Electromagnetic waves near perfect conductors. II. Casimir effect

The Casimir free energy of the electromagnetic field in regions bounded by thin perfect conductors with arbitrary smooth shapes is studied. This free energy is expressed as a convergent multiple scattering expansion, in which the wave is damped between scatterings taking place on conductors. The result depends neither on the large box needed to render the spectrum discrete, nor on the shape of the high frequency cut-off corresponding to slightly imperfect conductivity; this establishes the universal character of quantum electro-magnetic forces on perfectly conducting thin objects.General expansions are derived for the Casimir free energy in the low and high temperature limits. They exhibit a dependence on the topology of the conducting surfaces. At low temperature, the Casimir entropy vanishes as T³ for simply connected conductors, and as T for multiply connected ones. At high temperature, the free energy has the semiclassical behavior ?$\text{C}$$\text{T}\text{log}\text{(}\text{T}\text{h}\text{?}\text{cQ}\text{)}$. The constant $\text{C}$ = $\text{(128φ)}{}^{\mathrm{?1}}\text{?dθ(}\text{3}\text{R}{}_1{}^2\text{+}\text{3}\text{R}{}_2{}^2\text{+}\text{3}\text{R}{}_1\text{R}{}_2\text{)?n}$ may be interpreted as the number of additional modes of finite frequency created by introducing the conducting surface. It depends on the topology of the surface through its genus n, and on its local curvature radii R₁, R₂. The average wave number Q depends on the shape of the surface and its size. The symmetry between low and high temperatures is recovered for parallel plates.While a plane foil is stable against Casimir stresses at zero temperature, such stresses would tend at finite temperature to create wrinkles of dimensions larger thn 2.9 $\text{h}\text{?}\text{c}\text{T}$. The same wrinkling effect exists for an arbitrary conductor at high enough temperature. The presence of a curved conducting surface transfers the free energy of radiation from the concave to the convex side at zero temperature, and does the converse at high enough temperatures. Wedges may be considered for slightly imperfect conductors; the electro-magnetic energy is lowered at finite temperature by creation of wedges, and also at zero temperature by creation of multiple wedges as in a honeycomb structure.The formalism is also applied to evaluate Van der Waals forces and torques at arbitrary temperature between remote conductors, and to show that the Casimir energy of a cylinder at zero temperature vanishes to lowest order in the multiple scattering expansion.Finally, the sphere is used as a test for studying convergence of the multiple scattering expansions. Using radial and transverse combinations of vector spherical harmonics yields a simple expression for the electromagnetic Green functions. This expression is recovered by summation of the multiple scattering expansion; the convergence domain, in the plane kR = x + iy, of the expansion includes a neighborhood of the origin, as well as the whole region $\text{y >0.0348 |x|}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$. The numerical convergence of the Casimir free energy is rapid, the lowest order term already yields a result with less than 7% error.     

Multiple scattering of seismic waves

We present theory and numerical simulations to model seismic wave propagation in the Earth crust. We compare them to observations made in Mexico.     

Eikonal approach to scattering from coated perfect conductors in a spinor formalism

The eikonal approach is applied to the problem of the scattering of electromagnetic waves from an excluded volume in the presence of a weak external potential. The scattering of electromagnetic waves is treated in the spinor formalism previously developed by the author. The excluded volume is eventually taken to be a perfectly conducting cone, the external potential a coating of thickness , with complex dielectric constant , and permeability (tacitly assumed equal to 1). It is shown that to order (N–1), whereN=( )^1/2, the eikonal approach in the spinor formalism yields results equivalent to those obtained from the vector theory of Überall in the particular case of nose-on backscattering, using the eikonal function corresponding to straight-line propagation.Any views expressed in this paper are those of the author. They should not be interpreted as reflecting the views of The RAND Corporation or the official opinion or policy of any of its governmental or private research sponsors. Papers are reproduced by The RAND Corporation as a courtesy to members of its staff.     

Skin effect in massive conductors used in pulsed electrical devices: I. Electromagnetic field of massive conductors

The diffusion of a pulsed electromagnetic field into massive conductors with an arbitrary smooth surface is considered for the case where the field penetration depth is small. By using the boundary layer method, an asymptotic solution for the electromagnetic field is constructed. First-and second-order corrections to the limiting solution, which corresponds to the field distribution at an indefinitely high conductivity of the conductors, are found. Time dependences of the first-and second-order approximations to the electric field on the surface of the conductor are determined.     

Multiple expansions for energy and momenta carried by gravitational waves

We present expressions for the energy, linear momentum and angular momentum carried away from an isolated system by gravitational radiation based on spin-weighted spherical harmonics decomposition of the Weyl scalar Ψ ₄. We also show that the expressions derived are equivalent to the common expressions obtained when using a framework based on perturbations of a Schwazschild background. The main idea is to collect together all the different expressions in a uniform and consistent way. The formulae presented here are directly applicable to the calculation of the radiated energy, linear momentum and angular momentum starting from the gravitational waveforms which are typically extracted from numerical simulations. An erratum to this article can be found at     

Electromagnetic surface waves of multilayer stacks: coupling between guided modes and Bloch modes

The study of the dependence of surface mode dispersion on the termination of multilayer stacks reveals interesting features. For stratified media with high-refractive-index contrasts, surface modes can shift across several bandgaps if the thickness of the final layer is changed. The distance to the photonic band edge influences the decay length of the mode inside the multilayer stack. In the middle of the bandgap, the electromagnetic energy is concentrated in the final layer of the crystal, while near bandgap edges the decay length extends over several periods. Additional evidence suggests that surface modes behave like guided modes that can couple with the extended Bloch modes and give rise to evanescent field profiles oscillating along several periods.     

Semiclassical scattering theory with complex trajectories. I. Elastic waves

The WKB approximation to the one-particle Schrödinger equation is used to obtain the wave function at a given point as a sum of semiclassical terms, each of them corresponding to a different classical trajectory ending up at the same point. Besides the usual, real trajectories, also possible complex solutions of the classical equations of motion are considered. The simplicity of the method makes its use easy in practical cases and allows realistic calculations. The general solution of the one-dimensional WKB equations for an arbitrary number of complex turning points is given, and the solution is applied to calculate the position of the Regge poles of the scattering amplitude. The solution of the WKB equations in three dimensions for a central analytical potential is also obtained in a way that can be easily generalized to N-dimensions, provided the problem is separable. A multiple reflection series is derived, leading to a separation of the scattering amplitude into a smooth “background” term (single reflection approximation) that can be treated using classical but complex trajectories and a second resonating term that can be treated using the Sommerfeld-Watson transformation. The physical interpretation of the complex solutions of the classical equations of motion is given: they describe diffractive effects such as Fresnel, Fraunhofer diffraction, or the penetration of the quantal wave into shadow regions of caustics. They arise also in the scattering by a complex potential in an absorptive medium. The comparison with exact quantal calculations shows an astonishingly good agreement, and establishes the complex semiclassical approximation as a quantitative tool even in cases where the potential varies rapidly within a fraction of a wavelength. An approximate property of classical paths is discussed. The general pattern of the trajectories depends only on the product ? = EΘ, and not on energy and angle separately. This property is confirmed by experiments and besides the signature it gives for the semiclassical behavior, it simplifies considerably the search for all trajectories scattering through the same angle. Finally, a general classification of the different types of elastic heavy ion cross sections is given.     

畸形波电磁散射特性分析及其特征识别标识的研究

针对弱非线性的Longuet-Higgins模型在模拟强非线性畸形波海面时所存在的问题,采用修正的相位调制法模拟一维畸形波时间、空间波面,该方法能够实现畸形波的定时定点生成,并且其波形既能保持目标谱的频谱结构,又能较大程度地满足波浪序列的统计特性.同时,基于改进的双尺度(TSM)法及时域有限差分法建立畸形波的电磁散射模型,经过相对平均偏差和均方根偏差误差分析后,基于TSM法研究分析了畸形波及其背景海面波的归一化雷达散射截面(NRCS)的计算结果.实验表明,合成孔径雷达成像中畸形波的NRCS比背景波要小,即畸形波的合成孔径雷达图像成像比背景波要灰暗,因此可以将NRCS作为畸形波的特征识别标识.通过分析研究不同极化方式、入射角、入射频率条件下畸形波与背景波面的电磁散射特性实验数据得出:当二者的NRCS差值大于-11.8 dB及以上时,即认为产生畸形波,这为实际的工程应用提供了参照标准.     

Multiple Rayleigh scattering of electromagnetic waves: A partial summation

Abstract

This paper deals with the multiple scattering of electromagnetic waves by a homogeneous distribution of spherical Rayleigh scatterers. We use the Feynman diagrams symbolism and define the planar diagrams which describe the interactions with possible backscattering to a previously used scatterer (loops). Resummation over every level of overlapping loops is performed analytically. In terms of the absorption properties of the equivalent continuous medium, departure from Twersky's assumption is shown to occur only in dense media, and the first level of loops is a good approximation of the whole set. Results are compared with the scalar waves case.     

Multiple scattering in random media I. Restricted two-body additive approximation

We present a microscopic theory of the problem of finding the properties of a particle interacting with potentials located at random sites. The sites are governed by a general probability distribution. The starting point is the multiple scattering equations for the amplitude k ₁|T |k ₂ in terms of the individual scattering amplitudes k ₁|T |k ₂. We work with quantitiesA defined by k ₁|T |k ₂=k ₁|T |k ₂exp[i(k ₁–k ₂)R ]. The theory is based on a splitting of the fundamental equation forA into equations for the mean A and the fluctuationsAA . Neglect of the fluctuations yields the quasicrystalline approximation. We rearrange the equation forAA to isolate the collective part of the fluctuations. We then make the simplest microscopic truncation which is thatAA is a restricted two-body additive function of the site positions. With the contribution of the collective fluctuations, this yields results forA that are accurate to ordert ⁴.Work supported in part by the National Science Foundation under Contract No. NSF DMRWork supported in part by the National Science Foundation under Contract No. NSF DMR     

Multiple scattering of short waves at a system of interacting particles

A plane wave incident on a classical nonideal gas is considered. The mean field is derived up to terms for pair collisions of the scattering particles subject to certain assumptions about the wavelength and the interaction potential.I am indebted to V. E. Zuev and S. D. Tvorogov for many discussions.     

Multiple scattering of short waves in a system of interacting particles

The multiple scattering of a plane wave in a system of scatterers (a nonideal gas) is examined for certain constraints on the wavelength and interaction potential. For the ideal gas model, the radiation intensity is described by the kinetic transport equation. An analogous equation is obtained which accounts exactly for the pair collisions of the scatterers, and approximately for higher-order collisions.In conclusion, the author wishes to express his appreciation for valuable discussions with V. E. Zuev and S. D. Tvorogov.     

Multiple scattering of HF radio waves propagating across the sea surface

A problem that is of considerable importance to the HF communications and radar communities is the calculation of the spatio-temporal modulations imposed on a radio wave as a result of propagating across a rough sea surface. We derive equations, in the form of a multiple-scattering series, describing the spatio-temporal structure of a radio wave at a point on the rough sea surface after propagation from an antenna at another point on the surface. We present calculations illustrating the sensitivity of the resultant Doppler spectra to ocean surface conditions, as described by well known parametric models. The generality of the approach means that it can easily accommodate user-specified antenna patterns, arbitrary coastline geometries, arbitrary sea-wave spectrum models, wave-field inhomogeneity and a wide range of seawater conductivities.     

Electromagnetic energy flow near nanoparticles—I: single spheres

The electromagnetic (EM) energy flow near single spheres is investigated by applying Mie theory. From the patterns of the energy flow, the absorption and the scattering of light can be understood in the microscopic point of view. In the absorption profiles of metallic particles, most absorbed energy is consumed on the surface of the particles, which indicates that the resonance of surface plasmon is different from that of the bulk plasmon. Two mechanisms to enhanced local EM field are also distinguished. One is the surface plasmon resonance, and another one is the intensified energy flow.     

Instanton-centred expansions and elimination of zero modes

A general method of saddle-point expansions proposed on a basis of simple geometrical ideas allows us to avoid zero modes and develop a perturbation theory around a degenerate classical solution. Application to the Yang-Mills theory results in an instanton-centered perturbation theory in which the gluon propagator reduces to the scalar one. The BRST symmetry is extended to the instanton sector.     

Analysis of waveguides containing EMCs (electromagnetic conductors) or PEMCs (perfect electromagnetic conductors)

The concept of a perfect electromagnetic conductor (PEMC) was introduced to generalize and unify two well-known and apparently disjoint concepts in electromagnetics: the perfect electric conductor (PEC) and the perfect magnetic conductor (PMC). Although the PEMC has proven a fertile tool in electromagnetic analyses dealing with new and complex boundaries, its corresponding definition as a medium has, nevertheless, raised several problems. In fact, according to its initial 3D definition, the PEMC cannot be considered a unique and well-defined medium: it leads to extraneous fields without physical meaning. By using a previously published generalization of a PEMC that regards this concept both as a boundary and as a medium – which was dubbed an MIM (Minkowskian isotropic medium) and acts, in practice, as an actual electromagnetic conductor (EMC) – it is herein presented a straightforward analysis of waveguides containing PEMCs that readily and systematically follows from the general framework of waveguides containing EMCs.     

Electromagnetic waves scattering by cylindrical dielectric structures: Application to microwave devices design

A method which allows us to analyze the electromagnetic scattering characteristics of multiple discontinuities in shielded dielectric waveguides is presented. There are not restrictions both geometry of the cross section and electrical parameters of the dielectrics which are assumed to be linear, inhomogeneous, isotropic and free from losses. Each discontinuity is analyzed combining a modal matching technique with a generalized telegraphist's equations formulation; in this way, we obtain its scattering matrix. By using the concept of the generalized scattering matrix of two discontinuities, the equivalent generalized scattering matrix (EGSM) of the cascaded set is calculated. Theoretical and experimental results were obtained for different dielectric structures such as dielectric posts, isolated and coupled, as well as dielectric waveguides with circular cross section connected by means of abrupt and gradual transitions. The experimental values for the scattering properties show a good agrement with the theoretical ones. This study has shown the possibility of using cylindrical dielectric structures to design microwave devices such as: resonators, power-dividers and filters.