DOCSLIB.ORG

Electron-Spin Dynamics in Elliptically Polarized Light Waves

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Electron-Spin Dynamics in Elliptically Polarized Light Waves Electron-spin dynamics in elliptically polarized light waves 1, 1, 2 1, 2 Heiko Bauke, ∗ Sven Ahrens, and Rainer Grobe 1Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany 2Intense Laser Physics Theory Unit and Department of Physics, Illinois State University, Normal, Illinois 61790-4560 USA (Dated: November 4, 2014) We investigate the coupling of the spin angular momentum of light beams with elliptical polarization to the spin degree of freedom of free electrons. It is shown that this coupling, which is of similar origin as the well-known spin-orbit coupling, can lead to spin precession. The spin-precession frequency is proportional to the product of the laser-field’s intensity and its spin density. The electron-spin dynamics is analyzed by employing exact numerical methods as well as time-dependent perturbation theory based on the fully relativistic Dirac equation and on the nonrelativistic Pauli equation that is amended by a relativistic correction that accounts for the light’s spin density. PACS numbers: 03.65.Pm, 31.15.aj, 31.30.J– 1. Introduction quantum mechanical equations of motion, relativistic and non- relativistic. Subsequently, we solve these equations via time- dependent perturbation theory in Sec. 3 and by numerical meth- Novel light sources such as the ELI-Ultra High Field Facility, ods in Sec. 4. In Sec. 5 we estimate under which experimental for example, envisage to provide field intensities in excess of 20 2 conditions the predicted spin precession may be realized before 10 W/cm and field frequencies in the x-ray domain [1–6]. we summarize our results in Sec. 6. Such facilities open the possibility of probing light-matter inter- action in the ultra-relativistic regime [7–9]. In this regime, the quantum dynamical interaction of electrons, ions, or atoms can be modified due to relativistic corrections of the Dirac equation 2. Electrons in standing laser fields to the nonrelativistic Schrödinger equation. In certain setups, however, relativistic effects may change the dynamics not only quantitatively, but also qualitatively. As an example, spin-orbit 2.1. Laser fields with elliptical polarization interaction, that is, the interaction of a particle’s spin with its orbital angular momentum, can lead to a splitting of atomic We examine electrons in standing light waves formed by two energy levels, which are degenerated within the framework of counterpropagating laser beams with circular or elliptic polar- nonrelativistic quantum theory. At ultra-high laser intensities ization. The electric and magnetic field components of two also, spin effects can set in [10–12]. elliptically polarized laser fields counterpropagating in the di- In our recent publication [13], we demonstrated that relativis- rection of the x axis are given by tic effects may also modify the spin dynamics of an electron ! ! 2π(x ct) 2π(x ct) in external electromagnetic waves with elliptical polarization. E1;2(r; t) = Eˆ cos ∓ ey + cos ∓ η ez ; This effect can be attributed to a coupling of the spin density λ λ ± that is associated with the external laser field to the electron’s (1a) ! ! spin. The coupling between the electronic spin and the pho- Eˆ 2π(x ct) 2π(x ct) B (r; t) = cos ∓ η e cos ∓ ez : tonic spin is of similar origin as the well-known spin-orbit 1;2 c ∓ λ ± y ± λ effect. Both effects have in common that two angular momen- (1b) arXiv:1409.2647v2 [quant-ph] 3 Nov 2014 tum degrees of freedom are coupled that are of different origin. Other examples for the interaction between different kinds of Here, we have introduced the speed of light c, the position T angular momentum have been studied in very different contexts, vector r = (x; y; z) , the time t, and ex, ey, and ez denoting e. g., the angular momentum of phonons in a magnetic crystal unit vectors in the direction of the coordinate axes. The two and the angular momentum of the crystal [14] or the orbital electromagnetic waves (1) feature the same wavelength λ, the angular momentum of light and the internal degrees of motion same electric field amplitude Eˆ, and the same intensity in molecules [15], to mention two recent examples. 2 In this contribution, we elaborate and extend the theoreti- I = "0cEˆ (2) cal description of the electron’s spin dynamics as studied in but have opposite helicity. The parameter η ( π; π] deter- Ref. [13]. The article is organized as follows. In Sec. 2 we 2 − describe the considered laser setup and derive the electron’s mines the degree of ellipticity with η = 0 and η = π correspond- ing to linear polarization and η = π/2 to circular polarization. Introducing the wave number k =±2π/λ and the lasers’ angular frequency ! = kc, the total electric and magnetic fields of the ∗ [email protected] standing laser wave that is formed by superimposing the two 2 Note, that for plane wave fields the photonic spin density may be expressed solely by the electric field and the magnetic z y B vector potential, see (5), and many works utilize the expression " E A to define the photonic spin density, which is not equiv- x 0 × λ alent to "0(E A + cB C)/2 for none-plane-wave fields, e. g., standing waves.× The total× amount of spin angular momentum, E however, does not depend on which of both densities is chosen [22], this means Z Z " FIG. 1: (Color online) Schematic illustration of the electric (solid red " E A d3r = 0 (E A + cB C) d3r ; (7) line) and magnetic (dotted blue line) field components of the standing 0 × 2 × × laser wave (3) with η = π/2 that is a superposition of two counterprop- agating circularly polarized laser waves of opposite helicity and equal where the integration goes over all space regions with non- wavelength λ. The electric and magnetic components E and B are vanishing electromagnetic fields. parallel to each other and rotate around the propagation direction. Furthermore, note that the given expressions for the photonic spin density require that the electromagnetic vector potentials A and C are given in Coulomb gauge. If another gauge is electric and magnetic field components (1a) and (1b) follow as applied then A and C in the definition of the spin density have to be replaced by A? and C?, which are defined by splitting the vector potentials A and C such that A = A? + Ak and r A? = 0 E(r; t) = 2Eˆ cos kx (cos !t ey + cos(!t η) ez) ; (3a) · − and r Ak = 0 and similarly for the electric vector potential C 2Eˆ [23]. Thus× the Coulomb gauge is the most convenient gauge to B(r; t) = sin kx ( sin(!t η) e + sin !t ez) : (3b) c − − y study effects of the photonic spin and it is therefore employed here. All predicted effects of this contribution result from For circular fields (η = π/2), the electric and the magnetic gauge invariant quantum mechanical wave equations and are, components (3) are parallel to each other and rotate around the therefore, independent of the choice of the gauge. propagation direction as sketched in Fig. 1. The elliptically polarized fields (1) carry spin angular mo- mentum [16, 17]. The partition of a general light beam’s total angular momentum into orbital angular momentum and spin 2.2. Quantum dynamics in momentum angular momentum was debated in the literature. A definition space that seems to be well established now [18–21] utilizes Coulomb gauge vector potentials for both the magnetic as well as the Introducing the window function, which allows for a smooth electric field component. The Coulomb gauge vector potentials turn-on and turn-off of the laser field, A1;2 and C1;2 of the elliptically polarized fields (1) are 8 >sin2 πt if 0 t ∆T, > 2∆T ≤ ≤ Eˆ w(t) = <1 if ∆T t T ∆T, (8) C1;2(r; t) = ( sin(kx !t η) ey sin(kx !t) ez) ; > > 2 π(T t) ≤ ≤ − −! ± ∓ ± − ∓ :sin − if T ∆T t T, (4a) 2∆T − ≤ ≤ Eˆ the magnetic vector potential of the combined laser fields (3) A1;2(r; t) = ( sin(kx !t) ey sin(kx !t η) ez) : is given by −! ∓ ∓ ∓ ∓ ± (4b) 2w(t)Eˆ A(r; t) = cos kx (sin !t e + sin(!t η) ez) : (9) These potentials are related to the electromagnetic fields via − ! y − E = A˙ = cr C , B = r A = C˙ /c and 1;2 1;2 1;2 1;2 1;2 1;2 The parameters T and ∆T denote the total interaction time and the potentials− comply− × with the Coulomb× gauge as−r A = 1;2 the turn-on and turn-off intervals. The quantum mechanical r C = 0. With these quantities the spin density of each· laser 1;2 evolution of an electron of mass m and charge q = e in the field· is given by the expression " (E A + cB C )/2, 0 1;2 1;2 1;2 1;2 laser field with the vector potential (9) is governed by− the Dirac which reduces in the case of the plane× wave fields (1)× to equation ˆ2 "0 "0E λ sin η 2 (E A +cB C ) = " E A = ex : i~Ψ(˙ r; t) = cα i~r qA(r; t) + mc β Ψ(r; t) (10) 2 1;2 × 1;2 1;2 × 1;2 0 1;2 × 1;2 2πc · − − (5) T The total photonic spin density of the setup is with the Dirac matrices α = (αx; αy; αz) and β [24, 25]. The quasi one-dimensional sinusoidal structure of the vector poten- 2 "0 "0Eˆ λ sin η tial (9) allows us to cast the partial differential equation (10) (E A + cB C) = ex ; (6) 2 × × πc into a set of coupled ordinary differential equations [12].
Load more

Recommended publications
  • Lab 8: Polarization of Light
    Lab 8: Polarization of Light 1 Introduction Refer to Appendix D for photos of the appara- tus Polarization is a fundamental property of light and a very important concept of physical optics. Not all sources of light are polarized; for instance, light from an ordinary light bulb is not polarized. In addition to unpolarized light, there is partially polarized light and totally polarized light. Light from a rainbow, reflected sunlight, and coherent laser light are examples of po- larized light. There are three di®erent types of po- larization states: linear, circular and elliptical. Each of these commonly encountered states is characterized Figure 1: (a)Oscillation of E vector, (b)An electromagnetic by a di®ering motion of the electric ¯eld vector with ¯eld. respect to the direction of propagation of the light wave. It is useful to be able to di®erentiate between 2 Background the di®erent types of polarization. Some common de- vices for measuring polarization are linear polarizers and retarders. Polaroid sunglasses are examples of po- Light is a transverse electromagnetic wave. Its prop- larizers. They block certain radiations such as glare agation can therefore be explained by recalling the from reflected sunlight. Polarizers are useful in ob- properties of transverse waves. Picture a transverse taining and analyzing linear polarization. Retarders wave as traced by a point that oscillates sinusoidally (also called wave plates) can alter the type of polar- in a plane, such that the direction of oscillation is ization and/or rotate its direction. They are used in perpendicular to the direction of propagation of the controlling and analyzing polarization states.
    [Show full text]
  • Observation of Elliptically Polarized Light from Total Internal Reflection in Bubbles
    Observation of elliptically polarized light from total internal reflection in bubbles Item Type Article Authors Miller, Sawyer; Ding, Yitian; Jiang, Linan; Tu, Xingzhou; Pau, Stanley Citation Miller, S., Ding, Y., Jiang, L. et al. Observation of elliptically polarized light from total internal reflection in bubbles. Sci Rep 10, 8725 (2020). https://doi.org/10.1038/s41598-020-65410-5 DOI 10.1038/s41598-020-65410-5 Publisher NATURE PUBLISHING GROUP Journal SCIENTIFIC REPORTS Rights Copyright © The Author(s) 2020. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License. Download date 29/09/2021 02:08:57 Item License https://creativecommons.org/licenses/by/4.0/ Version Final published version Link to Item http://hdl.handle.net/10150/641865 www.nature.com/scientificreports OPEN Observation of elliptically polarized light from total internal refection in bubbles Sawyer Miller1,2 ✉ , Yitian Ding1,2, Linan Jiang1, Xingzhou Tu1 & Stanley Pau1 ✉ Bubbles are ubiquitous in the natural environment, where diferent substances and phases of the same substance forms globules due to diferences in pressure and surface tension. Total internal refection occurs at the interface of a bubble, where light travels from the higher refractive index material outside a bubble to the lower index material inside a bubble at appropriate angles of incidence, which can lead to a phase shift in the refected light. Linearly polarized skylight can be converted to elliptically polarized light with efciency up to 53% by single scattering from the water-air interface. Total internal refection from air bubble in water is one of the few sources of elliptical polarization in the natural world.
    [Show full text]
  • Ellipsometry
    AALBORG UNIVERSITY Institute of Physics and Nanotechnology Pontoppidanstræde 103 - 9220 Aalborg Øst - Telephone 96 35 92 15 TITLE: Ellipsometry SYNOPSIS: This project concerns measurement of the re- fractive index of various materials and mea- PROJECT PERIOD: surement of the thickness of thin films on sili- September 1st - December 21st 2004 con substrates by use of ellipsometry. The el- lipsometer used in the experiments is the SE 850 photometric rotating analyzer ellipsome- ter from Sentech. THEME: After an introduction to ellipsometry and a Detection of Nanostructures problem description, the subjects of polar- ization and essential ellipsometry theory are covered. PROJECT GROUP: The index of refraction for silicon, alu- 116 minum, copper and silver are modelled us- ing the Drude-Lorentz harmonic oscillator model and afterwards measured by ellipsom- etry. The results based on the measurements GROUP MEMBERS: show a tendency towards, but are not ade- Jesper Jung quately close to, the table values. The mate- Jakob Bork rials are therefore modelled with a thin layer of oxide, and the refractive indexes are com- Tobias Holmgaard puted. This model yields good results for the Niels Anker Kortbek refractive index of silicon and copper. For aluminum the result is improved whereas the result for silver is not. SUPERVISOR: The thickness of a thin film of SiO2 on a sub- strate of silicon is measured by use of ellip- Kjeld Pedersen sometry. The result is 22.9 nm which deviates from the provided information by 6.5 %. The thickness of two thick (multiple wave- NUMBERS PRINTED: 7 lengths) thin polymer films are measured. The polymer films have been spin coated on REPORT PAGE NUMBER: 70 substrates of silicon and the uniformities of the surfaces are investigated.
    [Show full text]
  • Understanding Polarization
    Semrock Technical Note Series: Understanding Polarization The Standard in Optical Filters for Biotech & Analytical Instrumentation Understanding Polarization 1. Introduction Polarization is a fundamental property of light. While many optical applications are based on systems that are “blind” to polarization, a very large number are not. Some applications rely directly on polarization as a key measurement variable, such as those based on how much an object depolarizes or rotates a polarized probe beam. For other applications, variations due to polarization are a source of noise, and thus throughout the system light must maintain a fixed state of polarization – or remain completely depolarized – to eliminate these variations. And for applications based on interference of non-parallel light beams, polarization greatly impacts contrast. As a result, for a large number of applications control of polarization is just as critical as control of ray propagation, diffraction, or the spectrum of the light. Yet despite its importance, polarization is often considered a more esoteric property of light that is not so well understood. In this article our aim is to answer some basic questions about the polarization of light, including: what polarization is and how it is described, how it is controlled by optical components, and when it matters in optical systems. 2. A description of the polarization of light To understand the polarization of light, we must first recognize that light can be described as a classical wave. The most basic parameters that describe any wave are the amplitude and the wavelength. For example, the amplitude of a wave represents the longitudinal displacement of air molecules for a sound wave traveling through the air, or the transverse displacement of a string or water molecules for a wave on a guitar string or on the surface of a pond, respectively.
    [Show full text]
  • Elliptical Polarization
    Lecture 5: Polarization and Related Antenna Parameters (Polarization of EM fields – revision. Polarization vector. Antenna polarization. Polarization loss factor and polarization efficiency.) 1. Polarization of EM fields. The polarization of the EM field describes the time variations of the field vectors at a given point. In other words, it describes the way the direction and magnitude the field vectors (usually E ) change in time. The polarization is the figure traced by the extremity of the time- varying field vector at a given point. According to the shape of the trace, three types of polarization exist for harmonic fields: linear, circular and elliptical: y y y E E E ω z x z x z x (a) linear polarization (b) circular polarization (c) elliptical polarization Any type of polarization can be represented by two orthogonal linear polarizations, ( EExy, )or(EEHV, ), whose fields are out of phase by an angle δ of L . • δ = π IfL 0 or n , then a linear polarization results. • δπ= = If L /2 (90) and EExy, then a circular polarization results. • In the most general case, elliptical polarization is defined. 1 It is also true that any type of polarization can be represented by a right-hand circular and a left-hand circular polarizations (EELR , ). We shall revise the above statements and definitions, while introducing the new concept of polarization vector. 2. Field polarization in terms of two orthogonal linearly polarized components. The polarization of any field can be represented by a suitable set of two orthogonal linearly polarized fields. Assume that locally a far field propagates along the z-axis, and the field vectors have only transverse components.
    [Show full text]
  • Polarized Light, It Is Important That the Azimuthal Angle of a Polarization Optic (I.E
    An Economical Means for Accurate Azimuthal Alignment of Polarization Optics © 2005 Meadowlark Optics, Inc. Joel R. Blum* Meadowlark Optics, 5964 Iris Parkway, PO Box 1000, Frederick, CO, USA 80530 ABSTRACT For many applications involving polarized light, it is important that the azimuthal angle of a polarization optic (i.e. polarizer, retarder, etc…) be accurately aligned to a physical datum or to an eigenaxis of another polarization optic. A simple opto-mechanical tool for azimuthal alignment can be used to perform accurate alignments and consists of two “rotatable” mounts. One mount holds a polarizer, while the other holds a half-wave retarder. The method of swings is used to aid in the azimuthal alignment of the polarization optic and is illustrated using the Poincaré Sphere. Additionally, imperfections in polarization optics are discussed. Keywords: alignment, polarization, optic, polarizer, azimuthal, kinematic, swings, waveplate 1. INTRODUCTION 1.1.Polarized Light Basics Polarized light can be divided into three different states: linear, elliptical, and circular. For monochromatic light, these three states can be represented using their electric field amplitudes and a phase lag term. The electric field amplitudes are broken down into their x and y components, which are aligned to a convenient reference frame. The phase lag term describes the phase relationship between the x and y components. A phase lag of ±90˚ describes circularly polarized light, while a phase lag of 0˚ describes linearly polarized light. Other phase lags between -180˚ and +180˚ describe elliptically polarized light. The ellipticity, ε, is defined as the ratio of the minor axis, b, over the major axis, a, and α is the angle of the major axis from the horizontal, as shown in Figure 1.
    [Show full text]
  • Modeling and Measuring the Polarization of Light: from Jones Matrices to Ellipsometry
    MODELING AND MEASURING THE POLARIZATION OF LIGHT: FROM JONES MATRICES TO ELLIPSOMETRY OVERALL GOALS The Polarization of Light lab strongly emphasizes connecting mathematical formalism with measurable results. It is not your job to understand every aspect of the theory, but rather to understand it well enough to make predictions in a variety of experimental situations. The model developed in this lab will have parameters that are easily experimentally adjustable. Additionally, you will refine your predictive models by accounting for systematic error sources that occur in the apparatus. The overarching goals for the lab are to: Model the vector nature of light. (Week 1) Model optical components that manipulate polarization (e.g., polarizing filters and quarter-wave plates). (Week 1) Measure a general polarization state of light. (Week 1) Model and measure the reflection and transmission of light at a dielectric interface. (Week 2) Perform an ellipsometry measurement on a Lucite surface. WEEK 1 PRELIMINARY OBSERVATIONS Question 1 Set up the optical arrangement shown in Figure 1. It consists of (1) a laser, (2) Polarizing filters, (3) A quarter-wave plate (4) A photodetector. Observe the variation in the photodetector voltage as you rotate the polarizing filters and/or quarter-wave plate. This is as complicated as the apparatus gets. The challenge of week 1 is to build accurate models of these components and model their combined effect in an optical system. An understanding of polarized light and polarizing optical elements forms the foundation of many optical applications including AMO experiments such as magneto-optical traps, LCD displays, and 3D projectors.
    [Show full text]
  • A Simple Method for Changing the State of Polarization from Elliptical Into Circular
    INVESTIGACION´ REVISTA MEXICANA DE FISICA´ 51 (5) 510–515 OCTUBRE 2005 A simple method for changing the state of polarization from elliptical into circular M. Montoya, G. Paez, D. Malacara-Hernandez,´ and J. Garc´ıa-Marquez´ Centro de Investigaciones en Optica,´ A.C., Loma del Bosque 115, Col. Lomas del Campestre, Leon,´ Gto., 37150 Mexico,´ e-mail: [email protected], [email protected], [email protected], [email protected] Recibido el 25 de mayo de 2005; aceptado el 7 de julio de 2005 Changes of polarization occur as a consequence of the interaction of light and the various optical elements through which it passes. A circularly polarized light beam may change its state to slightly elliptically polarized for many reasons. To correct this is not always easy but we show a very simple method for correcting circular polarization that has changed slightly into elliptic polarization. In this paper we propose to restore the circular state of polarization of an elliptically polarized light beam back to circular by means of a glass plate properly oriented while polarization is being measured. The basic idea is to modulate the transmittances of the electric field in both the major and minor axes of the ellipse of polarization. It is done by means of glass plates at non-normal incidence. Experimental results are consistent with theory. Keywords: Interference; polarization; polarizers; interferometers; interferometry. Al pasar la luz a traves´ de diferentes elementos opticos,´ ocurren cambios de polarizacion.´ Un haz circularmente polarizado puede cambiar ligeramente a el´ıpticamente polarizado por muchas razones. Corregir esto puede ser complicado.
    [Show full text]
  • Polarization of Light Thursday, 11/09/2006 Physics 158 Peter Beyersdorf
    Polarization of Light Thursday, 11/09/2006 Physics 158 Peter Beyersdorf Document info 1 Class Outline Polarization of Light Polarization basis’ Jones Calculus 16. 2 Polarization The Electric field direction defines the polarization of light Since light is a transverse wave, the electric field can point in any direction transverse to the direction of propagation Any arbitrary polarization state can be considered as a superposition of two orthogonal polarization states (i.e. it can be described in different bases) 16. 3 Electric Field Direction Light is a transverse electromagnetic wave so the electric (and magnetic) field oscillates in a direction transverse to the direction of propagation Possible states of electric field polarization are Linear electric field Circular plane wave Elliptical Random 4 Examples of polarization states right hand circular horizontal (CW as seen from observer) left hand circular vertical (CCW as seen from observer) linear polarization at an elliptical arbitrary angle 16. 5 Linear Polarization Basis Any polarization state can be described as the sum of two orthogonal linear polarization states i(kz ωt+φx) Ex(z, t) = E0xˆie − i(kz ωt+φy ) Ey(z, t) = E0yˆje − ! iφxˆ iφy ˆ i(kz ωt) E(z, t) = Ex(z,!t) + Ey(z, t) = E0xe i + E0ye j e − " # ! !E0y=0 ! E0x=0 E0y=E0x E0y=-E0x y y y y x x x horizontal vertical diagonal diagonal φ φ φx=φy+π x= y 16. 6 Circular Polarization iφx iφy i(kz ωt) E(z, t) = E0xe ˆi + E0ye ˆj e − " # Fo!r the case |φx-φy|=π/2 the magnitude of the field doesn’t change, but the direction sweeps out a circle The polarization is said Left-Handed Right-Handed to be right-handed if y y it progresses clockwise as seen by an observer looking into the light.
    [Show full text]
  • CHAPTER 8 INTERACTION of LIGHT and MATTER Point in the Direction of Propagation
    Chapter 8 Interaction of Light and Matter 8.1 Electromagnetic Waves at an Interface A beam of light (implicitly a plane wave) in vacuum or in an isotropic medium propa- gates in the particular fixed direction specified by its Poynting vector until it encoun- ters the interface with a different medium. The light causes the charges (electrons, atoms, or molecules) in the medium to oscillate and thus emit additional light waves thatcantravelinanydirection(overthesphereof4π steradians of solid angle). The oscillating particles vibrate at the frequency of the incident light and re-emit energy as light of that frequency (this is the mechanism of light “scattering”). If the emit- ted light is “out of phase” with the incident light (phase difference ∼= π radians), then the two waves interfere destructively and the original beam is attenuated.± If the attenuation is nearly complete, the incident light is said to be “absorbed.” Scattered light may interfere constructively with the incident light in certain directions, forming beamsthathavebeenreflected and/or transmitted. The constructive interference of the transmitted beam occurs at the angle that satisfies Snell’s law; while that after reflection occurs for θreflected = θincident. The mathematics are based on Maxwell’s equations for the three waves and the continuity conditions that must be satisfied at the boundary. The equations for these three electromagnetic waves are not diffi- cult to derive, though the process is somewhat tedious. The equations determine the properties of light on either side of the interface and lead to the phenomena of: 1. Equal angles of incidence and reflection; 2. Snell’s Law that relates the incident and refracted wave; 3.
    [Show full text]
  • Chapter 12: Polarization
    Chapter 12 Polarization In this chapter, we return to (9.46)-(9.48) and examine the consequences of Maxwell’s equa- tions in a homogeneous material for a general traveling electromagnetic plane wave. The extra complication is polarization. Preview Polarization is a general feature of transverse waves in three dimensions. The general elec- tromagnetic plane wave has two polarization states, corresponding to the two directions that the electric field can point transverse to the direction of the wave’s motion. This gives rise to much interesting physics. i. We introduce the idea of polarization in the transverse oscillations of a string. ii. We discuss the general form of electromagnetic waves and describe the polarization state in terms of a complex, two-component vector, Z. We compute the energy and momentum density as a function of Z and discuss the Poynting vector. We describe the varieties of possible polarization states of a plane wave: linear, circular and elliptical. iii. We describe “unpolarized light,” and explain how to generate and manipulate polarized light with polarizers and wave plates. We discuss the rotation of the plane of linearly polarized light by optically active substances. iv. We analyze the reflection and transmission of polarized light at an angle on a boundary between dielectrics. 333 334 CHAPTER 12. POLARIZATION 12.1 The String in Three Dimensions In most of our discussions of wave phenomena so far, we have assumed that the motion is taking place in a plane, so that we can draw pictures of the system on a sheet of paper. We have implicitly been restricting ourselves to two-dimensional waves.
    [Show full text]
  • Arxiv:1401.1911V1 [Astro-Ph.IM] 9 Jan 2014 Rpitpeae Yteauthor the by Prepared Preprint .PLRGNSS
    Journal of The Korean Astronomical Society (Preprint - no DOI assigned) 00: 1 25, 2013 December ISSN:1225-4614 Preprint∼ prepared by the author http://jkas.kas.org POLARIZATION AND POLARIMETRY: A REVIEW Sascha Trippe Department of Physics and Astronomy, Seoul National University, Seoul 151-742, South Korea E-mail: [email protected] (Received 30 August 2013; Revised 17 December 2013; Accepted 28 December 2013) ABSTRACT Polarization is a basic property of light and is fundamentally linked to the internal geometry of a source of radiation. Polarimetry complements photometric, spectroscopic, and imaging analyses of sources of radiation and has made possible multiple astrophysical discoveries. In this article I review (i) the phys- ical basics of polarization: electromagnetic waves, photons, and parameterizations; (ii) astrophysical sources of polarization: scattering, synchrotron radiation, active media, and the Zeeman, Goldreich- Kylafis, and Hanle effects, as well as interactions between polarization and matter (like birefringence, Faraday rotation, or the Chandrasekhar-Fermi effect); (iii) observational methodology: on-sky geom- etry, influence of atmosphere and instrumental polarization, polarization statistics, and observational techniques for radio, optical, and X/γ wavelengths; and (iv) science cases for astronomical polarime- try: solar and stellar physics, planetary system bodies, interstellar matter, astrobiology, astronomical masers, pulsars, galactic magnetic fields, gamma-ray bursts, active galactic nuclei, and cosmic microwave background radiation. Key words : Polarization — Methods: polarimetric — Radiation mechanisms: general Contents 3.8.4. Faraday Depolarization . 12 3.8.5. Polarization Conversion . 13 1. INTRODUCTION ................. 2 3.8.6. Chandrasekhar-Fermi Effect . 13 2. PHYSICALBASICS ................ 2 4. OBSERVATIONS ................. 13 2.1. Electromagnetic Waves . 2 4.1.
    [Show full text]