Abstract

The optical theorem provides a powerful tool for calculating the extinction cross section of a particle from a solution to Maxwell’s equations, relating the cross section to the scattering amplitude in the forward direction. The theorem has been generalized by a number of other workers to consider a particle near an interface between media with different refractive indices. Here we present a derivation of the generalized optical theorem that is valid for a particle embedded in the interface, as well as an incident beam undergoing total internal reflection. We also obtain an additional useful physical result: we show that the far-field scattered field must be zero in the direction parallel to the interface. Our results enable the verification of computations of scattering by particles embedded in interfaces and may be relevant to experiments on colloidal particles at fluid interfaces.

© 2013 Optical Society of America

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References

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  1. M. Born, E. Wolf, and A. B. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).
  2. H. van de Hulst, Light Scattering by Small Particles (Dover Publications, 1981).
  3. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  4. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [CrossRef]
  5. V. Villamizar and O. Rojas, “Time-dependent numerical method with boundary-conforming curvilinear coordinates applied to wave interactions with prototypical antennas,” J. Comput. Phys. 177, 1–36 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  8. M. I. Mishchenko, M. J. Berg, C. M. Sorensen, and C. V. van der Mee, “On definition and measurement of extinction cross section,” J. Quant. Spectrosc. Radiat. Transfer 110, 323–327 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  11. J. H. Kim, S. H. Ehrman, G. W. Mulholland, and T. A. Germer, “Polarized light scattering by dielectric and metallic spheres on silicon wafers,” Appl. Opt. 41, 5405–5412 (2002).
    [CrossRef]
  12. P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for scalar fields,” Phys. Rev. E 70, 36611 (2004).
    [CrossRef]
  13. D. R. Lytle, P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for electromagnetic fields,” Phys. Rev. E 71, 56610 (2005).
    [CrossRef]
  14. D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, “Physical ageing of the contact line on colloidal particles at liquid interfaces,” Nat. Mater. 11, 138–142 (2012).
    [CrossRef]
  15. P. S. Carney, “The optical cross-section theorem with incident fields containing evanescent components,” J. Mod. Opt. 46, 891–899 (1999).
  16. D. Torrungrueng, B. Ungan, and J. Johnson, “Optical theorem for electromagnetic scattering by a three-dimensional scatterer in the presence of a lossless half space,” IEEE Geosci. Remote Sens. Lett. 1, 131–135 (2004).
    [CrossRef]
  17. A. Doicu, R. Schuh, and T. Wriedt, “Scattering by particles on or near a plane surface,” in Light Scattering Reviews 3 (Springer, 2008), pp. 109–130.
  18. W. Lukosz and R. Kunz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. II. Radiation patterns of perpendicular oriented dipoles,” J. Opt. Soc. Am. 67, 1615–1619 (1977).
    [CrossRef]

2012 (1)

D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, “Physical ageing of the contact line on colloidal particles at liquid interfaces,” Nat. Mater. 11, 138–142 (2012).
[CrossRef]

2009 (1)

M. I. Mishchenko, M. J. Berg, C. M. Sorensen, and C. V. van der Mee, “On definition and measurement of extinction cross section,” J. Quant. Spectrosc. Radiat. Transfer 110, 323–327 (2009).
[CrossRef]

2008 (1)

2005 (1)

D. R. Lytle, P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for electromagnetic fields,” Phys. Rev. E 71, 56610 (2005).
[CrossRef]

2004 (2)

P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for scalar fields,” Phys. Rev. E 70, 36611 (2004).
[CrossRef]

D. Torrungrueng, B. Ungan, and J. Johnson, “Optical theorem for electromagnetic scattering by a three-dimensional scatterer in the presence of a lossless half space,” IEEE Geosci. Remote Sens. Lett. 1, 131–135 (2004).
[CrossRef]

2002 (2)

V. Villamizar and O. Rojas, “Time-dependent numerical method with boundary-conforming curvilinear coordinates applied to wave interactions with prototypical antennas,” J. Comput. Phys. 177, 1–36 (2002).
[CrossRef]

J. H. Kim, S. H. Ehrman, G. W. Mulholland, and T. A. Germer, “Polarized light scattering by dielectric and metallic spheres on silicon wafers,” Appl. Opt. 41, 5405–5412 (2002).
[CrossRef]

1999 (2)

L. Sung, G. W. Mulholland, and T. A. Germer, “Polarized light-scattering measurements of dielectric spheres upon a silicon surface,” Opt. Lett. 24, 866–868 (1999).
[CrossRef]

P. S. Carney, “The optical cross-section theorem with incident fields containing evanescent components,” J. Mod. Opt. 46, 891–899 (1999).

1994 (1)

1988 (1)

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

1987 (1)

1977 (1)

Berg, M. J.

M. I. Mishchenko, M. J. Berg, C. M. Sorensen, and C. V. van der Mee, “On definition and measurement of extinction cross section,” J. Quant. Spectrosc. Radiat. Transfer 110, 323–327 (2009).
[CrossRef]

M. J. Berg, C. M. Sorensen, and A. Chakrabarti, “Extinction and the optical theorem. Part I. Single particles,” J. Opt. Soc. Am. A 25, 1504–1513 (2008).
[CrossRef]

Bhatia, A. B.

M. Born, E. Wolf, and A. B. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Born, M.

M. Born, E. Wolf, and A. B. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

Brenner, M. P.

D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, “Physical ageing of the contact line on colloidal particles at liquid interfaces,” Nat. Mater. 11, 138–142 (2012).
[CrossRef]

Carney, P. S.

D. R. Lytle, P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for electromagnetic fields,” Phys. Rev. E 71, 56610 (2005).
[CrossRef]

P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for scalar fields,” Phys. Rev. E 70, 36611 (2004).
[CrossRef]

P. S. Carney, “The optical cross-section theorem with incident fields containing evanescent components,” J. Mod. Opt. 46, 891–899 (1999).

Chakrabarti, A.

Doicu, A.

A. Doicu, R. Schuh, and T. Wriedt, “Scattering by particles on or near a plane surface,” in Light Scattering Reviews 3 (Springer, 2008), pp. 109–130.

Draine, B. T.

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

Ehrman, S. H.

Germer, T. A.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Johnson, J.

D. Torrungrueng, B. Ungan, and J. Johnson, “Optical theorem for electromagnetic scattering by a three-dimensional scatterer in the presence of a lossless half space,” IEEE Geosci. Remote Sens. Lett. 1, 131–135 (2004).
[CrossRef]

Kaz, D. M.

D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, “Physical ageing of the contact line on colloidal particles at liquid interfaces,” Nat. Mater. 11, 138–142 (2012).
[CrossRef]

Kim, J. H.

Kunz, R.

Lukosz, W.

Lytle, D. R.

D. R. Lytle, P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for electromagnetic fields,” Phys. Rev. E 71, 56610 (2005).
[CrossRef]

Mackowski, D. W.

Mani, M.

D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, “Physical ageing of the contact line on colloidal particles at liquid interfaces,” Nat. Mater. 11, 138–142 (2012).
[CrossRef]

Manoharan, V. N.

D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, “Physical ageing of the contact line on colloidal particles at liquid interfaces,” Nat. Mater. 11, 138–142 (2012).
[CrossRef]

McGorty, R.

D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, “Physical ageing of the contact line on colloidal particles at liquid interfaces,” Nat. Mater. 11, 138–142 (2012).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, M. J. Berg, C. M. Sorensen, and C. V. van der Mee, “On definition and measurement of extinction cross section,” J. Quant. Spectrosc. Radiat. Transfer 110, 323–327 (2009).
[CrossRef]

Mulholland, G. W.

Nahm, K. B.

Rojas, O.

V. Villamizar and O. Rojas, “Time-dependent numerical method with boundary-conforming curvilinear coordinates applied to wave interactions with prototypical antennas,” J. Comput. Phys. 177, 1–36 (2002).
[CrossRef]

Schotland, J. C.

D. R. Lytle, P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for electromagnetic fields,” Phys. Rev. E 71, 56610 (2005).
[CrossRef]

P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for scalar fields,” Phys. Rev. E 70, 36611 (2004).
[CrossRef]

Schuh, R.

A. Doicu, R. Schuh, and T. Wriedt, “Scattering by particles on or near a plane surface,” in Light Scattering Reviews 3 (Springer, 2008), pp. 109–130.

Sorensen, C. M.

M. I. Mishchenko, M. J. Berg, C. M. Sorensen, and C. V. van der Mee, “On definition and measurement of extinction cross section,” J. Quant. Spectrosc. Radiat. Transfer 110, 323–327 (2009).
[CrossRef]

M. J. Berg, C. M. Sorensen, and A. Chakrabarti, “Extinction and the optical theorem. Part I. Single particles,” J. Opt. Soc. Am. A 25, 1504–1513 (2008).
[CrossRef]

Sung, L.

Torrungrueng, D.

D. Torrungrueng, B. Ungan, and J. Johnson, “Optical theorem for electromagnetic scattering by a three-dimensional scatterer in the presence of a lossless half space,” IEEE Geosci. Remote Sens. Lett. 1, 131–135 (2004).
[CrossRef]

Ungan, B.

D. Torrungrueng, B. Ungan, and J. Johnson, “Optical theorem for electromagnetic scattering by a three-dimensional scatterer in the presence of a lossless half space,” IEEE Geosci. Remote Sens. Lett. 1, 131–135 (2004).
[CrossRef]

van de Hulst, H.

H. van de Hulst, Light Scattering by Small Particles (Dover Publications, 1981).

van der Mee, C. V.

M. I. Mishchenko, M. J. Berg, C. M. Sorensen, and C. V. van der Mee, “On definition and measurement of extinction cross section,” J. Quant. Spectrosc. Radiat. Transfer 110, 323–327 (2009).
[CrossRef]

Villamizar, V.

V. Villamizar and O. Rojas, “Time-dependent numerical method with boundary-conforming curvilinear coordinates applied to wave interactions with prototypical antennas,” J. Comput. Phys. 177, 1–36 (2002).
[CrossRef]

Wolf, E.

D. R. Lytle, P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for electromagnetic fields,” Phys. Rev. E 71, 56610 (2005).
[CrossRef]

P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for scalar fields,” Phys. Rev. E 70, 36611 (2004).
[CrossRef]

M. Born, E. Wolf, and A. B. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

Wolfe, W. L.

Wriedt, T.

A. Doicu, R. Schuh, and T. Wriedt, “Scattering by particles on or near a plane surface,” in Light Scattering Reviews 3 (Springer, 2008), pp. 109–130.

Appl. Opt. (2)

Astrophys. J. (1)

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

IEEE Geosci. Remote Sens. Lett. (1)

D. Torrungrueng, B. Ungan, and J. Johnson, “Optical theorem for electromagnetic scattering by a three-dimensional scatterer in the presence of a lossless half space,” IEEE Geosci. Remote Sens. Lett. 1, 131–135 (2004).
[CrossRef]

J. Comput. Phys. (1)

V. Villamizar and O. Rojas, “Time-dependent numerical method with boundary-conforming curvilinear coordinates applied to wave interactions with prototypical antennas,” J. Comput. Phys. 177, 1–36 (2002).
[CrossRef]

J. Mod. Opt. (1)

P. S. Carney, “The optical cross-section theorem with incident fields containing evanescent components,” J. Mod. Opt. 46, 891–899 (1999).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Quant. Spectrosc. Radiat. Transfer (1)

M. I. Mishchenko, M. J. Berg, C. M. Sorensen, and C. V. van der Mee, “On definition and measurement of extinction cross section,” J. Quant. Spectrosc. Radiat. Transfer 110, 323–327 (2009).
[CrossRef]

Nat. Mater. (1)

D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, “Physical ageing of the contact line on colloidal particles at liquid interfaces,” Nat. Mater. 11, 138–142 (2012).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. E (2)

P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for scalar fields,” Phys. Rev. E 70, 36611 (2004).
[CrossRef]

D. R. Lytle, P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for electromagnetic fields,” Phys. Rev. E 71, 56610 (2005).
[CrossRef]

Other (4)

M. Born, E. Wolf, and A. B. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

H. van de Hulst, Light Scattering by Small Particles (Dover Publications, 1981).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

A. Doicu, R. Schuh, and T. Wriedt, “Scattering by particles on or near a plane surface,” in Light Scattering Reviews 3 (Springer, 2008), pp. 109–130.

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Figures (3)

Fig. 1.
Fig. 1.

(a) Schematic and (b) coordinate system for the particle and interface, as well as the directions of the incident, reflected, transmitted, and scattered fields.

Fig. 2.
Fig. 2.

Mutual orientations of Ei and Hs, and Es and Hi, for “incident” and scattered fields that are copropagating or counterpropagating. All vectors in each part exist at the same physical point, but have been separated for clarity. (a) The “incident” and scattered fields propagate in the same direction, so both Ei×Hs and Es×Hi point in the same direction. (b) The “incident” and scattered fields propagate in opposite directions; Ei×Hs and Es×Hi point in opposite directions.

Fig. 3.
Fig. 3.

Polar angle θ (measured with respect to the k vector) at the interface depends on the azimuthal angle ϕ.

Equations (18)

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×E=1cμHtik×E=ik0μH,H=kμk0×E.
Iinc=n12μ1.
Es=ξ(θ,ϕ)exp(ink0r)k0r=(ξθ(θ,ϕ)θ^+ξϕ(θ,ϕ)ϕ^)exp(ink0r)k0r,
Hs=nμr^×Es,
Wa=12Re(4π(E×H*)·r^r2dΩ)=12Re(4π(Ei×Hi*+Es×Hs*+Ei×Hs*+Es×Hi*)·r^r2dΩ),
Wa+Ws=Wext=12Re(4π(Ei×Hs*+Es×Hi*)·r^r2dΩ).
r^·(Ei×Hs*)=Ei·(r^×Hs*)=Ei·(r^×(r^×nμEs*))=nμEi·Es*.
r^·Es×Hi*=±nμEi*·Es,
mediumEi·Es*r2dΩ=ϕinterfacecosθ=11k0rξ*(θ,ϕ)·E0eink0r(1cosθ)r2dcosθdϕ,
mediumEi·Es*r2dΩ=ϕξ*(θ,ϕ)·E0eink0r(1cosθ)ink02|interfacecosθ=1dϕϕinterfacecosθ=11ink02ξ*(θ,ϕ)cosθ·E0eink0r(1cosθ)dcosθdϕ.
mediumEi·Es*r2dΩ=2πξ*(θ,ϕ)·E0ink02+interface term.
mediumEi·Es*r2dΩ=2πink02ξ*(θ,ϕ)·E0.
Wext=2πk02Re(1iμ2ξ*(kt)·E2,t+1iμ1ξ*(kr)·E1,r)=2πk02Im(1μ2ξ*(kt)·E2,t+1μ1ξ*(kr)·E1,r).
Wext=2πk02Im(1μ2ξ(kt)·E2,t+1μ1ξ(kr)·E1,r).
σe=4πn1μ1k02Im(1μ2ξ(kt)·E2,t+1μ1ξ(kr)·E1,r).
medium 2Ei·Es*r2dΩ=ϕ=02πcosθ=01eκrcosθeikxrsinθcosϕξj*(θ,ϕ)ein2k0rk0rr2d(cosθ)dϕ=z=0reκzϕ=02πξj*(θ,ϕ)ein2k0rk0eikxr2z2cosϕdϕdz,
σe=4πn1μ12k02Im(ξ*(kr)·E1,r).
2+n2(z)k02E=2z2E(kx2+ky2)E+k02n2(z)E=2z2E+k02(n2(z)kx2+ky2k02)=0.

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