1 Chapter 4: Polarization of light

2 Preliminaries and definitions Preliminaries and definitions Plane-wave approximation: E(r,t) and B(r,t) are uniform in the plane  k 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) 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 Similarly, polarization plane contains E(r,t) and k kBE

3 Simple polarization states Linear or plane polarization Linear or plane polarization Circular polarization Circular polarization Which one is LCP, and which is RCP ? 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

4 Simple polarization states Which one is LCP, and which is RCP? Which one is LCP, and which is RCP? Warning: optics definition is opposite to that in high-energy physics; helicity 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., edemo0.htm There are many helpful resources available on the web, including spectacular animations of various polarization states, e.g., edemo0.htm edemo0.htm edemo0.htm Go to Polarization Tutorial

5 More definitions LCP and RCP are defined w/o reference to a particular quantization axis LCP and RCP are defined w/o reference to a particular quantization axis Suppose we define a z-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

6 Elliptically polarized light a, b – semi-major axes a, b – semi-major axes

7 Unpolarized light ? Is similar to free lunch in that such thing, strictly speaking, does not exist Is similar to free lunch in that such thing, strictly speaking, does not exist Need to talk about non-monochromatic light 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 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 ! Anisotropic light (a light beam) cannot be unpolarized !

8 Angular momentum carried by light The simplest description is in the photon picture : The simplest description is in the photon picture : A photon is a particle with intrinsic angular momentum one ( ) A photon is a particle with intrinsic angular momentum one ( ) Orbital angular momentum Orbital angular momentum Orbital angular momentum and Laguerre- Gaussian Modes (theory and experiment) Orbital angular momentum and Laguerre- Gaussian Modes (theory and experiment)

9 Helical Light: Wavefronts

10 Formal description of light polarization The spherical basis : The spherical basis : E +1  LCP for light propagating along +z : E +1  LCP for light propagating along +z : Lagging by  /2 zyx  LCP

11 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

12 Polarization density matrix 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 For light propagating along z

13 Polarization density matrix DM is useful because it allows one to describe “unpolarized” … and “partially polarized” light Theorem: Pure polarization state  ρ 2 =ρ Examples: “Unpolarized”Pure circular polarization

14 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

15 Visualization of polarization Examples circularly polarized light propagating along z

16 Visualization of polarization Examples LCP light propagating along θ=  /6; φ=  /3 Need to rotate the DM; details are given, for example, in :  Result :

17 Visualization of polarization Examples LCP light propagating along θ=  /6; φ=  /3

18 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  /4 – I -  /4 Lin. pol.   /4 P 3 = I + – I - Circular pol. Another closely related representation is the Poincaré Sphere See

19 Description of polarization with Stokes parameters and Poincaré Sphere P 0 = I = I x + I y Total intensity P 1 = I x – I y Lin. pol. x-y P 2 = I  /4 – I -  /4 Lin. pol.   /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

20 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

21 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