###### Abstract

Diffusive random walks feature the surprising property that the average length
of all possible random trajectories that enter and exit a finite domain is
determined solely by the domain boundary. Changes in the diffusion constant or
the mean-free path, that characterize the diffusion process, leave the mean
path length unchanged. Here we demonstrate experimentally that this result can
be transferred to the scattering of waves, even when wave interference leads
to marked deviations from a diffusion process. Using a versatile microwave
setup, we establish the mean path length invariance for the crossover to
Anderson localization and for the case of a band gap in a photonic crystal. We
obtain these results on the mean path length solely based on a transmission
matrix measurement through a novel procedure that turns out to be more robust
to absorption and incomplete measurement in the localized regime as compared
to an assessment based on the full scattering matrix.

††preprint: APS/123-QED

A fundamental result of diffusion theory is that the mean path length of
particles diffusing across a certain region of space is entirely independent
of the characteristics of the diffusion process [1, 2, 3]. In fact, only the
shape of the outer boundary of this region determines the average length of
particle trajectories between entering and exiting this region; whether the
paths taken by the particles are straight lines or convoluted random walks is,
however, completely irrelevant. As such, this mean path length invariance
generalizes the so-called “mean chord length theorem” valid in the ballistic
limit [2] and encompasses applications in basically all research fields where
diffusion processes or random walks arise –- ranging from nuclear physics [1]
and solar energy harvesting [4] to the movement of bacteria [5].

The scope of this invariance property was recently expanded even further, when
it was shown that not only particles, but also waves that scatter through a
region of space are subject to the same invariance property [6]. The key
insight here is that the average “time-delay” associated with a scattering
process is directly linked to the density of states (DOS) inside the
scattering region [1, 2, 3, 4]. From the Weyl law [11, 12] we then know that
the DOS remains invariant when transforming a homogeneous medium in the
ballistic limit into a diffusively scattering disordered medium of the same
size [6]. A recent experimental implementation followed exactly this line of
thought by changing the turbidity of a liquid from nearly transparent to very
opaque and demonstrated that the mean path length of isotropically incoming
light inside the liquid stays, indeed, unchanged over nearly two orders of
magnitude in scattering strength [13]. In spite of its coherent nature, laser
light could thus be observed to obey the same universal invariance property as
particles when scattering ballistically or diffusively.

Coherent wave effects can, however, also lead to very strong deviations from
any of the transport regimes that particles can be in. Consider here, e.g.,
the regime of Anderson localization [14, 15] or the formation of a band gap in
a photonic crystal [16], to cite just two genuinely wave-like phenomena that
both rely on wave interference. The natural question to ask at this point is
whether any such effects going beyond a trajectory-based description may lead
to a violation of the mean path length invariance since they clearly fall
outside the scope of both the mean chord length theorem and a random walk
picture. More specifically, since both Anderson localization and a band gap
prevent incident waves from propagating inside the scattering region, one
naturally expects that the mean path length invariance should break down in
these cases. Insights into such questions are not just of academic interest:
consider, e.g., that the mean path length invariance is strongly linked to the
so-called “Yablonovitch limit” that imposes a cap on the optimal intensity
enhancement inside solar cells [17]. A break-down of the invariance property
may thus also provide a strategy for overcoming current limitations in solar
cell design [4].

Figure 1:  Experimental setup and path length spectra. a, Sketch of the
experimental setup, where the cavity is represented for a random arrangement
of 40 aluminum cylinders. The top plate has been removed to visualize the
interior of the sample. The embedded intensity distribution corresponds to the
highest transmitting transmission eigenstate in this scattering configuration
obtained from a numerical simulation. b, Experimental data for the variation
of the inverse of the statistical conductance,
$(1/g^{\prime})=3\mathrm{var}(s_{a})/2$ as a function of the conductance $g$
to estimate whether the experiment operates in the diffusive
$[(1/g^{\prime})<1]$ or in the localized regime $[(1/g^{\prime})>1]$ [18]. c,
Spectra of the mean path length $s(\omega)=s[2Q_{t}(\omega)]$ for a chaotic
cavity (blue line), a diffusive sample with $n_{\text{scat}}=100$ scatterers
(green line) and a localized sample with $n_{\text{scat}}=280$ scatterers (red
line). The values of $1/g^{\prime}$ corresponding to these samples are shown
with red, green and blue markers in b.

Here we will address these questions through an experiment that gives us
direct access to these scattering regimes beyond both the ballistic and the
diffusive limit. We test the invariance of the mean time-delay in microwave
measurements using a multichannel cavity for which the scattering strength of
a random sample can be tuned by changing the number of scatterers within the
cavity and for which an ordered arrangement of the scatterers mimics a
photonic crystal.

The scattering region is formed by an effectively two-dimensional cavity (see
Methods) of length $L=0.5$ m and width $W=0.25$ m with two arrays of $N=8$
antennas attached on the left and right interfaces, respectively [8]. The
antennas are single mode waveguides fully coupled to the cavity between 11 and
18 GHz (see Fig. 1a). Having full control over all these waveguides enables
the measurement of the frequency-dependent $N\times N$ transmission matrix
(TM) $t_{ba}(\omega)$, which contains the complex and flux-normalized
transmission coefficients between the two arrays. The TM is complete, but
strong internal reflections occur at the left and right interfaces of the
cavity as the spacing between two adjacent antennas is metallic (unlike in
open waveguides).

To induce a transition from the ballistic to the chaotic, diffusive and the
localized regime we gradually increase the scattering strength within the
cavity. Specifically, we measure the transmission matrix $t(\omega)$ for an
empty cavity (ballistic system), for a cavity with an aluminum semicircle of
53 mm diameter [chaotic system, see Supplementary Material (SM) for details]
and for disordered samples with $n_{\text{scat}}$ randomly distributed
aluminum cylinders of radius $r_{s}=3$ mm. For the disordered systems the
conductance $g=\langle\Sigma_{n=1}^{N}\tau_{n}\rangle$ (i.e., the frequency-
averaged sum of transmission eigenvalues $\tau_{n}$ of $t^{\dagger}t$) ranges
over more than two orders of magnitude from $4.36$ for the empty cavity to
$0.01$ for the sample with the strongest disorder ($n_{\text{scat}}=280$).

Because of dissipation within the system and strong internal reflections, the
scaling of the conductance $g$ may not reflect the crossover from diffusive to
localized waves found at $g=1$ in open random systems [20]. We exploit instead
the statistics of the transmitted intensity as a reliable indicator of the
localization transition even in the presence of absorption [18]. Specifically,
the variance of the normalized total transmission $s_{a}=T_{a}/\langle
T_{a}\rangle$, with $T_{a}=\Sigma_{b}|t_{ba}|^{2}$, is given for non-
dissipative diffusive samples by $\mathrm{var}(s_{a})=2/(3g)$. In the case of
finite dissipation the so-called statistical conductance
$g^{\prime}=2/[3\mathrm{var}(s_{a})]$ has been found to indicate localization
reliably when taking on values $(1/g^{\prime})>1$ [18]. In Fig. 1b, we thus
show the change of $1/g^{\prime}$ with the conductance $g$ and observe values
for $(1/g^{\prime})$ that considerably exceed 1 for $g<0.6$. This confirms
that the increased disorder in our two-dimensional system leads to wave
localization.

In multichannel systems, the mean path length can be estimated from
measurements of the Wigner-Smith (WS) time-delay operator
$Q=-iS^{-1}\partial_{\omega}S$ applied to the scattering matrix $S$ which
relates incoming and outgoing channels [21, 22, 23, 24, 25, 26, 27, 28, 29].
The operator $Q$ is a multichannel generalization of the phase derivative
$d\phi_{ba}/d\omega$, which provides the time-delay of a spectrally narrow
pulse between two channels $a,b$. Averaging over all channels leads to the
mean Wigner-Smith time-delay given by
$\bar{t}_{\textrm{WS}}(\omega)=\mathrm{Tr}[Q(\omega)]/(2N)$.

In principle, estimating the mean time-delay requires a measurement of the
complete scattering matrix $S(\omega)$ including the two reflection matrices
on the left and right sides of the sample, respectively. Experimentally, such
a measurement is highly challenging, however, since most setups provide access
either only to a one-sided reflection matrix or to the TM. To overcome this
difficulty, we show in the SM that the trace of $Q(\omega)$ and the trace of
the Wigner-Smith operator involving only the TM from left to right,
$Q_{t}(\omega)=-it^{-1}\partial_{\omega}t$, are connected through the
following equivalence relation,

$s[Q(\omega)]=s[2Q_{t}(\omega)]\ ,$ (1)

where $s(\mathcal{O})=c_{0}\mathrm{Re}[\mathrm{Tr}(\mathcal{O})/(2N)]$ is the
mean length obtained with an operator $\mathcal{O}$. The above simple relation
is proven in the SM for non-absorbing systems using the decomposition of the
TM into transmission eigenchannels. The key ingredient is the correspondence
between transmission and reflection eigenchannels as a consequence of the
unitarity of the scattering matrix, $S(\omega)S^{\dagger}(\omega)=\mathbb{1}$
[30, 31]. The transmission eigenchannel time-delay
$t_{n}^{(t)}(\omega)=d\theta_{n}/d\omega$, found from the derivative of a
composite phase shift $\theta_{n}$ of the singular vectors of the TM [31], is
then equal to the average of the corresponding reflection delay times at the
right and left sides of the sample,
$t_{n}^{(t)}(\omega)=[t_{n}^{(r)}(\omega)+t_{n}^{(r^{\prime})}(\omega)]/2$ (as
in 1D systems [32]), where a prime denotes the quantities at the other
waveguide port. Equation (1) thus provides access to the mean path length
$s[Q(\omega)]=c_{0}\bar{t}_{\textrm{WS}}(\omega)$ through transmission
measurements only.

Figure 2:  Mean path lengths with respect to the number of scatterers. a, Mean
path length evaluated as $\langle s(2Q_{t})\rangle$, see Eq. (1), obtained in
simulations without (black crosses) and with (black circles) absorption as a
function of the number of metallic cylinders within the cavity. The red solid
line is the theoretical length calculated from the Weyl law [see Eq. (2)].
Absorption reduces the estimated mean path length. Background colors in rose,
green, and white mark the localized, diffusive, and the ballistic regime as in
Fig. 1b. b, The same quantity, $\langle s(2Q_{t})\rangle$, found
experimentally (blue circles with error bars). The squares and triangles are
the mean path length in transmission and in reflection, respectively (see SM
for details).

In practice, evaluating $s[2Q_{t}(\omega)]$ requires that eigenchannels of
$t(\omega)$ with small transmission eigenvalues $\tau_{n}$ (typically
$\tau_{n}<10^{-6}$) are removed from the experimental data since these
eigenchannels may contain time-delays $t_{n}^{(t)}(\omega)$ that are corrupted
by the noise level of the experimental setup. In diffusive and localized
systems, eigenchannels with the smallest transmission values are typically
associated with a small intensity build-up inside the medium and therefore
with small time-delays [31]. Removing their contribution to
$s[2Q_{t}(\omega)]$ thus modifies the estimated length only weakly (see SM).

Spectra of the mean path length $s[2Q_{t}(\omega)]$ corresponding to a chaotic
cavity, a diffusive sample and a localized sample, are presented in Fig. 1c.
In contrast to the diffusive regime in which the overlap of many resonances
leads to small fluctuations of $s[2Q_{t}(\omega)]$, the peaks observed in the
localized regime correspond to the contributions of spectrally isolated
resonances associated with localized modes. To meaningfully compare the mean
path length obtained in the different propagation regimes, it is thus
necessary to average $s[2Q_{t}(\omega)]$ over a frequency range containing
several of these peaks.

In Fig. 2 we compare the mean path length $\langle s(2Q_{t})\rangle$ resulting
from a frequency-average in single configurations (see Methods) with the
theoretical predictions obtained using the Weyl law [11, 12, 6] given by,

$s_{\mathrm{theo}}(\omega)=\frac{1}{2Nc_{0}}\left(\omega
A+\frac{C-B}{2}c_{0}\right)\,.$ (2)

This Weyl prediction for the mean path length involves the scattering area
$A=LW+16lw-n_{\text{scat}}\pi r_{s}^{2}$ corresponding here to the surface of
the cavity, including the attached channels of length $l=38$ mm and aperture
$w=15.79$ mm, from which the area of the impenetrable metallic scatterers is
subtracted. A first-order correction term takes into account the external
boundaries $C=16w$ and the internal boundaries $B$ of the scattering system,
that include the metallic boundaries of the cavity and the circumferences of
the metallic cylinders [6]. Smaller values of $g$ are reached with an
increasing number of these cylinders, which reduces the effective scattering
area and increases the length of the internal boundaries. The theoretical
estimate of the mean path length $\langle s_{\mathrm{theo}}(\omega)\rangle$ is
then the average of $s_{\mathrm{theo}}(\omega)$ over the frequency range and
decreases from 2.05 m for an empty cavity to 1.76 m for a sample with 280
cylinders. The theoretical mean path length for an empty cavity $\langle
s_{\mathrm{theo}}(\omega)\rangle\sim 4L$ strongly exceeds its value for open
waveguides $\langle s_{\mathrm{theo}}(\omega)\rangle=\pi L/2$ [6]. This
enhancement is due to the metallic spacings between the antennas at the right
and left interfaces of the cavity. Numerical simulations presented in the SM
also show the existence of states with very long delay times in the empty
cavity, corresponding to path lengths of a few hundred meters. These states
are caused by bouncing orbits between the top and bottom interfaces of the
cavity (in $y$-direction).

Figure 2a shows perfect agreement between the Weyl prediction (see red line)
and the mean path length $\langle s(2Q_{t})\rangle$ obtained in numerical
simulations (see black crosses) of the experimental setup in absence of
absorption (see Methods). Even the small reduction in the mean path length
predicted for an increasing number of scatterers is well reproduced. This
confirms the validity of the mean path length invariance across the onset of
the localization transition.

Also the experimental data shown in Fig. 2b are in good agreement with
$\langle s_{\mathrm{theo}}(\omega)\rangle$ (see blue circles). How non-trivial
this invariance property is, can be appreciated when contrasting it with the
strong enhancement (reduction) of the transmission (reflection) time-delays
across the localization transition (see blue squares and triangles). We also
observe that the mean path length is slightly underestimated, especially in
the ballistic regime. Even though the DOS integrated over frequency is
independent of absorption [33], the presence of absorption within the cavity
makes the scattering matrix sub-unitary [34] and leads to a violation of the
mean path length invariance [6] so that this deviation comes as no surprise.
Indeed, the experimental data is well reproduced by numerical simulations when
dissipation is included. For the data shown in Fig. 2a (see black circles), we
introduce uniform absorption by adding an imaginary part to the effective
refractive index of the cavity. We here use for the empty cavity as well as
for all disorder configurations an average uniform imaginary part of the
refractive index ($n_{i}=2\times 10^{-4}$) found by comparing the frequency-
averaged transmission measured for the empty cavity with the numerical
simulations.

In addition to its detrimental effect on the mean path length invariance,
dissipation also breaks the unitarity of $S$ and therefore the correspondence
between $s[Q(\omega)]$ and $s[2Q_{t}(\omega)]$ in Eq. (1). In the diffusive
regime, all of these effects are sufficiently weak such that losses through
outgoing channels dominate over uniform absorption over the sample. From the
broadening of the average linewidth with respect to the number of ports
connected to the cavity, we estimate that the ratio of losses through ports
relative to uniform losses within the cavity is equal to 5.4 (see SM). With
the scattering matrix then still being sufficiently close to unitarity,
$\langle s(2Q_{t})\rangle$ provides a reliable estimator of the theoretical
value $\langle s_{\text{theo}}\rangle$ in absence of absorption.
Interestingly, the stronger deviations found for the empty cavity are a
consequence of very specific states that bounce many times between the top and
bottom cavity boundary (in $y$-direction). Due to their long cavity dwell
times, these states are very strongly affected by dissipation and therefore
lead to significant deviations from the mean path length invariance (see SM).
The existence of these states is reflected by the large fluctuations of the
mean path length in Fig. 2. In the diffusive regime, the disorder scattering
naturally leads to a suppression of such states with strongly enhanced time-
delays and thereby to a better agreement with the theoretical predictions.

Figure 3: Influence of absorption. a,b, Normalized transmission eigenchannel
contribution to the mean path length as a function of the index $n$, where
$n=1$ ($n=8$) corresponds to the highest (lowest) transmitting eigenchannel.
Increasing absorption leads to a redistribution of the transmission time-
delays which all converge to the length of the most direct path in the limit
of strong absorption (see SM). The horizontal black dashed line marks an
estimation of the most direct path in the empty system.

Surprisingly, we observe that our estimate for the mean path length, $\langle
s(2Q_{t})\rangle$, is more robust to absorption in the localized regime than
in the empty cavity even though Anderson localization also gives rise to the
large time-delays in the cavity and large fluctuations of the mean path length
(see Fig. 2b). Using a toy model, we demonstrate analytically in the SM that
in the limit of strong absorption $s[2Q_{t}(\omega)]$ converges towards the
most direct path in transmission $\langle L_{\text{direct}}\rangle$. As shown
in Fig. 3, all eigenchannels feature a quasi-ballistic propagation in this
limit providing the same contribution to the mean path length. In this way,
the flux in the empty system (with $n_{i}=10^{-2}$, see SM) becomes almost
perfectly aligned with the $x$-direction from left to right as strong
absorption suppresses all longer paths [35]. Indeed, the resulting estimate
for the most direct path $\langle
L_{\mathrm{direct}}^{\mathrm{empty}}\rangle\approx 0.31\langle
s_{\mathrm{theo}}\rangle$ is in very good agreement with the numerical value
of $\langle s(2Q_{t})\rangle=0.32\langle s_{\mathrm{theo}}\rangle$. In the
localized regime, the presence of metallic and hence impenetrable scatterers
elongates the direct path, which is found to be comparable to the mean path
length ($\langle s(2Q_{t})\rangle=0.66\langle s_{\mathrm{theo}}\rangle$ for
the sample with 280 scatterers and $n_{i}=10^{-2}$). The increased absorption-
stability of the mean path length in the localized regime as compared to the
ballistic regime is thus explained by the difference in the most direct
scattering contributions in these two cases. We emphasize that this robustness
of $s[2Q_{t}(\omega)]$ in the localized regime is in stark contrast to the
much stronger dependence on absorption we observe for a corresponding estimate
using the full Wigner-Smith matrix $s[Q(\omega)]$ (see SM). $Q(\omega)$
contains the full scattering matrix and thus depends on the sum of phase delay
times of both transmitted and reflected waves [36]. In analogy to our analysis
from above, the strong absorption reduces these contributions to those coming
from the shortest possible paths. However, since the shortest paths
contributing here are those that are directly reflected when entering the
cavity, their extremely short time-delay values will dominate in the strong
absorption limit (see SM). Compared to transmission eigenchannels whose direct
paths have to traverse the whole system, the convergence of $s[Q(\omega)]$ to
the direct paths in reflection leads eventually to a pronounced
underestimation of the mean path length for strongly scattering samples.

Figure 4: Path lengths in a photonic crystal. a,b, Frequency spectra a of the
transmission and b of the mean path length in a photonic crystal obtained in
measurements to which the free space contribution has been removed outside the
band gap (blue line) and in simulations with absorption (black line). A
transmission band gap is clearly observed for this sample with 15 longitudinal
layers of regularly spaced scatterers. c, Integral of the mean path length
normalized by its theoretical value over a frequency window spanning $\Delta
f$ around the center of the bandgap. The black dashed line is the result of
simulations in absence of absorption. The decrease of the mean path length
within the band gap is followed by a strong enhancement starting at the band
edges such that the frequency-averaged value progressively converges towards
the theoretical Weyl prediction. The experimental data (blue solid line) shows
the same trend, but stays slightly below the Weyl prediction for large $\Delta
f$ due to absorption (see agreement with the simulations including absorption,
black solid line).

After having explored the validity of the mean path length invariance in the
cross-over to very strong disorder, we will now consider the opposite limit of
a structured medium with periodic order. For this purpose we form a photonic
crystal (PC) by a periodic arrangement of alternating aluminum and Teflon
cylinders (see Methods). The structure with 15 layers of scatterers in
logitudinal direction exhibits a band gap centered at
$f_{0}=12~{}\mathrm{GHz}$ with a width $\Delta f_{0}\approx 2~{}\mathrm{GHz}$
as can be seen in Fig. 4a. Because the PC only fills the middle part of the
entire scattering area of the cavity, the mean path length includes the
contribution of the PC as well as the free space between the PC and the
interfaces of the cavity. To isolate the impact of the PC, we subtract the
theoretical free space contribution to $s[2Q_{t}(\omega)]$, except in the band
gap for which the transmission remains in the noise floor of the experimental
setup ($\approx 10^{-6}$) and we set $s[2Q_{t}(\omega)]=0$.

Numerical simulations are then performed in a cavity whose scattering region
has the same dimensions as the PC with small disorder in the scatterer
positions and absorption being added to mimic the situation in the experiment
(see Methods). The central frequency of the band gap is now
$f_{0}=12.6~{}\mathrm{GHz}$ with a width of 3.3 GHz, which is larger than in
measurements. We attribute these differences to tiny air gaps between the top
plate and the aluminum cylinders in the experimental setup, where scattering
at the top cylinder edge causes the excitation of evanescent modes. The
effective properties of the scatterers are thus modified locally due to the
coupling of such evanescent modes to neighboring Teflon scatterers. The mean
path length is also strongly reduced within the band gap in simulations (see
Fig. 4b), but $s[2Q_{t}(\omega)]$ is not vanishing as a consequence of the
finite length of the PC. We also note that the dwell time can be negative as a
result of absorption [37] and since the dwell time operator is related to the
time-delay operator and thus also to $s[2Q_{t}(\omega)]$, this explains the
negative values in Fig. 4b.

From such an observation one may be tempted to conclude that the mean path
length invariance does not hold in such periodic systems. Following prior
theoretical work on frequency sum rules [33, 38], we know, however, that in a
sufficiently broad spectral window, reductions and enhancements of the DOS
should compensate each other in arbitrary systems including the case of
photonic bandgap materials. Due to the connection between the DOS and the mean
path length, we should find, correspondingly, that the strong decrease of
$s[2Q_{t}(\omega)]$ within the band gap is compensated by a corresponding
enhancement right outside of the band gap. As can be seen in Fig. 4b, we
indeed observe such an enhancement of the mean path length close to the band
edges, with values for $s[2Q_{t}(\omega)]$ even exceeding
$3s_{\text{theo}}(\omega)$ in measurements and in simulations.

In Fig. 4c we now show the mean path length normalized by Weyl prediction,
$s[2Q_{t}(f)]/s_{\text{theo}}(f)$, integrated over a frequency window $\Delta
f$ centered around the middle of the corresponding band gap $f_{0}$ obtained
in the experiment or the simulation, $N_{s}(\Delta f)=\int_{f0-\frac{\Delta
f}{2}}^{f0+\frac{\Delta f}{2}}\,s[2Q_{t}(f)]/s_{\textrm{theo}}(f)\,df$. The
Weyl law prediction for this quantity is $N_{s}(\Delta f)=\Delta f$. Note that
the bandgap is not located in the center of our frequency range. Thus, once
the lower end of the integration frequency window has reached 9.5 GHz, we
continue the integration with only the higher frequency range. In the
experiment, this average ratio $N_{s}(\Delta f)$ now almost vanishes for a
spectral window $\Delta f$ smaller than the width of the band gap but then
increases rapidly and progressively converges towards $N_{s}(\Delta f)\approx
0.86\Delta f$ as $\Delta f=8.5~{}\mathrm{GHz}$. As in the case of the empty
system or the disordered configurations, this 14% deviation from the Weyl law
is due to absorption within the sample. This is confirmed by simulations
without absorption for which $N_{s}(\Delta f)$ indeed reaches $N_{s}(\Delta
f)\approx 0.975\Delta f$ . The convergence of $N_{s}(\Delta f)$ now
demonstrates that the pronounced enhancements of $s[2Q_{t}(\omega)]$ at the
edges of the band gap compensate the vanishing mean path length within the
band gap. The fact that we observe the experimental data and the simulations
with absorption to converge towards the same value of $N_{s}$ for large
$\Delta f$ in spite of the different band gap sizes for these two cases,
further substantiates the invariance property of the mean path length.

In summary, we experimentally demonstrated with microwave measurements that
the knowledge of the transmission matrix alone provides a robust way of
estimating the invariant mean path length, even in scattering systems with
strong wave-interference and weak absorption. Our results clearly show that
this invariance property reaches far beyond the diffusive regime and thus
provides a comprehensive bound on enhancement of the mean path length of
broadband light in a medium.

##  I Methods

Experimental setup. Sixteen antennas that are waveguide-to-coax adapters
operating in the Ku-band are attached to the system and fully coupled to the
cavity between 11 and 18 GHz. We operate in a frequency window smaller than
$c_{0}/2h=18.75~{}\mathrm{GHz}$ where only the fundamental mode in the
vertical $z$-direction is excited, making the cavity effectively two-
dimensional. Measurements of the frequency spectra of the TM are carried out
with two ports of a vector network analyzer connected to two $N\times 1$
electro-mechanical switches to successively excite each transmitting and
receiving antenna. The ports of the switches that are not excited are
terminated with 50 $\Omega$ loads so that the antennas mimic absorbing
boundary conditions. The metallic spacing between adjacent antennas generate
strong internal reflections at the interfaces of the cavity. The TM is
measured between 11 and 13 GHz in the ballistic and diffusive regimes and
between 11 and 12 GHz in the localized regime, with frequency steps of 0.4
MHz.

Photonic crystal. The period of the square lattice between a metallic and a
dielectric Teflon scatterer is $d=1.2$ cm in both the longitudinal and
transverse direction. A picture of the experimental setup is shown in the SM.
The photonic crystal consists of 15 longitudinal layers of scatterers. The
variations of the theoretical mean path length $s_{\text{theo}}(\omega)$ with
frequency for a sample of 300 cylinders is given by Eq. (2), where the area of
the Teflon cylinders has to be multiplied with their refractive index squared
to account for the increased density of states in dielectric materials.

Numerical simulations. We solve the two-dimensional scalar Helmholtz equation
$\left[\Delta+n^{2}(\mathbf{r})k_{0}^{2}\right]\psi(\mathbf{r})=0$ using a
finite element method [39, Schoberl1997, Schoberl2014]. Here, $\Delta$ is the
Laplacian in two dimensions, $n(\mathbf{r})$ is the refractive index
distribution, $\mathbf{r}=(x,y)$ is the position vector, $k_{0}=2\pi/\lambda$
is the vaccum wavenumber and $\psi(\mathbf{r})$ is the $z$-component of the
TE-polarized electric field. In our simulations we use the exact dimensions of
the experimental setup (see Fig. 1), where the single mode leads are
terminated with perfectly matched layers which absorb the outgoing waves
without any back-reflections and thus mimic semi-infinite leads. To account
for the global losses in the experiment, we add a frequency-independent
uniform imaginary part of $2\times 10^{-4}$ to the effective refractive index
of the cavity which yields the same frequency-averaged transmission in the
empty system as in the experiment.

In case of the disordered systems, we use – just like in the experiment – a
single random configuration for each number of scatterers and average the
calculated mean path lengths in the range of 11-13 GHz (11-12 GHz for 280
scatterers).

The simulations of the photonic crystal are performed in the frequency
interval of 9.5-17 GHz. To mimic experimental uncertainties, we introduce a
slight disorder to the positions of the scatterers, i.e., we displace
scatterers in the transverse and longitudinal direction by a random value
drawn from the interval $[-r_{s}/10,r_{s}/10]$. We then define our bandgap as
the frequency interval in which the transmission in the numerical simulation
reaches the experimental noise floor ($\approx 10^{-6}$).

To calculate the time-delay operators $Q$, $Q_{t}$, $Q_{r}$ and
$Q_{r^{\prime}}$ we have to invert the corresponding scattering, transmission
or reflection matrices. Since these matrices can be singular, we perform a
singular value decomposition and project our matrices onto subspaces
containing only singular vectors corresponding to singular values greater than
$10^{-10}$ which enables us to compute their pseudo-inverse (see Supplementary
Material of [28] or [42] for details).

##  II Acknowledgments

This publication was supported by the European Union through the European
Regional Development Fund (ERDF), by the French region of Brittany and Rennes
Métropole through the CPER Project SOPHIE/STIC & Ondes, and by the Austrian
Science Fund (FWF) through project P32300 (WAVELAND). The computational
results presented were achieved using the Vienna Scientific Cluster (VSC). M.
D. acknowledges the Institut Universitaire de France.

##  III Author contributions

Measurements and data evaluation were carried out by M.D. Numerical
simulations were carried out by M.K. under the supervision of S.R. Theoretical
tasks were carried out by M.D., M.K., S.R. and S.G. M.D., M.K. and S.R. wrote
the manuscript with input from all authors.

##  IV Competing interests

The authors declare no competing interests.

##  V Data availability

The data that support the plots within this paper and other findings of this
study are available from the corresponding authors on reasonable request.

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Supplementary Material for ’Mean path length invariance in wave-scattering
beyond the diffusive regime’

##  I Measurements

###  I.1 Experimental Setup

Figure S1:  a, Picture of the cavity with a semicircle. b, Picture of the
ordered arrangement of aluminum and teflon cylinders.

Figure S1a shows a picture of the chaotic cavity. The cavity is made chaotic
by including an aluminum semicircle of 53 mm diameter. The semicircle is
located near a boundary.

We also present in Fig. S1b a picture of the the structured medium forming a
photonic crystal. The periodic order is made by the arrangement of aluminum
and teflon cylinders. The distance between two cylinders is equal to $d=1.2$
cm in both the longitudinal and transverse direction. The number of layers of
scatterers is equal to 15.

###  I.2 Mean path lengths in transmission and reflection

In Fig. 2 of the main text, we compare the mean path length with the mean path
lengths in transmission $\langle s_{t}\rangle$ and in reflection $\langle
s_{r}\rangle$. In transmission, $\langle s_{t}\rangle$ is found using the
average of the single-channel delay times with

$\langle s_{t}\rangle=c_{0}\frac{\langle
T_{ba}\frac{d\phi_{ba}}{d\omega}\rangle}{\langle T_{ba}\rangle}.$ (S1)

Here $T_{ba}$ and $\phi_{ba}$ are the transmitted intensity and the phase
associated to the element $t_{ba}$ of the transmission matrix, i.e., the
transmission coefficient between incoming channel $a$ and outgoing channel
$b$, $t_{ba}=\sqrt{T_{ba}}\exp(i\phi_{ba})$. The averaging is performed over
incoming and outgoing channels as well as the frequency range and $c_{0}$ is
the speed of light.

In case of reflection, our experimental setup does not allow the measurement
of the full reflection matrix. However, we obtain its diagonal part with
elements $r_{aa}=\sqrt{R_{aa}}\exp(i\phi_{aa})$ using the $S_{11}$ parameters
of the Vector Network Analyzer. The mean reflection length is then computed
from

$\langle s_{r}\rangle=c_{0}\frac{\langle
R_{aa}\frac{d\phi_{aa}}{d\omega}\rangle}{\langle R_{aa}\rangle}.$ (S2)

##  II Derivation of Eq. (1) of the main text

In the following we give an analytical proof of Eq. (1) of the main text which
reduces to

$\mathrm{Tr}(Q)=2\mathrm{Re}\left[\mathrm{Tr}(Q_{t})\right]\ ,$ (S3)

with the time-delay operators

$\displaystyle Q=-iS^{-1}\frac{dS}{d\omega}\ ,$ (S4) $\displaystyle
Q_{t}=-it^{-1}\frac{dt}{d\omega}\ ,$ (S5)

in case of a two-port scattering matrix

$S=\begin{pmatrix}r&t^{\prime}\\\ t&r^{\prime}\\\ \end{pmatrix},$ (S6)

which is unitary, i.e., $S^{-1}=S^{\dagger}$ and thus
$S^{\dagger}S=\mathbb{1}$. Here, primed quantities denote the transmission or
reflection from the other port. Thus, we can write the $Q$-operator as

$Q=\begin{pmatrix}-ir^{\dagger}\frac{dr}{d\omega}-it^{\dagger}\frac{dt}{d\omega}&-ir^{\dagger}\frac{dt^{\prime}}{d\omega}-it^{\dagger}\frac{dr^{\prime}}{d\omega}\\\
-it^{\prime\,\dagger}\frac{dr}{d\omega}-ir^{\prime\,\dagger}\frac{dt}{d\omega}&-ir^{\prime\,\dagger}\frac{dr^{\prime\,\dagger}}{d\omega}-it^{\prime\,\dagger}\frac{dt^{\prime\,\dagger}}{d\omega}\end{pmatrix},$
(S7)

where its trace is given by

$\displaystyle\begin{split}\mathrm{Tr}(Q)=\mathrm{Tr}\bigg{(}&-it^{\dagger}\frac{dt}{d\omega}-it^{\prime\,\dagger}\frac{dt^{\prime\,\dagger}}{d\omega}\\\
&-ir^{\dagger}\frac{dr}{d\omega}-ir^{\prime\,\dagger}\frac{dr^{\prime\,\dagger}}{d\omega}\bigg{)}.\end{split}$
(S8)

To calculate the first term, we use a singular value decomposition (SVD) of
the transmission matrix $t=U\Sigma V^{\dagger}$, where
$U=(\vec{u}_{1},\vec{u}_{2},\ldots,\vec{u}_{N})$ and
$V=(\vec{v}_{1},\vec{v}_{2},\ldots,\vec{v}_{N})$ are the matrices which
contain column-wise the left and right singular vectors of $t$ and
$\Sigma=\mathrm{diag\\{\sigma_{n}\\}}$ is a diagonal matrix containing the
singular values $\sigma_{n}$. Making use of $U^{\dagger}U=\mathbb{1}$ and
$V^{\dagger}V=\mathbb{1}$ and the invariance of the trace with respect to
cyclic permutations we can write

$\displaystyle\begin{split}\mathrm{Tr}\left(-it^{\dagger}\frac{dt}{d\omega}\right)=\mathrm{Tr}\bigg{(}&-i\Sigma^{2}U^{\dagger}\frac{dU}{d\omega}\\\
&-i\Sigma^{2}\frac{dV^{\dagger}}{d\omega}V-i\Sigma\frac{d\Sigma}{d\omega}\bigg{)}.\end{split}$
(S9)

Using $V^{\dagger}V=\mathbb{1}$, we can rewrite the second term in the trace
as $\frac{dV^{\dagger}}{d\omega}V=-V^{\dagger}\frac{dV}{d\omega}$. Making
further use of the fact that the singular values of $t$ are the square root of
the eigenvalues of $t^{\dagger}t$ and $tt^{\dagger}$, i.e.,
$\sigma_{n}^{2}=\tau_{n}$ are the transmission values, we arrive at

$\displaystyle\begin{split}\mathrm{Tr}\left(-it^{\dagger}\frac{dt}{d\omega}\right)=\sum_{n}\tau_{n}\
\frac{d\theta_{n}^{(t)}}{d\omega}-i\sigma_{n}^{(t)}\frac{d\sigma_{n}^{(t)}}{d\omega}\
.\end{split}$ (S10)

Here we have introduced the channel delay times as

$\frac{d\theta_{n}^{(t)}}{d\omega}\equiv\frac{1}{i}\left(\vec{u}_{n}^{\,(t)\dagger}\frac{d\vec{u}_{n}^{\,(t)}}{d\omega}-\vec{v}_{n}^{\,(t)\dagger}\frac{d\vec{v}_{n}^{\,(t)}}{d\omega}\right)\
,$ (S11)

where the superscript denotes in the following to which matrix a singular
vector (or singular value) belongs to.

To calculate the remaining terms, we make use of the relations between the
singular vectors which follow directly from the unitarity of the scattering
matrix. For the right singular vectors $\vec{v}_{n}$ we get

$\displaystyle t^{\dagger}t+r^{\dagger}r=\mathbb{1}\quad$
$\displaystyle\Rightarrow\quad\vec{v}_{n}^{\,(t)}=\vec{v}_{n}^{\,(r)}\equiv\vec{v}_{n}^{\,(t,r)}\
,$ (S12) $\displaystyle
t^{\prime\,\dagger}t^{\prime}+r^{\prime\,\dagger}r^{\prime}=\mathbb{1}\quad$
$\displaystyle\Rightarrow\quad\vec{v}_{n}^{\,(t^{\prime})}=\vec{v}_{n}^{\,(r^{\prime})}\equiv\vec{v}_{n}^{\,(t^{\prime},r^{\prime})}\
,$ (S13)

The left singular vectors $\vec{u}_{n}$ are related as follows

$\displaystyle tt^{\dagger}+r^{\prime}r^{\prime\,\dagger}=\mathbb{1}\quad$
$\displaystyle\Rightarrow\quad\vec{u}_{n}^{\,(t)}=\vec{u}_{n}^{\,(r^{\prime})}\equiv\vec{u}_{n}^{\,(t,r^{\prime})}\
,$ (S14) $\displaystyle
t^{\prime}t^{\prime\,\dagger}+rr^{\dagger}=\mathbb{1}\quad$
$\displaystyle\Rightarrow\quad\vec{u}_{n}^{\,(t^{\prime})}=\vec{u}_{n}^{\,(r)}\equiv\vec{u}_{n}^{\,(t^{\prime},r)}\
.$ (S15)

Furthermore, we know that $t^{\prime}=t^{T}$ which implies that $t$ and
$t^{\prime}$ share the same eigenvalues. Thus,
$\sigma_{n}^{(t)}=\sigma_{n}^{(t^{\prime})}$ and since
${\sigma_{n}^{(t)}}^{2}=\tau_{n}$ it follows that
${\sigma_{n}^{(r)}}^{2}=\rho_{n}=1-\tau_{n}=1-\tau^{\prime}_{n}=\rho^{\prime}_{n}={\sigma_{n}^{(r^{\prime})}}^{2}$.
Using now Eq. (S10) and similar expressions for the terms in Eq. (S8)
containing $t^{\prime}$, $r$ and $r^{\prime}$ yields

$\displaystyle\begin{split}\mathrm{Tr}(Q)=\sum_{n}&\frac{1}{i}\Bigg{[}\vec{u}_{n}^{\,(t,r^{\prime})\dagger}\frac{d\vec{u}_{n}^{\,(t,r^{\prime})}}{d\omega}-\vec{v}_{n}^{\,(t,r)\dagger}\frac{d\vec{v}_{n}^{\,(t,r)}}{d\omega}\\\
&+\vec{u}_{n}^{\,(t^{\prime},r)\dagger}\frac{d\vec{u}_{n}^{\,(t^{\prime},r)}}{d\omega}-\vec{v}_{n}^{\,(t^{\prime},r^{\prime})\dagger}\frac{d\vec{v}_{n}^{\,(t^{\prime},r^{\prime})}}{d\omega}\Bigg{]}\\\
&-i\left[\frac{d{\sigma_{n}^{(t,t^{\prime})}}^{2}}{d\omega}+\frac{d{\sigma_{n}^{(r,r^{\prime})}}^{2}}{d\omega}\right],\end{split}$
(S16)

where we have already used the relations (S12)-(S15). The last term
corresponds now to the imaginary part of the $Q$-eigenvalues which has to
vanish because $Q$ is a Hermitian operator (if $S$ is unitary) featuring real
eigenvalues. Using $\tau_{n}^{(\prime)}+\rho_{n}^{(\prime)}=1$ which follows
from the unitarity of $S$ then indeed yields

$\frac{d{\sigma_{n}^{(t,t^{\prime})}}^{2}}{d\omega}+\frac{d{\sigma_{n}^{(r,r^{\prime})}}^{2}}{d\omega}=\frac{d(\tau_{n}^{(\prime)}+\rho_{n}^{(\prime)})}{d\omega}=0\
.$ (S17)

Eq. (S16) can now be written in two different ways:

$\displaystyle\begin{split}\mathrm{Tr}(Q)&=\sum_{n}\frac{d\theta_{n}^{(t)}}{d\omega}+\frac{d\theta_{n}^{(t^{\prime})}}{d\omega}\\\
&=\sum_{n}\frac{d\theta_{n}^{(r)}}{d\omega}+\frac{d\theta_{n}^{(r^{\prime})}}{d\omega}\
.\end{split}$ (S18)

Next, we want to connect this result to the trace of $Q_{t}$, which can be
simplified as follows:

$\displaystyle\begin{split}\mathrm{Tr}(Q_{t})&=\left(-it^{-1}\frac{dt}{d\omega}\right)\\\
&=\mathrm{Tr}\left(-iU^{\dagger}\frac{dU}{d\omega}+iV^{\dagger}\frac{dV}{d\omega}-i\Sigma^{-1}\frac{d\Sigma}{d\omega}\right)\\\
&=\sum_{n}\frac{d\theta_{n}^{(t)}}{d\omega}-i\frac{d\ln(\sigma_{n}^{(t)})}{d\omega}\
.\end{split}$ (S19)

Please note that the channel delay times $d\theta_{n}^{(t)}/d\omega$
correspond to the delay times of transmission eigenchannels and thus they
differ from the real part of the $Q_{t}$-eigenvalues. However, the invariance
of the trace under similarity transforms used in the derivation above keeps
their sum the same.

Since we know that $t^{\prime}=t^{T}$, it follows that
$Q_{t^{\prime}}=t^{\prime\,-1}Q_{t}^{T}t^{\prime}$. Thus, $Q_{t^{\prime}}$ is
a similarity transform of the transpose of $Q_{t}$ which implies that they
also share the same eigenvalues and thus their trace is the same. In case of
the reflection matrix we, however, do not have a direct relation between $r$
and $r^{\prime}$ and hence there is no direct relation between the eigenvalues
of $Q_{r}$ and $Q_{r^{\prime}}$, but unitarity still enforces that the sum of
the corresponding channel delay times in (S18) is equal to the sum of
transmission channel delay times. With the help of Eq. (S19) and similar trace
expresions for $Q_{t^{\prime}}$, $Q_{r}$ and $Q_{r^{\prime}}$, Eq. (S18) can
then finally be written as

$\displaystyle\begin{split}\mathrm{Tr}(Q)&=\mathrm{Re}\left[\mathrm{Tr}(Q_{r})+\mathrm{Tr}(Q_{r^{\prime}})\right]\\\
&=\mathrm{Re}\left[\mathrm{Tr}(Q_{t})+\mathrm{Tr}(Q_{t^{\prime}})\right]\\\
&=2\mathrm{Re}\left[\mathrm{Tr}(Q_{t})\right].\end{split}$ (S20)

It is important to note that there is no relation between the transmission and
reflection delay times for single eigenchannels. This can easily be seen by
comparing the expressions

$\frac{d\theta_{n}^{(t)}}{d\omega}=\frac{1}{i}\left[\vec{u}_{n}^{\,(t,r^{\prime})\dagger}\frac{d\vec{u}_{n}^{\,(t,r^{\prime})}}{d\omega}-\vec{v}_{n}^{\,(t,r)\dagger}\frac{d\vec{v}_{n}^{\,(t,r)}}{d\omega}\right]\neq\frac{d\theta_{n}^{(r)}}{d\omega}$
(S21)

since the left eigenvectors do not coincide, i.e.,
$\vec{u}_{n}^{\,(t)}~{}\neq~{}\vec{u}_{n}^{\,(r)}$. However, unitarity implies
that the sum of delay times in transmission for a single eigenchannel is equal
to the sum of delay times in reflection, i.e.,

$\frac{d\theta_{n}^{(t)}}{d\omega}+\frac{d\theta_{n}^{(t^{\prime})}}{d\omega}=\frac{d\theta_{n}^{(r)}}{d\omega}+\frac{d\theta_{n}^{(r^{\prime})}}{d\omega}\
.$ (S22)

##  III Influence of absorption in the experiment

###  III.1 Empty cavity

Figure S2: a, Semi-logarithmic plot of the mean path length spectrum for the
empty cavity found in simulations, where the black arrow denotes the frequency
at which the intensity distribution in b is shown. b, Intensity distribution
of a transversally trapped transmission eigenstate at 11.42 GHz giving rise to
a mean path length as large as 800 m.

We have seen in Fig. 2 of the main text that the estimated mean path length is
underestimated for an empty cavity, whereas it almost reaches the theoretical
prediction as soon as a few scatterers are added to our system. This
underestimation can be explained by the existence of very long delay times in
the empty cavity for which the impact of absorption is stronger. This is
illustrated by the spectrum of the mean path length found in simulations in
absence of absorption [see Fig. S2a]. Strong peaks corresponding to mean path
lengths of a few hundred meters are found which are caused by bouncing orbits
between the top and bottom interfaces of the cavity as shown in Fig. S2b. Due
to their very long dwell times in the cavity, weak absorption already affects
these states strongly giving rise to a dampened and statistically
inhomogeneous intensity distribution which results in an underestimation of
the mean path length.

###  III.2 Ballistic and diffusive regime

Figure S3: a, Variation of $N\langle s\rangle$ as a function of the number of
connected ports $N$ for diffusive waves with 20 scatterers inside the cavity.
b, Average total linewidth $\langle\Gamma_{n}\rangle$ as a function of the
number of connected ports $N$. The extrapolation for $N\rightarrow 0$ gives
the linewidth associated to uniform absorption over the volume of the sample,
$\Gamma_{a}$. c, Variation of the ratio
$\langle\tilde{\Gamma}_{n}\rangle/\Gamma_{a}$ with $N$, where
$\langle\tilde{\Gamma}_{n}\rangle=\langle 2N\Gamma_{c}\rangle$ with
$\langle\Gamma_{c}\rangle$ being the linewidth associated to the single port
losses.

In contrast to the empty cavity, the presence of scatters fully randomizes
incoming waves in the ballistic and diffusive regime which then does not allow
the existence of such orbits yielding a noticeably better estimate of the mean
path length in the presence of absorption. To further explain the
experimentally observed enhanced robustness to absorption in the diffusive
regime compared to the empty cavity, we consider a varying number of ports
attached to the system, increasing from $N=1$ to $N=8$. For $N<8$, the unused
ports are disconnected and behave as metallic boundary conditions. To
conveniently estimate the impact of absorption as a function of $N$, we
consider the quantity $N\langle
s\rangle=Nc_{0}\langle\bar{t}_{\textrm{WS}}\rangle$, which, in the absence of
absorption, should take the same value for all systems with a different number
of ports $N$.

In the diffusive regime, as shown in Fig. S3a, the values for
$Nc_{0}\langle\bar{t}_{\textrm{WS}}\rangle$ converge towards the theoretical
prediction only for $N=8$. This corresponds to the number of ports for which
losses through incoming and outgoing channels dominate the dissipative losses
within the sample such that $s[2Q_{t}(\omega)]$ provides a robust estimation
of the mean path length. This can be shown by separating these two sources of
losses using a decomposition of the average linewidths as a sum of losses
through the ports $\langle\tilde{\Gamma}_{n}\rangle$ and uniform absorption
$\Gamma_{a}$, i.e.,
$\langle\Gamma_{n}\rangle=\langle\tilde{\Gamma}_{n}\rangle+\Gamma_{a}$. The
average losses through the $2N$ ports connected to the cavity scale linearly
with the losses through a single channel $\langle\Gamma_{c}\rangle$, i.e.,
$\langle\tilde{\Gamma}_{n}\rangle=\langle 2N\Gamma_{c}\rangle$. To extract the
values of $\Gamma_{a}$ and $\langle\Gamma_{c}\rangle$, we use that the average
transmission through the sample in the time domain decreases exponentially
such as $\langle|t_{ba}(t)|^{2}\rangle=\exp(-\langle\Gamma_{n}\rangle t)$,
where $\langle|t_{ba}(t)|^{2}\rangle$ is found from an inverse Fourier
transform of spectra of transmission coefficients $t_{ba}(\omega)$. We
therefore find $\langle\Gamma_{n}\rangle$ for each $N$ and we fit this curve
as $\langle\Gamma_{n}\rangle=\langle 2N\Gamma_{c}\rangle+\Gamma_{a}$ [see Fig.
S3b] which yields $\langle\Gamma_{c}\rangle\sim 7.8$ MHz and $\Gamma_{a}\sim
23$ MHz. For $N=8$, $\langle 2N\Gamma_{c}\rangle=125$ MHz and the ratio
$\langle 2N{\Gamma}_{c}\rangle/\Gamma_{a}$ shown in Fig. S3c is equal to 5.4.
In this case, losses through ports indeed dominate so that the estimation of
the mean path length is robust to the uniform absorption present in the
experiment.

###  III.3 Localized regime

Just like in the diffusive regime, we have seen in the main text that the
transmission matrix provides a reliable estimate of the Weyl prediction also
in the localized regime. This is the case even though the probability
distribution of the eigenchannel delay times is substantially broadened, i.e.,
there exist very short as well as very long paths. In the following, we
provide an explanation of the robustness of the mean path length to the
absorption present in the experimental setup using its connection with the
density of states (DOS).

Figure S4:  Variation of $N\langle s\rangle$ as a function of the number of
connected ports $N$ for localized waves with 250 scatterers inside the cavity.

We employ the connection between the mean Wigner-Smith time delay operator and
the density of states (DOS) [1, 2, 3, 4]

$\mathrm{Tr}[Q(\omega)]=2\pi\rho(\omega)$ (S23)

to explain the robustness of Tr($Q_{t}$). The DOS $\rho(\omega)$ can be
decomposed into a superposition of individual contributions from quasi-normal
modes (QNM) of the system. These QNMs (or resonances) are associated with
complex eigenfrequencies $\tilde{\omega}_{n}=\omega_{n}-i\Gamma_{n}/2$, where
$\omega_{n}$ is the central frequency of the $n$-th mode and $\Gamma_{n}$ is
its linewidth, such that [5]

$\displaystyle\begin{split}\rho(\omega)&=\frac{1}{\pi}\sum_{n}\rho_{n}(\omega)\\\
&=\frac{1}{\pi}\sum_{n}\frac{\Gamma_{n}/2}{(\omega-\omega_{n})^{2}+(\Gamma_{n}/2)^{2}}\,.\end{split}$
(S24)

These eigenfrequencies $\tilde{\omega}_{n}$ are also the poles of the
scattering matrix. The contributions of a mode $\rho_{n}(\omega)$ satisfies
$\int_{0}^{\infty}d\omega\rho_{n}(\omega)=1$ so that the average DOS is
independent of absorption. Hence, adding absorption to our system results in
broadening of the linewidths, $\Gamma_{n}\rightarrow\Gamma_{n}+\Gamma_{a}$,
where $\Gamma_{a}$ is the linewidth associated to the losses within the
sample.

The transmitted speckle pattern at the output for a spectrally isolated mode
does not depend on the position of the excitation so that the transmission
matrix is of unit rank [6, 7, 8]. Measuring the transmission through a single
channel ($N=1$) then yields the DOS using its connection to Tr($Q_{t}$). When
several resonances overlap spectrally, however, the number of ports $N$ has to
be larger than the number of resonances contributing to the DOS. We now
quantify the number of ports that are required for an accurate estimation of
the DOS.

Following Breit-Wigner theory, the modal expansion of the transmission matrix
is

$t(\omega)=-iW_{R}(\omega-\tilde{\omega})^{-1}W_{L}^{T}\,$ (S25)

where $W_{R}$ and $W_{L}$ are the projection of the eigenstates of the open
system onto the right and the left channels. Furthermore,
$\tilde{\omega}_{n}=\omega_{n}-i\Gamma_{n}/2$ are the complex eigenfrequencies
of the eigenstates of the open system, i.e., the poles of the scattering
matrix, where $\omega_{n}$ is the central frequency of the $n$-th mode and
$\Gamma_{n}$ is the corresponding linewidth. The $W$-matrices are assumed to
be independent of frequency and of dimension $N\times M$, where $N$ is the
number of incoming channels and $M$ is the number of modes of the open system.
Denoting their pseudo-inverses as $W_{L}^{-1}$ and $W_{R}^{-1}$ we can write

$\displaystyle t^{-1}=i{W_{L}^{T}}^{-1}(\omega-\tilde{\omega})W_{R}^{-1}\ ,$
(S26)
$\displaystyle\frac{dt}{d\omega}=iW_{R}(\omega-\tilde{\omega})^{-2}W_{L}^{T}\
,$ (S27)

and thus

$Q_{t}=i{W_{L}^{T}}^{-1}(\omega-\tilde{\omega})^{-1}W_{L}^{T}\ ,$ (S28)

where we have assumed that $W_{R}^{-1}W_{R}=\mathbb{1}$, i.e., that
$W_{R}^{-1}$ is the left-inverse of $W_{R}$. The latter is only true if $M<N$
and rank$(W_{R})=M$ in which case $W_{R}$ features orthonormal columns which
results in $W_{R}^{-1}$ being the left inverse of $W_{R}$. Equation (S28) now
shows that modes with resonances far from $\omega$ only weakly contribute to
the transmission matrix. The size of $(\omega-\tilde{\omega})^{-1}$ can hence
be restricted to the participation ratio of modal contributions
$\rho_{n}(\omega)$ to the DOS,
$M^{\prime}=\left[\sum_{n}\rho_{n}(\omega)\right]^{2}/[\Sigma_{n}\rho_{n}^{2}(\omega)]$,
so that $W_{L}$ and $W_{R}$ become of dimension $N\times M^{\prime}$.
$M^{\prime}$ is obviously linked to the modal overlap
$\delta=\langle\Gamma_{n}\rangle/\Delta\omega$, where $\Delta\omega$ is the
level spacing, $\Delta\omega=\langle\omega_{n+1}-\omega_{n}\rangle$. For
$M^{\prime}<N$, the similarity-invariance of the trace operator gives

$\displaystyle\begin{split}\mathrm{Re}\left[\mathrm{Tr}(Q_{t})\right]&=-\mathrm{Im}\left[\mathrm{Tr}(\omega-\tilde{\omega})^{-1}\right]\\\
&=\sum_{n}\frac{\Gamma_{n}/2}{(\omega-\omega_{n})^{2}+(\Gamma_{n}/2)^{2}}=\pi\rho(\omega)\end{split}$
(S29)

where we have used in the first step that $W_{L}^{-1}$ is the left-inverse of
$W_{L}$ in case of $M^{\prime}<N$. Inserting now Eq. (S29) into Eq. (S23)
finally yields

$\mathrm{Tr}(Q)=2\mathrm{Re}\left[\mathrm{Tr}(Q_{t})\right].$ (S30)

This result now shows that the mean path length is not only well estimated
from measurement of the transmission matrix only, but can also be well
estimated in the presence of absorption as long as $M^{\prime}<N$. However,
$M^{\prime}$ increases with absorption and when $M^{\prime}>N$, the pseudo-
inverse coupling matrices are no longer left-inverse, i.e.,
$W_{L,R}^{-1}W_{L,R}\neq\mathbb{1}$, which results in an under-estimated mean
path length, i.e., $\mathrm{Re}\left[\mathrm{Tr}(Q_{t})\right]<\rho(\omega)$.
Last, we want emphasize that this result is not only valid in the presence for
absorption, but it also applies in the case of incomplete channel control.

In non-dissipative systems, the modal overlap $\delta$ of localized waves is
small and $\langle M^{\prime}\rangle\sim 1$. However, as already explained
above, absorption in our experimental setup can give rise to an increase of
these two quantities. Looking at our measurements through a cavity with 280
scatterers, we find that $\langle M^{\prime}\rangle\approx 3.2$ so that
$M^{\prime}<N$ and the estimation of the mean path length is robust. The
impact of the modal overlap is illustrated in Fig. S4 in which the
transmission matrix is measured with a number of ports attached to the system
increasing from $N=1$ to $N=8$. $s_{\text{theo}}$ is seen to be accurately
estimated for $N>3$, in agreement with $\langle M^{\prime}\rangle\approx 3.2$.
In this small modal overlap regime, the first eigenchannels correspond to the
contribution of single localized long-lived modes with isolated resonances.
For these eigenchannels, values as large as $t_{n}^{(t)}=100$ ns corresponding
to travel path of 30 m in a 0.5 m long-cavity are found. Despite such long
paths, we observe in Fig. S4 that the mean path length estimated with respect
to the number of connected ports, $Nc_{0}\langle\bar{t}_{\textrm{WS}}\rangle$,
reaches a plateau already when the number of connected ports to the cavity
satisfies $N>3$.

It is worth pointing out that the transmission eigenvalues $\tau_{n}$ do not
contribute to the real part of the eigenvalues of $Q_{t}$ since the latter
only contains the frequency-derivatives of the singular vectors of $t$ which
allows an accurate estimation of the mean path length even in such localizing
systems featuring very low transmission.

##  IV Convergence towards the most direct path in the limit of strong
absorption

To investigate the effect of strong absorption on all operators involved in
Eq. (S20), we consider the two-dimensional toy-model shown in Fig. S5 which
consists of two scattering regions with scattering matrices

$S_{1}=\begin{pmatrix}r_{1}&t_{1}^{\prime}\\\
t_{1}&r^{\prime}_{1}\end{pmatrix},\quad
S_{2}=\begin{pmatrix}r_{2}&t_{2}^{\prime}\\\
t_{2}&r^{\prime}_{2}\end{pmatrix}$ (S31)

separated by free space described by the scattering matrix
$S_{2}^{\mathrm{free}}$, where we also add a free space section described by
$S_{1}^{\mathrm{free}}$ ($S_{3}^{\mathrm{free}}$) in front of the first (after
the second) scattering region for generality. The corresponding scattering
matrices for these free space sections are given by

$S_{j}^{\mathrm{free}}=\begin{pmatrix}0&P_{j}\\\ P_{j}&0\end{pmatrix},$ (S32)

where the propagation matrices are defined as

$P_{j}=\mathrm{diag}\left(e^{ik_{x,n}L_{j}}\right)\ .$ (S33)

Here, $k_{x,n}=\sqrt{k^{2}-k_{y,n}^{2}}$ are the longitudinal propagation
constants of the waveguide modes, $k=2\pi/\lambda$ is the free space wave
vector, $k_{y,n}=n\pi/W$ are the transverse wave vectors of the modes in a
waveguide of width $W$ and $L_{j}$ are the lengths of the corresponding free
space sections.

Figure S5: Sketch of the two-dimensional toy-model, where
$S_{j}^{\mathrm{free}}$ with $j=1,2,3$ are the scattering matrices of the free
space sections and $S_{1}$ and $S_{2}$ are scattering matrices of the two
considered scattering regions.

###  IV.1 Trace of $Q_{t}$

To calculate the transmission matrix of such a system, we employ the Feynman
path integral formulation and sum over all possible paths which yields

$\displaystyle\begin{split}t=\,&P_{1}t_{1}P_{2}t_{2}P_{3}\\\
&+P_{1}t_{1}P_{2}r_{2}P_{2}r^{\prime}_{1}P_{2}t_{2}P_{3}\\\
&+P_{1}t_{1}P_{2}r_{2}P_{2}r^{\prime}_{1}P_{2}r_{2}P_{2}r^{\prime}_{1}P_{2}t_{2}P_{3}+\
\ldots\\\
=\,&P_{1}t_{1}P_{2}\sum_{n=0}^{\infty}(r_{2}P_{2}r^{\prime}_{1}P_{2})^{n}t_{2}P_{3}\\\
=\,&P_{1}t_{1}P_{2}(\mathbb{1}-r_{2}P_{2}r^{\prime}_{1}P_{2})^{-1}t_{2}P_{3}\
,\end{split}$ (S34)

Hence

$t^{-1}=P_{3}^{-1}t_{2}^{-1}(\mathbb{1}-r_{2}P_{2}r^{\prime}_{1}P_{2})P_{2}^{-1}t_{1}^{-1}P_{1}^{-1}$
(S35)

and using these expressions to calculate the trace of $Q_{t}=-it^{-1}dt/dk$
then gives

$\displaystyle\mathrm{Tr}(Q_{t})=\mathrm{Tr}(Q_{t})_{\mathrm{direct}}+\mathrm{Tr}(Q_{t})_{\mathrm{multi}}$
(S36)

with

$\displaystyle\mathrm{Tr}(Q_{t})_{\mathrm{direct}}=\mathrm{Tr}\bigg{[}$
$\displaystyle-iP_{1}^{-1}\frac{dP_{1}}{dk}-it_{1}^{-1}\frac{dt_{1}}{dk}$
(S37) $\displaystyle-
iP_{2}^{-1}\frac{dP_{2}}{dk}-it_{2}^{-1}\frac{dt_{2}}{dk}-iP_{3}^{-1}\frac{dP_{3}}{dk}\bigg{]},$

being the time-delay caused by the direct path and

$\displaystyle\begin{split}\mathrm{Tr}(Q_{t})_{\mathrm{multi}}=\mathrm{Tr}\bigg{[}-i(\mathbb{1}&-r_{2}P_{2}r^{\prime}_{1}P_{2})\\\
&\times\frac{d(\mathbb{1}-r_{2}P_{2}r^{\prime}_{1}P_{2})^{-1}}{dk}\bigg{]}\end{split}$
(S38)

corresponds to the contributions from the multiply scattered paths, where we
have used the linearity of the trace and its invariance under cyclic
permutations of matrices.

We now add uniform absorption by adding an imaginary part to the total wave
vector, i.e., $k\to k+i\kappa$, where $\kappa=n_{i}k$ with $n_{i}$ being the
uniform imaginary part of the refractive index. This imaginary part then
causes a change of the propagation constants

$k_{x,n}\to\tilde{k}_{x,n}\left(1+in_{i}k^{2}/\tilde{k}_{x,n}^{2}\right)$
(S39)

with $\tilde{k}_{x,n}=\sqrt{(1-n_{i}^{2})k^{2}-k_{y,n}^{2}}$ in case of
$n_{i}\ll 1$. Thus, we get

$P_{j}\to\mathrm{diag}\left(e^{i\tilde{k}_{x,n}L_{j}}e^{-n_{i}L_{j}k^{2}/\tilde{k}_{x,n}}\right)\
,$ (S40)

where its derivative evaluates to

$\frac{dP_{j}}{dk}\to\mathrm{diag}\left(\frac{ikL_{j}}{\tilde{k}_{x,n}}-n_{i}L_{j}\left[\frac{2k}{\tilde{k}_{x,n}}-\frac{k^{3}}{\tilde{k}_{x,n}^{3}}\right]\right)P_{j}\
.$ (S41)

Taylor-expanding now the inverse matrix expression in (S38) yields

$\displaystyle\begin{split}(\mathbb{1}-r_{2}P_{2}r^{\prime}_{1}P_{2})^{-1}=\mathbb{1}&+r_{2}P_{2}r^{\prime}_{1}P_{2}\\\
&+(r_{2}P_{2}r^{\prime}_{1}P_{2})^{2}+\ \ldots\end{split}$ (S42)

and due to (S41) we find that

$\frac{d(\mathbb{1}-r_{2}P_{2}r^{\prime}_{1}P_{2})^{-1}}{dk}\propto\mathcal{O}(P_{2}^{2})\
.$ (S43)

This reveals now that
$\mathrm{Tr}(Q_{t})_{\mathrm{multi}}\propto\mathcal{O}(P_{2}^{2})\to 0$ in the
limit of strong absorption due to the exponential decay in (S40) and thus

$\mathrm{Tr}(Q_{t})\to\mathrm{Tr}(Q_{t})_{\mathrm{direct}}\ .$ (S44)

This means that the transmission delay times converge to those of the most
direct paths in the limit of strong absorption which is already reached for
$n_{i}\ll 1$ due to the fast exponential decrease in (S40). In a less complex
one-dimensional system featuring barriers of finite height, it can be even
shown that $Q_{t}$ reduces to the optical path length of the direct path
through this system if absorption is strong. It is also worth pointing out
that $\mathrm{Tr}(Q_{t})_{\mathrm{direct}}$ is independent of absorption in
the sense that the exponential decay drops out in (S37), but absorption still
enters in form of a correction term in (S41). However, since usually $n_{i}\ll
1$, these corrections are small and one can approximate
$\tilde{k}_{x,n}\approx k_{x,n}$ and
$dP_{j}/dk\approx\mathrm{diag}(ikL_{j}/k_{x,n})$ where the latter expression
yields the free space propagation delay time in the absence of absorption in
(S37).

###  IV.2 Trace of $Q_{r}$

Next, we investigate the effect of absorption on $Q_{r}$. To set up the
reflection matrix of our toy-model system, we again sum over all possible
paths and get

$\displaystyle\begin{split}r=\,&P_{1}r_{1}P_{1}\\\
&+P_{1}t_{1}P_{2}r_{2}P_{2}t^{\prime}_{1}P_{1}\\\
&+P_{1}t_{1}P_{2}r_{2}P_{2}r^{\prime}_{1}P_{2}r_{2}P_{2}t^{\prime}_{1}P_{1}+\
\ldots\\\
=\,&P_{1}r_{1}P_{1}+P_{1}t_{1}P_{2}r_{2}P_{2}\sum_{n=0}^{\infty}(r^{\prime}_{1}P_{2}r_{2}P_{2})^{n}t^{\prime}_{1}P_{1}\\\
=\,&P_{1}r_{1}P_{1}+P_{1}t_{1}P_{2}r_{2}P_{2}(\mathbb{1}-r^{\prime}_{1}P_{2}r_{2}P_{2})^{-1}t^{\prime}_{1}P_{1}\end{split}$
(S45)

To calculate the inverse of $r$ we first rewrite the above expression as
follows

$\displaystyle\begin{split}r=\,P_{1}r_{1}P_{1}\big{[}\mathbb{1}+&P_{1}^{-1}r_{1}^{-1}t_{1}P_{2}r_{2}P_{2}\\\
&\times(\mathbb{1}-r^{\prime}_{1}P_{2}r_{2}P_{2})^{-1}t^{\prime}_{1}P_{1}\big{]}.\end{split}$
(S46)

Next, we assume strong absorption which causes the term in the square brackets
to be small thus enabling us to approximate its inverse by $(1+x)^{-1}\approx
1-x$ which yields

$\displaystyle\begin{split}r^{-1}\approx\big{[}\mathbb{1}-&P_{1}^{-1}r_{1}^{-1}t_{1}P_{2}r_{2}P_{2}\\\
&\times(\mathbb{1}-r^{\prime}_{1}P_{2}r_{2}P_{2})^{-1}t^{\prime}_{1}P_{1}\big{]}P_{1}^{-1}r_{1}^{-1}P_{1}^{-1}\
.\end{split}$ (S47)

Following the same strategy as above and calculating $dr/dk$ and then $Q_{r}$,
we find that

$\displaystyle\mathrm{Tr}(Q_{r})=\mathrm{Tr}(Q_{r})_{\mathrm{direct}}+\mathrm{Tr}(Q_{r})_{\mathrm{multi}}\
,$ (S48)

with the contributions from the direct path being

$\displaystyle\begin{split}\mathrm{Tr}(Q_{r})_{\mathrm{direct}}=\mathrm{Tr}\bigg{[}&-iP_{1}^{-1}\frac{dP_{1}}{dk}\\\
&-ir_{1}^{-1}\frac{dr_{1}}{dk}-iP_{1}^{-1}\frac{dP_{1}}{dk}\bigg{]},\end{split}$
(S49)

where the contributions from the mutliply scattered paths are again

$\displaystyle\mathrm{Tr}(Q_{r})_{\mathrm{multi}}=\mathcal{O}(P_{2}^{2})\ .$
(S50)

In the limit of strong absorption, we thus again find that
$\mathrm{Tr}(Q_{r})_{\mathrm{multi}}\to 0$ which yields

$\mathrm{Tr}(Q_{r})\to\mathrm{Tr}(Q_{r})_{\mathrm{direct}}\ .$ (S51)

Hence, for strong absorption and $n_{i}\ll 1$, the eigenchannel delay times in
reflection reduce to the delay times of the most direct paths in the absence
of absorption as was the case for the time delay operator containing the
transmission matrix.

Figure S6:  a,b, Normalized mean path lengths obtained in numerical
simulations by $\langle s(\mathcal{O})\rangle/\langle
s_{\mathrm{theo}}\rangle$ with $\mathcal{O}$ representing the corresponding
operators in the legend. The ratios of these quantities with the
dissipationless value $\langle s_{\text{theo}}\rangle$ obtained from Eq. (2)
monotonically decrease with increasing absorption strength both for the empty
cavity a ($\langle s_{\text{theo}}\rangle=2.05$ m) and in the localized regime
b ($n_{\text{scat}}=280$, $\langle s_{\text{theo}}\rangle=1.76$ m). The
vertical dashed lines indicate the absorption strength for which the
frequency-averaged conductance in the simulation is equal to the
experimentally measured one. c-f, Spatial distribution of the Poynting vector
for the highest transmitting eigenchannel of the empty cavity (c,e) and a
localized sample (c,e) in the the absence of absorption (c,d) and in case of
strong absorption (e, f). The colors correspond to the longitudinal component
of the Poynting vector normalized by its magnitude.

###  IV.3 Trace of $Q$

Last, we investigate the effect of absorption on Q which contains the full
scattering matrix of the system

$S=\begin{pmatrix}r&t^{\prime}\\\ t&r^{\prime}\end{pmatrix}.$ (S52)

The scattering matrix $S$ can then be inverted blockwise

$S^{-1}=\begin{pmatrix}r^{-1}+r^{-1}t^{\prime}Mtr^{-1}&-r^{-1}t^{\prime}M\\\
-Mtr^{-1}&M\end{pmatrix}$ (S53)

with $M=(r^{\prime}-tr^{-1}t^{\prime})^{-1}$ which is valid if $r$ and $M$ are
non-singular. The trace of $Q=-iS^{-1}dS/dk$ is then given by

$\displaystyle\mathrm{Tr}(Q)=\mathrm{Tr}\Bigg{[}$ $\displaystyle-
ir^{-1}\frac{dr}{dk}-ir^{-1}t^{\prime}Mtr^{-1}\frac{dr}{dk}$ (S54)
$\displaystyle+ir^{-1}t^{\prime}M\frac{dt}{dk}+iMtr^{-1}\frac{dt^{\prime}}{dk}-iM\frac{dr^{\prime}}{dk}\Bigg{]}.$

Since absorption causes an exponential decrease in transmission, strong
absorption will lead to $t\to 0$ and $t^{\prime}\to 0$. In this limit, $M\to
r^{\prime-1}$ and

$\mathrm{Tr}(Q)\to\mathrm{Tr}(Q_{r})+\mathrm{Tr}(Q_{r^{\prime}})\ .$ (S55)

###  IV.4 Numerical results

To investigate the influence of strong absorption in more detail, we perform
numerical simulations of the experimental setup between 11 and 13 GHz and
introduce uniform absorption of increasing strength by adding a progressively
increasing imaginary part to the effective refractive index of the cavity. The
results in the empty cavity and localized regime are shown in Fig. S6a,b,
where each data point is the result of a frequency- and configuration average
over 200 random configurations. In both cases, we observe that the path
lengths obtained with the different operators decrease with respect to
absorption.

In the empty system, the transmission matrix and reflection matrices as well
as the full scattering matrix provide mean path lengths that are much smaller
than Weyl prediction in the strong absorption limit. To understand the rapid
decrease of the path lengths for an increasing absorption strength, we show in
Fig. S6 the spatial Poynting vector distribution of the eigenstate featuring
the highest transmission in the empty cavity at 10 GHz. In the absence of
absorption, complex patterns in the Poynting vector distribution are visible
(see Fig. S6c). However adding strong absorption causes the flux to align with
the positive x-direction (red color) which illustrates the convergence towards
the most direct path from the input to the output ports (see Fig. S6e).

To estimate the most direct path in the empty cavity, we use that the time-
delay of a straight section of length $\ell$ without any scatterers is $\ell
k/k_{x,n}$ with $k=2\pi/\lambda$ being the total free space wavevectora and
$k_{x,n}=\sqrt{k^{2}-(n\pi/W)^{2}}$ being the propagation constant of the
$n$-th mode in a section of width $W$. In the limit of strong absorption, only
the most direct path will survive which is given by the time-delay of the
lowest transverse waveguide mode in the corresponding sections and thus our
estimate reads

$\langle
L_{\mathrm{direct}}^{\mathrm{empty}}(k)\rangle\approx\left\langle\frac{2L_{\mathrm{lead}}k}{k_{x,1}^{\mathrm{lead}}}+\frac{L_{\mathrm{scat}}k}{k_{x,1}^{\mathrm{scat}}}\right\rangle=0.63\
\mathrm{m}\ ,$ (S56)

where $L_{\mathrm{lead}}=38$ mm,
$k_{x,1}^{\mathrm{lead}}=\sqrt{k^{2}-(\pi/W_{\mathrm{lead}})^{2}}$,
$W_{\mathrm{lead}}=15.79$ mm, $L_{\mathrm{scat}}=500$ mm,
$k_{x,1}^{\mathrm{scat}}=\sqrt{k^{2}-(\pi/W_{\mathrm{scat}})^{2}}$,
$W_{\mathrm{scat}}=250$ mm and the average was performed over the frequency
range of 11-13 GHz. Normalizing it now with the corresponding average mean
path length in this frequency interval $\langle s_{\mathrm{theo}}\rangle=2.05$
m then yields

$\frac{\langle L_{\mathrm{direct}}^{\mathrm{empty}}\rangle}{\langle
s_{\mathrm{theo}}\rangle}\approx 0.31\ .$ (S57)

This value is in very good agreement with the value of $\langle
s(2Q_{t})\rangle/\langle s_{\mathrm{theo}}\rangle$ for the largest absorption
strength which is slightly larger (0.32) due to the fact the direct paths can
also contain contributions featuring a slight angle to the longitudinal axis
causing the waves to exit the system through one of the output ports not
exactly opposite to their input port. Moreover, this absorption strength might
still be not strong enough, i.e., we might not have fully converged to the
direct path yet.

In the localized sample, the transmission delay times in the limit of strong
absorption also reduce to the ones of the most direct paths which still have
to traverse the system. However, adding impenetrable metallic scatterers to
the scattering region increases the length of the most direct path as can be
seen by comparing the mean path length for strong absorption in Fig. S6b with
Fig. S6a. The distribution of the Poynting vector in Fig. S6f now shows a
complex pattern made of tortuous paths within the sample even in the strong
absorption regime. The transmission eigenchannel delay times hence converge to
a value much larger than the one associated to $\langle
L_{\mathrm{direct}}^{\mathrm{empty}}\rangle$. This elongation of the direct
path leads thus to an increased mean path length compared to the empty system
in the strong absorption limit and thus to a better estimate of the value
predicted by the Weyl law.

In contrast, the reflection delay times decrease and converge in the
localizing configurations to small values which resemble the direct paths in
reflection, i.e., direct reflections at the first scattering layers – as
predicted by Eq. (S51) – where the trace of $Q$ converges to the sum of
reflection delay times due to the almost vanishing transmission [see Eq.
(S55)]. Thus, estimating the mean path length using the transmission matrix
provides a much more reliable estimator of the Weyl prediction in the
localized regime than its estimation based on the full scattering matrix as
illustrated in Fig. S6b.

##  V Influence of the truncation of singular values of the TM

In this section, we take a closer look at the influence of the cut-off applied
to the singular values for computing the pseudo-inverse of the scattering,
transmission and reflection matrices. Figure S7 shows the mean path length
$\langle s(2Q_{t})\rangle$ obtained from the numerical simulations for
different cut-off values $\sigma_{\mathrm{cut}}$, where
$\sigma_{\mathrm{cut}}=10^{-10}$ corresponds to the value chosen in the main
text. In the empty system and in the diffusive regime, the transmission is
high enough such that the cut-off does not affect the mean path length.
However, in the localized regime we clearly see a decrease of the mean path
length with increasing cut-off, which is caused by low transmission values
$\sigma<\sigma_{\mathrm{cut}}$ that are cut away in our SVD projection
procedure.

Figure S7:  Mean path length $\langle s(2Q_{t})\rangle$ for different singular
value cut-offs $\sigma_{\mathrm{cut}}$ (black crosses and blue circles), where
the black crosses correspond to the ones in Fig. 2a of the main text for which
a cut-off of $\sigma_{\mathrm{cut}}=10^{-10}$ was chosen. The red line shows
the theoretical value $\langle s_{\mathrm{theo}}\rangle$ obtained using the
Weyl law (frequency average of Eq. 2 in the main text). The deviations
observed here from the theoretical values are typically decreasing, when we
average not only over frequency (as done here), but also over different
disorder configurations (not shown). To make sure that the system with 280
scatterers is clearly in the localized regime, the frequency range was here
reduced to 11-12 GHz, as compared to the range 11-13 GHz used for all the
other scatterer numbers.

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