Multiple-scattering model for the effective refractive index of dense suspensions of forward-scattering particles.

We present a multiple-scattering model for the effective refractive index of an arbitrarily dense suspension of forward-scattering particles. The model provides a very simple formula for the effective refractive index of such a suspension and reproduces with high accuracy available experimental results. Furthermore, the derivation we present herein is mathematically transparent and enables us to obtain information on the underlying physical processes rather than obscuring them. We also provide insight into the extent of the model's validity and a simple way to determine whether or not it will be valid for an arbitrary suspension. Due to its simplicity, analytical closedness, and wide range of applicability, we believe the model can be used as a diagnostic tool for complex materials of vastly different natures.

Method for measuring the extinction coefficient of fluorescing media within the emission band.

We propose and test a method for determining a fluorescent medium's absorption or extinction index while it is fluorescing. The method uses an optical arrangement that records changes in fluorescence intensity at a fixed viewing angle as a function of the angle of incidence of an excitation light beam. We tested the proposed method on polymeric films doped with Rhodamine 6G (R6G). We found a strong anisotropy in the fluorescence emission and, thus, limited the method to TE-polarized excitation light. The method proposed is model dependent, and we provide a simplified model for its use in this work. We report the extinction index of the fluorescing samples at a selected wavelength within the emission band of the fluorophore R6G. We found that the extinction index at the emission wavelengths in our samples is appreciably larger than the extinction index at the excitation wavelength, which is the opposite of what one might expect from measuring the absorption spectrum of the medium with a spectrofluorometer. The proposed method could be applied to fluorescent media with additional absorption other than by the fluorophore.

An efficient analysis of oblique reflection of airborne ultrasound beams from thin membranes for gas sensing.

The use of the angular spectrum method (ASM) to simulate the reflection of airborne ultrasound beams from a thin membrane separating air from a mixture of air and another gas is examined. The main advantage of this method is its high computing speed and efficiency for practical design calculations, suitable for sensing applications. The implemented ASM code is validated against custom Rayleigh integral code in a pure propagation simulation. In addition, ultrasound beam reflection calculations using ASM with finite element numerical results and experimental measurements are compared, finding good agreement in both cases. Then, ASM is used to estimate the sensitivity of specular reflection signals to variations in the composition of the incidence medium as a function of the angle of incidence. Conditions for which a reflection signal using inexpensive commercial ultrasound emitter/receiver at 40 kHz, in a simple configuration, offer a high enough sensitivity suitable for monitoring air quality indoors are found.

Visual Assessment of Blood Plasma versus Optical Transmittance and Refractive Index Measurements for Quantifying Lipemia.

Today, visual classification of the degree of lipemia in blood samples is frequently performed in clinical laboratories. However, achieving standardization of this classification at low cost and with fewer resources is an objective that is still under development. In this work, a comparison is made between the visual inspection and optical measurements of blood plasma for quantifying lipemia. The plasma refractive index's real part was measured using an Abbe refractometer and transmittance measurements were made at a 589 nm wavelength and wavelengths ranging from 320 to 1100 nm in the spectral region, respectively. Taking the slope of the transmittance spectrum at two specific wavelengths, it is possible to establish a more standardized selection criterion and implement it quickly using low-cost optical devices. Furthermore, using the proposed transmittance-spectrum-slope method, statistically significant differences (p < 0.05) were found between healthy blood samples and lipemia 1, 2, 3, and 4. However, there were no statistical differences between lipemia 1 and 2.

Analysis of wavelength-scale 1D depth-dependent refractive-index gradients at an interface by their effects on the internal reflectance near the critical angle.

Light's internal reflectivity near a critical angle is very sensitive to the angle of incidence and the optical properties of the external medium near the interface. Novel applications in biology and medicine of subcritical internal reflection are being pursued. In many practical situations, the refractive index of the external medium may vary with respect to its bulk value due to different physical phenomena at surfaces. Thus, there is a pressing need to understand the effects of a refractive-index gradient at a surface for near-critical-angle reflection. In this work, we investigate theoretically the reflectivity near the critical angle at an interface with glass assuming the external medium has a continuous depth-dependent refractive index. We present graphs of the internal reflectivity as a function of the angle of incidence, which exhibit the effects of a refractive-index gradient at the interface. We analyze the behavior of the reflectivity curves before total internal reflection is achieved. Our results provide insight into how one can recognize the existence of a refractive-index gradient at the interface and shed light on the viability of characterizing it.

Probing bio-tissue films by optical internal reflectivity: modeling and measurements.

In this paper, we develop a detailed theoretical model for the optical reflectivity of a bio-tissue film confined between two flat interfaces based on the anomalous-diffraction approximation. We consider bio-tissue films consisting of a few layers of spheroidal cells surrounded by extracellular medium. We explore numerically the predictions of our model and compare them with simple effective medium theories, sometimes used as a first attempt to understand the optical properties of biological media. Then, we fit the model to experimental reflectivity-versus-angle-of-incidence curves of confined whole-blood films measured in an internal reflection configuration. Measurements were performed by confining a drop of fresh blood between a prism and a coverslip. Our experimental results show that it is possible to measure the coherent reflectance with small enough error to infer microstructural parameters with a good precision. The errors in measuring the coherent reflectance depend on the reflectivity magnitude. For instance, for a reflectivity of about 0.3 the error is below 2%, and the refractive indices of cells and surrounding medium can be obtained with a precision better than 1%. These results also indicate that the present model can readily be used to figure out the physical changes experienced at the microscale in bio-tissue films during a physicochemical process.

Unambiguous derivation of the effective refractive index of biological suspensions and an extension to dense tissue such as blood.

The van de Hulst formula provides an expression for the effective refractive index or effective propagation constant of a suspension of particles of arbitrary shape, size, and refractive index in an optically homogeneous medium. However, its validity for biological matter, which often consists of very dense suspensions of cells, is unclear because existing derivations of the formula or similar results rely on far-field scattering and/or on the suspension in question being dilute. We present a derivation of the van de Hulst formula valid for suspensions of large, tenuous scatterers-the type biological suspensions are typically made of-that does not rely on these conditions, showing that they are not strictly necessary for the formula to be valid. We apply these results specifically to blood and epithelial tissue. Furthermore, we determine the true condition for the formula to be valid for these types of tissues. We finally provide a simple way to estimate-and, more importantly, correct-the error incurred by the van de Hulst formula when this condition is not met.

Optical reflectivity of an interface with random refractive-index-contrast patterns.

We develop simple models for the optical reflectivity of an interface in optical contact with random media consisting of discrete volumes of arbitrary form and different refractive indices. Examples of interest are surfaces sprinkled with microdroplets or an interface with biological cells adhered to it at random locations. We focus our attention to the case of internal reflectivity, in which the incidence medium has a larger refractive index than the refractive indices at the other side of the interface. Assuming an incident plane wave, we provide simple approximate expressions for the surface's coherent reflectance and for the surface's total reflectance. We compare predictions of the surface coherent-reflectance model with numerical simulations. Then we use the surface's reflectance models to interpret experimental measurements obtained with an optical prism and a thin vegetable tissue adhered to its base. In general, the surface reflectivity can be used to determine fractional contact area between the interface and microdroplets or biological cells and infer their refractive indices with an accuracy of about 0.5%.

Analytical approximation to the complex refractive index of nanofluids with extended applicability.

Currently, there are available a few simple analytical approximations to the complex effective refractive index that may be used for nanofluids. Namely, the Maxwell Garnett mixing formula with scattering corrections, the Maxell Garnett Mie approximation, the Foldy-Lax approximation and the small particle limit of the quasi-crystalline approximation. These approximations are valid either for very small nanoparticles (below a few nanometers in radius) or for very dilute nanofluids (below about 1% in particles' volume fractions) and therefore, do not cover the whole domain of particle suspensions referred to as nanofluids. Here we propose a new simple analytical approximation based on local field corrections to the Foldy-Lax approximation. The new mixing formula coincides with the mentioned approximations when they are expected to be valid and provides physically sound predictions when the mentioned approximations are no longer valid, within the realm of nanofluids. We compare predictions of the analytical approximations considered in this work with experimental data published earlier for nanofluids of polystyrene in water.

Requirements and applications of accurate modeling of the optical transmission of transparent conducting coatings.

A comprehensive model for the optical transmission is constructed and used to investigate the requirements for fitting accurately the experimental data of the optical transmittance at normal incidence of transparent conducting coatings of ZnO:Al deposited on glass substrates by ultrasonic spray pyrolysis. The model takes into account the Urbach tail absorption edge at the low wavelength region, the contribution of free carrier concentration to the weak absorption in the visible and near-infrared ranges, and the effect of scattering of light originated by the surface roughness of the films. The carrier concentration of the ZnO:Al films was measured experimentally by the Hall effect and dc-electrical conductivity measurements in the Van der Paw configuration. It is shown that all mentioned physical effects must be included in order to fit accurately the transmittance spectrum in the VIS-NIR spectral window. The full expression for the optical transmittance was used for choosing the optimal thickness of these films as transparent conductive contacts and the calculation of the figure of merit.

Optical sizing of nanoparticles in thin films of nonabsorbing nanocolloids.

We study an optical method to infer the size of nanoparticles in a thin film of a dilute nonabsorbing nanocolloid. It is based on determining the contribution of the nanoparticles to the complex effective refractive index of a suspension from reflectivity versus the angle of incidence curves in an internal reflection configuration. The method requires knowing only approximately the particles' refractive index and volume fraction. The error margin in the refractive index used to illustrate this technique was 2%. The method is applicable to sizing nanoparticles from a few tens of nanometers to about 200 nm in radius and requires a small volume of the sample, in the range of a few microliters. The method could be used to sense nanoparticle aggregation and is suitable to be integrated into microfluidic devices.

Optical Coherent Reflection from a Confined Colloidal Film: Modeling and Experiment.

We study the optical reflectivity of confined colloidal films as a function of the angle of incidence in an internal reflection configuration. Two effective medium models and an extended coherent-scattering model for thin colloidal films are compared against experimental measurements with gold, latex, and titanium dioxide colloids. A derivation of the coherent scattering model for confined colloidal films used in this work is presented in a comprehensive way. The model lies within the framework of the multiple-scattering theory and is valid for any angle of incidence and for colloids of small or large particles compared to the wavelength of light, however, only for small and moderately small particles' volume fraction. Reflectivity versus angle of incidence curves for an opaque colloidal film in an internal reflection configuration show the effects of two critical angles. Within the two critical angles, there is a high sensitivity to the presence of colloidal particles, while the volume of colloidal samples needed is in the microliter range. Upon comparing theory with experiment, no model fitting was done in any case. The experimental setup and its calibration procedure are discussed. The results provide physical insight into applications involving optical properties of colloidal systems.

Analytical modeling of optical reflectivity of random plasmonic nano-monolayers.

In this paper, we compare three different models that have been used to interpret reflectivity measurements of supported monolayers of nanoparticles. Two of them: (i) isotropic Maxwell Garnett and (ii) anisotropic two-dimensional-dipolar model are based on an effective-medium approach, while the third one (iii) coherent-scattering model, lies within the framework of multiple-scattering theory. First, we briefly review, on physical grounds, the foundations of each model and write down the corresponding formulas for the calculation of the reflectivity. In the two-dimensional-dipolar model, the dilute limit of the pair-correlation function (also called hole-correlation function) is always used in the calculation of the effective optical response. Then we use these formulas to plot and analyze graphs of the reflectivity of a monolayer of gold nanoparticles on a glass substrate, as a function of several relevant parameters, for two different commonly used experimental configurations. Finally, we discuss the importance of our results and how they can be used to infer the limits of validity of each model.

On the extinction coefficient of light in non-absorbing nanoparticle suspensions.

We assess the validity and possible use of a simple formula for the complex effective refractive index of a colloidal suspension of very small particles obtained from the quasi-crystalline approximation. This approximation takes into account the so-called dependent scattering effects, which are strongest, in relative terms, in suspensions of non-absorbing nanometer-sized particles. We test experimentally the predictions of the model for the extinction of light in dispersions of sodium dodecyl sulfate micelles in tri-distilled water. We find that the theoretical model can be fitted to the experimental data and provide estimates of the micelles' size and the minimum distance between each other in suspension.

Reflectivity and transmissivity of a surface covered by a disordered monolayer of large and tenuous particles: theory versus experiment.

The main objective of this paper is to endorse a recently derived theoretical model for the coherent reflectance and transmittance from a surface supporting a disordered monolayer of large and tenuous particles by comparison with experimental measurements. The model is based on the so-called anomalous-diffraction approximation and is assumed to be valid for small and moderate angles of incidence. We prepared disordered monolayers of spherical polystyrene particles of 1.8 µm diameter and of human red blood cells on glass microscope slides. In both cases, particles were immersed in a liquid of refractive index close to that of the particle. We measured the relative reflectivity and transmissivity of the samples versus the wavelength of light at normal incidence and the reflectivity and transmissivity versus the angle of incidence at a fixed wavelength, and compared with predictions of the anomalous-diffraction approximation model. For the polystyrene particle samples, we also compare the results with another available theoretical model developed some years ago to deal with disordered monolayers of highly scattering particles. In the case of red blood cell monolayers, we also present measurements with a hemolyzed sample and a disordered multilayer film for comparison. We find that the anomalous diffraction model can be fitted very well to the experimental curves, in some cases, even for high angles of incidence.

Collimated light reflection and transmission of a surface partially covered by large and tenuous particles.

We derive simple approximate expressions for the reflectivity and transmissivity of light from disordered monolayers of tenuous particles of dimensions larger than the wavelength and supported by a flat interface. The expressions derived can be used for different particle shapes and for moderate angles of incidence. We then investigate the effects of particle shape and orientation on reflectivity and transmissivity spectra of a monolayer of tenuous particles containing an optical chromophore in a solution in their interior. We also simulate the effects of a particle's shape and orientation on the angle dependence of the optical reflectivity and transmissivity. In our examples, we consider disordered monolayers of particles analogous to some biological cells.

Reflection of diffuse light from dielectric one-dimensional rough surfaces.

We study the reflection of diffuse light from 1D randomly rough dielectric interfaces. Results for the reflectance under diffuse illumination are obtained by rigorous numerical simulations and then contrasted with those obtained for flat surfaces. We also explore the possibility of using perturbation theories and conclude that they are limited for this type of study. Numerical techniques based on Kirchhoff approximation and reduced Rayleigh equations yield better results. We find that, depending on the refractive index contrast and nature of the irregularities, the roughness can increase or decrease the diffuse reflectance of the surface.

Experimental Test of Reflectivity Formulas for Turbid Colloids: Beyond the Fresnel Reflection Amplitudes.

We compare light reflectivity measurements as a function of the angle of incidence for an interface between an optical glass and a turbid suspension of small particles, with theoretical predictions for the coherent reflectance calculated with different available theoretical models. The comparisons are made only in a small range of angles of incidence around the critical angle of the interface between the glass and the matrix of the colloidal suspensions. The experimental setup and its calibration procedure are discussed. We considered two Fresnel-based approximations and another two based on a multiple-scattering approach, and we present results for monodisperse latex colloidal suspensions of polymeric spherical particles in water with particle diameters of 120 and 520 nm, polydisperse titanium dioxide (rutile) particles suspensions in water with a most probable diameter of 404 nm, and suspensions of copper particles in water with diameters of 500 nm. The comparisons between experiment and theory are made without fitting any parameters.

Extension of Fresnel's formulas for turbid colloidal suspensions: a rigorous treatment.

We provide new expressions for the reflection amplitudes of a half space of randomly located identical spherical particles that can be regarded as an extension of Fresnel's formulas when scattering is prominent. We derive them rigorously from Maxwell's equations by solving an integral equation for the electric field within the effective-field approximation. The integral equation is given in terms of the nonlocal conductivity tensor of an isolated sphere. Approximate expressions for the reflection amplitudes are also proposed and their accuracy is analyzed, first for the case of a self-sustained suspension of silver particles, and then for the more realistic situation of silver particles in water. In this latter case the integral equation is modified by introducing the half-space Green's function dyadic instead of the one in free-space, but the method of solution is analogous in both. This extension of Fresnel's formulas, together with the numerical comparison of the different approximations proposed here, is necessary for an accurate interpretation of reflection-spectroscopy measurements in dilute colloidal suspensions of practical interest. The connection between the nonlocal conductivity tensor and the T-matrix operator of scattering theory is also made manifest.

Optical reflectivity of a disordered monolayer of highly scattering particles: coherent scattering model versus experiment.

Recently a multiple-scattering model for the reflectivity of a disordered monolayer of scattering particles on a flat surface was put forth [J. Opt. Soc. Am.29, 1161 (2012)]. The approximate theoretical model provides relatively simple formulas for the reflectivity, but it was developed for a monodisperse monolayer. Here we extend the model to the case of a polydisperse monolayer and derive the appropriate formulas to calculate the optical transmissivity of the monolayer supported by a flat interface. A second objective of this paper is to test the approximate theoretical model against experimental data with highly scattering particles. We prepared monolayers of three different surface coverage fractions of polydisperse alumina and titanium dioxide particles deposited randomly on a glass slide. We measured their optical reflectivity and transmissivity versus the angle of incidence using a laser with a wavelength of 670 nm. Using the nominal values for the particles' most probable radius and refractive index, we fitted the theoretical model to the experimental curves and found that it reproduces very well the experimental curves. Interestingly, a dip in the reflectivity curves at large angles of incidence is present for the alumina monolayers but not in the titanium dioxide monolayers. The dip corresponds to a maximum in the scattering efficiency by the alumina monolayers. The theoretical model reproduces very well this behavior.