DOCSLIB.ORG

Crystal Optics with Intense Light Sources Solution of Exercise Sheet #3

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Crystal Optics with Intense Light Sources Solution of Exercise Sheet #3 FS 2020 - ETH Zurich Crystal Optics with Intense Light Sources Prof. Manfred Fiebig To be reviewed on 17.03.2020. Ques- Solution of Exercise sheet #3 tions and comments to be posted on https://crystaloptics.freeforums.net/ Exercise 3.1: Short questions (11 points) (a) Magnetic dipoles cannot follow the high frequencies of optical magnetic fields. Therefore the relative permeability is one at optical frequencies: µr = 1. Gyromagnetic resonances appear up to approximately 1 GHz. Optical frequencies are 1014 Hz. (b) Along the optical axis the light velocity/refractive index does not depend on the light polarization: The cut between the plane perpendicular to light propagation k and the indicatrix n(D) generally gives an oval, but for k parallel to the optical axis the cut is circular. This also means that the polarization of a beam travelling along an optical axis remains unchanged. For other directions differently polarized beams are refracted differently (and therefore travel in different directions depending on polarization). For biaxial crystals the optical axes must not be in agreement with the crystallographic axes (and are in general wavelength dependent leading to a rotation of the indicatrix)! In uniaxial materials the optical axis lies along the high-symmetry axis of the crystal (restricted by symmetry). (c) The indicatrix is a general ellipsoid defined by 2 2 2 x1 x2 x3 2 + 2 + 2 = 1 : n1 n2 n3 It allows a description of the refractive index n as a function of the displacement vector D and is independent of the wave vector k. Originally the indicatrix describes the local shape at a given point of a curved surface in three dimensional space. (d) For uniaxial crystals: The ordinary refractive index (no) is defined perpendicular to the optical axis (OA) no = n?, and therefore the polarization of ordinary rays (polarized perpendicular to the OA) remains unchanged (EjjD). njj = ne along the OA gives the extraordinary refractive index (ne). For arbitrary light polarizations (θ is the angle to the optical axis) one can use the indicatrix defined by 1 cos2 θ sin2 θ 2 = 2 + 2 : n (θ) no ne Biaxial crystals have two optical axes, leading to two possible ordinary refractive indices. A light beam can always be separated into an ordinary and an extraordinary ray, with the free choice of the ordinary direction. (e) TiO2 has a higher refractive index compared to glass (nT iO2 = 2.62, nglass = 1.54), such that thinner glasses can be manufactured. However it becomes more expensive. (f) As the wavelength of light is usually much larger than unit cell parameters, optical effects are generally given by macroscopic tensors and restricted only by the point symmetry of the crystal. Exceptions are optical activity (see formula in exercise before, the light rotation adds to a ma- croscopic value if the crystal has just one chirality), short wavelengths (x-ray), and maybe the predictions of the crystal field splitting (site symmetry more important than macroscopic crystal symmetry). (g) There are some points on the indicatrix with the same curvature known as isotropic points: for example, wavelength-dependent no and ne can obtain the same value for a certain wavelength. Other case one could think of would be a certain state of stress, such that no = ne. (h) The section between the indicatrix and the plane perpendicular to the light propagation direction k gives always an ellipse. The two main axes of this ellipse determine the two refractive indices, hence birefringent. (i) The three different refractive indices define the ellipsoid of the indicatrix (na 6= nb 6= nc). Only under two special directions the section between the ellipsoid and the plane of polarization of the light beam describes a circle, i.e. the optical axis. (j) 6 constants are needed. 3 indices of diffraction and 3 directions of the ellipsoid with respect to the crystal structure. p (k) Because n = rµr but also c0 = nc with c0 the speed of light in vacuum, one can rewrite these two equations as: c = p c0 to obtain the relationship. r µr Exercise 3.2: Birefringent and gyrotropic materials (8 points) (a) This depends on the crystal system: Cubic structures are isotropic/not birefringent; orthorhom- bic/monoclinic/triclinic crystals are biaxial birefringent, all other systems are uniaxial. diamond (F d3m, cubic): isotropic; ZnS (P 63mc, hexagonal wurtzite): uniaxial; ZnS (43m, cubic zinc-blende): isotropic; BBO (R3c, rhomboidal): uniaxial; KDP (I42d above TC , tetragonal): uniaxial; Topaz (P bmn, orthorhombic): biaxial. (b) Optical active materials must be chiral (handedness, i.e., lost mirror operation). The direction of polarization rotation depends on the chirality of the passed specimen. Fused silica is a glass and is therefore achiral, calcite is achiral (R32=c), as is BBO (R3c). α-quartz comes in enantiomorphs with sym- metries P3121 or P3221 (chirality vs. screw axes!). Hexagonal β-quartz is also chiral. Both quartz types are optically active. Sodium bro- mate (with chiral group P213) is optically ac- tive. Glucose/dextrose molecules show circu- lar birefringence: the name dexter\(right) co- " mes from the observed positive light rotation (also used to measure sugar concentration). Negative light rotation is observed in fructo- se/laevulose ( left\). " Also optical active: certain phases of liquid crystals, that can be aligned by applied electric fields (used for calculator displays). (c) The wavelength 1064 nm is specific for a Nd:YAG-laser, 532 nm and 355 nm correspond to the second and third harmonic, respectively. Following the graph (just an exponential fit to the data!) one gets the following results (brackets give thickness of commercially available polarization rotators): 1064 nm: ≈ 6.1◦/mm ) d =14.8 mm (14.2 mm). 532 nm: ≈ 27.5◦/mm ) d =3.3 mm (3.4 mm). 355 nm: ≈ 60.5◦/mm ) d =1.5 mm (0.7 mm). One sees that at lower wavelength the deviations are quite large { so just approximate data at high energies (lower wavelengths)! 3 0 1 3 0 1 2 0 1 1 0 Q u a r t z 1 0 0 f i t 9 0 m m / 8 0 ° n 7 0 o i t a ] t 6 0 o r m c i 5 0 f i c m e 4 0 / p 2 0 s ° [ 3 0 n 2 0 o i 1 0 t a 0 t 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 1 0 0 0 1 1 0 0 1 2 0 0 o r W a v e l e n g t h [ n m ] c i f i c e p s 1 0 d a t a f r o m : h t t p : / / w w w . k a y e l a b y . n p l . c o . u k / g e n e r a l _ p h y s i c s / 2 _ 5 / 2 _ 5 _ 1 0 . h t m l p o i n t a t 3 5 5 n m f r o m : h t t p : / / w w w . u - o p t i c . c o m / g i d _ 5 2 _ p r d i d _ 6 6 _ p i d _ 2 9 9 _ l _ 1 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 1 0 0 0 1 1 0 0 W a v e l e n g t h [ n m ] Exercise 3.3: Rotating light polarizations (3 points) (a) A waveplate or retarder is an optical device that alters the polarization state of a light wave travelling through it. A half-wave plate shifts the polarization direction of linearly polarized light. For example a half-wave plate at 45◦ will change an x-polarized light to y-polarized light and vice versa. ! 2π (b) The accumulated phase can be written as φ = k(n) · d, where k(n) = n = nk0 = n is the c0 λ light wavevector in the material and d its thickness. The phases can be written as: 2π φ = k d + k d0 = (n d + n d0) ; jj e o λ e o 2π φ = k d + k d0 = (n d + n d0) ; ? o e λ o e 2π ∆φ = φ − φ = (n − n )(d0 − d) : jj ? λ o e 0 If the thicknesses are the same (d = d ) both polarization components Ejj, E? experience the same retardation and thus the polarization remains unchanged. For a 90◦-rotation the accumulated phase must be ∆φ = π. The equation then reduces to (d0 − d) = λ/2∆n. For λ ≈ 600 nm, the difference in the refractive index is ∆n ≈ 0:01 which leads to (d0 − d) ≈ 30 µm, which can be well adjusted by sliding the quartz wedges using a micrometer screw. E|| E E . d d’ ..
Load more

Recommended publications
  • (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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.....................................................................................................
    [Show full text]
  • Quantum Mechanics Polarization Density
    Quantum Mechanics_polarization density In classical electromagnetism, polarization density (or electric polarization, or simplypolarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external Electric field, its molecules gain Electric dipole moment and the dielectric is said to be polarized. The electric dipole moment induced per unit volume of the dielectric material is called the electric polarization of the dielectric.[1][2] Polarization density also describes how a material responds to an applied electric field as well as the way the material changes the electric field, and can be used to calculate the forces that result from those interactions. It can be compared to Magnetization, which is the measure of the corresponding response of a material to a Magnetic field in Magnetism. TheSI unit of measure is coulombs per square metre, and polarization density is represented by a vectorP.[3] Definition The polarization density P is defined as the average Electric dipole moment d per unit volume V of the dielectric material:[4] which can be interpreted as a measure of how strong and how aligned the dipoles are in a region of the material. For the calculation of P due to an applied electric field, the electric susceptibility χ of the dielectric must be known (see below). Polarization density in Maxwell's equations The behavior of electric fields (E and D), magnetic fields (B, H), charge density (ρ) and current density (J) are described by Maxwell's equations in matter. The role of the polarization density P is described below.
    [Show full text]
  • Cornu Spiral, 64 Crystal Optics, 16, 84
    Index Absorption , 25, 117 Cornu spiral , 64 Amplitude , complex , 29 Crystal optics, 16, 84- 108 Aperture , 48 , 49 , 53 , 55 , 60 , 62 Crystals, 20 numerical , 61 optically active, 137 Axes , uniaxial , 90 optical ( optic ) , 90, 101 Current density, 109 principal , 85 ray , 105 Damped oscillations , 118 Damping , radiation , 121 Babinet ' s theorem , 49 Damping constant , 79 Bessel function , 57 Damping for ('e, 117 Bessel 's inequality , 32 Diaphragm , 49, 51, 52, 55. See also Binormals , 101 Aperture lliradials , 105 Dielectric constant , Birefringence , 108, 147 complex , 78- 82 Boundary conditions , 46, 47, 48, 50 principal , 86 Brewster ' s law , 75 Dielectric constant tensor , 84 - 86 Brun ' s law , 11 Differential equations for light rays , 10 , 19 , 22 Canonical equations , 15, 18, 20 Diffraction , inhomogeneous form of , 22 from circular aperture , 55 - 57 Charged particles , 109 Fraunhofer , 53 Circular frequency , 28 f ' resnel , 53 , 62 - 66 Clausius .Mosotti equation , 112 from grating , 57- 59 Coherent light sources , 61, 151 by slit , 54- 55 Coherent wave trains , 43 Diffraction angle , 48, 51, 55, 57, 59 Completeness relation , 32 Diffraction pattern , 58 , 59 , 61 Conductivity , electrical , 77, 80 Dipole moment , Cone , electric , 109 ray , 104 electric , of optically active wave , 105 crystals , 140 Conservation of energy , 16 magnetic , of optically active for electromagnetic waves , 69, crystals , 142 85 , 114 Discontinuity of E at surface , 70, 71 155 156 INDEX Dispersion law . 34 . 35 . 38 . 117 Frequency, for xrays , 128 circular , 28 Doppler effect , 133 Larmor , 145 Dual equations , 95, 96 I-sum rule , 152- 153 Dual variables , 92 Grating , 57, 58, 60, 61 step ., 62 Eikonal , 8 Green ' s theorem , 46 angular , 25 Group velocity , 37, 98, 115, 116 Eikonal equationt 8 Electrons , 113 , 119 , 121 - 131 , 145 Ha milton , W .
    [Show full text]
  • Arxiv:1709.07277V1 [Physics.Optics]
    On the Theory of Light Propagation in Crystalline Dielectrics Marius Dommermuth and Nils Schopohl∗ Institut f¨ur Theoretische Physik and CQ Center for Collective Quantum Phenomena and their Applications in LISA+, Eberhard-Karls-Universit¨at T¨ubingen, Auf der Morgenstelle 14, D-72076 T¨ubingen, Germany (Dated: September 22, 2017) A synoptic view on the long-established theory of light propagation in crystalline dielectrics is presented, where charges, tightly bound to atoms (molecules, ions) interact with the microscopic local electromagnetic field. Applying the Helmholtz-Hodge decomposition to the current density in Maxwell’s equations, two decoupled sets of equations result determining separately the divergence- free (transversal) and curl-free (longitudinal) parts of the electromagnetic field, thus facilitating the restatement of Maxwell’s equations as equivalent field-integral equations. Employing a suit- ably chosen basis system of Bloch functions we present for dielectric crystals an exact solution to the inhomogenous field-integral equations determining the local electromagnetic field that polar- izes individual atoms or ionic subunits in reaction to an external electromagnetic wave. From the solvability condition of the associated homogenous integral equation then the propagating modes and the photonic bandstructure ωn (q) for various crystalline symmetries Λ are found solving a small sized 3M × 3M matrix eigenvalue problem, with M denoting the number of polarizable atoms (ions) in the unit cell. Identifying the macroscopic electric field inside the sample with the spatially low-pass filtered microscopic local electric field, the dielectric 3 × 3-tensor εΛ (q,ω) of crys- tal optics emerges, relating the accordingly low-pass filtered microscopic polarization field to the macroscopic electric field, solely with the individual microscopic polarizabilities α (R,ω) of atoms (molecules, ions) at a site R and the crystalline symmetry as input into the theory.
    [Show full text]
  • Crystal Optics Product Portfolio
    Prosp_Kristalloptik_08.qxd:Prosp_Kristalloptik_08.qxd 27.10.2008 13:39 Uhr Seite 24 OPTICAL SYSTEMS I LASERS & MATERIAL PROCESSING I INDUSTRIAL METROLOGY I TRAFFIC SOLUTIONS I DEFENSE & CIVIL SYSTEMS Crystal optics Product portfolio JENOPTIK I Optical Systems Optical Systems Division JENOPTIK Laser, Optik, Systeme GmbH Göschwitzer Straße 25 07745 Jena I Germany Phone +49 364165-3148 I Fax -3677 [email protected] www.jenoptik-los.com Coastal Optical Systems, Inc. 16490 Innovation Drive Jupiter I FL 33478 I USA Tel. +1256859-1886 I Fax -5890 [email protected] www.coastalopt.com en Our designs and specifications are in constant development. We reserve the right to make changes that reflect technological progress. 017840-020-99-14-1008 Optics Prosp_Kristalloptik_08.qxd:Prosp_Kristalloptik_08.qxd 27.10.2008 13:39 Uhr Seite 2 Crystal optics Crystal optics for a wide range of applications Crystal optics fulfill a function in optical systems for In addition to the development and manufacturing of The basics of polarization optics Optical materials applications in laser technology, semiconductor industry, optical solutions, the Optical Systems Division also supplies Crystals feature optical properties that are responsible for Naturally grown crystals often occur in the polycrystalline metrology and medical technology, as well as applications customized crystal optics from its manufacturing site in reflection, diffraction and absorption of light, among form. In other words, the crystalline solids consist of many in security and defense and in aerospace industry. Giessen. All crystal optics, from a single component up other things, which makes crystal optics to an important tiny crystals, which are separated from each other by grain to assembled optical modules, are manufactured to the part of the photonics.
    [Show full text]
  • Polarization and Crystal Optics * - Polarization ↔ Time Course of the Direction of E (R,T) - Polarization Affects:  Amount of Light Reflected at Material Interfaces
    EE 485, Winter 2004, Lih Y. Lin Chapter 6 Polarization and Crystal Optics * - Polarization ↔ Time course of the direction of E (r,t) - Polarization affects: Amount of light reflected at material interfaces. Absorption in some materials. Scattering. Refractive index (thus velocity) of anisotropic materials. Optically active materials to rotate polarization. 6.1 Polarization of Light Consider (z,t) = Re{A exp(jω(t − z )} (6.1-1) E c = + With complex envelope A Axxˆ Ay yˆ (6.1-2) Trace the endpoint of E (z,t) at each position z as a function of time. The polarization ellipse = ϕ = ϕ Ax ax exp( j x ), Ay ay exp( j y ) = + E (z,t) Exxˆ Ey yˆ (6.1-3) = a cos[ω(t − z )+ ϕ ] (6.1-4a) Ex x c x = a cos[ω(t − z )+ ϕ ] (6.1-4b) Ey y c y 2 E 2 EE Ex + y − ϕ x y = 2 ϕ 2 2 2cos sin (6.1-5) ax ay axay ay → An ellipse. The shape of the ellipse depends on and ϕ . The size ax a2 + a2 of the ellipse determines intensity I = x y (η: impedance of the medium). 2η 1 EE 485, Winter 2004, Lih Y. Lin Rotation direction (viewed from the direction towards which the wave is approaching): ϕ > ϕ y x : Clockwise rotation ϕ < ϕ y x : Counter-clockwise rotation • Linearly-polarized light = → ax 0 Ey only = → ay 0 Ex only Or ϕ = 0 or π a = y for ϕ = 0 Ey a Ex → x = − ay ϕ = π Ey E for ax = → ± ax ay 45 polarization • Circularly-polarized light 2 EE 485, Winter 2004, Lih Y.
    [Show full text]
  • Crystal Optics of Visible Light
    Crystal Optics of Visible Light This can be a very helpful aspect of minerals in understanding the petrographic history of a rock. The manner by which light is transferred through a mineral is a means to describe and quantify mineral properties in situ as well as to describe the relationships between minerals in a rock. The interaction and behavior of light within minerals is a key diagnostic features that can be applied in multiple ways. To study minerals in thin section we use a polarizing light microscope or petrographic microscope Using a substage light source with a series of lenses, polarizer(s), and light intensity adjustments we can examine the properties of a mineral The path of light in a polarizing microscope The vernier graduated stage on a petrographic microscope can be freely rotated through 360° There are two condensers that can be used to vary the diameter of the beam of light interacting with the sample (helpful when using the Bertrand Lens for viewing interference figures) The substage diaphram is used for altering the light intensity (most useful for viewing relief and observing the Becke Line) Refraction (only really useful for studying optically isotropic mediums) ●The speed of light in a vacuum is 3 x 108 m/sec ●When light passes from one materials to another of a different density at an angle oblique to perpendicular the velocity of that ray of light must change ~ the result is also a change in direction Light velocity in material A sin α _____________________ = _____ Light velocity in material B sin β ●This is Snell’s Law: the propagation velocity of light in a medium A to its velocity in a denser medium B is equal to the ratio of the sines of the angle of incidence (α) at the interface and the angle of refraction (β) with B.
    [Show full text]