Journal of Quantitative Spectroscopy &
Radiative Transfer 73 (2002) 159–168 www.elsevier.com/locate/jqsrt
a
b
b
c ; ∗ a Department of Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey,
98 Brett Road, Piscataway, NJ 08854, USA b Department of Chemical Engineering and Chemistry, Polytechnic University, 6 Metrotech Center,
Brooklyn, NY 11201, USA c Department of Mechanical, Aerospace and Manufacturing Engineering, Polytechnic University,
6 Metrotech Center, Brooklyn, NY 11201, USA
Received 27 August 2001
Abstract
A three-dimensional Monte Carlo simulation of transient radiative transfer is performed for short pulse laser transport in scattering and absorbing media. Experimental results of a 60 ps pulse laser transmission in scattering media are presented and compared with simulation. Good agreement between the Monte Carlo simulation and experimental measurement is found. The refractive index of the scattering particles is found to in8uence strongly the prediction of transmitted pulse shape. The temporal shape of the transmittance is very weakly in8uenced by the output detector angle for a di:use medium. Scaled isotropic scattering modeling is shown to be insu;cient in transient three-dimensional radiative transfer for early times.
?
2002 Elsevier
Science Ltd. All rights reserved.
Keywords: Transient radiative transfer; Ultrafast lasers; Light scattering; Monte Carlo simulation; Experimental measurement
1. Introduction
The study of short pulse laser radiation transfer in a strongly scattering medium has received increasing attention during the past few years, partly as a result of its wide applications to medical diagnosis and thermal therapy [1,2], remote sensing [3], and laser material processing of microstructures [1,4]. With laser pulse widths in the femtosecond to picosecond range, the time-dependence of radiative transport due to the large but Cnite speed of radiation propagation must be incorporated.
∗ Corresponding author. Tel.: +1-718-260-3810; fax: +1-718-260-3532.
E-mail address: skumar@poly.edu (S. Kumar).
0022-4073/02/$ - see front matter ?
2002 Elsevier Science Ltd. All rights reserved.
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160 Z. Guo et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 159–168
The propagation of light in optically dense random media is characterized by multiple scattering, so that the di:usion approximation is often used to describe the transient radiation transport [5].
However, recent studies have demonstrated that the di:usion approximation fails to fully describe backscattering = re8ection and transmission characteristics. These drawbacks have been investigated both by analytical treatments and experimental studies [6,7].
Very few studies have addressed transient radiative transfer by considering the fully transient radiative transfer equation. Kumar et al. [8] and Kumar and Mitra [9] were among the Crst to consider the entire transient radiative equation by using a variety of models for short-pulse applications.
Previously, the transient radiative transfer equation with a source of constant strength at the boundaries had been solved by Rackmil and Buckius [10] using Laplace transform and adding–doubling methods. Recently, Mitra et al. [11] solved the wave-like transient radiative transfer equation using the P
1 approximation for a boundary-driven two-dimensional case. Guo and Kumar [12] evaluated the scaled isotropic scattering results with the anisotropic scattering in transient radiative transfer in a one-dimensional medium. Tan and Hsu [13] developed an integral equation formulation for transient radiative transfer. The radiation element method was extended to transient radiative transfer in plane-parallel inhomogeneous media [14]. The traditional discrete-ordinates solution was extended to the study of short-pulsed laser transport in two-dimensional turbid media [15].
The Monte Carlo (MC) method has been used extensively in steady-state simulations of radiative heat transfer [16]. Many researchers have also employed the MC method to address transient laser-tissue interactions [17,18]. However, an instantaneous, isotropic and point source was generally used to simulate the incident laser beam, and the system was generally restricted to a one-dimensional slab. Recently, Guo et al. [19] studied the characteristics of multi-dimensional transient radiative transfer using the MC method. They found that the MC method is very suitable for the study of pulse laser radiation transport in highly scattering media because of its ability to handle realistic physical conditions and its relatively inexpensive computation cost since only the incident laser photons are to be traced, even though the MC method is subject to statistical errors.
In the present study, a MC approach is developed to simulate transient radiative transport of short laser pulses incident upon three-dimensional highly scattering media. Realistic physical models, such as the temporal and spatial Gaussian distributions of the laser beam, the Fresnel re8ection at the media = air boundaries, the position and angle of the detector, and the anisotropic scattering of the media, are incorporated. The sensitivity and accuracy of the MC method are examined. The MC method is used to simulate a 60 ps laser pulse transport through a sample block containing scattering silica micro-spheres. The sample has optical scattering properties similar to biological tissue. The predicted temporal transmittance is compared with the experimental measurements. The in8uence of the refractive index of the scattering particles is discussed. The isotropic scaling law is examined in the transient domain for radiative transfer in a three-dimensional highly forward anisotropic scattering medium.
2. Monte Carlo formulation
The pulse laser source is normally incident upon the participating medium at the origin of the
Cartesian coordinates. The incident laser intensity is axisymmetric and has temporal and spatial

Z. Guo et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 159–168 161
Gaussian proCles expressed by
I c
( x; y; z = 0 ; t ) = I
0 exp − 4 ln 2 × t t p
− 2
2
× exp − r 2
2
; (1) where the pulse width t p
, is full-width at half-maximum (FWHM), r 2 = x 2 + y 2 , and 2 = 2 is the spatial variance of the Gaussian incident beam. The Cartesian coordinates are represented by x; y; z , where z is the direction of propagation of the incident beam. In the present study, the input pulse is simulated at the time range t ⊆ [0 ; 4 t p
] for Gaussian temporal proCle.
The Henyey–Greenstein scattering phase function is a reasonable approximation in highly scattering media [12,17,18], which is expressed as
( ) =
1 − g 2
(1 + g 2 − 2 g ) 3 = 2
; (2) where = cos( ) ; is the polar angle, and g is a dimensionless asymmetry factor that can vary from − 1 to 1.
Fresnel’s re8ection is taken into account at the medium = air boundaries, where Snell’s law is obeyed: n s sin i
= sin r
; (3) in which the incident angle is n s i
, and refraction angle is
¿ 1, total re8ection occurs at the boundaries when i r
. Since the refractive index of the medium
¿ sin − 1 (1 =n s
). When i
¡ sin − 1 re8ectance between the random walk medium and air is given by Fresnel’s equation as
(1 =n s
), the
=
1
2 tan tan
2
2 (
( i i
−
+ r r
)
)
+ sin 2 sin 2
(
( i i
−
+ r
) r
)
: (4)
For the Monte Carlo simulation the position ( x; y; ) of the incidence of the laser photon bundles on the z = 0 interface is selected as r = ln[1 − R (1 − exp( − r i
2 = 2 ))] − 1 ;
= 2 R;
(5)
(6) x = r cos ; y = r sin ; (7) in which r i is the beam radius of the incident laser, and is the circumferential angle measured from the x -axis, see Fig. 1. In the present study, R is the random number generated by a computer using random seeds.
The distance that any one bundle may travel before absorption or scattering is selected as
D = [ln(1 =R )] =; (8) where is the extinction coe;cient of the medium. The fraction of energy deposited in the medium by absorption is 1 − !
, where !
is the scattering albedo. For the temporal analysis, the total optical

162 Z. Guo et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 159–168
Fig. 1. The local coordinates for anisotropic scattering.
path length of each ray inside the medium is converted to time of 8ight by using the speed of light in the medium t = D=c = D × n s
=c
0
:
The direction of light propagation after a scattering event is determined by
= 2 R;
= cos − 1
1 + g 2 − [(1 − g 2 )
2 g
= (1 + g − 2 gR )] 2
;
(9)
(10) where the polar angle and the circumferential angle s is measured from an axis pointing in the incoming direction of the radiation is measured in a plane normal to the incoming direction. The new after anisotropic scattering must then be found by introducing a local system at the point of scattering, with ˆ pointing into its z -direction as shown in Fig. 1. The local x - and y -directions are given by e ˆ
1 where ˆ
= ˆ × s ˆ ; e ˆ
2
= − sin
= ˆ i ˆ + cos
× j ˆ e ˆ
1
;
. The new direction vector is expressed as
(11) s ˆ = sin (cos e ˆ
1
+ sin e ˆ
2
) + cos s ˆ : (12)
Other details about the implementation of the transient Monte Carlo model and the numerical procedure have been described in recent publications [12,19].
3. Experiment description
The experimental setup is shown in Fig. 2. The apparatus is composed of three primary parts: a short pulse laser source unit, a sample cell, and an ultrafast detecting unit. Laser pulses of 60 ps at a repetition rate of 76 MHz were generated from frequency-doubled Nd 3+ : YAG mode-locked laser. The beam diameter is 1 : 3 mm. A beamsplitter diverts 10% of the 532 nm, 60 ps, 76 MHz laser pulse train to a power meter to monitor the intensity of the incident pulse. The stationary

Z. Guo et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 159–168 163
Fig. 2. Experimental setup.
sample is mounted in the center of a rotary stage, with an optical Cber mount attached to the rotary platen. The scattered light is collected by an approximately 100 m optical Cber whose angular position relative to the incident laser beam’s axis can be accurately varied by rotating the mount, enabling monitoring of the angular distribution, intensity and temporal proCle of the scattered light by a Hamamatsu OOS-01 optical oscilloscope. The halfwave plate = polarizer combination controls both polarization and the incident power on the sample. An adjustable mask with a circular aperture placed on the output face of the sample limits the area whose scattered output is monitored.
The sample is a three-dimensional block and is made to model the optical properties of artiCcial biological tissues. The samples are made by dispersing amorphous silica micro-spheres of very regular shape, diameter 1000 nm ± 10%, in styrene, which is then polymerized. These cast solids, typically about 0.81% v = v-loaded in the present study, have the advantage of 8exibility of shape, minimal number of interfaces to account for in modeling, and lack of convective motion from localized heating due to absorption. Absorption is negligible in the polymer matrix for visible wavelengths, and can be adjusted by mixing dyes into the styrene before polymerization.
4. Results and discussion
At Crst, we examine the MC method in short pulse transport through a cube composed of a weakly absorbing and highly scattering medium. The side length of the cube is 10 mm. The medium properties are assumed to be n properties are t p
= 60 ps, and r i s
= 1 : 59 ;
= 1 : 3 mm.
= 10 mm − 1 , !
= 0 : 998, and g = 0 : 9. The incident pulse
Fig. 3 shows the temporal proCles of normalized transmittances at the center of the cube at the output surface, measured at di:erent angles from the z -axis. Various transmitted angles are selected from = 0 ◦ to 80 ◦ with increments of 10 ◦ . The hemispherical transmittance is also plotted for comparison. It is seen that the temporal shapes of the transmitted pulse at di:erent detector angles are very similar, so that a hemispherical transmittance shape could be used to represent the transmittance

164 Z. Guo et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 159–168
1
0.8
0.6
60 ps pulse passing a cube with
β
L = 100
ω
= 0.998
g = 0.90
0.4
0.2
0
100 200
Dotted or dashed lines represents detectors at azimuthal angles varying from 0, 10, 20, until 80 deg with increment = 10 deg
Hemispherical
300
Time (ps)
400 500 600
Fig. 3. Temporal proCles of normalized transmittance detected at various output angles.
4 10
-6
3 10
-6
2 10
-6
1 10
-6
0
0
20
L = 100,
ω
= 0.998, g = 0.90
From 30 Independent Runs
Ray Number per Run, nray = 10
6
15
10 nray = 3 X 10
6
5
400 600
Time (ps)
800
0
1000
Fig. 4. Numerical uncertainty of the mean hemispherical transmittance.
200 detected at speciCc angle. However, it should be noted that such a conclusion is only valid in the case of an optically thick medium, which is usually encountered in bioengineering applications [2].
The MC result is a sample mean calculated from 30 independent runs in the above example. The statistical errors (standard deviation of the mean) and relative statistical errors (standard deviation over mean) in hemispherical transmittance are drawn in Fig. 4. Two di:erent ray numbers are selected in each run for comparison, i.e., nray = 10 6 , and 3 × 10 6 , respectively. The proCle of the absolute statistical error follows the proCle of the transmittance. In other words, the absolute statistical error is smaller when the corresponding transmittance is smaller. However, the trend of relative statistical error is the reverse, i.e., the relative statistical error is smaller when the corresponding transmittance is larger. With the increase of the ray number in each run, the statistical errors become smaller.
Theoretically, the statistical error could be driven to near zero, but this would increase dramatically the computation time. The computation time is linearly proportional to the value of the ray number.
It is seen that, the statistical errors in the case of nray = 3 case of nray = 10 6
× 10 6 are generally smaller than the
. However, the rate of improvement in precision is less than the rate of increase in computational time. Thus, there should be a compromise in the trade-o: between precision and computation cost. In the present study, we mainly focus on the prediction of the transmitted pulse shape. From Fig. 4, it is seen that the relative uncertainty of the temporal shape in most ranges is less than 5%.
Fig. 5 shows the temporal proCles of relative transmittance for input pulses with various pulse widths. Other parameters are the same as aforementioned. The input pulses have a Gaussian temporal proCle as described in Eq. (1). The characteristic system radiation propagation time (= character length L= light speed c in the medium) for the cube is about 53 ps. The importance of transient e:ects in short pulse laser radiation transfer is clearly demonstrated when the input pulse width is smaller than the radiation propagation time in the medium, i.e., the output pulse width is signiCcantly enlarged. In the case of input pulse width t p
= 1 ps, the transmitted pulse has a fast response at early times, followed by a long decaying tail. The transmitted pulse width is at least two orders of magnitude larger than the original incident laser pulse width. With the increase of input pulse

Z. Guo et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 159–168
Transmitted pulses through a cube system propagation time = 53 ps with various input pulse widths t p
= 1, 10, 60, 100, 500 ps
10
60
100 t p
= 500
1
0.8
0.6
Experiment Measurement
Laser: 60 ps pulse, 532 nm, 1.45 W
Fiber: 100 micron, 3 cm distance
Sample size: 49 X 47.3 X 14.4
particles: silica, d = 1000 nm
0.81% vol. load detector at 0 deg detector at 30 deg
165
1
0.4
0.2
0 500 1000
Time (ps)
1500 2000
0
0
MC, R.I. = 1.417
MC, R.I. = 1.458
500
Time (ps)
1000 1500
Fig. 5. The broadening of transmitted pulse with di:erent input pulse widths.
Fig. 6. Experimental measurements and comparison with MC simulations for a 60 ps laser pulse passing through a purely scattering medium of dimensions
49 mm × 47 : 3 mm × 14 : 4 mm.
width to the same order of the system propagation time, the transmitted pulse shape does not change perceptibly, except that the initial rise in the transmission signal becomes slower. Even in the case of t p
= 100 ps, i.e., the input pulse width is about two times larger than the system propagation time, the broadening of the transmitted pulse is still obvious. However, when the input pulse width is signiCcantly larger than the system propagation time, e.g., t p
= 500 ps, the broadening of the transmitted pulse is very slight. From the viewpoint of imaging, the consequences are that shorter laser pulse widths are desirable. When the pulse width is very short the transmittance signal shape is unequivocally due to the scattering properties of the medium and re8ects the meandering paths that the photons have to take due to local scattering properties before they can exit the medium. For larger pulse widths the transmittance signal shape is due both to the input pulse and the scattering properties. Thus, the extraction of medium properties from the transmitted results will be more di;cult. For example, as an extreme case, if the incident pulse width is 1 min, the transmitted pulse width will also be virtually 1 min wide, irrespective of the scattering properties of the medium.
In Fig. 6, experimental results are presented and compared with MC simulations. The thickness of the sample is 14 : 4 mm. The two other dimensions are 49 and 47 : 3 mm. Measurements were performed at detector angles of 0 detectors at 0 ◦ and 30 ◦
◦ , 30 ◦ , 50 ◦ , and 70 ◦ , respectively. The results are shown only for
. The di:erence of the normalized temporal transmittance between di:erent angles is slight compared with the measurement uncertainty. In the experiment, we found that the uncertainty becomes larger with the increase of the detector angle. This is because the magnitude of the transmitted intensity decreases. We found that for all detector angles, there exists a plateau in the temporal transmittance in the time period of 1000 ps ¡ t ¡ 1100 ps. In several trials, this phenomenon did not disappear. It might be caused by the re-entry of backscattered pulse, but is most likely due to internal re8ection in the optical oscilloscope between the slit surface and the sampling streak tube.

166 Z. Guo et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 159–168
The bulk refractive index (RI) of the silica spheres is 1.458 at the wavelength of 589 nm [20], while a value of 1.417 was reported at the wavelength of 550 nm for individual silica beads by Firbank et al. [21]. When RI = 1 : 417, the scattering coe;cient and anisotropy of the sample are predicted by the Mie theory to be 17 : 052 mm − 1 and 0.96276, respectively. When RI = 1 : 458, the scattering coe;cient and anisotropy of the sample are 11 : 309 mm − 1 and 0.96649, respectively.
The scattering properties of the sample are strongly in8uenced by the refractive index of the scattering particles. Such a di:erence also a:ects the MC simulation. In Fig. 6, it is seen that the predicted temporal proCle of transmittance closely matches the experimental measurement in the case of RI = 1 : 417. In the case of RI = 1 : 458, the MC predicted transmitted pulse width is obviously narrower than that of the experimental measurement. Thus, RI = 1 : 417 is taken to be the best value for the individual silica micro-beads.
The slight mismatch between prediction and measurement in the range of 500 ¡ t ¡ 1000 ps can be explained by the di:erent detector conCguration used in experiment and simulation. The detector in the experiments is located 3 mm away from the output surface and its diameter is 100 m, and that in the simulation is located adjacent to the output surface with a diameter of 1500 m. The signal in the simulation is the hemispherical output, i.e., the aggregate 8ux of photons emerging in all angular directions from the output surface of 1500 m diameter adjacent to the detector. However, since the normalized values are being considered, the hemispherical output does not contribute to any variation in the shape of the normalized curve vs. the output from a detector located at speciCed angles (as discussed in previous paragraphs and in Fig. 3). In the experiment the cone of acceptance of the detector, expanding from the detector face towards the output surface of the sample, intersects a smaller area on the output surface than the area under consideration in the simulation. Therefore, the simulations accept photons exiting from locations farther from the optical axis, thereby broadening the simulated transmittance pulse [19].
The above results indicate that the measurement of temporal transmittance is a promising technique for measuring the refractive index of the medium. The results demonstrate the sensitivity of the temporal transmittance signature to the refractive index, and therefore to optical properties in general. Although the di:erence between the two refractive index values is less than
3%, the variation in the transmittance is signiCcant and can be easily discerned visually on the graph.
The CPU time spent in the MC simulation of the above experiment in the purely scattering sample was about 108 h on a PC with one Pentium III Xeon 500 MHz processor. In order to reduce the computation cost, one might exploit the use of the isotropic scattering scaling law, which scales the anisotropic scattering into an equivalent isotropic scattering. The isotropic scaling usually gives reasonable accuracy in steady-state radiative transfer. In the MC simulation of scaled isotropic scattering, the CPU time is 2 : 23 h, which is only 2.1% of the CPU time used in anisotropic scattering modeling. However, the comparison of the results as shown in Fig. 7 illustrates that the scaled isotropic scattering cannot predict the accurate temporal behavior of the transmittance. This is consistent with the one-dimensional study of Guo and Kumar [12], where it was found that the isotropic scaling is only suitable for optically very di:use medium (after isotropic scaling) with forward scattering and at the long times. At early times, the isotropic scaling allows a large fraction of direct transmission without the interaction with the medium. However, such a ballistic transmission is not visible in the simulation using the complete anisotropic scattering phase function because of the zig–zag behavior of multiple scattering events in photon transport. Thus, the equivalent isotropic

Z. Guo et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 159–168
0.004
0.003
0.002
0.001
MC Prediction at 0 deg
Scaled Isotropic Scattering
β
I
β -1
Anisotropic Scattering
β
= 17.052 mm
-1 g = 0.96276
167
0
0 200 400 600
Time (ps)
800 1000
Fig. 7. Comparison of temporal proCles of transmittance between predictions of anisotropic scattering model and scaled isotropic scattering model.
scattering models are not useful in the time-resolved analysis of short pulse laser transport through scattering media for earlier times.
5. Conclusions
The Monte Carlo method is formulated to simulate the short pulse laser radiation transport through three-dimensional scattering–absorbing medium. The simulations are compared with experimental measurements in strongly forward scattering media. It is found, both experimentally and by simulation, that when the optical thickness of the medium is large, the di:erence of temporal shapes of transmittance (but not magnitudes) between di:erent detector angles is slight. In other words, the normalized temporal proCle of the transmitted pulse is not a function of the output detector angle. This implies that it might be reasonable to evaluate the hemispherical transmittance rather than simulate the transmitted pulse at speciCc detector angles if only the temporal pulse shape is of consequence. The other important Cndings are that the transmitted signal shape and magnitude are very sensitive to the medium’s optical properties, and that the shorter pulse widths are more desirable for probing the medium’s properties. The use of an accurate value of the refractive index of the scattering particles is therefore very important in simulation, and conversely experimental transmittance signatures can yield accurate assessment of the medium properties. Di:erences of a few percentage points in optical scattering properties can be easily di:erentiated through the transmittance results. It is also demonstrated that the use of scaled isotropic scattering, which has the potential to dramatically save on computational cost, does not have su;cient accuracy at the early times in the transient domain.
Acknowledgements
The authors acknowledge partial support from the SANDIA-NSF joint grant AW-9963 administered by the Sandia National Labs, Albuquerque, NM, where Dr. Shawn P. Burns served as the project manager.

168 Z. Guo et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 159–168
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