Lecture 2: Dispersion in Materials. 5 nm

Transcription

1 Lectue : Dispesion in Mateials 5 nm

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3 Pevious Lectue: Maxwell + Wave Equation Speed of the EM wave: Compae E =µεε v 1 1 c = = µε ε ε E t and geneal wave Equation: ( t, ) U = 1 v U t ( t, ) Whee c = 1/(ε μ ) = 1/((8.85x1-1 C /m 3 kg) (4π x 1-7 m kg/c )) = ( 3. x 1 8 m/s) Optical efactive index Refactive index is defined by: n c = = ε = 1+ χ v Note: Including polaization esults in same wave equation with a diffeent ε c becomes v

4 Refactive Index Vaious Mateials Refactive index: n λ (µm)

5 z Dispesion elation: = (k) Deived fom wave equation Substitute: Result: Check this! k E c = k n Goup velocity: Phase velocity: A n c = v g v ph Dispesion Relations (, t) E = (, t) { } ( z, t) = Re E( z, ) exp( ikz + it ) n c E t v g d c c c = = = = dk k n ε 1+ χ k k

6 Absoption and Dispesion of EM Waves Tanspaent mateials can be descibed by a puely eal efactive index n { } EM wave: E( z, t) = Re E( z, ) exp( ikz + it ) c Dispesion elation = k k = ± n n c Absobing mateials can be descibed by a complex n: n = n' + in'' α It follows that: k =± ( n' + in'' ) =± n' + i n'' ± β i c c c Investigate + sign: E( ) E( ) Note: α zt, = Re z, exp iβz z+ it Taveling wave Decay β = n ' = kn ' n acts as a egula efactive index c α = n '' = kn '' α is the absoption coefficient c

7 Summay and Futue Diections Maxwell s Equations B D = ρ f B = E = t Cul Equations lead to E P E = µε + µ t t Linea, Homogeneous, and Isotopic Media (unde cetain conditions) D H = + t J P = εχe Wave Equation with v = c/n n E(, t) E(, t) = c t Today: In eal life the elation between P and E is dynamic (, ) = εχ ( ) E( k, ) P k Solutions: Mateials have a esponse time Loentz model: Helps to undestand the optical esponse of eal mateials χ(), ε (), n()

8 n and n vs χ and χ vs ε and ε All pais (n and n, χ and χ, ε and ε ) descibe the same physics Fo some poblems one set is pefeable fo othes anothe n and n used when discussing wave popagation α E( zt, ) = Re E( z, ) exp iβz z+ it whee β = kn ' and α = kn '' Phase popagation absoption χ and χ used when discussing micoscopic oigin of optical effects ε and ε As we will see today Inte elationships Example: n and ε Fom n = ε n' + in'' = ε ' + iε '' ( ) ( ) ε ' = n' n'' ε '' = nn ' '' and n' = n'' = ( ) ( ) ε ' + ε '' + ε ' ( ) ( ) ε ' + ε '' ε '

9 Light Popagation Dispesive Media Relation between P and E is dynamic The elation : (, t) = εχe(, t) P assumes an instantaneous esponse In eal life: + ε (, t) = dt ' x( t t ') E(, t ') P P esults fom esponse to E ove some chaacteistic time τ : Function x(t) is a scala function lasting a chaacteistic time τ : E(t ) x(t-t ) x(t-t ) = fo t > t (causality) t = t - τ t = t t

10 EM waves in Dispesive Media Relation between P and E is dynamic EM wave: + ε (, t) = dt ' x( t t ') E(, t ') P { } { } (, t) = Re (, ) exp( i + it) E E k k (, t) = Re (, ) exp( i + i t) P P k k Relation between complex amplitudes (, ) = εχ ( ) E( k, ) P k (Slow esponse of matte -dependent behavio) This follows by equation of the coefficients of exp(it)..check this! It also follows that: ( ) = + ( ) ε ε 1 χ Next: We will model the fequency dependence of χ, ε, and n

11 Behavio of bound electons in an electomagnetic field Optical popeties of insulatos ae detemined by bound electons Loentz model Linea Dielectic Response of Matte Chages in a mateial ae teated as hamonic oscillatos m a = F + F + F el E, Local Damping Sping d d m + mγ + C = ee exp L i t dt dt Guess a solution of the fom: ( ) p= p exp it ( ) The electic dipole moment of this system is: d p dp m + mγ + Cp= e E exp L i t dt dt dp ; = ip exp dt m p imγp + Cp = e E L (one oscillato) p= e ( ) ( it) d p dt E L C, γ ; p exp = Solve fo p (E L ) + Nucleus e -, m p = e ( it)

12 Atomic Polaizability Detemination of atomic polaizability Last slide: m p imγp + Cp = e E L C e iγ L p p + p = E (Divide by m) m m Define as (tuns out to be the esonance ) p = e 1 E iγ m L Define atomic polaizability: ( ) α Atomic polaizability (in SI units) p = E L e m 1 iγ Resonance fequency Damping tem

13 Chaacteistics of the Atomic Polaizability Response of matte (P) is not instantaneous -dependent esponse p e 1 Atomic polaizability: α( ) = = Aexp iθ( ) m iγ = E L Amplitude e 1 A = m ( ) + γ 1/ Amplitude smalle γ Phase lag of α with E: 18 θ = tan 1 γ Phase lag 9 smalle γ

14 Relation Atomic Polaizability (α) and χ: cases Case 1: Raified media (.. gasses) Dipole moment one atom, : Polaization vecto: Occus in Maxwell s equation.. sum ove all atoms p ( ) = α E L P 1 1 = = α = Nα V p V E E L L e 1 α( ) = m iγ P= Ne 1 L εχ L εχ m iγ E = E = E Micoscopic oigin susceptibility: Plasma fequency defined as: p ( ) χ Ne ε m E-field photon Ne 1 = ε m iγ = χ ( ) = p iγ Density

15 Remembe: ε and n follow diectly fom χ Fequency dependence ε Relation of ε to χ : ε = 1+ χ = 1+ p iγ ε + iε = 1+ χ + iχ = 1+ ' '' ' " p iγ ( ) ε ' = 1 + χ' = 1+ ( ) ε '' = χ'' = p ( ) ( ) + ( ) γ γ p + γ ε 1+ ( ) p 1 ε ' ε '' χ ' = χ = p p

16 Popagation of EM-waves: Need n and n Relation between n and ε n = ε ε ( ) ε '' ( ) ( ) ε ' = n' n'' ε ' χ ' = ε '' = nn ' '' 1 n( ) << : High n low v ph =c/n : Stong dependence v ph n ' n '' n ' = 1 Lage absoption (~ n ) >> : n = 1 v ph =c

17 χ = k Realistic Raefied Media Realistic atoms have many esonances Resonances occu due to motion of the atoms (low ) and electons (high ) Ne k 1 ε m iγ k Whee N k is the density of the electons/atoms with a esonance at k Example of a ealistic dependence of n and n α= k n Molecule otation Atomic vibation Electon excitation n n > 1 indicates pesence high oscillatos 1 3 Log ()

18 Back to Relation Atomic Polaizability (α) and χ: Case : Solids Solid p i E-field? Atom feels field fom: 1) Incident light beam ) Induced dipola field fom othe atoms, p i Local field: E = E + E L I Local field Field without matte Induced dipola field fom all the othe atoms

19 Local field Local field: Electic Susceptibility of a Solid E = E + E L I Local field Field without matte All the othe atoms Induced dipola field Example: Fo cubic symmety: P EL = E+ 3ε Polaization of a solid (Simila elations can be deived fo any solid) (Solid state Phys. Books, e.g. Kittel p381-39) Solid consists of atom type at a concentation N P P P= ε N α E = ε N α E+ = ε N α E+ N α 3 3 L ε p P 1 1 N α = ε N α 3 E P χ = = ε 1 E 1 3 N α N α

20 Clausius-Mossotti Relation Polaization of a solid Susceptibility: Limit of low atomic concentation:.o weak polaizability: petty good fo gasses and glasses Clausius-Mossotti By definition: ε = 1+ χ Reaanging I gives P χ = = ε 1 E 1 3 χ N α ε 1 1 = N α ε + 3 N α N α I II III Conclusion: Dielectic popeties solids elated to atomic polaizability This is vey geneal!!