Optically polarized atoms_ch_4
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Transcript Optically polarized atoms_ch_4
Chapter 4: Polarization of light
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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
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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
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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
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More definitions
LCP and RCP are defined w/o reference to
a particular quantization axis
Suppose we define a z-axis
-polarization : linear along z
+: LCP (!) light propagating along z
-
:
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
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Elliptically polarized light
a, b – semi-major axes
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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 !
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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)
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Helical Light: Wavefronts
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Formal description of light polarization
The spherical basis :
E+1 LCP for light propagating along +z :
y
x
z
Lagging by /2 LCP
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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
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Polarization density matrix
For light propagating along z
• Diagonal elements – intensities of light with corresponding polarizations
• Off-diagonal elements – correlations
• Hermitian:
• “Unit” trace:
Tr E
q
E
q *
| E |2
q
• We will be mostly using normalized DM where this factor is divided out
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Polarization density matrix
• DM is useful because it allows one to describe “unpolarized”
0
1/ 3 0
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
0 1 0 ; 2 0 1 0
3
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0 0 1
0 0 1
1
2
3
Pure circular polarization
1 0 0
1 0 0
0 0 0 ; 2 0 0 0
0 0 0
0 0 0
2
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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
Angular-momentum probability surface
Examples
• z-polarized light
sin 2
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Visualization of polarization
Examples
• circularly polarized light propagating along z
1 cos
2
1 cos
2
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Visualization of polarization
Examples
• LCP light propagating along θ=/6; φ= /3
• Need to rotate the DM; details are given, for example, in :
Result :
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Visualization of polarization
Examples
• LCP light propagating along θ=/6; φ= /3
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Description of polarization with
Stokes parameters
• P0 = I = Ix + Iy
Total intensity
• P1 = Ix – Iy
Lin. pol. x-y
• P2 = I/4 – I- /4
Lin. pol. /4
• P3 = I+ – I-
Circular pol.
Another closely related representation
is the Poincaré Sphere
See http://www.ipr.res.in/~othdiag/zeeman/poincare2.htm
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Description of polarization with
Stokes parameters and Poincaré Sphere
• P0 = I = Ix + Iy
Total intensity
• P1 = Ix – Iy
Lin. pol. x-y
• P2 = I/4 – I- /4
Lin. pol. /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
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Jones Calculus
• Consider polarized light propagating along z:
• This can be represented as a column (Jones) vector:
• Linear optical elements 22 operators (Jones matrices), for example:
• If the axis of an element is rotated, apply
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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
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