Electromagnetic optics!
|
|
- Hubert Morgan
- 6 years ago
- Views:
Transcription
1 1 EM theory Electromagnetic optics! EM waves Monochromatic light
2 2 Electromagnetic optics! Electromagnetic theory of light Electromagnetic waves in dielectric media Monochromatic light References: Fundamentals of Photonics, Ch. 5
3 3 From ray optics to quantum optics! Ray optics Wave optics Electromagnetic optics! Quantum optics
4 E-M fields are described by two related vector fields functions of ( r,t ) 4 Electric field E(r,t) [V/m] Magnetic field H(r,t) [A/m] Both fields are related by a set of PDE s: E H = ε 0 t H E = µ 0 t H = 0 E = 0 Maxwell s equations! (in free space)! Both E(r,t) and H(r,t) are real functions ε 0 1/36π 10-9 F/m is the electric permittivity μ 0 4π 10-7 H/m is the magnetic permeability
5 5 After 155 years, Maxwell s equations are still famous!
6 6 The wave equation connects E-M optics and wave optics! Each of the components (E x,e y,e z ) and (H x,h y,h z ) must satisfy 2 u 1 c u t 2 = 0 The wave equation! (in free space)! c 0 = 1 ε 0 µ m/s is the speed of light in vacuum (that also appears in wave optics). The (wave)function u(r,t) is any of the components.
7 7 Inside a medium two additional vector fields appear! Electric flux density D(r,t) [C/m 2 ] Magnetic flux density! B(r,t) [Wb/m 2 ] The 4 fields are related by Maxwell s equations in a source-free medium: H = D t E = B t B = 0 D = 0 Maxwell s equations! (source-free medium)! Polarization density P(r,t) Magnetization densitym(r,t) D = ε 0 E + P B = µ 0 H + µ 0 M (in free space P = M = 0)
8 8 Boundary conditions! Homogeneous medium: all components of E, H, D and B are continuous functions of position. Boundary between two different media: Dielectric Dielectric: E //, H //, D and B are continuous Dielectric Metal: E // = 0
9 9 The Poynting vector governs The direction of power flow is perpendicular to both E and H: the flow of power! The optical intensity I is the time-averaged magnitude of the Poynting vector. Calculating the divergence of S we obtain Poynting s theorem: the power flowing from a region equals the change in stored energy ( ) S = E H S = E H [E] = N/C = N/A.s [H] = A/m [S] = N/m.s = W/m 2 = t ( 1 ε 2 0E µ 2 0H 2 ) + E P t + µ 0 H M t energy density of electric and magnetic fields energy on the electric and magnetic dipoles
10 10 An electromagnetic wave also carries linear momentum! This results in radiation pressure on objects from which the wave reflects or scatters. In free space, the linear momentum density / unit volume is a vector ε 0 E B = 1 c 2 S Remember: EM waves carry Power Intensity Energy Momentum
11 11 Orbital angular momentum! Light beams with an azimuthal phase dependence carry orbital angular momentum. Such beams have: Laguerre-Gaussian amplitude! helical phase front!
12 12 A medium is characterized by the constitutive relations! These relate P, M to E, H but in most practical cases we only need to consider the dielectric properties, i.e. P to E. Type of dielectric medium! linear nondispersive Properties! P(r,t) is proportional to E(r,t) P(t) only depends on E(t) homogeneous independent of r! isotropic independent of direction of E(r,t) (so we must have P // E ) E(r,t) medium P(r,t)
13 13 Examples of different media! Optical glass BK7 Calcite GRIN lens KDP crystal
14 A) Linear, nondispersive, homogeneous, and isotropic media! 14 E χ P The relation at every (r,t ) is simply (χ is the electric susceptibility) D and E are also parallel (ε is the electric permittivity) H = ε E t E = µ H t P = ε 0 χe D = ε 0 E + P = ε 0 (1+ χ)e = εe B = µh H = 0 E = 0
15 15 The properties of the medium define the speed of light! Each component of E and H satisfies the wave equation 2 u 1 2 u c 2 t = 0 where c = 1 2 εµ The speed of light inside the medium is related to the refractive index and the susceptibility: n = c 0 c = εµ ε 0 µ 0 = 1+ χ Poynting s theorem has the form of a continuity equation, where W = energy density stored in the medium S = W t W = 1 2 εe µh2
16 B) Linear, nondispersive, inhomogeneous, isotropic media! 16 E(r) χ(r) P(r) The relations become (χ, ε and n become functions of position) P = ε 0 χ(r)e D = ε(r)e The wave equation gains a new term but for slowly varying ε(r) we may write 2 E 2 E 1 ε E µ ε 0 ε 2 E t = 0 2 c(r) = 1/ 1 c 2 (r) 2 E t 2 0 µ 0 ε(r) = c 0 / n(r)
17 C) Linear, nondispersive, homogeneous, anisotropic media! 17 E xyz χ jj P xyz The relation becomes dependent on the direction of the vector E! The dielectric properties of the medium are described by an array of 3 3 constants: the electric susceptibility tensor {χ ij } and the electric permittivity tensor {ε ij } P i = D = j j ε 0 χ ij E j ε ij E j
18 D) Linear, dispersive, homogeneous, isotropic media! 18 E(t ) P(t ) χ(t ) The response depends on the values of E(t ) for all t t : Since a correlation in the time domain is a product in the frequency domain: impulse response function (time domain) transfer function (frequency domain) P(t) = ε 0 χ(t t )E( t )d t P(ν) = ε 0 χ(ν)e(ν) ε 0 χ(t) ε 0 χ(ν)
19 E) Nonlinear, nondispersive, homogeneous, and isotropic media! 19 The relation is nonlinear; the previous Maxwell s equation in a linear medium and the wave equation are not valid.! P χ (1) E + χ (2) E 2 + χ (3) E 3 + Starting from the general Maxwell s eqns. in a medium, and considering a homogeneous and isotropic medium: E E = µ H t ( ) = µ H t general wave equation for a homogeneous, isotropic medium 2 E 1 c E t 2 = µ 0 2 P t 2 Most dielectric media are approximately linear, except when using very powerful, focused laser beams.
20 20 Nonlinear optics was born with the discovery of the laser! When the electric field exceeds the interatomic electric fields (~10 8 V/m) strange phenomena start happening
21 21 Monochromatic electromagnetic waves! Just as we did for wave optics, let s see some examples of solutions of the wave equation. Things are much simpler when we consider monochromatic waves. Introducing a complex representation: It s easy to calculate the time derivative of these waves: Maxwell s equations in a medium become: E(r,t) = Re{ E(r )e iωt } H(r,t) = Re{ H(r )e iωt } t t ( E(r )e iωt ) = iωe(r )e iωt ( H(r )e iωt ) = iωh(r )e iωt H = jωd B = 0 E = jωb D = 0 (D = ε 0 E + P)
22 22 Intensity and Helmholtz equation for monochromatic E-M waves! Writing the complex Poynting vector: The optical intensity is then the magnitude of the vector Re{S} S = Re{ Ee iωt } Re{ He iωt } S = = Re{S} 1 2 E H In a linear, nondispersive, homogeneous, isotropic medium we have Like in wave optics, a very simple equation can be now derived for U = (E x,y,z ) or (H x,y,z ): the well-known Helmholtz equation! D = εe B = µh 2 U + k 2 U = 0 ( k = nk 0 = ω εµ )
23 23 The Transverse E-M plane wave! Let s consider a monochromatic E-M wave whose E and H fields are plane waves: From wave optics, we already know that plane waves are solutions of the Helmholtz equation, with Replacing H, E in Maxwell s equations: It follows that: E is perpendicular to both H and k H is perpendicular to both E and k!! This is then a transverse electromagnetic (TEM) wave! H(r) = H o exp( ik r) E(r) = E o exp( ik r) k = k = nk 0 k H o = ωεe o k E o = ωµh 0
24 24 Intensity of a TEM wave! We have for the complex Poynting vector: Where we have introduced the parameter called impedance of the medium: (impedance of vacuum 377 Ω) The intensity is then: S = S = 1 2 E H η = E 0 H 0 = 1 2 E 0H 0 = E 0 2 / 2η I = E 0 2 / 2η µ ε The most powerful laser in the world has a peak intensity of ~10 22 W/cm 2. What is the corresponding magnitude of the electric field? Compare with the electric fields used at a particle accelerator (~10 7 V/m)
25 25 Electron acceleration in plasmas waves using lasers!
Summary of Beam Optics
Summary of Beam Optics Gaussian beams, waves with limited spatial extension perpendicular to propagation direction, Gaussian beam is solution of paraxial Helmholtz equation, Gaussian beam has parabolic
More informationOverview in Images. S. Lin et al, Nature, vol. 394, p , (1998) T.Thio et al., Optics Letters 26, (2001).
Overview in Images 5 nm K.S. Min et al. PhD Thesis K.V. Vahala et al, Phys. Rev. Lett, 85, p.74 (000) J. D. Joannopoulos, et al, Nature, vol.386, p.143-9 (1997) T.Thio et al., Optics Letters 6, 197-1974
More informationEE485 Introduction to Photonics. Introduction
EE485 Introduction to Photonics Introduction Nature of Light They could but make the best of it and went around with woebegone faces, sadly complaining that on Mondays, Wednesdays, and Fridays, they must
More informationElectromagnetic (EM) Waves
Electromagnetic (EM) Waves Short review on calculus vector Outline A. Various formulations of the Maxwell equation: 1. In a vacuum 2. In a vacuum without source charge 3. In a medium 4. In a dielectric
More information1 Fundamentals of laser energy absorption
1 Fundamentals of laser energy absorption 1.1 Classical electromagnetic-theory concepts 1.1.1 Electric and magnetic properties of materials Electric and magnetic fields can exert forces directly on atoms
More informationLecture 21 Reminder/Introduction to Wave Optics
Lecture 1 Reminder/Introduction to Wave Optics Program: 1. Maxwell s Equations.. Magnetic induction and electric displacement. 3. Origins of the electric permittivity and magnetic permeability. 4. Wave
More information1 Macroscopic Maxwell s equations
This lecture purports to the macroscopic Maxwell s equations in the differential forms and their revealation about the propagation of light in vacuum and in matter. Microscopic Maxwell s equations and
More informationH ( E) E ( H) = H B t
Chapter 5 Energy and Momentum The equations established so far describe the behavior of electric and magnetic fields. They are a direct consequence of Maxwell s equations and the properties of matter.
More informationOptics and Optical Design. Chapter 5: Electromagnetic Optics. Lectures 9 & 10
Optics and Optical Design Chapter 5: Electromagnetic Optics Lectures 9 & 1 Cord Arnold / Anne L Huillier Electromagnetic waves in dielectric media EM optics compared to simpler theories Electromagnetic
More informationChap. 4. Electromagnetic Propagation in Anisotropic Media
Chap. 4. Electromagnetic Propagation in Anisotropic Media - Optical properties depend on the direction of propagation and the polarization of the light. - Crystals such as calcite, quartz, KDP, and liquid
More informationE E D E=0 2 E 2 E (3.1)
Chapter 3 Constitutive Relations Maxwell s equations define the fields that are generated by currents and charges. However, they do not describe how these currents and charges are generated. Thus, to find
More informationLecture 2 Review of Maxwell s Equations and the EM Constitutive Parameters
Lecture 2 Review of Maxwell s Equations and the EM Constitutive Parameters Optional Reading: Steer Appendix D, or Pozar Section 1.2,1.6, or any text on Engineering Electromagnetics (e.g., Hayt/Buck) Time-domain
More informationLight in Matter (Hecht Ch. 3)
Phys 531 Lecture 3 9 September 2004 Light in Matter (Hecht Ch. 3) Last time, talked about light in vacuum: Maxwell equations wave equation Light = EM wave 1 Today: What happens inside material? typical
More informationCHAPTER 9 ELECTROMAGNETIC WAVES
CHAPTER 9 ELECTROMAGNETIC WAVES Outlines 1. Waves in one dimension 2. Electromagnetic Waves in Vacuum 3. Electromagnetic waves in Matter 4. Absorption and Dispersion 5. Guided Waves 2 Skip 9.1.1 and 9.1.2
More informationChap. 1 Fundamental Concepts
NE 2 Chap. 1 Fundamental Concepts Important Laws in Electromagnetics Coulomb s Law (1785) Gauss s Law (1839) Ampere s Law (1827) Ohm s Law (1827) Kirchhoff s Law (1845) Biot-Savart Law (1820) Faradays
More informationElectromagnetic Waves
May 7, 2008 1 1 J.D.Jackson, Classical Electrodynamics, 2nd Edition, Section 7 Maxwell Equations In a region of space where there are no free sources (ρ = 0, J = 0), Maxwell s equations reduce to a simple
More informationOverview in Images. 5 nm
Overview in Images 5 nm K.S. Min et al. PhD Thesis K.V. Vahala et al, Phys. Rev. Lett, 85, p.74 (000) J. D. Joannopoulos, et al, Nature, vol.386, p.143-9 (1997) S. Lin et al, Nature, vol. 394, p. 51-3,
More informationBasics of electromagnetic response of materials
Basics of electromagnetic response of materials Microscopic electric and magnetic field Let s point charge q moving with velocity v in fields e and b Force on q: F e F qeqvb F m Lorenz force Microscopic
More information(a) Show that the amplitudes of the reflected and transmitted waves, corrrect to first order
Problem 1. A conducting slab A plane polarized electromagnetic wave E = E I e ikz ωt is incident normally on a flat uniform sheet of an excellent conductor (σ ω) having thickness D. Assume that in space
More informationElectromagnetic fields and waves
Electromagnetic fields and waves Maxwell s rainbow Outline Maxwell s equations Plane waves Pulses and group velocity Polarization of light Transmission and reflection at an interface Macroscopic Maxwell
More informationAntennas and Propagation. Chapter 2: Basic Electromagnetic Analysis
Antennas and Propagation : Basic Electromagnetic Analysis Outline Vector Potentials, Wave Equation Far-field Radiation Duality/Reciprocity Transmission Lines Antennas and Propagation Slide 2 Antenna Theory
More informationIntroduction to Polarization
Phone: Ext 659, E-mail: hcchui@mail.ncku.edu.tw Fall/007 Introduction to Polarization Text Book: A Yariv and P Yeh, Photonics, Oxford (007) 1.6 Polarization States and Representations (Stokes Parameters
More informationMUDRA PHYSICAL SCIENCES
MUDRA PHYSICAL SCIENCES VOLUME- PART B & C MODEL QUESTION BANK FOR THE TOPICS:. Electromagnetic Theory UNIT-I UNIT-II 7 4. Quantum Physics & Application UNIT-I 8 UNIT-II 97 (MCQs) Part B & C Vol- . Electromagnetic
More information4. The interaction of light with matter
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,
More informationElectromagnetic Waves Retarded potentials 2. Energy and the Poynting vector 3. Wave equations for E and B 4. Plane EM waves in free space
Electromagnetic Waves 1 1. Retarded potentials 2. Energy and the Poynting vector 3. Wave equations for E and B 4. Plane EM waves in free space 1 Retarded Potentials For volume charge & current = 1 4πε
More informationEM waves: energy, resonators. Scalar wave equation Maxwell equations to the EM wave equation A simple linear resonator Energy in EM waves 3D waves
EM waves: energy, resonators Scalar wave equation Maxwell equations to the EM wave equation A simple linear resonator Energy in EM waves 3D waves Simple scalar wave equation 2 nd order PDE 2 z 2 ψ (z,t)
More informationOn Impossibility of Negative Refraction
On Impossibility of Negative Refraction Vadim A. Markel Radiology/Bioengeneering, UPenn, Philly REFERENCES: V.A.Markel, Correct definition of the Poynting vector in electrically and magnetically polarizable
More information4 Electric Fields in Matter
4 Electric Fields in Matter 4.1 Parity and Time Reversal: Lecture 10 (a) We discussed how fields transform under parity and time reversal. A useful table is Quantity Parity Time Reversal t Even Odd r Odd
More informationOptical Imaging Chapter 5 Light Scattering
Optical Imaging Chapter 5 Light Scattering Gabriel Popescu University of Illinois at Urbana-Champaign Beckman Institute Quantitative Light Imaging Laboratory http://light.ece.uiuc.edu Principles of Optical
More informationElectromagnetic Waves. Chapter 33 (Halliday/Resnick/Walker, Fundamentals of Physics 8 th edition)
PH 222-3A Spring 2007 Electromagnetic Waves Lecture 22 Chapter 33 (Halliday/Resnick/Walker, Fundamentals of Physics 8 th edition) 1 Chapter 33 Electromagnetic Waves Today s information age is based almost
More informationELECTROMAGNETISM SUMMARY
Review of E and B ELECTROMAGNETISM SUMMARY (Rees Chapters 2 and 3) The electric field E is a vector function. E q o q If we place a second test charged q o in the electric field of the charge q, the two
More informationChapter 33. Electromagnetic Waves
Chapter 33 Electromagnetic Waves Today s information age is based almost entirely on the physics of electromagnetic waves. The connection between electric and magnetic fields to produce light is own of
More informationMechanical effects of light spin and orbital angular momentum in liquid crystals
Mechanical effects of light spin and orbital angular momentum in liquid crystals E.Santamato Università degli Studi di Napoli Federico II Dipartimento di Scienze Fisiche Complesso di Monte S. Angelo via
More informationElectromagnetic Waves Across Interfaces
Lecture 1: Foundations of Optics Outline 1 Electromagnetic Waves 2 Material Properties 3 Electromagnetic Waves Across Interfaces 4 Fresnel Equations 5 Brewster Angle 6 Total Internal Reflection Christoph
More informationIV. Electromagnetic optics. Microscopic & macroscopic fields, potentials, waves FARADAY S LAW AMPÈRE S LAW (GENERALIZED) COULOMB S LAW
IV. Electromagnetic optics Microscopic & macroscopic fields, potentials, waves Heinrich Hertz (1857-1894) groundbreaking experiment in 1885 revealed that light is electromagnetic radiation, the theoretical
More informationin Electromagnetics Numerical Method Introduction to Electromagnetics I Lecturer: Charusluk Viphavakit, PhD
2141418 Numerical Method in Electromagnetics Introduction to Electromagnetics I Lecturer: Charusluk Viphavakit, PhD ISE, Chulalongkorn University, 2 nd /2018 Email: charusluk.v@chula.ac.th Website: Light
More information4. Energy, Power, and Photons
4. Energy, Power, and Photons Energy in a light wave Why we can often neglect the magnetic field Poynting vector and irradiance The quantum nature of light Photon energy and photon momentum An electromagnetic
More informationChapter Three: Propagation of light waves
Chapter Three Propagation of Light Waves CHAPTER OUTLINE 3.1 Maxwell s Equations 3.2 Physical Significance of Maxwell s Equations 3.3 Properties of Electromagnetic Waves 3.4 Constitutive Relations 3.5
More informationPH 222-2C Fall Electromagnetic Waves Lectures Chapter 33 (Halliday/Resnick/Walker, Fundamentals of Physics 8 th edition)
PH 222-2C Fall 2012 Electromagnetic Waves Lectures 21-22 Chapter 33 (Halliday/Resnick/Walker, Fundamentals of Physics 8 th edition) 1 Chapter 33 Electromagnetic Waves Today s information age is based almost
More informationfiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES
Content-ELECTRICITY AND MAGNETISM 1. Electrostatics (1-58) 1.1 Coulomb s Law and Superposition Principle 1.1.1 Electric field 1.2 Gauss s law 1.2.1 Field lines and Electric flux 1.2.2 Applications 1.3
More information9 The conservation theorems: Lecture 23
9 The conservation theorems: Lecture 23 9.1 Energy Conservation (a) For energy to be conserved we expect that the total energy density (energy per volume ) u tot to obey a conservation law t u tot + i
More informationECE357H1S ELECTROMAGNETIC FIELDS TERM TEST March 2016, 18:00 19:00. Examiner: Prof. Sean V. Hum
UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING The Edward S. Rogers Sr. Department of Electrical and Computer Engineering ECE357H1S ELECTROMAGNETIC FIELDS TERM TEST 2 21 March 2016, 18:00
More informationElectromagnetically Induced Flows in Water
Electromagnetically Induced Flows in Water Michiel de Reus 8 maart 213 () Electromagnetically Induced Flows 1 / 56 Outline 1 Introduction 2 Maxwell equations Complex Maxwell equations 3 Gaussian sources
More informationElectromagnetic Theory (Hecht Ch. 3)
Phys 531 Lecture 2 30 August 2005 Electromagnetic Theory (Hecht Ch. 3) Last time, talked about waves in general wave equation: 2 ψ(r, t) = 1 v 2 2 ψ t 2 ψ = amplitude of disturbance of medium For light,
More informationLecture 10 February 25, 2010
Lecture 10 February 5, 010 Last time we discussed a small scatterer at origin. Interesting effects come from many small scatterers occupying a region of size d large compared to λ. The scatterer j at position
More information1 Photon optics! Photons and modes Photon properties Photon streams and statistics
1 Photons and modes Photon properties Photon optics! Photon streams and statistics 2 Photon optics! Photons and modes Photon properties Energy, polarization, position, momentum, interference, time Photon
More informationII Theory Of Surface Plasmon Resonance (SPR)
II Theory Of Surface Plasmon Resonance (SPR) II.1 Maxwell equations and dielectric constant of metals Surface Plasmons Polaritons (SPP) exist at the interface of a dielectric and a metal whose electrons
More informationIntroduction to Nonlinear Optics
Introduction to Nonlinear Optics Prof. Cleber R. Mendonca http://www.fotonica.ifsc.usp.br Outline Linear optics Introduction to nonlinear optics Second order nonlinearities Third order nonlinearities Two-photon
More informationElectromagnetic Waves
Electromagnetic Waves Maxwell s equations predict the propagation of electromagnetic energy away from time-varying sources (current and charge) in the form of waves. Consider a linear, homogeneous, isotropic
More informationLecture Notes on Wave Optics (03/05/14) 2.71/2.710 Introduction to Optics Nick Fang
Outline: A. Electromagnetism B. Frequency Domain (Fourier transform) C. EM waves in Cartesian coordinates D. Energy Flow and Poynting Vector E. Connection to geometrical optics F. Eikonal Equations: Path
More informationMicroscopic-Macroscopic connection. Silvana Botti
relating experiment and theory European Theoretical Spectroscopy Facility (ETSF) CNRS - Laboratoire des Solides Irradiés Ecole Polytechnique, Palaiseau - France Temporary Address: Centre for Computational
More informationCBSE Sample Paper 8. c = ms -1 h = Js e = C
1 CBSE Sample Paper 8 General Instruction: 1. Answer all questions 2. Internal choices are provided for some questions 3. Question numbers 1 to 8 are very short answer questions and carry 1 mark each.
More informationGeneral review: - a) Dot Product
General review: - a) Dot Product If θ is the angle between the vectors a and b, then a b = a b cos θ NOTE: Two vectors a and b are orthogonal, if and only if a b = 0. Properties of the Dot Product If a,
More informationCourse Secretary: Christine Berber O3.095, phone x-6351,
IMPRS: Ultrafast Source Technologies Franz X. Kärtner (Umit Demirbas) & Thorsten Uphues, Bldg. 99, O3.097 & Room 6/3 Email & phone: franz.kaertner@cfel.de, 040 8998 6350 thorsten.uphues@cfel.de, 040 8998
More informationPhysics of Condensed Matter I
Physics of Condensed Matter I 1100-4INZ`PC Faculty of Physics UW Jacek.Szczytko@fuw.edu.pl Dictionary D = εe ε 0 vacuum permittivity, permittivity of free space (przenikalność elektryczna próżni) ε r relative
More informationElectrodynamics I Final Exam - Part A - Closed Book KSU 2005/12/12 Electro Dynamic
Electrodynamics I Final Exam - Part A - Closed Book KSU 2005/12/12 Name Electro Dynamic Instructions: Use SI units. Short answers! No derivations here, just state your responses clearly. 1. (2) Write an
More information1. Electricity and Magnetism (Fall 1995, Part 1) A metal sphere has a radius R and a charge Q.
1. Electricity and Magnetism (Fall 1995, Part 1) A metal sphere has a radius R and a charge Q. (a) Compute the electric part of the Maxwell stress tensor T ij (r) = 1 {E i E j 12 } 4π E2 δ ij both inside
More informationModern Optics Prof. Partha Roy Chaudhuri Department of Physics Indian Institute of Technology, Kharagpur
Modern Optics Prof. Partha Roy Chaudhuri Department of Physics Indian Institute of Technology, Kharagpur Lecture 08 Wave propagation in anisotropic media Now, we will discuss the propagation of electromagnetic
More informationChapter 1 Mathematical Foundations
Computational Electromagnetics; Chapter 1 1 Chapter 1 Mathematical Foundations 1.1 Maxwell s Equations Electromagnetic phenomena can be described by the electric field E, the electric induction D, the
More informationCharacterization of Left-Handed Materials
Characterization of Left-Handed Materials Massachusetts Institute of Technology 6.635 lecture notes 1 Introduction 1. How are they realized? 2. Why the denomination Left-Handed? 3. What are their properties?
More informationPHYS4210 Electromagnetic Theory Quiz 1 Feb 2010
PHYS4210 Electromagnetic Theory Quiz 1 Feb 2010 1. An electric dipole is formed from two charges ±q separated by a distance b. For large distances r b from the dipole, the electric potential falls like
More informationANTENNAS. Vector and Scalar Potentials. Maxwell's Equations. E = jωb. H = J + jωd. D = ρ (M3) B = 0 (M4) D = εe
ANTENNAS Vector and Scalar Potentials Maxwell's Equations E = jωb H = J + jωd D = ρ B = (M) (M) (M3) (M4) D = εe B= µh For a linear, homogeneous, isotropic medium µ and ε are contant. Since B =, there
More informationMassachusetts Institute of Technology Physics 8.03 Practice Final Exam 3
Massachusetts Institute of Technology Physics 8.03 Practice Final Exam 3 Instructions Please write your solutions in the white booklets. We will not grade anything written on the exam copy. This exam is
More informationarxiv: v1 [physics.optics] 30 Mar 2010
Analytical vectorial structure of non-paraxial four-petal Gaussian beams in the far field Xuewen Long a,b, Keqing Lu a, Yuhong Zhang a,b, Jianbang Guo a,b, and Kehao Li a,b a State Key Laboratory of Transient
More informationOptical Properties of Left-Handed Materials by Nathaniel Ferraro 01
Optical Properties of Left-Handed Materials by Nathaniel Ferraro 1 Abstract Recently materials with the unusual property of having a simultaneously negative permeability and permittivity have been tested
More information12. Nonlinear optics I
1. Nonlinear optics I What are nonlinear-optical effects and why do they occur? Maxwell's equations in a medium Nonlinear-optical media Second-harmonic generation Conservation laws for photons ("Phasematching")
More informationChapter 4 Wave Equations
Chapter 4 Wave Equations Lecture Notes for Modern Optics based on Pedrotti & Pedrotti & Pedrotti Instructor: Nayer Eradat Spring 2009 3/11/2009 Wave Equations 1 Wave Equation Chapter Goal: developing the
More informationElectrodynamics Qualifier Examination
Electrodynamics Qualifier Examination January 10, 2007 1. This problem deals with magnetostatics, described by a time-independent magnetic field, produced by a current density which is divergenceless,
More informationElectromagnetic Wave Propagation Lecture 8: Propagation in birefringent media
Electromagnetic Wave Propagation Lecture 8: Propagation in birefringent media Daniel Sjöberg Department of Electrical and Information Technology September 27, 2012 Outline 1 Introduction 2 Maxwell s equations
More informationarxiv: v1 [math-ph] 3 Nov 2011
Formalism of operators for Laguerre-Gauss modes A. L. F. da Silva (α), A. T. B. Celeste (β), M. Pazetti (γ), C. E. F. Lopes (δ) (α,β) Instituto Federal do Sertão Pernambucano, Petrolina - PE, Brazil (γ)
More informationElectromagnetic Theory for Microwaves and Optoelectronics
Keqian Zhang Dejie Li Electromagnetic Theory for Microwaves and Optoelectronics Translated by authors With 259 Figures Springer Contents 1 Basic Electromagnetic Theory 1 1.1 Maxwell's Equations 1 1.1.1
More informationFundamentals on light scattering, absorption and thermal radiation, and its relation to the vector radiative transfer equation
Fundamentals on light scattering, absorption and thermal radiation, and its relation to the vector radiative transfer equation Klaus Jockers November 11, 2014 Max-Planck-Institut für Sonnensystemforschung
More informationTypical anisotropies introduced by geometry (not everything is spherically symmetric) temperature gradients magnetic fields electrical fields
Lecture 6: Polarimetry 1 Outline 1 Polarized Light in the Universe 2 Fundamentals of Polarized Light 3 Descriptions of Polarized Light Polarized Light in the Universe Polarization indicates anisotropy
More informationChapter 31 Maxwell s Equations and Electromagnetic Waves. Copyright 2009 Pearson Education, Inc.
Chapter 31 Maxwell s Equations and Electromagnetic Waves Units of Chapter 31 Changing Electric Fields Produce Magnetic Fields; Ampère s Law and Displacement Current Gauss s Law for Magnetism Maxwell s
More informationLasers and Electro-optics
Lasers and Electro-optics Second Edition CHRISTOPHER C. DAVIS University of Maryland III ^0 CAMBRIDGE UNIVERSITY PRESS Preface to the Second Edition page xv 1 Electromagnetic waves, light, and lasers 1
More informationNonlinear optics: a back-to-basics primer Lecture 1: linear optics
Guoqing (Noah) Chang, October 9, 15 Nonlinear optics: a back-to-basics primer Lecture 1: linear optics 1 Suggested references Robert W. Boyd, Nonlinear optics (8) Geoffrey New, Introduction to nonlinear
More informationIntroduction to the School
Lucio Crivellari Instituto de Astrofísica de Canarias D.pto de Astrofísica, Universidad de La Laguna & INAF Osservatorio Astronomico di Trieste (Italy) Introduction to the School 10/11/17 1 Setting the
More informationRadio Propagation Channels Exercise 2 with solutions. Polarization / Wave Vector
/8 Polarization / Wave Vector Assume the following three magnetic fields of homogeneous, plane waves H (t) H A cos (ωt kz) e x H A sin (ωt kz) e y () H 2 (t) H A cos (ωt kz) e x + H A sin (ωt kz) e y (2)
More informationScattering of ECRF waves by edge density fluctuations and blobs
PSFC/JA-14-7 Scattering of ECRF waves by edge density fluctuations and blobs A. K. Ram and K. Hizanidis a June 2014 Plasma Science and Fusion Center, Massachusetts Institute of Technology Cambridge, MA
More informationarxiv: v1 [physics.class-ph] 8 Apr 2019
Representation Independent Boundary Conditions for a Piecewise-Homogeneous Linear Magneto-dielectric Medium arxiv:1904.04679v1 [physics.class-ph] 8 Apr 019 Michael E. Crenshaw 1 Charles M. Bowden Research
More informationSummary of Fourier Optics
Summary of Fourier Optics Diffraction of the paraxial wave is described by Fresnel diffraction integral, u(x, y, z) = j λz dx 0 dy 0 u 0 (x 0, y 0 )e j(k/2z)[(x x 0) 2 +(y y 0 ) 2 )], Fraunhofer diffraction
More informationMetamaterials. Peter Hertel. University of Osnabrück, Germany. Lecture presented at APS, Nankai University, China
University of Osnabrück, Germany Lecture presented at APS, Nankai University, China http://www.home.uni-osnabrueck.de/phertel Spring 2012 are produced artificially with strange optical properties for instance
More informationElectromagnetism II Lecture 7
Electromagnetism II Lecture 7 Instructor: Andrei Sirenko sirenko@njit.edu Spring 13 Thursdays 1 pm 4 pm Spring 13, NJIT 1 Previous Lecture: Conservation Laws Previous Lecture: EM waves Normal incidence
More informationHomework 1. Nano Optics, Fall Semester 2017 Photonics Laboratory, ETH Zürich
Homework 1 Contact: mfrimmer@ethz.ch Due date: Friday 13.10.2017; 10:00 a.m. Nano Optics, Fall Semester 2017 Photonics Laboratory, ETH Zürich www.photonics.ethz.ch The goal of this homework is to establish
More informationYOUR NAME Sample Final Physics 1404 (Dr. Huang)), Correct answers are underlined.
YOUR NAME Sample Final Physics 1404 (Dr. Huang)), Correct answers are underlined. Useful constants: e=1.6 10-19 C, m e =9.1 10-31 kg, m p =1.67 10-27 kg, ε 0 =8.85 10-12 C 2 /N m 2, c=3 10 8 m/s k e =8.99
More informationExam in TFY4240 Electromagnetic Theory Wednesday Dec 9, :00 13:00
NTNU Page 1 of 5 Institutt for fysikk Contact during the exam: Paul Anton Letnes Telephone: Office: 735 93 648, Mobile: 98 62 08 26 Exam in TFY4240 Electromagnetic Theory Wednesday Dec 9, 2009 09:00 13:00
More informationPHYS 408, Optics. Problem Set 1 - Spring Posted: Fri, January 8, 2015 Due: Thu, January 21, 2015.
PHYS 408, Optics Problem Set 1 - Spring 2016 Posted: Fri, January 8, 2015 Due: Thu, January 21, 2015. 1. An electric field in vacuum has the wave equation, Let us consider the solution, 2 E 1 c 2 2 E =
More informationMaxwell s Equations:
Course Instructor Dr. Raymond C. Rumpf Office: A-337 Phone: (915) 747-6958 E-Mail: rcrumpf@utep.edu Maxwell s Equations: Terms & Definitions EE-3321 Electromagnetic Field Theory Outline Maxwell s Equations
More informationProblem set 3. Electromagnetic waves
Second Year Electromagnetism Michaelmas Term 2017 Caroline Terquem Problem set 3 Electromagnetic waves Problem 1: Poynting vector and resistance heating This problem is not about waves but is useful to
More informationElectromagnetic Theory for Microwaves and Optoelectronics
Keqian Zhang Dejie Li Electromagnetic Theory for Microwaves and Optoelectronics Second Edition With 280 Figures and 13 Tables 4u Springer Basic Electromagnetic Theory 1 1.1 Maxwell's Equations 1 1.1.1
More informationELECTROMAGNETIC WAVES
UNIT V ELECTROMAGNETIC WAVES Weightage Marks : 03 Displacement current, electromagnetic waves and their characteristics (qualitative ideas only). Transverse nature of electromagnetic waves. Electromagnetic
More informationInvisible Random Media And Diffraction Gratings That Don't Diffract
Invisible Random Media And Diffraction Gratings That Don't Diffract 29/08/2017 Christopher King, Simon Horsley and Tom Philbin, University of Exeter, United Kingdom, email: cgk203@exeter.ac.uk webpage:
More informationand the radiation from source 2 has the form. The vector r points from the origin to the point P. What will the net electric field be at point P?
Physics 3 Interference and Interferometry Page 1 of 6 Interference Imagine that we have two or more waves that interact at a single point. At that point, we are concerned with the interaction of those
More informationb) Derive the charge-current continuity equation for free charge-density (, ) and free current-density (, ) from Maxwell s microscopic equations.
Fall 205 Written Comprehensive Exam Opti 50 System of units: MKSA 2Pts a) The charge-current continuity equation is written (, )+ (, ) =0. Explain in a few sentences the physical meaning of the equation
More informationLaser Beam Interactions with Solids In absorbing materials photons deposit energy hc λ. h λ. p =
Laser Beam Interactions with Solids In absorbing materials photons deposit energy E = hv = hc λ where h = Plank's constant = 6.63 x 10-34 J s c = speed of light Also photons also transfer momentum p p
More informationMassachusetts Institute of Technology Physics 8.03SC Fall 2016 Homework 9
Massachusetts Institute of Technology Physics 8.03SC Fall 016 Homework 9 Problems Problem 9.1 (0 pts) The ionosphere can be viewed as a dielectric medium of refractive index ωp n = 1 ω Where ω is the frequency
More information1. In Young s double slit experiment, when the illumination is white light, the higherorder fringes are in color.
TRUE-FALSE STATEMENTS: ELECTRICITY: 1. Electric field lines originate on negative charges. 2. The flux of the electric field over a closed surface is proportional to the net charge enclosed by the surface.
More informationAlong with C1 the magnetic field is also observed at location C 2 though no current is threading through this loop.
Displacement current British physicist James C. Maxwell gave final shape to all phenomenon connecting electricity and magnetism. He noticed an inconsistency in Ampere s Law connecting Electric current
More informationPhysics of Light and Optics
Physics of Light and Optics Justin Peatross and Harold Stokes Brigham Young University Department of Physics and Astronomy All Publication Rights Reserved (2001) Revised April 2002 This project is supported
More informationMultilayer Reflectivity
Multilayer Reflectivity John E. Davis jed@jedsoft.org January 5, 2014 1 Introduction The purpose of this document is to present an ab initio derivation of the reflectivity for a plane electromagnetic wave
More information