Abstract

A closed form formula is found for the Mie scattering coefficents of incident complex focus beams (which are a nonparaxial generalization of Gaussian beams) with any numerical aperture. This formula takes the compact form of multipoles evaluated at a single complex point. Included are the cases of incident scalar fields as well as electromagnetic fields with many polarizations, such as linear, circular, azimuthal and radial. Examples of incident radially and azimuthally polarized beams are presented.

© 2008 Optical Society of America

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References

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  1. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2807 (2004).
    [Crossref]
  2. D. Ganic, X. Gan, and M. Gu, “Optical trapping force with annular and doughnut laser beams based on vectorial diffraction,” Opt. Express 13, 1260–1265 (2005).
    [Crossref] [PubMed]
  3. Z. J. Smith and A. J. Berger, “Integrated raman- and angular- scattering microscopy,” Opt. Lett. 33, 714–716 (2008).
    [Crossref] [PubMed]
  4. G. Gouesbet and G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris)  13, 97–103 (1982).
    [Crossref]
  5. G. Gouesbet, G. Gréhan, and 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]
  6. G. Gouesbet, G. Gréhan, and B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris)  16, 83–93 (1985).
    [Crossref]
  7. R. Kant, “Generalized Lorenz-Mie scattering theory for focused radiation and finite solids of revolution: case i: symmetrically polarized beams,” J. Mod. Opt. 52, 2067–2092 (2005).
    [Crossref]
  8. A. S. van de Nes and P. Török, “Rigorous analysis of spheres in Gauss-Laguerre beams,” Opt. Express 15, 13,360–13,374 (2007).
    [Crossref]
  9. Y. A. Kravtsov, “Complex rays and complex caustics,” Radiophys. and Quant. Elec. 10, 719–730 (1967).
    [Crossref]
  10. F. A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. Royal Soc. Lond. A 366, 155–171 (1979).
    [Crossref]
  11. C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
    [Crossref]
  12. C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16, 1381–1386 (1999).
    [Crossref]
  13. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253, 358–379 (1959).
    [Crossref]
  14. C. J. R. Sheppard and S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24, 1543–1545 (1999).
    [Crossref]
  15. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
    [Crossref]
  16. J. D. Jackson, Classical Electrodynamics (Wiley, 1999), pp. 108–109.
  17. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic Press, 1969), pp. 39–54.
  18. J. D. Jackson, Classical Electrodynamics (Wiley, 1999), pp. 430–431.
  19. A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
    [Crossref]
  20. M. A. Alonso, R. Borghi, and M. Santarsiero, “New basis for rotationally symmetric nonparaxial fields in terms of spherical waves with complex foci,” Opt. Express 14, 6894–6905 (2006).
    [Crossref] [PubMed]

2008 (1)

2007 (1)

A. S. van de Nes and P. Török, “Rigorous analysis of spheres in Gauss-Laguerre beams,” Opt. Express 15, 13,360–13,374 (2007).
[Crossref]

2006 (1)

2005 (2)

D. Ganic, X. Gan, and M. Gu, “Optical trapping force with annular and doughnut laser beams based on vectorial diffraction,” Opt. Express 13, 1260–1265 (2005).
[Crossref] [PubMed]

R. Kant, “Generalized Lorenz-Mie scattering theory for focused radiation and finite solids of revolution: case i: symmetrically polarized beams,” J. Mod. Opt. 52, 2067–2092 (2005).
[Crossref]

2004 (1)

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2807 (2004).
[Crossref]

2000 (1)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

1999 (2)

1998 (1)

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[Crossref]

1988 (1)

1985 (1)

G. Gouesbet, G. Gréhan, and B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris)  16, 83–93 (1985).
[Crossref]

1982 (1)

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

1979 (1)

F. A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. Royal Soc. Lond. A 366, 155–171 (1979).
[Crossref]

1974 (1)

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

1967 (1)

Y. A. Kravtsov, “Complex rays and complex caustics,” Radiophys. and Quant. Elec. 10, 719–730 (1967).
[Crossref]

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253, 358–379 (1959).
[Crossref]

Alonso, M. A.

Berger, A. J.

Block, S. M.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2807 (2004).
[Crossref]

Borghi, R.

Cullen, F. A. L.

F. A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. Royal Soc. Lond. A 366, 155–171 (1979).
[Crossref]

Devaney, A. J.

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

Dorn, R.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

Gan, X.

Ganic, D.

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

Gouesbet, G.

G. Gouesbet, G. Gréhan, and 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, and B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris)  16, 83–93 (1985).
[Crossref]

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

Gréhan, G.

G. Gouesbet, G. Gréhan, and 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, and B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris)  16, 83–93 (1985).
[Crossref]

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

Gu, M.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999), pp. 430–431.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999), pp. 108–109.

Kant, R.

R. Kant, “Generalized Lorenz-Mie scattering theory for focused radiation and finite solids of revolution: case i: symmetrically polarized beams,” J. Mod. Opt. 52, 2067–2092 (2005).
[Crossref]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic Press, 1969), pp. 39–54.

Kravtsov, Y. A.

Y. A. Kravtsov, “Complex rays and complex caustics,” Radiophys. and Quant. Elec. 10, 719–730 (1967).
[Crossref]

Leuchs, G.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

Maheu, B.

G. Gouesbet, G. Gréhan, and 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, and B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris)  16, 83–93 (1985).
[Crossref]

Neuman, K. C.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2807 (2004).
[Crossref]

Quabis, S.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253, 358–379 (1959).
[Crossref]

Saghafi, S.

Santarsiero, M.

Sheppard, C. J. R.

Smith, Z. J.

Török, P.

A. S. van de Nes and P. Török, “Rigorous analysis of spheres in Gauss-Laguerre beams,” Opt. Express 15, 13,360–13,374 (2007).
[Crossref]

van de Nes, A. S.

A. S. van de Nes and P. Török, “Rigorous analysis of spheres in Gauss-Laguerre beams,” Opt. Express 15, 13,360–13,374 (2007).
[Crossref]

Wolf, E.

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253, 358–379 (1959).
[Crossref]

Yu, P. K.

F. A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. Royal Soc. Lond. A 366, 155–171 (1979).
[Crossref]

Appl. Opt. (1)

J. Math. Phys. (1)

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

J. Mod. Opt. (1)

R. Kant, “Generalized Lorenz-Mie scattering theory for focused radiation and finite solids of revolution: case i: symmetrically polarized beams,” J. Mod. Opt. 52, 2067–2092 (2005).
[Crossref]

J. Opt. (2)

G. Gouesbet, G. Gréhan, and B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris)  16, 83–93 (1985).
[Crossref]

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

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

Opt. Commun. (1)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. A (1)

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[Crossref]

Proc. Roy. Soc. A (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253, 358–379 (1959).
[Crossref]

Proc. Royal Soc. Lond. A (1)

F. A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. Royal Soc. Lond. A 366, 155–171 (1979).
[Crossref]

Radiophys. and Quant. Elec. (1)

Y. A. Kravtsov, “Complex rays and complex caustics,” Radiophys. and Quant. Elec. 10, 719–730 (1967).
[Crossref]

Rev. Sci. Instrum. (1)

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2807 (2004).
[Crossref]

Other (3)

J. D. Jackson, Classical Electrodynamics (Wiley, 1999), pp. 108–109.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic Press, 1969), pp. 39–54.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999), pp. 430–431.

Supplementary Material (2)

» Media 1: MOV (436 KB)     
» Media 2: MOV (420 KB)     

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Figures (2)

Fig. 1.
Fig. 1.

(a) Radially polarized beam with the specified focus and complex displacement, (b) total field (incident, scattered, and inside a scattering sphere of radius kR=5 and index n=2), (c) coefficients of the expansion and (d) radiant intensity. The animation shows the change of these plots as the focus of the beam moves laterally from the edge of the sphere to its center, then axially to the front of the sphere and finally as kq increases (i.e., the beam becomes more collimated). [Media 1]

Download Full Size | PPT Slide | PDF

Fig. 2.
Fig. 2.

(a) Azimuthally polarized beam with the specified focus and complex displacement, (b) total field (incident, scattered, and inside a scattering sphere of radius kR=5 and index n=2), (c) coefficients of the expansion and (d) radiant intensity. The animation shows the change of these plots as the focus of the beam moves laterally from the edge of the sphere to its center, then axially to the front of the sphere and finally as kq increases (i.e., the beam becomes more collimated). [Media 2]

Download Full Size | PPT Slide | PDF

Equations (38)

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U CF ( r ; ρ 0 ) = 4 π U 0 sin { k [ ( r − ρ 0 ) · ( r − ρ 0 ) ] 1 2 } k [ ( r − ρ 0 ) · ( r − ρ 0 ) ] 1 2 ,
E CF ( E ) ( r ; ρ 0 , p ) = [ p + ( p · ∇ ) ∇ k 2 ] U CF ( r ; ρ 0 ) ,
E CF ( B ) ( r ; ρ 0 , p ) = − ip × ∇ k U CF ( r ; ρ 0 ) ,
sin { k [ ( r a − r b ) · ( r a − r b ) ] 1 2 } k [ ( r a − r b ) · ( r a − r b ) ] 1 2 = 1 4 π ∑ l = 0 ∞ ∑ m = − l l [ Λ lm ( r b − r c ) ] * Λ lm ( r a − r c ) ,
Λ lm ( r ) = 4 π i l j l ( kr ) Y lm ( θ r , ϕ r ) .
Y lm ( θ , ϕ ) = 2 l + 1 ( l − m ) ! 4 π ( l + m ) ! P l m ( cos ⁡ θ ) exp ⁡ ( i m ϕ ) ,
U ( CF ) ( r ; ρ 0 ) = U 0 ∑ l , m [ Λ lm ( ρ 0 ) ] * Λ lm ( r ) ,
ρ 0 = ρ 0 x 2 + ρ 0 y 2 + ρ 0 z 2 ,
θ ρ = arccos ⁡ ρ 0 z ρ 0 ,
ϕ ρ = arctan ⁡ ρ 0 y ρ 0 x ,
U ( sc ) ( r ; ρ 0 ) = U 0 ∑ l , m c l [ Λ lm ( ρ 0 ) ] * Π lm ( r ) ,
Π lm ( r ) = 4 π i l h l ( 1 ) ( kr ) Y lm ( θ r , ϕ r ) .
c l ( k R , n ) = n j l ( k R ) j l ′ ( knR ) − j l ( knR ) j l ′ ( kR ) j l ( knR ) h l ( 1 ) ′ ( kR ) − nh l ( 1 ) ( k R ) j l ′ ( kn R ) ,
Λ lm ( I ) ( r ) = 1 k l ( l + 1 ) ∇ × ∇ × [ r Λ lm ( r ) ] = 4 π i l { l ( l + 1 ) kr j l ( kr ) Y lm ( θ r , ϕ r ) r ̂
− i [ j l ( kr ) kr + j l ′ ( kr ) ] Z lm ( θ r , ϕ r ) } ,
Λ lm ( II ) ( r ) = i l ( l + 1 ) ∇ × [ r Λ lm ( r ) ] = 4 π i l j l ( kr ) Y lm ( θ r , ϕ r ) ,
Y lm ( θ , ϕ ) = 1 l ( l + 1 ) L Y lm ( θ , ϕ ) ,
Z lm ( θ , ϕ ) = u × Y lm ( θ , ϕ ) ,
L = i θ ̂ sin ⁡ θ ∂ ∂ ϕ − i ϕ ̂ ∂ ∂ θ .
E ( CF ) ( E ) ( r ; ρ 0 , p ) = U 0 ∑ l = 1 ∞ ∑ m = − l l [ γ lm ( I ) ( ρ 0 , p ) Λ lm ( I ) ( r ) + γ lm ( II ) ( ρ 0 , p ) Λ lm ( II ) ( r ) ] ,
E ( CF ) ( B ) ( r ; ρ 0 , q ) = U 0 ∑ l = 1 ∞ ∑ m = − l l [ γ lm ( II ) ( ρ 0 , p ) Λ lm ( I ) ( r ) − γ lm ( I ) ( ρ 0 , p ) Λ lm ( II ) ( r ) ] ,
γ lm ( I , II ) ( ρ 0 , p ) = p . [ Λ lm ( I , II ) ( ρ 0 ) ] * .
E ( sc ) ( E ) ( r ; ρ 0 , p ) = U 0 ∑ l , m ⌈ a l γ lm ( I ) ( ρ 0 , p ) Π lm ( I ) ( r ) + b l γ lm ( II ) ( ρ 0 , p ) Π lm ( II ) ( r ) ⌉ ,
E ( sc ) ( B ) ( r ; ρ 0 , p ) = U 0 ∑ lm ⌈ a l γ lm ( II ) ( ρ 0 , p ) Π lm ( I ) ( r ) − b l γ lm ( I ) ( ρ 0 , p ) Π lm ( II ) ( r ) ⌉ ,
a l ( kR , n ) = j l ( kR ) [ j l ( knR ) knR + j ′ l ( knR ) ] − nj l ( knR ) [ j l ( kR ) kR + j ′ l ( kR ) ] nj l ( knR ) [ h l ( 1 ) ( kR ) kR + h l ( 1 ) ′ ( kR ) ] − h l ( 1 ) ( kR ) [ j l ( knR ) knR + j ′ l ( knR ) ] ,
b l ( kR , n ) = nj l ( kR ) [ j l ( knR ) knR + j ′ l ( knR ) ] − j l ( knR ) [ j l ( kR ) kR + j ′ l ( kR ) ] j l ( knR ) [ h l ( 1 ) ( kR ) kR + h l ( 1 ) ′ ( kR ) ] − nh l ( 1 ) ( kR ) [ j l ( knR ) knR + j ′ l ( knR ) ] .
sin ⁡ { k [ ( r a − r b ) · ( r a − r b ) ] 1 2 } k [ ( r a − r b ) · ( r a − r b ) ] 1 2 = 1 4 π ∫ 4 π exp ⁡ [ i k ( r a − r b ) · u ] d Ω
= 1 4 π ∬ 4 π δ ( u , u ′ ) exp ⁡ [ − i k ( r b − r c ) · u ′ ]
× exp ⁡ [ i k ( r a − r c ) · u ] d Ω d Ω ′ ,
δ ( u , u ′ ) = ∑ l , m Y lm * ( θ ′ , ϕ ′ ) Y lm ( θ , ϕ ) ,
Λ lm ( r ) = ∫ 4 π Y lm ( θ , ϕ ) exp ⁡ ( i k r · u ) d Ω ,
E ( CF ) ( E ) ( r ; ρ 0 , p ) = U 0 4 π ∫ 4 π exp ⁡ ( − i k u · ρ 0 ) [ p − ( p · u ) u ] exp ⁡ ( i k u · r ) d Ω
= U 0 4 π p · ∬ 4 π exp ⁡ ( − i k u ′ · ρ 0 ) δ ↔ ⊥ ( u , u ′ ) exp ⁡ ( i k u · r ) d Ω d Ω ′ ,
E ( CF ) ( B ) ( r ; ρ 0 , p ) = U 0 4 π ∫ 4 π exp ⁡ ( − i k u · ρ 0 ) u × p exp ⁡ ( i k u · r ) d Ω
= U 0 4 π p · ∬ 4 π exp ⁡ ( − i k u ′ · ρ 0 ) δ ↔ ⊥ ( u , u ′ ) × u exp ⁡ ( i k u · r ) d Ω d Ω ′ ,
δ ↔ ⊥ ( u , u ′ ) = ∑ l , m [ Z lm * ( θ ′ , ϕ ′ ) ⊗ Z lm ( θ , ϕ ) + Y lm * ( θ ′ , ϕ ′ ) ⊗ Y lm ( θ , ϕ ) ] ,
Λ lm ( I ) ( r ) = ∫ 4 π Z lm ( θ , ϕ ) exp ( i k r · u ) d Ω ,
Λ lm ( II ) ( r ) = ∫ 4 π Y lm ( θ , ϕ ) exp ( i k r · u ) d Ω ,

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