MIT 2.71/2.71010/12/05 wk6-b-1So far• Geometrical Optics– Reflection and refraction from planar and spherical interfaces– Imaging condition in the paraxial approximation– Apertures & stops– Aberrations (violations of the imaging condition due to terms oforder higher than paraxial or due to dispersion)• Limits of validity of geometrical optics: features of interest are much bigger than the wavelength λ– Problem: point objects/images are smaller than λ!!!– So light focusing at a single point is an artifact of our approximations– To understand light behavior at scales ~ λ we need to take into account the wave nature of light.MIT 2.71/2.71010/12/05 wk6-b-2Step #1 towards wave optics: electro-dynamics• Electromagnetic fields (definitions and properties) in vacuo• Electromagnetic fields in matter• Maxwell’s equations– Integral form– Differential form– Energy flux and the Poynting vector• The electromagnetic wave equationMIT 2.71/2.71010/12/05 wk6-b-3Electric and magnetic forces+ +rfree chargesCoulomb force2041rqq′=πεFqq´FdlFrII´rIIlπµ2dd0′=FMagnetic force(dielectric) permitivityof free space(magnetic) permeabilityof free spaceMIT 2.71/2.71010/12/05 wk6-b-4Note the units…20DistanceCharge1forceElectric⎟⎠⎞⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛ε2041rqq′=πεF20TimeChargeforceMagnetic⎟⎠⎞⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛µrIIlπµ2dd0′=F() ()SpeedTimeDistance2/100≡⎟⎠⎞⎜⎝⎛≡⇒µεMIT 2.71/2.71010/12/05 wk6-b-5Electric and magnetic fields+v⊗FBEelectricchargevelocityLorentzforceelectricfieldmagneticinduction()BvEF×+= qObservation Generation+Estatic charge:⇒electric fieldv++Belectric current(moving charges):⇒magnetic fieldqMIT 2.71/2.71010/12/05 wk6-b-6Gauss Law: electric fields+EdaE+∫∫ ∫∫∫=⋅AVVd 1d0ρεaEAVcharge density0ερ=⋅∇ EGauss theoremdaMIT 2.71/2.71010/12/05 wk6-b-7Gauss Law: magnetic fieldsBAV∫∫=⋅A0daB“magnetic charge” density0=⋅∇BGauss theoremthere are nomagneticchargesdaMIT 2.71/2.71010/12/05 wk6-b-8Faraday’s Law: electromotive forcedlEB(t) (in/de)creasing∫∫⋅−=⋅CAtl aBE ddddStokes theoremt∂∂−=×∇BECAMIT 2.71/2.71010/12/05 wk6-b-9Ampere’s Law: magnetic inductiondlBICA∫∫∫∫∫⎟⎟⎠⎞⎜⎜⎝⎛⋅∂∂+⋅=⋅CAAtl aEaJB ddd00εµcurrentcapacitordlBcurrent densityMaxwell’s extension,Displacement currentStokes theorem⎟⎠⎞⎜⎝⎛∂∂+=×∇tEJB00εµMIT 2.71/2.71010/12/05 wk6-b-10Maxwell’s equations∫∫ ∫∫∫=⋅AVVd 1d0ρεaE0ερ=⋅∇ E∫∫=⋅A0daB0=⋅∇ B∫∫⋅−=⋅CAtl aBE ddddt∂∂−=×∇BE∫∫∫⋅⎟⎠⎞⎜⎝⎛∂∂+=⋅CAtl aEJB dd00εµ⎟⎠⎞⎜⎝⎛∂∂+=×∇tEJB00εµGauss/electricGauss/magneticFaradayAmpere-Maxwell(in vacuo)MIT 2.71/2.71010/12/05 wk6-b-11Electric fields in dielectric media++++––––++++––––E−−++−=rrp qqatom in equilibriumatom under electric field:• charge neutrality is preserved• spatial distribution of chargesbecomes assymetric±±±–+–+–+–+–+–+–+–+–+–+–+–+Spatially variant polarizationinduces local charge imbalances(bound charges)P⋅−∇=boundρ∑= pPDipole momentPolarizationMIT 2.71/2.71010/12/05 wk6-b-12Electric displacementGauss Law:()boundfree0total011ρρερε+==⋅∇ E()P⋅∇−=free01ρε()free0ρε=+⋅∇PEPED+=0εElectric displacement field:freeρ=⋅∇DLinear, isotropic polarizability:EPχε0=()EEDεχε≡+=10MIT 2.71/2.71010/12/05 wk6-b-13General cases of polarizationLinear, isotropic polarizability:EPχε0=EPLinear, anisotropic polarizability:EP⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=3332312322211312110χχχχχχχχχεEPNonlinear, isotropic polarizability:...)2(00++= EEEPχεχεetc.MIT 2.71/2.71010/12/05 wk6-b-14Constitutive relationshipsE: electric fieldH: magnetic fieldD: electric displacementB: magnetic inductionPED+=0εpolarizationMBH −=0µmagnetizationMIT 2.71/2.71010/12/05 wk6-b-15Maxwell’s equations∫∫ ∫∫∫=⋅AVVd dfreeρaDfreeρ=⋅∇ D∫∫=⋅A0daB0=⋅∇ B∫∫⋅−=⋅CAtl aBE ddddt∂∂−=×∇BE∫∫∫⋅⎟⎠⎞⎜⎝⎛∂∂+=⋅CAtl aDJH ddfree0µ⎟⎠⎞⎜⎝⎛∂∂+=×∇tDJHfree0µGauss/electricGauss/magneticFaradayAmpere-Maxwell(in matter)MIT 2.71/2.71010/12/05 wk6-b-16Maxwell’s equations wave equation()0=⋅∇ Eε0=⋅∇ Bt∂∂−=×∇BE()t∂∂=×∇EBεµ0(in linear, anisotropic, non-magnetic matter, no free charges/currents)matter spatially andtemporally invariant0=⋅∇ E0=⋅∇ Bt∂∂−=×∇BEt∂∂=×∇EBεµ0()()220ttt ∂∂−=∂×∇∂−=×∇×∇⇒∂∂−=×∇EBEBEεµ()()EEE∇⋅∇−⋅∇∇=×∇×∇=002202=∂∂−∇tEEεµelectromagneticwave equationMIT 2.71/2.71010/12/05 wk6-b-17Maxwell’s equations wave equation(in linear, anisotropic, non-magnetic matter, no free charges/currents)02202=∂∂−∇tEEεµ0=⋅∇ E0=⋅∇ Bt∂∂−=×∇BEt∂∂=×∇EBεµ0201c≡εµwave velocity012222=∂∂−∇tcEEMIT 2.71/2.71010/12/05 wk6-b-18Light velocity and refractive index 12vacuum00c≡εµ()0201εεχεn≡+=22vacuum201ccn≡=εµn: index of refractioncvacuum: speed of lightin vacuumc≡cvacuum/n: speed of lightin medium of refr. index nMIT 2.71/2.71010/12/05 wk6-b-19Simplified (1D, scalar) wave equation0122222=∂∂−∂∂tEczE• E is a scalar quantity (e.g. the component Eyof an electric field E)• the geometry is symmetric in x, y ⇒ the x, yderivatives are zeroSolution:⎟⎠⎞⎜⎝⎛++⎟⎠⎞⎜⎝⎛−= tczgtczftzE ),(zztt+∆ttcz∆=∆0zzz∆+0⎟⎠⎞⎜⎝⎛−tczf0MIT 2.71/2.71010/12/05 wk6-b-20Special case: harmonic solutionλzzt=0t+∆t⎟⎠⎞⎜⎝⎛==λπzatzf 2cos)0,(()⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛−≡⎟⎠⎞⎜⎝⎛−= tczactzatzfπνλπ2cos2cos),(λν=cdispersionrelationΦλπct2=∆ΦMIT 2.71/2.71010/12/05 wk6-b-21Complex representation of waves()() ( )() ( )()()()() (){}() ()()phasor""or amplitudecomplex etionrepresenta complex e,ˆ aka ,ˆ,ˆRe, i.e.sincos,ˆcos,2 ,2 cos, 22cos,φφωφωφωφωπνωλπφωφπνλπitkziAAtzftzftzftzftkziAtkzAtzftkzAtzfktkzAtzftzAtzf−−−==−−+−−=−−===−−=⎟⎠⎞⎜⎝⎛−−=wave-numberangular frequencyMIT 2.71/2.71010/12/05 wk6-b-22Time reversalz()uf()tkzfω−z()uf()tkzfω+MIT 2.71/2.71010/12/05 wk6-b-23Superpositionz()uf()tkzfω−()uf()tkzfω+()()tkzftkzfωω++−⇒is also a solutionMore generally,()()tkzgtkzfωω++−is a solutionEven more