Lecture 7 TE and TM Reflections Brewster Angle

1. Lecture 7TE and TM ReflectionsBrewster Angle 6.013 ELECTROMAGNETICS AND APPLICATIONS Luca Daniel

2. 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

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

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

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

6. 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

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

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

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

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

11. TM Wave at Interface 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

12. Duality of Maxwell’s Equations Then we also have a solution to these equations If we 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!

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

14. |G|2 Critical angle |G|2 1 1 TE TE TM TM q q 0 0 90o 90o Brewster’s angle qB Brewster’s angle qB Brewster Angle (no reflection, total transmission)  1 TM q 0 90o TE -1 Horizontally polarized glasses cut glare Laser beam Brewster angle window Water/snow Noreflection at qB