## Alternate formulation of enhanced backscattering as phase conjugation and diffraction: derivation and experimental observation |

Optics Express, Vol. 19, Issue 13, pp. 11922-11931 (2011)

http://dx.doi.org/10.1364/OE.19.011922

Acrobat PDF (2221 KB)

### 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.

© 2011 OSA

## 1. Introduction

## 2. Concept and theory

### 2.1. Phase conjugation

*U*(

_{i}*x*) =

*A*(

_{i}*x*)

*e*

^{iϕi(x)}. If the path-pair samples this wavefront at two points

*x*

_{1}and

*x*

_{2}, the phase exiting the medium will be given by the sum of the incident phase and the phase

*ε*of the common path,

*ϕ*(

_{e}*x*

_{1}) =

*ϕ*(

_{i}*x*

_{2}) +

*ε*and

*ϕ*(

_{e}*x*

_{2}) =

*ϕ*(

_{i}*x*

_{1}) +

*ε*. Likewise, the amplitude will be proportional to the incident amplitude at the opposite point and also a factor

*, that varies randomly for each path-pair. We factor out a constant*α ˜

*e*

^{i(ϕi (x1)+ϕi (x2)+ε)}=

*e*and use Dirac delta functions to represent the sampling operation.

^{ic}*A*(

_{i}*x*

_{1}+

*x*

_{2}–

*x*) =

*A*(

_{i}*x*) then the exiting complex amplitude for each path-pair is proportional to

*U*(

_{i}*x*)

***. This can be regarded as phase conjugation of a filtered or sampled portion of the wavefront.

### 2.2. Diffraction from a virtual aperture

*A*(

_{i}*x*

_{1}) =

*A*(

_{i}*x*

_{2}) =

*A*. We can use

_{i}*A*(

_{i}*x*) =

*rect*(

*x*) to model a finite spot size, and for simplicity we assume the spot size is much broader than the function

*P*(

*s*) so that

*a*is the area of the illumination. Next, we treat the special case of a diverging spherical wave from a point source on the

*z*-plane at

*ξ*= 0, the incident phase is given by

*ϕ*=

_{i}*kx*2

*/*(2

*z*). Starting with Eq. (3), the

*x*

^{2}terms of the exponent exactly cancel, and the equation can be written in cosine form. Since

*P*(

*s*) is symmetric, the inner integral over

*x*

_{1}is a Fourier transform of

*P*(

*s*) about each point

*x*

_{2}, and the outer integral is then the sum of these Fourier transforms from each

*x*

_{2}. If we again make use of the fact that the spot size is much larger than

*P*(

*s*) and ignore the effect of the edge of the spot, the transform is independent of shift

*x*

_{2}and the outer integral reduces to a factor of

*a*. Ignoring edge effects, 〈

*I*〉 is then proportional to 1 +

_{z}*ℱ*{

*P*(

*s*)}. The Fourier transform can be thought of as representing diffraction by a virtual apodized aperture described by the function

*P*(

*s*).

### 2.3. Partial spatial coherence and extended sources

*P*(

*s*) ×

*C*(

*s*)], or equivalently, each point in the extended source can be thought to produce an independent (incoherent) peak. The resulting peak is the convolution of the coherent peak with the source function as shown by applying the superposition integral. The van Cittert–Zernike theorem shows that when

*C*(0) = 1, these two conceptualizations are mathematically equivalent: 〈

*I*(

*ξ*)〉 ∝ 1 +

*ℱ*{

*P*(

*s*) ×

*C*(

*s*)} = (1 +

*ℱ*{

*P*(

*s*)}) *

*ℱ*{

*C*(

*s*)}.

## 3. Experimental observation

*μm*diameter core and 0.22 Numerical Aperture (NA). Since the multi-mode fiber scrambles the phase and polarization, any type of illumination may be used. We have observed the peak using an arclamp, HeNe laser, or as with the experimental results shown here, a high-brightness LED with peak wavelength 635 nm and 20 nm bandwidth (FWHM). The fiber is tilted at an angle relative to the surface such that the no ray from the source is normally incident on the sample, ensuring that the specular reflection is separated from the peak. The diverging beam passes through the beam-splitter and illuminates the sample mounted on a stage to allow scanning in the z-direction. Backscattered light is reflected by the beam-splitter and the peak is formed at the point in space conjugate to the source.

*x*

_{1}=

*y*

_{1}= 0 and applying an experimentally determined factor of 0.43 to the peak to account for depolarization. The depolarization factor was obtained by taking the ratio of measured peaks with and without a linear polarizer sheet covering the sample.

*P*(

*s*) was obtained from Monte Carlo simulation [15

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]

*P*(

*s*). We then fit a second order polynomial for use in the analytical equation. Although polarization effects are significant for this particle size, we treat polarization indirectly by rotationally averaging the exit locations to compute

*P*(

*s*) and compare to unpolarized experimental results which include all polarization states. In this case, the primary effect of the medium dependent depolarization is to increase the incoherent baseline relative to the peak, and this is accounted for by the measured depolarization factor.

*μ*m a distance

*z*

_{0}away from the sample [16]. The numerical integration was performed out to a radius twice the illumination area which accounts for an increase in the baseline due to light that is returned without an associated path pair. By changing

*z*

_{0}, the spatial coherence length can be scanned,

*L*=

_{sc}*z*

_{0}/(

*kρ*) for source radius

*ρ*. The inset shows the dependence of the peak on sample distance with enhancement increasing with

*L*. The scaling is due to the change in the spatial coherence function,

_{sc}*C*(

*s*), relative to

*P*(

*s*).

*z*=

*z*–

*z*

_{0}was calculated using the objective’s longitudinal magnification and shown as Δ

*z*in Fig. 6.

*z*-dependence of the peak is dominated by the effect of the illumination NA. As shown earlier, the lateral shape of the peak is the result of diffraction by the virtual aperture. For the geometry in the this experiment, the size of the virtual aperture extends a few hundred microns. However, the illumination spot is comprised of many such virtual apertures, each of which forms a peak convergent at the conjugate plane. The Δ

*z*-dependence of the peak is then analogous to depth of focus given the angular extent of the illuminated spot. This was confirmed by taking measurements of the peak through Δ

*z*for a full illumination spot and for a reduced illumination spot where a black card with a small hole was placed over the sample to reduce the NA, shown on the right side of Fig. 6. To predict the depth of focus, the peak was approximated as Gaussian with 100

*μ*m width and convolved with a

*rect*() of width

*y*that results in half the maximum value at the center. The depth of focus is then

*d*≈

_{z}*y/NA*. For the full NA 0.22 (left),

*d*is estimated to be 900

_{z}*μ*m and for the reduced NA of 0.15 (right),

*d*is estimated to be 1300

_{z}*μ*m, which agrees qualitatively with the result in Fig. 6 although noise limited measurements for Δ

*z*larger than the range shown.

*P*(

*s*).

## 4. Conclusions

*P*(

*s*) that describes the probability distribution of path-pair separations; (4) an extended source forms a superposition of EBS peaks from each point in the source and can be calculated in Fourier space by multiplying

*P*(

*s*) with the coherence function

*C*(

*s*).

*P*(

*s*) due to local relative tilt between the wavefront and the sample surface at each location. Second, computational resources limited the integration of Eq. (3) plotted in Fig. 5 to a single entrance point at the center of the spot. Evaluating the integral for all entrance points including those closer to the edge of the spot may also reduce the peak enhancement. It is possible that these effects contribute to the scaling error between the predicted peak and the observed peak in Fig. 5 and further work will be needed to explore this.

*in vivo*measurements for biomedical applications like tissue characterization [15

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]

*P*(

*s*) that can take into account realistic experimental parameters such as finite spot size. Finally, we demonstrate the obvious but interesting consequence that a random medium can form an image of any source without the use of lenses.

## Acknowledgments

## References and links

1. | A. Dogariu and G. D. Boreman, “Enhanced backscattering in a converging-beam configuration,” Opt. Lett. |

2. | N. C. Bruce, “A comparison of the converging and diverging geometries for measuring enhanced backscatter,” J. Mod. Opt. |

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. |

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. |

5. | F. Reil and J. E. Thomas, “Observation of phase conjugation of light arising from enhanced backscattering in a random medium,” Phys. Rev. Lett. |

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. |

7. | E. Akkermans, P. Wolf, R. Maynard, and G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. France |

8. | P. Sheng, |

9. | M. I. Mishchenko, L. D. Travis, and A. A. Lacis, |

10. | E. Akkermans and G. Montambaux, |

11. | R. T. Deck and H. J. Simon, “Simple diffractive theory of enhanced backscattering,” J. Opt. Soc. Am. B |

12. | T. Okamoto and T. Asakura, “Enhanced backscattering of partially coherent light,” Opt. Lett. |

13. | Y. Kim, Y. Liu, V. Turzhitsky, H. Roy, R. Wali, and V. Backman, “Coherent backscattering spectroscopy,” Opt. Lett. |

14. | J. Goodman, |

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. | M. Born and E. Wolf, |

**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

(050.1940) Diffraction and gratings : Diffraction

(290.1350) Scattering : Backscattering

**ToC Category:**

Scattering

**History**

Original Manuscript: December 20, 2010

Revised Manuscript: April 28, 2011

Manuscript Accepted: May 27, 2011

Published: June 6, 2011

**Citation**

Jeremy D. Rogers, Valentina Stoyneva, Vladimir Turzhitsky, Nikhil N. Mutyal, Prabhakar Pradhan, İlker R. Çapoğlu, and Vadim Backman, "Alternate formulation of enhanced backscattering as phase conjugation and diffraction: derivation and experimental observation," Opt. Express **19**, 11922-11931 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-13-11922

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### References

- A. Dogariu and G. D. Boreman, “Enhanced backscattering in a converging-beam configuration,” Opt. Lett. 21, 1718–1720 (1996). [CrossRef] [PubMed]
- N. C. Bruce, “A comparison of the converging and diverging geometries for measuring enhanced backscatter,” J. Mod. Opt. 49, 2167–2181 (2002). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- R. T. Deck and H. J. Simon, “Simple diffractive theory of enhanced backscattering,” J. Opt. Soc. Am. B 10, 1000–1005 (1993). [CrossRef]
- T. Okamoto and T. Asakura, “Enhanced backscattering of partially coherent light,” Opt. Lett. 21, 369–371 (1996). [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]
- J. Goodman, Introduction to Fourier Optics (McGraw-Hill Science, Engineering & Mathematics, 1996).
- 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]
- M. Born and E. Wolf, Principles of Optics , 7th ed. (Cambridge University Press, 1999).

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