phy 712 electrodynamics 9-9:50 & 10-10:50 am olin 107 plan for lecture 16-17: read chapter 7

PHY 712 Spring 2014 -- Lecture 16-17 102/21/2014

PHY 712 Electrodynamics9-9:50 & 10-10:50 AM Olin 107

Plan for Lecture 16-17:Read Chapter 7

1. Plane polarized electromagnetic waves2. Reflectance and transmittance of electromagnetic

waves – extension to anisotropy and complexity3. Frequency dependence of dielectric materials;

Drude model4. Kramers-Kronig relationships


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 HE

EB

BE

0 B

Maxwell’s equations


0 2

22

2

t

ttBB

EBEB

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

ttEE

BEBE


2

20022

2

2

22

2

2

22

where

01

01

nccv

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



00

222

00

where

, ,

:equation wave tosolutions wavePlane

nc

nv

etet tiitii

k

ErEBrB rkrk

Note: , , 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

cn

t



0ˆ and where

, ˆ

,

: wavesneticelectromag plane ofSummary

000

222

00

Ekk

ErEEkrB rkrk

nc

nv

etec

nt tiitii

220

0

20

Poynting vector and energy density:

1ˆ ˆ2 2

12

avg

avg

nc

u

ES k E k

E

E0

B0k


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

’ ’

k’

kikR

i R

q


Reflection and refraction -- continued

’ ’

k’ki kRi R

q

tt

cnt

et

ttcnt

et

RRRRR

ctniRR

iiiii

ctniii

Rc

ic

,ˆ,ˆ,

,

,ˆ,ˆ,

,

: mediumIn

ˆ

0

ˆ

0

rEkrEkrB

ErE

rEkrEkrB

ErE

rk

rk

tt

cnt

et ctni c

,'ˆ'',''ˆ','

','

:'' mediumIn 'ˆ'

0

rEkrEkrB

ErE rk



inninxxnnn

Riix

x

yx

i

Ri

sinsin' : law sSnell'sinsin' ˆ'ˆ'

ˆsinˆsin'ˆ

ˆ0ˆˆ :planeboundary at factors phase matching -- law sSnell'

qq

q

rkrk

rkrk

rk

zyxr

’ ’

k’ki kRi R

q

z

x



continuous ˆ ,ˆcontinuous ˆ ,ˆ

:planeboundary at amplitudes field Matching

0 0

0 0:sources noith boundary wat equations Continuity

zEzHzBzD

BEDH

BD

tt

’ ’

k’ki kRi R

q

z

x



zEk

zEkEk

zBzEzEE

zD

ˆ''ˆ'

ˆˆˆ:continuous ˆ

ˆ''ˆ:continuous ˆ

:planeboundary at amplitudes field Matching

0

00

000

i

RRii

Ri

n

n

’ ’

k’ki kRi R

q

z

x

zEkzEkEk

zHzEzEE

zE

ˆ''ˆ'' ˆˆˆ

:continuous ˆˆ'ˆ

:continuous ˆ

000

000

iRRii

Ri

nn



’ ’

k’ki kRi R

q

z

x

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

zEkzEkEk

zHzEzEE

zE

ˆ''ˆ'' ˆˆˆ

:continuous ˆˆ'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

inEE

nin

nin

EE

ii

R

q

qq

q



’ ’

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

inEE

nin

nin

EE

ii

R

q

qq

q

zEkzEkEk

zHzEzEE

zD

ˆ''ˆ'' ˆˆˆ

:continuous ˆˆ''ˆ

:continuous ˆ

000

000

iRRii

Ri

nn



’ ’

k’ki kRi R

q

z

x

1TR that Note

coscos

'''

ˆˆ'T

ˆˆ

R

:ance transmitte,Reflectanc2

0

0

2

0

0

in

nEE

EE

iii

R

i

R q

zSzS

zSzS


sin'cos' : thatNote

cos''

cos

cos2' cos'

'cos

cos''

cos

222

0

0

0

0

innn

nin

inEE

nin

nin

EE

ii

R

q

qq

q

For s-polarization

sin'cos' : thatNote

coscos''

cos2' coscos'

'

coscos''

222

0

0

0

0

innn

nin

inEE

nin

nin

EE

ii

R

q

qq

q

For p-polarization


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

nn

nEE

nn

nn

EE

ii

R

''

2' '

'

''

0

0

0

0

''

''

2'

''T

''

''R

:ance transmitte,Reflectanc

2

2

0

0

2

2

0

0

nn

nn

nnn

EE

nn

nn

EE

i

i

R


Multilayer dielectrics (Problem #7.2)

n1 n2 n3

ki

kR

ktkb

ka

d


Extension of analysis to anisotropic media --


Consider the problem of determining the reflectance from an anisotropic medium with isotropic permeability 0 and anisotropic permittivity 0 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 00 00 0

xx

yy

zz

kk

k

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 t

cx y zE E E

E x y z


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 00 0i i

H κ EE H H κ Eò

2

2

2 2

00 0.

0 0 ( )

xx y x y x

x y yy x y

zz x y z

n n n En n n E

n n E

kk

k

( )

0

1 ˆ 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


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


Solution for s-polarization

2 2

( ) ( )

0

0

1ˆ ˆ ˆe ( ) ex 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

k

E z H x y

0 0 0 0 0 0

must be determined from the continuity conditions:

( ) cos ( ) in sz

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

E

n

0

0

cos.

cosy

y

i nEE i n


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

Ec n

kk

k

kk

k

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

Ek k

0

0

cos.

cosxx y

xx y

i nEE i n

kk


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


Paul Karl Ludwig Drude 1863-1906

http://photos.aip.org/history/Thumbnails/drude_paul_a1.jpg

http://photos.aip.org/history/Thumbnails/drude_paul_a1.jpg

PHY 712 Spring 2014 -- Lecture 16-17 27

iii

ti

ti

ti

eim

qq

im

qe

mmeqm

rrpP

PEED

Erp

Errr

rrEr

3

0

220

02

220

000

200

:fieldnt Displaceme

1

:dipole Induced

1 ,For

02/21/2014

http://img.tfd.com/ggse/d6/gsed_0001_0012_0_img2972.png

Drude model: Vibrations of charged particles near equilibrium:

r



rrEr 200 mmeqm ti

02/21/2014

Drude model: Vibration of particle of charge q and mass m near equilibrium:

r http://img.tfd.com/ggse/d6/gsed_0001_0012_0_img2972.png

Note that: > 0 represents dissipation of energy. 0 represents the natural frequency of

the vibration; 0=0 would represent a free (unbound) particle



rrEr 200 mmeqm ti

02/21/2014



eim

qq

im

qe

ti

ti

220

02

220

000

1

:dipole Induced

1 ,For

Erp

Errr



rrEr 200 mmeqm ti

02/21/2014



dipoles typeoffraction umedipole/volnumber

:fieldnt Displaceme

3

0

i fN

fN

i

iii

iii

prrpP

PEED




rhttp://img.tfd.com/ggse/d6/gsed_0001_0012_0_img2972.png

i iii

ii

ti

ti

iii

iii

iii

im qfN e

eim

q q

fN

220

2

00

220

2

00

11

1

:typermittivifor expression model Drude

EE

Erp

pEPEED



Drude model dielectric function:

i ii

i

i

ii

I

i ii

i

i

ii

R

IR

i iii

ii

m qfN

m qfN

i

im qfN

222220

2

0

22222

22

0

2

0

00

220

2

0

1

11


R

0

I

0

Drude model dielectric function:


Drude model dielectric function – some analytic properties:

2

2

0

2

20

220

2

0

1

11 For

11

P

i i

iii

i iii

ii

m qfN

ω

im qfN


Drude model dielectric function – some analytic properties:

00

000

20

00

220

2

0

000

220

2

0

1 11

mass , charge of particle free a ing(represent 0For

11

i

im qiNf

im qfN

mq

im qfN

b

i iii

ii

i iii

ii

im qNf

iit

it bb

b

00

20

01

:details Some

EEEDJH

EJED


Analytic properties of the dielectric function (in the Drude model or from “first principles” -- Kramers-Kronig transform

z-αf(z)dz

πifzfdz

zf

includes

21 0)(

:)(function analytican for formula integral sCauchy'Consider

Re(z)

Im(z)


Kramers-Kronig transform -- continued

z-αf(z)dz

-αz)f(zdz

πi

z-αf(z)dz

πif

restR

RR

includes

21

21

Re(z)

Im(z)

=0

f-αz

)f(zdzPπi

-αz

)f(zdzπi

fR

RR

R

RR )(

21

21

21



-αz)(zfdzPf

-αz)(zfdzPf

-αzzifzfdzP

πiiff

zifz fz f

f-αz

)f(zdzPπi

f

R

RRRI

R

RIRR

R

RIRRRIR

RIRRR

R

RR

1

1

21

21

: Suppose

)(21

21



-αz)(zfdzPf

-αz)(zfdzPf

R

RRRI

R

RIRR

1

1

for vanishes 2.

0for analytic is 1. : thatshowMust

1 when

function dielectric for the useful is transformKronig-Kramers This

0

zzfzzf

zf

I

R


auubub

au

u-udu

u-udu

u-udu P

ugduugduugduP

s

ss

s

b

u s

u

a s

b

a s

b

u

u

a

b

a

s

s

s

s

lnlnln0

lim

1 1 0

lim1

:Example

)( )( 0

lim)(

:nintegratio parts Principal ionsconsiderat practical Some

a us bu


II

RR

: function dielectricFor ionsconsiderat practical More

*

0)(for for vanishes 2.

0for analytic is 1. : thatshowMust

1 of treatment the

justifyhich function w dielectric theproperties Analytic

0

I

I

zzzfzzf

zzf


Analysis for Drude model dielectric function:

zm qfN

zzf

z

izzm qfN

zzf

im qfN

i i

ii

i

i iii

ii

i iii

ii

largeat vanishes 1

For

11 Let

11

0

2

2

220

2

0

220

2

0


Analysis for Drude model dielectric function – continued -- Analytic properties:

0for analytic is 0 that Note 22

0at poles has

11

22

22

220

2

0

PP

ii

iP

iPPiP

i iii

ii

zzfz

iz

izzzzf

izzm qfN

zzf


Kramers-Kronig transform – for dielectric function:

IIRR

RI

IR

-dP

-dP

;with

'1 1''1

'1 ''11

00

00


Further comments on analytic behavior of dielectric function

00

00

1

, ,,

:fields and between iprelationsh Causal""

ieGd

tGdtt rErErD

DE

phy 712 electrodynamics 9-9:50 & 10-10:50 am olin 107 plan for lecture 16-17: read chapter 7

Documents

analysis of maxwells

plane of incidence

anisotropic medium

plane interface

extension of analysis

maxwells equations02212014phy

xy plane

anisotropic media