Probing Anderson
localization of light via weak
non-linear effects
Christof Aegerter, Uni Zürich
Together with:
Tilo Sperling, Wolfgang Bührer, Mirco
Ackermann and Georg Maret
Waves and Disorder, 08.07.2014
Questions to ponder during the
next 40 minutes or so
• What are generic features of Anderson
localization?
• How can we probe this transition?
• What happens in the presence of nonlinearities?
• Can we probe the intensity distribution?
• Can we directly observe localization?
• What have we learned?
Transmission of a random walk Resistance in metals
L >> l*
T ~ l*/L
Photons
(Ohm´s law)
r2 ~ t
Same as Drude
conductance
Turbid medium
"A drunk man will find his way home, but
a drunk bird may get lost forever"
Polya, (1921)
So there is a transition to localization
only in three dimensions
Abrahams et al., PRL 42, 673 (1979)
So how does the wave nature of light
lead to Anderson localization? –
enhanced backscattering.
So how does the wave nature of light
lead to Anderson localization? –
enhanced backscattering.
So how does the wave nature of light
lead to Anderson localization? –
enhanced backscattering.
So how does the wave nature of light
lead to Anderson localization? –
enhanced backscattering.
So how does the wave nature of light
lead to Anderson localization? –
enhanced backscattering.
Now suppose you go inside the sample
and you get an interfering mode, which
gets enhanced
With decreasing mean free path,
these interfering modes will be
macroscopically populated...
So what are the resulting
experimental consequences?
• Transition to a breakdown of
transmission with disorder
• Long-time tail in time-resolved total
transmission
• Confinement of the spread of photons in
transmission
• Non-exponential distribution of speckle
intensities
Time resolved transmission gives the
diffusion coefficient and absorption.
D0 /L2
Watson et al. PRL 58, 945 (1987).
Non-exponential
decay indicates
D(t)
Fitting the data with localization theory
D(t) ~ 1/t
tloc
Störzer et al, PRL (2006)
Plot the fitted localization length –
yields kl*c = 4.2(2)
CMA et al. EPL (2006).
However, titania show non-linear optical
properties, mainly due to Kerr effect –
this gives Raman scattering
Evans et al Opt. Exp. (2013)
Non-linearities are small (<10-5)
They can be seen at long times
kl* = 5.7
Intensity dependence does not depend
on kl*
kl* = 2.7
Significant spectral broadening at long
times – high intensity on long paths
kl* = 2.7
At high kl* less to no spectral
broadening
kl* = 5.7
How does this fit into the localization
picture? – remember speckle statistics
Hu et al. Nature Phys 4, 945 (2008).
Another way to change turbidity wavelength dependence of kl*
Spectral dependence also seen in
time of flight measurements
Little spectral broadening at high kl*
More spectral broadening at low kl*
So what do we expect to see in TOF
data given the non-Rayleigh Intensity
distribution and Raman scattering?
Hu et al. Nature Phys 4, 945 (2008).
Evans et al Opt. Exp. (2013)
Significant spectral broadening at long
times – high intensity on long paths
kl* = 2.7
This still depends on absorption – can
we do better?
Hu et al. Nature Phys (2008), Cherroret et al. PRE (2010).
Gated camera with image intensifier –
allows for making „movies“ with a time
resolution of 500 ps
Directly watch the diffusive transport of
photons through the sample – measure
is independent of absorption!
Time snap-shots of light propagation
kl* = 5.7 gives normal diffusion
Sperling et al Nat. Phot. (2013)
Width levels off for kl* = 2.7
Sperling et al Nat. Phot. (2013)
Now we have to make sure that the
nonlinearities do not destroy
localization
Maret et al Nat. Phot. (2013)
Actually in transverse, localization is
even enhanced by non-linearities
Schwartz et al Nature (2007)
So what can we learn about the
transition to localization?
Sperling et al Nat. Phot. (2013)
Reminder for the corresponding scales
of kl*
So what can we learn about the
transition to localization?
Sperling et al Nat. Phot. (2013)
Localization length vs. kl*
Sperling et al Nat. Phot. (2013)
What have we learned?
• Long-time tail in time resolved transmission
indicates localization of light
• Non-linear optical properties lead to enhanced
spectral shifts in high intensity localized modes
• Long-time tails as well as spectral shifts show up
after a transition with increasing turbidity
• Direct determination of the spread of photons
shows that they are in fact localized
• Combining this with a tuning of turbidity, the
localization transition and critical exponent are
characterized
Exponent of increase
r2 ~ t
r2 = const
Width of the backscattering cone
gives kl* directly
FWHM =
0.95 (kl*)-1
Akkermans et al. PRL 56, 1471 (1986).
D0 = vTl*/3 –
yields vT
How to show it's an interference effect?
– Faraday rotation in a magnetic field
Faraday effect brakes reciprocity
of light propagation.....
...and destroys coherent backscattering
Erbacher et al. EPL 21, 551 (1993).
Index-matching gives higher kl* and
classical diffusion
kl* = 15
Aegerter et al. JMO 54, 2667 (2007).
Sample R700
Fit the profile with a Gaussian for s
Actually the width even decreases –
why could this be?
Characterization of particle size –
scanning electron microscopy
Sample R700 – diameter 250 nm