Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media

WAVE PROPAGATION AND LOCALIZATION VIA QUASI-NORMAL MODES AND TRANSMISSION EIGENCHANNELS

Field transmission coefficients for microwave radiation between arrays of points on the incident and output surfaces of random samples are analyzed to yield the underlying quasi-normal modes and transmission eigenchannels of each realization of the sample. The linewidths, central frequencies, and transmitted speckle patterns associated with each of the modes of the medium are found. Modal speckle patterns are found to be strongly correlated leading to destructive interference between modes. This explains distinctive features of transmission spectra and pulsed transmission. An alternate description of wave transport is obtained from the eigenchannels and eigenvalues of the transmission matrix. The maximum transmission eigenvalue, τ1 is near unity for diffusive waves even in turbid samples. For localized waves, τ1 is nearly equal to the dimensionless conductance, which is the sum of all transmission eigenvalues, g = Στn. The spacings between the ensemble averages of successive values of ln τn are constant and equal to the inverse of the bare conductance in accord with predictions by Dorokhov. The effective number of transmission eigenvalues Neff determines the contrast between the peak and background of radiation focused for maximum peak intensity. The connection between the mode and channel approaches is discussed.

Vector transmission matrix for the polarization behavior of light propagation in highly scattering media

Recently the optical transmission matrix (TM) has been shown to be useful in controlling the propagation of light in highly scattering media. In this paper, we present the vector transmission matrix (VTM) which, unlike the TM, captures both the intensity and polarization transmission property of the scattering medium. We present an experimental technique for measuring the absolute values of the VTM elements which is in contrast to existing techniques whereby the TM elements are measured to within a scaling factor. The usefulness of the VTM is illustrated by showing that it can be used to both predict and control the magnitude of the complex polarization ratio of the light focused through the scattering medium. To the best of our knowledge, this is the first study to show the possibility of controlling the polarization of the light transmitted through highly scattering media.

Focusing through dynamic scattering media

We demonstrate steady-state focusing of coherent light through dynamic scattering media. The phase of an incident beam is controlled both spatially and temporally using a reflective, 1020-segment MEMS spatial light modulator, using a coordinate descent optimization technique. We achieve focal intensity enhancement of between 5 and 400 for dynamic media with speckle decorrelation time constants ranging from 0.4 seconds to 20 seconds. We show that this optimization approach combined with a fast spatial light modulator enables focusing through dynamic media. The capacity to enhance focal intensity despite transmission through dynamic scattering media could enable advancement in biological microscopy and imaging through turbid environments.

Adaptive control of input field to achieve desired output intensity profile in multimode fiber with random mode coupling

We develop a method for synthesis of a desired intensity profile at the output of a multimode fiber (MMF) with random mode coupling by controlling the input field distribution using a spatial light modulator (SLM) whose complex reflectance is piecewise constant over a set of disjoint blocks. Depending on the application, the desired intensity profile may be known or unknown a priori. We pose the problem as optimization of an objective function quantifying, and derive a theoretical lower bound on the achievable objective function. We present an adaptive sequential coordinate ascent (SCA) algorithm for controlling the SLM, which does not require characterizing the full transfer characteristic of the MMF, and which converges to near the lower bound after one pass over the SLM blocks. This algorithm is faster than optimizations based on genetic algorithms or random assignment of SLM phases. We present simulated and experimental results applying the algorithm to forming spots of light at a MMF output, and describe how the algorithm can be applied to imaging.

Controlling the diffused nonlinear light generated in random materials

We describe a method for shaping the wavefront of the second-harmonic light generated in nonlinear materials with a disordered structure using a spatial light modulator on the fundamental beam. We show that, for the case of a transparent strontium-barium niobate crystal with a random distribution of antiparallel domains, the speckle generation can be concentrated into a single spot.

Focusing and scanning light through a multimode optical fiber using digital phase conjugation

We demonstrate for the first time to our knowledge a digital phase conjugation technique for generating a sharp focus point at the end of a multimode optical fiber. A sharp focus with a contrast of 1800 is experimentally obtained at the tip of a 105 μm core multimode fiber. Scanning of the focal point is also demonstrated by digital means. Effects from illumination and fiber bending are addressed.

Sensitivity of synthetic aperture laser optical feedback imaging

In this paper, we compare the sensitivity of two imaging configurations, both based on laser optical feedback imaging (LOFI). The first one is direct imaging, which uses conventional optical focalization on target, and the second one is made by a synthetic aperture (SA) laser, which uses numerical focalization. We show that SA configuration allows us to obtain good resolutions with high working distance and that the drawback of SA imagery is that it has a worse photometric balance in comparison to a conventional microscope. This drawback is partially compensated by the important sensitivity of LOFI. Another interest of SA relies on the capacity of getting three-dimensional information in a single x-y scan.

Nonimaging speckle interferometry for high-speed nanometer-scale position detection

We experimentally demonstrate a nonimaging approach to displacement measurement for complex scattering materials. By spatially controlling the wavefront of the light that incidents on the material, we concentrate the scattered light in a focus on a designated position. This wavefront acts as a unique optical fingerprint that enables precise position detection of the illuminated material by simply measuring the intensity in the focus. By combining two fingerprints we demonstrate position detection along one in-plane dimension with a displacement resolution of 2.1 nm. As our approach does not require an image of the scattered field, it is possible to employ fast nonimaging detectors to enable high-speed position detection of scattering materials.

Genetic algorithm optimization for focusing through turbid media in noisy environments

We introduce genetic algorithms (GA) for wavefront control to focus light through highly scattering media. We theoretically and experimentally compare GAs to existing phase control algorithms and show that GAs are particularly advantageous in low signal-to-noise environments.

Scattered light fluorescence microscopy in three dimensions

Recently, we have proposed a method to image fluorescent structures behind turbid layers at diffraction limited resolution using wave-front shaping and the memory effect. However, this was limited to a raster scanning of the wave-front shaped focus to a two dimensional plane. In applications, it can however be of great importance to be able to scan a three dimensional volume. Here we show that this can be implemented in the same setup. This is achieved by the addition of a parabolic phase shift to the shaped wave-front. Via the memory effect, this phase shift leads to a shift of the interference based focus in the z-direction, thus opening the possibility of three dimensional imaging using scattered light fluorescence microscopy. Here, we show an example of such a three dimensional image of fluorescent nano-beads taken behind a turbid layer more than 10 mean free paths thick. Finally, we discuss the differences of the scanning in the z-direction with that in the x-y plane and the corresponding possibilities and limitations of the technique.

A multi-mode fiber probe for holographic micromanipulation and microscopy

Holographic tweezers have revolutionized the way we do experiments at the micron scale. The possibility of applying controlled force fields on simultaneously trapped micro-particles has allowed to directly probe interactions and mechanical properties of colloids, macromolecules and living cells. Holographic micromanipulation requires the careful shaping of a laser beam that is then focused by a microscope objective onto a micro-hologram in the sample volume. The same objective is used for imaging. That approach is therefore limited to in vitro samples contained in transparent cells that are easily accessed optically. Here we demonstrate that the complex light propagator of a real multimode fiber can be directly measured. That allows to transmit a micro-hologram through a 1 metre long (60 μm core) optical fiber and produce dynamic arrays of focused spots at the fiber output. We show that those spots can be used for interactive holographic micromanipulation of micron sized beads facing the fiber tip. Scanning a single spot across the output fiber we can employ the same fiber as a probe for scanning fluorescence microscopy. Our findings open the way towards the fabrication of endoscopic probes which could be capable of seeing and manipulating single cells deep into biological tissues.

Three-dimensional scanning microscopy through thin turbid media

We demonstrate three-dimensional imaging through a thin turbid medium using digital phase conjugation of the second harmonic signal emitted from a beacon nanoparticle. The digitally phase-conjugated focus scans the volume in the vicinity of its initial position through numerically manipulated phase patterns projected onto the spatial light modulator. Accurate three dimensional images of a fluorescent sample placed behind a turbid medium are obtained.

Transmission eigenvalues and the bare conductance in the crossover to Anderson localization

We measure the field transmission matrix t for microwave radiation propagating through random waveguides in the crossover to Anderson localization. From these measurements, we determine the dimensionless conductance g and the individual eigenvalues τ(n) of the transmission matrix tt(†) whose sum equals g. In diffusive samples, the highest eigenvalue, τ(1), is close to unity corresponding to a transmission of nearly 100%, while for localized waves, the average of τ(1), is nearly equal to g. We find that the spacing between average values of lnτ(n) is constant and demonstrate that when surface interactions are taken into account it is equal to the inverse of the bare conductance.

High-speed scattering medium characterization with application to focusing light through turbid media

We introduce a phase-control holographic technique to characterize scattering media with the purpose of focusing light through it. The system generates computer-generated holograms implemented via a deformable mirror device (DMD) based on micro-electro-mechanical technology. The DMD can be updated at high data rates, enabling high speed wavefront measurements using the transmission matrix method. The transmission matrix of a scattering material determines the hologram required for focusing through the scatterer. We demonstrate this technique measuring a transmission matrix with 256 input modes and a single output mode in 33.8 ms and creating a focus with a signal to background ratio of 160. We also demonstrate focusing through a temporally dynamic, strongly scattering sample with short speckle decorrelation times.

Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis

We report on the experimental measurement of the backscattering matrix of a weakly scattering medium in optics, composed of a few dispersed gold nanobeads. The decomposition of the time-reversal operator is applied to this matrix and we demonstrate selective and efficient focusing on individual scatterers, even through an aberrating layer. Moreover, we show that this approach provides the decomposition of the scattering pattern of a single nanoparticle. These results open important perspectives for optical imaging, characterization, and selective excitation of nanoparticles.

Synthetic aperture microscopy for high resolution imaging through a turbid medium

We report on synthetic aperture microscopy through a highly turbid medium. We first recorded a transmission matrix for the turbid medium with an angular basis of 20,000 complex images covering 0.6 NA. This effectively converts the medium into a lens of the same NA. Distorted images of a target object are then taken at 500 different angles of illumination covering 0.6 NA. For each of the distorted images, the original object image is reconstructed from the transmission matrix by the recently developed turbid lens imaging (TLI) technique. All 500 reconstructed images are synthesized to enhance the NA to 1.2 and thereby generate an object image with twice the enhanced spatial resolution of the individual images. Our method of applying aperture synthesis for TLI makes it possible to enhance the resolving power without increasing the number of transmission matrix elements. This relieves the demand for data acquisition and processing that has impeded the practicality of TLI.

Hidden black: coherent enhancement of absorption in strongly scattering media

We show that a weakly absorbing, strongly scattering (white) medium can be made very strongly absorbing at any frequency within its strong-scattering bandwidth by optimizing the input electromagnetic field. For uniform absorption, results from random matrix theory imply that the reflectivity of the medium can be suppressed by a factor ∼(ℓ(a)/ℓ)N(-2), where N is the number of incident channels and ℓ, ℓ(a) are the elastic and absorption mean free paths, respectively. It is thus possible to increase absorption from a few percent to >99%. For a localized weak absorber buried in a nonabsorbing scattering medium, we find a large but bounded enhancement.

Time Reversal in Subwavelength-Scaled Resonant Media: Beating the Diffraction Limit

Time reversal is a physical concept that can focus waves both spatially and temporally regardless of the complexity of the propagation medium. Time reversal mirrors have been demonstrated first in acoustics, then with electromagnetic waves, and are being intensively studied in many fields ranging from underwater communications to sensing. In this paper, we will review the principles of time reversal and in particular its ability to focus waves in complex media. We will show that this focusing effect depends on the complexity of the propagation medium rather than on the time reversal mirror itself. A modal approach will be utilized to explain the physical mechanism underlying the concept. A particular focus will be given on the possibility to break the diffraction barrier from the far field using time reversal. We will show that finite size media made out of coupled subwavelength resonators support modes which can radiate efficiently in the far field spatial information of the near field of a source. We will show through various examples that such a process, due to reversibility, permits to beat the diffraction limit using far field time reversal, and especially that this result occurs owing to the broadband inherent nature of time reversal.

Overcoming the diffraction limit using multiple light scattering in a highly disordered medium

We report that disordered media made of randomly distributed nanoparticles can be used to overcome the diffraction limit of a conventional imaging system. By developing a method to extract the original image information from the multiple scattering induced by the turbid media, we dramatically increase a numerical aperture of the imaging system. As a result, the resolution is enhanced by more than 5 times over the diffraction limit, and the field of view is extended over the physical area of the camera. Our technique lays the foundation to use a turbid medium as a far-field superlens.

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.

Generating particlelike scattering states in wave transport

We introduce a procedure to generate scattering states which display trajectorylike wave function patterns in wave transport through complex scatterers. These deterministic scattering states feature the dual property of being eigenstates to the Wigner-Smith time-delay matrix Q and to the transmission matrix t(†)t with classical (noiseless) transmission eigenvalues close to 0 or 1. Our procedure to create such beamlike states is based solely on the scattering matrix and successfully tested numerically for regular, chaotic, and disordered cavities. These results pave the way for the experimental realization of highly collimated wave fronts in transport through complex media with possible applications such as secure and low-power communication.

Control of light transmission through opaque scattering media in space and time

We report the first experimental demonstration of combined spatial and temporal control of light transmission through opaque media. This control is achieved by solely manipulating spatial degrees of freedom of the incident wave front. As an application, we demonstrate that the present approach is capable of forming bandwidth-limited ultrashort pulses from the otherwise randomly transmitted light with a controllable interaction time of the pulses with the medium. Our approach provides a new tool for fundamental studies of light propagation in complex media and has the potential for applications for coherent control, sensing and imaging in nano- and biophotonics.

Time-reversed ultrasonically encoded optical focusing into scattering media

Light focusing plays a central role in biomedical imaging, manipulation, and therapy. In scattering media, direct light focusing becomes infeasible beyond one transport mean free path. All previous methods1-3 to overcome this diffusion limit lack a practical internal "guide star."4 Here we proposed and experimentally validated a novel concept, called Time-Reversed Ultrasonically Encoded (TRUE) optical focusing, to deliver light into any dynamically defined location inside a scattering medium. First, diffused coherent light is encoded by a focused ultrasonic wave to provide a virtual internal "guide star"; then, only the encoded light is time-reversed and transmitted back to the ultrasonic focus. The TRUE optical focus-defined by the ultrasonic wave-is unaffected by multiple scattering of light. Such focusing is especially desirable in biological tissue where ultrasonic scattering is ~1000 times weaker than optical scattering. Various fields including biomedical and colloidal optics can benefit from TRUE optical focusing.

Focusing light through random photonic media by binary amplitude modulation

We study the focusing of light through random photonic materials using wavefront shaping. We explore a novel approach namely binary amplitude modulation. To this end, the light incident to a random photonic medium is spatially divided into a number of segments. We identify the segments that give rise to fields that are out of phase with the total field at the intended focus and assign these a zero amplitude, whereas the remaining segments maintain their original amplitude. Using 812 independently controlled segments of light, we find the intensity at the target to be 75±6 times enhanced over the average intensity behind the sample. We experimentally demonstrate focusing of light through random photonic media using both an amplitude only mode liquid crystal spatial light modulator and a MEMS-based spatial light modulator. Our use of Micro Electro-Mechanical System (MEMS)-based digital micromirror devices for the control of the incident light field opens an avenue to high speed implementations of wavefront shaping.

Fluctuations of the local density of states probe localized surface plasmons on disordered metal films

We measure the statistical distribution of the local density of optical states (LDOS) on disordered semicontinuous metal films. We show that LDOS fluctuations exhibit a maximum in a regime where fractal clusters dominate the film surface. These large fluctuations are a signature of surface-plasmon localization on the nanometer scale.

Image transmission through an opaque material

Optical imaging relies on the ability to illuminate an object, collect and analyse the light it scatters or transmits. Propagation through complex media such as biological tissues was so far believed to degrade the attainable depth, as well as the resolution for imaging, because of multiple scattering. This is why such media are usually considered opaque. Recently, we demonstrated that it is possible to measure the complex mesoscopic optical transmission channels that allow light to traverse through such an opaque medium. Here, we show that we can optimally exploit those channels to coherently transmit and recover an arbitrary image with a high fidelity, independently of the complexity of the propagation.

Quantum interference and entanglement induced by multiple scattering of light

We report on the effects of quantum interference induced by the transmission of an arbitrary number of optical quantum states through a multiple-scattering medium. We identify the role of quantum interference on the photon correlations and the degree of continuous variable entanglement between two output modes. It is shown that quantum interference survives averaging over all ensembles of disorder and manifests itself as increased photon correlations due to photon antibunching. Furthermore, the existence of continuous variable entanglement correlations in a volume speckle pattern is predicted. Our results suggest that multiple scattering provides a promising way of coherently interfering many independent quantum states of light of potential use in quantum information processing.

Direct phase probing and mapping via spintronic Michelson interferometry

A spintronic approach is introduced to transform classic Michelson interferometry that probes the electromagnetic phase only. This method utilizes a nonlinear four-wave coherent mixing effect. A previously unknown striking relation between spin dynamics and the relative phase of electromagnetic waves is revealed. Spintronic Michelson interferometry allows direct probing of both the spin-resonance phase and the relative phase of electromagnetic waves via microspintronics. Thereby, it breaks new ground for cross-disciplinary applications with unprecedented capabilities, which we demonstrate via a powerful phase-resolved spin-resonance spectroscopy on magnetic materials and an on-chip technique for phase-resolved near-field microwave imaging.

Transmission matrices of random media: means for spectral polarimetric measurements

The field transmitted through disordered media is essentially a randomly sampled version of the incident field. Properties of the initial field can be recovered if this sampling function or transmission matrix is known. Here we demonstrate how the transmission matrix of a disordered material can be used to simultaneously measure the spectral and polarimetric properties of an optical field.

Scattered light fluorescence microscopy: imaging through turbid layers

A major limitation of any type of microscope is the penetration depth in turbid tissue. Here, we demonstrate a fundamentally novel kind of fluorescence microscope that images through optically thick turbid layers. The microscope uses scattered light, rather than light propagating along a straight path, for imaging with subwavelength resolution. Our method uses constructive interference to focus scattered laser light through the turbid layer. Microscopic fluorescent structures behind the layer were imaged by raster scanning the focus.