Jacopo Bertolotti

A Levy flight for light

A random walk is a stochastic process in which particles or waves travel along random trajectories. The first application of a random walk was in the description of particle motion in a fluid (brownian motion); now it is a central concept in statistical physics, describing transport phenomena such as heat, sound and light diffusion. Lévy flights are a particular class of generalized random walk in which the step lengths during the walk are described by a ‘heavy-tailed’ probability distribution. They can describe all stochastic processes that are scale invariant. Lévy flights have accordingly turned out to be applicable to a diverse range of fields, describing animal foraging patterns, the distribution of human travel and even some aspects of earthquake behaviour. Transport based on Lévy flights has been extensively studied numerically, but experimental work has been limited and, to date, it has not seemed possible to observe and study Lévy transport in actual materials. For example, experimental work on heat, sound, and light diffusion is generally limited to normal, brownian, diffusion. Here we show that it is possible to engineer an optical material in which light waves perform a Lévy flight. The key parameters that determine the transport behaviour can be easily tuned, making this an ideal experimental system in which to study Lévy flights in a controlled way. The development of a material in which the diffusive transport of light is governed by Lévy statistics might even permit the development of new optical functionalities that go beyond normal light diffusion.

Scattering Lens Resolves Sub-100 nm Structures with Visible Light

The smallest structures that conventional lenses are able to optically resolve are of the order of 200 nm. We introduce a new type of lens that exploits multiple scattering of light to generate a scanning nanosized optical focus. With an experimental realization of this lens in gallium phosphide we imaged gold nanoparticles at 97 nm optical resolution. Our work is the first lens that provides a resolution better than 100 nm at visible wavelengths.

Superpixel-based spatial amplitude and phase modulation using a digital micromirror device

We present a superpixel method for full spatial phase and amplitude control of a light beam using a digital micromirror device (DMD) combined with a spatial filter. We combine square regions of nearby micromirrors into superpixels by low pass filtering in a Fourier plane of the DMD. At each superpixel we are able to independently modulate the phase and the amplitude of light, while retaining a high resolution and the very high speed of a DMD. The method achieves a measured fidelity F = 0.98 for a target field with fully independent phase and amplitude at a resolution of 8 × 8 pixels per diffraction limited spot. For the LG10 orbital angular momentum mode the calculated fidelity is F = 0.99993, using 768 × 768 DMD pixels. The superpixel method reduces the errors when compared to the state of the art Lee holography method for these test fields by 50% and 18%, with a comparable light efficiency of around 5%. Our control software is publicly available.

Speckle correlation resolution enhancement of wide-field fluorescence imaging

High-resolution fluorescence imaging is essential in nanoscience and biological sciences. Due to the diffraction limit, conventional imaging systems can only resolve structures larger than 200 nm. Here, we introduce a new fluorescence imaging method that enhances the resolution by using a high-index scattering medium as an imaging lens. Simultaneously, we achieve a wide field of view. We develop a new image reconstruction algorithm that converges even for complex object structures. We collect two-dimensional fluorescence images of a collection of 100 nm diameter dye-doped nanospheres, and demonstrate a deconvolved Abbe resolution of 116 nm with a field of view of 10 μm×10  μm . Our method is robust against optical aberrations and stage drifts, and therefore is well suited to image nanostructures with high resolution under ambient conditions.Hasan Yılmaz, Elbert G. van Putten, Jacopo Bertolotti, Ad Lagendijk, Willem L. Vos, and Allard P. Mosk Complex Photonic Systems (COPS), MESA+ Institute for Nanotechnology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands Present address: Philips Research Laboratories, 5656 AE Eindhoven, The Netherlands Present address: Physics and Astronomy Department, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom

Wide-band transmission of nondistorted slow waves in one-dimensional optical superlattices

Few micron-thick one-dimensional optical superlattices were designed and grown, in which an optimized choice of external dielectric layers allows the formation of a wide and high transmission miniband of coupled cavity states. In such structures a reduction in light group velocity and minimal line shape distortion of propagating optical signal was observed. Group velocity reduction by a factor of 5, obtained both from phase (white-light interferometry) and from time-resolved measurements, is in reasonably good agreement with those calculated through a transfer matrix approach. Time-resolved experiments confirm the minimal line shape distortion for optical pulses of 1.8THz bandwidth at λ=1.5μm wavelength.

Weak Localization of Light in Superdiffusive Random Systems

Lévy flights constitute a broad class of random walks that occur in many fields of research, from biology to economy and geophysics. The recent advent of Lévy glasses allows us to study Lévy flights-and the resultant superdiffusion-using light waves. This raises several questions about the influence of interference on superdiffusive transport. Superdiffusive structures have the extraordinary property that all points are connected via direct jumps, which is expected to have a strong impact on interference effects such as weak and strong localization. Here we report on the experimental observation of weak localization in Lévy glasses and compare our results with a recently developed theory for multiple scattering in superdiffusive media. Experimental results are in good agreement with theory and allow us to unveil the light propagation inside a finite-size superdiffusive system.

Multiple scattering of light in superdiffusive media.

Light transport in superdiffusive media of finite size is studied theoretically. The intensity Greens function for a slab geometry is found by discretizing the fractional diffusion equation and employing the eigenfunction expansion method. Truncated step length distributions and complex boundary conditions are considered. The profile of a coherent backscattering cone is calculated in the superdiffusion approximation.

Optical transmission matrix as a probe of the photonic strength

We demonstrate that optical transmission matrices (TMs) provide a powerful tool to extract the photonic strength of disordered complex media, independent of surface effects. We measure the TM of a strongly scattering GaP nanowire medium and compare the singular value density of the measured TM to a random-matrix-based wave transport model. By varying the transport mean free path and effective refractive index in the model, we retrieve the photonic strength. From separate numerical simulations we conclude that the photonic strength derived from TM statistics is insensitive to the surface reflection at rear surface of the sample.

Quantum correlation of light scattered by disordered media

We study theoretically how multiple scattering of light in a disordered medium can spontaneously generate quantum correlations. In particular we focus on the case where the input state is Gaussian and characterize the correlations between two arbitrary output modes. As there is not a single all-inclusive measure of correlation, we characterise the output correlations with three measures: intensity fluctuations, entanglement, and quantum discord. We find that, while a coherent input state can not produce quantum correlations, any other Gaussian input will produce them in one form or another. This includes input states that are usually regarded as more classical than coherent ones, such as thermal states, which will produce a non-zero quantum discord.

Measurements on the optical transmission matrices of strongly scattering nanowire layers

Summary form only given. Light incident on a scattering medium is redistributed over transport channels that either transmit through or reflect from the medium. We perform experiments aiming at finding individual transport channels of extremely strongly scattering materials. A small number of transport channels in a scattering sample are open with transmission coefficient close to 1; field transmission mainly takes place through these channels [1-3]. This means that, even if two very different incident fields are sent to the sample, the corresponding transmitted fields are correlated. As the scattering becomes stronger, these correlations become more pronounced.One way to investigate these correlations is to construct a transmission matrix by measuring the fields transmitted through the medium in response to pre-determined incident fields. Recently, microwave and optical experiments have been performed on measurements on the transmission matrices of scattering materials. In these studies, knowledge of the transmission matrix have been used for focusing [4, 5] and enhancing the transmission [6] through disordered media, or to study predictions of random matrix theory [7]. An observation of correlations in the optical transmission matrices of strongly scattering materials have not been reported so far. We measure transmission matrices of strongly scattering layers of disordered GaP nanowires, which are among the strongest scattering materials for visible light. The samples under study have thicknesses varying between -1.5 μm and -6 μm and transport mean free path of -0.2 μm [8]. We investigate the correlations in the measured transmission matrices and compare our experimental findings to a numerical model in order to retrieve physical parameters such as the scattering strength of the samples.