Memory effects in propagation of optical waves through disordered media

Experimental confirmation of long-memory correlations in star-wander data

In this Letter we have analyzed the temporal correlations of the angle-of-arrival fluctuations of stellar images. Experimentally measured data were carefully examined by implementing multifractal detrended fluctuation analysis. This algorithm is able to discriminate the presence of fractal and multifractal structures in recorded time sequences. We have confirmed that turbulence-degraded stellar wavefronts are compatible with a long-memory correlated monofractal process. This experimental result is quite significant for the accurate comprehension and modeling of the atmospheric turbulence effects on the stellar images. It can also be of great utility within the adaptive optics field.

Speckle suppression via sparse representation for wide-field imaging through turbid media

Speckle suppression is one of the most important tasks in the image transmission through turbid media. Insufficient speckle suppression requires an additional procedure such as temporal ensemble averaging over multiple exposures. In this paper, we consider the image recovery process based on the so-called transmission matrix (TM) of turbid media for the image transmission through the media. We show that the speckle left unremoved in the TM-based image recovery can be suppressed effectively via sparse representation (SR). SR is a relatively new signal reconstruction framework which works well even for ill-conditioned problems. This is the first study to show the benefit of using the SR as compared to the phase conjugation (PC) a de facto standard method to date for TM-based imaging through turbid media including a live cell through tissue slice.

Rayleigh scattering of surface plasmons by sub-wavelength holes

We study the scattering of surface plasmons from sub-wavelength holes and find that it exhibits a stronger wavelength dependence than the traditional λ(-4) scaling found for Rayleigh scattering of light from small particles. This experimental observation is consistent with recent theoretical work and linked to the two-dimensional nature of the surface plasmon and the wavelength dependence of its spatial extent in the third dimension. The scattering cross sections are obtained with a frequency-correlation technique, which compares intensity speckle patterns observed behind various random structures of holes and recorded at different wavelengths. This powerful technique even allows us to distinguish between scattering of surface plasmons into photons and scattering into other surface plasmons.

Estimating wide-angle, spatially varying reflectance using time-resolved inversion of backscattered light

Imaging through complex media is a well-known challenge, as scattering distorts a signal and invalidates imaging equations. For coherent imaging, the input field can be reconstructed using phase conjugation or knowledge of the complex transmission matrix. However, for incoherent light, wave interference methods are limited to small viewing angles. On the other hand, time-resolved methods do not rely on signal or object phase correlations, making them suitable for reconstructing wide-angle, larger-scale objects. Previously, a time-resolved technique was demonstrated for uniformly reflecting objects. Here, we generalize the technique to reconstruct the spatially varying reflectance of shapes hidden by angle-dependent diffuse layers. The technique is a noninvasive method of imaging three-dimensional objects without relying on coherence. For a given diffuser, ultrafast measurements are used in a convex optimization program to reconstruct a wide-angle, three-dimensional reflectance function. The method has potential use for biological imaging and material characterization.

The single-channel regime of transport through random media

The propagation of light through samples with random inhomogeneities can be described by way of transmission eigenchannels, which connect incoming and outgoing external propagating modes. Although the detailed structure of a disordered sample can generally not be fully specified, these transmission eigenchannels can nonetheless be successfully controlled and used for focusing and imaging light through random media. Here we demonstrate that in deeply localized quasi-1D systems, the single dominant transmission eigenchannel is formed by an individual Anderson-localized mode or by a 'necklace state'. In this single-channel regime, the disordered sample can be treated as an effective 1D system with a renormalized localization length, coupled through all the external modes to its surroundings. Using statistical criteria of the single-channel regime and pulsed excitations of the disordered samples allows us to identify long-lived localized modes and short-lived necklace states at long and short time delays, respectively.

Polarization correlations in backscattering from media with different optical densities

Random electromagnetic fields have a number of distinctive statistical properties that may depend on their origin. We show that, when two mutually coherent fields overlap, their individual characteristics are not completely lost. If assumptions can be made regarding the coherence properties of one of the fields, then the correlation length of the second one can be retrieved using the higher-order polarization properties of the combined field. We demonstrate experimentally that colloidal particles of different sizes can be identified based on polarization correlations measured even in situations of strong multiple scattering.

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.

Recurrent scattering and memory effect at the Anderson localization transition

We report on ultrasonic measurements of the propagation operator in a strongly scattering mesoglass. The backscattered field is shown to display a deterministic spatial coherence due to a remarkably large memory effect induced by long recurrent trajectories. Investigation of the recurrent scattering contribution directly yields the probability for a wave to come back close to its starting spot. The decay of this quantity with time is shown to change dramatically near the Anderson localization transition. The singular value decomposition of the propagation operator reveals the dominance of very intense recurrent scattering paths near the mobility edge.

Direct imaging of fluorescent structures behind turbid layers

We present a method to directly image fluorescent structures inside turbid media. This is based on wave-front shaping to optimize the scattered light onto a single fluorescent particle, as the optical memory effect for a scanning image of the surroundings of this particle. We show that iterating the optimization leads to the focusing on a single particle whose surroundings are subsequently scanned. In combination with a parabolic phase pattern, this method can be extended to a three dimensional imaging method inside turbid media.

Semiclassical theory of speckle correlations

Coherent wave propagation in random media results in a characteristic speckle pattern, with spatial intensity correlations with short-range and long-range behavior. Here, we show how the speckle correlation function can be obtained from a ray picture for two representative geometries, namely a chaotic cavity and a random waveguide. Our calculation allows us to study the crossover between a "ray limit" and a "wave limit," in which the Ehrenfest time τ(E) is larger or smaller than the typical transmission time τ(D), respectively. Remarkably, long-range speckle correlations persist in the ray limit τ(E)≫τ(D).

Image restoration through thin turbid layers by correlation with a known object

A method to recover the image of an object behind thin turbid layers is developed by wavefront shaping technique. The optimized wavefront is generated by modulating the scattering light of a known object with a spatial light modulator. A Pearson Correlation Coefficient is introduced as a cost function for the optimization. A beam scanning method based on optical memory effect is proposed to further enlarge the Field-of-View (FOV). The experimental results show good fidelity and large FOV of the recovered image.

Exploiting multimode waveguides for pure fibre-based imaging

There has been an immense drive in modern microscopy towards miniaturization and fibre-based technology. This has been necessitated by the need to access hostile or difficult environments in situ and in vivo. Strategies to date have included the use of specialist fibres and miniaturized scanning systems accompanied by ingenious microfabricated lenses. Here we present a novel approach for this field by utilizing disordered light within a standard multimode optical fibre for lensless microscopy and optical mode conversion. We demonstrate the modalities of bright- and dark-field imaging and scanning fluorescence microscopy at acquisition rates that allow observation of dynamic processes such as Brownian motion of mesoscopic particles. Furthermore, we show how such control can realize a new form of mode converter and generate various types of advanced light fields such as propagation-invariant beams and optical vortices. These may be useful for future fibre-based implementations of super-resolution or light-sheet microscopy.

Fibre-based technologies provide miniaturization, flexibility and the capability to access hard to reach areas. Čižmár and Dholakia exploit disorder in multimode fibres to enable a variety of imaging modalities, including bright- and dark-field microscopy and fluorescent imaging, using a single waveguide.

Seeing through turbidity with harmonic holography [Invited]

The ability to see inside the body noninvasively is indispensable in modern biology and medicine. Optical approaches to such abilities are of rapidly growing interest because of their nonionizing nature and low cost. However, the problem of opacity due to the optical turbidity of tissues must be addressed before optical means become practical. Harmonic holography amalgamates the capability of holographic phase conjugation with the contrast-forming mechanism of second-harmonic generation, which provides a unique opportunity for imaging through a turbid medium. In this review we give accounts of the effort of imaging through turbid media using harmonic holographic phase conjugation.

Spectral control of broadband light through random media by wavefront shaping

A random medium can serve as a controllable arbitrary spectral filter with spectral resolution determined by the inverse of the interaction time of the light in the medium. We use wavefront shaping to implement an arbitrary spectral response at a particular point in the scattered field. We experimentally demonstrate this technique by selecting either a narrow band or dual bands with a width of 5.5 nm each.

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.

Correlations of polarization in random electro-magnetic fields

Random electromagnetic fields have a number of distinctive statistical properties that may depend on their origin. We show here that when two mutually coherent fields are overlapped, the individual characteristics are not completely lost. In particular, we demonstrate that if assumptions can be made regarding the coherence properties of one of the fields, both the relative average strength and the field correlation length of the second one can be retrieved using higher-order polarization properties of the combined field.

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.

Self-assembled photonic structures

Photonic crystals have proven their potential and are nowadays a familiar concept. They have been approached from many scientific and technological flanks. Among the many techniques devised to implement this technology self-assembly has always been one of great popularity surely due to its ease of access and the richness of results offered. Self-assembly is also probably the approach entailing more materials aspects owing to the fact that they lend themselves to be fabricated by a great many, very different methods on a vast variety of materials and to multiple purposes. To these well-known material systems a new sibling has been born (photonic glass) expanding the paradigm of optical materials inspired by solid state physics crystal concept. It is expected that they may become an important player in the near future not only because they complement the properties of photonic crystals but because they entice the researchers' curiosity. In this review a panorama is presented of the state of the art in this field with the view to serve a broad community concerned with materials aspects of photonic structures and more so those interested in self-assembly.

Coherent incident field information through thick random scattering media from speckle correlations over source position

With two nonoverlapping beams incident at different angles on a heavily scattering medium, the spatial correlation of speckle patterns over source position has a beat that is related to the incident angle difference. A model presented explains the measurement. The spatial correlation is shown to decorrelate faster than the beam intensity correlation function and to be sensitive to the incident field profile. Increased scatter results in more rapid decorrelation. This work suggests new opportunities for imaging through scattering media.

Imaging through turbid layers by scanning the phase conjugated second harmonic radiation from a nanoparticle

We demonstrate imaging through a turbid layer by using digital phase conjugation of the second harmonic field radiated from a beacon nanoparticle. We show that the phase-conjugated focus can be displaced from its initial position by illuminating the same region of the turbid layer with an angular offset. An image is obtained by scanning the phase-conjugated focus through the turbid layer in a region around the nanoparticle. We obtain a clear image of the target by measuring the light transmitted through it when scanning the focused beam.

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.

Finding the field transfer matrix of scattering media

When illuminated by temporally coherent light, multiply scattering media produce speckle patterns that in many situations are unpolarized on spatial averaging. As a result, the underlying field statistics are assumed to be Gaussian and information about them can be extracted from intensity-intensity correlations. However, such an approach cannot be applied to any scattering medium where the interaction leads to partially developed speckle patterns. We present a general procedure to directly measure the field transfer matrix of a linear medium without regard to the scattering regime. Experimental results demonstrate the ability of our procedure to correctly measure field transfer matrices and use them to recover the polarization state of incident illumination.

Speckle evolution of diffusive and localized waves

We show that while the statistics of static speckle patterns are generic, fluctuations in the change within speckle patterns are greatly enhanced in the localization transition. The probability distributions of the displacement of phase singularities and the standard deviations of the changes of phase and intensity with frequency shift of incident microwave radiation are given in terms of the same expression which describes the probability distribution of total transmission. This function depends only upon a single parameter, the variance of the corresponding variable. The changing statistics in the localization transition reflects the number of underlying electromagnetic modes with which the incident wave interacts.

Quantification of spatial intensity correlations and photodetector intensity fluctuations of coherent light reflected from turbid particle suspensions

We present a model for predicting the spatial intensity correlation function of dynamic speckle patterns formed by light backscattered from turbid suspensions, and an experimental validation of these predictions. The spatial correlation varies remarkably with multiple scattering. The provided computational scheme is a step towards correctly interpreting signals obtained from instruments based on the measurement of dynamic speckle patterns in the far field.

Spatial wave intensity correlations in quasi-one-dimensional wires

Spatial intensity correlations between waves transmitted through random media are analyzed within the framework of the random matrix theory of transport. Assuming that the statistical distribution of transfer matrices is isotropic, we found that the spatial correlation function can be expressed as the sum of three terms, with distinctive spatial dependences. This result coincides with the one obtained in the diffusive regime from perturbative calculations, but holds all the way from quasiballistic transport to localization. While correlations are positive in the diffusive regime, we predict a transition to negative correlations as the length of the system decreases.

Quantum noise frequency correlations of multiply scattered light

Frequency correlations in multiply scattered light that are present in quantum fluctuations are investigated. The speckle correlations for quantum and classical noise are compared and are found to depend markedly differently on optical frequency, which was confirmed in a recent experiment. Furthermore, novel mesoscopic correlations are predicted that depend on the photon statistics of the incoming light.

Spatial quantum correlations in multiple scattered light

We predict a new spatial quantum correlation in light propagating through a multiple scattering random medium. The correlation depends on the quantum state of the light illuminating the medium, is infinite in range, and dominates over classical mesoscopic intensity correlations. The spatial quantum correlation is revealed in the quantum fluctuations of the total transmission or reflection through the sample and should be readily observable experimentally.

Wave transport in two-dimensional random media: the ballistic to diffusive transition and the extrapolation length

By using a first-principles approach based on the Bethe-Salpeter equation, we study the behavior of wave propagation through a two-dimensional random slab as a function of thickness, L , in the region where L is much smaller than the localization length. A general two-dimensional vertex function for the ladder diagrams is derived from the Ward identity. We calculate both the static and the time-resolved transmitted intensities as functions of L/l , where l is the mean free path. When L is comparable to l , we study the ballistic to diffusive transition. A sharp crossover is observed when L(c) approximately = 6l, which is significantly larger than the crossover thickness of L(c) approximately = 3l found in three dimensions. When L>>l , we obtain the extrapolation length in two dimensions, i.e., z2De approximately = 0.82l, which is noticeably larger than the previously used value of z2De = pi/4 obtained by an analytical approach.

High-order correlations from rough-surface scattering

It is proposed to apply an optical setup of a randomly weak rough dielectric film on a reflecting metal substrate for the measurement of high-order correlations from rough-surface scattering. The angular amplitude and intensity correlations are measured. Because of multiple scattering, when the input laser beam size is comparatively small or close to the travel pass length inside the film, C(2) and C(3) are measured by subtraction of the amplitude correlation from the intensity correlation.

Mesoscopic correlation with polarization rotation of electromagnetic waves

Mesoscopic correlations are observed in the polarization of microwave radiation transmitted through a random waveguide. These measurements, supported by diagrammatic theory, permit an unambiguous decomposition of the intensity correlation function of a vector wave into short, long, and infinite range components. Infinite range correlation that leads to universal conductance fluctuations is measured and found to be in agreement with calculations. The long and infinite range components include nonuniversal frequency-independent terms associated with coupling into and out of the sample.

Speckle correlation measurement in a disordered medium observed through second-harmonics generation

We examined speckle correlation in a disordered medium consisting of LiNbO3 corpuscles, by observing the total intensity of the second-harmonics (SH) light generated by angular correlated two excitation beams. The total SH intensity showed a correlation peak at theta approximately 0, where theta was the angle between two excitation beams, and decreased as theta was increased. The observation could be understood as the effect of the constructive interference between two volume speckle patterns produced by the excitation beams inside the medium, and the peak could reflect the mutual correlation between two patterns. The experimental results showed good agreements with a theoretical analysis based on a diffusion approximation.

Electromagnetic scattering by spherical negative-refractive-index particles: Low-frequency resonance and localization parameters

The Mie scattering of electromagnetic waves of wave vector k by spherical negative-refractive-index particles of radius a exhibits an unusual resonance at ka-->0. The scattering enhancement from the ka-->0 resonance is insensitive to the size of scatterers, distinct from the Mie scattering resonances from positive-refractive-index particles. For media consisting of a collection of the negative-refractive-index particles, the unusual resonance results in a significant reduction of the localization parameter, providing a possibility to reach the light localization transition by reducing the wave vector k, in analogy to electronic systems.

Time-resolved pulse propagation in a strongly scattering material

Light transport in macroporous gallium phosphide, perhaps the strongest nonabsorbing scatterer of visible light, is studied using phase-sensitive femtosecond pulse interferometry. Phase statistics are measured at optical wavelengths in both reflection and transmission and compared with theory. The diffusion constant of light is measured in both reflection and transmission as a function of thickness and compared with theories for diffusive transport and localization. An unusually high energy velocity due to the bicontinuous structure of the porous network is reported. For such strongly scattering samples, we show that surface properties and the effective index of refraction need to be treated carefully.

Near-field short range correlation in optical waves transmitted through random media

Two-dimensional near-field images of light transmitted through a disordered dielectric structure have been measured for two probe wavelengths. From these data, the 2D spatial dependence of the intensity correlation function, C(deltaR-->), has been extracted. We observe that the spatial dependence of C is dominated by a rapidly varying feature determined by the wavelength of the probe light and the average refractive index of the material, as expected by theory. Finally, we deduce the absolute value of the effective refractive index by fitting the theoretical spatial dependence of C to our experimental results.

Wave transport through thin slabs of random media with internal reflection: Ballistic to diffusive transition

The static and dynamic properties of wave transport through thin slabs of random media in the presence of internal reflection are investigated by performing first-principles calculations. These results are compared with results from time-independent and time-dependent diffusion equations, respectively, where the effects due to internal reflection are incorporated into an average extrapolation length in the boundary conditions. For the static properties, we find an abrupt transition from ballistic to diffusive behavior when sample thickness is about three mean free paths, i.e., L approximately 3l. The diffusion approximation is valid when L>3l, independent of the amount of internal reflection. For the dynamic properties, both the peak arrival time at short times and the diffusion constant at long times of the transmitted pulse indicate that there is a region of anomalous diffusion when 3l<L<L(c). The diffusion constant in this region increases with decreasing L. It also increases with the amount of internal reflection. The physical origin of the existence of such an anomalous region is the resonance-induced wave focusing effect. Due to the presence of internal reflection, the wave energy tends to concentrate in the forward direction at output boundary. It makes direction randomization difficult in the scattered waves. A similar wave focusing effect has been found in resonant tunneling systems of electrons in the presence of elastic scattering. The diffusion approximation is valid when L>L(c). The value of L(c) is about ten times the average extrapolation length, i.e., L approximately 10z(e), where z(e) is a fast increasing function of the amount of internal reflection.

Finite-size effects on intensity correlations in random media

The correlations between waves transmitted through random media are analyzed by use of a random-matrix approach and numerical simulations of rough waveguides. Although the intensity and conductance fluctuations are practically independent of the sample length, the correlations present a strong dependence on the length of the disordered region. In waveguide geometries the long-range correlations C((2)) and C((3)), usually associated to intensity and conductance fluctuations, respectively, become negative as the length of the system decreases. Our results provide a new interpretation of recent optical experiments on disordered slab geometries.

Spatial-field correlation: the building block of mesoscopic fluctuations

The absence of self-averaging in mesoscopic systems is a consequence of long-range intensity correlations. Microwave measurements suggest, and diagrammatic calculations confirm, that the correlation function of the normalized intensity with displacement of the source and detector, Delta R and Delta r, respectively, can be expressed as the sum of three terms, with distinctive spatial dependences. Each term involves only the sum or the product of the square of the field correlation function, F identical with F(2)(E). The leading-order term is the product, F(Delta R)F(Delta r); the next term is proportional to the sum, F(Delta R)+F(Delta r); the third term is proportional to F(Delta R)F(Delta r)+[F(Delta R)+F(Delta r)]+1.

Field and intensity correlation in random media

We have obtained the spectral and spatial field correlation functions, C(E)(Deltaomega) and C(E)(Deltax), respectively, from measurement of the microwave field spectrum at a series of points along a line on the output of a random dielectric medium. C(E)(Deltaomega) and C(E)(Deltax) are shown to be the Fourier transforms, respectively, of the time of flight distribution, obtained from pulsed measurements, and of the specific intensity. Unlike C(E)(Deltaomega), the imaginary part of C(E)(Deltax) is shown to vanish as a result of the isotropy of the correlation function in the output plane. The complex square of the field correlation function gives the short-range or C1 contribution to the intensity correlation function C. Longer-range contributions to the intensity correlation function are obtained directly by subtracting C1 from C and are in good agreement with theory.

Speckle correlations around the backscattering direction in the double-passage configuration

Experimental results are presented for the angular correlation function of far-field speckle patterns scattered in the double passage of waves through a one-dimensional random-phase screen. The theoretical analysis of the symmetry of speckle patterns around the backscattering direction and the motion of the speckle as the source is moved, made by Escamilla et al. [Appl. Opt. 32, 2734 (1993)], are verified in this paper.

New effects in light scattering in disordered media and coherent backscattering cone: systems of magnetic particles

Single and multiple scattering of light by magnetic particles and their implications to the coherent backscattering effect are reported. Single scattering of light by small magnetic particles presents unusual features such as forward-backward asymmetry and resonance effects. In multiple scattering, this leads to a global decrease in the localization parameter kl(*), which exhibits an oscillatory dependence on the scatterer magnetic permeability. By considering magnetic scatterers following a Curie-Weiss susceptibility law, we suggest that kl(*) can be tuned by varying the temperature.

Detection of a small defect on a rough surface

We report experimental results that show that the far-field correlation function is sensitive to a small local change in rough-surface geometry, where the speckle spatial correlation rather than the sample ensemble average is adopted. The angular cross-correlation function of the far-field speckles scattered by one-dimensional random rough surfaces is measured when a polarized beam of light is incident upon the rough surface from vacuum, where one part of the surface used is a thin dielectric film deposited upon a glass substrate and the other part is identical to the first except for a localized defect. We envisage application of this sensitivity property to inspection of a sample with a defect by means of speckle mapping.

Angular correlation function of speckle patterns scattered from a one-dimensional rough dielectric film on a glass substrate

We present the experimental results of the angular correlation function of far-field speckle patterns scattered by a one-dimensionally random rough surface of a thin dielectric film on a glass substrate when a polarized beam of light is incident upon the rough surface from vacuum. This surface, which separates the vacuum and the dielectric, is rough enough that only diffused speckles are observed. The experiment for the correlation measurement was set up to make use of a CCD camera to obtain the image of the speckle pattern in the specular direction for each given angle of incidence; the cross-correlation function is then calculated from the digitized images. It is found that the intensity correlation functions exhibit two distinct maxima: one arises from the autocorrelation and the other from the reciprocity condition. It is also found that different scattering processes give rise to quite different correlation functions: multiple-scattering processes produce narrow peaks with secondary maxima and single-scattering processes produce relatively broad peaks.