MIT 2.71/2.71010/17/05 wk7-a-1Elementary waves: plane, sphericalMIT 2.71/2.71010/17/05 wk7-a-2The EM vector wave equation2222222zyx ∂∂+∂∂+∂∂=∇zyxEˆˆˆzyxEEE++=0101012z222z22z22z22y222y22y22y22x222x22x22x2=∂∂−∂∂+∂∂+∂∂=∂∂−∂∂+∂∂+∂∂=∂∂−∂∂+∂∂+∂∂tEczEyExEtEczEyExEtEczEyExE012222=∂∂−∇tcEExEzEyEEMIT 2.71/2.71010/17/05 wk7-a-3Harmonic solution in 3D: plane wavezyxΦ=0Φ=2πΦ=4πΦ=6πE(z=0,t=0)E(z=λ/2,t=0)E(z=λ,t=0)E(z=3λ/2,t=0)E(z=5λ/2,t=0)E(z=2λ,t=0)E(z=3λ,t=0)phase=constanton the planet=0MIT 2.71/2.71010/17/05 wk7-a-4Plane wave propagatingzyxΦ=0Φ=2πΦ=4πΦ=6πE(z=0,t=∆t)E(z=λ/2,t=∆t)E(z=λ,t=∆t)E(z=3λ/2,t=∆t)E(z=5λ/2,t=∆t)E(z=2λ,t=∆t)E(z=3λ,t=∆t)phase=constanton the planet=∆tc∆tpropagationdirectionMIT 2.71/2.71010/17/05 wk7-a-5Complex representation of 3D waves() ()()()()()()()()const.,, surface : Wavefront"",, wherephasor""or amplitudecomplex etionrepresenta complex e,,,ˆetc. ,cos2 cos,,, coscoscos2cos,,,0,,000=−++≡==−−++=⎟⎠⎞⎜⎝⎛−−++=−−−++zyxzkykxkzyxAAtzyxfktzkykxkAtzyxftzyxAtzyxfzyxzyxitzkykxkixzyxzyxφφφαλπφωφωγβαλπφφωMIT 2.71/2.71010/17/05 wk7-a-6Plane wavekkxkykzxyzwave-vectorMIT 2.71/2.71010/17/05 wk7-a-7Plane wavekkxkykzxyzwave-vector()()relation)n (dispersio iffequation wavesolves ˆˆˆ vector)coordinate (Cartesian ˆˆˆe 0ckkkzyxAazyxtiωω=++=++==−⋅kzyxkzyxrrrkMIT 2.71/2.71010/17/05 wk7-a-8Plane wavekkxkykzxyz()()plane a isfront - waveconst. :condition phaseconstant ˆˆˆ vector)coordinate (Cartesian ˆˆˆe 0⇒=−⋅++=++==−⋅tkkkzyxAazyxtiωωrkzyxkzyxrrrkconst.by described surface:wavefront""=⋅rkMIT 2.71/2.71010/17/05 wk7-a-9Plane wave propagatingκΦ=0Φ=2πΦ=4πΦ=6πE(κ=0,t=0)E(κ=λ/2,t=0)E(κ=λ,t=0)E(κ=3λ/2,t=0)E(κ=5λ/2,t=0)E(κ=2λ,t=0)E(κ=3λ,t=0)phase=constanton the planet=0const. plane=⋅rkwave-vectordirectionMIT 2.71/2.71010/17/05 wk7-a-10Plane wave propagatingyxΦ=0Φ=2πΦ=4πΦ=6πE(κ=0,t=∆t)E(κ=λ/2,t=∆t)E(κ=λ,t=∆t)E(κ=3λ/2,t=∆t)E(κ=5λ/2,t=∆t)E(κ=2λ,t=∆t)E(κ=3λ,t=∆t)t=∆tc∆tpropagationdirectionphase=constanton the planeconst. plane=⋅rkwave-vectordirectionκMIT 2.71/2.71010/17/05 wk7-a-11Spherical wave“point”sourceOutgoingraysOutgoingwavefrontsequation of wavefrontconstant=− tkRωR()RtkRAtzyxa2/cos),,,(πω+−=(){}iRtkRiAtzyxa exp),,,(ω−=⎭⎬⎫⎩⎨⎧++=zyxiziiRAzyxaλπλπ22 2 exp),,(exponentialnotationparaxialapproximationMIT 2.71/2.71010/17/05 wk7-a-12Spherical wave“point”sourceΦ=2πΦ=4πΦ=6πspherical wavefronts“point”sourceΦ=2πΦ=4πΦ=6πparabolic wavefrontsexactexactparaxial approximation/paraxial approximation//Gaussian beams/Gaussian beamsMIT 2.71/2.71010/17/05 wk7-a-13The role of lenses“point”image“point”sourcespherical wave(divergent)plane waveplane wave spherical wave(convergent)MIT 2.71/2.71010/17/05 wk7-a-14The role of lenses“point”image“point”sourceplane waveplane wavespherical wave(divergent)spherical wave(convergent)MIT 2.71/2.71010/17/05 wk7-a-15PolarizationMIT 2.71/2.71010/17/05 wk7-a-16Propagation and polarizationxyzkwave-vectorEelectric field vectorconst.efrontplanar wav=⋅rk) toparallelnot have couldone crystals, e.g.media, canisotropiin :(reminder0generally, More i.e.0etc.) glass, amorphousspace, free (e.g.media isotropicIn DEDkEkEk=⋅⊥=⋅MIT 2.71/2.71010/17/05 wk7-a-17Linear polarization (frozen time)zyxΦ=0Φ=2πE(z=0,t=0)E(z=λ/2,t=0)E(z=λ,t=0)phase=constanton the planet=0MIT 2.71/2.71010/17/05 wk7-a-18Linear polarization (fixed space)tyxΦ=0Φ=2πE(z=0,t=0)E(z=0,t=π/ω)E(z=0,t=2π/ω)phase=constanton the planez=0MIT 2.71/2.71010/17/05 wk7-a-19Circular polarization (frozen time)zyxΦ=0Φ=2πE(z=0,t=0)E(z=λ/2,t=0)E(z=λ,t=0)phase=constanton the planet=0MIT 2.71/2.71010/17/05 wk7-a-20Circular polarization: linear components+ExEyzzMIT 2.71/2.71010/17/05 wk7-a-21Circular polarization (fixed space)zyxE(z=0,t=0)E(z=0,t=π/ω)E(z=0,t=π/2ω)E(z=0,t=3π/2ω)rotationdirectionMIT 2.71/2.71010/17/05 wk7-a-22zλ/4 platebirefringentλ/4 plateLinearLinearpolarizationpolarizationCircularCircularpolarizationpolarizationMIT 2.71/2.71010/17/05 wk7-a-23zλ/2 platebirefringentλ/2 plateLinearLinearpolarizationpolarizationLinear (90Linear (90oo--rotated)rotated)polarizationpolarizationMIT 2.71/2.71010/17/05 wk7-a-24Think about thatbirefringentλ/4 plateLinearLinearpolarizationpolarizationzmirrorIncomingIncoming????????OutgoingOutgoingMIT 2.71/2.71010/17/05 wk7-a-25Relationship between E and BEBk()EkBkxEBErk×=⇒−≡∂∂×≡∇×⇒=∂∂−=×∇−⋅ωωω1 and eˆ where0itiEttiVectors k, E, B form aright-handed triad.Note:free space or isotropic media