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### Heuristic model for ballistic photon detection in collimated transmittance measurements

### MARTELLI, Fabrizio, BINZONI, Tiziano

**Abstract**

An heuristic model for ballistic photon detection in continuous-wave measurements of collimated transmittance through a slab is presented. The model is based on the small angle approximation and the diffusion equation and covers all the ranges of optical thicknesses of the slab from the ballistic to the diffusive regime. The performances of the model have been studied by means of comparisons with the results of gold standard Monte Carlo simulations for a wide range of optical thicknesses and two types of scattering functions. For a non-absorbing slab and field of view of the receiver less than 3° the model shows errors less than 15% for any value of the optical thickness. Even for an albedo value of 0.9, and field of view of the receiver less than 3° the model shows errors less than 20%. These results have been verified for a large set of scattering functions based on the Henyey-Greenstein model and Mie theory for spherical scatterers. The latter has also been used to simulate the scattering function of Intralipid, a diffusive material widely used as reference standard for tissue simulating phantoms. The proposed model [...]

### MARTELLI, Fabrizio, BINZONI, Tiziano. Heuristic model for ballistic photon detection in collimated transmittance measurements. *Optics Express* , 2018, vol. 26, no. 2, p. 744-761

### DOI : 10.1364/oe.26.000744 PMID : 29401955

Available at:

http://archive-ouverte.unige.ch/unige:102007

Disclaimer: layout of this document may differ from the published version.

1 / 1

**Heuristic model for ballistic photon detection in** **collimated transmittance measurements**

**F**

^{ABRIZIO}**M**

^{ARTELLI}^{1}

^{AND}**T**

^{IZIANO}**B**

^{INZONI}**,**

^{2,3}*1**Università degli Studi di Firenze, Dipartimento di Fisica e Astronomia, Via G. Sansone 1, 50019 Sesto*
*Fiorentino, Firenze, Italy*

*2**Département de Neurosciences Fondamentales, University of Geneva, Switzerland*

*3**Département de l’Imageries et des Sciences de l’Information Médicale, University Hospital, Geneva,*
*Switzerland*

****fabrizio.martelli@unifi.it*

**Abstract:** An heuristic model for ballistic photon detection in continuous-wave measurements
of collimated transmittance through a slab is presented. The model is based on the small angle
approximation and the diffusion equation and covers all the ranges of optical thicknesses of the
slab from the ballistic to the diffusive regime. The performances of the model have been studied
by means of comparisons with the results of gold standard Monte Carlo simulations for a wide
range of optical thicknesses and two types of scattering functions. For a non-absorbing slab and
field of view of the receiver less than 3^{◦}the model shows errors less than 15% for any value of the
optical thickness. Even for an albedo value of 0.9, and field of view of the receiver less than 3^{◦}the
model shows errors less than 20%. These results have been verified for a large set of scattering
functions based on the Henyey-Greenstein model and Mie theory for spherical scatterers. The
latter has also been used to simulate the scattering function of Intralipid, a diffusive material
widely used as reference standard for tissue simulating phantoms. The proposed model represents
an effective improvement compared to the existing literature.

© 2018 Optical Society of America under the terms of theOSA Open Access Publishing Agreement
**OCIS codes:**(110.0113) Imaging through turbid media; (110.7050) Turbid media; (290.7050) Turbid media; (170.7050)
Turbid media; (170.6510) Spectroscopy, tissue diagnostics; (170.5280) Photon migration.

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#309193 https://doi.org/10.1364/OE.26.000744

Journal © 2018 Received 16 Oct 2017; revised 10 Nov 2017; accepted 10 Nov 2017; published 9 Jan 2018

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**1.** **Introduction**

*Ballistic photons*are all detected photons that do not experience scattering interactions prior the
detection in a transmittance measurement [1, 2].*Quasi ballistic*or*snake photons*are photons
that have undergone few scattering interactions, but they are still traveling in approximatively
the same initial direction. Ballistic photons follow the shorter path-length and reach the detector
earlier than non-ballistic photons. They also show a better collimation than non-ballistic light
and this characteristic is often used for their detection. Ballistic light can be directly linked to
the extinction coefficient of the medium and, for imaging purposes, the photons paths can be
precisely identified without the need of complex reconstructions techniques [3–6]. Measurements
of ballistic light are largely used in spectrophotometric applications aimed to extract information
from the extinction coefficient of the medium and in imaging applications for media characterized
by small optical thicknesses (see [1] for some examples of application).

The actual crucial question beside*ballistic light*or*quasi-ballistic light*is the real detectability
limits of this radiation. The research activity on this field has spanned two decades but still
unrevealed all its possible perspectives as testified by very recent works dedicated to ballistic and
sub-diffusive regimes [7, 8].

Light propagating through common natural media experiences scattering and absorption interactions. The presence of scattering interactions can spoil the detection of ballistic photons since scattered light can enter the detector’s field of view and mixes with the ballistic contribution.

When the number of scattering interactions increases, light propagation tends to a diffusive

regime and the transition ballistic to diffusive is subjected to several factors such as the absorption coefficient of the medium, the scattering function of the medium and the characteristics of the receiver. A correct modeling of the transition from ballistic to diffusive is a key information for ballistic detection since it provides the conditions under which such detection is feasible.

This transition has been studied in the past by several groups [9–13] with different purposes and results. Actually, it is still missed a model that can describe such transition with enough accuracy in all the possible experimental conditions.

The errors in the measurement of the ballistic light have already been highlighted in the previous
literature [13–15]. Wind and Szymanski [14] implemented a correction to the Beer-Lambert’s law,
although it’s validity is only within single scattering conditions so that the other scattering orders
are ignored. Ben*et al.*[13] have recently proposed a simple model for the transition from the
ballistic to the diffusive regimes designed only for very specific experimental conditions and based
on the assumption that the detected radiation consists only of two terms: ballistic and diffusive.

The ballistic term is expressed by the Beer-Lambert’s law, while the diffusive is proportional to the field of view of the receiver and to the attenuation exponential factor of the intensity inside a diffusive infinite medium [13]. The model stands out for its extreme mathematical simplicity, while its accuracy shows relative errors usually larger than 20% and from this point of view there is margin for improvements. Thennadil and Chen [15] used an alternative measurement configurations for extracting bulk optical properties where the measurements of the collimated transmittance were replaced by reflectance measurements at different sample thicknesses. In this method they avoid collimated transmittance measurements because of the errors affecting the ballistic detection. All the cited examples testify the limitations introduced by errors on ballistic detection and how would be useful to provide an estimate of them.

With the present work, we propose a model for ballistic detection in the continuous wave (CW) domain that covers all the ranges of light propagation from the ballistic to the diffusive regime.

The proposed model is a substantial improvement in term of accuracy compared to previous models. In the range from ballistic to diffusive we can identify four regimes of propagation that characterize the detected light in a measurement of collimated transmittance: ballistic, quasi-ballistic, quasi-diffusive and diffusive. Thus, a rigorous modeling of ballistic detection should cover all the four ranges where are expected to be valid different characteristics of the transmitted light. In this work, we have basically divided the photon migration in three regimes:

ballistic, intermediate and diffusive. The ballistic regime in this context pertains to ballistic and quasi-ballistic detected light. The intermediate regime describes the detected scattered light that cannot yet be completely described by the diffusion approximation. Finally, the diffusive regime must be interpreted in its natural meaning. Further introductory information to the presented model is provided in the preamble of the theory (Sec. 2.1).

In Sec. 2 the model is described in all its characteristics. In Sec. 3 the model is validated by means of comparisons with the results of gold standard Monte Carlo (MC) simulations.

Discussion and conclusions are presented in Sec. 4.

**2.** **Theory and methods**
2.1. Preamble

An intuitive schematic for the detection of ballistic photons is given in Fig. 1, where a scattering
medium is illuminated by a CW pencil beam and where the transmitted collimated light is
collected by an optical receiver. As explained in Sec. 1, the main limitation of this measurement
is that the exact assessment of ballistic light is spoiled by a certain amount of scattered light that
reaches the detector. This fact alters the measurement of the actual extinction coefficient,µ_{e}, of
the investigated medium that is in principle given by the ratio between optical thickness,τ, and

geometrical thickness,s_{0}. The parameterτis usually experimentally assessed using
τ=−lnP_{0}

P_{e}, (1)

where P_{e} is the power emitted by the light source, injected inside the sample, and P_{0} =
P_{e}exp(−µ_{e}s_{0})=P_{e}exp(−τ)the ballistic power. Unfortunately, a measurement setup like in Fig.

1 allows only to obtain an apparent optical thickness
τ_{A}=−lnP_{R}

P_{e} =−lnP_{0}+P_{s}

P_{e} , (2)

whereP_{R}is the measured power by the detector andP_{s}represents the spurious scattered power
collected in the field of view of the receiver. Thus, the presence ofPsgenerates aτ_{A}different
from the desiredτ, and finally leads to a inexactµ_{e}=τ_{A}/s_{0},τ/s_{0}estimation. We note the fact

Laser Source Optical

Receiver

*s** _{0}*
Scattering a
Medium

*d*
*r*
*n*_{i}*n*_{e}

Fig. 1. Basic schematic for measurements of ballistic light in a collimated transmittance
configuration. Parameters are:r, radius of the entrance surface of the optical receiver;α,
semi-aperture of the field of view of the optical receiver;s_{0}, thickness of the scattering
medium slab;d, distance between the scattering medium and the optical receiver;n_{i}refractive
index of the scattering medium;neexternal refractive index (e.g. air).

that the the schematic of Fig. 1 pertains to a basic representation of a transmittance measurement, while in serial spectrophotometers the situation can be significantly more complex. However, the specific purpose of the figure is to highlight the main effects affecting this type of measurements.

In Fig. 1, the definitions of the geometrical symbols used in the text are given in the figure
caption. The scattering medium has scattering, reduced scattering, and absorption coefficients,µ_{s},
µ^{0}_{s}=µ_{s}(1−g), andµ_{a}, respectively; with scattering function (phase function)p(θ)(normalized
to 1 within the whole solid angle), anisotropy factorg, and albedo of single scatteringΛ= ^{µ}_{µ}^{s}_{e}.
We denote asτ=µ_{e}s_{0}the optical thickness of the medium,τ_{s}^{0}=µ^{0}_{s}s_{0}the reduced scattering
optical thickness and,τ_{s}=µ_{s}s_{0}andτ_{a}=µ_{a}s_{0}the scattering and absorption optical thicknesses
of the medium, respectively.

At this point, it is important to note that the apparent optical thicknessτ_{A}, obtained from
collimated transmittance measurements, when plotted versusτshows a typical behavior widely
known in the previous literature [16,17]. For explanatory purposes, making use of MC simulations,
we have reproduced the experimental results of [16] for a scattering cell (medium) constituted of a
suspension of polystyrene latex spheres in water. The spheres had a gaussian size distribution with
average diameterhφi=15.8µm and standard deviationSD=2.8 µm. The results are reported
in Fig. 2, whereτ_{A}is plotted versusτfor three values ofα. The calculations were forn_{i} =n_{e}
andΛ=1. To underline deviations with respect to the ideal measurement the straight lineτ_{A}=τ
is also shown. These results are paradigmatic of many experimental situations and thus provide a
general view of the problem. We will use Fig. 2 as tutorial for introducing the characteristics of a
typical measurement of collimated transmittance. For this purpose, in Fig. 2 we have emphasized

the three regimes of propagation that will be described with three complementary mathematical models in the next sections: ballistic, intermediate and diffusive (see Sec. 1).

The*ballistic regime*is typical for small values ofτ. The measured transmittance is dominated
by ballistic photons or, for scattering functions with very high forward scattering like in this
example, by photons that undergo only few scattering events with paths length very close to
s_{0}. These photons are usually denoted in literature as*early*or*snakes photons:* τ_{A} increases
proportionally toτwith a constant slope that can be significantly lower than 1 for media with
p(θ)highly forward peaked as in our example.

For largeτ,τ_{A}reaches a kind of saturation and photon migration is dominated by multiple
scattering. Thus, the amount of ballistic photons in the detected transmittance becomes negligible
and forτsufficiently large we are in the*diffusive regime*(Fig. 2, the region forτ >≈18).

Between the ballistics and the diffusive regime we have what is called the*intermediate regime.*

Note that the transition between intermediate regime and diffusive regime mainly depends on the geometry of the measurement (e.g.r,αandd) and onp(θ). So far, no previous mathematical models are available in the literature to describe the intermediate regime.

Note that by increasingµ_{a}we decrease the amount of scattering, because photons with large
paths length are preferentially absorbed and not detected by the receiver (s0is the shortest possible
path length). Thus, ideally, forµ_{s}µ_{a}we have thatτ_{A}tends toτ.

0 5 10 15 20 25 30 35 40

0 2 4 6 8 10 12 14 16 18 20

Fig. 2. Simulated MC experiments for a scattering medium constituted by a suspension of
polystyrene latex spheres in water (hφi=15.8µm). The figure showsτ_{A}versusτfor three
αvalues;s_{0} =10 cm,d=70 cm andr =2 cm. The straight line representing the ideal
measurement is also shown.

Eventually, the tutorial example gives us an idea on the complex dependence ofτ_{A}on the optical
parameters and highlights the difficulty to define a simple mathematical model that encompass
the three regimes. In the next sections, we will see how this modeling can be performed through
an effective heuristic approach.

In synthesis, in the present contribution, we propose an heuristic model that can describeτ_{A}
as a function ofτ for the three different regimes shown in Fig. 2: 1) the ballistic regime (τ_{s}

<≈4), where any deviation from the Beer-Lambert’s law can be described starting from the single
scattering contribution to the detected light; 2) the diffusive regime (forτ_{s}≥≈4 andτ_{s}^{0}>≈1.5); 3)
the intermediate regime between ballistic and diffusive. Note that the diffusive regime must be
defined making use ofτ_{s}^{0}since the diffusive propagation only depends onµ_{s}^{0}and not onµ_{s}and
p(θ)separately [19]. We will see that an extrapolation of the diffusive model toward the ballistic
regime allows us to effectively describe the intermediate regime. The model presented in this
work provides the correct functional dependence ofτ_{A}versusτ. We stress that this representation

of the data is related to the importance of the measuredτ_{A}whenever extinction measurements
are carried out.

The model is presented for the pencil beam source, ford =0 and for the refractive index
matched case (ni =n_{e}). In Sec. 4 these conditions are discussed to extend the solution to a more
general case.

2.2. Model for small optical thickness: ballistic regime

Source Receiver

*l*_{1}*dl** _{1}*
q

_{1}q_{2}*d*W_{1}*l*_{2}

*l*_{3}*dl**2*

*dl*_{3}

*dl*_{k}

W* _{k}*
q

_{k}*l*

_{k+1}*dl*

_{k-1}Fig. 3. Schematic of multiple scattering interactions in a transmittance measurement for a
pencil beam impinging onto the medium. Parameters are:l_{k}, steps between scattering events;

dl_{k}, infinitesimal step before a scattering event;dΩ_{k}, solid angle;θ_{k}, scattering angle. This
figure is a 2D projection, but must be imagined in 3D.

As explained in Secs. 1 and 2.1 the measured powerP_{R}is spoiled by presence of the scattered
lightP_{s}. Such termP_{s}can be expressed as a sum of detected light powers,P_{si}, received after
i=1,2· · ·kscattering events, i.e.,

P_{R}=P_{0}+P_{s}=P_{0}+P_{s1}+P_{s2}+· · ·+P_{sk}+· · · (3)
With reference to Fig. 3, the infinitesimal contribution to the light detected afterkscattering
eventsdP_{sk}can be calculated as [14, 19]

dP_{sk}=P_{e} exp(−µ_{e}l_{1})µ_{s}dl_{1}p(θ_{1})dΩ_{1}

×exp(−µ_{e}l_{2})µ_{s}dl_{2}p(θ_{2})dΩ_{2}· · ·exp(−µ_{e}l_{k})µ_{s}dl_{k}p(θ_{k})dΩ_{k} exp(−µ_{e}l_{k+1})

=P_{e}µ^{k}_{s}exp(−µ_{e}

k+1

Õ

i=1

l_{i})p(θ_{1})p(θ_{2}) · · ·p(θ_{k})dΩ_{1}dΩ_{2}· · ·dΩ_{k} dl_{1}dl_{2}· · ·dl_{k}. (4)
Equation (4) comes from an iterative calculation for each scattering contribution, starting from
the first order to the scattering of orderk. The termP_{e}exp(−µ_{e}l_{1})is the power’s fraction that
reaches the volume element of thicknessdl_{1} at distancel_{1} from the injection point dues to
scattering and absorption interactions throughl_{1}. Then, the power’s fraction scattered within
the infinitesimal solid angledΩ_{1}around the scattering directionθ_{1}is obtained multiplying by
µ_{s}dl_{1}p(θ_{1})dΩ_{1}. Finally, in accordance to the schematic of Fig. 3, the same procedure used for the
first order is iterated for the remaining scattering orders. Integrating Eq. (4) all over the angular
and spatial variables, returns the contribution of thek^{th}scattering event,P_{sk}, to the measured
transmittance. The importance of Eq. (4) is related to the fact that it shows the actual dependence
ofP_{sk}on the optical and geometrical properties of the medium. Indeed, the integration of Eq.

(4) is not affordable unless a MC method is used. For this reason, our approach is to provide a
solution for small scattering angles [18] (Small Angle Approximation, SAA) that is valid when
the dominant scattering angle per scattering event is small. Under such conditionsPskis mainly
given by photons that follow trajectories with smallθ_{i}so that the total length of a single trajectory
can be approximated toÍ_{k+1}

i=1 li s0, andPskcan be written as

Psk Peτ_{s}^{k}exp(−τ)K_{k}(p(θ),s0,d, α,r), (5)

where

K_{k}(p(θ),s_{0},d, α,r)=∫

p(θ_{1})p(θ_{2}) · · ·p(θ_{k})dΩ_{1}dΩ_{2}· · ·dΩ_{k} dl_{1}dl_{2}· · ·dl_{k} (6)
is a coefficient that depends on the geometry of the scatters, throughp(θ), and on the geometry
of the measurement (defined e.g. bys_{0},d,αorr) through the integration boundaries (see below
explicit calculations fork=1), but that, under the SAA, does not depend onµ_{e}andµ_{s}. Inserting
Eqs. (1) and (5) in Eq. (3), the total transmitted power becomes

P_{R}=P_{e}exp(−τ)
1+Õ

k

τ_{s}^{k}K_{k}(p(θ),s_{0},d, α,r). (7)
This equation implies that, starting fromP_{R}P_{0} =P_{e}exp(−τ)whenτ_{s}→0, the weight of the
scattered light increases with the increment ofµ_{s}andτ_{s}. Thus, increasingτ_{s}the propagation is
dominated by the higher scattering orders and the ballistic component (un-scattered light),P_{0},
decreases progressively. By taking the logarithm of Eq. (7) and by using Eq. (2), we find

τ_{A}=−lnP_{R}

P_{e} =τ−ln
1+Õ

k

τ_{s}^{k}K_{k}(p(θ),s_{0},d, α,r), (8)
with a relative error,ε, onτ_{A}

ε= τ−τ_{A}

τ =

ln[1+Í

k

τ_{s}^{k}Kk(p(θ),s0,d, α,r)]

τ . (9)

Whenτ_{s}1 inε, i.e. the number of scattering events tends to 0 due to a very smallµ_{s}or a very
smalls_{0}, survives only the single scattering contribution, i.e.,

τlims→0ε≡ε_{SS} =τ_{s}K1(p(θ),s0,d, α,r)

τ =ΛK1(p(θ),s0,d, α,r), (10) where Eq. (6) gives

K_{1}(p(θ),s_{0},d, α,r)=∫

p(θ_{1})dΩ_{1}dl_{1} =2π1
s0

∫ s0

0

∫ θ(l,s0,d,α,r) 0

p(θ)sinθdθdl, (11) with

θ(l,s0,d, α,r)=minα,arctan( r

s_{0}+d−l). (12)
The notation "SS" wants to point out the assumption that the single scattering contribution is
more important compared to the other scattering orders. Equation (12) holds in case of matched
refractive indexes between medium and external (for the mismatched case see Sec. 4.1). Equations.

(10-12) hold also whend ,0.

We note that if the receiver is sufficiently large (r ≥ (s_{0}+d)tanα) to detect all light scattered
withθ≤αthen Eq. (11) simply becomes

K_{1}(p(θ),s_{0},d, α,r)=2π∫ α
0

p(θ)sinθdθ, (13)

andK_{1}(.)represents the fraction of scattered light withinα. The proposed model always uses Eq.

(11).

Finally, using Eqs. (8), (9) and (10), a suitable model,τ_{A}_{S S}, for smallτcan be summarized as
τ_{A}_{S S} =τ(1−ε)τ(1−ε_{SS})=τ[1−ΛK1(p(θ),s0,d, α,r)]. (14)

The validity of Eq. (14) is supported by the experimental results in [16] and by MC results like
those in Fig. 2 in which for moderate values ofτ,τ_{A}linearly increases withτ. We must also note
that the validity of the SAA, necessary to derive Eq. (10), is well verified in many experimental
measurements of collimated transmittance where usually small values ofαare employed.

Equation (14) allows us to express the received powerP_{R}as

P_{R}=P_{e}exp(−τ_{A}_{S S})=P_{e}exp[−µ_{e}(1−ε_{SS})s_{0}], (15)
that can be considered as a modified Beer Lambert’s law with an apparent extinction coefficient,
µ_{e A}_{S S},

µ_{e A}_{S S} =µ_{e}(1−ε_{SS}), (16)

that practically is the extinction coefficient obtained when only the fraction of scattered light withθ > αis considered lost.

In conclusion, the proposed model makes use of the single scattering contribution to describe the detected light (ballistic regime). In practice, multiple reflections inside the medium may also occur for small optical thicknesses. However, such effects are usually of second order compared to those here analyzed, so that they become important only for extreme experimental situations.

Information for modeling the transmitted light in these situations can be found in [20].

2.3. Model for large optical thickness: diffusive regime

The behavior ofτ_{A}versusτ(Fig.2) shows a kind of saturation for large values ofτ. The low
increment ofτ_{A}versusτimplies that the ballistic component of the received light is reduced to
a negligible fraction compared to the scattered light. This fact suggests that a good candidate
to provide a model in such regime can be the diffusion equation (DE) and its solutions [19, 21]

since for high values ofτis well known to provide an excellent description of the spatial and angular distribution of the transmitted scattered light.

Analytical solutions of the DE are available for a CW pencil beam source and for a detector
placed on the external surface of a slab. The transmitted power,P_{DE}(r), that impinges onto the
receiver of radiusris given (obtained from Eq. (4.39) in [19])) by

P_{DE}(r)=P_{e}^{1}_{2}

m=+∞

Í

m=−∞

(

sgn(z1m)exp(−µ_{e f f}|z_{1m}|) −sgn(z2m)exp(−µ_{e f f}|z_{2m}|)+

−

z1mexp

−µe f f

√

r^{2}+z^{2}_{1m}

√

r^{2}+z^{2}_{1m} +

z2mexp

−µe f f

√

r^{2}+z_{2m}^{2}

√

r^{2}+z^{2}_{2m}

) ,

(17)

with

z_{1m}=(1−2m)s_{0}−4mze−z_{s}

z_{2m}=(1−2m)s_{0}− (4m−2)z_{e}+z_{s}, (18)
andµ_{e f f} =p3µ_{a}µ_{s}^{0},z_{s} =1/µ^{0}_{s},z_{e} =2AD,D=1/(3µ^{0}_{s}),Acoefficient for the extrapolated
boundary condition [19, 21] andP_{e}power injected inside the slab.P_{DE}(r)is the total diffuse
power light impinging onto the receiver. The power fraction actually detected is that outgoing
with anglesθ_{e}≤αand can be calculated by using the related formula obtained with the DE. The
angular dependence of light outgoing from a slab can be described by a functionF(θ_{e})as [19, 22]

F(θ_{e})=_{4π}^{1}

ne ni

2n

1−R_{F}h

arcsin(^{n}^{e}

ni sinθ_{e})i o n

2A+3 cosh

arcsin(^{n}^{e}

ni sinθ_{e})i o

, (19)

withR_{F}[.]the Fresnel reflection coefficient for unpolarized light,θ_{e}angle between the direction
of the outgoing radiation and the normal to the surface. The fraction of powerP_{R}(r, α)that

emerges withinαand that reaches the detector, is assumed to be the sum of the ballistic component
P_{0}=P_{e}exp(−τ)given by Beer-Lambert’s law, plus the fraction ofP_{DE}(r)withinα, i.e. [18, 19]

P_{R}(r, α)P_{0}+P_{DE}(r)χ(α), (20)

where

χ(α)=∫ α 0

F(θ_{e})cosθ_{e}2πsinθ_{e}dθ_{e}. (21)
We note that forn_{i} =n_{e}, Eq. (21) assumes the simple form

χ(α)=1−1

2(cos^{2}α+cos^{3}α). (22)

Finally, for largeτthe apparent optical thickness is
τ_{A}_{D E}_{α} =−lnP_{R}(r, α)

P_{e} . (23)

In this model the received power is given by the ballistic component plus the fraction of diffuse light. In a strict sense, this model when used with receivers with large surface and large field of viewαcould lead, for small optical thicknesses of the slab, to detect more light than that injected into the medium. However, this lack of the model only happens for very smallτvalues, where Eq. (20) should not be used anymore. The actual validity of Eq. (20) in diffusion conditions will be further proven in Sec. 3 by comparisons with the results of MC simulations.

2.4. Model for intermediate optical thickness: the intermediate regime

Equation (14) works pretty well forτ_{s} <≈ 4, whilst Eq. (23) works very well forτ_{s}^{0} >≈ 2
(see following sections). In what follows, we propose to calculate the transmitted light in
the intermediate regime between ballistic and diffusive making use of a formula obtained by
extrapolating the behavior of the DE solution for lowτ_{s}values. We note that the solution of the
DE (Eq. (17)) provides physically correct results only forτ_{s}^{0} >1 since it is obtained placing
an isotropic source inside the medium at a depth equal toz_{s}=1/µ^{0}_{s}. The extrapolated model
here presented can also work forτ_{s}^{0}<1, where the DE solution cannot be used. Therefore, we
implement an extrapolation of the DE solution forτ_{s}^{0}<≈1.5 and we represent the transmitted
scattered light,PsE x t(r, α), with a power law, i.e

P_{s}_{E x t}(r, α)=a(r, α)τ_{s}^{0}^{b(r,α)}, (τ_{s} ≥4, τ_{s}^{0}≤1.5), (24)

with the parametersa(r, α)andb(r, α)obtained by fitting Eq. (24) toP_{DE}(r)χ(α)in the range
1.2≤τ_{s}^{0}≤2. The choice of the power law forPsE x t(r, α)is motivated by the fact that a power
law also holds for the total transmittance through a diffusive slab. Indeed, in accordance to
the Ohm’s law for light [23] the transmitted light through a diffusive non-absorbing slab is
approximatively proportional to 1/τ_{s}^{0}. We also stress thata(r, α)andb(r, α)depend onµ_{a}, while
they are independent ofp(θ). As a matter of fact, since the dependence ofP_{DE}(r)χ(α)onαfor
d=0 is by a multiplying factor, thenb(r, α)actually is independent ofαwhend=0. In Tab.1
the values ofa(r, α)andb(r)have been shown for a slab 10 mm thick withn_{i} =n_{e},µ_{a}=0 and
d=0. Figure 4 shows examples ofP_{s}_{E x t}(r, α)obtained from the fitting ofP_{DE}(r)χ(α)for a slab
withµ_{a} =0,n_{i} =n_{e},α=90^{◦}andr =0.5, 1, 2, 4, 10 and∞mm. The results for lower values of
αcan be obtained simply by scaling the results forα=90^{◦}with the suitableχ(α). The values
obtained foraandbin the figure forP_{s}_{E x t}(r, α)are those shown in Tab.1 forα=90^{◦}.

Thus, the total received power,P_{R}(r, α), is assumed to be

PR(r, α)=P0+PsE x t(r, α), (25)

Table 1. Coefficientsaandbof Eq. (24) calculated for a slab 10 mm thick withni =ne, µa=0 andd=0.

α=0.5^{◦} α=1^{◦} α=2^{◦} α=3^{◦} α=90^{◦}

r (mm) a(r, α) a(r, α) a(r, α) a(r, α) a(r, α) b(r)

0.5 3.055×10^{−6} 1.222×10^{−5} 4.885×10^{−5} 1.098×10^{−4} 3.209×10^{−2} -3.790
1 1.036×10^{−5} 4.143×10^{−5} 1.656×10^{−4} 3.724×10^{−4} 1.088×10^{−1} -3.550
2 2.580×10^{−5} 1.032×10^{−4} 4.126×10^{−4} 9.277×10^{−4} 2.711×10^{−1} -2.902
4 4.122×10^{−5} 1.649×10^{−4} 6.592×10^{−4} 1.482×10^{−3} 4.331×10^{−1} -1.840
10 5.413×10^{−5} 2.165×10^{−4} 8.655×10^{−4} 1.946×10^{−3} 5.686×10^{−1} -0.745

∞ 6.884×10^{−5} 2.753×10^{−4} 1.101×10^{−3} 2.475×10^{−3} 7.232×10^{−1} -0.541

1.2 1.4 1.6 1.8 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Fig. 4. Examples ofP_{s}_{E x t}(r, α)obtained from the fitting ofP_{DE}(r)χ(α)for a slab 10 mm
thick withµa=0,n_{i}=ne,α=90^{◦}and for several values ofr.PsE x t(r, α)andP_{DE}(r)χ(α)
are normalized to the powerP_{e}injected inside the slab and thus they have not units.

and the corresponding apparent optical thickness,τ_{A}_{E x t},
τAE x t =−lnPR(r, α)

P_{e} . (26)

Even in this case, the validity of Eq. (25) will be proven in Sec. 3 by comparisons with the results of MC simulations.

2.5. Heuristic model ford =0

The proposed model for the calculation of the apparent optical thickness,τ_{A}, suggests to use
Eq. (14) (τ_{A}_{S S}) for small optical thicknesses (ballistic regime), Eq. (26) (τ_{A}_{E x t}) for intermediate
optical thicknesses (intermediate regime) and Eq. (23) (τ_{A}_{D E}) for large optical thicknesses
(diffusive regime). We can summarize the calculation ofτ_{A}in the three regimes as

τ_{A}=τ_{A}_{S S} forτ_{s}<4 (Eq. (14)), (27)

τ_{A}=min(τ_{A}_{S S}, τ_{A}_{E x t}) forτ_{s}≥4 andτ_{s}^{0}≤1.5 (Eq. (26)), (28)
τ_{A}=min(τ_{A}_{S S}, τ_{A}_{D E}) forτ_{s}≥4 andτ_{s}^{0}>1.5 (Eq. (23)). (29)

The choice of the thresholds delimiting the three regions (i.e. the choice of the limiting values
defined byτ_{s}andτ_{s}^{0}) has been done based on comparisons with the results of gold standard
MC simulations and on the experience gained with the solutions of the DE. The ranges of use
proposed have shown to be trustable for a high number of comparisons performed with the results
of MC simulations. We note that the case given by Eq. (28) never happens whenp(θ)is scarcely
forward peaked (e.g., forg <0.625 andτ_{s} >4 we haveτ_{s}^{0} >1.5). While, with p(θ)highly
forward peaked Eq. (28) is largely used. For instance, wheng=0.9 andg=0.98 the condition
is used for 4< τ_{s} <15 and 4< τ_{s}<75, respectively. The minimum function in Eqs. (28) and
(29) has been inserted on the basis that both experimental and simulated data show a slope of the
curves representingτ_{A}versusτthat never increases compared to that expected fromτ_{A}_{S S}.
2.6. Monte Carlo simulations

We have generated all the MC results presented in the next section making use of two types of scattering functions: the Henyey-Greenstein (HG) model and the Mie theory. Mie theory with λ=632.8nm has been used to simulate the scattering functions for polystyrene latex spheres in water of the experiment in [16] (spheres with average diameterhφi=0.305, 1.091, and 15.8 µm) and the scattering function of Intralipid by assuming the size distribution reported in [25].

Intralipid is a pharmaceutical product for parental nutrition that is also a diffusive material widely used as reference standard for tissue-simulating phantoms [26]. The HG model has been used to generate scattering functions with the asymmetry factorgranging from -0.9 to 0.95. Figure 5 shows some of the scattering functions used in the MC simulations.

The MC simulations have been carried out with a MC code specially designed starting from a
basic elementary code [19,24]. The peculiarity of our code is the use of exact scaling relationships
(see Sec. 3.4.1 in [19]) that allow to use the same simulated trajectory for simultaneously
calculating the detected signal from slabs of different optical thickness. Within the computation
time necessary to calculate the collimated transmittance for a singleτ_{0}, the signal for 30 different
values ofτ≤τ_{0}was reconstructed.

0° 20° 40° 60° 80° 100° 120° 140° 160° 180°

10^{-3}
10^{-2}
10^{-1}
10^{0}
10^{1}
10^{2}
10^{3}

Fig. 5. Examples of the scattering functions employed in the MC simulations.

Since the core of a MC program is actually the generation of the trajectories, the validation of the MC code has been performed by comparing the MC results for the statistics of the positions (e.g. the mean values, afterkscattering events, of the square distance from the source or of the depth), where different orders of scattering occur, with exact analytical expressions [19, 27].

Comparisons showed an excellent agreement, both for HG model and Mie theory. Discrepancies
were within the standard deviation of MC results, even when a large number of trajectories (10^{8})

was simulated, and the relative error was as small as 0.01%. The MC program was also previously compared versus experimental results obtaining excellent agreements [28].

**3.** **Results: comparison heuristic model versus MC**

In this section, the prediction of the heuristic model presented in the previous section is compared
with the results of gold standard MC simulations. This validation is necessary given the heuristic
and approximate nature of the model. In the comparisons we focus our attention to the apparent
optical thicknessτ_{A}for a non-absorbing slab. The caseµ_{a} ,0 is analyzed in Sec. 4.

We present a first set of comparisons for a pencil beam impinging onto a non-absorbing slab
withs_{0} =10 mm andn_{i} =n_{e}. The detector is coaxial atd =0 to the pencil-beam impinging
onto the slab. Figures 6-9 shows the comparisons forτ_{A}versusτup toτ=40 for a HG scattering
function withg=0, 0.8, 0.9, 0.95, respectively. The results are shown for a receiver of radius
r =1 mm for different values ofα(panels (a)) and forα=1^{◦}for different values ofr(panels
(b)). Forg=0 the model shows an excellent agreement with the MC results with an error less
than 1% forα <3^{◦}. Increasing the value ofgthe accuracy decreases showing errors that for
α <3^{◦}are less than 10% forg=0.8, 15% forg=0.9 and 20% forg=0.95. Thus, the model
has excellent performances in describing the collimated transmittance for smallαalthough its
accuracy decreases whenp(θ)becomes highly forward peaked.

We note that, for small values ofαandr, according to the DE we have thatP_{R} P_{s}∝α^{2}r^{2}.
If we look at the behavior in Fig. 6 for largeτ, it reflects this law and we have that the ratio
of two curves with differentαorris ln(^{α}_{α}^{1}

2)^{2}and ln(^{r}_{r}^{1}

2)^{2}respectively. It is also worth to point
out that forg =0 the intermediate regime disappears and the diffusive model starts to work for
τ_{s} >4. This is because with isotropicp(θ)a lower number of scattering events are needed before
diffusion conditions hold. At the same time we also note that, for larger values ofgthe width of
the intermediate regime increases and it shows the largest width for the largestgvalue.

In Fig. 10 the same kind of results have been plotted for the scattering function of Intralipid.

In this case we have an error of the model less than 4 % forα <3^{◦}. Finally, in Fig. 11 we have
plotted the results for polystyrene latex spheres having a Gaussian size distribution with average
diameterhφi=15.8µm and standard deviationSD=2.8µm as in [16]. In this case we have an
error of the model less than 15 % forα <3^{◦}.

The region where we note the largest lacks of the model is the intermediate due to the limitations introduced in the interpolation between diffusive and ballistic region. It is worth to stress that for g<0.625 the intermediate regime vanishes and the heuristic model switches directly from Eq.

(14) forτ_{s}<4 to Eq. (23) forτ_{s}>4 without using the extrapolated model.

Figure 6 and Figs. 7-11 show a different dependence onαin the ballistic regime. While this dependence is not visible in Fig. 6, it is noticeable in Figs. 7-11. Note that this effect does not depend only on thegvalue assumed byp(θ). Indeed, such effect is more related to the forward scattering valuep(θ=0).

In Tab. 2 the accuracy of the heuristic model for the comparisons shown in the previous
figures and for other comparisons done for other types of scattering functions is summarized
by showing the percentage relative error of the model calculated by using MC results as gold
standard values. Table 2 pertains to a pencil beam,d=0 and to a non absorbing medium with
n_{i} =n_{e}. For scattering functions withg<0.95 the discrepancies remain within 15% for small
αvalues (α <≈ 3^{◦}), i.e. theαvalues required for accurate measurements of the extinction
coefficient that depends on the amount light scattered within the field of view of the receiver. The
overview of these results confirms that the accuracy of the proposed model strongly depends
on the characteristics of the scattering function of the medium. The evident relation between
accuracy andgvalue of the scattering function stresses that the forward scattering characteristics
ofp(θ)affects the performances of the model. Table 2 also shows results for scattering functions
withg < 0. Also these cases have a physical meaning (see for instance the nanostructured

(a)

0 5 10 15 20

(b)

0 5 10 15 20 25 30 35 40

0 5 10 15 20

Fig. 6. Comparison heuristic model*vs.* MC results forτ_{A} versusτ in a transmittance
measurement through a non-absorbing slab withs_{0}=10 mm andn_{i} =ne. The detector is
coaxial to the pencil-beam source atd=0. A HG model withg=0 is used forp(θ). The
straight line represents the ideal measurement.

(a)

0 5 10 15 20

(b)

0 5 10 15 20 25 30 35 40

0 5 10 15 20

Fig. 7. As Fig. 6, but for a HG model withg=0.8.

(a)

0 5 10 15 20

(b)

0 5 10 15 20 25 30 35 40

0 5 10 15 20

Fig. 8. As Fig. 6, but for a HG model withg=0.9.

(a)

0 5 10 15

(b)

0 5 10 15 20 25 30 35 40

0 5 10 15

Fig. 9. As Fig. 6, but for a HG model withg=0.95.

(a)

0 5 10 15 20

(b)

0 5 10 15 20 25 30 35 40

0 5 10 15 20

Fig. 10. As Fig. 6, but for the scattering function of Intralipid atλ=632.8 nm.

(a)

0 2 4 6 8 10 12 14 16

(b)

0 5 10 15 20 25 30 35 40

0 2 4 6 8 10 12 14 16

Fig. 11. As Fig. 6, but for a water suspension of polystyrene latex spheres having a Gaussian size distribution withhφi=15.8µm andSD=2.8µm atλ=632.8 nm.

glass-ceramics described in [29, 30]).

All the presented results have been obtained for a slab 10 mm thick that is a typical thickness of a scattering cuvette used with spectrophotometric instrumentation. We point out that, making use of the scaling relationships for radiative transfer [19], these results are also valid for other slab thicknesses provided thatris properly scaled.

Table 2. Accuracy of the heuristic model for a slab 10 mm thick,d=0,µ_{a}=0, andn_{i}=n_{e}.
The percentage relative errors onτ_{A}calculated as percentage difference between the model
and MC simulations are shown for several scattering functionsp(θ). The values of the radius
of the receiver considered arer=0.5, 1, 2, 4, 10 and∞mm.

p(θ) g α **error**

HG -0.9 <3^{◦} <5%

HG -0.5 <3^{◦} <1%

HG 0 <3^{◦} <1%

HG 0.5 <3^{◦} <2%

HG 0.8 <3^{◦} <10%

HG 0.9 <3^{◦} <15%

HG 0.95 <3^{◦} <20%

Mie, Intralipid at 633 nm 0.73 <3^{◦} <4%

Mie,hφi=0.305µm andSD 0 0.665 <3^{◦} <2%

Mie,hφi=1.091µm andSD 0 0.924 <3^{◦} <15%

Mie,hφi=15.8µm andSD=2.8µm 0.915 <3^{◦} <15%

**4.** **Discussion and conclusions**

All the presented results were referred to the refractive index matched case, to the pencil beam source, to the cased =0 and without including a diaphragm on the exit surface. In this section, we first discuss what happen when some of the previous conditions are released.

4.1. Heuristic model forn_{i} ,n_{e}

For intermediate and diffusive regimes the DE model can already account for the casen_{i} ,n_{e}
since the DE solutions include Fresnel reflections [19] through the factorsA(see [19, 21]) and
F(see Eq. (19)) and with the termP_{DE}(r)χ(α). Differently, some changes must be introduced
to the model in the ballistic regime whenn_{i} ,n_{e}. We provide for this case a simple rule that
holds true for small angles under which the refraction due to a refractive index mismatch can be
approximatively accounted by an equivalent receiver with these characteristics:

r_{eqF} =r, d_{eqF} = n_{i}

n_{e}d α_{eqF} =n_{e}

n_{i}α. (30)

Comparisons with MC results showed the suitability of Eq. (30) to account for the effect of a
refractive index mismatch for moderate values ofα(α≤10^{◦}). The influence of the reflection
coefficient for smallαvalues is almost identical for both the ballistic and the single scattering
component of transmitted light.

4.2. Effect of the source

All the results presented pertain to a pencil beam source impinging onto the slab. In real experiments a source beam has finite width and divergence so that can be more precisely approximated by a cylindrical beam or a conical beam. Thus, a more realistic modeling could also include width and divergence of the impinging source beam. We have explored, by means of MC

simulations, the existing differences between the model for a pencil beam and for a cylindrical or
conical beam. We have found that, if the radius of the source beam (cylindrical or conical) is
lower thanrand the divergence of the source beam is lower thanα, then the calculated value for
τ_{A}does not differ significantly from the results obtained with the pencil beam. However, if a light
source with a large divergence is used (e.g. a light emitting diode), the spot of ballistic photons
can be much larger than the receiver area and/or the beam divergence much larger than the
receiver field of view. In this case only a small fraction of the emitted photons can be detected as
ballistic radiation, while all the emitted photons can give a contribution to the received scattered
radiationP_{s}. The apparent optical thicknessτ_{A}can be therefore significantly smaller and the
error on the measured extinction coefficient significantly larger with respect to the ideal case of
a pencil beam light source. It may also occur situations in which the decrease of the received
ballistic radiation due to an increase of the scattering coefficient is more than compensated by an
increase ofPs: in such case a negative extinction would be measured.

4.3. Heuristic model ford >0

When the receiver is at a distancedfrom the slab, the apparent optical thicknessτ_{A}for low
τcan be still calculated by Eq. (14) in the casen_{i} =n_{e}(for the casen_{i} ,n_{e}see Sec. 4.1),
withK1(p(θ),s0,d, α,r)given by Eq. (11), since this expression is still valid. Whilst, for the
intermediate and diffusive regimes we must introduce changes compared to the cased=0 (the
DE solution is only available for the receiver on the exit surface). Making use of geometrical
and physical considerations we state that both forr ≥dtanαandr≤dtanαthe expression for
P_{s}(r, α,d >0)can be approximated to

P_{s}(r, α,d >0)P_{s}(r_{eqd}, α_{eqd},d=0)=P_{DE}(r_{eqd},d =0)χ(α_{eqd}), (31)
with

reqd(α,r,d)=max(r,dtanα), and α_{eqd}(α,r,d)=min(α,arctan(r/d)). (32)
The above equation can be applied for the calculation ofτ_{A}for intermediate and diffusive regimes.

Comparisons with MC results showed that Eqs. (31) and (32) work very well. The discrepancies observed ford >0 were a little smaller than ford=0.

4.4. Use of the Heuristic model in presence of a diaphragm on the exit surface
A simple and widely used way to limit the effects of the scattered light on the measured power in
a transmittance measurement is to introduce a diaphragm at the exit surface of the sample in
order to reduce the actual surface from which scattered light can be received. This effect can be
directly included inside the heuristic model. The scattered power in the diffusive regime can still
be calculated by Eq. (31), withα_{eq}given by Eq. (32), but withr_{eq} given by

r_{eq}(α,r,d,R_{d})=min

R_{d},max(r,dtanα), (33)

with R_{d} radius of the diaphragm. We note that in the ballistic regime for the geometries of
practical interest the diaphragm does not affect the calculation ofτ_{A}.

4.5. Case of an absorbing medium

In order to test the accuracy of the model in presence of absorption we have done a set of
simulations withΛ=0.9 (data not shown). The presence of absorption reduces the amount ofP_{s}
and asµ_{a}increasesτ_{A}tends toτ(forΛ=0,τ_{A}=τ). For smallτthe model works still very
well, while for highτ, due to the intrinsic limitations of the DE in high absorbing media, the
accuracy decreases. ForΛ=0.9 andα <3^{◦}the model shows errors less than 20%, while we had
errors less than 15% forΛ=0.