Finite element ray tracing: a new method for ray tracing in gradient-index media

I apply the principle of finite elements, known in optics for calculating the beam propagation in waveguide structures, for calculation of meridional rays in inhomogeneous media. The plane is divided into finite elements that have a constant refractive index. The ray trajectory is calculated by a simple algorithm. Contrary to the existing methods, the model I propose in this research does not require an explicit formula for the index distribution. Only the numerical representation is sufficient, which can be a major advantage for calculation of the light propagation in real problems, such as the thermal lensing effect.     

Rainbows from inhomogeneous transparent spheres: a ray-theoretic approach

A ray-theoretic account of the passage of light through a radially inhomogeneous transparent sphere has been used to establish the existence of multiple primary rainbows for some refractive index profiles. The existence of such additional bows is a consequence of a sufficiently attractive potential in the interior of the drop, i.e., the refractive index gradient should be sufficiently negative there. The profiles for which this gradient is monotonically increasing do not result in this phenomenon, but nonmonotone profiles can do so, depending on the form of n. Sufficiently oscillatory profiles can lead to apparently singular behavior in the deviation angle (within the geometrical optics approximation) as well as multiple rainbows. These results also apply to systems with circular cylindrical cross sections, and may be of value in the field of rainbow refractometry.     

Two solvable problems of planar geometrical optics

In the framework of geometrical optics we consider a two-dimensional transparent inhomogeneous isotropic medium (dispersive or not). We show that (i) for any family belonging to a certain class of planar monoparametric families of monochromatic light rays given in the form f(x,y)=c of any definite color and satisfying a differential condition, all the refractive index profiles n=n(x,y) allowing for the creation of the given family can be found analytically (inverse problem) and that (ii) for any member of a class of two-dimensional refractive index profiles n=n(x,y) satisfying a differential condition, all the compatible families of light rays can be found analytically (direct problem). We present appropriate examples.     

Diffractive element design for resonant scanner angular correction: a beam retardation approach

A new approach for designing diffractive optical corrective elements with zooming capability to convert nonlinear sinusoidal scanning into linear scanning is proposed. Such a device will be useful for linearizing the angular scan of a resonant mirror scanner. The design methodology is to create a graded index of a refraction device as the reference design with its index of refraction parameters based on beam retardation through propagation in an inhomogeneous medium. The diffractive element is designed by utilizing a binarizing algorithm of the accumulated phase from transmission through the refractive element. In contrast to a prior approach, which was introduced based on the beam propagation through inhomogeneous media, the new approach takes beam diameters into consideration. This makes both the refractive element and its associated diffractive element more robust against beam fanning.     

Wigner distribution function in nonlinear optics

Transformation laws for the Wigner distribution function, the radiant intensity, the radiant emittance, and the first- and second-order moments of the Wigner distribution function through an inhomogeneous, Kerr-type medium have been derived as well as for the beam quality factor and the kurtosis parameter. It is shown that the inhomogeneous Kerr-type medium can be approximated from the Wigner-distribution-function transformation-law point of view with a symplectic ABCD matrix with elements depending on the field distribution.     

Optical remote sensing inside an inhomogeneous axisymmetric medium: the absorption field measurement

It is shown that the absorption field inside an inhomogeneous, rotationally symmetric medium with a spatially variable refractive index can be reconstructed by means of a tomographic technique. The classic Abel transform is extended to non-Euclidean optical media. The optical behavior of such a medium is described and, provided that the product of the refractive index with the radial distance is a monotonic function, an exact inverse formula is found. Both a numerical and an analytical test on a phantom function is carried out to prove the exactness of this formula. In contrast, when the assumption of a monotonic function is not true, it is shown that the reconstruction problem becomes subdeterminate because of the presence of annular regions, known as blind areas, inside of which no curved path reaches an extremum. The spatial localization and the size of these regions are related to the extrema of the index of refraction times the radial distance.     

A generalized two point ellipsoidal anisotropic ray tracing for converted waves

Rocks can be anisotropic due to a variety of reasons. When estimating rock velocities from seismic data, failure to introduce anisotropy into earth models could generate distortions in the final images that can have enormous economic impact. To estimate anisotropic earth velocities by tomographic methods, it is necessary to trace rays or to solve the wave equation in models where anisotropy has been properly considered. Thus, in this work we present a 3-D generalized ellipsoidal travel time formulation that allow us to trace rays in an anisotropic medium. We propose to trace rays in anisotropic media by solving a set of nonlinear optimization problems, where the group velocities for P and S wave propagation modes are 3-D ellipsoidal approximations that have been recently obtained. Moreover, we prove that this 3-D ellipsoidal anisotropic ray tracing formulation is a convex nonlinear optimization problem, and therefore any solution of the problem is a global minimum. Each optimization problem is solved by the global spectral gradient method, which requires first order information and has low computation and low storage requirements. Our approach for tracing rays in anisotropic media is a generalization in the sense that handles titled axis of symmetry and, close to the axis of symmetry, it is an accurate formulation for 2-D transversely isotropic media and 3-D orthorhombic media, depending on the input parameters. Moreover, this formulation gives the exact ray trajectories in 2-D and 3-D homogeneous isotropic media. The simplicity of the formulation and the low computational cost of the optimization method allow us to present a variety of numerical results that illustrate the behavior and computational advantages of the approach, and the difficulties when working in anisotropic media. Partially supported by Fonacit project UCV-97-003769     

Multidimensional calculation of ray path in a twisted nematic liquid crystal cell

In general, light propagating an inhomogeneous liquid crystal (LC) cell can be modeled as ‘bundle rays’ because the LC cell consists of many birefringence layers. In order to calculate the optical path of the propagating light in the inhomogeneous LC cell, we multidimensionally calculated the wavevector, k, and the Poynting vector, S, of an ordinary and an extraordinary ray at LC grid interfaces, which are isotropic to a uniaxial medium and a uniaxial-to-uniaxial medium, by using the phase matching method. Furthermore, we also investigated the transmission coefficients and transmittance of the ordinary and the extraordinary rays as a function of difference of the optical axes of the facing birefringence medium at the interface to obtain the significant rays in the LC cell. Finally, we could calculate the exact path of the significant rays in the inhomogeneous LC cell, and compared the ray path in an electrically controlled birefringence (ECB) mode and a twisted nematic (TN) LC mode.     

Refractive index profile modelling of dielectric inhomogeneous coatings using effective medium theories

The coatings having refractive index changing with the thickness present interesting optical performance, improved mechanical properties and smaller light scattering in comparison with classical multilayer stacks. Lot of theoretical work and experimental advances have been done for designing and production of mixture layers with such particular performances. The effective refractive index of the mixture coatings can be calculated by the use of effective medium theories. The refractive index profile characterization of inhomogeneous films that are mixtures of SiO₂ and Nb₂O₅ is presented. The composition is linearly changed through the thickness of the layers. Ex-situ spectrophotometric measurements, i.e. reflectance and transmittance at different incidence angles, are used for the precise characterization of the refractive index profiles. Linear, Maxwell-Garnet, Bruggeman and Lorentz-Lorenz effective medium theories are applied and quality and differences of the results are studied and analyzed. It is shown that the Lorentz-Lorenz model is the most appropriate for the given mixture, suggesting components are well mixed and there are no separated phases.     

Optical fiber design and the trapping of Cerenkov radiation

Cerenkov radiation is generated in optical fibers immersed in radiation fields and can interfere with signal transmission. We develop a theory for predicting the intensity of Cerenkov radiation generated within the core of a multimode optical fiber by using a ray optic approach and use it to make predictions of the intensity of radiation transmitted down the fiber in propagating modes. The intensity transmitted down the fiber is found to be dominated by bound rays with a contribution from tunneling rays. It is confirmed that for relativistic particles the intensity of the radiation that is transmitted along the fiber is a function of the angle between the particle beam and the fiber axis. The angle of peak intensity is found to be a function of the fiber refractive index difference as well as the core refractive index, with larger refractive index differences shifting the peak significantly toward lower angles. The angular range of the distribution is also significantly increased in both directions by increasing the fiber refractive index difference. The intensity of the radiation is found to be proportional to the cube of the fiber core radius in addition to its dependence on refractive index difference. As the particle energy is reduced into the nonrelativistic range the entire distribution is shifted toward lower angles. Recommendations on minimizing the quantity of Cerenkov light transmitted in the fiber optic system in a radiation field are given.     

Full-field digital gradient sensing method for evaluating stress gradients in transparent solids

A full-field digital gradient sensing method is proposed for measuring small angular deflections of light rays due to local stresses in transparent planar solids. The working principle of the method is explained, and the governing equations are derived. The analysis shows that angular deflections of light rays can be linked to nonuniform changes in thickness and refractive index of the material. In mechanically loaded planar solids, the angular deflections can be further related to spatial gradients of first invariant of stresses under plane stress conditions. The proposed method is first demonstrated by capturing the angular deflection fields in two orthogonal directions for a thin plano-convex lens. The measured contours of constant angular deflection of light rays are in good agreement with the expected ones for a spherical wavefront. The method is also successfully implemented to study a stress concentration problem involving a line load acting on an edge of a large planar sheet. Again, the stress gradients, measured simultaneously along and perpendicular to the loading directions, are in good agreement with the analytical predictions. The measured stress gradients have also been used to estimate stresses in the load point vicinity where plane stress results hold.     

Modeling of the rough-interface effect on a converging light beam propagating in a skin tissue phantom

Light distribution in a strong turbid medium such as skin tissue depends on both the bulk optical properties and the profiles of the interfaces where mismatch in the refractive index occurs. We present recent results of a numerical investigation on the light distribution inside a human skin tissue phantom for a converging laser beam with a wavelength near 1 mum and its dependence on the roughness of the interfaces and index mismatch. The skin tissue is modeled by a two-layer structure, and within each layer the tissue is considered macroscopically homogeneous. The two interfaces that separate the epidermis from the ambient medium and the dermis are considered randomly rough. With a recently developed method of Monte Carlo simulation capable of treating inhomogeneous boundary conditions, light distributions in various cases of interface roughness and index mismatch are obtained, and their relevance to the measurements of optical parameters of the skin tissue and laser surgery under the skin surface are discussed.     

Determination of the optical constants (n and k) of inhomogeneous thin films with linear index profiles

A new method for the determination of optical constants of absorbing inhomogeneous thin films is proposed. It requires measurements at normal incidence of the reflectance and transmittance of the film. In an inhomogeneous thin film, the optical constants vary along the thickness of the film. It has been reported in the literature that only the spatial integral value of the absorption index needs to be considered if its value is small. Therefore, in the proposed method, the mean value of the absorption index was used. The validity of this assumption was tested. On the other hand, the variation in the refractive index along the thickness of the film was taken into account. The method is discussed along with the nature of the solutions obtained and the effects of various parameters and assumptions. The method is applied successfully to inhomogeneous thin films of zirconium oxide.     

Hook method: recovery of density information from interferograms distorted by large spatial gradients

The hook method is a well-established technique for measuring the spatial distribution of species' densities in the gas phase, particularly in optically thick plasmas. However, in the presence of large density gradients (such as those occurring in a metal vapor laser plasma), the hook interferogram suffers severe distortion and the standard hook equation is invalid. By the use of a computer simulation of fringe formation, it is shown that this effect arises as a result of the strong wavelength-dependent lensing of probe rays in the test medium. On the basis of this lensing mechanism, a criterion has been derived for the maximum permissible density gradient, above which the standard hook analysis cannot be accurately applied. Finally, a new technique is presented that permits density data to be recovered from interferograms that are too distorted to analyze by the use of standard techniques. This technique is based on extracting density gradient values from distortion-free features of the fringe pattern. The new technique also permits density data to be obtained with an increased spatial resolution over that of the standard hook analysis.     

Manipulating the focal shift in a medium with electromagnetically induced transparency

By approximating the index distribution of a medium with electromagnetically induced transparency (EIT) as a gradient index (GRIN), propagation laws in the medium with EIT can be obtained. Transmission properties in an optical system with an EIT medium are analyzed. The results show that, unlike the case in the ordinary GRIN medium, the refractive index of EIT medium has the better controllability. Consequently, discussions are focused on how to conveniently manipulate the focal shift of the input in the EIT by means of controlling the index of the medium. Additionally, the speckle radius on the location of the actual focus can be diminished by adjusting some parameters in the EIT medium.     

Photometric immersion refractometry: a method for determining the refractive index of marine microbial particles from beam attenuation

Photometric immersion refractometry is a technique for determining the refractive index of particulate material. In this technique, the attenuation of light by a suspension of particles is measured as a function of the refractive index of the immersion medium. A minimum attenuation occurs at the refractive index of the medium equal to that of the particles. This technique can serve as a benchmark method for the refractive index determination because it is independent of assumptions invoked by other techniques, such as those based on the inversion of the spectral attenuation data. We present a rigorous model of the photometric immersion refractometry based on the anomalous diffraction approximation for the attenuation efficiency of particles. This model permits one to determine the average value of the real part of the refractive index of the particles, its variance, and the average imaginary part of the refractive index of the particles. In addition, the fourth moment of the particle size distribution can be determined if the concentration and shape of the particles are known. We analyze the sensitivity of this model to experimental errors and discuss the applicability of photometric immersion refractometry to marine microbial particles. We also present experimental results of this technique as applied to heterotrophic marine bacteria. The results indicate that the refractive index of these bacteria was narrowly distributed about the average value of 1.3886.     

Linear <Emphasis Type="Italic">s</Emphasis>-polarized surface waves in a medium inhomogeneous in one dimension

Linear s-polarized surface waves can exist at the boundary between an isotropic homogeneous medium and a medium inhomogeneous in one dimension (1D-inhomogeneous medium). This is related to deformation of the spatial envelope of the electric and magnetic components of the surface wave propagating in the 1D-inhomogeneous medium (in particular, in a plane-stratified medium). Such linear s-polarized surface waves can appear only provided that the refractive index of the inhomogeneous medium increases with the distance from the interface.     

Validity conditions for the radiative transfer equation

We compare the radiative transfer equation for media with constant refractive index with the radiative transfer equation for media with spatially varying refractive indices [J. Opt. A Pure App. Opt. 1, L1 (1999)] and obtain approximate conditions under which the former equation is accurate for modeling light propagation in scattering media with spatially varying refractive indices. These conditions impose restrictions on the variations of the refractive index, the gradient of the refractive index, the divergence of the rays, the frequency of modulation, and the widths of light pulses, which are related to the mean refractive index, the absorption coefficient, and the reduced scattering coefficient of the medium. Each condition is geometrically interpreted. Some implications of the results are discussed.     

Zero-order bows in radially inhomogeneous spheres: direct and inverse problems

Zero-order ray paths are examined in radially inhomogeneous spheres with differentiable refractive index profiles. It is demonstrated that zero-order and sometimes twin zero-order bows can exist when the gradient of refractive index is sufficiently negative. Abel inversion is used to "recover" the refractive index profiles; it is therefore possible in principle to specify the nature and type of bows and determine the refractive index profile that induces them. This may be of interest in the field of rainbow refractometry and optical fiber studies. This ray-theoretic analysis has direct similarities with the phenomenon of "orbiting" and other phenomena in scattering theory and also in seismological, surface gravity wave, and gravitational "lensing" studies. For completeness these topics are briefly discussed in the appendixes; they may also be of pedagogic interest.