# Light Diffusion

Introduction

The propagation of light through complex media is often diffusive. Take, for example, the scattering of sunlight through fog or dust in air, where it can be difficult to determine from where the light originates. This is a phenomenon that very often escapes our understanding but not our perception: light scatters by fat in a glass of milk or sunlight is dispersed through dust in air, the white of teeth or of chalk. Contrast this with photonic crystals, which transport light in a coherent manner. Disordered scattering elements may scramble an incident light beam, but we have found that substantial control of light transport can be achieved even in presence of strong disorder. Disorder – which is normally regarded as a weakening factor in photonic devices – plays a central role, and may make it possible to engineer the flow of diffuse light.

Transmitted intensity: the photonic Ohm’s law

Light transmitted through a disordered medium presents a granular pattern called speckle, like this one:

Ensamble average on the sample configuration, obtained simply moving the sample and recording the sum of multiple transmission images leads to an homogeneous pattern, which can often be analyzed with a diffusion model.

Light diffusion through an isotropic disordered media. On the left a Monte Carlo simulation, while on the right a macroscopic experiment.

In analogy with electron conduction, a photonic Ohm’s law applies, …

Static light transport

Figure schematically draws the set up used to perform static measurements. It consists of an integrating sphere which consists of a hollow cavity with its interior coated for high diffuse reflectance. Slabs photonic glasses with different thickness are placed on the integrating sphere and illuminated with light provided by a Tungsten lamp. If the sample is optically very thick, ballistic or unscattered light propagating through it is exponentially attenuated. We can therefore assume that only diffusive light comes out from the sample and enters in the integrating sphere.

A typical setup look like in figure:

 Cartoon of the experimental set up to perform static measurements. A slab of photonic glass with thickness L is placed in the entrance of an integrating sphere and illuminated with white light. Diffuse light is measured in the detector.

Dynamic light transport

The diffusion constant $\mathcal{D}(\lambda)$ is a dynamical quantity and therefore can be probed directly with a measurement of the spread in time, T(t), of a short pulse crossing the sample. The time profile of the transmitted light is measured with a streak camera, while the pulses are provided by a Ti:Al2 O 3 laser (2 ps pulse duration), tunable within 700-920 nm. For long times, the pulse spread is exponential with a time constant given by $\tau (\lambda) = (L + 2 z_e)2/\pi2 \mathcal{D}(\lambda)$ (in the absence of absorption), while the the full solution at all times (for δ-function light source) is:

$T(t,\lambda) = \frac{I_0 \, \exp(-t/\tau_i)}{ 4 \, t\,(4 \pi t \mathcal{D}(\lambda))^{3/2} } \, [ \sum^{+\infty}_{j = -\infty} A \exp(-A^2/4\mathcal{D}(\lambda)t)+ \sum^{+\infty}_{j = -\infty} B\exp(-B^2/4\mathcal{D}(\lambda)t) ]$

where $A = (1-2j)(L + 2 z_e)-2(z_p + \ell_t)$ and B = (2j + 1)(L + 2ze).

A typical setup look like in figure:

 Typical setup for pulse reshaping measurements. Frequency and time spreading can be measured at the same time, with ps resolution, thanks to the Streak camera.

while a typical measurement looks like:

 An example of a measurement with the setup shown above. In the image the vertical is the frequency, of few nm, while the horizontal is the time, of 2 ns.

The fit is obtained through a Mathlab routine, that includes the full solution, summing over a given number of orders (typically 20 or 30) and that includes the convolution with the laser pulse gaussian profile. Care has to be taken in using the right average reflectivity at the boundaries, as it is very different than the one obtained from Snell’s law with a geometrical average refractive index.

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