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

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 2

3. 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

4. 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:

5. 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

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 6

7. 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:

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):
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 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

12. 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!

13. 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

14. 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