Geometrical Optics Limit Research Articles

Gravitational scalar field coupled directly to the Maxwell field and its effect on solar-system experiments

The effect of the massless gravitational scalar field assumed to couple directly to the Maxwell field on the solar-system experiments is estimated. We start with discussing the theoretical significance of this coupling. Rather disappointingly, however, we find that the scalar-field parameters never affect the observation in the limit of geometric optics, indicating a marked difference from the well-known contribution through the spacetime metric.

Emissivity of Triangular Surfaces Determined by Differential Method: From Homogenization to Validity Limit of Geometrical Optics

Geometric optics approximation for emissivity from triangular surfaces was compared with exact scattering predictions from electromagnetic theory. Rigorous electromagnetic scattering theory was numerically formulated based on the differential method. We have used a numerical simulation of the emissivity of gold and tungsten for a wavelength equal 0.55 micron to explore the validity of the geometric optics. Surface parameter domains for the regions of accuracy of the geometric optics approximation are quantified and presented as functions of surface slope and roughness. Influence on the validity of the approximate method of multiple scattering, the shadowing effect and the cavity effect of metallic surface have been investigated. For the latter, our interest was focused on the mechanism that enhances the emissivity of an interface when ruling a grating. It has been seen that the mechanism responsible for the enhancement of the emissivity depends very much on the period of the grating. For gratings with a period much smaller than the wavelength, the roughness essentially behaves as a transition layer with a gradient of the optical index. For different period / wavelength ratio, we have found a good agreement between the differential method and the homogenization regime when the period was smaller.

Polynomial chaos expansions as extended Markov approximations in the propagation of uncertainty

The use of polynomial chaos for incorporating environmental variability into propagation models has been previously investigated where it was found that low-order expansions work quite well when the medium correlation length in the direction of propagation was small, i.e., the Markov approximation is roughly valid. It was suggested that an iterative scheme incorporating extensions of the Markov approximation might be useful. Using the method of successive approximations introduced by Klyatskin and Tatarskii (who used the method for investigating the validity of the Markov approximation), we show how the polynomial chaos equations can be written as a (smaller) set of equations that are basically nested expansions in the correlation function. This allows the use of a closure scheme, which does not involve truncating the expansion and which seems to be simpler to use. Comparisons are made between the approximation schemes in a geometrical optics limit.

Geometrodynamics in a spherically symmetric, static crossflow of null dust

The spherically symmetric, static space-time generated by a crossflow of noninteracting radiation streams, treated in the geometrical optics limit (null dust), is equivalent to an anisotropic fluid forming a radiation atmosphere of a star. This reference fluid provides a preferred/internal time, which is employed as a canonical coordinate. Among the advantages we encounter a new Hamiltonian constraint, which becomes linear in the momentum conjugate to the internal time (therefore yielding a functional Schr\odinger equation after quantization), and a strongly commuting algebra of the new constraints.

Relativistic effects on imaging by a rotating optical system

Context. High accuracy astrometric instruments like Gaia aiming at an accuracy of 1 microarcsecond cannot be considered as point-like observers in the framework of relativistic modelling of observable quantities. Aims. Special-relativistic effects on the imaging by a non-point-like arbitrarily moving optical instrument are discussed. Methods. A special-relativistic reflection law for a mirror of arbitrary shape and motion is derived in the limit of geometrical optics. The aberration patterns are computed with ray tracing using a full special-relativistic model for two simple rotating optical instruments. Results. The effect of special-relativistic reflection law on the photocenters of the aberration patterns of an optical system rotating with a moderate angular velocity of 60/s may be at the level of 1 microarcsecond if the system involves mirrors significantly inclined relative to the optical axis. Conclusions. Special-relativistic optical modelling of future astrometric instruments is indispensable if a level of a few microarcseconds is envisaged.

Thermal shield effect with uniaxial crystals

A model describing radiative transfers inside uniaxial anisotropic media is presented. The transport equations for each electromagnetic mode supported by these media are derived in the limit of geometrical optics and an analytical solution is obtained from a ray tracing method. The temperature field inside such a medium illuminated on both sides by a blackbody radiation is calculated and compared to the temperature field of an isotropic medium submitted to the same conditions. We show that the temperature field in the anisotropic medium is drastically smaller than its counterpart, even for weak anisotropy.

Rayleigh’s Hypothesis and the Geometrical Optics Limit

The Rayleigh hypothesis (RH) is often invoked in the theoretical and numerical treatment of rough surface scattering in order to decouple the analytical form of the scattered field. The hypothesis stipulates that the scattered field away from the surface can be extended down onto the rough surface even though it is formed by solely up-going waves. Traditionally this hypothesis is systematically used to derive the Volterra series under the small perturbation method which is equivalent to the low-frequency limit. In this Letter we demonstrate that the RH also carries the high-frequency or the geometrical optics limit, at least to first order. This finding has never been explicitly derived in the literature. Our result comforts the idea that the RH might be an exact solution under some constraints in the general case of random rough surfaces and not only in the case of small-slope deterministic periodic gratings.

Aspects of light propagation in anisotropic dielectric media

Some aspects of light propagation in local anisotropic nonlinear dielectric media at rest in the limit of geometrical optics are investigated. Natural and artificially induced anisotropies in dielectric materials are discussed. Analogies are proposed in such way that, as far as light is considered, kinematic aspects of some cosmological models are recovered. Particularly, analogue models for isotropic and anisotropic cosmologies are presented.

Bistatic scattering from an anisotropic sea surface: Numerical comparison between the first-order SSA and the TSM models

The first-order small-slope approximation (SSA-1) model is used for numerical predictions of the normalized radar cross section (NRCS) of an anisotropic ocean surface in bistatic configurations for the K u -band radar frequency. The calculations were made by assuming the Elfouhaily et al. surface-height spectrum for fully developed seas. In the forward–backward case, the SSA-1 presents an agreement with the geometric optics limit of the Kirchhoff approximation results in the near-specular directions where it is well known that the last model works well. In the fully bistatic case, SSA-1 numerical results are compared with those of the two-scale model in several configurations as a function of wind speed, wind direction, incident/scattering angles and for co-and cross-polarization states. Good agreement between the two models is noted in the co-and cross-polarization case with a small difference of about 1–2 dB. But in certain configurations, the SSA-1 model tends to overestimate the radar cross section peak behaviour. This irregularity is discussed and interpreted. The main purpose of this paper is to analyse NRCS predictions based on the SSA-1 model in a fully bistatic configuration.

Variable Magnification With Kirkpatrick-Baez Optics for Synchrotron X-Ray Microscopy.

We describe the distinction between the operation of a short focal length x-ray microscope forming a real image with a laboratory source (convergent illumination) and with a highly collimated intense beam from a synchrotron light source (Köhler illumination). We demonstrate the distinction with a Kirkpatrick-Baez microscope consisting of short focal length multilayer mirrors operating at an energy of 8 keV. In addition to realizing improvements in the resolution of the optics, the synchrotron radiation microscope is not limited to the usual single magnification at a fixed image plane. Higher magnification images are produced by projection in the limit of geometrical optics with a collimated beam. However, in distinction to the common method of placing the sample behind the optical source of a diverging beam, we describe the situation in which the sample is located in the collimated beam before the optical element. The ultimate limits of this magnification result from diffraction by the specimen and are determined by the sample position relative to the focal point of the optic. We present criteria by which the diffraction is minimized.

Electromagnetic light rays in local dielectrics

An approach to deal with the limit of geometrical optics of electromagnetic waves which propagate in moving nonlinear local dielectric media in the context of Maxwellian electrodynamics is here developed in order to apply to quite general material media. Fresnel equations for the light rays are generically found, and its solutions are intrinsically obtained. The multi-refringence problem is addressed, and no more than four monochromatic polarization modes are found to propagate there.

Optical transfer function of three-dimensional display systems

I investigate the optical transfer function of three-dimensional display systems. Moreover, I obtain an average sampled modulation transfer function describing discrete, sampled display systems and show that in the proper limit of geometrical optics it is equivalent to the shift-invariant optical transfer function. I apply the theory to describe holographic stereograms and discuss the effects of amplitude and phase filters on the optical resolution.

Radiative Properties of Dense Fibrous Medium Containing Fibers in the Geometric Limit

The aim of this paper is to investigate the dependent regime in dense fibrous materials with size parameters ranging in the geometric optics limit. We use a method based on a Monte Carlo procedure which permits one to identify the radiative properties of dispersed media. This method is applied to materials made of opaque or semitransparent randomly oriented long circular cylinders representing the fibers. The results permit us to investigate the limit of validity of independent scattering hypothesis and to analyze the evolution of the extinction coefficient, scattering albedo and phase function of the fibrous material with the porosity and the reflecting properties of the particles when the shadowing effect due to geometric sized objects is not negligible. We also propose a correlation to estimate the radiative properties in dependent regime from the results of the independent scattering hypothesis. Thereafter, the radiative characteristics obtained are compared to those predicted by previous authors.

Thin front propagation in random shear flows

Front propagation in time-dependent laminar flows is investigated in the limit of very fast reaction and very thin fronts--i.e., the so-called geometrical optics limit. In particular, we consider fronts stirred by random shear flows, whose time evolution is modeled in terms of Ornstein-Uhlembeck processes. We show that the ratio between the time correlation of the flow and an intrinsic time scale of the reaction dynamics (the wrinkling time tw) is crucial in determining both the front propagation speed and the front spatial patterns. The relevance of time correlation in realistic flows is briefly discussed in light of the bending phenomenon--i.e., the decrease of propagation speed observed at high flow intensities.

Interpretation of Backscattering Mechanisms from Non-gaussian Correlated Randomly Rough Surfaces

In this study we analyzed backscattering mechanism using a non-Gaussian correlation function, called exponential-like, for rough surface. The corresponding surface spectrum properties were investigated. It contains an excessive amount of high frequency spectral components. But unlike exponential one, it has rms slopes and allows to obtain the desired rms slopes. New insight into the physical implications of mechanism behind the backscattering behavior was obtained using the advanced integral equation model (AIEM). It is known that backscattering at large incident angles is proportional to the presence of small scale roughness or the high-frequency spectral components of the surface spectrum. Similarly, the peaking at near normal incidence is proportional to the large scale roughness or the low frequency spectral components of the surface spectrum. However, by increasing frequency while keeping the small roughness scale, scattering cannot approach the geometric optics at large angles of incidence where small roughness scales dominate scattering. There is condition where VV tends to HH returns without approaching the geometric optics limit. This is where we increase the surface rms slope. In fact, the presence of roughness scales, small or comparable to the incident wavelength, implies that the geometric optics condition is not satisfied. When there is a significant amount of backscattering in the large incident angle region, it indicates presence of high frequency spectral components in the surface spectrum. To demonstrate the applicability of the proposed correlation function, comparison of backscattering coefficients between model predictions and insitu measurements was made. Well match was obtained in terms of level and angular trend.

An improved geometrical optics model for bistatic GPS scattering from the ocean surface

This paper is concerned with the properties of bistatic microwave scattering from a randomly rough surface, and specifically its application to the study of global positioning system (GPS) satellite signals reflected from the ocean. We present a discussion of some recent refinements of Kirchhoff-type models based on second-order iterations of the surface-current integral equation, and the relationship between these models and their high-frequency (geometric optics) limit. In particular, we show that use of these refinements can extend the domain of applicability of the standard geometrical optics (GO) model. It is found that GO can be nearly as accurate as a Kirchhoff-based model provided that the wavenumber cutoff, at which the surface wave spectrum must be filtered in computing the required slope moments, depends on the roughness of the ocean surface (i.e., wind speed) as well as the incident angle and frequency of the radiation. We use a GO model refined in this way to analyze GPS surface reflection data collected from an aircraft equipped with two down-looking antennas for receiving both left- and right-hand circularly polarized reflections. Concurrent measurements of the local wind and wave conditions were collected from a nearby research vessel. Measured waveforms and mean Doppler widths at both polarizations are compared with predictions from our refined GO model, and discussion is given concerning the sensitivity of the reflected radiation to various geophysical parameters and the utility of GPS reflections for remote sensing applications.

Light Propagation in a Cole-Cole Nonlinear Medium via the Burgers-Hopf Equation

A new model of light propagation through a so-called weakly three-dimensional Cole-Cole nonlinear medium with short-range nonlocality was recently proposed. In particular, it was shown that in the geometric optics limit, the model is integrable and is governed by the dispersionless Veselov-Novikov (dVN) equation. The Burgers-Hopf equation can be obtained as a (1+1)-dimensional reduction of the dVN equation. We discuss its properties in the specific context of nonlinear geometric optics and consider an illustrative explicit example.

Physical and geometrical optics for 2-D rough surfaces with power-law height spectra

Random rough surfaces with slowly decaying power spectral density can have infinite slope variance. Such surfaces do not satisfy the classical curvature criterion for validity of the physical optics (PO) approximation, and the infinite frequency geometrical optics limit or specular point scattering model breaks down. We show for two-dimensional surfaces with infinite slope variance that the Gaussian form of the classical geometrical limit generalizes to a stable distribution function. We also show that the PO integral is insensitive to surface components with spatial frequency above a cutoff wavenumber, which explains past observations that PO can be accurate for surfaces with power law spectra. This result leads to a general validity condition for the PO approximation in the backscattering direction for power-law surfaces, which in the case of a k/sup -4/ spectrum requires that the significant slope of the surface be less than 0.03.

Birefringence in nonlinear anisotropic dielectric media

Light propagation is investigated in the context of local anisotropic nonlinear dielectric media at rest with the dielectric coefficients $\ensuremath{\epsilon}^{\ensuremath{\mu}}{}_{\ensuremath{\nu}}=\ensuremath{\epsilon}^{\ensuremath{\mu}}{}_{\ensuremath{\nu}}(\stackrel{\ensuremath{\rightarrow}}{E},\stackrel{\ensuremath{\rightarrow}}{B})$ and constant $\ensuremath{\mu}$, in the limit of geometrical optics. Birefringence was examined and the general conditions for its occurrence were presented. A toy model is exhibited, in which uniaxial birefringent media with nonlinear dielectric properties could be driven by external fields in such way that birefringence may be artificially controlled. The effective geometry interpretation is also addressed.

Actinic flux and photolysis in water droplets: Mie calculations and geometrical optics limit

Abstract. Photolysis of water-soluble components inside cloud droplets by ultraviolet/visible radiation may play an important role in atmospheric chemistry. Two earlier studies have suggested that the actinic flux and hence the photolysis frequency within spherical droplets is enhanced relative to that in the surrounding air, but have given different values for this enhancement. Here, we reconcile these discrepancies by noting slight errors in both studies that, when corrected, lead to consistent results. Madronich (1987) examined the geometric (large droplet) limit and concluded that refraction leads to an enhancement factor, averaged over all incident directions, of 1.56. However, the physically relevant quantity is the enhancement of the average actinic flux (rather than the average enhancement factor) which we show here to be 1.26 in the geometric limit. Ruggaber et al. (1997) used Mie theory to derive energy density enhancements slightly larger than 2 for typical droplet sizes, and applied these directly to the calculation of photolysis rates. However, the physically relevant quantity is the actinic flux (rather than the energy density) which is obtained by dividing the energy density by the refractive index of water, 1.33. Thus, the Mie-predicted enhancement for typical cloud droplet sizes is in the range 1.5, only coincidentally in agreement with the value originally given by Madronich. We also investigated the influence of resonances in the actinic flux enhancement. These narrow spikes which are resolved only by very high resolution calculations are orders of magnitude higher than the intermediate values but contribute only little to the actinic flux enhancement when averaged over droplet size distributions. Finally, a table is provided which may be used to obtain the actinic flux enhancement for the photolysis of any dissolved species.