Brownian Motion in a Speckle Light Field: Tunable Anomalous Diffusion and Selective Optical Manipulation

The motion of particles in random potentials occurs in several natural phenomena ranging from the mobility of organelles within a biological cell to the diffusion of stars within a galaxy. A Brownian particle moving in the random optical potential associated to a speckle pattern, i.e., a complex interference pattern generated by the scattering of coherent light by a random medium, provides an ideal model system to study such phenomena. Here, we derive a theory for the motion of a Brownian particle in a speckle field and, in particular, we identify its universal characteristic timescale. Based on this theoretical insight, we show how speckle light fields can be used to control the anomalous diffusion of a Brownian particle and to perform some basic optical manipulation tasks such as guiding and sorting. Our results might broaden the perspectives of optical manipulation for real-life applications.

nm, (c) R = 250 nm, (d) R = 500 nm and (e) R = 1000 nm. The force field acting on particles whose radius is smaller than the wavelength (b-d) is qualitatively the same as the one acting on the point dipole in (a). A change in the field happens only for much larger particles: the particle in (e) is not attracted to the maxima of intensities in the speckle field but to a minimum of intensity separating two or more speckle grains, as previously observed in periodic patterns 41 . The white circles represent the different particle's sizes. In every plot the length of the arrows representing the force is normalized to its maximum. The resolution of the fractionation is only limited by the size of the speckle pattern, i.e., the longer the speckle pattern the higher the sensitivity in particle's size.

Speckle pattern intensity distribution
A speckle pattern is an interference figure resulting from random scattering of coherent light by a complex medium. The probability density function of the speckle pattern intensity follows the negative exponential distribution 3,24 : where is the average speckle pattern intensity. Figure S4 shows the very good agreement between theoretical and numerical distributions of the speckle pattern intensities used in the simulations.

Speckle pattern correlation function and its Gaussian approximation
The normalized spatial autocorrelation function of the speckle pattern, which provides a measure of the average speckle grain size, is defined by the diffraction process that generates the speckle pattern itself 3,24 . For a fully developed speckle pattern, in the general case, this autocorrelation function can be approximated by a Gaussian function whose standard deviation depends on the size of the average speckle grain. In what follows, we treat the case of a speckle pattern generated by diffraction through a circular aperture, as used in the simulations. In this case, the autocorrelation function is the Airy disk ( Figure S5 So far, we have considered the case of 2-dimensional speckle patterns, while a full 3dimensional description of speckle patterns might be sometimes necessary. As for the case of the 2-dimensional speckle pattern, the statistical properties of the 3-dimensional patterns are also fully defined by the diffraction process that generates them, so that the average size of the 3-dimensional speckle grain can be defined at any point of observation away from the plane where the speckle pattern is generated 24 . 9

Force probability density function
In a speckle field, the optical forces exerted on a Brownian particle are proportional to the gradient of the speckle pattern intensity (see Methods). Since the speckle field is known, we can fully derive numerically the associated random force field and calculate its statistical properties. In particular, from the definition of variance, it can be shown numerically that he following property holds: where is the absolute value of the force and . This property will be useful to derive Equation (S10). In Figure S6, we plot the probability density function of the force for different particle' radii, and, as a guide for the eyes, we fit it to the following empirical function: where is a normalization factor.

Force correlation function
The aim is to calculate the force correlation function from the intensity correlation  Figure S7 shows two schematics of different possible setups that can be implemented at low cost with little alignment based on two processes to generate a speckle pattern, e.g., light propagation through a multimode optical fiber and light scattering from a complex medium, such as paper, paint, or many biological tissues 4 . The use of a multimode optical fiber, in particular, provides homogeneous speckle fields over controllable areas. Since the random potential of a speckle field results from optical forces, we expect the power requirements per unit of area to be comparable to the ones employed to achieve optical manipulation in periodic potentials [13][14][15][16][17][18][19]34,41 . As for other optical trapping techniques, the bigger the particles and the higher the refractive index, the lower is the need for power to achieve a certain level of average force. Therefore, the input power will greatly scale down when increasing the volume of the particles.