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Chapter 4: Polarization of light. B. E. k. Preliminaries and definitions Plane-wave approximation : E ( r , t ) and B ( r , t ) are uniform in the plane  k

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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) andk
simple polarization states
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

simple polarization states4
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.,

Go to Polarization Tutorial

more definitions
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

elliptically polarized light
Elliptically polarized light
  • a, b – semi-major axes
unpolarized light
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 !
angular momentum carried by light
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)
formal description of light polarization




Formal description of light polarization
  • The spherical basis :
  • E+1  LCP for light propagating along +z:

Lagging by /2


decomposition of an arbitrary vector e into spherical unit vectors
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

polarization density matrix
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
polarization density matrix13
Polarization density matrix
  • DM is useful because it allows one to describe “unpolarized”
  • … and “partially polarized” light
  • Theorem: Pure polarization state  ρ2=ρ
  • Examples:
  • “Unpolarized”Pure circular polarization
visualization of polarization
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
visualization of polarization15
Visualization of polarization
  • Examples
  • circularly polarized light propagating along z
visualization of polarization16
Visualization of polarization
  • Examples
  • LCP light propagating along θ=/6; φ= /3
  • Need to rotate the DM; details are given, for example, in :

 Result :

visualization of polarization17
Visualization of polarization
  • Examples
  • LCP light propagating along θ=/6; φ= /3
description of polarization with stokes parameters
Description of polarization withStokes parameters
  • P0 = I = Ix + Iy Total intensity
  • P1 = Ix – Iy Lin. pol. x-y
  • P2 = I/4 – I- /4Lin. pol.  /4
  • P3 = I+ – I- Circular pol.

Another closely related representation is the Poincaré Sphere


description of polarization with stokes parameters and poincar sphere
Description of polarization withStokes parameters and Poincaré Sphere
  • P0 = I = Ix + Iy Total intensity
  • P1 = Ix – Iy Lin. pol. x-y
  • P2 = I/4 – I- /4Lin. 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
jones calculus
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
jones calculus an example
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