyatt, E. Wu, M. Snajdr, A. Devilliers, Y. He, C. Hickman, P. Liu, D. de Lang, B. Geh, E. Byers, and S. Light, “Improving aberration control with application specific optimization using computational lithography,” Proc. SPIE 7640, 76400K (2010).
[CrossRef]

T. Nakashima, Y. Ohmura, T. Ogata, Y. Uehara, H. Nishinaga, and T. Matsuyama, “Thermal aberration control in projection lens,” Proc. SPIE 6924, 69241V (2008).
[CrossRef]

Other (5)

K. B. Doyle, V. L. Genberg, and G. J. Michels, Integrated Optomechanical Analysis (SPIE, 2002).

SigFit Reference Manual, version 2012R1d, Sigmadyne Inc.

HPFS Fused silica standard grade semiconductor optics, Corning Inc..

M. Bigelow and N. Harned, “Taking optical precision to the extreme,” oemagazine, 32–33 (2004).
[CrossRef]

P. R. Yoder, Mounting Optics in Optical Instruments, 2nd ed. (SPIE, 2008).

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Figures (10)

Fig. 1.
Fig. 1.

Setup of the three lens experimental system. (a) Diameter of the three lenses is between 125 and 152 mm, thickness is about 36 mm, the diameter of the six electrical film heaters is 20 mm, resistance is 31 Ω, thickness is 0.2 mm. (b) Photograph of the experimental system.

Fig. 2.
Fig. 2.

Arrangement of the temperature sensors and film electrical heaters on the three lenses.

Fig. 3.
Fig. 3.

(a) FEM model of the three-lens system. (b) Temperature distribution of the system. (c), (d) Displacement and stress distribution of lens nodes when use temperature results in (b) as temperature load.

Fig. 4.
Fig. 4.

Simulated results. (a) PV and rms values of the six surfaces (without power) and (b) system thermal aberration induced by surface deformations.

Fig. 5.
Fig. 5.

Definition of integration paths for OPD analysis.

Fig. 6.
Fig. 6.

Simulated results. (a) PV and rms wavefront values of the three lenses and (b) system thermal aberration induced by change of refractive index.

Fig. 7.
Fig. 7.

Experimental results. (a) System wavefront before heated, (b) system wavefront after heated, (c) system thermal aberration calculated by subtraction of interferogram, (d) Code V standard Zernike fitting of (c), (e) residual of Zernike polynomials fitting, and (f) FFT low pass filter of (e), and the cutoff period is 4 mm.

Fig. 8.
Fig. 8.

Z4Z45 Code V standard Zernike coefficients of the tested and simulated thermal aberrations. (a) Experimental result after data alignment, (b) simulated result combining both surface deformation and OPD effects, and (c) figure generated from corresponding Zernike term coefficient subtraction of (a) and (b).

Fig. 9.
Fig. 9.

Comparison of the tested and simulated temperature distribution on the three lenses.

Fig. 10.
Fig. 10.

Experimental verification of trefoil aberration. (a) 1 V heating voltage, (b) 2 V heating voltage, and (c) 3 V heating voltage.

Tables (2)

Tables Icon

Table 1. Main Physical Properties of Metals

Tables Icon

Table 2. System Thermal Aberration with Different Structure Materials

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

OPD=i=1Nδniδli,δni=T0TikdT.
Birefringence=R(σxσy),R=12n03(q11q12),
OPDstress=i=1N(Δn1+Δn22)iδli.

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