Radio Propagation Channels Exercise 2 with solutions. Polarization / Wave Vector

Size: px
Start display at page:

Download "Radio Propagation Channels Exercise 2 with solutions. Polarization / Wave Vector"

Transcription

1 /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) H 3 (t) H B cos (ωt kz) e x + H B sin (ωt kz) e y, (3) where H A > 0, H B > 0, H A < H B. Problem. Determine the phasor of the magnetic field H..2 What is the polarization type of the magnetic fields H (t) and H 2 (t)?.3 What is the polarization type of the superposition of H (t) and H 2 (t)?.4 What is the polarization type of the superposition of H (t) and H 3 (t)? Given is the following electric and magnetic field of a homogeneous, plane wave that propagates in a dielectric fluid with ε r 3 and µ r ( ) E E 2 e x e y + 2e z e jk r (4) Problem 2 H H ( 2e x + e y + e z ) e jk r. (5) 2. In which direction does the wave propagate? Determine the normalized vector n, where k k n (k is the wave number in the dielectric fluid). 2.2 Determine the wave vector k. 2.3 Determine the phase velocity v ph of a wave propagating in this dielectric fluid.

2 2/8 Solution of Problem (. Using sin (ωt) cos ωt π ) and H A > 0, H (t) can be rewritten as 2 ( H (t) H A cos (ωt kz) e x H A cos ωt kz π ) e y 2 } Re {H A e j(ωt kz) e x H A e j(ωt kz π 2 ) ey (HA ) } Re{ e jkz e x H A e j π 2 }{{} e jkz e y e jωt j { (HA ) Re e jkz e x + jh A e jkz e y e jωt} { } Re H A (e x + je y ) e jkz e jωt. }{{} H Therefore, the phasor of the magnetic field H is H H A (e x + je y ) e jkz..2 The period of H (t) H A cos (ωt kz) e x H A sin (ωt kz) e y is T 2π ω. Now consider the constant phase plane z 0 and the different time instants t 0, T 4, T 2, 3T 4, we can get with H A > 0 H (t 0, z 0) H A cos 0 e x H A sin 0 e y H A e x point at the positive x-axis, H (t T 4, z 0) H A cos π 2 e x H A sin π 2 e y H A e y point at the negative y-axis, H (t T 2, z 0) H A cos π e x H A sin π e y H A e x point at the negative x-axis, H (t 3T 4, z 0) H A cos 3π 2 e x H A sin 3π 2 e y H A e y point at the positive y-axis. Figure illustrates the polarization type of H (t). Observe the figure and note the clockwise rotation direction with the same length H A we can draw the following conclusion that H (t) is left circularly polarized.

3 3/8 y H (t 3T 4, z 0) H Ae y H (t T 2, z 0) H Ae x z 0 clockwise rotation x H (t 0, z 0) H A e x H (t T 4, z 0) H Ae y Figure : Polarization type of H (t). For magnetic field H 2 (t) H A cos (ωt kz) e x + H A sin (ωt kz) e y, analogously at the constant phase plane z 0 and t 0, T 4, T 2, 3T 4, we can get with H A > 0 H 2 (t 0, z 0) H A cos 0 e x + H A sin 0 e y H A e x point at the positive x-axis, H 2 (t T 4, z 0) H A cos π 2 e x + H A sin π 2 e y H A e y point at the positive y-axis, H 2 (t T 2, z 0) H A cos π e x + H A sin π e y H A e x point at the negative x-axis, H 2 (t 3T 4, z 0) H A cos 3π 2 e x + H A sin 3π 2 e y H A e y point at the negative y-axis. Note the anticlockwise rotation direction with the same length H A we can draw the following conclusion that H 2 (t) is right circularly polarized..3 H (t) + H 2 (t) 2H A cos (ωt kz) e x is also a solution of the wave equation. Consider the equiphase plane z 0 and t 0, T 4, T 2, 3T 4, we can get with H A > 0 H (t 0, z 0) + H 2 (t 0, z 0) 2H A cos 0 e x 2H A e x H (t T 4, z 0) + H 2(t T 4, z 0) 2H A cos π 2 e x 0 H (t T 2, z 0) + H 2(t T 2, z 0) 2H A cos π e x 2H A e x

4 4/8 H (t 3T 2, z 0) + H 2(t 3T 2, z 0) 2H A cos 3π 2 e x 0. Vectors always lie on the same line (x-axis) linearly polarized..4 H (t)+h 3 (t) (H A + H B ) cos (ωt kz) e x +(H B H A ) sin (ωt kz) e y. Consider the equiphase plane z 0 and time instants t 0, T 4, T 2, 3T, we can get with 4 H A > 0, H B > 0, H A < H B H B H A > 0 H (t 0, z 0) + H 3 (t 0, z 0) (H A + H B ) e x H (t T 4, z 0) + H 3(t T 4, z 0) (H B H A ) e y H (t T 2, z 0) + H 3(t T 2, z 0) (H A + H B ) e x H (t 3T 4, z 0) + H 3(t 3T 4, z 0) (H B H A ) e y. y (H B H A ) at (t T 4, z 0) (H A + H B ) at (t T 2, z 0) z 0 x (H A + H B ) at (t 0, z 0) (H B H A ) at (t 3T 4, z 0) Figure 2: Polarization type of H (t) + H 3 (t). Figure 2 illustrates the polarization type of H (t) + H 3 (t). Observe the figure and note the anticlockwise rotation direction with the different length we can draw the following conclusion that H (t) + H 3 (t) is right elliptically polarized.

5 5/8 Some Remarks on Planar Wave in an Arbitrary Direction From the lecture we know that a generalized planar harmonic wave propagating in an arbitrary direction, which is specified by its electric field, can be represented as E (E e + E 2 e 2 ) e jk r (6) and a time-dependent electric wave propagating along the direction of k can be expressed as E(t) Re { E e jωt} ] [E (t)e + E 2 (t)e 2 e jk r [Ê cos (ωt + ϕ ) e + Ê2 cos (ωt + ϕ 2 ) e 2 ] e jk r (7) with e k 0, e 2 k 0 and e e 2 0, i.e., they follow the right hand orthogonal rule, see Figure 3. e 2 e k Figure 3: Right hand orthogonal rule of e, e 2 and k. In the cartesian coordinate system, a location vector r can be represented as k is vector wave number and defined as r xe x + ye y + ze z. k k x e x + k y e y + k z e z with relationship to scalar wave number k k k k 2 k k 2 x + k 2 y + k 2 z. We have found that the fields of the electromagnetic wave are perpendicular to each other, and that they are also perpendicular (or transverse) to the direction of propagation k.

6 6/8 Electromagnetic power flows with the wave along the direction of propagation and it is also constant on the equiphase planes. The power density is described by the time dependent Poynting vector P(t) E(t) H(t). The Poynting vector is perpendicular to both field components, and is parallel to the direction of wave propagation. It means that the following relationships hold true. k E, k H, E H and k (E H P) Solution of Problem 2 2. A homogeneous planar wave one holds n E, n H, E H n (E H) as illustrated in Figure 4. H E k k n Figure 4: Right hand orthogonal rule of E, H and k k n. Since E and H are constants and e jk r denotes a certain phase, we can obtain (E H) ( ) 2 e x e y + 2e z ( 2e x + e y + e z ) (E H) e x e y e z ( 2 3e x e y + 3 ) 2 e z. 2 Hence, n ( 3e x e y + 3 ) 2 e z.

7 7/8 Normalization n yields (both propagation directions are possible!) n ± ( ) 2 ( ) (3e x e y + 3 ) 2 e z n ± 4 (2e x + 3e y + e z ). 2.2 k ω ε µ ω ε r ε 0 µ r µ 0 ε r µ r ω ε 0 µ 0 }{{}}{{} 3 k 0 where k 0 is the scalar wave number of vacuum. Therefore, 3k0, k k n 3 3k 0 n ± 4 k 0 (2e x + 3e y + e z ). 2.3 A time dependent electric field E (t) can be described as E (t) Re { E e jωt} Re { E e jk r e jωt} Re { E e j(ωt k r) } E cos (ωt k r), where E E ( 2 e x e y + 2e z ) and the planar electromagnetic wave propagates along the direction of k. Figure 5: Example of a homogeneous planar wave E + x (z, t), which is propagating along the positive z-axis in vacuum. Here, β 0 is the wave number of vacuum.

8 8/8 For reasons of simplicity, we assume the wave with the electric field E (t) E cos (ωt k z) E cos (ωt k z), which propagates along the positive z-axis. For the isotropic medium one can assume that the wave does not change behavior with its direction. See Figure 5. Looking at equiphase planes, one obtains Thus, we can get ωt k z ωt k z constant. v ph dz dt ω k ω ω ε µ ε0 µ 0 εr µ r c c. εr µ r 3

Electromagnetic Waves For fast-varying phenomena, the displacement current cannot be neglected, and the full set of Maxwell s equations must be used

Electromagnetic Waves For fast-varying phenomena, the displacement current cannot be neglected, and the full set of Maxwell s equations must be used Electromagnetic Waves For fast-varying phenomena, the displacement current cannot be neglected, and the full set of Maxwell s equations must be used B( t) E = dt D t H = J+ t D =ρ B = 0 D=εE B=µ H () F

More information

Waves. Daniel S. Weile. ELEG 648 Waves. Department of Electrical and Computer Engineering University of Delaware. Plane Waves Reflection of Waves

Waves. Daniel S. Weile. ELEG 648 Waves. Department of Electrical and Computer Engineering University of Delaware. Plane Waves Reflection of Waves Waves Daniel S. Weile Department of Electrical and Computer Engineering University of Delaware ELEG 648 Waves Outline Outline Introduction Let s start by introducing simple solutions to Maxwell s equations

More information

APPLIED OPTICS POLARIZATION

APPLIED OPTICS POLARIZATION A. La Rosa Lecture Notes APPLIED OPTICS POLARIZATION Linearly-polarized light Description of linearly polarized light (using Real variables) Alternative description of linearly polarized light using phasors

More information

Electromagnetic Wave Propagation Lecture 3: Plane waves in isotropic and bianisotropic media

Electromagnetic Wave Propagation Lecture 3: Plane waves in isotropic and bianisotropic media Electromagnetic Wave Propagation Lecture 3: Plane waves in isotropic and bianisotropic media Daniel Sjöberg Department of Electrical and Information Technology September 2016 Outline 1 Plane waves in lossless

More information

Uniform Plane Waves Page 1. Uniform Plane Waves. 1 The Helmholtz Wave Equation

Uniform Plane Waves Page 1. Uniform Plane Waves. 1 The Helmholtz Wave Equation Uniform Plane Waves Page 1 Uniform Plane Waves 1 The Helmholtz Wave Equation Let s rewrite Maxwell s equations in terms of E and H exclusively. Let s assume the medium is lossless (σ = 0). Let s also assume

More information

ELE 3310 Tutorial 10. Maxwell s Equations & Plane Waves

ELE 3310 Tutorial 10. Maxwell s Equations & Plane Waves ELE 3310 Tutorial 10 Mawell s Equations & Plane Waves Mawell s Equations Differential Form Integral Form Faraday s law Ampere s law Gauss s law No isolated magnetic charge E H D B B D J + ρ 0 C C E r dl

More information

APPLIED OPTICS POLARIZATION

APPLIED OPTICS POLARIZATION A. La Rosa Lecture Notes APPLIED OPTICS POLARIZATION Linearly-polarized light Description of linearly polarized light (using Real variables) Alternative description of linearly polarized light using phasors

More information

Light Waves and Polarization

Light Waves and Polarization Light Waves and Polarization Xavier Fernando Ryerson Communications Lab http://www.ee.ryerson.ca/~fernando The Nature of Light There are three theories explain the nature of light: Quantum Theory Light

More information

Plane Waves Part II. 1. For an electromagnetic wave incident from one medium to a second medium, total reflection takes place when

Plane Waves Part II. 1. For an electromagnetic wave incident from one medium to a second medium, total reflection takes place when Plane Waves Part II. For an electromagnetic wave incident from one medium to a second medium, total reflection takes place when (a) The angle of incidence is equal to the Brewster angle with E field perpendicular

More information

Electromagnetic Waves & Polarization

Electromagnetic Waves & Polarization Course Instructor Dr. Raymond C. Rumpf Office: A 337 Phone: (915) 747 6958 E Mail: rcrumpf@utep.edu EE 4347 Applied Electromagnetics Topic 3a Electromagnetic Waves & Polarization Electromagnetic These

More information

Electromagnetic Waves

Electromagnetic 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 information

Introduction to Polarization

Introduction 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 information

Multilayer Reflectivity

Multilayer 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

Electromagnetic 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 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 information

Electromagnetic (EM) Waves

Electromagnetic (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 information

Waves in Linear Optical Media

Waves in Linear Optical Media 1/53 Waves in Linear Optical Media Sergey A. Ponomarenko Dalhousie University c 2009 S. A. Ponomarenko Outline Plane waves in free space. Polarization. Plane waves in linear lossy media. Dispersion relations

More information

A Review of Basic Electromagnetic Theories

A Review of Basic Electromagnetic Theories A Review of Basic Electromagnetic Theories 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)

More information

Electromagnetic Waves

Electromagnetic Waves Chapter 32 Electromagnetic Waves PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson Goals for Chapter 32 To learn why a light

More information

Lecture 4: Polarisation of light, introduction

Lecture 4: Polarisation of light, introduction Lecture 4: Polarisation of light, introduction Lecture aims to explain: 1. Light as a transverse electro-magnetic wave 2. Importance of polarisation of light 3. Linearly polarised light 4. Natural light

More information

Chap. 1 Fundamental Concepts

Chap. 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 information

Electromagnetic Waves

Electromagnetic 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 information

Class 30: Outline. Hour 1: Traveling & Standing Waves. Hour 2: Electromagnetic (EM) Waves P30-

Class 30: Outline. Hour 1: Traveling & Standing Waves. Hour 2: Electromagnetic (EM) Waves P30- Class 30: Outline Hour 1: Traveling & Standing Waves Hour : Electromagnetic (EM) Waves P30-1 Last Time: Traveling Waves P30- Amplitude (y 0 ) Traveling Sine Wave Now consider f(x) = y = y 0 sin(kx): π

More information

Wavepackets. Outline. - Review: Reflection & Refraction - Superposition of Plane Waves - Wavepackets - ΔΔk Δx Relations

Wavepackets. Outline. - Review: Reflection & Refraction - Superposition of Plane Waves - Wavepackets - ΔΔk Δx Relations Wavepackets Outline - Review: Reflection & Refraction - Superposition of Plane Waves - Wavepackets - ΔΔk Δx Relations 1 Sample Midterm (one of these would be Student X s Problem) Q1: Midterm 1 re-mix (Ex:

More information

H ( E) E ( H) = H B t

H ( 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 information

Wave Propagation in Uniaxial Media. Reflection and Transmission at Interfaces

Wave Propagation in Uniaxial Media. Reflection and Transmission at Interfaces Lecture 5: Crystal Optics Outline 1 Homogeneous, Anisotropic Media 2 Crystals 3 Plane Waves in Anisotropic Media 4 Wave Propagation in Uniaxial Media 5 Reflection and Transmission at Interfaces Christoph

More information

Simple medium: D = ɛe Dispersive medium: D = ɛ(ω)e Anisotropic medium: Permittivity as a tensor

Simple medium: D = ɛe Dispersive medium: D = ɛ(ω)e Anisotropic medium: Permittivity as a tensor Plane Waves 1 Review dielectrics 2 Plane waves in the time domain 3 Plane waves in the frequency domain 4 Plane waves in lossy and dispersive media 5 Phase and group velocity 6 Wave polarization Levis,

More information

Electromagnetic Waves Across Interfaces

Electromagnetic 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 information

Jackson 7.6 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell

Jackson 7.6 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell Jackson 7.6 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell PROBLEM: A plane wave of frequency ω is incident normally from vacuum on a semi-infinite slab of material

More information

Class 15 : Electromagnetic Waves

Class 15 : Electromagnetic Waves Class 15 : Electromagnetic Waves Wave equations Why do electromagnetic waves arise? What are their properties? How do they transport energy from place to place? Recap (1) In a region of space containing

More information

Electromagnetic Waves. Chapter 33 (Halliday/Resnick/Walker, Fundamentals of Physics 8 th edition)

Electromagnetic 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 information

1 Electromagnetic concepts useful for radar applications

1 Electromagnetic concepts useful for radar applications Electromagnetic concepts useful for radar applications The scattering of electromagnetic waves by precipitation particles and their propagation through precipitation media are of fundamental importance

More information

EECS 117. Lecture 23: Oblique Incidence and Reflection. Prof. Niknejad. University of California, Berkeley

EECS 117. Lecture 23: Oblique Incidence and Reflection. Prof. Niknejad. University of California, Berkeley University of California, Berkeley EECS 117 Lecture 23 p. 1/2 EECS 117 Lecture 23: Oblique Incidence and Reflection Prof. Niknejad University of California, Berkeley University of California, Berkeley

More information

Typical anisotropies introduced by geometry (not everything is spherically symmetric) temperature gradients magnetic fields electrical fields

Typical 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 information

CHAPTER 4 ELECTROMAGNETIC WAVES IN CYLINDRICAL SYSTEMS

CHAPTER 4 ELECTROMAGNETIC WAVES IN CYLINDRICAL SYSTEMS CHAPTER 4 ELECTROMAGNETIC WAVES IN CYLINDRICAL SYSTEMS The vector Helmholtz equations satisfied by the phasor) electric and magnetic fields are where. In low-loss media and for a high frequency, i.e.,

More information

Chapter 33. Electromagnetic Waves

Chapter 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 information

remain essentially unchanged for the case of time-varying fields, the remaining two

remain essentially unchanged for the case of time-varying fields, the remaining two Unit 2 Maxwell s Equations Time-Varying Form While the Gauss law forms for the static electric and steady magnetic field equations remain essentially unchanged for the case of time-varying fields, the

More information

1 Fundamentals of laser energy absorption

1 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 information

ELE3310: Basic ElectroMagnetic Theory

ELE3310: Basic ElectroMagnetic Theory A summary for the final examination EE Department The Chinese University of Hong Kong November 2008 Outline Mathematics 1 Mathematics Vectors and products Differential operators Integrals 2 Integral expressions

More information

Electromagnetic Theory: PHAS3201, Winter Maxwell s Equations and EM Waves

Electromagnetic Theory: PHAS3201, Winter Maxwell s Equations and EM Waves Electromagnetic Theory: PHA3201, Winter 2008 5. Maxwell s Equations and EM Waves 1 Displacement Current We already have most of the pieces that we require for a full statement of Maxwell s Equations; however,

More information

Plane electromagnetic waves and Gaussian beams (Lecture 17)

Plane electromagnetic waves and Gaussian beams (Lecture 17) Plane electromagnetic waves and Gaussian beams (Lecture 17) February 2, 2016 305/441 Lecture outline In this lecture we will study electromagnetic field propagating in space free of charges and currents.

More information

Mathematical Tripos, Part IB : Electromagnetism

Mathematical Tripos, Part IB : Electromagnetism Mathematical Tripos, Part IB : Electromagnetism Proof of the result G = m B Refer to Sec. 3.7, Force and couples, and supply the proof that the couple exerted by a uniform magnetic field B on a plane current

More information

Antennas and Propagation

Antennas and Propagation Antennas and Propagation Ranga Rodrigo University of Moratuwa October 20, 2008 Compiled based on Lectures of Prof. (Mrs.) Indra Dayawansa. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation

More information

EM Waves. From previous Lecture. This Lecture More on EM waves EM spectrum Polarization. Displacement currents Maxwell s equations EM Waves

EM Waves. From previous Lecture. This Lecture More on EM waves EM spectrum Polarization. Displacement currents Maxwell s equations EM Waves EM Waves This Lecture More on EM waves EM spectrum Polarization From previous Lecture Displacement currents Maxwell s equations EM Waves 1 Reminders on waves Traveling waves on a string along x obey the

More information

EITN90 Radar and Remote Sensing Lecture 5: Target Reflectivity

EITN90 Radar and Remote Sensing Lecture 5: Target Reflectivity EITN90 Radar and Remote Sensing Lecture 5: Target Reflectivity Daniel Sjöberg Department of Electrical and Information Technology Spring 2018 Outline 1 Basic reflection physics 2 Radar cross section definition

More information

Lecture 2 Review of Maxwell s Equations and the EM Constitutive Parameters

Lecture 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 information

Physics 214 Course Overview

Physics 214 Course Overview Physics 214 Course Overview Lecturer: Mike Kagan Course topics Electromagnetic waves Optics Thin lenses Interference Diffraction Relativity Photons Matter waves Black Holes EM waves Intensity Polarization

More information

Matrix description of wave propagation and polarization

Matrix description of wave propagation and polarization Chapter Matrix description of wave propagation and polarization Contents.1 Electromagnetic waves................................... 1. Matrix description of wave propagation in linear systems..............

More information

Lecture 21 Reminder/Introduction to Wave Optics

Lecture 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 information

Propagation of Plane Waves

Propagation of Plane Waves Chapter 6 Propagation of Plane Waves 6 Plane Wave in a Source-Free Homogeneous Medium 62 Plane Wave in a Lossy Medium 63 Interference of Two Plane Waves 64 Reflection and Transmission at a Planar Interface

More information

PLANE WAVE PROPAGATION AND REFLECTION. David R. Jackson Department of Electrical and Computer Engineering University of Houston Houston, TX

PLANE WAVE PROPAGATION AND REFLECTION. David R. Jackson Department of Electrical and Computer Engineering University of Houston Houston, TX PLANE WAVE PROPAGATION AND REFLECTION David R. Jackson Department of Electrical and Computer Engineering University of Houston Houston, TX 7704-4793 Abstract The basic properties of plane waves propagating

More information

Chap. 2. Polarization of Optical Waves

Chap. 2. Polarization of Optical Waves Chap. 2. Polarization of Optical Waves 2.1 Polarization States - Direction of the Electric Field Vector : r E = E xˆ + E yˆ E x x y ( ω t kz + ϕ ), E = E ( ωt kz + ϕ ) = E cos 0 x cos x y 0 y - Role :

More information

PH 222-2C Fall Electromagnetic Waves Lectures Chapter 33 (Halliday/Resnick/Walker, Fundamentals of Physics 8 th edition)

PH 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 information

Sound Propagation through Media. Nachiketa Tiwari Indian Institute of Technology Kanpur

Sound Propagation through Media. Nachiketa Tiwari Indian Institute of Technology Kanpur Sound Propagation through Media Nachiketa Tiwari Indian Institute of Technology Kanpur LECTURE-17 HARMONIC PLANE WAVES Introduction In this lecture, we discuss propagation of 1-D planar waves as they travel

More information

Plane Waves GATE Problems (Part I)

Plane Waves GATE Problems (Part I) Plane Waves GATE Problems (Part I). A plane electromagnetic wave traveling along the + z direction, has its electric field given by E x = cos(ωt) and E y = cos(ω + 90 0 ) the wave is (a) linearly polarized

More information

Problem 1.1 Energy Conversion Between Majors. Problem 1.2 Energy Stored in a Toyota Prius

Problem 1.1 Energy Conversion Between Majors. Problem 1.2 Energy Stored in a Toyota Prius MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.007 Electromagnetic Energy: From Motors to Lasers Spring 2011 Problem Set 1: Energy Conversion Due Wednesday,February

More information

Electrodynamics HW Problems 06 EM Waves

Electrodynamics HW Problems 06 EM Waves Electrodynamics HW Problems 06 EM Waves 1. Energy in a wave on a string 2. Traveling wave on a string 3. Standing wave 4. Spherical traveling wave 5. Traveling EM wave 6. 3- D electromagnetic plane wave

More information

EECS 117 Lecture 20: Plane Waves

EECS 117 Lecture 20: Plane Waves University of California, Berkeley EECS 117 Lecture 20 p. 1/2 EECS 117 Lecture 20: Plane Waves Prof. Niknejad University of California, Berkeley University of California, Berkeley EECS 117 Lecture 20 p.

More information

Microscopic electrodynamics. Trond Saue (LCPQ, Toulouse) Microscopic electrodynamics Virginia Tech / 46

Microscopic electrodynamics. Trond Saue (LCPQ, Toulouse) Microscopic electrodynamics Virginia Tech / 46 Microscopic electrodynamics Trond Saue (LCPQ, Toulouse) Microscopic electrodynamics Virginia Tech 2015 1 / 46 Maxwell s equations for electric field E and magnetic field B in terms of sources ρ and j The

More information

2 u 1-D: 3-D: x + 2 u

2 u 1-D: 3-D: x + 2 u c 2013 C.S. Casari - Politecnico di Milano - Introduction to Nanoscience 2013-14 Onde 1 1 Waves 1.1 wave propagation 1.1.1 field Field: a physical quantity (measurable, at least in principle) function

More information

Chapter 3 Uniform Plane Waves Dr. Stuart Long

Chapter 3 Uniform Plane Waves Dr. Stuart Long 3-1 Chapter 3 Uniform Plane Waves Dr. Stuart Long 3- What is a wave? Mechanism by which a disturbance is propagated from one place to another water, heat, sound, gravity, and EM (radio, light, microwaves,

More information

: Imaging Systems Laboratory II. Laboratory 6: The Polarization of Light April 16 & 18, 2002

: Imaging Systems Laboratory II. Laboratory 6: The Polarization of Light April 16 & 18, 2002 151-232: Imaging Systems Laboratory II Laboratory 6: The Polarization of Light April 16 & 18, 22 Abstract. In this lab, we will investigate linear and circular polarization of light. Linearly polarized

More information

MIDSUMMER EXAMINATIONS 2001

MIDSUMMER EXAMINATIONS 2001 No. of Pages: 7 No. of Questions: 10 MIDSUMMER EXAMINATIONS 2001 Subject PHYSICS, PHYSICS WITH ASTROPHYSICS, PHYSICS WITH SPACE SCIENCE & TECHNOLOGY, PHYSICS WITH MEDICAL PHYSICS Title of Paper MODULE

More information

Electromagnetic fields and waves

Electromagnetic 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 information

The Calculus of Vec- tors

The Calculus of Vec- tors Physics 2460 Electricity and Magnetism I, Fall 2007, Lecture 3 1 The Calculus of Vec- Summary: tors 1. Calculus of Vectors: Limits and Derivatives 2. Parametric representation of Curves r(t) = [x(t), y(t),

More information

Maxwell s Equations and Electromagnetic Waves W13D2

Maxwell s Equations and Electromagnetic Waves W13D2 Maxwell s Equations and Electromagnetic Waves W13D2 1 Announcements Week 13 Prepset due online Friday 8:30 am Sunday Tutoring 1-5 pm in 26-152 PS 10 due Week 14 Friday at 9 pm in boxes outside 26-152 2

More information

Lecture 5: Polarization. Polarized Light in the Universe. Descriptions of Polarized Light. Polarizers. Retarders. Outline

Lecture 5: Polarization. Polarized Light in the Universe. Descriptions of Polarized Light. Polarizers. Retarders. Outline Lecture 5: Polarization Outline 1 Polarized Light in the Universe 2 Descriptions of Polarized Light 3 Polarizers 4 Retarders Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl ATI 2016,

More information

Chapter 1 - The Nature of Light

Chapter 1 - The Nature of Light David J. Starling Penn State Hazleton PHYS 214 Electromagnetic radiation comes in many forms, differing only in wavelength, frequency or energy. Electromagnetic radiation comes in many forms, differing

More information

Introduction to electromagnetic theory

Introduction to electromagnetic theory Chapter 1 Introduction to electromagnetic theory 1.1 Introduction Electromagnetism is a fundamental physical phenomena that is basic to many areas science and technology. This phenomenon is due to the

More information

Theoretische Physik 2: Elektrodynamik (Prof. A-S. Smith) Home assignment 9

Theoretische Physik 2: Elektrodynamik (Prof. A-S. Smith) Home assignment 9 WiSe 202 20.2.202 Prof. Dr. A-S. Smith Dipl.-Phys. Ellen Fischermeier Dipl.-Phys. Matthias Saba am Lehrstuhl für Theoretische Physik I Department für Physik Friedrich-Alexander-Universität Erlangen-Nürnberg

More information

Lecture Outline. Maxwell s Equations Predict Waves Derivation of the Wave Equation Solution to the Wave Equation 8/7/2018

Lecture Outline. Maxwell s Equations Predict Waves Derivation of the Wave Equation Solution to the Wave Equation 8/7/2018 Course Instructor Dr. Raymond C. Rumpf Office: A 337 Phone: (915) 747 6958 E Mail: rcrumpf@utep.edu EE 4347 Applied Electromagnetics Topic 3a Electromagnetic Waves Electromagnetic These notes Waves may

More information

Lecture 14 (Poynting Vector and Standing Waves) Physics Spring 2018 Douglas Fields

Lecture 14 (Poynting Vector and Standing Waves) Physics Spring 2018 Douglas Fields Lecture 14 (Poynting Vector and Standing Waves) Physics 6-01 Spring 018 Douglas Fields Reading Quiz For the wave described by E E ˆsin Max j kz t, what is the direction of the Poynting vector? A) +x direction

More information

Light Scattering Group

Light Scattering Group Light Scattering Inversion Light Scattering Group A method of inverting the Mie light scattering equation of spherical homogeneous particles of real and complex argument is being investigated The aims

More information

roth t dive = 0 (4.2.3) divh = 0 (4.2.4) Chapter 4 Waves in Unbounded Medium Electromagnetic Sources 4.2 Uniform plane waves in free space

roth t dive = 0 (4.2.3) divh = 0 (4.2.4) Chapter 4 Waves in Unbounded Medium Electromagnetic Sources 4.2 Uniform plane waves in free space Chapter 4 Waves in Unbounded Medium 4. lectromagnetic Sources 4. Uniform plane waves in free space Mawell s equation in free space is given b: H rot = (4..) roth = (4..) div = (4..3) divh = (4..4) which

More information

Electromagnetic Waves

Electromagnetic Waves Electromagnetic Waves Our discussion on dynamic electromagnetic field is incomplete. I H E An AC current induces a magnetic field, which is also AC and thus induces an AC electric field. H dl Edl J ds

More information

Guided waves - Lecture 11

Guided waves - Lecture 11 Guided waves - Lecture 11 1 Wave equations in a rectangular wave guide Suppose EM waves are contained within the cavity of a long conducting pipe. To simplify the geometry, consider a pipe of rectangular

More information

Dielectric Slab Waveguide

Dielectric Slab Waveguide Chapter Dielectric Slab Waveguide We will start off examining the waveguide properties of a slab of dielectric shown in Fig... d n n x z n Figure.: Cross-sectional view of a slab waveguide. { n, x < d/

More information

Introduction to the School

Introduction 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 information

PROPAGATION IN A FERRITE CIRCULAR WAVEGUIDE MAGNETIZED THROUGH A ROTARY FOUR-POLE MAGNETIC FIELD

PROPAGATION IN A FERRITE CIRCULAR WAVEGUIDE MAGNETIZED THROUGH A ROTARY FOUR-POLE MAGNETIC FIELD Progress In Electromagnetics Research, PIER 68, 1 13, 2007 PROPAGATION IN A FERRITE CIRCULAR WAVEGUIDE MAGNETIZED THROUGH A ROTARY FOUR-POLE MAGNETIC FIELD M. Mazur Analog Techniques Department Telecommunication

More information

Chapter 9. Electromagnetic waves

Chapter 9. Electromagnetic waves Chapter 9. lectromagnetic waves 9.1.1 The (classical or Mechanical) waves equation Given the initial shape of the string, what is the subsequent form, The displacement at point z, at the later time t,

More information

The concept of perfect electromagnetic conductor (PEMC) has been defined by medium conditions of the form [1, 2]

The concept of perfect electromagnetic conductor (PEMC) has been defined by medium conditions of the form [1, 2] Progress In Electromagnetics Research B, Vol. 5, 169 183, 2008 REFLECTION AND TRANSMISSION OF WAVES AT THE INTERFACE OF PERFECT ELECTROMAGNETIC CONDUCTOR PEMC I. V. Lindell and A. H. Sihvola Electromagnetics

More information

ELECTROMAGNETIC WAVES

ELECTROMAGNETIC WAVES Physics 4D ELECTROMAGNETIC WAVE Hans P. Paar 26 January 2006 i Chapter 1 Vector Calculus 1.1 Introduction Vector calculus is a branch of mathematics that allows differentiation and integration of (scalar)

More information

Lecture 38: FRI 24 APR Ch.33 Electromagnetic Waves

Lecture 38: FRI 24 APR Ch.33 Electromagnetic Waves Physics 2113 Jonathan Dowling Heinrich Hertz (1857 1894) Lecture 38: FRI 24 APR Ch.33 Electromagnetic Waves Maxwell Equations in Empty Space: E da = 0 S B da = 0 S C C B ds = µ ε 0 0 E ds = d dt d dt S

More information

- HH Why Can Light Propagate in Vacuum? Hsiu-Hau Lin (Apr 1, 2014)

- HH Why Can Light Propagate in Vacuum? Hsiu-Hau Lin (Apr 1, 2014) - HH0124 - Why Can Light Propagate in Vacuum? Hsiu-Hau Lin hsiuhau@phys.nthu.edu.tw (Apr 1, 2014) Sunlight is vital for most creatures in this warm and humid planet. From interference experiments, scientists

More information

Chapter Three: Propagation of light waves

Chapter 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 information

Today in Physics 218: electromagnetic waves in linear media

Today in Physics 218: electromagnetic waves in linear media Today in Physics 218: electromagnetic waves in linear media Their energy and momentum Their reflectance and transmission, for normal incidence Their polarization Sunrise over Victoria Falls, Zambezi River

More information

Principles of Mobile Communications

Principles of Mobile Communications Communication Networks 1 Principles of Mobile Communications University Duisburg-Essen WS 2003/2004 Page 1 N e v e r s t o p t h i n k i n g. Wave Propagation Single- and Multipath Propagation Overview:

More information

Summary of Beam Optics

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 information

Electromagnetic optics!

Electromagnetic optics! 1 EM theory Electromagnetic optics! EM waves Monochromatic light 2 Electromagnetic optics! Electromagnetic theory of light Electromagnetic waves in dielectric media Monochromatic light References: Fundamentals

More information

Optics and Optical Design. Chapter 6: Polarization Optics. Lectures 11-13

Optics and Optical Design. Chapter 6: Polarization Optics. Lectures 11-13 Optics and Optical Design Chapter 6: Polarization Optics Lectures 11-13 Cord Arnold / Anne L Huillier Polarization of Light Arbitrary wave vs. paraxial wave One component in x-direction y x z Components

More information

Exam in TFY4240 Electromagnetic Theory Wednesday Dec 9, :00 13:00

Exam 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 information

Basics of Wave Propagation

Basics of Wave Propagation Basics of Wave Propagation S. R. Zinka zinka@hyderabad.bits-pilani.ac.in Department of Electrical & Electronics Engineering BITS Pilani, Hyderbad Campus May 7, 2015 Outline 1 Time Harmonic Fields 2 Helmholtz

More information

Physics 3312 Lecture 9 February 13, LAST TIME: Finished mirrors and aberrations, more on plane waves

Physics 3312 Lecture 9 February 13, LAST TIME: Finished mirrors and aberrations, more on plane waves Physics 331 Lecture 9 February 13, 019 LAST TIME: Finished mirrors and aberrations, more on plane waves Recall, Represents a plane wave having a propagation vector k that propagates in any direction with

More information

4: birefringence and phase matching

4: birefringence and phase matching /3/7 4: birefringence and phase matching Polarization states in EM Linear anisotropic response χ () tensor and its symmetry properties Working with the index ellipsoid: angle tuning Phase matching in crystals

More information

Maxwell s Equations & Electromagnetic Waves. The Equations So Far...

Maxwell s Equations & Electromagnetic Waves. The Equations So Far... Maxwell s Equations & Electromagnetic Waves Maxwell s equations contain the wave equation Velocity of electromagnetic waves c = 2.99792458 x 1 8 m/s Relationship between E and B in an EM wave Energy in

More information

Course Updates. 2) This week: Electromagnetic Waves +

Course Updates.  2) This week: Electromagnetic Waves + Course Updates http://www.phys.hawaii.edu/~varner/phys272-spr1/physics272.html Reminders: 1) Assignment #11 due Wednesday 2) This week: Electromagnetic Waves + 3) In the home stretch [review schedule]

More information

Electromagnetic Wave Propagation Lecture 2: Uniform plane waves

Electromagnetic Wave Propagation Lecture 2: Uniform plane waves Electromagnetic Wave Propagation Lecture 2: Uniform plane waves Daniel Sjöberg Department of Electrical and Information Technology March 25, 2010 Outline 1 Plane waves in lossless media General time dependence

More information

Jones vector & matrices

Jones vector & matrices Jones vector & matrices Department of Physics 1 Matrix treatment of polarization Consider a light ray with an instantaneous E-vector as shown y E k, t = xe x (k, t) + ye y k, t E y E x x E x = E 0x e i

More information

Overview in Images. S. Lin et al, Nature, vol. 394, p , (1998) T.Thio et al., Optics Letters 26, (2001).

Overview 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 information

5 Electromagnetic Waves

5 Electromagnetic Waves 5 Electromagnetic Waves 5.1 General Form for Electromagnetic Waves. In free space, Maxwell s equations are: E ρ ɛ 0 (5.1.1) E + B 0 (5.1.) B 0 (5.1.3) B µ 0 ɛ 0 E µ 0 J (5.1.4) In section 4.3 we derived

More information

Clicker Question. Is the following equation a solution to the wave equation: y(x,t)=a sin(kx-ωt) (a) yes (b) no

Clicker Question. Is the following equation a solution to the wave equation: y(x,t)=a sin(kx-ωt) (a) yes (b) no Is the following equation a solution to the wave equation: y(x,t)=a sin(kx-ωt) (a) yes (b) no Is the following equation a solution to the wave equation: y(x,t)=a sin(kx-ωt) (a) yes (b) no Is the following

More information