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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 21 — Oct. 21, 2013
  • pp: 24969–24984
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3D-PSTD simulation and polarization analysis of a light pulse transmitted through a scattering medium

Fabrice Devaux and Eric Lantz  »View Author Affiliations


Optics Express, Vol. 21, Issue 21, pp. 24969-24984 (2013)
http://dx.doi.org/10.1364/OE.21.024969


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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

In a recent paper, we have presented results of ultrafast compensation of turbidity of ex-vivo biological tissues by type 2 three-wave-mixing phase conjugation (TWMPC), where images transmitted through biological tissues with thicknesses up to 5 mm were restored [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]

]. These results indicate that the scattering process in such biological samples is more or less independent of the polarization state of the light and that a polarization change of the phase conjugated wave with respect to the incident wave does not preclude the image restoration process. Indeed, the phase conjugated wave (i.e. the idler wave) that retraces the scattering path is not polarized as the scattered light exiting from the biological tissues (i.e. the signal), because in a type 2 three-wave-mixing process the signal and the idler wave are polarized in orthogonal directions. The hypothesis of polarization insensitivity, first confirmed by a simple numerical model based on the Monte Carlo method, deserves further study with a more efficient numerical model. In fact, we have to implement a numerical model able to simulate accurately, in time and in space, electromagnetic phenomena such as the propagation of light in a scattering medium and the non linear optical process of TWMPC. Although the Monte Carlo offers a flexible and accurate method approach to model the variation in time of the state of polarization of the light transmitted through scattering media [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]

], non linear optical phenomena and coherent effects of light propagation are not accessible with this method.

In this paper, we propose a 3D-PSTD algorithm to model the propagation of a light pulse through scattering media with realistic dimensions. The variation in time and the state of polarization of the transmitted light are analyzed for different characteristics of the scattering medium and different polarization states of the incident light. To validate our algorithm, we compare the results obtained from our simulations with a Monte carlo numerical method and with some previously reported results. 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 simple fact could explain the results 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]

].

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

In PSTD algorithms, Maxwell’s curl equations are calculated with discrete Fourier transforms in order to solve the spatial derivatives on an unstaggered, collocated grid [13

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

]. The Fast-Fourier-Transform (FFT) is used to implement these spatial derivatives and limitations of FFT, due to the periodic boundary conditions, are eliminated by using absorbing boundary conditions formulated for perfect matched layer (PML) in nonconductive media [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]

, 14

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

, 15

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

We consider here the propagation along the 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

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]

]. We give here the details of the time-variation calculations for the 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 :
{Dxt=Dxyt+Dxzt=1μ0[BzyByz]Ex=DxεBxt=Bxyt+Bxzt=Ezy+Eyz
(1)

Sx|jkln and Sy|jkln are the transverse amplitude components of the pulsed source term (delayed with a time t0 and with a time width σt) at the time step nΔt and at the spatial sampling point (jΔx, kΔy, lΔz). These components are given by:
{Sx|jkln=S0|jklcosψei(ωnΔt+φx)e(nΔtt0)22σt2Sy|jkln=S0|jklsinψei(ωnΔt+φy)e(nΔtt0)22σt2
(5)

ψ, φx and φy are the parameters used to define the polarization state of the electromagnetic wave emitted by the source. The spatial amplitude S0|jkl is designed with a Gaussian shape in the (x, y) transverse plane and with the optimized three-cells normalized pattern [ 14, 12, 14] along the 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]

]. Indeed, this optimized three-cells normalized pattern, according to the properties of Pascal’s triangle, has a spatial frequency spectrum which is a discrete decreasing function becoming null at the cutoff spatial frequency. Finally, the propagation of the wave in the increasing z direction is ensured by adding the source terms simultaneously on the magnetic field so that B=μ0εez×D. It is obtained as follows:
{Bxz|jkln+12=Bxz|jklnμ0εjklSy|jklnByz|jkln+12=Bxz|jkln+μ0εjklSx|jkln
(6)

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

The considered volume is sampled with nx × ny × nz points and sampling steps Δx = Δy = Δz = 0.3λ. 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 nx8, ny8 and nz8 pixels, are defined at the boundaries of the volume along each spatial dimension. The scattering medium is delimited by these absorbing layers along the 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 εm) randomly filled with dielectric spheres, with a radius rs, a relative permittivity εs > εm and a volume concentration βs. Figure 1(b) shows a typical distribution, in a (18λ)3 volume, of the dielectric spheres with a radius rs = 0.9λ and a volume concentration βs ≃ 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

17. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1983 Chap. 4, pp. 82–129).

] 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 Vs ≃ (30λ)2 × 56λ.

Fig. 1 (a) Section along the xz plane of the sampled volume. (b) Example of a scattering medium modelized by dielectric spheres randomly embedded in a homogeneous dielectric medium.

3. Numerical results

First, we investigate the temporal properties of the scattered light exiting from inhomogeneous media designed with different values of βs. 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

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]

]:
T(t)=(4πDν)12t32×eμavt×{(dls*)e(dls*)24Dvt(d+ls*)e(d+ls*)24Dvt+(3dls*)e(3dls*)24Dvt(3d+ls*)e(3d+ls*)24Dvt}
(7)

D=13[μa+(1g)μs] is the diffusion coefficient where μs, μa and 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 μa = 0. ls*=1(1g)μs is the reduced mean free path, 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]

].

Then, we analyze the time dependence of the polarization state of the scattered light for different polarization states of the incident light.

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

A short pulse polarized along the x axis and with a 27 fs duration ( 1e2 full width) is emitted by the source. The source has a very narrow circular gaussian shape in the transverse plane with a full width 6Δx = 1.8λ. For a radius rs = 0.9λ and a refractive index of the non-absorptive dielectric spheres ns=εs=1.34, the size parameter of the spheres is krs=2πnsrsλ=7.6, which implies that light scattering occurs in the Mie regime, with an anisotropy coefficient 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 = 1μm 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]

]. Considering that spheres are embedded in a lossless homogeneous medium with a refractive index nm=εm=1 and with volume concentrations of 5%, 7%, 9% and 11%, the scattering coefficients are calculated using the Mie theory. Table 1 gives the values of these coefficients with respect to the volume concentrations βs. μs and μs* are given in μm−1. Ns=μsd=dls and Ns*=μs*d=dls* are, respectively, the number of scattering events and the number of reduced scattering events calculated for a thickness d = 56λ = 56μm of the scattering medium.

Table 1. Scattering coefficients with respect to the volume concentrations βs of the dielectric spheres.

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Fig. 2 (a)–(d) Single-frame excerpts from the video ( Media 1) recording, in the xz plane and at the times 0, 70, 120 and 170 fs, the propagation of the 27 fs pulse in a scattering medium with βs = 7%. (a) The white contours show the locations and the shapes of the particles in the considered xz plane of the medium. (b)–(d) The white dotted lines represent the boundaries of the scattering medium. (e) Corresponding profiles along the z axis of the pulse intensity integrated in the (x, y) transverse plane.

In order to validate our numerical model, we compare systematically the temporal shape of the transmitted light given by our PSTD algorithm with the analytical expression of the time-resolved transmittance of a scattering medium given by Eq. (7) and with Monte-Carlo simulations [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]

]. The curves giving the results of the Monte Carlo simulations correspond to the histogram of the optical pathlengths calculated over 10000 trajectories. First, simulations are performed with scattering media only delimited by the absorbing layers. With the PTSD algorithm, the durations of the transmitted pulses are significantly shorter than the results given by the two other methods. We explain this discrepancy by the small transverse dimensions of the scattering mediium : indeed, a large part of the scattered light is lost in the absorbing layers. Therefore, in order to avoid these losses, we have added reflective layers at the boundaries of the transverse plane (see Fig. 1).

Figure 3 shows the different temporal shapes of the output light (green curves), obtained with the reflective layers, with respect to the thickness expressed in number of reduced mean free paths. Each curve is compared with the corresponding time-resolved transmittance curve (red curves) convolved with the input pulse (blue curves) and with the Monte-Carlo simulations (cyan curves). The 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 : 〈nsd. 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 ls* should be investigated in order to demonstrate a possible better agreement between the PSTD and the diffusion theory.

Fig. 3 Comparison of the time-shapes of the transmitted pulse given by the PSTD algorithm (green curves), Monte-Carlo simulations (cyan curves) and analytical transmittance convolved by the input pulse shape (red curves) for different values of Ns*. The x-axis is graduated in optical pathlength units. The blue curves represent the input pulse and the vertical dotted lines correspond to the optical length of the scattering medium.

From the time-integrated speckle patterns obtained with our PSTD algorithm, we have calculated the contrast σII, where 〈I〉 is the mean intensity and σI 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 τl = 27 fs ( 1e2 full width), transmitted through a medium with a thickness d=3ls*. Table 2 summarizes the values of the contrasts calculated with the different methods. τs and τ′l are, respectively, the average one-way traversal time (in fs) and the time full width of the transmitted pulse (in fs measured at 1e2. We can observe that all the methods give values of the contrast in good agreement even if the calculated one-way traversal times are significantly greater than the durations of the output pulses. The same measurements have been performed for increasing time durations of the incident pulse (from 27 to 253 fs). A scattering medium with a thickness d=2ls* that corresponds to an average one-way traversal time τs = 577 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.

Fig. 4 Normalized intensity of the output speckle pattern obtained with a 3ls* scattering medium. The contrast is C = 0.21.

Table 2. Comparison of the contrasts of the speckle patterns calculated with different methods and for different values of Ns*.

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The good fit between the results obtained with the PSTD and the Monte-Carlo simulations and the good agreement between the contrast values calculated by the different methods tend to prove that our PSTD algorithm correctly simulates the propagation of a light pulse through an inhomogeneous medium.

3.2. Time-resolved polarization analysis of scattered light

In this section, we study the space and time-resolved polarization state of the transmitted light with respect to the properties of the scattering medium and with respect to the polarization state of the input pulse. The time variation of the local polarization state of the transmitted light is characterized by calculating the components of the Stokes vector in the the output plane, at the time step nΔt and at the spatial sampling point (jΔx, kΔy) :
I|jkn=Ex|jknEx*|jkn+Ey|jknEy*|jknQ|jkn=Ex|jknEx*|jknEy|jknEy*|jknU|jkn=Ex|jknEy*|jkn+Ex*|jknEy|jknV|jkn=i(Ex|jknEy*|jknEx*|jknEy|jkn)
(9)

Fig. 5 Time-integrated components of the local Stokes vector and the local degree of polarization (DOP) of the light transmitted through a 2ls* scattering medium. Stokes parameters are normalized by the peak intensity of the speckle pattern.

For the different values of Ns*, we calculated the average values and the standard deviations of the Stokes parameters and of the DOP. Figure 6 and Table 3 summarize the statistical properties of these variables. 〈〉xyt denotes the integration in time and space of the considered variables. As the input pulse is linearly polarized in the x direction, its normalized Stokes vector reads Iin = 1, Qin = 1, Uin = 0 and Vin = 0 (DOPin = 1). When Ns* increases, we can observe that the average values of the 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 Ns*, a significant part of the transmitted light remains linearly polarized along the x axis even when the thickness of the scattering medium is larger than ls*.

Fig. 6 Histograms of the time-integrated local Stokes parameters and of the DOP for different values of Ns*.

Table 3. Average values and standard deviations of the space and time-integrated local Stokes parameter Q and DOP.

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Now, let’s us examine the evolution in time of the state of polarization of the light transmitted through the scattering medium. We have plotted the variation with time of the instantaneous values of the spatially integrated Stokes parameters for the different scattering media (Fig. 7). The x-axis (i.e. time axis) is still graduated in optical pathlengths. The first observation concerns the peak of the Q parameter. When Ns* increases, its value decreases relatively to the peak intensity but its position is less delayed than the peak intensity position. The rising edges of the Q and I variations are confounded and the time width of the Q component (τQ ∼ 100 fs) is much larger than the width of the input pulse (τl = 27 fs). Moreover, while the time width of the intensity profiles increases with Ns*, the time width of the 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.

Fig. 7 Time variation of the space-integrated components of the Stokes vector for the different media. x-axes are graduated in optical pathlength units.

We studied also the variation of the DOP during time. Figure 8(a) represents the transient regime of the space and time-integrated 〈DOPxyt 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 DOP at different times for the 2ls* scattering medium. In Fig. 8(b), that corresponds to the early time of the scattered light, the local 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).

Fig. 8 (a) Time variation of the time and space integrated DOP for the different media. (b)–(d) Time integrated local DOP at different time for the 2ls* medium. The vertical black dotted line represents the optical pathlength through the homogeneous medium.

The last numerical experiments concern the influence of the polarization state of light on the scattering process. Linear and circular polarizations of the light emitted by the source are considered. In both cases, the time-resolved variation of the Stokes components and of the DOP of the transmitted light are studied for different scattering media with increasing values of βs. We have modelized scattering media made of dielectric spheres, with a radius rs = 0.6λ and a refractive index ns = 1.34, embedded in a medium with a refractive index nm = 1. At λ = 1μm, the anisotropy coefficient is g = 0.81 and the size parameter of a sphere is krs=2πnsrsλ=5.0. With the volume concentrations of 7, 9 10 and 12%, the scattering media are characterized by the coefficients μs*=0.043, 0.053, 0.063 and 0.073 μm−1, and the thickness of the scattering media corresponds, respectively, to a number of reduced scattering events Ns*=2.4, 3, 3.6 and 4.1.

Fig. 9 Comparison of the time variations of the space integrated Stokes parameters (I, Q) or (I, V), respectively for linearly (blue lines) or circularly (red lines) polarized incident pulses for scattering media with volume concentrations (a) βs = 7% and (b) βs = 9%. (c) Time variation of the space integrated linear and circular degrees of polarization when the incident light is, respectively, linearly and circularly polarized with βs =7, 9, 10 and 12 %.

Because the mean values of the Stokes components (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 DOPLxy=QIxy and the degree of circular polarization DOPCxy=VIxy. Figure 9(c) depicts the time variation of the DOPL and the DOPC for scattering media with βs =7, 9, 10 and 12 %. The x-axis origin of the curves corresponds to the optical thickness of the media. In agreement with the Q and V variations,the DOPL decreases a little faster than the DOPC. Since the FWHM of the curves decreases approximately from 19 μm (i.e. 63 fs) to 16 μm (i.e. 53 fs) when βs 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 βs. 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 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

This work has been supported by the Agence Nationale de la Recherche (ICLM, project ANR-2011-BS04-017-03) and the Conseil Régional de Franche-Comté. Computations have been performed on the supercomputer facilities of the Mésocentre de calcul de Franche–Comté.

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. 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]

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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]

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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]

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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.

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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]

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

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

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]

17.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1983 Chap. 4, pp. 82–129).

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]

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]

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. 36, 3332–3334 (2011). [CrossRef] [PubMed]

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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


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References

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