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Chapter 4: Polarization of light

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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Chapter 4: Polarization of light

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  1. Chapter 4: Polarization of light

  2. 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

  3. 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

  4. 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

  5. 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

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

  7. 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 !

  8. 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)

  9. Helical Light: Wavefronts

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

  13. 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

  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 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

  19. 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

  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

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