Geometrical Optics Approach Research Articles

Nonlinear processes in microdroplets: a geometrical optics approach

We present a geometrical optics model for the calculation of the internal fields in microspheres with nonlinear optical properties, which occur, for example, in stimulated Raman scattering. Our approach is based on a ray interpretation of the nonlinear wave equations describing the fields of the Stokes modes in a microdroplet. The Stokes modes are assumed to be whispering gallery modes. For such rays the nonlinear wave equations are used to describe the growth of the field along the path of the ray and to calculate the internal fields belonging to various Stokes modes. The interaction between different-order Stokes fields is considered, by using an iteration over the different orders. Although our approach can be extended to stimulated Brillouin scattering as well, in this paper only stimulated Raman scattering is considered.

Absorption of a Micro-Particle for Laser

This paper investigated the absorption of a micro-particle irradiated by laser. Micro-particles usually appear within the plasma induced by a laser or powder in the process of laser cladding. These particles are assumed to be spherical and neutral (no surface charge). Laser-particle interactions involve scattering, refraction, and diffraction phenomena. Refraction and diffraction can enhance radiation absorption. The complex optical indexes of material and size parameters of micro-particles characterize the absorption of particles in these materials processing. The electromagnetic wave theory and geometrical optics approach were utilized to analyze the absorption in the particle. The errors between these two methods were discussed for different indexes of absorption and size parameters. The compatibility of geometrical optics approach for a small particle is also presented.

Comparison of time-dependent extended mild-slope equations for random waves

The extended mild-slope equations of Suh et al. [Suh, K.D., Lee, C., Park, W.S., 1997. Time-dependent equations for wave propagation on rapidly varying topography. Coastal Eng., 32, 91–117] and Lee et al. [Lee, C., Kim, G., Suh, K.D., 2003. Extended mild-slope equation for random waves. Coastal Eng., 48, 277–287] are compared analytically and numerically to determine their applicability to random wave transformation. The geometric optics approach is used to compare the two models analytically. In the model of Suh et al., the wave number of the component wave with a local angular frequency ω is approximated with an accuracy of O( ω − ω¯) at a constant water depth, where ω¯ is the carrier frequency of random waves. In the model of Suh et al., however, the diffraction effects and higher-order bottom effects are considered only for monochromatic waves, and the shoaling coefficient of random waves is not accurately approximated. This inaccuracy arises because the model of Suh et al. was derived for regular waves. In the model of Lee et al., all the parameters of random waves such as wave number, shoaling coefficient, diffraction effects, and higher-order bottom effects are approximated with an accuracy of O( ω − ω¯). This approximation is because the model of Lee et al. was developed using the Taylor series expansion technique for random waves. The result of dispersion relation analysis suggests the use of the peak and weighted-average frequencies as a carrier frequency for Suh et al. and Lee et al. models, respectively. All the analytical results are verified by numerical experiments of shoaling of random waves over a slightly inclined bed and diffraction of random waves through a breakwater gap on a flat bottom.

Adiabatic nanofocusing of plasmons by sharp metallic grooves: Geometrical optics approach

In this paper, we demonstrate the possibility of efficient adiabatic nanofocusing of gap plasmons by sharp metallic V grooves or dielectric wedges covered with metal. The geometrical optics approach and the approximation of continuous electrodynamics are used for the analysis. In particular, it is demonstrated that both the phase and group velocities of an incident symmetric (with respect to the magnetic field) plasmon tend to zero at the tip of the groove, and the plasmon adiabatically slows down, eventually dissipating in the metal. The amplitude of the plasmon strongly increases near the tip of the groove. However, unlike nanofocusing by a sharp metal conical tip, even in the absence of dissipation, the amplitude of the plasmon near the tip of a V groove remains finite. The dependence of the maximal local-field enhancement on structural parameters, dissipation in the metal, angle of incidence, etc., is analyzed. It is also shown that a symmetric gap plasmon can effectively be guided by the groove, forming a channel plasmon-polariton—a special plasmon mode propagating along the tip of a metallic V groove. An existence condition for these strongly localized plasmon-polaritons is derived and discussed.

Further considerations of defocus rangefinders

This article follows a previous one that used a geometrical optics approach to analyse the operation of a defocus rangefinder system. In it, procedures were developed to enable range estimation to be made from width measurements of the blur profile when the temporally incoherent light source on the object was small and of unknown finite width. In this article, it is shown that the same procedures can be used for spatially incoherent monochromatic and quasi-monochromatic light sources in the presence of diffraction provided some constraints are met.

Prediction of light-transmission losses in plastic optical fibers

A theoretical expression is derived, based on a geometrical optics approach, with which to predict light-transmission losses in multimode plastic optical fibers for office or home lighting. Two types of optical ray arrangement, meridional ray and skew ray, are evaluated, and five loss mechanisms are identified and considered. The meridional arrangement results in a lower overall loss of light than the skew ray arrangement. The theoretical results were compared with experimental measurements taken for a 0.5-cm-diameter polymer optical fiber. For optical rays entering the fiber at incident angles of less than 20 degrees, the theoretical results are in good agreement with the empirical results.

Interaction of noncollinear spatial solitons in a nonlinear optical medium

The effects caused by nonresonant nonlinear interaction between noncollinear self-focusing beams are considered in two-dimensional optical samples by use of multiscale analysis. An analytical expression for beam trajectory shifts that are due to mutual interaction is derived, and the range of parameters is given, beyond which the mentioned consideration fails. We compare our results with the naive geometrical-optics model. It is shown that these two approaches give the same results. This justifies use of the geometrical-optics approach to describe elastic and almost-elastic collision processes both in Kerr and saturable nonlinear media. The results we obtained could be useful for the design of phase independent nonlinear photonic switches and all-optical logic elements.

Estimates of the albedo of powderlike surfaces in the geometrical optics approximation

Results of computer simulation of light scattering by powderlike media composed of large semitransparent particles of different shapes are presented. The geometrical optics approach is used. The cases of particles of spherical, cubic, and random shapes are considered. The one-dimensional geometrical optics model of light scattering by powderlike media, which finds application in investigations of planetary surfaces, meteorites, and lunar samples, is briefly described. The numerical simulation allowed us to estimate the error of this model, which turned out to be within the limits of several percent. The albedo calculated by using the one-dimensional model is shown to be closest to the “three-dimensional” reflectance of the surfaces at a phase angle of about 60°. This albedo also approximates fairly well the integrated reflectance, which in laboratory measurements is determined with the help of an integrating sphere.

Geometrical optics approach in liquid crystal films with three-dimensional director variations.

A formal geometrical optics approach (GOA) to the optics of nematic liquid crystals whose optic axis (director) varies in more than one dimension is described. The GOA is applied to the propagation of light through liquid crystal films whose director varies in three spatial dimensions. As an example, the GOA is applied to the calculation of light transmittance for the case of a liquid crystal cell which exhibits the homeotropic to multidomainlike transition (HMD cell). Properties of the GOA solution are explored, and comparison with the Jones calculus solution is also made. For variations on a smaller scale, where the Jones calculus breaks down, the GOA provides a fast, accurate method for calculating light transmittance. The results of light transmittance calculations for the HMD cell based on the director patterns provided by two methods, direct computer calculation and a previously developed simplified model, are in good agreement.

Geometrical optical modeling of femtosecond setups having angular dispersion

A reliable and efficient method based on a geometrical optical approach is presented to model the propagation of ultrashort pulses in optical systems. It is shown that the commonly used method to determine the group delay of the spectral components of a pulse from their geometrical optical path lengths is only correct for aberration-free optical systems. In the case of systems with angular dispersion and imaging errors, a correction to the path values obtained from ray-tracing calculations must be applied, since for specific systems neglect of the correction causes significant errors. A technique for performing this correction is presented. Two optical arrangements used for the generation and detection of tunable THz radiation by the femtosecond tilted-pulse-front excitation technique are analyzed to demonstrate the indispensability of the correction.

Synthesis of broad-beam patterns through leaky-wave antennas with rectilinear geometry

The feasibility of broad-beam radiation-pattern synthesis through leaky-wave antennas (LWAs) with rectilinear geometry is shown for the first time. The synthesis is based on a geometrical-optics approach and is achieved by a nonstandard tapering of a straight LWA, which involves the variation of both phase and leakage constants of leaky-mode propagating along the structure. Such an approach is an alternative to that based on a curved LWA and allows for a simpler manufacturing and a significant bulk reduction of the structure. The relevant theory is provided together with examples of broad-beam pattern synthesis.

Catastrophe theory interpretation of multiple peaks produced by light scattering from very rough dielectric surfaces

Dielectric rough surfaces, which scatter light with several maxima of intensity, are investigated in the framework of the geometrical optics approach. We consider surfaces between two media of refractive indices n 1 and n 2+i k 2. In this case, previous works notice two maxima in the plane of incidence: near the Brewster angle (p-polarization), or near the critical angle (if n 1> n 2). New effects are predicted here: three maxima in the plane of incidence near the Brewster and the critical angles (with n 1> n 2 in p-polarization); and two maxima in the cross-polarized components, near the critical angle (if n 1> n 2). We show that such “multiple maxima” effects disappear when k 2 increases. We demonstrate that the transition is accounted for rigorously by Thom's catastrophe theory and corresponds to a cusp catastrophe.

Fresnel diffraction effects in the frontlight for a liquid-crystal display.

In a new type of illumination system for reflective liquid-crystal displays, the frontlight, unwanted shadows appear in certain viewing directions. It will be shown that for an accurate description of these shadows the geometrical optics approach is not satisfactory and that Fresnel diffraction has to be taken into account. A model for the diffraction effects was developed, and the predicted results correspond well to the measurements. In addition, some remarkable effects were explained from the theory of Fresnel zones.

Optical efficiency of image sensor pixels.

The ability to reproduce a high-quality image depends strongly on the image sensor light sensitivity. This sensitivity depends, in turn, on the materials, the circuitry, and the optical properties of the pixel. We calculate the optical efficiency of a complementary metal oxide semiconductor (CMOS) image sensor pixel by using a geometrical-optics phase-space approach. We compare the theoretical predictions with measurements made by using a CMOS digital pixel sensor, and we find them to be in agreement within 3%. Finally, we show how to use these optical efficiency calculations to trade off image sensor pixel sensitivity and functionality as CMOS process technology scales.

Radar backscattering from Gerstner's sea surface wave

In the framework of a two-scale scattering model, radar backscattering from the rough sea surface was considered. The sea surface was modelled as a superposition of a nonlinear, large-scale Gerstner's wave and small-scale resonant Bragg scattering ripples. The zero-order diffracted field was found by a geometrical optics approach, with shadowing taken into account, and by an ‘exact’ solution of the diffraction problem obtained numerically. For vertical and horizontal polarizations, the spatial distribution of specific scattering cross sections along the large-scale wave was obtained. The spatially averaged specific backscattering cross sections, as well as the mean Doppler frequency shifts at both polarizations, obtained by the geometrical optics approach are compared with those obtained by using the ‘exact’ solution of the large-scale diffraction problem. The roles of shadowing and multiple wave scattering processes are discussed, and qualitative explanations of the difference between these two approaches are given.

P‐78: Geometrical Optics Approach for LCDs with Three Dimensional Director Variations

AbstractThe geometrical optics approach (GOA) is applied to the propagation of light through liquid crystal films whose director varies in three spatial dimensions. As an example of application, the GOA is applied to the calculation of light transmittance for the case of a LCD cell which exhibits the homeotropic to multidomainlike (HMD) transition. Comparison with Jones calculus is also provided. For smaller geometries, where the Jones calculus breaks down, the GOA provides a fast, accurate method for calculating the optical performance of LCDs.

A study of the influence of absorption on the spatial distribution offluorescence intensity within large droplets using Mie theory, geometricaloptics and imaging experiments

The dependence of energy distributions within droplets oninternal absorption effects has been investigated by calculationsbased on Mie theory and the geometrical optics approximation andexperiments. The objective was to evaluate the accuracy of thegeometrical optics approximation in calculating droplet volume tofluorescence intensity proportionality, required for planardroplet sizing measurements in sprays based on Mie scattering andfluorescence intensity from droplets. A geometrical opticsapproach was used to calculate the energy-density distribution inthe meridional plane of a droplet and this was compared to the Mietheory solution for a range of imaginary refractive indicesmi = 1×10-5 to mi = 5×10-4. Integrationof the energy density distribution over the droplet volumeprovided a method to compare experimental and theoreticalresults. Good agreement was found for the energy density andvolume integrated energy distribution patterns obtained from bothcalculation methods and the experimental results. Quantitativecomparison of the volume integrated energy results shows that forthe investigated range of absorptivity Mie theory calculationslead to results that are ≈30% higher than in thegeometrical optics case. This discrepancy is independent of lightabsorption and droplet size. Tunnelling waves were identified asthe cause for the discrepancies between Mie theory andgeometrical optics; these contribute to high energy density in therim region of the droplet images.

Self-Shadowing Influence on the Reflection from a Random Stepped Surface

The beam reflection from a stepped surface having a stochastic size of steps has been scrutinized using a geometrical optics approach. The coefficient that accounts for the effect of self-shadowing this type of surface is elaborated through auxiliary functions, each of them corresponding to the solution of integro-differential equation. In this paper the exact solution of these equations is constructed for the exponential function of the step height distribution density. In contrast to the earlier investigations this solution is validated for the finite distances between a receiver and a source and a random surface.

Low-grazing-angle (LGA) sea-spike backscattering from plunging breaker crests

The low-grazing-angle (LGA) microwave backscattering from a series of wave profiles representing the time evolution of a plunging breaker water wave is numerically calculated and compared with modeled predictions. The crest regions of the waves are isolated to remove large-scale multiple back-reflection paths that give interference. The horizontally polarized backscatter (HH) significantly exceeds that at vertical polarization (VV) during the breaking, despite the lack of multipath. Existing scattering models with some heuristic corrections are applied to the profiles to identify the dominant scattering mechanisms. The large HH-to-VV ratio is predicted from single scattering using an extended geometrical optics (EGO) approach. The initial rise in the backscatter as the wave steepens is predicted as diffraction from inflection points in the surface curvature at the crest using a modification of the geometrical theory of diffraction. The calculations show that the LGA backscattering from breaking wave crests is very complicated even in the absence of multipath and simple optically based models that treat the crest cross sections as equal at the two polarizations will be inaccurate.

Wind speed measurement using forward scattered GPS signals

Instrumentation and retrieval algorithms are described which use the forward scattered range-coded signals from the global positioning system (GPS) radio navigation system for the measurement of sea surface roughness. This roughness has long been known to be dependent upon the surface wind speed. Experiments were conducted from aircraft along the TOPEX ground track and over experimental surface truth buoys. These flights used a receiver capable of recording the cross-correlation power in the reflected signal. The shape of this power distribution was then compared against analytical models, which employ a geometric optics approach. Two techniques for matching these functions were studied. The first recognized the most significant information content in the reflected signal is contained in the trailing edge slope of the waveform. The second attempted to match the complete shape of the waveform by approximating it as a series expansion and obtaining the nonlinear least squares estimate. Discussion is also presented on anomalies in the receiver operation and their identification and correction.