## Abstract

The lattice Boltzmann method (LBM) is extended to solve transient radiative transfer in one-dimensional slab containing scattering media subjected to a collimated short laser irradiation. By using a fully implicit backward differencing scheme to discretize the transient term in the radiative transfer equation, a new type of lattice structure is devised. The accuracy and computational efficiency of this algorithm are examined firstly. Afterwards, effects of the medium properties such as the extinction coefficient, the scattering albedo and the anisotropy factor, and the shapes of laser pulse on time-resolved signals of transmittance and reflectance are investigated. Results of the present method are found to compare very well with the data from the literature. For an oblique incidence, the LBM results in this paper are compared with those by Monte Carlo method generated by ourselves. In addition, transient radiative transfer in a two-Layer inhomogeneous media subjected to a short square pulse irradiation is investigated. At last, the LBM is further extended to study the transient radiative transfer in homogeneous medium with a refractive index discontinuity irradiated by the short pulse laser. Several trends on the time-resolved signals different from those for refractive index of 1 (i.e. refractive-index-matched boundary) are observed and analysed.

© 2013 Optical Society of America

## 1. Introduction

In the past twenty years, due to the availability of short-pulse lasers, transient radiative transfer (TRT) in participating media has received considerable attention in many emerging applications. The developments in micro/nano-scale systems [1, 2], short-pulsed laser in materials [3], optical tomography [4], laser therapy [5], particle detection and sizing [6] and other applications have indicated that TRT is an important process which requires rigorous study. A detailed review on various aspects of the transient radiative transfer behavior induced by a short-pulsed laser was presented by Kumar and Mitra [7].

An interaction of a short-pulse radiation with a participating medium results in temporal signals. The time-resolved radiative signals offer some unique characteristics that are not available from the steady-state light sources. Usually the temporal or time-resolved radiative signals (transmittance and/or reflectance) are used to determine the medium’s internal structure or/and radiative properties. Hence, sufficiently accurate and efficient solution methods are required.

In the recent decade, with the start of research in the area of transient radiative transfer, several numerical strategies have been developed, including the Monte Carlo method (MCM), discrete ordinate method (DOM), integral equation (IE) models, finite volume method FVM, and the discontinuous finite element method (DFEM).

The Monte Carlo method was used to model the transient radiative transfer by Schweiger et al. [8] and Guo et al. [9]. However, the Monte Carlo method requires a large number of energy bundles to obtain accurate and smooth results, and is so computationally expensive. Lu and Hsu [10, 11] developed a reverse Monte Carlo (RMC) method which shortened the computation time and improved the computational efficiency in the investigation of transient radiative transfer. Martinelli et al. [12] presented a theoretical analysis that provided a clear line of derivation from the RTE to the scaling relations which formed the basis of single Monte Carlo (sMC). Guo and Kumar [13] and Sakami et al. [14] extended applications of the DOM to the 2-D rectangular enclosure. Application of the DOM to the 3-D rectangular enclosure was extended by Guo and Kumar [15]. Wu et al. [16] applied the DOM with a first-order spatial scheme and a modified DOM (MDOM) to solve transient radiative transfer in a scattering slab with variable refractive index and diffuse substrate exposed to a diffuse strong irradiation at one of its boundaries. Tan and Hsu [17] developed an integral equation (IE) formulation to treat the general transient radiative transfer equation, and Wu [18] used a slightly different form of IE formulation to study the light pulse transport in a participating planar medium. Their solutions provided very accurate results and were verified with Monte Carlo algorithms [19]. Chai [20] introduced the FVM to solve the transient radiative transfer in a one-dimensional absorbing and isotropically scattering planar medium. Kim et al. [21] investigated the transient radiative heat transfer in one-dimensional slabs separated by a participating media using the FVM, and in the work the convection schemes in computational fluid dynamics (CFD) such as step, diamond, 2nd order upwind, QUICK, and CLAM were introduced to capture the physics of the radiative wave propagation. Ruan et al. [22] used the FVM to solve the transient radiative transfer problem in 1-D homogeneous and inhomogeneous media irradiated by the short pulse laser. A non-dimensional number was proposed to analyze the characteristics of the temporal transmittance and reflectance signals in [22]. Mishra et al. [23] provided the analysis of transient radiative transfer caused by a short-pulse laser irradiation on a participating media using the discrete transfer method (DTM), DOM, and FVM. Liu and associates [24, 25] extended the application of discontinuous finite element method (DFEM) based on the discrete ordinates equation to solving transient radiative transfer in absorbing, emitting, and scattering media, with the use of a time shift and superposition principle for improving the computational efficiency.

The lattice Boltzmann method (LBM) is a relatively new computational tool. In the recent decades, it has emerged as an efficient method to analyze a vast range of problems in fluid flow and heat transfer [26–28]. This surge in applications of the LBM is owing to its attractive properties of simple implementation on the computer, mesoscopic nature, ability to handle complex geometry and boundary conditions, capability of stable and accurate simulation, and the inherent parallel nature.

Over the years, the LBM has been applied to solve the energy equations of combined mode conduction and/or convection problems [29–32] involving volumetric radiation in which the radiative information has been computed using the conventional numerical methods like the DTM [29], FVM [30–32]. Recently, the LBM itself has been adopted for solving radiation transport problems [33–37] in which one-dimensional (1D) and two-dimensional (2D) examples of radiative transfer were discussed. Asinari and associates [33, 34] described the advantage of having common data structures for radiation intensity and fluid flow in radiative heat transfer and fluid mechanics problems. They extended the application of the LBM to solve a benchmark radiative equilibrium problem involving a 2-D rectangular enclosure, and the LBM was found to have an edge over the FVM. Based on the Chapman-Enskog method, Ma et al. [35] proposed the lattice Boltzmann model for one-dimensional radiative transfer from the Boltzmann equation. Bindra et al. [36] extended the LBM to solving the radiative or neutron transport equation with considering the scattering term. Mishra and associates [37] extended the LBM to analysis of transport of collimated radiation in a participating media.

However, to the best of our knowledge, as a promising numerical scheme, the LBM has not been used for the solution to transient radiative transfer in the participating medium irradiated by the short pulse laser. In this article we present a LBM for solving the transient radiative transfer in one-dimensional slab containing absorbing, scattering media subjected to the collimated short laser irradiation. No research works have been carried out to investigate the transient radiative transfer in homogeneous medium with a refractive index discontinuity irradiated by the short pulse laser. Thus, we further extend the application of LBM to studying this problem.

The outline of this paper is as below. In the following section, the framework of lattice Boltzmann method for solving the transient radiative transfer is formulated and the solution process about the implementation of LBM is presented. In section 3, the accuracy and efficiency of this algorithm are studied firstly. Afterwards, to show the flexibility of the LBM for different calculation conditions, several test examples are examined. For the refractive index matched boundary case, the effects of the angle of the incidence on temporal signals of transmittance and reflectance are studied.

## 2. Mathematical formulation

Consider a plane-parallel slab filled with a grey medium of finite thickness *L* as shown in Fig. 1(a). The medium is assumed to have azimuthal symmetry with constant physical properties. The refractive index of the medium *n* is homogeneous and could be either equal to or higher than those of the environment (in this paper, the refractive index of the environment is 1). The boundary at *x* = 0 is exposed to a collimated short-pulse irradiation with an incident angle *θ*_{0} and a radiation intensity *I*_{0} as illustrated in Fig. 1(a). The wave shape of the short pulse could be either square or Gaussian as shown in Fig. 1(b). Besides, it is assumed that the thermal emission of the medium is negligible as compared with the incident radiation. The propagation of light pulse in the semi-transparent medium is described by the transient radiative transfer equation (TRTE). In the Cartesian coordinate system, the TRTE for one-dimensional problem can be written as

*c*is the propagation speed of light in vacuum,

_{o}*n*is the refractive index of the medium,

*t*is the time,

*μ*is the direction cosine of the polar angle (−1 ≤

*μ*≤ 1),

*I*(

*x*,

*μ*,

*t*) is the radiative intensity, and

*κ*,

_{a}*σ*and

_{s}*β*=

*κ*+ σ

_{a}*are the absorption, scattering and extinction coefficients, respectively. Φ(*

_{s}*μ*,

*μ*) = 1 +

^{′}*aμμ*is the scattering phase function with

^{′}*a*denoting the anisotropically scattering coefficient.

When *n* = 1, the collimated radiation penetrates directly into the medium without changing the direction. In this case the intensity *I* within the medium can be divided into two components, viz., the collimated intensity *I _{c}* and the diffuse intensity

*I*.

_{d}The collimated component *I _{c}* within the medium is decreased exponentially according to Beer’s law [23]

Substituting Eqs. (2) and (3) in Eq. (1) yields

*S*=

_{t}*S*+

_{c}*S*is the total source term.

_{d}As the refractive index of the medium matches with the environment at the interface (with refractive index *n* = 1), the two boundaries at *x* = 0 and *x* = *L* are considered to be non-reflecting. Thus the radiative boundary condition can be written as

When the refractive index mismatches with the environment at the surface (*n* > 1) and the medium surfaces are supposed to be diffusely reflecting and semitransparent, the transient process of radiation transfer is quite different from the case of *n* = 1. The collimated pulse irradiation on the left boundary is divided into two parts, viz., the diffusely reflected intensity towards the environment and the diffusely transmitted intensity towards the medium inside. Therefore, only the diffuse intensity *I*_{d} can be obtained.

The external diffuse reflectivity *ρ _{O}* can be expressed as [38]:

*n′*=

*n*/1 =

*n*. Considering the effect of total reflection, the internal diffuse reflectivity is [38]

Thus for the diffusely reflecting semitransparent surface, the radiative boundary condition can be expressed readily as

According to Eq. (7), the (1−*ρ _{I}* )

*n*

^{2}

*I*

_{0}could be replaced by (1−

*ρ*)

_{O}*I*

_{0}.

In this paper, both the square and Gaussian pulses irradiation is considered. For the square pulse, the radiation intensity incident on the boundary at *x* = 0 may then be written as

*t*is the pulse duration,

_{p}*H*(

*t*) is the Heaviside step function,

*δ*is the Dirac delta function, and

*μ*

_{0}is the direction cosine of the angle of incidence.

While for the Gaussian pulse irradiation, the incident radiation intensity is expressed as

The temporal Gaussian pulse shape described in Fig. 1(b) is a truncated Gaussian distribution with the maximum at *t* = *t _{c}* and the half maxima at

*t*=

*t*±

_{c}*t*/2, where the pulse intensity exceeds one-half of the maximum intensity.

_{p}As for *n* = 1, the collimated intensity *I*_{c} in the medium for the square pulse and the Gaussian pulse are needed and can be derived from Eqs. (3), (9) and (10). The collimated remnant of the square pulse irradiation can be expressed as

*s*=

*x*/

*μ*

_{0}is the geometric distance in the incident direction,

*t**=

*βc*and

_{o}t*t*=

_{p}**βc*is the non-dimensional time and the non-dimensional pulse-width, respectively. The attenuation of the collimated Gaussian irradiation as it travels through the medium is given by

_{o}t_{p}In terms of the non-dimensional time *t**, the RTE given by Eq. (4) is now rewritten as

Using fully implicit backward differencing scheme in time, Eq. (13) can be written as

The above equation can be rearranged as

For the LB method, a pseudo time marching is performed with a *Μ* − velocity lattice model in 1D (D1QM) [37]. *M* is the total number of discrete directions. It is obviously that the speed of particle propagation along the *m*th discrete direction is ${e}_{m}=(\Delta x/{\mu}^{m})/\Delta {t}^{*}$. Using the spatial finite difference to discretize the left side of Eq. (16), the TRTE in the *m*th discrete direction can be expressed as

The evolution equation corresponding to Eq. (17) is given as

Using the standard LBM terminology [37], Eq. (17) can be rewritten as

Since the polar angle space is divided equally into *M* parts, the source term *S _{c}* and

*S*are computed from the following equations:

_{d}*m*.

^{′}For one-dimensional slab, the time-resolved signals at the boundary of incidence and opposite are termed as reflectance *q _{R}* and transmittance

*q*, respectively. In the analysis of the transient radiative transfer, the time-resolved reflectance and transmittance provide specific information about the media. Transmittance is defined as the dimensionless net radiative heat flux emerging out of the medium due to transmission, namely the dimensionless net radiative heat flux at the right boundary (

_{T}*x*=

*L*). Reflectance is the dimensionless net radiative heat flux at the boundary which is subjected to the laser irradiation, and in the present case, it is the dimensionless reflected heat flux at the left boundary (

*x*= 0).

For *n* = 1, the time-resolved signals can be expressed as

For *n* >1, they can be defined as

After the derivation of the D1QM lattice Boltzmann model for one-dimensional transient radiative transfer is completed, the implementation of LBM solution process can be carried out according to the following routine.

*Step 1*: Set the initial parameters, using appropriate number of lattices to mesh the solution domain.

*Step 2*: Confirm the time step Δ*t** and total calculation time span.

*Step 3*: Loop at each time step.

- (1) Loop for the global iterations.
- (a) For each discrete direction
*m*, implement the streaming and colliding processes according to Eq. (19), and update the radiative intensity. - (b) Impose boundary conditions on the boundary nodes.
- (c) Terminate the global iteration process if the stop criterion (the maximum relative error of source term
*S*is not bigger than a very small value) is satisfied. Otherwise, go back to step (a)._{t}

- (2) Compute the time-resolved reflectance and transmittance from Eqs. (23a) and (23b) or from Eqs. (24a) and (24b), respectively. If the total non-dimensional time reaches the total calculation time span, terminate the iteration process of the time loop, otherwise, go back to step (1).

## 3. Results and discussion

In the following section, to verify the accuracy of the LBM approach for solving transient radiative transfer problems in absorbing, scattering media under the irradiation of short-pulse lasers, the LBM formulation is validated firstly, and the investigation of its computational efficiency is also conducted. Following that, several test examples are shown.

In all these cases, the geometric thickness of the 1-D medium is *L* = 1.0 m. At a given time level, convergence is assumed to have been achieved when the change in source term *S _{t}* value at all points for the two consecutive iterations do not exceed 1 × 10

^{−7}. The present LBM for transient radiative heat transfer is coded using MATLAB. All runs were taken on Intel(R) Core(TM) i5-2320 processor with 3.00GHz CPU and 6GB RAM.

#### 3.1 The correctness and computational efficiency of the LBM

In this case, the isotropic scattering medium is contained in a slab with the optical thickness *τ _{L}* = 1 and the scattering albedo is

*ω*= 1. The pulse incident on the left side of the slab is a square pulse with duration

*t*= 1.0. With normal incidence of the collimated radiation (

_{p}**μ*= 0.0), the LBM results for the transient transmittance signal

_{0}*q*are shown in Fig. 2. The non-dimensional computation time span is taken as

_{T}*t**= 10. For grid and ray independent solutions, a maximum of 201 lattices and 12 directions are used. The time interval is chosen as Δ

*t**= 0.05. We can see that the LBM results agree with those obtained by FVM [23] very well.

Under the same calculation parameters, the CPU times (in second) taken by the LBM and FVM for different number of rays, lattices and non-dimensional time interval are presented in Table 1. It is seen from Table1 that the computational time used by the LBM is always less than the FVM. It can be concluded that the LBM is computationally efficient.

#### 3.2 Transient radiative transfer in isotropically scattering medium with short square pulse laser irratiation

In this case, the propagation of short square pulse laser in homogenous, isotropically scattering media is solved by the LBM. The time-resolved signals of transmittance and reflectance for one-dimensional slab are obtained, and the results are compared with those in the literature.

For *t _{p}** = 1.0, with normal incidence of the collimated radiation (

*μ*= 0.0), the curves of

_{0}*q*and

_{T}*q*have been plotted in Figs. 3(a)-3(f) for different scattering albedo and extinction coefficient. The effects of scattering albedo

_{R}*ω*, taken as 0.5 and 0.9, on the results are shown for extinction coefficient

*β*= 1, 5 and 10 m

^{−1}, respectively. It can be seen that the transmittance signals begin to appear just at

*t**=

*τ*and the reflectance signals remain available since the start of the transient process. For a given extinction coefficient

_{L}*β*, with decreases in

*ω*, the peaks of both the signals decrease and they last for a shorter duration. In all these cases, number of 12 rays and the time interval chosen as Δ

*t**= 0.05 are considered. These solutions have been obtained with lattices of 201 for

*β*= 1 m

^{−1}, 501 for

*β*= 5 m

^{−1}and 1001 for

*β*= 10.0 m

^{−1}. It can be seen that for

*β*= 1 m

^{−1}and 5 m

^{−1}with different scattering albedo, results by the LBM agree well with those obtained by the FVM [23]. In Fig. 3(e), for

*β*= 10.0 m

^{−1}and

*ω*= 0.9, an obviously deviation from the FVM and DTM solutions is found. While the LBM results agree well with those by Monte Carlo method (MCM) developed by ourselves.

Effect of the angle of incidence *θ _{0}* on

*q*and

_{T}*q*has been shown in Figs. 4(a)-4(f). For

_{R}*ω*= 1 and

*t*= 1.0, the transmittance and reflectance signals are presented for three values of

_{p}**β*, 1.0, 5.0 and 10.0 m

^{−1}, respectively. For each value of

*β*, results have been illustrated for

*θ*= 0°, 45° and 60°. Mishra et al. have investigated these problems [23], while the correctness of their results has not been verified. Here, for validation of the model with an oblique incidence of the pulse laser built by LBM, we compare the results for

_{0}*ω*= 1,

*β*= 1.0 m

^{−1}and

*θ*= 60° obtained by LBM with those by Monte Carlo method (MCM) developed by ourselves. In Figs. 4(a) and 4(b), it can been seen that the LBM results agree well with those by MCM.

_{0}The differences of the incident angle will change the path of the collimated light propagation, which makes influence to the time-resolved signals. It can be seen in Figs. 4(a), 4(c) and 4(e), with a bigger incident angle, the curve of the time-resolved reflectance in the time range of *t** = 0 to *t* = *t _{p}** increases faster to a maximum value at

*t**=

*t*, as a result of which, a higher peak value is obtained. The contribution of the pulse irradiation to the transient process is embodied in the source terms in Eq. (4). It can be concluded from the analytical solutions of the collimated intensity (see Eq. (11)) that as the incident angle increases, the source terms at all the position decrease, and as a result of which, the diffuse intensity

_{p}**I*

_{d}obtained decreases. Thus, the numerator in Eq. (22a) decreases. However, the denominator (${q}_{0}={I}_{0}\mathrm{cos}{\theta}_{0}$) also decreases at the same time. It can be concluded from Figs. 4(a), 4(c) and 4(e) that as the incident angle increases, the denominator decreases more than the numerator. It can be further observed that, after a period of time, the curves for different incident angle intersect with each other. The reason for this is that, for a bigger incident angle, the energy produced by pulse irradiation kept in the slab is lower, as a result of which after the time

*t*=

*t*the time-resolved reflectance decreases more quickly.

_{p}*It can be seen from Figs. 4(b), 4(d) and 4(f) that, for all the incident angle, the *q _{T}* signals begin to appear at

*t**=

*τ*, while

_{L}*q*signals remain available from the start of the process. For the cases of oblique incidence, the existence of the

_{R}*q*signals during the time period

_{T}*t**=

*τ*to

_{L}*t*=

_{s}**τ*cos

_{L}/*θ*is owing to the contribution of the diffuse radiation which reaches the right boundary before the collimated radiation. It can be further observed that the

_{0}*q*signals undergo a noticeable change at

_{T}*t*and

_{s}**t**=

*t*+

_{s}**t*. This behavior is owing to the fact that the collimated radiation combined with the diffuse radiation passes through the right boundary. Right after

_{p}**t**=

*t*+

_{s}**t*, only the diffuse radiation is at work.

_{p}*#### 3.3 Transient radiative transfer in anisotropically scattering medium with square pulse laser irratiation

In this case, the medium is contained in a slab with the optical thickness *τ _{L}* = 10 and the albedo

*ω*= 0.998. The incident pulse with normal incidence on the left side of the slab is a square signal with a duration of

*t*= 1.0. Three values of the anisotropically scattering coefficient

_{p}**a*are taken for the case. Backward scattering is considered for

*a*= – 0.9, isotropic scattering for

*a*= 0.0 and forward scattering for

*a*= + 0.9. Lu et al. [10] investigated this problem by Reverse Monte Carlo Method (RMCM).

The LBM results for the time-resolved reflectance and transmittance are presented in Fig. 5. For obtaining the convergent and stable solutions, a combination of 1001 lattices and 22 equally spaced directions are required in this case. The results are plotted for time span of *t** = 100. With the time interval setting as Δ*t** = 0.1, it takes about 389.899 CPU seconds for the calculations. It can be seen that in Fig. 5(a) the time-resolved results of transmittance obtained by LBM agree well with the data obtained by the RMCM [10]. It can been observed in Fig. 5(a), the maximum of transmittance signal is higher for the case of forward scattering, while its duration in time is longer for backward scattering. It is owing to the fact that, for the case of forward scattering, photons are pushed towards the emergence wall, while for case of backward scattering, photons travel for a longer time in the medium.

#### 3.4 Transient radiative transfer in purely scattering medium subjected to Gaussian pulse

In this case, the left boundary (at *x* = 0) of the slab is exposed normally to a laser Gaussian pulse with an incident radiation:

The optical thickness of the slab is *τ _{L}* = 1.0 and the albedo is

*ω*= 1.0. The LBM is used to solve the time-resolved reflectance and transmittance for the case of

*t*= 0.4. Lattices of 101 and 12 directions are used in this case. With the time step taken as Δ

_{p}**t**= 0.1, the CPU time cost for the calculation till

*t**= 10 is 17.73661 seconds. The results are shown in Figs. 6(a) and 6(b). The solutions obtained by LBM agree well with the data obtained by the time shift and superposition method in combination with the DFEM [25].

#### 3.5 Transient radiative transfer in two-layer nonhomogeneous media with short square pulse irradiation

As shown in Fig. 7(a), the short square pulse light with width of *t _{p}** = 0.3 irradiates normally on the left surface of the two-layer isotropic scattering media. In this case, the optical thicknesses of the two-layer media are all 0.5. By using the reverse Monte Carlo method, Lu and Hsu [11] investigated the transient radiative transfer problems with the scattering albedo of

*ω*

_{1}= 0.1 and

*ω*

_{2}= 0.9, or

*ω*

_{1}= 0.9 and

*ω*

_{2}= 0.1. In their work, the geometry length is set as

*L*

_{1}=

*L*

_{2}= 0.5 mm. In [22], Ruan et al. presented a new non-dimensional number

*ζ*(

*ζ*=

*ct*/

_{p}*L*=

*t*/

_{p}**τ*). The authors pointed out that, as the scattering albedo and the refractive index are kept the same, the temporal signals of the medium would overlap one another having different combinations of the pulse duration and the thickness of the medium with the same

*ζ*. It means that only if

*τ*is chosen and the non-dimensional pulse-width

*t*is the same, the temporal signals are kept unchanged with different geometry length. Therefore, the geometry length

_{p}**L*

_{1}=

*L*

_{2}= 0.5 m is used in this paper.

The LBM results for time-resolved reflectance are shown in Fig. 7(b). The non-dimensional time is used as horizontal ordinate. Here, 401 lattices and 14 equally spaced directions are considered. The total observed time span is *t** = 6. With the time interval setting as Δ*t** = 0.025, it takes about 72.3591 CPU seconds for the calculations. As shown in Fig. 7(b), our results are in good agreement compared with those in [11]. It can be observed from Fig. 7(b) that in the case of *ω*_{1} = 0.1 and *ω*_{2} = 0.9, the special ‘dual peak’ phenomenon occurs in the reflectance signals. Due to the strong scattering of the second layer, the local minimum in reflectance signal can be found at *t** = 1 moment. Just as is presented by Lu and Hsu [11].

#### 3.6 Transient radiative transfer in isotropically scattering media with refractive index mismatched at the boundary

In this case, the transient radiative transfer problem in a slab containing isotropically scattering medium with refractive-index mismatched boundary (*n* >1) is investigated. The two surfaces of the slab are diffusely reflecting and semitransparent. The normal incident pulse on the left side of the slab is a square pulse with duration of *t _{p}** = 1.0.

In Fig. 8, the time-resolved reflectance and transmittance are plotted for the cases with *n* = 1.5, *β* = 1.0 m^{−1} and *ω* = 1.0. The MCM results developed by ourselves are also presented for the comparison. It can be seen that the LBM results agree well with those obtained by the MCM.

The distributions of the diffuse intensities at different directions and different non-dimensional time *t** are presented in Figs. 9(a) and 9(b) for *x* = 0 and *L*, respectively.

In Fig. 8(a), it can be observed that, the transmittance signal begins to appear just at *t** = *τ _{L}n* (1.5). It is owing to the fact that, the diffuse radiation produced by the normal irradiation on the semitransparent surface takes

*t**=

*τ*at least to reach the right boundary. The

_{L}n*q*increases gradually to the peak value at the time

_{T}*t**=

*τ*(2.5). The noticeable change observed in the case of

_{L}n + t_{p}**n*= 1 doesn’t appear in the case of

*n*= 1.5. This is due to the fact that in the case with the refractive index bigger than one, only diffuse radiation transfers in the medium, and consequently the corresponding transmittance signal changes gradually inside the slab. Referring to Fig. 9(b), we can have an intuitive feeling for the increase of the

*q*. It can be observed that the intensities in the positive directions are great during this period of time

_{T}*t**= 1.5~2.5.

It can be observed in Fig. 8(b), the reflectance signal *q _{R}* is available from the start of the transient process. During the time span of

*t**= 1 to

*t*=

*t*it increases gradually, and its values are noticeable higher than those after

_{p}**t*=

*t*. This is because the reflectance signals during this period consist of two parts, viz., the diffuse radiation transmitting through the internal surface and the radiation diffusely reflected by the external surface. Owing to the existence of the pulse irradiation,

_{p}**q*increases during this period of time. Right after the time

_{R}*t*=

*t*, only the diffusely transmitted radiation make contributions to the

_{p}**q*. It can be further observed that,

_{R}*q*increases again during the time interval of

_{R}*t**≈3 to

*t**≈4.5. It is owing to the fact that, the diffuse energy produced by the pulse on the left interface takes time of

*τ*to travel to the right interface, and after being reflected it takes a total of at least time of 2

_{L}n*τ*(

_{L}n*t**≈3) to get back to the left interface . Referring to Fig. 9(a), we can also have an intuitive feeling for the increase of the

*q*in this period. It can be observed that the values of intensities in the negative directions are noticeable during this period.

_{R}With *β* = 1.0 m^{−1} and *ω* = 1.0, effects of the refractive index *n* on the time-resolved reflectance and transmittance have been shown in Figs. 10(a) and 10(b). With different *n*, the diffuse reflectivity on the semitransparent surface and the propagation speed of the light in the medium are different. Consequently, the time-resolved signals for different *n* are different.

The internal and external diffuse reflectivity of the semitransparent surface are presented in Table 2 for *n* = 1.2, 1.5 and 1.8. It can be seen that, with increases in *n*, both the internal and external diffuse reflectivity increase. It means that, for a higher *n*, pulsed energy transmitted through the left internal interface is lower and the energy inside the slab is reflected more by the internal interface.

From Fig. 10(a) we can observe the following trends of the time-resolved reflectance. (i) In the time range of *t** = 0 to *t** = *t _{p}**, the

*q*curve for the case with a higher

_{R}*n*is of higher values but has a lower increasing rate. As it can be seen in Eq. (24a) that the

*ρ*is added to the

_{O}*q*as a constant value during this period. For a higher

_{R}*n*, the value of

*ρ*is higher, which results in the higher value of the

_{O}*q*curve. The rise of the curve is owing to the diffuse radiation inside the slab. For a higher

_{R}*n*, based on the above analysis, the diffuse radiative energy inside the slab is lower, which results in the lower increasing rate of the

*q*curve. (ii) After the time of

_{R}*t*, a second peak value of

_{p}**q*appears for different

_{R}*n*. For a higher

*n*, the peak appears later, which is owing to the slower speed of light. (iii) For the higher

*n*, the diffuse radiative energy inside the slab is lower. However, the curve descends slower than those cases having lower

*n*after the second peak. It is owing to the fact that, the inner diffuse reflectivity is higher for a higher

*n*, which means that the energy could be kept longer inside the slab.

From Fig. 10(b) the following trends for the *q _{T}* may be observed. (i) As expected, the

*q*begins to appear just at

_{T}*t**=

*τ*. (ii) With a lower

_{L}n*n*, the peak of the curve is higher. For a lower

*n*, both the

*ρ*and

_{O}*ρ*are lower, and as a result of which, more pulsed energy is transmitted through the two interfaces of the slab. (ii) Similar to the curve of

_{I}*q*, with a higher

_{R}*n*, the

*q*curve decreases slower than those cases having lower

_{T}*n*after the peak.

## 4. Conclusion

The application of the LBM was extended to solve transient radiative transfer problem in a one-dimensional slab of participating medium subjected to collimated short pulse irradiation. Both the cases for refractive index matched (*n* = 1) or mismatched (*n*>1) semitransparent boundary were considered.

The accuracy and efficiency of this algorithm are studied. To show the flexibility of the LBM for different working conditions, we investigated the effects of the incident angle, scattering properties, pulse laser shapes and optical inhomogeneity on transmittance and reflectance signals. Results of the LBM were found to compare very well with those data from the published literatures. In brief, the extended LBM proposed in this paper is a simple and accurate solution scheme for the one-dimensional transient radiative transfer. Finally, the LBM was further extended to study the transient radiative transfer in homogeneous medium with a refractive index discontinuity irradiated by the short pulse laser. We observed and analyzed several interesting trends on the time-resolved signals different from those of the case considering refractive index matched boundary.

## Acknowledgments

This work was supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant No. 51121004) and the National Natural Science Foundation of China (Grant No. 51176040).

## References and links

**1. **A. Majumdar, “Microscale heat conduction in dielectric thin films,” J. Heat Transfer **115**(1), 7–16 (1993). [CrossRef]

**2. **J. Y. Murthy and S. R. Mathur, “Computation of sub-micron thermal transport using an unstructured finite volume method,” J. Heat Transfer **124**(6), 1176–1181 (2002). [CrossRef]

**3. **T. Q. Qiu and C. L. Tien, “Short-pulse laser heating on metals,” Int. J. Heat Mass Transfer **35**(3), 719–726 (1992). [CrossRef]

**4. **F. Liu, K. M. Yoo, and R. R. Alfano, “Ultrafast Laser-Pulse Transmission and Imaging Through Biological Tissues,” Appl. Opt. **32**(4), 554–558 (1993). [CrossRef] [PubMed]

**5. **M. C. van Gemert and A. J. Welch, “Clinical Use of Laser-Tissue Interactions,” IEEE Eng. Med. Biol. Mag. **8**(4), 10–13 (1989). [CrossRef] [PubMed]

**6. **K. J. Grant, J. A. Piper, D. J. Ramsay, and K. L. Williams, “Pulsed lasers in particle detection and sizing,” Appl. Opt. **32**(4), 416–417 (1993). [CrossRef] [PubMed]

**7. **S. Kumar and K. Mitra, “Microscale Aspects of Thermal Radiation and Laser Applications,” Adv. Heat Transfer **33**, 187–294 (1999). [CrossRef]

**8. **H. Schweiger, A. Oliva, M. Costa, and C. D. P. Segarra, “A Monte Carlo method for the simulation of transient radiation heat transfer: application to compound honeycomb transparent insulation,” Numer. Heat Transf. B **35**, 113–136 (2001).

**9. **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**(2-5), 159–168 (2002). [CrossRef]

**10. **X. D. Lu and P. F. Hsu, “Reverse Monte Carlo method for transient radiative transfer in participating media,” J. Heat Transfer **126**(4), 621–627 (2004). [CrossRef]

**11. **X. D. Lu and P. F. Hsu, “Reverse Monte Carlo simulations of light pulse propagation in nonhomogeneous media,” J. Quant. Spectrosc. Radiat. Transf. **93**(1-3), 349–367 (2005). [CrossRef]

**12. **M. Martinelli, A. Gardner, D. Cuccia, C. Hayakawa, J. Spanier, and V. Venugopalan, “Analysis of single Monte Carlo methods for prediction of reflectance from turbid media,” Opt. Express **19**(20), 19627–19642 (2011). [CrossRef] [PubMed]

**13. **Z. X. 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]

**14. **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**(2-5), 169–179 (2002). [CrossRef]

**15. **Z. X. Guo and S. Kumar, “Three-dimensional discrete ordinates method in transient radiative transfer,” J. Thermophys, Heat Transfer **16**, 289–296 (2002).

**16. **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**(19-20), 3799–3806 (2010). [CrossRef]

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

**18. **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]

**19. **P. F. Hsu, “Effects of multiple scattering and reflective boundary on the transient radiative transfer process,” Int. J. Therm. Sci. **40**(6), 539–549 (2001). [CrossRef]

**20. **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]

**21. **M. Y. Kim, S. Menon, and S. W. Baek, “On the transient radiative transfer in a one-dimensional planar medium subjected to radiative equilibrium,” Int. J. Heat Mass Transfer **53**(25-26), 5682–5691 (2010). [CrossRef]

**22. **L. M. Ruan, S. G. Wang, H. Qi, and D. L. Wang, “Analysis of the characteristics of time-resolved signals for transient radiative transfer in scattering participating media,” J. Quant. Spectrosc. Radiat. Transf. **111**(16), 2405–2414 (2010). [CrossRef]

**23. **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**(11-12), 1820–1832 (2006). [CrossRef]

**24. **L. H. Liu and L. J. Liu, “Discontinuous finite element approach for transient radiative transfer equation,” J. Heat Transfer **129**(8), 1069–1074 (2007). [CrossRef]

**25. **L. H. Liu and P. F. Hsu, “Time shift and superposition method for solving transient radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. **109**(7), 1297–1308 (2008). [CrossRef]

**26. **X. He, S. Chen, and R. A. Zhang, “Lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh–Taylor instability,” J. Comput. Phys. **152**(2), 642–663 (1999). [CrossRef]

**27. **S. Succi, *The Lattice Boltzmann Method for Fluid Dynamics and Beyond* (Oxford University, 2001).

**28. **W. S. Jiaung, J. R. Ho, and C. P. Kuo, “Lattice Boltzmann method for heat conduction problem with phase change,” Numer. Heat Transfer, Part B **39**, 167–187 (2001).

**29. **S. C. Mishra and A. Lankadasu, “Analysis of transient conduction and radiation heat transfer using the lattice Boltzmann method and the discrete transfer method,” Numer. Heat Transfer, Part A **47**, 935–954 (2005).

**30. **S. C. Mishra and H. K. Roy, “Solving transient conduction-radiation problems using the lattice Boltzmann method and the finite volume method,” J. Comput. Phys. **223**(1), 89–107 (2007). [CrossRef]

**31. **S. C. Mishra, T. B. Pavan Kumar, and B. Mondal, “Lattice Boltzmann method applied to the solution of energy equation of a radiation and non-Fourier heat conduction problem,” Numer. Heat Transfer, Part A **54**, 798–818 (2008).

**32. **B. Mondal and S. C. Mishra, “Simulation of natural convection in the presence of volumetric radiation using the lattice Boltzmann method,” Numer. Heat Transfer, Part A **55**, 18–41 (2009).

**33. **P. Asinari, S. C. Mishra, and R. Borchiellini, “A lattice Boltzmann formulation to the analysis of radiative heat transfer problems in a participating medium,” Numer. Heat Transfer, Part B **57**, 126–146 (2010).

**34. **A. F. D. Rienzo, P. Asinari, R. Borchiellini, and S. C. Mishra, “Improved angular discretization and error analysis of the lattice Boltzmann method for solving radiative heat transfer in a participating medium,” Int. J. Numer. Methods Heat Fluid Flow **21**(5), 640–662 (2011). [CrossRef]

**35. **Y. Ma, S. K. Dong, and H. P. Tan, “Lattice Boltzmann method for one-dimensional radiation transfer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **84**(1), 016704 (2011). [CrossRef] [PubMed]

**36. **H. Bindra and D. V. Patil, “Radiative or neutron transport modeling using a lattice Boltzmann equation framework,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **86**(1), 016706 (2012). [CrossRef] [PubMed]

**37. **S. C. Mishra and R. R. Vernekar, “Analysis of transport of collimated radiation in a participating media using the lattice Boltzmann method,” J. Quant. Spectrosc. Radiat. Transf. **113**(16), 2088–2099 (2012). [CrossRef]

**38. **R. Siegel, “Variable Refractive Index Effects on Radiation in Semitransparent Scattering Multilayered Regions,” J. Thermophys. Heat Transfer **7**, 624–630 (1993).