PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 10:
02/20/2015PHY 712 Spring 2015 -- Lecture 161 PHY 712 Electrodynamics 9-9:50 AM Olin 103 Plan for...
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Transcript of 02/20/2015PHY 712 Spring 2015 -- Lecture 161 PHY 712 Electrodynamics 9-9:50 AM Olin 103 Plan for...
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