Chapter 6: Polarization of light

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Chapter 6 Polarization of light
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First, review the chapter onAtomic Structure

• The elements

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• Preliminaries and definitions
• 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/sz
  ia/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 semi-major axis b semi-major axis

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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 Laguerre-Gaussian
  Modes (theory and experiment)

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Helical Light Wavefronts
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Formal description of light polarization

• The spherical basis

• e1 ? LCP for light propagating along 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

• ? 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

• and partially polarized light
• Theorem Pure polarization state ? ?2?
• Examples
• Unpolarized Pure circular polarization

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Visualization of polarization

• Treat light as spin-one particles
• Choose a spatial direction (?,f)
• Plot the probability of measuring
  spin-projection 1 on this direction

? Angular-momentum probability surface

• Examples
• z-polarized light

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Visualization of polarization

• Examples
• circularly polarized light propagating along z

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Visualization of polarization

• Examples
• LCP light propagating along ??/6 f ?/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 ??/6 f ?/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/poincare
2.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 ? Rlt1
• R degree of polarization

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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

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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
• The Jones matrix for the rotated wave plate is
• Ignore overall phase factor ?
• After the plate, we have
• Or
• expected circular polarization