fi 3221 electromagnetic interactions in...

18/01/2016

1

Alexander A. Iskandar

Physics of Magnetism and Photonics

FI 3221 ELECTROMAGNETIC

INTERACTIONS IN MATTER

• Maxwell eq.

in vacuum

• Wave eq.

and its

solution

• Propagation

of energy

• Maxwell eq.

in matter

• Interface

phenomena

REVIEW OF

ELECTROMAGNETISM AND

WAVES

Alexander A. Iskandar Electromagnetic Interactions in Matter 2

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2

Main

M. Dressel and G. Gruner : Section 2.1 and 2.2 and 2.4

S.A. Maier : Section 1.1


REFERENCES

In vacuum, the governing equations of electro-

magnetism are the Maxwell equations:


MAXWELL’S EQUATIONS IN VACUUM

0 B

t

BE

0

E

t

EJB

000

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If there are no sources ( = 0 and = 0), taking the

curl of the last two equations and recall that :

we arrive at the wave equation,


WAVE EQUATION IN VACUUM AND ITS

SOLUTIONS

J

001 v01

2

2

2

2

B

E

tv

B

E

B

E

2

The monochromatic/harmonic free wave has the

general form of

with the spatial part satisfying the Helmholtz eq.

is called the propagation vector it points to the

direction to where the wave travels.



SOLUTIONS

tiexB

xE

txB

txE

)(

)(

),(

),(

0)(

)(22

xB

xEk

vk

v

vk ˆ

k

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The free wave equation admits solutions of the

following specific forms :

harmonic plane waves

harmonic spherical waves

Alexander A. Iskandar 7


SOLUTIONS

Electromagnetic Interactions in Matter

and are constant vectors.

Wavefront (position that has the same phase) is a

plane

8

rktieB

E

trB

trE

0

0

,

,

r

k

constrk

r

r

k

planar wavefront /

phase front

0E

0B

PLANE WAVE SOLUTIONS

Alexander A. Iskandar Electromagnetic Interactions in Matter

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Wavefront is a spherical surface

9

k

rktieB

E

rtrB

trE

0

01

,

,

constkr

r

r

r

SPHERICAL WAVE SOLUTIONS


In matter, the charge can be classified in two kinds

And the current can be decomposed into

Further, the Ohm’s law gives


ELECTROMAGNETIC FIELDS IN MATTER

boundfree Pbound

t

PMJbound

boundcond JJJ

EJcond

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Constitutive relations, expressing material responses


ELECTROMAGNETIC FIELDS IN MATTER

)1(0 eEPED 0,

)1(0 mHMHB 0,

,0 EP e

χe = electric susceptibility

= 0, in vacuum

,HM m

χm = magnetic susceptibility

= 0 (=0), in vacuum and non magnetic medium

The Maxwell’s equations in matter becomes


MAXWELL’S EQUATION IN MATTER

0 B

t

BE

t

DJH cond

freeD

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Consider the case with no free sources (free = 0 and

= 0), taking the curl of the last two

equations yielded :

For uniform medium : , yields


WAVE EQUATION IN MATTER

0

H

E

01

2

2

2

2

H

E

tv

1v wave propagation speed

H

E

H

E

2

condJ

For non-uniform medium :

Refractive index is defined the ratio of propagation

speed in vacuum to that in matter,

For non-magnetic materials ( = 0)


WAVE EQUATION IN MATTER

02

22

H

E

t

en 12

00

v

cn

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Consider a (non-magnetic, m = 0) metal with no free

charges.

Taking the curl of the Faraday’s equation yields

And making use of the Ampere’s equation gives the

following wave equation

Solution and skin-depth.


WAVE ON (NON-MAGNETIC) METAL

Ht

EE

2

02

2

00

2

t

E

t

EE


PROPAGATION OF ENERGY

Poynting vector : energy flow

density

Time-average Poynting vector

Power transported through

surface-area S

S S

HES

2*

21 Re mWHES

WdSS S

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9


Applying the continuity condition of the wave vector,

we obtained the Snell’s law (independent of

polarization)

18

'1k

1'

1

2

1n

2n

1k

2k

x

zy

11

2211 sinsin nn 22

zjjx kkk zzz kkk 212,1j

REFLECTION AND TRANSMISSION


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TE (s) wave : plane of incidence (x) or //

(linearly polarized)

Boundary condition : continuities in and

19


E

y

yE xEy


)exp(ˆ10

2,1 xikzikEry xz

'1k

1'

1

2

1n

2n

1k

2k

H

H

H

x

z

)exp(ˆ 10 xikzikEy xz

)exp(ˆ20

2,1 xikzikEty xz

y

Resulted reflection and transmission coefficients :

20



;coscos

coscos

2211

2211

21

21

1

12,1

nn

nn

kk

kk

E

Er

xx

xx

2211

11

21

1

1

22,1

coscos

cos2

2

nn

n

kk

k

E

Et

xx

x

)exp(ˆ10

2,1 xikzikEry xz

'1k

1'

1

2

1n

2n

1k

2k

H

H

H

x

z

)exp(ˆ 10 xikzikEy xz

)exp(ˆ20

2,1 xikzikEty xz

y

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TM (p) wave : plane of incidence or equivalently

// the plane of incidence (draw your own figure in

close analogy with the previous one !)

Boundary condition : continuities in and

21



H

yHx

H

n

y

2

1

2112

11

2

2

11

2

2

122,1

//coscos

cos22

nn

n

knkn

knt

xx

x

2112

2112

2

2

11

2

2

2

2

11

2

22,1

//coscos

coscos

nn

nn

knkn

knknr

xx

xx

Transmission and reflection of energy flow across an

interface can be calculated from the Poynting vector

One can show that for non-absorptive materials,

energy is conserved


TRANSMITTANCE AND REFLECTANCE

2

11

22

cos

cos

ˆ

ˆt

n

n

Sx

Sx

I

IT

i

t

i

t

2

ˆ

ˆr

Sx

Sx

I

IR

i

r

i

r

1TR

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Show that the time-average Poynting vector is given as

Derive Fresnel formulas

Derive the transmittance and reflectance formulas

and conservation of energy formula


HOMEWORK

2*

21 Re mWHES

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