( ) + + REFLECTION FROM A METALLIC SURFACE

Size: px
Start display at page:

Download "( ) + + REFLECTION FROM A METALLIC SURFACE"

Transcription

1 REFLECTION FROM A METALLIC SURFACE For a metallc medum the delectrc functon and the ndex of refracton are complex valued functons. Ths s also the case for semconductors and nsulators n certan frequency ranges near and at absorpton bands. Fresnel's equatons are stll val but the angles n the equatons are now complex valued and do no longer have the obvous geometrcal nterpretaton. For normal ncence we have Ê n - n ˆ n n R Á Æ - Ë n + n n + n where ñ n + k n( + k ) ( ) + ( - ) ( ) + + n - n k k n + n ( k k) where k s the so called extncton coeffcent. For metallc systems n e e + 4ps w For normal ncence the refracted, or rather transmtted, wave wll vary as ( ) - ( - ) nkz E 0 -wt kkz 0 nkz 0 wt ~ e e e

2 Bo E. Sernelus 3: REFRACTION INTO A CONDUCTING MEDIUM Navely one would beleve that the plane wave nse the metal would vary lke n the precedng secton but n a new drecton. The dampng would be along the drecton of propagaton. Ths s however not so! The dampng s stll n the z-drecton only. Ths means that the surfaces of constant ampltude are no longer the same as the surfaces of constant phase. The surfaces of constant ampltude are parallel to the nterface and the surfaces of constant phase are perpendcular to the drecton of propagaton. The whole problem s a bt awkward. Approxmately the angle of refracton and the wavelength are determned by the real part of the ndex of refracton and the dampng n the z-drecton by the magnary part of the ndex of refracton. Note however that ths s just approxmate! The correct dependence s ( ) ( ) k E 0snq0 -x k0 n sn q 0 z wt ~ e ( ) k0 Im n -sn q 0 z k 0 snq 0 x k 0 Re n sn e e (- )+ - q 0 z-wt and the real transmsson angle s q real È sn q arctaní 0 Í ÎRe n - sn q 0 Note that Fresnel's coeffcents are stll val for complex valued refractve ndces but the angles are complex valued.

3 Bo E. Sernelus 3:3 TOTAL INTERNAL REFLECTION I wll just brefly touch upon ths subject, because of lack of tme. It s very mportant though. I treat ths n more detal n my other course, TFYY70 fundamentals of surface modes. If an electromagnetc wave s mpngng on an nterface from a materal of hgher refractve ndex to a one wth lower, the refracton angle s greater than the angle of ncence. If we let the angle of ncence ncrease we reach a crtcal angle, - q c sn ( n n ), when all energy s reflected. For all angles greater than ths crtcal angle we have total reflecton. The nterestng thng s that there s also a wave parallel to the nterface, whch decays exponentally away from the nterface. If we have a steady state condton and an nfnte nterface all energy s reflected. If we start the rradaton, at frst some energy s used to create ths evanescent wave but as soon as steady state condtons are reached no more energy s needed for ths. If however the nterface s fnte ths evanescent wave can radate out at the edge of the nterface and some energy s needed to compensate for ths. There can also be mperfectons at the nterface makng the wave radate. There can also be dampng or losses n e.g. a metallc system whch means that energy has to be fed nto the mode. There are other nterestng effects arsng from that the mode has felds extendng outse the nterface. If the nterface s a glass-prsm ar nterface the felds extend n the ar outse the prsm. Puttng another prsm close to the frst allows the mode to "jump" across the ar gap and be emtted through the second prsm. Smlar effects are utlzed n the so called ATR-experment, used to study surface modes.

4 Bo E. Sernelus 3:4 MULTILAYERS Now we wll study the reflecton and transmsson between two parallel nterfaces between three meda. It could for nstance be an ant reflecton coatng on an ar-glass nterface. We have such a problem to solve n the problem solvng sesson. I want to demonstrate that the problem can be solved n dfferent ways and also show how the soluton can be extended to the case of many layers. The frst approach s a straght forward extenson of what we have done for the sngle nterface. In the frst medum we have an ncomng wave and a reflected. In the sandwched layer we also have two waves, one gong n the postve z- drecton and one n the negatve z-drecton. In the thrd medum we only have one wave travellng n the postve z-drecton. We use the same type of boundary condtons as we used n the sngle nterface geometry at both boundares and fnd the ampltudes and angles for the waves. The procedure s straght forward and can be extended to a geometry wth more than two nterfaces. However, the soluton becomes ncreasngly cumbersome the more nterfaces we have. We wll use ths method n the problem solvng sesson and wll not dwell on t here. Instead we use another method. All the waves n our ansatz, except the ncomng wave on the frst nterface, can be vewed as a superposton of waves havng been multple reflected a varyng number of tmes between the nterfaces. The ampltudes of these waves, takng the phases nto account are added together and ths leads to nterference effects We wll here follow the outlne n Surface Modes n Physcs. Bo E. Sernelus, Surface Modes n Physc, Wley-VCH, Berln 00.

5 Bo E. Sernelus 3:5 Fresnel's coeffcents gve the relatve ampltudes of the reflected and transmtted waves at an nterface between two meda. We have derved these. They are: n tj s sn qj cosq cosq sn q + qj ncosq nj cosqj ( ) + ( ) n n rj s sn q - qj j j - ( + j ) cosq - cosq sn q q ncosq + nj cosqj n tj p sn q j cosq cosq sn q + qj cos q qj nj cosq + ncosqj ( ) ( - ) ( q - q ) n cosq ( q + qj ) - n cosq + rj p tan j tan j ncosqj j ncosqj where the superscrpts s and p represent s-polarzed and p-polarzed waves, respectvely and the angles q and qj are the angle of ncence and angle of transmsson, respectvely. The optcal propertes of each materal enter n the form of the refractve ndex, n. These coeffcents are val also for complex valued refractve ndces. In that case the angles are not to be nterpreted as the geometrcal angles. They are complex valued, and the sne and cosne of the angles are also complex valued. The sne functons are obtaned from Snell's law njsn qj nsn q and the cosne functons are obtaned from relatons of the type: cosq - sn q For s-polarzed waves the electrc feld vector s perpendcular to the plane of ncence and for p-polarzed waves t s n the plane of ncence. The plane of ncence s the plane defned by the ncomng wave and the normal to the surface.

6 Bo E. Sernelus 3:6 q j q j The nterface between two meda and j, dscussed n the text.

7 Bo E. Sernelus 3:7 TWO INTERFACES BETWEEN THREE MEDIA We treat two nterfaces as shown n the fgure below. We need to take multple reflectons n the mdle layer nto account. 3 Waves contrbutng to the total reflected and transmtted waves n a three layer structure as dscussed n the text. The total reflected ampltude, r, s obtaned from an nfnte summaton of waves due to the multple reflectons n the mdle layer. Each tme a wave mpnges on an nterface the Fresnel equatons are used and the phases of the waves are taken nto account. All coeffcents are complexvalued. The phase d s the phase dfference for two waves: one wave that s transmtted through the frst nterface, passng through layer, s reflected at the second nterface, passng through layer agan and s fnally transmtted through the frst nterface; the second wave s one that s reflected at the frst nterface. Ths second wave has furthermore traveled a longer dstance n layer before t mpnges on the nterface.

8 Bo E. Sernelus 3:8 The results are obtaned as follows: e r r r + t xt x e d ; r e 3 [ + rx] ; x 3 - e r3r e d t t yt y [ e d 3 + yr3e d ; re ]; y - e r3r pd d n cos( q ) l e r t t r - r e r r + e 3 3 r3tt r r + - e r3r - e r3r r + e r3( tt -rr) r + e r 3 - e d r3r - e r 3 r r + e r 3 + e rr3 e t t t 3 + d e rr3 We have made use of the followng useful relatons: r -r tt - rr These results are val for both polarzaton drectons. The reflecton, transmsson and absorpton are [( ) ] Ï * Re n Ô 3 cosq 3 t, p - polarzaton Ô * R r ; T Re[ ( n ) Ì cosq ] ; Ô ( n Ô Re 3 cosq 3) t, s - polarzaton ÓÔ Re( n cosq) eff n sn q sn q N 0 ; real real sn q sn q eff real eff real sn q N sn q ; Re ˆ w 0 k n c N cosq [ ] A -R -T

9 Bo E. Sernelus 3:9 GENERAL NUMBER OF LAYERS The procedure descrbed n the precedng secton can be extended to more layers, but becomes too cumbersome f we have many layers. Here we wll ntroduce a more sutable approach. Assume that we have N layers sandwched between medum 0 and N+. So we have N+ meda. Then n general layer n wll have an ncomng and a reflected wave on the left se. We denote these wth x n and y n, respectvely. On the rght hand se there wll be an ncomng and an outgong wave as well. These are reflected and ncomng waves on layer n+. x n y n+ y n x n+ n The waves on the two ses of the layer are related to each other. Ths relaton can be expressed as Ê xn ˆ Á Ë yn Ê xn+ ˆ M n Á Ë yn+ where

10 Bo E. Sernelus 3:0 Ê rn-, nˆ Ê - e n 0 ˆ M n Á tn, n r Ë n, n Á - - d Ë 0 e n Ê - e n rn, ne d n ˆ - Á t - n, n rn, ne n d e n - Ë - Each nterface s shared between two neghborng layers. We have let the left most nterface belong to the layer. Ths means that the felds to the rght n the fgure are the felds nse layer n just to the left of the rght nterface. To get the felds just nse layer n+ we have to multply from the rght wth the matrx Ê Á tnn, + Ërnn, + rnn, + ˆ We have N number of layers and N+ number of nterfaces n our problem. Ths means that we n the end have to multply wth a matrx of ths knd to take care of the rght most nterface. Ê x ˆ rnn xn Á n Ê, + ˆ N y Á Ë t NN rnn Ê + ˆ M M KM KM Á, + Ë, + Ë yn + Ê xn + ˆ M Á Ë yn + Thus f we know the felds on the rght hand se of our layers we get the felds on the left hand se. Ths s not exactly what we want. We want y and x N+ as functons of x when y N+ 0. Ths s not mpossble to fnd. We have: x MxN+ + MyN+ MxN+ y MxN+ + MyN+ MxN+ and

11 Bo E. Sernelus 3: x t N + x M y M r x M Let us see f we reproduce our prevous result wth one layer. Then we have M Ê - e r ˆ Ê e Á - Á t Ër e e t3 Ër3 r3 ˆ Ê e r + ˆ r3e r3e r e t Á - - t 3 Ër e + r 3e rr3e + e and M M and - e + rr3e tt3 - re + r3e tt3 t r tt3 - e + rr3e - re + r3e - e + rr3e e t t 3 + rr3e d r r e + 3 d + rr3e d

Boundaries, Near-field Optics

Boundaries, Near-field Optics Boundares, Near-feld Optcs Fve boundary condtons at an nterface Fresnel Equatons : Transmsson and Reflecton Coeffcents Transmttance and Reflectance Brewster s condton a consequence of Impedance matchng

More information

16 Reflection and transmission, TE mode

16 Reflection and transmission, TE mode 16 Reflecton transmsson TE mode Last lecture we learned how to represent plane-tem waves propagatng n a drecton ˆ n terms of feld phasors such that η = Ẽ = E o e j r H = ˆ Ẽ η µ ɛ = ˆ = ω µɛ E o =0. Such

More information

Frequency dependence of the permittivity

Frequency dependence of the permittivity Frequency dependence of the permttvty February 7, 016 In materals, the delectrc constant and permeablty are actually frequency dependent. Ths does not affect our results for sngle frequency modes, but

More information

ECE 107: Electromagnetism

ECE 107: Electromagnetism ECE 107: Electromagnetsm Set 8: Plane waves Instructor: Prof. Vtaly Lomakn Department of Electrcal and Computer Engneerng Unversty of Calforna, San Dego, CA 92093 1 Wave equaton Source-free lossless Maxwell

More information

Fresnel's Equations for Reflection and Refraction

Fresnel's Equations for Reflection and Refraction Fresnel's Equatons for Reflecton and Refracton Incdent, transmtted, and reflected beams at nterfaces Reflecton and transmsson coeffcents The Fresnel Equatons Brewster's Angle Total nternal reflecton Power

More information

CHAPTER II THEORETICAL BACKGROUND

CHAPTER II THEORETICAL BACKGROUND 3 CHAPTER II THEORETICAL BACKGROUND.1. Lght Propagaton nsde the Photonc Crystal The frst person that studes the one dmenson photonc crystal s Lord Raylegh n 1887. He showed that the lght propagaton depend

More information

Note: Please use the actual date you accessed this material in your citation.

Note: Please use the actual date you accessed this material in your citation. MIT OpenCourseWare http://ocw.mt.edu 6.13/ESD.13J Electromagnetcs and Applcatons, Fall 5 Please use the followng ctaton format: Markus Zahn, Erch Ippen, and Davd Staeln, 6.13/ESD.13J Electromagnetcs and

More information

Lecture 3. Interaction of radiation with surfaces. Upcoming classes

Lecture 3. Interaction of radiation with surfaces. Upcoming classes Radaton transfer n envronmental scences Lecture 3. Interacton of radaton wth surfaces Upcomng classes When a ray of lght nteracts wth a surface several nteractons are possble: 1. It s absorbed. 2. It s

More information

Supporting Information

Supporting Information Supportng Informaton Water structure at the ar-aqueous nterface of dvalent caton and ntrate solutons Man Xu, Rck Spnney, Heather C. Allen* allen@chemstry.oho-state.edu Fresnel factors and spectra normalzaton

More information

Homework 4. 1 Electromagnetic surface waves (55 pts.) Nano Optics, Fall Semester 2015 Photonics Laboratory, ETH Zürich

Homework 4. 1 Electromagnetic surface waves (55 pts.) Nano Optics, Fall Semester 2015 Photonics Laboratory, ETH Zürich Homework 4 Contact: frmmerm@ethz.ch Due date: December 04, 015 Nano Optcs, Fall Semester 015 Photoncs Laboratory, ETH Zürch www.photoncs.ethz.ch The goal of ths problem set s to understand how surface

More information

Section 8.3 Polar Form of Complex Numbers

Section 8.3 Polar Form of Complex Numbers 80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the

More information

Mathematical Preparations

Mathematical Preparations 1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the

More information

Problem 1: To prove that under the assumptions at hand, the group velocity of an EM wave is less than c, I am going to show that

Problem 1: To prove that under the assumptions at hand, the group velocity of an EM wave is less than c, I am going to show that PHY 387 K. Solutons for problem set #7. Problem 1: To prove that under the assumptons at hand, the group velocty of an EM wave s less than c, I am gong to show that (a) v group < v phase, and (b) v group

More information

Difference Equations

Difference Equations Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1

More information

Supplementary Information for Observation of Parity-Time Symmetry in. Optically Induced Atomic Lattices

Supplementary Information for Observation of Parity-Time Symmetry in. Optically Induced Atomic Lattices Supplementary Informaton for Observaton of Party-Tme Symmetry n Optcally Induced Atomc attces Zhaoyang Zhang 1,, Yq Zhang, Jteng Sheng 3, u Yang 1, 4, Mohammad-Al Mr 5, Demetros N. Chrstodouldes 5, Bng

More information

Introductory Optomechanical Engineering. 2) First order optics

Introductory Optomechanical Engineering. 2) First order optics Introductory Optomechancal Engneerng 2) Frst order optcs Moton of optcal elements affects the optcal performance? 1. by movng the mage 2. hgher order thngs (aberratons) The frst order effects are most

More information

Lecture 12: Discrete Laplacian

Lecture 12: Discrete Laplacian Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly

More information

Effect of Losses in a Layered Structure Containing DPS and DNG Media

Effect of Losses in a Layered Structure Containing DPS and DNG Media PIERS ONLINE, VOL. 4, NO. 5, 8 546 Effect of Losses n a Layered Structure Contanng DPS and DNG Meda J. R. Canto, S. A. Matos, C. R. Pava, and A. M. Barbosa Insttuto de Telecomuncações and Department of

More information

1 Matrix representations of canonical matrices

1 Matrix representations of canonical matrices 1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:

More information

Moments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.

Moments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is. Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these

More information

Tensor Analysis. For orthogonal curvilinear coordinates, ˆ ˆ (98) Expanding the derivative, we have, ˆ. h q. . h q h q

Tensor Analysis. For orthogonal curvilinear coordinates, ˆ ˆ (98) Expanding the derivative, we have, ˆ. h q. . h q h q For orthogonal curvlnear coordnates, eˆ grad a a= ( aˆ ˆ e). h q (98) Expandng the dervatve, we have, eˆ aˆ ˆ e a= ˆ ˆ a h e + q q 1 aˆ ˆ ˆ a e = ee ˆˆ ˆ + e. h q h q Now expandng eˆ / q (some of the detals

More information

Irregular vibrations in multi-mass discrete-continuous systems torsionally deformed

Irregular vibrations in multi-mass discrete-continuous systems torsionally deformed (2) 4 48 Irregular vbratons n mult-mass dscrete-contnuous systems torsonally deformed Abstract In the paper rregular vbratons of dscrete-contnuous systems consstng of an arbtrary number rgd bodes connected

More information

Unit 5: Quadratic Equations & Functions

Unit 5: Quadratic Equations & Functions Date Perod Unt 5: Quadratc Equatons & Functons DAY TOPIC 1 Modelng Data wth Quadratc Functons Factorng Quadratc Epressons 3 Solvng Quadratc Equatons 4 Comple Numbers Smplfcaton, Addton/Subtracton & Multplcaton

More information

Rate of Absorption and Stimulated Emission

Rate of Absorption and Stimulated Emission MIT Department of Chemstry 5.74, Sprng 005: Introductory Quantum Mechancs II Instructor: Professor Andre Tokmakoff p. 81 Rate of Absorpton and Stmulated Emsson The rate of absorpton nduced by the feld

More information

Structure and Drive Paul A. Jensen Copyright July 20, 2003

Structure and Drive Paul A. Jensen Copyright July 20, 2003 Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.

More information

One-sided finite-difference approximations suitable for use with Richardson extrapolation

One-sided finite-difference approximations suitable for use with Richardson extrapolation Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,

More information

Physics 443, Solutions to PS 7

Physics 443, Solutions to PS 7 Physcs 443, Solutons to PS 7. Grffths 4.50 The snglet confguraton state s χ ) χ + χ χ χ + ) where that second equaton defnes the abbrevated notaton χ + and χ. S a ) S ) b χ â S )ˆb S ) χ In sphercal coordnates

More information

Advanced Circuits Topics - Part 1 by Dr. Colton (Fall 2017)

Advanced Circuits Topics - Part 1 by Dr. Colton (Fall 2017) Advanced rcuts Topcs - Part by Dr. olton (Fall 07) Part : Some thngs you should already know from Physcs 0 and 45 These are all thngs that you should have learned n Physcs 0 and/or 45. Ths secton s organzed

More information

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons

More information

Night Vision and Electronic Sensors Directorate

Night Vision and Electronic Sensors Directorate Nght Vson and Electronc Sensors Drectorate RDER-NV-TR-67 A Note on the rewster Angle n Lossy Delectrc Meda Approved for Publc Release: Dstrbuton Unlmted Fo elvor, Vrgna 060-5806 Nght Vson and Electronc

More information

One Dimension Again. Chapter Fourteen

One Dimension Again. Chapter Fourteen hapter Fourteen One Dmenson Agan 4 Scalar Lne Integrals Now we agan consder the dea of the ntegral n one dmenson When we were ntroduced to the ntegral back n elementary school, we consdered only functons

More information

Lecture 13 APPROXIMATION OF SECOMD ORDER DERIVATIVES

Lecture 13 APPROXIMATION OF SECOMD ORDER DERIVATIVES COMPUTATIONAL FLUID DYNAMICS: FDM: Appromaton of Second Order Dervatves Lecture APPROXIMATION OF SECOMD ORDER DERIVATIVES. APPROXIMATION OF SECOND ORDER DERIVATIVES Second order dervatves appear n dffusve

More information

EEE 241: Linear Systems

EEE 241: Linear Systems EEE : Lnear Systems Summary #: Backpropagaton BACKPROPAGATION The perceptron rule as well as the Wdrow Hoff learnng were desgned to tran sngle layer networks. They suffer from the same dsadvantage: they

More information

AGC Introduction

AGC Introduction . Introducton AGC 3 The prmary controller response to a load/generaton mbalance results n generaton adjustment so as to mantan load/generaton balance. However, due to droop, t also results n a non-zero

More information

Implementation of the Matrix Method

Implementation of the Matrix Method Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena Computatonal Photoncs Semnar 0 Implementaton of the Matr Method calculaton of the transfer matr calculaton of reflecton and

More information

Comparative Study Between Dispersive and Non-Dispersive Dielectric Permittivity in Spectral Remittances of Chiral Sculptured Zirconia Thin Films

Comparative Study Between Dispersive and Non-Dispersive Dielectric Permittivity in Spectral Remittances of Chiral Sculptured Zirconia Thin Films Comparatve Study Between Dspersve and Non-Dspersve Delectrc Permttvty n Spectral emttances of Chral Sculptured Zrcona Thn Flms Ferydon Babae * and Had Savalon 2 Department of Physcs Unversty of Qom Qom

More information

CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE

CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng

More information

Lecture 8: Reflection and Transmission of Waves. Normal incidence propagating waves. Normal incidence propagating waves

Lecture 8: Reflection and Transmission of Waves. Normal incidence propagating waves. Normal incidence propagating waves /8/5 Lecture 8: Reflecton and Transmsson of Waves Instructor: Dr. Gleb V. Tcheslavsk Contact: gleb@ee.lamar.edu Offce Hours: Room 3 Class web ste: www.ee.lamar.edu/gleb/e m/index.htm So far we have consdered

More information

The photon model and equations are derived through timedomain mutual energy current

The photon model and equations are derived through timedomain mutual energy current The photon model and equatons are derved through tmedoman mutual energy current Shuang-ren Zhao, Kevn Yang, Kang Yang, Xngang Yang, (Imrecons Inc, London Ontaro, Canada) Xnte Yang (Avaton Academy, Northwestern

More information

Formal solvers of the RT equation

Formal solvers of the RT equation Formal solvers of the RT equaton Formal RT solvers Runge- Kutta (reference solver) Pskunov N.: 979, Master Thess Long characterstcs (Feautrer scheme) Cannon C.J.: 970, ApJ 6, 55 Short characterstcs (Hermtan

More information

Complex Numbers. x = B B 2 4AC 2A. or x = x = 2 ± 4 4 (1) (5) 2 (1)

Complex Numbers. x = B B 2 4AC 2A. or x = x = 2 ± 4 4 (1) (5) 2 (1) Complex Numbers If you have not yet encountered complex numbers, you wll soon do so n the process of solvng quadratc equatons. The general quadratc equaton Ax + Bx + C 0 has solutons x B + B 4AC A For

More information

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could

More information

Implementation of the Matrix Method

Implementation of the Matrix Method Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena Computatonal Photoncs Semnar 0 Implementaton of the Matr Method calculaton of the transfer matr calculaton of reflecton and

More information

Complex Numbers Alpha, Round 1 Test #123

Complex Numbers Alpha, Round 1 Test #123 Complex Numbers Alpha, Round Test #3. Wrte your 6-dgt ID# n the I.D. NUMBER grd, left-justfed, and bubble. Check that each column has only one number darkened.. In the EXAM NO. grd, wrte the 3-dgt Test

More information

Electrical double layer: revisit based on boundary conditions

Electrical double layer: revisit based on boundary conditions Electrcal double layer: revst based on boundary condtons Jong U. Km Department of Electrcal and Computer Engneerng, Texas A&M Unversty College Staton, TX 77843-318, USA Abstract The electrcal double layer

More information

Physics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1

Physics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1 P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the

More information

Introduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:

Introduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law: CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and

More information

More metrics on cartesian products

More metrics on cartesian products More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of

More information

Temperature. Chapter Heat Engine

Temperature. Chapter Heat Engine Chapter 3 Temperature In prevous chapters of these notes we ntroduced the Prncple of Maxmum ntropy as a technque for estmatng probablty dstrbutons consstent wth constrants. In Chapter 9 we dscussed the

More information

Physics 4B. A positive value is obtained, so the current is counterclockwise around the circuit.

Physics 4B. A positive value is obtained, so the current is counterclockwise around the circuit. Physcs 4B Solutons to Chapter 7 HW Chapter 7: Questons:, 8, 0 Problems:,,, 45, 48,,, 7, 9 Queston 7- (a) no (b) yes (c) all te Queston 7-8 0 μc Queston 7-0, c;, a;, d; 4, b Problem 7- (a) Let be the current

More information

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0 MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector

More information

Errors for Linear Systems

Errors for Linear Systems Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch

More information

Numerical Solution of Boussinesq Equations as a Model of Interfacial-wave Propagation

Numerical Solution of Boussinesq Equations as a Model of Interfacial-wave Propagation BULLETIN of the Malaysan Mathematcal Scences Socety http://math.usm.my/bulletn Bull. Malays. Math. Sc. Soc. (2) 28(2) (2005), 163 172 Numercal Soluton of Boussnesq Equatons as a Model of Interfacal-wave

More information

ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM

ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look

More information

Transverse angular shift in the reflection of light beams

Transverse angular shift in the reflection of light beams 1 August 000 Optcs Communcatons 18 000 1 10 www.elsever.comrlocateroptcom Transverse angular shft n the reflecton of lght beams Javer Alda ) Optcs Department. UnÕersty Complutense of Madrd, School of Optcs,

More information

Lecture 5.8 Flux Vector Splitting

Lecture 5.8 Flux Vector Splitting Lecture 5.8 Flux Vector Splttng 1 Flux Vector Splttng The vector E n (5.7.) can be rewrtten as E = AU (5.8.1) (wth A as gven n (5.7.4) or (5.7.6) ) whenever, the equaton of state s of the separable form

More information

Inductance Calculation for Conductors of Arbitrary Shape

Inductance Calculation for Conductors of Arbitrary Shape CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors

More information

Transfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system

Transfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng

More information

2 Finite difference basics

2 Finite difference basics Numersche Methoden 1, WS 11/12 B.J.P. Kaus 2 Fnte dfference bascs Consder the one- The bascs of the fnte dfference method are best understood wth an example. dmensonal transent heat conducton equaton T

More information

Chapter - 2. Distribution System Power Flow Analysis

Chapter - 2. Distribution System Power Flow Analysis Chapter - 2 Dstrbuton System Power Flow Analyss CHAPTER - 2 Radal Dstrbuton System Load Flow 2.1 Introducton Load flow s an mportant tool [66] for analyzng electrcal power system network performance. Load

More information

Title: Radiative transitions and spectral broadening

Title: Radiative transitions and spectral broadening Lecture 6 Ttle: Radatve transtons and spectral broadenng Objectves The spectral lnes emtted by atomc vapors at moderate temperature and pressure show the wavelength spread around the central frequency.

More information

8.6 The Complex Number System

8.6 The Complex Number System 8.6 The Complex Number System Earler n the chapter, we mentoned that we cannot have a negatve under a square root, snce the square of any postve or negatve number s always postve. In ths secton we want

More information

SECOND ORDER NONLINEAR PROCESSES AT SURFACES AND INTERFACES

SECOND ORDER NONLINEAR PROCESSES AT SURFACES AND INTERFACES SECOND ORDER NONLINEAR PROCESSES AT SURFACES AND INTERFACES C.Stancu, R.Ehlch /A1 Boundary condtons of a polarzed sheet Radaton from a polarzed sheet Surface nonlnear response Bulk nonlnear response The

More information

Polarization of light: Malus law, the Fresnel equations, and optical activity.

Polarization of light: Malus law, the Fresnel equations, and optical activity. Polarzaton of lght: Malus law, the Fresnel equatons, and optcal actvty. PHYS 3330: Experments n Optcs Department of Physcs and Astronomy, Unversty of Georga, Athens, Georga 3060 (Dated: Revsed August 0)

More information

SUPPLEMENTARY INFORMATION

SUPPLEMENTARY INFORMATION do: 0.08/nature09 I. Resonant absorpton of XUV pulses n Kr + usng the reduced densty matrx approach The quantum beats nvestgated n ths paper are the result of nterference between two exctaton paths of

More information

Applied Nuclear Physics (Fall 2004) Lecture 23 (12/3/04) Nuclear Reactions: Energetics and Compound Nucleus

Applied Nuclear Physics (Fall 2004) Lecture 23 (12/3/04) Nuclear Reactions: Energetics and Compound Nucleus .101 Appled Nuclear Physcs (Fall 004) Lecture 3 (1/3/04) Nuclear Reactons: Energetcs and Compound Nucleus References: W. E. Meyerhof, Elements of Nuclear Physcs (McGraw-Hll, New York, 1967), Chap 5. Among

More information

From Biot-Savart Law to Divergence of B (1)

From Biot-Savart Law to Divergence of B (1) From Bot-Savart Law to Dvergence of B (1) Let s prove that Bot-Savart gves us B (r ) = 0 for an arbtrary current densty. Frst take the dvergence of both sdes of Bot-Savart. The dervatve s wth respect to

More information

Thermal-Fluids I. Chapter 18 Transient heat conduction. Dr. Primal Fernando Ph: (850)

Thermal-Fluids I. Chapter 18 Transient heat conduction. Dr. Primal Fernando Ph: (850) hermal-fluds I Chapter 18 ransent heat conducton Dr. Prmal Fernando prmal@eng.fsu.edu Ph: (850) 410-6323 1 ransent heat conducton In general, he temperature of a body vares wth tme as well as poston. In

More information

PHY2049 Exam 2 solutions Fall 2016 Solution:

PHY2049 Exam 2 solutions Fall 2016 Solution: PHY2049 Exam 2 solutons Fall 2016 General strategy: Fnd two resstors, one par at a tme, that are connected ether n SERIES or n PARALLEL; replace these two resstors wth one of an equvalent resstance. Now

More information

Lecture Note 3. Eshelby s Inclusion II

Lecture Note 3. Eshelby s Inclusion II ME340B Elastcty of Mcroscopc Structures Stanford Unversty Wnter 004 Lecture Note 3. Eshelby s Incluson II Chrs Wenberger and We Ca c All rghts reserved January 6, 004 Contents 1 Incluson energy n an nfnte

More information

CONDUCTORS AND INSULATORS

CONDUCTORS AND INSULATORS CONDUCTORS AND INSULATORS We defne a conductor as a materal n whch charges are free to move over macroscopc dstances.e., they can leave ther nucle and move around the materal. An nsulator s anythng else.

More information

Level Crossing Spectroscopy

Level Crossing Spectroscopy Level Crossng Spectroscopy October 8, 2008 Contents 1 Theory 1 2 Test set-up 4 3 Laboratory Exercses 4 3.1 Hanle-effect for fne structure.................... 4 3.2 Hanle-effect for hyperfne structure.................

More information

THE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens

THE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of

More information

Polarization of light: Malus law, the Fresnel equations, and optical activity.

Polarization of light: Malus law, the Fresnel equations, and optical activity. Polarzaton of lght: Malus law, the Fresnel equatons, and optcal actvty. PHYS 3330: Experments n Optcs Department of Physcs and Astronomy, Unversty of Georga, Athens, Georga 3060 (Dated: Revsed August 0)

More information

COMPLEX NUMBERS AND QUADRATIC EQUATIONS

COMPLEX NUMBERS AND QUADRATIC EQUATIONS COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not

More information

A how to guide to second quantization method.

A how to guide to second quantization method. Phys. 67 (Graduate Quantum Mechancs Sprng 2009 Prof. Pu K. Lam. Verson 3 (4/3/2009 A how to gude to second quantzaton method. -> Second quantzaton s a mathematcal notaton desgned to handle dentcal partcle

More information

Comparison of the Population Variance Estimators. of 2-Parameter Exponential Distribution Based on. Multiple Criteria Decision Making Method

Comparison of the Population Variance Estimators. of 2-Parameter Exponential Distribution Based on. Multiple Criteria Decision Making Method Appled Mathematcal Scences, Vol. 7, 0, no. 47, 07-0 HIARI Ltd, www.m-hkar.com Comparson of the Populaton Varance Estmators of -Parameter Exponental Dstrbuton Based on Multple Crtera Decson Makng Method

More information

Physics 114 Exam 2 Fall 2014 Solutions. Name:

Physics 114 Exam 2 Fall 2014 Solutions. Name: Physcs 114 Exam Fall 014 Name: For gradng purposes (do not wrte here): Queston 1. 1... 3. 3. Problem Answer each of the followng questons. Ponts for each queston are ndcated n red. Unless otherwse ndcated,

More information

A new Approach for Solving Linear Ordinary Differential Equations

A new Approach for Solving Linear Ordinary Differential Equations , ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of

More information

Physics 2A Chapter 3 HW Solutions

Physics 2A Chapter 3 HW Solutions Phscs A Chapter 3 HW Solutons Chapter 3 Conceptual Queston: 4, 6, 8, Problems: 5,, 8, 7, 3, 44, 46, 69, 70, 73 Q3.4. Reason: (a) C = A+ B onl A and B are n the same drecton. Sze does not matter. (b) C

More information

A semi-analytic technique to determine the propagation constant of periodically segmented Ti:LiNbO 3 waveguide

A semi-analytic technique to determine the propagation constant of periodically segmented Ti:LiNbO 3 waveguide Avalable onlne at www.pelagaresearchlbrary.com Pelaga Research Lbrary Advances n Appled Scence Research, 011, (1): 16-144 ISSN: 0976-8610 CODEN (USA): AASRFC A sem-analytc technque to determne the propagaton

More information

This model contains two bonds per unit cell (one along the x-direction and the other along y). So we can rewrite the Hamiltonian as:

This model contains two bonds per unit cell (one along the x-direction and the other along y). So we can rewrite the Hamiltonian as: 1 Problem set #1 1.1. A one-band model on a square lattce Fg. 1 Consder a square lattce wth only nearest-neghbor hoppngs (as shown n the fgure above): H t, j a a j (1.1) where,j stands for nearest neghbors

More information

One Dimensional Axial Deformations

One Dimensional Axial Deformations One Dmensonal al Deformatons In ths secton, a specfc smple geometr s consdered, that of a long and thn straght component loaded n such a wa that t deforms n the aal drecton onl. The -as s taken as the

More information

Now that we have laws or better postulates we should explore what they imply

Now that we have laws or better postulates we should explore what they imply I-1 Theorems from Postulates: Now that we have laws or better postulates we should explore what they mply about workng q.m. problems -- Theorems (Levne 7.2, 7.4) Thm 1 -- egen values of Hermtan operators

More information

MATH 5630: Discrete Time-Space Model Hung Phan, UMass Lowell March 1, 2018

MATH 5630: Discrete Time-Space Model Hung Phan, UMass Lowell March 1, 2018 MATH 5630: Dscrete Tme-Space Model Hung Phan, UMass Lowell March, 08 Newton s Law of Coolng Consder the coolng of a well strred coffee so that the temperature does not depend on space Newton s law of collng

More information

Workshop: Approximating energies and wave functions Quantum aspects of physical chemistry

Workshop: Approximating energies and wave functions Quantum aspects of physical chemistry Workshop: Approxmatng energes and wave functons Quantum aspects of physcal chemstry http://quantum.bu.edu/pltl/6/6.pdf Last updated Thursday, November 7, 25 7:9:5-5: Copyrght 25 Dan Dll (dan@bu.edu) Department

More information

Gravitational Acceleration: A case of constant acceleration (approx. 2 hr.) (6/7/11)

Gravitational Acceleration: A case of constant acceleration (approx. 2 hr.) (6/7/11) Gravtatonal Acceleraton: A case of constant acceleraton (approx. hr.) (6/7/11) Introducton The gravtatonal force s one of the fundamental forces of nature. Under the nfluence of ths force all objects havng

More information

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle

More information

1 Derivation of Rate Equations from Single-Cell Conductance (Hodgkin-Huxley-like) Equations

1 Derivation of Rate Equations from Single-Cell Conductance (Hodgkin-Huxley-like) Equations Physcs 171/271 -Davd Klenfeld - Fall 2005 (revsed Wnter 2011) 1 Dervaton of Rate Equatons from Sngle-Cell Conductance (Hodgkn-Huxley-lke) Equatons We consder a network of many neurons, each of whch obeys

More information

Formulas for the Determinant

Formulas for the Determinant page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use

More information

arxiv: v1 [math.ho] 18 May 2008

arxiv: v1 [math.ho] 18 May 2008 Recurrence Formulas for Fbonacc Sums Adlson J. V. Brandão, João L. Martns 2 arxv:0805.2707v [math.ho] 8 May 2008 Abstract. In ths artcle we present a new recurrence formula for a fnte sum nvolvng the Fbonacc

More information

NUMERICAL DIFFERENTIATION

NUMERICAL DIFFERENTIATION NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the

More information

How Differential Equations Arise. Newton s Second Law of Motion

How Differential Equations Arise. Newton s Second Law of Motion page 1 CHAPTER 1 Frst-Order Dfferental Equatons Among all of the mathematcal dscplnes the theory of dfferental equatons s the most mportant. It furnshes the explanaton of all those elementary manfestatons

More information

Lecture 3: Probability Distributions

Lecture 3: Probability Distributions Lecture 3: Probablty Dstrbutons Random Varables Let us begn by defnng a sample space as a set of outcomes from an experment. We denote ths by S. A random varable s a functon whch maps outcomes nto the

More information

Density matrix. c α (t)φ α (q)

Density matrix. c α (t)φ α (q) Densty matrx Note: ths s supplementary materal. I strongly recommend that you read t for your own nterest. I beleve t wll help wth understandng the quantum ensembles, but t s not necessary to know t n

More information

Differentiating Gaussian Processes

Differentiating Gaussian Processes Dfferentatng Gaussan Processes Andrew McHutchon Aprl 17, 013 1 Frst Order Dervatve of the Posteror Mean The posteror mean of a GP s gven by, f = x, X KX, X 1 y x, X α 1 Only the x, X term depends on the

More information

8.592J: Solutions for Assignment 7 Spring 2005

8.592J: Solutions for Assignment 7 Spring 2005 8.59J: Solutons for Assgnment 7 Sprng 5 Problem 1 (a) A flament of length l can be created by addton of a monomer to one of length l 1 (at rate a) or removal of a monomer from a flament of length l + 1

More information

The equation of motion of a dynamical system is given by a set of differential equations. That is (1)

The equation of motion of a dynamical system is given by a set of differential equations. That is (1) Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence

More information

Supplementary materials for Self-induced back-action optical pulling force

Supplementary materials for Self-induced back-action optical pulling force Supplementar materals for Self-nduced back-acton optcal pullng force Tongtong Zhu, Yongn Cao, Ln Wang, Zhongquan Ne 2, Tun Cao 3, Fangku Sun, Zehu Jang, Manuel Neto-Vespernas 4, Yongmn Lu 5, Cheng-We Qu

More information

Grover s Algorithm + Quantum Zeno Effect + Vaidman

Grover s Algorithm + Quantum Zeno Effect + Vaidman Grover s Algorthm + Quantum Zeno Effect + Vadman CS 294-2 Bomb 10/12/04 Fall 2004 Lecture 11 Grover s algorthm Recall that Grover s algorthm for searchng over a space of sze wors as follows: consder the

More information