Chapter 4: Polarization of light

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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 ïž k

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B

E

k

• 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
• 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 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

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
• a, b – semi-major axes
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
• 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)

y

x

z

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

Lagging by /2

 LCP

Recipe for finding how much of a given basic polarization is contained in the field E

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 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
• 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 polarization
• Examples
• circularly polarized light propagating along z
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 polarization
• Examples
• LCP light propagating along θ=/6; φ= /3
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

See http://www.ipr.res.in/~othdiag/zeeman/poincare2.htm

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
• 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
• 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