Overview in Images. S. Lin et al, Nature, vol. 394, p , (1998) T.Thio et al., Optics Letters 26, (2001).

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1 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 (1997) T.Thio et al., Optics Letters 6, (001). S. Lin et al, Nature, vol. 394, p. 51-3, (1998) J.R. Krenn et al., Europhys.Lett. 60, (00)

2 Motivation Major breakthroughs are often materials related Stone Age, Iron Age, Si Age,. People realized the utility of naturally occurring materials Scientists are now able to engineer new functional nanostructured materials Is it possible to engineer new materials with useful optical properties Yes! Wonderful things happen when structural dimensions are λ light This course talk about what these things are and why they happen What are the smallest possible devices with optical functionality? Scientists have gone from big lenses, to optical fibers, to photonic crystals, to Does the diffraction set a fundamental limit? Possible solution: metal optics/plasmonics

3 Designing New Functional Devices We need to be able to solve the following problem: ε h ε 3 ε 6 ε 1 ε 4 ε 5?

4 Maxwell s Equations Light Interaction with Matter Divergence equations D = ρ f B =0 Curl equations B E = t D H = + J D = Electric flux density E = Electric field vector ρ = charge density B = Magnetic flux density H = Magnetic field vector J = current density

5 Constitutive Relations Constitutive relations relate flux density to polarization of a medium Electric E ( ) D= ε E+ P E = ε E Electric polarization vector Material dependent!! ε 0 = Dielectric constant of vacuum = C N -1 m - [F/m] ε = Material dependent dielectric constant When P is proportional to E Total electric flux density = Flux from external E-field + flux due to material polarization Magnetic 0 0 ( ) B = μ H + μ M H Magnetic flux density Magnetic field vector Magnetic polarization vector μ 0 = permeability of free space = 4πx10-7 H/m Note: For now, we will focus on materials for which M = 0 B = μ0h

6 Divergence Equations D = ρ How did people come up with:? Coulomb Charges of same sign repel each other (+ and + or and -) Charges of opposite sign attract each other (+ and -) He explained this using the concept of an electric field : F = qe Every charge has some field lines associated with it + - He found: Larger charges give rise to stronger forces between charges Coulomb explained this with a stronger field (more field lines)

7 Divergence Equations Gauss s Law (Gauss ) E D ds = εe ds = ρdv A A V ds E-field related to enclosed charge Gauss s Theorem (very general) A F ds = Fdv V Combining the Gauss s D ds = Ddv= ρdv D = ρ A V V The other divergence eq. B = 0 is derived in a similar way from B d S = 0 A

8 Curl Equations D How did people come up with: H = + J? A H J D H C J D increasing when charging the capacitor Ampere ( ) C D H dl = + J ds Magnetic field induced by: i A Changes in el. flux Electrical currents

9 Ampere: Stokes theorem: C D H dl = + J ds i C Other curl eq. A A Curl Equations ( ) i F d l = F d S B E = t Derived in a similar way from i C A A C D H dl = ( H) ds = + J ds i C A A B E dl = ds B E dl = ( E) ds = ds i Stokes A D H = + J A B E C

10 Summary Maxwell s Equations Divergence equations D = ρ Curl equations B E = t B =0 D H = + J Flux lines start and end on charges or poles Changes in fluxes give rise to fields Currents give rise to H-fields Note: No constants such as μ 0 ε 0, μ ε, c, χ,. appear when Eqs are written this way.

11 The Wave Equation Plausibility argument for existence of EM waves H H E E E. Curl equations: Changing E-field results in changing H-field results in changing E- field. The real thing Goal: Derive a wave equation: Solution: Waves propagating with a (phase) velocity v ( rt, ) = U 1 v { 0 } (, t) = Re ( ) exp( iωt) U r U r U ( rt, ) for E and H Position Time Starting point: The curl equations

12 The Wave Equation for the E-field Goal: Curl Eqs: ( rt, ) = E a) b) 1 v B E = = μ0 D H = + J E ( rt, ) H (Materials with M = 0 only) Step 1: Try and obtain partial differential equation that just depends on E Apply curl on both side of a) ( H ) H E = μ0 μ = 0 Step : Substitute b) into a) D J E P J E = μ μ = μ ε μ μ D= ε 0 E+ P Cool!...looks like a wave equation already

13 The Wave Equation for the E-field Compare: E = 1 v E With: Use vector identity: E P J E = με 0 0 μ 0 μ 0 Verify that E = 0 when 1) ρ f = 0 E = ( E) E ) ε(r) does not vary significantly within a λ distance! Result: E = με μ 0 + μ 0 E P J In order to solve this we need: 1) Find P(E) ) Find J(E) something like Ohm s law: J(E) = σe we will look at this later..for now assume: J(E) = 0

14 P linearly proportional to E: P 0 Dielectric Media Linear, Homogeneous, and Isotropic Media = ε χ E χ is a scalar constant called the electric susceptibility E = με μ 0 + μ 0 E P J E E E E = με με 0 0χ = με 0 0( 1+ χ) Define relative dielectric constant as: ε 1 r = + χ All the materials properties E E = μεε 0 0 r Results from P Note 1 : In anisotropic media P and E are not necessarily parallel: P = ε0χ E Note : In non-linear media: i ij j j P = εχe+ εχ E + εχ E + () (3)

15 Properties of EM Waves in Bulk Materials We have derived a wave equation for EM waves! E E = μεε 0 0 r Euh. Now what? Let s look at some of their properties

16 Speed of the EM wave: Speed of an EM Wave in Matter Compare E E = μεε 0 0 r and E = 1 v E v 1 1 c = = μ ε ε ε r r Where c 0 = 1/(ε 0 μ 0 ) = 1/((8.85x10-1 C /m 3 kg) (4π x 10-7 m kg/c )) = ( 3.0 x 10 8 m/s) Optical refractive index Refractive index is defined by: n c = = ε r = 1+ χ v Note: Including polarization results in same wave equation with a different ε r c becomes v

17 Refractive Index Various Materials Refractive index: n λ (μm)

18 Dispersion relation: ω = ω(k) Derived from wave equation Substitute: E Dispersion Relation (, t) = E r (, t) { } ( zt, ) = Re E( z, ω) exp( ikr+ iωt) n c E r Result: k Check this! n c ω = ω c ω = k n Group velocity: Phase velocity: v g v ph dω ω c c c = = = = dk k n ε 1+ ω k r v g k χ

19 Solution to: Electromagnetic Waves (, t) = E r n c (, t) E r { } Monochromatic waves: E( r, t) = Re E( k, ω) exp( ik r+ iωt) { } (, t) = Re (, ω) exp( i + iωt) H r H k k r Check these are solutions! TEM wave Optical intensity Symmetry Maxwell s Equations result in E H propagation direction Time average of Poynting vector: S ( r, t) = E( r, t) H( r, t)

20 Light Propagation Dispersive Media Relation between P and E is dynamic (, t) = εχe( r, t) P r The relation : assumes an instantaneous response 0 In real life: + ε 0 (, t) = dt' x( t t' ) E( r, t' ) P r P results from response to E over some characteristic time τ : Function x(t) is a scalar function lasting a characteristic time τ : E(t ) x(t-t ) x(t-t ) = 0 for t > t (causality) t = t - τ t = t t

21 EM waves in Dispersive Media Relation between P and E is dynamic EM wave: + ε 0 (, t) = dt' x( t t' ) E( r, t' ) P r { } { ω ω } (, t) = Re (, ω) exp( i + iωt) E r E k k r (, t) = Re (, ) exp( i + i t) P r P k k r Relation between complex amplitudes (, ω) = εχω ( ) E( k, ω) P k 0 (Slow response of matter ω-dependent behavior) This follows by equation of the coefficients of exp(iωt)..check this! It also follows that: ( ) = + ( ) ε ω ε 1 χ ω 0

22 Absorption and Dispersion of EM Waves Transparent materials can be described by a purely real refractive index n { } EM wave: E( zt, ) = Re E( z, ω) exp( ikz+ iωt) c ω Dispersion relation ω = k k =± n n c Absorbing materials can be described by a complex n: n= n' + in'' ω ω ω α It follows that: k =± ( n' + in'' ) =± n' + i n'' ± β i c c c Investigate + sign: E( ) E( ) α zt, = Re z, ω exp iβz z+ iωt Traveling wave Decay ω Note: β = n' = k0n' n act as a regular refractive index c ω α = n'' = k0n'' α is the absorption coefficient c

23 Absorption and Dispersion of EM Waves n is derived quantity from χ (next lecture we determine χ for different materials) Complex n results from a complex χ: χ = χ' + iχ'' n = 1+ χ n= n' + in'' = 1+ χ = 1 + χ' + iχ'' α = kn'' 0 α n= n' i = 1 + χ ' + iχ '' k 0 Weakly absorbing media When χ <<1 and χ << 1: 1 + χ ' + iχ'' 1 + ( χ' + iχ'' ) Refractive index: 1 n ' = 1 + χ ' α kn = '' = kχ '' Absorption coefficient: 0 0 1

24 Summary Maxwell s Equations B D = ρ f B = 0 E = t Curl Equations lead to E P E = με μ 0 (under certain conditions) Linear, Homogeneous, and Isotropic Media D H = + J P = ε χ E 0 Wave Equation with v = c/n n E ( r, t) E( r, t) = c In real life: Relation between P and E is dynamic + P( r, t) = ε dt' x( t t' ) E( r, t' ) P( k, ω) = εχω 0 ( ) E( k, ω) 0 This will have major consequences!!!

25 Next Lectures Real and imaginary part of χ are linked Kramers-Kronig Origin frequency dependence of χ in real materials Derivation of χ for a range of materials Insulators (Lattice absorption, Urbach tail, color centers ) Semiconductors (Energy bands, excitons ) Metals (Plasmons, plasmon-polaritons, )

26 Useful Equations and Valuable Relations Maxwell s Equations Constitutive relations: Gauss s Law Maxwell (also) Handy Math Rules Vector identities: Divergence Equations D = ρ B =0 ( ) D = ε E+ P = ε + ε χe = ε + χ E = ε ε E o r ( 1 ) Curl Equations B = μ H + μ M = μ H + μ χ H = μ + χ H = μ μ H m 0 m 0 r D ds = εe ds = ρdv A A V C D H dl = + J ds i A E = ( E) E ε E = ε E+ E ε + ε 0 D H = + J B E = t Dynamic relation between P and E: P ( r, t) = dt' x( t t' ) E( r, t' ) and P( k, ω) = εχω ( ) E( k, ω) Dispersive and absorbing materials: α E( zt, ) = Re E( z, ω) exp iβz z+ iωt ω ω where β = n' = k0n',absorption coefficient α = n'' = k0n'' c c 0 Gauss theorem Stokes C A F ds = Fdv V A ( ) i F d l = F d S