Abstract

Behind periodic amplitude or phase objects, the object transmittance is repeated at the so-called Talbot distances. In these planes perpendicular to the propagation direction, Talbot self-images are formed. In the case of plane wave illumination, the distances between the self-images are equally spaced. A periodic pattern called optical carpet or Talbot carpet is formed along the propagation direction. We show theoretically how the presence of spherical particles (10 to 100 μm in diameter) behind gratings of 20 and 50 μm period affects the formation of Talbot carpets and Talbot self-images at 633 nm illumination wavelength. The scattering of the particles is modeled by the Fresnel diffraction of its geometrical shadow. We analytically calculate the interference of the diffraction orders of rectangular and sinusoidal amplitude gratings disturbed by the presence of particles. To verify our model, we present measurements of Talbot carpets perturbed with both opaque disks and transparent spheres, and discuss the effects for various size parameters. We present an approach to simulate the movement of particles within the Talbot pattern in real time. We simulate and measure axial and lateral particle movements within a probe volume and evaluate the effect on the signal formation in a Talbot interferometric setup. We evaluate the best system parameters in terms of grating period and particle-detector-distance for a prospective measuring setup to determine characteristics of flowing suspensions, such as particle volume concentration or particle size distribution.

© 2012 Optical Society of America

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References

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2011 (1)

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5d imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid Pub. 6, 11034 (2011).
[CrossRef]

2010 (2)

M. Bech, T. H. Jensen, O. Bunk, T. Donath, C. David, T. Weitkamp, G. Le Duc, A. Bravin, P. Cloetens, and F. Pfeiffer, “Advanced contrast modalities for x-ray radiology: phase-contrast and dark-field imaging using a grating interferometer,” Z. Med. Phys. 20, 7–16 (2010).

M. Hofmann, X. Ma, J. Schneider, and S. Sinzinger, “Highly integrated optical microsystem for particle concentration measurement,” Proc. SPIE 7716, 77160T (2010).
[CrossRef]

2009 (1)

2008 (1)

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grunzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nature Materials 7, 134–137 (2008).
[CrossRef]

2007 (3)

H. Knuppertz, J. Jahns, and R. Grunwald, “Temporal impulse response of the Talbot interferometer,” Opt. Commun. 277, 67–73 (2007).
[CrossRef]

D. Chicea, “Biospeckle size and contrast measurement application in particle sizing and concentration assessment,” Rom. J. Phys. 52, 625–632 (2007).

I. Ayranci, G. Pinguet, D. Escudié, N. Selcuk, R. Vaillon, and F. Andre, “Effect of particle polydispersity on particle concentration measurement by using laser Doppler anemometry,” Exp. Therm. Fluid. Sci. 31, 839–847 (2007).
[CrossRef]

2006 (1)

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

1999 (1)

M. Breitenstein, U. Kräuter, and U. Riebel, “The fundamentals of particle size analysis by transmission fluctuation spectrometry,” Part. Part. Syst. Charact. 16, 249–256 (1999).
[CrossRef]

1998 (1)

1997 (1)

1996 (1)

B. Wessely, J. Altmann, and S. Ripperger, “The use of statistical properties of transmission signals for particle characterization,” Chem. Eng. Technol. 19, 438–442 (1996).
[CrossRef]

1992 (1)

H.-H. Qiu and M. Sommerfeld, “A reliable method for determining the measurement volume size and particle mass fluxes using phase-Doppler anemometry,” Exp. Fluids 13, 393–404 (1992).
[CrossRef]

1990 (2)

1989 (1)

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1–108 (1989).
[CrossRef]

1985 (1)

J. Gregory, “Turbidity fluctuations in flowing suspensions,” J. Colloid Interface Sci. 105, 357–371 (1985).
[CrossRef]

1981 (1)

1980 (2)

O. Gencelia, J. Schemm, and C. Vest, “Measurement of size and concentration of scattering particles by speckle photography,” J. Opt. Soc. Am. 70, 1212–1218 (1980).
[CrossRef]

R. Zollars, “Turbidimetric method for on-line determination of latex particle number and particle size distribution,” J. Colloid Interface Sci. 74, 163–172 (1980).
[CrossRef]

1971 (1)

A. Lohmann and D. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971).
[CrossRef]

1881 (1)

L. Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag. 11, 196 (1881).

1836 (1)

W. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–407 (1836).

Altmann, J.

B. Wessely, J. Altmann, and S. Ripperger, “The use of statistical properties of transmission signals for particle characterization,” Chem. Eng. Technol. 19, 438–442 (1996).
[CrossRef]

Andre, F.

I. Ayranci, G. Pinguet, D. Escudié, N. Selcuk, R. Vaillon, and F. Andre, “Effect of particle polydispersity on particle concentration measurement by using laser Doppler anemometry,” Exp. Therm. Fluid. Sci. 31, 839–847 (2007).
[CrossRef]

Arndt, M.

Ayranci, I.

I. Ayranci, G. Pinguet, D. Escudié, N. Selcuk, R. Vaillon, and F. Andre, “Effect of particle polydispersity on particle concentration measurement by using laser Doppler anemometry,” Exp. Therm. Fluid. Sci. 31, 839–847 (2007).
[CrossRef]

Bech, M.

M. Bech, T. H. Jensen, O. Bunk, T. Donath, C. David, T. Weitkamp, G. Le Duc, A. Bravin, P. Cloetens, and F. Pfeiffer, “Advanced contrast modalities for x-ray radiology: phase-contrast and dark-field imaging using a grating interferometer,” Z. Med. Phys. 20, 7–16 (2010).

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grunzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nature Materials 7, 134–137 (2008).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Bravin, A.

M. Bech, T. H. Jensen, O. Bunk, T. Donath, C. David, T. Weitkamp, G. Le Duc, A. Bravin, P. Cloetens, and F. Pfeiffer, “Advanced contrast modalities for x-ray radiology: phase-contrast and dark-field imaging using a grating interferometer,” Z. Med. Phys. 20, 7–16 (2010).

Breitenstein, M.

M. Breitenstein, U. Kräuter, and U. Riebel, “The fundamentals of particle size analysis by transmission fluctuation spectrometry,” Part. Part. Syst. Charact. 16, 249–256 (1999).
[CrossRef]

Bronnimann, C.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grunzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nature Materials 7, 134–137 (2008).
[CrossRef]

Bunk, O.

M. Bech, T. H. Jensen, O. Bunk, T. Donath, C. David, T. Weitkamp, G. Le Duc, A. Bravin, P. Cloetens, and F. Pfeiffer, “Advanced contrast modalities for x-ray radiology: phase-contrast and dark-field imaging using a grating interferometer,” Z. Med. Phys. 20, 7–16 (2010).

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grunzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nature Materials 7, 134–137 (2008).
[CrossRef]

Case, W.

Chicea, D.

D. Chicea, “Biospeckle size and contrast measurement application in particle sizing and concentration assessment,” Rom. J. Phys. 52, 625–632 (2007).

Cloetens, P.

M. Bech, T. H. Jensen, O. Bunk, T. Donath, C. David, T. Weitkamp, G. Le Duc, A. Bravin, P. Cloetens, and F. Pfeiffer, “Advanced contrast modalities for x-ray radiology: phase-contrast and dark-field imaging using a grating interferometer,” Z. Med. Phys. 20, 7–16 (2010).

David, C.

M. Bech, T. H. Jensen, O. Bunk, T. Donath, C. David, T. Weitkamp, G. Le Duc, A. Bravin, P. Cloetens, and F. Pfeiffer, “Advanced contrast modalities for x-ray radiology: phase-contrast and dark-field imaging using a grating interferometer,” Z. Med. Phys. 20, 7–16 (2010).

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grunzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nature Materials 7, 134–137 (2008).
[CrossRef]

Deachapunya, S.

Donath, T.

M. Bech, T. H. Jensen, O. Bunk, T. Donath, C. David, T. Weitkamp, G. Le Duc, A. Bravin, P. Cloetens, and F. Pfeiffer, “Advanced contrast modalities for x-ray radiology: phase-contrast and dark-field imaging using a grating interferometer,” Z. Med. Phys. 20, 7–16 (2010).

Eikenberry, E. F.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grunzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nature Materials 7, 134–137 (2008).
[CrossRef]

Escudié, D.

I. Ayranci, G. Pinguet, D. Escudié, N. Selcuk, R. Vaillon, and F. Andre, “Effect of particle polydispersity on particle concentration measurement by using laser Doppler anemometry,” Exp. Therm. Fluid. Sci. 31, 839–847 (2007).
[CrossRef]

Gencelia, O.

Goodman, J.

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

Gregory, J.

J. Gregory, “Turbidity fluctuations in flowing suspensions,” J. Colloid Interface Sci. 105, 357–371 (1985).
[CrossRef]

Grunwald, R.

H. Knuppertz, J. Jahns, and R. Grunwald, “Temporal impulse response of the Talbot interferometer,” Opt. Commun. 277, 67–73 (2007).
[CrossRef]

Grunzweig, C.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grunzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nature Materials 7, 134–137 (2008).
[CrossRef]

Healy, J. J.

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5d imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid Pub. 6, 11034 (2011).
[CrossRef]

Hennelly, B. M.

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5d imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid Pub. 6, 11034 (2011).
[CrossRef]

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Hofmann, M.

M. Hofmann, X. Ma, J. Schneider, and S. Sinzinger, “Highly integrated optical microsystem for particle concentration measurement,” Proc. SPIE 7716, 77160T (2010).
[CrossRef]

Jahns, J.

H. Knuppertz, J. Jahns, and R. Grunwald, “Temporal impulse response of the Talbot interferometer,” Opt. Commun. 277, 67–73 (2007).
[CrossRef]

M. Testorf and J. Jahns, “Planar-integrated Talbot array illuminators,” Appl. Opt. 37, 5399–5407 (1998).
[CrossRef]

Jensen, T. H.

M. Bech, T. H. Jensen, O. Bunk, T. Donath, C. David, T. Weitkamp, G. Le Duc, A. Bravin, P. Cloetens, and F. Pfeiffer, “Advanced contrast modalities for x-ray radiology: phase-contrast and dark-field imaging using a grating interferometer,” Z. Med. Phys. 20, 7–16 (2010).

Kelly, D. P.

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5d imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid Pub. 6, 11034 (2011).
[CrossRef]

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light (Academic, 1969).

Knuppertz, H.

H. Knuppertz, J. Jahns, and R. Grunwald, “Temporal impulse response of the Talbot interferometer,” Opt. Commun. 277, 67–73 (2007).
[CrossRef]

Kourti, T.

T. Kourti, “Turbidimetry in particle size analysis,” in Encyclopedia of Analytical Chemistry: Instrumentation and Applications (Wiley, 2000).

Kraft, P.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grunzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nature Materials 7, 134–137 (2008).
[CrossRef]

Kräuter, U.

M. Breitenstein, U. Kräuter, and U. Riebel, “The fundamentals of particle size analysis by transmission fluctuation spectrometry,” Part. Part. Syst. Charact. 16, 249–256 (1999).
[CrossRef]

Le Duc, G.

M. Bech, T. H. Jensen, O. Bunk, T. Donath, C. David, T. Weitkamp, G. Le Duc, A. Bravin, P. Cloetens, and F. Pfeiffer, “Advanced contrast modalities for x-ray radiology: phase-contrast and dark-field imaging using a grating interferometer,” Z. Med. Phys. 20, 7–16 (2010).

Lohmann, A.

A. Lohmann and D. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971).
[CrossRef]

Lohmann, A. W.

Ma, X.

M. Hofmann, X. Ma, J. Schneider, and S. Sinzinger, “Highly integrated optical microsystem for particle concentration measurement,” Proc. SPIE 7716, 77160T (2010).
[CrossRef]

Nefedov, A.

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1–108 (1989).
[CrossRef]

Petrov, O.

Pfeiffer, F.

M. Bech, T. H. Jensen, O. Bunk, T. Donath, C. David, T. Weitkamp, G. Le Duc, A. Bravin, P. Cloetens, and F. Pfeiffer, “Advanced contrast modalities for x-ray radiology: phase-contrast and dark-field imaging using a grating interferometer,” Z. Med. Phys. 20, 7–16 (2010).

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grunzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nature Materials 7, 134–137 (2008).
[CrossRef]

Pinguet, G.

I. Ayranci, G. Pinguet, D. Escudié, N. Selcuk, R. Vaillon, and F. Andre, “Effect of particle polydispersity on particle concentration measurement by using laser Doppler anemometry,” Exp. Therm. Fluid. Sci. 31, 839–847 (2007).
[CrossRef]

Qiu, H.-H.

H.-H. Qiu and M. Sommerfeld, “A reliable method for determining the measurement volume size and particle mass fluxes using phase-Doppler anemometry,” Exp. Fluids 13, 393–404 (1992).
[CrossRef]

Rayleigh, L.

L. Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag. 11, 196 (1881).

Rhodes, W. T.

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Riebel, U.

M. Breitenstein, U. Kräuter, and U. Riebel, “The fundamentals of particle size analysis by transmission fluctuation spectrometry,” Part. Part. Syst. Charact. 16, 249–256 (1999).
[CrossRef]

Ripperger, S.

B. Wessely, J. Altmann, and S. Ripperger, “The use of statistical properties of transmission signals for particle characterization,” Chem. Eng. Technol. 19, 438–442 (1996).
[CrossRef]

Schemm, J.

Schneider, J.

M. Hofmann, X. Ma, J. Schneider, and S. Sinzinger, “Highly integrated optical microsystem for particle concentration measurement,” Proc. SPIE 7716, 77160T (2010).
[CrossRef]

Selcuk, N.

I. Ayranci, G. Pinguet, D. Escudié, N. Selcuk, R. Vaillon, and F. Andre, “Effect of particle polydispersity on particle concentration measurement by using laser Doppler anemometry,” Exp. Therm. Fluid. Sci. 31, 839–847 (2007).
[CrossRef]

Sheridan, J. T.

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5d imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid Pub. 6, 11034 (2011).
[CrossRef]

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Silva, D.

A. Lohmann and D. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971).
[CrossRef]

Sinzinger, S.

M. Hofmann, X. Ma, J. Schneider, and S. Sinzinger, “Highly integrated optical microsystem for particle concentration measurement,” Proc. SPIE 7716, 77160T (2010).
[CrossRef]

Sommargren, G. E.

Sommerfeld, M.

H.-H. Qiu and M. Sommerfeld, “A reliable method for determining the measurement volume size and particle mass fluxes using phase-Doppler anemometry,” Exp. Fluids 13, 393–404 (1992).
[CrossRef]

Southwell, W. H.

Talbot, W.

W. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–407 (1836).

Testorf, M.

Thomas, J. A.

Tomandl, M.

Vaillon, R.

I. Ayranci, G. Pinguet, D. Escudié, N. Selcuk, R. Vaillon, and F. Andre, “Effect of particle polydispersity on particle concentration measurement by using laser Doppler anemometry,” Exp. Therm. Fluid. Sci. 31, 839–847 (2007).
[CrossRef]

van de Hulst, H.

H. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

Vaulina, O.

Vest, C.

Watson, G.

G. Watson, A Treatise on the Theory of Bessel Functions (Cambridge, 1922).

Weaver, H. J.

Weitkamp, T.

M. Bech, T. H. Jensen, O. Bunk, T. Donath, C. David, T. Weitkamp, G. Le Duc, A. Bravin, P. Cloetens, and F. Pfeiffer, “Advanced contrast modalities for x-ray radiology: phase-contrast and dark-field imaging using a grating interferometer,” Z. Med. Phys. 20, 7–16 (2010).

Wessely, B.

B. Wessely, J. Altmann, and S. Ripperger, “The use of statistical properties of transmission signals for particle characterization,” Chem. Eng. Technol. 19, 438–442 (1996).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Zollars, R.

R. Zollars, “Turbidimetric method for on-line determination of latex particle number and particle size distribution,” J. Colloid Interface Sci. 74, 163–172 (1980).
[CrossRef]

Appl. Opt. (4)

Chem. Eng. Technol. (1)

B. Wessely, J. Altmann, and S. Ripperger, “The use of statistical properties of transmission signals for particle characterization,” Chem. Eng. Technol. 19, 438–442 (1996).
[CrossRef]

Exp. Fluids (1)

H.-H. Qiu and M. Sommerfeld, “A reliable method for determining the measurement volume size and particle mass fluxes using phase-Doppler anemometry,” Exp. Fluids 13, 393–404 (1992).
[CrossRef]

Exp. Therm. Fluid. Sci. (1)

I. Ayranci, G. Pinguet, D. Escudié, N. Selcuk, R. Vaillon, and F. Andre, “Effect of particle polydispersity on particle concentration measurement by using laser Doppler anemometry,” Exp. Therm. Fluid. Sci. 31, 839–847 (2007).
[CrossRef]

J. Colloid Interface Sci. (2)

R. Zollars, “Turbidimetric method for on-line determination of latex particle number and particle size distribution,” J. Colloid Interface Sci. 74, 163–172 (1980).
[CrossRef]

J. Gregory, “Turbidity fluctuations in flowing suspensions,” J. Colloid Interface Sci. 105, 357–371 (1985).
[CrossRef]

J. Eur. Opt. Soc. Rapid Pub. (1)

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5d imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid Pub. 6, 11034 (2011).
[CrossRef]

J. Opt. Soc. Am. (2)

Nature Materials (1)

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grunzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nature Materials 7, 134–137 (2008).
[CrossRef]

Opt. Commun. (2)

H. Knuppertz, J. Jahns, and R. Grunwald, “Temporal impulse response of the Talbot interferometer,” Opt. Commun. 277, 67–73 (2007).
[CrossRef]

A. Lohmann and D. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971).
[CrossRef]

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D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Opt. Express (1)

Part. Part. Syst. Charact. (1)

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[CrossRef]

Philos. Mag. (2)

W. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–407 (1836).

L. Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag. 11, 196 (1881).

Proc. SPIE (1)

M. Hofmann, X. Ma, J. Schneider, and S. Sinzinger, “Highly integrated optical microsystem for particle concentration measurement,” Proc. SPIE 7716, 77160T (2010).
[CrossRef]

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[CrossRef]

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Figures (12)

Fig. 1.
Fig. 1.

Schematic of the measuring setup.

Fig. 2.
Fig. 2.

(a) Scattering functions for different values of size parameter α. (b) Far field intensity distribution calculated with diffraction and Mie theory (spheres with radius 50 μm with different refractive indices in water (n=1.33) at 633 nm wavelength).

Fig. 3.
Fig. 3.

Diffraction geometry.

Fig. 4.
Fig. 4.

Phase and squared amplitude of a 100 μm opaque sphere.

Fig. 5.
Fig. 5.

Simulation of Talbot carpets (squared amplitude) behind a sine grating (left column), and a rectangular grating (right column). (a) and (b) show the undisturbed carpet. In (c) and (d), a particle is inserted at z=0; in (e) and (f), the particle is shifted in z, and in (g) and (h), the particle is shifted in z and x.

Fig. 6.
Fig. 6.

Calculation scheme. (a) Example of the phases of the first order, and (b) the resulting field considering the three orders of a sinusoidal grating.

Fig. 7.
Fig. 7.

Setup for the measurement of Talbot carpets.

Fig. 8.
Fig. 8.

Measurement and simulation of the disturbed Talbot carpet (scenario 1). (a) Sphere, (b) disk, and (c) simulation.

Fig. 9.
Fig. 9.

Measurement and simulation of the disturbed Talbot carpet (scenario 2). (a) Sphere, (b) disk, and (c) simulation.

Fig. 10.
Fig. 10.

(a) Undisturbed, and (b) disturbed self-image.

Fig. 11.
Fig. 11.

(a) Normalized detector signal at 2.5zT while the particle moves towards it, (b) maximum intensity, and (c) particle-detector-distance.

Fig. 12.
Fig. 12.

Simulation and measurement of the detector signal for (a) an axial particle movement, and (b) a lateral particle movement (z=6.5mm). Grating period 50 μm, particle diameter 100 μm.

Tables (2)

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Table 1. Scenarios of Measurement

Tables Icon

Table 2. Detector Parameters

Equations (22)

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Up(x,y,z)=exp(jkz)jλzU0(x0,y0)exp{jπλz[(xx0)2+(yy0)2]}dx0dy0.
Up(ρ,φ,z)=exp(jkz)jλzexp(jπλzρ2)02πaexp(jπλzR2)exp[j2πλzρRcos(φθ)]RdRdθ.
J0(a)=12π02πexp[jacos(φθ)]dθ,
Up(ρ,z)=kjzexp(jkz)exp(jk2zρ2)aexp(jk2zR2)J0(kzρR)RdR.
Up(ρ,z)=1juejkzexp(jk2zρ2)1exp(jut22)J0(vt)tdt.
Un(u,v)=m=0(1)m(uv)n+2mJn+2m(v),
Vn(u,v)=m=0(1)m(vu)n+2mJn+2m(v).
Up(|ρ|<a,z)=exp(jkz)exp(jk2zρ2)exp(ju2)[V0(u,v)jV1(u,v)]
Up(|ρ|>a,z)=exp(jkz)exp(jk2zρ2)×[exp(jv22u)+jexp(ju2)U1(u,v)+exp(ju2)U2(u,v)]
Up(ρ=a,z)=exp(jkz)exp(jk2za2)exp(ju2){12[J0(v)+cos(v)]12jsin(v)},
Up(ρ=0,z)=exp(jkz)exp(ju2).
T(x)=ncnexp(2πjnνx).
U(x,z)=exp(jkz)jλzT(x0)exp[jπλz(xx0)2]dx0.
Ug(x,z)=cnexp{jπλz[x2(xnλνz)2]}×{exp(jkz)jλzexp[jπλz(x0x+nλνz)2]dx0}.
Ug(x,z)=cnexp{jπλz[x2(xnλνz)2]}=ncnexp[jπ(nν)2λz]exp(2πjnνx).
zT=2p2λ,
Ug(x,d)=ncnexp[jπ(nν)2λd]exp(2πjnνx)
U(x,z)=exp(jkz)jλzncnexp[jπ(nν)2λd]exp(2πjnνx0)exp[jπλz(xx0)2]dx0.
U(x,z)=ncnexp[jπ(nν)2λd]exp(jkz)jλzexp(2πjnνx0)exp[jπλz(xx0)2]dx0=ncnexp[jπ(nν)2λd]exp[jπ(nν)2λz]exp(2πjnνx)×exp(jkz)jλzexp[jπλz(x0x+nλνz)2]dx0
U(x,z)=nexp[jπ(nν)2λd]{cnexp[jπ(nν)2λz]exp(2πjnνx)}×(exp(jkz)jλzexp{jπλz[(xnλνz)x0]2}dx0).
U(x,z)=n=77Φn(d)Ugn(x,z)Upn(xnλνz,z).
U(x,z)=n=77Ugn(x,z+d)Upn(xnλνz,z).

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