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
18/01/2016
2
Main
M. Dressel and G. Gruner : Section 2.1 and 2.2 and 2.4
S.A. Maier : Section 1.1
Alexander A. Iskandar Electromagnetic Interactions in Matter 3
REFERENCES
In vacuum, the governing equations of electro-
magnetism are the Maxwell equations:
Alexander A. Iskandar Electromagnetic Interactions in Matter 4
MAXWELL’S EQUATIONS IN VACUUM
0 B
t
BE
0
E
t
EJB
000
18/01/2016
3
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,
Alexander A. Iskandar Electromagnetic Interactions in Matter 5
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.
Alexander A. Iskandar Electromagnetic Interactions in Matter 6
WAVE EQUATION IN VACUUM AND ITS
SOLUTIONS
tiexB
xE
txB
txE
)(
)(
),(
),(
0)(
)(22
xB
xEk
vk
v
vk ˆ
k
18/01/2016
4
The free wave equation admits solutions of the
following specific forms :
harmonic plane waves
harmonic spherical waves
Alexander A. Iskandar 7
WAVE EQUATION IN VACUUM AND ITS
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
18/01/2016
5
Wavefront is a spherical surface
9
k
rktieB
E
rtrB
trE
0
01
,
,
constkr
r
r
r
SPHERICAL WAVE SOLUTIONS
Alexander A. Iskandar Electromagnetic Interactions in Matter
In matter, the charge can be classified in two kinds
And the current can be decomposed into
Further, the Ohm’s law gives
Alexander A. Iskandar Electromagnetic Interactions in Matter 10
ELECTROMAGNETIC FIELDS IN MATTER
boundfree Pbound
t
PMJbound
boundcond JJJ
EJcond
18/01/2016
6
Constitutive relations, expressing material responses
Alexander A. Iskandar Electromagnetic Interactions in Matter 11
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
Alexander A. Iskandar Electromagnetic Interactions in Matter 12
MAXWELL’S EQUATION IN MATTER
0 B
t
BE
t
DJH cond
freeD
18/01/2016
7
Consider the case with no free sources (free = 0 and
= 0), taking the curl of the last two
equations yielded :
For uniform medium : , yields
Alexander A. Iskandar Electromagnetic Interactions in Matter 13
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)
Alexander A. Iskandar Electromagnetic Interactions in Matter 14
WAVE EQUATION IN MATTER
02
22
H
E
t
en 12
00
v
cn
18/01/2016
8
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.
Alexander A. Iskandar Electromagnetic Interactions in Matter 15
WAVE ON (NON-MAGNETIC) METAL
Ht
EE
2
02
2
00
2
t
E
t
EE
Alexander A. Iskandar Electromagnetic Interactions in Matter 16
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
18/01/2016
9
Alexander A. Iskandar Electromagnetic Interactions in Matter 17
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
Alexander A. Iskandar Electromagnetic Interactions in Matter
18/01/2016
10
TE (s) wave : plane of incidence (x) or //
(linearly polarized)
Boundary condition : continuities in and
19
REFLECTION AND TRANSMISSION
E
y
yE xEy
Alexander A. Iskandar Electromagnetic Interactions in Matter
)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
REFLECTION AND TRANSMISSION
Alexander A. Iskandar Electromagnetic Interactions in Matter
;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
18/01/2016
11
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
REFLECTION AND TRANSMISSION
Alexander A. Iskandar Electromagnetic Interactions in Matter
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
Alexander A. Iskandar Electromagnetic Interactions in Matter 22
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
18/01/2016
12
Show that the time-average Poynting vector is given as
Derive Fresnel formulas
Derive the transmittance and reflectance formulas
and conservation of energy formula
Alexander A. Iskandar Electromagnetic Interactions in Matter 23
HOMEWORK
2*
21 Re mWHES