Abstract

This paper describes a polarized-light imaging interferometer to measure the rotation field of reflecting surfaces. This setup is based on a homemade prism featuring a birefringence gradient. The arrangement is presented before focusing on the homemade prism and its manufacturing process. The dependence of the measured optical phase on the rotation of the surface is derived, thus highlighting the key parameters driving the sensitivity. The system’s capabilities are illustrated by imaging the rotation field at the surface of a tip-loaded polymer specimen.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  5. P. Vairac and B. Cretin, “Electromechanical resonator in scanning microdeformation microscopy: theory and experiment,” Surf. Interface Anal. 27, 588–591 (1999).
    [CrossRef]
  6. P. Vairac, S. Ballandras, and B. Cretin, “Finite element analysis of the behavior of the scanning microdeformation microscope,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48, 895–899 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  9. R. Feng and R. J. Farris, “The characterization of thermal and elastic constants for an epoxy photoresist SU8 coating,” J. Mater. Sci. 37, 033509 (2002).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  20. W. Gao, P. S. Huang, T. Yamada, and S. Kiyono, “A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers,” Precis. Eng. 26, 396–404 (2002).
    [CrossRef]
  21. M. Xiao, S. Jujo, S. Takahashi, and K. Takamasu, “Nanometer profile measurement of large aspheric optical surface by scanning deflectometry with rotatable devices—Uncertainty propagation analysis and experiments,” Precis. Eng. 36, 91–96 (2012).
    [CrossRef]
  22. Y. Surrel, N. Fournier, M. Grédiac, and P.-A. Paris, “Phase-stepped deflectometry applied to shape measurement of bent plates,” Exp. Mech. 39, 66–70 (1999).
    [CrossRef]
  23. J.-R. Lee, J. Molimard, A. Vautrin, and Y. Surrel, “Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension,” Compos. Part A Appl. Sci. Manufact. 35, 849–859 (2004).
    [CrossRef]
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    [CrossRef]

2012

J. Le Rouzic, P. Delobelle, B. Cretin, P. Vairac, and F. Amiot, “Simultaneous measurement of Young’s modulus and Poisson’s ratio at microscale with two-modes scanning microdeformation microscopy,” Mater. Lett. 68, 370–373 (2012).
[CrossRef]

M. Xiao, S. Jujo, S. Takahashi, and K. Takamasu, “Nanometer profile measurement of large aspheric optical surface by scanning deflectometry with rotatable devices—Uncertainty propagation analysis and experiments,” Precis. Eng. 36, 91–96 (2012).
[CrossRef]

2011

A. Boisen, S. Dohn, S. S. Keller, S. Schmid, and M. Tenje, “Cantilever-like micromechanical sensors,” Rep. Prog. Phys. 74, 036101 (2011).
[CrossRef]

2009

J. Le Rouzic, P. Delobelle, P. Vairac, and B. Cretin, “Comparison of three different scales techniques for the dynamic mechanical characterization of two polymers (PDMS and SU8),” Eur. Phys. J. Appl. Phys. 48, 11201 (2009).
[CrossRef]

2007

D. C. Hurley and J. A. Turner, “Measurement of Poisson’s ratio with contact-resonance atomic force microscopy,” J. Appl. Phys. 102, 033509 (2007).
[CrossRef]

2006

2004

J.-R. Lee, J. Molimard, A. Vautrin, and Y. Surrel, “Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension,” Compos. Part A Appl. Sci. Manufact. 35, 849–859 (2004).
[CrossRef]

2002

W. Gao, P. S. Huang, T. Yamada, and S. Kiyono, “A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers,” Precis. Eng. 26, 396–404 (2002).
[CrossRef]

R. Feng and R. J. Farris, “The characterization of thermal and elastic constants for an epoxy photoresist SU8 coating,” J. Mater. Sci. 37, 033509 (2002).

2001

M. J. Bamber, K. E. Cooke, A. B. Mann, and B. Derby, “Accurate determination of Young’s modulus and Poisson’s ratio of thin films by a combination of acoustic microscopy and nanoindentation,” Thin Solid Films 398–399, 299–305 (2001).
[CrossRef]

P. Vairac, S. Ballandras, and B. Cretin, “Finite element analysis of the behavior of the scanning microdeformation microscope,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48, 895–899 (2001).
[CrossRef]

C. J. Tay, C. Quan, S. H. Wang, and H. M. Shang, “Determination of a micromirror angular rotation using laser interferometric method,” Opt. Commun. 195, 71–77 (2001).
[CrossRef]

2000

A. Dubois, J. Selb, L. Vabre, and A. C. Boccara, “Phase measurements with wide-aperture interferometers,” Appl. Opt. 39, 2326–2331 (2000).
[CrossRef]

U. Rabe, S. Amelio, E. Kester, V. Scherer, S. Hirsekorn, and W. Arnold, “Quantitative determination of contact stiffness using atomic force acoustic microscopy,” Ultrasonics 38, 430–437 (2000).
[CrossRef]

1999

P. Vairac and B. Cretin, “Electromechanical resonator in scanning microdeformation microscopy: theory and experiment,” Surf. Interface Anal. 27, 588–591 (1999).
[CrossRef]

Y. Surrel, N. Fournier, M. Grédiac, and P.-A. Paris, “Phase-stepped deflectometry applied to shape measurement of bent plates,” Exp. Mech. 39, 66–70 (1999).
[CrossRef]

1998

B. Cretin and P. Vairac, “Measurement of cantilever vibrations with a new heterodyne laser probe—application to scanning microdeformation microscopy,” Appl. Phys. A. 66, S235–S238 (1998).

1993

B. Cretin and F. Sthal, “Scanning microdeformation microscopy,” Appl. Phys. Lett. 62, 829–831 (1993).
[CrossRef]

1992

W. C. Oliver and G. M. Pharr, “An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments,” J. Mater. Res. 7, 1564–1583 (1992).
[CrossRef]

1988

1974

1970

Amelio, S.

U. Rabe, S. Amelio, E. Kester, V. Scherer, S. Hirsekorn, and W. Arnold, “Quantitative determination of contact stiffness using atomic force acoustic microscopy,” Ultrasonics 38, 430–437 (2000).
[CrossRef]

Amiot, F.

J. Le Rouzic, P. Delobelle, B. Cretin, P. Vairac, and F. Amiot, “Simultaneous measurement of Young’s modulus and Poisson’s ratio at microscale with two-modes scanning microdeformation microscopy,” Mater. Lett. 68, 370–373 (2012).
[CrossRef]

F. Amiot and J. P. Roger, “Nomarski imaging interferometry to measure the displacement field of micro-electro-mechanical systems,” Appl. Opt. 45, 7800–7810 (2006).
[CrossRef]

Arnold, W.

U. Rabe, S. Amelio, E. Kester, V. Scherer, S. Hirsekorn, and W. Arnold, “Quantitative determination of contact stiffness using atomic force acoustic microscopy,” Ultrasonics 38, 430–437 (2000).
[CrossRef]

Ballandras, S.

P. Vairac, S. Ballandras, and B. Cretin, “Finite element analysis of the behavior of the scanning microdeformation microscope,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48, 895–899 (2001).
[CrossRef]

Bamber, M. J.

M. J. Bamber, K. E. Cooke, A. B. Mann, and B. Derby, “Accurate determination of Young’s modulus and Poisson’s ratio of thin films by a combination of acoustic microscopy and nanoindentation,” Thin Solid Films 398–399, 299–305 (2001).
[CrossRef]

Bhatia, A. B.

M. Born, E. Wolf, and A. B. Bhatia, “Geometrical theory of optical imaging,” in Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1959), pp. 133–202.

Boccara, A. C.

Boisen, A.

A. Boisen, S. Dohn, S. S. Keller, S. Schmid, and M. Tenje, “Cantilever-like micromechanical sensors,” Rep. Prog. Phys. 74, 036101 (2011).
[CrossRef]

Born, M.

M. Born, E. Wolf, and A. B. Bhatia, “Geometrical theory of optical imaging,” in Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1959), pp. 133–202.

Boussinesq, M. J.

M. J. Boussinesq, “Valeurs des déplacements, des déformations et de pressions intérieures, quand les potentiels se réduisent à un seul de leurs éléments,” in Application des potentiels à l’étude de l’équilibre et du mouvement des solides élastiques (Gauthier-Villars, 1885), pp. 81–108 (in French).

Chapman, G. D.

Cooke, K. E.

M. J. Bamber, K. E. Cooke, A. B. Mann, and B. Derby, “Accurate determination of Young’s modulus and Poisson’s ratio of thin films by a combination of acoustic microscopy and nanoindentation,” Thin Solid Films 398–399, 299–305 (2001).
[CrossRef]

Cretin, B.

J. Le Rouzic, P. Delobelle, B. Cretin, P. Vairac, and F. Amiot, “Simultaneous measurement of Young’s modulus and Poisson’s ratio at microscale with two-modes scanning microdeformation microscopy,” Mater. Lett. 68, 370–373 (2012).
[CrossRef]

J. Le Rouzic, P. Delobelle, P. Vairac, and B. Cretin, “Comparison of three different scales techniques for the dynamic mechanical characterization of two polymers (PDMS and SU8),” Eur. Phys. J. Appl. Phys. 48, 11201 (2009).
[CrossRef]

P. Vairac, S. Ballandras, and B. Cretin, “Finite element analysis of the behavior of the scanning microdeformation microscope,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48, 895–899 (2001).
[CrossRef]

P. Vairac and B. Cretin, “Electromechanical resonator in scanning microdeformation microscopy: theory and experiment,” Surf. Interface Anal. 27, 588–591 (1999).
[CrossRef]

B. Cretin and P. Vairac, “Measurement of cantilever vibrations with a new heterodyne laser probe—application to scanning microdeformation microscopy,” Appl. Phys. A. 66, S235–S238 (1998).

B. Cretin and F. Sthal, “Scanning microdeformation microscopy,” Appl. Phys. Lett. 62, 829–831 (1993).
[CrossRef]

Delobelle, P.

J. Le Rouzic, P. Delobelle, B. Cretin, P. Vairac, and F. Amiot, “Simultaneous measurement of Young’s modulus and Poisson’s ratio at microscale with two-modes scanning microdeformation microscopy,” Mater. Lett. 68, 370–373 (2012).
[CrossRef]

J. Le Rouzic, P. Delobelle, P. Vairac, and B. Cretin, “Comparison of three different scales techniques for the dynamic mechanical characterization of two polymers (PDMS and SU8),” Eur. Phys. J. Appl. Phys. 48, 11201 (2009).
[CrossRef]

Derby, B.

M. J. Bamber, K. E. Cooke, A. B. Mann, and B. Derby, “Accurate determination of Young’s modulus and Poisson’s ratio of thin films by a combination of acoustic microscopy and nanoindentation,” Thin Solid Films 398–399, 299–305 (2001).
[CrossRef]

Dohn, S.

A. Boisen, S. Dohn, S. S. Keller, S. Schmid, and M. Tenje, “Cantilever-like micromechanical sensors,” Rep. Prog. Phys. 74, 036101 (2011).
[CrossRef]

Dubois, A.

Farris, R. J.

R. Feng and R. J. Farris, “The characterization of thermal and elastic constants for an epoxy photoresist SU8 coating,” J. Mater. Sci. 37, 033509 (2002).

Feng, R.

R. Feng and R. J. Farris, “The characterization of thermal and elastic constants for an epoxy photoresist SU8 coating,” J. Mater. Sci. 37, 033509 (2002).

Fournier, N.

Y. Surrel, N. Fournier, M. Grédiac, and P.-A. Paris, “Phase-stepped deflectometry applied to shape measurement of bent plates,” Exp. Mech. 39, 66–70 (1999).
[CrossRef]

Françon, M.

M. Françon and S. Mallick, “Compensated polarization interferometers for the observation of phase objects,” in Polarization Interferometers: Applications in Microscopy and Macroscopy (Wiley-Interscience, 1971), pp. 55–67.

Gao, W.

W. Gao, P. S. Huang, T. Yamada, and S. Kiyono, “A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers,” Precis. Eng. 26, 396–404 (2002).
[CrossRef]

Grédiac, M.

Y. Surrel, N. Fournier, M. Grédiac, and P.-A. Paris, “Phase-stepped deflectometry applied to shape measurement of bent plates,” Exp. Mech. 39, 66–70 (1999).
[CrossRef]

Harris, O.

Hirsekorn, S.

U. Rabe, S. Amelio, E. Kester, V. Scherer, S. Hirsekorn, and W. Arnold, “Quantitative determination of contact stiffness using atomic force acoustic microscopy,” Ultrasonics 38, 430–437 (2000).
[CrossRef]

Huang, P. S.

W. Gao, P. S. Huang, T. Yamada, and S. Kiyono, “A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers,” Precis. Eng. 26, 396–404 (2002).
[CrossRef]

Hurley, D. C.

D. C. Hurley and J. A. Turner, “Measurement of Poisson’s ratio with contact-resonance atomic force microscopy,” J. Appl. Phys. 102, 033509 (2007).
[CrossRef]

Jujo, S.

M. Xiao, S. Jujo, S. Takahashi, and K. Takamasu, “Nanometer profile measurement of large aspheric optical surface by scanning deflectometry with rotatable devices—Uncertainty propagation analysis and experiments,” Precis. Eng. 36, 91–96 (2012).
[CrossRef]

Keller, S. S.

A. Boisen, S. Dohn, S. S. Keller, S. Schmid, and M. Tenje, “Cantilever-like micromechanical sensors,” Rep. Prog. Phys. 74, 036101 (2011).
[CrossRef]

Kester, E.

U. Rabe, S. Amelio, E. Kester, V. Scherer, S. Hirsekorn, and W. Arnold, “Quantitative determination of contact stiffness using atomic force acoustic microscopy,” Ultrasonics 38, 430–437 (2000).
[CrossRef]

Kiyono, S.

W. Gao, P. S. Huang, T. Yamada, and S. Kiyono, “A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers,” Precis. Eng. 26, 396–404 (2002).
[CrossRef]

Le Rouzic, J.

J. Le Rouzic, P. Delobelle, B. Cretin, P. Vairac, and F. Amiot, “Simultaneous measurement of Young’s modulus and Poisson’s ratio at microscale with two-modes scanning microdeformation microscopy,” Mater. Lett. 68, 370–373 (2012).
[CrossRef]

J. Le Rouzic, P. Delobelle, P. Vairac, and B. Cretin, “Comparison of three different scales techniques for the dynamic mechanical characterization of two polymers (PDMS and SU8),” Eur. Phys. J. Appl. Phys. 48, 11201 (2009).
[CrossRef]

Lee, J.-R.

J.-R. Lee, J. Molimard, A. Vautrin, and Y. Surrel, “Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension,” Compos. Part A Appl. Sci. Manufact. 35, 849–859 (2004).
[CrossRef]

Malacara, D.

Mallick, S.

M. Françon and S. Mallick, “Compensated polarization interferometers for the observation of phase objects,” in Polarization Interferometers: Applications in Microscopy and Macroscopy (Wiley-Interscience, 1971), pp. 55–67.

Mann, A. B.

M. J. Bamber, K. E. Cooke, A. B. Mann, and B. Derby, “Accurate determination of Young’s modulus and Poisson’s ratio of thin films by a combination of acoustic microscopy and nanoindentation,” Thin Solid Films 398–399, 299–305 (2001).
[CrossRef]

Molimard, J.

J.-R. Lee, J. Molimard, A. Vautrin, and Y. Surrel, “Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension,” Compos. Part A Appl. Sci. Manufact. 35, 849–859 (2004).
[CrossRef]

Oliver, W. C.

W. C. Oliver and G. M. Pharr, “An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments,” J. Mater. Res. 7, 1564–1583 (1992).
[CrossRef]

Paris, P.-A.

Y. Surrel, N. Fournier, M. Grédiac, and P.-A. Paris, “Phase-stepped deflectometry applied to shape measurement of bent plates,” Exp. Mech. 39, 66–70 (1999).
[CrossRef]

Pharr, G. M.

W. C. Oliver and G. M. Pharr, “An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments,” J. Mater. Res. 7, 1564–1583 (1992).
[CrossRef]

Quan, C.

C. J. Tay, C. Quan, S. H. Wang, and H. M. Shang, “Determination of a micromirror angular rotation using laser interferometric method,” Opt. Commun. 195, 71–77 (2001).
[CrossRef]

Rabe, U.

U. Rabe, S. Amelio, E. Kester, V. Scherer, S. Hirsekorn, and W. Arnold, “Quantitative determination of contact stiffness using atomic force acoustic microscopy,” Ultrasonics 38, 430–437 (2000).
[CrossRef]

Roger, J. P.

Scherer, V.

U. Rabe, S. Amelio, E. Kester, V. Scherer, S. Hirsekorn, and W. Arnold, “Quantitative determination of contact stiffness using atomic force acoustic microscopy,” Ultrasonics 38, 430–437 (2000).
[CrossRef]

Schmid, S.

A. Boisen, S. Dohn, S. S. Keller, S. Schmid, and M. Tenje, “Cantilever-like micromechanical sensors,” Rep. Prog. Phys. 74, 036101 (2011).
[CrossRef]

Selb, J.

Shang, H. M.

C. J. Tay, C. Quan, S. H. Wang, and H. M. Shang, “Determination of a micromirror angular rotation using laser interferometric method,” Opt. Commun. 195, 71–77 (2001).
[CrossRef]

Shi, P.

Sthal, F.

B. Cretin and F. Sthal, “Scanning microdeformation microscopy,” Appl. Phys. Lett. 62, 829–831 (1993).
[CrossRef]

Stijns, E.

Surrel, Y.

J.-R. Lee, J. Molimard, A. Vautrin, and Y. Surrel, “Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension,” Compos. Part A Appl. Sci. Manufact. 35, 849–859 (2004).
[CrossRef]

Y. Surrel, N. Fournier, M. Grédiac, and P.-A. Paris, “Phase-stepped deflectometry applied to shape measurement of bent plates,” Exp. Mech. 39, 66–70 (1999).
[CrossRef]

Takahashi, S.

M. Xiao, S. Jujo, S. Takahashi, and K. Takamasu, “Nanometer profile measurement of large aspheric optical surface by scanning deflectometry with rotatable devices—Uncertainty propagation analysis and experiments,” Precis. Eng. 36, 91–96 (2012).
[CrossRef]

Takamasu, K.

M. Xiao, S. Jujo, S. Takahashi, and K. Takamasu, “Nanometer profile measurement of large aspheric optical surface by scanning deflectometry with rotatable devices—Uncertainty propagation analysis and experiments,” Precis. Eng. 36, 91–96 (2012).
[CrossRef]

Tay, C. J.

C. J. Tay, C. Quan, S. H. Wang, and H. M. Shang, “Determination of a micromirror angular rotation using laser interferometric method,” Opt. Commun. 195, 71–77 (2001).
[CrossRef]

Tenje, M.

A. Boisen, S. Dohn, S. S. Keller, S. Schmid, and M. Tenje, “Cantilever-like micromechanical sensors,” Rep. Prog. Phys. 74, 036101 (2011).
[CrossRef]

Turner, J. A.

D. C. Hurley and J. A. Turner, “Measurement of Poisson’s ratio with contact-resonance atomic force microscopy,” J. Appl. Phys. 102, 033509 (2007).
[CrossRef]

Vabre, L.

Vairac, P.

J. Le Rouzic, P. Delobelle, B. Cretin, P. Vairac, and F. Amiot, “Simultaneous measurement of Young’s modulus and Poisson’s ratio at microscale with two-modes scanning microdeformation microscopy,” Mater. Lett. 68, 370–373 (2012).
[CrossRef]

J. Le Rouzic, P. Delobelle, P. Vairac, and B. Cretin, “Comparison of three different scales techniques for the dynamic mechanical characterization of two polymers (PDMS and SU8),” Eur. Phys. J. Appl. Phys. 48, 11201 (2009).
[CrossRef]

P. Vairac, S. Ballandras, and B. Cretin, “Finite element analysis of the behavior of the scanning microdeformation microscope,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48, 895–899 (2001).
[CrossRef]

P. Vairac and B. Cretin, “Electromechanical resonator in scanning microdeformation microscopy: theory and experiment,” Surf. Interface Anal. 27, 588–591 (1999).
[CrossRef]

B. Cretin and P. Vairac, “Measurement of cantilever vibrations with a new heterodyne laser probe—application to scanning microdeformation microscopy,” Appl. Phys. A. 66, S235–S238 (1998).

Vautrin, A.

J.-R. Lee, J. Molimard, A. Vautrin, and Y. Surrel, “Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension,” Compos. Part A Appl. Sci. Manufact. 35, 849–859 (2004).
[CrossRef]

Wang, S. H.

C. J. Tay, C. Quan, S. H. Wang, and H. M. Shang, “Determination of a micromirror angular rotation using laser interferometric method,” Opt. Commun. 195, 71–77 (2001).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, and A. B. Bhatia, “Geometrical theory of optical imaging,” in Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1959), pp. 133–202.

Xiao, M.

M. Xiao, S. Jujo, S. Takahashi, and K. Takamasu, “Nanometer profile measurement of large aspheric optical surface by scanning deflectometry with rotatable devices—Uncertainty propagation analysis and experiments,” Precis. Eng. 36, 91–96 (2012).
[CrossRef]

Yamada, T.

W. Gao, P. S. Huang, T. Yamada, and S. Kiyono, “A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers,” Precis. Eng. 26, 396–404 (2002).
[CrossRef]

Appl. Opt.

Appl. Phys. A.

B. Cretin and P. Vairac, “Measurement of cantilever vibrations with a new heterodyne laser probe—application to scanning microdeformation microscopy,” Appl. Phys. A. 66, S235–S238 (1998).

Appl. Phys. Lett.

B. Cretin and F. Sthal, “Scanning microdeformation microscopy,” Appl. Phys. Lett. 62, 829–831 (1993).
[CrossRef]

Compos. Part A Appl. Sci. Manufact.

J.-R. Lee, J. Molimard, A. Vautrin, and Y. Surrel, “Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension,” Compos. Part A Appl. Sci. Manufact. 35, 849–859 (2004).
[CrossRef]

Eur. Phys. J. Appl. Phys.

J. Le Rouzic, P. Delobelle, P. Vairac, and B. Cretin, “Comparison of three different scales techniques for the dynamic mechanical characterization of two polymers (PDMS and SU8),” Eur. Phys. J. Appl. Phys. 48, 11201 (2009).
[CrossRef]

Exp. Mech.

Y. Surrel, N. Fournier, M. Grédiac, and P.-A. Paris, “Phase-stepped deflectometry applied to shape measurement of bent plates,” Exp. Mech. 39, 66–70 (1999).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control

P. Vairac, S. Ballandras, and B. Cretin, “Finite element analysis of the behavior of the scanning microdeformation microscope,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48, 895–899 (2001).
[CrossRef]

J. Appl. Phys.

D. C. Hurley and J. A. Turner, “Measurement of Poisson’s ratio with contact-resonance atomic force microscopy,” J. Appl. Phys. 102, 033509 (2007).
[CrossRef]

J. Mater. Res.

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Figures (8)

Fig. 1.
Fig. 1.

Schematic view of the interferometric imaging setup.

Fig. 2.
Fig. 2.

Arrangement used to establish a stress gradient in the sample. The prism is cut out of the specimen gauge section.

Fig. 3.
Fig. 3.

Ray tracing in the prism.

Fig. 4.
Fig. 4.

Ray tracing through the objective for the two emerging rays of Fig. 3.

Fig. 5.
Fig. 5.

Tilting sample interferograms. (a) For Y=183μm and for δγ ranging from 5° to 5°. (b) For Y={62.7,183,303}μm and for δγ ranging from 2° to 3°.

Fig. 6.
Fig. 6.

Phase map obtained when pressing a tip onto a PDMS sample.

Fig. 7.
Fig. 7.

Phase along YT axis. Circles, experimental phase far from the tip; crosses, experimental phase close to the tip; solid lines, theoretical phases far and close to the tip.

Fig. 8.
Fig. 8.

(a) Ray tracing illustrating the light collection as a function of the tilt of the sample. (b) Collected light in the plane of the pupil.

Tables (3)

Tables Icon

Table 1. Estimated Parameters

Tables Icon

Table 2. Fitted Global Parameters

Tables Icon

Table 3. Fitted Local Parameters

Equations (66)

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I=I0+Acos(ϕ+π)
σ̲̲=(Gy00000000)(x,y,z),
nTE=n*+CbGy=n*+cTEy,
sinθe=nTMsinθaTM=nTE(ye)sinθaTE,
nTMsinθbTM=sinθoTM=sinθe,
nTE(yoTE)sinθbTE=sinθoTE,
θbTE=θaTE+ϵ+o(θaTE3,ϵ3)
ϵ=cTEenTE(ye)3×104rad.
yoTM=ye+etanθaTM.
yoTE=ye+e(θaTE+ϵ2)+o(θaTE3,ϵ3).
θoTEθoTM=cTEe+o(θe3,ϵ3)104rad.
z=tan(θPAS)ye[112n*]+o(θe2,ϵ2,(cTEye)2)
θPAS=ecTE+2θe2n*2+o(θe2,ϵ2,(cTEye)2).
LTM=n*ecosθaTM=n*2en*2sin2θe.
LTE=e0nTE(yTE(z))cos(θTE(z))dz
yTE(z)=yez(θenTE(ye)ϵz2e)+o(θe3,ϵ3),
θTE(z)=θenTE(ye)ϵze+o(θe3,ϵ3).
LTE=e[nTE(ye)(1+ϵ26)+cTEe6ϵ+ϵθe+θe22nTE(ye)]+o(θe3,ϵ3).
YPASi=YPAS2fo2+YPAS2foγi2[αPAS+(1+1n*2)θe+n*ϵ+ecTE2n*2]δPAS+o(αPAS2,θe2,θPAS2,ϵ2,γi2),
ϕp(γTE)=2πλ{[LTEforth+LTEback(γTE)][LTMforth+LTMback]}.
ϕp=ϕγTE+ϕθe+ϕΔ+ϕr
ϕγTE=4πλcTEefo2+Ye2foγTE+o(γTE2),
ϕθe=2πλcTEe[2(1+1n*2)δPAS+en*]θe+o(θe2,θPAS2),
ϕΔ=4πλcTEeΔ+o(θPAS2),
ϕr=2πλcTEe[(2αPAS+cTEen*2)δPAS+(2n*δPASe3)ϵ]+o(αPAS2,θPAS2,ϵ2),
d=focTEe+o(θe2,θPAS2,αPAS2,ϵ2).
ϕo=4πλΔZcosα,
ϕ(K,γd+δγ,Ψ,α)=K[1+sin2(α)][γd+δγ]+Ψ
K=4πλcTEefo,
γd+δγ=γTE=γTM,
Ψ=ϕθe+ϕΔ+ϕr.
Ψ=ΨaY+Ψb
ΨaY=ϕθe,Ψb=ϕΔ+ϕr.
I(K,γd+δγ,Ψ,γc,NA,m)=I0+AF(γd+δγ,γc,NA,m,K,Ψ)
γc=γcaY+γcb.
R02(p,I0(i,j),A(i,j))=δγ{Iexp(i,j,δγ)[I0(i,j)+A(i,j)F(p,δγ)]}2.
R12(p,i,j)=minI0(i,j),A(i,j)R02(p,I0(i,j),A(i,j)).
R22(p)=i,jR12(p,i,j)i,jδγ[Iexp(i,j,δγ)]2.
R32=minpR22(p).
Ψb=ϕΔ+δΔ+ϕr=Ψb+ϕδΔ=Ψb+sΔδΔ.
sΔth=4πλcTEe8.36×103radm1,
sΔexp=ΨbΨbδΔ6.66×103radm1.
Kth=4πλcTEefo167radrad1,
Kexp147radrad1,
Kth|sΔth|=fo=20mm,
Kexp|sΔexp|22.1mm.
Kthdth=4πλ20.0μm1,
Kexpdexp18.7μm1.
Iexp(i,j,Δk)=I0(i,j)+A(i,j)cos[ϕ(i,j)+sΔΔk].
ϕfar from the tip=ϕθe+ϕΔ+ϕr=ΨaY+Ψb=Ψ.
ϕtip=ϕθe+ϕΔ+ϕr+ϕγTE+ϕo=ΨaY+Ψb+ϕγTEγTE(XT,YT)+ϕΔZΔZ(XT,YT,d)
sin(α)=Yefo+o(θe2,αPAS2,θPAS2),
ϕ(γTE,α)=4πλcTEefo[1+sin2(α)]γTE4πλΔZcosα+ϕθe+ϕΔ+ϕr+o(γTE2).
I(γTE,γc,NA,m)=I0+AF(γTE,γc,NA,m).
F(γTE,γc,NA,m)={2θ1πα1α2(θ)f1(γTE,α,m)dθdα+20θ2α1α3(θ)f1(γTE,α,m)dθdα}/{2θ1πα1α2(θ)f2(α)dθdα+20θ2α1α3(θ)f2(α)dθdα},
F(γTE,γc,NA,m)={20θ1α2(θ)α1f1(γTE,α,m)dθdα+2θ2πα3(θ)α1f1(γTE,α,m)dθdα}/{20θ1α2(θ)α1f2(α)dθdα+2θ2πα3(θ)α1f2(α)dθdα}
f1(γTE,α,m)=cos[ϕ(γTE,α)]Pm(α)sinα,
f2(α)=P1(α)sinα,
θ1=arccos(2Y0pDpup),
θ2=arccos(2Y0pDpup),
α1=arcsin(Y0pfo),
α2(θ)=arcsin(NAcosθ),
α3(θ)=arcsin(NAcosθ2Y0pfo),
Y0p=g2=Dpup21+NA22NA(γTE+γc)+o(γTE2,γc2).
sin(α)=rcos(θ)fo=rcos(θ)gfo.
Pm(α)=[cos(α)]m.

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