DOCSLIB.ORG

Plasma Waves

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Plasma Waves Plasma Waves S.M.Lea January 2007 1 General considerations To consider the different possible normal modes of a plasma, we will usually begin by assuming that there is an equilibrium in which the plasma parameters such as density and magnetic field are uniform and constant in time. We will then look at small perturbations away from this equilibrium, and investigate the time and space dependence of those perturbations. The usual notation is to label the equilibrium quantities with a subscript 0, e.g. n0, and the pertrubed quantities with a subscript 1, eg n1. Then the assumption of small perturbations is n /n 1. When the perturbations are small, we can generally ignore j 1 0j ¿ squares and higher powers of these quantities, thus obtaining a set of linear equations for the unknowns. These linear equations may be Fourier transformed in both space and time, thus reducing the differential equations to a set of algebraic equations. Equivalently, we may assume that each perturbed quantity has the mathematical form n = n exp i~k ~x iωt (1) 1 ¢ ¡ where the real part is implicitly assumed. Th³is form descri´bes a wave. The amplitude n is in ~ general complex, allowing for a non•zero phase constant φ0. The vector k, called the wave vector, gives both the direction of propagation of the wave and the wavelength: k = 2π/λ; ω is the angular frequency. There is a relation between ω and ~k that is determined by the physical properties of the system. The function ω ~k is called the dispersion relation for the wave. ³ ´ A point of constant phase on the wave form moves so that dφ = ~k ~v ω = 0 dt ¢ Á ¡ where the wave phase velocity is ω ~vÁ = k^ (2) k The phase speed of a wave may (and often does) exceed the speed of light, since no information is carried at vÁ. Information is carried by a modulation of the wave, in either amplitude or frequency. For example, a wave pulse may have a Gaussian envelope. For any 1 physical quantity u, x2 /a2 ikx iωt u (x, t) = Ae¡ e ¡ Thus can be Fourier•transformed to yield a Gaussian in k space. ¡ A general disturbance of the system may be written as a superposition of plane waves. With x axis chosen along the direction of propagation for simplicity: ¡ + 1 1 ikx i!t u (x, t) = A (k) e ¡ dk p2π Z¡1 But ω is related to k through the dispersion relation: d! ω = ω (k) = ω + (k k ) + 0 ¡ 0 dk ¢ ¢ ¢ ¯k 0 ¯ where k is the wave number at which A (k) peaks. Then¯the integral becomes: 0 ¯ + 1 1 d! ikx i! 0t u (x, t) = A (k) e ¡ exp i (k k0) t dk p2π ¡ ¡ dk Z " ¯k 0 # ¡1 ¯ + ¯ 1 d! 1 ¯ d! = exp i k0 !0 t A (k) exp ik x t dk p2¼ dk ¡ ¡ dk " Ã ¯k 0 ! # Z " Ã ¯k0 !# ¯ ¡1 ¯ d! ¯ d! ¯ = exp i k ¯! t u x t; 0 ¯ 0 dk ¡ 0 ¡ dk " Ã ¯k 0 ! # Ã ¯k 0 ! ¯ ¯ Thus, apart from an overall ph¯ase factor, the disturbance p¯ ropagates at the group speed ¯ ¯ d! vg = (3) dk ¯k 0 ¯ Information travels at this group speed. ¯ ¯ Additional terms we have neglected in the exponent (involving the higher derivatives dn!=dkn ) give rise to pulse spreading and other factors generally referred to as dispersion. (More on this in Phys 704!) Hence the name "dispersion relation". Our goal will be to identify the different wave modes that occur in the plasma, and to find the dispersion relation ! ~k for each. If the frequency has an imaginary part, that indicates damping or growth o³f t´he wave. 2 Plasma oscillations In this wave we assume that the initial condition is a cold, uniform, unmagnetized plasma with n = n ; T 0 and ~v = 0: Then the equation of motion is: 0 ´ 0 @ mn ~v + ~v ~ ~v = en E~ 0 @t 1 1 ¢ r 1 ¡ 0 µ ³ ´ ¶ We ignore the second term on the left, because it involves a small quantity squared. Then we use the assumed wave form to obtain i!m~v = eE~ (4) ¡ 1 ¡ 2 Next the continuity equation gives @ (n + n ) + ~ (n + n ) ~v = 0 @t 0 1 r ¢ 0 1 1 @ n + n ~ ~v + ~ (n ~v ) = 0 @t 1 0r ¢ 1 r ¢ 1 1 ~ We used the facts that @n0=@t and n0 are zero. Again we ignore the last term because it involves the product of two small qurantities. Then: i!n + n ik~ ~v = 0 (5) ¡ 1 0 ¢ 1 Or, solving for n1; k~ ~v n = n ¢ 1 (6) 1 0 ! ~ Thus the density is not perturbed unless ~v1 has a component along k: This means that ~ transverse waves with ~v1 perpendicular to k do not perturb the plasma density. Finally we use Poissonzs equation, since this is a high•frequency wave: ~ ~ e e E = [n0 (n0 + n1)] = n1 r ¢ ¡ "0 ¡ ¡"0 This equation is already linear, and we get: ~ ~ e ik E = n1 (7) ¢ ¡ "0 Now we dot equation (4) with k~ and substitute in from equations (5) and (7). i!mk~ ~v = e~k E~ ¡ ¢ 1 ¡ ¢ !n e i!m 1 = e n ¡ n ¡ ¡ i" 1 0 µ 0 ¶ 2 !n1 e !m = n1 n0 "0 Now either n1 = 0 (not the result we want) or 2 2 n0e !p = (8) "0m The frequency of the disturbance is the plasma frequency !p: Notice that the result is independent of k: There is no phase speed or group speed: the wave is an oscillation that does not propagate. As we shall see, introduction of a non•zero temperature causes the wave to propagate. We can also see from equation (7) that the electric ®eld is out of phase with the density by ¼=2; while the velocity is in phase (equation 6). This oscillation is a fundamental mode of the plasma and has many ramifications. Our next step is to begin to investigate its importance. 3 3 Electromagnetic waves in an unmagnetized plasma To understand the importance of the plasma frequency a bit more, letzs look at the propagation of electromagnetic waves in the plasma. Maxwellzs equations are already linear. We have Poissonsz equation for E~ (7) and B~, i~k B~ = 0 (9) ¢ 1 Faradayzs Law i~k E~ = i!B~ (10) £ 1 and Amperezs law i~k B~ = ¹ ~j + i!" E~ (11) £ 0 0 The current in the plasma is due to the electron³and ion mo´tions: ~j = n e~v (n + n ) e~v 0 i ¡ 0 1 e For high frequency waves we may neglect the ion motion (~v 0) since the massive ions i ' do not respond rapidly enough as the driving force due to E~ changes. We also ignore the product of the two small quantities n1 and ve to get: ~j = n e~v (12) ¡ 0 e so that Amperezs law becomes: ik~ B~ = ¹ n e~v + i!" E~ (13) £ 1 0 ¡ 0 e 0 Finally we need the equation of motion for ³the electrons: ´ i!m~v = e E~ + ~v B~ (14) ¡ 1 ¡ e £ 1 which reduces to equation (4) when we ignore³the non•linear´term. Equation (6) shows that the density variations are zero for a transverse wave ~v perpendicular to ~k , and Poissonzs equation becomes k~ E~ = 0; which is consistent with ¢ t³he assumption of a tr´ansverse wave with ~v parallel to E~ (equation 14) . So let us look for transverse waves. Cross equation (13) with ~k to get: ik~ ~k B~ = ¹ n e~k ~v + i!" k~ E~ £ £ 1 0 ¡ 0 £ e 0 £ ³ ´ ³n e ´ ~k ~k B~ k2B~ = ¹ 0 ~k ~v ¹ !" !B~ ¢ 1 ¡ 1 ¡ 0 i £ e ¡ 0 0 1 where we used Far³adayzs l´aw. Further substitutions from equations (9) and (4) give: ~ 2 ~ n0e ~ eE 2 ~ k B1 = ¹0 k ¹0"0! B1 ¡ ¡ i £ Ã i!m ! ¡ e2 = ¹ n !B~ ¹ " !2B~ 0 0 !m ¡ 0 0 1 = ¹ " !2 !2 B~ (15) 0 0 p ¡ 1 ¡ ¢ 4 ~ ~ Again we argue that the solution B1 = 0 is uninteresting. The solution with non•zero B has 2 2 2 2 ! = k c + !p (16) 2 2 2 ! = k c + !p (17) Thus at high frquencies (! ! ) we have thqe usual vacuum relation ! = ck: À p 3 2.5 2 omega1.5 1 0.5 0 0.5 1 1.5ck 2 2.5 3 However, the wave number 2 2 ! !p k = ¡ q c becomes imaginary (k = i°) when ! < !p: The wave form becomes exp [i (i°) x i!t] = e °xei!t ¡ ¡ The wave does not propagate, and the disturbance damps as exp ( °x). The waves have a ¡ cut•off at ! = !p: An electromagnetic wave propagating at ! ! is able to excite plasma oscillations in · p the plasma, thus draining energy out of the wave and into the motion of plasma particles. Hence the wave damps. In the ionosphere with n 1012 m 3; the plasma frequency is » ¡ 12 3 19 2 1 (10 m¡ ) (1:6 10¡ C) fp = £ = 9: MHz 2¼ s (8:85 10 12 F/m) (9 10 31 kg) £ ¡ £ ¡ and waves at lower frequencies do not propagate through the ionosphere. This is an intrinsic limit on our ability to do radio astronomy from the Earthzs surface. If we do build a base on the moon, a radio observatory will quickly follow! The refractive index of the plasma is 2 c ck !p n = = = 1 2 (18) vÁ ! r ¡ ! and is less than one for all ! < : 1 5 1 0.8 n 0.6 0.4 0.2 0 1 2 omega 3 4 5 Snellsz law thus predicts the phenomenon of total external reflection when an EM wave of frequency ! < !p is incident on a plasma.
Load more

Recommended publications
  • Plasma Oscillation in Semiconductor Superlattice Structure
    Memoirs of the Faculty of Engineering,Okayama University,Vol.23, No, I, November 1988 Plasma Oscillation in Semiconductor Superlattice Structure Hiroo Totsuji* and Makoto Takei* (Received September 30, 1988) Abstract The statistical properties of two-dimensional systems of charges in semiconductor superlattices are analyzed and the dispersion relation of the plasma oscillation is ealculated. The possibility to excite these oscillations by applying the electric field parallel to the structure is discussed. ]. Introduction The layered structure of semiconductors with thickness of the order of lO-6 cm or less is called semiconductor superlattice. The superlattice was first proposed by Esaki and Tsu [11 as a structure which has a Brillouin zone of reduced size and therefore allows to apply the negative mass part of the band structure to electronic devices through conduction of carriers perpendicular to the structure. The superlattice has been realized by subsequent developments of technologies such as molecular beam epitaxy (MBE) and metal-organic chemical vapor deposition (MOCVD) in fabricating controlled fine structures. At the same time, many interesting and useful physical phenomena related to the parallel conduction have also been revealed in addition to the parallel conduction. From the view point of application to devices, the enhancement of the carrier mobility due to separation of channels from Ionized impurities may be one of the most important progresses. Some high speed devices are based on this technique. The superlattlce structure
    [Show full text]
  • Carbon Fiber Skeleton/Silver Nanowires Composites with Tunable Negative
    EPJ Appl. Metamat. 8, 1 (2021) © Y. An et al., published by EDP Sciences, 2021 https://doi.org/10.1051/epjam/2020019 Available online at: epjam.edp-open.org Metamaterial Research Updates from China RESEARCH ARTICLE Carbon fiber skeleton/silver nanowires composites with tunable negative permittivity behavior Yan An1, Jinyuan Qin1, Kai Sun1,*, Jiahong Tian1, Zhongyang Wang2,*, Yaman Zhao1, Xiaofeng Li1, and Runhua Fan1 1 College of Ocean Science and Engineering, Shanghai Maritime University, Shanghai 201306, PR China 2 State Key Lab of Metal Matrix Composites, School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China Received: 13 November 2020 / Accepted: 25 December 2020 Abstract. With the development of periodic metamaterials, more attention has been paid to negative permittivity behavior due to great potential applications. In this paper, silver nanowires (AgNWs) were introduced to the porous carbon fibers (CFS) by an impregnation process to prepare CFS/AgNWs composites with different content of AgNWs and the dielectric property was investigated. With the formation of conductive network, the Drude-like negative permittivity was observed in CFS/AgNWs composites. With the increase of AgNWs, the connectivity of conductive network became enhanced, the conductivity gradually increases, and the absolute value of the negative dielectric constant also increases to 8.9 Â 104, which was ascribed to the enhancement of electron density of the composite material. Further investigation revealed that the inductive characteristic was responsible for the negative permittivity. Keywords: Negative permittivity / metacomposites / plasma oscillation / inductive characteristic 1 Introduction capacitors [10,11]. The electromagnetic metamaterials have a great value in the fields of wireless communication In the past decades, electromagnetic metamaterials with [12], electromagnetic absorption [13] and shielding [14], etc.
    [Show full text]
  • Free Electron Lasers and High-Energy Electron Cooling*
    FREE ELECTRON LASERS AND HIGH-ENERGY ELECTRON COOLING* Vladimir N. Litvinenko, BNL, Upton, Long Island, NY, USA# Yaroslav S. Derbenev, TJNAF, Newport News, VA, USA) Abstract The main figure of merit of any collider is its average Cooling intense high-energy hadron beams remains a luminosity, i.e., its average productivity for an appropriate major challenge in modern accelerator physics. branch of physics. Cooling hadron beams at top energy Synchrotron radiation of such beams is too feeble to may further this productivity. provide significant cooling: even in the Large Hadron For a round beam, typical for hadron colliders, the Collider (LHC) with 7 TeV protons, the longitudinal luminosity is given by a simple expression: ' * damping time is about thirteen hours. Decrements of N1N2 & s traditional electron cooling decrease rapidly as the high L = fc * % h) * , (1) 4"# $ ( # + power of beam energy, and an effective electron cooling of protons or antiprotons at energies above 100 GeV where N1, N2 are the number of particles per bunch, fc is seems unlikely. Traditional stochastic cooling still cannot their collision frequency, !* is the transverse !-function catch up with the challenge of cooling high-intensity at the collision point, " is the transverse emittance of the bunched proton beams - to be effective, its bandwidth b!ea m, #s is the bunch length, and h ! 1 is a coefficient must be increased by about two orders-of-magnitude. accounting for the so-called hourglass effect [1]: Two techniques offering the potential to cool high- " 1/ x 2 h(x) = e erfc(1/ x). energy hadron beams are optical stochastic cooling (OSC) x and coherent electron cooling (CEC) – the latter is the The hourglass effect is caused by variations in the beam’s 2 focus of this paper.
    [Show full text]
  • Optimization of Ion Flow Rates in a Helicon Injected IEC Fusion System
    Aeronautics and Aerospace Open Access Journal Research Article Open Access Optimization of ion flow rates in a helicon injected IEC fusion system Abstract Volume 4 Issue 2 - 2020 The Helicon Injected Inertial Electrostatic Confinement (IEC) offers an attractive D-D Qiheng Cai, George H Miley neutron source for neutron commercial and homeland security activation neutron analysis. Department of Nuclear, Plasma & Radiological Engineering, Designs with multiple injectors also provide a potential route to an attractive small fusion University of Illinois at Champaign-Urbana, USA reactor. Use of such a reactor has also been studied for deep space propulsion. In addition, a non-fusion design been studied for use as an electric thruster for near-term space Correspondence: Qiheng Cai, Department of Nuclear, Plasma applications. The Helicon Inertial Plasma Electrostatic Rocket (HIIPER) is an advanced & Radiological Engineering, University of Illinois at Champaign- space plasma thruster coupling the helicon and a modified IEC. A key aspect for all of these Urbana, Urbana, IL, 61801, Tel +15712679353, systems is to develop efficient coupling between the Helicon plasma injector and the IEC. Email This issue is under study and will be described in this presentation. To analyze the coupling efficiency, ion flow rates (which indicate how many ions exit the helicon and enter the IEC Received: May 01, 2020 | Published: May 21, 2020 device per second) are investigated by a global model. In this simulation particle rate and power balance equations are solved to investigate the time evolution of electron density, neutral density and electron temperature in the helicon tube. In addition to the Helicon geometry and RF field design, the use of a potential bias plate at the gas inlet of the Helicon is considered.
    [Show full text]
  • Plasma Physics I APPH E6101x Columbia University Fall, 2016
    Lecture 1: Plasma Physics I APPH E6101x Columbia University Fall, 2016 1 Syllabus and Class Website http://sites.apam.columbia.edu/courses/apph6101x/ 2 Textbook “Plasma Physics offers a broad and modern introduction to the many aspects of plasma science … . A curious student or interested researcher could track down laboratory notes, older monographs, and obscure papers … . with an extensive list of more than 300 references and, in particular, its excellent overview of the various techniques to generate plasma in a laboratory, Plasma Physics is an excellent entree for students into this rapidly growing field. It’s also a useful reference for professional low-temperature plasma researchers.” (Michael Brown, Physics Today, June, 2011) 3 Grading • Weekly homework • Two in-class quizzes (25%) • Final exam (50%) 4 https://www.nasa.gov/mission_pages/sdo/overview/ Launched 11 Feb 2010 5 http://www.nasa.gov/mission_pages/sdo/news/sdo-year2.html#.VerqQLRgyxI 6 http://www.ccfe.ac.uk/MAST.aspx http://www.ccfe.ac.uk/mast_upgrade_project.aspx 7 https://youtu.be/svrMsZQuZrs 8 9 10 ITER: The International Burning Plasma Experiment Important fusion science experiment, but without low-activation fusion materials, tritium breeding, … ~ 500 MW 10 minute pulses 23,000 tonne 51 GJ >30B $US (?) DIII-D ⇒ ITER ÷ 3.7 (50 times smaller volume) (400 times smaller energy) 11 Prof. Robert Gross Columbia University Fusion Energy (1984) “Fusion has proved to be a very difficult challenge. The early question was—Can fusion be done, and, if so how? … Now, the challenge lies in whether fusion can be done in a reliable, an economical, and socially acceptable way…” 12 http://lasco-www.nrl.navy.mil 13 14 Plasmasphere (Image EUV) 15 https://youtu.be/TaPgSWdcYtY 16 17 LETTER doi:10.1038/nature14476 Small particles dominate Saturn’s Phoebe ring to surprisingly large distances Douglas P.
    [Show full text]
  • The Effect Pressure on Wavelengths of Plasma Oscillations in Argon And
    AN ABSTRACT OF TI-lE TRESIS OF - ¡Idgar for Gold.an the ri,, s. in (Name) (Degree) (Najor) Date Thesis presented ia. i9i__ Title flTO PSDEo LPLASMA OSeILLATIos..iN - Abstract Approved (-(Najor Professor) Experiments on argon and. nitrogen are described. which show that the wave length of electromaetjc radiation emitted from a gas plasma in a magnetic field is a function of the pressure of the gas. This result is Consistent with the theory of plasma oscillation developed by Tonics and Langniir if the assumption is made that plasma electron density is a function of gas pressure. A lower limit of wave length plasma of oscillation is indicated by the experiments in qualitative agreement with the Debye length equation, No difference in the wave length-tressn, relationship between the two gases was observed. The experimental tube, which consisted of a gas filled cylindrical anode with an axial tungsten filament) was motmted. in a magnetic field parallel to the axis of the anode. Gas pressures between i and 50 microns and. magnetic fields between 500 and 1,000 gauss were used. The anode voltage was aplied. in short pulses in order to mini- mize heating of the cathode by ion bombardment and to make possible the use of alternating voltage amplifiers in the receiver. The receiver of electromagnetic radiation consisted. of a crystal- detector dipole, constructed from a 1N26 crystal cartridge, followed by a video amp1ifier The amplifier had a maximum gain of 1.6 X iO7, and. a bandwidth of 2 megacycles Wave lengths were measured by means of an interferometer.
    [Show full text]
  • Double Negative Dispersion Relations from Coated Plasmonic Rods∗
    MULTISCALE MODEL. SIMUL. c 2013 Society for Industrial and Applied Mathematics Vol. 11, No. 1, pp. 192–212 DOUBLE NEGATIVE DISPERSION RELATIONS FROM COATED PLASMONIC RODS∗ YUE CHEN† AND ROBERT LIPTON‡ Abstract. A metamaterial with frequency dependent double negative effective properties is constructed from a subwavelength periodic array of coated rods. Explicit power series are developed for the dispersion relation and associated Bloch wave solutions. The expansion parameter is the ratio of the length scale of the periodic lattice to the wavelength. Direct numerical simulations for finite size period cells show that the leading order term in the power series for the dispersion relation is a good predictor of the dispersive behavior of the metamaterial. Key words. metamaterials, dispersion relations, Bloch waves, simulations AMS subject classifications. 35Q60, 68U20, 78A48, 78M40 DOI. 10.1137/120864702 1. Introduction. Metamaterials are artificial materials designed to have elec- tromagnetic properties not generally found in nature. One contemporary area of research explores novel subwavelength constructions that deliver metamaterials with both a negative bulk dielectric constant and bulk magnetic permeability across cer- tain frequency intervals. These double negative materials are promising materials for the creation of negative index superlenses that overcome the small diffraction limit and have great potential in applications such as biomedical imaging, optical lithography, and data storage. The early work of Veselago [39] identified novel ef- fects associated with hypothetical materials for which both the dielectric constant and magnetic permeability are simultaneously negative. Such double negative media support electromagnetic wave propagation in which the phase velocity is antiparallel to the direction of energy flow and other unusual electromagnetic effects, such as the reversal of the Doppler effect and Cerenkov radiation.
    [Show full text]
  • Introduction to Metamaterials
    FEATURE ARTICLE INTRODUCTION TO METAMATERIALS BY MAREK S. WARTAK, KOSMAS L. TSAKMAKIDIS AND ORTWIN HESS etamaterials (MMs) are artificial structures LEFT-HANDED MATERIALS designed to have properties not available in To describe the basic properties of metamaterials, let us nature [1]. They resemble natural crystals as recall Maxwell’s equations M they are build from periodically arranged (e.g., square) unit cells, each with a side length of a. The ∂H ∂E ∇×E =−μμand ∇×H = εε (1) unit cells are not made of physical atoms or molecules but, 00rr∂tt∂ instead, contain small metallic resonators which interact μ ε with an external electromagnetic wave that has a wave- where the r and r are relative permeability and permit- λ ⎡ ∂ ∂ ∂ ⎤ length . The manner in which the incident light wave tivity, respectively, and L =⎢ ,,⎥ . From the above ⎣⎢∂xy∂ ∂z ⎦⎥ interacts with these metallic “meta-atoms” of a metamate- equations, one obtains the wave equation rial determines the medium’s electromagnetic proper- ties – which may, hence, be made to enter highly unusual ∂2E ∇=−2E εμεμ (2) regimes, such as one where the electric permittivity and 00rr∂t 2 the magnetic permeability become simultaneously (in the ε μ same frequency region) negative. If losses are ignored and r and r are considered as real numbers, then one can observe that the wave equation is ε The response of a metamaterial to an incident electromag- unchanged when we simultaneously change signs of r and μ netic wave can be classified by ascribing to it an effective r . (averaged over the volume of a unit cell) permittivity ε ε ε μ μ μ To understand why such materials are also called left- eff = 0 r and effective permeability eff = 0 r .
    [Show full text]
  • Dispersion Relation & Index Ellipsoids
    8/27/2020 Advanced Electromagnetics: 21st Century Electromagnetics Dispersion Relation & Index Ellipsoids Lecture Outline • Dispersion relation • Dispersion surfaces • Index ellipsoids Slide 2 1 8/27/2020 Dispersion Relation Slide 3 The Wave Vector The wave vector (wave momentum) is a vector quantity that conveys two pieces of information: 1. Wavelength and Refractive Index –The magnitude of the wave vector conveys the spatial period (i.e. wavelength) of the wave inside the material. When the frequency is known, the magnitude conveys the material’s refractive index n (more to be said later). 22 n k 0 free space wavelength 0 2. Direction –The direction of the wave is perpendicular to the wave fronts (more to be said later). ˆ kkakbkcabcˆˆ Slide 4 2 8/27/2020 The Dispersion Relation The dispersion relation for a material relates the wave vector to frequency . Essentially, it sets a rule for the values of as a function of direction and frequency. For an ordinary linear, homogeneous and isotropic (LHI) material, the dispersion relation is: 2 222n kkkabc c0 222 kkkabc 2 This can also be written as: 2 kk000 nc0 Slide 5 How to Derive the Dispersion Relation (1 of 2) The wave equation in a linear homogeneous anisotropic material is: 2 Assume no magnetic response i. e. 1 . Ek 00 r E 0 The solution to this equation is still a plane wave, but the allowed values for (modes) are more complicated. jk r ˆ E Ee00 E Eaabcˆˆ Eb Ec Substituting this solution into the wave equation leads to the following relation: 2 kk E000r0 kE k E 0 This equation has the form: abcˆˆ ˆ 0 Each (•••) term has the form: EEEabc 0 Each vector component must be set to zero independently.
    [Show full text]
  • 7 Plasmonics
    7 Plasmonics Highlights of this chapter: In this chapter we introduce the concept of surface plasmon polaritons (SPP). We discuss various types of SPP and explain excitation methods. Finally, di®erent recent research topics and applications related to SPP are introduced. 7.1 Introduction Long before scientists have started to investigate the optical properties of metal nanostructures, they have been used by artists to generate brilliant colors in glass artefacts and artwork, where the inclusion of gold nanoparticles of di®erent size into the glass creates a multitude of colors. Famous examples are the Lycurgus cup (Roman empire, 4th century AD), which has a green color when observing in reflecting light, while it shines in red in transmitting light conditions, and church window glasses. Figure 172: Left: Lycurgus cup, right: color windows made by Marc Chagall, St. Stephans Church in Mainz Today, the electromagnetic properties of metal{dielectric interfaces undergo a steadily increasing interest in science, dating back in the works of Gustav Mie (1908) and Rufus Ritchie (1957) on small metal particles and flat surfaces. This is further moti- vated by the development of improved nano-fabrication techniques, such as electron beam lithographie or ion beam milling, and by modern characterization techniques, such as near ¯eld microscopy. Todays applications of surface plasmonics include the utilization of metal nanostructures used as nano-antennas for optical probes in biology and chemistry, the implementation of sub-wavelength waveguides, or the development of e±cient solar cells. 208 7.2 Electro-magnetics in metals and on metal surfaces 7.2.1 Basics The interaction of metals with electro-magnetic ¯elds can be completely described within the frame of classical Maxwell equations: r ¢ D = ½ (316) r ¢ B = 0 (317) r £ E = ¡@B=@t (318) r £ H = J + @D=@t; (319) which connects the macroscopic ¯elds (dielectric displacement D, electric ¯eld E, magnetic ¯eld H and magnetic induction B) with an external charge density ½ and current density J.
    [Show full text]
  • Problems for the Course F5170 – Introduction to Plasma Physics
    Problems for the Course F5170 { Introduction to Plasma Physics Jiˇr´ı Sperka,ˇ Jan Vor´aˇc,Lenka Zaj´ıˇckov´a Department of Physical Electronics Faculty of Science Masaryk University 2014 Contents 1 Introduction5 1.1 Theory...............................5 1.2 Problems.............................6 1.2.1 Derivation of the plasma frequency...........6 1.2.2 Plasma frequency and Debye length..........7 1.2.3 Debye-H¨uckel potential.................8 2 Motion of particles in electromagnetic fields9 2.1 Theory...............................9 2.2 Problems............................. 10 2.2.1 Magnetic mirror..................... 10 2.2.2 Magnetic mirror of a different construction...... 10 2.2.3 Electron in vacuum { three parts............ 11 2.2.4 E × B drift........................ 11 2.2.5 Relativistic cyclotron frequency............. 12 2.2.6 Relativistic particle in an uniform magnetic field... 12 2.2.7 Law of conservation of electric charge......... 12 2.2.8 Magnetostatic field.................... 12 2.2.9 Cyclotron frequency of electron............. 12 2.2.10 Cyclotron frequency of ionized hydrogen atom.... 13 2.2.11 Magnetic moment.................... 13 2.2.12 Magnetic moment 2................... 13 2.2.13 Lorentz force....................... 13 3 Elements of plasma kinetic theory 14 3.1 Theory............................... 14 3.2 Problems............................. 15 3.2.1 Uniform distribution function.............. 15 3.2.2 Linear distribution function............... 15 3.2.3 Quadratic distribution function............. 15 3.2.4 Sinusoidal distribution function............. 15 3.2.5 Boltzmann kinetic equation............... 15 1 CONTENTS 2 4 Average values and macroscopic variables 16 4.1 Theory............................... 16 4.2 Problems............................. 17 4.2.1 RMS speed........................ 17 4.2.2 Mean speed of sinusoidal distribution........
    [Show full text]
  • What Is Plasma? Its Characteristics Plasma Oscillations Plasmon And
    Topics covered here are: What is Plasma? Its Characteristics Plasma Oscillations Plasmon and plasmon frequency Sources used here are: http://farside.ph.utexas.edu/teaching/plasma/Plasmahtml/node1.html https://www.britannica.com/science/plasma-state-of-matter/Plasma-oscillations- and-parameters What is Plasma The electromagnetic force is generally observed to create structure: e.g., stable atoms and molecules, crystalline solids. Structured systems have binding energies larger than the ambient thermal energy. Placed in a sufficiently hot environment, they decompose: e.g., crystals melt, molecules disassociate. At temperatures near or exceeding atomic ionization energies, atoms similarly decompose into negatively charged electrons and positively charged ions. These charged particles are by no means free: in fact, they are strongly affected by each others' electromagnetic fields. Nevertheless, because the charges are no longer bound, their assemblage becomes capable of collective motions of great vigor and complexity. Such an assemblage is termed a plasma. Of course, bound systems can display extreme complexity of structure: e.g., a protein molecule. Complexity in a plasma is somewhat different, being expressed temporally as much as spatially. It is predominately characterized by the excitation of an enormous variety of collective dynamical modes. Its occurrence and characteristics Since thermal decomposition breaks interatomic bonds before ionizing, most terrestrial plasmas begin as gases. In fact, a plasma is sometimes defined as a gas that is sufficiently ionized to exhibit plasma-like behavior. Note that plasma-like behavior ensues after a remarkably small fraction of the gas has undergone ionization. Thus, fractionally ionized gases exhibit most of the exotic phenomena characteristic of fully ionized gases.
    [Show full text]