02/21/2014PHY 712 Spring 2014 -- Lecture 16-171 PHY 712 Electrodynamics 9-9:50 & 10-10:50 AM Olin...

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 71. Plane polarized electromagnetic waves

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

3. Frequency dependence of dielectric materials; Drude model

4. 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 H

E

EB

BE

0 B

Maxwell’s equations


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


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



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



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


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


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



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



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



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



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



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



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


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


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


Multilayer dielectrics (Problem #7.2)

n1 n2n3

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 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


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


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 ( ) 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


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


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:

dr



rrEr 200 mmeqm ti

02/21/2014

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

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

Note that: g > 0 represents dissipation of energy. w0 represents the natural frequency of

the vibration; w0=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 f

N

fN

i

iii

iii

prrpP

PEED




drhttp://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

ii

ti

t bb

b

00

20

0

1

:details Some

EE

ED

JH

EJED


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

z-α

f(z)dz

πifzfdz

zf

includes

2

1 0)(

:)(function analytican for formula integral sCauchy'Consider

Re(z)

Im(z)

a


Kramers-Kronig transform -- continued

z-α

f(z)dz

-αz

)f(zdz

πi

z-α

f(z)dz

πif

restR

RR

includes

2

1

2

1

Re(z)

Im(z)

a

=0

f-αz

)f(zdzP

πi

-αz

)f(zdz

πi f

R

RR

R

RR )(

2

1

2

1

2

1



-αz

)(zfdzPf

-αz

)(zfdzPf

-αz

zifzfdzP

πiiff

zifz fz f

f-αz

)f(zdzP

πi f

R

RRRI

R

RIRR

R

RIRRRIR

RIRRR

R

RR

1

1

2

1

2

1

: Suppose

)(2

1

2

1



-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

zzf

zzf

zf

I

R


au

ubub

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

zzzf

zzf

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

0

0

1

, ,,

:fields and between iprelationsh Causal""

ieGd

tGdtt rErErD

DE

02/21/2014PHY 712 Spring 2014 -- Lecture 16-171 PHY 712 Electrodynamics 9-9:50 & 10-10:50 AM Olin...

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Transcript of 02/21/2014PHY 712 Spring 2014 -- Lecture 16-171 PHY 712 Electrodynamics 9-9:50 & 10-10:50 AM Olin...