## 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]

**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.

*c*

_{0}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

**Z(Ω**',

**Ω)**is the scattering phase matrix for scattering from an incoming direction

**Ω**' to an outgoing direction

**Ω**,

**I**

_{b}= (

*I*

_{b}, 0, 0, 0)

^{T}and

*I*

_{b}is the black body emission.

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

**Ω**” denotes the corresponding incident direction of the reflected beam of direction

**Ω**, and

**n**

_{w}denotes the unit outward normal vector of the boundary.

**R**

_{s}and

**R**

_{d}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 *N _{θ}* ×

*N*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 ${I}_{-}^{n}$ and ${I}_{+}^{n}$ are assumed to be different, the adjacent elements are connected by modeling the boundary flux [48]

_{φ}*n*' denotes the

*n*th discrete direction, ${\overline{I}}^{n}$ and ${\overline{){\rm I}}}^{n}$ are defined as

As for the treatment for the one-dimensional TVRTE in [48], the dimensionless time *t** = *βc*_{0}*t* 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 *k*th time step can be written in the matrix form as

*ϕ*is the shape function,

*Δt** is the dimensionless time step, the subscript '

*k*' denotes the

*k*th 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.

## 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 *N _{x}* ×

*N*= 20 × 20 quadrilateral elements and the angular space is divided into

_{y}*N*×

_{θ}*N*= 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

_{φ}*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**

_{c}denotes the collimated part and

**I**

_{d}denotes the diffuse part of the Stokes vector. The collimated part

**I**

_{c}within the medium decreases exponentially according to Beer's law and can be derived mathematically as

The diffuse part **I**_{d} inside the medium can be obtained by solving Eq. (1) after modifying the right-hand source term as

**Ω**

^{0}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 **I**_{0} = *π* × (1, 0.5, 0.5, 0.5)^{T}, is incident on the left boundary in the direction *μ*_{0} = cos*θ*_{0} = 0, *φ*_{0} = 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.

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.

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*.

To evaluate the energy distribution within the computational domain, the incident radiation is transformed into a false temperature defined by *T*_{f} = (*G*/4*σ*)^{0.25} with *G* denoting the incident radiation and *σ* = 5.67 × 10^{−8} W/(m^{2} K^{4}) 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.

#### 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 T_{g} = 1000 K at *t** = 0 and maintained at that level.

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 *q _{x}*/

*σ*T

_{g}

^{4}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.

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.

## 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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