Abstract

Experimental laser beam profiles often deviate somewhat from the ideal Gaussian shape of the TEM00 laser mode. In order to take these deviations into account when calculating light scattering, we propose a method for approximating the beam shape coefficients in the partial wave expansion of an experimental laser beam. We then compute scattering by a single dielectric spherical particle placed on the beam’s axis using this method and compare our results to laboratory data. Our model calculations fit the laboratory data well.

© 1996 Optical Society of America

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References

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  1. E. E. M. Khaled, S. C. Hill, P. W. Barber, D. Q. Chowdhury, “Near-resonance excitation of dielectric spheres with plane waves and off-axis Gaussian beams,” Appl. Opt. 31, 1166–1169 (1992).
    [CrossRef] [PubMed]
  2. E. E. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
    [CrossRef]
  3. G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).
    [CrossRef]
  4. G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formalism,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
    [CrossRef]
  5. J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
    [CrossRef]
  6. J. T. Hodges, C. Presser, G. Gréhan, G. Gouesbet, “Forward scattering of a Gaussian beam by a nonabsorbing sphere,” Appl. Opt. 34, 2120–2132 (1995).
    [CrossRef] [PubMed]
  7. 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]
  8. J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693–706 (1993).
    [CrossRef]
  9. J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Section 17.5.
  10. G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. (Paris) 16, 239–247 (1985).
    [CrossRef]
  11. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1964), Section 10.3.
  12. G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985), Eq. (11.158).
  13. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Section 12.32.
  14. Ref. 13, Section 9.22.
  15. Ref. 12, Section 15.1.
  16. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  17. 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]
  18. J. P. Chevaillier, J. Fabre, G. Gréhan, G. Gouesbet, “Comparison of diffraction theory and generalized Lorenz-Mie theory for a sphere located on the axis of a laser beam,” Appl. Opt. 29, 1293–1298 (1990).
    [CrossRef] [PubMed]
  19. F. Guilloteau, G. Gréhan, G. Gouesbet, “Optical levitation experiments to assess the validity of the generalized Lorenz–Mie theory,” Appl. Opt. 31, 2942–2951 (1992).
    [CrossRef] [PubMed]

1995

1993

J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693–706 (1993).
[CrossRef]

E. E. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

1992

1990

1988

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

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

1986

1985

G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. (Paris) 16, 239–247 (1985).
[CrossRef]

1982

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

1979

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

Alexander, D. R.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985), Eq. (11.158).

Barber, P. W.

E. E. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

E. E. M. Khaled, S. C. Hill, P. W. Barber, D. Q. Chowdhury, “Near-resonance excitation of dielectric spheres with plane waves and off-axis Gaussian beams,” Appl. Opt. 31, 1166–1169 (1992).
[CrossRef] [PubMed]

Barton, J. P.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Chevaillier, J. P.

Chevaillier, J.-P.

Chowdhury, D. Q.

Christy, R. W.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Section 17.5.

Davis, L. W.

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

Fabre, J.

Gouesbet, G.

Gréhan, G.

Guilloteau, F.

Hamelin, P.

Hill, S. C.

E. E. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

E. E. M. Khaled, S. C. Hill, P. W. Barber, D. Q. Chowdhury, “Near-resonance excitation of dielectric spheres with plane waves and off-axis Gaussian beams,” Appl. Opt. 31, 1166–1169 (1992).
[CrossRef] [PubMed]

Hodges, J. T.

Khaled, E. E.

E. E. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

Khaled, E. E. M.

Lock, J. A.

Maheu, B.

Milford, F. J.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Section 17.5.

Presser, C.

Reitz, J. R.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Section 17.5.

Schaub, S. A.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Section 12.32.

Appl. Opt.

IEEE Trans. Antennas Propag.

E. E. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

J. Appl. Phys.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

J. Opt. (Paris)

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. (Paris) 16, 239–247 (1985).
[CrossRef]

J. Opt. Soc. Am. A

Phys. Rev. A

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

Other

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Section 17.5.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1964), Section 10.3.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985), Eq. (11.158).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Section 12.32.

Ref. 13, Section 9.22.

Ref. 12, Section 15.1.

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Figures (3)

Fig. 1
Fig. 1

Focused laser beam incident on a spherical particle whose center is at the origin of coordinates. The center of the beam focal waist is at (0, 0, z 0) with respect to the particle, and the detector is located at (x d , y d , z d ). The distance from the origin to the detector is r, and the distance from the center of the focal waist to the detector is r′.

Fig. 2
Fig. 2

Intensity as a function of angle for a focused Ar+ laser beam at a distance of z d = 146 ± 1.5 mm beyond the beam focal waist. The detector is in the far zone of the beam.

Fig. 3
Fig. 3

Beam-plus-scattered intensity as a function of scattering angle θ for a beam waist particle spacing of (a) z 0 = −4.0 mm, (b) z 0 = −14.5 mm, (c) z 0 = −25.0 mm. The experimental data are from Figs. 10(b), 10(d), and 10(f) of Ref. 6, respectively, and the theoretical intensity is from Eqs. (10)(13) and (20) and approximation (22). The single-Gaussian fit to the data for w 0 = 18 μm from Ref. 6 is given by the dashed curves.

Equations (29)

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k = 2 π / λ ,
E beam ( r , θ , ϕ ) = i E 0 sin θ cos ϕ k 2 r 2 u ˆ r l = 1 ( 2 l + 1 ) i l g l J l ( k r ) π l ( θ ) + E 0 cos ϕ k r u ˆ θ l = 1 i l ( 2 l + 1 ) l ( l + 1 ) g l × [ J l ( k r ) π l ( θ ) i J l ( k r ) τ l ( θ ) ] E 0 sin ϕ k r u ˆ ϕ l = 1 i l ( 2 l + 1 ) l ( l + 1 ) g l × [ J l ( k r ) τ l ( θ ) i J l ( k r ) π l ( θ ) ] , B beam ( r , θ , ϕ ) = i E 0 sin θ sin ϕ k 2 r 2 u ˆ r l = 1 i l ( 2 l + 1 ) g l J l ( k r ) π l ( θ ) + E 0 sin ϕ k r u ˆ θ l = 1 i l ( 2 l + 1 ) l ( l + 1 ) g l × [ J l ( k r ) π l ( θ ) i J l ( k r ) τ l ( θ ) ] E 0 cos ϕ k r u ˆ ϕ l = 1 i l ( 2 l + 1 ) l ( l + 1 ) g l × [ J l ( k r ) τ l ( θ ) i J l ( k r ) π l ( θ ) ] .
J l ( k r ) = k r j l ( k r ) .
π l ( θ ) = 1 sin θ P l 1 ( cos θ ) , τ l ( θ ) = d P l 1 ( cos θ ) .
lim k r J l ( k r ) = sin ( k r l π 2 ) ,
lim θ 1 π l ( θ ) = l ( l + 1 ) 2 [ J 0 ( u ) + J 2 ( u ) ] , lim θ 1 τ l ( θ ) = l ( l + 1 ) 2 [ J 0 ( u ) J 2 ( u ) ] ,
u = ( l + 1 / 2 ) θ .
E beam ( r , θ , ϕ ) = i E 0 ( cos ϕ u ˆ θ sin ϕ u ˆ ϕ ) exp ( i k r ) k r × l = 1 ( l + 1 / 2 ) g l J 0 ( u ) + i E 0 ( cos ϕ u ˆ θ + sin ϕ u ˆ ϕ ) × exp ( i k r ) k r l = 1 ( 1 ) l ( l + 1 / 2 ) g l J 2 ( u ) , B beam ( r , θ , ϕ ) = i E 0 c ( sin ϕ u ˆ θ + cos ϕ u ˆ ϕ ) exp ( i k r ) k r × l = 1 ( l + 1 / 2 ) g l J 0 ( u ) + i E 0 c ( sin ϕ u ˆ θ cos ϕ u ˆ ϕ ) × exp ( i k r ) k r l = 1 ( 1 ) l ( l + 1 / 2 ) g l J 2 ( u )
E beam ( r , θ , ϕ ) i E 0 u ˆ x exp ( i k r ) k r l = 1 ( l + 1 / 2 ) g l J 0 ( u ) , B beam ( r , θ , ϕ ) i E 0 c u ˆ y exp ( i k r ) k r l = 1 ( l + 1 / 2 ) g l J 0 ( u ) .
lim r E scattered ( r , θ , ϕ ) = i E 0 k r exp ( i k r ) [ S 2 ( θ ) cos ϕ u ˆ θ + S 1 ( θ ) sin ϕ u ˆ ϕ ] , lim r B scattered ( r , θ , ϕ ) = i E 0 c k r exp ( i k r ) [ S 1 ( θ ) sin ϕ u ˆ θ S 2 ( θ ) cos ϕ u ˆ ϕ ] .
S 1 ( θ ) = l = 1 ( 2 l + 1 ) l ( l + 1 ) g l [ a l π l ( θ ) + b l τ l ( θ ) ] , S 2 ( θ ) = l = 1 ( 2 l + 1 ) l ( l + 1 ) g l [ a l τ l ( θ ) + b l π l ( θ ) ] ,
E total = E beam + E scattered , B total = B beam + B scattered
E beam radial = E 0 exp ( i k r cos θ ) f ( k r , θ ) sin θ cos ϕ , B beam radial = E 0 c exp ( i k r cos θ ) f ( k r , θ ) sin θ sin ϕ ,
g l = ( i ) l 1 2 k r j l ( k r ) 1 l ( l + 1 ) 0 π sin 2 θdθ f ( k r , θ ) × exp ( i k r cos  θ ) P l ( cos  θ ) .
E beam = i E 0 u ˆ x  exp  ( i k r ) k r M ( θ ) .
r z d + ( x d 2 + y d 2 ) 2 z d , θ ( x d 2 + y d 2 ) 1 / 2 z d , r z d z 0 + ( x d 2 + y d 2 ) 2 ( z d z 0 ) z d z 0 + ( x d 2 + y d 2 ) 2 z d + ( x d 2 + y d 2 ) 2 z d z 0 , θ ( x d 2 + y d 2 ) 1 / 2 z d z 0 θ ( 1 z 0 z d ) .
E beam i E 0 u ˆ x  exp ( i k r ) k r M ( θ ) exp  ( i k z 0 ) exp  ( i k z 0 θ 2 / 2 ) .
exp  ( i k z 0 ) exp  ( i k z 0 θ 2 / 2 ) M ( θ ) l = 1 ( l + 1 / 2 ) g l J 0 [ ( l + 1 / 2 ) θ ]
( 2 μ 0 c ) 1 / 2 k r E 0 I beam ( θ ) 1 / 2 exp  ( i k z 0 ) exp  ( i k z 0 θ 2 / 2 ) l = 1 ( l + 1 / 2 ) g l J 0 [ ( l + 1 / 2 ) θ ] ,
I beam ( θ ) = E 0 2 μ 0 c k 2 r 2 M 2 ( θ ) .
( 2 μ 0 c ) 1 / 2 k r E 0 exp  ( i k z 0 ) I beam ( θ ) 1 / 2 exp  ( i k z 0 θ 2 / 2 ) 0 l d l g ( l ) J 0 ( l θ ) .
g ( l ) ( 2 μ 0 c ) 1 / 2 k r E 0 exp  ( i k z 0 ) 0 θdθ I beam ( θ ) 1 / 2 × exp  ( i k z 0 θ 2 / 2 ) J 0 ( l θ ) .
E Davis ( x , y , z = z 0 ) = E 0 exp  [ ( x 2 + y 2 ) / w 0 2 ] u ˆ x
lim k r E Davis ( r , θ , ϕ ) = i E 0 k r k 2 w 0 2 2 exp  ( i k r ) exp  ( k 2 w 0 2 θ 2 4 ) u ˆ x .
I Davis ( θ ) E 0 2 2 μ 0 c 1 k 2 r 2 k 4 w 0 4 4 exp  ( k 2 w 0 2 θ 2 2 ) .
g Davis ( l ) = 1 1 2 i z 0 / k w 0 2 exp  ( i k z 0 ) × exp  ( l 2 / k 2 w 0 2 1 2 i z 0 / k w 0 2 ) .
I Gaussian ( θ ) = exp  ( θ 2 / 4 s 2 ) for 0 θ θ max 
s = 1 k w 0 ,
l max  4 / s .

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