Lecture 7 TE and TM Reflections Brewster Angle 6.013 ELECTROMAGNETICS AND APPLICATIONS Luca Daniel Lecture 7 TE and TM Reflections Brewster Angle

Today’s Outline Review of Fundamental Electromagnetic Laws Electromagnetic Waves in Media and Interfaces The EM waves in homogenous Media Electromagnetic Power and Energy EM Fields at Interfaces between Different Media EM Waves Incident “Normally” to a Different Medium EM Waves Incident at General Angle UPW in arbitrary direction TE wave at planar interface Phase Matching and Snell’s Law Critical Angle Total Reflection and Evanescent Waves Reflection and Transmission Coefficients Duality TM wave at planar interface No Reflection - Brewster Angle Digital & Analog Communications Today 2

Wave Front Shapes at Boundaries (Case kt<ki) Standard refraction: i < c “Phase Matching” at boundary i Phase fronts glass Glass z Air oz Lines of constant phase t o Beyond the critical angle, i > c: Total reflection & evanescence i x i > c glass Glass z t = 90° o = oz

Total Reflection and Evanescent Waves When qi > qc, ktz > kt and: x e.g., glass Since: Therefore: ki ki i r z e.g., air kiz>kt t kt where: Fields when q > qc:

Total Reflection and Evanescent Waves Standard refraction: i < c “Phase Matching” at boundary i Phase fronts glass Glass z Air oz Lines of constant phase t o Beyond the critical angle, i > c: Total reflection & evanescence i x glass i > c z ex Lines of constant amplitude evanescent region t = 90° o = oz

Today’s Outline Review of Fundamental Electromagnetic Laws Electromagnetic Waves in Media and Interfaces The EM waves in homogenous Media Electromagnetic Power and Energy EM Fields at Interfaces between Different Media EM Waves Incident “Normally” to a Different Medium EM Waves Incident at General Angle UPW in arbitrary direction TE wave at planar interface Phase Matching and Snell’s Law Critical Angle Total Reflection and Evanescent Waves Reflection and Transmission Coefficients Duality TM wave at planar interface Brewster Angle Digital & Analog Communications Today 6

TE UPW At Planar Boundary Case 1: TE Wave “Transverse Electric” x kiz i x kix r i i,i z y kz t,t y z t Trial Solutions:

TE Wave: H at Boundary Case 1: TE Wave x i i i i , z y t,t t

Impose Boundary Conditions are continuous at x = 0: Continuity of tangential E at x=0 for all z (last time): last time: phase matching Continuity of tangential H at x=0 for all z: where

TE Reflection and Transmission Coefficients We found: Solving yields where Check special case normal incidence: qi = 0, cosqi = 1, qt = 0, cosqt = 1

TM Wave at Interface Case 2: TM Wave Any incoming UPW can be decomposed into TE and TM components Case 2: TM Wave x i i i,i z y t,t t Option A: Repeat method for TE (write field expressions with unknown G and T; impose boundary conditions; solve for G and T) Option B: Use duality to map TE solution to TM case

Duality of Maxwell’s Equations If we have a solution to these equations Then we also have a solution to these equations Which we get by making these substitutions: Claim: the solutions to the second set of equations satisfy Maxwell’s Equations. Why? Because the second set of equations ARE Maxwell’s Equations... just reordered!

Duality: TM Wave Solutions For TE waves we found: For TM waves : Zero reflection at Brewster’s Angle for TM z x i t y i,i t,t 90o

Brewster Angle (no reflection, total transmission)  q |G|2 90o TM TE 1 Brewster’s angle qB q |G|2 90o TM TE 1 Brewster’s angle qB Critical angle 1 TM q 90o TE -1 gas laser beam: need glass window to keep the gas inside, but we want full transmission though the window: use brester angle Laser beam Horizontally polarized glasses cut glare Brewster angle window Water/snow No reflection at qB