kz Ö electromagnetic waves and polarization e b maxwell’s
TRANSCRIPT
Physics topic handout – Electromagnetic Waves, Fresnel Equations & Polarization Dr Andrew French. www.eclecticon.info PAGE 1
Electromagnetic waves and polarization
Electromagnetic waves (of a particular amplitude, and wavelength) comprise of sinusoidally varying vector components of electric E and magnetic B fields.
Maxwell’s Equations, which describe the relationships between electric and magnetic fields (and charge) predict the following:
1. If an electromagnetic wave propagates in direction parallel to vector k, the electric and magnetic field are both perpendicular to this direction. In other words
(E,B,k) forms a right handed set* in a Cartesian (x,y,z) sense. No vector component of E or B is parallel to the direction of propagation.
2. Electromagnetic waves travel at a finite speed through a medium. This is independent of any coordinate system, so you can never ‘catch up’ with an electromagnetic wave, no
matter how fast you move. This idea is the main reason (in Special Relativity) behind the need to modify space and time as one approaches the speed of light. The speed of
electromagnetic waves is c/n where c = 2.998 x 108ms-1 and n is the refractive index. For a vacuum, n is unity. A magnitude less than unity is impossible*
3. At an interface between media of differing refractive index, vector components of B perpendicular to the interface surface must be continuous across the boundary. Also,
components of the E and H fields which are parallel to the surface, must be continuous across the boundary.
4. For an electromagnetic wave:
*Actually, it is (E,H,k). But for isotropic media, B is parallel to H
For isotropic
media:
n
8 -1
0 0
12.998 10 msc
7 -1
0
12 -1
0
4 10 Hm
8.85 10 Fm
0 B H
0
( )
0
0
x
i kz t
y
E
E e
E
c
nE B
Relative permeability. Unity for non magnetic materials. Magnetic materials such as iron
have a relative permeability of about 5000. Ferrite is 640, Nickel is 100.
Relative permittivity. Unity for vacuum and approximately for air. Water is about 1.77,
glass 3.7-10, diamond 5.5-10, sapphire 8.9-11.1
0
( )
0
0
y
i kz t
x
En
E ec
B
2k
2 f
ck
n wavespeed
c
n
x
y
z
E
H
k
ˆwavevector kk z
0
( )
0
00
y
i kz t
x
En
E ec
H
The polarization of an electromagnetic wave describes the relationship between the electric field
vector components and how they vary with time t and propagation distance z.
0 ( )
0
i i kz t
i
aE e e
be
E
Note the actual E and B fields
are the real parts of these
complex quantities.
12
cos sin
1 0
i
i i
e i
e i e
i.e. 0
0
iy
x
E be
E a
i
a
be
is called the Jones vector. Different values of a,b and phase
give rise to linear, circular and elliptical polarizations.
This is because the time variation of the electric field vector.
Re( ) Re i t
i
ae
be
E will follow a linear, circular or elliptical
trajectory depending on the values set. 1
0
0
1
1
2
1
1
1
2
1
i
1
2
1
i
Right hand circular Left hand circular
Linear 45o x linear
y linear
Note De Moivre’s Theorem **But it is possible to
have a negative or
indeed complex
refractive index!
(metamaterials &
metals respectively).
ˆ n
c
EB k
0
2
max
0 0
212
1
1
377
n
c
nS
S E× B
S E B E
E
Poynting
vector
i.e. Power / m2
Power of an EM wave varies as the square of electric field strength 2 2
1 12 20
0
1u
E BEnergy per unit volume
stored in E and B fields is:
The effect of an electromagnetic wave passing through a polariser (e.g. a material which will modify the wave in
different ways depending on the polarisation of the electric field) can be modelled by the matrix multiplication of the
Jones vector J for an incident wave by a 2 x 2 Jones matrix.
i
a
be
J J MJ
1 0
0 0
0 0
0 1
M
M
1
2
1
2
1
1
1
1
i
i
i
i
M
M
Linear // x
Linear //y
Linear 45o
Linear -45o
Right circular
Left circular
Quarter wave
plate (“fast” x)
Quarter wave
plate (“fast” y)
0 0( ) ( )
0 0
i ii kz t i kz t
i
aE e e E e e
be
E J
1
2
1
2
1 1
1 1
1 1
1 1
M
M
14
14
1 0
0
1 0
0
i
i
ei
ei
M
M
13
1
3
12
1
2
1 1
1 1
ie
J
M
1
2
1
2
1
1
1
1
i
i
J
M
1
2
1
1
J 1 1
2 2
1 1
1 1
i
i
J 1
3
1
3
1
2i
e
J
13
1 12 3
11 1
1 1 2i
e
J
2 2
2tan 2 cos
ab
a b
For elliptical polarization, the
tilt of the ellipse () is given by
a
b
In this example, elliptical
polarisation becomes 45o
linear following application of
the polariser.
In this example, 45o
linear polarisation becomes
Right circular following
application of the polariser
The Jones matrix defines the output polarization.
The amount of transmission loss depends on how
close the original polarization was to the desired
output.
( 0, )z tE
Physics topic handout – Electromagnetic Waves, Fresnel Equations & Polarization Dr Andrew French. www.eclecticon.info PAGE 2
tE
rE
iE
ik
rk
tk
iB r
B
tB
i
i
t
i
i
t
Note if linear polarised light is incident upon a
linear polariser with polarisation direction tilted
by from the polarisation of the incident light
Intensity
0I
watts m-1
Intensity
I
watts m-1 Linear polariser
Malus’s Law states 2
0cosI I
If an electromagnetic wave meets a change in
refractive index, the general response will be for
reflected and transmitted (i.e. refracted) waves
to be created. The power of the incident wave will
be shared between these. The balance of power
depends on a number of factors, which are
modelled via the Fresnel Equations.
Augustin-Jean Fresnel
1788-1927
The Fresnel Equations
0
( )
0
0
x
i kz t
y
E
E e
E
0
( )
0
0
y
i kz t
x
En
E ec
B
0
( )
0
00
y
i kz t
x
En
E ec
H
Maxwell’s Equations tell us: “At an interface between media of differing refractive index, vector components of B
perpendicular to the interface surface must be continuous across the boundary. Also, components of the E and H
fields which are parallel to the surface, must be continuous across the boundary.”
Let us consider two scenarios separately:
Let us start with the E,B
and H fields associated
with an electromagnetic
wave in isotropic media
(see previous pages)
Case 1: Electric field vector is parallel to the plane containing the
incident, reflected and transmitted wave propagation directions
1 1,n
2 2,n
E continuity:
H continuity:
cos cos cosi i r i t t
E E E
i
i
1 21
i i r r t t
i tr
i r t
E E E
n E n En E
c c c
E E E
B B B
1 1 2
1 0 1 0 2 0
1 2
2 1
i r t
t i r
n n nE E E
c c c
nE E E
n
Law of
reflection
0
1
H B
1 2
2 1
1 2 1 2
2 1 2 1
1 2 2 1
1 2 2 1
1 2
1 2
2 1
2 1
cos cos cos
cos cos cos cos
cos cos cos cos
cos cos
cos cos
i i r i i r t
i i t r i t
i t i r i t
t i
r
ii t
nE E E E
n
n nE E
n n
n n n nE E
n n
Er
n nE
1 2
2 1
1 2
2 1
1 2
1 2 1 2
2 12 1
2 1
2
1 2 2
2 12 1
2 1
1
cos cos
1
cos cos
2cos
cos cos
t i r
t
i
t t
i t
i
i t
nE E E
n
E nt r
E n
n n
nt
n nn
n
nt
n nn
1
1
2 1
2 1
2cos
cos cos
i
i t
n
tn n
Power coefficients are:
2
2
1
R r
T r
This scenario is commonly known
as ‘P’ polarisation (P for Parallel)
since the projection of
polariser on electric
field polarisation is cos
Physics topic handout – Electromagnetic Waves, Fresnel Equations & Polarization Dr Andrew French. www.eclecticon.info PAGE 3
tE
rEi
E
ik
rk
tk
iB
rB
tB
i
i
t
i
i
t
Case 2: Electric field vector is perpendicular to the
plane containing the incident, reflected and transmitted
wave propagation directions
1 1,n
2 2,n
i r tE E E
1 1 2
1 0 1 0 2 0
cos cos cosi i r i t t
n n nE E E
c c c
Power coefficients are:
2
21
R r
T r
1 21
i i r r t t
i tr
i r t
E E E
n E n En E
c c c
E E E
B B B
0
1
H B
i
i
E continuity:
H continuity:
1 1 2
1 1 2
1 2 1 2
1 2 1 2
1 2
1 2
1 2
1 2
cos cos cos
cos cos cos cos
cos cos
cos cos
i i r i i r t
i t i r i t
i t
i t
n n nE E E E
n n n nE E
n n
rn n
1 2
1 2
1 2
1 2
1
1
1 2
1 2
1
cos cos
1
cos cos
2cos
cos cos
i r t
t
i
i t
i t
i
i t
E E E
Et r
E
n n
tn n
n
tn n
This scenario is commonly known
as ‘S’ polarisation (S from senkrecht,
which is German for perpendicular)
ik r
k
tk
i
i
t
1n
2n
Law of
reflection
1
2
sin0 1i
n
n
Real solutions for the
transmitted ray angle
when
Always true if 2 1n n
1 2
1 2
1
sini
n n
n
n
So EM waves propagating
from a high refractive index
medium to a lower
refractive index medium
will be internally reflected
(i.e. no transmission) when
the angle of incidence
exceeds a critical angle c
1 2
1
sinc
n
n
Snell’s Law of refraction
1 2
1 1
2
sin sin
sinsin
i t
i
t
n n
n
n
ik r
k
tk
i
i
t
1n
2n
Law of
reflection
Brewster’s Angle
If a scenario arises where the angle between reflected
and transmitted waves is a right angle, this means the
reflected waves can only be S polarised.
P-polarised transmitted waves result from electrons
oscillating parallel to the plane at the interface of the
two mediums. However, no radiation occurs in the
direction of he polarisation. Since the transmitted
wave polarisation direction is parallel to the
reflected wavevector, this means no P polarised
radiation is reflected.
S polarised
only
rE
o o
o
1 2
o
1 2
1 2
2
1
1 2
1
90 180
90
sin sin
sin sin 90
sin cos
tan
tan
i t
t i
i t
i i
i i
i
B
n n
n n
n n
n
n
n
n
This effect is used in the design of glare reducing
optical devices such as sunglasses, car windows and
photographic lens filters.
From the geometry of the
above diagram
Snell’s Law
Brewster’s
Angle Sir David Brewster
1781-1869
1
2
sin0 1i
n
n
Critical angle
Physics topic handout – Electromagnetic Waves, Fresnel Equations & Polarization Dr Andrew French. www.eclecticon.info PAGE 4
ik r
k
tk
i
i
t
1n
2n
Law of
reflection
Brewster’s Angle from the Fresnel equations
S polarised
only
rE
1 2
1 2
2 1
2 1
cos cos
cos cos
t i
r
ii t
n n
Er
n nE
For P-polarised EM waves
Assuming the non-magnetic media
1 21
1 2
1 2
2
2 21
2
2
2 21
2
0 cos cos 0
cos cos
cos cos
1 sin cos
t i
t i
t i
t i
r n n
n n
n
n
n
n
1 2
2
2 21
2
sin sin
sin sin
i t
t i
n n
n
n
Snell’s Law
2 2
2 21 1
2 2
2 2
21 1
2
2 2
2 2
2 21 1
2 2
2 2
21 2
2 1
2 2
22 1
2
2
1 sin cos
1tan 1
cos
1 tan tan 1
1 tan 1
tan
i i
i
i
i i
i
n n
n n
n n
n n
n n
n n
n n
n n
n n
n
2 2
2 1
2
1
2
1
tan
i
i
n n
n
n
n
1 2
1
tanB
n
n
The power coefficient for
P-polarised is zero (and also a
minima) at the Brewster angle.
2
2
11 tan
cos
Physics topic handout – Electromagnetic Waves, Fresnel Equations & Polarization Dr Andrew French. www.eclecticon.info PAGE 5
Physics topic handout – Fresnel Equations & Polarization Dr Andrew French. www.eclecticon.info PAGE 6
1 2
1
sinc
n
n
1 2
1
tanB
n
n
1 2
1 2
2 1
2 1
cos cos
cos cos
t i
r
ii t
n n
Er
n nE
1
1
2 1
2 1
2cos
cos cos
i
i t
n
tn n
1 2
1 2
1 2
1 2
cos cos
cos cos
i t
i t
n n
rn n
1
1
1 2
1 2
2cos
cos cos
i
i t
n
tn n
P-polarised Electric field
(parallel to plane containing
wave vectors)
S-polarised Electric field
(perpendicular to plane
containing wave vectors)
Total internal reflection
beyond the critical angle
of incidence
2R r
2
R r
1T R
1T R