Electromagnetic Wave Propagation Lecture 2: Time harmonic dependence, constitutive relations
|
|
- Anastasia Poole
- 5 years ago
- Views:
Transcription
1 Electromagnetic Wave Propagation Lecture 2: Time harmonic dependence, constitutive relations Daniel Sjöberg Department of Electrical and Information Technology September 2, 2010
2 Outline 1 Harmonic time dependence 2 Constitutive relations, time domain 3 Constitutive relations, frequency domain 4 Examples of material models 5 Bounds on metamaterials 6 Conclusions
3 Outline 1 Harmonic time dependence 2 Constitutive relations, time domain 3 Constitutive relations, frequency domain 4 Examples of material models 5 Bounds on metamaterials 6 Conclusions
4 Scope The theory given in this lecture (and the entire course) is applicable to the whole electromagnetic spectrum. However, different processes are dominant in different bands, making the material models different. Today, you learn what restrictions are imposed by the requirements 1. Linearity 2. Causality 3. Time translational invariance 4. Passivity
5 Electromagnetic spectrum, c 0 = fλ = m/s Band Frequency Wavelength ELF Extremely Low Frequency Hz 1 10 Mm VF Voice Frequency Hz km VLF Very Low Frequency 3 30 khz km LF Low Frequency khz 1 10 km MF Medium Frequency khz m HF High Frequency 3 30 MHz m VHF Very High Frequency MHz 1 10 m UHF Ultra High Frequency MHz cm SHF Super High Frequency 3 30 GHz 1 10 cm EHF Extremely High Frequency GHz 1 10 mm Submillimeter GHz µm Infrared THz µm Visible THz nm Ultraviolet 750 THz 30 PHz nm X-ray 30 PHz 3 EHz 10 nm 100 pm γ-ray >3 EHz <100 pm
6 Three ways of introducing time harmonic fields Fourier transform (finite energy fields, ω = 2πf) E(r, ω) = E(r, t) = 1 2π E(r, t)e jωt dt E(r, ω)e jωt dω Laplace transform (causal fields, zero for t < 0, s = α + jω) E(r, s) = 0 E(r, t) = 1 2πj E(r, t)e st dt α+j α j E(r, s)e st ds Real-value convention (purely harmonic cos ωt, preserves units) E(r, t) = Re{E(r, ω)e jωt }
7 Some examples Unit step function: u(t) = { 0 t < 0 1 t > 0 E(r, t) Fourier Laplace Real-value e αt2 π ω2 αe 4α cos(ω 0 t) π(δ(ω + ω 0 ) + δ(ω ω 0 )) 1 sin(ω 0 t) jπ(δ(ω + ω 0 ) δ(ω ω 0 )) j e at u(t) 1 jω+a e at cos(ω 0 t)u(t) e at sin(ω 0 t)u(t) jω+a (jω+a) 2 +ω 2 0 ω 0 (jω+a) 2 +ω s+a s+a (s+a) 2 +ω 2 0 ω 0 (s+a) 2 +ω 2 0 δ(t) 1 1
8 Different time conventions Different traditions: Engineering: Time dependence e jωt, plane wave factor e j(ωt k r). Physics: Time dependence e iωt, plane wave factor e i(k r ωt). If you use j and i consistently, all results can be translated between conventions using the simple rule j = i In this course we follow Orfanidis choice e jωt.
9 Outline 1 Harmonic time dependence 2 Constitutive relations, time domain 3 Constitutive relations, frequency domain 4 Examples of material models 5 Bounds on metamaterials 6 Conclusions
10 The need for material models Maxwell s equations are B(r, t) E(r, t) = t D(r, t) H(r, t) = J(r, t) + t This is 2 3 = 6 equations for at least 4 3 = 12 unknowns. Something is needed! We choose E and H as our fundamental fields (partly due to conformity with boundary conditions), and search for constitutive relations on the form { D B } = F ({ E H }) This would provide the missing 6 equations.
11 Examples of models Linear, isotropic materials ( standard media ): D = ɛe, B = µh Linear, anisotropic materials: D = (ɛ x ˆxˆx+ɛ y ŷŷ+ɛ z ẑẑ) E Linear, dispersive materials: D x ɛ x 0 0 E x D y = 0 ɛ y 0 E y D z 0 0 ɛ z E z Debye material Gyrotropic material D = ɛ 0 E + P B = µ 0 (H + M) P t = ɛ 0αE P /τ M t = ω S ẑ (βm H) The models are direct results of the physical processes in the material.
12 Basic assumptions To simplify, we formulate our assumptions for a non-magnetic material where D = F (E) and B = µ 0 H. We require the mapping F to satisfy four basic physical principles: Linearity: For each α, β, E 1, and E 2 we have F (αe 1 + βe 2 ) = αf (E 1 ) + βf (E 2 ) Causality: For all fields E such that E(t) = 0 when t < τ, we have F (E)(t) = 0 for t < τ Time translational invariance: If D 1 = F (E 1 ), D 2 = F (E 2 ), and E 2 (t) = E 1 (t τ), we have D 2 (t) = D 1 (t τ) Passivity: The material is not a source of electromagnetic energy, that is, S 0.
13 The general linear model The result of the assumptions is that all such materials can be modeled as t ] D(t) = ɛ 0 [E(t) + χ e (t t ) E(t ) dt t + ξ(t t ) H(t ) dt B(t) = t ζ(t t ) E(t ) dt t ] + µ 0 [H(t) + χ m (t t ) H(t ) dt The dyadic convolution kernels χ e (t), ξ(t), ζ(t), and χ m (t) model the induced polarization and magnetization.
14 Instantaneous response Some physical processes in the material may be considerably faster than the others This means the susceptibility function can be split in two parts χ(t) = χ 1 (t) + χ 2 (t)
15 Instantaneous response, continued Assume that E(t) does not vary considerably on the time scale of χ 1 (t). We then have t D(t)/ɛ 0 = E(t) + [ t = E(t) + [ = E(t) + [χ 1 (t t ) + χ 2 (t t )]E(t ) dt ] χ 1 (t t ) dt E(t) + 0 t χ 2 (t t )E(t ) dt ] t χ 1 (t ) dt E(t) + χ 2 (t t )E(t ) dt The quantity ɛ = χ 1 (t ) dt is called the instantaneous response (or momentaneous response, or optical response). Thus, there is some freedom of choice how to model the material, depending on the time scale!
16 Instantaneous response, example 1 E(t) P(t) D(t) χ(t) time
17 Outline 1 Harmonic time dependence 2 Constitutive relations, time domain 3 Constitutive relations, frequency domain 4 Examples of material models 5 Bounds on metamaterials 6 Conclusions
18 Constitutive relations in the frequency domain Applying a Fourier transform to the convolutions implies D(ω) = ɛ 0 [ɛ E(ω) + χ e (ω) E(ω)] + ξ(ω) H(ω) B(ω) = ζ(ω) E(ω) + µ 0 [µ H(ω) + χ m (ω) η 0 H(ω)] or ( ) D(ω) = B(ω) ( ) ɛ(ω) ξ(ω) ζ(ω) µ(ω) ( ) E(ω) H(ω) where we introduced the permittivity and permeability dyadics ɛ(ω) = ɛ 0 [ɛ + χ e (ω)] µ(ω) = µ 0 [µ + χ m (ω)] This is a fully bianisotropic material model.
19 Classification of materials Type ɛ, µ ξ, ζ Isotropic Both I Both 0 An-isotropic Some not I Both 0 Bi-isotropic Both I Both I Bi-an-isotropic All other cases
20 Modeling arbitriness Assume the models J(ω) = σ(ω)e(ω), D(ω) = ɛ(ω)e(ω) The total current in Maxwell s equations can then be written J(ω) + jωd(ω) = [σ(ω) + jωɛ(ω)] E(ω) = J (ω) }{{} =σ (ω) [ ] σ(ω) = jω jω + ɛ(ω) E(ω) = jωd (ω) }{{} =ɛ (ω) where σ (ω) and ɛ (ω) are equivalent models for the material. Thus, there is an arbitrariness in how to model dispersive materials, either by a conductivity model (σ (ω)) or by a permittivity model (ɛ (ω)), or any combination.
21 When to use what? Consider the total current as a sum of a conduction current J c and a displacement current J d : J tot (ω) = σ c (ω)e + jωɛ }{{} d (ω)e }{{} J c(ω) J d (ω) The ratio can take many different values (using f = 1 GHz) J c (ω) J d (ω) = σ c(ω) ωɛ d (ω) = 10 9 copper (σ = S/m and ɛ = ɛ 0 ) 1 seawater (σ = 4 S/m and ɛ = 72ɛ 0 ) 10 9 glass (σ = S/m and ɛ = 2ɛ 0 ) 18 orders of magnitude in difference! Conductivity model good when J c J d, permittivity model good when J d J c.
22 Poynting s theorem in the frequency domain In the time domain we had (E(t) H(t)) + H(t) B(t) t + E(t) D(t) t + E(t) J(t) = 0 For time harmonic fields, we consider the time average over one period (where f(t) = 1 t+t T t f(t ) dt ): E(t) H(t) + H(t) B(t) + E(t) D(t) + E(t) J(t) = 0 t t The time average of a product of two harmonic signals is f(t)g(t) = 1 2 Re{f(ω)g(ω) }.
23 The different terms are Poynting s theorem, continued E(t) H(t) = 1 2 Re{E(ω) H(ω) } H(t) B(t) = 1 t 2 Re { jωh(ω) B(ω) } E(t) D(t) = 1 t 2 Re { jωe(ω) D(ω) } E(t) J(t) = 1 2 Re {E(ω) J(ω) } For a purely dielectric material, we have D(ω) = ɛ(ω) E(ω) and 2 Re { jωe(ω) D(ω) } = jωe(ω) [ɛ(ω) E(ω)] + jωe(ω) ɛ(ω) E(ω) = jωe(ω) [ɛ(ω) ɛ(ω) ] E(ω)
24 Poynting s theorem, final version Using a permittivity model (J = 0), we have S(t) = jω ( ) ( ) ( ) E(ω) ɛ(ω) ɛ(ω) ξ(ω) ζ(ω) E(ω) 4 H(ω) ζ(ω) ξ(ω) µ(ω) µ(ω) H(ω) Passive material: S(t) 0 Active material: S(t) > 0 Lossless material: S(t) = 0 = Definitions! This boils down to conditions on the material matrix ( ɛ(ω) ɛ(ω) ξ(ω) ζ(ω) jω ) {( )} ɛ(ω) ξ(ω) ζ(ω) ξ(ω) µ(ω) µ(ω) = 2ω Im ζ(ω) µ(ω) If it is positive we have a lossy material, if it is zero we have a lossless material.
25 Example: Standard media With the material model D = ɛe, J = σe, B = µh we have ( ) (( ) ) ɛ(ω) ξ(ω) ɛ + σ = jω I 0 ζ(ω) µ(ω) 0 µi and {( )} ɛ(ω) ξ(ω) ω Im = ζ(ω) µ(ω) ( ) σi This model is lossy with electric fields present, but not with pure magnetic fields. In wave propagation, we always have both E and H fields.
26 Lossless media The condition on lossless media, ( ɛ(ω) ɛ(ω) ξ(ω) ζ(ω) ) ζ(ω) ξ(ω) µ(ω) µ(ω) = 0 can also be written ( ) ( ) ɛ(ω) ξ(ω) ɛ(ω) ξ(ω) = ζ(ω) µ(ω) ζ(ω) µ(ω) That is, the matrix should be hermitian symmetric.
27 Isotropic materials A bi-isotropic material is described by ( ) ( ) ɛ(ω) ξ(ω) ɛ(ω)i ξ(ω)i = ζ(ω) µ(ω) ζ(ω)i µ(ω)i The passivity requirement implies that all eigenvalues of the matrix ( ɛ(ω) ɛ(ω) ξ(ω) ζ(ω) jω ) ζ(ω) ξ(ω) µ(ω) µ(ω) are positive. If ξ = ζ = 0 it is seen that this requires (using that ɛ(ω) ɛ(ω) = 2j Im ɛ(ω)) ω Im ɛ(ω) > 0, ω Im µ(ω) > 0 and if ξ and ζ are nonzero we also require (after more algebra) ξ(ω) ζ(ω) 2 < 4 Im ɛ(ω) Im µ(ω)
28 Kramers-Kronig dispersion relations The causality requirement implies the Kramers-Kronig dispersion relations (writing χ(ω) = χ r (ω) jχ i (ω) for the real and imaginary part) χ r (ω) = 1 π P χ i (ω) = 1 π P χ i (ω ) ω ω dω χ r (ω ) ω ω dω where the principal part of a singular integral is P χ i (ω [ ) ω δ ω ω dω = lim δ 0 χ i (ω ) ω ω dω + ω+δ χ i (ω ] ) ω ω dω The Kramers-Kronig relations prohibit the existence of a lossless frequency dependent material.
29 Proof of the Kramers-Kronig relations Orfanidis gives several proofs, one of which is based on the analyticity. Causality requires χ(t) = 0 for t < 0, so χ(ω) = e jωt χ(t) dt = 0 e jωt χ(t) dt is analytic in the lower half of the ω-plane, since ω w = ω jα implies exponential decay of the integrand e jwt = e jωt αt. The Kramers-Kronig relations then follow from Cauchy s integral theorem χ(w) = 1 2πj C χ(w ) w w dw The integral along C goes to zero, and as the point w = ω jɛ approaches ω the Kramers-Kronig relations remain when writing χ = χ r jχ i.
30 Some notes The Kramers-Kronig relations restrict the possible frequency behavior of any causal material. It requires a model χ(ω) for all frequencies. Usually, our models are derived or measured only in a finite frequency interval, ω 1 < ω < ω 2. We need to extrapolate the models to zero and infinite frequencies, ω 0 and ω. Not a trivial task!
31 Outline 1 Harmonic time dependence 2 Constitutive relations, time domain 3 Constitutive relations, frequency domain 4 Examples of material models 5 Bounds on metamaterials 6 Conclusions
32 Randomly oriented dipoles Consider a medium consisting of randomly oriented electric dipoles, for instance water. The polarization is i P = lim p i V 0 V A typical situation is depicted below.
33 Physical processes We now consider two processes: 1. The molecules strive to align with an imposed electric field, at the rate ɛ 0 αe. 2. Thermal motion tries to disorient the polarization. With τ being the relaxation time for this process, the rate of changes in P are proportional to P /τ. This results in the following differential equation: P (t) t = ɛ 0 αe(t) P (t) τ
34 Physical processes We now consider two processes: 1. The molecules strive to align with an imposed electric field, at the rate ɛ 0 αe. 2. Thermal motion tries to disorient the polarization. With τ being the relaxation time for this process, the rate of changes in P are proportional to P /τ. This results in the following differential equation: P (t) t = ɛ 0 αe(t) P (t) τ This is an ordinary differential equation with the solution (assuming P = 0 at t = ) P (t) = ɛ 0 t αe (t t )/τ E(t ) dt
35 Dispersion or conductivity model From the solution we identify the susceptibility function χ(t) = u(t)αe t/τ This is monotonically decaying without oscillations.
36 Dispersion or conductivity model From the solution we identify the susceptibility function χ(t) = u(t)αe t/τ This is monotonically decaying without oscillations. Including all the effects in the D-field results in t ) D(t) = ɛ 0 (E(t) + αe (t t )/τ E(t ) dt J(t) = 0
37 Dispersion or conductivity model From the solution we identify the susceptibility function χ(t) = u(t)αe t/τ This is monotonically decaying without oscillations. Including all the effects in the D-field results in t ) D(t) = ɛ 0 (E(t) + αe (t t )/τ E(t ) dt J(t) = 0 and shifting it to J results in D(t) = ɛ 0 E(t) α J(t) = ɛ 0 αe(t) ɛ 0 τ t e (t t )/τ E(t ) dt Both versions have the same total current J(t) + D(t) t.
38 Response to different excitations Consider two different excitation functions: One square pulse E(t) = 1 for 0 < t < T, and zero elsewhere. A damped sine function, E(t) = e tν sin(ωt). The response P (t) can be calculated by numerically performing the convolution integral P (t) = ɛ 0 t χ(t t )E(t ) dt
39 Debye model, square pulse excitation 1.4 E(t) P(t) D(t) χ(t) time
40 Debye model, sine excitation 0.8 E(t) P(t) D(t) χ(t) time
41 Debye material in frequency domain The susceptibility kernel is χ(t) = αe t/τ u(t), with the Fourier transform χ(ω) = 0 αe t/τ e jωt dt = α jω + 1/τ = ατ 1 + jωτ The frequency dependent relative permittivity is ɛ r (ω) = 1 + χ(ω) = ɛ jɛ, with typical behavior as below:
42 Harmonic oscillator The archetypical material behavior is derived from an electron orbiting a positively charged nucleus. The typical forces on the electron are: 1. An electric force F 1 = qe from the applied electric field. 2. A restoring force proportional to the displacement F 2 = mω 2 0 r, where ω 0 is the harmonic frequency. 3. A frictional force proportional to the velocity, F 3 = mν r/ t.
43 Harmonic oscillator, continued Newton s acceleration law now gives m 2 r t 2 = F 1 + F 2 + F 3 = qe mω0r 2 mν r t Introducing the polarization as P = Nqr, where N is the number of charges per unit volume, we have 2 P (t) t 2 P (t) + ν + ω 2 t 0P (t) = Nq2 m E(t) This is an ordinary differential equation, with the solution (assuming P = 0 for t = ) ω 2 t p P (t) = ɛ 0 e (t t )ν/2 sin(ν 0 (t t ))E(t ) dt ν 0 with ω p = Nq 2 /(mɛ 0 ) and ν 2 0 = ω2 0 ν2 /4.
44 Interpretation The susceptibility function of the Lorentz model is with typical behavior as below. χ(t) = u(t) ω2 p ν 0 e tν/2 sin(ν 0 t)
45 Lorentz model, square pulse excitation 3 E(t) P(t) D(t) χ(t) time
46 Lorentz model, sine excitation E(t) P(t) D(t) χ(t) time
47 Lorentz model, resonant excitation 0.8 E(t) P(t) D(t) χ(t) time
48 Lorentz material in frequency domain The susceptibility kernel is χ(t) = ω2 p ν 0 e νt/2 sin(ν 0 t)u(t), with the Fourier transform χ(ω) = 0 ω 2 p ν 0 e νt/2 sin(ν 0 t)e jωt dt = ω 2 p ω 2 + ω jων The frequency dependent relative permittivity is ɛ r (ω) = 1 + χ(ω) = ɛ jɛ, with typical behavior as below:
49 Example: permittivity of water Microwave properties (one Debye model): Light properties (many Lorentz resonances):
50 Outline 1 Harmonic time dependence 2 Constitutive relations, time domain 3 Constitutive relations, frequency domain 4 Examples of material models 5 Bounds on metamaterials 6 Conclusions
51 What is a metamaterial? Engineered materials, designed to have unusual properties Periodic structures Resonant inclusions Negative refractive index Negative or near zero permittivity/permeability M. Lapine and S. Tretyakov, IET Microw. Antennas Propag., 2007, 1, (1), pp
52 Some examples
53 Bounds on metamaterials The requirements of linearity, causality, time translational invariance, and passivity, can be used to show bounds on metamaterial behavior in terms of relative bandwidth B: max ɛ(ω) ɛ m ω B B 1 + B/2 (ɛ ɛ m ) { 1/2 lossy case 1 lossless case
54 Bounds on metamaterials Physically realizable materials with target value ɛ m for some frequency band. ɛ < ɛ m < ɛ s ɛ m < ɛ < ɛ s No limits! Strong bounds!
55 Outline 1 Harmonic time dependence 2 Constitutive relations, time domain 3 Constitutive relations, frequency domain 4 Examples of material models 5 Bounds on metamaterials 6 Conclusions
56 Conclusions Constitutive relations are necessary in order to fully solve Maxwell s equations. Their form is restricted by physical principles such as linearity, causality, time translational invariance, and passivity. A Debye model is suitable for dipoles aligning with an imposed field (relaxation model). A Lorentz model is suitable for bound charges (resonance model). There are restrictions on what kind of frequency behavior is physically possible. If you want more extreme behavior, you get less bandwidth.
Electromagnetic Wave Propagation Lecture 2: Time harmonic dependence, constitutive relations
Electromagnetic Wave Propagation Lecture 2: Time harmonic dependence, constitutive relations Daniel Sjöberg Department of Electrical and Information Technology September 2015 Outline 1 Harmonic time dependence
More informationElectromagnetic Wave Propagation Lecture 3: Plane waves in isotropic and bianisotropic media
Electromagnetic Wave Propagation Lecture 3: Plane waves in isotropic and bianisotropic media Daniel Sjöberg Department of Electrical and Information Technology September 2016 Outline 1 Plane waves in lossless
More information3 Constitutive Relations: Macroscopic Properties of Matter
EECS 53 Lecture 3 c Kamal Sarabandi Fall 21 All rights reserved 3 Constitutive Relations: Macroscopic Properties of Matter As shown previously, out of the four Maxwell s equations only the Faraday s and
More informationElectromagnetic Wave Propagation Lecture 5: Propagation in birefringent media
Electromagnetic Wave Propagation Lecture 5: Propagation in birefringent media Daniel Sjöberg Department of Electrical and Information Technology April 15, 2010 Outline 1 Introduction 2 Wave propagation
More informationElectromagnetic Wave Propagation Lecture 8: Propagation in birefringent media
Electromagnetic Wave Propagation Lecture 8: Propagation in birefringent media Daniel Sjöberg Department of Electrical and Information Technology September 27, 2012 Outline 1 Introduction 2 Maxwell s equations
More informationElectromagnetic Wave Propagation Lecture 2: Uniform plane waves
Electromagnetic Wave Propagation Lecture 2: Uniform plane waves Daniel Sjöberg Department of Electrical and Information Technology March 25, 2010 Outline 1 Plane waves in lossless media General time dependence
More informationChapter 11: Dielectric Properties of Materials
Chapter 11: Dielectric Properties of Materials Lindhardt January 30, 2017 Contents 1 Classical Dielectric Response of Materials 2 1.1 Conditions on ɛ............................. 4 1.2 Kramer s Kronig
More informationCausality. but that does not mean it is local in time, for = 1. Let us write ɛ(ω) = ɛ 0 [1 + χ e (ω)] in terms of the electric susceptibility.
We have seen that the issue of how ɛ, µ n depend on ω raises questions about causality: Can signals travel faster than c, or even backwards in time? It is very often useful to assume that polarization
More informationLecture 21 Reminder/Introduction to Wave Optics
Lecture 1 Reminder/Introduction to Wave Optics Program: 1. Maxwell s Equations.. Magnetic induction and electric displacement. 3. Origins of the electric permittivity and magnetic permeability. 4. Wave
More informationLight in Matter (Hecht Ch. 3)
Phys 531 Lecture 3 9 September 2004 Light in Matter (Hecht Ch. 3) Last time, talked about light in vacuum: Maxwell equations wave equation Light = EM wave 1 Today: What happens inside material? typical
More informationCharacterization of Left-Handed Materials
Characterization of Left-Handed Materials Massachusetts Institute of Technology 6.635 lecture notes 1 Introduction 1. How are they realized? 2. Why the denomination Left-Handed? 3. What are their properties?
More informationLecture 2 Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell
Lecture Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell 1. Dispersion Introduction - An electromagnetic wave with an arbitrary wave-shape
More informationPolynomial Chaos Approach for Maxwell s Equations in Dispersive Media
Polynomial Chaos Approach for Maxwell s Equations in Dispersive Media Prof. Nathan L. Gibson Department of Mathematics Applied Mathematics and Computation Seminar March 15, 2013 Prof. Gibson (OSU) PC-FDTD
More informationSummary of Beam Optics
Summary of Beam Optics Gaussian beams, waves with limited spatial extension perpendicular to propagation direction, Gaussian beam is solution of paraxial Helmholtz equation, Gaussian beam has parabolic
More informationEECS 117. Lecture 22: Poynting s Theorem and Normal Incidence. Prof. Niknejad. University of California, Berkeley
University of California, Berkeley EECS 117 Lecture 22 p. 1/2 EECS 117 Lecture 22: Poynting s Theorem and Normal Incidence Prof. Niknejad University of California, Berkeley University of California, Berkeley
More informationPhysics 506 Winter 2004
Physics 506 Winter 004 G. Raithel January 6, 004 Disclaimer: The purpose of these notes is to provide you with a general list of topics that were covered in class. The notes are not a substitute for reading
More informationFourier transforms, Generalised functions and Greens functions
Fourier transforms, Generalised functions and Greens functions T. Johnson 2015-01-23 Electromagnetic Processes In Dispersive Media, Lecture 2 - T. Johnson 1 Motivation A big part of this course concerns
More informationENERGY DENSITY OF MACROSCOPIC ELECTRIC AND MAGNETIC FIELDS IN DISPERSIVE MEDIUM WITH LOSSES
Progress In Electromagnetics Research B, Vol. 40, 343 360, 2012 ENERGY DENSITY OF MACROSCOPIC ELECTRIC AND MAGNETIC FIELDS IN DISPERSIVE MEDIUM WITH LOSSES O. B. Vorobyev * Stavropol Institute of Radiocommunications,
More informationMicroscopic-Macroscopic connection. Silvana Botti
relating experiment and theory European Theoretical Spectroscopy Facility (ETSF) CNRS - Laboratoire des Solides Irradiés Ecole Polytechnique, Palaiseau - France Temporary Address: Centre for Computational
More information9 The conservation theorems: Lecture 23
9 The conservation theorems: Lecture 23 9.1 Energy Conservation (a) For energy to be conserved we expect that the total energy density (energy per volume ) u tot to obey a conservation law t u tot + i
More informationDrude theory & linear response
DRAFT: run through L A TEX on 9 May 16 at 13:51 Drude theory & linear response 1 Static conductivity According to classical mechanics, the motion of a free electron in a constant E field obeys the Newton
More informationWaves. Daniel S. Weile. ELEG 648 Waves. Department of Electrical and Computer Engineering University of Delaware. Plane Waves Reflection of Waves
Waves Daniel S. Weile Department of Electrical and Computer Engineering University of Delaware ELEG 648 Waves Outline Outline Introduction Let s start by introducing simple solutions to Maxwell s equations
More informationCHAPTER 9 ELECTROMAGNETIC WAVES
CHAPTER 9 ELECTROMAGNETIC WAVES Outlines 1. Waves in one dimension 2. Electromagnetic Waves in Vacuum 3. Electromagnetic waves in Matter 4. Absorption and Dispersion 5. Guided Waves 2 Skip 9.1.1 and 9.1.2
More informationIntroduction to electromagnetic theory
Chapter 1 Introduction to electromagnetic theory 1.1 Introduction Electromagnetism is a fundamental physical phenomena that is basic to many areas science and technology. This phenomenon is due to the
More informationFor the magnetic field B called magnetic induction (unfortunately) M called magnetization is the induced field H called magnetic field H =
To review, in our original presentation of Maxwell s equations, ρ all J all represented all charges, both free bound. Upon separating them, free from bound, we have (dropping quadripole terms): For the
More informationMetamaterials. Peter Hertel. University of Osnabrück, Germany. Lecture presented at APS, Nankai University, China
University of Osnabrück, Germany Lecture presented at APS, Nankai University, China http://www.home.uni-osnabrueck.de/phertel Spring 2012 are produced artificially with strange optical properties for instance
More informationElectromagnetic Wave Propagation Lecture 13: Oblique incidence II
Electromagnetic Wave Propagation Lecture 13: Oblique incidence II Daniel Sjöberg Department of Electrical and Information Technology October 15, 2013 Outline 1 Surface plasmons 2 Snel s law in negative-index
More informationOverview in Images. S. Lin et al, Nature, vol. 394, p , (1998) T.Thio et al., Optics Letters 26, (2001).
Overview in Images 5 nm K.S. Min et al. PhD Thesis K.V. Vahala et al, Phys. Rev. Lett, 85, p.74 (000) J. D. Joannopoulos, et al, Nature, vol.386, p.143-9 (1997) T.Thio et al., Optics Letters 6, 197-1974
More informationMaxwell s Equations. 1.1 Maxwell s Equations. 1.2 Lorentz Force. m dv = F = q(e + v B) (1.2.2) = m v dv dt = v F = q v E (1.2.3)
2 1. Maxwell s Equations 1 Maxwell s Equations the receiving antennas. Away from the sources, that is, in source-free regions of space, Maxwell s equations take the simpler form: E = B t H = D t D = (source-free
More informationChemistry 24b Lecture 23 Spring Quarter 2004 Instructor: Richard Roberts. (1) It induces a dipole moment in the atom or molecule.
Chemistry 24b Lecture 23 Spring Quarter 2004 Instructor: Richard Roberts Absorption and Dispersion v E * of light waves has two effects on a molecule or atom. (1) It induces a dipole moment in the atom
More informationLINEAR RESPONSE THEORY
MIT Department of Chemistry 5.74, Spring 5: Introductory Quantum Mechanics II Instructor: Professor Andrei Tokmakoff p. 8 LINEAR RESPONSE THEORY We have statistically described the time-dependent behavior
More informationOverview - Previous lecture 1/2
Overview - Previous lecture 1/2 Derived the wave equation with solutions of the form We found that the polarization of the material affects wave propagation, and found the dispersion relation ω(k) with
More informationElectromagnetic Relaxation Time Distribution Inverse Problems in the Time-domain
Electromagnetic Relaxation Time Distribution Inverse Problems in the Time-domain Prof Nathan L Gibson Department of Mathematics Joint Math Meeting Jan 9, 2011 Prof Gibson (OSU) Inverse Problems for Distributions
More informationElectromagnetic Wave Propagation Lecture 13: Oblique incidence II
Electromagnetic Wave Propagation Lecture 13: Oblique incidence II Daniel Sjöberg Department of Electrical and Information Technology October 2016 Outline 1 Surface plasmons 2 Snel s law in negative-index
More informationElectrical and optical properties of materials
Electrical and optical properties of materials John JL Morton Part 4: Mawell s Equations We have already used Mawell s equations for electromagnetism, and in many ways they are simply a reformulation (or
More informationElectromagnetic Theory (Hecht Ch. 3)
Phys 531 Lecture 2 30 August 2005 Electromagnetic Theory (Hecht Ch. 3) Last time, talked about waves in general wave equation: 2 ψ(r, t) = 1 v 2 2 ψ t 2 ψ = amplitude of disturbance of medium For light,
More informationECEN 420 LINEAR CONTROL SYSTEMS. Lecture 2 Laplace Transform I 1/52
1/52 ECEN 420 LINEAR CONTROL SYSTEMS Lecture 2 Laplace Transform I Linear Time Invariant Systems A general LTI system may be described by the linear constant coefficient differential equation: a n d n
More informationJ10M.1 - Rod on a Rail (M93M.2)
Part I - Mechanics J10M.1 - Rod on a Rail (M93M.2) J10M.1 - Rod on a Rail (M93M.2) s α l θ g z x A uniform rod of length l and mass m moves in the x-z plane. One end of the rod is suspended from a straight
More information1 Fundamentals of laser energy absorption
1 Fundamentals of laser energy absorption 1.1 Classical electromagnetic-theory concepts 1.1.1 Electric and magnetic properties of materials Electric and magnetic fields can exert forces directly on atoms
More informationOptics and Optical Design. Chapter 5: Electromagnetic Optics. Lectures 9 & 10
Optics and Optical Design Chapter 5: Electromagnetic Optics Lectures 9 & 1 Cord Arnold / Anne L Huillier Electromagnetic waves in dielectric media EM optics compared to simpler theories Electromagnetic
More informationParameter Estimation Versus Homogenization Techniques in Time-Domain Characterization of Composite Dielectrics
Parameter Estimation Versus Homogenization Techniques in Time-Domain Characterization of Composite Dielectrics H. T. Banks 1 V. A. Bokil and N. L. Gibson, 3 Center For Research in Scientific Computation
More informationTheory and Applications of Dielectric Materials Introduction
SERG Summer Seminar Series #11 Theory and Applications of Dielectric Materials Introduction Tzuyang Yu Associate Professor, Ph.D. Structural Engineering Research Group (SERG) Department of Civil and Environmental
More informationE E D E=0 2 E 2 E (3.1)
Chapter 3 Constitutive Relations Maxwell s equations define the fields that are generated by currents and charges. However, they do not describe how these currents and charges are generated. Thus, to find
More informationMacroscopic dielectric theory
Macroscopic dielectric theory Maxwellʼs equations E = 1 c E =4πρ B t B = 4π c J + 1 c B = E t In a medium it is convenient to explicitly introduce induced charges and currents E = 1 B c t D =4πρ H = 4π
More informationElectromagnetic optics!
1 EM theory Electromagnetic optics! EM waves Monochromatic light 2 Electromagnetic optics! Electromagnetic theory of light Electromagnetic waves in dielectric media Monochromatic light References: Fundamentals
More informationElectromagnetic Waves Across Interfaces
Lecture 1: Foundations of Optics Outline 1 Electromagnetic Waves 2 Material Properties 3 Electromagnetic Waves Across Interfaces 4 Fresnel Equations 5 Brewster Angle 6 Total Internal Reflection Christoph
More informationSimple medium: D = ɛe Dispersive medium: D = ɛ(ω)e Anisotropic medium: Permittivity as a tensor
Plane Waves 1 Review dielectrics 2 Plane waves in the time domain 3 Plane waves in the frequency domain 4 Plane waves in lossy and dispersive media 5 Phase and group velocity 6 Wave polarization Levis,
More informationTHE FOURIER TRANSFORM (Fourier series for a function whose period is very, very long) Reading: Main 11.3
THE FOURIER TRANSFORM (Fourier series for a function whose period is very, very long) Reading: Main 11.3 Any periodic function f(t) can be written as a Fourier Series a 0 2 + a n cos( nωt) + b n sin n
More informationSeries FOURIER SERIES. Graham S McDonald. A self-contained Tutorial Module for learning the technique of Fourier series analysis
Series FOURIER SERIES Graham S McDonald A self-contained Tutorial Module for learning the technique of Fourier series analysis Table of contents Begin Tutorial c 24 g.s.mcdonald@salford.ac.uk 1. Theory
More information3.3 Energy absorption and the Green function
142 3. LINEAR RESPONSE THEORY 3.3 Energy absorption and the Green function In this section, we first present a calculation of the energy transferred to the system by the external perturbation H 1 = Âf(t)
More informationAdvanced Quantum Mechanics
Advanced Quantum Mechanics Rajdeep Sensarma sensarma@theory.tifr.res.in Quantum Dynamics Lecture #2 Recap of Last Class Schrodinger and Heisenberg Picture Time Evolution operator/ Propagator : Retarded
More informationA system that is both linear and time-invariant is called linear time-invariant (LTI).
The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Lecture Notes: Time, Frequency & Transform Domains February 28, 2012 Signals & Systems Signals are mapped
More informationElectrodynamics I Final Exam - Part A - Closed Book KSU 2005/12/12 Electro Dynamic
Electrodynamics I Final Exam - Part A - Closed Book KSU 2005/12/12 Name Electro Dynamic Instructions: Use SI units. Short answers! No derivations here, just state your responses clearly. 1. (2) Write an
More informationWave Phenomena Physics 15c. Lecture 11 Dispersion
Wave Phenomena Physics 15c Lecture 11 Dispersion What We Did Last Time Defined Fourier transform f (t) = F(ω)e iωt dω F(ω) = 1 2π f(t) and F(w) represent a function in time and frequency domains Analyzed
More informationremain essentially unchanged for the case of time-varying fields, the remaining two
Unit 2 Maxwell s Equations Time-Varying Form While the Gauss law forms for the static electric and steady magnetic field equations remain essentially unchanged for the case of time-varying fields, the
More informationLight and Matter. Thursday, 8/31/2006 Physics 158 Peter Beyersdorf. Document info
Light and Matter Thursday, 8/31/2006 Physics 158 Peter Beyersdorf Document info 3. 1 1 Class Outline Common materials used in optics Index of refraction absorption Classical model of light absorption Light
More informationSecond Order Systems
Second Order Systems independent energy storage elements => Resonance: inertance & capacitance trade energy, kinetic to potential Example: Automobile Suspension x z vertical motions suspension spring shock
More informationANDERS KARLSSON. and GERHARD KRISTENSSON MICROWAVE THEORY
ANDERS KARLSSON and GERHARD KRISTENSSON MICROWAVE THEORY Rules for the -operator (1) (ϕ + ψ) = ϕ + ψ () (ϕψ) = ψ ϕ + ϕ ψ (3) (a b) = (a )b + (b )a + a ( b) + b ( a) (4) (a b) = (a b) + (b )a + a ( b) +
More information1. Reminder: E-Dynamics in homogenous media and at interfaces
0. Introduction 1. Reminder: E-Dynamics in homogenous media and at interfaces 2. Photonic Crystals 2.1 Introduction 2.2 1D Photonic Crystals 2.3 2D and 3D Photonic Crystals 2.4 Numerical Methods 2.5 Fabrication
More informationLecture 2 Review of Maxwell s Equations and the EM Constitutive Parameters
Lecture 2 Review of Maxwell s Equations and the EM Constitutive Parameters Optional Reading: Steer Appendix D, or Pozar Section 1.2,1.6, or any text on Engineering Electromagnetics (e.g., Hayt/Buck) Time-domain
More informationSolution Set 2 Phys 4510 Optics Fall 2014
Solution Set Phys 4510 Optics Fall 014 Due date: Tu, September 16, in class Scoring rubric 4 points/sub-problem, total: 40 points 3: Small mistake in calculation or formula : Correct formula but calculation
More informationECE 3084 QUIZ 2 SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING GEORGIA INSTITUTE OF TECHNOLOGY APRIL 2, Name:
ECE 3084 QUIZ 2 SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING GEORGIA INSTITUTE OF TECHNOLOGY APRIL 2, 205 Name:. The quiz is closed book, except for one 2-sided sheet of handwritten notes. 2. Turn off
More informationElectromagnetic Waves in Materials
Electromagnetic Waves in Materials Outline Review of the Lorentz Oscillator Model Complex index of refraction what does it mean? TART Microscopic model for plasmas and metals 1 True / False 1. In the Lorentz
More informationElectromagnetic Waves
Electromagnetic Waves Our discussion on dynamic electromagnetic field is incomplete. I H E An AC current induces a magnetic field, which is also AC and thus induces an AC electric field. H dl Edl J ds
More informationNONLINEAR OPTICS. Ch. 1 INTRODUCTION TO NONLINEAR OPTICS
NONLINEAR OPTICS Ch. 1 INTRODUCTION TO NONLINEAR OPTICS Nonlinear regime - Order of magnitude Origin of the nonlinearities - Induced Dipole and Polarization - Description of the classical anharmonic oscillator
More informationSpectral Analysis of Random Processes
Spectral Analysis of Random Processes Spectral Analysis of Random Processes Generally, all properties of a random process should be defined by averaging over the ensemble of realizations. Generally, all
More informationUniform Plane Waves Page 1. Uniform Plane Waves. 1 The Helmholtz Wave Equation
Uniform Plane Waves Page 1 Uniform Plane Waves 1 The Helmholtz Wave Equation Let s rewrite Maxwell s equations in terms of E and H exclusively. Let s assume the medium is lossless (σ = 0). Let s also assume
More informationElectromagnetic (EM) Waves
Electromagnetic (EM) Waves Short review on calculus vector Outline A. Various formulations of the Maxwell equation: 1. In a vacuum 2. In a vacuum without source charge 3. In a medium 4. In a dielectric
More information10. Optics of metals - plasmons
1. Optics of metals - plasmons Drude theory at higher frequencies The Drude scattering time corresponds to the frictional damping rate The ultraviolet transparency of metals Interface waves - surface plasmons
More informationECE357H1S ELECTROMAGNETIC FIELDS TERM TEST March 2016, 18:00 19:00. Examiner: Prof. Sean V. Hum
UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING The Edward S. Rogers Sr. Department of Electrical and Computer Engineering ECE357H1S ELECTROMAGNETIC FIELDS TERM TEST 2 21 March 2016, 18:00
More informationFORTH. Essential electromagnetism for photonic metamaterials. Maria Kafesaki. Foundation for Research & Technology, Hellas, Greece (FORTH)
FORTH Essential electromagnetism for photonic metamaterials Maria Kafesaki Foundation for Research & Technology, Hellas, Greece (FORTH) Photonic metamaterials Metamaterials: Man-made structured materials
More informationH ( E) E ( H) = H B t
Chapter 5 Energy and Momentum The equations established so far describe the behavior of electric and magnetic fields. They are a direct consequence of Maxwell s equations and the properties of matter.
More informationSignal and systems. Linear Systems. Luigi Palopoli. Signal and systems p. 1/5
Signal and systems p. 1/5 Signal and systems Linear Systems Luigi Palopoli palopoli@dit.unitn.it Wrap-Up Signal and systems p. 2/5 Signal and systems p. 3/5 Fourier Series We have see that is a signal
More informationElectromagnetic Waves
Electromagnetic Waves Maxwell s equations predict the propagation of electromagnetic energy away from time-varying sources (current and charge) in the form of waves. Consider a linear, homogeneous, isotropic
More informationContents. 1 Basic Equations 1. Acknowledgment. 1.1 The Maxwell Equations Constitutive Relations 11
Preface Foreword Acknowledgment xvi xviii xix 1 Basic Equations 1 1.1 The Maxwell Equations 1 1.1.1 Boundary Conditions at Interfaces 4 1.1.2 Energy Conservation and Poynting s Theorem 9 1.2 Constitutive
More informationin Electromagnetics Numerical Method Introduction to Electromagnetics I Lecturer: Charusluk Viphavakit, PhD
2141418 Numerical Method in Electromagnetics Introduction to Electromagnetics I Lecturer: Charusluk Viphavakit, PhD ISE, Chulalongkorn University, 2 nd /2018 Email: charusluk.v@chula.ac.th Website: Light
More information2.161 Signal Processing: Continuous and Discrete Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 2.6 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS
More informationD. S. Weile Radiation
Radiation Daniel S. Weile Department of Electrical and Computer Engineering University of Delaware ELEG 648 Radiation Outline Outline Maxwell Redux Maxwell s Equation s are: 1 E = jωb = jωµh 2 H = J +
More informationGeneral review: - a) Dot Product
General review: - a) Dot Product If θ is the angle between the vectors a and b, then a b = a b cos θ NOTE: Two vectors a and b are orthogonal, if and only if a b = 0. Properties of the Dot Product If a,
More informationANALOG AND DIGITAL SIGNAL PROCESSING CHAPTER 3 : LINEAR SYSTEM RESPONSE (GENERAL CASE)
3. Linear System Response (general case) 3. INTRODUCTION In chapter 2, we determined that : a) If the system is linear (or operate in a linear domain) b) If the input signal can be assumed as periodic
More informationThe Generation of Ultrashort Laser Pulses II
The Generation of Ultrashort Laser Pulses II The phase condition Trains of pulses the Shah function Laser modes and mode locking 1 There are 3 conditions for steady-state laser operation. Amplitude condition
More informationBasics of electromagnetic response of materials
Basics of electromagnetic response of materials Microscopic electric and magnetic field Let s point charge q moving with velocity v in fields e and b Force on q: F e F qeqvb F m Lorenz force Microscopic
More informationEITN90 Radar and Remote Sensing Lecture 5: Target Reflectivity
EITN90 Radar and Remote Sensing Lecture 5: Target Reflectivity Daniel Sjöberg Department of Electrical and Information Technology Spring 2018 Outline 1 Basic reflection physics 2 Radar cross section definition
More informationMCQs E M WAVES. Physics Without Fear.
MCQs E M WAVES Physics Without Fear Electromagnetic Waves At A Glance Ampere s law B. dl = μ 0 I relates magnetic fields due to current sources. Maxwell argued that this law is incomplete as it does not
More informationEvanescent modes stored in cavity resonators with backward-wave slabs
arxiv:cond-mat/0212392v1 17 Dec 2002 Evanescent modes stored in cavity resonators with backward-wave slabs S.A. Tretyakov, S.I. Maslovski, I.S. Nefedov, M.K. Kärkkäinen Radio Laboratory, Helsinki University
More informationSet 5: Classical E&M and Plasma Processes
Set 5: Classical E&M and Plasma Processes Maxwell Equations Classical E&M defined by the Maxwell Equations (fields sourced by matter) and the Lorentz force (matter moved by fields) In cgs (gaussian) units
More informationAntennas and Propagation. Chapter 2: Basic Electromagnetic Analysis
Antennas and Propagation : Basic Electromagnetic Analysis Outline Vector Potentials, Wave Equation Far-field Radiation Duality/Reciprocity Transmission Lines Antennas and Propagation Slide 2 Antenna Theory
More informationPhysics of Condensed Matter I
Physics of Condensed Matter I 1100-4INZ`PC Faculty of Physics UW Jacek.Szczytko@fuw.edu.pl Dictionary D = εe ε 0 vacuum permittivity, permittivity of free space (przenikalność elektryczna próżni) ε r relative
More informationHomework 4. May An LTI system has an input, x(t) and output y(t) related through the equation y(t) = t e (t t ) x(t 2)dt
Homework 4 May 2017 1. An LTI system has an input, x(t) and output y(t) related through the equation y(t) = t e (t t ) x(t 2)dt Determine the impulse response of the system. Rewriting as y(t) = t e (t
More informationPhonons and lattice dynamics
Chapter Phonons and lattice dynamics. Vibration modes of a cluster Consider a cluster or a molecule formed of an assembly of atoms bound due to a specific potential. First, the structure must be relaxed
More informationSignals and Systems Spring 2004 Lecture #9
Signals and Systems Spring 2004 Lecture #9 (3/4/04). The convolution Property of the CTFT 2. Frequency Response and LTI Systems Revisited 3. Multiplication Property and Parseval s Relation 4. The DT Fourier
More informationCloaking The Road to Realization
Cloaking The Road to Realization by Reuven Shavit Electrical and Computer Engineering Department Ben-Gurion University of the Negev 1 Outline Introduction Transformation Optics Laplace s Equation- Transformation
More informationII Theory Of Surface Plasmon Resonance (SPR)
II Theory Of Surface Plasmon Resonance (SPR) II.1 Maxwell equations and dielectric constant of metals Surface Plasmons Polaritons (SPP) exist at the interface of a dielectric and a metal whose electrons
More informationINTERACTION OF LIGHT WITH MATTER
INTERACTION OF LIGHT WITH MATTER Already.the speed of light can be related to the permittivity, ε and the magnetic permeability, µ of the material by Rememberε = ε r ε 0 and µ = µ r µ 0 where ε 0 = 8.85
More information20 Poynting theorem and monochromatic waves
0 Poynting theorem and monochromatic waves The magnitude of Poynting vector S = E H represents the amount of power transported often called energy flux byelectromagneticfieldse and H over a unit area transverse
More informationPlasmonics: elementary excitation of a plasma (gas of free charges) nano-scale optics done with plasmons at metal interfaces
Plasmonics Plasmon: Plasmonics: elementary excitation of a plasma (gas of free charges) nano-scale optics done with plasmons at metal interfaces Femius Koenderink Center for Nanophotonics AMOLF, Amsterdam
More informationLecture 3 Fiber Optical Communication Lecture 3, Slide 1
Lecture 3 Optical fibers as waveguides Maxwell s equations The wave equation Fiber modes Phase velocity, group velocity Dispersion Fiber Optical Communication Lecture 3, Slide 1 Maxwell s equations in
More informationTypical anisotropies introduced by geometry (not everything is spherically symmetric) temperature gradients magnetic fields electrical fields
Lecture 6: Polarimetry 1 Outline 1 Polarized Light in the Universe 2 Fundamentals of Polarized Light 3 Descriptions of Polarized Light Polarized Light in the Universe Polarization indicates anisotropy
More informationDSP-I DSP-I DSP-I DSP-I
NOTES FOR 8-79 LECTURES 3 and 4 Introduction to Discrete-Time Fourier Transforms (DTFTs Distributed: September 8, 2005 Notes: This handout contains in brief outline form the lecture notes used for 8-79
More informationarxiv: v2 [physics.class-ph] 14 Aug 2014
Transmission Lines Emulating Moving Media arxiv:1406.1624v2 [physics.class-ph] 14 Aug 2014 J. Vehmas 1, S. Hrabar 2, and S. Tretyakov 1 1 Department of Radio Science and Engineering/SMARAD Center of Excellence,
More information