Abstract

Transient/time-dependent radiative transfer in a two-dimensional scattering medium is numerically solved by the discontinuous finite element method (DFEM). The time-dependent term of the transient vector radiative transfer equation is discretized by the second-order central difference scheme and the space domain is discretized into non-overlapping quadrilateral elements by using the discontinuous finite element approach. The accuracy of the transient DFEM model for the radiative transfer equation considering the polarization effect is verified by comparing the time-resolved Stokes vector component distributions against the steady solutions for a polarized radiative transfer problem in a two-dimensional rectangular enclosure filled with a scattering medium. The transient polarized radiative transfer problems in a scattering medium exposed to an external beam and in an irregular emitting medium are solved. The distributions of the time-resolved Stokes vector components are presented and discussed.

© 2017 Optical Society of America

1. Introduction

The research field of ultra-fast optics [1] has attracted considerable attention in recent years and the transient/time-dependent radiative transfer (TRT) in a scattering/turbid medium has become the focus of current research due to its wide applications in the field of optical tomography and imaging [2–4]. The transient radiative transfer behavior and its applications in various aspects have been presented in a detailed review paper by Kumar and Mitra [5].The propagation process of the transient radiative transfer problem in a scattering medium is described by the radiative transfer equation (RTE) which is a complex integral-differential equation and is difficult to solve analytically. In the last decade, several numerical strategies have been developed for solving the TRT, including the discrete ordinate method (DOM) [6–12], the integral equation (IE) method [13–15], the finite volume method (FVM) [16,17], the lattice Boltzmann method (LBM) [18, 19], the distribution of ratios of energy scattered by the medium or reflected by the boundary surface (DRESOR) method [20, 21], the discrete transfer method (DTM) [22], and the Monte Carlo method (MCM) [23–27]. By coupling LBM and DTM, Mishra et al. [28] solved the transient conduction-radiation heat transfer in participating media and they further compared the performance of DOM, DTM, and FVM for solving transient radiative transfer problems in a participating medium [29]. However, the previous work on transient radiative transfer problems only considered the scalar radiative transfer equation and ignored the polarization characteristic of the radiated light.

Radiative transfer of polarized light in a scattering medium has considerable significance in the fields of remote sensing [30], optical imaging [31], and astronomy [32]. A recent work by Brown [33] dealt with light scattering in a coherent fashion for the full polarization state and established the mathematical equivalence relations linking laboratory, LIDAR and planetary Mueller matrix schemes. Biomedical diagnosis and fast remote sensing also benefit from the additional information provided by the time-dependent polarization characteristics. The polarized radiative transfer process can be accurately described by the vector radiative transfer equation (VRTE). Researchers have been attempting various numerical methods for polarized radiative transfer problems but most of them only focused on the steady-state cases [34–43].

Due to the complexity of the transient vector radiative transfer equation (TVRTE) which takes both the time effect and the polarization effect into account, only a few studies [44–47] on time-dependent polarized radiative transfer problems have been published. Ishimaru et al. [44] solved the time-dependent vector radiative equation for a plane-parallel Mie scattering medium by using DOM. Wang et al. [45] studied the polarized light transmitting through a randomly scattering medium by using the experimental measurement and the Monte Carlo method. Sakami et al. [46] applied the DOM to solve the transient vector radiative transfer and investigate the propagation of a polarized pulse in a random medium. Ilyushin et al. [47] studied the propagation of short pulses of light in a scattering medium by proposing the small angle approximation of the radiative transfer theory and discussed the limitations and applications of their new approach to the polarized light propagation problems.

Most recently, we introduced the discontinuous finite element method (DFEM) to solve the TVRTE in a one-dimensional scattering medium, and our results show that the DFEM is effective and accurate [48]. In this paper, we extend the application of the DFEM to solve the transient radiative transfer in a two-dimensional scattering medium considering the polarization effect. The remainder of this paper is divided into three sections. The mathematical formulation and the DFEM discretization of the TVRTE are presented in the next section. The DFEM applications to the time-dependent polarized radiative transfer problem in a rectangular scattering medium with an external radiation source and in an irregular emitting medium are discussed in section 3, which is followed by the remarkable conclusions in section 4.

2. Mathematical formulation and DFEM discretization

Based on the incoherent addition principle of Stokes parameters, the TVRTE in a two-dimensional emitting, absorbing and scattering medium can be written as [49, 50]

1c0I(r,Ω,t)t+ΩI(r,Ω,t)+βI(r,Ω,t)=S(r,Ω,t),
where I = (I, Q, U, V)T is the Stokes vector, the superscript 'T' denotes the transposition of the matrix, the Stokes component I is the radiation intensity, Q is the linear polarization aligned parallel or perpendicular to the propagation direction, U is the linear polarization aligned ± 45° to the propagation direction, and V is the circular polarization. c0 is the light speed in the vacuum, t is time, r is the location, Ω is the direction, β is the extinction coefficient matrix, and S is the source term written as
S(z,Ω,t)=κaIb+κs4π4πZ(Ω,Ω)I(r,Ω,t)dΩ,
where Z(Ω', Ω) is the scattering phase matrix for scattering from an incoming direction Ω' to an outgoing direction Ω, Ib = (Ib, 0, 0, 0)T and Ib is the black body emission.

The boundary condition for Eq. (1) considering the specular and diffuse reflection is given as

I(rw,Ω,t)=RsI(rw,Ω,t)+1πnwΩn>0RdI(rw,Ω,t)|nwΩ|dΩ,
where the subscript 'w' denotes the variables on the global boundary, Ω” denotes the corresponding incident direction of the reflected beam of direction Ω, and nw denotes the unit outward normal vector of the boundary. Rs and Rd are the specular and diffuse reflection matrix [36].

For the DFEM application for the time-dependent radiative transfer in a two-dimensional medium, the computational domain is divided into finite non-overlapping elements as shown in Fig. 1 and the angular space is discretized into Nθ × Nφ directions. The adjacent elements, which share the same non-physical boundary in the conventional finite element method, are separated from each other in the DFEM approach. The boundary variables inside and outside the element denoted by In and I+n are assumed to be different, the adjacent elements are connected by modeling the boundary flux [48]

[In]=ΩnI¯n+|Ωn|InnK,
where the subscript 'n' denotes the nth discrete direction, I¯n and Ιn are defined as

 

Fig. 1 Sketch of elements, element boundaries and the radiation values on the boundaries for the two-dimensional discrete elements.

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I¯n=12(In+I+n),In=12(InI+n).

As for the treatment for the one-dimensional TVRTE in [48], the dimensionless time t* = βc0t is used and the transient term is discretized by the second-order Crank-Nicolson (time central difference) scheme [51] in this paper. Following the DFEM application for TVRTE presented in [48] to avoid the duplication, the final discretization over the two-dimensional element e and at kth time step can be written in the matrix form as

KknIkn=Hkn,
where the matrix Κkn and Hknare defined as
Kk,jin=eϕiΩnϕjdA+12e(Ωnne+|Ωn|ϕiϕj)ds+eβ˜ϕiϕjdA,
Hk,jn=eS˜knϕjdA12e(Ωnne|Ωn|)Ik,+nϕjds,
where
β˜=(2Δt*+1)β,
S˜kn(r,Ω)=Sk(r,Ω)+Sk1(r,Ω)ΩIk1(r,Ω)(12Δt*)βIk1(r,Ω),
where ϕ is the shape function, Δt* is the dimensionless time step, the subscript 'k' denotes the kth time step, the subscript ' + ' denotes the Stokes vector at the boundary outside element e as shown in Fig. 1, and all the symbol definitions appeared in the equations can be seen in Table 1 in Appendix A.

Tables Icon

Table 1. Nomenclature.

3. Results and discussions

A Matlab program was constructed to calculate the matrix terms and solve the transient polarized radiative transfer equation iteratively. After the field variables on each DFEM elements are solved by Eq. (4), the radiative intensity at a global node is obtained by averaging the intensities on the DFEM nodes having the same global coordinate. The DFEM was then applied to transient polarized radiative transfer problems in a rectangular scattering medium with an external radiation source and a self-emitting medium respectively. In all the following cases, the computation domain is discretized into Nx × Ny = 20 × 20 quadrilateral elements and the angular space is divided into Nθ × Nφ = 40 × 60 sub-angles for the application of the DFEM and grid-independent results [39, 48]. The simulations in this paper are all taken on a personal laptop with the configuration of an Intel Core (TM) i7-6700 processor with 3.40 GHz CPU and 8 GB RAM. The dimensionless time step is set as Δt* = 0.01 which is found to be sufficient to obtain stable and satisfactory results. At a given time step, when the change in the incident radiation at all nodes for the two consecutive iterations is less than 10−6, the convergence is assumed to be achieved and the program goes to the next time step.

3. 1 Transient polarized radiative transfer in a scattering medium with an external radiation source

In this case, we consider a two-dimensional enclosure in the x-o-z plane as shown in Fig. 2, the dimension in the y-direction (perpendicular to the paper) is assumed to be infinite. An external collimated beam is incident on the left boundary of the cold medium at time t* = 0. As for the treatment of the steady cases, the transient Stokes vector inside the medium is decomposed into a collimated part and a diffuse part [2, 8]

I=Ic+Id,
where Ic denotes the collimated part and Id denotes the diffuse part of the Stokes vector. The collimated part Ic within the medium decreases exponentially according to Beer's law and can be derived mathematically as

 

Fig. 2 Physical model of the two-dimensional scattering medium exposed to an external collimated beam illumination.

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Ic(r,Ω0,t*)=I0exp(βx).

The diffuse part Id inside the medium can be obtained by solving Eq. (1) after modifying the right-hand source term as

S(r,Ω,t*)=κs4π[Z(r,Ω0,Ω)Ic(r,Ω0,t*)+4πZ(r,ΩΩ)Id(r,Ω,t*)dΩ],
where Ω0 is the incident direction of the beam.

First of all, the accuracy of the transient DFEM model is verified. The time-dependent polarized radiative transfer emphasis on the Stokes vector distribution before the steady-state is achieved. As no accurate benchmark results for two-dimensional TVRTE, to the authors’ best knowledge, have been published, the accuracy of the developed DFEM for transient polarized radiative transfer problem is verified in a round-about way. From the physical nature of the time-dependent problem, it is seen that the energy distribution within the medium will reach a steady-state as time proceeds. The comparison between the time-resolved solutions against the published steady-state solutions, which is a general and acceptable strategy to verify a new developed transient model [12], can be used to examine the validity of our present scheme. As shown in Fig. 2, the medium considered in this case is a two-dimension rectangular atmosphere above the sea with L × H = 30 m × 30 m. The medium is non-emitting, Rayleigh scattering with a single scattering albedo ω = 0.99, and an optical thickness τ = βL = 1.0. The bottom boundary of the rectangle is specular and the other boundaries are transparent. The collimated beam illumination with linear polarization state and a flux intensity I0 = π × (1, 0.5, 0.5, 0.5)T, is incident on the left boundary in the direction μ0 = cosθ0 = 0, φ0 = 0 (x-direction). The transient polarized radiative transfer process during the dimensionless time t* = 0~6.0 is investigated. The computation time for the first time step is 7.64 minutes. As the relative error between two adjacent intervals becomes smaller with time proceeds, the computation time for a later time step is less than that for the first time step and the total computation time for the 600 time steps is 44.58 hours. The steady solutions for this case have been obtained by the MCM [43] and the present time-resolved solutions of the Stokes vector component distributions at a long enough time interval are compared with the steady solutions.

The radiative flux density of the Stokes vector components along the top boundary are plotted in Fig. 3 for different transient times, namely t* = 1.0, 1.5, 2.0, 5.0 and 6.0. The benchmark MCM solutions in steady state are also shown in Fig. 3 for comparison. It is seen that the I and Q heat flux increases with time and all the Stokes radiative flux distributions after time t* = 5.0 (t* = 5.0 and 6.0) remain constant and show an excellent match (the biggest relative error based on the MCM results is 1.42%) with the steady solutions which means the steady state has been reached at t* = 5.0. The time-dependent results approach and finally overlap with the steady solutions of the radiative flux of the Stokes vector components which we believe demonstrates the accuracy of the correctness of the transient DFEM model developed in this paper for solving TVRTE in the two-dimensional scattering medium with an external radiation source.

 

Fig. 3 Comparisons of the time-resolved Stokes vector radiative fluxes along the top boundary against the steady solution.

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To better highlight the transient effect during the polarized radiative transfer process, the time-resolved polarized radiative fluxes of Stokes vector components along the top boundary are shown in Fig. 4. Five dimensionless times, t* = 0.25, 0.5, 1.0, 2.0 and 5.0 are considered. For the time t* = 0.25, the wave front of the incident irradiation can be obtained analytically by x/L(t* = 0.25) = ct/L = c × (t*/βLc) = 0.25. From Fig. 4 it can be seen that radiative fluxes for t* = 0.25 and 0.5 are only appreciable before x/L = 0.25 and 0.5 respectively, which coincides with the analytical results. From the results for t* = 0.25, 0.5 and 1.0 we find that the DFEM accurately captures the radiative flux at the penetration front. This is due to the fact that continuity at inter-element boundaries is relaxed in DFEM discretization, field variables are calculated element by element and the propagation of the incident beam can be precisely simulated. The radiative flux of the Stokes vector component I and Q increase with time as there will be more radiation energy inside the medium at a later time. Due to the attenuation of the collimated source term based on Eq. (9) along the x-direction, the radiative flux of the Stokes vector component I has a peak value at the location (x/L = 0.35) near the left boundary. The radiative flux of the Stokes vector component Q has a similar trend with that of the radiative flux of the Stokes vector component I and has a peak value at x/L = 0.30, while the radiative fluxes of the Stokes vector component Q and V along the bottom boundary increase monotonically and reach the maximum values at x/L = 1.0.

 

Fig. 4 Time-resolved Stokes vector radiative flux along the top boundary.

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Figure 5 plots the polarized radiative flux along the right boundary. Compared with the radiative flux along the top boundary, the radiative flux along the right boundary depends not only on the diffuse (scattering) part but also the collimated part of the incident radiation. As a result, the polarized radiative fluxes along the right boundary are relatively larger than those along the top boundary. As shown in Fig. 5, all the four polarized radiative fluxes have peak values at the middle location z/H = 0.5, which is due to the symmetry of the simulation domain. However, as the boundary condition is not symmetric about the center line z/H = 0.5 (the bottom boundary is specular while the top boundary is totally transparent), therefore the radiative fluxes are not symmetric about the center point of the right boundary. Due to the contribution of the radiation energy reflected by the bottom boundary, the radiative fluxes at locations near the bottom boundary are larger than those at locations near the top boundary. From Fig. 5 we can also find that the radiative fluxes of Stokes vector component U and V at t* = 3.0, 4.0 and 5.0 overlap with each other, this demonstrates a steady state for U and V distribution has been reached at t* = 3.0 which is earlier than that for I and Q.

 

Fig. 5 Time-resolved Stokes vector radiative flux along the right boundary.

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To evaluate the energy distribution within the computational domain, the incident radiation is transformed into a false temperature defined by Tf = (G/4σ)0.25 with G denoting the incident radiation and σ = 5.67 × 10−8 W/(m2 K4) denoting the Stefan-Boltzmann constant [52]. The false temperature distributions for different transient times t* = 0.25, 0.5, 1.0, 2.0, 3.0 and 5.0 are plotted in Fig. 6. The propagation of the incident radiation can be seen from the false temperature distributions for t* = 0.25, 0.5, and 1.0. The energy within the medium increases with time due to the continuous external radiation source, and the maximum of the false temperature increases before the steady state comes at t* = 5.0, the general trend can be seen from Fig. 6 and the maxima of the temperature are at 27.35, 31.77, 36.57, 40.28, 41.13, and 41.41 K for dimensionless time t* = 0.25, 0.5, 1.0, 2.0, 3.0, and 5.0. For t* = 0.25, the distance of the penetration front is x/L = 0.25 and the temperature in the area x/L > 0.25 remains zero as no energy has been reached, a similar phenomenon can be found for t* = 0.5 and 1.0. When the steady state has been reached at t* = 5.0, an elliptic area with a center coordinate (x, z) = (0.35, 0.5) exists and the temperature decreases quickly at the corners of the rectangle, especially for the top right corner because radiation near this corner area escapes easily as both of the top and right boundaries are transparent.

 

Fig. 6 Distributions of the false temperature within the rectangular atmosphere at different times.

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3.2 Transient polarized radiative transfer in an irregular emitting medium

In this case, the transient DFEM model is applied to the polarized radiative transfer in an irregular enclosure. The geometric coordinate in meters and the grid discretization are shown in Fig. 7(a). The enclosure is filled with an emitting, absorbing and scattering medium with absorption coefficient β = 1.0 m−1 and a single scattering albedo ω = 0.5. Rayleigh scattering is considered and all the boundaries are assumed to be black and cold. The medium temperature is suddenly increased to Tg = 1000 K at t* = 0 and maintained at that level.

 

Fig. 7 (a): The geometric coordinate in meters and the grid discretization, and (b): time-resolved radiative heat flux along the bottom boundary of the irregular emitting medium.

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The accuracy of the DFEM for the irregular medium is first verified by comparing the DFEM solutions with the published results [17] for a transient scalar radiative transfer problem. The polarization can be ignored and the polarized DFEM model can be simplified to a scalar one by setting all off-diagonal elements of the phase matrix Z to be zero. Figure 7(b) compares the DFEM and FVM solutions [17] for dimensionless radiative heat flux qx/σTg4 along the bottom boundary at different time t* = 0.25, 0.5, 1.0, and 2.0. It is seen that the DFEM results agree with the FVM results very well, the biggest relative error based on the FVM results is 1.85%. The DFEM results at t* = 3.0 are also plotted in Fig. 7(b) and they are found to overlap with the results at t* = 2.0, which means the steady state has been reached at t* = 2.0 for the transient scalar radiative transfer problem in the irregular medium.

The transient polarized radiative transfer process during the dimensionless time t* = 0~8.0 is then simulated by our DFEM model. In this case, the computation time for the first time step is 5.31 minutes and the total time for the 800 time steps is 36.93 hours, which is less than that of the rectangular medium, this is due to the fact that the scattering coefficient in this case is smaller than that of the rectangular medium. The time-resolved dimensionless radiative flux of Stokes vector components I and Q along the bottom boundary are plotted in Fig. 8, as the Stokes vector components U and V remains zero for the medium with only a self-emitting radiation source [36]. As shown in Fig. 8(a), for the selected time t* = 0.25, 0.5, and 0.75, a platform can be found in the radiative flux curve of the Stokes vector component I, for the time t* = 1.0, 2.0, 4.0 and 6.0, a peak at about the center of the bottom boundary can be found for radiative flux curve of the component I. The radiative flux of I increases with time and the results at t* = 4.0 and t* = 6.0 has overlapped, which means the steady state has been reached at t* = 4.0 for the component I. However, for the radiative flux of Stokes vector component Q as shown in Fig. 8(b), the radiative flux distribution at t* = 4.0 has obvious differences with that at t* = 6.0, which means the component Q has not reached the steady state at dimensionless time t* = 4.0. The distributions of Stokes vector component Q at t* = 6.0 and t* = 8.0 coincide and this demonstrates the steady state for Q is achieved after time t* = 8.0. The trend of the radiative flux of Stokes vector component Q is relatively complex due to the multiple scattering and the radiative flux curve of component Q, and has three peaks at x = 0.11, 0.94, and 1.98, respectively.

 

Fig. 8 Polarized radiative flux distributions of Stokes vector component (a): I and (b): Q on the bottom boundary of the irregular medium at different times.

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The false temperature for this case is plotted in Fig. 9 for selected dimensionless time t* = 0.25, 0.5, 0.75, 1.0, 2.0 and 4.0. The temperature increases with time before the steady state for the Stokes vector component I at t* = 4.0, which is due to the multiple scattering of the radiation and the increasing accumulation of the incident radiation. Near the beginning of the simulation, at t* = 0.25 for example, the inner area of the emitting medium has not been influenced much by the black and cold walls and most of the inner area has the same false temperature. The false temperature distribution becomes smooth as the simulation proceeds and finally reaches a steady state with a maximum of 810.04 K, the details of the temperature distributions at other times can be seen from Fig. 9.

 

Fig. 9 False temperature distributions within the emitting irregular medium at different times.

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4. Conclusions

The discontinuous finite method has been extended to solve the transient/time-dependent radiative transfer problems in a scattering medium considering the polarization effect of the scattered light. The time-resolved DFEM solutions of the Stokes vector component distributions for the scattering medium exposed to an external beam approaches and finally overlaps the published steady results, which verifies the accuracy of our transient DFEM model for solving transient vector radiative transfer equation in the two-dimensional scattering medium. For the rectangular medium exposed to an external radiation source, our results show the DFEM can accurately capture the penetration front of the incident beam. It takes dimensionless time t* = βct = 5.0 for all the Stokes vector components within the computation domain to reach a steady state, while it only takes t* = 3.0 for the Stokes vector components U and V along the right boundary to reach the steady state. The application of the DFEM was then extended to an irregular emitting and scattering medium, the time-resolved Stokes vector component distributions along the bottom boundary were presented and discussed, our results show that in this scenario it takes longer for the energy inside the medium to reach a steady state. In summary, the transient DFEM model developed in this paper is accurate for transient radiative transfer problems considering the polarization effect in a two-dimensional scattering medium, the time-resolved results of the Stokes vector components presented in this paper can be used to analyze the radiation distributions and to verify the accuracy of other numerical methods designed to solve transient vector radiative transfer equation.

Appendix A Nomenclature

Funding

National Natural Science Foundation of China (NSFC) (51422602).

Acknowledgments

We would like to specially acknowledge the editors and referees who made important comments that helped us to improve this paper.

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33. A. J. Brown, “Equivalence relations and symmetries for laboratory, LIDAR, and planetary Müeller matrix scattering geometries,” J. Opt. Soc. Am. A 31(12), 2789–2794 (2014). [CrossRef]   [PubMed]  

34. R. D. M. Garcia and C. E. Siewert, “The FN method for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 41(2), 117–145 (1989). [CrossRef]  

35. C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 64(3), 227–254 (2000). [CrossRef]  

36. K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991). [CrossRef]  

37. E. R. Sommersten, J. K. Lotsberg, K. Stamnes, and J. J. Stamnes, “Discrete ordinate and Monte Carlo simulations for polarized radiative transfer in a coupled system consisting of two media with different refractive indices,” J. Quant. Spectrosc. Radiat. Transf. 111(4), 616–633 (2010). [CrossRef]  

38. G. N. Plass, G. W. Kattawar, and F. E. Catchings, “Matrix operator theory of radiative transfer. 1: rayleigh scattering,” Appl. Opt. 12(2), 314–329 (1973). [CrossRef]   [PubMed]  

39. C. H. Wang, H. L. Yi, and H.-P. Tan, “Discontinuous finite element method for vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 189, 383–397 (2017). [CrossRef]  

40. R. Vaillona, B. T. Wong, and M. P. Mengüç, “Polarized radiative transfer in a particle-laden semi-transparent medium via a vector Monte Carlo method,” J. Quant. Spectrosc. Radiat. Transf. 84(4), 383–394 (2004). [CrossRef]  

41. A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Z. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12), 1931–1946 (2010). [CrossRef]  

42. C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015). [CrossRef]  

43. J. M. Zhao, J. Y. Tan, and L. H. Liu, “Monte Carlo method for polarized radiative transfer in gradient-index media,” J. Quant. Spectrosc. Radiat. Transf. 152, 114–126 (2015). [CrossRef]  

44. A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, “Polarized pulse waves in random discrete scatterers,” Appl. Opt. 40(30), 5495–5502 (2001). [CrossRef]   [PubMed]  

45. X. Wang, L. V. Wang, C. W. Sun, and C. C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8(4), 608–617 (2003). [CrossRef]   [PubMed]  

46. M. Sakami and A. Dogariu, “Polarized light-pulse transport through scattering media,” J. Opt. Soc. Am. A 23(3), 664–670 (2006). [CrossRef]   [PubMed]  

47. Y. A. Ilyushin and V. P. Budak, “Analysis of the propagation of the femtosecond laser pulse in the scattering medium,” Comput. Phys. Commun. 182(4), 940–945 (2011). [CrossRef]  

48. C. H. Wang, H. L. Yi, and H. P. Tan, “Transient polarized radiative transfer analysis in a scattering medium by a discontinuous finite element method,” Opt. Express 25(7), 7418–7442 (2017). [CrossRef]   [PubMed]  

49. S. Chandrasekhar, Radiative transfer (Dover, 1960).

50. D. H. Goldstein, Polarized light (Dekker, 2003).

51. J. Crank, P. Nicolson, and D. R. Hartree, “A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type,” Math. Proc. Camb. Philos. Soc. 43(1), 50–67 (1947). [CrossRef]  

52. J. M. Zhao, J. Y. Tan, and L. H. Liu, “A second order radiative transfer equation and its solution by meshless method with application to strongly inhomogeneous media,” J. Comput. Phys. 232(1), 431–455 (2013). [CrossRef]  

References

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  1. K. E. Sheetz and J. Squier, “Ultrafast optics: Imaging and manipulating biological systems,” J. Appl. Phys. 105(5), 051101 (2009).
    [Crossref]
  2. M. Sakami, K. Mitra, and T. Vo-Dinh, “Analysis of short-pulse laser photon transport through tissues for optical tomography,” Opt. Lett. 27(5), 336–338 (2002).
    [Crossref] [PubMed]
  3. T. D. O’Sullivan, A. E. Cerussi, D. J. Cuccia, and B. J. Tromberg, “Diffuse optical imaging using spatially and temporally modulated light,” J. Biomed. Opt. 17(7), 071311 (2012).
    [PubMed]
  4. J. Wu, Y. Xu, J. Xu, X. Wei, A. C. Chan, A. H. Tang, A. K. Lau, B. M. Chung, H. C. Shum, E. Y. Lam, K. K. Wong, and K. K. Tsia, “Ultrafast laser-scanning time-stretch imaging at visible wavelengths,” Light Sci. Appl. 6(1), e16196 (2017).
    [Crossref]
  5. S. Kumar and K. Mitra, “Microscale aspects of thermal radiation transport and laser applications,” Adv. Heat Transf. 33, 187–294 (1999).
    [Crossref]
  6. K. Mitra, M. S. Lai, and S. Kumar, “Transient radiation transport in participating media within a rectangular enclosure,” AIAA J. Thermophys. Heat Transfer 11(3), 409–414 (1997).
    [Crossref]
  7. K. Mitra and S. Kumar, “Development and comparison of models for light-pulse transport through scattering-absorbing media,” Appl. Opt. 38(1), 188–196 (1999).
    [Crossref] [PubMed]
  8. Z. Guo and S. Kumar, “Discrete-ordinates solution of short-pulsed laser transport in two-dimensional turbid media,” Appl. Opt. 40(19), 3156–3163 (2001).
    [Crossref] [PubMed]
  9. M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light-pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 169–179 (2002).
    [Crossref]
  10. Z. Guo and K. Kim, “Ultrafast-laser-radiation transfer in heterogeneous tissues with the discrete-ordinates method,” Appl. Opt. 42(16), 2897–2905 (2003).
    [Crossref] [PubMed]
  11. C. Y. Wu and N. R. Ou, “Differential approximation for transient radiative transfer through a participating medium exposed to collimated irradiation,” J. Quant. Spectrosc. Radiat. Transf. 73(1), 111–120 (2002).
    [Crossref]
  12. J. M. Wang and C. Y. Wu, “Transient radiative transfer in a scattering slab with variable refractive index and diffuse substrate,” Int. J. Heat Mass Transfer 53(19), 3799–3806 (2010).
    [Crossref]
  13. Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” ASME J. Heat Transfer 123(3), 466–475 (2001).
    [Crossref]
  14. C. Y. Wu, “Propagation of scattered radiation in a participating planar medium with pulse irradiation,” J. Quant. Spectrosc. Radiat. Transf. 64(5), 537–548 (2000).
    [Crossref]
  15. C. Y. Wu and S. H. Wu, “Integral equation formulation for transient radiative transfer in an anisotropically scattering medium,” Int. J. Heat Mass Transfer 43(11), 2009–2020 (2000).
    [Crossref]
  16. J. C. Chai, “One-dimensional transient radiation heat transfer modeling using a finite-volume method,” Numer. Heat Transf. B 44(2), 187–208 (2003).
    [Crossref]
  17. J. C. Chai, “Transient radiative transfer in irregular two-dimensional geometries,” J. Quant. Spectrosc. Radiat. Transf. 84(3), 281–294 (2004).
    [Crossref]
  18. Y. Zhang, H. Yi, and H. Tan, “One-dimensional transient radiative transfer by lattice Boltzmann method,” Opt. Express 21(21), 24532–24549 (2013).
    [Crossref] [PubMed]
  19. S. C. Mishra, A. Lankadasu, and K. N. Beronov, “Application of the lattice Boltzmann method for solving the energy equation of a 2-D transient conduction–radiation problem,” Int. J. Heat Mass Transfer 48(17), 3648–3659 (2005).
    [Crossref]
  20. Q. Cheng, H. C. Zhou, Z. F. Huang, Y. L. Yu, and D. X. Huang, “The solution of transient radiative transfer with collimated incident serial pulse in a plane-parallel medium by the DRESOR method,” ASME J. Heat Transfer 130(10), 102701 (2008).
    [Crossref]
  21. Z. C. Wang, Q. Cheng, and H. C. Zhou, “The DRESOR method for transient radiation transfer in 1-D graded index medium with pulse irradiation,” Int. J. Therm. Sci. 68, 127–135 (2013).
    [Crossref]
  22. P. Rath, S. C. Mishra, P. Mahanta, U. K. Saha, and K. Mitra, “Discrete transfer method applied to transient radiative transfer problems in participating medium,” Numer. Heat Transf. A 44(2), 183–197 (2003).
    [Crossref]
  23. C. H. Wang, Q. Ai, H. L. Yi, and H. P. Tan, “Transient radiative transfer in a graded index medium with specularly reflecting surfaces,” Numer. Heat Transf. A 67(11), 1232–1252 (2015).
    [Crossref]
  24. Z. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 159–168 (2002).
    [Crossref]
  25. Z. Guo and S. Kumar, “Equivalent isotropic scattering formulation for transient short-pulse radiative transfer in anisotropic scattering planar media,” Appl. Opt. 39(24), 4411–4417 (2000).
    [Crossref] [PubMed]
  26. M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light-pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 169–179 (2002).
    [Crossref]
  27. C. H. Wang, Y. Zhang, H. L. Yi, and H. P. Tan, “Transient radiative transfer in two-dimensional graded index medium by Monte Carlo method combined with the time shift and superposition principle,” Numer. Heat Transf. A 69(6), 574–588 (2016).
    [Crossref]
  28. S. C. Mishra and A. Lankadasu, “Transient conduction-radiation heat transfer in participating media using the lattice Boltzmann method and the discrete transfer method,” Numer. Heat Transf. A 47(9), 935–954 (2005).
    [Crossref]
  29. S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer 49(11), 1820–1832 (2006).
    [Crossref]
  30. M. Chami, R. Santer, and E. Dilligeard, “Radiative transfer model for the computation of radiance and polarization in an ocean-atmosphere system: polarization properties of suspended matter for remote sensing,” Appl. Opt. 40(15), 2398–2416 (2001).
    [Crossref] [PubMed]
  31. A. J. Brown, “Spectral bluing induced by small particles under the Mie and Rayleigh regimes,” Icarus 239, 85–95 (2014).
    [Crossref]
  32. A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, G. Videen, and C. J. Grund, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transf. 153, 131–143 (2015).
    [Crossref]
  33. A. J. Brown, “Equivalence relations and symmetries for laboratory, LIDAR, and planetary Müeller matrix scattering geometries,” J. Opt. Soc. Am. A 31(12), 2789–2794 (2014).
    [Crossref] [PubMed]
  34. R. D. M. Garcia and C. E. Siewert, “The FN method for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 41(2), 117–145 (1989).
    [Crossref]
  35. C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 64(3), 227–254 (2000).
    [Crossref]
  36. K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991).
    [Crossref]
  37. E. R. Sommersten, J. K. Lotsberg, K. Stamnes, and J. J. Stamnes, “Discrete ordinate and Monte Carlo simulations for polarized radiative transfer in a coupled system consisting of two media with different refractive indices,” J. Quant. Spectrosc. Radiat. Transf. 111(4), 616–633 (2010).
    [Crossref]
  38. G. N. Plass, G. W. Kattawar, and F. E. Catchings, “Matrix operator theory of radiative transfer. 1: rayleigh scattering,” Appl. Opt. 12(2), 314–329 (1973).
    [Crossref] [PubMed]
  39. C. H. Wang, H. L. Yi, and H.-P. Tan, “Discontinuous finite element method for vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 189, 383–397 (2017).
    [Crossref]
  40. R. Vaillona, B. T. Wong, and M. P. Mengüç, “Polarized radiative transfer in a particle-laden semi-transparent medium via a vector Monte Carlo method,” J. Quant. Spectrosc. Radiat. Transf. 84(4), 383–394 (2004).
    [Crossref]
  41. A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Z. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12), 1931–1946 (2010).
    [Crossref]
  42. C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
    [Crossref]
  43. J. M. Zhao, J. Y. Tan, and L. H. Liu, “Monte Carlo method for polarized radiative transfer in gradient-index media,” J. Quant. Spectrosc. Radiat. Transf. 152, 114–126 (2015).
    [Crossref]
  44. A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, “Polarized pulse waves in random discrete scatterers,” Appl. Opt. 40(30), 5495–5502 (2001).
    [Crossref] [PubMed]
  45. X. Wang, L. V. Wang, C. W. Sun, and C. C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8(4), 608–617 (2003).
    [Crossref] [PubMed]
  46. M. Sakami and A. Dogariu, “Polarized light-pulse transport through scattering media,” J. Opt. Soc. Am. A 23(3), 664–670 (2006).
    [Crossref] [PubMed]
  47. Y. A. Ilyushin and V. P. Budak, “Analysis of the propagation of the femtosecond laser pulse in the scattering medium,” Comput. Phys. Commun. 182(4), 940–945 (2011).
    [Crossref]
  48. C. H. Wang, H. L. Yi, and H. P. Tan, “Transient polarized radiative transfer analysis in a scattering medium by a discontinuous finite element method,” Opt. Express 25(7), 7418–7442 (2017).
    [Crossref] [PubMed]
  49. S. Chandrasekhar, Radiative transfer (Dover, 1960).
  50. D. H. Goldstein, Polarized light (Dekker, 2003).
  51. J. Crank, P. Nicolson, and D. R. Hartree, “A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type,” Math. Proc. Camb. Philos. Soc. 43(1), 50–67 (1947).
    [Crossref]
  52. J. M. Zhao, J. Y. Tan, and L. H. Liu, “A second order radiative transfer equation and its solution by meshless method with application to strongly inhomogeneous media,” J. Comput. Phys. 232(1), 431–455 (2013).
    [Crossref]

2017 (3)

J. Wu, Y. Xu, J. Xu, X. Wei, A. C. Chan, A. H. Tang, A. K. Lau, B. M. Chung, H. C. Shum, E. Y. Lam, K. K. Wong, and K. K. Tsia, “Ultrafast laser-scanning time-stretch imaging at visible wavelengths,” Light Sci. Appl. 6(1), e16196 (2017).
[Crossref]

C. H. Wang, H. L. Yi, and H.-P. Tan, “Discontinuous finite element method for vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 189, 383–397 (2017).
[Crossref]

C. H. Wang, H. L. Yi, and H. P. Tan, “Transient polarized radiative transfer analysis in a scattering medium by a discontinuous finite element method,” Opt. Express 25(7), 7418–7442 (2017).
[Crossref] [PubMed]

2016 (1)

C. H. Wang, Y. Zhang, H. L. Yi, and H. P. Tan, “Transient radiative transfer in two-dimensional graded index medium by Monte Carlo method combined with the time shift and superposition principle,” Numer. Heat Transf. A 69(6), 574–588 (2016).
[Crossref]

2015 (4)

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, G. Videen, and C. J. Grund, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transf. 153, 131–143 (2015).
[Crossref]

C. H. Wang, Q. Ai, H. L. Yi, and H. P. Tan, “Transient radiative transfer in a graded index medium with specularly reflecting surfaces,” Numer. Heat Transf. A 67(11), 1232–1252 (2015).
[Crossref]

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
[Crossref]

J. M. Zhao, J. Y. Tan, and L. H. Liu, “Monte Carlo method for polarized radiative transfer in gradient-index media,” J. Quant. Spectrosc. Radiat. Transf. 152, 114–126 (2015).
[Crossref]

2014 (2)

2013 (3)

Y. Zhang, H. Yi, and H. Tan, “One-dimensional transient radiative transfer by lattice Boltzmann method,” Opt. Express 21(21), 24532–24549 (2013).
[Crossref] [PubMed]

Z. C. Wang, Q. Cheng, and H. C. Zhou, “The DRESOR method for transient radiation transfer in 1-D graded index medium with pulse irradiation,” Int. J. Therm. Sci. 68, 127–135 (2013).
[Crossref]

J. M. Zhao, J. Y. Tan, and L. H. Liu, “A second order radiative transfer equation and its solution by meshless method with application to strongly inhomogeneous media,” J. Comput. Phys. 232(1), 431–455 (2013).
[Crossref]

2012 (1)

T. D. O’Sullivan, A. E. Cerussi, D. J. Cuccia, and B. J. Tromberg, “Diffuse optical imaging using spatially and temporally modulated light,” J. Biomed. Opt. 17(7), 071311 (2012).
[PubMed]

2011 (1)

Y. A. Ilyushin and V. P. Budak, “Analysis of the propagation of the femtosecond laser pulse in the scattering medium,” Comput. Phys. Commun. 182(4), 940–945 (2011).
[Crossref]

2010 (3)

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Z. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12), 1931–1946 (2010).
[Crossref]

E. R. Sommersten, J. K. Lotsberg, K. Stamnes, and J. J. Stamnes, “Discrete ordinate and Monte Carlo simulations for polarized radiative transfer in a coupled system consisting of two media with different refractive indices,” J. Quant. Spectrosc. Radiat. Transf. 111(4), 616–633 (2010).
[Crossref]

J. M. Wang and C. Y. Wu, “Transient radiative transfer in a scattering slab with variable refractive index and diffuse substrate,” Int. J. Heat Mass Transfer 53(19), 3799–3806 (2010).
[Crossref]

2009 (1)

K. E. Sheetz and J. Squier, “Ultrafast optics: Imaging and manipulating biological systems,” J. Appl. Phys. 105(5), 051101 (2009).
[Crossref]

2008 (1)

Q. Cheng, H. C. Zhou, Z. F. Huang, Y. L. Yu, and D. X. Huang, “The solution of transient radiative transfer with collimated incident serial pulse in a plane-parallel medium by the DRESOR method,” ASME J. Heat Transfer 130(10), 102701 (2008).
[Crossref]

2006 (2)

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer 49(11), 1820–1832 (2006).
[Crossref]

M. Sakami and A. Dogariu, “Polarized light-pulse transport through scattering media,” J. Opt. Soc. Am. A 23(3), 664–670 (2006).
[Crossref] [PubMed]

2005 (2)

S. C. Mishra and A. Lankadasu, “Transient conduction-radiation heat transfer in participating media using the lattice Boltzmann method and the discrete transfer method,” Numer. Heat Transf. A 47(9), 935–954 (2005).
[Crossref]

S. C. Mishra, A. Lankadasu, and K. N. Beronov, “Application of the lattice Boltzmann method for solving the energy equation of a 2-D transient conduction–radiation problem,” Int. J. Heat Mass Transfer 48(17), 3648–3659 (2005).
[Crossref]

2004 (2)

J. C. Chai, “Transient radiative transfer in irregular two-dimensional geometries,” J. Quant. Spectrosc. Radiat. Transf. 84(3), 281–294 (2004).
[Crossref]

R. Vaillona, B. T. Wong, and M. P. Mengüç, “Polarized radiative transfer in a particle-laden semi-transparent medium via a vector Monte Carlo method,” J. Quant. Spectrosc. Radiat. Transf. 84(4), 383–394 (2004).
[Crossref]

2003 (4)

X. Wang, L. V. Wang, C. W. Sun, and C. C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8(4), 608–617 (2003).
[Crossref] [PubMed]

J. C. Chai, “One-dimensional transient radiation heat transfer modeling using a finite-volume method,” Numer. Heat Transf. B 44(2), 187–208 (2003).
[Crossref]

Z. Guo and K. Kim, “Ultrafast-laser-radiation transfer in heterogeneous tissues with the discrete-ordinates method,” Appl. Opt. 42(16), 2897–2905 (2003).
[Crossref] [PubMed]

P. Rath, S. C. Mishra, P. Mahanta, U. K. Saha, and K. Mitra, “Discrete transfer method applied to transient radiative transfer problems in participating medium,” Numer. Heat Transf. A 44(2), 183–197 (2003).
[Crossref]

2002 (5)

M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light-pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 169–179 (2002).
[Crossref]

Z. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 159–168 (2002).
[Crossref]

C. Y. Wu and N. R. Ou, “Differential approximation for transient radiative transfer through a participating medium exposed to collimated irradiation,” J. Quant. Spectrosc. Radiat. Transf. 73(1), 111–120 (2002).
[Crossref]

M. Sakami, K. Mitra, and T. Vo-Dinh, “Analysis of short-pulse laser photon transport through tissues for optical tomography,” Opt. Lett. 27(5), 336–338 (2002).
[Crossref] [PubMed]

M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light-pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 169–179 (2002).
[Crossref]

2001 (4)

2000 (4)

C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 64(3), 227–254 (2000).
[Crossref]

Z. Guo and S. Kumar, “Equivalent isotropic scattering formulation for transient short-pulse radiative transfer in anisotropic scattering planar media,” Appl. Opt. 39(24), 4411–4417 (2000).
[Crossref] [PubMed]

C. Y. Wu, “Propagation of scattered radiation in a participating planar medium with pulse irradiation,” J. Quant. Spectrosc. Radiat. Transf. 64(5), 537–548 (2000).
[Crossref]

C. Y. Wu and S. H. Wu, “Integral equation formulation for transient radiative transfer in an anisotropically scattering medium,” Int. J. Heat Mass Transfer 43(11), 2009–2020 (2000).
[Crossref]

1999 (2)

K. Mitra and S. Kumar, “Development and comparison of models for light-pulse transport through scattering-absorbing media,” Appl. Opt. 38(1), 188–196 (1999).
[Crossref] [PubMed]

S. Kumar and K. Mitra, “Microscale aspects of thermal radiation transport and laser applications,” Adv. Heat Transf. 33, 187–294 (1999).
[Crossref]

1997 (1)

K. Mitra, M. S. Lai, and S. Kumar, “Transient radiation transport in participating media within a rectangular enclosure,” AIAA J. Thermophys. Heat Transfer 11(3), 409–414 (1997).
[Crossref]

1991 (1)

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991).
[Crossref]

1989 (1)

R. D. M. Garcia and C. E. Siewert, “The FN method for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 41(2), 117–145 (1989).
[Crossref]

1973 (1)

1947 (1)

J. Crank, P. Nicolson, and D. R. Hartree, “A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type,” Math. Proc. Camb. Philos. Soc. 43(1), 50–67 (1947).
[Crossref]

Aber, J.

Z. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 159–168 (2002).
[Crossref]

Ai, Q.

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S. C. Mishra, A. Lankadasu, and K. N. Beronov, “Application of the lattice Boltzmann method for solving the energy equation of a 2-D transient conduction–radiation problem,” Int. J. Heat Mass Transfer 48(17), 3648–3659 (2005).
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A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, G. Videen, and C. J. Grund, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transf. 153, 131–143 (2015).
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Y. A. Ilyushin and V. P. Budak, “Analysis of the propagation of the femtosecond laser pulse in the scattering medium,” Comput. Phys. Commun. 182(4), 940–945 (2011).
[Crossref]

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Z. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12), 1931–1946 (2010).
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A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, G. Videen, and C. J. Grund, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transf. 153, 131–143 (2015).
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Cerussi, A. E.

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J. Wu, Y. Xu, J. Xu, X. Wei, A. C. Chan, A. H. Tang, A. K. Lau, B. M. Chung, H. C. Shum, E. Y. Lam, K. K. Wong, and K. K. Tsia, “Ultrafast laser-scanning time-stretch imaging at visible wavelengths,” Light Sci. Appl. 6(1), e16196 (2017).
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Z. C. Wang, Q. Cheng, and H. C. Zhou, “The DRESOR method for transient radiation transfer in 1-D graded index medium with pulse irradiation,” Int. J. Therm. Sci. 68, 127–135 (2013).
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Q. Cheng, H. C. Zhou, Z. F. Huang, Y. L. Yu, and D. X. Huang, “The solution of transient radiative transfer with collimated incident serial pulse in a plane-parallel medium by the DRESOR method,” ASME J. Heat Transfer 130(10), 102701 (2008).
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S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer 49(11), 1820–1832 (2006).
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J. Wu, Y. Xu, J. Xu, X. Wei, A. C. Chan, A. H. Tang, A. K. Lau, B. M. Chung, H. C. Shum, E. Y. Lam, K. K. Wong, and K. K. Tsia, “Ultrafast laser-scanning time-stretch imaging at visible wavelengths,” Light Sci. Appl. 6(1), e16196 (2017).
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A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Z. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12), 1931–1946 (2010).
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A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, G. Videen, and C. J. Grund, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transf. 153, 131–143 (2015).
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C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
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A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Z. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12), 1931–1946 (2010).
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J. Crank, P. Nicolson, and D. R. Hartree, “A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type,” Math. Proc. Camb. Philos. Soc. 43(1), 50–67 (1947).
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Cuccia, D. J.

T. D. O’Sullivan, A. E. Cerussi, D. J. Cuccia, and B. J. Tromberg, “Diffuse optical imaging using spatially and temporally modulated light,” J. Biomed. Opt. 17(7), 071311 (2012).
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Dogariu, A.

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C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
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A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Z. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12), 1931–1946 (2010).
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Evans, F.

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
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K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991).
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R. D. M. Garcia and C. E. Siewert, “The FN method for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 41(2), 117–145 (1989).
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Garetz, B. A.

Z. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 159–168 (2002).
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Grund, C. J.

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, G. Videen, and C. J. Grund, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transf. 153, 131–143 (2015).
[Crossref]

Guo, Z.

Hartree, D. R.

J. Crank, P. Nicolson, and D. R. Hartree, “A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type,” Math. Proc. Camb. Philos. Soc. 43(1), 50–67 (1947).
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Hsu, P. F.

M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light-pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 169–179 (2002).
[Crossref]

M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light-pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 169–179 (2002).
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Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” ASME J. Heat Transfer 123(3), 466–475 (2001).
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Huang, D. X.

Q. Cheng, H. C. Zhou, Z. F. Huang, Y. L. Yu, and D. X. Huang, “The solution of transient radiative transfer with collimated incident serial pulse in a plane-parallel medium by the DRESOR method,” ASME J. Heat Transfer 130(10), 102701 (2008).
[Crossref]

Huang, Z. F.

Q. Cheng, H. C. Zhou, Z. F. Huang, Y. L. Yu, and D. X. Huang, “The solution of transient radiative transfer with collimated incident serial pulse in a plane-parallel medium by the DRESOR method,” ASME J. Heat Transfer 130(10), 102701 (2008).
[Crossref]

Ilyushin, Y. A.

Y. A. Ilyushin and V. P. Budak, “Analysis of the propagation of the femtosecond laser pulse in the scattering medium,” Comput. Phys. Commun. 182(4), 940–945 (2011).
[Crossref]

Ishimaru, A.

Jaruwatanadilok, S.

Katsev, I. L.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Z. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12), 1931–1946 (2010).
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Kattawar, G. W.

Kim, K.

Klyukov, D. A.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Z. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12), 1931–1946 (2010).
[Crossref]

Kokhanovsky, A. A.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Z. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12), 1931–1946 (2010).
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Korkin, S.

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
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Korkin, S. V.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Z. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12), 1931–1946 (2010).
[Crossref]

Kuga, Y.

Kumar, P.

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer 49(11), 1820–1832 (2006).
[Crossref]

Kumar, S.

Z. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 159–168 (2002).
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Z. Guo and S. Kumar, “Discrete-ordinates solution of short-pulsed laser transport in two-dimensional turbid media,” Appl. Opt. 40(19), 3156–3163 (2001).
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Z. Guo and S. Kumar, “Equivalent isotropic scattering formulation for transient short-pulse radiative transfer in anisotropic scattering planar media,” Appl. Opt. 39(24), 4411–4417 (2000).
[Crossref] [PubMed]

K. Mitra and S. Kumar, “Development and comparison of models for light-pulse transport through scattering-absorbing media,” Appl. Opt. 38(1), 188–196 (1999).
[Crossref] [PubMed]

S. Kumar and K. Mitra, “Microscale aspects of thermal radiation transport and laser applications,” Adv. Heat Transf. 33, 187–294 (1999).
[Crossref]

K. Mitra, M. S. Lai, and S. Kumar, “Transient radiation transport in participating media within a rectangular enclosure,” AIAA J. Thermophys. Heat Transfer 11(3), 409–414 (1997).
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Labonnote, L. C.

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
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Lai, M. S.

K. Mitra, M. S. Lai, and S. Kumar, “Transient radiation transport in participating media within a rectangular enclosure,” AIAA J. Thermophys. Heat Transfer 11(3), 409–414 (1997).
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Lam, E. Y.

J. Wu, Y. Xu, J. Xu, X. Wei, A. C. Chan, A. H. Tang, A. K. Lau, B. M. Chung, H. C. Shum, E. Y. Lam, K. K. Wong, and K. K. Tsia, “Ultrafast laser-scanning time-stretch imaging at visible wavelengths,” Light Sci. Appl. 6(1), e16196 (2017).
[Crossref]

Lankadasu, A.

S. C. Mishra, A. Lankadasu, and K. N. Beronov, “Application of the lattice Boltzmann method for solving the energy equation of a 2-D transient conduction–radiation problem,” Int. J. Heat Mass Transfer 48(17), 3648–3659 (2005).
[Crossref]

S. C. Mishra and A. Lankadasu, “Transient conduction-radiation heat transfer in participating media using the lattice Boltzmann method and the discrete transfer method,” Numer. Heat Transf. A 47(9), 935–954 (2005).
[Crossref]

Lau, A. K.

J. Wu, Y. Xu, J. Xu, X. Wei, A. C. Chan, A. H. Tang, A. K. Lau, B. M. Chung, H. C. Shum, E. Y. Lam, K. K. Wong, and K. K. Tsia, “Ultrafast laser-scanning time-stretch imaging at visible wavelengths,” Light Sci. Appl. 6(1), e16196 (2017).
[Crossref]

Liu, L. H.

J. M. Zhao, J. Y. Tan, and L. H. Liu, “Monte Carlo method for polarized radiative transfer in gradient-index media,” J. Quant. Spectrosc. Radiat. Transf. 152, 114–126 (2015).
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J. M. Zhao, J. Y. Tan, and L. H. Liu, “A second order radiative transfer equation and its solution by meshless method with application to strongly inhomogeneous media,” J. Comput. Phys. 232(1), 431–455 (2013).
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Lotsberg, J. K.

E. R. Sommersten, J. K. Lotsberg, K. Stamnes, and J. J. Stamnes, “Discrete ordinate and Monte Carlo simulations for polarized radiative transfer in a coupled system consisting of two media with different refractive indices,” J. Quant. Spectrosc. Radiat. Transf. 111(4), 616–633 (2010).
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Lyapustin, A.

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
[Crossref]

Macke, A.

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
[Crossref]

Mahanta, P.

P. Rath, S. C. Mishra, P. Mahanta, U. K. Saha, and K. Mitra, “Discrete transfer method applied to transient radiative transfer problems in participating medium,” Numer. Heat Transf. A 44(2), 183–197 (2003).
[Crossref]

Mayer, B.

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
[Crossref]

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Z. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12), 1931–1946 (2010).
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Mengüç, M. P.

R. Vaillona, B. T. Wong, and M. P. Mengüç, “Polarized radiative transfer in a particle-laden semi-transparent medium via a vector Monte Carlo method,” J. Quant. Spectrosc. Radiat. Transf. 84(4), 383–394 (2004).
[Crossref]

Michaels, T. I.

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, G. Videen, and C. J. Grund, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transf. 153, 131–143 (2015).
[Crossref]

Min, Q.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Z. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12), 1931–1946 (2010).
[Crossref]

Mishra, S. C.

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer 49(11), 1820–1832 (2006).
[Crossref]

S. C. Mishra and A. Lankadasu, “Transient conduction-radiation heat transfer in participating media using the lattice Boltzmann method and the discrete transfer method,” Numer. Heat Transf. A 47(9), 935–954 (2005).
[Crossref]

S. C. Mishra, A. Lankadasu, and K. N. Beronov, “Application of the lattice Boltzmann method for solving the energy equation of a 2-D transient conduction–radiation problem,” Int. J. Heat Mass Transfer 48(17), 3648–3659 (2005).
[Crossref]

P. Rath, S. C. Mishra, P. Mahanta, U. K. Saha, and K. Mitra, “Discrete transfer method applied to transient radiative transfer problems in participating medium,” Numer. Heat Transf. A 44(2), 183–197 (2003).
[Crossref]

Mitra, K.

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer 49(11), 1820–1832 (2006).
[Crossref]

P. Rath, S. C. Mishra, P. Mahanta, U. K. Saha, and K. Mitra, “Discrete transfer method applied to transient radiative transfer problems in participating medium,” Numer. Heat Transf. A 44(2), 183–197 (2003).
[Crossref]

M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light-pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 169–179 (2002).
[Crossref]

M. Sakami, K. Mitra, and T. Vo-Dinh, “Analysis of short-pulse laser photon transport through tissues for optical tomography,” Opt. Lett. 27(5), 336–338 (2002).
[Crossref] [PubMed]

M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light-pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 169–179 (2002).
[Crossref]

K. Mitra and S. Kumar, “Development and comparison of models for light-pulse transport through scattering-absorbing media,” Appl. Opt. 38(1), 188–196 (1999).
[Crossref] [PubMed]

S. Kumar and K. Mitra, “Microscale aspects of thermal radiation transport and laser applications,” Adv. Heat Transf. 33, 187–294 (1999).
[Crossref]

K. Mitra, M. S. Lai, and S. Kumar, “Transient radiation transport in participating media within a rectangular enclosure,” AIAA J. Thermophys. Heat Transfer 11(3), 409–414 (1997).
[Crossref]

Nakajima, T.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Z. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12), 1931–1946 (2010).
[Crossref]

Nicolson, P.

J. Crank, P. Nicolson, and D. R. Hartree, “A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type,” Math. Proc. Camb. Philos. Soc. 43(1), 50–67 (1947).
[Crossref]

O’Sullivan, T. D.

T. D. O’Sullivan, A. E. Cerussi, D. J. Cuccia, and B. J. Tromberg, “Diffuse optical imaging using spatially and temporally modulated light,” J. Biomed. Opt. 17(7), 071311 (2012).
[PubMed]

Ota, Y.

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
[Crossref]

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Z. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12), 1931–1946 (2010).
[Crossref]

Ou, N. R.

C. Y. Wu and N. R. Ou, “Differential approximation for transient radiative transfer through a participating medium exposed to collimated irradiation,” J. Quant. Spectrosc. Radiat. Transf. 73(1), 111–120 (2002).
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Plass, G. N.

Prikhach, A. S.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Z. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12), 1931–1946 (2010).
[Crossref]

Rath, P.

P. Rath, S. C. Mishra, P. Mahanta, U. K. Saha, and K. Mitra, “Discrete transfer method applied to transient radiative transfer problems in participating medium,” Numer. Heat Transf. A 44(2), 183–197 (2003).
[Crossref]

Rozanov, V. V.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Z. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12), 1931–1946 (2010).
[Crossref]

Saha, U. K.

P. Rath, S. C. Mishra, P. Mahanta, U. K. Saha, and K. Mitra, “Discrete transfer method applied to transient radiative transfer problems in participating medium,” Numer. Heat Transf. A 44(2), 183–197 (2003).
[Crossref]

Sakami, M.

M. Sakami and A. Dogariu, “Polarized light-pulse transport through scattering media,” J. Opt. Soc. Am. A 23(3), 664–670 (2006).
[Crossref] [PubMed]

M. Sakami, K. Mitra, and T. Vo-Dinh, “Analysis of short-pulse laser photon transport through tissues for optical tomography,” Opt. Lett. 27(5), 336–338 (2002).
[Crossref] [PubMed]

M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light-pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 169–179 (2002).
[Crossref]

M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light-pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 169–179 (2002).
[Crossref]

Santer, R.

Sheetz, K. E.

K. E. Sheetz and J. Squier, “Ultrafast optics: Imaging and manipulating biological systems,” J. Appl. Phys. 105(5), 051101 (2009).
[Crossref]

Shum, H. C.

J. Wu, Y. Xu, J. Xu, X. Wei, A. C. Chan, A. H. Tang, A. K. Lau, B. M. Chung, H. C. Shum, E. Y. Lam, K. K. Wong, and K. K. Tsia, “Ultrafast laser-scanning time-stretch imaging at visible wavelengths,” Light Sci. Appl. 6(1), e16196 (2017).
[Crossref]

Siewert, C. E.

C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 64(3), 227–254 (2000).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “The FN method for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 41(2), 117–145 (1989).
[Crossref]

Sommersten, E. R.

E. R. Sommersten, J. K. Lotsberg, K. Stamnes, and J. J. Stamnes, “Discrete ordinate and Monte Carlo simulations for polarized radiative transfer in a coupled system consisting of two media with different refractive indices,” J. Quant. Spectrosc. Radiat. Transf. 111(4), 616–633 (2010).
[Crossref]

Squier, J.

K. E. Sheetz and J. Squier, “Ultrafast optics: Imaging and manipulating biological systems,” J. Appl. Phys. 105(5), 051101 (2009).
[Crossref]

Stamnes, J. J.

E. R. Sommersten, J. K. Lotsberg, K. Stamnes, and J. J. Stamnes, “Discrete ordinate and Monte Carlo simulations for polarized radiative transfer in a coupled system consisting of two media with different refractive indices,” J. Quant. Spectrosc. Radiat. Transf. 111(4), 616–633 (2010).
[Crossref]

Stamnes, K.

E. R. Sommersten, J. K. Lotsberg, K. Stamnes, and J. J. Stamnes, “Discrete ordinate and Monte Carlo simulations for polarized radiative transfer in a coupled system consisting of two media with different refractive indices,” J. Quant. Spectrosc. Radiat. Transf. 111(4), 616–633 (2010).
[Crossref]

Stephens, G. L.

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991).
[Crossref]

Sun, C. W.

X. Wang, L. V. Wang, C. W. Sun, and C. C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8(4), 608–617 (2003).
[Crossref] [PubMed]

Sun, W.

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, G. Videen, and C. J. Grund, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transf. 153, 131–143 (2015).
[Crossref]

Tan, H.

Tan, H. P.

C. H. Wang, H. L. Yi, and H. P. Tan, “Transient polarized radiative transfer analysis in a scattering medium by a discontinuous finite element method,” Opt. Express 25(7), 7418–7442 (2017).
[Crossref] [PubMed]

C. H. Wang, Y. Zhang, H. L. Yi, and H. P. Tan, “Transient radiative transfer in two-dimensional graded index medium by Monte Carlo method combined with the time shift and superposition principle,” Numer. Heat Transf. A 69(6), 574–588 (2016).
[Crossref]

C. H. Wang, Q. Ai, H. L. Yi, and H. P. Tan, “Transient radiative transfer in a graded index medium with specularly reflecting surfaces,” Numer. Heat Transf. A 67(11), 1232–1252 (2015).
[Crossref]

Tan, H.-P.

C. H. Wang, H. L. Yi, and H.-P. Tan, “Discontinuous finite element method for vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 189, 383–397 (2017).
[Crossref]

Tan, J. Y.

J. M. Zhao, J. Y. Tan, and L. H. Liu, “Monte Carlo method for polarized radiative transfer in gradient-index media,” J. Quant. Spectrosc. Radiat. Transf. 152, 114–126 (2015).
[Crossref]

J. M. Zhao, J. Y. Tan, and L. H. Liu, “A second order radiative transfer equation and its solution by meshless method with application to strongly inhomogeneous media,” J. Comput. Phys. 232(1), 431–455 (2013).
[Crossref]

Tan, Z. M.

Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” ASME J. Heat Transfer 123(3), 466–475 (2001).
[Crossref]

Tang, A. H.

J. Wu, Y. Xu, J. Xu, X. Wei, A. C. Chan, A. H. Tang, A. K. Lau, B. M. Chung, H. C. Shum, E. Y. Lam, K. K. Wong, and K. K. Tsia, “Ultrafast laser-scanning time-stretch imaging at visible wavelengths,” Light Sci. Appl. 6(1), e16196 (2017).
[Crossref]

Titus, T. N.

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, G. Videen, and C. J. Grund, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transf. 153, 131–143 (2015).
[Crossref]

Tromberg, B. J.

T. D. O’Sullivan, A. E. Cerussi, D. J. Cuccia, and B. J. Tromberg, “Diffuse optical imaging using spatially and temporally modulated light,” J. Biomed. Opt. 17(7), 071311 (2012).
[PubMed]

Tsia, K. K.

J. Wu, Y. Xu, J. Xu, X. Wei, A. C. Chan, A. H. Tang, A. K. Lau, B. M. Chung, H. C. Shum, E. Y. Lam, K. K. Wong, and K. K. Tsia, “Ultrafast laser-scanning time-stretch imaging at visible wavelengths,” Light Sci. Appl. 6(1), e16196 (2017).
[Crossref]

Vaillona, R.

R. Vaillona, B. T. Wong, and M. P. Mengüç, “Polarized radiative transfer in a particle-laden semi-transparent medium via a vector Monte Carlo method,” J. Quant. Spectrosc. Radiat. Transf. 84(4), 383–394 (2004).
[Crossref]

Videen, G.

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, G. Videen, and C. J. Grund, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transf. 153, 131–143 (2015).
[Crossref]

Vo-Dinh, T.

Wang, C. H.

C. H. Wang, H. L. Yi, and H.-P. Tan, “Discontinuous finite element method for vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 189, 383–397 (2017).
[Crossref]

C. H. Wang, H. L. Yi, and H. P. Tan, “Transient polarized radiative transfer analysis in a scattering medium by a discontinuous finite element method,” Opt. Express 25(7), 7418–7442 (2017).
[Crossref] [PubMed]

C. H. Wang, Y. Zhang, H. L. Yi, and H. P. Tan, “Transient radiative transfer in two-dimensional graded index medium by Monte Carlo method combined with the time shift and superposition principle,” Numer. Heat Transf. A 69(6), 574–588 (2016).
[Crossref]

C. H. Wang, Q. Ai, H. L. Yi, and H. P. Tan, “Transient radiative transfer in a graded index medium with specularly reflecting surfaces,” Numer. Heat Transf. A 67(11), 1232–1252 (2015).
[Crossref]

Wang, J. M.

J. M. Wang and C. Y. Wu, “Transient radiative transfer in a scattering slab with variable refractive index and diffuse substrate,” Int. J. Heat Mass Transfer 53(19), 3799–3806 (2010).
[Crossref]

Wang, L. V.

X. Wang, L. V. Wang, C. W. Sun, and C. C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8(4), 608–617 (2003).
[Crossref] [PubMed]

Wang, X.

X. Wang, L. V. Wang, C. W. Sun, and C. C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8(4), 608–617 (2003).
[Crossref] [PubMed]

Wang, Z. C.

Z. C. Wang, Q. Cheng, and H. C. Zhou, “The DRESOR method for transient radiation transfer in 1-D graded index medium with pulse irradiation,” Int. J. Therm. Sci. 68, 127–135 (2013).
[Crossref]

Wei, X.

J. Wu, Y. Xu, J. Xu, X. Wei, A. C. Chan, A. H. Tang, A. K. Lau, B. M. Chung, H. C. Shum, E. Y. Lam, K. K. Wong, and K. K. Tsia, “Ultrafast laser-scanning time-stretch imaging at visible wavelengths,” Light Sci. Appl. 6(1), e16196 (2017).
[Crossref]

Wendisch, M.

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
[Crossref]

Wolff, M. J.

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, G. Videen, and C. J. Grund, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transf. 153, 131–143 (2015).
[Crossref]

Wong, B. T.

R. Vaillona, B. T. Wong, and M. P. Mengüç, “Polarized radiative transfer in a particle-laden semi-transparent medium via a vector Monte Carlo method,” J. Quant. Spectrosc. Radiat. Transf. 84(4), 383–394 (2004).
[Crossref]

Wong, K. K.

J. Wu, Y. Xu, J. Xu, X. Wei, A. C. Chan, A. H. Tang, A. K. Lau, B. M. Chung, H. C. Shum, E. Y. Lam, K. K. Wong, and K. K. Tsia, “Ultrafast laser-scanning time-stretch imaging at visible wavelengths,” Light Sci. Appl. 6(1), e16196 (2017).
[Crossref]

Wu, C. Y.

J. M. Wang and C. Y. Wu, “Transient radiative transfer in a scattering slab with variable refractive index and diffuse substrate,” Int. J. Heat Mass Transfer 53(19), 3799–3806 (2010).
[Crossref]

C. Y. Wu and N. R. Ou, “Differential approximation for transient radiative transfer through a participating medium exposed to collimated irradiation,” J. Quant. Spectrosc. Radiat. Transf. 73(1), 111–120 (2002).
[Crossref]

C. Y. Wu, “Propagation of scattered radiation in a participating planar medium with pulse irradiation,” J. Quant. Spectrosc. Radiat. Transf. 64(5), 537–548 (2000).
[Crossref]

C. Y. Wu and S. H. Wu, “Integral equation formulation for transient radiative transfer in an anisotropically scattering medium,” Int. J. Heat Mass Transfer 43(11), 2009–2020 (2000).
[Crossref]

Wu, J.

J. Wu, Y. Xu, J. Xu, X. Wei, A. C. Chan, A. H. Tang, A. K. Lau, B. M. Chung, H. C. Shum, E. Y. Lam, K. K. Wong, and K. K. Tsia, “Ultrafast laser-scanning time-stretch imaging at visible wavelengths,” Light Sci. Appl. 6(1), e16196 (2017).
[Crossref]

Wu, S. H.

C. Y. Wu and S. H. Wu, “Integral equation formulation for transient radiative transfer in an anisotropically scattering medium,” Int. J. Heat Mass Transfer 43(11), 2009–2020 (2000).
[Crossref]

Xu, J.

J. Wu, Y. Xu, J. Xu, X. Wei, A. C. Chan, A. H. Tang, A. K. Lau, B. M. Chung, H. C. Shum, E. Y. Lam, K. K. Wong, and K. K. Tsia, “Ultrafast laser-scanning time-stretch imaging at visible wavelengths,” Light Sci. Appl. 6(1), e16196 (2017).
[Crossref]

Xu, Y.

J. Wu, Y. Xu, J. Xu, X. Wei, A. C. Chan, A. H. Tang, A. K. Lau, B. M. Chung, H. C. Shum, E. Y. Lam, K. K. Wong, and K. K. Tsia, “Ultrafast laser-scanning time-stretch imaging at visible wavelengths,” Light Sci. Appl. 6(1), e16196 (2017).
[Crossref]

Yang, C. C.

X. Wang, L. V. Wang, C. W. Sun, and C. C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8(4), 608–617 (2003).
[Crossref] [PubMed]

Yi, H.

Yi, H. L.

C. H. Wang, H. L. Yi, and H.-P. Tan, “Discontinuous finite element method for vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 189, 383–397 (2017).
[Crossref]

C. H. Wang, H. L. Yi, and H. P. Tan, “Transient polarized radiative transfer analysis in a scattering medium by a discontinuous finite element method,” Opt. Express 25(7), 7418–7442 (2017).
[Crossref] [PubMed]

C. H. Wang, Y. Zhang, H. L. Yi, and H. P. Tan, “Transient radiative transfer in two-dimensional graded index medium by Monte Carlo method combined with the time shift and superposition principle,” Numer. Heat Transf. A 69(6), 574–588 (2016).
[Crossref]

C. H. Wang, Q. Ai, H. L. Yi, and H. P. Tan, “Transient radiative transfer in a graded index medium with specularly reflecting surfaces,” Numer. Heat Transf. A 67(11), 1232–1252 (2015).
[Crossref]

Yokota, T.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Z. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12), 1931–1946 (2010).
[Crossref]

Yu, Y. L.

Q. Cheng, H. C. Zhou, Z. F. Huang, Y. L. Yu, and D. X. Huang, “The solution of transient radiative transfer with collimated incident serial pulse in a plane-parallel medium by the DRESOR method,” ASME J. Heat Transfer 130(10), 102701 (2008).
[Crossref]

Zege, E. P.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Z. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12), 1931–1946 (2010).
[Crossref]

Zhang, Y.

C. H. Wang, Y. Zhang, H. L. Yi, and H. P. Tan, “Transient radiative transfer in two-dimensional graded index medium by Monte Carlo method combined with the time shift and superposition principle,” Numer. Heat Transf. A 69(6), 574–588 (2016).
[Crossref]

Y. Zhang, H. Yi, and H. Tan, “One-dimensional transient radiative transfer by lattice Boltzmann method,” Opt. Express 21(21), 24532–24549 (2013).
[Crossref] [PubMed]

Zhao, J. M.

J. M. Zhao, J. Y. Tan, and L. H. Liu, “Monte Carlo method for polarized radiative transfer in gradient-index media,” J. Quant. Spectrosc. Radiat. Transf. 152, 114–126 (2015).
[Crossref]

J. M. Zhao, J. Y. Tan, and L. H. Liu, “A second order radiative transfer equation and its solution by meshless method with application to strongly inhomogeneous media,” J. Comput. Phys. 232(1), 431–455 (2013).
[Crossref]

Zhou, H. C.

Z. C. Wang, Q. Cheng, and H. C. Zhou, “The DRESOR method for transient radiation transfer in 1-D graded index medium with pulse irradiation,” Int. J. Therm. Sci. 68, 127–135 (2013).
[Crossref]

Q. Cheng, H. C. Zhou, Z. F. Huang, Y. L. Yu, and D. X. Huang, “The solution of transient radiative transfer with collimated incident serial pulse in a plane-parallel medium by the DRESOR method,” ASME J. Heat Transfer 130(10), 102701 (2008).
[Crossref]

Adv. Heat Transf. (1)

S. Kumar and K. Mitra, “Microscale aspects of thermal radiation transport and laser applications,” Adv. Heat Transf. 33, 187–294 (1999).
[Crossref]

AIAA J. Thermophys. Heat Transfer (1)

K. Mitra, M. S. Lai, and S. Kumar, “Transient radiation transport in participating media within a rectangular enclosure,” AIAA J. Thermophys. Heat Transfer 11(3), 409–414 (1997).
[Crossref]

Appl. Opt. (7)

ASME J. Heat Transfer (2)

Q. Cheng, H. C. Zhou, Z. F. Huang, Y. L. Yu, and D. X. Huang, “The solution of transient radiative transfer with collimated incident serial pulse in a plane-parallel medium by the DRESOR method,” ASME J. Heat Transfer 130(10), 102701 (2008).
[Crossref]

Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” ASME J. Heat Transfer 123(3), 466–475 (2001).
[Crossref]

Comput. Phys. Commun. (1)

Y. A. Ilyushin and V. P. Budak, “Analysis of the propagation of the femtosecond laser pulse in the scattering medium,” Comput. Phys. Commun. 182(4), 940–945 (2011).
[Crossref]

Icarus (1)

A. J. Brown, “Spectral bluing induced by small particles under the Mie and Rayleigh regimes,” Icarus 239, 85–95 (2014).
[Crossref]

Int. J. Heat Mass Transfer (4)

C. Y. Wu and S. H. Wu, “Integral equation formulation for transient radiative transfer in an anisotropically scattering medium,” Int. J. Heat Mass Transfer 43(11), 2009–2020 (2000).
[Crossref]

J. M. Wang and C. Y. Wu, “Transient radiative transfer in a scattering slab with variable refractive index and diffuse substrate,” Int. J. Heat Mass Transfer 53(19), 3799–3806 (2010).
[Crossref]

S. C. Mishra, A. Lankadasu, and K. N. Beronov, “Application of the lattice Boltzmann method for solving the energy equation of a 2-D transient conduction–radiation problem,” Int. J. Heat Mass Transfer 48(17), 3648–3659 (2005).
[Crossref]

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer 49(11), 1820–1832 (2006).
[Crossref]

Int. J. Therm. Sci. (1)

Z. C. Wang, Q. Cheng, and H. C. Zhou, “The DRESOR method for transient radiation transfer in 1-D graded index medium with pulse irradiation,” Int. J. Therm. Sci. 68, 127–135 (2013).
[Crossref]

J. Appl. Phys. (1)

K. E. Sheetz and J. Squier, “Ultrafast optics: Imaging and manipulating biological systems,” J. Appl. Phys. 105(5), 051101 (2009).
[Crossref]

J. Biomed. Opt. (2)

T. D. O’Sullivan, A. E. Cerussi, D. J. Cuccia, and B. J. Tromberg, “Diffuse optical imaging using spatially and temporally modulated light,” J. Biomed. Opt. 17(7), 071311 (2012).
[PubMed]

X. Wang, L. V. Wang, C. W. Sun, and C. C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8(4), 608–617 (2003).
[Crossref] [PubMed]

J. Comput. Phys. (1)

J. M. Zhao, J. Y. Tan, and L. H. Liu, “A second order radiative transfer equation and its solution by meshless method with application to strongly inhomogeneous media,” J. Comput. Phys. 232(1), 431–455 (2013).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Quant. Spectrosc. Radiat. Transf. (16)

R. D. M. Garcia and C. E. Siewert, “The FN method for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 41(2), 117–145 (1989).
[Crossref]

C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 64(3), 227–254 (2000).
[Crossref]

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991).
[Crossref]

E. R. Sommersten, J. K. Lotsberg, K. Stamnes, and J. J. Stamnes, “Discrete ordinate and Monte Carlo simulations for polarized radiative transfer in a coupled system consisting of two media with different refractive indices,” J. Quant. Spectrosc. Radiat. Transf. 111(4), 616–633 (2010).
[Crossref]

Z. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 159–168 (2002).
[Crossref]

M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light-pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 169–179 (2002).
[Crossref]

M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light-pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf. 73(2), 169–179 (2002).
[Crossref]

J. C. Chai, “Transient radiative transfer in irregular two-dimensional geometries,” J. Quant. Spectrosc. Radiat. Transf. 84(3), 281–294 (2004).
[Crossref]

C. Y. Wu, “Propagation of scattered radiation in a participating planar medium with pulse irradiation,” J. Quant. Spectrosc. Radiat. Transf. 64(5), 537–548 (2000).
[Crossref]

C. Y. Wu and N. R. Ou, “Differential approximation for transient radiative transfer through a participating medium exposed to collimated irradiation,” J. Quant. Spectrosc. Radiat. Transf. 73(1), 111–120 (2002).
[Crossref]

A. J. Brown, T. I. Michaels, S. Byrne, W. Sun, T. N. Titus, A. Colaprete, M. J. Wolff, G. Videen, and C. J. Grund, “The case for a modern multi-wavelength, polarization-sensitive LIDAR in orbit around Mars,” J. Quant. Spectrosc. Radiat. Transf. 153, 131–143 (2015).
[Crossref]

C. H. Wang, H. L. Yi, and H.-P. Tan, “Discontinuous finite element method for vector radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 189, 383–397 (2017).
[Crossref]

R. Vaillona, B. T. Wong, and M. P. Mengüç, “Polarized radiative transfer in a particle-laden semi-transparent medium via a vector Monte Carlo method,” J. Quant. Spectrosc. Radiat. Transf. 84(4), 383–394 (2004).
[Crossref]

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Z. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12), 1931–1946 (2010).
[Crossref]

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
[Crossref]

J. M. Zhao, J. Y. Tan, and L. H. Liu, “Monte Carlo method for polarized radiative transfer in gradient-index media,” J. Quant. Spectrosc. Radiat. Transf. 152, 114–126 (2015).
[Crossref]

Light Sci. Appl. (1)

J. Wu, Y. Xu, J. Xu, X. Wei, A. C. Chan, A. H. Tang, A. K. Lau, B. M. Chung, H. C. Shum, E. Y. Lam, K. K. Wong, and K. K. Tsia, “Ultrafast laser-scanning time-stretch imaging at visible wavelengths,” Light Sci. Appl. 6(1), e16196 (2017).
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Figures (9)

Fig. 1
Fig. 1 Sketch of elements, element boundaries and the radiation values on the boundaries for the two-dimensional discrete elements.
Fig. 2
Fig. 2 Physical model of the two-dimensional scattering medium exposed to an external collimated beam illumination.
Fig. 3
Fig. 3 Comparisons of the time-resolved Stokes vector radiative fluxes along the top boundary against the steady solution.
Fig. 4
Fig. 4 Time-resolved Stokes vector radiative flux along the top boundary.
Fig. 5
Fig. 5 Time-resolved Stokes vector radiative flux along the right boundary.
Fig. 6
Fig. 6 Distributions of the false temperature within the rectangular atmosphere at different times.
Fig. 7
Fig. 7 (a): The geometric coordinate in meters and the grid discretization, and (b): time-resolved radiative heat flux along the bottom boundary of the irregular emitting medium.
Fig. 8
Fig. 8 Polarized radiative flux distributions of Stokes vector component (a): I and (b): Q on the bottom boundary of the irregular medium at different times.
Fig. 9
Fig. 9 False temperature distributions within the emitting irregular medium at different times.

Tables (1)

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Table 1 Nomenclature.

Equations (13)

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1 c 0 I(r,Ω,t) t +ΩI(r,Ω,t)+βI(r,Ω,t)=S(r,Ω,t) ,
S(z,Ω,t)= κ a I b + κ s 4π 4π Z( Ω ,Ω)I(r, Ω ,t) d Ω ,
I( r w ,Ω,t)= R s I( r w , Ω ,t)+ 1 π n w Ω n >0 R d I( r w , Ω ,t)| n w Ω |d Ω ,
[ I n ]= Ω n I ¯ n +| Ω n | I n n K ,
I ¯ n = 1 2 ( I n + I + n ), I n = 1 2 ( I n I + n ) .
K k n I k n = H k n ,
K k,ji n = e ϕ i Ω n ϕ j dA + 1 2 e ( Ω n n e +| Ω n | ϕ i ϕ j )ds + e β ˜ ϕ i ϕ j dA ,
H k,j n = e S ˜ k n ϕ j dA 1 2 e ( Ω n n e | Ω n |) I k,+ n ϕ j ds ,
β ˜ =( 2 Δ t * +1)β ,
S ˜ k n (r,Ω)= S k (r,Ω)+ S k1 (r,Ω)Ω I k1 (r,Ω)(1 2 Δ t * )β I k1 (r,Ω) ,
I= I c + I d ,
I c (r, Ω 0 , t * )= I 0 exp(βx) .
S(r,Ω, t * )= κ s 4π [ Z(r, Ω 0 ,Ω) I c (r, Ω 0 , t * )+ 4π Z(r, Ω Ω) I d (r, Ω , t * )d Ω ] ,

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