Showing papers on "Electromagnetic field published in 2006"

Controlling Electromagnetic Fields

Abstract: Using the freedom of design that metamaterials provide, we show how electromagnetic fields can be redirected at will and propose a design strategy. The conserved fields-electric displacement field D, magnetic induction field B, and Poynting vector B-are all displaced in a consistent manner. A simple illustration is given of the cloaking of a proscribed volume of space to exclude completely all electromagnetic fields. Our work has relevance to exotic lens design and to the cloaking of objects from electromagnetic fields.

Nonlinear collective effects in photon-photon and photon-plasma interactions

Abstract: Strong-field effects in laboratory and astrophysical plasmas and high intensity laser and cavity systems are considered, related to quantum electrodynamical (QED) photon-photon scattering. Current state-of-the-art laser facilities are close to reaching energy scales at which laboratory astrophysics will become possible. In such high energy density laboratory astrophysical systems, quantum electrodynamics will play a crucial role in the dynamics of plasmas and indeed the vacuum itself. Developments such as the free-electron laser may also give a means for exploring remote violent events such as supernovae in a laboratory environment. At the same time, superconducting cavities have steadily increased their quality factors, and quantum nondemolition measurements are capable of retrieving information from systems consisting of a few photons. Thus, not only will QED effects such as elastic photon-photon scattering be important in laboratory experiments, it may also be directly measurable in cavity experiments. Here implications of collective interactions between photons and photon-plasma systems are described. An overview of strong field vacuum effects is given, as formulated through the Heisenberg-Euler Lagrangian. Based on the dispersion relation for a single test photon traveling in a slowly varying background electromagnetic field, a set of equations describing the nonlinear propagation of an electromagnetic pulse on a radiation plasma is derived. The stability of the governing equations is discussed, and it is shown using numerical methods that electromagnetic pulses may collapse and split into pulse trains, as well as be trapped in a relativistic electron hole. Effects, such as the generation of novel electromagnetic modes, introduced by QED in pair plasmas is described. Applications to laser-plasma systems and astrophysical environments are also discussed.

Full-wave simulations of electromagnetic cloaking structures.

Abstract: Pendry et al. have reported electromagnetically anisotropic and inhomogeneous shells that, in theory, completely shield an interior structure of arbitrary size from electromagnetic fields without perturbing the external fields. Neither the coordinate transformation-based analytical formulation nor the supporting ray-tracing simulation indicate how material perturbations and full-wave effects might affect the solution. We report fully electromagnetic simulations of the cylindrical version of this cloaking structure using ideal and nonideal (but physically realizable) electromagnetic parameters that show that the low-reflection and power-flow bending properties of the electromagnetic cloaking structure are not especially sensitive to modest permittivity and permeability variations. The cloaking performance degrades smoothly with increasing loss, and effective low-reflection shielding can be achieved with a cylindrical shell composed of an eight- (homogeneous) layer approximation of the ideal continuous medium. An imperfect but simpler version of the cloaking material is derived and is shown to reproduce the ray bending of the ideal material in a manner that may be easier to experimentally realize.

Giant Gyrotropy due to Electromagnetic-Field Coupling in a Bilayered Chiral Structure

Abstract: We report experimental evidence that electromagnetic coupling between physically separated planar metal patterns located in parallel planes provides for extremely strong polarization rotatory power if one pattern is twisted with respect to the other, creating a chiral object. In terms of a rotary power per sample thickness equal to one wavelength, the bilayered structure rotates 5 orders of magnitude stronger than a gyrotropic crystal of quartz in the visible spectrum.

Numerical solution of critical state in superconductivity by finite element software

Abstract: A numerical method is proposed to analyse the electromagnetic behaviour of systems including high-temperature superconductors (HTSCs) in time-varying external fields and superconducting cables carrying AC transport current. The E–J constitutive law together with an H-formulation is used to calculate the current distribution and electromagnetic fields in HTSCs, and the magnetization of HTSCs; then the forces in the interaction between the electromagnet and the superconductor and the AC loss of the superconducting cable can be obtained. This numerical method is based on solving the partial differential equations time dependently and is adapted to the commercial finite element software Comsol Multiphysics 3.2. The advantage of this method is to make the modelling of the superconductivity simple, flexible and extendable.

Plasmonic field enhancement and SERS in the effective mode volume picture

Abstract: The controlled creation of nanometric electromagnetic field confinement via surface plasmon polariton excitations in metal/insulator/metal heterostructures is described via the concept of an effective electromagnetic mode volume Veff. Extensively used for the description of dielectric microcavities, its extension to plasmonics provides a convenient figure of merit and allows comparisons with dielectric counterparts. Using a one-dimensional analytical model and three-dimensional finite-difference time-domain simulations, it is shown that plasmonic cavities with nanometric dielectric gaps indeed allow for physical as well as effective mode volumes well below the diffraction limit in the gap material, despite significant energy penetration into the metal. In this picture, matter-plasmon interactions can be quantified in terms of quality factor Q and Veff, enabling a resonant cavity description of surface enhanced Raman scattering.

Dynamics of Charged Particles and their Radiation Field

Abstract: The motion of a charged particle interacting with its own electromagnetic field is an area of research that has a long history; this problem has never ceased to fascinate its investigators. On the one hand the theory ought to be straightforward to formulate: one has Maxwell's equations that tell the field how to behave (given the motion of the particle), and one has the Lorentz-force law that tells the particle how to move (given the field). On the other hand the theory is fundamentally ambiguous because of the field singularities that necessarily come with a point particle. While each separate sub-problem can easily be solved, to couple the field to the particle in a self-consistent treatment turns out to be tricky. I believe it is this dilemma (the theory is straightforward but tricky) that has been the main source of the endless fascination. For readers of Classical and Quantum Gravity, the fascination does not end there. For them it is also rooted in the fact that the electromagnetic self-force problem is deeply analogous to the gravitational self-force problem, which is of direct relevance to future gravitational wave observations. The motion of point particles in curved spacetime has been the topic of a recent Topical Review [1], and it was the focus of a recent Special Issue [2]. It is surprising to me that radiation reaction is a subject that continues to be poorly covered in the standard textbooks, including Jackson's bible [3]. Exceptions are Rohrlich's excellent text [4], which makes a very useful introduction to radiation reaction, and the Landau and Lifshitz classic [5], which contains what is probably the most perfect summary of the foundational ideas (presented in characteristic terseness). It is therefore with some trepidation that I received Herbert Spohn's book, which covers both the classical and quantum theories of a charged particle coupled to its own field (the presentation is limited to flat spacetime). Is this the text that graduate students and researchers should turn to in order to get a complete and accessible education in radiation reaction? My answer is that while the book does indeed contain a lot of useful material, it is not a very accessible source of information, and it is certainly not a student-friendly textbook. Instead, the book presents a technical account of the author's personal take on the theory, and represents a culminating summary of the author's research contributions over more than a decade. The book is written in a fairly mathematical style (the author is Professor of Mathematical Physics at the Technische Universitat in Munich), and it very much emphasises mathematical rigour. This makes the book less accessible than I would wish it to be, but this is perhaps less a criticism than a statement about my taste, expectation, and attitude. The presentation of the classical theory begins with a point particle, but Spohn immediately smears the charge distribution to eliminate the vexing singularities of the retarded field. He considers both the nonrelativistic Abraham model (in which the extended particle is spherically symmetric in the laboratory frame) and the relativistic Lorentz model (in which the particle is spherical in its rest frame). In Spohn's work, the smearing of the charge distribution is entirely a mathematical procedure, and I would have wished for a more physical discussion. A physically extended body, held together against electrostatic repulsion by cohesive forces (sometimes called Poincar? stresses) would make a sound starting point for a classical theory of charged particles, and would have nicely (and physically) motivated the smearing operation adopted in the book. Spohn goes on to derive energy?momentum relations for the extended objects, and to obtain their equations of motion. A compelling aspect of his presentation is that he formally introduces the 'adiabatic limit', the idea that the external fields acting on the charged body should have length and time scales that are long compared with the particle's internal scales (respectively the electrostatic classical radius and its associated time scale). As a consequence, the equations of motion do not involve a differentiated acceleration vector (as is the case for the Abraham?Lorentz?Dirac equations) but are proper second-order differential equations for the position vector. In effect, the correct equations of motion are obtained from the Abraham?Lorentz?Dirac equations by a reduction-of-order procedure that was first proposed (as far as I know) by Landau and Lifshitz [5]. In Spohn's work this procedure is not {\it ad hoc}, but a natural consequence of the adiabatic approximation. An aspect of the classical portion of the book that got me particularly excited is Spohn's proposal for an experimental test of the predictions of the Landau?Lifshitz equations. His proposed experiment involves a Penning trap, a device that uses a uniform magnetic field and a quadrupole electric field to trap an electron for very long times. Without radiation reaction, the motion of an electron in the trap is an epicycle that consists of a rapid (and small) cyclotron orbit superposed onto a slow (and large) magnetron orbit. Spohn shows that according to the Landau?Lifshitz equations, the radiation reaction produces a damping of the cyclotron motion. For reasonable laboratory situations this damping occurs over a time scale of the order of 0.1 second. This experiment might well be within technological reach. The presentation of the quantum theory is based on the nonrelativistic Abraham model, which upon quantization leads to the well-known Pauli-Fierz Hamiltonian of nonrelativistic quantum electrodynamics. This theory, an approximation to the fully relativistic version of QED, has a wide domain of validity that includes many aspects of quantum optics and laser-matter interactions. As I am not an expert in this field, my ability to review this portion of Spohn's book is limited, and I will indeed restrict myself to a few remarks. I first admit that I found Spohn's presentation to be tough going. Unlike the pair of delightful books by Cohen-Tannoudji, Dupont-Roc, and Grynberg [6, 7], this is not a gentle introduction to the quantum theory of a charged particle coupled to its own electromagnetic field. Instead, Spohn proceeds rather quickly through the formulation of the theory (defining the Hamiltonian and the Hilbert space) and then presents some applications (for example, he constructs the ground states of the theory, he examines radiation processes, and he explores finite-temperature aspects). There is a lot of material in the eight chapters devoted to the quantum theory, but my insufficient preparation and the advanced nature of Spohn's presentation were significant obstacles; I was not able to draw much appreciation for this material. One of the most useful resources in Spohn's book are the historical notes and literature reviews that are inserted at the end of each chapter. I discovered a wealth of interesting articles by reading these, and I am grateful that the author made the effort to collect this information for the benefit of his readers. References [1] Poisson E 2004 Radiation reaction of point particles in curved spacetime Class. Quantum Grav 21 R153?R232 [2] Lousto C O 2005 Special issue: Gravitational Radiation from Binary Black Holes: Advances in the Perturbative Approach, Class. Quantum Grav22 S543?S868 [3] Jackson J D 1999 Classical Electrodynamics Third Edition (New York: Wiley) [4] Rohrlich F 1990 Classical Charged Particles (Redwood City, CA: Addison?Wesley) [5] Landau L D and Lifshitz E M 2000 The Classical Theory of Fields Fourth Edition (Oxford: Butterworth?Heinemann) [6] Cohen-Tannoudji C Dupont-Roc J and Grynberg G 1997 Photons and Atoms - Introduction to Quantum Electrodynamics (New York: Wiley-Interscience) [7] Cohen-Tannoudji C, Dupont-Roc J and G Grynberg G 1998 Atom?Photon Interactions: Basic Processes and Applications (New York: Wiley-Interscience)

Compensating losses in negative-index metamaterials by optical parametric amplification.

Abstract: Optical parametric amplification controlled by the auxiliary electromagnetic field enables transparency, amplification, and oscillation with no cavity in strongly absorbing negative-index metamaterials The opposite directions of the wave vector and the Poynting vector in such materials result in extraordinary optical properties, including "backward" phase matching and the generation of entangled pairs of left- and right-handed counterpropagating photons

Optical interactions in a plasmonic particle coupled to a metallic film

Abstract: The interplay between localized surface plasmon (LSP) and surface plasmon-polariton (SPP) is studied in detail in a system composed of a three-dimensional gold particle located at a short distance from a gold thin film. Important frequency shifts of the LSP associated with the particle are observed for spacing distances between 0 and 50 nm. Beyond this distance the LSP and SPP resonances overlap, although some cavity effects between the particle and the film can still be observed. In particular, when the spacing increases the field in the cavity decreases more slowly than one would expect from a simple image dipole interpretation. For short separations the coupling between the particle and the film can produce a dramatic enhancement of the electromagnetic field in the space between them, where the electric field intensity can reach 5000 times that of the illumination field. Several movies show the spectral and time evolutions of the field distribution in the system both in and out of resonance. The character of the different modes excited in the system is studied. They include dipolar and quadrupolar modes, the latter exhibiting essentially a magnetic response.

Fast focus field calculations.

Abstract: We present a fast calculation of the electromagnetic field near the focus of an objective with a high numerical aperture (NA). Instead of direct integration, the vectorial Debye diffraction integral is evaluated with the fast Fourier transform for calculating the electromagnetic field in the entire focal region. We generalize this concept with the chirp z transform for obtaining a flexible sampling grid and an additional gain in computation speed. Under the conditions for the validity of the Debye integral representation, our method yields the amplitude, phase and polarization of the focus field for an arbitrary paraxial input field on the objective. We present two case studies by calculating the focus fields of a 40×1.20 NA water immersion objective for different amplitude distributions of the input field, and a 100×1.45 NA oil immersion objective containing evanescent field contributions for both linearly and radially polarized input fields.

A large-area flexible wireless power transmission sheet using printed plastic MEMS switches and organic field-effect transistors

Abstract: We have successfully manufactured a large-area power transmission sheet by using printing technologies. The position of electronic objects on this sheet can be contactlessly sensed by electromagnetic coupling using an organic transistor active matrix. Power is selectively fed to the objects by an electromagnetic field using a plastic MEMS-switching matrix.

Small ultra wideband antenna having unidirectional radiation pattern

Abstract: A small ultra wideband (UWB) antenna designed to have a unidirectional radiation pattern is disclosed. The UWB antenna includes a substrate; a power feeding part, provided on an upper surface of the substrate, for receiving a supply of an external electromagnetic energy; a dipole radiator excited by the electromagnetic energy fed through the power feeding part and radiating electromagnetic waves in one and the other directions of the substrate; and an active loop radiator excited by the electromagnetic energy fed through the power feeding part, respectively enhancing and canceling the electromagnetic fields produced in one or the other directions of the substrate by the dipole radiator.

Light diffraction by a strong standing electromagnetic wave.

Abstract: The nonlinear quantum interaction of a linearly polarized x-ray probe beam with a focused intense standing laser wave is studied theoretically. Because of the tight focusing of the standing laser pulse, diffraction effects arise for the probe beam as opposed to the corresponding plane wave scenario. A quantitative estimate for realistic experimental conditions of the ellipticity and the rotation of the main polarization plane acquired by the x-ray probe after the interaction shows that the implementation of such vacuum effects is feasible with future X-ray Free Electron Laser light.

An FDTD model for low and high altitude lightning-generated EM fields

Abstract: To explore lightning-generated electromagnetic wave behavior and lightning-related ionospheric phenomena, a full-wave two-dimensional cylindrical finite-difference time-domain (FDTD) model was developed to simulate lightning-generated electromagnetic wave propagation in the ionosphere with high altitude and long distance capabilities. This FDTD model removes the approximations made in other similar models to extend its applicability, and incorporates a variety of existing methods and new techniques. A dispersive and anisotropic realization of the nearly perfectly matched layer (NPML) absorbing boundary condition is adopted in this numerical model for ease of implementation. Earth curvature is included in the model through the modified refractive index method. The surface impedance boundary condition is adopted to treat arbitrary but homogeneous ground parameters. We quantify the errors through dispersion relations, and the solution convergence is analyzed. Comparisons between our simulation, numerical waveguide mode theory, and experimental data validate this model and show its capabilities compared to other methods. Although this FDTD model was developed for the lightning-generated electromagnetic field simulation, it is also applicable for other very low frequency (VLF, 3-30 kHz) and extremely low frequency (ELF, 3-3000 Hz) wave propagation problems

Consequences of Dirac Theory of the Positron

Abstract: According to Dirac's theory of the positron, an electromagnetic field tends to create pairs of particles which leads to a change of Maxwell's equations in the vacuum. These changes are calculated in the special case that no real electrons or positrons are present and the field varies little over a Compton wavelength.

Vibration of PM Brushless Machines Having a Fractional Number of Slots Per Pole

Abstract: In this paper, a method for predicting the electromagnetic vibration of permanent magnet (PM) brushless motors is presented. It calculates the electromagnetic force on individual stator teeth by the Maxwell stress method, based on two-dimensional finite-element analysis of the electromagnetic field. The resulting radial force contour is then used in an analytical shell model to predict the stator vibration. The method is validated experimentally, and is used to investigate the vibration in various PM brushless motors with different fractional slot/pole number combinations. The relationship between the slot/pole number combination and the dominant vibration frequency and mode order is also established

Dynamics of high‐energy electrons interacting with whistler mode chorus emissions in the magnetosphere

Abstract: [1] Close examination of whistler mode chorus emissions reveals that a chorus emission is a coherent monochromatic wave typically with a fast rising tone. The frequency of the emission increases rapidly along with growth of the wave amplitude. We first consider the generation mechanism of whistler mode chorus emissions. The essential mechanism of the frequency change is critically related to the inhomogeneity of the geomagnetic field in the equatorial region. The rising tone emission is only possible when the coherent wave propagates away from the equator interacting with a sufficient flux of counterstreaming resonant electrons. Depletion of the resonant trapped electrons from the wave phase space results in formation of an electromagnetic electron hole, which gives rise to a transverse resonant current causing both wave growth and frequency increase. The wave growth of a rising tone can elongate the nonlinear trapping zone, which works as an effective wave train that guides a fraction of the resonant electrons moving toward the equator. We perform test particle simulations where we solve the relativistic equations of motion for high-energy electrons under the action of the electromagnetic field of a coherent whistler mode wave and the assumed dipole geomagnetic field. We find that a fraction of weakly relativistic electrons are trapped by a coherent wave and that the trapped electrons are effectively accelerated by the chorus wave with a rising tone.

Spatial mapping of the internal and external electromagnetic fields of negative index metamaterials.

Abstract: We perform an experimental study of the phase and amplitude of microwaves interacting with and scattered by two-dimensional negative index metamaterials. The measurements are performed in a parallel plate waveguide apparatus at X-band frequencies (8–12 GHz), thus constraining the electromagnetic fields to two dimensions. A detection antenna is fixed to one of the plates, while a second plate with a fixed source antenna or waveguide is translated relative to the first plate. The detection antenna is inserted into, but not protruding below, the stationary plate so that fields internal to the metamaterial samples can be mapped. From the measured mappings of the electric field, the interplay between the microstructure of the metamaterial lattice and the macroscopic averaged response is revealed. For example, the mapped phase fronts within a metamaterial having a negative refractive index are consistent with a macroscopic phase—in accordance with the effective medium predictions—which travels in a direction opposite to the direction of propagation. The field maps are in excellent agreement with finite element numerical simulations performed assuming homogeneous metamaterial structures.

Thin shell wormholes in higher dimensional Einstein–Maxwell theory

Abstract: We construct thin shell Lorentzian wormholes in higher dimensional Einstein–Maxwell theory applying the ‘Cut and Paste’ technique proposed by Visser. The linearized stability is analyzed under radial perturbations around some assumed higher dimensional spherically symmetric static solution of the Einstein field equations in presence of Electromagnetic field. We determine the total amount of exotic matter, which is concentrated at the wormhole throat.

The Vlasov-Maxwell-Boltzmann System in the Whole Space

Abstract: The Vlasov–Maxwell–Boltzmann system is one of the most fundamental models to describe the dynamics of dilute charged particles, where particles interact via collisions and through their self-consistent electromagnetic field. We prove existence of global in time classical solutions to the Cauchy problem near Maxwellians.

Role of the electromagnetic field in the formation of domains in the process of symmetry-breaking phase transitions

Abstract: In the framework of quantum field theory we discuss the emergence of a phase locking among the electromagnetic modes and the matter components on an extended space-time region. We discuss the formation of extended domains exhibiting in their fundamental states nonvanishing order parameters, whose existence is not included in the Lagrangian. Our discussion is motivated by the interest in the study of the general problem of the stability of mesoscopic and macroscopic complex systems arising from fluctuating quantum components in connection with the problem of defect formation during the process of non-equilibrium symmetry breaking phase transitions characterized by an order parameter.

Drag force in SYM plasma with B field from AdS/CFT

Abstract: We investigate drag force in a thermal plasma of N=4 super Yang-Mills theory via both fundamental and Dirichlet strings under the influence of non-zero NSNS B-field background. In the description of AdS/CFT correspondence the endpoint of these strings correspondes to an external monopole or quark moving with a constant electromagnetic field. We demonstrate how the configuration of string tail as well as the drag force obtains corrections in this background.

Hamiltonian treatment of the electromagnetic field in dispersive and absorptive structured media

Abstract: We introduce a Hamiltonian formulation of electromagnetic fields in dispersive and absorptive structured media of arbitrary dimensionality; the Kramers-Kronig relations are satisfied by construction. Our method is based on an identification of the photonic component of the polariton modes of the system. Although the medium degrees of freedom are introduced in an oscillator model, only the susceptibility of the medium appears in the derived eigenvalue equation for the polaritons; the theory is applicable to both classical and quantum optics. A discrete polariton spectrum is obtained in the transparent regime below the absorption cutoff frequency, and the normalization condition contains the material dispersion in a simple way. In the absorptive regime, a continuous polariton spectrum is obtained. The expressions for the full electromagnetic field of the system can be written in terms of the modes of a limiting, nondispersive, nonabsorptive system, so the theory is well suited to studying the effect of dispersion or absorption on photonic dispersion relations and mode structure. Available codes for dispersive photonic modes can easily be leveraged to obtain polariton modes in both the transparent and absorptive regimes.

Signatures of the Unruh effect from electrons accelerated by ultrastrong laser fields.

Abstract: We calculate the radiation resulting from the Unruh effect for strongly accelerated electrons and show that the photons are created in pairs whose polarizations are perfectly correlated. Apart from the photon statistics, this quantum radiation can further be discriminated from the classical (Larmor) radiation via the different spectral and angular distributions. The signatures of the Unruh effect become significant if the external electromagnetic field accelerating the electrons is not too far below the Schwinger limit and might be observable with future facilities. Finally, the corrections due to the birefringent nature of the QED vacuum at such ultrahigh fields are discussed.

SU(2) × U(1) unified theory for charge, orbit and spin currents

Abstract: Spin and charge currents in systems with Rashba or Dresselhaus spin–orbit couplings are formulated in a unified version of four-dimensional SU(2) × U(1) gauge theory, with U(1) being the Maxwell field and SU(2) being the Yang–Mills field. While the bare spin current is non-conserved, it is compensated by a contribution from the SU(2) gauge field, which gives rise to a spin torque in the spin transport, consistent with the semi-classical theory of Culcer et al. Orbit current is shown to be non-conserved in the presence of electromagnetic fields. Similar to the Maxwell field inducing forces on charge and charge current, we derive forces acting on spin and spin current induced by the Yang–Mills fields such as the Rashba and Dresselhaus fields and the sheer strain field. The spin density and spin current may be considered as a source generating Yang–Mills field in certain condensed matter systems.

Heating of Metal Particles in a Single-Mode Microwave Applicator

Abstract: Microwave (MW) heating behavior of various metal particles was investigated using a single-mode applicator. Considering the distributions of the electromagnetic fields in the wave guide, specimens were placed at four specific positions with respect to the electric and the magnetic fields of MW. They were heated at conditions of constant power input. It was demonstrated that iron particles were heated well in the magnetic field, and that ferro-magnetic metal particles having the higher Curie point was heated the better. It was possible to heat iron bulk particles (� 3 mm) in a magnetic field without occurrence of electric discharge. In the range of nickel particle size between 45 and 150 mm, the particles with the smaller size were heated the better. Nickel oxide (NiO) was heated well only in the position of large electric field, which indicates that the heating was caused by the different (dielectric heating) mechanism from the metal particles. From these results, contribution of magnetic field to heating metal particles was discussed, considering the heating mechanisms of the magnetic loss and the eddy current loss. The dependence of the heating rate of metal particles on their size was discussed in terms of the heat transfer rate.

Revisiting the anomalous RF field penetration into a warm plasma

Abstract: Radio frequency (RF) waves do not penetrate into a plasma and are damped within it. The electric field of the wave and plasma current are concentrated near the plasma boundary in a skin layer. Electrons can transport the plasma current away from the skin layer due to their thermal motion. As a result, the width of the skin layer increases when electron thermal velocity is taken into account. This phenomenon is called the anomalous skin effect. The anomalous penetration of the RF electromagnetic field occurs not only for the electric field parallel to the plasma boundary (inductively coupled plasmas), but also for the electric field normal to the plasma boundary (capacitively coupled plasmas). Such anomalous penetration of the RF field modifies the structure of the RF sheath in capacitive coupled plasma. Recent advances in the nonlinear, nonlocal theory of the capacitive sheath are reported. It is shown that separating the electric field profile into exponential and nonexponential parts yields an efficient qualitative and quantitative description of the anomalous RF field penetration in both inductively and capacitively coupled plasmas