Lecture 5: Linear Polarizers
Outline
1 Jones and Mueller Matrices for Linear Polarizers 2 Wire Grid Polarizers 3 Polaroid-type Polarizers Polarizers 4 Crystal-based Polarizers polarizer: optical element that produces (at least partially) 5 Thin-Film Polarizers polarized light when the input light beam is unpolarized polarizer can be linear, circular, or in general, elliptical, depending 6 Polarizer Selection Guide on the type of polarization that emerges linear polarizers by far the most common large variety of polarizers that all have their respective advantages and disadvantages
Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 1 Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 2
Mueller Matric for Linear Polarizer Jones Matrix for Linear Polarizers 2 2 2 2 px + py px − py 0 0 linear polarizer described by its transmittance of electrical field in 2 2 2 2 1 px − py px + py 0 0 two orthogonal directions Mp = 2 0 0 2px py 0 Jones matrix for linear polarizer: 0 0 0 2px py px 0 Jp = unpolarized incoming beam will always be linearly polarized 0 py emerging Stokes vector only completely polarized if 2 2 2 2 real values 0 ≤ px ≤ 1 and 0 ≤ py ≤ 1 are transmission factors for px + py = px − py x and y-components of electric field partial linear polarizer produces partially polarized beam from unpolarized light E0 = p E , E0 = p E x x x y y y real polarizers always only partial polarizers
px = 1, py = 0: linear polarizer in +Q direction polarized incoming beam ⇒ emerging beam is, in general, elliptically polarized because of non-zero diagonal terms 2px py px = 0, py = 1: linear polarizer in −Q direction totally polarized beam remains totally polarized even when px = py : neutral density filter passing partial linear polarizer ⇒ ideal partial polarizer does not depolarize
Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 3 Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 4 Mueller Matrix for Ideal Linear Polarizer at Angle θ Characterizing Linear Polarizers 1 cos 2θ sin 2θ 0 2 parameters describe linear polarizer performance: 1 cos 2θ cos2 2θ sin 2θ cos 2θ 0 k1: (intensity) transmittance of polarizer for fully linearly polarized Mpol (θ) = beam at angle that maximize transmitted intensity 2 sin 2θ sin 2θ cos 2θ sin2 2θ 0 k : minimum transmittance for incoming linearly polarized beam 0 0 0 0 2 2 2 k1 = px , k2 = py if px > py Poincare Sphere ratio of k1 to k2 is called extinction ratio contrast k1 and k2 are functions of wavelength can be determined from transmittances for unpolarized light of parallel and crossed identical polarizers
1 2 2 Tparallel = 2 k1 + k2 Tcrossed = k1k2
also used: degree of polarizability or polarizance defined by polarizer is a point on the Poincaré sphere transmitted intensity: cos2(l/2), l is arch length of great circle k − k P = 1 2 between incoming polarization and polarizer on Poincaré sphere k1 + k2
Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 5 Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 6
Wire Grid Polarizers Wire Grid Polarizers for the Infrared
parallel conducting wires, spacing d λ act as polarizer . rule of thumb: plane of polarization perpendicular to wires is transmitted d < λ/2 ⇒ strong polarization because electric field component parallel to wires induces d λ ⇒ high transmission of both polarization states (weak electrical currents in wires, which strongly attenuates transmitted polarization) electric field parallel to wires mostly used in infrared because wire spacing becomes very small induced electrical current such that polarization parallel to wires is at visible wavelengths reflected made by depositing thin-film metallic grid on substrate can make polarizing beam-splitter with wire grid polarizer, reflects, free-standing wire grid for longer wavelengths transmits orthogonal linear polarization states
Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 7 Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 8 Wire-Grid Polarizers for the Visible MOXTEK Inc. ProFlux Polarcor (Corning) Corning Polarcor excellent performance at high glass polarizer with high light/energy levels performance for 600 to 90% of energy reflected instead 2300 nm of absorbed borosilicate glass containing good in high heat environment aligned silver nano-particles in surface layers trade-off between transmission and contrast elongated, conducting silver particles act as small wires polarization occurs in 25 to 50 µm top layer maximum diameter currently limited to 20 mm contrast ratio > 10000
Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 9 Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 10
Dichroic Materials
dichroic materials preferentially absorb O-type and E-type Dichroic Polarizers one polarization state O-type polarizers transmit ordinary ray (k = 0), attenuate absorption depends on wavelength ⇒ o extraordinary ray (k > 0) different colors depending on angles of e illumination and viewing O-type polarizers transmit independent of angle of incidence because index of refraction of the ordinary beam is independent arises from anisotropy of complex index of the angle of incidence of refraction E-type polarizers transmit extraordinary ray and absorb ordinary natural dichroic crystals: tourmaline, ray herapathite E-type polarizers attenuate light in any direction except for waves W.B. Herapath discovered in 1852 salt of propagating perpendicular to c-axis quinine with polarizing properties; made first artificial crystals large enough to O-type dichroic polarizers much preferred over E-type polarizers study under microscope difficult to produce uniform, large dichroic crystals
Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 11 Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 12 Polaroid-type Polarizers
Field-of-View of Dichroic Sheet Polarizers conducting, needle-like particles aligned on common axis perpendicular to surface normal model them with uniaxially anisotropic medium with complex H-type sheet polarizers: stretched polyvynil alcohol (PVA) sheet, indices of refraction laminated to sheet of cellulose acetate butyrate, treated with transmitted polarization state sees real index of refraction iodine absorbed polarization state sees complex index of refraction with different H-type polarizers have different amounts of iodine in PVA imaginary part large enough to reduce intensity by orders of PVA-iodine complex analogous to short, conducting wire magnitude Polaroid names (e.g. HN-38) identify overall type (H) color dichroic polarizers have limited field of view, largely a geometrical (N=neutral), approximate transmittance for unpolarized light effect inherent to uniaxial medium K-type similar to H-type, but environmentally more stable HR-type based on a PVA-polyvinylene-iodine complex, works well from 0.7 to 2.3 µm
Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 13 Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 14
Beam Displacer, Savart and Modified Savart Plates
Crystal-Based Polarizers Beam Displacer uniaxial crystals are basis of highest quality polarizers precise arrangement of atom/molecules and anisotropy separate incoming beam into two beams with precisely orthogonal polarization polarizing beam-splitters (both beams usable) and polarizing prisms (only one useful state) most simple crystal polarizer: polarizing beam displacer prisms can be cemented or air spaced single uniaxial crystal block with optic axis at ∼45◦ air-spaced: good for short wavelengths, high power densities ordinary ray passes without deflection cemented: much better optical quality extraordinary ray deflected by dispersion angle α calcite is most often used in crystal-based polarizers because of very large birefringence, low absorption in visible beam separation d as function of block length D: many other suitable materials (n2 − n2) tan θ d = D tan α = D e o 2 2 2 ne + no tan θ
Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 15 Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 16 Focus Difference and Astigmatism converging beam focus difference between ordinary and extraordinary rays: Beam Displacer Problems 2 2 sin 2θ ne − no = 2 2 2 2 2 no sin θ + ne cos θ
θ is angle of optic axis with normal to interface astigmatism leads to two focus positions longitudinal astigmatism
D tan θ tan αno ordinary and extraordinary beams have different path lengths l = q 2 2 2 2 extraordinary ray suffers from crystal astigmatism ne no sin θ + ne cos θ for imaging system, ’smearing’ of image given by transverse l astigmatism t = F where F is the F-number of the beam rule of thumb for calcite: astigmatic focus difference l ∼5% of thickness
Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 17 Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 18
Beam Separation and Astigmatism Savart Plate
beam separation d (solid), longitudinal astigmatism l (dashed) of extraordinary beam for 10-mm calcite beam displacer at 630 nm avoid difference of focal point position problem by splitting calcite into two pieces crossed at 90◦ optic axis orientation of 45◦ close to optimum exchanges ordinary and extraordinary beams half-way ⇒ both beams have same optical path length beam separation increases almost linearly for small optic axis √ angles splitting reduced by 2 for same total length of calcite astigmatism increases much more slowly both beams now show crystal astigmatism, oriented in opposite directions; amount of astigmatism half of single-piece beam possible to trade off beam separation versus astigmatism displacer
Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 19 Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 20 Cylinder Lens Compensating Crystal Astigmatism Modified Savart Plate
additional cylindrical lens can compensate astigmatism both calcite blocks in same orientation only works over a limited wavelength range because of wavlength half-wave plate at 45◦ between them to exchange ordinary and dependence of half-wave retarder ◦ extraordinary beams deviation from half-wave retardance or 45 orientation leads to same direction of astigmatism in both beams ghost beam between oppositely polarized beams other aberrations such as spherical aberration occur also developed for SOLIS VSM, excellent performance
Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 21 Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 22
Glan-Thompson Prism Glan-Thompson Properties
2 versions differ in prism angles, index of cement long version: acceptance angle of ∼26◦ short form: acceptance angle of ∼15◦ 5 7 two calcite prisms with cement (index nc) in between typical extinction ratios 10 to 10 surrounding dielectric medium with refractive index n somewhat reduced extinction rations when used from “wrong” side ordinary beam undergoes TIR, absorbed by black paint on side Glan-Thompson gives most uniform rotation of plane of critical cut angles for TIR for ordinary and extraordinary beams polarization for conical beam of light plane of vibration of transmitted beam may vary if optic axis is not n n sin Ω = c , sin Ω = c in the end face o n e n o e if prism is illuminated with convergent light, different parts of beam cut angle Ω such that acceptable range of angles of incidence is will have slightly different orientations of linear polarization ⇒ symmetric around normal incidence strong reduction in extinction ratio for convergent beam
Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 23 Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 24 Other Glan Prism Polarizers Rochon Prism
Glan or Glan-Foucault: Glan-Thompson with air gap absorption of the deflected beam above 2 µm oldest polarizing beam-splitter (1783) ◦ ◦ field of view much reduced, between 13 and 7.5 first prism: beam parallel to optic axis ⇒ ordinary and acceptance angle rotationally symmetric only at single wavelength extraordinary beams see same index n TIR of ordinary ray responsible for polarization o transmission is not as high as Glan-Thompson because of second prism: optic axis perpendicular to beam reflection losses at calcite-air interfaces ordinary beam sees again same index, passes without deviation ◦ Glan-Taylor: orientation of optic axis 90 rotated with respect to extraordinary beam sees extraordinary index ⇒ refracted Glan-Thompson according to Snell’s law of importance only air-spaced version main advantage over Glan-Foucault type prisms is improved deviation of extraordinary ray at exit face due to Snell’s law transmission and much smaller amount of multiple reflections Sénarmont prism similar to Rochon, but ordinary and between the two prism halfs extraordinary beams have opposite polarization states
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Wollaston Prism Wollaston Beam Deviation at refracting interface, Snell’s law for the ordinary and extraordinary 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 Wollaston prisms most common beam-deviating polarizers 0 0 2 beams with orthogonal linear polarizations, parallel and ne sin (Ω − θo) = n sin θo and no sin (Ω − θe) = n sin θe perpendicular to refracting edge both rays refracted away from normal of exit face n is index of external dielectric medium first half: ordinary and extraordinary rays see their respective equations are not completely symmetrical ⇒ two beams will have indices of refraction slightly different absolute angles upon exiting indices are reversed in second half of prism beam deviation depends on wavelength-dependent birefringence made of calcite or quartz prisms cemented together in converging beam, two beams do not come to focus at the same distance behind the exit face usable spectral range typically 300 nm to 2200 nm each beam is astigmatic angle of divergence determined by wedge angle (∼ 15 − 45◦)
Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 27 Christoph U. Keller, Utrecht University, [email protected] Lecture 5: Linear Polarizers 28 Thin-Film Polarizers
Polarizer Selection Guide type extinction transmission wavelength bandpass acceptance size cost ratio (polarized) range (nm) (nm) angle (◦) mm
thin-film polarizers mostly used in cube beam-splitters Glan > 105 > 84% 300-2700 full 8 < 40 $$$ Glan-Thompson > 106 > 92% 300-2700 full 15-25 < 30 $$$ 2 orthogonally polarized beams emerge at right angles Wollaston > 106 > 92% 300-2200 full 20 < 50 $$ Polarcor > 104 > 80% 630-2300 150 > 20 < 25 $$ thin-film stack between 2 cemented glass prisms Polaroid 150 − 104 > 75% 310-2000 200 > 20 > 200 $ Polarizing cube > 500 > 90% 400-1600 200-400 10 70 $$ total internal reflection at Brewster angle within thin film Wire Grid > 100 > 90% 4 · 102 − 106 ∼ λ > 20 > 70 $$ limited extinction ratio and wavelength range extinction ratio of transmitted beam much better than reflected beam produced cheaply even for large apertures (5–10 cm) also possible on surface of oblique glass plate
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