Abstract

Scattered light patterns from a spherical particle located on the axis of a Gaussian beam are computed with localized interpretation of the generalized Lorenz-Mie theory and compared with diffraction theory results as well as experimental results.

© 1990 Optical Society of America

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References

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  1. L. Lorenz, “Lysbevaegelsen i og uden for en haf plane lysbolger belyst kulge,” Vidensk. Selk. Skr. 6, 1–62 (1890).
  2. G. Mie, “Beitrage zur optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377–452 (1908).
    [CrossRef]
  3. J. P. Chevaillier, J. Fabre, P. Hamelin, J. L. Lesne, “Forward Scattering Signature of a Spherical Particle Crossing a Laser Beam out of the Beam Waist,” Part. Part. Syst. Characterization 5, 9–12, (1988).
    [CrossRef]
  4. B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the Generalized Lorenz-Mie Theory for Arbitrary Location of the Scatterer in an Arbitrary Incident Profile,” J. Opt. 19, 59–67 (1988).
    [CrossRef]
  5. 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]
  6. G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian Beam by a Mie Scatter Center Using a Bromwich Formalism,” J. Opt. 16, 83–93 (1985).
    [CrossRef]
  7. J. P. Chevaillier, J. Fabre, P. Hamelin, “Forward Scattered Light Intensities by a Sphere Located Anywhere in a Gaussian Beam,” Appl. Opt. 25, 1222–1225, (1986).
    [CrossRef] [PubMed]
  8. G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gn Coefficients in the Generalized Lorenz-Mie Theory Using Three Different Methods,” Appl. Opt. 27, 4874–4883 (1988).
    [CrossRef] [PubMed]
  9. F. Slimani, G. Gréhan, G. Gouesbet, D. Allano, “Near-Field Lorenz-Mie Theory and its Application to Microholography,” Appl. Opt. 23, 4140–4148 (1984).
    [CrossRef] [PubMed]
  10. H. Kogelnik, “On the Propagation of Gaussian Beams of Light Through Lenslike Media Including those with a Loss or Gain Variation,” Appl. Opt. 4, 1562–1569 (1965).
    [CrossRef]
  11. H. C. Van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  12. M. Born, E. Wolf, Principles of Optics, Pergamon, New York (1980).
  13. J. P. Chevaillier, J. Fabre, P. Hamelin, “Scattering Properties of Spherical Particles Situated in a Laser Beam and Application for Sizing,” in Particle Size Analysis 1985, P. J. Lloyd Ed (Wiley, New York, 1987).
  14. P. Hamelin, “Application de la diffusion lumineuse à la métrologie des particules en écoulement diphasique dispersé,” Thesis, Institut National Polytechnique de Toulouse; published in Bulletin de la Direction des Etudes et Recherches d’Electricité de France, A, 3/4 (1986).
  15. A. Ungut, G. Gréhan, G. Gouesbet, “Comparisons between Geometrical Optics and Lorenz-Mie Theory,” Appl. Opt. 20, 2911–2918 (1981).
    [CrossRef] [PubMed]

1988

J. P. Chevaillier, J. Fabre, P. Hamelin, J. L. Lesne, “Forward Scattering Signature of a Spherical Particle Crossing a Laser Beam out of the Beam Waist,” Part. Part. Syst. Characterization 5, 9–12, (1988).
[CrossRef]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the Generalized Lorenz-Mie Theory for Arbitrary Location of the Scatterer in an Arbitrary Incident Profile,” J. Opt. 19, 59–67 (1988).
[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, “Computations of the gn Coefficients in the Generalized Lorenz-Mie Theory Using Three Different Methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

1986

1985

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian Beam by a Mie Scatter Center Using a Bromwich Formalism,” J. Opt. 16, 83–93 (1985).
[CrossRef]

1984

1981

1965

1908

G. Mie, “Beitrage zur optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

1890

L. Lorenz, “Lysbevaegelsen i og uden for en haf plane lysbolger belyst kulge,” Vidensk. Selk. Skr. 6, 1–62 (1890).

Allano, D.

Born, M.

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

Chevaillier, J. P.

J. P. Chevaillier, J. Fabre, P. Hamelin, J. L. Lesne, “Forward Scattering Signature of a Spherical Particle Crossing a Laser Beam out of the Beam Waist,” Part. Part. Syst. Characterization 5, 9–12, (1988).
[CrossRef]

J. P. Chevaillier, J. Fabre, P. Hamelin, “Forward Scattered Light Intensities by a Sphere Located Anywhere in a Gaussian Beam,” Appl. Opt. 25, 1222–1225, (1986).
[CrossRef] [PubMed]

J. P. Chevaillier, J. Fabre, P. Hamelin, “Scattering Properties of Spherical Particles Situated in a Laser Beam and Application for Sizing,” in Particle Size Analysis 1985, P. J. Lloyd Ed (Wiley, New York, 1987).

Fabre, J.

J. P. Chevaillier, J. Fabre, P. Hamelin, J. L. Lesne, “Forward Scattering Signature of a Spherical Particle Crossing a Laser Beam out of the Beam Waist,” Part. Part. Syst. Characterization 5, 9–12, (1988).
[CrossRef]

J. P. Chevaillier, J. Fabre, P. Hamelin, “Forward Scattered Light Intensities by a Sphere Located Anywhere in a Gaussian Beam,” Appl. Opt. 25, 1222–1225, (1986).
[CrossRef] [PubMed]

J. P. Chevaillier, J. Fabre, P. Hamelin, “Scattering Properties of Spherical Particles Situated in a Laser Beam and Application for Sizing,” in Particle Size Analysis 1985, P. J. Lloyd Ed (Wiley, New York, 1987).

Gouesbet, G.

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the Generalized Lorenz-Mie Theory for Arbitrary Location of the Scatterer in an Arbitrary Incident Profile,” J. Opt. 19, 59–67 (1988).
[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, “Computations of the gn Coefficients in the Generalized Lorenz-Mie Theory Using Three Different Methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian Beam by a Mie Scatter Center Using a Bromwich Formalism,” J. Opt. 16, 83–93 (1985).
[CrossRef]

F. Slimani, G. Gréhan, G. Gouesbet, D. Allano, “Near-Field Lorenz-Mie Theory and its Application to Microholography,” Appl. Opt. 23, 4140–4148 (1984).
[CrossRef] [PubMed]

A. Ungut, G. Gréhan, G. Gouesbet, “Comparisons between Geometrical Optics and Lorenz-Mie Theory,” Appl. Opt. 20, 2911–2918 (1981).
[CrossRef] [PubMed]

Gréhan, G.

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, “Computations of the gn Coefficients in the Generalized Lorenz-Mie Theory Using Three Different Methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the Generalized Lorenz-Mie Theory for Arbitrary Location of the Scatterer in an Arbitrary Incident Profile,” J. Opt. 19, 59–67 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian Beam by a Mie Scatter Center Using a Bromwich Formalism,” J. Opt. 16, 83–93 (1985).
[CrossRef]

F. Slimani, G. Gréhan, G. Gouesbet, D. Allano, “Near-Field Lorenz-Mie Theory and its Application to Microholography,” Appl. Opt. 23, 4140–4148 (1984).
[CrossRef] [PubMed]

A. Ungut, G. Gréhan, G. Gouesbet, “Comparisons between Geometrical Optics and Lorenz-Mie Theory,” Appl. Opt. 20, 2911–2918 (1981).
[CrossRef] [PubMed]

Hamelin, P.

J. P. Chevaillier, J. Fabre, P. Hamelin, J. L. Lesne, “Forward Scattering Signature of a Spherical Particle Crossing a Laser Beam out of the Beam Waist,” Part. Part. Syst. Characterization 5, 9–12, (1988).
[CrossRef]

J. P. Chevaillier, J. Fabre, P. Hamelin, “Forward Scattered Light Intensities by a Sphere Located Anywhere in a Gaussian Beam,” Appl. Opt. 25, 1222–1225, (1986).
[CrossRef] [PubMed]

J. P. Chevaillier, J. Fabre, P. Hamelin, “Scattering Properties of Spherical Particles Situated in a Laser Beam and Application for Sizing,” in Particle Size Analysis 1985, P. J. Lloyd Ed (Wiley, New York, 1987).

P. Hamelin, “Application de la diffusion lumineuse à la métrologie des particules en écoulement diphasique dispersé,” Thesis, Institut National Polytechnique de Toulouse; published in Bulletin de la Direction des Etudes et Recherches d’Electricité de France, A, 3/4 (1986).

Kogelnik, H.

Lesne, J. L.

J. P. Chevaillier, J. Fabre, P. Hamelin, J. L. Lesne, “Forward Scattering Signature of a Spherical Particle Crossing a Laser Beam out of the Beam Waist,” Part. Part. Syst. Characterization 5, 9–12, (1988).
[CrossRef]

Lorenz, L.

L. Lorenz, “Lysbevaegelsen i og uden for en haf plane lysbolger belyst kulge,” Vidensk. Selk. Skr. 6, 1–62 (1890).

Maheu, B.

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the Generalized Lorenz-Mie Theory for Arbitrary Location of the Scatterer in an Arbitrary Incident Profile,” J. Opt. 19, 59–67 (1988).
[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, “Computations of the gn Coefficients in the Generalized Lorenz-Mie Theory Using Three Different Methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian Beam by a Mie Scatter Center Using a Bromwich Formalism,” J. Opt. 16, 83–93 (1985).
[CrossRef]

Mie, G.

G. Mie, “Beitrage zur optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

Slimani, F.

Ungut, A.

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Wolf, E.

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

Ann. Phys.

G. Mie, “Beitrage zur optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

Appl. Opt.

J. Opt, Soc. Am. A

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]

J. Opt.

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian Beam by a Mie Scatter Center Using a Bromwich Formalism,” J. Opt. 16, 83–93 (1985).
[CrossRef]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the Generalized Lorenz-Mie Theory for Arbitrary Location of the Scatterer in an Arbitrary Incident Profile,” J. Opt. 19, 59–67 (1988).
[CrossRef]

Part. Part. Syst. Characterization

J. P. Chevaillier, J. Fabre, P. Hamelin, J. L. Lesne, “Forward Scattering Signature of a Spherical Particle Crossing a Laser Beam out of the Beam Waist,” Part. Part. Syst. Characterization 5, 9–12, (1988).
[CrossRef]

Vidensk. Selk. Skr.

L. Lorenz, “Lysbevaegelsen i og uden for en haf plane lysbolger belyst kulge,” Vidensk. Selk. Skr. 6, 1–62 (1890).

Other

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

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

J. P. Chevaillier, J. Fabre, P. Hamelin, “Scattering Properties of Spherical Particles Situated in a Laser Beam and Application for Sizing,” in Particle Size Analysis 1985, P. J. Lloyd Ed (Wiley, New York, 1987).

P. Hamelin, “Application de la diffusion lumineuse à la métrologie des particules en écoulement diphasique dispersé,” Thesis, Institut National Polytechnique de Toulouse; published in Bulletin de la Direction des Etudes et Recherches d’Electricité de France, A, 3/4 (1986).

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Figures (12)

Fig. 1
Fig. 1

Coordinate system.

Fig. 2
Fig. 2

Comparison between scattered light intensity profiles computed with the diffraction theory (DT) and with the generalized Lorenz-Mie theory for transparent particle (m = 1.5; GLMT0) and for opaque particle (m = 1.5 − 10 i; GLMT1). A particle of radius R = 10.4 μm is located at beam waist z0 = 0, λ = 632,8 nm.

Fig. 3
Fig. 3

Same as Fig. 2 but with R = 21.3 μm.

Fig. 4
Fig. 4

Same as Fig. 2 but with R = 34.8 um.

Fig. 5
Fig. 5

Same as Fig. 2 but with z0 = 4.5 mm.

Fig. 6
Fig. 6

Same as Fig. 3 but with z0 = 4.5 mm.

Fig. 7
Fig. 7

Same as Fig. 4 but with z0 = 4.5 mm.

Fig. 8
Fig. 8

Scattered light intensity profiles comput ed with the diffraction theory (DT) and with the generalized Lorenz-Mie theory (GLMT0), for R → 0. Particles of radius R = 10μm, R = 1 μm. and R = 0.1 μm are located at the beam waist, λ = 632,8 nm.

Fig. 9
Fig. 9

Comparison between experimental measurements (EXP) and numerical results of the diffraction theory (DT) and the generalized Lorenz-Mie theory (GLMT0) with transparent sphere of radius R = 10.4 μm located at the beam waist, λ = 632,8 nm.

Fig. 10
Fig. 10

Same as Fig. 9 but with z0 = 4.5 mm.

Fig. 11
Fig. 11

Same as Fig. 9 but with a particle radius R = 34.8 μm.

Fig. 12
Fig. 12

Same as Fig. 10 but with a particle radius R = 34.8 μm.

Equations (21)

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E θ s = i E 0 exp ( - i k r ) k r cos φ n = 1 2 n + 1 n ( n + 1 ) × g n { a n τ n ( cos θ ) + b n π n ( cos θ ) } ,
E φ s = - i E 0 exp ( - i k r ) k r sin φ n = 1 2 n + 1 n ( n + 1 ) × g n { a n π n ( cos θ ) + b n τ n ( cos θ ) } ,
H θ s = i H 0 exp ( - i k r ) k r sin φ n = 1 2 n + 1 n ( n + 1 ) × g n { a n π n ( cos θ ) + b n τ n ( cos θ ) } ,
H φ s = i H 0 exp ( - i k r ) k r cos φ n = 1 2 n + 1 n ( n + 1 ) × g n { a n τ n ( cos θ ) + b n π n ( cos θ ) } ,
g n = i Q ¯ exp ( - i Q ¯ { ( n + 1 / 2 ) k w O } 2 ) exp ( i k z 0 ) ,
Q ¯ = 1 1 - 2 z 0 k w 0 2 ,
g n = w 0 w exp { i k ( z 0 + ( n + 1 / 2 k ) 2 1 2 R c - ψ k ) } × exp ( - { n + 1 / 2 k w } 2 )
w = w 0 [ 1 + ( 2 z 0 k w 0 2 ) 2 ] 1 / 2 .
R c = z 0 [ 1 + ( k w 0 2 2 z 0 ) 2 ] ,
ψ = tan - 1 ( 2 z 0 k w 0 2 ) .
E ( x , y ; d ) = - i k 2 π d exp ( i k d ) exp [ i k x 2 + y 2 2 d ] J ,
J = C ( P ) E ( x , y ; 0 ) exp ( - i k x x + y y d ) d x d y ,
E ( x , y ; 0 ) = w O w exp [ i k ( - z 0 + x 2 + y 2 2 R c - ψ k ) ] exp ( - x 2 + y 2 w 2 ) .
E ( ρ , φ ; d ) = - i k 2 π d [ exp ( i k d ) ( I 0 - I D ) ] ,
I 0 = π w 0 2 exp [ - ( k w 0 ρ 2 d ) 2 ] exp [ i k ( ρ 2 2 d ) ] ,
I D = w 0 w exp [ i k ( ρ 2 2 d - ψ k ) ] R 2 0 2 π I ( φ ) d φ ,
β = R 2 w 2 ,
γ = - k R ρ cos ( φ - φ ) d ,
δ = k R 2 2 R c ,
I ( φ ) = 0 1 exp ( - β t 2 ) exp [ i γ t + δ t 2 ] t d t .
R w 1 ,             R ( R c k ) 1 / 2 .

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