1. 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,

2. 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 non-linearities?
Can we probe the intensity distribution?
Can we directly observe localization?
What have we learned?

3. Transmission of a random walk - Resistance in metals
L >> l*
T ~ l*/L
(Ohm´s law)
Photons
r2 ~ t
Same as Drude conductance
Turbid medium

4. "A drunk man will find his way home, but a drunk bird may get lost forever"
Polya, (1921)

5. So there is a transition to localization only in three dimensions
Abrahams et al., PRL 42, 673 (1979)

6. So how does the wave nature of light lead to Anderson localization
So how does the wave nature of light lead to Anderson localization? – enhanced backscattering.

7. So how does the wave nature of light lead to Anderson localization
So how does the wave nature of light lead to Anderson localization? – enhanced backscattering.

8. So how does the wave nature of light lead to Anderson localization
So how does the wave nature of light lead to Anderson localization? – enhanced backscattering.

9. So how does the wave nature of light lead to Anderson localization
So how does the wave nature of light lead to Anderson localization? – enhanced backscattering.

10. So how does the wave nature of light lead to Anderson localization
So how does the wave nature of light lead to Anderson localization? – enhanced backscattering.

11. Now suppose you go inside the sample and you get an interfering mode, which gets enhanced

12. With decreasing mean free path, these interfering modes will be macroscopically populated...

13. 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

14. Time resolved transmission gives the diffusion coefficient and absorption.
D0 /L2
Non-exponential decay indicates D(t)
Watson et al. PRL 58, 945 (1987).

15. Fitting the data with localization theory
D(t) ~ 1/t
tloc
Störzer et al, PRL (2006)

16. Plot the fitted localization length – yields kl*c = 4.2(2)
CMA et al. EPL (2006).

17. However, titania show non-linear optical properties, mainly due to Kerr effect –this gives Raman scattering
Evans et al Opt. Exp. (2013)

18. Non-linearities are small (<10-5)

19. They can be seen at long times
kl* = 5.7

20. Intensity dependence does not depend on kl*

21. Significant spectral broadening at long times – high intensity on long paths
kl* = 2.7

22. At high kl* less to no spectral broadening

23. How does this fit into the localization picture
How does this fit into the localization picture? – remember speckle statistics
Hu et al. Nature Phys 4, 945 (2008).

24. Another way to change turbidity - wavelength dependence of kl*

25. Spectral dependence also seen in time of flight measurements

26. Little spectral broadening at high kl*

27. More spectral broadening at low kl*

28. 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)

29. Significant spectral broadening at long times – high intensity on long paths
kl* = 2.7

30. This still depends on absorption – can we do better?
Hu et al. Nature Phys (2008), Cherroret et al. PRE (2010).

31. 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!

32. Time snap-shots of light propagation

33. kl* = 5.7 gives normal diffusion
Sperling et al Nat. Phot. (2013)

34. Width levels off for kl* = 2.7
Sperling et al Nat. Phot. (2013)

35. Now we have to make sure that the nonlinearities do not destroy localization
Maret et al Nat. Phot. (2013)

36. Actually in transverse, localization is even enhanced by non-linearities
Schwartz et al Nature (2007)

37. So what can we learn about the transition to localization?
Sperling et al Nat. Phot. (2013)

38. Reminder for the corresponding scales of kl*

39. So what can we learn about the transition to localization?
Sperling et al Nat. Phot. (2013)

40. Localization length vs. kl*
Sperling et al Nat. Phot. (2013)

41. 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

42. Exponent of increase
r2 ~ t
r2 = const

43. Width of the backscattering cone gives kl* directly
FWHM =
0.95 (kl*)-1
D0 = vTl*/3 –
yields vT
Akkermans et al. PRL 56, 1471 (1986).

44. How to show it's an interference effect
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).

45. Index-matching gives higher kl* and classical diffusion
Aegerter et al. JMO 54, 2667 (2007).
Sample R700

46. Fit the profile with a Gaussian for s

47. Actually the width even decreases – why could this be?

48. Characterization of particle size – scanning electron microscopy
Sample R700 – diameter 250 nm