lecture 4: polarization - leiden observatorykeller/teaching/... · lecture 4: polarization content...

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

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


13.7 billion year old temperature fluctuations from WMAP

BICEP2 results and Planck Dust Polarization Map


Scattering Polarization 1


Scattering Polarization 2


The Power of Polarized Light Measurements

T Tauri in intensity MWC147 in intensity


The Power of Polarimetry

T Tauri in Linear Polarization MWC147 in Linear Polarization


Solar Magnetic Field Maps from Longitudinal Zeeman Effect


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


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


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


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


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


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


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


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


Linear Polarization

horizontal:

1100

vertical:

1−100

45◦:

1010

Circular Polarization

left:

1001

right:

100−1


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


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


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


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


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


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


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)


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


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


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


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


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


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


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


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


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


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


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


Wollaston Prism

Savart Plate

Foster Prism


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


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


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


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


Quarter-Wave Plate on the Poincaré Sphere

retarder eigenvector (fast axis) in Poincaré spherepoints on sphere are rotated around retarder axis by amount ofretardation


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


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)


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


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


lecture 4: polarization - leiden observatorykeller/teaching/... · lecture 4: polarization content...

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