## 3D-PSTD simulation and polarization analysis of a light pulse transmitted through a scattering medium |

Optics Express, Vol. 21, Issue 21, pp. 24969-24984 (2013)

http://dx.doi.org/10.1364/OE.21.024969

Acrobat PDF (1768 KB)

### Abstract

A tridimensional pseudo-spectral time domain (3D-PSTD) algorithm, that solves the full-wave Maxwell’s equations by using Fourier transforms to calculate the spatial derivatives, has been applied to determine the time characteristics of the propagation of electromagnetic waves in inhomogeneous media. Since the 3D simulation gives access to the full-vector components of the electromagnetic fields, it allowed us to analyse the polarization state of the scattered light with respect to the characteristics of the scattering medium and the polarization state of the incident light. We show that, while the incident light is strongly depolarized on the whole, the light that reaches the output face of the scattering medium is much less depolarized. This fact is consistent with our recently reported experimental results, where a rotation of the polarization does not preclude the restoration of an image by phase conjugation.

© 2013 OSA

## 1. Introduction

1. F. Devaux and E. Lantz, “Real time suppression of turbidity of biological tissues in motion by three-wave mixing phase conjugation,” J. of Biomed. Opt. **18**, 111405 (2013). [CrossRef]

2. X. Wang, L. V. Wang, C.W. Sun, and C.C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiements,” J. of Biomed. Opt. **8**, 608–617 (2003). [CrossRef]

3. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Transactions on Antennas and Propagation **14**, 302–307 (1966). [CrossRef]

4. Q. H. Liu, “The PSTD algorithm: a time-domain method requiring only two cells per wavelength,” Microw. Opt. Technol. Lett. **15**, 158–165 (1997). [CrossRef]

*to 8*

^{D}*where*

^{D}*D*is the dimensionality of the problem) and with a better accuracy than FDTD methods [4

4. Q. H. Liu, “The PSTD algorithm: a time-domain method requiring only two cells per wavelength,” Microw. Opt. Technol. Lett. **15**, 158–165 (1997). [CrossRef]

6. T. W. Lee and S. C. Hagness, “Pseudospectral time-domain methods for modeling optical wave propagation in second-order nonlinear materials,” J. Opt. Soc. Am. B **21**, 330–342 (2004). [CrossRef]

7. X. Liu and Y. Chen, “Applications of transformed-space non-uniform PSTD (TSNU-PSTD) in scattering analysis with the use of the non-uniform FFT,” Microw. Opt. Technol. Lett. **38**, 16–21 (2003). [CrossRef]

8. C. Liu, R. L. Panetta, and P. Yang, “Application of the pseudo-spectral time domain method to compute particle single-scattering properties for size parameters up to 200,” J. Quant. Spectrosc. Transfer **113**, 1728–1740 (2012). [CrossRef]

9. S. H. Tseng, A. Taflove, D. Maitland, and V. Backman, “Pseudospectral time simulations of mutiple light scattering in three-dimensional macroscopic random media,” Radio Science **41**, RS4009 (2006). [CrossRef]

10. C. Liu, L. Bi, R. L. Panetta, P. Yang, and M. A. Yurkin, “Comparison between the pseudospectral time domain method and the discrete dipole approximation for light scattering simulations,” Opt. Express **20**, 16763–16776 (2012). [CrossRef]

11. S. H. Tseng and C. Yang, “2-D PSTD simulation of optical phase conjugation for turbidity suppression,” Opt. Express **15**, 1605–1616 (2007). [CrossRef]

12. S. H. Tseng, “PSTD simulation of optical phase conjugation of light propagating long optical paths,” Opt. Express **17**, 5490–5495 (2009). [CrossRef] [PubMed]

1. F. Devaux and E. Lantz, “Real time suppression of turbidity of biological tissues in motion by three-wave mixing phase conjugation,” J. of Biomed. Opt. **18**, 111405 (2013). [CrossRef]

## 2. Principle of the tri-dimensional pseudo-spectral time domain method

13. Q. H. liu and G. Zhao, “Review of PSTD methods for transient electromagnetics,” Int. J. Numer. Model. **17**, 299–323 (2004). [CrossRef]

4. Q. H. Liu, “The PSTD algorithm: a time-domain method requiring only two cells per wavelength,” Microw. Opt. Technol. Lett. **15**, 158–165 (1997). [CrossRef]

14. J. -P. Berenger, “A perfect matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. **114**, 185–200 (1994). [CrossRef]

15. W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations,” Microw. Opt. Technol. Lett. **7**, 599–604 (1994). [CrossRef]

### 2.1. Tri-dimensional pseudo-spectral time domain scheme

*z*axis of a transverse, pulsed electromagnetic wave in a linear, lossless, non dispersive, non conductive and inhomogeneous medium. Because a non dispersive medium is considered, we assume that light is monochromatic or, more precisely, that its Gaussian time envelope is Fourier transform. According to the Maxwell’s equations, the components along the (

*x*,

*y*,

*z*) axes of the electric displacement field

*D⃗*, the electric field

*E⃗*and the magnetic field

*B⃗*are calculated using the time-stepping iterations [4

**15**, 158–165 (1997). [CrossRef]

*x*components of the different fields. The other components can be deduced by a simple circular permutation of the

*x*,

*y*,

*z*indices. The equations governing electromagnetic fields in the medium are given by :

*t*

_{0}and with a time width

*σ*) at the time step

_{t}*n*Δ

*t*and at the spatial sampling point (

*j*Δ

*x*,

*k*Δ

*y*,

*l*Δ

*z*). These components are given by:

*ψ*,

*φ*and

_{x}*φ*are the parameters used to define the polarization state of the electromagnetic wave emitted by the source. The spatial amplitude

_{y}*S*

_{0}|

*is designed with a Gaussian shape in the (*

_{jkl}*x*,

*y*) transverse plane and with the optimized three-cells normalized pattern [

*z*axis in order to suppress the aliasing errors [16

16. Z. Li, “The optimal spatially-smoothed source patterns for the pseudospectral time-domain method,” IEEE Transactions on Antennas and Propagation **58**, 227–229 (2010). [CrossRef]

*z*direction is ensured by adding the source terms simultaneously on the magnetic field so that

### 2.2. Characteristics of the sampled volume and of the scattering medium

*n*×

_{x}*n*×

_{y}*n*points and sampling steps Δ

_{z}*x*= Δ

*y*= Δ

*z*= 0.3

*λ*. Figure 1(a) shows a slice along the

*xz*plane of the sampled volume. This volume is made up of several domains. Absorbing layers, with widths of

*x*and

*y*dimensions. Along the propagation axis (

*z*axis), the input face, corresponding to the center of the three-cells source, and the output face, where measurements are made, are distant respectively of three and two pixels from the absorbing layers. Finally, in some cases the scattering medium will be surrounded by reflecting layers along the

*x*and

*y*directions; this point is discussed in section 3.1. The scattering medium is constituted by an homogeneous dielectric material (with a relative permittivity

*ε*) randomly filled with dielectric spheres, with a radius

_{m}*r*, a relative permittivity

_{s}*ε*>

_{s}*ε*and a volume concentration

_{m}*β*. Figure 1(b) shows a typical distribution, in a (18

_{s}*λ*)

^{3}volume, of the dielectric spheres with a radius

*r*= 0.9

_{s}*λ*and a volume concentration

*β*≃ 5%. In this figure, we can observe that some dielectric spheres overlap and form aggregates with sizes greater than that of a sphere. Moreover, the spatial sampling step introduces some distortions in the spherical shape of the spheres. Although aggregates and the imperfect shape of the spheres introduce some small uncertainties, we do not observe significative variations in our numerical results when different realizations of the scattering medium with the same characteristics are performed. Moreover, numerical results are consistent with the estimation of the scattering coefficients calculated from the Mie theory [17] for spherical particles. In the presented results, the total volume is sampled with 128 × 128 × 256 points and the volume of the scattering medium is

_{s}*V*≃ (30

_{s}*λ*)

^{2}× 56

*λ*.

## 3. Numerical results

*β*. In order to validate our numerical model, the temporal profile of the output light intensity is compared with the time-resolved transmittance function of a semi-infinite scattering medium given by Patterson et al. [18

_{s}18. M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. **28**, 2331–2336 (1989). [CrossRef] [PubMed]

*μ*,

_{s}*μ*and

_{a}*g*= 〈cos

*θ*〉 are, respectively, the scattering coefficient, the linear absorption coefficient and the anisotropy coefficient. Because non-absorptive particles are considered, calculations are performed with

*μ*= 0.

_{a}*d*is the thickness of the scattering medium,

*t*is the time and

*v*is the light velocity in the medium. We also compare the results of the PSTD algorithm with Monte Carlo simulations. In the Monte-Carlo method for photon transport, the propagation distance between two scattering events and the scattering direction cosines are randomly generated using the inverse distribution laws method [19

19. L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Monte Carlo modeling of photon transport in multi-layered tissues,” Computer Methods and Programs in Biomedicine **47**, 131–146 (1995). [CrossRef]

### 3.1. Time-resolved propagation of a short pulse in scattering media

*x*axis and with a 27

*fs*duration (

*x*= 1.8

*λ*. For a radius

*r*= 0.9

_{s}*λ*and a refractive index of the non-absorptive dielectric spheres

*g*= 0.85. Although the scattering properties of a medium mostly depend on the relative size of the spheres with respect to the wavelength, we perform calculations by considering a wavelength

*λ*= 1

*μm*corresponding approximately to the wavelength used in our experiments reported in [1

1. F. Devaux and E. Lantz, “Real time suppression of turbidity of biological tissues in motion by three-wave mixing phase conjugation,” J. of Biomed. Opt. **18**, 111405 (2013). [CrossRef]

*β*.

_{s}*μ*and

_{s}*μm*

^{−1}.

*d*= 56

*λ*= 56

*μm*of the scattering medium.

19. L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Monte Carlo modeling of photon transport in multi-layered tissues,” Computer Methods and Programs in Biomedicine **47**, 131–146 (1995). [CrossRef]

*x*-axis is graduated in units of optical pathlength and the vertical dotted lines correspond to the optical pathlength corresponding to a straight propagation in the scattering medium : 〈

*n*〉

_{s}*d*. Each curve is normalized by its peak value. Though Monte-Carlo simulations are calculated over 10000 trajectories, some oscillations remain in the presented results, but these oscillations are progressively smoothed out when the number of trajectories increases. We can observe that the results of the PSTD algorithm fit with a very good agreement the Monte-Carlo simulations but not the transmittance function. This discrepancy can be explained by the fact that the diffusion model used to calculate the transmission function is valid for scattering media with a large number of reduced mean free paths, which is not the case here. In future works, scattering media with a larger numbers of

*I*〉 is the mean intensity and

*σ*is the standard deviation of the intensity fluctuations. Then, we have compared these contrast values with the contrast calculated with Eq. (8) as well as with the contrasts calculated from measurements of the time widths of the transmitted pulses. Figure 4 shows the typical output speckle pattern obtained with an input pulse, with a duration

_{I}*τ*= 27

_{l}*fs*(

*τ*and

_{s}*τ′*are, respectively, the average one-way traversal time (in

_{l}*fs*) and the time full width of the transmitted pulse (in

*fs*measured at

*fs*). A scattering medium with a thickness

*τ*= 577

_{s}*fs*is considered. As expected, the contrast of the speckle patterns observed at the output of the scattering medium increases from 0.24 to 0.52 with the time width of the input pulse and the contrasts calculated with the different methods give results in good agreement.

### 3.2. Time-resolved polarization analysis of scattered light

*n*Δ

*t*and at the spatial sampling point (

*j*Δ

*x*,

*k*Δ

*y*) :

*DOP*. Figure 6 and Table 3 summarize the statistical properties of these variables. 〈〉

*denotes the integration in time and space of the considered variables. As the input pulse is linearly polarized in the*

_{xyt}*x*direction, its normalized Stokes vector reads

*I*= 1,

_{in}*Q*= 1,

_{in}*U*= 0 and

_{in}*V*= 0 (

_{in}*DOP*= 1). When

_{in}*U*and

*V*components remain null with a large standard deviation and the average values of the

*Q*parameter and of the

*DOP*decrease. More surprisingly, we find that, although the rate of depolarized light increases with

*x*axis even when the thickness of the scattering medium is larger than

*x*-axis (i.e. time axis) is still graduated in optical pathlengths. The first observation concerns the peak of the

*Q*parameter. When

*Q*and

*I*variations are confounded and the time width of the

*Q*component (

*τ*∼ 100

_{Q}*fs*) is much larger than the width of the input pulse (

*τ*= 27

_{l}*fs*). Moreover, while the time width of the intensity profiles increases with

*Q*component remains constant as well as its temporal shape. We can also observe that the light is completely depolarized when the optical pathlength of the scattered light is greater than twice the optical thickness of the medium. These results confirm that the early transmitted light (corresponding to the so-called ballistic photons) is preferentially vertically polarized as well as the weakly scattered light corresponding to the so-called snake photons. Moreover, it shows that the time arrival of the weakly scattered light is not determined by the scattering coefficient of the medium. Future works should be done in order to characterize more precisely this property with respect to the thickness of the scattering medium, the size and the shape of the scattering particles.

*DOP*during time. Figure 8(a) represents the transient regime of the space and time-integrated 〈

*DOP*〉

*obtained with the different scattering media. The curves exhibit maxima very close to 1 at the early times and decrease up to the final values given in Table 3. Figures 8(b)–8(d) show the local time-integrated*

_{xyt}*DOP*at different times for the

*DOP*is close to 1 in an area located at the center of the transverse plane and with transverse dimensions close to the size of the source. This corresponds to the light that travels in a straight line across the medium. At the time corresponding to the peak value (Fig. 8(c)), the

*DOP*is close to 1 in the whole transverse section. For increasing times, the local

*DOP*exhibits some increasing fluctuations until it reaches its final state depicted by the Fig. 8(d).

*DOP*of the transmitted light are studied for different scattering media with increasing values of

*β*. We have modelized scattering media made of dielectric spheres, with a radius

_{s}*r*= 0.6

_{s}*λ*and a refractive index

*n*= 1.34, embedded in a medium with a refractive index

_{s}*n*= 1. At

_{m}*λ*= 1

*μm*, the anisotropy coefficient is

*g*= 0.81 and the size parameter of a sphere is

*μm*

^{−1}, and the thickness of the scattering media corresponds, respectively, to a number of reduced scattering events

*β*, the time-resolved variations of the space-integrated Stokes components (〈

_{s}*I*〉

*, 〈*

_{xy}*Q*〉

*) and (〈*

_{xy}*I*〉

*, 〈*

_{xy}*V*〉

*) of the transmitted light initially linearly and circularly polarized. These components are normalized by the peak values of the intensity. In both figures, we can observe that the time variations of the intensities are confounded. It shows that the intensity temporal shape of the transmitted light is independent of the incident polarization state [2*

_{xy}2. X. Wang, L. V. Wang, C.W. Sun, and C.C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiements,” J. of Biomed. Opt. **8**, 608–617 (2003). [CrossRef]

*Q*of the linearly polarized light decreases a little faster than the component

*V*of the circularly polarized light. This result is consistent with previous works showing that, in the Mie regime, linearly polarized light is more rapidly depolarized than circularly polarized light [2

2. X. Wang, L. V. Wang, C.W. Sun, and C.C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiements,” J. of Biomed. Opt. **8**, 608–617 (2003). [CrossRef]

22. D. Bicout, C. Brosseau, A. S. martinez, and J. M. Schmitt, “Depolarization of multiply scattered waves by spherical difFusers: Influence of the size parameter,” Phys. Rev. E **49**, 1767–1770 (1994). [CrossRef]

*Q*and

*V*is also clearly exhibited, with a decay constant of about 0.25

*μm*

^{−1}with the time axis graduated in optical pathlength units (corresponding to a time decay constant of about 0.075

*fs*

^{−1}). With the

*x*-axis graduated in number of scattering events unit (

*c*(

*t*−

*t*

_{0}) + 〈

*n*〉

_{s}*d*)

*U*,

*V*) and (

*Q*,

*U*) are null, respectively for the linear and the circular polarization state of the incident light, we calculated the corresponding instantaneous time variation of the space integrated degree of linear polarization

*DOP*and the

_{L}*DOP*for scattering media with

_{C}*β*=7, 9, 10 and 12 %. The

_{s}*x*-axis origin of the curves corresponds to the optical thickness of the media. In agreement with the

*Q*and

*V*variations,the

*DOP*decreases a little faster than the

_{L}*DOP*. Since the FWHM of the curves decreases approximately from 19

_{C}*μm*(i.e. 63

*fs*) to 16

*μm*(i.e. 53

*fs*) when

*β*increases, the FWHM of the DOP curves expressed in optical pathlength unit divided by the reduced mean free paths increases from 0.8 to 1.1, if we take into account the reduced mean free paths calculated for the different values of

_{s}*β*. It confirms that the exiting scattered light has undergone, on average, one reduced scattering event before being depolarized. In Fig. 9(b) weak oscillations (they would be almost invisible with a linear scale) can be observed in the time variation of the Stokes parameters

_{s}*V*for an incident wave circularly polarized (red dashed curve) between optical pathlengths of 80 and 90

*μm*. These oscillations, traducing a fluctuation of the instantaneous polarization state of light, are reproducible but with different shapes for different realizations of the scattering medium. Similar oscillations can be observed in Fig. 7 for the Stokes parameters

*U*and

*V*.

## 4. Conclusion

## Acknowledgments

## References and links

1. | F. Devaux and E. Lantz, “Real time suppression of turbidity of biological tissues in motion by three-wave mixing phase conjugation,” J. of Biomed. Opt. |

2. | X. Wang, L. V. Wang, C.W. Sun, and C.C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiements,” J. of Biomed. Opt. |

3. | K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Transactions on Antennas and Propagation |

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

5. | Z. Tang and Q. H. Liu, “The 2.5D FDTD and Fourier PSTD methods and applications,” Microw. Opt. Technol. Lett. |

6. | T. W. Lee and S. C. Hagness, “Pseudospectral time-domain methods for modeling optical wave propagation in second-order nonlinear materials,” J. Opt. Soc. Am. B |

7. | X. Liu and Y. Chen, “Applications of transformed-space non-uniform PSTD (TSNU-PSTD) in scattering analysis with the use of the non-uniform FFT,” Microw. Opt. Technol. Lett. |

8. | C. Liu, R. L. Panetta, and P. Yang, “Application of the pseudo-spectral time domain method to compute particle single-scattering properties for size parameters up to 200,” J. Quant. Spectrosc. Transfer |

9. | S. H. Tseng, A. Taflove, D. Maitland, and V. Backman, “Pseudospectral time simulations of mutiple light scattering in three-dimensional macroscopic random media,” Radio Science |

10. | C. Liu, L. Bi, R. L. Panetta, P. Yang, and M. A. Yurkin, “Comparison between the pseudospectral time domain method and the discrete dipole approximation for light scattering simulations,” Opt. Express |

11. | S. H. Tseng and C. Yang, “2-D PSTD simulation of optical phase conjugation for turbidity suppression,” Opt. Express |

12. | S. H. Tseng, “PSTD simulation of optical phase conjugation of light propagating long optical paths,” Opt. Express |

13. | Q. H. liu and G. Zhao, “Review of PSTD methods for transient electromagnetics,” Int. J. Numer. Model. |

14. | J. -P. Berenger, “A perfect matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

15. | W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations,” Microw. Opt. Technol. Lett. |

16. | Z. Li, “The optimal spatially-smoothed source patterns for the pseudospectral time-domain method,” IEEE Transactions on Antennas and Propagation |

17. | C. F. Bohren and D. R. Huffman, |

18. | M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. |

19. | L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Monte Carlo modeling of photon transport in multi-layered tissues,” Computer Methods and Programs in Biomedicine |

20. | N. Curry, P. Bondareff, M. Leclerq, N. K. Van Hulst, R. Sapienza, S. Gigan, and S. Grésillon, “Direct determination of diffusion properties of random media from speckle contrast,” Opt. Lett. |

21. | R. Landauer and M. Büttiker, “Diffusive traversal time: Effective area in magnetically induced interference,” Phys. Rev. B |

22. | D. Bicout, C. Brosseau, A. S. martinez, and J. M. Schmitt, “Depolarization of multiply scattered waves by spherical difFusers: Influence of the size parameter,” Phys. Rev. E |

23. | F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B |

24. | M. Cui, E. J. McDowell, and C. Yang, “Observation of polarization-gate based reconstruction quality improvement during the process of turbidity suppression by optical phase conjugation,” Appl. Phys. Lett. |

**OCIS Codes**

(290.4210) Scattering : Multiple scattering

(290.2558) Scattering : Forward scattering

(290.5855) Scattering : Scattering, polarization

**ToC Category:**

Scattering

**History**

Original Manuscript: July 18, 2013

Revised Manuscript: September 16, 2013

Manuscript Accepted: September 24, 2013

Published: October 11, 2013

**Citation**

Fabrice Devaux and Eric Lantz, "3D-PSTD simulation and polarization analysis of a light pulse transmitted through a scattering medium," Opt. Express **21**, 24969-24984 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-21-24969

Sort: Year | Journal | Reset

### References

- F. Devaux and E. Lantz, “Real time suppression of turbidity of biological tissues in motion by three-wave mixing phase conjugation,” J. of Biomed. Opt.18, 111405 (2013). [CrossRef]
- X. Wang, L. V. Wang, C.W. Sun, and C.C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiements,” J. of Biomed. Opt.8, 608–617 (2003). [CrossRef]
- K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Transactions on Antennas and Propagation14, 302–307 (1966). [CrossRef]
- Q. H. Liu, “The PSTD algorithm: a time-domain method requiring only two cells per wavelength,” Microw. Opt. Technol. Lett.15, 158–165 (1997). [CrossRef]
- Z. Tang and Q. H. Liu, “The 2.5D FDTD and Fourier PSTD methods and applications,” Microw. Opt. Technol. Lett.36, 430–436 (2003). [CrossRef]
- T. W. Lee and S. C. Hagness, “Pseudospectral time-domain methods for modeling optical wave propagation in second-order nonlinear materials,” J. Opt. Soc. Am. B21, 330–342 (2004). [CrossRef]
- X. Liu and Y. Chen, “Applications of transformed-space non-uniform PSTD (TSNU-PSTD) in scattering analysis with the use of the non-uniform FFT,” Microw. Opt. Technol. Lett.38, 16–21 (2003). [CrossRef]
- C. Liu, R. L. Panetta, and P. Yang, “Application of the pseudo-spectral time domain method to compute particle single-scattering properties for size parameters up to 200,” J. Quant. Spectrosc. Transfer113, 1728–1740 (2012). [CrossRef]
- S. H. Tseng, A. Taflove, D. Maitland, and V. Backman, “Pseudospectral time simulations of mutiple light scattering in three-dimensional macroscopic random media,” Radio Science41, RS4009 (2006). [CrossRef]
- C. Liu, L. Bi, R. L. Panetta, P. Yang, and M. A. Yurkin, “Comparison between the pseudospectral time domain method and the discrete dipole approximation for light scattering simulations,” Opt. Express20, 16763–16776 (2012). [CrossRef]
- S. H. Tseng and C. Yang, “2-D PSTD simulation of optical phase conjugation for turbidity suppression,” Opt. Express15, 1605–1616 (2007). [CrossRef]
- S. H. Tseng, “PSTD simulation of optical phase conjugation of light propagating long optical paths,” Opt. Express17, 5490–5495 (2009). [CrossRef] [PubMed]
- Q. H. liu and G. Zhao, “Review of PSTD methods for transient electromagnetics,” Int. J. Numer. Model.17, 299–323 (2004). [CrossRef]
- J. -P. Berenger, “A perfect matched layer for the absorption of electromagnetic waves,” J. Comput. Phys.114, 185–200 (1994). [CrossRef]
- W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations,” Microw. Opt. Technol. Lett.7, 599–604 (1994). [CrossRef]
- Z. Li, “The optimal spatially-smoothed source patterns for the pseudospectral time-domain method,” IEEE Transactions on Antennas and Propagation58, 227–229 (2010). [CrossRef]
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1983 Chap. 4, pp. 82–129).
- M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt.28, 2331–2336 (1989). [CrossRef] [PubMed]
- L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Monte Carlo modeling of photon transport in multi-layered tissues,” Computer Methods and Programs in Biomedicine47, 131–146 (1995). [CrossRef]
- N. Curry, P. Bondareff, M. Leclerq, N. K. Van Hulst, R. Sapienza, S. Gigan, and S. Grésillon, “Direct determination of diffusion properties of random media from speckle contrast,” Opt. Lett.36, 3332–3334 (2011). [CrossRef] [PubMed]
- R. Landauer and M. Büttiker, “Diffusive traversal time: Effective area in magnetically induced interference,” Phys. Rev. B36, 6255–6210 (1987). [CrossRef]
- D. Bicout, C. Brosseau, A. S. martinez, and J. M. Schmitt, “Depolarization of multiply scattered waves by spherical difFusers: Influence of the size parameter,” Phys. Rev. E49, 1767–1770 (1994). [CrossRef]
- F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B40, 9342–9345 (1989). [CrossRef]
- M. Cui, E. J. McDowell, and C. Yang, “Observation of polarization-gate based reconstruction quality improvement during the process of turbidity suppression by optical phase conjugation,” Appl. Phys. Lett.95, 123702 (2009). [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.