## Electric field Monte Carlo simulation of coherent backscattering of polarized light by a turbid medium containing Mie scatterers

Optics Express, Vol. 16, Issue 8, pp. 5728-5738 (2008)

http://dx.doi.org/10.1364/OE.16.005728

Acrobat PDF (296 KB)

### Abstract

A method for directly simulating coherent backscattering of polarized light by a turbid medium has been developed based on the Electric field Monte Carlo (EMC) method. Electric fields of light traveling in a pair of time-reversed paths are added coherently to simulate their interference. An efficient approach for computing the electric field of light traveling along a time-reversed path is derived and implemented based on the time-reversal symmetry of electromagnetic waves. Coherent backscattering of linearly and circularly polarized light by a turbid medium containing Mie scatterers is then investigated using this method.

© 2008 Optical Society of America

## 1. Introduction

5. P. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. **55**, 2696–2699 (1985). [CrossRef] [PubMed]

6. M. P. Van Albada and Ad Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. **55**, 2692–2695 (1985). [CrossRef] [PubMed]

**E**

_{out}+

**E**

^{Reν}

_{out}|

^{2}, where

**E**

_{out},

**E**

^{Reν}

_{out}are the electric fields of the two partial waves, roughly double the intensity |

**E**

_{out}|

^{2}+|

**E**

^{Reν}

_{out}|

^{2}when backscattered light is assumed to be incoherent.

*E*,

_{x}_{y}are the complex electric field components perpendicular to each other and the propagation direction of light. The superscript

*T*refers to the transpose of the matrix and 〈 〉 denotes ensemble average. Recently, the Electric field Monte Carlo (EMC) method was developed by us to simulate polarized light propagating through turbid media [12

12. M. Xu, “Electric field Monte Carlo simulation of polarized light propagation through turbid media,” Opt. Express **12**, 6530–6539 (2004). [CrossRef] [PubMed]

13. K. G. Philips, M. Xu, S. K. Gayen, and R. R. Alfano, “Time-resolved ring structures of circularly polarized beams backscattered from forward scattering media,” Opt. Express **13**, 7954–7969 (2005). [CrossRef]

## 2. Theory

**E**is the complex electric field with components parallel and perpendicular to the previous scattering plane,

**E**′ is the complex electric field with components parallel and perpendicular to the present scattering plane,

*S*is the amplitude scattering matrix dependent on the scattering angle

*θ*between the incoming propagation direction and outgoing propagation direction, and

*R*is the rotation matrix dependent on the angle

*ϕ*between the incoming perpendicular electric field component and the outgoing perpendicular electric field component. The rotation matrix

*R*rotates the reference frame

*ϕ*degrees azimuthally to align the incoming perpendicular electric field component to the normal of the present scattering plane. Equation (1) can be explicitly written as the following for Mie scatterers:

*E*

_{1}and

*E*

_{2}are the complex parallel and perpendicular components of the incoming electric field and

*E*′

_{1}and

*E*′

_{2}are the complex electric field components after the scattering event. The properties of the scattering particle are contained in the

*S*matrix:

*S*

_{1}and

*S*

_{2}dependent on the refractive indices inside and outside the particle as well as the size of the particle and the wavelength of the incident light.

## 2.1 Electric field of the partial waves in the forward and time-reversed paths

**m**,

**n**, and

**s**is rotated after every scattering event. Here

**m**is the direction of the parallel electric field component and

**n**is the direction of the perpendicular electric field component. The vector

**n**is perpendicular to the scattering plane spanned by the previous propagation direction and the current propagation direction,

**s**.

**s**

_{1}direction after the first scattering event;

*ϕ*

_{0}is the angle formed between the incoming perpendicular electric field component and

**n**

_{1}=

**s**

_{0}×

**s**

_{1}/|

**s**

_{0}×

**s**

_{1}. After a total of

*n*scattering events, light is scattered into the direction

**s**

_{n}and escapes the medium.

*ϕ*

_{n-1}is the angle formed between

**n**

_{n-1}=

**s**

_{n-2}×

**s**

_{n-1}/|

**s**

_{n-2}×

**s**

_{n-1}| and

**n**

_{n}=

**s**

_{n-1}×

**s**

_{n}/|

**s**

_{n-1}×

**s**

_{n}|. When the path is reversed, the first scattering event occurs at the

*n*scatterer, where light is scattered from

^{th}**s**

_{0}to -

**s**

_{n-1}.

*ϕ*

_{n}is the angle formed between the incoming perpendicular electric field component and

**n**′=

**s**

_{0}×(-

**s**

_{n-1})/|

**s**

_{0}×(-

**s**

_{n-1})| and

*ϕ*′

_{n}is the angle formed between

**n**′ and -

**n**

_{n-1}=(-

**s**

_{n-1})×(-

**s**

_{n-2})/|(-

**s**

_{n-1})×(-

**s**

_{n-2})|. After the

*ϕ*′

_{n}rotation, the second scattering event in the time-reversed path occurs at the

*n*-1 scatterer, followed by a rotation of

*ϕ*

_{n-2}azimuthally along -

**s**

_{n-2}from -

**n**

_{n-1}to -

**n**

_{n-2}. This continues until the last site that scatters light in the reverse path, which is the first site in the forward path. At this last scattering event, light is scattered into

**s**

_{n}from -

**s**

_{1}.

*ϕ*

_{1}is the angle between -

**n**

_{2}=(-

**s**

_{2})×(-

**s**

_{1})/|(-

**s**

_{2})×(-

**s**

_{1})| and -

**n**

_{1}=(-

**s**

_{1})×(-

**s**

_{0})/|(-

**s**

_{1})×(-

**s**

_{0})| and

*ϕ*′

_{1}is the angle between -

**n**

*1*and

**n**″=(-

**s**

_{1})×

**s**

_{n}/|(-

**s**

_{1})×

**s**

_{n}|. The angle

*ϕ*

_{1}′=0 if

**s**

_{n}=-

**s**

_{0}. After a total of

*n*scattering events the electric field in the forward path is hence given by:

*S*denotes the site at which the scattering event takes place.

## 2.2 Special azimuthal rotations in the time-reversed path

*n*” (the first scatterer in the reverse path) in order to align the reference frame to the scattering plane spanned by -

**s**

_{n-1}and -

**s**

_{n-2}. Let

**m**

_{0}and

**n**

_{0}be the directions of the incoming parallel electric field and the incoming perpendicular electric field, respectively. In the reverse path, the first two rotations can be described by the following expression:

*R*(

*ϕ*

_{n}), aligns the perpendicular electric field of the incident beam to the normal of the scattering plane. The second rotation,

*R*(

*ϕ*′

_{n}), aligns

**m**′ to

**m**

_{n-1}, and

**n**′ to -

**n**

_{n-1}. The direction, -

**n**

_{n-1}, is the normal of the scattering plane for the upcoming scattering event at the second scatterer “

*n*-1” in the reverse path.

**m**

_{1},-

**n**

_{1},-

**s**

_{1})

^{T}. The rotation

*R*(

*ϕ*′

_{1}) aligns the perpendicular electric field component to the normal of the upcoming scattering plane spanned by -

**s**

_{1}and

**s**

_{n}.

**m**″ and

**n**″ are the directions of the outgoing parallel and perpendicular electric field components, respectively.

## 2.3 Relation between electric fields in the forward and time-reversed paths

## 3. Results and discussion

12. M. Xu, “Electric field Monte Carlo simulation of polarized light propagation through turbid media,” Opt. Express **12**, 6530–6539 (2004). [CrossRef] [PubMed]

## 3.1 Intensity and enhancement of circularly polarized light around the exact backscattering direction

_{+}+I

_{-}, middle: I

_{+}, right: I

_{-}). I

_{±}is the intensity of the backscattered light of the same (or opposite) helicity as that of the incident beam. The second row displays the corresponding intensities for incident incoherent light of circular polarization. As one can see from Fig. 3, concentric circles of equal intensity are visible with or without coherence. The intensity also drops off as the zenith angle θ

_{b}of detection increases in all cases. This azimuthal angle ϕ

_{b}independence is expected as circularly polarized light has by definition centric symmetry.

_{+}is greater than I

_{-}close to exact backscattering direction and is smaller than I

_{-}for all other angles; for incoherent backscattered intensities, I

_{+}is less than I

_{-}for all θ

_{b}simulated. Enhancement is higher for I

_{+}near the exact escape direction but falls off more sharply than I

_{_}as θ

_{b}increases. Incoherent backscattered light is stronger in the negative helicity channel than in the positive helicity one (I

_{+}>I

_{_}) because the helicity asymmetry is negative for backscattered light by small Mie scatterers in a turbid medium [16]. Multiply scattered light loses coherence and helicity simultaneously upon scattering. The enhancement factor is larger for the helicity preserved channel than the helicity flipped channel near the exact backscattering direction. This boosts the intensity of the coherent backscattered I

_{+}beyond I

_{_}within a narrow angular range around the exact backscattering direction.

## 3.2 Intensity and enhancement of linearly polarized light around the exact backscattering direction

_{x}+I

_{y}, I

_{x}, and I

_{y}for incident coherent light polarized along the

*x*direction. The second row is the same except the incident light is incoherent. The third row is the same as the first row but the zenith detection angle goes from 0 degrees to 2.25 degrees instead. Figure 6 displays the angular profiles for coherent backscattering light, incoherent backscattering light, and the enhancement factor along three directions corresponding to the azimuthal angle of 0, 45, and 90 degrees, respectively.

*x*,

*y*axes, but it no longer has circular symmetry. The intensity of the backscattered light of

*x*polarization from the incident coherent or incoherent

*x*-polarized light is, in general, greater than that of

*y*polarization because the preference in linear polarization only gets lost after multiple scattering. As with circularly polarized coherent light, backscattering of linearly polarized coherent light is enhanced relative to that of incoherent light within the backscattering cone. I

_{x}is enhanced greater than I

_{y}because photons contributing to I

_{x}preserve the original linear polarization and suffer less scattering events than photons contributing to I

_{y}and hence maintain coherence better. Unlike the case involving circularly polarized light, the magnitude of the intensity of linearly polarized light, I

_{x}+I

_{y}, is no longer circular symmetric. This shows that there is a dependence on the azimuthal angle ϕ

_{b}for backscattering of linearly polarized light. As to understand why the patterns for I

_{x}’s and I

_{x}+I

_{y}’s are elongated along the

*y*-axis and the pattern for I

_{y}’s is elongated along the

*x*-axis, we point out that light tends to be scattered preferably into directions out of the plane of its polarization when scattered by a Mie scatterer. Thus the

*x*-component is elongated along the

*y*-axis and squeezed in along the

*x*-axis and the

*y*-component is elongated along the

*x*-axis and squeezed in along the

*y*-axis. Both coherent and incoherent I

_{x}’s increase slightly with the zenith angle, mostly owing to the influence of single scattered light.

_{x}+I

_{y}and I

_{x}appear to now be elongated along the

*x*-axis and squeezed slightly at the

*y*-axis due to the much stronger enhancement factor for backscattered light remitting from along the

*x*-axis [see Fig. 6(f)]. Also, I

_{y}displays interesting 4-fold “X” symmetry closer to the exact escape direction, which is characteristic of light being multiply scattered by Rayleigh-like particles.

## 4. Conclusion

## Acknowledgments

## References and links

1. | A. Ishimaru, |

2. | A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today |

3. | S. K. Gayen and R. R. Alfano, “Emerging optical biomedical imaging techniques,” Opt. Photonics News 7, 17–22 (1996). |

4. | S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. |

5. | P. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. |

6. | M. P. Van Albada and Ad Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. |

7. | I. Lux and L. Koblinger, |

8. | S. Bartel and A. H. Hielscher, “Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media,” Appl. Opt. |

9. | H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, “Monte Carlo and multicomponent approximation methods for vector radiative transfer by use of effective Mueller matrix calculations,” Appl. Opt. |

10. | B. Kaplan, G. Ledanois, and B. Villon, “Mueller matrix of dense polystyrene latex sphere suspensions: Measurements and Monte Carlo simulation,” Appl. Opt. |

11. | X. Wang and L. V. Wang, “Propagation of polarized light in birefringent turbid media: A Monte Carlo study,” J. Biomed. Opt. |

12. | M. Xu, “Electric field Monte Carlo simulation of polarized light propagation through turbid media,” Opt. Express |

13. | K. G. Philips, M. Xu, S. K. Gayen, and R. R. Alfano, “Time-resolved ring structures of circularly polarized beams backscattered from forward scattering media,” Opt. Express |

14. | H. C. van de Hulst, |

15. | D. S. Saxon, “Tensor scattering matrix for the electromagnetic field,” Phys. Rev. |

16. | M. Xu and R. R. Alfano, “Circular polarization memory of light,” Phys. Rev. E 72 , 065061(R) (2005). |

**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

(030.5620) Coherence and statistical optics : Radiative transfer

(290.1350) Scattering : Backscattering

(290.4210) Scattering : Multiple scattering

(290.7050) Scattering : Turbid media

**ToC Category:**

Scattering

**History**

Original Manuscript: December 7, 2007

Revised Manuscript: January 27, 2008

Manuscript Accepted: February 6, 2008

Published: April 9, 2008

**Virtual Issues**

Vol. 3, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

John Sawicki, Nikolas Kastor, and Min Xu, "Electric field Monte Carlo simulation of coherent backscattering of polarized light by a turbid medium containing Mie scatterers," Opt. Express **16**, 5728-5738 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5728

Sort: Year | Journal | Reset

### References

- A. Ishimaru, Wave Propagation and Scattering in Random Media, I and II (Academic, New York, 1978).
- A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 38-40 (1995). [CrossRef]
- S. K. Gayen and R. R. Alfano, "Emerging optical biomedical imaging techniques," Opt. Photon. News 7, 17-22 (1996).
- S. R. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, R41-R93 (1999). [CrossRef]
- P. Wolf and G. Maret, "Weak localization and coherent backscattering of photons in disordered media," Phys. Rev. Lett. 55, 2696-2699 (1985). [CrossRef] [PubMed]
- M. P. Van Albada and Ad Lagendijk, "Observation of weak localization of light in a random medium," Phys. Rev. Lett. 55, 2692-2695 (1985). [CrossRef] [PubMed]
- I. Lux and L. Koblinger, Monte Carlo Particle Transport Methods: Neutron and Photon Calculations (CRC Press, Boca Raton, Fla., 1991).
- S. Bartel and A. H. Hielscher, "Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media," Appl. Opt. 39, 1580-1588 (2000). [CrossRef]
- H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, "Monte Carlo and multicomponent approximation methods for vector radiative transfer by use of effective Mueller matrix calculations," Appl. Opt. 40, 400-412 (2001). [CrossRef]
- B. Kaplan, G. Ledanois, and B. Villon, "Mueller matrix of dense polystyrene latex sphere suspensions: Measurements and Monte Carlo simulation," Appl. Opt. 40, 2769-2777 (2001). [CrossRef]
- X. Wang and L. V. Wang, "Propagation of polarized light in birefringent turbid media: A Monte Carlo study," J. Biomed. Opt. 7, 279-290 (2002). [CrossRef] [PubMed]
- M. Xu, "Electric field Monte Carlo simulation of polarized light propagation through turbid media," Opt. Express 12, 6530-6539 (2004). [CrossRef] [PubMed]
- K. G. Philips, M. Xu, S. K. Gayen, and R. R. Alfano, "Time-resolved ring structures of circularly polarized beams backscattered from forward scattering media," Opt. Express 13, 7954-7969 (2005). [CrossRef]
- H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
- D. S. Saxon, "Tensor scattering matrix for the electromagnetic field," Phys. Rev. 100, 1771-1775 (1955). [CrossRef]
- M. Xu and R. R. Alfano, "Circular polarization memory of light," Phys. Rev. E 72, 065061(R) (2005).

## 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.