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Reflection/Transmission

Reflection/Transmission.  2 E = me ∂ 2 E/∂t 2. Wave equation for fields in Free space.  2 B = me ∂ 2 B/∂t 2. Recap: EM wave equations. What about potentials?. Ans: Poisson ’ s equations become wave equations… … with an obvious twist. .A + 1/c 2 V/t = 0. .B = 0

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Reflection/Transmission

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  1. Reflection/Transmission

  2. 2E = me∂2E/∂t2 Wave equation for fields in Free space 2B = me∂2B/∂t2 Recap: EM wave equations What about potentials? Ans: Poisson’s equations become wave equations… … with an obvious twist

  3. .A + 1/c2V/t = 0 .B = 0  x B = J + me∂E/∂t B =  x A E = -V - ∂A/∂t Choosing the reference for A such that Back to Maxwell Eqns .E = -r/e  x E = - ∂B/∂t Potentials Plug above,

  4. (2V - me∂2V/∂t2) = -r/e (2A - me∂2A/∂t2) = -mJ Wave equation for potentials (extensions of Poisson) Wave equations for potentials PROVIDED ..

  5. From Poisson to waves Recall 2V = -r/e Solution V(r) = 1/4pe ∫dv’/r x rv(r’) Also recall for waves (2V - me∂2V/∂t2) = -r/e Solution must be function of t-r/c, where c = 1/√me

  6. Consequence of speed of light V(r,t) = 1/4pe ∫dv’/r x rv(r ’,t-r/c) Delay due to finite speed of light Dt = r/c V(r’ + r,t’+Dt) = V(r,t) r(r’,t’) Dt = r/c So t’ = t-Dt = t-r/c c=1/me

  7. Consequence of speed of light Just familiar Poisson solutions with time-delay

  8. E = E0ej(wt-b.r) •  -jb ∂/∂tjw 2E = me∂2E/∂t2 Wave equation l Solution in free space: Plane Wave Wavefront: Points of equal phase A snapshot at fixed time t gives b.r = bxx + byy + bzz = constant This is the equation to a plane with direction cosines (bx,by, bz) b=w/v

  9. E = E0ej(wt-b.r) •  -jb ∂/∂tjw As time varies, the constant in the equation to the plane varies, and the wavefront moves to the right l Plane Wave The next wavefront lags behind by one wavelength b=w/v

  10. E = ER/R.ej(wt-bR) 2E = me∂2E/∂t2 Solution in free space: Spherical Wave 1/Rd2/dR2(RER) = me∂2ER/∂t2 ie, d2/dR2(RER) = me∂2(RER)/∂t2 ‹ Thus, (RER) satisfies a 1-D plane wave

  11. E = ER/R.ej(wt-bR) Wavefront: Points of equal phase A snapshot at fixed time t gives b.R = constant = C ie, R2 = (x2+y2+z2) = C2/b2 This is the equation to a sphere with radius (C/b) As time varies, the radius of the sphere increases Spherical Wave ‹

  12. Polarization Ex = axcos(wt-kz) Ey = aycos(wt-kz + d) d = p d = -p/2 d = 0 d = p/2 y y y y x x x x x,y rise and fall together x,-y rise and fall together As x decreases -y increases As x decreases y increases

  13. Linear Polarization d = 0 Ex = axcos(wt-kz) Ey = aycos(wt-kz) Ey= mEx where m = ay/ax y x x,y rise and fall together

  14. Linear Polarization d = p Ex = axcos(wt-kz) Ey = -aycos(wt-kz) Ey= mEx where m = -ay/ax y x x,-y rise and fall together

  15. Left Circular Polarization • = p/2 ax = ay = a Ex = acos(wt-kz) Ey = asin(wt-kz) Ex2 + Ey2 = a2 y x Set z=0 and track E Ex = acos(wt) ~ a for small t Ey = asin(wt) ~awt for small t As t increases, Ex remains same, Ey becomes more positive Curls along right fingers if thumb is along propagation z  RCP As x decreases -y increases

  16. Right Circular Polarization • = -p/2 ax = ay = a Ex = acos(wt-kz) Ey = -asin(wt-kz) Ex2 + Ey2 = a2 y Set z=0 and track E Ex = acos(wt) ~ a for small t Ey = -asin(wt) ~-awt for small t As t increases, Ex remains same, Ey becomes more negative Curls along left fingers if thumb is along propagation z  LCP x As x decreases y increases

  17. Circular Polarization Projection of sinusoids onto a 2-d circle: RCP Projection of 2-d circle into 3-d helix: LCP

  18. Polarizer and Analyzer

  19. Polarizer and Analyzer

  20. Polarizer and Analyzer Analyzer Polarizer

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