Fresnel Wave Research Articles

Detection of an evanescent field scattered by silicon tips in an apertureless scanning near-field optical microscope

We describe how a commercial silicon cantilever can be used both as a force probe and a local scatterer of light. This tip can thus be used as the key element of an apertureless Scanning Near-field Optical Microscope (SNOM) combined with an Atomic Force Microscope (AFM). We use a Total Internal Reflection (TIR) illumination to provide a Fresnel evanescent wave with well-defined characteristics. We show how the weak signal issued from the near-field scattered by the tip can be extracted from the background noise. Finally we present how a signal related to the near field can also be obtained when the tip is oscillated at a frequency close to the resonance frequency of the cantilever. These results are compared with a simulation of the lock-in detection. Finally we describe the role of the vibration amplitude on the signal.

Description of propagation, dispersion, and scattering in optics

The Fresnel wave zone method is used with the radiation law for an induced electric dipole in a unified technique applied to a simplified model description of the interactions of secondary light waves with a dispersing and scattering medium, which enables one to derive simple dispersion and scattering formulas.

Phase-space localization and discrete representations of wave fields

The high-frequency behavior of wave fields in free space is characterized by localization in phase space, exemplified by the description of geometrical optics in which distinct wave vectors are associated with the rays passing through each spatial point. I explore the manifestation of phase-space localization in discrete representations of the wave field, in particular the discrete Fourier transform (DFT) and the Gabor representation. A number of auxiliary concepts, such as spectral truncation, the Lagrange manifold, and the Landau–Pollak (LP) theorem, are described and exploited in the process of understanding the behavior of high-frequency fields, and it is shown that the Lagrange manifold is the source in phase space of the dominant contributions to these discrete representations in a number of specially selected examples. The LP theorem specifies the number of discrete degrees of freedom required for a given field to be approximated to a prescribed accuracy, and the theorem controls both the number of samples for DFT implementation and the number of Gabor coefficients required. The behavior of the Gabor coefficients away from the Lagrange manifold is studied for a Fresnel wave (quadratic phase variation on a line), and it is shown that these coefficients decay exponentially, signifying the localization of the Gabor representation. The number of operations required for computing the Gabor representation is compared with the fast-Fourier-transform (FFT) implementation of the DFT, with the LP dimension used as the common cardinality of the two representations, the result being that the FFT is asymptotically more efficient than the Gabor representation even after one allows for the localization of the latter. However, the Gabor representation can be used in circumstances in which the FFT is inapplicable, when the explicit localization is a significant advantage.

An adaptive approach to the numerical solution of Fresnel's wave equation

An adaptive approach to the numerical solution of the wave propagation in integrated optics devices with 1-D cross sections is described. Fresnel's approximation of the exact wave equation resulting from Maxwell's equations is considered. A criterion to estimate the validity of this approximation is derived. Discretization in longitudinal direction with step-size control leads to a stationary subproblem for the transversal field distribution, which is then handled by an adaptive finite-element method. Thus, full adaptivity of the algorithm is realized. The numerical examples focus on waveguide tapers. >

Ampère's Invention of Equilibrium Apparatus: A Response to Experimental Anomaly

André-Marie Ampère's contributions to electrodynamics came at a late stage in an unconventional career. In 1820, he had reached the age of forty-five and had not as yet done any systematic research in physics. As a member of the mathematics section of the Académie des Sciences, his only significant contributions to the physical sciences had been some constructive criticisms of Fresnel's wave theory of light and three memoirs on chemical classification and gas theory. Meanwhile, his longstanding interests in metaphysics and epistemology had resulted in pointed methodological and philosophical attitudes which both motivated and structured his subsequent work in electrodynamics. Not surprisingly, events during the first few months of Ampère's research included many unplanned and unexpected encounters as this wildly enthusiastic theoretician grappled for the first time with the recalcitrant complexity of actual experimentation in an uncharted new domain.

Ionospheric scintillation calculations based on in situ irregularity spectra

Recent results from rocket, satellite, and radar experiments have greatly increased our understanding of equatorial spread‐F and its effects upon communication systems. In situ measurements have shown that the typical irregularity spectrum is a power law with a one‐dimensional index of −2. Previous applications of scintillation theory to such irregularities have utilized an anisotropic power law, in an effort to model elongation of irregularities along the magnetic field, and have found approximate solutions for the scintillation index S4, the rms phase deviation, and the characteristic scale size of the scintillation spectrum. We present here rigorous closed‐form solutions for these quantities which are valid to the extent that weak scattering thin screen theory allows. In addition we introduce a second “hybrid” form for irregularity spectra which is gaussian along the magnetic field and a power law in the perpendicular plane. The index is chosen in such a way that a one‐dimensional spectrum obtained on a spacecraft whose velocity makes a reasonable angle to the magnetic field and varies as k−2. Such a spectrum is introduced since it seems likely that at least at long wavelengths equatorial spread‐F is an interchange instability and that the density variation along is that of the zero‐order density variation. One‐dimensional spectra for small angles between and would hence be steeper than k−2 at intermediate k values, a result consistent with some in situ measurements. If particle precipitation is responsible for high latitude irregularities at long wavelengths, the hybrid spectrum might also be more appropriate for their characterization. At longer k, waves with small but finite kz such as drift waves might be important and hence the anisotropic power law more appropriate. Using both spectral forms, curves are presented for S4 and the characteristic scale as a function of x = 2 k20/k2ƒ, where k0 is the outer scale wave number of the irregularity spectrum and kƒ the Fresnel wave number. The frequency dependence of S4 based on the power law spectrum is also plotted in the regime where the theory is applicable.

George Gabriel Stokes on Stellar Aberration and the Luminiferous Ether

Acceptance of Augustin Fresnel's wave theory of light posed numerous questions for early nineteenth-century physicists. Among the most pressing was the problem of the properties of the luminiferous ether. Fresnel had shown that light waves were transverse. Therefore, since, among ordinary materials, only solids support transverse vibrations, there existed striking likenesses between highly tangible solids and the highly intangible ether. Accordingly, such men as Augustin-Louis Cauchy, James MacCullagh, Franz Neumann, and George Green constructed various theories of an elastic-solid ether.1 At the same time, however, the disconcerting implausibilities of an all-pervasive solid provoked considerable apprehension in regard to the elastic-solid tradition. Thomas Young found the concept ‘perfectly appalling’ and argued that ‘the hypothesis [that fluids can support transverse vibrations] remains completely open for discussion, notwithstanding the apparent difficulties attending it.’2 John Herschel, probably the most important English advocate of the wave theory, regarded the concept of a solid ether as only a temporary device, useful ‘till the real truth shall be discovered.’3 Consequently, despite the accomplishments which helped to make the elastic-solid theory ‘the most celebrated special form of the wave theory’,4 there were important voices of reservation.

Mechanical Interpretation of Maxwell's Equations

THROUGHOUT the nineteenth century there were intensive efforts to develop mechanical models of electromagnetic phenomena based on the hypothesis of a linear aether continuum pervading all space1. One of the most interesting models was the gyrostatic aether proposed by MacCullagh in 1837, a medium characterized by resistance to rotation but not to compression or distortion, and designed to secure a dynamical foundation for Fresnel's wave theory of light2. This proposal lay dormant until taken up again many years later by Fitzgerald (1880), followed by Kelvin (1892) and Larmor (1894), who assimilated it to Maxwell's field equations. Larmor pictured the electron as a nucleus of rotational strain in the gyrostatic aether, thereby partially anticipating Volterra's3 dislocation singularities which may exist in an elastic continuum. Although Larmor's electron as it stands is incapable of motion, it marked a distinct advance over alternative conceptions, including that of Lorentz (1892), which insisted on an inexplicable dichotomy between the aether and ordinary matter. This supposed dichotomy was responsible for many paradoxes and, in spite of Larmor's suggestion, prevented a rapprochement with the new relativity physics in 1905. An aether with mechanical properties completely imaging electromagnetic phenomena, and containing mobile singularities imaging material particles, would be perfectly compatible with the Special Theory of Relativity. It would, however, go beyond that theory in also providing a rational theory of matter. The purpose of this communication is to introduce the gyrostatic aether on a fresh mathematical basis, capable of assimilating charges and currents as well as fields, and thereby fulfilling the primary condition of any acceptable aether hypothesis. An extended development will be presented elsewhere.

Enhancing HF received fields with large planar and cylindrical ground screens

Users of extended range HF (3 to 30 MHz) radar and communication systems employing the ionosphere desire signal reception at incidence angles near-grazing to the local earth tangent. For vertical polarization, the vanishing received fields at low incidence angles over dielectric earth may be increased by using large ground screens. In this paper a ground-screen formulation based on scattering techniques is developed. The ground screen is viewed as a scatterer in free space, excited by a plane wave. A Fresnel image wave is added to establish the air-earth interface. Formulations are developed for the semi-infinite screen and for the circular-cylinder surface segment screen. The semi-infinite screen is representative in performance to a round screen of radius equal to the distance from the edge of the semi-infinite screen at which the field is computed or measured. For a ground screen on a hill the cylindrical segment is appropriate. Computations for a simulated earth were made and corroborated by experimental simulation with scale models. Improvements in field strength of 7 to 14 dB or more can be achieved with large screens over no screen. "Relatively small" tilted or raised flat screens, and cylindrical segment screens, can give improvements equal to very large flat screens on the earth.

Convergent polarized light and hertz's problem for a uniaxial material

The paper gives a mathematical treatment of the behaviour of convergent polarized light passing through a uniaxial crystal, together with some practical details relating to the production of rings and brushes. The Fresnel wave surface is shown to be the isophasic for a system of waves diverging from a point source in the crystal. Explicit expressions are given, in terms of elementary functions, for the electric and magnetic fields and the energy-stream, both far from and close to the source; and the validity of Fresnel's construction is discussed.

LVI. On a mode of deducing the equation of Fresnel's Wave

LVI. <i>On a mode of deducing the equation of</i> Fresnel's <i>Wave</i>