Lecture 9 Polarization of Light

Introduction Part I: Different polarization states of light Part II: Stokes parameters, Mueller matrices Part III: Optical components for polarimetry Part IV: Polarimeters

Part I: Different polarization states of light Light as an electromagnetic wave Mathematical and graphical descriptions of polarization Linear, circular, elliptical light Polarized, unpolarized light

Light as an electromagnetic wave Part I: Polarization states Light as an electromagnetic wave Light is a transverse wave, an electromagnetic wave

Mathematical description of the EM wave Part I: Polarization states Mathematical description of the EM wave Light wave that propagates in the z direction:

Graphical representation of the EM wave (I) Part I: Polarization states Graphical representation of the EM wave (I) One can go from: to the equation of an ellipse (using trigonometric identities, squaring, adding):

Graphical representation of the EM wave (II) Part I: Polarization states Graphical representation of the EM wave (II) An ellipse can be represented by 4 quantities: size of minor axis size of major axis orientation (angle) sense (CW, CCW) Light can be represented by 4 quantities...

Vertically polarized light Part I: Polarization states, linear polarization Vertically polarized light If there is no amplitude in x (E0x = 0), there is only one component, in y (vertical).

Polarization at 45º (I) If there is no phase difference (=0) and Part I: Polarization states, linear polarization Polarization at 45º (I) If there is no phase difference (=0) and E0x = E0y, then Ex = Ey

Part I: Polarization states, linear polarization Polarization at 45º (II)

Linear polarization E-field magnitude oscillates Direction fixed Arbitrary polarization angle superposition of x and y polarized waves real numbers Time evolution Example 45 ° linear polarization

Circular polarization (I) Part I: Polarization states, circular polarization Circular polarization (I) If the phase difference is = 90º and E0x = E0y then: Ex / E0x = cos  , Ey / E0y = sin  and we get the equation of a circle:

Circular polarization (II) Part I: Polarization states, circular polarization Circular polarization (II)

Circular polarization (III) Part I: Polarization states, circular polarization Circular polarization (III)

Circular polarization (IV) Part I: Polarization states, circular polarization... see it now? Circular polarization (IV)

Circular polarization E-field magnitude constant Direction rotates Complex superposition of x and y polarizations x and y in quadrature Time evolution Example: right circular polarization

Elliptical polarization Part I: Polarization states, elliptical polarization Elliptical polarization Linear + circular polarization = elliptical polarization

Elliptical polarization General case polarization partly linear and partly circular E-field sweeps out ellipse both magnitude and direction change with time Superposition of L and R states Superposition of L and R Elliptical polarization

Polarization summary Decompose into x and y polarizations Linear -- real superposition Circular -- quadrature superposition Linear polarizations Circular polarizations Ey E+45 i Ey Ex ERight Ex Ey ELeft E-45 i Ey Ex Ex

Unpolarized light (natural light) Part I: Polarization states, unpolarized light Unpolarized light (natural light)

Producing linear polarization -- 1 Induce loss for one polarization direction Examples: wire grid, polaroid filter

Producing linear polarization -- Separation in birefringent crystal -- ex: calcite Ordinary wave -- behaves as expected Extra-ordinary wave -- behaves differently example -- E-field not perpendicular to propagation direction Input light converted to two polarized beams

Producing linear polarization -- Brewster’s angle Only one polarization reflected Reflected light polarized Works with most surfaces Good way to calibrate polarizers Refracted beam creates dipoles in medium Brewster angle: dipole field zero perpendicular to reflection prop. direction

Polarizing cubes Uses fact that total internal reflection close to Brewster’s angle reflection large for other polarization Multi-layer coating enhances effect reflections from multiple surfaces -- resonance Brewster polarization reflection always zero

Waveplates Polarization converters One linear polarization direction propagates faster Half wave plate -- phase delay 180° rotate linear polarization up to 90° fast axis at 45° to input polarization direction Quarter wave plate -- phase delay 90° convert linear to circular polarization R or L for fast axis +45 or -45 to input pol. Rotate linear pol. by angle 2q Create circular polarization Retardation of one polarization

Other circular polarizers Use phase shift for total internal reflection 45° over broad range of angles Two reflections give 90° Converts linear to circular polarization

Part II: Stokes parameters and Mueller matrices Stokes parameters, Stokes vector Stokes parameters for linear and circular polarization Stokes parameters and polarization P Mueller matrices, Mueller calculus Jones formalism

Part II: Stokes parameters Stokes parameters (I) 1852: Sir George Gabriel Stokes took a very different approach and discovered that polarization can be described in terms of observables using an experimental definition The polarization ellipse is only valid at a given instant of time (function of time): To get the Stokes parameters, do a time average (integral over time) and a little bit of algebra...

Stokes parameters (II) described in terms of the electric field Part II: Stokes parameters Stokes parameters (II) described in terms of the electric field The 4 Stokes parameters are:

Stokes parameters (III) described in geometrical terms Part II: Stokes parameters Stokes parameters (III) described in geometrical terms

Part II: Stokes parameters, Stokes vectors The Stokes parameters can be arranged in a Stokes vector: Linear polarization Circular polarization Fully polarized light Partially polarized light Unpolarized light

Pictorial representation of the Stokes parameters Part II: Stokes parameters Pictorial representation of the Stokes parameters

Stokes vectors for linearly polarized light Part II: Stokes parameters, examples Stokes vectors for linearly polarized light LHP light LVP light +45º light -45º light

Stokes vectors for circularly polarized light Part II: Stokes parameters, examples Stokes vectors for circularly polarized light RCP light LCP light

Part II: Stokes parameters, Mueller matrices If light is represented by Stokes vectors, optical components are then described with Mueller matrices: [output light] = [Muller matrix] [input light]

Mueller calculus (I) I’ = M3 M2 M1 I Element 1 Element 2 Element 3 Part II: Stokes parameters, Mueller matrices Mueller calculus (I) Element 1 Element 2 Element 3 I’ = M3 M2 M1 I

Part II: Stokes parameters, Mueller matrices Mueller calculus (II) Mueller matrix M’ of an optical component with Mueller matrix M rotated by an angle : M’ = R(- ) M R() with:

Part II: Stokes parameters, Jones formalism, not that important here... Stokes vectors and Mueller matrices cannot describe interference effects. If the phase information is important (radio-astronomy, masers...), one has to use the Jones formalism, with complex vectors and Jones matrices: Jones vectors to describe the polarization of light: Jones matrices to represent optical components: BUT: Jones formalism can only deal with 100% polarization...

Part III: Optical components for polarimetry Complex index of refraction Polarizers Retarders

Complex index of refraction Part III: Optical components Complex index of refraction The index of refraction is actually a complex quantity: real part optical path length, refraction: speed of light depends on media birefringence: speed of light also depends on P imaginary part absorption, attenuation, extinction: depends on media dichroism/diattenuation: also depends on P

Part III: Optical components, polarizers Polarizers absorb one component of the polarization but not the other. The input is natural light, the output is polarized light (linear, circular, elliptical). They work by dichroism, birefringence, reflection, or scattering.

Wire-grid polarizers (I) [dichroism] Part III: Optical components, polarizers Wire-grid polarizers (I) [dichroism] Mainly used in the IR and longer wavelengths Grid of parallel conducting wires with a spacing comparable to the wavelength of observation Electric field vector parallel to the wires is attenuated because of currents induced in the wires

Wide-grid polarizers (II) [dichroism] Part III: Optical components, polarizers Wide-grid polarizers (II) [dichroism]

Dichroic crystals [dichroism] Part III: Optical components, polarizers Dichroic crystals [dichroism] Dichroic crystals absorb one polarization state over the other one. Example: tourmaline.

Polaroids [dichroism] Part III: Optical components, polarizers – Polaroids, like in sunglasses! Polaroids [dichroism] Made by heating and stretching a sheet of PVA laminated to a supporting sheet of cellulose acetate treated with iodine solution (H-type polaroid). Invented in 1928.

Crystal polarizers (I) [birefringence] Part III: Optical components, polarizers Crystal polarizers (I) [birefringence] Optically anisotropic crystals Mechanical model: the crystal is anisotropic, which means that the electrons are bound with different ‘springs’ depending on the orientation different ‘spring constants’ gives different propagation speeds, therefore different indices of refraction, therefore 2 output beams

Crystal polarizers (II) [birefringence] Part III: Optical components, polarizers Crystal polarizers (II) [birefringence] isotropic crystal (sodium chloride) anisotropic (calcite) The 2 output beams are polarized (orthogonally).

Crystal polarizers (IV) [birefringence] Part III: Optical components, polarizers Crystal polarizers (IV) [birefringence] Crystal polarizers used as: Beam displacers, Beam splitters, Polarizers, Analyzers, ... Examples: Nicol prism, Glan-Thomson polarizer, Glan or Glan-Foucault prism, Wollaston prism, Thin-film polarizer, ...

Part III: Optical components, retarders In retarders, one polarization gets ‘retarded’, or delayed, with respect to the other one. There is a final phase difference between the 2 components of the polarization. Therefore, the polarization is changed. Most retarders are based on birefringent materials (quartz, mica, polymers) that have different indices of refraction depending on the polarization of the incoming light.

Part III: Optical components, retarders Half-Wave plate (I) Retardation of ½ wave or 180º for one of the polarizations. Used to flip the linear polarization or change the handedness of circular polarization.

Part III: Optical components, retarders Half-Wave plate (II)

Quarter-Wave plate (I) Part III: Optical components, retarders Quarter-Wave plate (I) Retardation of ¼ wave or 90º for one of the polarizations Used to convert linear polarization to elliptical.

Quarter-Wave plate (II) Part III: Optical components, retarders Quarter-Wave plate (II) Special case: incoming light polarized at 45º with respect to the retarder’s axis Conversion from linear to circular polarization (vice versa)

(Back to polarizers, briefly) Circular polarizers Part III: Optical components, polarizers (Back to polarizers, briefly) Circular polarizers Input light: unpolarized --- Output light: circularly polarized Made of a linear polarizer glued to a quarter-wave plate oriented at 45º with respect to one another.

Part III: Optical components, retarders Other retarders Soleil-Babinet: variable retardation to better than 0.01 waves Nematic liquid crystals... Liquid crystal variable retarders... Ferroelectric liquid crystals... Piezo-elastic modulators... Pockels and Kerr cells...

Part IV: Polarimeters Polaroid-type polarimeters Dual-beam polarimeters

Polaroid-type polarimeter for linear polarimetry (I) Part IV: Polarimeters, polaroid-type Polaroid-type polarimeter for linear polarimetry (I) Use a linear polarizer (polaroid) to measure linear polarization . Polarization percentage and position angle:

Dual-beam polarimeters Principle Part IV: Polarimeters, dual-beam type Dual-beam polarimeters Principle Instead of cutting out one polarization and keeping the other one (polaroid), split the 2 polarization states and keep them both Use a Wollaston prism as an analyzer Disadvantages: need 2 detectors (PMTs, APDs) or an array; end up with 2 ‘pixels’ with different gain Solution: rotate the Wollaston or keep it fixed and use a half-wave plate to switch the 2 beams

Dual-beam polarimeters Switching beams Part IV: Polarimeters, dual-beam type Dual-beam polarimeters Switching beams Unpolarized light: two beams have identical intensities whatever the prism’s position if the 2 pixels have the same gain To compensate different gains, switch the 2 beams and average the 2 measurements

Dual-beam polarimeters Switching beams by rotating the prism Part IV: Polarimeters, dual-beam type Dual-beam polarimeters Switching beams by rotating the prism rotate by 180º

Dual-beam polarimeters Switching beams using a ½ wave plate Part IV: Polarimeters, dual-beam type Dual-beam polarimeters Switching beams using a ½ wave plate Rotated by 45º

Faraday effect Magnetic field induces polarization rotation Orients electron spins in medium Angular momenta of electrons and photons interact R and L have different propagation delays Useful for magnetometers example: rubidium vapor

Kerr effect Electro optic effect Polarization in direction of applied field changes propagation speed Input linear polarization In direction of applied field -- phase modulator Perpendicular to applied field -- nothing 45° to applied field -- variable waveplate output polarizer gives intensity modulator

Pockels effect Similar to Kerr effect Apply electric field along propagation direction Crystal with no center of symmetry -- also piezoelectric Delay linear in applied field -- Kerr effect quadratic

Liquid crystals Electric field changes average orientation of molecules Delay depends on polarization direction Phase modulator or variable waveplate Intensity modulator needs polarizers Used for displays -- ex: computer monitors

Isolators -- 1 Polarizer and quarter waveplate Double pass through quarter wave plate same as half wave plate rotate polarization by up to 45° Polarizer blocks reflected light Quarter wave Polarizer Reflecting element

Isolators -- 2 Faraday effect non-reciprocal Opposite for different propagation directions Put passive polarization rotator and Faraday rotator in series One direction -- no effect Opposite direction -- rotate polarization 90° Polarizer blocks reflections Used for fiber optics, laser diodes performance good -- 10 - 30 dB Passive rotator Reflecting element Faraday crystal Polarizer Ex: optically active crystal magnet

Mathematical description of polarization Stokes vectors Elements give: 1/2 total intensity horizontal linear +45° linear right circular Jones vectors Elements give E-field x-component E-field y-component Only applicable to polarized light Unpolarized state only I0 is non-zero