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Electromagnetic Waves

by: Brandy Koch

Electromagnetic Waves OPTI 6104

Brandy Koch
GPA 3.92

Tsing-Hua Her

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Tsing-Hua Her
Class Notes
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This 12 page Class Notes was uploaded by Brandy Koch on Sunday October 25, 2015. The Class Notes belongs to OPTI 6104 at University of North Carolina - Charlotte taught by Tsing-Hua Her in Fall. Since its upload, it has received 15 views. For similar materials see /class/228885/opti-6104-university-of-north-carolina-charlotte in Optical Studies at University of North Carolina - Charlotte.


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Date Created: 10/25/15
Constitutive Relation amp Forces Driven Oscillation Tuesclay February 12 2008 3907 PM The optical properties of material is summarized in the constitutive relations which states the relation between Dfield and Bfield in terms of E and H field Constitutive relation however does not tell you the physics of what is going on It can merely a phenomenological result Depending on the complexity ofthe material constitutive relations can be classified as follows 0 Isotropic 6b ii u Hm O Anisotropic 3 fl Em 39 B N A In H w 0 Bianisotropic Bu gm Eadi ihv Elwlgm 139 F firm In this course I will only touch isotropic material When EM fields or waves are present inside materials the fields induce changes in their electronic or structural properties and produce induced electrical dipoles to the first order approximation those are dominating terms from localized source approximation These dipoles are driven by the incident fields and represent source terms in the Maxwell s equations that radiate at the same frequency as incident light The reemitted light interferes with the incident light and thus modifies its behavior This constitutes the heart of material response of EM fields amp waves 1 tram In W W M WMAW I Consider the case most common case in which the electrical dipole does not vanish and therefore dominate the material response Optical Properties of Material Page 1 a r e6 at surf g 1 i VHi ru 7xz VIE vane 59745 1 199 9 39VIVx39E v 3 al 3 V V15 Mame MM c VIE M E u F 4quot WWI r L M t m o EMA Eo are Ar kibmti MM W 7 VG 116331F F 4W WWMMAEMM o The above equation is similar to the wellknown forced driven harmonic oscillator There are two solutions to it one is the homogeneous solution which is the independent of the source and the other is the inhomogeneous solution that is the radiation driven by induced polarization The interference between the incident field amp the radiation from the induced dipole yields the macroscopic field we measured Therefore it is not surprising that waves are modified inside the material 0 The next question is how the electric polarization depends on electric field and this is the business of electrical susceptibility as discussed later Optical Properties of Material Page 2 o In case whether magnetic response cannot be ignored the induced magnetic dipole moment can be related to Hfield through magnetic susceptibility This could become increasingly important in light of magnetic metamaterials which are artificial engineering materials 0 Constitutive relations can also be expressed in the time domains Timedomain approach is I 39 39 39 to frequencydomain nnnmnrh amp they turns handy 39 quot on problems at hand and aive us d erent physical pictures I will cover this in class For interested reader please refer to The Elements of Nonlinear Optics by PN Butcher amp D Cotter Optical Properties of Material Page 3 Ori in o o tical res onse a materials Tuesday Februaly 12 2008 5 PM Reading JDJs75 Mark Fox 221223 Ignore local field effect which usually applied for lowdensity material such as gas phase but in general not true for solid state The electron will be driven by external field through Lorentz force law ignoring magnetic field since 1c smaller than Efield EM ux 3quotE mc 32 e Frant g39 assuring time harmonic field expimt induced dipole moment by this electron per atom is F 2 Nquot J a u o Suppose N atomic density unit volume 2 electronsatoms and i wmmivaum 7 sin W uML a X oa39 fi WW WWWM4 7 N7 quots 7 I 1 6E f 7quot Dielectric constant or relative permittivity is 6m 2 5 We v39iw 6 39 5 4 Optical Properties of Material Page 4 quotquotl 7 411 K Near individuai resonance the dispersion oftne reai and imaginary part of dieiectric constant goes iike iM MJaumw W i 9 V Wquot 500 ns E s Tm QE c 14 ENif Alc f quot Grim H111 sm Alfutimr f Mixu Y 141 4 7r u A A I n 1r quot2 I39L I Y 1744 1 11 Me f 141 14quot an 2b 81 4 7Cquotlu quot1quot quot r 539 quot0 h 3 1 The shape of 139 and 1 near resonance can be shown to have Lorentzian shape as snownintne ieftfigureFordetaiispiease 7 referto Fox section 221 0 quotManon17 40 5TH7 0 i 100 120 140 a THz Anornaious dispersion the region where the refractive index increases with frequency is Optical Properties ofMaten39al Page 5 normal dispersion This corresponds to region outside the resonance Within resonance refractive index decreases as frequency increases and is called anomalous Dielectricfunctions for atoms with multiple resonances Vlbmlional hands eleclmnlc mmsiunns Refraclx 8 Index 3 Absorption inl39mrcd vismlc umvmlcl Xmy mu 0 on mm W mm 10 m Frequency Hz Optical Properties ofMaterial Page 6 mquot quot Limitamp quot39 39 39r uw 1 TueEday February 12 2008 3915 PM Afw ear 63941 ea I llIflds 5 39139 6w 1 way 1 quotA 5 I Grin 101ijamp 1A end 5 lab 0 erM lv39q 17 h WW W W m N39zo W 1 6190 teem 2f v5 6n Optical Properties of Material Page 7 Con the con ductors and insulators are different only at low frequency at AC or high frequency they are same Another way to look at this is that at higher frequency free electrons which make an ductor conducting at low frequency contribute very little to the dielectric constant they all originate from bound electrons 0 Separate the contribution of free electrons fuv a V L v 239 quot 6 l l39 5M J u 4W5 9 L x olwi 2 4 yak114d 139 w u 1 qu um I MM Liu 2v all 4 i1 Ne Wmmxwr er hand from an electrical point of view 0 Ontheoth VVF T 72 s39p39 g 39j39 oE vm F 1 Mu VVH 39Atof F 4 E kw gt1 Comparing this macroscopic model of electron conduction and microscopic picture of LorentzDrude model we can link electrical conductivity a macroscopic quantity to damping rate a microscopic quantity Optical Properties of Material Page 8 quotI N 2quot 9 a w 71 This relation has become the method to determine the collision rate in material from the measurement of electrical conductivity at lowfreduency For ex mple C as a s N 7gtlt107 mho at low frequency With N 8x10 m393 we can derive DC conductivity accordingto o 1quot N e quot In 1 I la LaxD514 n 3 7 Other common electrical conductivity of material the unit SmSmithmeter m o TABLE 34 Conductivilies of same common materials Englund 194A Material 1 Sim vacuum 0 can water 35 physiological saline 0 15 4 distilled water 2 x 104 silver 614 x 107 copper 580 x 107 iron 1 x lo7 dry earth 10 105 glass 10 2 p 10 14 m Ia 0 DC approximation of electrical conductivity works well up to THz uVlo39Ul ltlt Y quot3 Cwr fr amy This is an amazing fact electrons behave just like DC even at looGHz oscillation In another word frequency dependence of electron conductivity is not present until above 10THz Optical Properties ofMaterial Page 9 gt o rw I I39 it o The dispersive properties of material can be attributed as well to a complex dielectric constant as to a AC conductivity plus a dielectric constant Optical Properties of Material Page 10 HighFrequencx Limit amp Plasma Tuesday Februaw 12 2008 515 PM em a quot k Ne 1 z s l 4 4w 6 39 em a a I l I to Mfrb 4 U39u39 a439 u 4 J L at Hw 3 it r WK WP Mia w 51 What this means is that the contribution of all the bound electrons to dielectric constant at very high frequency is essentially like collisionless free electron s contribution to dielectric constant at all frequency This is not surprised bc free electron has resonance frequency at zero Notice that this is not saying these bound electrons are free at high frequency at very high frequency bound electrons cannot respond to such rapid oscillation amp all the retardness effect come in just like free electrons respond to almost every frequency 0 Plasma is a collection of negative and positive charges that are electrically neutral and one of which can move with respect to the other For example metals are plasma since it is neural electrically and electrons can move freely while u ions are too heavier to move At plasma frequency electrons move collectively with respect to the ions It correspond to a longitudinal wave the derivation of plasma frequency can be seen in Fox 75 included in class handout 3 Optical Properties of Material Page 11 OQticaI Proeerties of Metals in the quotOpticalquot Domain Tuesday February 12 2003 515 PM We have shown that dielectric property of metals can be expressed in terms of free electrons contribution and bound electron contribution and free electron contribution is essentially the same as AC conductivity DC conductivity dominates the freeelectron contribution for up to actually very high frequency For copper this is close to 10 THz That is to say the highfrequency approximation or collisionless plasma approximation of free electrons is not valid until at frequency gtgt 10 THz em a quot 6quot Ve v wz w l 6 39 em 3quot a 1 rlou 2 LL mu twin 0 At optical frequency N 1005 THz the collisionless plasma approximation of free electrons is justified and we can express dielectric constant as follows I H min In my 03 m 0 1007quot L W gt Y a c 2 ur r rm mm 4 um IF 1quot ML Met39f T 1 ur P 391 Fr 2 um l l T J 6H lT J u I HMlt0 W Optical Properties of Material Page 12


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