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Chapter 6 Polarization of light

First, review the chapter onAtomic Structure

- The elements

- 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

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/sz

ia/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 semi-major axis b semi-major axis

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)

Helical Light Wavefronts

Formal description of light polarization

- The spherical basis

- e1 ? LCP for light propagating along z

Lagging by ?/2

? LCP

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

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 ? ?2?
- Examples
- Unpolarized Pure circular polarization

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

Visualization of polarization

- Examples
- circularly polarized light propagating along z

Visualization of polarization

- Examples
- LCP light propagating along ??/6 f ?/3
- Need to rotate the DM details are given, for

example, in

? Result

Visualization of polarization

- Examples
- LCP light propagating along ??/6 f ?/3

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

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

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

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