Chapter 4: Polarization of light
1
„
Preliminaries and definitions
B
E
k
„
„
„
Plane-wave approximation: E(r,t) and B(r,t)
are uniform in the plane ^ k
We will say that light polarization vector is
along E(r,t) (although it was along B(r,t) in
classic optics literature)
Similarly, polarization plane contains E(r,t)
and k
2
Simple polarization states
„
Linear or plane polarization
Circular polarization
„
Which one is LCP, and which is RCP ?
„
Electric-field vector is seen
rotating counterclockwise by
an observer getting hit in their
eye by the light (do not try
this with lasers !)
Electric-field vector is seen
rotating clockwise by the said
observer
3
Simple polarization states
„
„
„
Which one is LCP, and which is RCP?
Warning: optics definition is opposite
to that in high-energy physics;
helicity
There are many helpful resources
available on the web, including
spectacular animations of various
polarization states, e.g.,
http://www.enzim.hu/~szia/cddemo/
edemo0.htm
Go to
Polarization
Tutorial
4
More definitions
„
„
LCP and RCP are defined w/o reference to
a particular quantization axis
Suppose we define a z-axis
¾ p-polarization
: linear along z
¾ s+:
LCP (!) light propagating along z
¾ s- :
RCP (!) light propagating along z
If, instead of light, we had a right-handed wood screw, it would move
opposite to the light propagation direction
5
Elliptically polarized light
„
a, b – semi-major axes
6
Unpolarized light ?
„
„
„
„
Is similar to free lunch in that such thing,
strictly speaking, does not exist
Need to talk about non-monochromatic light
The three-independent light-source model (all
three sources have equal average intensity, and
emit three orthogonal polarizations
Anisotropic light (a light beam) cannot be
unpolarized !
7
Angular momentum carried by light
„
„
„
„
The simplest description is in the photon picture :
A photon is a particle with intrinsic angular
momentum one ( = )
Orbital angular momentum
Orbital angular momentum and LaguerreGaussian Modes (theory and experiment)
8
Helical Light: Wavefronts
9
Formal description of light polarization
„
The spherical basis :
„
E+1 ¨ LCP for light propagating along +z :
y
x
z
Lagging by p/2 ï LCP
10
Decomposition of an arbitrary
vector E into spherical unit vectors
Recipe for
finding how
much of a
given basic
polarization is
contained in
the field E
11
Polarization density matrix
For light propagating along z
• Diagonal elements – intensities of light with corresponding polarizations
• Off-diagonal elements – correlations
• Hermitian:
• “Unit” trace:
ρ+ = ρ
Tr ρ = ∑ E
q
(E
)
q *
=| E |2
q
• fl We will be mostly using normalized DM where this factor is divided out
12
Polarization density matrix
• DM is useful because it allows one to describe “unpolarized”
0 ⎞
⎛ 1/ 3 0
ρ = ⎜⎜ 0 1/ 3 0 ⎟⎟
⎜ 0
⎟
0
1/
3
⎝
⎠
•… and “partially polarized” light
• Theorem: Pure polarization state ¨ ρ2=ρ
• Examples:
“Unpolarized”
⎛1 0 0⎞
⎛1 0 0⎞
1
1
ρ = ⎜⎜ 0 1 0 ⎟⎟ ; ρ 2 = ⎜⎜ 0 1 0 ⎟⎟
3⎜
9⎜
⎟
⎟
⎝0 0 1⎠
⎝0 0 1⎠
1
2
ρ = ρ≠ρ
3
Pure circular polarization
⎛ 1 0 0⎞
⎛ 1 0 0⎞
ρ = ⎜⎜ 0 0 0 ⎟⎟ ; ρ 2 = ⎜⎜ 0 0 0 ⎟⎟
⎜ 0 0 0⎟
⎜ 0 0 0⎟
⎝
⎠
⎝
⎠
ρ2 = ρ
13
Visualization of polarization
• Treat light as spin-one particles
• Choose a spatial direction (θ,φ)
• Plot the probability of measuring spin-projection =1 on this direction
fl
Angular-momentum probability surface
Examples
• z-polarized light
∝ sin 2 θ
14
Visualization of polarization
Examples
• circularly polarized light propagating along z
∝ (1 − cos θ )
2
∝ (1 + cos θ )
2
15
Visualization of polarization
Examples
• LCP light propagating along θ=p/6; φ= p/3
• Need to rotate the DM; details are given, for example, in :
fl Result :
16
Visualization of polarization
Examples
• LCP light propagating along θ=p/6; φ= p/3
17
Description of polarization with
Stokes parameters
• P0 = I = Ix + Iy
Total intensity
• P1 = Ix – Iy
Lin. pol. x-y
• P2 = Ip/4 – I- p/4
Lin. pol. ≤ p/4
• P3 = I+ – I-
Circular pol.
Another closely related representation
is the Poincaré Sphere
See http://www.ipr.res.in/~othdiag/zeeman/poincare2.htm
18
Description of polarization with
Stokes parameters and Poincaré
Sphere
• P0 = I = Ix + Iy Total intensity
• P1 = Ix – Iy
Lin. pol. x-y
• P2 = Ip/4 – I- p/4
Lin. pol. ≤ p/4
• P3 = I+ – I-
Circular pol.
• Cartesian coordinates on the Poincaré Sphere are normalized Stokes parameters:
P1/P0, P2/P0 , P3/P0
• With some trigonometry, one can see that a state of arbitrary polarization is represented by
a point on the Poincaré Sphere of unit radius:
• Partially polarized light ⇒ R<1
• R ≡ degree of polarization
R=
P12 + P22 + P32
=1
P0
19
Jones Calculus
• Consider polarized light propagating along z:
• This can be represented as a column (Jones) vector:
• Linear optical elements ⇒ 2×2 operators (Jones matrices), for example:
• If the axis of an element is rotated, apply
20
Jones Calculus: an example
• x-polarized light passes through quarter-wave plate whose axis is at 45° to x
• Initial Jones vector:
⎛1⎞
⎜ 0⎟
⎝ ⎠
• The Jones matrix for the rotated wave plate is:
• Ignore overall phase factor ⇒
• After the plate, we have:
• Or:
= expected circular polarization
21