Femtosecond laser pulses
|
|
- Εὐνίκη Γιάγκος
- 5 χρόνια πριν
- Προβολές:
Transcript
1 Femosecon laser pulses Inroucion on femosecon lasers Numerical analysis Compuer conrolle experimens Lab wor
2 Pulse shaping Femosecon laser pulses : lab wor Time omain eraher specroscopy Specral inerferomery Average e Curren na Time ps Wavelengh
3 Femosecon laser pulses Sessions 1 an 3 : Inroucion o basic noions - Femosecon laser pulses - Use of Mlbf Malab for aa processing Sessions 4-9 : experimenal wor a Laboraoire Opique e Biosciences hp:// manuel.joffre@polyechnique.fr
4 Orers of magniue Perio Frequency Wavelengh λ λ ct 3 as 3 PH X rays 1 nm ν 33 as 3 PH 1 nm Ulra-viole as 3 PH 1 nm 3 fs 33 fs 333 fs Visible 3 TH 1 µm 3 TH 3TH Infrare 1 µm 1 µm 1fs 1 1 as 1 c -18 s s.3µm/fs c 3 ps 33 ps 3 GH Micro waves 1 mm 3 GH 1 mm Raio waves
5 Inroucion o femosecon lasers 1. Represenaion of a femosecon pulse. Linear propagaion 3. Nonlinear propagaion 4. Femosecon laser sources
6 Represenaion of a femosecon pulse fs E Fourier ransform Joseph Fourier E E exp iϕ E Frequency Ampliue Phase
7 Properies of Fourier ransforms - 1 Parseval Plancherel heorem Complex conjugaion TF Derivaion TF TF
8 Properies of Fourier ransforms - Proucs an convoluion proucs TF TF avec Transforms of usual funcions δ f f
9 Represenaion of a femosecon pulse E E * + E E exp i Elecric fiel π E E exp iϕ E Specral ampliue ϕ Specral phase I E Specral linensiy i or specrum E real E E * E + *
10 φ E From real fiel o analyic represenaion E * Θ E + * E Θ exp iφ Complex fiel or analyic represenaion Θ [ [E]] cf Hilber ransform * + E Re φ Temporal phase I Temporal inensiy
11 Average values in ime an frequency We wrie A so ha is normalie : 1 π Δ Δ ² ² Pulse cener Cenral frequency carrier frequency π Δ Pulse uraion RMS Δ Specral wih RMS Δ Δ 1 Δ Δ
12 Group elay i i * π * i exp ϕ i exp ϕ ϕ i i + π ϕ π i + ϕ ϕ τ g Group elay ime of arrival of frequency componen τ g Group elay ime of arrival of frequency componen τ Pulse ime of arrival τ g Pulse ime of arrival
13 RMS pulse uraion i π exp ϕ i exp ϕ ϕ i i + π ϕ π + π π τ g + ϕ g τ g τ τ ϕ g g + Δ ϕ g g g τ ϕ Δ + Δ Δ
14 Pulse uraion an specral phase iϕ exp ϕ : specral phase τ g ϕ : group elay Δ Δ ϕ + Δτ g For a given he shores possible pulse is achieve when τ g is frequency inepenen Δτ g. This is a so-calle Fourier ransform limie pulse. Conversely when τ g varies wih frequency here is a frequency chirp.
15 1 ϕ ϕ Quaraic specral phase : Pulse lenghening an chirp ϕ τ ϕ g Δ Δϕ + ϕ Various frequency componens arrive one a a ime : frequency chirp. Δ ϕ τ g g E
16 Ranom specral phase : incoheren fiel Ch hamp élec crique SPECTRE Fréquence [TH] Δ Δϕ + Δτ ϕ g 5 Phase [r] Champ élecrique Inensié PROFIL TEMPOREL Temps [fs] Temps [fs]
17 Specral wih an emporal phase iφ exp φ : emporal phase π φ φ Ω : insananeous frequency φ Δ Δ φ + ΔΩ Δ φ Ω E ² Δ
18 Linear propagaion of a femosecon pulse where : Propagaion equaion : n. c + Soluion : exp i Thus : ϕ ϕ + Group elay: ϕ ϕ τ g + τ g + V g Group velociy V g 1 V g
19 Linear ispersion in a ransparen meium n α Specre e l'impulsion L IR Zone e ransparence UV à 8 nm L 1cm Quaniy Definiion Uni SiO SF1 Phase inex n c / Group inex n g c Group velociy V 1/ c / µm/fs g n g GVD Group Velociy Dispersion i f²/ fs²/cm Group elay τ ϕ L fs 49E3 58E3 g GDD Group Delay Dispersion τ ϕ LL fs² g TOD Thir Orer Dispersion τ ϕ L fs g
20 Linear ispersion in a ransparen meium n α Specre e l'impulsion L IR Zone e ransparence UV ϕ ϕ ϕ τ + + g τ g The pulse acquires a quaraic specral phase resuling from group velociy ispersion GVD hus a frequency chirp....
21 Pulse lenghening ϕ ϕ + + τ ϕ τ g g + + τ τ Δ + Δ Δ Δ Δ Relaion hols for any pulse shape
22 Pulse envelope u is by efiniion he pulse envelope. u exp i u exp i + u is always cenere a. u is always cenere a. 1 exp i u exp i u
23 From he soluion o he equaion i u u 1 exp p u i u u u u i NB : Slowly varying envelope approximaion i.e. conra-propagaing wave no aen ino accoun aen ino accoun.
24 Analogy wih he propagaion of a wavepace in quanum mechanics in quanum mechanics x ψ u x x m x i ψ ψ h h u u i x m i ψ ψ h h u u i m i ψ h u i exp i ϕ ψ ψ exp ϕ i u u m h ϕ 1 ϕ
25 Linear propagaion of a femosecon pulse Meium : fuse silica. n n g fs²/cm. Gaussian pulse. λ 8 nm.
26 Linear propagaion of a femosecon pulse Meium : fuse silica. n n g fs²/cm. Hypergaussian pulse. λ 8 nm.
27 3. Nonlinear propagaion
28 Nonlinear opics Linear regime Non-linear regime
29 Opical Kerr effec Thir-orer polariaion n I n + n I
30 Auofocusing nr n I n + n I ϕ x y π n + n I x y L λ
31 Generaion of specral coninuum n n + ni φ φ + n + n I v c c φ φ Φ Ω
32 Generaion of specral coninuum Ecole Polyechnique phoo G. Labroille P. Lavialle M. Joffre NB : Opical Kerr effec is only one of several nonlinear effecs conribuing o coninuum generaion.
33 4. Femosecon laser sources
34 Superposiion of longiuinal moes Frequency omain δ δ Time omain π/δ
35 General scheme for a femosecon laser Broaban amplifying meium L cav Energy gy source
36 A evice wih negaive ϕ" : he prism pair Dispersive Milieu ispersif meium GVD GVD compensaion
37 Anoher evice wih negaive ϕ" : he chirpe mirror
38 General scheme for a femosecon laser Group Velociy Dispersion Compensaion To achieve moe locing i is sufficien o mae sure ha losses are greaer when he specral phase is no fla i.e. when he pulse is longer. Moe-locing mechanism L cav Broaban amplifying meium Energy gy source
39 Dye laser Moe-locing mechanism : saurable absorban Mécanisme e blocage es moes : absorban saurable 6 nm.1 nj 3 fs 1 MH R6G AS Laser Argon
40 Tianium:Sapphire P.F. Moulon JOSAB
41 Tianium:Sapphire laser Ti:S 8 nm nj 1 fs 1 MH Pump laser Moe-locing mechanism : Opical Kerr effe nr Ti:S
42 Tianium:Sapphire laser elivering ulrashor pulses 5.4 fs U. Morgner F. X. Kärner S. H. Cho Y. Chen H. A. Haus J. G. Fujimoo E. P. Ippen V. Scheurer G. Angelow T. Tschui Sub-wo-cycle pulses from a Kerr-lens moe-loce Ti:sapphire laser Op. Le
43 Commercially available femosecon oscillaors 1 fs.7 1µm fs µm hp:// 1 fs 8 nm hp:// 5 fs 1.55 µm hp:// hp://
44 LOB Examples of femosecon amplifiers A. Bonvale e al. 1 mj / 1 fs 1 1 H
45 Examples of femosecon amplifiers Laboraoire Opique Appliquée ENSTA Ecole Polyechnique 1J / 3 fs 3 1 H
46 Examples of femosecon amplifiers Laboraoire Livermore Eas-Unis 1J/1ps 1 1PW
3 Frequency Domain Representation of Continuous Signals and Systems
3 Frequency Domain Represenaion of Coninuous Signals and Sysems 3. Fourier Series Represenaion of Periodic Signals............. 2 3.. Exponenial Fourier Series.................... 2 3..2 Discree Fourier
Διαβάστε περισσότεραLecture 12 Modulation and Sampling
EE 2 spring 2-22 Handou #25 Lecure 2 Modulaion and Sampling The Fourier ransform of he produc of wo signals Modulaion of a signal wih a sinusoid Sampling wih an impulse rain The sampling heorem 2 Convoluion
Διαβάστε περισσότεραAppendix. The solution begins with Eq. (2.15) from the text, which we repeat here for 1, (A.1)
Aenix Aenix A: The equaion o he sock rice. The soluion egins wih Eq..5 rom he ex, which we reea here or convenience as Eq.A.: [ [ E E X, A. c α where X u ε, α γ, an c α y AR. Take execaions o Eq. A. as
Διαβάστε περισσότεραFourier transform of continuous-time signals
Fourier ransform of coninuous-ime signals Specral represenaion of non-periodic signals Fourier ransform: aperiodic signals repeiion of a finie-duraion signal x()> periodic signals. x x T x kt x kt k k
Διαβάστε περισσότεραω = radians per sec, t = 3 sec
Secion. Linear and Angular Speed 7. From exercise, =. A= r A = ( 00 ) (. ) = 7,00 in 7. Since 7 is in quadran IV, he reference 7 8 7 angle is = =. In quadran IV, he cosine is posiive. Thus, 7 cos = cos
Διαβάστε περισσότερα16. 17. r t te 2t i t 1. 18 19 Find the derivative of the vector function. 19. r t e t cos t i e t sin t j ln t k. 31 33 Evaluate the integral.
SECTION.7 VECTOR FUNCTIONS AND SPACE CURVES.7 VECTOR FUNCTIONS AND SPACE CURVES A Click here for answers. S Click here for soluions. Copyrigh Cengage Learning. All righs reserved.. Find he domain of he
Διαβάστε περισσότερα( ) ( t) ( 0) ( ) dw w. = = β. Then the solution of (1.1) is easily found to. wt = t+ t. We generalize this to the following nonlinear differential
Periodic oluion of van der Pol differenial equaion. by A. Arimoo Deparmen of Mahemaic Muahi Iniue of Technology Tokyo Japan in Seminar a Kiami Iniue of Technology January 8 9. Inroducion Le u conider a
Διαβάστε περισσότεραd dt S = (t)si d dt R = (t)i d dt I = (t)si (t)i
d d S = ()SI d d I = ()SI ()I d d R = ()I d d S = ()SI μs + fi + hr d d I = + ()SI (μ + + f + ())I d d R = ()I (μ + h)r d d P(S,I,) = ()(S +1)(I 1)P(S +1, I 1, ) +()(I +1)P(S,I +1, ) (()SI + ()I)P(S,I,)
Διαβάστε περισσότερα( ) ( ) ( ) Fourier series. ; m is an integer. r(t) is periodic (T>0), r(t+t) = r(t), t Fundamental period T 0 = smallest T. Fundamental frequency ω
Fourier series e jm when m d when m ; m is an ineger. jm jm jm jm e d e e e jm jm jm jm r( is periodi (>, r(+ r(, Fundamenal period smalles Fundamenal frequeny r ( + r ( is periodi hen M M e j M, e j,
Διαβάστε περισσότερα6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
Διαβάστε περισσότεραAnti-aliasing Prefilter (6B) Young Won Lim 6/8/12
ni-aliasing Prefiler (6B) Copyrigh (c) Young W. Lim. Permission is graned o copy, disribue and/or modify his documen under he erms of he GNU Free Documenaion License, Version. or any laer version published
Διαβάστε περισσότεραSpace-Time Symmetries
Chapter Space-Time Symmetries In classical fiel theory any continuous symmetry of the action generates a conserve current by Noether's proceure. If the Lagrangian is not invariant but only shifts by a
Διαβάστε περισσότεραD-Wave D-Wave Systems Inc.
D-Wave D-Wave sems Inc. Anaol Yu. mirnov D-Wave sems Inc. Vancouver Briish Columbia HE QUANUM COMPUING COMPANY M Decoherence and Noise Conrol in rongl Driven uperconducing Quanum Bis Collaboraion: M. Grajcar
Διαβάστε περισσότεραUniversity of Washington Department of Chemistry Chemistry 553 Spring Quarter 2010 Homework Assignment 3 Due 04/26/10
Universiy of Washingon Deparmen of Chemisry Chemisry 553 Spring Quarer 1 Homework Assignmen 3 Due 4/6/1 v e v e A s ds: a) Show ha for large 1 and, (i.e. 1 >> and >>) he velociy auocorrelaion funcion 1)
Διαβάστε περισσότεραThe Euler Equations! λ 1. λ 2. λ 3. ρ ρu. E = e + u 2 /2. E + p ρ. = de /dt. = dh / dt; h = h( T ); c p. / c v. ; γ = c p. p = ( γ 1)ρe. c v.
hp://www.nd.ed/~gryggva/cfd-corse/ The Eler Eqaions The Eler Eqaions The Eler eqaions for D flow: + + p = x E E + p where Define E = e + / H = h + /; h = e + p/ Gréar Tryggvason Spring 3 Ideal Gas: p =
Διαβάστε περισσότεραBandPass (4A) Young Won Lim 1/11/14
BandPass (4A) Copyright (c) 22 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later version
Διαβάστε περισσότερα6.003: Signals and Systems
6.3: Signals and Sysems Modulaion December 6, 2 Communicaions Sysems Signals are no always well mached o he media hrough which we wish o ransmi hem. signal audio video inerne applicaions elephone, radio,
Διαβάστε περισσότεραFractional Calculus. Student: Manal AL-Ali Dr. Abdalla Obeidat
Fracional Calculu Suen: Manal AL-Ali Dr. Aballa Obeia Deignaion Deignaion mean inegraion an iffereniaion of arbirary orer, In oher ereion i mean ealing wih oeraor like,, i arbirary real or Comle value.
Διαβάστε περισσότεραFourier Transform. Fourier Transform
ECE 307 Z. Aliyziioglu Eleril & Compuer Engineering Dep. Cl Poly Pomon The Fourier rnsform (FT is he exension of he Fourier series o nonperiodi signls. The Fourier rnsform of signl exis if sisfies he following
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότερα6.4 Superposition of Linear Plane Progressive Waves
.0 - Marine Hydrodynamics, Spring 005 Lecture.0 - Marine Hydrodynamics Lecture 6.4 Superposition of Linear Plane Progressive Waves. Oblique Plane Waves z v k k k z v k = ( k, k z ) θ (Looking up the y-ais
Διαβάστε περισσότεραFourier Series. Fourier Series
ECE 37 Z. Aliyazicioglu Elecrical & Compuer Egieerig Dep. Cal Poly Pomoa Periodic sigal is a fucio ha repeas iself every secods. x() x( ± ) : period of a fucio, : ieger,,3, x() 3 x() x() Periodic sigal
Διαβάστε περισσότεραJesse Maassen and Mark Lundstrom Purdue University November 25, 2013
Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering
Διαβάστε περισσότεραRiemann Hypothesis: a GGC representation
Riemann Hypohesis: a GGC represenaion Nicholas G. Polson Universiy of Chicago Augus 8, 8 Absrac A GGC Generalized Gamma Convoluion represenaion for Riemann s reciprocal ξ-funcion is consruced. This provides
Διαβάστε περισσότεραPhys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
Διαβάστε περισσότεραCHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Διαβάστε περισσότεραMath 6 SL Probability Distributions Practice Test Mark Scheme
Math 6 SL Probability Distributions Practice Test Mark Scheme. (a) Note: Award A for vertical line to right of mean, A for shading to right of their vertical line. AA N (b) evidence of recognizing symmetry
Διαβάστε περισσότεραWhat happens when two or more waves overlap in a certain region of space at the same time?
Wave Superposition What happens when two or more waves overlap in a certain region of space at the same time? To find the resulting wave according to the principle of superposition we should sum the fields
Διαβάστε περισσότεραSecond Order RLC Filters
ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor
Διαβάστε περισσότεραSolutions - Chapter 4
Solutions - Chapter Kevin S. Huang Problem.1 Unitary: Ût = 1 ī hĥt Û tût = 1 Neglect t term: 1 + hĥ ī t 1 īhĥt = 1 + hĥ ī t ī hĥt = 1 Ĥ = Ĥ Problem. Ût = lim 1 ī ] n hĥ1t 1 ī ] hĥt... 1 ī ] hĥnt 1 ī ]
Διαβάστε περισσότεραReservoir modeling. Reservoir modelling Linear reservoirs. The linear reservoir, no input. Starting up reservoir modeling
Reservoir modeling Reservoir modelling Linear reservoirs Paul Torfs Basic equaion for one reservoir:) change in sorage = sum of inflows minus ouflows = Q in,n Q ou,n n n jus an ordinary differenial equaion
Διαβάστε περισσότεραwave energy Superposition of linear plane progressive waves Marine Hydrodynamics Lecture Oblique Plane Waves:
3.0 Marine Hydrodynamics, Fall 004 Lecture 0 Copyriht c 004 MIT - Department of Ocean Enineerin, All rihts reserved. 3.0 - Marine Hydrodynamics Lecture 0 Free-surface waves: wave enery linear superposition,
Διαβάστε περισσότερα6.003: Signals and Systems. Modulation
6.3: Signals and Sysems Modulaion December 6, 2 Subjec Evaluaions Your feedback is imporan o us! Please give feedback o he saff and fuure 6.3 sudens: hp://web.mi.edu/subjecevaluaion Evaluaions are open
Διαβάστε περισσότεραLecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
Διαβάστε περισσότερα= e 6t. = t 1 = t. 5 t 8L 1[ 1 = 3L 1 [ 1. L 1 [ π. = 3 π. = L 1 3s = L. = 3L 1 s t. = 3 cos(5t) sin(5t).
Worked Soluion 95 Chaper 25: The Invere Laplace Tranform 25 a From he able: L ] e 6 6 25 c L 2 ] ] L! + 25 e L 5 2 + 25] ] L 5 2 + 5 2 in(5) 252 a L 6 + 2] L 6 ( 2)] 6L ( 2)] 6e 2 252 c L 3 8 4] 3L ] 8L
Διαβάστε περισσότεραECE 308 SIGNALS AND SYSTEMS FALL 2017 Answers to selected problems on prior years examinations
ECE 308 SIGNALS AND SYSTEMS FALL 07 Answers to selected problems on prior years examinations Answers to problems on Midterm Examination #, Spring 009. x(t) = r(t + ) r(t ) u(t ) r(t ) + r(t 3) + u(t +
Διαβάστε περισσότεραFinite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Διαβάστε περισσότεραk A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +
Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b
Διαβάστε περισσότερα4.4 Superposition of Linear Plane Progressive Waves
.0 Marine Hydrodynamics, Fall 08 Lecture 6 Copyright c 08 MIT - Department of Mechanical Engineering, All rights reserved..0 - Marine Hydrodynamics Lecture 6 4.4 Superposition of Linear Plane Progressive
Διαβάστε περισσότερα4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Διαβάστε περισσότεραDurbin-Levinson recursive method
Durbin-Levinson recursive method A recursive method for computing ϕ n is useful because it avoids inverting large matrices; when new data are acquired, one can update predictions, instead of starting again
Διαβάστε περισσότεραMATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)
1. MATH43 String Theory Solutions 4 x = 0 τ = fs). 1) = = f s) ) x = x [f s)] + f s) 3) equation of motion is x = 0 if an only if f s) = 0 i.e. fs) = As + B with A, B constants. i.e. allowe reparametrisations
Διαβάστε περισσότεραMath221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Διαβάστε περισσότεραLecture 2. Soundness and completeness of propositional logic
Lecture 2 Soundness and completeness of propositional logic February 9, 2004 1 Overview Review of natural deduction. Soundness and completeness. Semantics of propositional formulas. Soundness proof. Completeness
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραPositive dispersion: 2 n. ω > 0, 2 n. Negative dispersion: ω < 0, 2 n
Positive dispersion: 2 n ω > 0, 2 n 2 λ > 0 2 Negative dispersion: 2 n ω < 0, 2 n 2 λ < 0 2 φ(z,ω) = k ( n ω )z E( z,t)= 1 2π E ( z = 0,ω )e iωt iφ z,ω e ( ) dω φ(z,ω) = k ( n ω )z φ( ω )= φ 0 + ω ω 0
Διαβάστε περισσότεραPart 4 RAYLEIGH AND LAMB WAVES
Part 4 RAYLEIGH AND LAMB WAVES Rayleigh Surfae Wave x x 1 x 3 urfae wave x 1 x 3 Partial Wave Deompoition Diplaement potential: u = ϕ + ψ Wave equation: 1 ϕ 1 ψ ϕ = = k ϕ an ψ = = k t t ψ Wave veloitie:
Διαβάστε περισσότεραF19MC2 Solutions 9 Complex Analysis
F9MC Solutions 9 Complex Analysis. (i) Let f(z) = eaz +z. Then f is ifferentiable except at z = ±i an so by Cauchy s Resiue Theorem e az z = πi[res(f,i)+res(f, i)]. +z C(,) Since + has zeros of orer at
Διαβάστε περισσότεραFourier transform, STFT 5. Continuous wavelet transform, CWT STFT STFT STFT STFT [1] CWT CWT CWT STFT [2 5] CWT STFT STFT CWT CWT. Griffin [8] CWT CWT
1,a) 1,2,b) Continuous wavelet transform, CWT CWT CWT CWT CWT 100 1. Continuous wavelet transform, CWT [1] CWT CWT CWT [2 5] CWT CWT CWT CWT CWT Irino [6] CWT CWT CWT CWT CWT 1, 7-3-1, 113-0033 2 NTT,
Διαβάστε περισσότεραΧρονοσειρές Μάθημα 3
Χρονοσειρές Μάθημα 3 Ασυσχέτιστες (λευκός θόρυβος) και ανεξάρτητες (iid) παρατηρήσεις Chafield C., The Analysis of Time Series, An Inroducion, 6 h ediion,. 38 (Chaer 3): Some auhors refer o make he weaker
Διαβάστε περισσότερα9.1 Introduction 9.2 Lags in the Error Term: Autocorrelation 9.3 Estimating an AR(1) Error Model 9.4 Testing for Autocorrelation 9.
9.1 Inroducion 9.2 Lags in he Error Term: Auocorrelaion 9.3 Esimaing an AR(1) Error Model 9.4 Tesing for Auocorrelaion 9.5 An Inroducion o Forecasing: Auoregressive Models 9.6 Finie Disribued Lags 9.7
Διαβάστε περισσότεραCalculating the propagation delay of coaxial cable
Your source for quality GNSS Networking Solutions and Design Services! Page 1 of 5 Calculating the propagation delay of coaxial cable The delay of a cable or velocity factor is determined by the dielectric
Διαβάστε περισσότεραA. Two Planes Waves, Same Frequency Visible light
Interference 1 A. Two Planes Waves, Same Frequency EE 1 rr, tt = EE 0,1 cccccc αα 1 ωω tt αα 1 kk 1. rr + εε 1 EE 2 rr, tt = EE 0,2 cccccc αα 2 ωω tt αα 2 kk 2. rr + εε 2 ωω = 4.3 7.5 10 14 HHHH Visible
Διαβάστε περισσότεραUniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
Διαβάστε περισσότεραExercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.
Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given
Διαβάστε περισσότερα6.003: Signals and Systems. Modulation
6.003: Signals and Systems Modulation May 6, 200 Communications Systems Signals are not always well matched to the media through which we wish to transmit them. signal audio video internet applications
Διαβάστε περισσότεραOutline Analog Communications. Lecture 05 Angle Modulation. Instantaneous Frequency and Frequency Deviation. Angle Modulation. Pierluigi SALVO ROSSI
Outline Analog Communications Lecture 05 Angle Modulation 1 PM and FM Pierluigi SALVO ROSSI Department of Industrial and Information Engineering Second University of Naples Via Roma 9, 81031 Aversa (CE),
Διαβάστε περισσότεραSupporting Information
Supporting Information Wiley-VC 007 9 Weinheim, Germany ew ear Infrared Dyes and Fluorophores Based on Diketopyrrolopyrroles Dipl.-Chem. Georg M. Fischer, Dipl.-Chem. Andreas P. Ehlers, Prof. Dr. Andreas
Διαβάστε περισσότεραLASER και ΕΦΑΡΜΟΓΕΣ ΣΤΗΝ ΤΕΧΝΟΛΟΓΙΑ ΤΗΣ ΑΠΕΙΚΟΝΙΣΗΣ
ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΑΤΡΩΝ ΤΜΗΜΑ ΦΥΣΙΚΗΣ ΤΟΜΕΑΣ ΗΛΕΚΤΡΟΝΙΚΗΣ & ΥΠΟΛΟΓΙΣΤΩΝ ΕΡΓΑΣΤΗΡΙΟ LASER http://www.physics.upatras.gr/laserlab LASER και ΕΦΑΡΜΟΓΕΣ ΣΤΗΝ ΤΕΧΝΟΛΟΓΙΑ ΤΗΣ ΑΠΕΙΚΟΝΙΣΗΣ ΟΜΙΛΗΤΗΣ Πέτρος Περσεφόνης
Διαβάστε περισσότερα2. Laser Specifications 2 1 Specifications IK4301R D IK4401R D IK4601R E IK4101R F. Linear Linear Linear Linear
2. Laser Specifications 2 1 Specifications IK4301R D IK4401R D IK4601R E IK4101R F 441.6 441.6 441.6 441.6 30 50 70 100 TEM00 TEM00 TEM00 TEM00 BEAM DIAMETER ( 1/e2) 1.1 1.1 1.2 1.2 0.5 0.5 0.5 0.4 RATIO
Διαβάστε περισσότεραReview: Molecules = + + = + + Start with the full Hamiltonian. Use the Born-Oppenheimer approximation
Review: Molecules Start with the full amiltonian Ze e = + + ZZe A A B i A i me A ma ia, 4πε 0riA i< j4πε 0rij A< B4πε 0rAB Use the Born-Oppenheimer approximation elec Ze e = + + A A B i i me ia, 4πε 0riA
Διαβάστε περισσότεραΣΗΜΑΤΑ ΔΙΑΚΡΙΤΟΥ ΧΡΟΝΟΥ
ΣΗΜΑΤΑ ΔΙΑΚΡΙΤΟΥ ΧΡΟΝΟΥ y t x Α. ΣΚΟΔΡΑΣ ΣΗΜΑΤΑ ΚΑΙ ΣΥΣΤΗΜΑΤΑ ΙΙ (22Y603) ΕΝΟΤΗΤΑ 1 ΔΙΑΛΕΞΗ 2 ΔΙΑΦΑΝΕΙΑ 1 ΤΥΠΟΙ ΣΗΜΑΤΩΝ Analog: Continuous Time & Continuous Amplitude Sampled: Discrete Time & Continuous
Διαβάστε περισσότεραSection 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Διαβάστε περισσότεραAppendix to On the stability of a compressible axisymmetric rotating flow in a pipe. By Z. Rusak & J. H. Lee
Appendi to On the stability of a compressible aisymmetric rotating flow in a pipe By Z. Rusak & J. H. Lee Journal of Fluid Mechanics, vol. 5 4, pp. 5 4 This material has not been copy-edited or typeset
Διαβάστε περισσότεραInverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότεραApproximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
Διαβάστε περισσότεραECE Spring Prof. David R. Jackson ECE Dept. Notes 2
ECE 634 Spring 6 Prof. David R. Jackson ECE Dept. Notes Fields in a Source-Free Region Example: Radiation from an aperture y PEC E t x Aperture Assume the following choice of vector potentials: A F = =
Διαβάστε περισσότεραNowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that
Διαβάστε περισσότεραAssignment 1 Solutions Complex Sinusoids
Assignment Solutions Complex Sinusoids ECE 223 Signals and Systems II Version. Spring 26. Eigenfunctions of LTI systems. Which of the following signals are eigenfunctions of LTI systems? a. x[n] =cos(
Διαβάστε περισσότεραHartree-Fock Theory. Solving electronic structure problem on computers
Hartree-Foc Theory Solving electronic structure problem on computers Hartree product of non-interacting electrons mean field molecular orbitals expectations values one and two electron operators Pauli
Διαβάστε περισσότεραCT Correlation (2B) Young Won Lim 8/15/14
CT Correlation (2B) 8/5/4 Copyright (c) 2-24 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any
Διαβάστε περισσότεραThe Student s t and F Distributions Page 1
The Suden s and F Disribuions Page The Fundamenal Transformaion formula for wo random variables: Consider wo random variables wih join probabiliy disribuion funcion f (, ) simulaneously ake on values in
Διαβάστε περισσότεραDESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.
DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec
Διαβάστε περισσότεραITU-R P (2012/02) khz 150
(0/0) khz 0 P ii (IPR) (ITU-T/ITU-R/ISO/IEC) ITU-R http://www.itu.int/itu-r/go/patents/en http://www.itu.int/publ/r-rec/en BO BR BS BT F M P RA RS S SA SF SM SNG TF V ITU-R 0 ITU 0 (ITU) khz 0 (0-009-00-003-00-994-990)
Διαβάστε περισσότεραOutline. Detection Theory. Background. Background (Cont.)
Outlie etectio heory Chapter7. etermiistic Sigals with Ukow Parameters afiseh S. Mazloum ov. 3th Backgroud Importace of sigal iformatio Ukow amplitude Ukow arrival time Siusoidal detectio Classical liear
Διαβάστε περισσότεραSCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Διαβάστε περισσότεραΨηφιακή Επεξεργασία Φωνής
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Ψηφιακή Επεξεργασία Φωνής Ενότητα 1η: Ψηφιακή Επεξεργασία Σήματος Στυλιανού Ιωάννης Τμήμα Επιστήμης Υπολογιστών CS578- Speech Signal Processing Lecture 1: Discrete-Time
Διαβάστε περισσότεραSolutions to the Schrodinger equation atomic orbitals. Ψ 1 s Ψ 2 s Ψ 2 px Ψ 2 py Ψ 2 pz
Solutions to the Schrodinger equation atomic orbitals Ψ 1 s Ψ 2 s Ψ 2 px Ψ 2 py Ψ 2 pz ybridization Valence Bond Approach to bonding sp 3 (Ψ 2 s + Ψ 2 px + Ψ 2 py + Ψ 2 pz) sp 2 (Ψ 2 s + Ψ 2 px + Ψ 2 py)
Διαβάστε περισσότεραencouraged to use the Version of Record that, when published, will replace this version. The most /BCJ BIOCHEMICAL JOURNAL
Biochemical Journal: this is an Accepted Manuscript, not the final Version of Record. You are encouraged to use the Version of Record that, when published, will replace this version. The most up-to-date
Διαβάστε περισσότεραGalatia SIL Keyboard Information
Galatia SIL Keyboard Information Keyboard ssignments The main purpose of the keyboards is to provide a wide range of keying options, so many characters can be entered in multiple ways. If you are typing
Διαβάστε περισσότεραCRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter
Διαβάστε περισσότεραDiracDelta. Notations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation
DiracDelta Notations Traditional name Dirac delta function Traditional notation x Mathematica StandardForm notation DiracDeltax Primary definition 4.03.02.000.0 x Π lim ε ; x ε0 x 2 2 ε Specific values
Διαβάστε περισσότεραAPPENDIX A DERIVATION OF JOINT FAILURE DENSITIES
APPENDIX A DERIVAION OF JOIN FAILRE DENSIIES I his Appedi we prese he derivaio o he eample ailre models as show i Chaper 3. Assme ha he ime ad se o ailre are relaed by he cio g ad he sochasic are o his
Διαβάστε περισσότεραJ. of Math. (PRC) u(t k ) = I k (u(t k )), k = 1, 2,, (1.6) , [3, 4] (1.1), (1.2), (1.3), [6 8]
Vol 36 ( 216 ) No 3 J of Mah (PR) 1, 2, 3 (1, 4335) (2, 4365) (3, 431) :,,,, : ; ; ; MR(21) : 35A1; 35A2 : O17529 : A : 255-7797(216)3-591-7 1 d d [x() g(, x )] = f(, x ),, (11) x = ϕ(), [ r, ], (12) x(
Διαβάστε περισσότεραCoupling of a Jet-Slot Oscillator With the Flow-Supply Duct: Flow-Acoustic Interaction Modeling
1th AIAA/CEAS Aeroacoustics Conference, May 006 interactions Coupling of a Jet-Slot Oscillator With the Flow-Supply Duct: Interaction M. Glesser 1, A. Billon 1, V. Valeau, and A. Sakout 1 mglesser@univ-lr.fr
Διαβάστε περισσότεραΔΙΑΚΡΙΤΟΣ ΜΕΤΑΣΧΗΜΑΤΙΣΜΟΣ FOURIER - Discrete Fourier Transform - DFT -
ΔΙΑΚΡΙΤΟΣ ΜΕΤΑΣΧΗΜΑΤΙΣΜΟΣ FOURIER - Discrete Fourier Transform - DFT - Α. ΣΚΟΔΡΑΣ ΣΗΜΑΤΑ ΚΑΙ ΣΥΣΤΗΜΑΤΑ ΙΙ (22Y603) ΕΝΟΤΗΤΑ 4 ΔΙΑΛΕΞΗ 1 ΔΙΑΦΑΝΕΙΑ 1 Διαφορετικοί Τύποι Μετασχηµατισµού Fourier Α. ΣΚΟΔΡΑΣ
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Διαβάστε περισσότεραThe ε-pseudospectrum of a Matrix
The ε-pseudospectrum of a Matrix Feb 16, 2015 () The ε-pseudospectrum of a Matrix Feb 16, 2015 1 / 18 1 Preliminaries 2 Definitions 3 Basic Properties 4 Computation of Pseudospectrum of 2 2 5 Problems
Διαβάστε περισσότεραγ c = rl = lt R ~ e (g l)t/t R Intensität 0 e γ c t Zeit, ns
There is however one main difference in this chapter compared to many other chapters. All loss and gain coefficients are given for the intensity and not the amplitude and are therefore a factor of 2 larger!
Διαβάστε περισσότερα14.5mm 14.5mm
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 1.119/JETCAS.218.289582,
Διαβάστε περισσότεραSOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr T t N n) Pr max X 1,..., X N ) t N n) Pr max
Διαβάστε περισσότεραSpectrum Representation (5A) Young Won Lim 11/3/16
Spectrum (5A) Copyright (c) 2009-2016 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later
Διαβάστε περισσότεραBounding Nonsplitting Enumeration Degrees
Bounding Nonsplitting Enumeration Degrees Thomas F. Kent Andrea Sorbi Università degli Studi di Siena Italia July 18, 2007 Goal: Introduce a form of Σ 0 2-permitting for the enumeration degrees. Till now,
Διαβάστε περισσότεραMulti-dimensional Central Limit Theorem
Mult-dmensonal Central Lmt heorem Outlne () () () t as () + () + + () () () Consder a sequence of ndependent random proceses t, t, dentcal to some ( t). Assume t 0. Defne the sum process t t t t () t ();
Διαβάστε περισσότεραConcrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
Διαβάστε περισσότεραEngineering Tunable Single and Dual Optical. Emission from Ru(II)-Polypyridyl Complexes. Through Excited State Design
Engineering Tunable Single and Dual Optical Emission from Ru(II)-Polypyridyl Complexes Through Excited State Design Supplementary Information Julia Romanova 1, Yousif Sadik 1, M. R. Ranga Prabhath 1,,
Διαβάστε περισσότεραμ μ μ s t j2 fct T () = a() t e π s t ka t e e j2π fct j2π fcτ0 R() = ( τ0) xt () = α 0 dl () pt ( lt) + wt () l wt () N 2 (0, σ ) Time-Delay Estimation Bias / T c 0.4 0.3 0.2 0.1 0-0.1-0.2-0.3 In-phase
Διαβάστε περισσότεραSupporting Information
Supporting Information Aluminum Complexes of N 2 O 2 3 Formazanate Ligands Supported by Phosphine Oxide Donors Ryan R. Maar, Amir Rabiee Kenaree, Ruizhong Zhang, Yichen Tao, Benjamin D. Katzman, Viktor
Διαβάστε περισσότεραPARTIAL NOTES for 6.1 Trigonometric Identities
PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot
Διαβάστε περισσότερα