## Simulation of enhanced backscattering of light by numerically solving Maxwell’s equations without heuristic approximations

Optics Express, Vol. 13, Issue 10, pp. 3666-3672 (2005)

http://dx.doi.org/10.1364/OPEX.13.003666

Acrobat PDF (921 KB)

### Abstract

We report what we believe to be the first simulation of enhanced backscattering (EBS) of light by numerically solving Maxwell’s equations without heuristic approximations. Our simulation employs the pseudospectral time-domain (PSTD) technique, which we have previously shown enables essentially exact numerical solutions of Maxwell’s equations for light scattering by millimeter-volume random media consisting of micrometer-scale inhomogeneities. We show calculations of EBS peaks of random media in the presence of speckle; in addition, we demonstrate speckle reduction using a frequency-averaging technique. More generally, this new technique is sufficiently robust to permit the study of EBS phenomena for random media of arbitrary geometry not amenable to simulation by other approaches, especially with regard to extension to full-vector electrodynamics in three dimensions.

© 2005 Optical Society of America

## 1. Introduction

1. E. Akkermans and G. Montambaux, “Mesoscopic physics of photons” J. Opt. Soc. Am. B **21**, 101–112 (2004). [CrossRef]

5. I.V. Meglinski, V.L. Kuzmin, D.Y. Churmakov, and D.A. Greenhalgh, “Monte Carlo simulation of coherent effects in multiple scattering” Proc. of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences , **461**: 43–53 (2005). [CrossRef]

6. R. Lenke, R. Tweer, and G. Maret, “Coherent backscattering of turbid samples containing large Mie spheres” J. Opt. A **4**, 293–298 (2002). [CrossRef]

7. V.L. Kuzmin and I.V. Meglinski, “Coherent multiple scattering effects and Monte Carlo method” JETP Lett. **79**, 109–112 (2004). [CrossRef]

8. E. Amic, J.M. Luck, and T.M. Nieuwenhuizen, “Anisotropic multiple scattering in diffusive media” J. Phys. A: Math. Gen. **29**, 4915–4955 (1996). [CrossRef]

8. E. Amic, J.M. Luck, and T.M. Nieuwenhuizen, “Anisotropic multiple scattering in diffusive media” J. Phys. A: Math. Gen. **29**, 4915–4955 (1996). [CrossRef]

9. J. Pearce and D.M. Mittleman, “Propagation of single-cycle terahertz pulses in random media” Opt. Lett. **26**, 2002–2004 (2001). [CrossRef]

11. L. Marti-Lopez, J. Bouza-Dominguez, J.C. Hebden, S.R. Arridge, and R.A. Martinez-Celorio, “Validity conditions for the radiative transfer equation” J. Opt. Soc. Am. A **20**, 2046–2056 (2003). [CrossRef]

12. Q.H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain (PSTD) algorithm” IEEE Transactions on Geoscience and Remote Sensing , **37**, 917–926 (1999). [CrossRef]

14. S.H. Tseng, J.H. Greene, A. Taflove, D. Maitland, V. Backman, and J.T. Walsh, “Exact solution of Maxwell’s equations for optical interactions with a macroscopic random medium” Opt. Lett. **29**, 1393–1395 (2004); **30**: 56–57 (2005). [CrossRef] [PubMed]

## 2. Methods

**F**and

**F**

^{-1}denote, respectively, the forward and inverse discrete Fourier transforms, and

*k*̃

_{x}is the Fourier transform variable representing the

*x*-component of the numerical wavevector. The spatial derivatives {(

*∂V/∂x*)

_{i}} can be calculated in one step. In multiple dimensions, this process is repeated for each cut parallel to the major axes of the space lattice. PSTD techniques have been shown to possess spectral accuracy; that is, errors due to spatial sampling decrease exponentially as the meshing density increases beyond the Nyquist rate. With trigonometric basis functions, this permits the PSTD meshing density to approach two samples per wavelength in each spatial dimension.

15. S.D. Gedney, “An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices” IEEE transactions on Antennas and Propagation , **44**: 1630–1639 (1996). [CrossRef]

_{0}ranging from 1 µm up to 600 µm) can be obtained in a single simulation.

## 3. PSTD Simulation of EBS

*N*randomly positioned, closely packed, infinitely long, dielectric cylinders with refractive index n=1.25. The cluster geometry is created by randomly positioning scatterers in free space, subject to the constraint of a minimum distance of 2.4 µm allowed between cylinders.

_{0}=1 µm, the transport mean free path,

*l*’, is 37.74 µm and 5.59 µm, for

_{s}*N*=10,000 and

*N*=20,000, respectively. We use a PSTD grid having a uniform spatial resolution of 0.33 µm, equivalent to 0.42

*λ*

_{d}(

*λ*

_{d}: optical wavelength in dielectric material) at frequency 300 THz for a cylinder refractive index

*n*=1.25. In each simulation, the cluster is illuminated by a coherent plane wave at a 15° incident angle to avoid specular reflection. Both the incident light and backscattered light are polarized perpendicular to the plane of incidence, equivalent to collinear detection in EBS experiments.

*f*

_{0}. This is similar to experimental observations of EBS using non-monochromatic illumination with a temporal coherence length of 10 µm. (The 50 PSTD-computed scattered intensities correspond to 50 frequencies evenly spaced between 1.05*

*f*

_{0}and 0.95*

*f*

_{0}. An estimated 16 modes are averaged in the process to suppress speckle.) As shown in Fig. 2(c), following frequency-averaging, the EBS peak is clearly visible, showing a significant correlation with experimental results reported by Tomita and Ikari [17

17. M. Tomita and H. Ikari, “Influence of Finite Coherence Length of Incoming Light on Enhanced Backscattering” Physical Review B , **43**: 3716–3719 (1991). [CrossRef]

18. E. Akkermans, P.E. Wolf, and R. Maynard, “Coherent Backscattering of Light by Disordered Media-Analysis of the Peak Line-Shape” Phys. Rev. Lett. **56**, 1471–1474 (1986). [CrossRef] [PubMed]

*l*=transport mean free path length;

*q*=2

*πθ/λ*; z

_{0}=2

*l*3; * denotes a convolution; and

*SF(θ)*is the far-field scattering function for a homogeneous slab of the same size illuminated by a plane wave. We observe a good agreement between the PSTD simulations and the benchmark EBS theory.

## 6. Summary and discussion

## Acknowledgments

## References and links

1. | E. Akkermans and G. Montambaux, “Mesoscopic physics of photons” J. Opt. Soc. Am. B |

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

3. | R. Sapienza, S. Mujumdar, C. Cheung, A.G. Yodh, and D. Wiersma, “Anisotropic weak localization of light” Phys. Rev. Lett.92 (2004). [CrossRef] [PubMed] |

4. | G. Labeyrie, D. Delande, C.A. Muller, C. Miniatura, and R. Kaiser, “Coherent backscattering of light by cold atoms: Theory meets experiment” Europhys. Lett. |

5. | I.V. Meglinski, V.L. Kuzmin, D.Y. Churmakov, and D.A. Greenhalgh, “Monte Carlo simulation of coherent effects in multiple scattering” Proc. of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences , |

6. | R. Lenke, R. Tweer, and G. Maret, “Coherent backscattering of turbid samples containing large Mie spheres” J. Opt. A |

7. | V.L. Kuzmin and I.V. Meglinski, “Coherent multiple scattering effects and Monte Carlo method” JETP Lett. |

8. | E. Amic, J.M. Luck, and T.M. Nieuwenhuizen, “Anisotropic multiple scattering in diffusive media” J. Phys. A: Math. Gen. |

9. | J. Pearce and D.M. Mittleman, “Propagation of single-cycle terahertz pulses in random media” Opt. Lett. |

10. | M. Haney and R. Snieder, “Breakdown of wave diffusion in 2D due to loops” Phys. Rev. Lett.91, (2003). [CrossRef] [PubMed] |

11. | L. Marti-Lopez, J. Bouza-Dominguez, J.C. Hebden, S.R. Arridge, and R.A. Martinez-Celorio, “Validity conditions for the radiative transfer equation” J. Opt. Soc. Am. A |

12. | Q.H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain (PSTD) algorithm” IEEE Transactions on Geoscience and Remote Sensing , |

13. | Q.H. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength” Microwave Opt. Technol. Lett. |

14. | S.H. Tseng, J.H. Greene, A. Taflove, D. Maitland, V. Backman, and J.T. Walsh, “Exact solution of Maxwell’s equations for optical interactions with a macroscopic random medium” Opt. Lett. |

15. | S.D. Gedney, “An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices” IEEE transactions on Antennas and Propagation , |

16. | A. Taflove and S.C. Hagness, |

17. | M. Tomita and H. Ikari, “Influence of Finite Coherence Length of Incoming Light on Enhanced Backscattering” Physical Review B , |

18. | E. Akkermans, P.E. Wolf, and R. Maynard, “Coherent Backscattering of Light by Disordered Media-Analysis of the Peak Line-Shape” Phys. Rev. Lett. |

**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

(290.1350) Scattering : Backscattering

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 25, 2005

Revised Manuscript: April 18, 2005

Published: May 16, 2005

**Citation**

Snow Tseng, Young Kim, Allen Taflove, Duncan Maitland, Vadim Backman, and Joseph Walsh, Jr., "Simulation of enhanced backscattering of light by numerically solving Maxwell�??s equations without heuristic approximations," Opt. Express **13**, 3666-3672 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-10-3666

Sort: Journal | Reset

### References

- E. Akkermans and G. Montambaux, "Mesoscopic physics of photons" J. Opt. Soc. Am. B 21, 101-112 (2004). [CrossRef]
- Y.L. Kim, Y. Liu, V.M. Turzhitsky, H.K. Roy, R.K. Wali, and V. Backman, "Coherent backscattering spectroscopy" Opt. Lett. 29, 1906-1908 (2004). [CrossRef] [PubMed]
- R. Sapienza, S. Mujumdar, C. Cheung, A.G. Yodh, and D. Wiersma, "Anisotropic weak localization of light" Phys. Rev. Lett. 92 (2004). [CrossRef] [PubMed]
- G. Labeyrie, D. Delande, C.A. Muller, C. Miniatura, and R. Kaiser, "Coherent backscattering of light by cold atoms: Theory meets experiment" Europhys. Lett. 61, 327-333 (2003). [CrossRef]
- I.V. Meglinski, V.L. Kuzmin, D.Y. Churmakov, and D.A. Greenhalgh, "Monte Carlo simulation of coherent effects in multiple scattering" Proc. of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences, 461: 43-53 (2005). [CrossRef]
- R. Lenke, R. Tweer, and G. Maret, "Coherent backscattering of turbid samples containing large Mie spheres" J. Opt. A 4, 293-298 (2002). [CrossRef]
- V.L. Kuzmin and I.V. Meglinski, "Coherent multiple scattering effects and Monte Carlo method" JETP Lett. 79, 109-112 (2004). [CrossRef]
- E. Amic, J.M. Luck, and T.M. Nieuwenhuizen, "Anisotropic multiple scattering in diffusive media" J. Phys. A: Math. Gen. 29, 4915-4955 (1996). [CrossRef]
- J. Pearce and D.M. Mittleman, "Propagation of single-cycle terahertz pulses in random media" Opt. Lett. 26, 2002-2004 (2001). [CrossRef]
- M. Haney and R. Snieder, "Breakdown of wave diffusion in 2D due to loops" Phys. Rev. Lett. 91, (2003). [CrossRef] [PubMed]
- L. Marti-Lopez, J. Bouza-Dominguez, J.C. Hebden, S.R. Arridge, and R.A. Martinez-Celorio, "Validity conditions for the radiative transfer equation" J. Opt. Soc. Am. A 20, 2046-2056 (2003). [CrossRef]
- Q.H. Liu, "Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain (PSTD) algorithm" IEEE Transactions on Geoscience and Remote Sensing, 37, 917-926 (1999). [CrossRef]
- Q.H. Liu, "The PSTD algorithm: A time-domain method requiring only two cells per wavelength" Microwave Opt. Technol. Lett. 15, 158-165 (1997). [CrossRef]
- S.H. Tseng, J.H. Greene, A. Taflove, D. Maitland, V. Backman, and J.T. Walsh, "Exact solution of Maxwell's equations for optical interactions with a macroscopic random medium" Opt. Lett. 29, 1393-1395 (2004); 30: 56-57 (2005). [CrossRef] [PubMed]
- S.D. Gedney, "An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices" IEEE Transactions on Antennas and Propagation, 44: 1630-1639 (1996). [CrossRef]
- A. Taflove and S.C. Hagness, Computational Electrodynamics: the finite-difference time-domain method. 2000: Artech House. 852.
- M. Tomita and H. Ikari, "Influence of Finite Coherence Length of Incoming Light on Enhanced Backscattering" Physical Review B, 43: 3716-3719 (1991). [CrossRef]
- E. Akkermans, P.E. Wolf, and R. Maynard, "Coherent Backscattering of Light by Disordered Media �?? Analysis of the Peak Line-Shape" Phys. Rev. Lett. 56, 1471-1474 (1986). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.