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Theoretical Development. The physical ... The P1 model is developed when we consider the ... where u is the average intensity over all angles and q is the heat flux: u x, t. 1. 4. 4 ...... general collocation software for partial differential equations.
Development and comparison of models for light-pulse transport through scattering–absorbing media Kunal Mitra and Sunil Kumar

We examine the transport of short light pulses through scattering–absorbing media through different approximate mathematical models. It is demonstrated that the predicted optical signal characteristics are significantly influenced by the various models considered, such as PN expansion, two-flux, and discrete ordinates. The effective propagation speed of the scattered radiation, the predicted magnitudes of the transmitted and backscattered fluxes, and the temporal shape and spread of the optical signals are functions of the models used to represent the intensity distributions. A computationally intensive direct numerical integration scheme that does not utilize approximations is also implemented for comparison. Results of some of the models asymptotically approach those of direct numerical simulation if the order of approximation is increased. In this study therefore we identify the importance of model selection in analyzing short-pulse laser applications such as optical tomography and remote sensing and highlight the parameters, such as wave speed, that must be examined before a model is adopted for analysis. © 1999 Optical Society of America OCIS codes: 030.5620, 110.7050, 140.7090, 170.6930, 290.4210.

1. Introduction

Rapid progress in technology is making it possible to create and observe thermophysical phenomena on increasingly shorter time scales. One such phenomenon of interest is the interaction of pulsed light with scattering and absorbing media. For example, short-pulse lasers ~pulse widths of the order of femtosecond to picosecond! can currently be created,1 and time-resolved measurements of the scattered signals induced by the interaction of these pulses with scattering medium can be used to infer medium properties.2 The study of transient radiative transport necessary for analyzing such phenomena had not been considered in depth in the literature except for a brief discussion,3 primarily because no applications existed to provide the impetus for related research. Only recently has the complete transient radiativetransfer equation been considered for pulsed light propagation,4 where the present authors have devel-

K. Mitra is with the Mechanical Engineering Program, Florida Institute of Technology, 150 West University Boulevard, Melbourne, Florida 32901. S. Kumar is with the Department of Mechanical Engineering, Polytechnic University, 6 Metrotech Center, Brooklyn, New York 11201. Received 16 March 1998; revised manuscript received 10 August 1998. 0003-6935y99y010188-09$15.00y0 © 1999 Optical Society of America 188

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oped a damped-wave hyperbolic model by considering the P1 model of the intensity distribution. In this paper several solution methodologies for evaluating the transient radiative-transfer equation are examined. Previously3,4 it was pointed out that the transient hyperbolic radiative-transfer equation obtained by invoking the P1 model yields an effective speed of propagation of the scattered radiation that is less than the speed of light. This lower propagation speed could have the potential drawback of predicting results that have a significant temporal mismatch with observed data. It was also shown that some physically unrealistic results are obtained at the initial stages of the interaction of the pulse with the medium.3 It is therefore imperative that various other methods be studied and their relative accuracies investigated to establish which of them could be used for modeling short-pulse applications. The results of the present study indicate that the magnitudes of the transient scattered and transmitted signals are functions of the solution methodology. This variation is to be expected because similar differences are also observed in the analyses of steadystate radiation transport.5 In addition, for the transient case it is also seen that the model selected has a significant impact on the effective propagation speed of the scattered radiation field and on the temporal shape and spread of the radiation signals. Different simplifications exhibit different effective speeds, most of which are significantly less that the

speed of light. This variation in propagation speeds therefore is a hurdle in time-resolved analysis and must be addressed. Increasing the order of approximations in most cases, where possible, leads to speeds of propagation that approach that of the speed of light and are thus more physically realistic. To isolate and highlight some of the effects associated with different models, a simplification is also considered where the laser is assumed to be completely scattered by the boundary and is treated as a diffuse radiation-boundary condition. This boundary-driven problem eliminates the parameters associated with the speed of propagation of the original laser pulse through the medium, and the only speed that influences the exhibited results is the effective propagation speed of the scattered radiation field. Applications where these transient methods are important are in probing the characteristics of the particulate medium by examining the transmitted or backscattered transients; evaluating the size distribution of particles6 and droplets in fuel-spray combustion and thermal plasma spraying; characterizing combustion products; imaging of biomedical systems for diagnostic purposes3,7; monitoring cryopreservation by tracking the freezingythawing of organs; monitoring of the oxygenation state of living tissues8; remote sensing of turbid media, ocean,9,10 and atmosphere; and many others.11 2. Theoretical Development

The physical case under consideration is an onedimensional scattering and absorbing medium of length L having azimuthal symmetry and constant properties. For simplicity the boundaries of the medium are considered to be nonreflecting and nonrefracting. This geometry is the simplest possible and therefore is chosen in order for one to examine the effects of various models with the least additional mathematical complexity. A pulsed radiation beam from an external source is incident on the x 5 0 face of the medium. As the beam enters and propagates through the medium it loses energy by absorption and scattering phenomena. The outscattered radiation from this pulse forms the source for the scattered radiation field I~x, m, t! that is created inside the medium. This source of radiation for the scattered field is depicted by S~ x, m, t! in the mathematical development below. The radiative-transfer equation for the scattered intensity field I~ x, m, t! in this geometry, assuming azimuthal symmetry and constant properties, is written as12–14 1 ]I~x, m, t! ]I~x, m, t! 1m 5 2se I~x, m, t! c ]t ]x ss 1 2

*

1

I~x, m9,t!p~m9 3 m!dm9 1 S~x,m,t!, (1)

21

where I is the scattered intensity; c is the speed of light in the medium; x is the Cartesian distance; t is

the time; s is the radiative coefficient; m is the cosine of u, where u is the polar angle measured from the positive x axis; and p is the scattering phase function. The above is an integrodifferential equation where the partial differentials correspond to a hyperbolic differential equation that yields wave solutions. The wave speed of the scattered radiation intensity I~m! along the positive x direction is cm. The phase function in the azimuthally symmetric case can be represented as a series of Legendre polynomials Pk by15 K

p~m9 3 m! 5

( a P ~m9! P ~m!, k

k

(2)

k

n50

where K is the order of anisotropy and ak are the coefficients in the expansion. The nature of the phase-function distribution depends strongly on the order of anisotropy used to represent the function. In the quasi-steady ~i.e., traditional! transient radiative-transport analysis, the first term on the left in Eq. ~1! is neglected because of the large value of c. The intensity I remains time dependent, but the time variation in the quasi-steady model is introduced only through the time-dependent boundary conditions or the time-dependent source. Scaling analyses that consider the time scales associated with the short-pulse laser applications of the kind considered in this study preclude the elimination of the transient term from the intensity equation, Eq. ~1!, and the full equation is used. The equation of transfer is complicated because of the integral on the right side corresponding to the inscattering gain term. To reduce the integral to a simpler form, different models have been presented in the literature. These models have so far been used for equations that do not contain the time derivative. Some of these models developed for the steady-state case in the literature are adapted here for the complete transient case. A.

P1 Model

The P1 model is developed when we consider the scattered intensity to be a linear function of m4: I~x, m, t! 5 u~x, t! 1

3 q~x, t!m, 4p

(3)

where u is the average intensity over all angles and q is the heat flux: u~x, t! 5

q~x, t! 5

1 4p

*

4p

*

4p

I~x, m, t!dV 5

1 2

I~x, m, t!mdV 5 2p

*

1

I~x, m, t!dm,

21

*

1

(4a) I~x, m, t!mdm,

21

(4b)

where V is the solid angle. The linear representation of the intensity from Eq. ~3! is then substituted into Eq. ~1!, and Eq. ~1! is ~a! integrated with respect 1 January 1999 y Vol. 38, No. 1 y APPLIED OPTICS

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dV and ~b! multiplied by m and then integrated with respect to dV. The definitions of u and q from Eq. ~4! are used to obtain the two resultant equations that are given by4

C.

Two-Flux Model

With the two-flux method we consider the scattered intensity to be constant over the forward- and the backward-facing hemispheres17:

4p ]u~x, t! ]q~x, t! 1 1 4psa u~x, t! c ]t ]x

*

5 2p

1

S~x, m, t!dm, (5a)

I~x, t, m! 5 I1~x, t!,

2p p , u , ~m . 0!, 2 2

(8a)

I~x, t, m! 5 I2~x, t!,

3p p ,u, ~m , 0!, 2 2

(8b)

21

1 ]q~x, t! 4p ]u~x, t! 1 1 se q~x, t! 2 ssp# q~x, t! c ]t 3 ]x 5 2p

*

1

S~x, m, t!mdm, (5b)

21

where I1 is the scattered intensity in the forwardfacing hemisphere and I2 is the intensity in the backward hemisphere. This model is used to simplify the integral in Eq. ~1!, the transient equation of transfer, which is then integrated separately over the backward- and the forward-facing hemispheres. The resultant equations are

where p# is a weighted average of the phase function4 whose value is a1y3, where a1 is the expansion coefficient in the representation of the phase function given by Eq. ~2!. The set of equations, Eqs. ~5a! and ~5b!, forms a hyperbolic set of equations. The wave nature of the equations and the wave speed can be easily identified if these equations are combined, which yields the following hyperbolic equation4: 3 ]2u ]2u 3 ]u 2 1 ~sa 1 se 2 ssp# ! 1 3~se 2 ssp# !sa u c2 ]t2 ]x2 c ]t 5 ~se 2 ssp# !

3 2

*

1

3 2

*

1

3 2

Sdm 2

21

*

1 ]S dm. (6) c ]t 21 1

B5

]u ] u 3 1 3~se 2 ssp# !sa u ~sa 1 se 2 ssp# ! 2 c ]t ]x2

*

1

21

*

(9a)

0

21

Sdm, (9b)

1 2

** 1

0

0

p~m9 3 m!dm9dm.

21

D.

PN Model

The general PN method models the intensity by expanding it as a series of Legendre polynomials of m as5,18 N

2

3 2

Sdm,

0

Equations ~9! for I1 and I2 are hyperbolic with a wave speed of cy2.

P1 Parabolic Approximation

5 ~se 2 ssp# !

*

1

where

]S mdm ]x 21

The parabolic approximation of the P1 method is obtained by neglecting the first term on the left-hand side of Eq. ~5b! compared with other terms of the equation. The procedure of obtaining the equation is the same as that used for deriving Eqs. ~5! and ~6!. This parabolic form of the equation is widely used in neutron transport16 and has now been adopted by researchers in laser optical tomography applications8 and is given by4

190

1 ]I2 1 ]I2 2 1 ~sa 1 ss B!I2 2 ss BI1 5 c ]t 2 ]x

1

Equation ~6! indicates that the propagation speed along the x direction of the resultant hyperbolic wave of u is cy=3. B.

1 ]I1 1 ]I1 1 1 ~sa 1 ss B!I1 2 ss BI2 5 c ]t 2 ]x

3 2

*

1

3 2

Sdm 2

]S mdm ]x 21 1

*

1 ]S dm. (7) c ]t 21 1

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I~x, m, t! 5

(I

m

~x, t! Pm~m!,

(10)

m50

which is then substituted into Eq. ~1!. Equation ~1! is subsequently multiplied by a Legendre polynomial Pk of order k less than or equal to N and integrated with respect to dm. Using the orthogonality of Legendre polynomials leads to Eq. ~11! for intensity coefficient Ik, 0 # k # N: 1 ]Ik k 1 1 ]Ik11 k ]Ik21 1 1 c ]t 2k 1 3 ]x 2k 2 1 ]x

S

1 se 2 ss

D

2k 1 1 ak Ik 5 2k 1 1 2

*

1

SPkdm. (11)

21

N 1 1 coupled hyperbolic equations are obtained, one for each k, 0 # k # N. The wave speed for the PN method is not obvious unless the equations are combined or solved numerically.

E.

Discrete-Ordinates Model

The method of discrete ordinates replaces the integral in Eq. ~1! by a quadrature, such as Gaussian, Lobatto, Chebyshev, or Fiveland.5,19,20 If the mi values are the quadrature points between the limits of integration, 21 to 1, corresponding to a 2M-order quadrature, and the wi values are the corresponding weights, the equation is reduced to the following system of coupled hyperbolic partial differential equations where the ith equation corresponds to the quadrature direction mi : 1 ]Ii ~x, t! ]Ii ~x, t! 1 mi 1 se Ii~x, t! c ]t ]x 5

ss M wj Ij~x, t!p~mj 3 mi ! 1 S~x, mi , t!, j Þ 0, 2 j52M

(

i 5 2M, . . . , 21, 1, . . . , M. (12) Here Ii ~x, t! 5 I~x, mi , t!. The order 2M of the quadrature contains an even number of points to avoid the value m 5 0. For simplicity the values of i between 2M and 21 correspond to the negative m and between 1 and M to the positive. The hyperbolic wave speed of Ii along the x direction corresponding to the discrete ordinate mi has the magnitude of the absolute value of mi c. F.

Direct Numerical Solution

The direct numerical integration solution is obtained without simplifying assumptions applied to the radiative-transfer equation given by Eq. ~1!. A different number of equally spaced angular nodes are used to directly evaluate the integral in the radiative transfer equation. G.

Fig. 1. Schematic of the boundary conditions for the scattered intensity field: ~a! Propagating pulse that decays spatially owing to absorption and scattering. Outscattering from the pulse is the source for the scattered intensity field in the medium. ~b! Boundary-driven problem.

component of the intensity Ic that has a value Iincident at the x 5 0 surface decays owing to the scattering and the absorption phenomena encountered in the medium. It is assumed that the incident pulse is directed along the x axis; i.e., it is incident perpendicular to the surface of the medium. Solving the corresponding radiative-transfer equation for the collimated component yields Ic~x, m, t! 5 Iincident exp~2se x!@H~t 2 xyc! 2 H~t 2 tp 2 xyc!#d~m 2 1!,

Quasi-steady Method

The quasi-steady ~i.e., traditional! transient method involves solving the radiative-transfer equation by entirely neglecting the time-derivative term from Eq. ~1!. The time dependency of the radiation intensity is introduced only through the source term andyor boundary conditions. This is in contrast with the other methods discussed above in which the temporal variation of the radiation intensity occurs through the transient term in the radiative-transfer equation as well as the temporal dependence of the source term and the boundary conditions. 3. Numerical Examples

The pulsed radiation incident on the medium ideally is a pulsed laser with a Gaussian temporal distribution. But for simplicity the incident intensity is assumed to be a square pulse ~i.e., of constant temporal magnitude! with a temporal duration, or pulse width, tp. The magnitude of the intensity of the square pulse before it enters the medium is Iincident. The intensity in the medium consists of a collimated component Ic corresponding to the pulsed radiation that originally entered the medium and a scattered intensity I that is described by Eq. ~1!. The collimated

(13)

where H~t! is the Heaviside step function and d is the Dirac delta function. The source function S in the radiative-transfer equation for the scattered intensity field, Eq. ~1!, is then given by S~x, m, t! 5 5

ss 2

*

1

Ic~x, m9, t!p~m9 3 m!dm9

(14a)

21

ss Iincident exp~2se x!@H~t 2 xyc! 2 2 H~t 2 tp 2 xyc!#p~1 3 m!.

(14b)

The boundary conditions are such that the scattered intensities pointing inward at x 5 0 and at x 5 L are zero @see Fig. 1~a!#, yielding I~ x 5 0, m . 0, t! 5 I~x 5 L, m , 0, t! 5 0.

(15)

These boundary conditions correspond to nonreflecting and nonrefracting boundaries. 4. Simplified Source

In the previous case of an external collimated radiation beam incident on the medium, the results are 1 January 1999 y Vol. 38, No. 1 y APPLIED OPTICS

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influenced by parameters related to the source temporal pulse width, the propagation speed of the incident source in the medium, and the rate of decay of the incident source inside the medium as it traverses the medium. To reduce the number of parameters and to isolate the effects of different models on the evolution of the scattered intensity field, we consider a boundary-driven case. By considering only a step change in the boundary ~or two sequential step changes corresponding to on and off conditions! and examining the temporal and the spatial evolution of the scattered intensity field I, we can obtain insights into the different methods. In this case the radiation is introduced as a diffuse boundary condition of the medium at the x 5 0 boundary @see Fig. 1~b!#. The source term S in Eq. ~1! and in all the models is dropped or set to zero. The corresponding boundary conditions are given by I~x 5 L, m , 0, t! 5 0,

(16)

I~x 5 0, m . 0, t! 5 Ib H~t!, step input,

(17a)

I~ x 5 0, m . 0, t! 5 Ib@H~t! 2 H~t 2 tp!#, on– off input, (17b) where Ib~5Iincidenty2p! is the directionally uniform boundary intensity. The time period over which the boundary condition is applied is tp in Eq. ~17b!. The boundary-driven cases are considered to isolate the effects of propagation speed and magnitude of the scattered radiation field without the partially masking effect of the parameters associated with the original radiation pulse propagating through the medium. By considering only step changes in the boundary conditions and examining the temporal evolution of the scattered radiation field in response to these step changes, we obtained insights into the different methods more easily. The step-change boundary condition, Eq. ~17a!, also facilitates the evaluation of the numerical accuracy of the models since at large times the solutions must asymptote to steady-state results that can be computed by a variety of methods. 5. Numerical Results

The different models considered, namely, two-flux, PN, and discrete ordinates, and the quasi-steady ~i.e., traditional! transient method are solved numerically to evaluate the transient radiative transfer. The numerical solutions of the equations for P1 and the two-flux method are obtained by the method of characteristics,4 and that of the P3, P5, and discreteordinates methods are obtained by using the subroutine PDECOL21 and DPDES or DMOLCH from International Mathematical and Statistical Libraries ~IMSL!. The quasi-steady ~i.e., traditional! transient method is solved with the routine DVCPR or DBVPFD from IMSL libraries. The P1 parabolic approximation is solved by the fully implicit finitedifference technique. For all of these cases the grid sizes for both time and space variables are varied 192

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Fig. 2. Transmittance through a medium having D 5 10.0, v 5 0.998, tp 5 1 ps as a function of time for hyperbolic P1, P3, P5, two-flux, and 12-quadrature-discrete-ordinates models.

within a range of 2 orders of magnitude, and the results are found to be stable and converging. Although the P1 method and its parabolic approximation have been considered in great depth previously,4 some of the results are repeated here for comparison with the other models developed in the study. The radiation properties used in the examples considered in this paper correspond to highly scattering media such as biological material in optical tomography applications. A forward-peaked, 16-term, phase function is used. The coefficients of this phase function are not presented here for brevity and can be found elsewhere in the literature.5,20 In addition, the integral inscattering term in Eq. ~1!, which is the main reason why different models are needed, is rendered more prominent when the scattering albedo ~defined to be equal to the ratio of the scattering coefficient to the extinction coefficient! is high, and thus this selection of properties highlights the differences between the various methods. The highscattering albedo cases are more difficult to evaluate numerically and are likely to have convergence problems. Therefore, if a method converges for such properties, there is a very high likelihood that it will also converge for all others. The results are presented in terms of nondimensional transmissivities and reflectivities. These terms are defined as the ratios of transmitted or reflected fluxes to the incident-radiation flux. The differences in the transmitted and the backscattered fluxes between different models are discussed first. Keeping the scattering albedo v~5ssy se! constant at 0.998 and pulse width tp at 1 ps, we consider two different optical depths D~5se L!, 10.0 and 30.0, in Figs. 2 and 3, respectively. For optical depths D of 10.0 and 30.0 the P1, P3, P5, two-flux, and discrete-ordinates methods where 12 Gaussian quadrature points ~i.e., 2M 5 12! are used show transmittance results that are quite similar at large times. But at earlier time periods each model predicts a different temporal shape and magnitude of the transmitted flux or transmissivity. The propagation wave speed of the scattered radiation of the 12 quadrature discrete ordinates is the highest, and therefore it achieves the peak value of the transmit-

Fig. 3. Transmittance through a medium having D 5 30.0, v 5 0.998, tp 5 1 ps as a function of time for hyperbolic P1, P3, P5, two-flux, 12-quadrature-discrete-ordinates models, and parabolic and quasi-steady methods.

tance value first. The wave speed of the two-flux method is the lowest, and therefore the peak is reached after a large value of time. The effective signal propagation speed of the scattered radiation of the P1 model is cy=3 and that of the two-flux method is cy2. The temporal variations in the transmittance results reflect these differences in speeds. As seen in Figs. 2 and 3 the values of all transmitted signals are zero for all models until the value of time reaches that corresponding to the time taken for the exponentially decaying original source pulse to traverse the medium at speed c, the speed of light in the medium. The scattered radiation traverses the medium at a slower pace, depending on the model selected, and therefore the earliest arriving photons correspond to those from the original laser pulse. This corresponds to a nondimensional time ctse equal to the optical thickness se L. After this instant the transmittance due to the scattered radiation rises rapidly for the P3, P5, and discrete-ordinates models and slowly for P1 and the two-flux models because of the corresponding relative effective speeds of propagation of the scattered fields. Figure 3 also shows the transmitted flux distribution for P1 parabolic and quasi-steady ~i.e., traditional! formulations. The results obtained by these methods show a significant difference in the nature of the transmitted flux. Because transmissivity values are predicted at times shorter than the time taken for the original laser to traverse the medium, the parabolic model predicts a speed of propagation of the transmitted signal that is greater than the speed of light. The parabolic model has also been compared with the experimental results in the literature, and the physically unrealistic propagation speed predicted by the model has been highlighted.22 The transmissivity obtained by quasi-steady ~i.e., traditional! transient analysis dies out instantaneously as the source pulse exits the medium, in contrast with the predictions of the models developed where transmittance values are observed for long times, even after the source pulse is shut off, and this helps in providing additional information about the nature of the optical properties of the medium. The temporal

Fig. 4. Reflectivity through a medium having D 5 30.0, v 5 0.998, tp 5 1 ps as a function of time for hyperbolic P1, P3, P5, two-flux, and 12-quadrature-discrete-ordinates models.

shape of the experimental signal has been shown to change with optical properties.7 In Fig. 4 the backscattered flux at the x 5 0 surface is plotted for the same parameters as in Fig. 3 in the form of reflectivity of the medium. The P1 model gives an unrealistic negative value at small times and is clearly not an appropriate model. Similar negative values have been reported in the literature for backscattering from a slab of scatterers when a P1 model is used.3 At large times the P1, P3, P5, twoflux, and discrete-ordinates results match one another, but as in Figs. 1–3 the shapes and spreads at earlier time periods vary with the model selected. 6. Simplified Source Numerical Results

As seen in Figs. 2 and 3 the earliest-arriving photons in the predicted transmittances correspond to those scattered from the original pulse as it reaches the boundary where the transmittance is measured. To isolate and highlight the effects of the differences in the effective propagation speed of the scattered radiation, the source of radiation is considered to be a diffuse boundary condition as discussed above. For this boundary-driven problem a boundary step input is first considered. The initial characteristics of the increase of the transmission signals from zero values when the first photons reach the detector are directly related to the propagation speed of the scattered field. Therefore the temporal examination of the earlierarriving photons can be used to measure the propagation speeds. The steady-state value attained is obtained by letting the computations proceed to large times until the difference in magnitude of nondimensional backscattered and transmitted flux between two consecutive time steps is less than 1027. These large time values are also checked by solving a steady-state problem ~no time derivatives in the equation! and comparing the two. The results of the two are found to match, instilling confidence in the numerical results. In Fig. 5 we depict the transmission results for different optical depths, keeping the albedo constant and considering the step-change boundary condition. The higher the optical depth of the medium, the 1 January 1999 y Vol. 38, No. 1 y APPLIED OPTICS

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Fig. 5. Transmittance as a function of time for different values of optical depth for 12 discrete ordinates. The albedo is kept constant for a boundary-driven problem.

smaller is the magnitude of transmission. This is expected since the radiation intensities travel across large distances for higher optical depths and therefore lose more energy by absorption and scattering. The results also indicate the time when the first significant transmission signal through the medium is observed. The value of this time is directly proportional to the optical depth of the medium. Figure 6 highlights the two main differences that arise from different models, namely, differences in magnitude of signal and differences in effective speed of propagation of the scattered radiation signal. The figure shows the transmitted results for the different methods used: P1, P3, P5, 12 discrete ordinates, and two-flux for optical depth D of 30.0 and scattering albedo v equal to 0.998. The P1 and the two-flux results yield a significantly different speed for the earliest-arriving signal ~cy=3 for P1 and cy2 for the two-flux method!. The magnitude of the transmitted flux obtained by the discrete-ordinate, P3, and P5 methods are very close to one another. The P1 parabolic model result is also shown in the figure to highlight the physically unrealistic magnitude of the transmitted signal being observed even before any optical signal could have traversed depth D of the medium under consideration with the speed of light. We also evaluated the transient radiative-transfer equation by direct numerical integration of the inte-

Fig. 7. Comparison of transmittance between 12 discrete ordinates and the numerical integration method by using a different number of nodes for a medium having D 5 1.0, v 5 0.998 for a boundary-driven problem.

gral appearing in Eq. ~1! by using a different number of equally spaced angular nodes. Figure 7 shows a comparison of transmittance for 16 discrete ordinates and a different number of angular nodes evaluated numerically with optical depth D of 1.0 and albedo v of 0.9. As the number of angular nodes increases, the numerically evaluated transmission results approach the results of the method of 16 discrete ordinates. Table 1 highlights the difference in effective propagation speed predicted by different methods. The theoretical values are obtained from the wave-speed value, except for P3 and P5, which cannot be evaluated analytically. The numerical values are evaluated based on the observation of the first significant deviation of the transmission value through the medium from zero to a value larger than the value of 10210. Then the corresponding ratios are calculated and tabulated in Table 1. It can be seen from Table 1 that as the number of discrete ordinates increases the effective speed becomes closer in value to the physically expected value, which is the actual speed c of light. For all the above cases considered in this section the boundary source is a step change at time t 5 0. Previous figures for the step-change boundary condition show the characteristics of the rise in the radiation signals due to the boundary source being turned on, Eq. ~17a!. To examine the signal falloff after the Table 1. Ratio of Predicted Effective Propagation Speed of the Scattered Flux to Actual Speed of Light

Fig. 6. Transmittance through a medium having D 5 30.0, v 5 0.998, as a function of nondimensional time for hyperbolic P1, P3, P5, two-flux, and 12-quadrature-discrete-ordinates models for the case of a boundary-driven problem. 194

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Method

Theoretical

Numerically Evaluated

P1 P3 P5 Two-flux 6 Discrete ordinates 8 Discrete ordinates 12 Discrete ordinates 16 Discrete ordinates Direct numerical integration

0.577350 — — 0.500000 0.932469 0.960290 0.981561 0.989401 1.000000

0.577928 0.862069 0.909090 0.501253 0.945180 0.972762 0.994035 1.000000 1.000000

Fig. 8. Transmittance through a medium having D 5 30.0, v 5 0.998 for a boundary source on for 1 ps for hyperbolic, parabolic, and a quasi-steady P1 approximation.

The speed of propagation of the signal as formulated by the different models is a mathematical resultant, or effective, speed that one obtains by averaging over different lines of sight oriented at different angles from the primary direction of propagation. In the discrete-ordinate model it is the effective speed along the primary direction of the signal traveling along a line of sight corresponding to the discrete ordinate. To ensure that the wave front of the signal travels with a speed c along the primary direction of propagation, the discrete ordinates must include the forward and the backward directions specified by m 5 61. This inclusion is possible if Lobatto quadrature is used since it includes these values of m.20 8. Conclusions

boundary source is shut off, we consider another case in which the boundary source is on for a finite time tp only, Eq. ~17b!. To analyze the difference in the transmittance once the source is shut off, we plotted the transmittance for the P1 method, its parabolic approximation, and the quasi-steady ~i.e., traditional! transient method by using the P1 expansion of the intensity in Fig. 8 for the case in which the source is shut off after 1 ps. The transmittance for the P1 quasi-steady case drops to zero as soon as the source is shut off, whereas the transient P1 and its parabolic approximation predict finite values of transmission for long time periods even after the source is shut off. Figures 8 and 3 can also be used to compare the two methods of modeling an incident laser pulse, either as a boundary condition or as an exponentially decaying source in the medium. The magnitude of transmittance is significantly smaller for the boundary source model because the entire source energy is located at the incident boundary, which is farthest from the detection boundary. When the laser pulse is considered to propagate through the medium and the outscattering from the pulse serves as the source, the magnitude of the transmitted flux is higher. Similarly, the peak of the signal occurs slightly later for the boundary source compared with the propagating source. 7. Discussion

The formulations corresponding to the different models listed above exhibit very different radiationtransport characteristics. The speed of propagation of the signal varies depending on the model selected. For the two-flux model the speed is cy2, for P1 it is cy=3, and that of the discrete ordinates is cmM. The speed of propagation for P3 and P5 can be obtained only if the equations are solved numerically. None of these speeds is equal to c, which is the speed of propagation of radiation through the medium. Thus, if the boundary condition is changed from one state to the next, the change will propagate into the medium at different speeds depending on the approximate model selected, and the speed predicted by the model will not necessarily be equal to the actual speed c.

To the best of the authors’ knowledge, the present study is the first in the literature to examine transient radiative transfer comprehensively through different models. In the study we identify parameters of importance in short-pulse radiation transport that have not been anticipated in the literature. New and previously unexplored consequences of the implementation of different models are highlighted, such as the different effective propagation speeds that span a wide range of values, most of them significantly less than the physically acceptable value, i.e., the speed of light. The dramatic differences in the predicted temporal shapes of the transient radiative field, from sharply peaked features to the gradual variations based on the model selected, have been revealed through this study. Thus the temporal shape and the spread of some of the predicted signals can be significantly different from the experimental values. If experimentally measured temporal signals are used to ascertain the characteristics of the scattering–absorbing media, model selection would play an important role since it would affect the results significantly. The study is of great significance in applications of short-pulse lasers in time-resolved optical tomography, remote sensing, and others. The physical case considered mimics the interaction of laser pulses with scattering–absorbing media, albeit by consideration of a simple slab geometry and a temporally square plane-wave pulse. The simple geometry and square pulse serve to reduce the input parameters and facilitate the isolation of the effects of the different models. Further enhancements are needed for one to consider more complicated geometries, but the model development for such cases can benefit from the insights developed through this study. Appendix A: Nomenclature ak B c D H I

terms in expansion of phase function, integrated phase function, speed of light in the medium, nondimensional optical thickness ~5se L!, Heaviside step function, scattered radiation intensity, 1 January 1999 y Vol. 38, No. 1 y APPLIED OPTICS

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Ic Ib K L M p Pk q S t tp u w x m u sa se ss v V

collimated component of intensity, directionally uniform boundary intensity, degree of anisotropy, thickness of the medium, half of quadrature, scattering phase function, Legendre polynomials, radiative flux, source, time, pulse width, intensity averaged over all directions, weights of quadrature, Cartesian coordinate, cos u, polar angle measured from the positive x axis, absorption coefficient, extinction coefficient ~5sa 1 ss!, scattering coefficient, scattering albedo ~5ssyse!, solid angle.

The authors thank N. K. Madsen of Lawrence Livermore Laboratory for providing the PDECOL subroutine. Partial support from a grant ~AW 9963! from the Sandia National Laboratory, N. Mex., is acknowledged.

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