Chapter 4: Polarization of light 1
Preliminaries and definitions B E 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 k 2
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 3
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 4
More definitions LCP and RCP are defined w/o reference to a particular quantization axis Suppose we define a z-axisz p-polarizationpolarization : linear along z s + : LCP (!)( ) light propagating along z s - : 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 5
Elliptically polarized light a, b semi-major major axes 6
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! 7
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) 8
Helical Light: Wavefronts 9
Formal description of light polarization The spherical basis : E +1 LCP for light propagating along +z + : y x z Lagging by p/2 ï LCP 10
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 11
Polarization density matrix For light propagating along z Diagonal elements intensities of light with corresponding polarizations Off-diagonal elements correlations Hermitian: ρ + = ρ Unit trace: q Tr E E q ( q) * E ρ = = 2 fl We will be mostly using normalized DM where this factor is divided out 12
Polarization density matrix DM is useful because it allows one to describe unpolarized and partially polarized light 1/3 0 0 ρ = 0 1/3 0 0 0 1/3 Theorem: Pure polarization state ρ 2 =ρ Examples: Unpolarized Pure circular polarization 1 0 0 1 0 0 1 0 0 1 0 0 1 2 1 2 ρ = 0 1 0 ; ρ 0 1 0 ρ 0 0 0 ; ρ 0 0 0 3 = 9 = = 0 0 1 0 0 1 0 0 0 0 0 0 2 1 2 ρ = ρ ρ ρ = ρ 3 13
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 fl z-polarized light Angular-momentum probability surface Examples 2 sin θ 14
Visualization of polarization Examples circularly polarized light propagating along z ( 1 cosθ ) 2 ( 1+ cosθ ) 2 15
Visualization of polarization Examples LCP light propagating along θ=p/6; φ= p/3 Need to rotate the DM; details are given, for example, in : fl Result : 16
Visualization of polarization Examples LCP light propagating along θ=p/6; φ= p/3 17
Description of polarization with Stokes parameters P 0 = I = I x + I y Total intensity P 1 = I x I y Lin. pol. x-y P 2 = I p/4 I - p/4 Lin. pol. p/4 P 3 = I + I - Circular pol. Another closely related representation is the Poincaré Sphere See http://www.ipr.res.in/~othdiag/zeeman/poincare2.htm 18
Description of polarization with Stokes parameters and Poincaré P 0 = I = I x + I y P 1 = I x I y Sphere Total intensity Lin. pol. x-y P 2 = I p/4 I - p/4 Lin. pol. p/4 P 3 = I + I - Circular pol. Cartesian coordinates on the Poincaré Sphere are normalized Stokes parameters: P 1 /P 0, P 2 /P 0, P 3 /P 0 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 2 2 2 P1 + P2 + P3 R = = 1 P 0 19
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 20
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 21