Disorder-Enhanced Imaging with Spatially Controlled Light

Disordered zinc oxide layers 45
strongly position dependent, it is better to adopt an effective refractive index to
characterize the material.
The effective refractive index n e of a strongly scattering material can be found
by measuring the angular resolved transmission of light.[15] In the diffusion approximation,
the angular resolved transmission probability P (µ e ) is equal to[16]
( ) 2 (
P (µ e ) ne 1 + R
=
µ e n 0 1 − R + 3 )
2 µ [1 − R(µ)] , (3.3)
where µ = cos θ and µ e = cos θ e are the projections of the incident angle θ and
the transmitted angle θ e perpendicular to the sample surface. The specific and
angle averaged reflectivity at the surface, R(µ) and R, are directly calculated
from the Fresnel coefficients. These reflectivities solely depend on the refractive
index contrast n e /n 0 between the interior and the exterior of the sample and
the polarization. An experimentally acquired P (µ e ) for a fixed polarization can
therefore be used to determine the refractive index contrast.
We use a broadband angle-dependent light scattering setup equivalent to the
apparatus that is described in Ref. [17]. The setup is placed in transmission
mode to measure the angular resolved transmission through our ZnO samples.
The sample is illuminated by a supercontinuum light source, which enables a measurement
of the effective refractive index over a wavelength range from 500 nm
up to 900 nm.
The experimental results are shown in Fig. 3.3. In panel a we see the spectrally
resolved transmission as function of the outgoing angle θ e . This data is
the p-polarized transmission averaged over 6 measurements on different sample
positions. We plotted P (µ e )/µ e for a wavelength of 532 nm in panel b. By fitting
this data with the refractive index contrast as the only free parameter, we find
n e = 1.34. We repeated this procedure for all measured wavelengths to determine
the spectral dependence of n e , which is shown in panel c. The gray area denotes
the error that we estimated by comparing the fits to the data for different values
of the refractive index.
Transport mean free path
In heterogeneous structures light is scattered as it propagates. The average distance
light travels before it completely looses direction is defined as the transport
mean free path l. In our samples that consist of isotropic scatterers, the transport
mean free path is equal to the scattering mean free path. The amount of
unscattered transmitted light reduces as e −L/l , where L is the layer thickness.
Therefore, a sample with a thickness of at least a few times the mean free path
is optically thick and opaque.
The transport mean free path can be obtained from a measurement of the
total diffusive transmission T as function of the layer thickness L. The total
transmission is inversely proportional to[18]
1
T = L + z e1 + z e2
l + z e1
, (3.4)