Lorentz and Drude Models - EMPossible

1 9/19/20161 ECE 5322 21stCentury ElectromagneticsInstructor:Office:Phone: E Mail:Dr. RaymondC. RumpfA 337(915) 747 Properties of Materials Part ILorentz and Drude ModelsLecture #2 Lecture 21 Lecture Outline High level picture of dielectric response Resonance Lorentz model for dielectrics Lorentz model for permeability Drude model for metals Generalizations Other materials modelsLecture 229/19/20162 High Level Picture of Dielectric ResponseLecture 24 Dielectric SlabWe wish to understand why a dielectric exhibits an electromagnetic 25 Atoms at RestWithout an applied electric field, the electron clouds around the nuclei are symmetric and at 26 Applied WaveThe electric field of a electromagnetic wave pushes the electrons away from the nuclei producing clouds that are 27 Secondary WavesThe motion of the charges emits secondary waves that interfere with the applied wave to produce an overall slowing effect on the Resonance Low Frequency Can push object to modulate amplitude Displacement is in phase with driving force DC offsetLecture 29 Visualizing Resonance on Resonance Can push object to large amplitude Displacement and driving force are 90 out of phase Peaks of push correspond to nulls of displacementLecture 2109/19/20166 Visualizing Resonance High Frequency Vanishing

2 Amplitude Displacement is 180 out of phaseLecture 211A Harmonic OscillatorAmplitudePhase Lag180 0 0> 0>> 0res 90 Lecture 2129/19/20167 Impulse Response of a Harmonic OscillatorAmplitudeTime, tExcitationBall DisplacementDamping lossLecture 213 Moving Charges Radiate Wavesoutward travelling waveLecture 2149/19/20168 Lorentz Modelfor DielectricsLorentz Oscillator ModelLecture 2169/19/20169 Maxwell s Equations with Material PolarizationLecture 2170 DEP Material polarization is incorporated into the constitutive E Response of materialResponse of free spaceConstitutive relation in terms of relative permittivity and susceptibility. 000000 1 1rrDEP EEPEDEE Comparing the above equations, we see thatEquation of Motion2202rrmm mrqEtt electric forcefrictional forcerestoring forceacceleration force0Km naturalfrequencyemm mass of an electron damping rate (loss/sec)Lecture 2189/19/201610 Fourier Transform2202rrmm mrqEtt 220mj rm jrmrqE 220mjmmr qE Fourier transformSimplifyLecture 219 Displacement 220eEqrmj r 220mjmmr qE Lecture 2209/19/201611 Dipole Moment qr r Definition of Dipole Moment:** Sorry for the confusing notation, but here is NOT permeability.

3 2220eEqmj Lecture 221chargedistance from centerLorentz Polarizability, E Definition of Polarizability:** Sorry for the confusing notation, but here is NOT absorption. 22201eqmj ( ) is a tensor quantity for anisotropic materials. For simplicity, we will use the scalar is the Lorentz polarizabilityfor a single 2229/19/201612 Polarization per Unit Volume 1iVPV Definition:Average dipole moment over all atoms in a billions and trillions of them!!! PN There is some randomness to the polarized atoms so a statistical approach is taken to compute the of atoms per unit volumeStatistical averageN UnpolarizedPolarized with some randomness Equivalent uniform polarizationLecture 223 AppliedE FieldAppliedE FieldSusceptibility (1 of 2) 022201ePNEEqmj Recall the following:This leads to an expression for the susceptibility: 2220001eNNqmj Lecture 2249/19/201613 Susceptibility (2 of 2)Susceptibility of a dielectric material: 2220pj 220peNqm Note this is the susceptibility of a dielectric which has only one resonance.

4 The location of atoms is important because they can influence each other. We ignored this. Real materials have many sources of resonance and all of these must be added 10 10 F 10 kgeqm Lecture 225 The Dielectric FunctionRecall that, 22201prj 220peNqm 00 0001rDEEPE EE Therefore, 1r The dielectric function for a material with a single resonance is then,The ~ symbol indicates the quantity is complexLecture 226vacuummaterial9/19/2016141. We wrote the equation of motion by comparing bound charges to a mass on a We performed a Fourier transform to solve this equation for We calculated the electric dipole moment of the charge displaced by We calculated the volume averaged dipole moment to derive the material We calculated the material We calculated the dielectric of Derivation2202ee errmm mrqEtt 220eEqrmj 2220eEqmj 1iVPN 2222200 ppeNqjm 22201prj Lecture 227 Real and Imaginary Parts of 22222001 prpeNqjm Split into real and imaginary parts 22202222002202222 22022220222 2222 222001 1 1prr rpppjjjjjj

5 22220222 2222 222001 rprp Lecture 2289/19/201615 Complex Refractive Index Refractive is like a density to an electromagnetic wave. It quantifies the speed of an electromagnetic wave through a material. Waves travel slower through materials with higher refractive index. 11rrmennj rnnj For now, we will ignore the magnetic between dielectric function and refractive index. 222222rrrrrrrrnjjnjjnjn jnjnjnj 222rrnn rrnjj ordinary refractive indexextinction coefficientn Lecture 229 Absorption Coefficient (1 of 2)From Maxwell s equations, a plane wave in a linear, homogeneous, isotropic (LHI) medium 0jkzEzEe 0002 kknk The wave number isSubstituting the complex refractive index into this equation leads 00000jk n j zkzjknzEzEeEe e Oscillatory termenvelope termThe absorption coefficient is defined in terms of the field intensity.

6 0zIzIe Lecture 2309/19/201616 Absorption Coefficient (2 of 2)The field intensity is related to the field amplitude through 2 IzEz 000022002200202kzjknzzkzzkzzIz EzIeEeeIeE eeek Substituting expressions from the previous slide, the absorption coefficient can be calculated from 0022kc Lecture 231 Reflectance (normal incidence)The amplitude reflection coefficient rquantifies the amplitude and phase of reflected power reflection coefficient (reflectance) is always positive and between 0 and 1 (for materials without gain). 11njrnj 22*2211nRrrn incidentreflectedtransmittedairmaterialL ecture 232 Loss contributes to reflections!9/19/201617 Kramers-Kroenig Relations (1 of 3)The susceptibility is essentially the impulse response of a material to an applied electric field.

7 0t Et Pt 00ePtEtdPE Causality requires that (t=0)=P(t=0)=0 From linear system theory, if (t) is a causal, than the real and imaginary parts of its Fourier transform are Hilbert transform pairs. j 11dd This means that and are not independent. If we can measure one, we can calculate the how do we measure at negative frequencies? Lecture 233 Kramers-Kroenig Relations (2 of 3)From Fourier theory, if (t) is purely real then is an even function is an odd function Applying this symmetry principle to the relations on the previous page leads to 22022022dd These equations can be applied to measurements taken over just positive 2349/19/201618 Kramers-Kroenig Relations (3 of 3)For dilute media with weak susceptibility ( and are small), the complex refractive index can be approximated from the susceptibility 122njj Comparing the real and imaginary components of and leads to the Kramers Kroenigrelations for the refractive index and absorption coefficient.

8 0220022011cndncd For dilute mediaLecture 235 Summary of Properties 222202222 2222 22001 rprp Dielectric Function The dielectric function is most fundamental to Maxwell s equations. Imaginary part only exists if there is loss. When there is loss, the real part contributes. Perhaps more difficult to extract physical meaning from the real and imaginary Index The refractive index is more closely related to wave propagation. It quantifies both velocity and loss. Real part is solely related to phase velocity. Imaginary part is solely related to loss. In many ways, refractive index is a more physically meaningful parameter. 11rrmennj Lecture 2369/19/201619 Typical Lorentz Model for Dielectrics04 2 r r n R Lecture 237 Example Salt WaterM.

9 R. Querry, R. C. Waring, W. E. Holland, M. Hale, W. Nijm, Optical Constants in the Infrared for Aqueous Solutions of NaCl, J. Opt. Soc. Am. 62(7), 849 855 (1972)Lecture 2389/19/201620 TART TransmissiveAbsorptiveReflectiveTransmis sive02 0022 02p p 04 2 r r n RTTRAThe TART regions are most distinct for small and large pLecture 239 Dispersion Re Im Frequency Complex dielectric function 0 Anomalous and negative dispersionLecture 2409/19/201621 Observation #1 Loss is very high near resonance. 04 2 Lecture 241 Observation #2 Damping rate determines width of 2 r r Lecture 2429/19/201622 Observation #3 Far from resonance, loss is very low.

10 04 2 Lecture 243 Observation #4 Material has no response at frequencies far above 2 r r Lecture 244 Materials, including metals, tend to become transparent at very high frequencies ( x rays).9/19/201623 Observation #5 Dielectric constant has a DC offset below 2 r r Lecture 245At frequencies well below the resonance, we can replace the Lorentz equation with just a simple constant. 22001pr Observation #6 Dielectric constant can be negative and/or less than 2 r r Lecture 2469/19/201624 Observation #7 Refractive index can be less than one. 04 2 Lecture 247 Lorentz Modelfor Permeability9/19/201625 Magnetic Response of Ordinary MaterialsEquilibrium StatePolarized StateB NucleusElectronMagnetic DipoleSlide 49 Lecture 2 Lorentz Model for PermeabilityLecture 250 2mp22m0m1rj mpm0mmagnetic plasma frequencymagnetic resonant frequencymagnetic damping rate Boardman, Allan D.