Eric Tinet, Sigrid Avrillier, and Jean Michel Tualle, "Fast semianalytical Monte Carlo simulation for time-resolved light propagation in turbid media," J. Opt. Soc. Am. A 13, 1903-1915 (1996)

The statistical estimator concept, created in the nuclear engineering field, has been adapted to the elaboration of a new and fast semianalytical Monte Carlo numerical simulation for time-resolved light-scattering problems. This concept has also been generalized to the case of unmatched boundaries. The model, discussed in detail in this paper, contains two stages. The first stage is the information generator in which, for each scattering event, the contribution to the total reflectance and transmittance is evaluated and subtracted from the photon current energy. This procedure reduces the number of photons required to produce a given accuracy, which makes it possible to store all event positions and energies. In the second stage, called the information processor, the results of the first stage are used to calculate analytically any desired result. Examples are given for scattering slabs of isotropic or anisotropic scatterers when collimated-beam incidence is used. Reflections at the boundaries are taken into account. The results obtained either with this new method or with classical Monte Carlo methods are very similar. However, the convergence of our new model is much better and, because of the separation into two stages, any quantity related to the problem can be easily calculated afterward without recomputing the simulation.

Tanner Young-Schultz, Stephen Brown, Lothar Lilge, and Vaughn Betz Biomed. Opt. Express 10(9) 4711-4726 (2019)

References

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Total Time- and Space-Integrated Reflectance (R) and Transmittance (T) of a Slab (in Percent)^{a}

Source of Results

ω_{0} = 0.99

g

μ_{t}d = 0.5

μ_{t}d = 1

μ_{t}d = 2

μ_{t}d = 4

μ_{t}d = 8

μ_{t}d = 16

VdH

0.875

R = 1.57

R = 3.27

R = 6.91

R = 14.17

R = 25.84

R = 37.53

T = 97.89

T = 95.58

T = 90.56

T = 80.01

T = 60.81

T = 35.21

0.75

R = 3.65

R = 7.60

R = 15.44

R = 28.43

R = 43.30

R = 53.00

T = 95.78

T = 91.12

T = 81.60

T = 64.65

T = 41.63

T = 19.30

0.5

R = 8.79

R = 17.07

R = 30.53

R = 46.98

R = 60.01

R = 65.61

T = 90.57

T = 81.45

T = 66.03

T = 45.27

T = 24.38

T = 8.60

0.0

R = 19.89

R = 33.29

R = 49.75

R = 64.50

R = 72.88

R = 75.13

T = 79.40

T = 65.10

T = 46.57

T = 27.55

T = 12.21

T = 2.96

CMC

0.875

R = 1.57

R = 3.27

R = 6.90

R = 14.15

R = 25.88

R = 37.46

T = 97.89

T = 95.58

T = 90.57

T = 80.03

T = 60.77

T = 35.32

0.75

R = 3.65

R = 7.60

R = 15.44

R = 28.43

R = 43.31

R = 52.98

T = 95.77

T = 91.12

T = 81.60

T = 64.65

T = 41.64

T = 19.37

0.5

R = 8.79

R = 17.08

R = 30.53

R = 47.03

R = 60.00

R = 65.54

T = 90.56

T = 81.45

T = 66.03

T = 45.21

T = 24.42

T = 8.68

0.0

R = 19.88

R = 33.22

R = 49.70

R = 64.47

R = 72.91

R = 75.11

T = 79.42

T = 65.17

T = 46.63

T = 27.59

T = 12.18

T = 2.96

MC3

0.875

R = 1.57

R = 3.26

R = 6.92

R = 14.17

R = 25.83

R = 37.54

T = 97.89

T = 95.59

T = 90.55

T = 80.02

T = 60.81

T = 35.17

0.75

R = 3.64

R = 7.60

R = 15.44

R = 28.44

R = 43.29

R = 52.98

T = 95.78

T = 91.12

T = 81.59

T = 64.64

T = 41.65

T = 19.36

0.5

R = 8.79

R = 17.08

R = 30.53

R = 46.99

R = 59.99

R = 65.56

T = 90.57

T = 81.44

T = 66.02

T = 45.26

T = 24.40

T = 8.66

0.0

R = 19.89

R = 33.29

R = 49.77

R = 64.48

R = 72.87

R = 75.11

T = 79.41

T = 65.10

T = 46.55

T = 27.57

T = 12.20

T = 2.96

Comparison of numerical data from van de Hulst24 (VdH) and the results obtained with a classical Monte Carlo (CMC) simulator and our MC3 simulator. The parameters are the albedo ω_{0} = 0.99, the anisotropy factor g, and the optical thickness μ_{t}d. There is index matching. 10^{6} photons are used in the Monte Carlo simulators to produce an accuracy better than 10^{−3}.

Total Time- and Space-Integrated Reflectance (R) and Transmittance (T) of a Slab (in percent)^{a}

Source of Results

ω_{0} = 0.99

g

μ_{t}d = 0.5

μ_{t}d = 1

μ_{t}d = 2

μ_{t}d = 4

μ_{t}d = 8

μ_{t}d = 16

MCML

0.875

R = 7.92

R = 10.47

R = 15.05

R = 21.79

R = 27.69

R = 30.16

T = 90.99

T = 87.20

T = 79.79

T = 66.78

T = 48.87

T = 28.99

0.75

R = 10.86

R = 15.81

R = 23.53

R = 31.88

R = 37.71

R = 40.97

T = 87.83

T = 81.35

T = 70.25

T = 54.93

T = 37.25

T = 18.50

0.5

R = 16.03

R = 23.86

R = 33.69

R = 42.70

R = 49.36

R = 52.51

T = 82.40

T = 72.80

T = 59.31

T = 43.28

T = 25.51

T = 9.47

0.0

R = 22.42

R = 33.11

R = 45.07

R = 55.13

R = 61.16

R = 62.93

T = 75.85

T = 63.30

T = 47.63

T = 30.76

T = 14.70

T = 3.67

CMC

0.875

R = 7.96

R = 10.45

R = 15.01

R = 21.80

R = 27.73

R = 30.23

T = 90.95

T = 87.21

T = 79.83

T = 66.77

T = 48.77

T = 28.88

0.75

R = 10.94

R = 15.83

R = 23.50

R = 31.79

R = 37.71

R = 41.06

T = 87.75

T = 81.34

T = 70.27

T = 55.01

T = 37.23

T = 18.30

0.5

R = 16.09

R = 23.95

R = 33.75

R = 42.73

R = 49.37

R = 52.55

T = 82.32

T = 72.71

T = 59.24

T = 43.23

T = 25.46

T = 9.39

0.0

R = 22.47

R = 33.11

R = 45.00

R = 55.09

R = 61.28

R = 63.02

T = 75.79

T = 63.32

T = 47.69

T = 30.77

T = 14.59

T = 3.61

MC3

0.875

R = 7.94

R = 10.47

R = 15.10

R = 21.82

R = 27.80

R = 30.23

T = 90.97

T = 87.25

T = 79.80

T = 66.78

T = 48.91

T = 28.91

0.75

R = 10.93

R = 15.90

R = 23.57

R = 31.90

R = 37.69

R = 40.83

T = 87.79

T = 81.36

T = 70.30

T = 55.00

T = 37.26

T = 18.44

0.5

R = 16.08

R = 24.01

R = 33.83

R = 42.69

R = 49.38

R = 52.49

T = 82.37

T = 72.75

T = 59.20

T = 43.31

T = 25.50

T = 9.51

0.0

R = 22.45

R = 33.11

R = 44.88

R = 54.63

R = 61.12

R = 63.00

T = 75.81

T = 63.30

T = 47.66

T = 30.81

T = 15.07

T = 3.69

Comparison of the results obtained with a classical Monte Carlo (MCML) simulator25 and with our CMC and MC3 simulators. The parameters are the albedo ω_{0} = 0.99, the anisotropy factor g, and the optical thickness μ_{t}d. There is no index matching (n_{int} = 1.4, n_{ext} = 1). 10^{6} photons are used, and the accuracy is approximately 10^{−3}.

Total Time- and Space-Integrated Reflectance (R) and Transmittance (T) of a Slab (in Percent)^{a}

Source of Results

ω_{0} = 0.99

g

μ_{t}d = 0.5

μ_{t}d = 1

μ_{t}d = 2

μ_{t}d = 4

μ_{t}d = 8

μ_{t}d = 16

VdH

0.875

R = 1.57

R = 3.27

R = 6.91

R = 14.17

R = 25.84

R = 37.53

T = 97.89

T = 95.58

T = 90.56

T = 80.01

T = 60.81

T = 35.21

0.75

R = 3.65

R = 7.60

R = 15.44

R = 28.43

R = 43.30

R = 53.00

T = 95.78

T = 91.12

T = 81.60

T = 64.65

T = 41.63

T = 19.30

0.5

R = 8.79

R = 17.07

R = 30.53

R = 46.98

R = 60.01

R = 65.61

T = 90.57

T = 81.45

T = 66.03

T = 45.27

T = 24.38

T = 8.60

0.0

R = 19.89

R = 33.29

R = 49.75

R = 64.50

R = 72.88

R = 75.13

T = 79.40

T = 65.10

T = 46.57

T = 27.55

T = 12.21

T = 2.96

CMC

0.875

R = 1.57

R = 3.27

R = 6.90

R = 14.15

R = 25.88

R = 37.46

T = 97.89

T = 95.58

T = 90.57

T = 80.03

T = 60.77

T = 35.32

0.75

R = 3.65

R = 7.60

R = 15.44

R = 28.43

R = 43.31

R = 52.98

T = 95.77

T = 91.12

T = 81.60

T = 64.65

T = 41.64

T = 19.37

0.5

R = 8.79

R = 17.08

R = 30.53

R = 47.03

R = 60.00

R = 65.54

T = 90.56

T = 81.45

T = 66.03

T = 45.21

T = 24.42

T = 8.68

0.0

R = 19.88

R = 33.22

R = 49.70

R = 64.47

R = 72.91

R = 75.11

T = 79.42

T = 65.17

T = 46.63

T = 27.59

T = 12.18

T = 2.96

MC3

0.875

R = 1.57

R = 3.26

R = 6.92

R = 14.17

R = 25.83

R = 37.54

T = 97.89

T = 95.59

T = 90.55

T = 80.02

T = 60.81

T = 35.17

0.75

R = 3.64

R = 7.60

R = 15.44

R = 28.44

R = 43.29

R = 52.98

T = 95.78

T = 91.12

T = 81.59

T = 64.64

T = 41.65

T = 19.36

0.5

R = 8.79

R = 17.08

R = 30.53

R = 46.99

R = 59.99

R = 65.56

T = 90.57

T = 81.44

T = 66.02

T = 45.26

T = 24.40

T = 8.66

0.0

R = 19.89

R = 33.29

R = 49.77

R = 64.48

R = 72.87

R = 75.11

T = 79.41

T = 65.10

T = 46.55

T = 27.57

T = 12.20

T = 2.96

Comparison of numerical data from van de Hulst24 (VdH) and the results obtained with a classical Monte Carlo (CMC) simulator and our MC3 simulator. The parameters are the albedo ω_{0} = 0.99, the anisotropy factor g, and the optical thickness μ_{t}d. There is index matching. 10^{6} photons are used in the Monte Carlo simulators to produce an accuracy better than 10^{−3}.

Total Time- and Space-Integrated Reflectance (R) and Transmittance (T) of a Slab (in percent)^{a}

Source of Results

ω_{0} = 0.99

g

μ_{t}d = 0.5

μ_{t}d = 1

μ_{t}d = 2

μ_{t}d = 4

μ_{t}d = 8

μ_{t}d = 16

MCML

0.875

R = 7.92

R = 10.47

R = 15.05

R = 21.79

R = 27.69

R = 30.16

T = 90.99

T = 87.20

T = 79.79

T = 66.78

T = 48.87

T = 28.99

0.75

R = 10.86

R = 15.81

R = 23.53

R = 31.88

R = 37.71

R = 40.97

T = 87.83

T = 81.35

T = 70.25

T = 54.93

T = 37.25

T = 18.50

0.5

R = 16.03

R = 23.86

R = 33.69

R = 42.70

R = 49.36

R = 52.51

T = 82.40

T = 72.80

T = 59.31

T = 43.28

T = 25.51

T = 9.47

0.0

R = 22.42

R = 33.11

R = 45.07

R = 55.13

R = 61.16

R = 62.93

T = 75.85

T = 63.30

T = 47.63

T = 30.76

T = 14.70

T = 3.67

CMC

0.875

R = 7.96

R = 10.45

R = 15.01

R = 21.80

R = 27.73

R = 30.23

T = 90.95

T = 87.21

T = 79.83

T = 66.77

T = 48.77

T = 28.88

0.75

R = 10.94

R = 15.83

R = 23.50

R = 31.79

R = 37.71

R = 41.06

T = 87.75

T = 81.34

T = 70.27

T = 55.01

T = 37.23

T = 18.30

0.5

R = 16.09

R = 23.95

R = 33.75

R = 42.73

R = 49.37

R = 52.55

T = 82.32

T = 72.71

T = 59.24

T = 43.23

T = 25.46

T = 9.39

0.0

R = 22.47

R = 33.11

R = 45.00

R = 55.09

R = 61.28

R = 63.02

T = 75.79

T = 63.32

T = 47.69

T = 30.77

T = 14.59

T = 3.61

MC3

0.875

R = 7.94

R = 10.47

R = 15.10

R = 21.82

R = 27.80

R = 30.23

T = 90.97

T = 87.25

T = 79.80

T = 66.78

T = 48.91

T = 28.91

0.75

R = 10.93

R = 15.90

R = 23.57

R = 31.90

R = 37.69

R = 40.83

T = 87.79

T = 81.36

T = 70.30

T = 55.00

T = 37.26

T = 18.44

0.5

R = 16.08

R = 24.01

R = 33.83

R = 42.69

R = 49.38

R = 52.49

T = 82.37

T = 72.75

T = 59.20

T = 43.31

T = 25.50

T = 9.51

0.0

R = 22.45

R = 33.11

R = 44.88

R = 54.63

R = 61.12

R = 63.00

T = 75.81

T = 63.30

T = 47.66

T = 30.81

T = 15.07

T = 3.69

Comparison of the results obtained with a classical Monte Carlo (MCML) simulator25 and with our CMC and MC3 simulators. The parameters are the albedo ω_{0} = 0.99, the anisotropy factor g, and the optical thickness μ_{t}d. There is no index matching (n_{int} = 1.4, n_{ext} = 1). 10^{6} photons are used, and the accuracy is approximately 10^{−3}.