Comparison of the filtered backpropagation and the filtered backprojection algorithms for quantitative tomography

We compare the filtered backpropagation algorithm with the filtered backprojection algorithm for reconstructing the complex refractive-index distribution of semitransparent, cylindrical objects. Before reconstruction, the recorded scattered light is propagated back to the reconstruction area by inverse diffraction. Our comparison is based on computer-simulated data, and experimental optical data obtained from fibers with step-index, graded-index, and uniform-index distributions. The results show that both the filtered backpropagation algorithm and the filtered backprojection algorithm can produce accurate reconstructions of the complex refractive-index distribution as long as the weak-scattering approximation is valid. The good agreement between the results obtained from these two reconstruction algorithms indicates that the errors introduced by the assumption of straight-line propagation inside the object are negligible compared with those introduced by the weak-scattering approximation.     

Single-material inhomogeneous optical filters based on microstructural gradients in plasma-deposited silicon nitride

Transparent hydrogenated amorphous silicon nitride (SiNx:H) coatings were prepared by dual-mode microwave-radio-frequency plasma-enhanced chemical vapor deposition. By controlling the effects of plasma density and ion energy on the film growth, it was possible to modify the microstructure of the coatings and hence the refractive index n. Using this method, we were able to vary n from 1.6 to 2.0, at 550 nm, by adjusting the power levels of the radio-frequency and microwave components while keeping the gas composition (SiH4, N2) and pressure constant. An inhomogeneous bandpass filter with a controlled refractive-index depth profile was fabricated, and its optical performance was compared with that of its multilayer counterpart. Besides the attractive optical features of such single-material rugate filters, we found that the mechanical resistance of inhomogeneous films is superior to that of multilayer systems.     

非均匀柱对称介质中的余弦光束

在一般情况下证明了余弦光束是柱对称介质中SVA近似下波动方程的本征解,并给出了折射率分布函数,对其特殊情况进行了讨论。     

Sharp bends with low losses in dielectric optical waveguides

A new method is proposed for making sharp bends with low radiation losses in dielectric optical waveguides. By modifying the transverse refractive-index profile at curved sections both the pure bend and the transition losses can be minimized. The optimum gradient-index profile requires an inhomogeneous medium. But in practice this can be replaced by a layered medium. By using four homogeneous layers the permitted radius of curvature of a slab waveguide can be reduced, e.g., from 6400 to only 100 wavelengths.     

Nonparaxial diffraction-limited propagation of light in a graded-index planar medium with a hyperbolic secant refractive-index profile

Nonparaxial diffraction-limited propagation of light with amplitude distribution in hyperbolic functions through an inhomogeneous planar medium with a hyperbolic secant refractive-index profile is studied by means of the stationary phase method. The irradiance distribution at geometrical shadow, edge of shadow, and a geometrically illuminated region is analyzed for a particular case.     

Effects of refractive index on near-infrared tomography of the breast

Near infrared (NIR) optical tomography is an imaging technique in which internal images of optical properties are reconstructed with the boundary measurements of light propagation through the medium. Recent advances in instrumentation and theory have led to the use of this method for the detection and characterization of tumors within the female breast tissue. Most image reconstruction approaches have used the diffusion approximation and have assumed that the refractive index of the breast is constant, with a bulk value of approximately 1.4. We have applied a previously reported modified diffusion approximation, in which the refractive index for different tissues can be modeled. The model was used to generate NIR data from a realistic breast geometry containing a localized anomaly. Using this simulated data, we have reconstructed optical images, both with and without correct knowledge of the refractive-index distribution to show that the modified diffusion approximation can accurately recover the anomaly given a priori knowledge of refractive index. But using a reconstruction algorithm without the use of correct a priori information regarding the refractive-index distribution is shown as recovering the anomaly but with a degraded quality, depending on the degree of refractive index mismatch. The results suggest that provided the refractive index of breast tissue is approximately 1.3-1.4, their exclusion will have minimal effect on the reconstructed images.     

Influence of small inhomogeneities on the spectral characteristics of single thin films

It is well known that the spectral transmittance and reflectance of a thin film can be influenced by even small inhomogeneities or variations in its complex refractive-index profile. Formulas are derived that describe the theoretical variation of the spectral characteristics for small changes in the refractive index and the extinction coefficient of a homogeneous thin film. These formulas, accurate to the first order in the change in the complex refractive index, are compared with exact calculations for a number of different types of inhomogeneities. It is shown that specific qualitative features in the refractive-index profile of a nearly homogeneous thin film frequently can be determined from an examination of the change in the spectral transmittance and reflectance at normal incidence.     

Fresnel zones in tapered gradient-index media

The free propagation of a wave front in an inhomogeneous medium with parabolic refractive-index profile and the division of the wave front into Fresnel zones are studied. We determine the radius and the area of each zone as well as the zone contribution to the total wave at an observation point inside the medium. We find the condition that the optical path must fulfill from each zone to that point so that the disturbance due to successive zones will be in phase opposition. Once this condition is settled the concept of zone plate in gradient-index media is introduced.     

Limits of the small-angle approximation to the radiative transport equation

The small-angle approximation to the radiative transport equation is used extensively in imaging models in which the transport medium is optically thick. The small-angle approximation is generally considered valid when the particles are very large compared with the wavelength, when the refractive-index ratio of the particle to the medium is close to 1, and when the optical thickness is not too large. We report results showing the limits of the validity of the small-angle approximation as a function of particle size and concentration for a particle-to-medium fixed refractive-index ratio of 1.196. This refractive-index ratio is comparable with that of minerals or diatoms suspended in water.     

Index profiling of anisotropic graded-index planar waveguides from effective indices

We present a method for recovering the refractive-index profile of an anisotropic graded-index waveguide from the effective indices by using a cubic spline interpolation function. The first and second derivatives of cubic splines are continuous to ensure a smooth index profile, which is consistent with practical graded-index distributions. A straightforward iteration with an exact dispersion equation to verify the interpolated profile makes this method easy and reliable for application. This approach is proved by numerical analysis of several typical index distributions and experimental examples showing that the refractive-index profiles in anisotropy can be reconstructed close to the exact profile. Waveguides with both more modes (more than four guiding modes) and fewer modes (two to four) can be universally profiled with good accuracy.     

Second-order iterative approach to inverse scattering: numerical results

We introduce an iterative algorithm for the reconstruction of dielectric profile functions from scattered field data, in which each step corresponds to the solution of a quadratic inversion problem. This means that, at each iteration, we perform a second-order approximation of the scattering operator connecting the unknown profile to the data about a reference profile function. This procedure is then compared with a linear iterative inversion algorithm, and it is pointed out that, within a prescribed class of profile functions, the linear iterative inversion does not converge to the actual solution, whereas the proposed approach does. This feature can be explained by reference not only to the improved approximation provided by the addition of a further term for profile functions of a larger norm but also to the different classes of functions that can be reconstructed by either the linear or the quadratic model. Numerical examples of profile reconstruction in the scalar two-dimensional geometry, with far-zone scattered field data at a fixed frequency, confirm the theoretical analysis.     

Analysis of inhomogeneous optical systems by the use of ray tracing. I. Planar systems

We describe a novel approach to refractive-index reconstruction in two-dimensional systems with no special symmetry, based on observation of traces of rays that travel through the optical system. The mathematical model of ray-tracing analysis is presented in detail, and both the analytical and numerical solutions are given. Methods of data processing in the presence of experimental errors are developed and applied to model problems.     

Letter Fibre-as-phase-object approximation in light scattering experiments

Abstract

The use of an optical fibre scattering pattern for measurements of its refractive-index profile and diameter through the fibre-as-phase-object method is discussed. The exact solution from electromagnetic theory and the phase-object approximation are compared. A great discrepancy between the exact solution and the fibre-as-phase-object approximation used in recently published papers is demonstrated and is critically analysed.     

Refractive-index distributions generating as light rays a given family of curves lying on a surface

In the framework of geometrical optics, we consider the inverse problem consisting in obtaining refractive-index distributions n=n(u,v) of a two-dimensional transparent inhomogeneous isotropic medium from a known family f(u,v)=c of monochromatic light rays, lying on a given regular surface. Using some basic concepts of differential geometry, we establish a first-order linear partial differential equation relating the assigned family of light rays with all possible refractive-index profiles compatible with this family. In particular, we study the refractive-index distribution producing, as light rays, a given family of geodesic lines on some remarkable surfaces. We give appropriate examples to explain the theory.     

Rainbow refractometry applied to radially inhomogeneous spheres: the critical case of evaporating droplets

A detailed study of rainbow thermometry and its application to droplets in reactive systems is presented. To this end the light-scattering history of a vaporizing droplet under unsteady conditions is discussed. Unlike in previous papers, the reduction of the droplet's diameter is also taken into account in addition to the variation of the refractive-index profile. A finely stratified sphere model with thousands of layers (i.e., 20,000) is used to compute the scattering patterns of a radially inhomogeneous evaporating droplet at different heating times and therefore with different diameters and refractive-index profiles. In the studied case the temperatures inferred from rainbow thermometry do not represent the actual temperatures inside the droplet. They do not represent an average internal temperature or even the surface or the core temperature. For droplets with a temperature that increases from the core to the surface, the inferred values are always lower than the minimum temperature inside the droplet. Therefore the rainbow technique should be applied with caution in all cases in which droplet inhomogeneities are suspected. In addition, a careful analysis of the scattering in the rainbow region is presented. Because of the physical structure of the rainbow, a marked uncertainty in the inferred temperatures also has to be considered in the case of homogenous droplets. For inhomogeneous spheres this intrinsic uncertainty has to be added to the effects caused by the internal profiles of the refractive index.     

Rigorous coupled-wave analysis for practical planar dielectric gratings: 1. Thickness-changed holograms and some characteristics of diffraction efficiency

A new and useful method for obtaining diffraction efficiencies from holograms manufactured practically is presented. Applying the rigorous coupled-wave analysis, we express each difference between the practical and the ideal as a mathematical component that can be easily integrated. In Part 1 the effects due to thickness change in the hologram layer (observed frequently after the development process) are treated. Although uniform swelling or shrinking causes a simple reconstruction wavelength or incidence-angle shift, nonuniform thickness extends the capacity of the Bragg condition matching, creating a diffraction efficiency curve in the asymmetric profile. Other characteristics of diffraction are also maintained. A refractive-index change has an effect that is similar to the thickness change. Higher-order terms in permittivity modulation create negligible effects in general holograms when used at or near the simple first-order Bragg condition.     

Thermal conductivity of graded composites: Numerical simulations and an effective medium approximation

We describe two methods for modeling the thermal conductivity and temperature profile in a graded composite film. The film consists of a random binary composite, whose concentration varies in the direction perpendicular to the film surface, and a fixed temperature difference is applied across the film. In the first method, the temperature profile is modeled directly, using a finite element technique in which the film is represented as a discrete network of thermal conductances, randomly distributed according to the assumed composition profile. The temperature at each node, and the effective thermal conductance, is then obtained by a transfer matrix technique. In the second approach, the film is treated by an effective-medium approximation, suitably generalized to account for the composition gradient. The methods are in rough agreement with each other, and suggest that thermophysical properties of the film can be treated reasonably well by approaches generalized from those which succeed in conventional composites.     

Mode-independent attenuation in evanescent-field sensors

Generally, the total power attenuation in multimode evanescent-field sensor waveguides is nonproportional to the bulk absorbance because the modal attenuation constants differ. Hence a direct measurement is difficult and is additionally aggravated because the waveguide absorbance is highly sensitive to the specific launching conditions at the waveguide input. A general asymptotic formula for the modal power attenuation in strongly asymmetric inhomogeneous planar waveguides with arbitrarily distributed weak absorption in the low-index superstrate is derived. Explicit expressions for typical refractive-index profiles are given. Except when very close to the cutoff, the predicted asymptotic attenuation behavior agrees well with exact calculations. The ratio of TM versus TE absorption has been derived to be (2 - n(0)(2)/n(f)(2)) for arbitrary profiles. Waveguides with a linear refractive-index profile show mode-independent attenuation coefficients within each polarization. Further, the asymptotic sensitivity is independent of the wavelength, so that it should be possible to directly measure the spectral variation of the bulk absorption. The mode independence of the attenuation has been verified experimentally for a second-order polynomial profile, which is close to a linear refractive-index distribution. In contrast, the attenuation in the step-profile waveguide has been found to depend strongly on the mode number, as predicted by theory. A strong spread of the modal attenuation coefficients is also predicted for the parabolic-profile waveguide sensor.     

Recovery of the refractive-index profile from the wigner distribution of an optical waveguide

We propose a new method for the recovery of the refractive-index profile of a single-mode or multimode optical guided structure. We solve the inverse problem using the Wigner distribution and reduce it to the solution of a linear system of equations.