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