Abstract

We propose using multiple superimposed noninterfering probes (SNIPs) of the same wavelength but different beam angles to extend the capabilities of phase Doppler anemometry. When a particle is moving in a SNIP the Doppler signals that are produced exhibit multiple Doppler frequencies and phase shifts. The resolution of the measurements of particle size (i.e., by fringe spacing and Doppler frequency) increases with beam angle. Then, with the solution proposed, even with only two detectors several measurements of size can be obtained for the same particle with increasing resolution if we consider higher frequencies in the signal. Several optical solutions to produce SNIPs as well as a signal-processing algorithm to treat the multiple-frequency Doppler signals are proposed. Experimental validations of the sizing of spherical and cylindrical particles demonstrate the applicability of this technique for particle measurement. We believe that this new technique can be of great interest when high resolution of size, velocity, and even refractive index is required.

© 2002 Optical Society of America

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References

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  1. F. Durst, M. Zaré, “Laser Doppler measurements in two-phase flows,” in The Accuracy of Flow Measurements by Laser Doppler Methods—Proceedings of LDA-Symposium, Copenhagen, Denmark, 1975, P. Buchhave, J. M. Delhaye, F. Durst, W. K. George, K. Refslund, J. H. Whitelaw, eds. (Proceedings LDA-Symposium, Skovlunde, Denmark, 1976), pp. 403–429.
  2. W. D. Bachalo, M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).
    [CrossRef]
  3. K. Bauckhage, H. H. Floegel, U. Fritsching, R. Hiller, “The phase Doppler difference method, a new laser Doppler technique for simultaneous size and velocity measurements. 2. Optical particle characteristics as a base for a new diagnostic technique,” Part. Part. Syst. Charact. 5, 66–71 (1988).
    [CrossRef]
  4. E. D. Hirleman, “History of development of the phase Doppler-sizing velocimeter,” Part. Part. Syst. Charact. 13, 59–67 (1996).
    [CrossRef]
  5. A. Naqwi, F. Durst, “Light scattering applied to LDA and PDA measurements. I. Theory and numerical treatments,” Part. Part. Syst. Charact. 8, 245–258 (1991).
    [CrossRef]
  6. G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Particle trajectory effects in phase Doppler systems: computations and experiments,” Part. Part. Syst. Charact. 10, 332–338 (1993).
    [CrossRef]
  7. R. J. Adrian, R. J. Goldstein, “Analysis of a laser-Doppler anemometer,” J. Phys. E 4, 505–511 (1971).
    [CrossRef]
  8. F. Durst, A. Melling, J. H. Whitelaw, Principles and Practice of Laser-Doppler Anemometry (Academic, London, 1981).
  9. W. Lauterborn, T. Kurz, M. Wiesenfeldt, Optique Cohérente (Masson, Paris, 1997).
  10. S. G. Lipson, H. Lipson, D. S. Tannhauser, Optical Physics (Cambridge U. Press, Cambridge, 1995).
    [CrossRef]
  11. J. Domnick, H. Ertel, C. Tropea, “Processing of phase-Doppler signals using the cross-spectral density function,” in Proceedings of the Third European Symposium on Particle Characterization, E. H. K. Leschonski, ed. (NurnbergMesse, Nurnberg, Germany, 1988), pp. 473–483.
  12. F. Onofri, L. Bergounoux, J.-L. Firpo, J. Mesguish-Ripault, “Velocity, size, and concentration measurements of optically inhomogeneous cylindrical and spherical particles,” Appl. Opt. 38, 4681–4690 (1999).
    [CrossRef]
  13. F. Onofri, A. Lenoble, “Sizing of single fibers under torsional stress,” in Proceedings of the Sixth International Congress on Optical Particle Characterization, A. R. Jones, ed. (Institute of Physics, Bristol, UK, 2001), paper 7.7.
  14. W. M. Farmer, “Measurement of particle size, number, density, and velocity using a laser interferometer,” Appl. Opt. 11, 2603–2609 (1972).
    [CrossRef] [PubMed]
  15. TSI Incorporated, 500 Cardigan Road, Saint Paul, Minn. 55126.
  16. F. Onofri, “Prise en compte de la dimension finie des faisceaux d’éclairage en granulométrie optique: anémométrie phase Doppler—diagnostics des milieux diphasiques,” Ph.D. dissertation (Université de Rouen, Rouen, France, 1995).
  17. F. Onofri, G. Gréhan, G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113–7124 (1995).
    [CrossRef] [PubMed]
  18. M. Mischenko, “Shape and Structure,” in Proceedings of the Sixth International Congress on Optical Particle Characterization, A. R. Jones, ed. (Institute of Physics, Bristol, UK, 2001), paper 1.1.
  19. A. A. Naqwi, R. P. A. Hartman, J. C. M. Marijnissen, “Basic studies of electrohydrodynamic atomization using phase Doppler measurement technique, “Part. Part. Syst. Charact. 13, 143–149 (1996).
    [CrossRef]
  20. A. Naqwi, “Sizing of irregular particles using a phase Doppler system,” Part. Part. Syst. Charact. 8, 343–349 (1996).
    [CrossRef]
  21. F. Onofri, “Etude numérique et expérimentale de la sensibilité de l’interfrométrie phase Doppler à l’état de surface des particles détectées,” in Proceedings of the Septième Congrés Francophone de Vélocimétrie Laser, M. Elena, ed. (Université de Provence, Marseille, France, 2000), pp. 335–342.
  22. C. Tropea, T.-H. Xu, F. Onofri, G. Gréhan, P. Haugen, “Dual-mode phase Doppler anemometer,” Part. Part. Syst. Charact. 13, 165–170 (1995).
    [CrossRef]

1999

1996

E. D. Hirleman, “History of development of the phase Doppler-sizing velocimeter,” Part. Part. Syst. Charact. 13, 59–67 (1996).
[CrossRef]

A. A. Naqwi, R. P. A. Hartman, J. C. M. Marijnissen, “Basic studies of electrohydrodynamic atomization using phase Doppler measurement technique, “Part. Part. Syst. Charact. 13, 143–149 (1996).
[CrossRef]

A. Naqwi, “Sizing of irregular particles using a phase Doppler system,” Part. Part. Syst. Charact. 8, 343–349 (1996).
[CrossRef]

1995

C. Tropea, T.-H. Xu, F. Onofri, G. Gréhan, P. Haugen, “Dual-mode phase Doppler anemometer,” Part. Part. Syst. Charact. 13, 165–170 (1995).
[CrossRef]

F. Onofri, G. Gréhan, G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113–7124 (1995).
[CrossRef] [PubMed]

1993

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Particle trajectory effects in phase Doppler systems: computations and experiments,” Part. Part. Syst. Charact. 10, 332–338 (1993).
[CrossRef]

1991

A. Naqwi, F. Durst, “Light scattering applied to LDA and PDA measurements. I. Theory and numerical treatments,” Part. Part. Syst. Charact. 8, 245–258 (1991).
[CrossRef]

1988

K. Bauckhage, H. H. Floegel, U. Fritsching, R. Hiller, “The phase Doppler difference method, a new laser Doppler technique for simultaneous size and velocity measurements. 2. Optical particle characteristics as a base for a new diagnostic technique,” Part. Part. Syst. Charact. 5, 66–71 (1988).
[CrossRef]

1984

W. D. Bachalo, M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).
[CrossRef]

1972

1971

R. J. Adrian, R. J. Goldstein, “Analysis of a laser-Doppler anemometer,” J. Phys. E 4, 505–511 (1971).
[CrossRef]

Adrian, R. J.

R. J. Adrian, R. J. Goldstein, “Analysis of a laser-Doppler anemometer,” J. Phys. E 4, 505–511 (1971).
[CrossRef]

Bachalo, W. D.

W. D. Bachalo, M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).
[CrossRef]

Bauckhage, K.

K. Bauckhage, H. H. Floegel, U. Fritsching, R. Hiller, “The phase Doppler difference method, a new laser Doppler technique for simultaneous size and velocity measurements. 2. Optical particle characteristics as a base for a new diagnostic technique,” Part. Part. Syst. Charact. 5, 66–71 (1988).
[CrossRef]

Bergounoux, L.

Domnick, J.

J. Domnick, H. Ertel, C. Tropea, “Processing of phase-Doppler signals using the cross-spectral density function,” in Proceedings of the Third European Symposium on Particle Characterization, E. H. K. Leschonski, ed. (NurnbergMesse, Nurnberg, Germany, 1988), pp. 473–483.

Durst, F.

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Particle trajectory effects in phase Doppler systems: computations and experiments,” Part. Part. Syst. Charact. 10, 332–338 (1993).
[CrossRef]

A. Naqwi, F. Durst, “Light scattering applied to LDA and PDA measurements. I. Theory and numerical treatments,” Part. Part. Syst. Charact. 8, 245–258 (1991).
[CrossRef]

F. Durst, A. Melling, J. H. Whitelaw, Principles and Practice of Laser-Doppler Anemometry (Academic, London, 1981).

F. Durst, M. Zaré, “Laser Doppler measurements in two-phase flows,” in The Accuracy of Flow Measurements by Laser Doppler Methods—Proceedings of LDA-Symposium, Copenhagen, Denmark, 1975, P. Buchhave, J. M. Delhaye, F. Durst, W. K. George, K. Refslund, J. H. Whitelaw, eds. (Proceedings LDA-Symposium, Skovlunde, Denmark, 1976), pp. 403–429.

Ertel, H.

J. Domnick, H. Ertel, C. Tropea, “Processing of phase-Doppler signals using the cross-spectral density function,” in Proceedings of the Third European Symposium on Particle Characterization, E. H. K. Leschonski, ed. (NurnbergMesse, Nurnberg, Germany, 1988), pp. 473–483.

Farmer, W. M.

Firpo, J.-L.

Floegel, H. H.

K. Bauckhage, H. H. Floegel, U. Fritsching, R. Hiller, “The phase Doppler difference method, a new laser Doppler technique for simultaneous size and velocity measurements. 2. Optical particle characteristics as a base for a new diagnostic technique,” Part. Part. Syst. Charact. 5, 66–71 (1988).
[CrossRef]

Fritsching, U.

K. Bauckhage, H. H. Floegel, U. Fritsching, R. Hiller, “The phase Doppler difference method, a new laser Doppler technique for simultaneous size and velocity measurements. 2. Optical particle characteristics as a base for a new diagnostic technique,” Part. Part. Syst. Charact. 5, 66–71 (1988).
[CrossRef]

Goldstein, R. J.

R. J. Adrian, R. J. Goldstein, “Analysis of a laser-Doppler anemometer,” J. Phys. E 4, 505–511 (1971).
[CrossRef]

Gouesbet, G.

F. Onofri, G. Gréhan, G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113–7124 (1995).
[CrossRef] [PubMed]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Particle trajectory effects in phase Doppler systems: computations and experiments,” Part. Part. Syst. Charact. 10, 332–338 (1993).
[CrossRef]

Gréhan, G.

F. Onofri, G. Gréhan, G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113–7124 (1995).
[CrossRef] [PubMed]

C. Tropea, T.-H. Xu, F. Onofri, G. Gréhan, P. Haugen, “Dual-mode phase Doppler anemometer,” Part. Part. Syst. Charact. 13, 165–170 (1995).
[CrossRef]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Particle trajectory effects in phase Doppler systems: computations and experiments,” Part. Part. Syst. Charact. 10, 332–338 (1993).
[CrossRef]

Hartman, R. P. A.

A. A. Naqwi, R. P. A. Hartman, J. C. M. Marijnissen, “Basic studies of electrohydrodynamic atomization using phase Doppler measurement technique, “Part. Part. Syst. Charact. 13, 143–149 (1996).
[CrossRef]

Haugen, P.

C. Tropea, T.-H. Xu, F. Onofri, G. Gréhan, P. Haugen, “Dual-mode phase Doppler anemometer,” Part. Part. Syst. Charact. 13, 165–170 (1995).
[CrossRef]

Hiller, R.

K. Bauckhage, H. H. Floegel, U. Fritsching, R. Hiller, “The phase Doppler difference method, a new laser Doppler technique for simultaneous size and velocity measurements. 2. Optical particle characteristics as a base for a new diagnostic technique,” Part. Part. Syst. Charact. 5, 66–71 (1988).
[CrossRef]

Hirleman, E. D.

E. D. Hirleman, “History of development of the phase Doppler-sizing velocimeter,” Part. Part. Syst. Charact. 13, 59–67 (1996).
[CrossRef]

Houser, M. J.

W. D. Bachalo, M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).
[CrossRef]

Kurz, T.

W. Lauterborn, T. Kurz, M. Wiesenfeldt, Optique Cohérente (Masson, Paris, 1997).

Lauterborn, W.

W. Lauterborn, T. Kurz, M. Wiesenfeldt, Optique Cohérente (Masson, Paris, 1997).

Lenoble, A.

F. Onofri, A. Lenoble, “Sizing of single fibers under torsional stress,” in Proceedings of the Sixth International Congress on Optical Particle Characterization, A. R. Jones, ed. (Institute of Physics, Bristol, UK, 2001), paper 7.7.

Lipson, H.

S. G. Lipson, H. Lipson, D. S. Tannhauser, Optical Physics (Cambridge U. Press, Cambridge, 1995).
[CrossRef]

Lipson, S. G.

S. G. Lipson, H. Lipson, D. S. Tannhauser, Optical Physics (Cambridge U. Press, Cambridge, 1995).
[CrossRef]

Marijnissen, J. C. M.

A. A. Naqwi, R. P. A. Hartman, J. C. M. Marijnissen, “Basic studies of electrohydrodynamic atomization using phase Doppler measurement technique, “Part. Part. Syst. Charact. 13, 143–149 (1996).
[CrossRef]

Melling, A.

F. Durst, A. Melling, J. H. Whitelaw, Principles and Practice of Laser-Doppler Anemometry (Academic, London, 1981).

Mesguish-Ripault, J.

Mischenko, M.

M. Mischenko, “Shape and Structure,” in Proceedings of the Sixth International Congress on Optical Particle Characterization, A. R. Jones, ed. (Institute of Physics, Bristol, UK, 2001), paper 1.1.

Naqwi, A.

A. Naqwi, “Sizing of irregular particles using a phase Doppler system,” Part. Part. Syst. Charact. 8, 343–349 (1996).
[CrossRef]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Particle trajectory effects in phase Doppler systems: computations and experiments,” Part. Part. Syst. Charact. 10, 332–338 (1993).
[CrossRef]

A. Naqwi, F. Durst, “Light scattering applied to LDA and PDA measurements. I. Theory and numerical treatments,” Part. Part. Syst. Charact. 8, 245–258 (1991).
[CrossRef]

Naqwi, A. A.

A. A. Naqwi, R. P. A. Hartman, J. C. M. Marijnissen, “Basic studies of electrohydrodynamic atomization using phase Doppler measurement technique, “Part. Part. Syst. Charact. 13, 143–149 (1996).
[CrossRef]

Onofri, F.

F. Onofri, L. Bergounoux, J.-L. Firpo, J. Mesguish-Ripault, “Velocity, size, and concentration measurements of optically inhomogeneous cylindrical and spherical particles,” Appl. Opt. 38, 4681–4690 (1999).
[CrossRef]

F. Onofri, G. Gréhan, G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113–7124 (1995).
[CrossRef] [PubMed]

C. Tropea, T.-H. Xu, F. Onofri, G. Gréhan, P. Haugen, “Dual-mode phase Doppler anemometer,” Part. Part. Syst. Charact. 13, 165–170 (1995).
[CrossRef]

F. Onofri, “Etude numérique et expérimentale de la sensibilité de l’interfrométrie phase Doppler à l’état de surface des particles détectées,” in Proceedings of the Septième Congrés Francophone de Vélocimétrie Laser, M. Elena, ed. (Université de Provence, Marseille, France, 2000), pp. 335–342.

F. Onofri, “Prise en compte de la dimension finie des faisceaux d’éclairage en granulométrie optique: anémométrie phase Doppler—diagnostics des milieux diphasiques,” Ph.D. dissertation (Université de Rouen, Rouen, France, 1995).

F. Onofri, A. Lenoble, “Sizing of single fibers under torsional stress,” in Proceedings of the Sixth International Congress on Optical Particle Characterization, A. R. Jones, ed. (Institute of Physics, Bristol, UK, 2001), paper 7.7.

Tannhauser, D. S.

S. G. Lipson, H. Lipson, D. S. Tannhauser, Optical Physics (Cambridge U. Press, Cambridge, 1995).
[CrossRef]

Tropea, C.

C. Tropea, T.-H. Xu, F. Onofri, G. Gréhan, P. Haugen, “Dual-mode phase Doppler anemometer,” Part. Part. Syst. Charact. 13, 165–170 (1995).
[CrossRef]

J. Domnick, H. Ertel, C. Tropea, “Processing of phase-Doppler signals using the cross-spectral density function,” in Proceedings of the Third European Symposium on Particle Characterization, E. H. K. Leschonski, ed. (NurnbergMesse, Nurnberg, Germany, 1988), pp. 473–483.

Whitelaw, J. H.

F. Durst, A. Melling, J. H. Whitelaw, Principles and Practice of Laser-Doppler Anemometry (Academic, London, 1981).

Wiesenfeldt, M.

W. Lauterborn, T. Kurz, M. Wiesenfeldt, Optique Cohérente (Masson, Paris, 1997).

Xu, T.-H.

C. Tropea, T.-H. Xu, F. Onofri, G. Gréhan, P. Haugen, “Dual-mode phase Doppler anemometer,” Part. Part. Syst. Charact. 13, 165–170 (1995).
[CrossRef]

Zaré, M.

F. Durst, M. Zaré, “Laser Doppler measurements in two-phase flows,” in The Accuracy of Flow Measurements by Laser Doppler Methods—Proceedings of LDA-Symposium, Copenhagen, Denmark, 1975, P. Buchhave, J. M. Delhaye, F. Durst, W. K. George, K. Refslund, J. H. Whitelaw, eds. (Proceedings LDA-Symposium, Skovlunde, Denmark, 1976), pp. 403–429.

Appl. Opt.

J. Phys. E

R. J. Adrian, R. J. Goldstein, “Analysis of a laser-Doppler anemometer,” J. Phys. E 4, 505–511 (1971).
[CrossRef]

Opt. Eng.

W. D. Bachalo, M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).
[CrossRef]

Part. Part. Syst. Charact.

K. Bauckhage, H. H. Floegel, U. Fritsching, R. Hiller, “The phase Doppler difference method, a new laser Doppler technique for simultaneous size and velocity measurements. 2. Optical particle characteristics as a base for a new diagnostic technique,” Part. Part. Syst. Charact. 5, 66–71 (1988).
[CrossRef]

E. D. Hirleman, “History of development of the phase Doppler-sizing velocimeter,” Part. Part. Syst. Charact. 13, 59–67 (1996).
[CrossRef]

A. Naqwi, F. Durst, “Light scattering applied to LDA and PDA measurements. I. Theory and numerical treatments,” Part. Part. Syst. Charact. 8, 245–258 (1991).
[CrossRef]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Particle trajectory effects in phase Doppler systems: computations and experiments,” Part. Part. Syst. Charact. 10, 332–338 (1993).
[CrossRef]

A. A. Naqwi, R. P. A. Hartman, J. C. M. Marijnissen, “Basic studies of electrohydrodynamic atomization using phase Doppler measurement technique, “Part. Part. Syst. Charact. 13, 143–149 (1996).
[CrossRef]

A. Naqwi, “Sizing of irregular particles using a phase Doppler system,” Part. Part. Syst. Charact. 8, 343–349 (1996).
[CrossRef]

C. Tropea, T.-H. Xu, F. Onofri, G. Gréhan, P. Haugen, “Dual-mode phase Doppler anemometer,” Part. Part. Syst. Charact. 13, 165–170 (1995).
[CrossRef]

Other

F. Durst, M. Zaré, “Laser Doppler measurements in two-phase flows,” in The Accuracy of Flow Measurements by Laser Doppler Methods—Proceedings of LDA-Symposium, Copenhagen, Denmark, 1975, P. Buchhave, J. M. Delhaye, F. Durst, W. K. George, K. Refslund, J. H. Whitelaw, eds. (Proceedings LDA-Symposium, Skovlunde, Denmark, 1976), pp. 403–429.

F. Onofri, “Etude numérique et expérimentale de la sensibilité de l’interfrométrie phase Doppler à l’état de surface des particles détectées,” in Proceedings of the Septième Congrés Francophone de Vélocimétrie Laser, M. Elena, ed. (Université de Provence, Marseille, France, 2000), pp. 335–342.

M. Mischenko, “Shape and Structure,” in Proceedings of the Sixth International Congress on Optical Particle Characterization, A. R. Jones, ed. (Institute of Physics, Bristol, UK, 2001), paper 1.1.

TSI Incorporated, 500 Cardigan Road, Saint Paul, Minn. 55126.

F. Onofri, “Prise en compte de la dimension finie des faisceaux d’éclairage en granulométrie optique: anémométrie phase Doppler—diagnostics des milieux diphasiques,” Ph.D. dissertation (Université de Rouen, Rouen, France, 1995).

F. Onofri, A. Lenoble, “Sizing of single fibers under torsional stress,” in Proceedings of the Sixth International Congress on Optical Particle Characterization, A. R. Jones, ed. (Institute of Physics, Bristol, UK, 2001), paper 7.7.

F. Durst, A. Melling, J. H. Whitelaw, Principles and Practice of Laser-Doppler Anemometry (Academic, London, 1981).

W. Lauterborn, T. Kurz, M. Wiesenfeldt, Optique Cohérente (Masson, Paris, 1997).

S. G. Lipson, H. Lipson, D. S. Tannhauser, Optical Physics (Cambridge U. Press, Cambridge, 1995).
[CrossRef]

J. Domnick, H. Ertel, C. Tropea, “Processing of phase-Doppler signals using the cross-spectral density function,” in Proceedings of the Third European Symposium on Particle Characterization, E. H. K. Leschonski, ed. (NurnbergMesse, Nurnberg, Germany, 1988), pp. 473–483.

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Figures (13)

Fig. 1
Fig. 1

Schematics of (a) PDA and (b) SNIP PDA techniques according to the heuristic fringe model.

Fig. 2
Fig. 2

Coordinate system Oxyz at the beam crossing.

Fig. 3
Fig. 3

(a) Calculation of the fringe intensity profile along Ox for z = y = 0. (b) Fringe intensity pattern in the Oxy plane when two coherent pairs of beams are superimposed with i p /i q = 3.

Fig. 4
Fig. 4

The same as in Figs. 3(a) and 3(b) but for two incoherent pairs of beams superimposed.

Fig. 5
Fig. 5

High-pass filtered signals simulated for n = 5 SNIPs with ν n /ν 1 = Δϕ12(ν n )/Δϕ12(ν 1) = n, where ν 1 = 20 kHz and Δϕ12(ν 1) = 20°: (a) complete signals, (b) zoom of (a), (c) signals with white noise added (SNR, signal-to-noise ratio).

Fig. 6
Fig. 6

(a) Phase and (b) modulus spectra of the CSD of signals from Fig. 5 with noise added.

Fig. 7
Fig. 7

Optical setups to produce a SNIP: (a) system (A) with a retardation line, (b) system (B) with two independent laser sources, (c) system (C) with cross-polarized probes. (1) He-Ne laser, (2) focusing lens, (3) transmission diffraction gratings, (4) collimating lens, (5) beam stop, (6) mirrors, (7) neutral-density filter, (8) half-wave retardation plate, (9) transmitting lens, and (10) SNIPs.

Fig. 8
Fig. 8

Measurement with system (A) of the coherence length of a 2.5-mW He-Ne laser with a cavity length of L c = 200 mm and λ = 0.6328 µm.

Fig. 9
Fig. 9

Image of the projection (Oxy plane) of the fringe pattern of system (B). The pair of beams I (left) and II (middle) is shown individually. Right, the two pairs of beams are superimposed.

Fig. 11
Fig. 11

(a) Typical experimental SNIP signals obtained for Fig. 10 when the frequency shift ratio is 1.3, (b) corresponding CSD phase, (c) CSD modulus spectrum.

Fig. 10
Fig. 10

Evolution of the phase-shift difference measured for small copper fibers fixed in the SNIP volume of system (B) versus the ratio of the frequency shifts of the two pairs of beams.

Fig. 12
Fig. 12

Size distributions measured for glass beads falling from a vibrating tank with (a) a classic PDA and (b), (c) two SNIP PDA systems, (B), and (A).

Fig. 13
Fig. 13

Comparison of the size measurements of small copper fibers with a precision micrometer and with system (C).

Equations (15)

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

C12πλmm-1 αψrad/μm.
Enr, t=E0 exp-jkn·r+φnexp-j2πνt,
Ir  n=12N En2=m=12N Emn=12N En*.
Ir  2E02N+n=12Nm>n2NcosKnm·r+φnm.
Knm·r=2πλcosαn-cosαmsinαn-sinαmzx.
φnm=2πfλ1/cosαn-1/cosαm.
E2n-1r, t=E0 exp-jk2n-1·r+φ2n×exp-j2πνt-n-1τ, E2nr, t=E0 exp-jk2n·r+φ2n×exp-j2πνt-n-1τ.
Ir  2E02N+n=12Nm>n2NγnmNnmτ×cosKnm·r+φnm+Λnm.
Ir  2E02N+n=1NcosK2n-12n·r.
E2n-1r, t=E0cos δnsin δnexp-jk2n-1·r+φ2n×exp-j2πνnt, E2nr, t=E0cos δnsin δnexp-jk2n·r+φ2n×exp-j2πνnt,
Ir=2E02N+n=12cosK2n-12n·r.
S1νS2ν*=G12νexp-jθ12ν=n=1Nδ2ν-νnexp-jΔϕn12.
St=n=1N Pn1+Vn cos2πνnt+ϕn,
G12νexp-jθ12ν=I02τ024n=1NVn exp-π ν-νnτ02exp-jϕn1× n=1NVn exp-πν-νnτ02exp+jϕn2.
G12νexp-jθ12ν=I02τ024n=1Nm>nNVn2 exp-2πτ02×ν-νn2 exp-jΔϕn12+2VnVm exp-2πτ02×ν-ν¯nm2+Δνnm2exp-jϕn1+ϕm2.

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