lecture 4: polarization - leiden observatorykeller/teaching/... · lecture 4: polarization content...
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Lecture 4: Polarization
Content
1 Polarized Light in the Universe2 Polarization Ellipse3 Jones Formalism4 Stokes and Mueller Formalisms5 Poincaré Sphere6 Polarizers7 Retarders
Christoph U. Keller, Leiden University, [email protected] Lecture 4: Polarization 1
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Polarized Light in the Universe
Polarization indicates anisotropy⇒ not all directions are equal
Typical anisotropies introduced bygeometry (not everything is spherically symmetric)temperature gradientsdensity gradientsmagnetic fieldselectrical fields
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13.7 billion year old temperature fluctuations from WMAP
BICEP2 results and Planck Dust Polarization Map
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Scattering Polarization 1
Christoph U. Keller, Leiden University, [email protected] Lecture 4: Polarization 4
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Scattering Polarization 2
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The Power of Polarized Light Measurements
T Tauri in intensity MWC147 in intensity
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The Power of Polarimetry
T Tauri in Linear Polarization MWC147 in Linear Polarization
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Solar Magnetic Field Maps from Longitudinal Zeeman Effect
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Summary of Polarization Origin
Plane Vector Wave ansatz ~E = ~E0ei(~k ·~x−ωt)
spatially, temporally constant vector ~E0 lays in planeperpendicular to propagation direction ~krepresent ~E0 in 2-D basis, unit vectors ~ex and ~ey , bothperpendicular to ~k
~E0 = Ex~ex + Ey~ey .
Ex , Ey : arbitrary complex scalars
damped plane-wave solution with given ω, ~k has 4 degrees offreedom (two complex scalars)additional property is called polarizationmany ways to represent these four quantitiesif Ex and Ey have identical phases, ~E oscillates in fixed planesum of plane waves is also a solution
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Polarization Ellipse
Polarization Ellipse Polarization
~E (t) = ~E0ei(~k ·~x−ωt)
~E0 = |Ex |eiδx~ex + |Ey |eiδy~ey
wave vector in z-direction~ex , ~ey : unit vectors in x , y|Ex |, |Ey |: (real) amplitudesδx ,y : (real) phases
Polarization Description2 complex scalars not the most useful descriptionat given ~x , time evolution of ~E described by polarization ellipseellipse described by axes a, b, orientation ψ
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Jones Formalism
Jones Vectors
~E0 = Ex~ex + Ey~ey
beam in z-direction~ex , ~ey unit vectors in x , y -directioncomplex scalars Ex ,y
Jones vector~e =
(ExEy
)phase difference between Ex , Ey multiple of π, electric field vectoroscillates in a fixed plane⇒ linear polarizationphase difference ±π
2 ⇒ circular polarization
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Summing and Measuring Jones Vectors
~E0 = Ex~ex + Ey~ey , ~e =
(ExEy
)
Maxwell’s equations linear⇒ sum of two solutions again asolutionJones vector of sum of two waves = sum of Jones vectors ofindividual waves if wave vectors ~k the samewaves must have the same wave vector and direction ofpropagationaddition of Jones vectors: coherent superposition of waveselements of Jones vectors are not observed directlyobservables always depend on products of elements of Jonesvectors, i.e. intensity I = ~e · ~e∗ = exe∗x + eye∗y
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Jones matricesinfluence of medium on polarization described by 2× 2 complexJones matrix J
~e′ = J~e =
(J11 J12J21 J22
)~e
assumes that medium not affected by polarization statedifferent media 1 to N in order of wave direction⇒ combinedinfluence described by
J = JNJN−1 · · · J2J1
order of matrices in product is crucialJones calculus describes coherent superposition of polarized light
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Linear Polarization
horizontal:(
10
)vertical:
(01
)45◦: 1√
2
(11
)
Circular Polarization
left: 1√2
(1i
)right: 1√
2
(1−i
)
Notes on Jones FormalismJones formalism operates on amplitudes, not intensitiescoherent superposition important for coherent light (lasers,interference effects)Jones formalism describes 100% polarized light
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Stokes and Mueller Formalisms
Stokes Vectorformalism to describe polarization of quasi-monochromatic lightdirectly related to measurable intensitiesStokes vector fulfills these requirements
~I =
IQUV
=
ExE∗x + EyE∗yExE∗x − EyE∗yExE∗y + EyE∗x
i(ExE∗y − EyE∗x
) =
|Ex |2 + |Ey |2|Ex |2 − |Ey |2
2|Ex ||Ey | cos δ2|Ex ||Ey | sin δ
Jones vector elements Ex ,y , real amplitudes |Ex ,y |, phasedifference δ = δy − δx
I2 ≥ Q2 + U2 + V 2
can describe unpolarized (Q = U = V = 0) light
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Stokes Vector Interpretation
~I =
IQUV
=
intensity
linear 0◦ − linear 90◦
linear 45◦ − linear 135◦
circular left− right
degree of polarization
P =
√Q2 + U2 + V 2
I
1 for fully polarized light, 0 for unpolarized lightsumming of Stokes vectors = incoherent adding ofquasi-monochromatic light waves
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Linear Polarization
horizontal:
1100
vertical:
1−100
45◦:
1010
Circular Polarization
left:
1001
right:
100−1
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Mueller Matrices4× 4 real Mueller matrices describe (linear) transformationbetween Stokes vectors when passing through or reflecting frommedia
~I′ = M~I ,
M =
M11 M12 M13 M14M21 M22 M23 M24M31 M32 M33 M34M41 M42 M43 M44
N optical elements, combined Mueller matrix is
M′ = MNMN−1 · · ·M2M1
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Rotating Mueller Matricesoptical element with Mueller matrix MMueller matrix of the same element rotated by θ around the beamgiven by
M(θ) = R(−θ)MR(θ)
with
R (θ) =
1 0 0 00 cos 2θ sin 2θ 00 − sin 2θ cos 2θ 00 0 0 1
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Poincaré Sphere
Relation to Stokes Vectorfully polarized light:I2 = Q2 + U2 + V 2
for I2 = 1: sphere inQ, U, V coordinatesystempoint on Poincarésphere representsparticular state ofpolarizationgraphicalrepresentation offully polarized light
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Polarizers
polarizer: optical element that produces polarized light fromunpolarized input lightlinear, circular, or in general elliptical polarizer, depending on typeof transmitted polarizationlinear polarizers by far the most commonlarge variety of polarizers
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Jones Matrix for Linear Polarizers
Jones matrix for linear polarizer: Jp =
(px 00 py
)0 ≤ px ≤ 1 and 0 ≤ py ≤ 1, real: transmission factors for x ,y -components of electric field: E ′x = pxEx , E ′y = pyEy
px = 1, py = 0: linear polarizer in +Q directionpx = 0, py = 1: linear polarizer in −Q directionpx = py : neutral density filter
Mueller Matrix for Linear Polarizers
Mhorizontal =12
1 1 0 01 1 0 00 0 0 00 0 0 0
, Mvertical =12
1 −1 0 0−1 1 0 00 0 0 00 0 0 0
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Mueller Matrix for Ideal Linear Polarizer at Angle θ
Mpol (θ) =12
1 cos 2θ sin 2θ 0
cos 2θ cos2 2θ sin 2θ cos 2θ 0sin 2θ sin 2θ cos 2θ sin2 2θ 0
0 0 0 0
Poincare Sphere
polarizer is a point on the Poincaré spheretransmitted intensity: cos2(l/2), l is arch length of great circlebetween incoming polarization and polarizer on Poincaré sphere
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Wire Grid Polarizers
parallel conducting wires, spacing d . λ act as polarizerelectric field parallel to wires induces electrical currents in wiresinduced electrical current reflects polarization parallel to wirespolarization perpendicular to wires is transmittedrule of thumb:
d < λ/2⇒ strong polarizationd � λ⇒ high transmission of both polarization states (weakpolarization)
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Polaroid-type Polarizers
developed by Edwin Land in 1938⇒ Polaroidsheet polarizers: stretched polyvynil alcohol (PVA) sheet,laminated to sheet of cellulose acetate butyrate, treated withiodinePVA-iodine complex analogous to short, conducting wirecheap, can be manufactured in large sizes
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Crystal-Based Polarizerscrystals make highest-quality polarizersprecise arrangement of atom/molecules and anisotropy of index ofrefraction separate incoming beam into two beams with preciselyorthogonal linear polarization stateswork well over large wavelength rangemany different configurationscalcite most often used in crystal-based polarizers because oflarge birefringence, low absorption in visiblemany other suitable crystals
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Indices of Refraction of Crystalin anisotropic material: dielectric constant is a tensorMaxwell equations imply symmetric dielectric tensor
ε = εT =
ε11 ε12 ε13ε12 ε22 ε23ε13 ε23 ε33
symmetric tensor of rank 2⇒ Cartesian coordinate system existswhere tensor is diagonal3 principal indices of refraction in coordinate system spanned byprincipal axes
~D =
n2x 0 0
0 n2y 0
0 0 n2z
~E
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Uniaxial Materials
anisotropic materials:nx 6= ny 6= nz
uniaxial materials: nx = ny 6= nz
optic axis is axis that has differentindex of refraction, also called c orcrystallographic axisordinary index: no = nx = ny
extraordinary index: ne = nz
fast axis: axis with smallest indexrotation of coordinate systemaround z has no effectmost materials used in polarimetryare (almost) uniaxial, e.g. calcite
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Plane Waves in Anisotropic Media
no net charges,∇ · ~D = 0⇒ ~D · ~k = 0⇒ ~D ⊥ ~k~D 6‖ ~E ⇒ ~E 6⊥ ~kscalar µ, no current density⇒ ~H ‖ ~B∇ · ~H = 0⇒ ~H ⊥ ~k∇× ~H = 1
c∂~D∂t ⇒ ~H ⊥ ~D
∇× ~E = −µc∂~H∂t ⇒ ~H ⊥ ~E
~D, ~E , and ~k all in one plane~H, ~B perpendicular to that planePoynting vector ~S = c
4π~E × ~H
perpendicular to ~E and ~H ⇒ ~S (ingeneral) not parallel to ~kenergy (in general) not transported indirection of wave vector ~k
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Wave Propagation in Uniaxial Mediatwo solutions to wave equation with orthogonal linear polarizationsordinary ray propagates as in isotropic medium with index no
extraordinary ray sees direction-dependent index of refraction
n2 (θ) =none√
n2o sin2 θ + n2
e cos2 θ
n2 direction-dependent index of refraction of the extraordinary rayno ordinary index of refractionne extraordinary index of refractionθ angle between extraordinary wave vector and optic axis
for θ = 0 n2 = no, for θ = 90◦ n2 = ne
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Energy Propagation in Uniaxial Media
ordinary ray propagates along wave vector ~k with electric fieldperpendicular to c-axisextraordinary ray and wave vector make dispersion angle α
tanα =(n2
e − n2o) tan θ
n2e + n2
o tan2 θ
dispersion angle α = 0 for θ = 0 or θ = 90◦
extraordinary electric field in plane of wave vector and optic axis
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Crystal-Based Polarizing Beamsplitter
one linear polarization goes straight through as in isotropicmaterial (ordinary ray)perpendicular linear polarization propagates at an angle(extraordinary ray)different optical path lengthscrystal aberrations
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Brewster Angle Polarizer
rp = tan(θi−θt )tan(θi+θt )
= 0 when θi + θt =π2
corresponds to Brewster angle of incidence of tan θB = n2n1
occurs when reflected wave is perpendicular to transmitted wavereflected light is completely s-polarizedtransmitted light is moderately polarized
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Total Internal Reflection (TIR)
Snell’s law: sin θt =n1n2
sin θi
high-index to lower index medium (e.g. glass to air): n1/n2 > 1right-hand side > 1 for sin θi >
n2n1
all light is reflected in high-index medium⇒ total internal reflectiontransmitted wave has complex phase angle⇒ damped wavealong interface
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Total Internal Reflection (TIR) in Crystalsno 6= ne ⇒ one beam can be totallyreflected while other is transmittedprincipal of most crystal polarizerscalcite at 632.8 nm: no = 1.6558,ne = 1.4852requirement for total reflectionn2sinθ > 1entrance: extraordinary ray notrefracted, two rays propagateaccording to indices no,ne
exit: rays (and wave vectors) at 40◦
to surfacefor θ = 40◦ extraordinary ray istransmitted, ordinary ray undergoesTIR
no, n
e
e
o
40°
40°
c
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Wollaston Prism
Savart Plate
Foster Prism
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Linear Retarders or Wave Platesuniaxial crystal, optic axis parallel to surface (θ = 90◦)fast axis (f) has lowest index, slow axis (s) has highest indexexample: halfwave retarder
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Phase Delay between Ordinary and Extraordinary Raysordinary and extraordinary wave propagate in same directionordinary ray propagates with speed c
no
extraordinary beam propagates at different speed cne
~Eo, ~Ee perpendicular to each other⇒ plane wave with arbitrarypolarization can be (coherently) decomposed into componentsparallel to ~Eo and ~Ee
2 components will travel at different speeds(coherently) superposing 2 components after distance d ⇒ phasedifference between 2 components ω
c (ne − no)d radiansphase difference⇒ change in polarization statebasis for constructing linear retarders
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Retarder Propertiesdoes not change intensity or degree of polarizationcharacterized by two (not identical, not trivial) Stokes vectors ofincoming light that are not changed by retarder⇒ eigenvectors ofretarderdepending on polarization described by eigenvectors, retarder is
linear retardercircular retarderelliptical retarder
linear, circular retarders are special cases of elliptical retarderscircular retarders sometimes called rotators since they rotate theorientation of linearly polarized lightlinear retarders by far the most common type of retarderretardation depends strongly on wavelengthachromatic retarders: combinations of different materials or thesame materials with different fast axis directions
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Jones Matrix for Linear Retarderslinear retarder with fast axis at 0◦ characterized by Jones matrix
Jr (δ) =
(eiδ 00 1
), Jr (δ) =
(ei δ2 00 e−i δ2
)
δ: phase shift between two linear polarization components (inradians)absolute phase does not matter
Mueller Matrix for Linear Retarder
Mr =
1 0 0 00 1 0 00 0 cos δ − sin δ0 0 sin δ cos δ
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Quarter-Wave Plate on the Poincaré Sphere
retarder eigenvector (fast axis) in Poincaré spherepoints on sphere are rotated around retarder axis by amount ofretardation
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Phase Change on Total Internal Reflection (TIR)
TIR induces phase changethat depends on polarizationcomplex ratios:rs,p = |rs,p|eiδs,p
phase change δ = δs − δp
tanδ
2=
cos θi
√sin2 θi −
(n2n1
)2
sin2 θi
relation valid between criticalangle and grazing incidenceat critical angle and grazingincidence δ = 0
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Variable Retarderssensitive polarimeters requires retarders whose properties(retardance, fast axis orientation) can be varied quickly(modulated)retardance changes (change of birefringence):
liquid crystalsFaraday, Kerr, Pockels cellspiezo-elastic modulators (PEM)
fast axis orientation changes (change of c-axis direction):rotating fixed retarderferro-electric liquid crystals (FLC)
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Liquid Crystals
liquid crystals: fluids with elongated moleculesat high temperatures: liquid crystal is isotropicat lower temperature: molecules become ordered in orientationand sometimes also space in one or more dimensionsliquid crystals can line up parallel or perpendicular to externalelectrical field
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Liquid Crystal Retarders
dielectric constant anisotropy often large⇒ very responsive tochanges in applied electric fieldbirefringence δn can be very large (larger than typical crystalbirefringence)liquid crystal layer only a few µm thickbirefringence shows strong temperature dependence
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