Abstract

Phase is a common wave parameter used to describe the consequences of light–matter interaction. When the scattering center is smaller than the wavelength, a simplistic geometric interpretation of the measured phase is insufficient. Using energetic arguments, we show that the appropriate description of measurements performed far away from subwavelength objects involves the duration and the effective volume of the elastic scattering event. Because the evanescent fields contribute significantly to coupling and releasing of energy, the effective interaction volume is significantly larger than the physical size of subwavelength particles. Our results also provide means to describe scattering phenomena in dense media where scatterers are in close proximity to each other.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. A. Sommerfeld, Optics. Lectures on Theoretical Physics (Academic, 1964), Vol. IV.
  2. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
  3. N. B. Sawyer, S. P. Morgan, M. G. Somekh, C. W. See, E. Astrakharchik-Farrimond, and B. Y. Shekunov, “Amplitude and phase microscopy for sizing of spherical particles,” Appl. Opt. 42, 4488–4498 (2003).
    [Crossref]
  4. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2008).
  5. E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. 98, 145–147 (1955).
    [Crossref]
  6. C. A. de Carvalho and H. M. Nussenzveig, “Time delay,” Phys. Rep. 364, 83–174 (2002).
    [Crossref]
  7. H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: a proposed resolution of an old paradox,” Phys. Rep. 436, 1–69 (2006).
    [Crossref]
  8. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
    [Crossref]
  9. R. Carminati, J. J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
    [Crossref]
  10. D. N. Pattanayak and E. Wolf, “General form and a new interpretation of the Ewald-Oseen extinction theorem,” Opt. Commun. 6, 217–220 (1972).
    [Crossref]
  11. F. T. Smith, “Lifetime matrix in collision theory,” Phys. Rev. 118, 349–356 (1960).
    [Crossref]
  12. S. Rotter and S. Gigan, “Light fields in complex media: mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).
    [Crossref]
  13. M. I. Mishchenko and L. D. Travis, “Gustav Mie and the evolving discipline of electromagnetic scattering by particles,” Bull. Am. Meteorol. Soc. 89, 1853–1862 (2008).
    [Crossref]
  14. A. Lagendijk and B. A. Van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
    [Crossref]
  15. J. Schäfer, S. C. Lee, and A. Kienle, “Calculation of the near fields for the scattering of electromagnetic waves by multiple infinite cylinders at perpendicular incidence,” J. Quantum Spectrosc. Radiat. Transfer 113, 2113–2123 (2012).
    [Crossref]
  16. J. Schäfer, “MatScat,” 2019, https://www.mathworks.com/matlabcentral/fileexchange/36831-matscat .
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    [Crossref]
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    [Crossref]
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    [Crossref]
  20. R. Rezvani Naraghi and A. Dogariu, “Phase transitions in diffusion of light,” Phys. Rev. Lett. 117, 263901 (2016).
    [Crossref]
  21. P. M. Saulnier, M. P. Zinkin, and G. H. Watson, “Scatterer correlation effects on photon transport in dense random media,” Phys. Rev. B 42, 2621–2623 (1990).
    [Crossref]
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    [Crossref]
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    [Crossref]
  25. P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
    [Crossref]
  26. M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quantum Spectrosc. Radiat. Transfer 55, 535–575 (1996).
    [Crossref]
  27. G. S. Agarwal, “Relation between Waterman’s extended boundary condition and the generalized extinction theorem,” Phys. Rev. D 14, 1168–1171 (1976).
    [Crossref]

2017 (1)

S. Rotter and S. Gigan, “Light fields in complex media: mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).
[Crossref]

2016 (1)

R. Rezvani Naraghi and A. Dogariu, “Phase transitions in diffusion of light,” Phys. Rev. Lett. 117, 263901 (2016).
[Crossref]

2015 (1)

R. Rezvani Naraghi, S. Sukhov, J. J. Saenz, and A. Dogariu, “Near-field effects in mesoscopic light transport,” Phys. Rev. Lett. 115, 203903 (2015).
[Crossref]

2014 (1)

M. Potenza and P. Milani, “Free nanoparticle characterization by optical scattered field analysis: opportunities and perspectives,” J. Nanopart. Res. 16, 2680 (2014).
[Crossref]

2012 (1)

J. Schäfer, S. C. Lee, and A. Kienle, “Calculation of the near fields for the scattering of electromagnetic waves by multiple infinite cylinders at perpendicular incidence,” J. Quantum Spectrosc. Radiat. Transfer 113, 2113–2123 (2012).
[Crossref]

2008 (1)

M. I. Mishchenko and L. D. Travis, “Gustav Mie and the evolving discipline of electromagnetic scattering by particles,” Bull. Am. Meteorol. Soc. 89, 1853–1862 (2008).
[Crossref]

2006 (2)

R. Carminati, J. J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[Crossref]

H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: a proposed resolution of an old paradox,” Phys. Rep. 436, 1–69 (2006).
[Crossref]

2003 (1)

2002 (1)

C. A. de Carvalho and H. M. Nussenzveig, “Time delay,” Phys. Rep. 364, 83–174 (2002).
[Crossref]

1999 (1)

1996 (2)

A. Lagendijk and B. A. Van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[Crossref]

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quantum Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[Crossref]

1994 (2)

1990 (1)

P. M. Saulnier, M. P. Zinkin, and G. H. Watson, “Scatterer correlation effects on photon transport in dense random media,” Phys. Rev. B 42, 2621–2623 (1990).
[Crossref]

1987 (1)

1982 (1)

1976 (1)

G. S. Agarwal, “Relation between Waterman’s extended boundary condition and the generalized extinction theorem,” Phys. Rev. D 14, 1168–1171 (1976).
[Crossref]

1972 (1)

D. N. Pattanayak and E. Wolf, “General form and a new interpretation of the Ewald-Oseen extinction theorem,” Opt. Commun. 6, 217–220 (1972).
[Crossref]

1965 (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[Crossref]

1960 (1)

F. T. Smith, “Lifetime matrix in collision theory,” Phys. Rev. 118, 349–356 (1960).
[Crossref]

1955 (1)

E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. 98, 145–147 (1955).
[Crossref]

Agarwal, G. S.

G. S. Agarwal, “Relation between Waterman’s extended boundary condition and the generalized extinction theorem,” Phys. Rev. D 14, 1168–1171 (1976).
[Crossref]

Astrakharchik-Farrimond, E.

Balgi, G.

Banerjee, S.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2008).

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Bott, A.

Carminati, R.

R. Carminati, J. J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[Crossref]

de Carvalho, C. A.

C. A. de Carvalho and H. M. Nussenzveig, “Time delay,” Phys. Rep. 364, 83–174 (2002).
[Crossref]

Dogariu, A.

R. Rezvani Naraghi and A. Dogariu, “Phase transitions in diffusion of light,” Phys. Rev. Lett. 117, 263901 (2016).
[Crossref]

R. Rezvani Naraghi, S. Sukhov, J. J. Saenz, and A. Dogariu, “Near-field effects in mesoscopic light transport,” Phys. Rev. Lett. 115, 203903 (2015).
[Crossref]

Draine, B. T.

Flatau, P. J.

Fung, A. K.

Gibbs, D.

Gigan, S.

S. Rotter and S. Gigan, “Light fields in complex media: mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).
[Crossref]

Greffet, J. J.

R. Carminati, J. J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[Crossref]

Henkel, C.

R. Carminati, J. J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[Crossref]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2008).

Ishimaru, A.

Kienle, A.

J. Schäfer, S. C. Lee, and A. Kienle, “Calculation of the near fields for the scattering of electromagnetic waves by multiple infinite cylinders at perpendicular incidence,” J. Quantum Spectrosc. Radiat. Transfer 113, 2113–2123 (2012).
[Crossref]

Kuga, Y.

Lagendijk, A.

A. Lagendijk and B. A. Van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[Crossref]

Lee, S. C.

J. Schäfer, S. C. Lee, and A. Kienle, “Calculation of the near fields for the scattering of electromagnetic waves by multiple infinite cylinders at perpendicular incidence,” J. Quantum Spectrosc. Radiat. Transfer 113, 2113–2123 (2012).
[Crossref]

Mackowski, D. W.

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quantum Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[Crossref]

Milani, P.

M. Potenza and P. Milani, “Free nanoparticle characterization by optical scattered field analysis: opportunities and perspectives,” J. Nanopart. Res. 16, 2680 (2014).
[Crossref]

Mishchenko, M. I.

M. I. Mishchenko and L. D. Travis, “Gustav Mie and the evolving discipline of electromagnetic scattering by particles,” Bull. Am. Meteorol. Soc. 89, 1853–1862 (2008).
[Crossref]

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quantum Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[Crossref]

Morgan, S. P.

Nussenzveig, H. M.

C. A. de Carvalho and H. M. Nussenzveig, “Time delay,” Phys. Rep. 364, 83–174 (2002).
[Crossref]

Pattanayak, D. N.

D. N. Pattanayak and E. Wolf, “General form and a new interpretation of the Ewald-Oseen extinction theorem,” Opt. Commun. 6, 217–220 (1972).
[Crossref]

Pierce, J.

Potenza, M.

M. Potenza and P. Milani, “Free nanoparticle characterization by optical scattered field analysis: opportunities and perspectives,” J. Nanopart. Res. 16, 2680 (2014).
[Crossref]

Reynolds, J.

Rezvani Naraghi, R.

R. Rezvani Naraghi and A. Dogariu, “Phase transitions in diffusion of light,” Phys. Rev. Lett. 117, 263901 (2016).
[Crossref]

R. Rezvani Naraghi, S. Sukhov, J. J. Saenz, and A. Dogariu, “Near-field effects in mesoscopic light transport,” Phys. Rev. Lett. 115, 203903 (2015).
[Crossref]

Richter, S.

Rotter, S.

S. Rotter and S. Gigan, “Light fields in complex media: mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).
[Crossref]

Saenz, J. J.

R. Rezvani Naraghi, S. Sukhov, J. J. Saenz, and A. Dogariu, “Near-field effects in mesoscopic light transport,” Phys. Rev. Lett. 115, 203903 (2015).
[Crossref]

Saulnier, P. M.

P. M. Saulnier, M. P. Zinkin, and G. H. Watson, “Scatterer correlation effects on photon transport in dense random media,” Phys. Rev. B 42, 2621–2623 (1990).
[Crossref]

Sawyer, N. B.

Schäfer, J.

J. Schäfer, S. C. Lee, and A. Kienle, “Calculation of the near fields for the scattering of electromagnetic waves by multiple infinite cylinders at perpendicular incidence,” J. Quantum Spectrosc. Radiat. Transfer 113, 2113–2123 (2012).
[Crossref]

See, C. W.

Sevick-Muraca, E.

Shekunov, B. Y.

Shinde, R.

Smith, F. T.

F. T. Smith, “Lifetime matrix in collision theory,” Phys. Rev. 118, 349–356 (1960).
[Crossref]

Somekh, M. G.

Sommerfeld, A.

A. Sommerfeld, Optics. Lectures on Theoretical Physics (Academic, 1964), Vol. IV.

Sukhov, S.

R. Rezvani Naraghi, S. Sukhov, J. J. Saenz, and A. Dogariu, “Near-field effects in mesoscopic light transport,” Phys. Rev. Lett. 115, 203903 (2015).
[Crossref]

Travis, L. D.

M. I. Mishchenko and L. D. Travis, “Gustav Mie and the evolving discipline of electromagnetic scattering by particles,” Bull. Am. Meteorol. Soc. 89, 1853–1862 (2008).
[Crossref]

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quantum Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[Crossref]

Tsang, L.

Van Tiggelen, B. A.

A. Lagendijk and B. A. Van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[Crossref]

Vigoureux, J. M.

R. Carminati, J. J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[Crossref]

Waterman, P. C.

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[Crossref]

Watson, G. H.

P. M. Saulnier, M. P. Zinkin, and G. H. Watson, “Scatterer correlation effects on photon transport in dense random media,” Phys. Rev. B 42, 2621–2623 (1990).
[Crossref]

West, R.

Wigner, E. P.

E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. 98, 145–147 (1955).
[Crossref]

Winful, H. G.

H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: a proposed resolution of an old paradox,” Phys. Rep. 436, 1–69 (2006).
[Crossref]

Wolf, E.

D. N. Pattanayak and E. Wolf, “General form and a new interpretation of the Ewald-Oseen extinction theorem,” Opt. Commun. 6, 217–220 (1972).
[Crossref]

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Zdunkowski, W.

Zinkin, M. P.

P. M. Saulnier, M. P. Zinkin, and G. H. Watson, “Scatterer correlation effects on photon transport in dense random media,” Phys. Rev. B 42, 2621–2623 (1990).
[Crossref]

Appl. Opt. (2)

Bull. Am. Meteorol. Soc. (1)

M. I. Mishchenko and L. D. Travis, “Gustav Mie and the evolving discipline of electromagnetic scattering by particles,” Bull. Am. Meteorol. Soc. 89, 1853–1862 (2008).
[Crossref]

J. Nanopart. Res. (1)

M. Potenza and P. Milani, “Free nanoparticle characterization by optical scattered field analysis: opportunities and perspectives,” J. Nanopart. Res. 16, 2680 (2014).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Quantum Spectrosc. Radiat. Transfer (2)

J. Schäfer, S. C. Lee, and A. Kienle, “Calculation of the near fields for the scattering of electromagnetic waves by multiple infinite cylinders at perpendicular incidence,” J. Quantum Spectrosc. Radiat. Transfer 113, 2113–2123 (2012).
[Crossref]

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quantum Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[Crossref]

Opt. Commun. (2)

R. Carminati, J. J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[Crossref]

D. N. Pattanayak and E. Wolf, “General form and a new interpretation of the Ewald-Oseen extinction theorem,” Opt. Commun. 6, 217–220 (1972).
[Crossref]

Phys. Rep. (3)

A. Lagendijk and B. A. Van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–215 (1996).
[Crossref]

C. A. de Carvalho and H. M. Nussenzveig, “Time delay,” Phys. Rep. 364, 83–174 (2002).
[Crossref]

H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: a proposed resolution of an old paradox,” Phys. Rep. 436, 1–69 (2006).
[Crossref]

Phys. Rev. (2)

E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. 98, 145–147 (1955).
[Crossref]

F. T. Smith, “Lifetime matrix in collision theory,” Phys. Rev. 118, 349–356 (1960).
[Crossref]

Phys. Rev. B (1)

P. M. Saulnier, M. P. Zinkin, and G. H. Watson, “Scatterer correlation effects on photon transport in dense random media,” Phys. Rev. B 42, 2621–2623 (1990).
[Crossref]

Phys. Rev. D (1)

G. S. Agarwal, “Relation between Waterman’s extended boundary condition and the generalized extinction theorem,” Phys. Rev. D 14, 1168–1171 (1976).
[Crossref]

Phys. Rev. Lett. (2)

R. Rezvani Naraghi, S. Sukhov, J. J. Saenz, and A. Dogariu, “Near-field effects in mesoscopic light transport,” Phys. Rev. Lett. 115, 203903 (2015).
[Crossref]

R. Rezvani Naraghi and A. Dogariu, “Phase transitions in diffusion of light,” Phys. Rev. Lett. 117, 263901 (2016).
[Crossref]

Proc. IEEE (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[Crossref]

Rev. Mod. Phys. (1)

S. Rotter and S. Gigan, “Light fields in complex media: mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).
[Crossref]

Other (4)

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2008).

A. Sommerfeld, Optics. Lectures on Theoretical Physics (Academic, 1964), Vol. IV.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

J. Schäfer, “MatScat,” 2019, https://www.mathworks.com/matlabcentral/fileexchange/36831-matscat .

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Figures (7)

Fig. 1.
Fig. 1. Geometric delay Δ g and phase delay Δ ϕ as a function of size a of subwavelength particles ( k = 2 π / λ is the vacuum wavenumber). Also shown is time delay Δ d corresponding to dipolar scattering (see text). The inset shows the ratio Δ ϕ / Δ g evaluated over the range indicated by the green dashed line.
Fig. 2.
Fig. 2. Scattering event describes the transformation between (a) initial and (b) final field configurations. The outcome of scattering is usually quantified after the wave has propagated in the far field, as indicated by the red dotted lines. Explaining how this transformation occurred requires a (c) near-field examination.
Fig. 3.
Fig. 3. Interaction volume V defining the dwell time τ w . (a) Free-space propagation. (b) Scattering by a small particle.
Fig. 4.
Fig. 4. Dwell time τ w as a function of size of the interaction volume for spherical particles of polystyrene (PS) and TiO 2 in air and with sizes as indicated.
Fig. 5.
Fig. 5. Comparison between Δ ϕ and Δ w estimated for different sizes of the interaction volume l . The panels illustrate the case PS and TiO 2 particles of different sizes k a .
Fig. 6.
Fig. 6. Volumes associated with the scattering event estimated based on the scattering cross section σ sc 3 / 2 (planar interception) and evaluated based on the scattering dwell time l opt 3 (volumetric interaction). Also shown is the physical volume a 3 of the subwavelength particle.
Fig. 7.
Fig. 7. Deviations of measured transport mean free path l * from the values l ISA * estimated using ISA. Arrows indicate the threshold volume fractions for which the interaction volumes l opt 3 of neighboring particles will start to intersect. The experimental data are taken from different literature reports and correspond to (a) PS particles illuminated by 632.8 nm [22], (b) PS particles illuminated by 670 nm [23], and (c) Teflon particles illuminated by 632.8 nm [24].

Equations (3)

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

τ w = ( V ) u ( r ) d r 3 ( A ) S out ( r ) · n ( r ) d r 2 ,
τ w ( V ) u ( r ) d r 3 ϕ i ( ) .
τ w cube u ( r ) d r 3 l 2 | S i | .