# Mean path length inside non-scattering refractive objects

Matt Majic<EMAIL_ADDRESS>Walter R. C. Somerville
<EMAIL_ADDRESS>Eric C. Le Ru<EMAIL_ADDRESS>The MacDiarmid
Institute for Advanced Materials and Nanotechnology, School of Chemical and
Physical Sciences, Victoria University of Wellington, PO Box 600 Wellington,
New Zealand

###### Abstract

It has recently been argued that the geometric-optics mean path length of rays
inside a refractive object under Lambertian illumination is independent of the
scattering strength of the medium [Savo et al., Science 358, 765 (2017)]. We
here show that it is in fact different in the case of zero-scattering. We
uncover and explain the role of trapped ray trajectories in creating this
unexpected discontinuity from zero- to low-scattering. This allows us to
derive new analytic results for the zero-scattering mean path length of simple
refractive shapes. This work provides a fresh perspective on the study of path
length inside refractive objects, with possible applications in for example
the study of scattering by large particles or the design of optical systems.

Finding the mean chord length for a random distribution of lines in a given
object is a natural question in many areas of physics. It is a seemingly
complex task from a mathematical perspective, since one should consider the
spatial and angular distribution of lines as well as how they intersect the
surface of the object. For convex bodies the answer is however surprisingly
simple; given by the mean chord length theorem, which has been known for more
than a century [1]. It states that the mean chord length $\langle C\rangle$ is
independent of the shape of the object, and only depends on the ratio of
volume $V$ to surface area $\Sigma$ as $\langle C\rangle=4V/\Sigma$. Proofs
from various perspectives have been given [2, 3, 4]. It was only fairly
recently shown that this theorem can be generalized further to the study of
random walks in diffusive objects. The mean path length theorem [5] states
that the mean path length is still simply $\langle L\rangle=4V/\Sigma$; this
is independent of both shape and the scattering/diffusive properties of the
medium. The validity extends across many fields as it is valid for any random
walk inside an object, and is particularly relevant to geometric optics within
a closed scattering medium. One important condition for this theorem is that
the entrance point and initial direction are uniformly and isotropically
distributed, which in optics is equivalent to a Lambertian illumination [2].

Path length distributions and mean path length are central to the design of
many optical systems where a ray optics description can be used. They can be
used for calculating the optical properties of absorbing and scattering media
[6, 7], refractive granular media in pharmaceutical powders [8], for solar
cell design [9, 10, 11], random lasing [12], and integrating spheres [13, 14].
Ray tracing can also be combined with diffraction effects to calculate the
electromagnetic scattering properties of large particles in models such as the
geometric optics approximation and physical optics model [15, 16, 17, 18, 19,
20], or anomalous diffraction theory [21, 22, 23]. These have for example been
applied to the study of ice crystals [17, 18, 20] for climate modeling. The
zero-scattering mean path length is directly related to the orientation-
averaged absorption cross-sections of large absorbing particles [24, 25, 26].
Path length distributions have also been used to derive the scattering phase
function of an object analytically [27].

In most of these applications, the object has a different refractive index
than the surrounding medium. Rays are then refracted at the boundary and may
also be reflected internally or externally, which may increase the path length
of some internal rays. Even then, it has been argued recently [28, 29] that
the mean path length invariance remains valid for scattering samples, and is
simply modified by a factor $s^{2}$:

$\displaystyle\langle L\rangle=\frac{4V}{\Sigma}s^{2}$ (1)

for any 3D convex body, independent of the scattering properties of the
sample. A justification of Eq. (1) was given assuming a thermodynamic
equilibrium/equipartition [28, 29], and following energy conservation
arguments drawing on the discussion in Ref. [30]. In this letter, we focus on
the mean path length in the low-scattering limit. The equipartition assumption
and Eq. (1) also apply, but we will show that the zero scattering case is
different, resulting in a discontinuous transition between low- and zero-
scattering for some geometries. This is related to the existence of trapped
paths (similar to the whispering gallery trajectories inside a sphere), which
cannot be populated in the strict absence of scattering. To further support
this argument, we have derived a number of new analytic expressions for the
zero-scattering mean path length inside simple 2D and 3D refractive objects.
These demonstrate the discontinuity at zero-scattering and their derivations
support the physical interpretations in terms of trapped rays. Less symmetric
geometries are also studied using Monte Carlo ray tracing simulations [29],
allowing us to discuss the generality of this discontinuity, its sensitivity
to imperfections, and its relevance to applications.

Mean path length in a refractive sample.

Figure 1: A light ray being refracted as it passes from medium 1 to medium 2.
The darker region in medium 2 is not accessible from rays entering from medium
1.

We consider as shown in Fig. 1 a non-absorbing convex body $V_{2}$ embedded in
an outside medium $V_{1}$ with refractive indices $n_{2}>n_{1}$ respectively
and define $s=n_{2}/n_{1}>1$. At this stage, we do not exclude the possibility
that medium 2 is a scattering medium. We study the trajectory of light rays
within the geometric-optics approximation, as they undergo stochastic
refraction/reflection at the interface. Consider a ray incident on the
surface, with an angle of incidence $\theta_{1}$ to the surface normal, and
rotated by $\phi$ around it. The ray may be refracted at an angle $\theta_{2}$
with a probability $T_{12}(\theta_{1})$ with $\sin\theta_{1}=s\sin\theta_{2}$
(Snell’s law), while the azimuthal angle $\phi$ is unaffected. By optical
reciprocity, the probabilities of transmission at complementary angles are
identical: $T_{12}(\theta_{1})=T_{21}(\theta_{2})$. Similar laws apply to
internal rays hitting the object surface with internal $\theta^{\prime}_{2}$
and external $\theta^{\prime}_{1}$ angles. Since $n_{2}>n_{1}$, angles
$\theta^{\prime}_{2}$ above the critical angle $\theta_{c}=\mathrm{asin}(1/s)$
have zero transmission: there is total internal reflection (TIR).

As for the mean path (or chord) length theorems, we assume that the
illumination of the object from the outside is Lambertian. The surface
irradiance is therefore uniform and the incident angles follow the probability
distribution $p(\theta_{1})=2\cos\theta_{1}\sin\theta_{1}=\sin(2\theta_{1})$
($0\leq\theta_{1}<\pi/2$) in 3D [2]. Because Lambertian illumination maximizes
entropy and energy is conserved in the scattering process, the ensemble of
external rays reflecting and internal rays exiting must also follow a
Lambertian distribution (otherwise entropy would have been decreased). Then,
since incident and outgoing rays have the same distribution, for every
internal ray making an angle $\theta^{\prime}_{2}<\theta_{c}$ to the normal
(and reflected with a probability $p_{2}=1-T_{21}(\theta^{\prime}_{2})$),
there is an incident external ray that is externally reflected at the Snell-
matching $\theta^{\prime}_{1}$ with the same probability
$p_{1}=1-T_{12}(\theta^{\prime}_{1})=p_{2}$, i.e. there is a one-to-one
correspondence between these rays (see Sec. S.I [31] for more detail). This
allows us to ignore all internally reflected rays with
$\theta^{\prime}_{2}<\theta_{c}$ in mean path length calculations, if we also
ignore any reflection from the outside (dashed rays in Fig. 1). These
externally reflected rays would normally have zero path length, but if they
were to transmit inwards, they would have exactly the path length and angular
distribution of the rays that reflect from the inside for
$\theta^{\prime}_{2}<\theta_{c}$. See also Sec. S.II for an explicit example
of this cancellation in simple geometries. This is a crucial result for
refractive objects, as it simplifies dramatically the calculations. Note
however that the contribution of rays with $\theta^{\prime}_{2}>\theta_{c}$
(TIRs), whose distribution is not specified, must still be accounted for. This
result also highlights that Eq. (1) for scattering media assumes that these
externally reflected rays with $L=0$ are included in the statistics, an
important point that was not made explicit in Refs. [28, 29]. The mean path
length not counting $L=0$ rays can be simply deduced as $\langle
L_{L>0}\rangle={\langle L\rangle}/{\bar{T}_{12}}$, where $\bar{T}_{12}$ is the
Lambertian-averaged transmission [32].

Following these considerations, we may express the mean path length in the
object as

$\displaystyle\langle
L\rangle=\int_{\Sigma}\frac{\mathrm{d}\Sigma}{\Sigma}\int\frac{\mathrm{d}\phi}{2\pi}\int_{0}^{\frac{\pi}{2}}\mathrm{d}\theta_{1}L(\mathbf{r},\theta_{1},\phi)\sin(2\theta_{1}).$
(2)

where $L(\mathbf{r},\theta_{1},\phi)$ denotes the total ray path length for a
given entry point $\mathbf{r}$ and incidence angles, including possible total
internal reflections until it reaches the surface with
$\theta^{\prime}_{2}<\theta_{c}$ (thanks to the cancellation between
internal/external reflections). In the absence of scattering and total
internal reflections, $L$ then coincides with the chord length $C$. In the
presence of scattering $L$ should be understood as an average over all
possible scattering paths, which renders this ray approach difficult.

We have applied this method to several standard geometries in the non-
scattering case and obtained analytical results for simple shapes and
numerical results for more complex shapes. The most surprising outcome is that
the zero-scattering mean path length $\langle L^{0}\rangle$ is different
(smaller) than the scattering mean path length $\langle L\rangle$ for some
geometries, which results in a discontinuous transition at zero-scattering. We
first discuss further this counter-intuitive result as it will provide
physical insight into its origin and provide an alternative method of
calculating $\langle L^{0}\rangle$ in special cases.

Transition between the low- and zero-scattering regimes. To understand how
this discontinuous transition arises, we will attempt to connect the two
different approaches for the scattering (thermodynamic/equipartition) and non-
scattering (ray optics) cases. Specifically, we here derive the mean path
length $\langle L\rangle$ in the low-scattering limit from $\langle
L^{0}\rangle$ using a ray-optics argument. At the center of this discussion is
the existence of trapped rays. These undergo successive TIRs and cannot
escape, similar to propagating modes in an optical fiber or whispering gallery
modes in dielectric spheres. Note that these trajectories may be repeating (as
the optical modes) or chaotic. Because of reciprocity, these rays cannot be
excited from outside in the ray-optics framework and therefore are irrelevant
to $\langle L^{0}\rangle$. 111Note that arbitrarily long path lengths may
still exist, for example in the 2D ellipse, where rays that refract in at
almost the critical angle at the tips will undergo a large number of total
internal reflections before inevitably refracting out. In terms of the
elliptical billiard table [33], the rays would enter at a trough on the phase
portrait that touches the line at the critical angle. However, these are not
strictly trapped. But if scattering is present, then there is a probability
that some rays are scattered into and out of trapped trajectories. For very
low scattering, this probability is small and one might expect that it does
not affect the mean path length. However, trapped rays exhibit very long path
lengths because scattering is low. It is this product of a small probability
by a large path length that may result in a finite, non-zero contribution to
$\langle L\rangle$ even in the limit of zero-scattering, but not for zero-
scattering, hence the discontinuity.

To be more quantitative, we denote the scattering coefficient $\alpha$ and
assume that the scattering mean free path $l_{\alpha}=1/\alpha$ is much
greater than $\langle L^{0}\rangle$, i.e. low-scattering limit. For
simplicity, we will here consider special cases where the probability of
scattering into a trapped trajectory, denoted $P_{T}$, is independent of the
position of the scattering event. A more general case is discussed in Sec.
S.III. The average probability of a ray scattering is $\alpha\langle
L^{0}\rangle\ll 1$. Since the average path length for non-trapped rays is of
order $\langle L^{0}\rangle$, scattering into them results in negligible
changes to path length (of order $\alpha\langle L^{0}\rangle^{2}$). In
contrast, a ray will only escape a trapped path if it is scattered again,
which results in an average path length $l_{\alpha}\gg\langle L^{0}\rangle$
for trapped rays. Moreover, scattering may occur into another trapped path
with probability $P_{T}$, which increases the path length by $l_{\alpha}$
again until the next scattering event. Summing, we obtain the mean path length
for trapped rays as:

$\displaystyle\langle
L^{T}\rangle=l_{\alpha}+P_{T}l_{\alpha}+P_{T}^{2}l_{\alpha}+\ldots=\frac{l_{\alpha}}{1-P_{T}}.$
(3)

$\langle L^{T}\rangle\gg\langle L^{0}\rangle$ but this is compensated by the
small probability $\alpha\langle L^{0}\rangle P_{T}$ of scattering into a
trapped trajectory. We can now add this contribution to $\langle L^{0}\rangle$
to obtain the low-scattering mean path-length:

$\displaystyle\langle L\rangle\approx\langle L^{0}\rangle+\left[\alpha\langle
L^{0}\rangle P_{T}\right]\langle L^{T}\rangle=\frac{\langle
L^{0}\rangle}{1-P_{T}}.$ (4)

This derivation provides an explanation for the discontinuity at zero-
scattering, which is due to the second term, related to trapped trajectories.
Eq. (4) moreover provides a simple method of deducing $\langle L^{0}\rangle$
analytically for objects where $P_{T}$ is independent of position and angle,
which includes many objects with faceted sides. An important special case is
for objects where no trapped rays can be supported ($P_{T}=0$), for which
$\langle L^{0}\rangle=\langle L\rangle$. Amongst these are objects with a
smaller refractive index than the embedding medium ($s<1$). The consideration
of trapped paths also suggests a link between this problem and the theory of
“billiards” in classical mechanics [33]. In particular, ergodic shapes will
also automatically have $P_{T}=0$ (since every ray samples the entire phase
space) and therefore $\langle L^{0}\rangle=\langle L\rangle$.

Analytic results. To further illustrate this discussion, we now provide a
collection of newly derived analytic results for $\langle L^{0}\rangle$ in
simple 3D and 2D geometries. The main 3D geometries that we considered are
summarized in Fig. 2, where their parameters are defined. The advantage of
these ideal geometries is that the derivations illustrate how concepts such as
trapped rays affect the mean path length. The derived $\langle L^{0}\rangle$
as a function of $s$ for all 3D geometries are summarized and compared to the
scattering case in Fig. 3. To calculate $\langle L^{0}\rangle$, we use Eq.
(2), rewritten in terms of the inside angle as:

$\displaystyle\langle
L^{0}\rangle=s^{2}\int_{\Sigma}\frac{\mathrm{d}\Sigma}{\Sigma}\int\frac{\mathrm{d}\phi}{2\pi}\int_{0}^{\theta_{c}}\mathrm{d}\theta_{2}L(\mathbf{r},\theta_{2},\phi)\sin(2\theta_{2}).$
(5)

Figure 2: Main geometries considered in this work, (a) the 2D strip and 3D
slab, (b) the circle and sphere, (c) the infinite cylinder and (d) the cube
and cuboid. We also treat the square, rectangle and infinite square rod.

We start with the simplest case of an infinite slab of width $a$, Fig. 2(a).
In this case, $L(\mathbf{r},\theta_{2},\phi)$ only depends on $\theta_{2}$.
Moreover, since we can ignore probabilistic reflections and it is not possible
to excite TIRs from the outside, $L$ is the same as the chord length so we
have $L=a/\cos\theta_{2}$ and the mean is calculated as

$\displaystyle\langle
L_{\text{slab}}^{0}\rangle=2as^{2}(1-\cos\theta_{c})=2as^{2}\left(1-\sqrt{1-\frac{1}{s^{2}}}\right).$
(6)

This could also have been deduced from Eq. (4) since the trapping probability
is uniform: $P_{T}=\cos\theta_{c}$. $\langle L_{\text{slab}}^{0}\rangle$
decreases with $s$ (see Fig. 3) and is less than the mean path length for a
slab with scattering, $\langle L_{\text{slab}}\rangle=2as^{2}$.

For a sphere of radius $a$, Fig. 2(b), $L(\mathbf{r},\theta_{2},\phi)$ again
depends on $\theta_{2}$ only and it is not possible to excite TIRs from the
outside, so we have $L=2a\cos\theta_{2}$ (the chord length) and integrating
Eq. (5):

$\displaystyle\langle L_{\text{sphere}}^{0}\rangle=$
$\displaystyle\frac{4a}{3}s^{2}\left[1-\left(1-\frac{1}{s^{2}}\right)^{\nicefrac{{3}}{{2}}}\right].$
(7)

This expression appears for example in the absorption cross section for large
weakly absorbing spheres [25, 19]. For comparison, the mean path length for a
sphere with scattering is $\langle L_{\text{sphere}}\rangle=(4/3)as^{2}$. We
cannot here use Eq. (4) because the probability of trapping $P_{T}$ depends on
position: trapping is more likely for scattering events close to the sphere
surface.

Figure 3: Comparison of the $s$ dependence of the zero-scattering mean path
length $\langle L^{0}\rangle$ for 3D objects for which analytical expressions
were derived. All values are normalized to the mean chord length $\langle
C\rangle=4V/\Sigma$. The scattering case $\langle L\rangle=s^{2}\langle
C\rangle$ is shown as a dashed line.

For the cube, Fig. 2(d), there are three regimes depending on $s$. First, we
can show that for $s\leq\sqrt{3/2}$, no trapped rays exist, hence $P_{T}=0$
and

$\displaystyle\langle L_{\text{cube}}^{0}(s\leq\sqrt{3/2})\rangle=\langle
L_{\text{cube}}\rangle=\frac{2}{3}as^{2}.$ (8)

For $s\geq\sqrt{2}$, we may use Eq. (5); the problem is simplified by the fact
that all rays exit the opposite face of the cube. Some will total-internally
reflect on an adjacent face (see Fig. 2(d)), some will exit straight through,
but in both cases, the path length is given as $L=a/\cos\theta_{2}$, therefore
we obtain

$\displaystyle\langle
L^{0}_{\text{cube}}(s\geq\sqrt{2})\rangle=2as^{2}\bigg{(}1-\sqrt{1-\frac{1}{s^{2}}}\bigg{)}.$
(9)

In the intermediate case, $\sqrt{3/2}<s<\sqrt{2}$, calculating $\langle
L_{\text{cube}}^{0}\rangle$ via Eq. (5) is rather technical, see Sec. S.IV. We
present in Sec. S.V a simpler derivation using Eq. (4), which applies because
the trapping probability $P_{T}$ is again independent of the location of the
scattering event. Both result in:

$\displaystyle\langle L_{\text{cube}}^{0}(\sqrt{3/2}<s<\sqrt{2})\rangle=$ (10)
$\displaystyle{\frac{4as^{2}}{\pi}}\left(\sin^{-1}(s^{2}-1)-\sqrt{1-\frac{1}{s^{2}}}~{}\sin^{-1}(2s^{2}-3)\right).$

A similar derivation can be carried out for a cuboid of edges $a,b,c$ and the
results are the same with the transformation:

$\displaystyle a\rightarrow\frac{3abc}{ab+bc+ca},$ (11)

i.e. rescaled by the relative factor $V/\Sigma$ for each shape. The cases of
an infinite circular cylinder (Fig. 2(c)) and an infinite square rod are also
derived and discussed in Secs. S.VI and S.VII.

Finally, 2D objects can be treated with a similar approach. We have obtained
analytic expressions for $\langle L^{0}\rangle$ for an infinite strip, a
circle, a square, and a rectangle. The results and derivations are provided in
Sec. S.VIII, along with a graphical summary. The conclusions are similar to
those for 3D objects.

Figure 4: Comparison of the $s$ dependence of the zero-scattering mean path
length $\langle L^{0}\rangle$ for less symmetric objects. All values are
normalized to the mean chord length $\langle C\rangle$, and the scattering
case ($\langle L\rangle$) is shown as a solid line. (a) 2D objects: ellipse
and stadium of aspect ratio 2, Pascal’s Limaçon of polar equation
$r(\theta)=b+a\cos\theta$, with $b/a=3$. (b) 3D objects: same as 2D objects
with symmetry of revolution around $z$.

Application to physical systems. To investigate the generality of these
results, we now consider less symmetric geometries, for which numerical
calculations (Monte Carlo ray tracing [29]) can be used to derive $\langle
L^{0}\rangle$. Figure 4 summarizes these results. We consider explicitly in
Fig. 4(a) 2D shapes with different refractive index and decreasing symmetry:
ellipse, stadium, and a convex Limaçon. The latter two do not show any
discontinuity within our numerical accuracy, i.e. $\langle
L^{0}\rangle=\langle L\rangle$, even at high refractive index, which suggests
that the probability of scattering into a trapped trajectory is zero. Note
that trapped rays may still exist (such as the one depicted for the stadium in
Fig. 4(a)), but they correspond to unstable orbits with vanishingly small
probabilities of scattering into. The situation is different for ellipses
where a larger number of rays may be trapped, in agreement with the theory of
elliptical billiards [33]. Interestingly, corresponding 3D objects with
symmetry of revolution all show $\langle L^{0}\rangle<\langle L\rangle$,
likely because of the trapped trajectories in the planes perpendicular to the
revolution axis. It would be interesting to further link these results to the
theory of classical billiard and chaos/ergodicity but this is outside the
scope of this letter.

Figure 4 overall suggests that the mean path length discontinuity is a special
property of geometries with higher symmetry. These special shapes are
nevertheless commonly used as model systems in many applications. As an
example, ice crystals are often taken as high-symmetry objects to derive their
optical properties for atmospheric models [17, 18, 20]. Within the geometric-
optics approximation [17, 19], the absorption cross-section $C_{\mathrm{abs}}$
of weakly absorbing objects is directly proportional to the zero-scattering
mean-path-length. The expressions we obtained (and more that could be derived
using the same approach) can then be used to derive an analytic expression.
For example, for a 5 $\mu$m-wide ice cube at $\lambda=1\,\mu$m
($s=s^{\prime}+is^{\prime\prime}=1.3+1.6\times 10^{-6}i$), we find that the
analytic prediction:

$\displaystyle\langle C_{\mathrm{abs}}\rangle\approx\frac{4\pi
s^{\prime\prime}}{\lambda}\langle L_{\text{cube}}^{0}(s^{\prime})\rangle$
$\displaystyle\approx\frac{16as^{\prime
2}s^{\prime\prime}}{\lambda}\left(\sin^{-1}(s^{\prime
2}-1)-\sqrt{1-\frac{1}{s^{\prime 2}}}~{}\sin^{-1}(2s^{\prime 2}-3)\right)$
(12)

agrees within $\pm 10\%$ with numerical calculations. This approach is valid
for particle sizes much larger than the wavelength, but smaller than the
characteristic absorption length, i.e.
$1\ll(2\pi/\lambda)a\lessapprox(1/s^{\prime\prime})$. Together with the
approximate extinction cross-section $\langle C_{\mathrm{ext}}\rangle\approx
3a^{2}$, these provide simple analytical inputs for atmospheric models over a
large size range, replacing the time-consuming ray-tracing simulations
otherwise required. This approach could be generalized to more realistic ice-
crystal shapes and to other weakly absorbing atmospheric aerosols.

Apart from such applications, whilst the zero-scattering discontinuity is
interesting from a fundamental point of view, we should also consider its
relevance to real physical systems. Firstly, all physical media are imperfect
and should exhibit a small, but non-zero, scattering coefficient. In addition,
surface imperfections are unavoidable, be it roughness or small deviation from
ideal shapes (likely to make the object non-convex). Thirdly, the ray-optics
description is only an approximation and wave effects can affect
reflection/refraction, in particular resulting in a small probability of
outcoupling during TIR events, which would preclude the existence of strict
trapped trajectories. Because of these effects, one could argue that the zero-
scattering discontinuity is irrelevant and the general formula (Eq. (1))
applies instead. However, one should also consider that any physical medium
has non-zero absorption coefficient. This small absorption negates the
contribution of extremely long-lived trapped rays for low scattering, so that
the relevant experimental mean path length is in fact the zero-scattering one,
as long as the scattering is smaller than the absorption coefficient. This
argument is developed more qualitatively in Sec. S.IX.

Conclusion. We have examined how shape affects the mean path length of rays in
non-scattering refractive objects, providing the theoretical groundwork to
derive the mean path length analytically, and applying it to simple shapes.
Crucial to being able to derive these results was the fact that all
probabilistic reflections below the critical angle can be discounted if they
are ignored from both the inside and outside. We believe that some other
geometries will be able to be treated using the same approach. These analytic
results also highlight explicitly the discontinuous transition from non-
scattering to scattering media and demonstrate that it is due to the existence
of trapped trajectories that cannot be occupied without scattering.

We believe this work is an important contribution to the resurgent study of
path length invariance in media. The derived analytic expressions will also be
useful in other theoretical contexts where mean path length, or path length
distributions, are studied, as refractive non-scattering objects are central
to many applications.

###### Acknowledgements.

The authors are grateful to the MacDiarmid Institute, NZ, for financial
support.

SUPPLEMENTARY INFORMATION

##  I Derivation of reflection cancellation

In the main text we used the fact that all probabilistic (non-TIR) reflections
can be ignored when calculating the mean path length. Here we provide a more
detailed explanation of this phenomenon with diagrams (Fig. 5). First we noted
that due to retaining entropy, there is a one-to-one correspondence between
ingoing and outgoing rays: for each ingoing external ray, there is an outgoing
ray at the same angle at that surface point. This in turn means that for each
ray incoming at an angle $\theta_{1}$ to the normal, there is an internal ray
headed toward the surface at the Snell matching angle $\theta_{2}$ where
$\sin\theta_{1}=s\sin\theta_{2}$. In terms of Fig. 5, this means that the red
and blue solid lines represent the same density or number of rays. As shown in
the top diagrams in Fig. 5, for each externally reflected ray at an angle
$\theta_{1}$, which was reflected with a probability
$p_{1}=1-T_{12}(\theta_{1})$, there is an internally reflected ray, which was
reflected inward at $\theta_{2}$ with the same probability
$p_{2}=1-T_{21}(\theta_{2})=p_{1}$. This is because optical reciprocity
ensures $T_{12}(\theta_{1})=T_{21}(\theta_{2})$. These cases are added
together in the center diagram of Fig. 5, showing that the outgoing rays (red
with blue dots) are composed of external rays undergoing reflection (with
probability $p_{1}=1-T_{12}(\theta_{1})$), or internal rays undergoing
refraction (with probability $T_{21}(\theta_{2})$), while the ingoing rays
(blue with red dots) are composed similarly. In total there is a relative
density of 1 outgoing at $\theta_{1}$ and a density of 1 ingoing at
$\theta_{2}$. As shown in the lower diagrams of Fig. 5, the situation is
equivalent to a scenario where both the internal and external reflections are
replaced with refractions, because this results in the same density of 1 for
outgoing and ingoing rays. We say equivalent meaning that the number density
of rays is conserved, as well as ray angles and therefore also the mean path
length. So in analytic derivations only reflections for internal rays with
$\theta^{\prime}_{2}>\theta_{c}$ (TIR) need to be considered, as these
reflections have no correspondence with external reflections.

Figure 5: Schematics illustrating the effective cancellation of external and
internal probabilistic (non-TIR) reflections. Top: the possible paths of
external (blue) and internal (red) rays incident on the surface, where
$\theta_{1}$ and $\theta_{2}$ are Snell-matching angles. The dashed lines
represent the fractional probability of the outcomes. Center: the addition of
these two cases, where the origins of the rays are indicated by the coloring
of the dashes. Bottom: equivalent scenario where probabilistic reflections are
ignored, which leads to the same total number of rays leaving at each angle,
so preserves the mean path length.

##  II Explicit example of reflection cancellation

We can demonstrate the cancellation between internal and external reflections
explicitly for simple geometries where the chord length and inside angle
$\theta_{2}$ of a light ray after each internal reflection is identical –
these include the 2D infinite strip, circle, 3D infinite slab, infinite
cylinder, and sphere. In these shapes, a light ray may internally reflect $n$
times, each time with a probability $1-T_{21}(\theta_{2})$ so the path length
is increased by a factor

$\displaystyle
f=\sum_{n=0}^{\infty}\left[1-T_{21}(\theta_{2})\right]^{n}=\frac{1}{T_{21}(\theta_{2})}$
(13)

The fraction of rays incident with an angle $\theta_{1}$ transmitted into the
sample is $T_{12}(\theta_{1})=T_{21}(\theta_{2})$, thereby canceling the
factor in Eq. (13).

##  III Effect of non-uniform probability of trapping $P_{T}$

In an object like a sphere, the density of trapped trajectories varies and is
larger closer to the surface. The trapping probability $P_{T}$ then depends on
the location of the scattering event. We may still define an average trapping
probability $\bar{P}_{T,1}$. But for any ray that gets “trapped”, there is
also a probability that the next scattering event will result in another
trapped ray, and so on. If $P_{T}$ is non-uniform, these subsequent average
probabilities $\bar{P}_{T,n}$ may be different to that of the first trapping
event. For a sphere for example, one expects $\bar{P}_{T,2}>\bar{P}_{T,1}$ as
the scattering event for a ray already in a trapped trajectory is more likely
to occur closer to the surface. Let $P_{n}$ be the average probability that a
ray gets trapped exactly $n$ times, with corresponding average path length
$nl_{\alpha}$. Then the mean path length may be approximated in the low
scattering limit by

$\displaystyle\langle L\rangle=\sum_{n\geq 0}P_{n}\langle L_{n}\rangle.$ (14)

We are ignoring as in the main text any terms of order $\alpha\langle
L_{0}\rangle$ or less. In the low scattering limit the probability of
scattering is $P_{S}\approx\alpha\langle L_{0}\rangle$. Then the probability
of a trapping event occurring is $P_{S}\bar{P}_{T,1}$. The probability of no
trapping is then $P_{0}=1-\alpha\langle L_{0}\rangle\bar{P}_{T,1}\approx 1$.
The probability of exactly one trapping event is $P_{1}=\alpha\langle
L_{0}\rangle\bar{P}_{T,1}(1-\bar{P}_{T,2})$. Following this logic for $n$
trappings gives

$\displaystyle P_{n}=\alpha\langle
L_{0}\rangle(1-\bar{P}_{T,n+1})\prod_{k=1}^{n}\bar{P}_{T,k}.$ (15)

Plugging all this into Eq. 14, using $\alpha l_{\alpha}=1$, and assuming that
the probability of trapping converges to $\bar{P}_{T,\infty}$ as the number of
scatterings approaches $\infty$:

$\displaystyle\langle L\rangle$ $\displaystyle=\langle L_{0}\rangle+\langle
L_{0}\rangle(1-\bar{P}_{T,\infty})\sum_{n=1}^{\infty}n\prod_{k=1}^{n}\bar{P}_{T,k}.$
(16)

As expected, this expression reduces to Eq. (4) in the main text when
$\bar{P}_{T,n}=P_{T}$.

##  IV Explicit integral calculation of $\langle L^{0}\rangle$ for a cube

Figure 6: A light ray refracting into the “front” face of a cube, reflecting
off the top face, and refracting out the opposite face.

We start from Eq. 5 in the main text. We integrate over the two angles of
entry and over a vertically oriented face of the cube, which covers $0\leq
z\leq 1$ and $0\leq x\leq 1$, where $z=0$ at the top and $x=0$ on the right,
as per Fig. 6. The integral may be split into four:

$\displaystyle\langle
L_{\text{cube}}^{0}\rangle=\frac{4s^{2}}{\pi}(I_{t}+I_{o1}+I_{o2}+I_{r}),$
(17)

where $I_{t}$ counts the rays that refract towards the top face then refract
out:

$\displaystyle I_{t}=$ $\displaystyle\int_{a_{1}}^{b_{1}}\int_{0}^{\xi_{\rm
TRE}}\int_{0}^{1}\int_{0}^{1}L_{t}(\theta_{2},\phi,z)p(\theta_{2})\mathrm{d}z\mathrm{d}x\mathrm{d}\phi\mathrm{d}\theta_{2},$
(18)

$I_{o1}$ counts the rays that refract towards the top face then reflect
towards the opposite face (depicted in Fig. 6), or leave the opposite face
directly:

$\displaystyle I_{o1}=$ $\displaystyle\int_{a_{2}}^{b_{2}}\int_{0}^{\xi_{\rm
TRE}}\int_{0}^{1}\int_{0}^{1}L_{o}(\theta_{2})p(\theta_{2})\mathrm{d}z\mathrm{d}x\mathrm{d}\phi\mathrm{d}\theta_{2},$
(19)

$I_{o2}$ counts the rays that refract towards the top face, reflect off both
the top and right faces, and leave out the opposite face:

$\displaystyle I_{o2}=$ $\displaystyle\int_{a_{3}}^{b_{3}}\int_{0}^{\xi_{\rm
TRE}}\int_{0}^{1}\int_{0}^{1}L_{o}(\theta_{2})p(\theta_{2})\mathrm{d}z\mathrm{d}x\mathrm{d}\phi\mathrm{d}\theta_{2},$
(20)

and $I_{r}$ counts the rays that refract towards the top face, reflect off the
top face, and leave out the right face:

$\displaystyle I_{r}=$ $\displaystyle\int_{a_{4}}^{b_{4}}\int_{0}^{\xi_{\rm
TRE}}\int_{0}^{1}\int_{0}^{1}L_{r}(\theta_{2},\phi,x)p(\theta_{2})\mathrm{d}z\mathrm{d}x\mathrm{d}\phi\mathrm{d}\theta_{2}.$
(21)

$p(\theta_{2})=2\cos\theta_{2}\sin\theta_{2}$, and the distances to the top
face $(L_{t})$, opposite face $(L_{o})$ and right face $(L_{r})$ are

$\displaystyle L_{o}=$ $\displaystyle\frac{a}{\cos\theta_{2}}$ $\displaystyle
L_{t}=$ $\displaystyle\frac{z}{\sin\theta_{2}\cos\phi}$ $\displaystyle L_{r}=$
$\displaystyle\frac{x}{\sin\theta_{2}\sin\phi}.$

$\phi$ is measured clockwise from the vertical as shown in Fig. 6. By symmetry
we can assume that $\phi\geq 0$ and that all rays head towards the top face
($\phi<\xi_{\rm TRE}$), where

$\displaystyle\xi_{\rm TRE}=\tan^{-1}\dfrac{x}{z}$ (22)

is the top-right edge. The integral bounds for $\theta_{2}$ are

$\displaystyle a_{1}=$ $\displaystyle\min(\theta_{c},\max(\xi_{\rm
TOE},\xi_{\rm TIRT}))$ $\displaystyle b_{1}=$ $\displaystyle\theta_{c}$
$\displaystyle a_{2}=$ $\displaystyle 0$ $\displaystyle b_{2}=$
$\displaystyle\min(\theta_{c},\min(\xi_{\rm ORE},\max(\xi_{\rm TOE},\xi_{\rm
TIRT})))$ $\displaystyle a_{3}=$ $\displaystyle\min(\theta_{c},\xi_{\rm ORE})$
$\displaystyle b_{3}=$ $\displaystyle\min(\theta_{c},\max(\xi_{\rm
ORE},\xi_{\rm TIRR}))$ $\displaystyle a_{4}=$
$\displaystyle\min(\theta_{c},\max(\xi_{\rm ORE},\xi_{\rm TIRR}))$
$\displaystyle b_{4}=$ $\displaystyle\min(\theta_{c},\max(\xi_{\rm
ORE},\xi_{\rm TIRT})),$

where $\xi_{\rm TOE}$ is the top-opposite edge:

$\displaystyle\xi_{\rm TOE}=\tan^{-1}\frac{z}{\cos\phi},$ (23)

$\xi_{\rm ORE}$ is the opposite-right edge:

$\displaystyle\xi_{\rm ORE}=\tan^{-1}\frac{x}{\cos\phi},$ (24)

$\xi_{\rm TIRT}$ is the condition for total internal reflection off the top
face:

$\displaystyle\xi_{\rm TIRT}=\sin^{-1}\frac{\cos\theta_{c}}{\cos\phi},$ (25)

$\xi_{\rm TIRR}$ is the condition for total internal reflection off the right
face (this can only occur if the ray first undergoes TIR off the top face):

$\displaystyle\xi_{\rm TIRR}=\min\bigg{(}\xi_{\rm
TIRT},\sin^{-1}\frac{\cos\theta_{c}}{\sin\phi}\bigg{)}.$ (26)

In order to evaluate these integrals analytically, they can be broken down
into regions of $x,z,\phi$ where the max and min functions are not necessary.
This results in about 50 4-dimensional integrals, most of which can be
evaluated as sums of trigonometric and logarithmic functions and elliptic
integrals. We do not provide details here, but a similar simpler derivation is
given in the case of the square in Sec. VIII. On adding all integrals together
the result simplifies to Eq. (17). This remarkable simplification highlights
the importance of the alternative approach based on trapping probability
$P_{T}$, developed in the next section.

##  V Calculation of $\langle L^{0}\rangle$ for a cube using Eq. (4)

We here use instead Eq. (4), which applies because the trapping probability
$P_{T}$ is independent of the location of the scattering event. We can
formulate the condition for these trapped rays by requiring TIR off all six
faces. Without loss of generality, let our trapped ray bounce off the front
face with $\theta>\theta_{c}$ and some $\phi$. Then the condition for TIR off
the top face (and also the bottom face) is
$\sin\theta<\frac{\cos\theta_{c}}{\cos\phi}$. And similarly the condition for
TIR off the right and left faces is
$\sin\theta<\frac{\cos\theta_{c}}{\sin\phi}$. Finding trapped trajectories is
equivalent to finding $\theta\geq\theta_{c}$ and $\phi$ fulfilling these two
constraints. The boundary of solutions lies at $\theta=\theta_{c}$. The other
two conditions are satisfied if $\tan\theta_{c}$ is less than the maximum
value that $1/\cos\phi$ and $1/\sin\phi$ can have simultaneously. This occurs
at $\phi=\pi/4$ where $1/\cos\phi=1/\sin\phi=\sqrt{2}$. So trapped
trajectories exist for $\tan\theta_{c}<\sqrt{2}$, or equivalently
$s>\sqrt{3/2}$. Moreover, the probability of trapping a ray is given by (using
the 8-fold symmetry about $\phi$):

$\displaystyle P_{T}=$
$\displaystyle\frac{2}{\pi}\int_{\cos^{-1}\cot\theta_{c}}^{\pi/4}\mathrm{d}\phi\int_{\theta_{c}}^{\sin^{-1}\tfrac{\cos\theta_{c}}{\cos\phi}}\sin\theta\mathrm{d}\theta.$
(27)

Then Eq. (4)gives after integration:

$\displaystyle\langle L_{\text{cube}}^{0}(\sqrt{3/2}<s<\sqrt{2})\rangle=$ (28)
$\displaystyle{\frac{4as^{2}}{\pi}}\left(\sin^{-1}(s^{2}-1)-\sqrt{1-\frac{1}{s^{2}}}~{}\sin^{-1}(2s^{2}-3)\right).$

##  VI Infinite circular cylinder

Figure 7: A ray refracting through a cylinder. The angle $\phi$ is not shown
as it comes out of the page; it is measured from the the vertical to the
projection of the ray to the plane coming out of the page.

For a cylinder of radius $a$, every incident point is equivalent, and the
angle to the normal between internal bounces is invariant, a consequence of
the reflectional and rotational symmetries of the cylinder. As a result, it is
not possible to excite total internal reflections from the outside, and hence
the mean path length and mean chord length are identical. The chord length $L$
can be derived from the cosine rule $c^{2}=L^{2}+a^{2}-2aL\cos\theta_{2}$,
where $c^{2}=h^{2}+a^{2}$, $h$ being the height along the cylinder that the
ray travels; see Fig. 7. $h$ is related to $L$ through
$h=L\sin\theta_{2}\cos\phi$. Putting this together gives

$\displaystyle L_{\rm
cylinder}=\frac{2a\cos\theta_{2}}{1-\sin^{2}\theta_{2}\cos^{2}\phi},$ (29)

and for the mean:

$\displaystyle\langle L^{0}_{\text{cylinder}}\rangle=$
$\displaystyle\frac{s^{2}}{\pi}\int_{0}^{\theta_{c}}\int_{0}^{2\pi}\frac{2a\cos^{2}\theta_{2}\sin\theta_{2}}{1-\sin^{2}\theta_{2}\cos^{2}\phi}~{}\mathrm{d}\phi\mathrm{d}\theta_{2}$
$\displaystyle=$ $\displaystyle
4as^{2}\int_{0}^{\theta_{c}}\sin\theta_{2}\cos\theta_{2}~{}\mathrm{d}\theta_{2}$
$\displaystyle=$ $\displaystyle 2a,$ (30)

which, interestingly, is independent of $s$. The cylinder qualitatively
combines a slab along the axial direction and a circle along the radial one.
The opposite effect of these shapes on $s$ seems to cancel perfectly. For
comparison, the scattering mean path length is $\langle
L_{\text{cylinder}}\rangle=2as^{2}$. The discrepancy between the scattering
and non-scattering cases may be attributed to trapped rays which run down the
length of the cylinder.

##  VII Infinite square rod

For the infinite square rod, the derivations are similar to that of the cube
and 2D square (see below) so are not repeated; there are two regimes:

$\displaystyle\langle L_{\text{square rod}}^{0}(s<\sqrt{2})\rangle=$ (31)
$\displaystyle\frac{as^{2}}{\pi}\left(\cos^{-1}\left(1-s^{2}\right)-\sqrt{1-\frac{1}{s^{2}}}~{}\cos^{-1}\left(3-2s^{2}\right)\right),$
$\displaystyle\langle L_{\text{square rod}}^{0}(s\geq\sqrt{2})\rangle=$
$\displaystyle as^{2}\bigg{(}1-\sqrt{1-\frac{1}{s^{2}}}\bigg{)}.$ (32)

These results are both less than the mean path length with scattering:
$\langle L_{\text{square rod}}\rangle=as^{2}$, due to trapped rays which run
down the length of the rod.

##  VIII 2D objects

We consider the problem of calculating the mean path length for a convex 2D
object with refractive index and no scattering. The diffuse external radiation
creates a Lambertian incidence at all points on the surface, where the angles
of rays incident are distributed in a 2D problem as
$\tfrac{1}{2}\cos\theta_{1}$. As in the 3D case, the average path length is
obtained from integrating over all possible incident points and angles:

$\displaystyle\langle L_{2D}^{0}\rangle$
$\displaystyle=\frac{1}{2P}\int_{P}\int_{-\pi/2}^{\pi/2}L(\theta_{1},r)\cos\theta_{1}\mathrm{d}\theta_{1}\mathrm{d}r$
(33)

where $P$ is the object’s perimeter, and $\int_{P}$ is the integral around the
perimeter, so that $P=\int_{P}\mathrm{d}r$. Since $L$ is more naturally
expressed in terms of $\theta_{2}$, it will be convenient to parametrize the
integral as

$\displaystyle\langle L_{2D}^{0}\rangle$
$\displaystyle=\frac{s}{2P}\int_{P}\int_{-\theta_{c}}^{\theta_{c}}L(\theta_{2},r)\cos\theta_{2}\mathrm{d}\theta_{2}\mathrm{d}r.$
(34)

We derive analytic expressions in the following for simple 2D geometries,
namely the infinite strip, cicle, and square. The results are summarized and
compared to the scattering case in Fig. 8.

Figure 8: Comparison of the $s$ dependence of the zero-scattering mean path
length $\langle L^{0}\rangle$ for 2D objects for which analytical expressions
were derived. All values are normalized to the mean chord length $\langle
C\rangle=\pi A/P$. The scattering case $\langle L\rangle=s\langle C\rangle$ is
shown as a dashed line.

###  VIII.1 Infinite strip

We start with the simplest case of an infinite strip of width $a$. The
derivation and results are very similar to the infinite slab presented in the
main text. The chord length is

$\displaystyle L_{\rm strip}=\frac{a}{\cos\theta_{2}}.$ (35)

Like the slab, there is no TIR and all surface points are identical. The
integral (34) then reduces to

$\displaystyle\langle L_{\text{strip}}^{0}\rangle$
$\displaystyle=\frac{as}{2}\int_{-\theta_{c}}^{\theta_{c}}\mathrm{d}\theta_{2}$
$\displaystyle=as\theta_{c}.$ (36)

This is less than the mean path length including scattering:

$\displaystyle\langle L_{\text{strip}}\rangle=as\frac{\pi}{2}.$ (37)

The result (36) can also be obtained via Eq. (4), since the probability of
trapping is simply the fraction of angles greater than $\theta_{c}$:
$P_{T}=1-2\theta_{c}/\pi$.

###  VIII.2 Circle

Figure 9: Light ray refracting through a circle. $L_{\rm min}$ is the minimum
path length due to refraction.

For a circle of radius $a$, which is analogous to the 3D sphere in many
respects, the chord length is again independent of the point of entry:

$\displaystyle L_{\rm circle}=2a\cos\theta_{2},$ (38)

and the mean is then:

$\displaystyle\langle L_{\text{circle}}^{0}\rangle$
$\displaystyle=as\int_{-\theta_{c}}^{\theta_{c}}\cos^{2}\theta_{2}\mathrm{d}\theta_{2}$
$\displaystyle=a\left[s\theta_{c}+\cos\theta_{c}\right],$ (39)

which is less than the scattering mean path length:

$\displaystyle\langle L_{\text{circle}}\rangle=\frac{as}{2}.$ (40)

###  VIII.3 Square

For a square of side length $a$ there are two cases, $s\leq\sqrt{2}$ and
$s\geq\sqrt{2}$. For $s\leq\sqrt{2}$, then $\theta_{c}\geq\pi/4$ and there are
no trapped rays; if a ray with angle $\theta_{2}$ undergoes TIR off one face -
i.e. $\theta_{2}>\theta_{c}$ relative to that face’s normal, then the ray then
hits an adjacent side with an angle $\pi/2-\theta_{2}<\theta_{c}$. The mean
path length is therefore identical to that for the scattering case, i.e.

$\displaystyle\langle L_{\text{square}}^{0}(s\leq\sqrt{2})\rangle=\langle
L_{\text{square}}\rangle=\frac{\pi}{4}as.$ (41)

For $s\geq\sqrt{2}$, since as explained in the main text reflections for
$\theta<\theta_{c}$ may be ignored, the problem simplifies in that all rays
that enter the square leave the opposite face, either due to hitting the
opposite face directly, or via totally internally reflecting off an adjacent
face, which is guaranteed if $\theta_{c}\leq\pi/4$. Then $L=a/\cos\theta_{2}$
regardless of the incident angle or point of entry, and the integral (34)
simplifies to

$\displaystyle\langle
L_{\text{square}}^{0}(s\geq\sqrt{2})\rangle=as\theta_{c}.$ (42)

this expression coincides with (41) when $s=\sqrt{2}$.

Figure 10: Three different cases of light rays (in red) crossing a square,
broken down into different angular regions. This square has $s\leq\sqrt{2}$,
($\theta_{c}\geq\frac{\pi}{4}$).

Alternatively, these results can be found, albeit with more difficulty, from
the integral (34) for $s\leq\sqrt{2}$ as follows. Consider a test ray entering
from the right along $0\leq z\leq a$. By symmetry we may let
$0\leq\theta_{2}\leq\theta_{c}$. As shown in Fig. 10, the ray can either hit
the opposite face directly, with a length $L=a/\cos\theta_{2}$, hit the top
face and TIR and leave out the opposite face (giving a total path of
$L={a}/{\cos\theta_{2}}$ again), or hit the top face and leave with
$L={z}/{\sin\theta_{2}}$. This depends on the angle from the point of entry to
the opposite top corner, $\theta_{t}=\tan^{-1}({z}/{a})$. Note that
$\theta_{t}$ is always less than $\theta_{c}$, but $\theta_{t}$ may be greater
or less than ${\pi}/{2}-\theta_{c}$. Explicitly:

$\displaystyle L_{\rm square}=\left\\{\begin{aligned}
&\frac{a}{\cos\theta_{2}}\quad&0\leq\theta_{2}\leq\theta_{t},\\\
&\frac{a}{\cos\theta_{2}}&\theta_{t}\leq\theta_{2}\leq\max(\tfrac{\pi}{2}-\theta_{c},\theta_{t}),\\\
&\frac{z}{\sin\theta_{2}}&\max(\tfrac{\pi}{2}-\theta_{c},\theta_{t})\leq\theta_{2}\leq\theta_{c}.\end{aligned}\right.$
(43)

Inserting this into the integral (34) gives

$\displaystyle\langle L_{\text{square}}^{0}(s\leq\sqrt{2})\rangle=$
$\displaystyle\frac{s}{a}\int_{0}^{a}\bigg{[}\int_{0}^{\max(\tfrac{\pi}{2}-\theta_{c},\theta_{t})}a\mathrm{d}\theta_{2}$
$\displaystyle+\int_{\max(\tfrac{\pi}{2}-\theta_{c},\theta_{t})}^{\theta_{c}}\frac{z}{\sin\theta_{2}}\cos\theta_{2}\mathrm{d}\theta_{2}\bigg{]}\mathrm{d}z.$
(44)

This can be evaluated analytically by splitting the integrals at the point
$\theta_{t}=\pi/2-\theta_{c}$, which is equivalent to $z=a\cot\theta_{c}$. If
$z<a\cot\theta_{c}$, then $\theta_{t}<\pi/2-\theta_{c}$, and vice versa. Then

$\displaystyle\langle L_{\text{square}}^{0}(s\leq\sqrt{2})\rangle=$
$\displaystyle\frac{s}{a}\bigg{\\{}\int_{0}^{a\cot\theta_{c}}\bigg{[}\int_{0}^{\tfrac{\pi}{2}-\theta_{c}}a\mathrm{d}\theta_{2}$
$\displaystyle+\int_{\tfrac{\pi}{2}-\theta_{c}}^{\theta_{c}}\frac{z}{\sin\theta_{2}}\cos\theta_{2}\mathrm{d}\theta_{2}\bigg{]}\mathrm{d}z$
$\displaystyle+\int_{a\cot\theta_{c}}^{a}\bigg{[}\int_{0}^{\theta_{t}}a\mathrm{d}\theta_{2}$
$\displaystyle+\int_{\theta_{t}}^{\theta_{c}}\frac{z}{\sin\theta_{2}}\cos\theta_{2}\mathrm{d}\theta_{2}\bigg{]}\mathrm{d}z\bigg{\\}}.$
(45)

The integrals individually evaluate to logarithmic and trigonometric
functions, which together cancel to give $\langle
L_{\text{square}}^{0}(s\leq\sqrt{2})\rangle=\pi as/4$.

###  VIII.4 Rectangle

For a rectangle of sides $a,b$, we can use the same arguments as for the
square. For $s\leq\sqrt{2}$ there are no trapped rays:

$\displaystyle\langle L^{0}_{\text{rect}}(s<\sqrt{2})\rangle=\langle
L_{\text{rect}}\rangle=\frac{\pi abs}{2(a+b)}.$ (46)

And for $s\geq\sqrt{2}$, again all rays exit the opposite face, so the
integral (34) reduces to

$\displaystyle\langle
L^{0}_{\text{rect}}(s\geq\sqrt{2})\rangle=\frac{2abs\theta_{c}}{a+b}.$ (47)

##  IX Absorption in a low scattering sample

One important application of the study of path length is to study the optical
absorption of a refractive object. If we define the absorption $A$ as the
fraction of rays that are absorbed compared to the total number of incident
rays, it can be expressed in terms of the path length distribution as [5, 29]:

$\displaystyle A=1-\int_{0}^{\infty}e^{-\alpha_{a}L}p(L)\mathrm{d}L.$ (48)

where $p(L)$ is the probability distribution of path length in the absence of
absorption and $\alpha_{a}$ is the absorption coefficient. In the low
absorption limit, $\alpha_{a}\rightarrow 0$, the exponential may be expanded
in a Taylor series, giving approximately

$\displaystyle A=\alpha_{a}\langle L\rangle+\mathcal{O}(\alpha_{a}^{2}\langle
L\rangle^{2}).$ (49)

This common approximation highlights the importance of the mean path length,
but would also suggest that the measured absorption could be discontinuous at
zero scattering when $\langle L^{0}\rangle\neq\langle L\rangle$. In practical
situations ideal zero-scattering is not possible, so we here analyze the
absorption in the limit of low scattering. An implicit assumption in deriving
Eq. (49) is that there are no significant contributions in $p(L)$ from $L$
comparable or larger than $1/\alpha_{a}$, since $\alpha_{a}L\ll 1$ is not
valid for those $L$. This is no longer true in cases where there exists
trapped, or even just long-lived, trajectories.

For objects with refractive index and a very low scattering coefficient
$\alpha_{s}$, it is appropriate to split the path length distribution $p(L)$
into:

  * •

Rays that are directly reflected off the surface with probability
$\bar{R}_{12}=1-\bar{T}_{12}$ (with zero path length), where $\bar{R}_{12}$
and $\bar{T}_{12}$ are the average (over a Lambertian distribution) reflection
and transmission coefficients.

  * •

Rays that enter, propagate inside, but do not scatter before exiting, with
probability $P_{0}^{\prime}$ and (renormalized) path length distribution
$p_{0}(L)$.

  * •

Rays that enter, scatter and exit without undergoing total internal
reflection, with probability $P_{S}^{\prime}$ and path length distribution
$p_{S}(L)$.

  * •

Rays that get trapped in long-lived trajectories, with probability
$P_{T}^{\prime}$ and path length distribution $p_{T}(L)$.

We can write explicitly:

$\displaystyle p(L)=$
$\displaystyle\bar{R}_{12}\delta(L)+P_{0}^{\prime}p_{0}(L)+P_{S}^{\prime}p_{S}(L)+P_{T}^{\prime}p_{T}(L).$
(50)

The probabilities in the low scattering limit are, for a general object:

$\displaystyle P_{0}^{\prime}=$ $\displaystyle\bar{T}_{12}-\alpha_{s}\langle
L_{0}\rangle$ $\displaystyle P_{S}^{\prime}=$
$\displaystyle(1-P_{T})\alpha_{s}\langle L_{0}\rangle$ $\displaystyle
P_{T}^{\prime}=$ $\displaystyle\alpha_{s}\langle L_{0}\rangle P_{T},$ (51)

where $\langle L_{0}\rangle$ is the mean path length for zero-scattering and
$P_{T}$ is the average probability of a scattered ray entering a trapped
trajectory. Note that:

$\displaystyle\bar{R}_{12}+P_{0}^{\prime}+P_{S}^{\prime}+P_{T}^{\prime}=1.$
(52)

To derive low order approximate expressions, the absorption may be split into
contributions from different types of ray trajectories, following Eq. 50:

$\displaystyle A=P_{0}^{\prime}A_{0}+P_{S}^{\prime}A_{S}+P_{T}^{\prime}A_{T},$
(53)

where

$\displaystyle A_{0}=1-\int_{0}^{\infty}p_{0}(L)e^{-\alpha_{a}L}\mathrm{d}L$
$\displaystyle A_{S}=1-\int_{0}^{\infty}p_{S}(L)e^{-\alpha_{a}L}\mathrm{d}L$
$\displaystyle A_{T}=1-\int_{0}^{\infty}p_{T}(L)e^{-\alpha_{a}L}\mathrm{d}L.$
(54)

In the low absorbance limit, $A_{0}$ and $A_{S}$ are simplified by the fact
that the path length distributions $p_{0}(L)$ and $p_{S}(L)$ are confined
mostly to small $L$ relative to the absorption mean free path $1/\alpha_{a}$,
so

$\displaystyle A_{0}\approx$ $\displaystyle\alpha_{a}\frac{\langle
L_{0}\rangle}{\bar{T}_{12}},$ $\displaystyle A_{S}\approx$
$\displaystyle\alpha_{a}\frac{\langle
L_{S}\rangle}{\bar{T}_{12}}+\mathcal{O}\Big{(}\frac{\alpha_{s}}{\alpha_{a}}\Big{)}$
(55)

where $\langle L_{0}\rangle/\bar{T}_{12}$ is the mean path length of rays that
enter, and $L_{S}/\bar{T}_{12}$ is the mean path length of rays that enter and
scatter but do not get trapped into long-lived trajectories.

In order to simplify further, we now consider the limit of low absorption and
lower scattering, i.e. the scattering coefficient $\alpha_{s}$ is much less
than the absorption coefficient, and both are small,
$\alpha_{s}\ll\alpha_{a}\ll 1/\langle L_{0}\rangle$. $A_{T}$ can then be
simplified to first order by recognizing that all trapped rays get absorbed
since the absorption coefficient is much higher than the scattering
coefficient, therefore:

$\displaystyle A_{T}\approx 1-\mathcal{O}(\alpha_{s}/\alpha_{a}).$ (56)

Then altogether the absorption is

$\displaystyle A\approx$ $\displaystyle\bigg{(}1-\alpha_{s}\frac{\langle
L_{0}\rangle}{\bar{T}_{12}}\bigg{)}\alpha_{a}\langle L_{0}\rangle$
$\displaystyle+(1-P_{T})\alpha_{s}\langle L_{0}\rangle\alpha_{a}\frac{\langle
L_{S}\rangle}{\bar{T}_{12}}$ $\displaystyle+\alpha_{s}\langle L_{0}\rangle
P_{T}.$ (57)

In fact the terms containing the product $\alpha_{a}\alpha_{s}$ are second
order, and neglecting them leaves

$\displaystyle A$ $\displaystyle\approx\alpha_{a}\langle
L_{0}\rangle+\alpha_{s}\langle L_{0}\rangle
P_{T}\qquad\left[\alpha_{s}\ll\alpha_{a}\ll 1/\langle L_{0}\rangle\right].$

If $\alpha_{s}\ll\alpha_{a}$, it then reduces to $A\approx\alpha_{a}\langle
L_{0}\rangle$. This demonstrates that for realistic objects with residual
scattering, the zero-scattering mean path length is the relevant quantity in
terms of optical absorption, providing that absorption dominates over
scattering. Although outside the scope of this work, the effect of other
imperfections, such as the wave nature of light or surface scattering from
imperfection, can be incorporated using similar arguments. In these cases we
also deduce that a small non-zero absorption coefficient will dramatically
limit the contribution of trapped and long-lived trajectories, resulting again
in the zero-scattering mean path length being the relevant quantity.

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