Heuristic model for ballistic photon detection in collimated transmittance measurements

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.


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][4][5][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][10][11][12][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][14][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.

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 parameter τ is usually experimentally assessed using 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 where P R is the measured power by the detector and P s represents the spurious scattered power collected in the field of view of the receiver.Thus, the presence of P s generates a τ A different from the desired τ, and finally leads to a inexact µ e = τ A /s 0 τ/s 0 estimation.We note the fact 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 s = µ s (1 − g), and µ a , respectively; with scattering function (phase function) p(θ) (normalized to 1 within the whole solid angle), anisotropy factor g, and albedo of single scattering Λ = µ s µ e .We denote as τ = µ e s 0 the optical thickness of the medium, τ s = µ s s 0 the reduced scattering optical thickness and, τ s = µ s s 0 and τ a = µ a s 0 the scattering and absorption optical thicknesses of the medium, respectively.

Laser
At this point, it is important to note that the apparent optical thickness τ A , obtained from collimated transmittance measurements, when plotted versus τ shows 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 diameter φ = 15.8 µm and standard deviation SD = 2.8 µm.The results are reported in Fig. 2, where τ A is plotted versus τ for three values of α.The calculations were for n i = n e and Λ =1.To underline deviations with respect to the ideal measurement the straight line τ 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 of τ.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 to τ with a constant slope that can be significantly lower than 1 for media with p(θ) highly forward peaked as in our example.
For large τ, 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 for τ sufficiently large we are in the diffusive regime (Fig. 2, the region for τ >≈ 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, α and d) and on p(θ).So far, no previous mathematical models are available in the literature to describe the intermediate regime.
Note that by increasing µ a we decrease the amount of scattering, because photons with large paths length are preferentially absorbed and not detected by the receiver (s 0 is the shortest possible path length).Thus, ideally, for µ s µ a we have that τ A tends to τ.Eventually, the tutorial example gives us an idea on the complex dependence of τ 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 describe τ A as a function of τ 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 (for τ s ≥≈4 and τ s >≈1.5); 3) the intermediate regime between ballistic and diffusive.Note that the diffusive regime must be defined making use of τ s since the diffusive propagation only depends on µ s and not on µ 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 of τ A versus τ.We stress that this representation of the data is related to the importance of the measured τ A whenever extinction measurements are carried out.
The model is presented for the pencil beam source, for d = 0 and for the refractive index matched case (n i = n e ).In Sec. 4 these conditions are discussed to extend the solution to a more general case.

Source
Receiver 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 power P R is spoiled by presence of the scattered light P s .Such term P s can be expressed as a sum of detected light powers, P si , received after i = 1, 2 • • • k scattering events, i.e., With reference to Fig. 3, the infinitesimal contribution to the light detected after k scattering events dP sk can be calculated as [14,19] dP sk = P e exp(−µ e l 1 )µ s dl Equation ( 4) comes from an iterative calculation for each scattering contribution, starting from the first order to the scattering of order k.The term P e exp(−µ e l 1 ) is the power's fraction that reaches the volume element of thickness dl 1 at distance l 1 from the injection point dues to scattering and absorption interactions through l 1 .Then, the power's fraction scattered within the infinitesimal solid angle dΩ 1 around the scattering direction θ 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 the k 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 of P 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 conditions P sk is mainly given by photons that follow trajectories with small θ i so that the total length of a single trajectory can be approximated to k+1 i=1 l i s 0 , and P sk can be written as where is a coefficient that depends on the geometry of the scatters, through p(θ), and on the geometry of the measurement (defined e.g. by s 0 , d, α or r) through the integration boundaries (see below explicit calculations for k = 1), but that, under the SAA, does not depend on µ e and µ s .Inserting Eqs. ( 1) and ( 5) in Eq. ( 3), the total transmitted power becomes This equation implies that, starting from P R P 0 = P e exp(−τ) when τ s → 0, the weight of the scattered light increases with the increment of µ s and τ s .Thus, increasing τ 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 with a relative error, ε, When τ s 1 in ε, i.e. the number of scattering events tends to 0 due to a very small µ s or a very small s 0 , survives only the single scattering contribution, i.e., lim where Eq. ( 6) gives 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)(11)(12) hold also when d 0.
We note that if the receiver is sufficiently large (r ≥ (s 0 + d) tan α) to detect all light scattered with θ ≤ α then Eq. ( 11) simply becomes and K 1 (.) represents the fraction of scattered light within α.The proposed model always uses Eq. (11).Finally, using Eqs.( 8), ( 9) and ( 10), a suitable model, τ A S S , for small τ can be summarized as 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 of τ, τ A linearly increases with τ.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 of α are employed.Equation ( 14) allows us to express the received power P R as that can be considered as a modified Beer Lambert's law with an apparent extinction coefficient, that practically is the extinction coefficient obtained when only the fraction of scattered light with θ > α is 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].

Model for large optical thickness: diffusive regime
The behavior of τ A versus τ (Fig. 2) shows a kind of saturation for large values of τ.The low increment of τ A versus τ implies 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 of τ is 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 radius r is given (obtained from Eq. (4.39) in [19])) by with and µ e f f = 3µ a µ s , z s = 1/µ s , z e = 2AD, D = 1/(3µ s ), A coefficient for the extrapolated boundary condition [19,21] and P 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 angles θ e ≤ α and 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 function F(θ e ) as [19,22] F(θ e ) = 1 4π with R 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 power P R (r, α) that emerges within α and 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 of P DE (r) within α, i.e. [18,19] P R (r, α) where We note that for n i = n e , Eq. ( 21) assumes the simple form Finally, for large τ the apparent optical thickness is 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 view α could 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 small τ values, 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.

Model for intermediate optical thickness: the intermediate regime
Equation ( 14) works pretty well for τ s <≈ 4, whilst Eq. ( 23) works very well for τ s >≈ 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 low τ s values.We note that the solution of the DE (Eq.( 17)) provides physically correct results only for τ s > 1 since it is obtained placing an isotropic source inside the medium at a depth equal to z s = 1/µ s .The extrapolated model here presented can also work for τ s < 1, where the DE solution cannot be used.Therefore, we implement an extrapolation of the DE solution for τ s <≈ 1.5 and we represent the transmitted scattered light, P s E x t (r, α), with a power law, i.e with the parameters a(r, α) and b(r, α) obtained by fitting Eq. ( 24) to P DE (r) χ(α) in the range 1.2 ≤ τ s ≤ 2. The choice of the power law for P s E 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 .We also stress that a(r, α) and b(r, α) depend on µ a , while they are independent of p(θ).As a matter of fact, since the dependence of P DE (r) χ(α) on α for d = 0 is by a multiplying factor, then b(r, α) actually is independent of α when d = 0.In Tab.1 the values of a(r, α) and b(r) have been shown for a slab 10 mm thick with n i = n e , µ a = 0 and d = 0. Figure 4 shows examples of P s E x t (r, α) obtained from the fitting of P DE (r) χ(α) for a slab with µ a = 0, n i = n e , α = 90 • and r = 0.5, 1, 2, 4, 10 and ∞ mm.The results for lower values of α can be obtained simply by scaling the results for α = 90 • with the suitable χ(α).The values obtained for a and b in the figure for P s E x t (r, α) are those shown in Tab.1 for α = 90 • .Thus, the total received power, P R (r, α), is assumed to be Table 1.Coefficients a and b of Eq. ( 24) calculated for a slab 10 mm thick with n i = n e , µ a = 0 and d = 0. and the corresponding apparent optical thickness, τ A E x t , Even in this case, the validity of Eq. ( 25) will be proven in Sec. 3 by comparisons with the results of MC simulations.

Heuristic model for d = 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 of τ A in the three regimes as for τ s < 4 (Eq.( 14)), ( 27) for τ s ≥ 4 and τ s ≤ 1.5 (Eq.( 26)), ( 28) for τ s ≥ 4 and τ s > 1.5 (Eq.( 23)).( 29) The choice of the thresholds delimiting the three regions (i.e. the choice of the limiting values defined by τ s and τ s ) 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 when p(θ) is scarcely forward peaked (e.g., for g < 0.625 and τ s > 4 we have τ s > 1.5).While, with p(θ) highly forward peaked Eq. ( 28) is largely used.For instance, when g = 0.9 and g = 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 representing τ A versus τ that never increases compared to that expected from τ A S S .

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 = 632.8nmhas been used to simulate the scattering functions for polystyrene latex spheres in water of the experiment in [16] (spheres with average diameter φ = 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 factor g ranging 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 single τ 0 , the signal for 30 different values of τ ≤ τ 0 was reconstructed.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, after k scattering 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].

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 thickness τ A for a non-absorbing slab.The case µ a 0 is analyzed in Sec. 4.
We present a first set of comparisons for a pencil beam impinging onto a non-absorbing slab with s 0 = 10 mm and n i = n e .The detector is coaxial at d = 0 to the pencil-beam impinging onto the slab.Figures 6-9 shows the comparisons for τ A versus τ up to τ =40 for a HG scattering function with g =0, 0.8, 0.9, 0.95, respectively.The results are shown for a receiver of radius r = 1 mm for different values of α (panels (a)) and for α = 1 • for different values of r (panels (b)).For g =0 the model shows an excellent agreement with the MC results with an error less than 1% for α < 3 • .Increasing the value of g the accuracy decreases showing errors that for α < 3 • are less than 10% for g = 0.8, 15% for g = 0.9 and 20% for g = 0.95.Thus, the model has excellent performances in describing the collimated transmittance for small α although its accuracy decreases when p(θ) becomes highly forward peaked.
We note that, for small values of α and r, according to the DE we have that P R P s ∝ α 2 r 2 .If we look at the behavior in Fig. 6 for large τ, it reflects this law and we have that the ratio of two curves with different α or r is ln( α 1 α 2 ) 2 and ln( r 1 r 2 ) 2 respectively.It is also worth to point out that for g = 0 the intermediate regime disappears and the diffusive model starts to work for τ s > 4.This is because with isotropic p(θ) a lower number of scattering events are needed before diffusion conditions hold.At the same time we also note that, for larger values of g the width of the intermediate regime increases and it shows the largest width for the largest g value.
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 % for α < 3 • .Finally, in Fig. 11 we have plotted the results for polystyrene latex spheres having a Gaussian size distribution with average diameter φ = 15.8 µm and standard deviation SD =2.8 µm as in [16].In this case we have an error of the model less than 15 % for α < 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) for τ s <4 to Eq. ( 23) for τ s >4 without using the extrapolated model.
Figure 6 and Figs.7-11 show a different dependence on α in 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 the g value assumed by p(θ).Indeed, such effect is more related to the forward scattering value p(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 with g < 0.95 the discrepancies remain within 15% for small α values (α <≈ 3 • ), i.e. the α values 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 and g value of the scattering function stresses that the forward scattering characteristics of p(θ) affects the performances of the model.Table 2 also shows results for scattering functions with g < 0. Also these cases have a physical meaning (see for instance the nanostructured    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 that r is properly scaled.
Table 2. Accuracy of the heuristic model for a slab 10 mm thick, d = 0, µ a = 0, and n i = n e .The percentage relative errors on τ A calculated as percentage difference between the model and MC simulations are shown for several scattering functions p(θ).The values of the radius of the receiver considered are r = 0.5, 1, 2, 4, 10 and ∞ mm.

Discussion and conclusions
All the presented results were referred to the refractive index matched case, to the pencil beam source, to the case d = 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.

Heuristic model for n i n e
For intermediate and diffusive regimes the DE model can already account for the case n i n e since the DE solutions include Fresnel reflections [19] through the factors A (see [19,21]) and F (see Eq. ( 19)) and with the term P DE (r) χ(α).Differently, some changes must be introduced to the model in the ballistic regime when n 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: Comparisons with MC results showed the suitability of Eq. ( 30) to account for the effect of a refractive index mismatch for moderate values of α (α ≤ 10 • ).The influence of the reflection coefficient for small α values is almost identical for both the ballistic and the single scattering component of transmitted light.

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 than r and the divergence of the source beam is lower than α, 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 radiation P s .The apparent optical thickness τ 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 of P s : in such case a negative extinction would be measured.

Heuristic model for d > 0
When the receiver is at a distance d from the slab, the apparent optical thickness τ A for low τ can be still calculated by Eq. ( 14) in the case n i = n e (for the case n i n e see Sec. 4.1), with K 1 (p(θ), s 0 , 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 case d = 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 for r ≥ d tan α and r ≤ d tan α the expression for P s (r, α, d > 0) can be approximated to The above equation can be applied for the calculation of τ A for intermediate and diffusive regimes.
Comparisons with MC results showed that Eqs.(31) and (32) work very well.The discrepancies observed for d > 0 were a little smaller than for d = 0.

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), with α eq given by Eq. ( 32), but with r eq given by 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 of τ A .

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 with Λ = 0.9 (data not shown).The presence of absorption reduces the amount of P s and as µ a increases τ A tends to τ (for Λ = 0, τ A = τ).For small τ the model works still very well, while for high τ, due to the intrinsic limitations of the DE in high absorbing media, the accuracy decreases.For Λ = 0.9 and α < 3 • the model shows errors less than 20%, while we had errors less than 15% for Λ = 0.

Conclusions
We have presented a heuristic CW model that can calculate the apparent optical thickness τ A measured by a receiver with radius r, field of view α and placed at a distance d from a slab of scattering material illuminated by a pencil beam.The model can also account for the refractive index mismatch between slab and external (n i n e ) and the presence of a diaphragm placed at the external surface of the slab.We can summarize the performances of the model as follow.
Model for the ballistic regime (τ s < 4).The model is based on the calculation of the single scattering contribution to the detected power.It requires a detained information on the scattering function of the medium in the forward direction and can rigorously account of the distance d.The effect of reflections is only accounted with the refraction at the external boundary by an equivalent receiver of radius r eqF that shows excellent performances for α <≈ 10 • .The model works only for a pencil beam.
Model for diffusive regime (τ s ≥ 4 and τ s > 1.5).The model is based on the solution of the DE for the received power by a receiver placed onto the external surface of the slab.The model also works when n i n e .The effect of the distance d is accounted by an equivalent receiver placed at d = 0 with r eqd and α eqd given by Eq. (32).Although the reasoning used to introduce the equivalent receiver is approximated, the comparisons with the results of MC simulations have shown that the performances of the model for d > 0 are even better than for d = 0.The model can also account for a diaphragm placed at the external surface of the slab.
Model for the intermediate regime (τ s ≥ 4 and τ s ≤ 1.5).In this case, we still use the DE solution, but only extrapolated results from larger optical thicknesses are indeed used.For what concern the validity of the model for d > 0, n i n e and for other kinds of source beam, the same comments done for large optical thicknesses are still valid.This calculation is used for forward peaked scattering functions and it plays a relevant role only for g >≈ 0.9.
Thus, the strength points of the model can be so summarized: a) It works in a wide range of experimental conditions that embrace a large set of geometrical configurations, optical thicknesses of the slab (the whole regime of propagation) and scattering functions; b) It is largely more accurate than the previous models of [13,14] and it holds in a wider regime of propagation; c) The mathematical formulas are simple especially in the intermediate regime.
This work emphasizes that the transition ballistic-diffusive is strongly related to the characteristics of the experimental setup through parameters such as α and r for instance.Actually, it is impossible to provide a general rule or characteristic for such transition since it is intrinsically affected by p(θ) and thus deeply related to the single scattering properties of the medium investigated.This means that the actual transition from ballistic to diffusive of our results must be identified given the geometry of the setup and the scattering properties of the medium.The proposed model can be used for such purpose.In our opinion, this point was not clearly considered in the previous literature dedicated to the transition ballistic-diffusive.
The model can also be used to estimate real detectability limits of ballistic radiation in a wide range of experimental conditions and can thus become a practical tool for many experimental investigations.Finally, a mention on the computation time that is negligible for all the regimes of propagation as we can argue from the mathematical structure of the proposed formulas.

Fig. 1 .
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; n e external refractive index (e.g.air).

Fig. 2 .
Fig. 2. Simulated MC experiments for a scattering medium constituted by a suspension of polystyrene latex spheres in water ( φ = 15.8 µm).The figure shows τ A versus τ for three α values; s 0 = 10 cm, d = 70 cm and r = 2 cm.The straight line representing the ideal measurement is also shown.

Fig. 4 .
Fig.4.Examples of P s E x t (r, α) obtained from the fitting of P DE (r) χ(α) for a slab 10 mm thick with µ a = 0, n i = n e , α = 90 • and for several values of r.P s E x t (r, α) and P DE (r) χ(α) are normalized to the power P e injected inside the slab and thus they have not units.

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

Fig. 6 .
Fig.6.Comparison heuristic model vs. MC results for τ A versus τ in a transmittance measurement through a non-absorbing slab with s 0 = 10 mm and n i = n e .The detector is coaxial to the pencil-beam source at d = 0.A HG model with g = 0 is used for p(θ).The straight line represents the ideal measurement.