ELECTROMAGNETICS AND APPLICATIONS Lecture 7 TE and TM Reflections Brewster Angle Luca Daniel
Feb 05, 2016
ELECTROMAGNETICS AND APPLICATIONS
Lecture 7TE and TM Reflections
Brewster Angle
Luca Daniel
L7-2
• Review of Fundamental Electromagnetic Laws
• Electromagnetic Waves in Media and Interfaceso The EM waves in homogenous Mediao Electromagnetic Power and Energyo EM Fields at Interfaces between Different Mediao EM Waves Incident “Normally” to a Different Mediumo 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’s Outline
TodayToday
L7-3
Wave Front Shapes at Boundaries (Case kt<ki)
Standard refraction: i < c
Beyond the critical angle, i > c: Total reflection & evanescence
i
Glass
Phase fronts
Airt o
oz
glass
zLines of
constant phase
t = 90°
i > c
o = oz
z
glass
ix
“Phase Matching” at boundary
Glass
L7-4
Total Reflection and Evanescent Waves
Since:
Therefore:2 2
tx tz tk j j k k
When > c, ktz > kt and:
where:
2 2 2 2tz i i iiz2 2t t t
k k sin
k
Fields when > c:
tx tz tzjk x jk z x jk zt 0 0ˆ ˆE y T E e y T E e (x < 0)
2 2 2 0tx t tzk k k
2 2 2 2t tx tz t tk k k
z
x
ki i
ki
kt
t
kiz>kt
r
e.g., glass
e.g., air
xt 0 tzˆE (t,z) y T E e cos( t-k z) (x < 0)
L7-5
Standard refraction: i < c
Beyond the critical angle, i > c: Total reflection & evanescence
i
Glass
Phase fronts
Airt o
oz
glass
zLines of
constant phase
t = 90°
i > c
o = oz
zevanescent
region
Lines of constant amplitude
glass
ix
ex
“Phase Matching” at boundary
Total Reflection and Evanescent Waves
L7-6
• Review of Fundamental Electromagnetic Laws
• Electromagnetic Waves in Media and Interfaceso The EM waves in homogenous Mediao Electromagnetic Power and Energyo EM Fields at Interfaces between Different Mediao EM Waves Incident “Normally” to a Different Mediumo 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’s Outline
TodayToday
L7-7
Case 1: TE Wave
ix iz
ix iz
tx tz
jk x jk zi o
jk x - jk zr o
+jk x - jk zt o
ˆIncident: E y E e
ˆReflected: E y E e
ˆTransmitted: E y T E e
“Transverse Electric”
E Plane of incidencex
i
r
t
zy
kix
kiz
i,i
t,t
iH
iE
i i ik
tEtk
i
rE
Trial Solutions:
ik
kz
rkE
ik
TE UPW At Planar Boundary
E tangential to boundaryx
y z
L7-8
ix iz
ix iz
ix iz
tx tz
jk x jk zi o
jk x jk zoi i i
ijk x jk zo
r i ii
+jk x - jk zot t t
t
ˆIncident: E yE e
Eˆ ˆ H ( x sin zcos )e
Eˆ ˆReflected: H ( x sin zcos )e
TEˆ ˆTransmitted: H ( x sin zcos ) e
TE Wave: H at Boundary
Case 1: TE Wave
x
i i
t
zy,t,t
iH
iE
tH
i
rE
ik
rH
i
tE
L7-9
tx tzix iz ix iz +jk x - jk zjk x - jk z jk x - jk zo o oˆ ˆ ˆyE e y E e y T E e
Impose Boundary Conditions
are continuous at x = 0: and E H i r t i r tE + E = E , and H H H for all y and z.
0 0
Continuity of tangential H at x=0 for all z:
TEiTEt
1 T
0
1 T
Continuity of tangential E at x=0 for all z (last time):
iz tzfor all z k klast time: phase matching
TE tt
t
TE ii
i
cos
cos
where
tx tzix iz ix iz +jk x - jk zjk x jk z jk x jk zo o oi i t
i i t
E E TEˆ ˆ ˆzcos e zcos e zcos e
L7-10
TEiTE
t
1 T
1 T
TEt
TE TEt i
TE TEt iTE TEt i
2T
Solving yields
TE Reflection and Transmission Coefficients
Check special case normal incidence: i = 0, cosi = 1, t = 0, cost = 1
t i
it i
( 0)
We found:
TE tt
t
TE ii
i
cos
cos
where
L7-11
TM Wave at Interface
Option A: Repeat method for TE (write field expressions with unknown and T; impose boundary conditions; solve for and T)
Case 2: TM Wave
x
i i
t
zyi,i
t,t
iEiH
tH
rE
ikrH
tE
Option B: Use duality to map TE solution to TM case
Any incoming UPW can be decomposed into TE and TM components
L7-12
Duality of Maxwell’s Equations
TMTE
TE TM
TE TMTE TM
TE TM
TE TM
EHE H
t t
E H H Et t
E 0 H 0
H 0 E 0
TE TM
TE TM
E H
H E
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!
L7-13
TE 1where cos
Duality: TM Wave Solutions
For TE waves we found:
TM 1 1where cos
For TM waves :
Zero reflection at Brewster’s Angle for TM
i i t t
i i t t
cos cos
k sin k sin
t ifor
TM TMTM t i
TM TMt i
TE TETE t i
TE TEt i
1Duality swaps
2
ti
i
i t
tan
rE
rH
z
x
i i
t
y
i,i
t,t
iEiH
tH
ik
tE
90o
L7-14
Brewster Angle (no reflection, total transmission)
090o
TM
TE1
Brewster’s angle B
090o
TE
TM
1
-1
090o
TM
TE1
Brewster’s angle B
Critical angle
No reflection at B
Laser beam
Brewster angle window Water/snow
Horizontally polarized glasses cut glare