Abstract

We introduce a relatively simple and efficient optical technique to measure nanoscale displacement based on visibility variations of the Fresnel diffraction fringes from a two-dimensional phase step. In this paper we use our technique to measure electromechanical expansions by a thin piezoelectric ceramic and also thermal changes in the diameter of a tungsten wire. Early results provide convincing evidence that sensitivity up to a few nanometers can be achieved, and our technique has the potential to be used as a nanodisplacement probe.

© 2012 Optical Society of America

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References

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  1. J. Fraden, Handbook of Modern Sensors: Physics, Designs and Applications (Springer, 2010).
  2. S. Soloman, Sensors Handbook2nd ed. (McGraw-Hill, 2009).
  3. D. S. Nyce, Linear Position Sensors: Theory and Application (Wiley–IEEE, 2004).
  4. D. Laing, “A look into the linear displacement and rotary position sensor markets,” Sensors 27 (2010), http://www.sensorsmag.com/sensors/position-presence-proximity/a-look-linear-displacement-and-rotary-position-sensor-market-7699.
  5. W. Gao, Precision Nanometrology: Sensors and Measuring Systems for Nanomanufacturing (Springer, 2010).
  6. O. V. Angelsky, S. G. Hanson, C. Y. Zenkova, M. P. Gorsky, and N. V. Gorodyns’ka, “On polarization metrology (estimation) of the degree of coherence of optical waves,” Opt. Express 17, 15623–15634 (2009).
    [Crossref]
  7. L. K. Baxter, Capacitive Sensors: Design and Applications (IEEE, 1997).
  8. S. A. Solin, “Magnetic field nanosensors,” Sci. Am. (July), 71–77 (2004).
  9. P. Günther, T. Pfister, and J. Czarske, “Non-incremental interferometric displacement measurement using a laser Doppler sensor with phase coding,” Opt. Lasers Eng. 49, 1190–1193 (2011).
    [Crossref]
  10. J. H. Chen, X. G. Huang, J. R. Zhao, J. Tao, W. X. He, and S. H. Liu, “Fabry–Perot interference-based fiber-optic sensor for small displacement measurement,” Opt. Commun. 283, 3315–3319 (2010).
    [Crossref]
  11. S. Zhao, C. Hou, J. Bai, G. Yang, and F. Tian, “Nanometer-scale displacement sensor based on phase-sensitive diffraction grating,” Appl. Opt. 50, 1413–1416 (2011).
    [Crossref]
  12. N. Lagakos, T. Litovitz, P. Macedo, R. Mohr, and R. Meister, “Multimode optical fiber displacement sensor,” Appl. Opt. 20, 167–168 (1981).
    [Crossref]
  13. J. S. Wilson, ed., Sensors Technology Handbook (Elsevier, 2005).
  14. L. Perret, L. Chassagne, S. Topu, P. Ruaux, B. Cagneau, and Y. Alayli, “Fiber optics sensor for sub-nanometric displacement and wide bandwidth systems,” Sensors Actuators A 165, 189–193 (2011).
    [Crossref]
  15. I. Iordache, M. Bojan, D. Apostol, V. Damian, F. Garoi, P. C. Logof, R. Muller, and B. Savu, “Optical encoder measurement technology,” Proc. SPIE 6635, 663506 (2007).
  16. M. T. Tavassoly, M. Amiri, A. Darudi, R. Aalipour, A. Saber, and A. R. Moradi, “Optical diffractometry,” J. Opt. Soc. Am. A 26, 540–547 (2009).
    [Crossref]
  17. M. Amiri and M. T. Tavassoly, “Fresnel diffraction from 1D and 2D phase steps in reflection and transmission modes,” Opt. Commun. 272, 349–361 (2007).
    [Crossref]
  18. M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, “Spectral modification by line singularity in Fresnel diffraction from 1D phase step,” Opt. Commun. 255, 23–34 (2005).
    [Crossref]
  19. A. Sabatyan and M. T. Tavassoly, “Determination of refractive indices of liquids by Fresnel diffraction,” Opt. Laser Technol. 41, 892–896 (2009).
    [Crossref]
  20. A. Sabatyan and M. T. Tavassoly, “Application of Fresnel diffraction to nondestructive measurement of the refractive index of optical fibers,” Opt. Eng. 46, 1–7 (2006).
    [Crossref]
  21. M. T. Tavassoly, I. M. Haghighi, and K. Hassani, “Application of Fresnel diffraction from a phase step to the measurement of film thickness,” Appl. Opt. 48, 5497–5501 (2009).
    [Crossref]
  22. M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 2007).
  23. O. R. George, J. B. DeVelis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE–American Institute of Physics, 1989).
  24. K. D. Mielenz, “Algorithms for Fresnel diffraction at rectangular and circular apertures,” J. Res. Natl. Inst. Stand. Technol. 103, 497–509 (1998).
    [Crossref]
  25. E. C. J. von Lommel, “Die Beugungserscheinungen einer kreisrunden Öffnung und eines kreisrunden Schirmchens,” Abh. der Math. Phys. Classe der Kön. Bayer. Akad. der Wiss 15, 229–328 (1886).
    [Crossref]
  26. F. W. J. Olver, D. W. Lozier, and R. F. Boisvert, eds., NIST Handbook of Mathematical Functions (Cambridge University, 2010).
  27. “Tutorial: Piezoelectrics in nanopositioning, designing with piezoelectric actuators,” http://www.physikinstrumente.com/tutorial .

2011 (3)

P. Günther, T. Pfister, and J. Czarske, “Non-incremental interferometric displacement measurement using a laser Doppler sensor with phase coding,” Opt. Lasers Eng. 49, 1190–1193 (2011).
[Crossref]

S. Zhao, C. Hou, J. Bai, G. Yang, and F. Tian, “Nanometer-scale displacement sensor based on phase-sensitive diffraction grating,” Appl. Opt. 50, 1413–1416 (2011).
[Crossref]

L. Perret, L. Chassagne, S. Topu, P. Ruaux, B. Cagneau, and Y. Alayli, “Fiber optics sensor for sub-nanometric displacement and wide bandwidth systems,” Sensors Actuators A 165, 189–193 (2011).
[Crossref]

2010 (2)

J. H. Chen, X. G. Huang, J. R. Zhao, J. Tao, W. X. He, and S. H. Liu, “Fabry–Perot interference-based fiber-optic sensor for small displacement measurement,” Opt. Commun. 283, 3315–3319 (2010).
[Crossref]

D. Laing, “A look into the linear displacement and rotary position sensor markets,” Sensors 27 (2010), http://www.sensorsmag.com/sensors/position-presence-proximity/a-look-linear-displacement-and-rotary-position-sensor-market-7699.

2009 (4)

2007 (2)

M. Amiri and M. T. Tavassoly, “Fresnel diffraction from 1D and 2D phase steps in reflection and transmission modes,” Opt. Commun. 272, 349–361 (2007).
[Crossref]

I. Iordache, M. Bojan, D. Apostol, V. Damian, F. Garoi, P. C. Logof, R. Muller, and B. Savu, “Optical encoder measurement technology,” Proc. SPIE 6635, 663506 (2007).

2006 (1)

A. Sabatyan and M. T. Tavassoly, “Application of Fresnel diffraction to nondestructive measurement of the refractive index of optical fibers,” Opt. Eng. 46, 1–7 (2006).
[Crossref]

2005 (1)

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, “Spectral modification by line singularity in Fresnel diffraction from 1D phase step,” Opt. Commun. 255, 23–34 (2005).
[Crossref]

2004 (1)

S. A. Solin, “Magnetic field nanosensors,” Sci. Am. (July), 71–77 (2004).

1998 (1)

K. D. Mielenz, “Algorithms for Fresnel diffraction at rectangular and circular apertures,” J. Res. Natl. Inst. Stand. Technol. 103, 497–509 (1998).
[Crossref]

1981 (1)

1886 (1)

E. C. J. von Lommel, “Die Beugungserscheinungen einer kreisrunden Öffnung und eines kreisrunden Schirmchens,” Abh. der Math. Phys. Classe der Kön. Bayer. Akad. der Wiss 15, 229–328 (1886).
[Crossref]

Aalipour, R.

Alayli, Y.

L. Perret, L. Chassagne, S. Topu, P. Ruaux, B. Cagneau, and Y. Alayli, “Fiber optics sensor for sub-nanometric displacement and wide bandwidth systems,” Sensors Actuators A 165, 189–193 (2011).
[Crossref]

Amiri, M.

M. T. Tavassoly, M. Amiri, A. Darudi, R. Aalipour, A. Saber, and A. R. Moradi, “Optical diffractometry,” J. Opt. Soc. Am. A 26, 540–547 (2009).
[Crossref]

M. Amiri and M. T. Tavassoly, “Fresnel diffraction from 1D and 2D phase steps in reflection and transmission modes,” Opt. Commun. 272, 349–361 (2007).
[Crossref]

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, “Spectral modification by line singularity in Fresnel diffraction from 1D phase step,” Opt. Commun. 255, 23–34 (2005).
[Crossref]

Angelsky, O. V.

Apostol, D.

I. Iordache, M. Bojan, D. Apostol, V. Damian, F. Garoi, P. C. Logof, R. Muller, and B. Savu, “Optical encoder measurement technology,” Proc. SPIE 6635, 663506 (2007).

Bai, J.

Baxter, L. K.

L. K. Baxter, Capacitive Sensors: Design and Applications (IEEE, 1997).

Bojan, M.

I. Iordache, M. Bojan, D. Apostol, V. Damian, F. Garoi, P. C. Logof, R. Muller, and B. Savu, “Optical encoder measurement technology,” Proc. SPIE 6635, 663506 (2007).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 2007).

Cagneau, B.

L. Perret, L. Chassagne, S. Topu, P. Ruaux, B. Cagneau, and Y. Alayli, “Fiber optics sensor for sub-nanometric displacement and wide bandwidth systems,” Sensors Actuators A 165, 189–193 (2011).
[Crossref]

Chassagne, L.

L. Perret, L. Chassagne, S. Topu, P. Ruaux, B. Cagneau, and Y. Alayli, “Fiber optics sensor for sub-nanometric displacement and wide bandwidth systems,” Sensors Actuators A 165, 189–193 (2011).
[Crossref]

Chen, J. H.

J. H. Chen, X. G. Huang, J. R. Zhao, J. Tao, W. X. He, and S. H. Liu, “Fabry–Perot interference-based fiber-optic sensor for small displacement measurement,” Opt. Commun. 283, 3315–3319 (2010).
[Crossref]

Czarske, J.

P. Günther, T. Pfister, and J. Czarske, “Non-incremental interferometric displacement measurement using a laser Doppler sensor with phase coding,” Opt. Lasers Eng. 49, 1190–1193 (2011).
[Crossref]

Damian, V.

I. Iordache, M. Bojan, D. Apostol, V. Damian, F. Garoi, P. C. Logof, R. Muller, and B. Savu, “Optical encoder measurement technology,” Proc. SPIE 6635, 663506 (2007).

Darudi, A.

DeVelis, J. B.

O. R. George, J. B. DeVelis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE–American Institute of Physics, 1989).

Fraden, J.

J. Fraden, Handbook of Modern Sensors: Physics, Designs and Applications (Springer, 2010).

Gao, W.

W. Gao, Precision Nanometrology: Sensors and Measuring Systems for Nanomanufacturing (Springer, 2010).

Garoi, F.

I. Iordache, M. Bojan, D. Apostol, V. Damian, F. Garoi, P. C. Logof, R. Muller, and B. Savu, “Optical encoder measurement technology,” Proc. SPIE 6635, 663506 (2007).

George, O. R.

O. R. George, J. B. DeVelis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE–American Institute of Physics, 1989).

Gorodyns’ka, N. V.

Gorsky, M. P.

Günther, P.

P. Günther, T. Pfister, and J. Czarske, “Non-incremental interferometric displacement measurement using a laser Doppler sensor with phase coding,” Opt. Lasers Eng. 49, 1190–1193 (2011).
[Crossref]

Haghighi, I. M.

Hanson, S. G.

Hassani, K.

He, W. X.

J. H. Chen, X. G. Huang, J. R. Zhao, J. Tao, W. X. He, and S. H. Liu, “Fabry–Perot interference-based fiber-optic sensor for small displacement measurement,” Opt. Commun. 283, 3315–3319 (2010).
[Crossref]

Hou, C.

Huang, X. G.

J. H. Chen, X. G. Huang, J. R. Zhao, J. Tao, W. X. He, and S. H. Liu, “Fabry–Perot interference-based fiber-optic sensor for small displacement measurement,” Opt. Commun. 283, 3315–3319 (2010).
[Crossref]

Iordache, I.

I. Iordache, M. Bojan, D. Apostol, V. Damian, F. Garoi, P. C. Logof, R. Muller, and B. Savu, “Optical encoder measurement technology,” Proc. SPIE 6635, 663506 (2007).

Karimi, E.

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, “Spectral modification by line singularity in Fresnel diffraction from 1D phase step,” Opt. Commun. 255, 23–34 (2005).
[Crossref]

Khalesifard, H. R.

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, “Spectral modification by line singularity in Fresnel diffraction from 1D phase step,” Opt. Commun. 255, 23–34 (2005).
[Crossref]

Lagakos, N.

Laing, D.

D. Laing, “A look into the linear displacement and rotary position sensor markets,” Sensors 27 (2010), http://www.sensorsmag.com/sensors/position-presence-proximity/a-look-linear-displacement-and-rotary-position-sensor-market-7699.

Litovitz, T.

Liu, S. H.

J. H. Chen, X. G. Huang, J. R. Zhao, J. Tao, W. X. He, and S. H. Liu, “Fabry–Perot interference-based fiber-optic sensor for small displacement measurement,” Opt. Commun. 283, 3315–3319 (2010).
[Crossref]

Logof, P. C.

I. Iordache, M. Bojan, D. Apostol, V. Damian, F. Garoi, P. C. Logof, R. Muller, and B. Savu, “Optical encoder measurement technology,” Proc. SPIE 6635, 663506 (2007).

Macedo, P.

Meister, R.

Mielenz, K. D.

K. D. Mielenz, “Algorithms for Fresnel diffraction at rectangular and circular apertures,” J. Res. Natl. Inst. Stand. Technol. 103, 497–509 (1998).
[Crossref]

Mohr, R.

Moradi, A. R.

Muller, R.

I. Iordache, M. Bojan, D. Apostol, V. Damian, F. Garoi, P. C. Logof, R. Muller, and B. Savu, “Optical encoder measurement technology,” Proc. SPIE 6635, 663506 (2007).

Nyce, D. S.

D. S. Nyce, Linear Position Sensors: Theory and Application (Wiley–IEEE, 2004).

Parrent, G. B.

O. R. George, J. B. DeVelis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE–American Institute of Physics, 1989).

Perret, L.

L. Perret, L. Chassagne, S. Topu, P. Ruaux, B. Cagneau, and Y. Alayli, “Fiber optics sensor for sub-nanometric displacement and wide bandwidth systems,” Sensors Actuators A 165, 189–193 (2011).
[Crossref]

Pfister, T.

P. Günther, T. Pfister, and J. Czarske, “Non-incremental interferometric displacement measurement using a laser Doppler sensor with phase coding,” Opt. Lasers Eng. 49, 1190–1193 (2011).
[Crossref]

Ruaux, P.

L. Perret, L. Chassagne, S. Topu, P. Ruaux, B. Cagneau, and Y. Alayli, “Fiber optics sensor for sub-nanometric displacement and wide bandwidth systems,” Sensors Actuators A 165, 189–193 (2011).
[Crossref]

Sabatyan, A.

A. Sabatyan and M. T. Tavassoly, “Determination of refractive indices of liquids by Fresnel diffraction,” Opt. Laser Technol. 41, 892–896 (2009).
[Crossref]

A. Sabatyan and M. T. Tavassoly, “Application of Fresnel diffraction to nondestructive measurement of the refractive index of optical fibers,” Opt. Eng. 46, 1–7 (2006).
[Crossref]

Saber, A.

Savu, B.

I. Iordache, M. Bojan, D. Apostol, V. Damian, F. Garoi, P. C. Logof, R. Muller, and B. Savu, “Optical encoder measurement technology,” Proc. SPIE 6635, 663506 (2007).

Solin, S. A.

S. A. Solin, “Magnetic field nanosensors,” Sci. Am. (July), 71–77 (2004).

Soloman, S.

S. Soloman, Sensors Handbook2nd ed. (McGraw-Hill, 2009).

Tao, J.

J. H. Chen, X. G. Huang, J. R. Zhao, J. Tao, W. X. He, and S. H. Liu, “Fabry–Perot interference-based fiber-optic sensor for small displacement measurement,” Opt. Commun. 283, 3315–3319 (2010).
[Crossref]

Tavassoly, M. T.

A. Sabatyan and M. T. Tavassoly, “Determination of refractive indices of liquids by Fresnel diffraction,” Opt. Laser Technol. 41, 892–896 (2009).
[Crossref]

M. T. Tavassoly, I. M. Haghighi, and K. Hassani, “Application of Fresnel diffraction from a phase step to the measurement of film thickness,” Appl. Opt. 48, 5497–5501 (2009).
[Crossref]

M. T. Tavassoly, M. Amiri, A. Darudi, R. Aalipour, A. Saber, and A. R. Moradi, “Optical diffractometry,” J. Opt. Soc. Am. A 26, 540–547 (2009).
[Crossref]

M. Amiri and M. T. Tavassoly, “Fresnel diffraction from 1D and 2D phase steps in reflection and transmission modes,” Opt. Commun. 272, 349–361 (2007).
[Crossref]

A. Sabatyan and M. T. Tavassoly, “Application of Fresnel diffraction to nondestructive measurement of the refractive index of optical fibers,” Opt. Eng. 46, 1–7 (2006).
[Crossref]

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, “Spectral modification by line singularity in Fresnel diffraction from 1D phase step,” Opt. Commun. 255, 23–34 (2005).
[Crossref]

Thompson, B. J.

O. R. George, J. B. DeVelis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE–American Institute of Physics, 1989).

Tian, F.

Topu, S.

L. Perret, L. Chassagne, S. Topu, P. Ruaux, B. Cagneau, and Y. Alayli, “Fiber optics sensor for sub-nanometric displacement and wide bandwidth systems,” Sensors Actuators A 165, 189–193 (2011).
[Crossref]

von Lommel, E. C. J.

E. C. J. von Lommel, “Die Beugungserscheinungen einer kreisrunden Öffnung und eines kreisrunden Schirmchens,” Abh. der Math. Phys. Classe der Kön. Bayer. Akad. der Wiss 15, 229–328 (1886).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 2007).

Yang, G.

Zenkova, C. Y.

Zhao, J. R.

J. H. Chen, X. G. Huang, J. R. Zhao, J. Tao, W. X. He, and S. H. Liu, “Fabry–Perot interference-based fiber-optic sensor for small displacement measurement,” Opt. Commun. 283, 3315–3319 (2010).
[Crossref]

Zhao, S.

Abh. der Math. Phys. Classe der Kön. Bayer. Akad. der Wiss (1)

E. C. J. von Lommel, “Die Beugungserscheinungen einer kreisrunden Öffnung und eines kreisrunden Schirmchens,” Abh. der Math. Phys. Classe der Kön. Bayer. Akad. der Wiss 15, 229–328 (1886).
[Crossref]

Appl. Opt. (3)

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

J. Res. Natl. Inst. Stand. Technol. (1)

K. D. Mielenz, “Algorithms for Fresnel diffraction at rectangular and circular apertures,” J. Res. Natl. Inst. Stand. Technol. 103, 497–509 (1998).
[Crossref]

Opt. Commun. (3)

M. Amiri and M. T. Tavassoly, “Fresnel diffraction from 1D and 2D phase steps in reflection and transmission modes,” Opt. Commun. 272, 349–361 (2007).
[Crossref]

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, “Spectral modification by line singularity in Fresnel diffraction from 1D phase step,” Opt. Commun. 255, 23–34 (2005).
[Crossref]

J. H. Chen, X. G. Huang, J. R. Zhao, J. Tao, W. X. He, and S. H. Liu, “Fabry–Perot interference-based fiber-optic sensor for small displacement measurement,” Opt. Commun. 283, 3315–3319 (2010).
[Crossref]

Opt. Eng. (1)

A. Sabatyan and M. T. Tavassoly, “Application of Fresnel diffraction to nondestructive measurement of the refractive index of optical fibers,” Opt. Eng. 46, 1–7 (2006).
[Crossref]

Opt. Express (1)

Opt. Laser Technol. (1)

A. Sabatyan and M. T. Tavassoly, “Determination of refractive indices of liquids by Fresnel diffraction,” Opt. Laser Technol. 41, 892–896 (2009).
[Crossref]

Opt. Lasers Eng. (1)

P. Günther, T. Pfister, and J. Czarske, “Non-incremental interferometric displacement measurement using a laser Doppler sensor with phase coding,” Opt. Lasers Eng. 49, 1190–1193 (2011).
[Crossref]

Proc. SPIE (1)

I. Iordache, M. Bojan, D. Apostol, V. Damian, F. Garoi, P. C. Logof, R. Muller, and B. Savu, “Optical encoder measurement technology,” Proc. SPIE 6635, 663506 (2007).

Sci. Am. (1)

S. A. Solin, “Magnetic field nanosensors,” Sci. Am. (July), 71–77 (2004).

Sensors (1)

D. Laing, “A look into the linear displacement and rotary position sensor markets,” Sensors 27 (2010), http://www.sensorsmag.com/sensors/position-presence-proximity/a-look-linear-displacement-and-rotary-position-sensor-market-7699.

Sensors Actuators A (1)

L. Perret, L. Chassagne, S. Topu, P. Ruaux, B. Cagneau, and Y. Alayli, “Fiber optics sensor for sub-nanometric displacement and wide bandwidth systems,” Sensors Actuators A 165, 189–193 (2011).
[Crossref]

Other (10)

J. S. Wilson, ed., Sensors Technology Handbook (Elsevier, 2005).

W. Gao, Precision Nanometrology: Sensors and Measuring Systems for Nanomanufacturing (Springer, 2010).

J. Fraden, Handbook of Modern Sensors: Physics, Designs and Applications (Springer, 2010).

S. Soloman, Sensors Handbook2nd ed. (McGraw-Hill, 2009).

D. S. Nyce, Linear Position Sensors: Theory and Application (Wiley–IEEE, 2004).

L. K. Baxter, Capacitive Sensors: Design and Applications (IEEE, 1997).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 2007).

O. R. George, J. B. DeVelis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE–American Institute of Physics, 1989).

F. W. J. Olver, D. W. Lozier, and R. F. Boisvert, eds., NIST Handbook of Mathematical Functions (Cambridge University, 2010).

“Tutorial: Piezoelectrics in nanopositioning, designing with piezoelectric actuators,” http://www.physikinstrumente.com/tutorial .

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Figures (7)

Fig. 1.
Fig. 1.

Fresnel diffraction from an arbitrary aperture: light from a point source S is incident on a plane σ at distance Z0 from the source, which contains an aperture A. Diffracted rays from different points on the aperture then interfere on another plane, Σ at distance z and form the Fresnel diffraction pattern.

Fig. 2.
Fig. 2.

Circular step: the circular ring has inner radius a and outer radius b. A circular disk of radius a is assumed to be at height h with respect to the ring.

Fig. 3.
Fig. 3.

Simulation of Fresnel diffraction patterns by a circular step of height (a), (b) λ4 and (c), (d) λ1.8. Right panels show cross sections along a diameter. Visibility varies between 0 and 1 as the step height changes by λ4.

Fig. 4.
Fig. 4.

Experimental setup for Fresnel diffraction from a variable circular step: BE, beam expander; M, mirror; BS, beam splitter; STEP, circular step; OBJ, moving object; CCD, CCD camera.

Fig. 5.
Fig. 5.

(a) Experimental 2D Fresnel diffraction pattern from the circular step. (b) Intensity graph along the radial direction from data inside the box shown in the 2D plot. The data are circularly averaged to reduce background noise.

Fig. 6.
Fig. 6.

(a) Visibility curves obtained by a piezostack. Points and the solid curve are experimental data. Simulation visibility curve is fitted to the data and plotted with a dashed curve. (b) Visibility curves for a single PZT ceramic 0.1 mm thick when the driving voltage is increased in 0.5 V steps. The resulting displacement steps are estimated to be of the order of two nanometers.

Fig. 7.
Fig. 7.

Left: visibility curve for radial thermal expansion of a tungsten wire with diameter equal to 1.5 mm. Average step size obtained by fitting the data to the simulation (dashed curve) is about 3.2 nm. Right: schematic drawing of the experimental setup. The wire is firmly supported by two rigid supports S and heated by a heater freely positioned in the middle of it. Radial thermal expansion is applied to the sensor via a hard tip H.

Equations (21)

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

E(P)=iλAscos(θ0)eik(r0+r)r0rAeikΔdξdη,
Δ(SQ+QP)(SO+OP).
E(P)=U(P)α(P)cos(θ0),
U(P)Aseik(r0+r)r0+r,
α(P)iλ(r0+r)r0rAeikΔdξdη.
α(P)=iu01dρρJ0(νρ)e12iuρ2,
ρqa,uka2(r0+r)r0r,νkacr.
cr(ll0)2+(mm0)2,
lxr,myr,
l0x0r0,m0y0r0.
αL=u2M(u,v)iu2L(u,v).
u2L(u,v)=sin(ν22u)+V0(u,v)sin(u2)V1(u,v)cos(u2),
u2M(u,v)=cos(ν22u)V0(u,v)cos(u2)V1(u,v)sin(u2),
u2L(u,v)=U1(u,v)cos(u2)+U2(u,v)sin(u2),
u2M(u,v)=U1(u,v)sin(u2)U2(u,v)cos(u2).
V0(u,v)J0(ν)(νu)2J2(ν)+(νu)4J4(ν),
V1(u,v)(νu)J1(ν)(νu)3J3(ν)+(νu)5J5(ν),
U1(u,v)(uν)J1(ν)(uν)3J3(ν)+(uν)5J5(ν),
U2(u,v)(uν)2J2(ν)(uν)4J4(ν)+(uν)6J6(ν).
E(P)=Eb(P)Ea(P)+Ea(P)eiϕ,
VImaxL+ImaxR2IminImaxL+ImaxR2+Imin,

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