Abstract

Radiation pressure exerted on a spherical particle by one extremely focused Gaussian beam is investigated by the use of generalized Lorenz–Mie theory (GLMT). Particular attention is devoted to reverse radiation pressure. GLMT predictions for different descriptions of the incident beam are compared with electrostriction predictions when the particle size is smaller than the wavelength and with geometric-optics predictions when the particle size is larger than the wavelength.

© 1996 Optical Society of America

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  1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
    [CrossRef]
  2. G. Roosen, “La lévitation optique de sphères,” J. Can. Phys. 57, 1260–1279 (1979).
    [CrossRef]
  3. A. Ashkin, J. M. Dziedzic, “Observation of light scattering from nonspherical particles using optical levitation,” Appl. Opt. 19, 660–668 (1980).
    [CrossRef] [PubMed]
  4. A. Ashkin, J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987).
    [CrossRef] [PubMed]
  5. K. F. Ren, G. Gréhan, G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz–Mie theory, and associated resonance effects,” Opt. Commun. 108, 343–354 (1994).
    [CrossRef]
  6. G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
    [CrossRef]
  7. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
    [CrossRef] [PubMed]
  8. T. C. B. Schut, G. Hesselink, B. G. Grooth, J. Greve, “Experimental and theoretical investigations on the validity of geometrical optics model for calculating the stability of optical traps,” Cytometry 12, 479–485 (1991).
    [CrossRef] [PubMed]
  9. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
    [CrossRef] [PubMed]
  10. R. Gussgard, T. Lindmo, I. Brevik, “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. B 9, 1922–1930 (1992).
    [CrossRef]
  11. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1965).
    [CrossRef]
  12. J. A. Lock, G. Gouesbet, “A rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz–Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
    [CrossRef]
  13. G. Gouesbet, J. A. Lock, “A rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
    [CrossRef]
  14. G. Gouesbet, J. A. Lock, G. Gréhan, “Do you know what a laser beam is?”, presented at the Seventh Workshop on Two-Phase Flows, Erlangen, Germany, April 11–14, 1994.
  15. G. Gouesbet, J. A. Lock, G. Gréhan, “Partial wave representation of laser beams for use in light scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).
    [CrossRef] [PubMed]
  16. G. Gouesbet, B. Maheu, 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]
  17. G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
    [CrossRef]
  18. G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
    [CrossRef] [PubMed]
  19. G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
    [CrossRef]
  20. K. F. Ren, G. Gréhan, G. Gouesbet, “Localized approximation of generalized Lorenz–Mie theory: faster algorithm for computations of beam shape coefficients, gnm,” Part. Part. Syst. Charact. 9, 144–150 (1992).
    [CrossRef]
  21. K. F. Ren, G. Gréhan, G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in the generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).
    [CrossRef]
  22. J. A. Lock, “Improved Gaussian beam-scattering algorithm,” Appl. Opt. 34, 559–570 (1995).
    [CrossRef] [PubMed]
  23. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. 19, 1177–1179 (1979).
    [CrossRef]
  24. J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
    [CrossRef]

1995 (3)

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
[CrossRef]

G. Gouesbet, J. A. Lock, G. Gréhan, “Partial wave representation of laser beams for use in light scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).
[CrossRef] [PubMed]

J. A. Lock, “Improved Gaussian beam-scattering algorithm,” Appl. Opt. 34, 559–570 (1995).
[CrossRef] [PubMed]

1994 (4)

1992 (3)

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

R. Gussgard, T. Lindmo, I. Brevik, “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. B 9, 1922–1930 (1992).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Localized approximation of generalized Lorenz–Mie theory: faster algorithm for computations of beam shape coefficients, gnm,” Part. Part. Syst. Charact. 9, 144–150 (1992).
[CrossRef]

1991 (1)

T. C. B. Schut, G. Hesselink, B. G. Grooth, J. Greve, “Experimental and theoretical investigations on the validity of geometrical optics model for calculating the stability of optical traps,” Cytometry 12, 479–485 (1991).
[CrossRef] [PubMed]

1990 (1)

1989 (1)

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

1988 (2)

G. Gouesbet, B. Maheu, 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]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

1987 (1)

A. Ashkin, J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987).
[CrossRef] [PubMed]

1986 (2)

1980 (1)

1979 (2)

G. Roosen, “La lévitation optique de sphères,” J. Can. Phys. 57, 1260–1279 (1979).
[CrossRef]

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. 19, 1177–1179 (1979).
[CrossRef]

1970 (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[CrossRef]

1965 (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1965).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

Angelova, M. I.

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
[CrossRef]

Ashkin, A.

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, “Observation of light scattering from nonspherical particles using optical levitation,” Appl. Opt. 19, 660–668 (1980).
[CrossRef] [PubMed]

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[CrossRef]

Barton, J. P.

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

Bjorkholm, J. E.

Brevik, I.

Chu, S.

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. 19, 1177–1179 (1979).
[CrossRef]

Dziedzic, J. M.

Gouesbet, G.

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
[CrossRef]

G. Gouesbet, J. A. Lock, G. Gréhan, “Partial wave representation of laser beams for use in light scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).
[CrossRef] [PubMed]

J. A. Lock, G. Gouesbet, “A rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz–Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
[CrossRef]

G. Gouesbet, J. A. Lock, “A rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz–Mie theory, and associated resonance effects,” Opt. Commun. 108, 343–354 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in the generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Localized approximation of generalized Lorenz–Mie theory: faster algorithm for computations of beam shape coefficients, gnm,” Part. Part. Syst. Charact. 9, 144–150 (1992).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

G. Gouesbet, B. Maheu, 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]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

G. Gouesbet, J. A. Lock, G. Gréhan, “Do you know what a laser beam is?”, presented at the Seventh Workshop on Two-Phase Flows, Erlangen, Germany, April 11–14, 1994.

Gréhan, G.

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
[CrossRef]

G. Gouesbet, J. A. Lock, G. Gréhan, “Partial wave representation of laser beams for use in light scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in the generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz–Mie theory, and associated resonance effects,” Opt. Commun. 108, 343–354 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Localized approximation of generalized Lorenz–Mie theory: faster algorithm for computations of beam shape coefficients, gnm,” Part. Part. Syst. Charact. 9, 144–150 (1992).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

G. Gouesbet, B. Maheu, 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]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

G. Gouesbet, J. A. Lock, G. Gréhan, “Do you know what a laser beam is?”, presented at the Seventh Workshop on Two-Phase Flows, Erlangen, Germany, April 11–14, 1994.

Greve, J.

T. C. B. Schut, G. Hesselink, B. G. Grooth, J. Greve, “Experimental and theoretical investigations on the validity of geometrical optics model for calculating the stability of optical traps,” Cytometry 12, 479–485 (1991).
[CrossRef] [PubMed]

Grooth, B. G.

T. C. B. Schut, G. Hesselink, B. G. Grooth, J. Greve, “Experimental and theoretical investigations on the validity of geometrical optics model for calculating the stability of optical traps,” Cytometry 12, 479–485 (1991).
[CrossRef] [PubMed]

Gussgard, R.

Hesselink, G.

T. C. B. Schut, G. Hesselink, B. G. Grooth, J. Greve, “Experimental and theoretical investigations on the validity of geometrical optics model for calculating the stability of optical traps,” Cytometry 12, 479–485 (1991).
[CrossRef] [PubMed]

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1965).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1965).
[CrossRef]

Lindmo, T.

Lock, J. A.

Maheu, B.

Martinot-Lagarde, G.

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
[CrossRef]

Pouligny, B.

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
[CrossRef]

Ren, K. F.

K. F. Ren, G. Gréhan, G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz–Mie theory, and associated resonance effects,” Opt. Commun. 108, 343–354 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in the generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Localized approximation of generalized Lorenz–Mie theory: faster algorithm for computations of beam shape coefficients, gnm,” Part. Part. Syst. Charact. 9, 144–150 (1992).
[CrossRef]

Roosen, G.

G. Roosen, “La lévitation optique de sphères,” J. Can. Phys. 57, 1260–1279 (1979).
[CrossRef]

Schut, T. C. B.

T. C. B. Schut, G. Hesselink, B. G. Grooth, J. Greve, “Experimental and theoretical investigations on the validity of geometrical optics model for calculating the stability of optical traps,” Cytometry 12, 479–485 (1991).
[CrossRef] [PubMed]

Appl. Opt. (4)

Biophys. J. (1)

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

Cytometry (1)

T. C. B. Schut, G. Hesselink, B. G. Grooth, J. Greve, “Experimental and theoretical investigations on the validity of geometrical optics model for calculating the stability of optical traps,” Cytometry 12, 479–485 (1991).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. Can. Phys. (1)

G. Roosen, “La lévitation optique de sphères,” J. Can. Phys. 57, 1260–1279 (1979).
[CrossRef]

J. Opt. (Paris) (1)

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

J. Opt. Soc. Am. A (5)

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

Opt. Commun. (1)

K. F. Ren, G. Gréhan, G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz–Mie theory, and associated resonance effects,” Opt. Commun. 108, 343–354 (1994).
[CrossRef]

Opt. Lett. (1)

Part. Part. Syst. Charact. (1)

K. F. Ren, G. Gréhan, G. Gouesbet, “Localized approximation of generalized Lorenz–Mie theory: faster algorithm for computations of beam shape coefficients, gnm,” Part. Part. Syst. Charact. 9, 144–150 (1992).
[CrossRef]

Phys. Rev. (1)

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. 19, 1177–1179 (1979).
[CrossRef]

Phys. Rev. Lett. (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[CrossRef]

Proc. IEEE (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1965).
[CrossRef]

Pure Appl. Opt. (1)

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
[CrossRef]

Science (1)

A. Ashkin, J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987).
[CrossRef] [PubMed]

Other (1)

G. Gouesbet, J. A. Lock, G. Gréhan, “Do you know what a laser beam is?”, presented at the Seventh Workshop on Two-Phase Flows, Erlangen, Germany, April 11–14, 1994.

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Figures (13)

Fig. 1
Fig. 1

Geometry under study (see text for definitions of coordinates).

Fig. 2
Fig. 2

Radiation pressure computed for different orders of the beam description for a Rayleigh particle (a = 0.01 μm, m = 1.5, w 0 = 0.25 μm, λ = 0.5 μm).

Fig. 3
Fig. 3

(a) C pr, z and (b) intensity gradient for three beam-waist radii (w 0 = 0.25, 0.5, 1.0 μm), where the wavelength is λ = 0.5 μm. For clarity, not all computed points are displayed.

Fig. 4
Fig. 4

Evolution of gradient (g 1) versus z 0. The parameter is the beam-waist radius, and the order of description is LA 1.

Fig. 5
Fig. 5

Evolution of gradient (g n ) versus z 0. The parameters are n (= 2 or 5) and w 0 equal to 0.25, 0.5, or 1 μm; λ = 0.5 μm, and the order of description is LA 1.

Fig. 6
Fig. 6

Radiation pressure on particles of radii a = 0.001, 0.01, 0.06, and 0.1 μm, and refractive index m = 1.5, exerted by a Gaussian beam of waist radius w 0 = 0.25 μm, with wavelength λ = 0.5 μm. The filled symbols correspond to a standard beam, and the empty symbols correspond to LA 1 predictions.

Fig. 7
Fig. 7

Comparison of GLMT with the predictions of Figure 4 of Schut et al.8: (a) Longitudinal radiation pressure C pr, z versus z 0. The parameter is the particle refractive index. (b) Longitudinal radiation pressure C pr, z versus z 0 for a strongly focused beam. The parameters are the refractive index and the wavelength.

Fig. 8
Fig. 8

Radiation pressure computed for different beam descriptions (standard, LA 1, and LA 5), for two Mie particles (a = 2.0 μm and a = 1.78 μm). The other parameters are m = 1.5, w 0 = 0.25 μm, and λ = 0.5 μm.

Fig. 9
Fig. 9

Radiation pressure for particles of radii a = 1.78, 1.79, 2.0, and 5.0 μm (refractive index m = 1.5), illuminated by a LA 5 beam with w 0 = 0.25 μm, and λ = 0.5 μm.

Fig. 10
Fig. 10

Scattering diagrams for a 4-μm-diameter particle symmetrically located in an extremely focused beam. The relative importance of the different g n terms can be seen. (a) diverging part of the beam (z 0 = −1.29 μm), (b) converging part of the beam (z 0 = 1.29 μm).

Fig. 11
Fig. 11

Evolution of the radiation pressure versus the number of terms in the series for the same parameter as in Fig. 10.

Fig. 12
Fig. 12

Scattering diagrams for a 4-μm-diameter particle symmetrically located in an extremely focused beam. A comparison is made among predictions for a LA 1, LA 3, or a LA 5 beam. (a) diverging part of the beam (z 0 = −1.29 μm), (b) converging part of the beam (z 0 = 1.29 μm).

Fig. 13
Fig. 13

Comparison of the reconstructed field for beam-shape coefficients g n computed for a LA 1 and a LA 5 beam, w 0 = 0.25 μm, λ = 0.5 μm. The computations are for x 0 = y 0 = 0 and z 0 = 1.29 μm.

Equations (17)

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C pr , z = λ 2 2 π n = 1 { 2 n + 1 n ( n + 1 ) g n 2 Re ( a n + b n - 2 a n b n * ) + n ( n + 2 ) n + 1 Re [ g n g n + 1 * ( a n + b n + a n + 1 * + b n + 1 * - 2 a n a n + 1 * - 2 b n b n + 1 * ) ] ,
A i = ( A u , 0 , 0 ) ,
A u = Ψ ( u , v , w ) exp ( - i k w ) ,
Δ A u + k 2 A u = 0.
Ψ = Ψ 0 + s 2 Ψ 2 + s 4 Ψ 4 + ,
g n k = j = 0 j + 2 l = 0 l = 2 k + 1 ( - 2 i s z 0 w 0 ) j ( - 1 ) l s 2 l × ( l + j ) ! l ! j ! 1 l ! ( n - 1 ) ! ( n - 1 - l ) ! × ( n + 1 + l ) ! ( n + 1 ) ! exp ( i k z 0 ) ,
g n = j = 0 l = 0 ( - 2 i s z 0 w 0 ) j ( - 1 ) l s 2 l ( l + j ) ! l ! j ! 1 l ! × ( n - 1 ) ! ( n - 1 - l ) ! ( n + 1 + l ) ! ( n + 1 ) ! exp ( i k z 0 ) .
E r 1 = [ E 0 exp ( - i k r cos θ ) cos φ sin θ ] i Q × exp ( - i Q r 2 sin 2 θ w 0 2 ) ( 1 - 2 Q s r cos θ w 0 ) ,
Q = 1 i + 2 r cos θ / l ,
E r 1 E 0 = exp ( - i k w ) F ( R , θ ) sin θ cos φ ,
L ^ F ( R , θ ) = F ( R = n + 1 / 2 , θ = π / 2 ) .
L ^ F ( R , θ ) = F { R = [ ( n - 1 ) ( n + 2 ) ] 1 / 2 , θ = π / 2 } ,
( g n ¯ ) = ( 1 + 2 i s z 0 w 0 ) - 1 exp ( i k z 0 ) exp [ - s 2 ( n - 1 ) ( n + 2 ) ( 1 + 2 i ( s z 0 / w 0 ) ] .
g n ¯ = i Q ¯ exp ( - i Q ¯ 2 ρ n 2 ) exp ( i k z 0 ) × [ 1 - 2 s 2 Q ¯ 2 ρ n 2 ( 3 - i Q ¯ ρ n 2 ) + 4 s 4 Q ¯ 4 ρ n 4 ( 10 - 5 i Q ¯ ρ n 2 - 0.5 Q ¯ 2 ρ n 4 ) ] ,
Q ¯ = 1 i - 2 ( s z 0 / w 0 ) ,             ρ n = [ ( n - 1 ) ( n + 2 ) s ] 1 / 2 ,
k = 1 , 2 = 4 = 0 , k = 3 , 2 = 1 , 4 = 0 , k = 5 , 2 = 4 = 1.
E 2 z = z ( I 0 l 2 l 2 + 4 z 2 ) = - I 0 8 l 2 z ( l 2 + 4 z 2 ) 2 ,

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