Abstract

Experimental laser beam profiles often deviate somewhat from the ideal Gaussian shape of the axisymmetric TEM00 laser mode. To take these deviations into account when calculating light scattering of an off-axis beam by a spherical particle, we use our phase-modeling method to approximate the beam-shape coefficients in the partial wave expansion of an experimental laser beam. We then use these beam-shape coefficients to compute the near-forward direction scattering of the off-axis beam by the particle. Our results are compared with laboratory data, and we give a physical interpretation of the various features observed in the angular scattering patterns.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. 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]
  2. 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]
  3. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  4. J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
    [CrossRef]
  5. 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]
  6. B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
    [CrossRef]
  7. 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]
  8. J. A. Lock, G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz–Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
    [CrossRef]
  9. G. Gouesbet, J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
    [CrossRef]
  10. J. T. Hodges, G. Gréhan, G. Gouesbet, C. Presser, “Forward scattering of a Gaussian beam by a nonabsorbing sphere,” Appl. Opt. 34, 2120–2132 (1995).
    [CrossRef] [PubMed]
  11. J. A. Lock, J. T. Hodges, “Far-field scattering of an axisymmetric laser beam of arbitrary profile by an on-axis spherical particle,” Appl. Opt. 35, 4283–4290 (1996).
    [CrossRef] [PubMed]
  12. 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]
  13. 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]
  14. 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]
  15. J. A. Lock, “Improved Gaussian beam scattering algorithm,” Appl. Opt. 34, 559–570 (1995).
    [CrossRef] [PubMed]
  16. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), pp. 795–797.
  17. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 718, no. 6.633.2.
  18. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 208–209, 210–214.
  19. Ref. 14, Appendix A.
  20. H. M. Nussenzveig, W. J. Wiscombe, “Complex angular momentum approximation to hard-core scattering,” Phys. Rev. A 43, 2093–2112 (1991).
    [CrossRef] [PubMed]
  21. G. Gouesbet, J. A. Lock, G. Gréhan, “Partial-wave representations of laser beams for use in light-scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).
    [CrossRef] [PubMed]
  22. A. Doicu, Department of Fine Mechanics and Optics, Politehnica, University of Bucharest, Romania (personal communication, 1994).

1996

1995

1994

1993

1992

1991

H. M. Nussenzveig, W. J. Wiscombe, “Complex angular momentum approximation to hard-core scattering,” Phys. Rev. A 43, 2093–2112 (1991).
[CrossRef] [PubMed]

1990

1989

B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
[CrossRef]

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

1988

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. 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

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, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[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]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), pp. 795–797.

Barton, J. P.

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

Chevaillier, J.-P.

Davis, L. W.

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

Doicu, A.

A. Doicu, Department of Fine Mechanics and Optics, Politehnica, University of Bucharest, Romania (personal communication, 1994).

Fabre, J.

Gouesbet, G.

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

J. T. Hodges, G. Gréhan, G. Gouesbet, C. Presser, “Forward scattering of a Gaussian beam by a nonabsorbing sphere,” Appl. Opt. 34, 2120–2132 (1995).
[CrossRef] [PubMed]

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

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

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]

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]

B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
[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. 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]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 718, no. 6.633.2.

Gréhan, G.

Guilloteau, F.

Hamelin, P.

Hodges, J. T.

Lock, J. A.

Maheu, B.

Nussenzveig, H. M.

H. M. Nussenzveig, W. J. Wiscombe, “Complex angular momentum approximation to hard-core scattering,” Phys. Rev. A 43, 2093–2112 (1991).
[CrossRef] [PubMed]

Presser, C.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 718, no. 6.633.2.

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), pp. 208–209, 210–214.

Wiscombe, W. J.

H. M. Nussenzveig, W. J. Wiscombe, “Complex angular momentum approximation to hard-core scattering,” Phys. Rev. A 43, 2093–2112 (1991).
[CrossRef] [PubMed]

Appl. Opt.

J. Appl. Phys.

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

J. Opt. Soc. Am. A

Opt. Commun.

B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
[CrossRef]

Phys. Rev. A

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

H. M. Nussenzveig, W. J. Wiscombe, “Complex angular momentum approximation to hard-core scattering,” Phys. Rev. A 43, 2093–2112 (1991).
[CrossRef] [PubMed]

Other

A. Doicu, Department of Fine Mechanics and Optics, Politehnica, University of Bucharest, Romania (personal communication, 1994).

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), pp. 795–797.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 718, no. 6.633.2.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 208–209, 210–214.

Ref. 14, Appendix A.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Intensity as a function of angle θ′ for a focused Ar+ laser beam a distance of z d ′ = 146.2 ± 1.5 mm beyond the beam focal waist. The detector is in the far zone of the beam.

Fig. 3
Fig. 3

Laser beam far-zone intensity profile compared with the intensity profile of a focused Gaussian beam with a focal plane electric field half-width w 0 of (a) 22.4 and (b) 18.5 μm.

Fig. 4
Fig. 4

Beam-plus-scattered intensity as a function of scattering angle θ for a beam waist-particle spacing of z 0 = −14.5 mm, a beam off-axis position of x 0 = 0, and (a) y 0 = −114 μm, (b) y 0 = −74 μm, (c) y 0 = −52 μm, (d) y 0 = −27 μm, (e) y 0 = −7 μm, (f) y 0 = 11 μm, (g) y 0 = 31 μm, (h) y 0 = 48 μm, (i) y 0 = 66 μm.

Fig. 5
Fig. 5

Beam-plus-scattered intensity as a function of scattering angle θ for the experimental data of Fig. 4(e) and our on-axis model with z 0 = −14.5 mm and x 0 = y 0 = 0.

Fig. 6
Fig. 6

Scattered intensity and beam intensity as a function of scattering angle θ for z 0 = −14.5 mm, x 0 = 0, and (a) y 0 = −50 μm, (b) y 0 = 0, (c) y 0 = 50 μm.

Equations (30)

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

E beam ( r , θ , ϕ ) = - i E 0 u ^ r ( k r ) 2 l = 1 m = - l l i l ( l + 1 2 ) × A l m G l ( k r ) P l m ( cos θ ) exp ( i m ϕ ) - i E 0 u ^ θ k r l = 1 m = - l l i l ( 2 l + 1 ) 2 l ( l + 1 ) × [ A l m G l ( k r ) τ l m ( θ ) exp ( i m ϕ ) - B l m G l ( k r ) m π l m ( θ ) exp ( i m ϕ ) ] + E 0 u ^ ϕ k r l = 1 m = - l l i l ( 2 l + 1 ) 2 l ( l + 1 ) × [ A l m G l ( k r ) m π l m ( θ ) exp ( i m ϕ ) - B l m G l ( k r ) τ l m ( θ ) exp ( i m ϕ ) ] .
π l m ( θ ) = 1 sin θ P l m [ cos ( θ ) ] π l m ( θ ) = d d θ P l m [ cos ( θ ) ] ,
A l m = { A l 0 for             m = 0 A l m + for             m 1 A l m - for             m - 1 , B l m = { B l 0 for             m = 0 B l m + for             m 1 B l m - for             m - 1.
lim k r G l ( k r ) = sin ( k r - l π 2 ) = 1 2 i [ ( - i ) l exp ( i k r ) - ( i ) l exp ( - i k r ) ] ,
lim θ 1 m π l m ( θ ) = ( l + m ) ! 2 ( l - m ) ! ( l + 1 / 2 ) m - 1 × [ J m - 1 ( u ) + J m + 1 ( u ) ] , lim θ 1 τ l m ( θ ) = ( l + m ) ! 2 ( l - m ) ! ( l + 1 / 2 ) m - 1 × [ J m - 1 ( u ) + J m + 1 ( u ) ] ,
u ( l + 1 / 2 ) θ .
E beam ( r , θ , ϕ ) = E 0 u ^ θ k r exp ( i k r ) ( i 2 l = 1 ( l + 1 / 2 ) 2 l ( l + 1 ) A l 0 J 1 ( u ) - i 2 m = 1 l = m ( l + 1 / 2 ) 2 l ( l + 1 ) ( l + m ) ! ( l - m ) ! 1 ( l + 1 / 2 ) m × { [ R l m + J m - 1 ( u ) - S l m + J m + 1 ( u ) ] exp ( i m ϕ ) + [ R l m - J m - 1 ( u ) - S l m - J m + 1 ( u ) ] × exp ( - i m ϕ ) } ) + E 0 u ^ ϕ k r exp ( i k r ) × ( - i 2 l = 1 ( l + 1 / 2 ) 2 l ( l + 1 ) B l 0 J 1 ( u ) + 1 2 m = 1 l = m × ( l + 1 / 2 ) 2 l ( l + 1 ) ( l + m ) ! ( l - m ) ! 1 ( l + 1 / 2 ) m + { [ R l m + J m - 1 ( u ) + S l m + J m + 1 ( u ) ] exp ( i m ϕ ) - [ R l m - J m - 1 ( u ) + S l m - J m + 1 ( u ) ] × exp ( - i m ϕ ) } ) ;
R l m ± 1 / 2 ( A l m ± ± i B l m ± ) ,             S l m ± 1 / 2 ( A l m ± i B l m ± ) .
( l + 1 / 2 ) 2 l ( l + 1 ) ( l + m ) ! ( l - m ) ! 1 ( l + 1 / 2 ) m l m ,
E beam ( r , θ , ϕ ) E 0 u ^ θ k r exp ( i k r ) { i 2 0 d l A 0 ( l ) J 1 ( l θ ) - i 2 m = 1 0 l m d l [ R m + ( l ) J m - 1 ( l θ ) - S m + ( l ) J m + 1 ( l θ ) ] exp ( i m ϕ ) - i 2 m = 1 0 l m d l [ R m - ( l ) J m - 1 ( l θ ) - S m - ( l ) J m + 1 ( l θ ) ] exp ( - i m ϕ ) } + E 0 u ^ ϕ k r { - i 2 0 d l B 0 ( l ) J 1 ( l θ ) + 1 2 m = 1 0 l m d l [ R m + ( l ) J m - 1 ( l θ ) + S m + ( l ) J m + 1 ( l θ ) ] exp ( i m ϕ ) - 1 2 m = 1 0 l m d l [ R m - ( l ) J m - 1 ( l θ ) + S m - ( l ) J m + 1 ( l θ ) ] exp ( - i m ϕ ) } .
E beam ( r , θ , ϕ ) = - i E 0 k r M ( θ ) exp ( i k r ) u ^ x ,
I beam ( r , θ , ϕ ) = E 0 2 2 μ 0 c M 2 ( θ ) k 2 r 2 ,
r = [ ( x - x 0 ) 2 + ( y - y 0 ) 2 + ( z - z 0 ) 2 ] 1 / 2 , tan θ = [ ( x - x 0 ) 2 + ( y - y 0 ) 2 ] 1 / 2 ( z - z 0 ) , tan ϕ = ( y - y 0 ) ( x - x 0 )
r = ( x 2 + y 2 + z 2 ) 1 / 2 ,             tan θ = ( x 2 + y 2 ) 1 / 2 z ,             tan ϕ = y x .
r r - z 0 - x x 0 z - y y 0 z + x 2 z 0 2 z 2 + y 2 z 0 2 z 2 + .
E beam ( r , θ , ϕ ) - i E 0 k r M ( x d , y d ) exp ( - i k z 0 ) u ^ x × exp ( i k r ) exp [ ( i k z 0 / 2 ) sin 2 θ ] × exp [ i k sin θ ( x 0 cos ϕ ) + y 0 sin ϕ ) ] .
ϕ = ϕ = ± π / 2.
θ ( θ 2 - 2 θ y 0 z ) 1 / 2 ( 1 + z 0 z ) .
u ^ x = cos ϕ u ^ θ - sin ϕ u ^ ϕ = 1 / 2 exp ( i ϕ ) ( u ^ θ - i u ^ ϕ ) + 1 / 2 exp ( - i ϕ ) ( u ^ θ + i u ^ ϕ ) ,
E beam ( r , θ , ϕ ) = - i E 0 k r exp ( i k r ) M ( x d , y d ) exp ( - i k z 0 ) × exp [ ( i k z 0 / 2 ) θ 2 ] u ^ θ { - i cos ϕ 0 J 1 ( Q ) + i 2 m = 1 ( - i ) m - 1 exp [ - i ( m - 1 ) ϕ 0 ] × exp ( i m ϕ ) [ J m - 1 ( Q ) - exp ( - 2 i ϕ 0 ) × J m + 1 ( Q ) ] + 1 2 m = 1 ( - i ) m - 1 exp [ i ( m - 1 ) ϕ 0 ] × exp ( - i m ϕ ) [ J m - 1 ( Q ) - exp ( 2 i ϕ 0 ) × J m + 1 ( Q ) ] } - i E 0 k r exp ( i k r ) × M ( x d , y d ) exp ( - i k z 0 ) exp [ ( i k z 0 / 2 ) θ 2 ] u ^ ϕ × { i sin ϕ 0 J 1 ( Q ) + i 2 m = 1 exp [ - i ( m - 1 ) ϕ 0 ] × exp ( i m ϕ ) [ J m - 1 ( Q ) + exp ( - 2 i ϕ 0 ) × J m + 1 ( Q ) ] - i 2 m = 1 exp [ + i ( m - 1 ) ϕ 0 ] × exp ( - i m ϕ ) [ J m - 1 ( Q ) + exp ( 2 i ϕ 0 ) × J m + 1 ( Q ) ] } ,
exp ( ± i ϕ 0 ) = x 0 ± i y 0 ( x 0 2 + y 0 2 ) 1 / 2 ,
Q k θ ( x 0 2 + y 0 2 ) 1 / 2 .
A 0 ( l ) = 2 i l cos ϕ 0 exp ( - i k z 0 ) H 1 ( l ) , A m ± ( l ) = exp ( - i k z 0 ) [ - i exp ( i ϕ 0 ) l ] m - 1 × [ H m - 1 ( l ) + exp ( 2 i ϕ 0 ) H m + 1 ( l ) ] , B 0 ( l ) = 2 i l sin ϕ 0 exp ( - i k z 0 ) H 1 ( l ) , B m ± ( l ) = exp ( - i k z 0 ) ( ± 1 i ) [ - i exp ( i ϕ 0 ) l ] m - 1 × [ H m - 1 ( l ) - exp ( 2 i ϕ 0 ) H m + 1 ( l ) ] ,
H p ( l ) 0 θ d θ M ( x d , y d ) exp ( i k z 0 θ 2 / 2 ) J p ( Q ) J p ( l θ ) .
M ( θ ) = 1 2 s 2 exp ( - θ 2 / 4 s 2 ) ,
s = 1 / k w 0 .
H m ( l ) = 1 2 s 2 0 θ d θ exp [ - θ 2 4 s 2 ( 1 - 2 i s z 0 w 0 ) ] × J m [ k ( x 0 2 + y 0 2 ) 1 / 2 θ ] J m ( l θ ) = ( 1 - 2 i s z 0 w 0 ) - 1 × exp [ - ( x 0 2 + y 0 2 ) w 0 2 ( 1 - 2 i s z 0 w 0 ) ] × I m ( 2 s l ( x 0 2 + y 0 2 ) w 0 2 1 ( 1 - 2 i s z 0 w 0 ) ) ,
E scattered ( r , θ , ϕ ) = i exp ( i k r ) k r [ S 2 ( θ , ϕ ) u ^ θ - S 1 ( θ , ϕ ) u ^ ϕ ] ,
S 1 ( θ , ϕ ) = l = 1 2 l + 1 2 l ( l + 1 ) B l 0 b l τ l 0 ( θ ) + m = 1 l = m 2 l + 1 2 l ( l + 1 ) i a l m π l m ( θ ) × [ - A l m + exp ( i m ϕ ) + A l m - exp ( - i m ϕ ) ] + m = 1 l = m 2 l + 1 2 l ( l + 1 ) b l τ l m ( θ ) [ B l m + exp ( i m ϕ ) + B l m - exp ( - i m ϕ ) ] S 2 ( θ , ϕ ) = l = 1 2 l + 1 2 l ( l + 1 ) A l 0 a l τ l 0 ( θ ) + m = 1 l = m 2 l + 1 2 l ( l + 1 ) a l τ l m ( θ ) [ A l m + exp ( i m ϕ ) + A l m - exp ( - i m ϕ ) ] + m = 1 l = m 2 l + 1 2 l ( l + 1 ) i b l m π l m ( θ ) × [ B l m + exp ( i m ϕ ) - B l m - exp ( - i m ϕ ) ] ,
E total ( r , θ , ϕ ) = E beam ( r , θ , ϕ ) + E scattered ( r , θ , ϕ ) .

Metrics