Quantum Mechanics Polarization Density
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
P142 Lecture 19
P142 Lecture 19 SUMMARY Vector derivatives: Cartesian Coordinates Vector derivatives: Spherical Coordinates See Appendix A in “Introduction to Electrodynamics” by Griffiths P142 Preview; Subject Matter P142 Lecture 19 • Electric dipole • Dielectric polarization • Electric fields in dielectrics • Electric displacement field, D • Summary Dielectrics: Electric dipole The electric dipole moment p is defined as p = q d where d is the separation distance between the charges q pointing from the negative to positive charge. Lecture 5 showed that Forces on an electric dipole in E field Torque: Forces on an electric dipole in E field Translational force: Dielectric polarization: Microscopic Atomic polarization: d ~ 10-15 m Molecular polarization: d ~ 10-14 m Align polar molecules: d ~ 10-10 m Competition between alignment torque and thermal motion or elastic forces Linear dielectrics: Three mechanisms typically lead to Electrostatic precipitator Linear dielectric: Translational force: Electrostatic precipitator Extract > 99% of ash and dust from gases at power, cement, and ore-processing plants Electric field in dielectrics: Macroscopic Electric field in dielectrics: Macroscopic Polarization density P in dielectrics: Polarization density in dielectrics: Polarization density in dielectrics: The dielectric constant κe • Net field in a dielectric Enet = EExt/κe • Most dielectrics linear up to dielectric strength Volume polarization density Electric Displacement Field Electric Displacement Field Electric Displacement Field Electric Displacement Field Summary; 1 Summary; 2 Refraction of the electric field at boundaries with dielectrics Capacitance with dielectrics Capacitance with dielectrics Electric energy storage • Showed last lecture that the energy stored in a capacitor is given by where for a dielectric • Typically it is more useful to express energy in terms of magnitude of the electric field E. -
(Physics), June 2003, Pp 18-19
ICO topical meeting on Polarization Optics, Joensuu. eds: Asher A. Friesem and Jari Turunen, Published by University of Joensuu (Physics), June 2003, pp 18-19 The singularities of crystal optics Michael Berry and Mark Dennis H H Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, United Kingdom http://www.phy.bris.ac.uk/staff/berry_mv.html This is a summary of our recent paper [1], where we reconstruct the classical theory of crystal optics [2] in a way that emphasises and extends recent ideas of singular optics [3, 4]. We concentrate on the general case where the crystal is chiral, i.e. optically active (gy- rotropic) and absorbing (dichroic) as well as biaxially anisotropic, and describe the proper- ties of the crystal as a function of propagation direction s. The refractive indices and po- larizations of plane waves are eigenvalues and eigenfunctions of the 2x2 part of the recip- rocal dielectric tensor h perpendicular to s. For transparent anisotropic crystals, h is real symmetric; addition of chirality makes h complex hermitian; addition of dichroism makes h complex nonhermitian. These different possibilities greatly influence the refractive indi- ces and eigenpolarizations. Using a new formalism involving projection from the sphere of directions s to the stereo- graphic plane R=X,Y, and associated complex variables Z=X+iY, we obtain convenient ex- plicit formulas for the two refractive indices and polarizations. Particular use is made of a representation of the polarization states as the complex ratios of their vector components. This enables three types of polarization singularity to classified and explored. First are sin- gular axes, which are degeneracies where the two refractive indices are equal. -
Experience with Ray-Tracing Simulations at the European Synchrotron Radiation Facility
Experience with ray-tracing simulations at the European Synchrotron Radiation Facility Manuel Sánchez del Río European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble Cedex 9 (France) (Presented on 18 October 1995) The ESRF is the first operational third-generation synchrotron radiation hard-x-ray source. Since the beginning of its construction (1988), the ray-tracing technique proved to be an essential computer tool for the beamline optics design. The optical systems of most beamlines have been simulated by ray tracing in order to optimize the optics, fully understand their properties, and check if operation performances were as expected. In this paper, a short compilation of the experience with ray tracing and optics simulation codes at the ESRF, as well as some other in-house developments, is presented. © 1996 American Institute of Physics. I. INTRODUCTION position), the energy resolution of the monochromatized beam, and the flux or power transmitted by the beamline. The construction of the ESRF synchrotron radiation source started in 1988 as a joint project of 12 European countries. The facility is designed and optimized to supply high II. MIRROR OPTICS brilliance x-rays from insertion devices. It consists of a 200-MeV electron LINAC, a 6-GeV fast cycling booster Mirrors are used essentially to focus or collimate the x-ray synchrotron and a 6-GeV low emittance storage ring. At beam. They can also be used as x-ray filters to avoid high present, 18 beamlines constructed by the ESRF and 4 built heat load on other devices (monochromators) or as harmonic by the Collaborating Research Groups (CRG) are operational rejecters. -
Modelling the Optics of High Resolution Liquid Crystal Devices by the Finite Differences in the Frequency Domain Method
Modelling the optics of high resolution liquid crystal devices by the finite differences in the frequency domain method Mengyang Yang, Sally E. Day and F. Aníbal Fernández Department of Electronic and Electrical Engineering University College London, London WC1E 7JE, UK [email protected] Abstract—A procedure combining accurate liquid crystal and electromagnetic modelling is developed for the analysis of wave II. LIQUID CRYSTAL AND ELECTROMAGNETIC WAVE propagation through liquid crystal devices. This is required to PROPAGATION MODELLING study the optics of high resolution liquid crystal cells or cells containing very small features, where diffraction effects occur. It A. Liquid crystal modelling is also necessary for the study of optical waveguiding devices The liquid crystal modelling used in this work consists of a using liquid crystal as variable permittivity substrates. An variational approach to the Landau-de Gennes theory accurate finite element modelling program is used to find the implemented with the formalism of Qian and Sheng [1], [5]. permittivity tensor distribution, which is then used to find the response of the device to an excitation electromagnetic field by This is a continuum theory in which the liquid crystal means of a finite difference in the frequency domain (FDFD) orientation is defined by the order tensor Q that not only approach. describes the local orientation but also takes into account order variations. This is adequate for the accurate modelling of liquid Keywords— Liquid crystal modelling; Finite differences crystal structures containing defects or other regions of order method; Wave propagation in anisotropic inhomogeneous media; parameter variation. The permittivity distribution is then found Liquid crystal optics as εε=+Δ⊥ Inn ε where nˆ is the liquid crystal director, ε ⊥ is the relative permittivity calculated in the direction I. -
Modeling Optical Metamaterials with Strong Spatial Dispersion
Fakultät für Physik Institut für theoretische Festkörperphysik Modeling Optical Metamaterials with Strong Spatial Dispersion M. Sc. Karim Mnasri von der KIT-Fakultät für Physik des Karlsruher Instituts für Technologie (KIT) genehmigte Dissertation zur Erlangung des akademischen Grades eines DOKTORS DER NATURWISSENSCHAFTEN (Dr. rer. nat.) Tag der mündlichen Prüfung: 29. November 2019 Referent: Prof. Dr. Carsten Rockstuhl (Institut für theoretische Festkörperphysik) Korreferent: Prof. Dr. Michael Plum (Institut für Analysis) KIT – Die Forschungsuniversität in der Helmholtz-Gemeinschaft Erklärung zur Selbstständigkeit Ich versichere, dass ich diese Arbeit selbstständig verfasst habe und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt habe, die wörtlich oder inhaltlich über- nommenen Stellen als solche kenntlich gemacht und die Satzung des KIT zur Sicherung guter wissenschaftlicher Praxis in der gültigen Fassung vom 24. Mai 2018 beachtet habe. Karlsruhe, den 21. Oktober 2019, Karim Mnasri Als Prüfungsexemplar genehmigt von Karlsruhe, den 28. Oktober 2019, Prof. Dr. Carsten Rockstuhl iv To Ouiem and Adam Thesis abstract Optical metamaterials are artificial media made from subwavelength inclusions with un- conventional properties at optical frequencies. While a response to the magnetic field of light in natural material is absent, metamaterials prompt to lift this limitation and to exhibit a response to both electric and magnetic fields at optical frequencies. Due tothe interplay of both the actual shape of the inclusions and the material from which they are made, but also from the specific details of their arrangement, the response canbe driven to one or multiple resonances within a desired frequency band. With such a high number of degrees of freedom, tedious trial-and-error simulations and costly experimen- tal essays are inefficient when considering optical metamaterials in the design of specific applications. -
Physics 112: Classical Electromagnetism, Fall 2013 Birefringence Notes
Physics 112: Classical Electromagnetism, Fall 2013 Birefringence Notes 1 A tensor susceptibility? The electrons bound within, and binding, the atoms of a dielectric crystal are not uniformly dis- tributed, but are restricted in their motion by the potentials which confine them. In response to an applied electric field, they may therefore move a greater or lesser distance, depending upon the strength of their confinement in the field direction. As a result, the induced polarization varies not only with the strength of the applied field, but also with its direction. The susceptibility{ and properties which depend upon it, such as the refractive index{ are therefore anisotropic, and cannot be characterized by a single value. The scalar electric susceptibility, χe, is defined to be the coefficient which relates the value of the ~ ~ local electric field, Eloc, to the local value of the polarization, P : ~ ~ P = χe0Elocal: (1) As we discussed in the last seminar we can `promote' this relation to a tensor relation with χe ! (χe)ij. In this case the dielectric constant is also a tensor and takes the form ij = [δij + (χe)ij] 0: (2) Figure 1: A cartoon showing how the electron is held by anisotropic springs{ causing an electric susceptibility which is different when the electric field is pointing in different directions. How does this happen in practice? In Fig. 1 we can see that in a crystal, for instance, the electrons will in general be held in bonds which are not spherically symmetric{ i.e., they are anisotropic. 1 Therefore, it will be easier to polarize the material in certain directions than it is in others. -
Circular Birefringence in Crystal Optics
Circular birefringence in crystal optics a) R J Potton Joule Physics Laboratory, School of Computing, Science and Engineering, Materials and Physics Research Centre, University of Salford, Greater Manchester M5 4WT, UK. Abstract In crystal optics the special status of the rest frame of the crystal means that space- time symmetry is less restrictive of electrodynamic phenomena than it is of static electromagnetic effects. A relativistic justification for this claim is provided and its consequences for the analysis of optical activity are explored. The discrete space-time symmetries P and T that lead to classification of static property tensors of crystals as polar or axial, time-invariant (-i) or time-change (-c) are shown to be connected by orientation considerations. The connection finds expression in the dynamic phenomenon of gyrotropy in certain, symmetry determined, crystal classes. In particular, the degeneracies of forward and backward waves in optically active crystals arise from the covariance of the wave equation under space-time reversal. a) Electronic mail: [email protected] 1 1. Introduction To account for optical activity in terms of the dielectric response in crystal optics is more difficult than might reasonably be expected [1]. Consequently, recourse is typically had to a phenomenological account. In the simplest cases the normal modes are assumed to be circularly polarized so that forward and backward waves of the same handedness are degenerate. If this is so, then the circular birefringence can be expanded in even powers of the direction cosines of the wave normal [2]. The leading terms in the expansion suggest that optical activity is an allowed effect in the crystal classes having second rank property tensors with non-vanishing symmetrical, axial parts. -
A Beautiful Approach: Transformational Optics
A beautiful approach: “Transformational optics” [ several precursors, but generalized & popularized by Ward & Pendry (1996) ] warp a ray of light …by warping space(?) [ figure: J. Pendry ] Euclidean x coordinates transformed x'(x) coordinates amazing Solutions of ordinary Euclidean Maxwell equations in x' fact: = transformed solutions from x if x' uses transformed materials ε' and μ' Maxwell’s Equations constants: ε0, μ0 = vacuum permittivity/permeability = 1 –1/2 c = vacuum speed of light = (ε0 μ0 ) = 1 ! " B = 0 Gauss: constitutive ! " D = # relations: James Clerk Maxwell #D E = D – P 1864 Ampere: ! " H = + J #t H = B – M $B Faraday: ! " E = # $t electromagnetic fields: sources: J = current density E = electric field ρ = charge density D = displacement field H = magnetic field / induction material response to fields: B = magnetic field / flux density P = polarization density M = magnetization density Constitutive relations for macroscopic linear materials P = χe E ⇒ D = (1+χe) E = ε E M = χm H B = (1+χm) H = μ H where ε = 1+χe = electric permittivity or dielectric constant µ = 1+χm = magnetic permeability εµ = (refractive index)2 Transformation-mimicking materials [ Ward & Pendry (1996) ] E(x), H(x) J–TE(x(x')), J–TH(x(x')) [ figure: J. Pendry ] Euclidean x coordinates transformed x'(x) coordinates J"JT JµJT ε(x), μ(x) " ! = , µ ! = (linear materials) det J det J J = Jacobian (Jij = ∂xi’/∂xj) (isotropic, nonmagnetic [μ=1], homogeneous materials ⇒ anisotropic, magnetic, inhomogeneous materials) an elementary derivation [ Kottke (2008) ] consider× -
4 Lightdielectrics.Pdf
4. The interaction of light with matter The propagation of light through chemical materials is described by a wave equation similar to the one that describes light travel in a vacuum (free space). Again, using E as the electric field of light, v as the speed of light in a material and z as its direction of propagation. !2 1 ! 2E ! 2E 1 ! 2E # n2 & ! 2E # & . 2 E= 2 2 " 2 = % 2 ( 2 = 2 2 !z c !t !z $ v ' !t $% c '( !t (Read the variation in the electric field with respect distance traveled is proportional to its variation with respect to time.) The refractive index, n, (also represented η) describes how matter affects light propagation: through the electric permittivity, ε, and the magnetic permeability, µ. ! µ n = !0 µ0 These properties describe how well a medium supports (permits the transmission of) electric and magnetic fields, respectively. The terms ε0 and µ0 are reference values: the permittivity and permeability of free space. Consequently, the refractive index for a vacuum is unity. In chemical materials ε is always larger than ε0, reflecting the interaction of the electric field of the incident beam with the electrons of the material. During this interaction, the energy from the electric field is transiently stored in the medium as the electrons in the material are temporarily aligned with the field. This phenomenon is referred to as polarization, P, in the sense that the charges of the medium are temporarily separated. (This must not be confused with the polarization, which refers to the orientation or behavior of the electric field.) This stored energy is re-radiated, but the beam travel is slowed by interaction with the material. -
Crystal Optics Homogeneous, Anisotropic Media
Lecture 3: Crystal Optics Homogeneous, Anisotropic Media Introduction Outline material equations for homogeneous anisotropic media ~ ~ 1 Homogeneous, Anisotropic Media D = E B~ = µH~ 2 Crystals tensors of rank 2, written as 3 by 3 matrices 3 Plane Waves in Anisotropic Media : dielectric tensor 4 Wave Propagation in Uniaxial Media µ: magnetic permeability tensor examples: 5 Reflection and Transmission at Interfaces crystals, liquid crystals external electric, magnetic fields acting on isotropic materials (glass, fluids, gas) anisotropic mechanical forces acting on isotropic materials Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 1 Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 2 Properties of Dielectric Tensor Uniaxial Materials Maxwell equations imply symmetric dielectric tensor isotropic materials: n = n = n 0 1 x y z 11 12 13 for any coordinate system T = = @ 12 22 23 A anisotropic materials: 13 23 33 nx 6= ny 6= nz symmetric tensor of rank 2 ) coordinate system exists where uniaxial materials: nx = ny 6= nz tensor is diagonal ordinary index of refraction: orthogonal axes of this coordinate system: principal axes no = nx = ny elements of diagonal tensor: principal dielectric constants extraordinary index of refraction: n = n 3 principal indices of refraction in coordinate system spanned by e z principal axes rotation of coordinate system 0 2 1 around z does not change nx 0 0 ~ 2 ~ anything D = @ 0 ny 0 A E 2 0 0 nz most materials used in polarimetry are (almost) uniaxial x, y, z because principal axes form Cartesian coordinate system Christoph U. -
Bent Crystal X-Ray Optics for the Diagnosis and Applications of Laser-Produced Plasmas
Bent crystal X-ray optics for the diagnosis and applications of laser-produced plasmas D issertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) seit 1558 vorgelegt dem Rat der Physikalisch-Astronomischen Fakultät der Friedrich- Schiller-U niversit ät von Dipl.-Phys. Robert Lötzsch geboren am 18.08.1980 in Jena Gutachter: 1. Prof. Dr. Eckhart Förster Institut für Optik und Quantenelektronik Friedrich-Schiller-Universität Jena 2. Prof. Dr. Helmut Zacharias Physikalisches Institut Westfälische Wilhelms-Universität Münster 3. Prof. Dr. Georg Pretzier Institut für Laser- und Plasmaphysik Heinrich-Heine-Universität Düsseldorf Tag der Disputation: 8.11.2012 Contents Motivation 3 1. Introduction 5 1.1. X-ray optics with toroidally bent crystals...................................................... 5 1.2. Interaction of short pulse lasers with so lid s................................................... 12 1.2.1. Target Normal Sheath Acceleration................................................... 14 1.2.2. Generation of ultrashort X-ray bursts................................................. 16 2. Laterally varying reflection properties of bent crystals 19 2.1. Measurement of laterally varying Bragg angles............................................. 21 2.1.1. Measurement sch em e............................................................................ 21 2.1.2. Analysis of the accuracy of the measurement system....................... 24 2.1.3. Results..................................................................................................... -
EM Dis Ch 5 Part 2.Pdf
1 LOGO Chapter 5 Electric Field in Material Space Part 2 iugaza2010.blogspot.com Polarization(P) in Dielectrics The application of E to the dielectric material causes the flux density to be grater than it would be in free space. D oE P P is proportional to the applied electric field E P e oE Where e is the electric susceptibility of the material - Measure of how susceptible (or sensitive) a given dielectric is to electric field. 3 D oE P oE e oE oE(1 e) oE( r) D o rE E o r r1 e o permitivity of free space permitivity of dielectric relative permitivity r 4 Dielectric constant or(relative permittivity) εr Is the ratio of the permittivity of the dielectric to that of free space. o r permitivity of dielectric relative permitivity r o permitivity of free space 5 Dielectric Strength Is the maximum electric field that a dielectric can withstand without breakdown. o r Material Dielectric Strength εr E(V/m) Water(sea) 80 7.5M Paper 7 12M Wood 2.5-8 25M Oil 2.1 12M Air 1 3M 6 A parallel plate capacitor with plate separation of 2mm has 1kV voltage applied to its plate. If the space between the plate is filled with polystyrene(εr=2.55) Find E,P V 1000 E 500 kV / m d 2103 2 P e oE o( r1)E o(1 2.55)(500k) 6.86 C / m 7 In a dielectric material Ex=5 V/m 1 2 and P (3a x a y 4a z )nc/ m 10 Find (a) electric susceptibility e (b) E (c) D 1 (3) 10 (a)P e oE e 2.517 o(5) P 1 (3a x a y 4a z ) (b)E 5a x 1.67a y 6.67a z e o 10 (2.517) o (c)D E o rE o rE o( e1)E 2 139.78a x 46.6a y 186.3a z pC/m 8 In a slab of dielectric