Sec. 16. Reflection and Refraction of Light at the Interface Between Two Dielectrics
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Ec6503 - Transmission Lines and Waveguides Transmission Lines and Waveguides Unit I - Transmission Line Theory 1
EC6503 - TRANSMISSION LINES AND WAVEGUIDES TRANSMISSION LINES AND WAVEGUIDES UNIT I - TRANSMISSION LINE THEORY 1. Define – Characteristic Impedance [M/J–2006, N/D–2006] Characteristic impedance is defined as the impedance of a transmission line measured at the sending end. It is given by 푍 푍0 = √ ⁄푌 where Z = R + jωL is the series impedance Y = G + jωC is the shunt admittance 2. State the line parameters of a transmission line. The line parameters of a transmission line are resistance, inductance, capacitance and conductance. Resistance (R) is defined as the loop resistance per unit length of the transmission line. Its unit is ohms/km. Inductance (L) is defined as the loop inductance per unit length of the transmission line. Its unit is Henries/km. Capacitance (C) is defined as the shunt capacitance per unit length between the two transmission lines. Its unit is Farad/km. Conductance (G) is defined as the shunt conductance per unit length between the two transmission lines. Its unit is mhos/km. 3. What are the secondary constants of a line? The secondary constants of a line are 푍 i. Characteristic impedance, 푍0 = √ ⁄푌 ii. Propagation constant, γ = α + jβ 4. Why the line parameters are called distributed elements? The line parameters R, L, C and G are distributed over the entire length of the transmission line. Hence they are called distributed parameters. They are also called primary constants. The infinite line, wavelength, velocity, propagation & Distortion line, the telephone cable 5. What is an infinite line? [M/J–2012, A/M–2004] An infinite line is a line where length is infinite. -
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. -
Waveguides Waveguides, Like Transmission Lines, Are Structures Used to Guide Electromagnetic Waves from Point to Point. However
Waveguides Waveguides, like transmission lines, are structures used to guide electromagnetic waves from point to point. However, the fundamental characteristics of waveguide and transmission line waves (modes) are quite different. The differences in these modes result from the basic differences in geometry for a transmission line and a waveguide. Waveguides can be generally classified as either metal waveguides or dielectric waveguides. Metal waveguides normally take the form of an enclosed conducting metal pipe. The waves propagating inside the metal waveguide may be characterized by reflections from the conducting walls. The dielectric waveguide consists of dielectrics only and employs reflections from dielectric interfaces to propagate the electromagnetic wave along the waveguide. Metal Waveguides Dielectric Waveguides Comparison of Waveguide and Transmission Line Characteristics Transmission line Waveguide • Two or more conductors CMetal waveguides are typically separated by some insulating one enclosed conductor filled medium (two-wire, coaxial, with an insulating medium microstrip, etc.). (rectangular, circular) while a dielectric waveguide consists of multiple dielectrics. • Normal operating mode is the COperating modes are TE or TM TEM or quasi-TEM mode (can modes (cannot support a TEM support TE and TM modes but mode). these modes are typically undesirable). • No cutoff frequency for the TEM CMust operate the waveguide at a mode. Transmission lines can frequency above the respective transmit signals from DC up to TE or TM mode cutoff frequency high frequency. for that mode to propagate. • Significant signal attenuation at CLower signal attenuation at high high frequencies due to frequencies than transmission conductor and dielectric losses. lines. • Small cross-section transmission CMetal waveguides can transmit lines (like coaxial cables) can high power levels. -
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. -
Wave Guides Summary and Problems
ECE 144 Electromagnetic Fields and Waves Bob York General Waveguide Theory Basic Equations (ωt γz) Consider wave propagation along the z-axis, with fields varying in time and distance according to e − . The propagation constant γ gives us much information about the character of the waves. We will assume that the fields propagating in a waveguide along the z-axis have no other variation with z,thatis,the transverse fields do not change shape (other than in magnitude and phase) as the wave propagates. Maxwell’s curl equations in a source-free region (ρ =0andJ = 0) can be combined to give the wave equations, or in terms of phasors, the Helmholtz equations: 2E + k2E =0 2H + k2H =0 ∇ ∇ where k = ω√µ. In rectangular or cylindrical coordinates, the vector Laplacian can be broken into two parts ∂2E 2E = 2E + ∇ ∇t ∂z2 γz so that with the assumed e− dependence we get the wave equations 2E +(γ2 + k2)E =0 2H +(γ2 + k2)H =0 ∇t ∇t (ωt γz) Substituting the e − into Maxwell’s curl equations separately gives (for rectangular coordinates) E = ωµH H = ωE ∇ × − ∇ × ∂Ez ∂Hz + γEy = ωµHx + γHy = ωEx ∂y − ∂y ∂Ez ∂Hz γEx = ωµHy γHx = ωEy − − ∂x − − − ∂x ∂Ey ∂Ex ∂Hy ∂Hx = ωµHz = ωEz ∂x − ∂y − ∂x − ∂y These can be rearranged to express all of the transverse fieldcomponentsintermsofEz and Hz,giving 1 ∂Ez ∂Hz 1 ∂Ez ∂Hz Ex = γ + ωµ Hx = ω γ −γ2 + k2 ∂x ∂y γ2 + k2 ∂y − ∂x w W w W 1 ∂Ez ∂Hz 1 ∂Ez ∂Hz Ey = γ + ωµ Hy = ω + γ γ2 + k2 − ∂y ∂x −γ2 + k2 ∂x ∂y w W w W For propagating waves, γ = β,whereβ is a real number provided there is no loss. -
Uniform Plane Waves
38 2. Uniform Plane Waves Because also ∂zEz = 0, it follows that Ez must be a constant, independent of z, t. Excluding static solutions, we may take this constant to be zero. Similarly, we have 2 = Hz 0. Thus, the fields have components only along the x, y directions: E(z, t) = xˆ Ex(z, t)+yˆ Ey(z, t) Uniform Plane Waves (transverse fields) (2.1.2) H(z, t) = xˆ Hx(z, t)+yˆ Hy(z, t) These fields must satisfy Faraday’s and Amp`ere’s laws in Eqs. (2.1.1). We rewrite these equations in a more convenient form by replacing and μ by: 1 η 1 μ = ,μ= , where c = √ ,η= (2.1.3) ηc c μ Thus, c, η are the speed of light and characteristic impedance of the propagation medium. Then, the first two of Eqs. (2.1.1) may be written in the equivalent forms: ∂E 1 ∂H ˆz × =− η 2.1 Uniform Plane Waves in Lossless Media ∂z c ∂t (2.1.4) ∂H 1 ∂E The simplest electromagnetic waves are uniform plane waves propagating along some η ˆz × = ∂z c ∂t fixed direction, say the z-direction, in a lossless medium {, μ}. The assumption of uniformity means that the fields have no dependence on the The first may be solved for ∂zE by crossing it with ˆz. Using the BAC-CAB rule, and transverse coordinates x, y and are functions only of z, t. Thus, we look for solutions noting that E has no z-component, we have: of Maxwell’s equations of the form: E(x, y, z, t)= E(z, t) and H(x, y, z, t)= H(z, t). -
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. -
Microstrip Solutions for Innovative Microwave Feed Systems
Examensarbete LiTH-ITN-ED-EX--2001/05--SE Microstrip Solutions for Innovative Microwave Feed Systems Magnus Petersson 2001-10-24 Department of Science and Technology Institutionen för teknik och naturvetenskap Linköping University Linköpings Universitet SE-601 74 Norrköping, Sweden 601 74 Norrköping LiTH-ITN-ED-EX--2001/05--SE Microstrip Solutions for Innovative Microwave Feed Systems Examensarbete utfört i Mikrovågsteknik / RF-elektronik vid Tekniska Högskolan i Linköping, Campus Norrköping Magnus Petersson Handledare: Ulf Nordh Per Törngren Examinator: Håkan Träff Norrköping den 24 oktober, 2001 'DWXP $YGHOQLQJ,QVWLWXWLRQ Date Division, Department Institutionen för teknik och naturvetenskap 2001-10-24 Ã Department of Science and Technology 6SUnN 5DSSRUWW\S ,6%1 Language Report category BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB Svenska/Swedish Licentiatavhandling X Engelska/English X Examensarbete ISRN LiTH-ITN-ED-EX--2001/05--SE C-uppsats 6HULHWLWHOÃRFKÃVHULHQXPPHUÃÃÃÃÃÃÃÃÃÃÃÃ,661 D-uppsats Title of series, numbering ___________________________________ _ ________________ Övrig rapport _ ________________ 85/I|UHOHNWURQLVNYHUVLRQ www.ep.liu.se/exjobb/itn/2001/ed/005/ 7LWHO Microstrip Solutions for Innovative Microwave Feed Systems Title )|UIDWWDUH Magnus Petersson Author 6DPPDQIDWWQLQJ Abstract This report is introduced with a presentation of fundamental electromagnetic theories, which have helped a lot in the achievement of methods for calculation and design of microstrip transmission lines and circulators. The used software for the work is also based on these theories. General considerations when designing microstrip solutions, such as different types of transmission lines and circulators, are then presented. Especially the design steps for microstrip lines, which have been used in this project, are described. Discontinuities, like bends of microstrip lines, are treated and simulated. -
A Study of Reflection and Transmission of Birefringent Retarders
JUST, Vol. IV, No. 1, 2016 Trent University A study of reflection and transmission of birefringent retarders James Godfrey Keywords Optics — Computational Physics — Theoretical Physics Champlain College 1. Interference in Rotating Waveplates The transmitted intensity of along each axis must be con- sidered separately, since equation (1-1) dictates that the re- 1.1 Research Goal flectance of the fast- and slow-axis components of the incident The objective of this project was to obtain a realistic theoreti- light will be different. The reflectance along the fast axis (Ro) cal prediction of the transmission curve for linearly-polarized and slow axis (Rs) are may be expressed: light incident normal on a [birefringent] retarder that behaves 2 as a Fabry-Perot etalon. The birefringent retarder considered n f − ni is a slab of birefringent crystal cut with the optic axis in the Ro = n f + ni face of the slab, and with parallel faces, and the waveplate has 2 no coating of any kind. ns − ni Rs = (1-2) Results of this project hoped to possibly explain the results ns + ni of work done by a previous student, Nolan Woodley, for his Physics project course in the 2013/14 academic year. In his Assuming the sides of the etalon are essentially paral- project, Woodley took many polarimeter scans which had a lel, the retarder may now be treated as a low-finesse (low- roughly sinusoidal shape [as they should], but adjacent peaks reflectance, R << 1) Fabry-Perot etalon. Their coefficients of of different heights. These differing heights were completely finesse (Fo and Fs, respectively) will also be different: unexplained, and not predicted by theory. -
Polarization Lecture Outline
1/31/20 Polarization Lecture outline • Why is polarization important? • Classification of polarization • Four ways to polarize EM waves • Polarization in active remote sensing systems 1 1/31/20 Definitions • Polarization is the phenomenon in which waves of light or other radiation are restricted in direction of vibration • Polarization is a property of waves that describes the orientation of their oscillations Coherence and incoherence • Coherent radiation originates from a single oscillator, or a group of oscillators in perfect synchronization (phase- locked, or with a constant phase offset) • e.g., microwave ovens, radars, lasers, radio towers (i.e., artificial sources) • Incoherent radiation originates from independent oscillators that are not phase-locked. Natural radiation is incoherent. 2 1/31/20 Why is polarization important? • Bohren (2006): ‘the only reason the polarization state of light is worth contemplating is that two beams, otherwise identical, may interact differently with matter if their polarization states are different.’ • Interference only occurs when EM waves have a similar frequency, constant phase offset and the same polarization (i.e., they are coherent) • Used by active remote sensing systems (radar, lidar) Classification of polarization • Linear • Circular • Elliptical • ‘Random’ or unpolarized 3 1/31/20 Linear polarization • Plane EM wave – linearly polarized • Trace of electric field vector is linear • Also called plane-polarized light • Convention is to refer to the electric field vector • Weather radars usually