A Study of Reflection and Transmission of Birefringent Retarders
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Polarization of Light: Malus' Law, the Fresnel Equations, and Optical Activity
Polarization of light: Malus' law, the Fresnel equations, and optical activity. PHYS 3330: Experiments in Optics Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602 (Dated: Revised August 2012) In this lab you will (1) test Malus' law for the transmission of light through crossed polarizers; (2) test the Fresnel equations describing the reflection of polarized light from optical interfaces, and (3) using polarimetry to determine the unknown concentration of a sucrose and water solution. I. POLARIZATION III. PROCEDURE You will need to complete some background reading 1. Position the fixed \V" polarizer after mirror \M" before your first meeting for this lab. Please carefully to establish a vertical polarization axis for the laser study the following sections of the \Newport Projects in light. Optics" document (found in the \Reference Materials" 2. Carefully adjust the position of the photodiode so section of the course website): 0.5 \Polarization" Also the laser beam falls entirely within the central dark read chapter 6 of your text \Physics of Light and Optics," square. by Peatross and Ware. Your pre-lab quiz cover concepts presented in these materials AND in the body of this 3. Plug the photodiode into the bench top voltmeter write-up. Don't worry about memorizing equations { the and observe the voltage { it should be below 200 quiz should be elementary IF you read these materials mV; if not, you may need to attenuate the light by carefully. Please note that \taking a quick look at" these [anticipating the validity off Malus' law!] inserting materials 5 minutes before lab begins will likely NOT be the polarizer on a rotatable arm \R" upstream of adequate to do well on the quiz. -
Lecture 26 – Propagation of Light Spring 2013 Semester Matthew Jones Midterm Exam
Physics 42200 Waves & Oscillations Lecture 26 – Propagation of Light Spring 2013 Semester Matthew Jones Midterm Exam Almost all grades have been uploaded to http://chip.physics.purdue.edu/public/422/spring2013/ These grades have not been adjusted Exam questions and solutions are available on the Physics 42200 web page . Outline for the rest of the course • Polarization • Geometric Optics • Interference • Diffraction • Review Polarization by Partial Reflection • Continuity conditions for Maxwell’s Equations at the boundary between two materials • Different conditions for the components of or parallel or perpendicular to the surface. Polarization by Partial Reflection • Continuity of electric and magnetic fields were different depending on their orientation: – Perpendicular to surface = = – Parallel to surface = = perpendicular to − cos + cos − cos = cos + cos cos = • Solve for /: − = !" + !" • Solve for /: !" = !" + !" perpendicular to cos − cos cos = cos + cos cos = • Solve for /: − = !" + !" • Solve for /: !" = !" + !" Fresnel’s Equations • In most dielectric media, = and therefore # sin = = = = # sin • After some trigonometry… sin − tan − = − = sin + tan + ) , /, /01 2 ) 45/ 2 /01 2 * = - . + * = + * )+ /01 2+32* )+ /01 2+32* 45/ 2+62* For perpendicular and parallel to plane of incidence. Application of Fresnel’s Equations • Unpolarized light in air ( # = 1) is incident -
The Fresnel Equations and Brewster's Law
The Fresnel Equations and Brewster's Law Equipment Optical bench pivot, two 1 meter optical benches, green laser at 543.5 nm, 2 10cm diameter polarizers, rectangular polarizer, LX-02 photo-detector in optical mount, thick acrylic block, thick glass block, Phillips multimeter, laser mount, sunglasses. Purpose To investigate polarization by reflection. To understand and verify the Fresnel equations. To explore Brewster’s Law and find Brewster’s angle experimentally. To use Brewster’s law to find Brewster’s angle. To gain experience working with optical equipment. Theory Light is an electromagnetic wave, of which fundamental characteristics can be described in terms of the electric field intensity. For light traveling along the z-axis, this can be written as r r i(kz−ωt) E = E0e (1) r where E0 is a constant complex vector, and k and ω are the wave number and frequency respectively, with k = 2π / λ , (2) λ being the wavelength. The purpose of this lab is to explore the properties the electric field in (1) at the interface between two media with indices of refraction ni and nt . In general, there will be an incident, reflected and transmitted wave (figure 1), which in certain cases reduce to incident and reflected or incident and transmitted only. Recall that the angles of the transmitted and reflected beams are described by the law of reflection and Snell’s law. This however tells us nothing about the amplitudes of the reflected and transmitted Figure 1 electric fields. These latter properties are defined by the Fresnel equations, which we review below. -
Polarized Light 1
EE485 Introduction to Photonics Polarized Light 1. Matrix treatment of polarization 2. Reflection and refraction at dielectric interfaces (Fresnel equations) 3. Polarization phenomena and devices Reading: Pedrotti3, Chapter 14, Sec. 15.1-15.2, 15.4-15.6, 17.5, 23.1-23.5 Polarization of Light Polarization: Time trajectory of the end point of the electric field direction. Assume the light ray travels in +z-direction. At a particular instance, Ex ˆˆEExy y ikz() t x EEexx 0 ikz() ty EEeyy 0 iixxikz() t Ex[]ˆˆEe00xy y Ee e ikz() t E0e Lih Y. Lin 2 One Application: Creating 3-D Images Code left- and right-eye paths with orthogonal polarizations. K. Iizuka, “Welcome to the wonderful world of 3D,” OSA Optics and Photonics News, p. 41-47, Oct. 2006. Lih Y. Lin 3 Matrix Representation ― Jones Vectors Eeix E0x 0x E0 E iy 0 y Ee0 y Linearly polarized light y y 0 1 x E0 x E0 1 0 Ẽ and Ẽ must be in phase. y 0x 0y x cos E0 sin (Note: Jones vectors are normalized.) Lih Y. Lin 4 Jones Vector ― Circular Polarization Left circular polarization y x EEe it EA cos t At z = 0, compare xx0 with x it() EAsin tA ( cos( t / 2)) EEeyy 0 y 1 1 yxxy /2, 0, E00 EA Jones vector = 2 i y Right circular polarization 1 1 x Jones vector = 2 i Lih Y. Lin 5 Jones Vector ― Elliptical Polarization Special cases: Counter-clockwise rotation 1 A Jones vector = AB22 iB Clockwise rotation 1 A Jones vector = AB22 iB General case: Eeix A 0x A B22C E0 i y bei B iC Ee0 y Jones vector = 1 A A ABC222 B iC 2cosEE00xy tan 2 22 EE00xy Lih Y. -
Chapter 6 Light at Interfaces – Law of Reflection and Refraction
Chapter 6 Light at Interfaces – Law of Reflection and Refraction Part I Introduction Law of Reflection and Refraction Law of Reflection and Refraction normal Law of reflection: i r i r n1 n2 Law of refraction “Snell’s Law”: sin n t i 2 sint n1 Incident, reflected, refracted, and surface normal in same plane Huygens’ Principle Every point on a wavefront may be regarded as a secondary source of wavelets Huygens’ Principle Law of refraction Fermat’s Principle The path a beam of light takes between two points is the one which is traversed in the least time Law of reflection Fermat’s Principle Law of refraction SO OP t vi vt h2 x2 b2 (a x)2 t vi vt To minimize t(x) with respect to x, dt x (a x) dx 2 2 2 2 vi h x vt b (a x) 0. sin sin Thus, i t , vi vt so, ni sini nt sint Fermat’s Principle For a ray passing through a multilayer (m layers) material s s s t 1 2 m v1 v2 vm Then 1 m t ni si , c i1 The summation is known as the optical path length (OPL) traversed by the ray. P (OPL) S n(s)ds. The route light travels having the smallest optical path length EM Wave Point of View Light is an electromagnetic wave, obeys the Maxwell’s equations Plane of incidence: formed by k and the normal of the interface plane normal EB k EM Wave Point of View i(tkr) E E0e i(tkr) B B0e E cB 1. -
Optical Modeling of Nanostructures
Optical modeling of nanostructures Phd Part A Report Emil Haldrup Eriksen 20103129 Supervisor: Peter Balling, Søren Peder Madsen January 2017 Department of Physics and Astronomy Aarhus University Contents 1 Introduction 3 2 Maxwell’s Equations 5 2.1 Macroscopic form ................................. 5 2.2 The wave equation ................................ 6 2.3 Assumptions .................................... 7 2.4 Boundary conditions ............................... 7 2.5 Polarization conventions ............................. 7 3 Transfer Matrix Method(s) 9 3.1 Fresnel equations ................................. 9 3.2 The transfer matrix method ........................... 9 3.3 Incoherence .................................... 11 3.4 Implementation .................................. 14 4 The Finite Element Method 15 4.1 Basic principle(s) ................................. 15 4.2 Scattered field formulation ............................ 17 4.3 Boundary conditions ............................... 17 4.4 Probes ....................................... 19 4.5 Implementation .................................. 19 5 Results 20 5.1 Nanowrinkles ................................... 20 5.2 The two particle model .............................. 24 6 Conclusion 28 6.1 Outlook ....................................... 28 Bibliography 29 Front page illustration: Calculated field enhancement near a gold nanostar fabricated by EBL. The model geometry was constructed in 2D from a top view SEM image using edge detection and extruded to 3D. 1 Introduction Solar -
Electromagnetic Waves
Electromagnetic Waves Lecture34: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay Electromagnetic wave at the interface between two dielectric media We have so far discussed the propagation of electromagnetic wave in an isotropic, homogeneous, dielectric medium, such as in air or vacuum. In this lecture, we woulddiscuss what happens when a plane electromagnetic wave is incident at the interface between two dielectric media. For being specific, we will take one of the medium to be air or vacuum and the other to be a dielectric such as glass. We have come across this in school in connection with the reflection and transmission of light waves at such an interface. In this lecture, we would investigate this problem from the point of view of electromagnetic theory. Let us choose the interface to be the xy plane (z=0). The angles of incidence, reflection and refraction are the angles made by the respective propagation vectors with the common normal at the interface. 1 We have indicated the propation vectors in the apprpriate medium by capital letters I, R and T so as not to confuse with the notation for the position vector and time t. The principle that we use to establish the laws of reflection⃗ and refraction is the continuity of the tangential components of the electric field at the interface, as discussed extensively during the course of these lectures. Let us represent the component of the electric field parallel to the interface by the superscript . We then have, ∥ 0 exp + 0 exp = 0 exp ∥ ∥ This equation� ���⃗ ⋅ must⃗ − remain� valid at� all����⃗ points⋅ ⃗ − in the� interface � and ����⃗ at⋅ ⃗all− times.� That is obviously possible if the exponential factors is the same for all the three terms or if they differe at best by a constant phase factor. -
Experiments for B. Tech. 1St Year Physics Laboratory Department Of
Experiments for B. Tech. 1st Year Physics Laboratory Experiment No. 12 Brewster’s Angle Aim : To determine the of Brewster’s angle for glass using a polarized monochromatic light source. Apparatus Required: He laser, dial fitted polarizer, photo detector, micro ammeter, rotational mount, glass plate, constant power supply . Formula Used : Brewsters angle for a given pair of medium is 푛 θB = 휃 = 푡푎푛 ( ) 푛 Where n1 and n2 are refractive index of medium 1 and 2 respectively Principle: When light moves between two media of differing refractive index (n), some of the light is reflected from the surface of the denser material. This reflected ray’s intensity changes with change in the incident angle (휃) at the interface of two mediums. At one specific angle of incidence of light only perpendicular vibrations of electric field vectors are reflected whereas parallel vibrations are restricted or polarized. This loss in light intensity is due to polarization by reflection and the angle of incidence for which reflected ray is polarized is called the Brewster's angle θ (also known as the Polarization angle). B This phenomenon of polarization by reflection is illustrated in the figure below. Figure1. Polarization by reflection and Brewster's angle (θ ) B Department of Physics, IIT Roorkee © 1 Experiments for B. Tech. 1st Year Physics Laboratory The fraction of the incident light that is reflected depends on both the angle of incidence and the polarization direction of the incident light. The functions that describe the reflection of light polarized parallel and perpendicular to the plane of incidence are called the Fresnel Equations. -
EXPERIMENT 3 Fresnel Reflection
EXPERIMENT 3 Fresnel Reflection 1. Reflectivity of polarized light The reflection of a polarized beam of light from a dielectric material such as air/glass was described by Augustin Jean Fresnel in 1823. While his derivation was based on an elastic theory of light waves, the same results are found with electromagnetic theory. The ratio of the reflected intensity to the incident intensity is called the reflectivity of the surface. It depends on the polarization of the incident light wave. Let be the angle of incidence and be the angle of transmission. Snell’s law relates these according to the refractive index in each media and : ( ) ( ) (1) The reflectivity for light polarized parallel to the plane of incidence (known as p-polarized light) is ( ) (2) ( ) but for light polarized perpendicular to the plane of incidence (known as s-polarized light) it is ( ) (3) ( ) Notice that for p-polarized light the denominator of the right hand side will be infinite when the sum ( ) . The angle of 1 incidence when this happens is called Brewster’s angle, . For light polarized in the plane of incidence, no energy is reflected at Brewster’s angle, i.e. 2. Making the measurements Getting started: A He-Ne laser (632.8 nm) beam is reflected from the front face of a prism on a rotating table. You can read the angle of rotation of the table from the precision index and vernier. It is very important to locate the center of the prism over the axis of rotation. Do this as best you can by eye. -
Foundations of Optics
Lecture 1: Foundations of Optics Outline 1 Electromagnetic Waves 2 Material Properties 3 Electromagnetic Waves Across Interfaces 4 Fresnel Equations Christoph U. Keller, Leiden University, [email protected] Lecture 1: Foundations of Optics 1 Electromagnetic Waves Electromagnetic Waves in Matter Maxwell’s equations ) electromagnetic waves optics: interaction of electromagnetic waves with matter as described by material equations polarization of electromagnetic waves are integral part of optics Maxwell’s Equations in Matter Symbols D~ electric displacement r · D~ = 4πρ ρ electric charge density ~ 1 @D~ 4π H magnetic field r × H~ − = ~j c @t c c speed of light in vacuum ~j electric current density 1 @B~ r × E~ + = 0 E~ electric field c @t B~ magnetic induction r · B~ = 0 t time Christoph U. Keller, Leiden University, [email protected] Lecture 1: Foundations of Optics 2 Linear Material Equations Symbols dielectric constant D~ = E~ µ magnetic permeability B~ = µH~ σ electrical conductivity ~j = σE~ Isotropic and Anisotropic Media isotropic media: and µ are scalars anisotropic media: and µ are tensors of rank 2 isotropy of medium broken by anisotropy of material itself (e.g. crystals) external fields (e.g. Kerr effect) mechanical deformation (e.g. stress birefringence) assumption of isotropy may depend on wavelength (e.g. CaF2 for semiconductor manufacturing) Christoph U. Keller, Leiden University, [email protected] Lecture 1: Foundations of Optics 3 Wave Equation in Matter static, homogeneous medium with no net charges: ρ = 0 for most materials: µ = 1 combine Maxwell, material equations ) differential equations for damped (vector) wave µ @2E~ 4πµσ @E~ r2E~ − − = 0 c2 @t2 c2 @t µ @2H~ 4πµσ @H~ r2H~ − − = 0 c2 @t2 c2 @t damping controlled by conductivity σ E~ and H~ are equivalent ) sufficient to consider E~ interaction with matter almost always through E~ but: at interfaces, boundary conditions for H~ are crucial Christoph U. -
3.3.3 Fresnel Equations Snell's Law Describes How the Angles Of
3.3.3 Fresnel equations Snell’s law describes how the angles of incidence and transmission are related through the refractive indices of the respective media. But what about the intensity of transmitted and reflected light? We are looking at a picture of the Montana glacier parks. White mountain tops are reflected in a perfectly still lake Mcdonald. It is a beautiful example of this video’s subject. Not only do we see the mountain tops reflected in the water, we can also find the each and every pebble lying on the bottom of the lake. A fraction of the light, reflected by the mountain tops, is reflected off the surface of the water. Another part is transmitted into the water and reflects off the rocks and pebbles. The Fresnel equations describe the intensity of the reflected and transmitted fractions. In this video we will learn which properties of light and matter affect the intensity of the transmitted and reflected light waves. We will introduce the concept of light polarization. Then we will discuss the fresnel equations and, finally, we will apply them to improve our solar cells. Let's look into the parameters that affect the fractions of light that are transmitted and reflected. This figure shows a light beam incident on the interface of two media. The refractive indices of both media, that describe how easily light propagates through a material, play an important role. The angle of incidence is also an important parameter. With the refractive indices and the angle of incidence we can calculate the angle of transmission, according to Snell’s law. -
Investigation of the Reflective Properties of a Left-Handed Metamaterial
Wright State University CORE Scholar Browse all Theses and Dissertations Theses and Dissertations 2007 Investigation of the Reflective Properties of a Left-Handed Metamaterial Amanda Durham Wright State University Follow this and additional works at: https://corescholar.libraries.wright.edu/etd_all Part of the Physics Commons Repository Citation Durham, Amanda, "Investigation of the Reflective Properties of a Left-Handed Metamaterial" (2007). Browse all Theses and Dissertations. 92. https://corescholar.libraries.wright.edu/etd_all/92 This Thesis is brought to you for free and open access by the Theses and Dissertations at CORE Scholar. It has been accepted for inclusion in Browse all Theses and Dissertations by an authorized administrator of CORE Scholar. For more information, please contact [email protected]. INVESTIGATION OF THE REFLECTIVE PROPERTIES OF A LEFT-HANDED METAMATERIAL A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science By AMANDA DURHAM B.S., Embry-Riddle Aeronautical University, 2004 2007 Wright State University WRIGHT STATE UNIVERSITY SCHOOL OF GRADUATE STUDIES March 29, 2007 I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPERVISION BY Amanda Durham ENTITLED Investigation of the Reflective Properties of a Left-handed Metamaterial BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science. _______________________ Lok Lew Yan Voon, Ph.D. Thesis Advisor _______________________ Lok Lew Yan Voon, Ph.D. Department Chair Committee on Final Examination ____________________________ Lok Lew Yan Voon, Ph.D. ____________________________ Gregory Kozlowski, Ph.D. ____________________________ Douglas Petkie, Ph.D. ____________________________ Joseph F. Thomas, Jr., Ph.D. Dean, School of Graduate Studies Abstract Durham, Amanda, M.S., Department of Physics, Wright State University, 2007.