FORTH. Essential electromagnetism for photonic metamaterials. Maria Kafesaki. Foundation for Research & Technology, Hellas, Greece (FORTH)

Transcription

1 FORTH Essential electromagnetism for photonic metamaterials Maria Kafesaki Foundation for Research & Technology, Hellas, Greece (FORTH)

2 Photonic metamaterials Metamaterials: Man-made structured materials (composites) with properties different than those of the constituent media result of the structuring (shape and size of their components). Properties mostly non-existent in natural materials Photonic Metamaterials (electromagnetic metamaterials): metamaterials aimed to control photons (electromagnetic (EM) waves) Essentials to study materials for photons? Maxwell s equations - determine the propagation of EM waves Electromagnetic response of a material

3 Outline Definitions of the essential electromagnetic quantities Maxwell s equations Wave equation waves in dielectrics and metals Electromagnetic response (dispersive properties) of materials Constitutive relations materials classifications Linear, isotropic, homogeneous, non-magnetic materials

4 Basic electromagnetic quantities: Definitions

5 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 = qe+ qv b F m Lorenz force Microscopic electric field Microscopic magnetic field Electric field, e: Electric force (F e ) per unit charge Magnetic field (induction), b: proportional to magnetic force (F m ) exerted at a moving charge Electric force (F e ) parallel to electric field; direction depends on charge Magnetic force (F m ) perpendicular to magnetic field; exerted at a currents F m + b v

6 Macroscopic electric and magnetic field Macroscopic quantities: Averaged quantities over volume V such as Picture by S. Linden Lattice constant a V a << V << λ 3 3 Wavelength λ Electric & magnetic fields: E 1 = edv V V B 1 = bdv V V L

7 Charge density and dipole moment Charge density, ρ: Charge per unit volume ρ 1 1 ρmdv V V = = V i q i r i q i V Dipole moment, p, of a dipole (e.g. atom): qr Displacement of the negative in respect to the positive charge, caused by an external field p = + q p r 1 - r -q E Dipole moment of a system of charges: p = q i i r i

8 Polarizability, polarization, susceptibility Polarizability, α : The ability of an atom/molecule to be polarized Polarization, P: (polarization density) P p = α E loc + 1 = V i p i Local electric field acting on an atom, i.e. external field and field scattered by other atoms r 1 r - q p -q E loc Dipole moment per unit volume how much a system of dipoles is polarized (on the average) Electric susceptibility, χ : P = ε χ E How susceptible is a system to an overall charge displacement Picture by ε 1 = Farads/m

9 Current density - magnetic dipole moment - magnetization Local current density, j : Current density, J : Current per unit area Magnetic dipole moment, m: Magnetization, M: M 1 = V i 1 J = ρv ( = qivi) V 1 m = r j Re-orientation of an atom as to be aligned with an external magnetic field m On the average reorientation of the atoms parallel to the magnetic field strength of the interaction with B j = qv i i j r m B

10 Displacement field and magnetic field Displacement field, D: D = ε E+ P Magnetic field, H: 1 H = ( B M ) μ B = μ H + M Fields with sources only external (or free) charges and currents Note: The actual total, measurable fields are E and B

11 Dielectric function, Conductivity Dielectric function/permittivity, ε : Relative permittivity, ε r : How much the total field is reduced compared to Etotal = Eapplied / ε r applied field ε = ε (1 + χ) ε = ε r / ε D = εe E s =E induced Conductivity, σ: J = σ E The ability of a material to conduct an electric current Current is caused by an E-field exerting forces on charge carriers Current density J depends linearly on E Current density J depends linearly on σ Resistivity, ρ: 1 ρ = σ

12 Magnetic susceptibility, permeability Magnetic susceptibility, χ m : M = μ χ m How susceptible is a system to an overall magnetization B Magnetic permeability, μ: B = μh μ 7 = 4π 1 Henries/m Relative permeability, μ r : How much the total field is changed compared to applied field Paramagnetic: μ r >1 Diamagnetic: μ r <1 (magnetic dipoles anti-allign) B μ μ / μ r = = μ B total r applied Paramagnetic dipoles B applied = μ H applied

13 Picture by Electric of an electric and magnetic dipole Energy of a dipole in an external E field: Energy of a magnetic dipole in an external B field: U d = p E U m = m B Charge and current density in a medium Total charge density: ρtot = ρ free + ρbound + ρexternal Free electrons Bound electrons external (source) charges Total current density: Jtot = J free + Jbound + Jexternal

14 Maxwell s equations in matter

15 Divergence and curl of a vector field S F ds = FdV V Divergence definition: F 1 F ds Gauss (divergence) theorem = lim V V S Large divergence: large normal to S components - strong sources S C F dl = ( F) ds Stokes theorem Curl definition: S 1 F F dl = lim S S C Large curl: large tangential to C components y C x

16 Maxwell s div equations in matter E = ρ ε tot Gauss law Volume integration and use of div theorem Diverging field lines from a point indicate the presence of electric charge at that point This charge can be detected by surrounding the point with a surface and observing the flux through the surface Q E d S = ε S B = No magnetic charges (magnetic monopoles)

17 Maxwell s curl equations in matter E B = με + μj t Ampere s law Magnetic fields are produced by currents time-varying electric fields B E = t Faraday's law Surface integration and use of curl theorem Time varying magnetic field produces an electric field Φ E d l = Φ= B d S Magnetic flux t Electromotive force

18 Maxwell s equations through D and H Two equivalent descriptions appear in literature Description 1: Distinction between free and bound electrons (preferred in microwaves) Dielectric function polarization displacement field describe Bound charges response P Jb = t D= εe Conductivity describe Free charges response J f = σ E Note: P J = t P = ρ (Result easily from the definition equations)

19 Maxwell s equations through D and H Description : No distinction between free and bound electrons (preferred in optics) Dielectric function polarization displacement field conductivity describe Total charges response P = ( ρ + ρ ) b tot ε = εtotal = ε + i σ ω f P J = Jb + J f = σ tote= t D= ε tot E

20 Maxwell s equations through D and H D = ρ B = Gauss law Sources of D are only the free (if no external) charges Description 1 ρ = ρ + ρ f ext J = J + J f ext D H = + t J Ampere s law Sources of H are only the free (if no external) currents Description ρ = ρ ext B E = t Faraday's law J = J ext J f J free

21 Ampere's law D H = + t J f Picture by V. Shalaev Displacement current D E P J D = = ε + In high frequencies t t t J D becomes larger Picture by D E iσ H = + J f = ε + σe = iωεe+ σe = iω( ε + ) E t t ω ε tot

22 Electromagnetic Waves

23 From Maxwell s equations to Wave equation Maxwell s curl equations in absence of external and free charges and currents: H E E = μ H = ε t t H H E ( E = μ ) E = μ = εμ t t t = E t E εμ = E ( E) E E H = ε t Wave equation 1 υ Same for H f t f = Wave velocity, υ: 1 c υ = = εμ n 1 c = ε μ c = 3. X 1 8 m/s Refractive index, n: n = εμ ε μ

24 Electromagnetic waves: Definitions (1) E t E εμ = Solutions are called plane waves, since the wavefonts (contours of maximum field) are planes Wave equation has solutions of the form E( r, t) = E cos( k r ωt + ϕ ) Wave vector Angular frequency Phase lag Picture by R. Trebino Dispersion relation: k = ω n c

25 Electromagnetic waves: Definitions () E= E cos( k r ωt + ϕ ) k = π λ Wavelength, λ: distance between any successive identical parts of a wave Dispersion relation k = ω n λ f = c c n Wavefunction Wavefunction ω = π f = π τ Frequency f: number of vibrations per unit time r Pictures by R. Trebino

26 Electromagnetic waves and complex notation Plane wave E= E cos( k r ωt + ϕ ) Since E= E e iϑ = cosϑ + isin i( kr ωt+ ϕ) Re[ e ] ϑ we adopt complex notation, incorporating phase into a complex amplitude E. Thus E= i E= Ee ϕ E e i( kr ωt) Note! Physical fields are the real parts

27 Electromagnetic (EM) waves features E = E e i( kr ωt) H = H e i( kr ωt) Substituting in Maxwell s equations: E = k E = H = k H = H E = μ k E = μωh t EM waves are transverse Magnetic and electric fields are perpendicular and in phase H k = E μω

28 Plane EM waves Slide by S. Linden Wavelength λ k r = const E and B are in phase!

29 Individual waves Sum Phase and Group velocity Adding plane waves of different wavelength produces beats: Picture by R. Trebino Envelope The points of constant phase (e.g. maxima) move with the phase velocity, c ph d( kx ωt) dx ω kx ωt = const = = dt dt k The envelop moves with the group velocity, v g c ph v g = dx dt = ω k d = ω dk

30 Linearly polarized wave the electric field oscillates along only one direction Circularly polarized wave Plane wave polarization the electric field amplitude writes a circle as the wave propagates Produced by two fields of: Equal magnitude, normal direction, phase difference 9o E() t = E ˆ ˆ [cos( ωt) x+ cos( ωt + π /) y]

31 Energy density and poynting vector S= Re( E) Re( H) Poynting vector (energy flow of a wave) t+τ 1 Re ( ) < S>= Sdt = E H t Time averaged over one period The magnitude of S denotes the intensity of the electromagnetic field: I 1 = S = ε εc E w 1 Re ( ED HB) = + 4 Energy density averaged over one period Energy per unit volume carried by an EM wave propagate in non-dispersive media

32 EM waves in metals E t E εμ = E = E e i( kr ωt) ε = ε + i σ ω Dispersion relation: k = ω n c but εμ σ ωε n = = 1 + i = nr + ini ε μ k = k + ik Waves of the form: Skin depth, δ: r Traveling distance required for intensity reduction e -1 i E = E e 1 1 δ = σμ ω k i i( kr x ωt) e kx i In 1 dimension for simplicity Attenuated waves For good conductors

33 1 1n n Electromagnetic fields at interfaces From Maxwell s equations in integral form Picture by E tangential H tangential D normal B normal have to be continous across charge- and current-free interfaces 1 1 D = 1t ε E =ε E ε ε 1 Subscript t = tangential Subscript n = normal 1 1 μ 1H1n = μh n Bt = B1 t μ μ1 D t

34 Refraction at a plane interface: Definitions Perpendicular ( S ) polarization perpendicular to the plane of incidence Plane of incidence is the plane that contains the incident and reflected k-vectors Parallel ( P ) polarization lies parallel to the plane of incidence k i k Interface Plane of the interface (here the yz plane) (perpendicular to page) z y Incident medium x E i θ i θ r θ t Slide by R. Trebino E r E t Transmitting medium k t r n i n t

35 Refraction at a plane interface: Definitions Reflection and transmission amplitudes: r E / E t E E = r i = t / i for the perpendicular polarization Picture by S. Linden r = E / E t = E / E // r i // t i for the parallel polarization Their calculation leads to Fresnel formulas Continuity of parallel k θ =θ i n sin r ( ) n sin ( ) i i t t E i, E r, and E t are the field complex amplitudes calculated by applying boundary conditions θ = θ Snell s law

36 Reflection and transmission coefficients Reflection coefficient R R Reflected poynting vector = = r Incident poynting vector Components perpendicular to the interface Transmission coefficient T T Transmitted poynting vector = = t Incident poynting vector ε ε 1 For lossless media T+R=1

37 Electromagnetic response (dispersive properties) of materials Input white beam Dispersed beam Prism Picture by R. Trebino Dispersion is the tendency of optical properties to depend on frequency

38 Relations among the fields The material properties in Maxwell s equations enter via the constitutive relations D= εe, B= μh Dr (,) t = εer (,)?? t NO Pr (,) t = ε χer (,)?? t Actual relations for time varying fields are not instantaneous D() t = d t ε ( t t ) Ε( t ) Relations are neither local P() t = ε d t χ ( t t ) Ε( t ) Polarization (and thus D) is affected by the field at previous times Polarization affected by the neighborhood of point r 3 D( r, t) = dt d r ε ( r, r ; t t ) Ε( r, t )

39 After Fourier transform Relations among the fields E (ω ) = E(t ) exp( iω t ) dt D( ω ) = ε( ω) Ε( ω) P( ω ) = ε χ( ω) Ε( ω) Material parameters are functions of frequency I.e., slow response of matter results to frequency dependent ε For long-range interactions are also functions of wavenumber

40 Calculation of the material response for dielectrics A bound electron in an electric field is treated as a forced harmonic oscillator Picture by V. Shalaev nd Newton s law d r m = F + F + F dt e damping restoring external d r dr m m m e i t dt dt e + eγ + eωr = E Lexp( ω ) Damping factor p = er = p exp( iω t) Resonance frequency Dipole moment of the resulting dipole p = e m e 1 E ω ω + iωγ L Atomic polarizability, a E

41 The forced oscillator Force Oscillator p = e m e 1 E ω ω + iωγ L Below resonance Weak vibration. In phase. ω << ω a E On resonance ω = ω Strong vibration.9 out of phase. Slide by. Above resonance ω >> ω Weak vibration. 18 out of phase. Slide by R. Trebino

42 Polarization and susceptibility of a system of atoms e p= m e 1 E ω ω + iωγ For a system of n atoms per unit volume, of one electron per atom, one can calculate polarization, as 1 ne 1 i ε i e P = p = ( ) E V ε m ω ω + iωγ For rare systems, local field E L equals averaged field E L L 1 ne 1 i ε i e P = p = ( ) E V ε m ω ω + iωγ This is the Lorenz model χe susceptibility

43 Dielectric function of a system of atoms ε = ε (1 ) + χe p ω ω ε = ε (1 ) ω ω + iωγ p = ne ε m Realistic atoms have many resonance frequencies e Relative ε Lorenz-type ε What ω and γ represent? ω : frequency of any atomic or molecular transition, between electronic, oscillating, or rotating levels γ : losses due to collisions, spontaneous emission Re ω For γ=

44 Dielectric function of a system of atoms Molecular rotations Atomic vibrations Electronic transitions For high frequencies ε 1 Picture from Wikipedia

45 Refractive index of a system of atoms Molecular rotations Atomic vibrations Anomalous dispersion: negative slope Picture by R. Trebino Electronic resonances usually occur in the UV; vibrational and rotational resonances occur in the IR; inner-shell electronic resonances occur in the x-ray region

46 Electric response of metals? Forces acting on a free electron: d r m = F + F + F dt e damping restoring external With same procedure as for bind electrons ω p ε = ε(1 ) ω + iωγ = m e ω r Drude-type permittivity ω p = ne ε m e Plasma frequency Conductivity σ: σ ω p ε = ε + i σ = ω γ iω

47 Constitutive relations Materials classification D( ω) = εω ( ) E( ω) B( ω) = μω ( ) H( ω)?? Only for isotropic media

48 Constitutive relations Materials classification The most general relations D= ε E+ ξ H B= ζ E+ μh In matrix form D ε ξ E = B ζ μ H ε, μ, ξ, ζ 3 3 matrices, i.e. up to 36 independent parameters Dx ε xx ε xy ε xz Ex ξ xx ξxy ξxz H x D = ε ε ε E + ξ ξ ξ H y yx yy yz y yx yy yz y D z zx zy zz E z zx zy zz H ε ε ε ξ ξ ξ z Such a medium is a Bianisotropic medium Bi = each flux depends on both (two) fields Anisotropic = at least one of the parameters is tensor Magnetoelectric coupling (coupling of electric and magnetic response)

49 Constitutive relations Materials classification In general a medium can be Bi- (non-bi) Isotropic Anisotropic Bi- D= ε E+ ξ H B= ζ E+ μh (non-bi) D= ε E B= μh

50 Isotropic medium ALL the parameters ε, μ are diagonal tensors of the form Thus D B ε ε = ε ε = ε E = μh AND D B μ μ = μ μ = ε E = μh i.e. D and E parallel and of the same ratio independently of the direction of wave propagation same for B and H Greek: isos=equal, tropos=way, i.e. isotropic=behave in equal way for all directions

51 E.g. D= ε E Anisotropic medium At least one of the parameters ε, μ (and ξ, ζ if non zero) is tensor Dx εxx εxy ε xz Ex D = ε ε ε E y yx yy yz y D z zx zy zz E ε ε ε z i.e. D and E not parallel and their relation depends on the direction Often ε or μ are diagonal, i.e. Greek: anisos=unequal, tropos=way Dx ε xx E x D y = ε yy E y D z zz E ε z

52 Constitutive relations Materials classification In general a medium can be D= ε E+ ξ H B= ζ E+ μh D= ε E B= μh Bi- (non-bi) Isotropic Anisotropic Independent parameters required for the knowledge of the material: Bianisotropic: up to 36 Biisotropic: 4 Anisotropic: up to 18 Isotropic: Causality requires*: ε ω ε ω μ ω μ ω * * * * ( ) = ( ), ( ) = ( ) Same for ξζ, *Since e.g. εω ( ) = ε( te ) iωt dt

53 Subclasses of anisotropic media: Uniaxial Uniaxial ε ε = ε ε z z: optical axis e.g. some crystals Birefringence effect: Two refractive indices: One for light polarized along the optic axis (n e ) and another for light polarized in either of the two directions (n o ) n o n e o-ray e-ray Light polarized along the optic axis is called the extraordinary ray, and light polarized perpendicular to it is called the ordinary ray Picture by R. Trebino

54 Subclasses of anisotropic media Gyroelectric ε iεg ε = iεg ε ε z e.g. electron plasma in magnetic field Bz Gyromagnetic μ iμg μ = iμg μ μ z e.g. ferrites in magnetic field Bz Greek: gyro=around

55 Subclasses of biisotropic media: Chiral Chiral media (no identical with their mirror images) D= εe+ iκ H B= iκ E+ μh In matrix form: e.g. DNA Greek: cheri=hand κ=chirality parameter D ε iκ E = B iκ μ H Chiral media have different refractive indices for right and left circularly polarized waves

56 Extreme idealized media Perfect electric conductor (PEC) Perfect magnetic conductor (PMC) Perfect electromagnetic conductor (PEMC) PEC ε, μ then μ σ, z = ε Small impedance ε z = PMC, μ then μ ε Infinite impedance

57 Main references used for this talk Books J. D. Jackson, Classical Electrodynamics, Academic Press D. J. Griffiths, Introduction to Electrodynamics, John Wiley E. N. Economou, Solid State Physics, in press P. Markos and C. M. Soukoulis, Wave Propagation: From Electrons to Photonic Crystals and Left-Handed Materials, Princeton Univ. Press J. A. Kong, Electromagnetic Wave Theory, EMW Publishing Internet lectures by Martin Savage, Univ. of Washington, ( Rick Trebino, Georgia Inst. of Technology, ( Stefan Linden & Martin Wegener, Univ.of Karlsruhe, (private commun.) Vladimir Shalaev, Purdue Univ., ( Sergei Tretyakov and Ari Shivola, Helsinki Univ. of Technology, ( Wikipedia

58 Thank you!!