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**Outline**• Stokes Vectors, Jones Calculus and Mueller Calculus • Optics of Crystals: Birefringence • Common polarization devices for the laboratory and for astronomical instruments • Principles of Polarimetry: Modulation and Analysis. Absolute and Relative Polarimetry • Principles of Polarimetry: Spatial modulation, Temporal modulation, Spectral modulation • Principles of Polarimetry: Noise and errors • Spurious sources of polarization**A. López Ariste**Stokes Vector, Jones Calculus,Mueller Calculus playing around with matrices**Assumptions:**• A plane transverse electromagnetic wave • Quasi-monochromatic • Propagating in a well defined direction z**Jones Vector:**It is actually a complex vector with 3 free parameters It transforms under the Pauli matrices. It is a spinor**The Jones matrix of an optical device**In group theory: SL(2,C)**From the quantum-mechanical point of view, the wave function**cannot be measured directly. Observables are made of quadratic forms of the wave function: J is a density matrix : The coherence matrix**Like Jones matrices, J also belongs to the SL(2,C) group,**and can be decomposed in the basis of the Pauli matrices. Is the Stokes Vector**The Stokes vector is the quadractic form of a spinor. It is**a bi-spinor, or also a 4-vector**The Minkowski space**I Partially polarized light Cone of (fully polarized) light Fully polarized light V Q**From group theory, the Mueller matrix belongs to a group of**transformations which is the square of SL(2,C) Actually a subgroup of this general group called O+(3,1) or Lorentz group**The cone of (fully polarized) light**I Lorentz boost = de/polarizer, attenuators, dichroism V Q**The cone of (fully polarized) light**I 3-d rotation = retardance, optical rotation V Q**Mueller Calculus**• Any macroscopic optical device that transforms one input Stokes vector to an output Stokes vector can be written as a Mueller matrix • Lorentz group is a group under matrix multiplication: A sequence of optical devices has as Mueller matrix the product of the individual matrices**Mueller Calculus: 3 basic operations**• Absorption of one component • Retardance of one component respect to the other • Rotation of the reference system**Mueller Calculus: 3 basic operations**• Absorption of one component**Mueller Calculus: 3 basic operations**• Absorption of one component • Retardance of one component respect to the other**Mueller Calculus: 3 basic operations**• Absorption of one component • Retardance of one component respect to the other • Rotation of the reference system**A. López Ariste**Optics of Crystals: Birefringence**Three types of crystals**A spherical wavefront**Three types of crystals**• Two apparent waves propagating at different speeds: • An ordinary wave, with a spherical wavefront propagating • at ordinary speed vo • An extraordinary wave with an elliptical wavefront, its speed • depends on direction with characteristic values vo and ve**z**s The ellipsoïd of D in uniaxial crystals De The two propagating waves are linearly polarized and orthogonal one to each other Do**Typical birefringences**• Quartz +0.009 • Calcite -0.172 • Rutile +0.287 • Lithium Niobate -0.085**A. López Ariste**Common polarization devices for the laboratory and for astronomical instruments**Glan-Taylor Polarizer**• Glan-Taylor.jpg