Abstract

Enhanced backscattering (EBS), also known as weak localization of light, is derived using the Huygens–Fresnel principle and backscattering is generally shown to be the sum of an incoherent baseline and a phase conjugated portion of the incident wave that forms EBS. The phase conjugated portion is truncated by an effective aperture described by the probability function P(s) of coherent path-pair separations. P(s) is determined by the scattering properties of the medium and so characterization of EBS can be used for metrology of scattering materials. A three dimensional intensity peak is predicted in free space at a point conjugate to the source and is experimentally observed.

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References

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  1. A. Dogariu and G. D. Boreman, “Enhanced backscattering in a converging-beam configuration,” Opt. Lett. 21, 1718–1720 (1996).
    [CrossRef] [PubMed]
  2. N. C. Bruce, “A comparison of the converging and diverging geometries for measuring enhanced backscatter,” J. Mod. Opt. 49, 2167–2181 (2002).
    [CrossRef]
  3. P. Gross, M. Störzer, S. Fiebig, M. Clausen, G. Maret, and C. Aegerter, “A Precise method to determine the angular distribution of backscattered light to high angles,” Rev. Sci. Instrum. 78, 033105 (2007).
    [CrossRef] [PubMed]
  4. M. A. Noginov, S. U. Egarievwe, H. J. Caulfield, N. E. Noginova, M. Curley, P. Venkateswarlu, A. Williams, and J. Paitz, “Diffusion and pseudo-phase-conjugation effects in coherent backscattering from nd0.5la0.5al3(bo3)4 ceramic,” Opt. Mater. 10, 1–7 (1998).
    [CrossRef]
  5. F. Reil and J. E. Thomas, “Observation of phase conjugation of light arising from enhanced backscattering in a random medium,” Phys. Rev. Lett. 95, 143903 (2005).
    [CrossRef] [PubMed]
  6. Y. L. Kim, P. Pradhan, M. H. Kim, and V. Backman, “Circular polarization memory effect in low-coherence enhanced backscattering of light,” Opt. Lett. 31, 2744–2746 (2006).
    [CrossRef] [PubMed]
  7. E. Akkermans, P. Wolf, R. Maynard, and G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. France 49, 77–98 (1988).
    [CrossRef]
  8. P. Sheng, Scattering and localization of classical waves in random media (World Scientific Pub Co Inc, 1990).
  9. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University Press, 2006).
  10. E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, 2007).
    [CrossRef]
  11. R. T. Deck and H. J. Simon, “Simple diffractive theory of enhanced backscattering,” J. Opt. Soc. Am. B 10, 1000–1005 (1993).
    [CrossRef]
  12. T. Okamoto and T. Asakura, “Enhanced backscattering of partially coherent light,” Opt. Lett. 21, 369–371 (1996).
    [CrossRef] [PubMed]
  13. Y. Kim, Y. Liu, V. Turzhitsky, H. Roy, R. Wali, and V. Backman, “Coherent backscattering spectroscopy,” Opt. Lett. 29, 1906–1908 (2004).
    [CrossRef] [PubMed]
  14. J. Goodman, Introduction to Fourier Optics (McGraw-Hill Science, Engineering & Mathematics, 1996).
  15. V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, “Characterization of light transport in scattering media at subdiffusion length scales with low-coherence enhanced backscattering,” IEEE J. Sel. Top. Quantum Electron. 16, 619–626 (2010).
    [CrossRef] [PubMed]
  16. M. Born and E. Wolf, Principles of Optics , 7th ed. (Cambridge University Press, 1999).

2010 (1)

V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, “Characterization of light transport in scattering media at subdiffusion length scales with low-coherence enhanced backscattering,” IEEE J. Sel. Top. Quantum Electron. 16, 619–626 (2010).
[CrossRef] [PubMed]

2007 (1)

P. Gross, M. Störzer, S. Fiebig, M. Clausen, G. Maret, and C. Aegerter, “A Precise method to determine the angular distribution of backscattered light to high angles,” Rev. Sci. Instrum. 78, 033105 (2007).
[CrossRef] [PubMed]

2006 (1)

2005 (1)

F. Reil and J. E. Thomas, “Observation of phase conjugation of light arising from enhanced backscattering in a random medium,” Phys. Rev. Lett. 95, 143903 (2005).
[CrossRef] [PubMed]

2004 (1)

2002 (1)

N. C. Bruce, “A comparison of the converging and diverging geometries for measuring enhanced backscatter,” J. Mod. Opt. 49, 2167–2181 (2002).
[CrossRef]

1998 (1)

M. A. Noginov, S. U. Egarievwe, H. J. Caulfield, N. E. Noginova, M. Curley, P. Venkateswarlu, A. Williams, and J. Paitz, “Diffusion and pseudo-phase-conjugation effects in coherent backscattering from nd0.5la0.5al3(bo3)4 ceramic,” Opt. Mater. 10, 1–7 (1998).
[CrossRef]

1996 (2)

1993 (1)

1988 (1)

E. Akkermans, P. Wolf, R. Maynard, and G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. France 49, 77–98 (1988).
[CrossRef]

Aegerter, C.

P. Gross, M. Störzer, S. Fiebig, M. Clausen, G. Maret, and C. Aegerter, “A Precise method to determine the angular distribution of backscattered light to high angles,” Rev. Sci. Instrum. 78, 033105 (2007).
[CrossRef] [PubMed]

Akkermans, E.

E. Akkermans, P. Wolf, R. Maynard, and G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. France 49, 77–98 (1988).
[CrossRef]

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, 2007).
[CrossRef]

Asakura, T.

Backman, V.

V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, “Characterization of light transport in scattering media at subdiffusion length scales with low-coherence enhanced backscattering,” IEEE J. Sel. Top. Quantum Electron. 16, 619–626 (2010).
[CrossRef] [PubMed]

Y. L. Kim, P. Pradhan, M. H. Kim, and V. Backman, “Circular polarization memory effect in low-coherence enhanced backscattering of light,” Opt. Lett. 31, 2744–2746 (2006).
[CrossRef] [PubMed]

Y. Kim, Y. Liu, V. Turzhitsky, H. Roy, R. Wali, and V. Backman, “Coherent backscattering spectroscopy,” Opt. Lett. 29, 1906–1908 (2004).
[CrossRef] [PubMed]

Boreman, G. D.

Born, M.

M. Born and E. Wolf, Principles of Optics , 7th ed. (Cambridge University Press, 1999).

Bruce, N. C.

N. C. Bruce, “A comparison of the converging and diverging geometries for measuring enhanced backscatter,” J. Mod. Opt. 49, 2167–2181 (2002).
[CrossRef]

Caulfield, H. J.

M. A. Noginov, S. U. Egarievwe, H. J. Caulfield, N. E. Noginova, M. Curley, P. Venkateswarlu, A. Williams, and J. Paitz, “Diffusion and pseudo-phase-conjugation effects in coherent backscattering from nd0.5la0.5al3(bo3)4 ceramic,” Opt. Mater. 10, 1–7 (1998).
[CrossRef]

Clausen, M.

P. Gross, M. Störzer, S. Fiebig, M. Clausen, G. Maret, and C. Aegerter, “A Precise method to determine the angular distribution of backscattered light to high angles,” Rev. Sci. Instrum. 78, 033105 (2007).
[CrossRef] [PubMed]

Curley, M.

M. A. Noginov, S. U. Egarievwe, H. J. Caulfield, N. E. Noginova, M. Curley, P. Venkateswarlu, A. Williams, and J. Paitz, “Diffusion and pseudo-phase-conjugation effects in coherent backscattering from nd0.5la0.5al3(bo3)4 ceramic,” Opt. Mater. 10, 1–7 (1998).
[CrossRef]

Deck, R. T.

Dogariu, A.

Egarievwe, S. U.

M. A. Noginov, S. U. Egarievwe, H. J. Caulfield, N. E. Noginova, M. Curley, P. Venkateswarlu, A. Williams, and J. Paitz, “Diffusion and pseudo-phase-conjugation effects in coherent backscattering from nd0.5la0.5al3(bo3)4 ceramic,” Opt. Mater. 10, 1–7 (1998).
[CrossRef]

Fiebig, S.

P. Gross, M. Störzer, S. Fiebig, M. Clausen, G. Maret, and C. Aegerter, “A Precise method to determine the angular distribution of backscattered light to high angles,” Rev. Sci. Instrum. 78, 033105 (2007).
[CrossRef] [PubMed]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill Science, Engineering & Mathematics, 1996).

Gross, P.

P. Gross, M. Störzer, S. Fiebig, M. Clausen, G. Maret, and C. Aegerter, “A Precise method to determine the angular distribution of backscattered light to high angles,” Rev. Sci. Instrum. 78, 033105 (2007).
[CrossRef] [PubMed]

Kim, M. H.

Kim, Y.

Kim, Y. L.

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University Press, 2006).

Liu, Y.

Maret, G.

P. Gross, M. Störzer, S. Fiebig, M. Clausen, G. Maret, and C. Aegerter, “A Precise method to determine the angular distribution of backscattered light to high angles,” Rev. Sci. Instrum. 78, 033105 (2007).
[CrossRef] [PubMed]

E. Akkermans, P. Wolf, R. Maynard, and G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. France 49, 77–98 (1988).
[CrossRef]

Maynard, R.

E. Akkermans, P. Wolf, R. Maynard, and G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. France 49, 77–98 (1988).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University Press, 2006).

Montambaux, G.

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, 2007).
[CrossRef]

Mutyal, N. N.

V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, “Characterization of light transport in scattering media at subdiffusion length scales with low-coherence enhanced backscattering,” IEEE J. Sel. Top. Quantum Electron. 16, 619–626 (2010).
[CrossRef] [PubMed]

Noginov, M. A.

M. A. Noginov, S. U. Egarievwe, H. J. Caulfield, N. E. Noginova, M. Curley, P. Venkateswarlu, A. Williams, and J. Paitz, “Diffusion and pseudo-phase-conjugation effects in coherent backscattering from nd0.5la0.5al3(bo3)4 ceramic,” Opt. Mater. 10, 1–7 (1998).
[CrossRef]

Noginova, N. E.

M. A. Noginov, S. U. Egarievwe, H. J. Caulfield, N. E. Noginova, M. Curley, P. Venkateswarlu, A. Williams, and J. Paitz, “Diffusion and pseudo-phase-conjugation effects in coherent backscattering from nd0.5la0.5al3(bo3)4 ceramic,” Opt. Mater. 10, 1–7 (1998).
[CrossRef]

Okamoto, T.

Paitz, J.

M. A. Noginov, S. U. Egarievwe, H. J. Caulfield, N. E. Noginova, M. Curley, P. Venkateswarlu, A. Williams, and J. Paitz, “Diffusion and pseudo-phase-conjugation effects in coherent backscattering from nd0.5la0.5al3(bo3)4 ceramic,” Opt. Mater. 10, 1–7 (1998).
[CrossRef]

Pradhan, P.

Reil, F.

F. Reil and J. E. Thomas, “Observation of phase conjugation of light arising from enhanced backscattering in a random medium,” Phys. Rev. Lett. 95, 143903 (2005).
[CrossRef] [PubMed]

Rogers, J. D.

V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, “Characterization of light transport in scattering media at subdiffusion length scales with low-coherence enhanced backscattering,” IEEE J. Sel. Top. Quantum Electron. 16, 619–626 (2010).
[CrossRef] [PubMed]

Roy, H.

Roy, H. K.

V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, “Characterization of light transport in scattering media at subdiffusion length scales with low-coherence enhanced backscattering,” IEEE J. Sel. Top. Quantum Electron. 16, 619–626 (2010).
[CrossRef] [PubMed]

Sheng, P.

P. Sheng, Scattering and localization of classical waves in random media (World Scientific Pub Co Inc, 1990).

Simon, H. J.

Störzer, M.

P. Gross, M. Störzer, S. Fiebig, M. Clausen, G. Maret, and C. Aegerter, “A Precise method to determine the angular distribution of backscattered light to high angles,” Rev. Sci. Instrum. 78, 033105 (2007).
[CrossRef] [PubMed]

Thomas, J. E.

F. Reil and J. E. Thomas, “Observation of phase conjugation of light arising from enhanced backscattering in a random medium,” Phys. Rev. Lett. 95, 143903 (2005).
[CrossRef] [PubMed]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University Press, 2006).

Turzhitsky, V.

V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, “Characterization of light transport in scattering media at subdiffusion length scales with low-coherence enhanced backscattering,” IEEE J. Sel. Top. Quantum Electron. 16, 619–626 (2010).
[CrossRef] [PubMed]

Y. Kim, Y. Liu, V. Turzhitsky, H. Roy, R. Wali, and V. Backman, “Coherent backscattering spectroscopy,” Opt. Lett. 29, 1906–1908 (2004).
[CrossRef] [PubMed]

Venkateswarlu, P.

M. A. Noginov, S. U. Egarievwe, H. J. Caulfield, N. E. Noginova, M. Curley, P. Venkateswarlu, A. Williams, and J. Paitz, “Diffusion and pseudo-phase-conjugation effects in coherent backscattering from nd0.5la0.5al3(bo3)4 ceramic,” Opt. Mater. 10, 1–7 (1998).
[CrossRef]

Wali, R.

Williams, A.

M. A. Noginov, S. U. Egarievwe, H. J. Caulfield, N. E. Noginova, M. Curley, P. Venkateswarlu, A. Williams, and J. Paitz, “Diffusion and pseudo-phase-conjugation effects in coherent backscattering from nd0.5la0.5al3(bo3)4 ceramic,” Opt. Mater. 10, 1–7 (1998).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics , 7th ed. (Cambridge University Press, 1999).

Wolf, P.

E. Akkermans, P. Wolf, R. Maynard, and G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. France 49, 77–98 (1988).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, “Characterization of light transport in scattering media at subdiffusion length scales with low-coherence enhanced backscattering,” IEEE J. Sel. Top. Quantum Electron. 16, 619–626 (2010).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

N. C. Bruce, “A comparison of the converging and diverging geometries for measuring enhanced backscatter,” J. Mod. Opt. 49, 2167–2181 (2002).
[CrossRef]

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

J. Phys. France (1)

E. Akkermans, P. Wolf, R. Maynard, and G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. France 49, 77–98 (1988).
[CrossRef]

Opt. Lett. (4)

Opt. Mater. (1)

M. A. Noginov, S. U. Egarievwe, H. J. Caulfield, N. E. Noginova, M. Curley, P. Venkateswarlu, A. Williams, and J. Paitz, “Diffusion and pseudo-phase-conjugation effects in coherent backscattering from nd0.5la0.5al3(bo3)4 ceramic,” Opt. Mater. 10, 1–7 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

F. Reil and J. E. Thomas, “Observation of phase conjugation of light arising from enhanced backscattering in a random medium,” Phys. Rev. Lett. 95, 143903 (2005).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

P. Gross, M. Störzer, S. Fiebig, M. Clausen, G. Maret, and C. Aegerter, “A Precise method to determine the angular distribution of backscattered light to high angles,” Rev. Sci. Instrum. 78, 033105 (2007).
[CrossRef] [PubMed]

Other (5)

P. Sheng, Scattering and localization of classical waves in random media (World Scientific Pub Co Inc, 1990).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University Press, 2006).

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, 2007).
[CrossRef]

M. Born and E. Wolf, Principles of Optics , 7th ed. (Cambridge University Press, 1999).

J. Goodman, Introduction to Fourier Optics (McGraw-Hill Science, Engineering & Mathematics, 1996).

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Figures (7)

Fig. 1
Fig. 1

Diagram illustrating the notation used in the derivation. The incident wave has complex amplitude Ui (x) at the plane of the x-axis which represents the boundary or surface of the scattering medium. A time reversed path-pair begins and ends at points x 1 and x 2 with black arrows following counter-clockwise propagation and gray arrows following clockwise propagation. The phase ε of the common path is acquired by both counter-propagating waves. The exiting wavelets then propagate to the observation plane at distance z with coordinate ξ.

Fig. 2
Fig. 2

Schematic of experimental setups for observation of EBS. Left: a typical configuration with a collimated beam and lens-detector system that measures angular intensity distribution. Right: our system comprised of simply a source, beam-splitter, and detector. Optionally, the peak is relayed to the detector using a microscope objective. In both cases, the reflected wavefront can be represented as a diffuse background plus a superposition of phase conjugated wavelets diffracted by the virtual aperture P(s) to form the intensity peak at a point in space conjugate to the source.

Fig. 3
Fig. 3

The peak is measured for two different fiber positions. The first peak is normalized by the second resulting in a flat baseline, a true positive peak, and an inverted peak.

Fig. 4
Fig. 4

Monte Carlo simulations were used to determine the probability P(s) of path-pair separations for 0.2 μm diameter polystyrene beads suspended in water. The rays exiting the suspension were binned according to the exit radius from the entrance point and plotted above. The empirical equation shown was fit to the simulation and used in the numerical integration of Eq. (3).

Fig. 5
Fig. 5

Peak from 0.2 μm diameter microsphere suspension with transport mean free path l s = 100 μ m . The baseline approaches one for all peaks, but small changes in intensity of the two measurements cause variation in the baseline values on the order of a few percent. The curves were therefore normalized to unity at a large radius for comparison. The simulated peak agrees well with the experimental data aside from a scaling factor of 0.76 which may be due to experimental error or simplifications in the theory (see conclusions). The inset shows the change in enhancement as a function of fiber–sample distance. As z 0 increases, Lsc increases and C(s) broadens relative to P(s) resulting in greater enhancement.

Fig. 6
Fig. 6

Intensity through focus for a full spot size (left) and reduced spot size (right). The energy is weakly localized in three dimensions as expected. The smaller spot size results in a longer distribution along Δz analogous to an increase in depth of focus from a reduced NA. The line indicates the half-maximum contour.

Fig. 7
Fig. 7

Peak formed by a pair of fibers at two different separations. On the right, direct images of the fibers are shown. Scales are μm.

Equations (5)

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A e ( x ) e i ϕ e ( x ) = α ˜ δ ( x x 1 ) A i ( x 2 ) e i ( ϕ i ( x 2 ) + ɛ ) + α ˜ δ ( x x 2 ) A i ( x 1 ) e i ( ϕ i ( x 1 ) + ɛ ) = α ˜ e ic [ δ ( x x 1 ) + δ ( x x 2 ) ] A i ( x 1 + x 2 x ) e i ϕ i ( x )
U z ( ξ ; x 1 , x 2 ) = α ˜ e ic z i λ [ A i ( x 2 ) e ik z 2 + ( ξ x 1 ) 2 i ϕ i ( x 1 ) z 2 + ( ξ x 1 ) 2 + A i ( x 1 ) e ik z 2 + ( ξ x 2 ) 2 i ϕ i ( x 2 ) z 2 + ( ξ x 2 ) 2 ]
I z ( ξ ) = P ( s ) λ 2 z 2 [ A i 2 ( x 1 ) + A i 2 ( x 2 ) + A i ( x 1 ) A i ( x 2 ) × ( e i k 2 z ( ( ξ x 2 ) 2 ( ξ x 1 ) 2 ) + i ϕ i ( x 1 ) i ϕ i ( x 2 ) + e i k 2 z ( ( ξ x 2 ) 2 ( ξ x 1 ) 2 i ϕ i ( x 1 ) + i ϕ i ( x 2 ) ) ) ] d x 1 d x 2
I z ( ξ ) = 2 a A i 2 λ 2 z 2 [ 1 + 1 a A i P ( s ) cos ( k z ξ ( x 2 x 1 ) ) dx 1 dx 2 ]
I z ( ξ ) = 2 a A i 2 λ 2 z 2 [ 1 + P ( s ) cos ( k z ξ s ) ds ]

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