Abstract

Optical holographic interferometry (HI) is realized by two well-known techniques: double exposure holographic interferometry (DEHI) and real-time holographic interferometry (RTHI). However, the digital version of HI is typically realized numerically by DEHI. The main problem in digital implementation of RTHI is the lack of commercially available cameras and spatial light modulators with the same pixel size. This mismatch results in lateral and transversal magnifications of an object wavefront reconstruction. In real-time digital HI the reconstruction of an object in an initial state has to be superimposed on top of the loaded object. In this work, we present and analyze five approaches to overcome the mismatch problem, and the performance of these procedures is numerically quantified and compared. The experimental suitability of these approaches is investigated.

© 2014 Optical Society of America

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References

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  1. C. M. Vest, Holographic Interferometry (Wiley, 1979).
  2. P. K. Rastogi, Holographic Interferometry: Principles and Methods (Springer, 1994).
  3. T. Kreis, Handbook of Holographic Interferometry—Optical and Digital Methods (Wiley, 2005).
  4. G. Lazarev, A. Hermerschmidt, S. Kruger, and S. Osten, “LCOS spatial light modulators: trends and applications,” in Optical Imaging and Metrology: Selected Topics, W. Osten and N. Reingand, eds. (Wiley-VCH, 2012), pp. 1–29.
  5. A. Michałkiewicz, M. Kujawińska, R. Lymarenko, O. Budryk, X. Wang, and P. J. Bos, “Simulation, registration, and reconstruction of digital holograms of arbitrary objects by means of liquid crystal on silicon spatial light modulator,” Proc. SPIE 5947, 59470G (2005).
    [CrossRef]
  6. M. Kujawinska, R. Porras-Aguilar, and W. Zaperty, “LCoS spatial light modulators as active phase elements of full-field measurement systems and sensors,” Metrology and Measurement Systems 19, 445–458 (2012).
    [CrossRef]
  7. T. Kreis, Holographic Interferometry, Principles and Methods, 1st ed., Akademie Verlag Series in Optical Metrology (Wiley-VCH, 1996).
  8. M. Kujawińska, G. Finke, P. Garbat, C. Falldorf, and M. B. Hennelly, “Wide angle digital holographic interferometry with real-time optical reconstruction,” Photon. Lett. Poland 4, 48–50 (2012).
    [CrossRef]
  9. P. Ferraro and W. Osten, “Digital holography and its application in MEMS/MOEMS inspection,” in Optical Inspection of Microsystems, W. Osten, ed. (CRC, 2006), Chap. 12.
  10. A. Michałkiewicz, M. Kujawińska, and K. Stasiewicz, “Digital holographic camera and data processing for remote monitoring and measurements of mechanical parts,” Opto-Electron. Rev. 16, 68–75 (2008).
    [CrossRef]
  11. M. K. Kim, Digital Holographic Microscopy: Principles, Techniques, and Applications (Springer, 2011).
  12. Y. Fu, G. Pedrini, and W. Osten, “Vibration measurement by temporal Fourier analyses of digital hologram sequence,” Appl. Opt. 46, 5719–5727 (2007).
    [CrossRef]
  13. C. Quan, W. Chen, and C. J. Tay, “Shape measurement by multi-illumination method in digital holographic interferometry,” Opt. Commun. 281, 3957–3964 (2008).
    [CrossRef]
  14. Z. Füzessy, F. Gyímesi, J. Kornis, B. Ráczkevi, V. Borbély, and B. Gombkötő, “Analogue and digital development for project DISCO at Budapest University of Technology and Economics,” Proc. SPIE 5457, 610–620 (2004).
    [CrossRef]
  15. CCD Detectors—Astrosurf, www.astrosurf.com/re/chip.html .
  16. Holoeye Photonics, AG, www.holoeye.com .
  17. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997).
    [CrossRef]
  18. M. Servin and A. Gonzalez, “Linear analysis of the 4-step Carré phase shifting algorithm: spectrum, signal-to-noise ratio, and harmonics response,” arXiv:1203.1947 (2012).
  19. T. Kozacki, K. Falaggis, and M. Kujawinska, “Computation of diffracted fields for the case of high numerical aperture using the angular spectrum method,” Appl. Opt. 51, 7080–7088 (2012).
    [CrossRef]
  20. U. Schnars and W. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
    [CrossRef]
  21. M. Kujawinska and T. Kozacki, “Holographic television: status and future,” in Optical Imaging and Metrology: Selected Topics, W. Osten and N. Reingand, eds. (Wiley-VCH, 2012).
  22. F. Dubois, O. Monnom, C. Yourassowsky, and J.-C. Legros, “Border processing in digital holography by extension of the digital hologram and reduction of the higher spatial frequencies,” Appl. Opt. 41, 2621–2626 (2002).
    [CrossRef]
  23. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, The Art of Scientific Computing, 2nd ed. (Cambridge University, 2002).
  24. G. Dahlquist and A. Björk, Equidistant Interpolation and the Runge Phenomenon, Numerical Methods (Courier Dover, 1974).

2012 (3)

M. Kujawinska, R. Porras-Aguilar, and W. Zaperty, “LCoS spatial light modulators as active phase elements of full-field measurement systems and sensors,” Metrology and Measurement Systems 19, 445–458 (2012).
[CrossRef]

M. Kujawińska, G. Finke, P. Garbat, C. Falldorf, and M. B. Hennelly, “Wide angle digital holographic interferometry with real-time optical reconstruction,” Photon. Lett. Poland 4, 48–50 (2012).
[CrossRef]

T. Kozacki, K. Falaggis, and M. Kujawinska, “Computation of diffracted fields for the case of high numerical aperture using the angular spectrum method,” Appl. Opt. 51, 7080–7088 (2012).
[CrossRef]

2008 (2)

C. Quan, W. Chen, and C. J. Tay, “Shape measurement by multi-illumination method in digital holographic interferometry,” Opt. Commun. 281, 3957–3964 (2008).
[CrossRef]

A. Michałkiewicz, M. Kujawińska, and K. Stasiewicz, “Digital holographic camera and data processing for remote monitoring and measurements of mechanical parts,” Opto-Electron. Rev. 16, 68–75 (2008).
[CrossRef]

2007 (1)

2005 (1)

A. Michałkiewicz, M. Kujawińska, R. Lymarenko, O. Budryk, X. Wang, and P. J. Bos, “Simulation, registration, and reconstruction of digital holograms of arbitrary objects by means of liquid crystal on silicon spatial light modulator,” Proc. SPIE 5947, 59470G (2005).
[CrossRef]

2004 (1)

Z. Füzessy, F. Gyímesi, J. Kornis, B. Ráczkevi, V. Borbély, and B. Gombkötő, “Analogue and digital development for project DISCO at Budapest University of Technology and Economics,” Proc. SPIE 5457, 610–620 (2004).
[CrossRef]

2002 (2)

1997 (1)

Björk, A.

G. Dahlquist and A. Björk, Equidistant Interpolation and the Runge Phenomenon, Numerical Methods (Courier Dover, 1974).

Borbély, V.

Z. Füzessy, F. Gyímesi, J. Kornis, B. Ráczkevi, V. Borbély, and B. Gombkötő, “Analogue and digital development for project DISCO at Budapest University of Technology and Economics,” Proc. SPIE 5457, 610–620 (2004).
[CrossRef]

Bos, P. J.

A. Michałkiewicz, M. Kujawińska, R. Lymarenko, O. Budryk, X. Wang, and P. J. Bos, “Simulation, registration, and reconstruction of digital holograms of arbitrary objects by means of liquid crystal on silicon spatial light modulator,” Proc. SPIE 5947, 59470G (2005).
[CrossRef]

Budryk, O.

A. Michałkiewicz, M. Kujawińska, R. Lymarenko, O. Budryk, X. Wang, and P. J. Bos, “Simulation, registration, and reconstruction of digital holograms of arbitrary objects by means of liquid crystal on silicon spatial light modulator,” Proc. SPIE 5947, 59470G (2005).
[CrossRef]

Chen, W.

C. Quan, W. Chen, and C. J. Tay, “Shape measurement by multi-illumination method in digital holographic interferometry,” Opt. Commun. 281, 3957–3964 (2008).
[CrossRef]

Dahlquist, G.

G. Dahlquist and A. Björk, Equidistant Interpolation and the Runge Phenomenon, Numerical Methods (Courier Dover, 1974).

Dubois, F.

Falaggis, K.

Falldorf, C.

M. Kujawińska, G. Finke, P. Garbat, C. Falldorf, and M. B. Hennelly, “Wide angle digital holographic interferometry with real-time optical reconstruction,” Photon. Lett. Poland 4, 48–50 (2012).
[CrossRef]

Ferraro, P.

P. Ferraro and W. Osten, “Digital holography and its application in MEMS/MOEMS inspection,” in Optical Inspection of Microsystems, W. Osten, ed. (CRC, 2006), Chap. 12.

Finke, G.

M. Kujawińska, G. Finke, P. Garbat, C. Falldorf, and M. B. Hennelly, “Wide angle digital holographic interferometry with real-time optical reconstruction,” Photon. Lett. Poland 4, 48–50 (2012).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, The Art of Scientific Computing, 2nd ed. (Cambridge University, 2002).

Fu, Y.

Füzessy, Z.

Z. Füzessy, F. Gyímesi, J. Kornis, B. Ráczkevi, V. Borbély, and B. Gombkötő, “Analogue and digital development for project DISCO at Budapest University of Technology and Economics,” Proc. SPIE 5457, 610–620 (2004).
[CrossRef]

Garbat, P.

M. Kujawińska, G. Finke, P. Garbat, C. Falldorf, and M. B. Hennelly, “Wide angle digital holographic interferometry with real-time optical reconstruction,” Photon. Lett. Poland 4, 48–50 (2012).
[CrossRef]

Gombköto, B.

Z. Füzessy, F. Gyímesi, J. Kornis, B. Ráczkevi, V. Borbély, and B. Gombkötő, “Analogue and digital development for project DISCO at Budapest University of Technology and Economics,” Proc. SPIE 5457, 610–620 (2004).
[CrossRef]

Gonzalez, A.

M. Servin and A. Gonzalez, “Linear analysis of the 4-step Carré phase shifting algorithm: spectrum, signal-to-noise ratio, and harmonics response,” arXiv:1203.1947 (2012).

Gyímesi, F.

Z. Füzessy, F. Gyímesi, J. Kornis, B. Ráczkevi, V. Borbély, and B. Gombkötő, “Analogue and digital development for project DISCO at Budapest University of Technology and Economics,” Proc. SPIE 5457, 610–620 (2004).
[CrossRef]

Hennelly, M. B.

M. Kujawińska, G. Finke, P. Garbat, C. Falldorf, and M. B. Hennelly, “Wide angle digital holographic interferometry with real-time optical reconstruction,” Photon. Lett. Poland 4, 48–50 (2012).
[CrossRef]

Hermerschmidt, A.

G. Lazarev, A. Hermerschmidt, S. Kruger, and S. Osten, “LCOS spatial light modulators: trends and applications,” in Optical Imaging and Metrology: Selected Topics, W. Osten and N. Reingand, eds. (Wiley-VCH, 2012), pp. 1–29.

Juptner, W.

U. Schnars and W. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

Kim, M. K.

M. K. Kim, Digital Holographic Microscopy: Principles, Techniques, and Applications (Springer, 2011).

Kornis, J.

Z. Füzessy, F. Gyímesi, J. Kornis, B. Ráczkevi, V. Borbély, and B. Gombkötő, “Analogue and digital development for project DISCO at Budapest University of Technology and Economics,” Proc. SPIE 5457, 610–620 (2004).
[CrossRef]

Kozacki, T.

T. Kozacki, K. Falaggis, and M. Kujawinska, “Computation of diffracted fields for the case of high numerical aperture using the angular spectrum method,” Appl. Opt. 51, 7080–7088 (2012).
[CrossRef]

M. Kujawinska and T. Kozacki, “Holographic television: status and future,” in Optical Imaging and Metrology: Selected Topics, W. Osten and N. Reingand, eds. (Wiley-VCH, 2012).

Kreis, T.

T. Kreis, Handbook of Holographic Interferometry—Optical and Digital Methods (Wiley, 2005).

T. Kreis, Holographic Interferometry, Principles and Methods, 1st ed., Akademie Verlag Series in Optical Metrology (Wiley-VCH, 1996).

Kruger, S.

G. Lazarev, A. Hermerschmidt, S. Kruger, and S. Osten, “LCOS spatial light modulators: trends and applications,” in Optical Imaging and Metrology: Selected Topics, W. Osten and N. Reingand, eds. (Wiley-VCH, 2012), pp. 1–29.

Kujawinska, M.

M. Kujawińska, G. Finke, P. Garbat, C. Falldorf, and M. B. Hennelly, “Wide angle digital holographic interferometry with real-time optical reconstruction,” Photon. Lett. Poland 4, 48–50 (2012).
[CrossRef]

M. Kujawinska, R. Porras-Aguilar, and W. Zaperty, “LCoS spatial light modulators as active phase elements of full-field measurement systems and sensors,” Metrology and Measurement Systems 19, 445–458 (2012).
[CrossRef]

T. Kozacki, K. Falaggis, and M. Kujawinska, “Computation of diffracted fields for the case of high numerical aperture using the angular spectrum method,” Appl. Opt. 51, 7080–7088 (2012).
[CrossRef]

A. Michałkiewicz, M. Kujawińska, and K. Stasiewicz, “Digital holographic camera and data processing for remote monitoring and measurements of mechanical parts,” Opto-Electron. Rev. 16, 68–75 (2008).
[CrossRef]

A. Michałkiewicz, M. Kujawińska, R. Lymarenko, O. Budryk, X. Wang, and P. J. Bos, “Simulation, registration, and reconstruction of digital holograms of arbitrary objects by means of liquid crystal on silicon spatial light modulator,” Proc. SPIE 5947, 59470G (2005).
[CrossRef]

M. Kujawinska and T. Kozacki, “Holographic television: status and future,” in Optical Imaging and Metrology: Selected Topics, W. Osten and N. Reingand, eds. (Wiley-VCH, 2012).

Lazarev, G.

G. Lazarev, A. Hermerschmidt, S. Kruger, and S. Osten, “LCOS spatial light modulators: trends and applications,” in Optical Imaging and Metrology: Selected Topics, W. Osten and N. Reingand, eds. (Wiley-VCH, 2012), pp. 1–29.

Legros, J.-C.

Lymarenko, R.

A. Michałkiewicz, M. Kujawińska, R. Lymarenko, O. Budryk, X. Wang, and P. J. Bos, “Simulation, registration, and reconstruction of digital holograms of arbitrary objects by means of liquid crystal on silicon spatial light modulator,” Proc. SPIE 5947, 59470G (2005).
[CrossRef]

Michalkiewicz, A.

A. Michałkiewicz, M. Kujawińska, and K. Stasiewicz, “Digital holographic camera and data processing for remote monitoring and measurements of mechanical parts,” Opto-Electron. Rev. 16, 68–75 (2008).
[CrossRef]

A. Michałkiewicz, M. Kujawińska, R. Lymarenko, O. Budryk, X. Wang, and P. J. Bos, “Simulation, registration, and reconstruction of digital holograms of arbitrary objects by means of liquid crystal on silicon spatial light modulator,” Proc. SPIE 5947, 59470G (2005).
[CrossRef]

Monnom, O.

Osten, S.

G. Lazarev, A. Hermerschmidt, S. Kruger, and S. Osten, “LCOS spatial light modulators: trends and applications,” in Optical Imaging and Metrology: Selected Topics, W. Osten and N. Reingand, eds. (Wiley-VCH, 2012), pp. 1–29.

Osten, W.

Y. Fu, G. Pedrini, and W. Osten, “Vibration measurement by temporal Fourier analyses of digital hologram sequence,” Appl. Opt. 46, 5719–5727 (2007).
[CrossRef]

P. Ferraro and W. Osten, “Digital holography and its application in MEMS/MOEMS inspection,” in Optical Inspection of Microsystems, W. Osten, ed. (CRC, 2006), Chap. 12.

Pedrini, G.

Porras-Aguilar, R.

M. Kujawinska, R. Porras-Aguilar, and W. Zaperty, “LCoS spatial light modulators as active phase elements of full-field measurement systems and sensors,” Metrology and Measurement Systems 19, 445–458 (2012).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, The Art of Scientific Computing, 2nd ed. (Cambridge University, 2002).

Quan, C.

C. Quan, W. Chen, and C. J. Tay, “Shape measurement by multi-illumination method in digital holographic interferometry,” Opt. Commun. 281, 3957–3964 (2008).
[CrossRef]

Ráczkevi, B.

Z. Füzessy, F. Gyímesi, J. Kornis, B. Ráczkevi, V. Borbély, and B. Gombkötő, “Analogue and digital development for project DISCO at Budapest University of Technology and Economics,” Proc. SPIE 5457, 610–620 (2004).
[CrossRef]

Rastogi, P. K.

P. K. Rastogi, Holographic Interferometry: Principles and Methods (Springer, 1994).

Schnars, U.

U. Schnars and W. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

Servin, M.

M. Servin and A. Gonzalez, “Linear analysis of the 4-step Carré phase shifting algorithm: spectrum, signal-to-noise ratio, and harmonics response,” arXiv:1203.1947 (2012).

Stasiewicz, K.

A. Michałkiewicz, M. Kujawińska, and K. Stasiewicz, “Digital holographic camera and data processing for remote monitoring and measurements of mechanical parts,” Opto-Electron. Rev. 16, 68–75 (2008).
[CrossRef]

Tay, C. J.

C. Quan, W. Chen, and C. J. Tay, “Shape measurement by multi-illumination method in digital holographic interferometry,” Opt. Commun. 281, 3957–3964 (2008).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, The Art of Scientific Computing, 2nd ed. (Cambridge University, 2002).

Vest, C. M.

C. M. Vest, Holographic Interferometry (Wiley, 1979).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, The Art of Scientific Computing, 2nd ed. (Cambridge University, 2002).

Wang, X.

A. Michałkiewicz, M. Kujawińska, R. Lymarenko, O. Budryk, X. Wang, and P. J. Bos, “Simulation, registration, and reconstruction of digital holograms of arbitrary objects by means of liquid crystal on silicon spatial light modulator,” Proc. SPIE 5947, 59470G (2005).
[CrossRef]

Yamaguchi, I.

Yourassowsky, C.

Zaperty, W.

M. Kujawinska, R. Porras-Aguilar, and W. Zaperty, “LCoS spatial light modulators as active phase elements of full-field measurement systems and sensors,” Metrology and Measurement Systems 19, 445–458 (2012).
[CrossRef]

Zhang, T.

Appl. Opt. (3)

Meas. Sci. Technol. (1)

U. Schnars and W. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

Metrology and Measurement Systems (1)

M. Kujawinska, R. Porras-Aguilar, and W. Zaperty, “LCoS spatial light modulators as active phase elements of full-field measurement systems and sensors,” Metrology and Measurement Systems 19, 445–458 (2012).
[CrossRef]

Opt. Commun. (1)

C. Quan, W. Chen, and C. J. Tay, “Shape measurement by multi-illumination method in digital holographic interferometry,” Opt. Commun. 281, 3957–3964 (2008).
[CrossRef]

Opt. Lett. (1)

Opto-Electron. Rev. (1)

A. Michałkiewicz, M. Kujawińska, and K. Stasiewicz, “Digital holographic camera and data processing for remote monitoring and measurements of mechanical parts,” Opto-Electron. Rev. 16, 68–75 (2008).
[CrossRef]

Photon. Lett. Poland (1)

M. Kujawińska, G. Finke, P. Garbat, C. Falldorf, and M. B. Hennelly, “Wide angle digital holographic interferometry with real-time optical reconstruction,” Photon. Lett. Poland 4, 48–50 (2012).
[CrossRef]

Proc. SPIE (2)

Z. Füzessy, F. Gyímesi, J. Kornis, B. Ráczkevi, V. Borbély, and B. Gombkötő, “Analogue and digital development for project DISCO at Budapest University of Technology and Economics,” Proc. SPIE 5457, 610–620 (2004).
[CrossRef]

A. Michałkiewicz, M. Kujawińska, R. Lymarenko, O. Budryk, X. Wang, and P. J. Bos, “Simulation, registration, and reconstruction of digital holograms of arbitrary objects by means of liquid crystal on silicon spatial light modulator,” Proc. SPIE 5947, 59470G (2005).
[CrossRef]

Other (13)

M. Kujawinska and T. Kozacki, “Holographic television: status and future,” in Optical Imaging and Metrology: Selected Topics, W. Osten and N. Reingand, eds. (Wiley-VCH, 2012).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, The Art of Scientific Computing, 2nd ed. (Cambridge University, 2002).

G. Dahlquist and A. Björk, Equidistant Interpolation and the Runge Phenomenon, Numerical Methods (Courier Dover, 1974).

CCD Detectors—Astrosurf, www.astrosurf.com/re/chip.html .

Holoeye Photonics, AG, www.holoeye.com .

M. K. Kim, Digital Holographic Microscopy: Principles, Techniques, and Applications (Springer, 2011).

M. Servin and A. Gonzalez, “Linear analysis of the 4-step Carré phase shifting algorithm: spectrum, signal-to-noise ratio, and harmonics response,” arXiv:1203.1947 (2012).

P. Ferraro and W. Osten, “Digital holography and its application in MEMS/MOEMS inspection,” in Optical Inspection of Microsystems, W. Osten, ed. (CRC, 2006), Chap. 12.

T. Kreis, Holographic Interferometry, Principles and Methods, 1st ed., Akademie Verlag Series in Optical Metrology (Wiley-VCH, 1996).

C. M. Vest, Holographic Interferometry (Wiley, 1979).

P. K. Rastogi, Holographic Interferometry: Principles and Methods (Springer, 1994).

T. Kreis, Handbook of Holographic Interferometry—Optical and Digital Methods (Wiley, 2005).

G. Lazarev, A. Hermerschmidt, S. Kruger, and S. Osten, “LCOS spatial light modulators: trends and applications,” in Optical Imaging and Metrology: Selected Topics, W. Osten and N. Reingand, eds. (Wiley-VCH, 2012), pp. 1–29.

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Figures (15)

Fig. 1.
Fig. 1.

Schemes of real-time digital holography interferometric system in the configuration during (a) capture and (b) reconstruction.

Fig. 2.
Fig. 2.

Scheme of the algorithm for the capturing procedure at an initial state of the object.

Fig. 3.
Fig. 3.

Scheme of the reconstruction procedure. Δ1 and Δ2 are the camera and SLM pixel size, respectively. The mismatch compensation approaches A, B, C, D, and E are described in Section 3.B.

Fig. 4.
Fig. 4.

Plot of the horizontal cross section of the wavefront deformation due to the mismatch between the capture and reconstruction systems for (a) F=0.75, (b) F=1.08, and (c) F=2.7 at z1=z2=210mm. [Please note that the scale at the x axis in plot (c) is different].

Fig. 5.
Fig. 5.

Phase shift induced when the reconstruction distance is increased by Δz=z2z1. This term can be removed before the optical field is propagated.

Fig. 6.
Fig. 6.

Cross section of the reconstructed phase of a tilted plane (2.5 fringes) at the object plane. R1 reconstruction for Δ1=Δ2=7.4μm. A, B, C, D, and E are the reconstructions for the evaluated approaches for Δ2=8μm.

Fig. 7.
Fig. 7.

Cross section of the reconstruction R1 and approaches A, B, C, D, and E for 2.5 fringes tilted mirror having a noise level of 4π peak to valley. The other simulation parameters are the same as in Fig. 4.

Fig. 8.
Fig. 8.

Significance level of similarity calculated by the K-S test for the evaluation of the compensation algorithms for a linear object phase.

Fig. 9.
Fig. 9.

Cross section of the reconstruction of an arbitrary continuous function for the case of no noise. As in Fig. 4, R1 reconstruction for Δ1=Δ2=7.4μm. A, B, C, D, and E are the reconstructions for the evaluated approaches for Δ2=8μm.

Fig. 10.
Fig. 10.

Cross section of the reconstruction of an arbitrary continuous function as in Fig. 8 but having a noise level of 4π peak to valley.

Fig. 11.
Fig. 11.

Significance level of similarity calculated by the K-S test for the various compensating algorithms using an arbitrary object phase distribution.

Fig. 12.
Fig. 12.

Phase difference between the initial and loaded state at the camera plane (1024×1024pixels).

Fig. 13.
Fig. 13.

Numerical reconstruction and compensation of experimental phase objects.

Fig. 14.
Fig. 14.

Intensity distribution of the interference between an experimental object wavefront and the reconstruction obtained by approach A. In the plot the average of this pattern in the direction x is shown.

Fig. 15.
Fig. 15.

Intensity distribution of the interference between an interpolated experimental object wavefront and the reconstruction obtained by approach E. In the plot the average of this pattern in the direction x is shown.

Tables (1)

Tables Icon

Table 1. Requirements, Advantages, and Disadvantages of the Compensation Approaches

Equations (18)

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

Er(x,y,z)=E^r(x,y,z)eiθr(x,y,z),
Eo1(x,y,z)=E^o1(x,y,z)eiθo1(x,y,z),
Ern(x,y,z)=E^r(x,y,z)eiθr(x,y,z)+nα,
In(x,y,z1)=Ian+Ibncos(θo1(x,y,z1)+nα),
hn(x,y)=o(x,y)+βτIn(x,y),
θo1CCD(x,y)=tan1[(h1h2)(h0h3)]·(h1h2+h0h3)h1h2+h0h3.
Eo1LCoS(x¯,y¯,z1)=Er(x¯,y¯,z)eiθo1LCoS(x¯,y¯,z1).
Eo2(x,y,0)=E^o2(x,y,0)eiθo2(x,y,0).
I(x,y,z)=Ic+Idcos(Δθ(x,y,z)),
z2=Δ12Δ22z1,
U(x¯,y¯)=exp(iθ01LCoS),
U(x¯,y¯)=exp(iθ01LCoSikΔz).
UΔz(x¯,y¯)=Propagated[exp(iθ01LCoSikΔz)].
θo1LCoS(x¯,y¯)=Interpolation[θo1CCD(x,y)].
θo1CCD=sqrt[Re(UCCD)2+Im(UCCD)2].
In=1+cos(θ+nα),n=0,3.
I=2|F1||F2|·cos(φ1φ2)+|F1|2+|F2|2,
I=2|F1||F2|·cos(φ1φ2)+|F1|2+|F2|2,

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