Abstract

It is proposed to use three interfering and coplanar laser beams to form the probe volume of laser Doppler systems. This allows us to obtain, for each particle crossing this probe volume, a Doppler signal whose frequency amplitude spectrum exhibits two characteristic peaks. Electromagnetic calculations and experimental validations clearly demonstrate that we can estimate simultaneously, from the analysis of these two frequency peaks, the particle position along the optical axis and one velocity component. This technique is expected to have great potentialities for velocity profile measurements in microfluidic or boundary layer flows, as well as for the sizing of spherical particles.

© 2006 Optical Society of America

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References

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  1. F. Durst, A. Melling, and J. H. Whitelaw, Principles and Practice of Laser-Doppler Anemometry (Academic, 1981).
  2. H. E. Albrecht, N. Damaschke, M. Borys, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques (Springer, 2003).
  3. L. Buettner and J. Czarske, "A multimode-fibre laser-Doppler anemometer for highly spatially resolved velocity measurements using low-coherence light," Meas. Sci. Technol. 12, 1891-1903 (2001).
    [CrossRef]
  4. A. Naqwi, X.-Z. Liu, and F. Durst, "Dual-cylindrical wave method for particle sizing," Part. Part. Syst. Charact. 7, 45-53 (1996).
    [CrossRef]
  5. F. Onofri, A. Lenoble, and S. Radev, "Superimposed noninterfering probes to extend the phase Doppler anemometry capabilities," Appl. Opt. 41, 3590-3600 (2002).
    [CrossRef] [PubMed]
  6. W. H. Stevenson, "Optical frequency shifting by means of a rotating diffraction grating," Appl. Opt. 9, 649-652 (1970).
    [CrossRef] [PubMed]
  7. F. Onofri, "Modélisation de la réponse d'un interféromètre Doppler laser a trois faisceaux cohérents," IUSTI-UMR CNRS 6595, Technopole de Château Gombert, 13453 Marseille, France (personal communication, 2003).
  8. G. Gouesbet, B. Maheu, and G. Gréhan, "Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation," J. Opt. Soc. Am. A 5, 1427-1443 (1988).
    [CrossRef]
  9. F. Onofri, G. Gréhan, and G. Gouesbet, "Electromagnetic scattering from a multilayered sphere located in an arbitrary beam," Appl. Opt. 34, 7113-7124 (1995).
    [CrossRef] [PubMed]
  10. Y. Aizu, J. Domnick, F. Durst, G. Gréhan, F. Onofri, H. H. Qiu, M. Sommerfeld, T.-H. Xu, and M. Ziema, "A new generation of phase Doppler instruments for particle velocity, size and concentration measurements," Part. Part. Syst. Charact. 2, 43-54 (1994).
    [CrossRef]
  11. F. Onofri, A. Lenoble, S. Radev, H. Bultynck, P.-H. Guering, and N. Marsault, "Interferometric sizing of single-axis birefringent glass fibres," Part. Part. Syst. Charact. 20, 171-182 (2003).
    [CrossRef]
  12. A. Naqwi and F. Durst, "Analysis of the laser light-scattering interferometric devices for the in-line diagnostic of moving particles," Appl. Opt. 32, 4003-4018 (1993).
    [PubMed]
  13. F. Durst and M. Zaré, "Laser Doppler measurements in two-phase flows," in The Accuracy of Flow Measurements by Laser Doppler Methods—Proceedings of the LDA Symposium, P.Buchhave, J.M.Delhaye,F.Durst, W.K.George, K.Refslund, and J.H.Whitelaw, eds. (1976), pp. 403-429.
  14. W. D. Bachalo and M. J. Houser, "Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions," Opt. Eng. 23, 583-590 (1984).
  15. K. Bauckhage, H. H. Floegel, U. Fritsching, and R. Hiller, "The phase Doppler difference method, a new laser Doppler technique for simultaneous size and velocity measurements," Part. Part. Syst. Charact. 5, 66-71 (1988).
    [CrossRef]
  16. A. Lenoble, "Caractérisation optique et étude de la stabilité d'un procédé de fibrage du verre," Ph.D. dissertation (Université de Provence, Marseille, France, 2004).
  17. P. Lemaître-Auger, A. Cartellier, P. Benech, and I. Schanen-Duport, "Integrated laser Doppler anemometer made by ion-exchange in glass substrate," in Developments in Laser Techniques and Fluid Mechanics, R.J.Adrian, D.F. G.Durão, F.Durst, M.V.Heitor, M.Maeda, and J.H.Whitelaw, eds. (Springer, 1998), pp. 39-51.

2003 (1)

F. Onofri, A. Lenoble, S. Radev, H. Bultynck, P.-H. Guering, and N. Marsault, "Interferometric sizing of single-axis birefringent glass fibres," Part. Part. Syst. Charact. 20, 171-182 (2003).
[CrossRef]

2002 (1)

2001 (1)

L. Buettner and J. Czarske, "A multimode-fibre laser-Doppler anemometer for highly spatially resolved velocity measurements using low-coherence light," Meas. Sci. Technol. 12, 1891-1903 (2001).
[CrossRef]

1996 (1)

A. Naqwi, X.-Z. Liu, and F. Durst, "Dual-cylindrical wave method for particle sizing," Part. Part. Syst. Charact. 7, 45-53 (1996).
[CrossRef]

1995 (1)

1994 (1)

Y. Aizu, J. Domnick, F. Durst, G. Gréhan, F. Onofri, H. H. Qiu, M. Sommerfeld, T.-H. Xu, and M. Ziema, "A new generation of phase Doppler instruments for particle velocity, size and concentration measurements," Part. Part. Syst. Charact. 2, 43-54 (1994).
[CrossRef]

1993 (1)

1988 (2)

G. Gouesbet, B. Maheu, and G. Gréhan, "Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation," J. Opt. Soc. Am. A 5, 1427-1443 (1988).
[CrossRef]

K. Bauckhage, H. H. Floegel, U. Fritsching, and R. Hiller, "The phase Doppler difference method, a new laser Doppler technique for simultaneous size and velocity measurements," Part. Part. Syst. Charact. 5, 66-71 (1988).
[CrossRef]

1984 (1)

W. D. Bachalo and M. J. Houser, "Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions," Opt. Eng. 23, 583-590 (1984).

1970 (1)

Aizu, Y.

Y. Aizu, J. Domnick, F. Durst, G. Gréhan, F. Onofri, H. H. Qiu, M. Sommerfeld, T.-H. Xu, and M. Ziema, "A new generation of phase Doppler instruments for particle velocity, size and concentration measurements," Part. Part. Syst. Charact. 2, 43-54 (1994).
[CrossRef]

Albrecht, H. E.

H. E. Albrecht, N. Damaschke, M. Borys, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques (Springer, 2003).

Bachalo, W. D.

W. D. Bachalo and M. J. Houser, "Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions," Opt. Eng. 23, 583-590 (1984).

Bauckhage, K.

K. Bauckhage, H. H. Floegel, U. Fritsching, and R. Hiller, "The phase Doppler difference method, a new laser Doppler technique for simultaneous size and velocity measurements," Part. Part. Syst. Charact. 5, 66-71 (1988).
[CrossRef]

Benech, P.

P. Lemaître-Auger, A. Cartellier, P. Benech, and I. Schanen-Duport, "Integrated laser Doppler anemometer made by ion-exchange in glass substrate," in Developments in Laser Techniques and Fluid Mechanics, R.J.Adrian, D.F. G.Durão, F.Durst, M.V.Heitor, M.Maeda, and J.H.Whitelaw, eds. (Springer, 1998), pp. 39-51.

Borys, M.

H. E. Albrecht, N. Damaschke, M. Borys, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques (Springer, 2003).

Buettner, L.

L. Buettner and J. Czarske, "A multimode-fibre laser-Doppler anemometer for highly spatially resolved velocity measurements using low-coherence light," Meas. Sci. Technol. 12, 1891-1903 (2001).
[CrossRef]

Bultynck, H.

F. Onofri, A. Lenoble, S. Radev, H. Bultynck, P.-H. Guering, and N. Marsault, "Interferometric sizing of single-axis birefringent glass fibres," Part. Part. Syst. Charact. 20, 171-182 (2003).
[CrossRef]

Cartellier, A.

P. Lemaître-Auger, A. Cartellier, P. Benech, and I. Schanen-Duport, "Integrated laser Doppler anemometer made by ion-exchange in glass substrate," in Developments in Laser Techniques and Fluid Mechanics, R.J.Adrian, D.F. G.Durão, F.Durst, M.V.Heitor, M.Maeda, and J.H.Whitelaw, eds. (Springer, 1998), pp. 39-51.

Czarske, J.

L. Buettner and J. Czarske, "A multimode-fibre laser-Doppler anemometer for highly spatially resolved velocity measurements using low-coherence light," Meas. Sci. Technol. 12, 1891-1903 (2001).
[CrossRef]

Damaschke, N.

H. E. Albrecht, N. Damaschke, M. Borys, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques (Springer, 2003).

Domnick, J.

Y. Aizu, J. Domnick, F. Durst, G. Gréhan, F. Onofri, H. H. Qiu, M. Sommerfeld, T.-H. Xu, and M. Ziema, "A new generation of phase Doppler instruments for particle velocity, size and concentration measurements," Part. Part. Syst. Charact. 2, 43-54 (1994).
[CrossRef]

Durst, F.

A. Naqwi, X.-Z. Liu, and F. Durst, "Dual-cylindrical wave method for particle sizing," Part. Part. Syst. Charact. 7, 45-53 (1996).
[CrossRef]

Y. Aizu, J. Domnick, F. Durst, G. Gréhan, F. Onofri, H. H. Qiu, M. Sommerfeld, T.-H. Xu, and M. Ziema, "A new generation of phase Doppler instruments for particle velocity, size and concentration measurements," Part. Part. Syst. Charact. 2, 43-54 (1994).
[CrossRef]

A. Naqwi and F. Durst, "Analysis of the laser light-scattering interferometric devices for the in-line diagnostic of moving particles," Appl. Opt. 32, 4003-4018 (1993).
[PubMed]

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

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

Floegel, H. H.

K. Bauckhage, H. H. Floegel, U. Fritsching, and R. Hiller, "The phase Doppler difference method, a new laser Doppler technique for simultaneous size and velocity measurements," Part. Part. Syst. Charact. 5, 66-71 (1988).
[CrossRef]

Fritsching, U.

K. Bauckhage, H. H. Floegel, U. Fritsching, and R. Hiller, "The phase Doppler difference method, a new laser Doppler technique for simultaneous size and velocity measurements," Part. Part. Syst. Charact. 5, 66-71 (1988).
[CrossRef]

Gouesbet, G.

Gréhan, G.

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

Y. Aizu, J. Domnick, F. Durst, G. Gréhan, F. Onofri, H. H. Qiu, M. Sommerfeld, T.-H. Xu, and M. Ziema, "A new generation of phase Doppler instruments for particle velocity, size and concentration measurements," Part. Part. Syst. Charact. 2, 43-54 (1994).
[CrossRef]

G. Gouesbet, B. Maheu, and G. Gréhan, "Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation," J. Opt. Soc. Am. A 5, 1427-1443 (1988).
[CrossRef]

Guering, P.-H.

F. Onofri, A. Lenoble, S. Radev, H. Bultynck, P.-H. Guering, and N. Marsault, "Interferometric sizing of single-axis birefringent glass fibres," Part. Part. Syst. Charact. 20, 171-182 (2003).
[CrossRef]

Hiller, R.

K. Bauckhage, H. H. Floegel, U. Fritsching, and R. Hiller, "The phase Doppler difference method, a new laser Doppler technique for simultaneous size and velocity measurements," Part. Part. Syst. Charact. 5, 66-71 (1988).
[CrossRef]

Houser, M. J.

W. D. Bachalo and M. J. Houser, "Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions," Opt. Eng. 23, 583-590 (1984).

Lemaître-Auger, P.

P. Lemaître-Auger, A. Cartellier, P. Benech, and I. Schanen-Duport, "Integrated laser Doppler anemometer made by ion-exchange in glass substrate," in Developments in Laser Techniques and Fluid Mechanics, R.J.Adrian, D.F. G.Durão, F.Durst, M.V.Heitor, M.Maeda, and J.H.Whitelaw, eds. (Springer, 1998), pp. 39-51.

Lenoble, A.

F. Onofri, A. Lenoble, S. Radev, H. Bultynck, P.-H. Guering, and N. Marsault, "Interferometric sizing of single-axis birefringent glass fibres," Part. Part. Syst. Charact. 20, 171-182 (2003).
[CrossRef]

F. Onofri, A. Lenoble, and S. Radev, "Superimposed noninterfering probes to extend the phase Doppler anemometry capabilities," Appl. Opt. 41, 3590-3600 (2002).
[CrossRef] [PubMed]

A. Lenoble, "Caractérisation optique et étude de la stabilité d'un procédé de fibrage du verre," Ph.D. dissertation (Université de Provence, Marseille, France, 2004).

Liu, X.-Z.

A. Naqwi, X.-Z. Liu, and F. Durst, "Dual-cylindrical wave method for particle sizing," Part. Part. Syst. Charact. 7, 45-53 (1996).
[CrossRef]

Maheu, B.

Marsault, N.

F. Onofri, A. Lenoble, S. Radev, H. Bultynck, P.-H. Guering, and N. Marsault, "Interferometric sizing of single-axis birefringent glass fibres," Part. Part. Syst. Charact. 20, 171-182 (2003).
[CrossRef]

Melling, A.

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

Naqwi, A.

A. Naqwi, X.-Z. Liu, and F. Durst, "Dual-cylindrical wave method for particle sizing," Part. Part. Syst. Charact. 7, 45-53 (1996).
[CrossRef]

A. Naqwi and F. Durst, "Analysis of the laser light-scattering interferometric devices for the in-line diagnostic of moving particles," Appl. Opt. 32, 4003-4018 (1993).
[PubMed]

Onofri, F.

F. Onofri, A. Lenoble, S. Radev, H. Bultynck, P.-H. Guering, and N. Marsault, "Interferometric sizing of single-axis birefringent glass fibres," Part. Part. Syst. Charact. 20, 171-182 (2003).
[CrossRef]

F. Onofri, A. Lenoble, and S. Radev, "Superimposed noninterfering probes to extend the phase Doppler anemometry capabilities," Appl. Opt. 41, 3590-3600 (2002).
[CrossRef] [PubMed]

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

Y. Aizu, J. Domnick, F. Durst, G. Gréhan, F. Onofri, H. H. Qiu, M. Sommerfeld, T.-H. Xu, and M. Ziema, "A new generation of phase Doppler instruments for particle velocity, size and concentration measurements," Part. Part. Syst. Charact. 2, 43-54 (1994).
[CrossRef]

F. Onofri, "Modélisation de la réponse d'un interféromètre Doppler laser a trois faisceaux cohérents," IUSTI-UMR CNRS 6595, Technopole de Château Gombert, 13453 Marseille, France (personal communication, 2003).

Qiu, H. H.

Y. Aizu, J. Domnick, F. Durst, G. Gréhan, F. Onofri, H. H. Qiu, M. Sommerfeld, T.-H. Xu, and M. Ziema, "A new generation of phase Doppler instruments for particle velocity, size and concentration measurements," Part. Part. Syst. Charact. 2, 43-54 (1994).
[CrossRef]

Radev, S.

F. Onofri, A. Lenoble, S. Radev, H. Bultynck, P.-H. Guering, and N. Marsault, "Interferometric sizing of single-axis birefringent glass fibres," Part. Part. Syst. Charact. 20, 171-182 (2003).
[CrossRef]

F. Onofri, A. Lenoble, and S. Radev, "Superimposed noninterfering probes to extend the phase Doppler anemometry capabilities," Appl. Opt. 41, 3590-3600 (2002).
[CrossRef] [PubMed]

Schanen-Duport, I.

P. Lemaître-Auger, A. Cartellier, P. Benech, and I. Schanen-Duport, "Integrated laser Doppler anemometer made by ion-exchange in glass substrate," in Developments in Laser Techniques and Fluid Mechanics, R.J.Adrian, D.F. G.Durão, F.Durst, M.V.Heitor, M.Maeda, and J.H.Whitelaw, eds. (Springer, 1998), pp. 39-51.

Sommerfeld, M.

Y. Aizu, J. Domnick, F. Durst, G. Gréhan, F. Onofri, H. H. Qiu, M. Sommerfeld, T.-H. Xu, and M. Ziema, "A new generation of phase Doppler instruments for particle velocity, size and concentration measurements," Part. Part. Syst. Charact. 2, 43-54 (1994).
[CrossRef]

Stevenson, W. H.

Tropea, C.

H. E. Albrecht, N. Damaschke, M. Borys, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques (Springer, 2003).

Whitelaw, J. H.

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

Xu, T.-H.

Y. Aizu, J. Domnick, F. Durst, G. Gréhan, F. Onofri, H. H. Qiu, M. Sommerfeld, T.-H. Xu, and M. Ziema, "A new generation of phase Doppler instruments for particle velocity, size and concentration measurements," Part. Part. Syst. Charact. 2, 43-54 (1994).
[CrossRef]

Zaré, M.

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

Ziema, M.

Y. Aizu, J. Domnick, F. Durst, G. Gréhan, F. Onofri, H. H. Qiu, M. Sommerfeld, T.-H. Xu, and M. Ziema, "A new generation of phase Doppler instruments for particle velocity, size and concentration measurements," Part. Part. Syst. Charact. 2, 43-54 (1994).
[CrossRef]

Appl. Opt. (4)

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

Meas. Sci. Technol. (1)

L. Buettner and J. Czarske, "A multimode-fibre laser-Doppler anemometer for highly spatially resolved velocity measurements using low-coherence light," Meas. Sci. Technol. 12, 1891-1903 (2001).
[CrossRef]

Opt. Eng. (1)

W. D. Bachalo and M. J. Houser, "Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions," Opt. Eng. 23, 583-590 (1984).

Part. Part. Syst. Charact. (4)

K. Bauckhage, H. H. Floegel, U. Fritsching, and R. Hiller, "The phase Doppler difference method, a new laser Doppler technique for simultaneous size and velocity measurements," Part. Part. Syst. Charact. 5, 66-71 (1988).
[CrossRef]

A. Naqwi, X.-Z. Liu, and F. Durst, "Dual-cylindrical wave method for particle sizing," Part. Part. Syst. Charact. 7, 45-53 (1996).
[CrossRef]

Y. Aizu, J. Domnick, F. Durst, G. Gréhan, F. Onofri, H. H. Qiu, M. Sommerfeld, T.-H. Xu, and M. Ziema, "A new generation of phase Doppler instruments for particle velocity, size and concentration measurements," Part. Part. Syst. Charact. 2, 43-54 (1994).
[CrossRef]

F. Onofri, A. Lenoble, S. Radev, H. Bultynck, P.-H. Guering, and N. Marsault, "Interferometric sizing of single-axis birefringent glass fibres," Part. Part. Syst. Charact. 20, 171-182 (2003).
[CrossRef]

Other (6)

F. Onofri, "Modélisation de la réponse d'un interféromètre Doppler laser a trois faisceaux cohérents," IUSTI-UMR CNRS 6595, Technopole de Château Gombert, 13453 Marseille, France (personal communication, 2003).

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

H. E. Albrecht, N. Damaschke, M. Borys, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques (Springer, 2003).

A. Lenoble, "Caractérisation optique et étude de la stabilité d'un procédé de fibrage du verre," Ph.D. dissertation (Université de Provence, Marseille, France, 2004).

P. Lemaître-Auger, A. Cartellier, P. Benech, and I. Schanen-Duport, "Integrated laser Doppler anemometer made by ion-exchange in glass substrate," in Developments in Laser Techniques and Fluid Mechanics, R.J.Adrian, D.F. G.Durão, F.Durst, M.V.Heitor, M.Maeda, and J.H.Whitelaw, eds. (Springer, 1998), pp. 39-51.

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

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Figures (10)

Fig. 1
Fig. 1

Geometry of the probe volume and the scattering models.

Fig. 2
Fig. 2

(a) Analytical model: normalized intensity distribution in the probe volume formed by three plane waves. (b) Generalized-Lorenz–Mie-theory-based model: normalized intensity scattered by a small water droplet moving in the probe volume formed by three focused laser beams (2ω0 = 200 μm).

Fig. 3
Fig. 3

Left, Doppler signals obtained when the particle trajectory is perpendicular to the fringes and for various positions along the optical axis z. Right, corresponding amplitude frequency spectra.

Fig. 4
Fig. 4

Evolution of the amplitude ratio of the two Doppler peaks versus the laser beam-waist diameters.

Fig. 5
Fig. 5

Influence of the SNR on the envelope and the amplitude frequency spectrum of the Doppler signals for z = 0.

Fig. 6
Fig. 6

Estimated particle position along the optical axis versus the nominal particle position and for various levels of SNR. The error bars correspond to the standard deviation calculated for 500 particle trajectories.

Fig. 7
Fig. 7

Evolution of the amplitude ratio of the Doppler peaks and the estimated particle position versus the particle nominal position for three half-beam angles, a SNR of 10 dB, and the distance range Δ z = | 0.458 z - 0.958 z | 100 μm .

Fig. 8
Fig. 8

Influence of the particle trajectory angle γ of the amplitude ratio of the Doppler peaks and the estimated particle position.

Fig. 9
Fig. 9

Schematic of the experimental setup to measure the velocity profile in a microchannel flow. APD, avalanche photodiode.

Fig. 10
Fig. 10

Raw velocity profiles measured within the microchannel for three Reynolds numbers (Re).

Tables (1)

Tables Icon

Table 1 Optical, Particle, and Sampling Parameters Used for the Simulations

Equations (25)

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E T ( y , z , t ) = E 1 exp { j [ k 1 r 2 π ( ν + ν s ) t ] } + E 1 exp { j [ k 1 r 2 π ( ν ν s ) t ] } + E 0 exp [ j ( k 0 r 2 πν t ) ] .
I ( y , z , t ) 2 E 1     2 [ 1 + cos ( 4 π y δ 2 4 π ν s t ) ] + [ 1 + 4 E 1 E 0 cos ( 2 π y δ 1 2 π ν s t ) cos ( β z ) ] ,
I S ( y = V y t , z , t ) 2 E 2 [ 3 2 + cos ( 2 π ν 2 t ) + 2 cos ( 2 π ν 1 t ) cos β z ] .
| S ( ν 0 ) | = 3 G ( ν ) δ ( 0 ) + G ( ν ν 2 ) δ ( ν ν 2 ) + 2 G ( ν ν 1 ) δ ( ν ν 1 ) | cos β z | ,
| G ( ν ) | = E 2 ( 0 ) π 2 ω 0 V y exp [ 1 2 ( π ω 0 ν V y ) 2 ] .
R ν ( z ) max [ S ( ν 1 ) ] max [ S ( ν 2 ) ] = 2 | cos β z | .
z E = 1 β { cos 1 [ 1 2 R ν ( z ) ] ± n π } .
E θ = i E k r exp [ i ( k r 2 πνt ) ] S 2 ,
E φ = E k r exp [ i ( k r 2 πνt ) ] S 1 .
S 1 = n = 1 m = n + n 2 n + 1 n ( n + 1 ) [ m A n g n , TM m π n     | m | ( cos θ ) + i B n g n ,TE m τ n     | m | ( cos θ ) ] exp ( i m φ ) ,
S 2 = n = 1 m = n + n 2 n + 1 n ( n + 1 ) [ A n g n ,TM m τ n     | m | ( cos θ ) + i m B n g n ,TE m π n     | m | ( cos θ ) ] exp ( i m φ ) ,
E T ( r , θ , φ = π / 2 , t ) = [ E + 1 θ ( θ + α ) + E 1 θ ( θ α ) + E 0 θ ( θ ) ] e θ ,
| S ( r , θ , π / 2 , t ) | = k 2 π ν ¯ μ 0 [ | E + 1 θ | 2 + | E 1 θ | 2 + | E 0 θ | 2 + 2 Re ( E + 1 θ E 1 θ * + E + 1 θ E 0 θ * + E 1 θ E 0 θ * ) ] .
I S ( r , t ) = Ω | S ( r , θ , π / 2 , t ) | d Ω .
P 1 = k 2 π ν ¯ μ 0 1 2 | E 1 θ | 2 + 1 2 | E + 1 θ | 2 + | E 0 θ | 2 Ω ,
H 1 Ω = E + 1 θ E ¯ 0 θ + E 1 θ E ¯ 0 θ Ω ,
V 1 = 2 | H 1 Ω | / P 1 ,
ϕ 1 = tan 1 [ Im ( H 1 Ω ) / Re ( H 1 Ω ) ] ,
P 2 = k 2 π ν ¯ μ 0 1 2 | E 1 θ | 2 + 1 2 | E + 1 θ | 2 Ω ,
H 2 Ω = E + 1 θ E ¯ 1 θ Ω ,
V 2 = 2 | H 2 Ω | / P 2 ,
ϕ 2 = tan 1 [ Im ( H 2 Ω ) / Re ( H 2 Ω ) ] .
I ( r , t ) = P 1 [ 1 + V 1 cos ( 2 π ν 1 t + ϕ 1 ) ] + P 2 [ 1 + V 2 cos ( 2 π ν 2 t + ϕ 2 ) ] .
R ν ( z ) = β 1 cos β 3 ( β 2 z ) ,
z E = 1 β 2 cos 1 [ R ν ( z ) β 1 ] 1 / β 3 .

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