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Lecture 7: Linear Polarizers

Outline

1 Jones and Mueller Matrices for Linear Polarizers2 Wire Grid Polarizers3 Polaroid-type Polarizers4 Crystal-based Polarizers5 Thin-Film Polarizers6 Polarizer Selection Guide

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 1

Polarizerspolarizer: optical element producing (at least partially) polarizedlight from unpolarized input beamlinear, circular, or in general, elliptical polarizermost common: linear polarizerlarge variety of types

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 2

Jones Matrix for Linear Polarizerslinear polarizer described by transmittance of electrical field in 2orthogonal directionsJones matrix for linear polarizer:

Jp =

(px 00 py

)real values 0 ≤ px ≤ 1 and 0 ≤ py ≤ 1: transmission factors for xand 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

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 3

Mueller Matric for Linear Polarizer

Mp =12

p2

x + p2y p2

x − p2y 0 0

p2x − p2

y p2x + p2

y 0 00 0 2pxpy 00 0 0 2pxpy

unpolarized incoming beam always linearly polarizedemerging Stokes vector only completely polarized ifp2

x + p2y =

∣∣p2x − p2

y∣∣

partial linear polarizer produces partially polarized beam fromunpolarized lightreal polarizers always only partial polarizerspolarized incoming beam⇒ elliptically polarized output beambecause of non-zero diagonal terms 2pxpy

totally polarized beam remains totally polarized even whenpassing partial linear polarizer (no depolarization)

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 4

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

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 5

Characterizing Linear Polarizers

2 parameters describing performance:k1: transmittance for fully linearly polarized beam at angle thatmaximizes transmitted intensityk2: minimum transmittance for incoming linearly polarized beam

k1 = p2x , k2 = p2

y if px > py

extinction ratio k1/k2 contrastk1, k2 depend on wavelengthdetermined from transmittances for unpolarized light of paralleland crossed identical polarizers

Tparallel = 12

(k2

1 + k22)

Tcrossed = k1k2

also used: degree of polarizability or polarizance defined by

P =k1 − k2

k1 + k2

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 6

Wire Grid Polarizers

parallel conducting wires, spacing d . λ act as polarizerelectric field component parallel to wires induces electricalcurrents in wires, attenuates transmitted electric field parallel towiresinduced electrical current ’reflects’ polarization parallel to wires⇒polarizing beam-splitter

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 7

Wire Grid Polarizers for the Infrared

rule of thumb:d < λ/2⇒ strong polarizationd λ⇒ high transmission of both polarization states (weakpolarization)

mostly used in infrared because wire spacing becomes very smallat visible wavelengthsmade by depositing thin-film metallic grid on substratefree-standing wire grid for longer wavelengths

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 8

Wire-Grid Polarizers for the Visibleexcellent performance at highlight/energy levels90% of energy reflectedgood in high heat/fluxenvironmenttrade-off between transmissionand contrast

MOXTEK Inc. ProFlux

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 9

Polarcor (Corning)glass polarizer with highperformance for 600 to2300 nmglass with aligned silvernano-particles in surfacelayerselongated, conducting silverparticles act as small wiresmaximum diameter currentlylimited to 20 mmcontrast ratio > 10000

Corning Polarcor

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 10

Dichroic Materials

absorb one polarization stateabsorption depends on wavelength⇒different colors depending on angles ofillumination and viewingarises from anisotropy of complex indexof refractionnatural dichroic crystals: tourmaline,herapathiteW.B. Herapath discovered in 1852 salt ofquinine with polarizing properties; madefirst artificial crystals large enough tostudy under microscopedifficult to produce uniform, large dichroiccrystals

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 11

Polaroid-type Polarizers

H-type sheet polarizers: stretched polyvynil alcohol (PVA) sheet,laminated to sheet of cellulose acetate butyrate, treated withiodinedifferent H-type polarizers have different amounts of iodine in PVAPVA-iodine complex analogous to short, conducting wirePolaroid names (e.g. HN-38) identify overall type (H), color(N=neutral), approximate transmittance for unpolarized lightK-type similar to H-type, but environmentally more stableHR-type based on a PVA-polyvinylene-iodine complex, works wellfrom 0.7 to 2.3 µm

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 12

Field-of-View of Dichroic Polarizersconducting, needle-like particles aligned on common axis parallelto surfaceuniaxially anisotropic medium with complex indices of refractiontransmitted polarization state sees real index of refractionabsorbed polarization state sees complex index of refraction withimaginary part large enough to reduce intensity by orders ofmagnitudedichroic polarizers have limited field of view, largely a geometricaleffect inherent to uniaxial medium

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 13

Crystal-Based Polarizershighest quality polarizersprecise arrangement of atom/molecules and anisotropy separateincoming beam into two beams with precisely orthogonalpolarizationpolarizing beam-splitters (both beams usable) and polarizingprisms (only one useful state)prisms can be cemented or air spacedair-spaced: good for short wavelengths, high power densitiescemented: much better optical qualitycalcite is most often used in crystal-based polarizers because ofvery large birefringence, low absorption in visiblemany other suitable materials

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 14

Beam Displacer, Savart and Modified Savart Plates

Beam Displacer

most simple crystal polarizer: polarizing beam displacersingle uniaxial crystal block with optic axis at ∼45

ordinary ray passes without deflectionextraordinary ray deflected by dispersion angle αbeam separation d as function of block length D:

d = D tanα = D(n2

e − n2o) tan θ

n2e + n2

o tan2 θ

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 15

Beam Displacer Problems

ordinary and extraordinary beams have different path lengthsextraordinary ray suffers from crystal astigmatism

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 16

Focus Difference and Astigmatism

converging beam focus difference ε between ordinary andextraordinary rays:

ε =sin 2θ

(n2

e − n2o)

2(

n2o sin2 θ + n2

e cos2 θ)

θ is angle of optic axis with normal to interfaceastigmatism leads to two focus positions longitudinal astigmatism

εl =D tan θ tanαno

ne

√n2

o sin2 θ + n2e cos2 θ

for imaging system, ’smearing’ of image given by transverseastigmatism εt = εl

F where F is the F-number of the beamrule of thumb for calcite: astigmatic focus difference εl ∼5% ofthickness

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 17

Beam Separation and Astigmatism

beam separation d (solid),longitudinal astigmatism εl(dashed) of extraordinary beamfor 10-mm calcite beam displacerat 630 nm

optic axis orientation of 45 close to optimumbeam separation increases almost linearly for small optic axisanglesastigmatism increases much more slowlypossible to trade off beam separation versus astigmatism

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 18

Savart Plate

avoid difference of focal point position problem by splitting calciteinto two pieces crossed at 90

exchanges ordinary and extraordinary beams half-way⇒ bothbeams have same optical path lengthsplitting reduced by

√2 for same total length of calcite

both beams now show crystal astigmatism, oriented in oppositedirections; amount of astigmatism half of single-piece beamdisplacer

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 19

Modified Savart Plate

both calcite blocks in same orientationhalf-wave plate at 45 between them to exchange ordinary andextraordinary beamssame direction of astigmatism in both beams

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 20

Cylinder Lens Compensating Crystal Astigmatism

additional cylindrical lens compensates astigmatismlimited wavelength range because of wavlength dependence ofhalf-wave retarderdeviation from half-wave retardance or 45 orientation leads toghost beam between oppositely polarized beamsother aberrations such as spherical aberration occur alsodeveloped for SOLIS VSM, excellent performance

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 21

Glan-Thompson Prism

two calcite prisms with cement (index nc) in betweensurrounding dielectric medium with refractive index nordinary beam undergoes TIR, absorbed by black paint on sidecritical cut angles for TIR for ordinary and extraordinary beams

sin Ωo =nc

no, sin Ωe =

nc

ne

cut angle Ω such that acceptable range of angles of incidence issymmetric around normal incidence

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 22

Glan-Thompson Properties

2 versions differ in prism angles, index of cementlong version: acceptance angle of ∼26

short form: acceptance angle of ∼15

typical extinction ratios 105 to 107

somewhat reduced extinction rations when used from “wrong” sideif prism is illuminated with convergent light, different parts of beamwill have slightly different orientations of linear polarization⇒strong reduction in extinction ratio for convergent beam

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 23

Wollaston Prism

Wollaston prisms most common beam-deviating polarizers2 beams with orthogonal linear polarizations, parallel andperpendicular to refracting edgeboth rays refracted away from normal of exit facefirst half: ordinary and extraordinary rays see their respectiveindices of refractionindices are reversed in second half of prismmade of calcite or quartz prisms cemented togetherusable spectral range typically 300 nm to 2200 nmangle of divergence determined by wedge angle (∼ 15− 45)

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 24

Wollaston Beam Deviationat refracting interface, Snell’s law for the ordinary andextraordinary beams

no sin Ω = ne sin θo and ne sin Ω = no sin θe

at exit face, two beams are refracted differently; with Snell’s law

ne sin (Ω− θo) = n sin θ′o and no sin (Ω− θe) = n sin θ′e

n is index of external dielectric mediumequations are not completely symmetrical⇒ two beams will haveslightly different absolute angles upon exitingbeam deviation depends on wavelength-dependent birefringencein converging beam, two beams do not come to focus at the samedistance behind the exit faceeach beam is astigmatic

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 25

Thin-Film Polarizers

thin-film polarizers mostly used in cube beam-splitters2 orthogonally polarized beams emerge at right anglesthin-film stack between 2 cemented glass prismstotal internal reflection at Brewster angle within thin filmlimited extinction ratio and wavelength rangeextinction ratio of transmitted beam much better than reflectedbeamproduced cheaply even for large apertures (5–10 cm)also possible on surface of oblique glass plate

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 26

Polarizer Selection Guidetype extinction transmission wavelength bandpass acceptance size cost

ratio (polarized) range (nm) (nm) angle () mm

Glan > 105 > 84% 300-2700 full 8 < 40 $$$Glan-Thompson > 106 > 92% 300-2700 full 15-25 < 30 $$$Wollaston > 106 > 92% 300-2200 full 20 < 50 $$Polarcor > 104 > 80% 630-2300 150 > 20 < 25 $$Polaroid 150− 104 > 75% 310-2000 200 > 20 > 200 $Polarizing cube > 500 > 90% 400-1600 200-400 10 70 $$Wire Grid > 100 > 90% 4 · 102 − 106 ∼ λ > 20 > 70 $$

Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 7: Linear Polarizers 27