Abstract

WWe report a modeling framework for evaluating the performance of piezoelectrically actuated MEMS tunable lenses. It models the static opto-electromechanical coupling for symmetric configurations of piezoelectric actuators based on the laminated-plate theory, linear piezoelectricity, and ray tracing. With these assumptions, it helps to find geometrical parameters for actuators on clamped square or circular diaphragms that give a diffraction-limited tunable lens with minimum F-number. The tunable lens' optical performance and its focusing capability, alone and in combination with a paraxial fixed lens, were calculated in terms of object distance and actuation voltage. Using the modeling framework, we confirmed that the modulation transfer function for objects located at different distances remains the same after voltage adjustment.

© 2016 Optical Society of America

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References

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  1. Y. Tseng, “Voice coil motor apparatus,” US Patent7,400,068 (2008).
  2. C. Zhao, Ultrasonic Motors: Technologies and Applications (Science PressBeijing and Springer-VerlagBerlin Heidelberg, 2011).
    [Crossref]
  3. M. Ye, B. Wang, and S. Sato, “Liquid-crystal lens with a focal length that is variable in a wide range,” Appl. Opt. 43, 6407–6412 (2004).
    [Crossref] [PubMed]
  4. N. Chronis, G. Liu, K.-H. Jeong, and L. Lee, “Tunable liquid-filled microlens array integrated with microfluidic network,” Opt. Express 11, 2370–2378 (2003).
    [Crossref] [PubMed]
  5. A. Werber and H. Zappe, “Tunable microfluidic microlenses,” Appl. Opt. 44, 3238–3245 (2005).
    [Crossref] [PubMed]
  6. S. Kuiper, B. H. Hendriks, L. J. Huijbregts, A. M. Hirschberg, C. A. Renders, and M. A. van As, “Variable-focus liquid lens for portable applications,” Proc. SPIE 5523, 100–109 (2004).
    [Crossref]
  7. U. Wallrabe, “Axicons et al. - highly aspherical adaptive optical elements for the life sciences,” in “2015 18th International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS), 2015 Transducers,” (2015), pp. 251–256.
  8. K. Haugholt, D. Wang, F. Tyholdt, W. Booij, and I. Johansen, “Polymer lens,” US Patent8,199,410 (2012).
  9. V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Appl. Opt. 33, 8121–8124 (1994).
    [Crossref] [PubMed]
  10. M. Deshpande and L. Saggere, “An analytical model and working equations for static deflections of a circular multi-layered diaphragm-type piezoelectric actuator,” Sensors and Actuators A: Physical 136, 673–689 (2007).
    [Crossref]
  11. S. I. E. Lin, “Investigation on packaging parameters of a circular multi-layered diaphragm-type piezoelectric actuator,” Computers and Structures 89, 371–379 (2011).
    [Crossref]
  12. J. N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd ed. (CRC Press, 2004).
  13. J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices (Oxford University Press, 1985).
  14. V. Birman, Plate Structures, vol. 178 of Solid Mechanics and its Applications (Springer Science+Bussiness Media, 2011).
    [Crossref]
  15. E. Tadmor and G. Kosa, “Electromechanical coupling correction for piezoelectric layered beams,” Journal of Microelectromechanical Systems 12, 899–906 (2003).
    [Crossref]
  16. H. F. Tiersten, “Hamilton’s principle for linear piezoelectric media,” in Proceedings of the IEEE55 (IEEE, 1967), pp. 1523–1524.
  17. J. Reddy and J. Mitchell, “On refined nonlinear theories of laminated composite structures with piezoelectric laminae,” Sadhana 20, 721–747 (1995).
    [Crossref]
  18. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  19. P. Laura, E. Romanelli, and R. Rossi, “Transverse vibrations of simply supported rectangular plates with rectangular cutouts,” Journal of Sound and Vibration 202, 275–283 (1997).
    [Crossref]
  20. B. Boyerinas, C. Mo, and W. W. Clark, “Behavior of unimorph rectangular piezoelectric diaphragm actuators,” Proc. SPIE 6173, 617302 (2006).
    [Crossref]
  21. R. L. Taylor and S. Govindjee, “Solution of clamped rectangular plate problems,” Communications in Numerical Methods in Engineering 20, 757–765 (2004).
    [Crossref]
  22. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge Univeristy Press, New York, 2010).
  23. N. Ledermann, P. Muralt, J. Baborowski, S. Gentil, K. Mukati, M. Cantoni, A. Seifert, and N. Setter, “1 0 0-textured, piezoelectric Pb(ZrxTi1−x)O3 thin films for MEMS: integration, deposition and properties,” Sensors and Actuators A: Physical 105, 162–170 (2003).
    [Crossref]
  24. T. Bakke, A. Vogl, O. Żero, F. Tyholdt, I.-R. Johansen, and D. Wang, “A novel ultra-planar, long-stroke and low-voltage piezoelectric micromirror,” Journal of Micromechanics and Microengineering 20, 064010 (2010).
    [Crossref]
  25. Zemax LLC, Zemax User’s Manual (2015).
  26. A. Maréchal, “Etude des influences conjuguées des aberrations et de la diffraction sur l’image d’un point,” Ph.D. thesis, Faculté des Sciences des Paris (1947).

2011 (1)

S. I. E. Lin, “Investigation on packaging parameters of a circular multi-layered diaphragm-type piezoelectric actuator,” Computers and Structures 89, 371–379 (2011).
[Crossref]

2010 (1)

T. Bakke, A. Vogl, O. Żero, F. Tyholdt, I.-R. Johansen, and D. Wang, “A novel ultra-planar, long-stroke and low-voltage piezoelectric micromirror,” Journal of Micromechanics and Microengineering 20, 064010 (2010).
[Crossref]

2007 (1)

M. Deshpande and L. Saggere, “An analytical model and working equations for static deflections of a circular multi-layered diaphragm-type piezoelectric actuator,” Sensors and Actuators A: Physical 136, 673–689 (2007).
[Crossref]

2006 (1)

B. Boyerinas, C. Mo, and W. W. Clark, “Behavior of unimorph rectangular piezoelectric diaphragm actuators,” Proc. SPIE 6173, 617302 (2006).
[Crossref]

2005 (1)

2004 (3)

M. Ye, B. Wang, and S. Sato, “Liquid-crystal lens with a focal length that is variable in a wide range,” Appl. Opt. 43, 6407–6412 (2004).
[Crossref] [PubMed]

R. L. Taylor and S. Govindjee, “Solution of clamped rectangular plate problems,” Communications in Numerical Methods in Engineering 20, 757–765 (2004).
[Crossref]

S. Kuiper, B. H. Hendriks, L. J. Huijbregts, A. M. Hirschberg, C. A. Renders, and M. A. van As, “Variable-focus liquid lens for portable applications,” Proc. SPIE 5523, 100–109 (2004).
[Crossref]

2003 (3)

N. Chronis, G. Liu, K.-H. Jeong, and L. Lee, “Tunable liquid-filled microlens array integrated with microfluidic network,” Opt. Express 11, 2370–2378 (2003).
[Crossref] [PubMed]

N. Ledermann, P. Muralt, J. Baborowski, S. Gentil, K. Mukati, M. Cantoni, A. Seifert, and N. Setter, “1 0 0-textured, piezoelectric Pb(ZrxTi1−x)O3 thin films for MEMS: integration, deposition and properties,” Sensors and Actuators A: Physical 105, 162–170 (2003).
[Crossref]

E. Tadmor and G. Kosa, “Electromechanical coupling correction for piezoelectric layered beams,” Journal of Microelectromechanical Systems 12, 899–906 (2003).
[Crossref]

1997 (1)

P. Laura, E. Romanelli, and R. Rossi, “Transverse vibrations of simply supported rectangular plates with rectangular cutouts,” Journal of Sound and Vibration 202, 275–283 (1997).
[Crossref]

1995 (1)

J. Reddy and J. Mitchell, “On refined nonlinear theories of laminated composite structures with piezoelectric laminae,” Sadhana 20, 721–747 (1995).
[Crossref]

1994 (1)

Baborowski, J.

N. Ledermann, P. Muralt, J. Baborowski, S. Gentil, K. Mukati, M. Cantoni, A. Seifert, and N. Setter, “1 0 0-textured, piezoelectric Pb(ZrxTi1−x)O3 thin films for MEMS: integration, deposition and properties,” Sensors and Actuators A: Physical 105, 162–170 (2003).
[Crossref]

Bakke, T.

T. Bakke, A. Vogl, O. Żero, F. Tyholdt, I.-R. Johansen, and D. Wang, “A novel ultra-planar, long-stroke and low-voltage piezoelectric micromirror,” Journal of Micromechanics and Microengineering 20, 064010 (2010).
[Crossref]

Birman, V.

V. Birman, Plate Structures, vol. 178 of Solid Mechanics and its Applications (Springer Science+Bussiness Media, 2011).
[Crossref]

Boisvert, R. F.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge Univeristy Press, New York, 2010).

Booij, W.

K. Haugholt, D. Wang, F. Tyholdt, W. Booij, and I. Johansen, “Polymer lens,” US Patent8,199,410 (2012).

Boyerinas, B.

B. Boyerinas, C. Mo, and W. W. Clark, “Behavior of unimorph rectangular piezoelectric diaphragm actuators,” Proc. SPIE 6173, 617302 (2006).
[Crossref]

Cantoni, M.

N. Ledermann, P. Muralt, J. Baborowski, S. Gentil, K. Mukati, M. Cantoni, A. Seifert, and N. Setter, “1 0 0-textured, piezoelectric Pb(ZrxTi1−x)O3 thin films for MEMS: integration, deposition and properties,” Sensors and Actuators A: Physical 105, 162–170 (2003).
[Crossref]

Chronis, N.

Clark, C. W.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge Univeristy Press, New York, 2010).

Clark, W. W.

B. Boyerinas, C. Mo, and W. W. Clark, “Behavior of unimorph rectangular piezoelectric diaphragm actuators,” Proc. SPIE 6173, 617302 (2006).
[Crossref]

Deshpande, M.

M. Deshpande and L. Saggere, “An analytical model and working equations for static deflections of a circular multi-layered diaphragm-type piezoelectric actuator,” Sensors and Actuators A: Physical 136, 673–689 (2007).
[Crossref]

Gentil, S.

N. Ledermann, P. Muralt, J. Baborowski, S. Gentil, K. Mukati, M. Cantoni, A. Seifert, and N. Setter, “1 0 0-textured, piezoelectric Pb(ZrxTi1−x)O3 thin films for MEMS: integration, deposition and properties,” Sensors and Actuators A: Physical 105, 162–170 (2003).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Govindjee, S.

R. L. Taylor and S. Govindjee, “Solution of clamped rectangular plate problems,” Communications in Numerical Methods in Engineering 20, 757–765 (2004).
[Crossref]

Haugholt, K.

K. Haugholt, D. Wang, F. Tyholdt, W. Booij, and I. Johansen, “Polymer lens,” US Patent8,199,410 (2012).

Hendriks, B. H.

S. Kuiper, B. H. Hendriks, L. J. Huijbregts, A. M. Hirschberg, C. A. Renders, and M. A. van As, “Variable-focus liquid lens for portable applications,” Proc. SPIE 5523, 100–109 (2004).
[Crossref]

Hirschberg, A. M.

S. Kuiper, B. H. Hendriks, L. J. Huijbregts, A. M. Hirschberg, C. A. Renders, and M. A. van As, “Variable-focus liquid lens for portable applications,” Proc. SPIE 5523, 100–109 (2004).
[Crossref]

Huijbregts, L. J.

S. Kuiper, B. H. Hendriks, L. J. Huijbregts, A. M. Hirschberg, C. A. Renders, and M. A. van As, “Variable-focus liquid lens for portable applications,” Proc. SPIE 5523, 100–109 (2004).
[Crossref]

Jeong, K.-H.

Johansen, I.

K. Haugholt, D. Wang, F. Tyholdt, W. Booij, and I. Johansen, “Polymer lens,” US Patent8,199,410 (2012).

Johansen, I.-R.

T. Bakke, A. Vogl, O. Żero, F. Tyholdt, I.-R. Johansen, and D. Wang, “A novel ultra-planar, long-stroke and low-voltage piezoelectric micromirror,” Journal of Micromechanics and Microengineering 20, 064010 (2010).
[Crossref]

Kosa, G.

E. Tadmor and G. Kosa, “Electromechanical coupling correction for piezoelectric layered beams,” Journal of Microelectromechanical Systems 12, 899–906 (2003).
[Crossref]

Kuiper, S.

S. Kuiper, B. H. Hendriks, L. J. Huijbregts, A. M. Hirschberg, C. A. Renders, and M. A. van As, “Variable-focus liquid lens for portable applications,” Proc. SPIE 5523, 100–109 (2004).
[Crossref]

Laura, P.

P. Laura, E. Romanelli, and R. Rossi, “Transverse vibrations of simply supported rectangular plates with rectangular cutouts,” Journal of Sound and Vibration 202, 275–283 (1997).
[Crossref]

Ledermann, N.

N. Ledermann, P. Muralt, J. Baborowski, S. Gentil, K. Mukati, M. Cantoni, A. Seifert, and N. Setter, “1 0 0-textured, piezoelectric Pb(ZrxTi1−x)O3 thin films for MEMS: integration, deposition and properties,” Sensors and Actuators A: Physical 105, 162–170 (2003).
[Crossref]

Lee, L.

Lin, S. I. E.

S. I. E. Lin, “Investigation on packaging parameters of a circular multi-layered diaphragm-type piezoelectric actuator,” Computers and Structures 89, 371–379 (2011).
[Crossref]

Liu, G.

Lozier, D. W.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge Univeristy Press, New York, 2010).

Mahajan, V. N.

Maréchal, A.

A. Maréchal, “Etude des influences conjuguées des aberrations et de la diffraction sur l’image d’un point,” Ph.D. thesis, Faculté des Sciences des Paris (1947).

Mitchell, J.

J. Reddy and J. Mitchell, “On refined nonlinear theories of laminated composite structures with piezoelectric laminae,” Sadhana 20, 721–747 (1995).
[Crossref]

Mo, C.

B. Boyerinas, C. Mo, and W. W. Clark, “Behavior of unimorph rectangular piezoelectric diaphragm actuators,” Proc. SPIE 6173, 617302 (2006).
[Crossref]

Mukati, K.

N. Ledermann, P. Muralt, J. Baborowski, S. Gentil, K. Mukati, M. Cantoni, A. Seifert, and N. Setter, “1 0 0-textured, piezoelectric Pb(ZrxTi1−x)O3 thin films for MEMS: integration, deposition and properties,” Sensors and Actuators A: Physical 105, 162–170 (2003).
[Crossref]

Muralt, P.

N. Ledermann, P. Muralt, J. Baborowski, S. Gentil, K. Mukati, M. Cantoni, A. Seifert, and N. Setter, “1 0 0-textured, piezoelectric Pb(ZrxTi1−x)O3 thin films for MEMS: integration, deposition and properties,” Sensors and Actuators A: Physical 105, 162–170 (2003).
[Crossref]

Nye, J. F.

J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices (Oxford University Press, 1985).

Olver, F. W. J.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge Univeristy Press, New York, 2010).

Reddy, J.

J. Reddy and J. Mitchell, “On refined nonlinear theories of laminated composite structures with piezoelectric laminae,” Sadhana 20, 721–747 (1995).
[Crossref]

Reddy, J. N.

J. N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd ed. (CRC Press, 2004).

Renders, C. A.

S. Kuiper, B. H. Hendriks, L. J. Huijbregts, A. M. Hirschberg, C. A. Renders, and M. A. van As, “Variable-focus liquid lens for portable applications,” Proc. SPIE 5523, 100–109 (2004).
[Crossref]

Romanelli, E.

P. Laura, E. Romanelli, and R. Rossi, “Transverse vibrations of simply supported rectangular plates with rectangular cutouts,” Journal of Sound and Vibration 202, 275–283 (1997).
[Crossref]

Rossi, R.

P. Laura, E. Romanelli, and R. Rossi, “Transverse vibrations of simply supported rectangular plates with rectangular cutouts,” Journal of Sound and Vibration 202, 275–283 (1997).
[Crossref]

Saggere, L.

M. Deshpande and L. Saggere, “An analytical model and working equations for static deflections of a circular multi-layered diaphragm-type piezoelectric actuator,” Sensors and Actuators A: Physical 136, 673–689 (2007).
[Crossref]

Sato, S.

Seifert, A.

N. Ledermann, P. Muralt, J. Baborowski, S. Gentil, K. Mukati, M. Cantoni, A. Seifert, and N. Setter, “1 0 0-textured, piezoelectric Pb(ZrxTi1−x)O3 thin films for MEMS: integration, deposition and properties,” Sensors and Actuators A: Physical 105, 162–170 (2003).
[Crossref]

Setter, N.

N. Ledermann, P. Muralt, J. Baborowski, S. Gentil, K. Mukati, M. Cantoni, A. Seifert, and N. Setter, “1 0 0-textured, piezoelectric Pb(ZrxTi1−x)O3 thin films for MEMS: integration, deposition and properties,” Sensors and Actuators A: Physical 105, 162–170 (2003).
[Crossref]

Tadmor, E.

E. Tadmor and G. Kosa, “Electromechanical coupling correction for piezoelectric layered beams,” Journal of Microelectromechanical Systems 12, 899–906 (2003).
[Crossref]

Taylor, R. L.

R. L. Taylor and S. Govindjee, “Solution of clamped rectangular plate problems,” Communications in Numerical Methods in Engineering 20, 757–765 (2004).
[Crossref]

Tiersten, H. F.

H. F. Tiersten, “Hamilton’s principle for linear piezoelectric media,” in Proceedings of the IEEE55 (IEEE, 1967), pp. 1523–1524.

Tseng, Y.

Y. Tseng, “Voice coil motor apparatus,” US Patent7,400,068 (2008).

Tyholdt, F.

T. Bakke, A. Vogl, O. Żero, F. Tyholdt, I.-R. Johansen, and D. Wang, “A novel ultra-planar, long-stroke and low-voltage piezoelectric micromirror,” Journal of Micromechanics and Microengineering 20, 064010 (2010).
[Crossref]

K. Haugholt, D. Wang, F. Tyholdt, W. Booij, and I. Johansen, “Polymer lens,” US Patent8,199,410 (2012).

van As, M. A.

S. Kuiper, B. H. Hendriks, L. J. Huijbregts, A. M. Hirschberg, C. A. Renders, and M. A. van As, “Variable-focus liquid lens for portable applications,” Proc. SPIE 5523, 100–109 (2004).
[Crossref]

Vogl, A.

T. Bakke, A. Vogl, O. Żero, F. Tyholdt, I.-R. Johansen, and D. Wang, “A novel ultra-planar, long-stroke and low-voltage piezoelectric micromirror,” Journal of Micromechanics and Microengineering 20, 064010 (2010).
[Crossref]

Wallrabe, U.

U. Wallrabe, “Axicons et al. - highly aspherical adaptive optical elements for the life sciences,” in “2015 18th International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS), 2015 Transducers,” (2015), pp. 251–256.

Wang, B.

Wang, D.

T. Bakke, A. Vogl, O. Żero, F. Tyholdt, I.-R. Johansen, and D. Wang, “A novel ultra-planar, long-stroke and low-voltage piezoelectric micromirror,” Journal of Micromechanics and Microengineering 20, 064010 (2010).
[Crossref]

K. Haugholt, D. Wang, F. Tyholdt, W. Booij, and I. Johansen, “Polymer lens,” US Patent8,199,410 (2012).

Werber, A.

Ye, M.

Zappe, H.

Zero, O.

T. Bakke, A. Vogl, O. Żero, F. Tyholdt, I.-R. Johansen, and D. Wang, “A novel ultra-planar, long-stroke and low-voltage piezoelectric micromirror,” Journal of Micromechanics and Microengineering 20, 064010 (2010).
[Crossref]

Zhao, C.

C. Zhao, Ultrasonic Motors: Technologies and Applications (Science PressBeijing and Springer-VerlagBerlin Heidelberg, 2011).
[Crossref]

Appl. Opt. (3)

Communications in Numerical Methods in Engineering (1)

R. L. Taylor and S. Govindjee, “Solution of clamped rectangular plate problems,” Communications in Numerical Methods in Engineering 20, 757–765 (2004).
[Crossref]

Computers and Structures (1)

S. I. E. Lin, “Investigation on packaging parameters of a circular multi-layered diaphragm-type piezoelectric actuator,” Computers and Structures 89, 371–379 (2011).
[Crossref]

Journal of Microelectromechanical Systems (1)

E. Tadmor and G. Kosa, “Electromechanical coupling correction for piezoelectric layered beams,” Journal of Microelectromechanical Systems 12, 899–906 (2003).
[Crossref]

Journal of Micromechanics and Microengineering (1)

T. Bakke, A. Vogl, O. Żero, F. Tyholdt, I.-R. Johansen, and D. Wang, “A novel ultra-planar, long-stroke and low-voltage piezoelectric micromirror,” Journal of Micromechanics and Microengineering 20, 064010 (2010).
[Crossref]

Journal of Sound and Vibration (1)

P. Laura, E. Romanelli, and R. Rossi, “Transverse vibrations of simply supported rectangular plates with rectangular cutouts,” Journal of Sound and Vibration 202, 275–283 (1997).
[Crossref]

Opt. Express (1)

Proc. SPIE (2)

S. Kuiper, B. H. Hendriks, L. J. Huijbregts, A. M. Hirschberg, C. A. Renders, and M. A. van As, “Variable-focus liquid lens for portable applications,” Proc. SPIE 5523, 100–109 (2004).
[Crossref]

B. Boyerinas, C. Mo, and W. W. Clark, “Behavior of unimorph rectangular piezoelectric diaphragm actuators,” Proc. SPIE 6173, 617302 (2006).
[Crossref]

Sadhana (1)

J. Reddy and J. Mitchell, “On refined nonlinear theories of laminated composite structures with piezoelectric laminae,” Sadhana 20, 721–747 (1995).
[Crossref]

Sensors and Actuators A: Physical (2)

N. Ledermann, P. Muralt, J. Baborowski, S. Gentil, K. Mukati, M. Cantoni, A. Seifert, and N. Setter, “1 0 0-textured, piezoelectric Pb(ZrxTi1−x)O3 thin films for MEMS: integration, deposition and properties,” Sensors and Actuators A: Physical 105, 162–170 (2003).
[Crossref]

M. Deshpande and L. Saggere, “An analytical model and working equations for static deflections of a circular multi-layered diaphragm-type piezoelectric actuator,” Sensors and Actuators A: Physical 136, 673–689 (2007).
[Crossref]

Other (12)

H. F. Tiersten, “Hamilton’s principle for linear piezoelectric media,” in Proceedings of the IEEE55 (IEEE, 1967), pp. 1523–1524.

J. N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd ed. (CRC Press, 2004).

J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices (Oxford University Press, 1985).

V. Birman, Plate Structures, vol. 178 of Solid Mechanics and its Applications (Springer Science+Bussiness Media, 2011).
[Crossref]

U. Wallrabe, “Axicons et al. - highly aspherical adaptive optical elements for the life sciences,” in “2015 18th International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS), 2015 Transducers,” (2015), pp. 251–256.

K. Haugholt, D. Wang, F. Tyholdt, W. Booij, and I. Johansen, “Polymer lens,” US Patent8,199,410 (2012).

Y. Tseng, “Voice coil motor apparatus,” US Patent7,400,068 (2008).

C. Zhao, Ultrasonic Motors: Technologies and Applications (Science PressBeijing and Springer-VerlagBerlin Heidelberg, 2011).
[Crossref]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Zemax LLC, Zemax User’s Manual (2015).

A. Maréchal, “Etude des influences conjuguées des aberrations et de la diffraction sur l’image d’un point,” Ph.D. thesis, Faculté des Sciences des Paris (1947).

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge Univeristy Press, New York, 2010).

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Figures (9)

Fig. 1
Fig. 1 (a) Schematic view showing tunable lens’s principle of operation; both at rest position when Vp = 0 and at focus when Vp is nonzero. (b) Cross-sectional view of tunable lens showing dimensions.
Fig. 2
Fig. 2 Planar view of possible study cases of piezoelectrically actuated MEMS tunable lenses. A clamped square diaphragm with circular opening: (a) case I and (b) case II ring actuator with opaque covering outside the ring till the diaphragm edges. A clamped circular diaphragm with circular opening: (c) case III actuator and (d) case IV ring actuator with opaque covering outside the ring till the diaphragm edges.
Fig. 3
Fig. 3 Even Gegenbauer-polynomial basis functions ϕm (X) on [−1, 1].
Fig. 4
Fig. 4 (a) Displacement profiles in xz−plane from FEM and the variational tool (N = 28) for square diaphragm with case I actuator at different values of ratio γ for piezoelectric material at Vp = −10V. (b) l2 relative error norm versus polynomial order N.
Fig. 5
Fig. 5 Displacement profiles in xz−plane from FEM and the variational tool (N = 28) for square diaphragm with case II actuator at different values of ratio γ1 when (a) γ2 = 0.9 and (b) γ2 = 0.5 at Vp = −10V.
Fig. 6
Fig. 6 Displacement profiles in xz−plane at Vp = −10V from FEM and the analytical model for (a) case III actuator at different values of γ and (b) and (c) case IV actuator at different values of γ1 while γ2 equals to 0.9 and 0.5, respectively.
Fig. 7
Fig. 7 (a) Tunable lens arrangement for on-axis optical simulation. (b) Reciprocal F# versus ratio (γ or γ1) for all tunable lense cases and different γ2, all with Vp = −10V and λ = 550nm.
Fig. 8
Fig. 8 (a) Focal length f and RMS wavefront error versus applied voltage on the piezoelectric stack with optimum geometrical values for tunable lens for square/circular diaphragm. Wavefront map using exit pupil shape at Vp = −10V for (b) square and (c) circular diaphragm.
Fig. 9
Fig. 9 (a) Arrangement of tunable lens with a paraxial fixed lens and image sensor in Zemax for on-axis optical simulation. Sagittal (tangential) MTF at image plane for tunable lens for optimum cases I and III at different actuation voltage Vp.

Tables (2)

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Table 1 Table contains minimum F# at different geometrical parameter of PZT actuators at Vp = −10V. The optimum cases for square and circular diaphragms are written in bold font.

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Table 2 Table contains values for PZT and glass material parameters.

Equations (36)

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[ T x x T y y T x y D 3 ] = [ Q ¯ 11 Q ¯ 12 0 Q ¯ 12 Q ¯ 22 0 0 0 Q ¯ 66 e ¯ 31 e ¯ 32 0 ] [ S x x S y y γ x y ] [ e ¯ 31 e ¯ 32 0 ϵ ¯ 33 S ] E 3
[ D 1 D 2 ] = [ e ¯ 15 0 0 e ¯ 24 ] [ γ y z γ x z ] + [ ϵ ¯ 11 S 0 0 ϵ ¯ 22 S ] [ E 1 E 2 ]
[ N x x N y y N x y M x x M y y M x y ] = [ A 11 A 12 0 B 11 B 12 0 A 12 A 22 0 B 21 B 22 0 0 0 A 66 0 0 B 66 B 11 B 12 0 D 11 D 12 0 B 12 B 22 0 D 21 D 22 0 0 0 B 66 0 0 D 66 ] [ S x x 0 S y y 0 γ x y 0 S x x 1 S y y 1 γ x y 1 ] [ N x x E N y y E N x y E M x x E M y y E M x y E ]
A i j = k = 1 2 Q ¯ i j ( k ) ( h k + 1 h k ) , B i j = 1 2 k = 1 2 Q ¯ i j ( k ) ( h k + 1 2 h k 2 ) and D i j = 1 3 k = 1 2 Q ¯ i j ( k ) ( h k + 1 3 h k 3 ) .
E 3 ( x , y , z ) = V p h p e ¯ 31 ϵ 33 S ( S x x 1 + S y y 1 ) ( z z ¯ p ) ,
v ( x , y , z ) = V p h p ( z h 2 ) + e ¯ 31 ϵ 33 S ( S x x 1 + S y y 1 ) ( ( z 2 h 2 2 ) 2 + z ¯ p ( h 2 z ) )
N x x E = N y y E = e ¯ 31 V p , N x y E = 0 ,
M x x E = M y y E = e ¯ 31 [ e ¯ 31 ϵ 33 S ( S x x 1 + S y y 1 ) ( ( h 3 h 2 ) 3 12 ) + V p z ¯ p ] , M x y E = 0 .
[ M x x M y y M x y ] = [ D 11 * D 12 * 0 D 21 * D 22 * 0 0 0 D 66 * ] [ S x x 1 S y y 1 γ x y 1 ] + e ¯ 31 V p z ¯ p [ 1 1 0 ]
D i j * = D i j g l + D i j p = k = 1 2 Q ¯ i j ( k ) [ 1 12 ( 1 + χ i j ( k ) ) ( h k + 1 h k ) 3 ] and χ i j ( k ) = e ¯ 3 i ( k ) e ¯ 3 j ( k ) Q ¯ i j ( k ) ϵ ¯ 33 S ; ( k ) .
δ H δ W = 0
0 = Ω ( M x x δ S x x 1 + M y y δ S y y 1 + M x y δ γ x y 1 ) d x d y + Γ Ω ( M ^ n n δ w 0 n ^ + M ^ n s δ w 0 s ^ ) d s + Ω q δ w 0 d s
Ω { ( D 11 * 2 w 0 x 2 + D 12 * 2 w 0 y 2 ) ( 2 δ w 0 x 2 ) + ( D 12 * 2 w 0 x 2 + D 22 * 2 w 0 y 2 ) ( 2 δ w 0 y 2 ) + ( 2 D 66 * 2 w 0 x y ) ( 2 2 δ w 0 x y ) } d x d y = e ¯ 31 V p z ¯ p Ω p { ( 2 w 0 x 2 ) + ( 2 δ w 0 y 2 ) } d x d y = e ¯ 31 V p z ¯ p Ω p x , y 2 δ w 0 d x d y
D i j * = D i j gl + { D i j p [ ( X , Y ) circ ( r γ ) ] Case I actuator D i j p [ circ ( r γ 2 ) circ ( r γ 1 ) ] Case II actuator
Ξ gl + [ Ξ p ϑ , γ p ] = β p ζ , γ p Case I actuator
Ξ gl + [ ϑ , γ 2 p ϑ , γ 1 p ] = ζ , γ 2 p ζ , γ 1 p Case II actuator
w 0 ( X , Y , 0 ) w N ( X , Y , 0 ) = m = 1 N n = 1 N C m n Φ m n ( X , Y )
Φ m n ( X , Y ) = ϕ m ( X ) ϕ n ( Y ) = ( 1 X 2 ) ( α 1 / 2 ) 2 ( 1 Y 2 ) ( α 1 / 2 ) 2 Weight factor enforcing BC G m ( α ) ( X ) G n ( α ) ( Y )
[ R m n p q ] N 2 × N 2 [ C m n ] N 2 × 1 = [ F p q ] N 2 × 1
Φ 00 ( r , θ ) = i = 0 N s j = 0 N s k m n i j Z i j ( r , θ ) = ( 3 γ 8 640 γ 6 16 + 5 γ 4 12 γ 2 + 1 ) Z 0 0 + γ 2 320 ( 3 γ 3 36 γ 4 + 200 γ 2 320 ) Z 2 0 + γ 4 1344 ( 9 γ 4 84 γ 2 + 280 ) Z 4 0 + γ 4 672 ( 15 γ 4 + 140 γ 2 168 ) Z 4 4 + γ 6 1280 ( 3 γ 2 16 ) Z 6 0 + γ 6 384 ( 3 γ 2 + 16 ) Z 6 4 + 3 γ 8 8960 Z 8 0 γ 8 896 Z 8 4 + γ 8 128 Z 8 8
ϵ l 2 = ( w FEM w N ) 2 w FEM 2 .
w i n t ( R ) = C 2 R 2 + C 0 = C 2 γ 2 ( a / 2 ) 2 2 3 k 20 3 ( 2 r 2 1 ) Z 2 0 + C 0 + C 2 γ 2 ( a / 2 ) 2 2 3 k 00
σ w = j = 2 a j 2
Q ¯ i j = C ¯ i j , C ¯ 11 = s 11 E ( s 11 E + s 12 E ) ( s 11 E s 12 E ) , C ¯ 12 = s 12 E ( s 11 E + s 12 E ) ( s 11 E s 12 E )
C ¯ 66 = 1 s 66 E , e ¯ 31 = e ¯ 32 = d 31 s 11 E + s 12 E , ϵ ¯ 33 S = ϵ 33 S 2 d 31 2 s 11 E + s 12 E
G n ( α ) ( X ) = k = 0 n / 2 ( 1 ) k Γ ( n k + α ) Γ ( α ) k ! ( n 2 k ) ! ( 2 X ) n 2 k ,
( 2 α + i ) G i ( α ) ( X ) = ( i + 1 ) X G i + 1 ( α ) ( X ) + 2 α ( 1 X 2 ) G i ( α + 1 ) ( X ) ,
G i ( α ) ( X ) X = 2 α G i 1 ( α + 1 ) ( X ) .
ϕ i X = ( 1 X 2 ) ( α 5 / 2 ) 2 [ ( α + i 1 / 2 ) X G i ( α ) ( X ) + ( 2 α + i 1 ) G i 1 ( α ) ( X ) ] ,
2 ϕ i X 2 = ( 1 X 2 ) ( α 9 / 2 ) 2 [ ( α + i 1 / 2 ) ( ( α + i 3 / 2 ) X 2 1 ) G i ( α ) ( X ) ( 2 α + i 1 ) ( 2 α + 2 i 4 ) X G i 1 ( α ) ( X ) + ( 2 α + i 1 ) ( 2 α + i 2 ) G i 2 ( α ) ] .
ϕ i X ϕ p X = ( 1 X 2 ) ( α + 5 / 2 ) [ ( α + i 1 / 2 ) ( α + p 1 / 2 ) X 2 G i ( α ) ( X ) G p ( α ) ( X ) + ( 2 α + i 1 ) ( 2 α + p 1 ) G i 1 ( α ) ( X ) G p 1 ( α ) ( X ) ( α + i 1 / 2 ) ( 2 α + p 1 ) X G i ( α ) G p 1 ( α ) ( α + i 1 / 2 ) ( 2 α + i 1 ) X G i 1 ( α ) G p ( α ) ] .
( 1 2 X 2 + X 4 ) X β G i ( α ) ( X ) G p ( α ) ( X ) d X = W T X
W ( k ) ( l , s ) = ( 1 ) l + s Γ ( i l + α ) Γ ( p s + α ) [ Γ ( α ) ] 2 l ! ( i 2 l ) ! s ! ( p 2 s ) ! ( 2 ) η
X ( k ) ( l , s ) = X η + β + 1 [ 1 η + β + 1 2 X 2 η + β + 3 + X 4 η + β + 5 ]
m = ( i / 2 + 1 ) ( p / 2 + 1 ) , η = i + p 2 ( l + s ) ,
l = 0 , 1 , , i / 2 , s = 0 , 1 , , p / 2 and k = 0 , 1 , , m .

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