The phase of multiply scattered light has recently attracted considerable interest. Coherent backscattering is a striking phenomenon of multiple scattered light in which the coherence of light survives multiple scattering in a random medium and is observable in the direction space as an enhancement of the intensity of backscattered light within a cone around the retroreflection direction. Reciprocity also leads to enhancement of backscattering light in the spatial space. The random medium be- haves as a reciprocity mirror which robustly converts a diverging incident beam into a converging backscattering one focusing at a conjugate spot in space. Here we first analyze theoretically this coherent backscattering mirror (CBM) phenomenon and then demonstrate the capability of CBM compensating and correcting both static and dynamic phase distortions occurring along the optical path.CBMmay offer novel approaches for high speed dynamic phase corrections in optical systems and find appli- cations in sensing and navigation.

Light reflectance has been widely used to diagnose random media in both in situ and in vivo applications. The quantification of the phase function of the medium from reflectance measurements, however, remains elusive due to the lack of an explicit connection between the light reflectance profile and the phase function. Here we first present an analytical model for reflectance of scattered light at an arbitrary source-detector separation by forward-peaked scattering media such as biological tissue and cells. The model incorporates the improved small-angle scattering approximation (SAA) to radiative transfer for sub-diffusive light reflectance and expresses the dependence of the light reflectance on the phase function of the scattering medium in a closed form. A spreading length scale, l, is found to characterise subdiffusive light reflectance at the high spatial frequency (close separation) limit. After validation by Monte Carlo simulations, we then demonstrate the application of the model in accurate determination of the complete set of optical properties and the phase function of a turbid medium from the profile of subdiffusive and diffusive light reflectance.

Critical-zone reactions involve inorganic and biogenic colloids in a cation-rich environment. The present research defines the rates and structure of purified Mg-montmorillonite aggregates formed in the presence of monovalent (K+) and divalent (Ca2+,Mg2+) cations using light-extinction measurements. Time evolution of turbidity was employed to determine early-stage aggregation rates. Turbidity spectra were used to measure the fractal dimension at later stages. The power law dependence of the stability ratios on cation concentration was found to vary with the reciprocal of the valence rather than the predicted reciprocal of valence-squared, indicating that the platelet structuremay be a factor influencing aggregation rates. The critical coagulation concentrations (CCC) (3 mM for CaCl2, 4 mM for MgCl2, and 70 mM for KCl) were obtained from the stability ratios. At a later time and above a minimal cation concentration, turbidity reached a quasi-stable state, indicating the formation of large aggregates. Under this condition, an approximate turbidity forward-scattering correction factor was applied and the fractal dimension was determined from the extinction spectra. For the divalent cations, the fractal dimensions were 1.650.3 for Ca2+ and 1.750.3 for Mg2+ and independent of cation concentrations above the CCC. For the monovalent cation, the fractal dimension increased with K+ concentration from 1.35 to 1.95, indicating a transition to a face-to-face geometry from either an edge-to-edge or edge-to-face orientation.

Diffuse optical imaging (DOI) for detecting and locating targets in a highly scattering turbid medium is treated as a blind source separation (BSS) problem. Three matrix decomposition methods, independent component analysis (ICA), principal component analysis (PCA), and nonnegative matrix factorization (NMF) were used to study the DOI problem. The efficacy of resulting approaches was evaluated and compared using simulated and experimental data. Samples used in the experiments included Intralipid-10% or Intralipid-20% suspension in water as the medium with absorptive or scattering targets embedded.

Optical extinction and diffuse reflection spectra of cancerous and normal prostate tissues in the 750 to 860 nm spectral range were measured. Optical extinction measurements using thin ex vivo prostate tissue samples were used to determine the scattering coefficient (μ(s)), while diffuse reflection measurements using thick prostate tissue samples were used to extract the absorption coefficient (μ(a)) and the reduced scattering coefficient (μ's ). The anisotropy factor (g) was obtained using the extracted values of μ(s) and μ's. The values of fractal dimension (D(f)) of cancerous and normal prostate tissues were obtained by fitting to the wavelength dependence of μ's. The number of scattering particles contributing to μ(s) as a function of particle size and the cutoff diameter D(max) as a function of g were investigated using the fractal soft tissue model and Mie theory. Results show that D(max) of the normal tissue is larger than that of the cancerous tissue. The cutoff diameter D(max) is observed to agree with the nuclear size for the normal tissues and the nucleolar size for the cancerous tissues. Transmission spectral polarization imaging measurements were performed that could distinguish the cancerous prostate tissue samples from the normal tissue samples based on the differences between their absorption and scattering parameters.

Diffuse optical imaging (DOI) for detecting and locating targets in a highly scattering turbid medium is treated as a blind source separation (BSS) problem. Three matrix decomposition methods, independent component analysis (ICA), principal component analysis (PCA), and nonnegative matrix factorization (NMF) were used to study the DOI problem. The efficacy of resulting approaches was evaluated and compared using simulated and experimental data. Samples used in the experiments included Intralipid-10% or Intralipid-20% suspension in water as the medium with absorptive or scattering targets embedded.

The scattering-phase theorem states that the values of scattering and reduced scattering coefficients of the bulk random media are proportional to the variance of the phase and the variance of the phase gradient, respectively, of the phase map of light passing through one thin slice of the medium. We report a new derivation of the scattering phase theorem and provide the correct form of the relation between the variance of phase gradient and the reduced scattering coefficient. We show the scattering-phase theorem is the consequence of anomalous diffraction by a thin slice of forward-peaked scattering media. A new set of scattering-phase relations with relaxed requirement on the thickness of the slice are provided. The condition for the scattering-phase theorem to be valid is discussed and illustrated with simulated data. The scattering-phase theorem is then applied to determine the scattering coefficient μs, the reduced scattering coefficient μ′s, and the anisotropy factor g for polystyrene sphere and Intralipid-20% suspensions with excellent accuracy from quantitative phase imaging of respective thin slices. The spatially-resolved μs, μ′s and g maps obtained via such a scattering-phase relationship may find general applications in the characterization of the optical property of homogeneous and heterogeneous random media.

Scanning backscattering imaging and independent component analysis (ICA) are used to probe targets hidden in the subsurface of a turbid medium. A new correction procedure is proposed and used to synthesize a "clean" image of a homogeneous host medium numerically from a set of raster-scanned "dirty" backscattering images of the medium with embedded targets. The independent intensity distributions on the surface of the medium corresponding to individual targets are then unmixed using ICA of the difference between the set of dirty images and the clean image. The target positions are localized by a novel analytical method, which marches the target to the surface of the turbid medium until a match with the retrieved independent component is accomplished. The unknown surface property of the turbid medium is automatically accounted for by this method. Employing clean image synthesis and target numerical marching, three-dimensional (3D) localization of objects embedded inside a turbid medium using independent component analysis in a backscattering geometry is demonstrated for the first time, using as an example, imaging a small piece of cancerous prostate tissue embedded in a host consisting of normal prostate tissue.

A time reversal optical tomography (TROT) method for near-infrared (NIR) diffuse optical imaging of targets embedded in a highly scattering turbid medium is presented. TROT combines the basic symmetry of time reversal invariance and subspace-based signal processing for retrieval of target location. The efficacy of TROT is tested using simulated data and data obtained from NIR imaging experiments on absorptive and scattering targets embedded in Intralipid-20% suspension in water, as turbid medium. The results demonstrate the potential of TROT for detecting and locating small targets in a turbid medium, such as, breast tumors in early stages of growth.

The characterization of the initial stages of complex aggregate flocculation is a central problem of colloidal science and environmental studies. A fast, white light scattering technique for monitoring aggregation kinetics is used to provide a unique probe of the changes that transpire during flocculation. The time evolution of an accreting smectite clay aggregate is monitored on the basis of changes in the fractal dimension (Df) initiated by the addition of an electrolyte (MgCl2). The evaluation of the Df change reveals for the first time three distinct stages in aggregate formation that characterize smectite flocculation: a rapid, initial jump; followed by a slow decrease; and a final gradual increase, eventually reaching a plateau. The three stages in the time evolution of Df correlate with (1) collapse of loosely packed face-to-edge structures and the formation of compact face-to-face composite structures; (2) the merging of the face-to-face composites; and (3) aggregation of the composites. Stage 3 eventually reaches a quasi-equilibrium condition. The quasi-equilibrium value of Df varies from 1.86 in 1 mM MgCl2—corresponding to diffusion-limited colloidal aggregation—to 2.20 in 57 mM MgCl2—corresponding to reaction-limited colloidal aggregation. These unexpected results originate from the formation of intermediate composite particles during aggregation and the increase in the compactness of the intermediate composite particles with electrolyte concentration. The new white light optical technique demonstrated here is non-invasive and has significant implications in the investigation of the growth dynamics of colloidal systems.

An analytical theory for coherent backscattering (CBS) of low-coherence light is presented. An expression linking the CBS profile to the radial distribution of the incoherent backscattered light is derived when the incident light is partially spatially coherent. The backscattered snake light, which has experienced exactly two large-angle scatterings, is taken into account together with the diffuse light in the analysis. Monte Carlo simulations demonstrate that the model describes well the CBS profile as long as the spatial coherence length, Lc , of the incident beam is larger than the scattering mean free path of light in the medium. The intensity of the enhanced backscattered light in the exact backscattering direction and the width of the CBS cone are found to be proportional to Lc and Lc^-1, respectively, in the limit of small Lc.

A unified Mie and fractal model for light scattering by biological cells is presented. This model is shown to provide an excellent global agreement with the angular dependent elastic light scattering spectroscopy of cells over the whole visible range (400 to 700 nm) and at all scattering angles (1.1 to 165 deg) investigated. Mie scattering from the bare cell and the nucleus is found to dominate light scattering in the forward directions, whereas the random fluctuation of the background refractive index within the cell, behaving as a fractal random continuous medium, is found to dominate light scattering at other angles. Angularly dependent elastic light scattering spectroscopy aided by the unified Mie and fractal model is demonstrated to be an effective noninvasive approach to characterize biological cells and their internal structures. The acetowhitening effect induced by applying acetic acid on epithelial cells is investigated as an example. The changes in morphology and refractive index of epithelial cells, nuclei, and subcellular structures after the application of acetic acid are successfully probed and quantified using the proposed approach. The unified Mie and fractal model may serve as the foundation for optical detection of precancerous and cancerous changes in biological cells and tissues based on light scattering techniques.

A superposition rule for light scattering by composite particles is presented that expresses the scattering amplitude of a composite particle as a superposition of that of the host particle and those of the shadowed inclusions. The superposition rule is derived for a soft composite particle but also provides insight into light scattering by a general composite scatterer. Favorable comparison with an exact numerical method demonstrates the usefulness of the rule in analyzing light scattering by composite particles such as biological cells.

A new approach for optical imaging and localization of objects in turbid media that makes use of the independent component analysis (ICA) from information theory is demonstrated. Experimental arrangement realizes a multi-source illumination of a turbid medium with embedded objects, and a multi-detector acquisition of transmitted light on the medium boundary. The resulting spatial diversity and multiple angular observations provide robust data for three-dimensional localization and characterization of absorbing and scattering inhomogeneities embedded in a tissue-like turbid medium. ICA of the perturbations in the spatial intensity distribution on the medium boundary sorts out the embedded objects and their locations are obtained from Green's function analysis based on any appropriate light propagation model. Imaging experiments were carried out on two human breast tissue simulating homogeneous turbid media of thickness approximately 50 times the transport mean free path of the medium with two embedded absorptive objects in one, and four scattering objects in the other. An independent component separation of the signal, in conjunction with diffusive photon migration theory was used to locate the embedded inhomogeneities. In both the cases, improved lateral and axial localization of the objects over the result obtained using common photon migration reconstruction algorithms were achieved. The approach is applicable for different medium geometries, can be used with any suitable photon propagation model, and is amenable to near real time imaging applications.

A new imaging approach for three-dimensional localization and characterization of objects in a turbid medium using independent component analysis (ICA) from information theory is developed and demonstrated using simulated data. This approach uses a multi-source and multi-detector signal acquisition scheme. Independent component analysis of the perturbations in the spatial intensity distribution measured on the medium boundary sorts out the embedded objects. The locations and optical characteristics of the embedded objects are obtained from a Green's function analysis based on any appropriate model for light propagation in the background medium. This approach is shown to locate and characterize absorptive and scattering inhomogeneities within highly scattering medium to a high degree of accuracy. In particular, we show this approach can discriminate between absorptive and scattering inhomogeneities, and can locate and characterize complex inhomogeneities which is both absorptive and scattering. The influence of noise and uncertainty in background absorption or scattering on the performance of this approach is investigated.

We study the propagation of polarized light in turbid media as a random walk of vector photons. The second order statistics of the polarization and propagation directions of polarized light vs the number n of scattering events in the direction space is shown to be characterized by two eigenvalues λ for Mie scatterers of arbitrary size and refractive index. Both light polarization and directionality anisotropies decay as a powerlaw of the larger eigenvalue λ₊ⁿ when n1/ln(λ₊)/(λ_-). The characteristic length for polarized light to become isotropic in its linear polarization and propagation directions is given by l_p=l_s/ln(1)/(λ₊) where l_s is the scattering mean free path.

We use fractal continuous random media to model visible and near infrared light scattering by biological tissue and cell suspensions. The powerlaw of the reduced scattering coefficient, the anisotropy factor of scattering, and the phase function are derived with good agreement with experimental results. Implications on spectroscopic tissue diagnosis are discussed.

We study the circular polarization memory of light multiply scattered by Mie particles of arbitrary size and refractive index. The loss of helicity of multiply scattered light is shown to be characterized by one parameter λ_x taking into account of both mechanisms (randomization of direction and randomization of helicity) for circular depolarization. The decay of helicity asymmetry follows a powerlaw λ_xⁿ with the increase of the number n of scattering events when n1. The characteristic length for circular polarized light to lose its helicity is given by l_x=l_s/ln(1)/(λ_x) where l_s is the mean scattering free path.

The backscattering of circularly polarized light at normal incidence to a half-space of scattering particles is studied using the Electric Field Monte Carlo (EMC) method. The spatial distribution of the backscattered light intensity is examined for both the time-resolved and continuous wave cases for large particles with anisotropy factor, g, in the range 0.8 to 0.97. For the time-resolved case, the backscattered light with the same helicity as that of the incident beam (co-polarized) is found to form a ring centered on the point of incidence. The ring expands and simultaneously grows weak as time increases. The intensity of backscattered light with helicity opposite to that of the incident beam (cross-polarized) is found to exhibit a ring behavior for g >=0.85, with significant backscattering at the point of incidence. For the continuous-wave case no such ring pattern is observed in backscattered light for either helicity. The present EMC study suggests that the ring behavior can only be observed in the time domain, in contrast to previous studies of light backscattered from forward scattering media based on the scalar time-independent Fokker-Planck approximation to the radiative transfer equation. The time-dependent ring structure of backscattered light may have potential use in subsurface imaging applications.

Light extinction measurements in the wavelength range of 400 to 1000 nm were performed in situ, on Bacillus subtilis endospores during heat shock induced activation. Simultaneous information on particle size and refractive indices during activation were calculated from the transmission spectra using the Gaussian Ray Approximation of anomalous diffraction theory. During activation, the refractive index of the core decreased from 1.51 to 1.39. while the size increased from 0.38 to 0.6 μm.

An analytical expression of the particle distribution based on an analytical cumulant solution of the time-dependent elastic Boltzmann transport equation BTE is presented. This expression improves upon the previous second order cumulant solution of the BTE described by a Gaussian distribution in two aspects: (1) separating the ballistic component from the scattered component to ensure that the summation in expressions is convergent; and (2) enforcing the causality condition to ensure that no particle travels faster than the free speed of the particles. Time-resolved profiles obtained using the analytical form are compared with those obtained by the Monte Carlo simulation, for both transmission and backscattering. The calculating time using our analytical form is much faster than that using the Monte Carlo approach.

Electric Field Monte Carlo (EMC) method for simulating polarized light propagation in turbid media is presented. Scattering of light by the particle is described by a rotation of the local coordinate system (m,n,s) with m and n the normal and binormal directions perpendicular to the photon propagation direction s and an update of the electric field E=E₁m+E₂n by the amplitude scattering matrix. The phase of light is accumulated according to the optical path. EMC can be used to study both coherent and incoherent properties of multiple scattering of light in turbid media. Backscattered speckle pattern and Muller matrix from a aqueous solution of polystyrene spheres in a slab geometry are computed by EMC as examples.

Multiple passages of light through an absorption inhomogeneity of finite size deep within a turbid medium are analyzed for optical imaging by use of the self-energy diagram. The nonlinear correction becomes more important for an inhomogeneity of a larger size and with greater contrast in absorption with respect to the host background. The nonlinear correction factor agrees well with that from Monte Carlo simulations for cw light. The correction is approximately 50%-75% in the near infrared for an absorption inhomogeneity with the typical optical properties found in tissues and five times the size of the transport mean free path.

From the geometrical path statistics of rays in an anomalous-diffraction theory (ADT) Opt. Lett. 28 , 179 (2003) closed-form expressions for the geometrical path distribution of rays and analytical formulas for the optical efficiencies of finite circular cylinders oriented in an arbitrary direction with respect to the incident light are derived. The characteristics of the shapes of the cylinders produce unique features in the geometrical path distributions of the cylinders compared with spheroids. Gaussian ray approximations, which depend only on the mean and the mean-squared geometrical paths of rays, of the optical efficiencies of finite circular cylinders and spheroids are compared with the exact optical efficiencies in ADT. The influence of the difference in shape between cylinders and spheroids on the optical efficiencies in ADT is illustrated by their respective geometrical path distributions of rays.

The anomalous-diffraction theory (ADT) of extinction of light by soft particles is shown to be determined by a statistical distribution of the geometrical paths of individual rays inside the particles. Light extinction depends on the mean and the mean-squared geometrical paths of the rays. Analytical formulas for optical efficiencies from a Gaussian distribution of the geometrical paths of rays are derived. This Gaussian ray approximation reduces to the exact ADT in the intermediate case of light scattering for an arbitrary soft particle and describes well the extinction of light from a system of randomly oriented and (or) polydisperse particles. The implications for probing of the sizes and shapes of particles by light extinction are discussed.

The powerlaw patterns in Mie scattering (the normalized light intensity I()/I(0) vs. the dimensionless qR where is the magnitude of the wave vector transfer at the scattering angle for wavelength , and R is the radius of the nonabsorbing sphere with a relative refractive index m>1) are analyzed using the geometrical optics approximation for particles of a large size parameter. The (qR)?4 powerlaw regime is shown to be present only in Mie scattering of soft particles. The (qR)?2 powerlaw regime occurs at the scattering angles of the p=1 geometrical ray (refracted without internal reflections) from the portion of the incident beam with an incidence angle around /4 upon the particle. The (qR)?2 powerlaw regimes from particles sharing one common relative refractive index but differing in size parameters are collinear. Simple analytical expressions are derived to describe these powerlaw regimes of Mie scattering.

Light extinction and angular scattering measurements were performed on three species of bacteria with different sizes and shapes ( Pseudomonas aeruginosa, Staphylococcus aureus, and Bacillus subtilis). The Gaussian ray approximation of anomalous diffraction theory was used to determine the average bacteria size from transmission measurements. A rescaled spectra combining multiple angular data was analyzed in the framework of the Rayleigh-Gans theory of light scattering. Particle shape and size distribution is then obtained from the rescale spectra. Particle characteristics (size and/or shape) retrieved from both methods are in good agreement with size and shape measured under scanning electron microscopy. These results demonstrate that light scattering may be able to detect and identify microbial contamination in the environment.

The photon distribution, as a function of position, angle, and time, is computed using the analytical cumulant solution of the Boltzmann radiative transfer equation (RTE). A linear forward model for light propagation in turbid media for three-dimensional (3-D) optical tomography is formed based on this solution. The model can be used with time resolved, continuous wave (CW), and frequency-domain measurements in parallel geometries. This cumulant forward model (CFM) is more accurate than that based on the diffusion approximation of RTE. An inverse algorithm that incorporates this CFM is developed, based on a fast 3-D hybrid-dual-Fourier tomographic approach using multiple detectors and multiple sources in parallel geometries. The inverse algorithm can produce a 3-D image of a turbid medium with more than 20 000 voxels in 1-2 min using a personal computer. A 3-D image reconstructed from simulated data is presented.

A photon transport model for light migration in turbid media based on a cumulant approximation to radiative transfer is presented for image reconstruction inside an infinite medium or a bounded medium with a planar geometry. This model treats weak inhomogeneities through a Born approximation of the Boltzmann radiative transfer equation and uses the second-order cumulant solution of photon density to the Boltzmann equation as the Green's function for the uniform background. It provides the correct behavior of photon migration at early times and reduces at long times to the center-moved diffusion approximation. At early times, it agrees much better with the result from the Monte Carlo simulation than the diffusion approximation. Both approximations agree well with the Monte Carlo simulation at later times. The weight function for image reconstruction under this proposed model is shown to have a strong dependence at both early and later times on absorption and/or scattering inhomogeneities located in the propagation direction of and close to the source, or in the field of view of and close to the detector. This effect originates from the initial ballistic motion of incident photons, which is substantially underestimated by the diffusion approximation.

Time-resolved Fourier optical diffuse tomography is a novel approach for imaging of objects in a highly scattering turbid medium with use of an incident (near) plane wave. The theory of the propagation of spatial Fourier components of the scattered wave field is presented, along with a fast algorithm for three-dimensional reconstruction in a parallel planar geometry. Examples of successful reconstructions of simulated hidden absorptive or scattering objects embedded inside a human-tissue-like semi-infinite turbid medium are provided.

A photon-transport forward model for image reconstruction in turbid media is derived that treats weak inhomogeneities through a Born approximation of the Boltzmann radiative transfer equation. This model can conveniently replace the commonly used diffusion approximation in optical tomography. An analytical expression of the background Green's function is obtained from the cumulant solution of the Boltzmann equation. Our model provides the correct behavior of photon migration at early times and reduces at long times to the center-moved diffusion approximation. Numerical comparisons between this model and the standard and center-moved diffusion models are presented.

Optical imaging and localization of objects inside a highly scattering medium, such as a tumor in the breast, is a challenging problem with many practical applications. Conventional imaging methods generally provide only two-dimensional (2-D) images of limited spatial resolution with little diagnostic ability. Here we present an inversion algorithm that uses time-resolved transillumination measurements in the form of a sequence of picosecond-duration intensity patterns of transmitted ultrashort light pulses to reconstruct three-dimensional (3-D) images of an absorbing object located inside a slab of a highly scattering medium. The experimental arrangement used a 3-mm-diameter collimated beam of 800-nm, 150-fs, 1-kHz repetition rate light pulses from a Ti:sapphire laser and amplifier system to illuminate one side of the slab sample. An ultrafast gated intensified camera system that provides a minimum FWHM gate width of 80 ps recorded the 2-D intensity patterns of the light transmitted through the opposite side of the slab. The gate position was varied in steps of 100 ps over a 5-ns range to obtain a sequence of 2-D transmitted light intensity patterns of both less-scattered and multiple-scattered light for image reconstruction. The inversion algorithm is based on the diffusion approximation of the radiative transfer theory for photon transport in a turbid medium. It uses a Green s function perturbative approach under the Rytov approximation and combines a 2-D matrix inversion with a one-dimensional Fourier-transform inversion to achieve speedy 3-D image reconstruction. In addition to the lateral position, the method provides information about the axial position of the object as well, whereas the 2-D reconstruction methods yield only lateral position.