A multimode fiber (ø = 62.5 μm) guides the reflected light from both arms to a photodiode. Signal processing after acquisition, which is described in detail in Ref. ^{1}, results in averaged spectra S(ℓ) with 8-nm resolution [∼500 averages per ℓ, to avoid any spectral modulations on S(ℓ) caused by interference between scattering particles]. We describe S(ℓ) with a single exponential decay model (Ref. ^{2}) S(ℓ) = S_{0}·T·Δℓ·
$\rm{{\bf{\umu}}_{\bf{b,NA}}}$
$\mu b,NA$·exp(−
$\rm{{\bf{\umu}}_{\bf{t}}}$
$\mu t$·ℓ),
^{2} where
S_{0} is the source power spectrum and
T is the system coupling efficiency. When
S(ℓ) is dominated by a single backscattered light,
$\rm{{\bf{\umu}}_{\bf{t}}}$
$\mu t$ is the attenuation coefficient of the sample and
$\rm{{\bf{\umu}}_{\bf{t}}}$
$\mu t$ equals
$\rm{{\bf{\umu}}_{\bf{s}}}$
$\mu s$ for nonabsorbing samples (this study). The system dependent parameters will be denoted by
ζ =
S_{0}·
T·Δℓ. The spectra
S(ℓ) are collected over the detection numerical aperture (
NA) of the system, therefore, we define the measured backscattering coefficient
$\rm{{\bf{\umu}}_{\bf{b,NA}}}$
$\mu b,NA$ as the product of
$\rm{{\bf{\umu}}_{\bf{s}}}$
$\mu s$ and the phase function
p(
$\bm\theta$
$\theta $), integrated over the solid angle of the
NA in the medium:
Display Formula1\documentclass[12pt]{minimal}\begin{document}\begin{eqnarray}
{\bf {\rm{\umu _{\bf b,NA} = \umu _{\bf s}} \cdot 2\pi \int\nolimits_{{{{\bm\theta} = {\bm \pi} - {\bf{NA}}}}}^\pi {{{\bf p}\left( {\bm{\theta}} \right) \cdot {\rm{sin}}\left( {\bm{\theta}} \right) \cdot d{\bm{\theta}} }}}}.
\end{eqnarray}\end{document}
$\mu b,NA=\mu s\xb72\pi \u222b\theta =\pi \u2212NA\pi p\theta \xb7 sin \theta \xb7d\theta .$ We measured the wavelength dependent point spread function in the medium
^{7} and derived the
NA (ranging from 0.035 to 0.045 between 480 to 700 nm) from the resulting Rayleigh length of the system. The terms
$\rm{{{\bm \uzeta\cdot\umu}}_{\bf{b,NA}}}$
$\zeta \xb7\mu b,NA$ and
$\rm{{\bf{\umu}}_{\bf{s}}}$
$\mu s$ are obtained by fitting a two-parameter (amplitude and decay, respectively) exponential function to
S(ℓ) versus ℓ. Uncertainties are estimated by the 95% confidence intervals (c.i.) of the fitted parameters.
^{1} The model is fitted to the measured
S(ℓ) up to a path length in the sample of five times the mean free path (5/
$\rm{{\bf{\umu}}_{\bf{s}}}$
$\mu s$ from Mie theory at 480 nm, varying from 100 to 1950 μm). Spectra acquired from ℓ < 50 μm suffer from boundary artifacts and are therefore excluded from the fits. Prior to fitting the model to
S(ℓ), a noise level is subtracted from
S(ℓ), which is the sum of the dc spectra of the sample and reference arm. Now,
$\rm{{\bf{\umu}}_{\bf{b,NA}}}$
$\mu b,NA$ can be calculated from the fitted amplitude
$\rm{{\bm{\uzeta\cdot\umu}}_{\bf{b,NA}}}$
$\zeta \xb7\mu b,NA$, if
ζ is determined in a separate calibration measurement in which
$\rm{{\bf{\umu}}_{\bf{b,NA}}}$
$\mu b,NA$ is exactly known from Mie theory and Eq.
1. To this end, we used National Institute of Standards and Technology (NIST)-certified polystyrene spheres of ø = 409±9 nm (diameter±SD, Thermo Scientific, USA). The obtained
ζ was used to determine
$\rm{{\bf{\umu}}_{\bf{b,NA}}}$
$\mu b,NA$ in subsequent measurements.