PHY 712 Spring 2015 -- Lecture 16 102/20/2015

PHY 712 Electrodynamics9-9:50 AM Olin 103

Plan for Lecture 16:

Read Chapter 71. Plane polarized electromagnetic waves

2. Reflectance and transmittance of electromagnetic waves – extension to anisotropy and complexity

PHY 712 Spring 2015 -- Lecture 16 202/20/2015

PHY 712 Spring 2015 -- Lecture 16 302/20/2015

For linear isotropic media and no sources: ;

Coulomb's law: 0

Ampere-Maxwell's law: 0

Faraday's law: 0

No magnetic monopoles:

t

t

D E B H

E

EB

BE

0 B

Maxwell’s equations

PHY 712 Spring 2015 -- Lecture 16 402/20/2015

0 2

22

2

t

tt

BB

EB

EB

Analysis of Maxwell’s equations without sources -- continued:

0 :monopoles magnetic No

0 :law sFaraday'

0 :law sMaxwell'-Ampere

0 :law sCoulomb'

B

BE

EB

E

t

t

0 2

22

2

t

tt

EE

BE

BE

PHY 712 Spring 2015 -- Lecture 16 502/20/2015

2

20022

2

2

22

2

2

22

where

01

01

n

ccv

tv

tv

EE

BB

Analysis of Maxwell’s equations without sources -- continued: Both E and B fields are solutions to a wave equation:

tiitii etet rkrk ErEBrB 00 , ,

:equation wave tosolutions wavePlane

PHY 712 Spring 2015 -- Lecture 16 602/20/2015

Analysis of Maxwell’s equations without sources -- continued:

00

222

00

where

, ,

:equation wave tosolutions wavePlane

nc

n

v

etet tiitii

k

ErEBrB rkrk

Note: , e m, n, k can all be complex; for the moment we will assume that they are all real (no dissipation).

0ˆ and 0ˆ :note also

ˆ

0 :law sFaraday' from

t;independennot are and that Note

00

000

00

BkEk

EkEkB

BE

BE

c

n

t

PHY 712 Spring 2015 -- Lecture 16 702/20/2015

Analysis of Maxwell’s equations without sources -- continued:

0ˆ and where

, ˆ

,

: wavesneticelectromag plane ofSummary

000

222

00

Ekk

ErEEk

rB rkrk

nc

n

v

etec

nt tiitii

220

0

2

0

Poynting vector and energy density:

1ˆ ˆ2 2

1

2

avg

avg

n

c

u

ES k E k

E

E0

B0

k

PHY 712 Spring 2015 -- Lecture 16 802/20/2015

Reflection and refraction of plane electromagnetic waves at a plane interface between dielectrics (assumed to be lossless)

m’ e’

m e

k’

kikR

i R

q

PHY 712 Spring 2015 -- Lecture 16 902/20/2015

Reflection and refraction -- continued

m’ e’

me

k’ki kRi R

q

tt

c

nt

et

ttc

nt

et

RRRRR

ctniRR

iiiii

ctniii

Rc

ic

,ˆ,ˆ,

,

,ˆ,ˆ,

,

: mediumIn

ˆ

0

ˆ

0

rEkrEkrB

ErE

rEkrEkrB

ErE

rk

rk

tt

c

nt

et ctni c

,'ˆ'',''ˆ','

','

:'' mediumIn 'ˆ'

0

rEkrEkrB

ErE rk

PHY 712 Spring 2015 -- Lecture 16 1002/20/2015

Reflection and refraction -- continued

inn

inxxnnn

Riix

x

yx

i

Ri

sinsin' : law sSnell'

sinsin' ˆ'ˆ'

ˆsinˆ

sin'ˆ

ˆ0ˆˆ

:planeboundary at factors phase matching -- law sSnell'

rkrk

rkrk

rk

zyxr

m’ e’

me

k’ki kRi R

q

z

x

PHY 712 Spring 2015 -- Lecture 16 1102/20/2015

Reflection and refraction -- continued

continuous ˆ ,ˆ

continuous ˆ ,ˆ

:planeboundary at amplitudes field Matching

0 0

0 0

:sources noith boundary wat equations Continuity

zEzH

zBzD

BE

DH

BD

tt

m’ e’

me

k’ki kRi R

q

z

x

PHY 712 Spring 2015 -- Lecture 16 1202/20/2015

Reflection and refraction -- continued

zEk

zEkEk

zB

zEzEE

zD

ˆ''ˆ'

ˆˆˆ

:continuous ˆ

ˆ''ˆ

:continuous ˆ

:planeboundary at amplitudes field Matching

0

00

000

i

RRii

Ri

n

n

m’ e’

me

k’ki kRi R

q

z

x

zEkzEkEk

zH

zEzEE

zE

ˆ''ˆ'

' ˆˆˆ

:continuous ˆ

ˆ'ˆ

:continuous ˆ

000

000

iRRii

Ri

nn

PHY 712 Spring 2015 -- Lecture 16 1302/20/2015

Reflection and refraction -- continued

m’ e’

me

k’ki kRi R

q

z

x

s-polarization – E field “polarized” perpendicular to plane of incidence

zEkzEkEk

zH

zEzEE

zE

ˆ''ˆ'

' ˆˆˆ

:continuous ˆ

ˆ'ˆ

:continuous ˆ

000

000

iRRii

Ri

nn

sin'cos' : thatNote

cos''

cos

cos2'

cos''

cos

cos''

cos

222

0

0

0

0

innn

nin

in

E

E

nin

nin

E

E

ii

R

PHY 712 Spring 2015 -- Lecture 16 1402/20/2015

Reflection and refraction -- continued

m’ e’

me

k’ki kRi R

q

z

x

p-polarization – E field “polarized” parallel to plane of incidence

sin'cos' : thatNote

coscos''

cos2'

coscos''

coscos''

222

0

0

0

0

innn

nin

in

E

E

nin

nin

E

E

ii

R

zEkzEkEk

zH

zEzEE

zD

ˆ''ˆ'

' ˆˆˆ

:continuous ˆ

ˆ''ˆ

:continuous ˆ

000

000

iRRii

Ri

nn

PHY 712 Spring 2015 -- Lecture 16 1502/20/2015

Reflection and refraction -- continued

m’ e’

me

k’ki kRi R

q

z

x

1TR that Note

cos

cos

'

''ˆ

ˆ'T

ˆ

ˆR

:ance transmitte,Reflectanc2

0

0

2

0

0

in

n

E

E

E

E

iii

R

i

R

zS

zS

zS

zS

PHY 712 Spring 2015 -- Lecture 16 1602/20/2015

sin'cos' : thatNote

cos''

cos

cos2'

cos''

cos

cos''

cos

222

0

0

0

0

innn

nin

in

E

E

nin

nin

E

E

ii

R

For s-polarization

sin'cos' : thatNote

coscos''

cos2'

coscos''

coscos''

222

0

0

0

0

innn

nin

in

E

E

nin

nin

E

E

ii

R

For p-polarization

PHY 712 Spring 2015 -- Lecture 16 1702/20/2015

Special case: normal incidence (i=0, q=0)

nn

n

E

E

nn

nn

E

E

ii

R

''

2'

''

''

0

0

0

0

'

'

''

2

'

''T

''

''

R

:ance transmitte,Reflectanc

2

2

0

0

2

2

0

0

n

n

nn

n

n

n

E

E

nn

nn

E

E

i

i

R

PHY 712 Spring 2015 -- Lecture 16 1802/20/2015

Multilayer dielectrics (Problem #7.2)

n1 n2n3

ki

kR

ktkb

ka

d

PHY 712 Spring 2015 -- Lecture 16 1902/20/2015

Extension of analysis to anisotropic media --

PHY 712 Spring 2015 -- Lecture 16 2002/20/2015

Consider the problem of determining the reflectance from an anisotropic medium with isotropic permeability m0 and anisotropic permittivity e0 k where:

By assumption, the wave vector in the medium isconfined to the x-y plane and will be denoted by

The electric field inside the medium is given by:

0 0

0 0

0 0

xx

yy

zz

κ

and ˆ ˆ( ), where are to be determine d.t yx y xn n n nc

k x y

( )ˆ ˆ ˆ( )e .

x yi n x n y i tc

x y zE E E

E x y z

PHY 712 Spring 2015 -- Lecture 16 2102/20/2015

Inside the anisotropic medium, Maxwell’s equations are:

After some algebra, the equation for E is:

From E, H can be determined from

0 0

0 0

0 0i i H κ E

E H H κ Eò

2

2

2 2

0

0 0.

0 0 ( )

xx y x y x

x y yy x y

zz x y z

n n n E

n n n E

n n E

( )

0

1ˆ ˆ ˆ( ) ( ) e .

x yi n x n y i tc

z y x y x x yE n n E n E nc

H x y z

PHY 712 Spring 2015 -- Lecture 16 2202/20/2015

The fields for the incident and reflected waves are the same as for the isotropic case.

Note that, consistent with Snell’s law:Continuity conditions at the y=0 plane must be applied for the following fields:

There will be two different solutions, depending of the polarization of the incident field.

ˆ ˆ(sin cos ),

ˆ ˆ(sin cos ).

i

R

i ic

i ic

k x y

k x y

sinxn i

( ,0, , ), ( ,0, , ), ( ,0, , ), and ( ,0, , ).x z yx z t E x z t E x z t D x z tH

PHY 712 Spring 2015 -- Lecture 16 2302/20/2015

Solution for s-polarization

2 2

( ) ( )

0

0

1ˆ ˆ ˆe ( ) e

x y x y

x y y zz x

i n x n y i t i n x n y i tc c

z z y x

E E n n

E E n nc

E z H x y

0 0 0 0 0 0

must be determined from the continuity conditions:

( ) cos ( ) in s

z

z z y z xE E E E E i E n E E i E

E

n

0

0

cos.

cosy

y

i nE

E i n

PHY 712 Spring 2015 -- Lecture 16 2402/20/2015

Solution for p-polarization2 2

( )

( )

0

0 ( ).

ˆ ˆ e .

ˆe .

x y

x y

xxy yy x

yy

i n x n y i txx x c

xyy y

i n x n y i tx xx c

z

y

E n n

nE

n

E

c n

E x y

H z

0 0 0 0 0 0( ) cos ( ) ( )sin .

must be determi

ned from the continuity conditions:

x

xx xx xx x x

y y

nE E i E E E E E E i E

n n

E

0

0

cos.

cosxx y

xx y

i nE

E i n

PHY 712 Spring 2015 -- Lecture 16 2502/20/2015

Extension of analysis to complex dielectric functions

rkrkrk EErE

ˆˆ

0

ˆ

0

2/122

2/122

2

0

0

,

2

2

:complex isfunction dielectric theSuppose

that assume simplicityFor

IcRcc nctnictni

IR

IRIR

eeet

nn

iinni