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Polarization

The document discusses polarization of light, including: 1) Natural or unpolarized light consists of randomly oriented electromagnetic waves from many emitters. Monochromatic planar waves can be linearly, circularly, or elliptically polarized depending on their wave properties. 2) Several methods can achieve polarization, including dichroism via materials that selectively absorb certain orientations, scattering, reflection using Brewster's angle, and birefringence in crystals. 3) Key polarization components are described like polarizers using dichroic materials, wire grids, reflection, and birefringent prisms made of crystals like calcite or quartz. Polarization ellipses represent the tip trajectory of the oscillating electric field

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1 POLARIZATION SOLO HERMELIN Updated 27.05.06http://www.solohermelin.com
POLARIZATION History SOLO TABLE OF CONTENT Natural or Unpolarized Light Monochromatic Planar Wave Equations Linear Polarization or Plane-Polarization Circular Polarization Elliptically Polarization Methods of Achieving Polarization Polarization Ellipse Degenerated States of Polarization Ellipse The Stokes Polarization Parameters Measuring the Stokes Parameters Poincaré Sphere The Mueller Matrices for Polarizing Components The Jones Polarization Parameters
3 POLARIZATIONSOLO TABLE OF CONTENT (Continue – 1) Faraday Effect Electro – Optical Effects ( Pockels, Kerr) References Optics Polarization
4 POLARIZATION Erasmus Bartholinus, doctor of medicine and professor of mathematics at the University of Copenhagen, showed in 1669 that crystals of “Iceland spar” (which we now call calcite, CaCO3) produced two refracted rays from a single incident beam. One ray, the “ordinary ray”, followed Snell’s law, while the other, the “extraordinary ray”, was not always even in the plan of incidence. SOLO History Erasmus Bartholinus 1625-1698 http://www.polarization.com/history/history.html
5 POLARIZATIONSOLO History Étienne Louis Malus 1775-1812 Etienne Louis Malus, military engineer and captain in the army of Napoleon, published in 1809 the Malus Law of irradiance through a Linear polarizer: I(θ)=I(0) cos2 θ. In 1810 he won the French Academy Prize with the discovery that reflected and scattered light also possessed “sidedness” which he called “polarization”.
6 POLARIZATION Arago and Fresnel investigated the interference of polarized rays of light and found in 1816 that two rays polarized at right angles to each other never interface. SOLO History (continue) Dominique François Jean Arago 1786-1853 Augustin Jean Fresnel 1788-1827 Arago relayed to Thomas Young in London the results of the experiment he had performed with Fresnel. This stimulate Young to propose in 1817 that the oscillations in the optical wave where transverse, or perpendicular to the direction of propagation, and not longitudinal as every proponent of wave theory believed. Thomas Young 1773-1829 Return to Table of Content
7 POLARIZATIONSOLO The Natural Light is emitted by excitation of material atoms. Each excited atom radiates a wave train for roughly 10-8 sec. The Natural light is the result of radiation of a large collection of such atoms. New waves are emitted in a completely unpredictable fashion and the light is actually composed of a succession of short living polarization states. Natural or Unpolarized Light A light source consists of a very large number of randomly oriented atoms emitters. We say that the Natural light is compose of a collection of monochromatic (polichromatic) unpolarized rays. One mathematical description of a monochromatic unpolarized ray moving in z direction, at a certain location in space, is by a Electric Intensity phasor of constant amplitude and a random phase: ( ) ( )( ) ( )( ) yx tzktjtzktj yx eAeAtE 11 ∧ +− ∧ +− += δωδω zyx 111 ,, ∧∧∧ are orthogonal unit vectors and δx (t) are δy (t) are randomly phase angles.
8 POLARIZATIONSOLO Light is a transverse electromagnetic wave; i.e. the Electric and Magnetic Intensities are perpendicular to each other and oscillate perpendicular to the direction of propagation. For the natural light the direction of the Electric Intensity vector changes randomly from time to time. We say that the natural light is Unpolarized. A Planar wave (in which the Electric Intensity propagates remaining in a plane – containing the propagation direction) is said to be Linearly Polarized or Plane-Polarized. If light is composed of two plane waves of equal amplitude but differing in phase by 90° then the light is aid to be Circular Polarized. If light is composed of two plane waves of different amplitudes and/or the difference in phase is different than 0,90,180,270° then the light is said to be Elliptically Polarized. E  Return to Table of Content
9 ELECTROMAGNETICSSOLO To satisfy the Maxwell equations for a source free media we must have: Monochromatic Planar Wave Equations we haveUsing: 1ˆˆ&ˆˆ 0 =⋅== kkknkkk εµω         =⋅∇ =⋅∇ −=×∇ =×∇ 0 0 H E HjE EjH ωµ ωε            =⋅ =⋅ =× −=× 0ˆ 0ˆ ˆ ˆ 0 0 00 00 Hk Ek HEk EHk ε µ µ ε        =⋅− =⋅− −=×− =×− ⇒ ⋅− ⋅− ⋅−⋅− ⋅−⋅− −=∇ ⋅−⋅− 0 0 0 0 00 00 rkj rkj rkjrkj rkjrkj ekje eHkj eEkj eHjeEkj eEjeHkj rkjrkj           ωµ ωε ( ) * 2 * 2 & 2 ˆ 2 ˆ HHwEEwwcn k wwcn k S meme µε ====+=  Time Average Poynting Vector of the Planar Wave ( ) ( )rktjrktj eHHeEE  ⋅−⋅− == ωω 00 & Return to Table of Content
10 POLARIZATIONSOLO A Planar wave (in which the Electric Intensity propagates remaining in a plane – containing the propagation direction) is said to be Linearly Polarized or Plane-Polarized. http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html http://www.enzim.hu/~szia/cddemo/edemo0.htm)Andras Szilagyi( Linear Polarization or Plane-Polarization ( ) yyzktj y eAE 1 ∧ +− = δω Return to Table of Content
11 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html If light is composed of two plane waves of equal amplitude but differing in phase by 90° then the light is said to be Circular Polarized. http://www.optics.arizona.edu/jcwyant/JoseDiaz/Polarization-Circular.htm ( ) ( ) yx xx zktjzktj eAeAE 11 2/ ∧ ++− ∧ +− += πδωδω Circular Polarization Return to Table of Content
12 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html If light is composed of two plane waves of different amplitudes and/or the difference in phase is different than 0,90,180,270° then the light is said to be Elliptically Polarized. ( ) ( ) yx yx zktj y zktj x eAeAE 11 ∧ +− ∧ +− += δωδω Elliptically Polarization Return to Table of Content
13 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Methods of Achieving Polarization Polarization is based on one of four fundamental physical mechanisms: 1. Dichroism or selective absorbtion 3. Reflection 2. Scattering 4. Birefrigerence To obtain Polarization we must have some asymmetry in the optical process.
14 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Methods of Achieving Polarization (continue – 1) Polarization by Dichroism A dichroic material has different absorption properties for perpendicular incident planes. An example of a dichroic material is the tourmaline that is a class of borone silicate. The tourmaline has a unique optic axis, and any electronic field normal to it is strongly absorbed.
15 POLARIZATION SOLO Polaroids (Dichroism) http://en.wikipedia.org/wiki/Edwin_H._Land Edwin H. Land 1909-1991 In 1928 Edwin H. Land undergraduate at Harvard College invented the Polaroid J-sheet. It consists of many microscopic Crystals of iodoquinine sulphate embeded in a transparent Nitrocellulose polymer film. The sunglasses use polaroid material that uses dichroism to achieve absorption..
16 Methods of Achieving Polarization (continue – 1) Wire-Grid Polarizer (Dichroism) POLARIZATIONSOLO Grid of parallel conducting wires with a spacing comparable to the wavelength of the electromagnetic wave. The Electric Field vector parallel to the wires is attenuated because of the currents induced in the wires.
17 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Methods of Achieving Polarization (continue – 2) Polarization by Scattering
18 POLARIZATIONSOLO Methods of Achieving Polarization (continue – 3) Polarization by Scattering
19 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Methods of Achieving Polarization (continue – 4) Polarization by Reflection
20 POLARIZATIONSOLO Methods of Achieving Polarization (continue – 5) Polarization by Reflection The Pile-of-plate Polarizer The problem encounter using the Brewster Effect is that the reflected beam although completely polarized is weak and the refracted beam is only partially polarized. The solution is to use a pile-of- plates polarizer as in Figure. This was invented by F.J. Arago in 1812. Dominique François Jean Arago 1786-1853
21 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Polarization by Birefrigerence (continue – 4) Polarization can be achieved with crystalline materials which have a different index of refraction in different planes. Such materials are said to be birefringent or doubly refracting. Nicol Prism The Nicol Prism (1828) is made up from two prisms of calcite cemented with Canada balsam. The ordinary ray can be made to totally reflect off the prism boundary, leving only the extraordinary ray. William Nicol(1768 ?– 1851) Scottish physicist Methods of Achieving Polarization (continue – 6)
22 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Polarization by Birefrigerence (continue – 4) Polarization can be achieved with crystalline materials which have a different index of refraction in different planes. Such materials are said to be birefringent or doubly refracting. Wollaston Prism William Hyde Wollaston 1766-1828 Methods of Achieving Polarization (continue – 7)
23 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Polarization by Birefrigerence (continue – 4) Polarization can be achieved with crystalline materials which have a different index of refraction in different planes. Such materials are said to be birefringent or doubly refracting. Glan-Foucault Polarizer Methods of Achieving Polarization (continue – 8)
24 POLARIZATION Methods of Achieving Polarization (continue – 9)
25 POLARIZATION Methods of Achieving Polarization (continue – 10)
26 POLARIZATION SOLO Polarizations Prisms Overview http://www.unitedcrystals.com/POverview.html Methods of Achieving Polarization (continue – 11)
27 POLARIZATION SOLO Polarizations Prisms Overview http://www.unitedcrystals.com/POverview.html Methods of Achieving Polarization (continue – 12) Return to Table of Content
28 POLARIZATIONSOLO The Polarization of a monochromatic planar wave is defined in terms of the behavior of the tip of the phasor vector as a function of timeE ( ) ( ) yx yx zktj y zktj x eAeAE 11 ∧ +− ∧ +− += δωδω Tacking the real part we obtain: ( )x x x zkt A E δω +−= cos ( ) ( ) δδωδδωδδδω δ sinsincoscoscos xxxyx y y zktzktzkt A E +−−+−=         −++−=  δδ sin1cos 2         −= x x x x y y A E A E A E  δδ 2 22 sin1cos                 −=         − x x x x y y A E A E A E δδ 2 22 sincos2 =         +−        y y y y x x x x A E A E A E A E 1 sinsin cos 2 sin 2 2 2 =         +−        δδ δ δ y y y y x x x x A E A E A E A E ellipse ( )y y y zkt A E δω +−= cos Polarization Ellipse
29 POLARIZATION SOLO Let transform the ellipse equation to canonical form by using a linear transformation through an angle ψ (to be defined) 1 sinsin cos 2 sin 2 2 2 =         +−        δδ δ δ y y y y x x x x A E A E A E A E               − =        ⇔              − =        η ξ η ξ ψψ ψψ ψψ ψψ E E E E E E E E y x y x cossin sincos cossin sincos 1 sin cossin cos sin cossin sin sincos 2 sin sincos 22 =         + +         +         − −        − δ ψψ δ δ ψψ δ ψψ δ ψψ ηξηξηξηξ yyxx A EE A EE A EE A EE ( ) 1sincos sin cos2 sin cossin2 sin cossin2 cos sin cos sin sin 2 sin cos sin sin cos sin sin sin cos 2 sin sin sin cos 0 22 22222 22 2 22 2 2 22 2 22 2 2 =         −−+−+         +++         −+    ψ ηξ ηξ ψψ δ δ δ ψψ δ ψψ δ δ ψ δ ψ δ ψ δ ψ δ δ ψ δ ψ δ ψ δ ψ definingby yxyx yxyxyxyx AAAA EE AAAA E AAAA E Choose ψ such that the last term is zero, or 02cos cos211 2sin 22 =−         − ψ δ ψ yxxy AAAA ( ) δαδδψ α cos2tancos /1 / cos 2 2tan /:tan 222 xy AA xy xy yx yx AA AA AA AA = = − = − = Polarization Ellipse (continue – 1)
30 POLARIZATION SOLO 1 22 =         +         η η ξ ξ A E A E Define (see Figure) δ δ ψ δ ψ δ ψ δ ψ δ δ ψ δ ψ δ ψ δ ψ χ ξ η cos sin cos sin sin 2 sin cos sin sin cos sin sin sin cos 2 sin sin sin cos :tan 22 2 22 2 22 2 22 2 2 2 yxyx yxyx AAAA AAAA A A ++ −+ =         = ( ) ( ) ( ) ( ) δψψαψψα δψψαψψα δψψψψ δψψψψ χ α coscossintan2cossintan coscossintan2sincostan coscossin/2cossin/ coscossin/2sincos/ tan 222 222tan/ 222 222 2 ++ −+ ++ −+ = = =xy EE xyxy xyxy EEEE EEEE ( )( ) ( ) α δψααψ α δψψααψψ χ χ χ 2 2 2 222 2 2 tan1 cos2sintan2tan12cos tan1 coscossintan4tan1sincos tan1 tan1 2cos + +− = + +−− = + − = From δαψ cos2tan2tan = α ψ δ 2tan 2tan cos = and α α α α α α 2 2 2 tan1 tan1 2cos& tan1 tan2 2tan + − = − = ( ) ψ α ψ ψ ψ α α α α ψ ψααψ χ ψ α 2cos 2cos 2cos 2sin 2cos tan1 tan1 tan1 2tan 2tan 2sintan2tan12cos 2cos 2cos/1 2 2cos 2 2 2 2 =      + + − = + +− =    δαψ cos2tan2tan = Polarization Ellipse (continue – 2)
31 POLARIZATIONSOLO 1 22 =         +         η η ξ ξ A E A E χ α ψ 2cos 2cos 2cos = δαψ cos2tan2tan = χ δ αδα χ α ψψψ 2cos cos 2sincos2tan 2cos 2cos 2tan2cos2sin === Therefore 1cos sin cos sin sin 2 sin cos sin sin cos sin sin sin cos 2 sin sin sin cos 22 2 22 2 2 22 2 22 2 2 =         +++         −+ δ δ ψ δ ψ δ ψ δ ψ δ δ ψ δ ψ δ ψ δ ψ ηξ yxyxyxyx AAAA E AAAA E Also δ δ ψ δ ψ δ ψ δ ψ δ δ ψ δ ψ δ ψ δ ψ η ξ cos sin cos sin sin 2 sin cos sin sin1 cos sin sin sin cos 2 sin sin sin cos1 22 2 22 2 2 22 2 22 2 2 yxyx yxyx AAAAA AAAAA ++= −+= δηξ 22222 sin 11111         +=+ yx AAAA Polarization Ellipse (continue – 3) To Stokes Parameters
32 POLARIZATION SOLO Let compute the area of the Polarization Ellipse ( ) ( )yyyyy xxxxx AzktAEy AzktAEx δτδω δτδω τ τ +=         +−== +=         +−== coscos: coscos:   ( ) ( ) ( ) ( ) δπτδδτδδ τδτδτ π δ π sin2sinsin 2 1 sincos 2 0 2 0 yxyxxyyx xxyy AAdAA dAAdxyArea =         ++−−= ++−== ∫ ∫∫  But the area of the Polarization Ellipse is1 22 =         +         η η ξ ξ A E A E ηξπ AAArea = Therefore δηξ sinyx AAAA = Using δηξ 22222 sin 11111         +=+ yx AAAA δηξ ηξ 222 22 22 22 sin 1 yx yx AA AA AA AA + = + 2222 yx AAAA +=+ ηξ Energy Equation Polarization Ellipse (continue – 4)
33 POLARIZATION SOLO δηξ sinyx AAAA = 2222 yx AAAA +=+ ηξ Energy Equation ξ η χ A A =:tanwe defined Therefore δαδ δ χ χ χ ηξ ηξ ξ η ξ η sin2sinsin 1 2 sin22 1 2 tan1 tan2 2sin 2222222 =         + = + = + =         + = + = y x y x yx yx A A A A AA AA AA AA A A A A δαχ sin2sin2sin = Polarization Ellipse (continue – 5) δαψ cos2tan2tan =We also found that ψ χ ψχψ χ αψ χ αψ αχ δ δ δ 2sin 2tan 2cos2cos2tan 2sin 2cos2tan 2sin 2tan/2tan 2sin/2sin cos sin tan = ⋅⋅ = ⋅ === ψ χ δ 2sin 2tan tan = ψχα 2cos2cos2cos ⋅= To Stokes Parameters
34 POLARIZATION SOLO Summary Polarization Ellipse (continue – 6) δηξ sinyx AAAA = 2222 yx AAAA +=+ ηξ    = = δαχ δαψ sin2sin2sin cos2tan2tan      = ⋅= ψ χ δ ψχα 2sin 2tan tan 2cos2cos2cos       ⇐       χ ψ δ α       ⇐       δ α χ ψ ξ η χ A A =:tan x y A A =:tanα 1 sinsin cos 2 sin 2 2 2 =         +−        δδ δ δ y y y y x x x x A E A E A E A E 1 22 =         +         η η ξ ξ A E A E               − =        η ξ ψψ ψψ E E E E y x cossin sincos xy δδδ −=( ) ( ) yxyx yx zktj y zktj xyx eAeAEEE 1111 ∧ +− ∧ +− ∧∧ +=+= δωδω Return to Table of Content
35 POLARIZATIONSOLO ( ) ( ) yx yx zktj y zktj x eAeAE 11 ∧ +− ∧ +− += δωδω Linearly Horizontally Polarized (LHP): ( )x x x zkt A E δω +−= cos Degenerated States of Polarization Ellipse ( ) 01 == ∧ +− y zktj x AeAE xxδω http://www.enzim.hu/~szia/cddemo/edemo0.htm (Andras Szilagyi) Linearly Vertically Polarized (LVP): ( ) 01 == ∧ +− x zktj y AeAE yyδω ( )y y y zkt A E δω +−= cos
36 POLARIZATIONSOLO Linear + 45° Polarized (L+45P) Degenerated States of Polarization Ellipse ( )tzkj yx eAAE yx ω−− ∧∧       += 11 ( )tzkj yx eAAE yx ω−− ∧∧       −= 11 Plane Polarization Mixed 0=−= xy δδδ πδδδ =−= xy Linear - 45 ° Polarized (L-45P) http://www.enzim.hu/~szia/cddemo/edemo0.htm (Andras Szilagyi) ( ) ( ) yx yx zktj y zktj x eAeAE 11 ∧ +− ∧ +− += δωδω
37 POLARIZATIONSOLO ( )x x zkt A E δω +−= cos ( )x y zkt A E δω +−−= sin Degenerated States of Polarization Ellipse Right Circular Polarization (RCP) AAA yxxy ===−= &2/πδδδ 1 22 =      +      A E A E yx http://www.enzim.hu/~szia/cddemo/edemo0.htm (Andras Szilagyi) ( ) ( ) yx yx zktj y zktj x eAeAE 11 ∧ +− ∧ +− += δωδω ( )x x zkt A E δω +−= cos ( )x y zkt A E δω +−= sin Left Circular Polarization (LCP) AAA yxxy ===−= & 2 3π δδδ
38 POLARIZATIONSOLO ( )x x zkt A E δω +−= cos ( )x y zkt A E δω +−= sin Degenerated States of Polarization Ellipse Left Circular Polarization (LCP) AAA yxxy ===−= & 2 3π δδδ 1 22 =      +      A E A E yx http://www.enzim.hu/~szia/cddemo/edemo0.htm (Andras Szilagyi) Superposition of Two Circular Polarizations ( ) ( ) yx yx zktj y zktj x eAeAE 11 ∧ +− ∧ +− += δωδω Return to Table of Content
39 POLARIZATION SOLO The Stokes Polarization Parameters George Gabriel Stokes 1819-1903 G.G. Stokes, “On the Composition and Resolution of Streams of Polarized Light from different Sources”, Trans. Cambridge Phil. Soc., Vol.9, 1852, pp.399-416 ( ) ( ) yx yyxx zktAzktAE 11 coscos ∧∧ +−++−= δωδω  where ( ) ( ) ( ) ( ) δδ 2 22 sin , cos ,, 2 , =         +−        y y y y x x x x A tzE A tzE A tzE A tzE The Polarization Ellipse is All the information of polarization is contained in this equation. In order to observe the quantities involved let take the time average <… > of the time dependent quantities in the Polarization Ellipse equation. ( ) ( ) ( ) ( ) δδ 2 2 2 2 2 sin , cos ,, 2 , =+− y y yx yx x x A tzE AA tzEtzE A tzE ( ) ( ) ( ) ( ) yxjidttzEtzE T tzEtzE T ji T ji ,,,, 1 :,, 0 lim == ∫→∞ ( ) ( ) ( ) ( ) ( )22222 sin2,4cos,,8,4 δδ yxyxyxyxxy AAtzEAtzEtzEAAtzEA =+−
40 POLARIZATION SOLO The Stokes Polarization Parameters (continue – 1) We obtain ( ) ( ) ( ) ( ) ( )22222 sin2,4cos,,8,4 δδ yxyxyxyxxy AAtzEAtzEtzEAAtzEA =+− ( ) ( ) ( ) ( ) ( ) ( ) δδδωδδ δωδω δ cos 2 22coscos 1 2 coscos 1 ,, 0 0 lim lim yx T xyyx T yx T yxyx T zx AA dtzkt T AA dtzktzktAA T tzEtzE =         ++−−−= +−+−= ∫ ∫ →∞ →∞  ( ) ( ) ( )[ ] 2 2cos1 1 2 cos 1 , 2 0 2 0 222 limlim x T x T x T xx T x A dtzkt T A dtzktA T tzE =+−−=+−= ∫∫ →∞→∞ δωδω ( ) ( ) 2 cos 1 , 2 0 222 lim y T yy T y A dtzktA T tzE =+−= ∫→∞ δω ( ) ( )222222 sin22cos22 δδ yxyxyxyx AAAAAAAA =+− By adding and subtracting on the left side of this equation we obtain ( ) ( ) ( ) ( )22222222 sin2cos2 δδ yxyxyxyx AAAAAAAA =−−−+ 44 2 yx AA +
41 POLARIZATION SOLO The Stokes Polarization Parameters (continue – 2) The Stokes Parameters are defined as ( ) ( ) ( ) ( )22222222 sin2cos2 δδ yxyxyxyx AAAAAAAA ++−=+ δ δ sin2 cos2 3 2 22 1 22 0 yx yx yx yx AAVS AAUS AAQS AAIS == == −== +== 2 3 2 2 2 1 2 0 SSSS ++= The Stokes Vector is defined as                 − + =               =               = δ δ sin2 cos2 22 22 3 2 1 0 yx yx yx yx AA AA AA AA V U Q I S S S S S  The Stokes Parameters are observable using a proper experiment.
42 POLARIZATION SOLO The Stokes Polarization Parameters (continue – 3) Consider a quasi-monochromatic wave of mean frequency ω propagating in z direction ( ) ( )( ) ( ) ( )( ) ( ) ( ) yxyx ztEztEetAetAE yx ttzkj y ttzkj x xy 1111 ,, ∧∧∧ +−− ∧ +−− +=+= δωδω For monochromatic waves Ax, Ay, δx, δy, ω are constant. Quasi-monochromatic Light For quasi-monochromatic waves Ax, Ay, δx, δy , ω are slowly changing with time. We have ( ) ( ) ( ) ( ) yxjidttzEtzE T tzEtzE T ji T ji ,,,, 1 :,, 0 lim == ∫ ∗ →∞ ∗ ( ) ( ) 2 0 21 :,, lim x T x T xx AdtA T tzEtzE == ∫→∞ ∗ ( ) ( ) 2 0 21 :,, lim y T y T yy AdtA T tzEtzE == ∫→∞ ∗ ( ) ( ) ( )yxyx j yx T j y j x T yx eAAdteAeA T tzEtzE δδδδ −− →∞ ∗ == ∫ 0 1 :,, lim ( ) ( ) ( )yxxy j yx T j x j y T xy eAAdteAeA T tzEtzE δδδδ −−− →∞ ∗ == ∫ 0 1 :,, lim
43 POLARIZATION SOLO The Stokes Polarization Parameters (continue – 3) ( ) ( )( ) ( ) ( )( ) ( ) ( ) yxyx ztEztEetAetAE yx ttzkj y ttzkj x xy 1111 ,, ∧∧∧ +−− ∧ +−− +=+= δωδω Quasi-monochromatic Light The Stokes Parameters are defined as ( ) ( ) ( ) ( ) ∗∗ ∗∗ ∗∗ ∗∗ −=−= +=−= −=−= +=+= xyyxxyyx xyyxxyyx yyxxyx yyxxyx EEEEjAAS EEEEAAS EEEEAAS EEEEAAS δδ δδ sin2 cos2 3 2 22 1 22 0 ( ) ( ) 2 0 21 :,, lim x T x T xx AdtA T tzEtzE == ∫→∞ ∗ ( ) ( ) 2 0 21 :,, lim y T y T yy AdtA T tzEtzE == ∫→∞ ∗ ( ) ( ) ( )yxyx j yx T j y j x T yx eAAdteAeA T tzEtzE δδδδ −− →∞ ∗ == ∫ 0 1 :,, lim ( ) ( ) ( )yxxy j yx T j x j y T xy eAAdteAeA T tzEtzE δδδδ −−− →∞ ∗ == ∫ 0 1 :,, lim
44 POLARIZATION SOLO The Stokes Polarization Parameters (continue – 4) Stokes Vector for Different Polarization Types LHP               = 0 0 1 1 0IS  2 0 0 x y AI A = = LVP 2 0 0 y x AI A = =               − = 0 0 1 1 0IS                = 0 1 0 1 0IS  L+45P 2 0 2 0 x yx AI AA = = =δ               − = 0 1 0 1 0IS  L-45P 2 0 2 x yx AI AA = = = πδ               = 1 0 0 1 0IS  RCP 2 0 2 2 x yx AI AA = = = π δ               − = 1 0 0 1 0IS  2 0 2 2 3 x yx AI AA = = = π δ LCP δ δ sin2 cos2 3 2 22 1 22 0 yx yx yx yx AAS AAS AAS AAS = = −= += 2 3 2 2 2 1 2 0 SSSS ++= ( ) ( ) yxyx yx zktj y zktj xyx eAeAEEE 1111 ∧ +− ∧ +− ∧∧ +=+= δωδω
45 POLARIZATION SOLO The Stokes Polarization Parameters (continue – 5) The Stokes Parameters are defined as δ δ sin2 cos2 3 2 22 1 22 0 yx yx yx yx AAS AAS AAS AAS = = −= += 2 3 2 2 2 1 2 0 SSSS ++= δαχ sin2sin2sin = αψχ 2cos2cos2cos =⋅ δαψχ cos2sin2sin2cos ⋅=⋅ We found 22 22 2 2 222 tan1 tan1 2cos& tan1 tan 2sin yx yx yx yx AA AA AA AA + − = + − = + = + = α α α α α α x y A A =:tanα 0 1 22 22 2cos2cos S S AA AA yx yx = + − =⋅ ψχ 0 2 22 cos cos2sin2sin2cos S S AA AA yx yx = + ⋅ =⋅=⋅ δ δαψχ 0 3 22 sin sin2sin2sin S S AA AA yx yx = + ⋅ == δ δαχ       =       = − − 0 31 1 21 sin 2 1 tan 2 1 S S S S χ ψ Return to Table of Content
46 POLARIZATION SOLO Consider a quasi-monochromatic wave of mean frequency ω propagating in z direction composed of a Unpolarized component AUP with random phases δrx and δry and a Polarized component Ax, δx , Ay, δy ( ) ( ) ( )tzkjj yP j UP j xP j UPyx eeAeAeAeAEEE yxyx yyrxxr ωδδδδ −− ∧∧∧∧     +++=+= 1111 Measuring the Stokes Parameters Pass the beam through a waveplate that induces a wave retardation of φ and a polarizer with a transmission axis at an angle β relative to x axis ( ) ( ) ( ) ( ) ( )tzkjj yP j UP j xP j UP eeAeAeAeAE yx yyrxxr ωϕδδϕδδ −− ∧ − ∧ +     +++= 11 2/2/ ' ( ) ( ) ( ) ( )[ ] ( )tzkjj yP j UP j xP j UP eeAeAeAeAE yyrxxr ωϕδδϕδδ ββ −−−+ +++= sincos" 2/2/ The waveplate that induces a wave retardation of φ between the phases of x and y components of the polarized light but will not affect the random phase of the unpolarized light. The polarizer will transmit only the component along the transmission axis
47 POLARIZATION SOLO Measuring the Stokes Parameters (continue – 1) ( ) ( ) ( ) ( )[ ] ( )tzkjj yP j UP j xP j UP eeAeAeAeAE yyrxxr ωϕδδϕδδ ββ −−−+ +++= sincos" 2/2/ ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] zz yyrxxryyrxxr j yP j UP j xP j UP j yP j UP j xP j UP eAeAeAeAeAeAeAeAcnkEEcnkS 11 sincossincos"", 2/2/2/2/ ∧ −−−+−−−+ ∧ ∗ +++⋅+++=⋅= ββββϕβ ϕδδϕδδϕδδϕδδ  ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) βββ βββ βββ βββ ββββ ϕδδϕδδϕδδ δϕδδϕδδδ ϕδδϕδδδϕδ δδδϕδδϕδ ϕδδϕδδϕδδϕδδ 22 0 2/ 0 2/ 2 0 2/2 0 2/ 0 2 0 2/22 0 2/ 00 2/22 0 2/2 2/2/2/2/ sincossin sin2/cossin cossincos cossincos2/ sincossincos         ++         ++         ++         ++         ++         ++         ++         += +++⋅+++ −−−−−−− −−−+− −−+−−+ −−−−+ −−−+−−−+ yP jj yPUP j yPxP jj yPUP jj yPUPUP jj UPxP jj UP j xPyP jj xPUPxP jj xPUP jj UPyP jj UP jj xPUPUP j yP j UP j xP j UP j yP j UP j xP j UP AeeAAeAAeeAA eeAAAeeAAeeA eAAeeAAAeeAA eeAAeeAeeAAA eAeAeAeAeAeAeAeA yxpxyyxr yyyrxyrrr xyxyrxx xryxryxrx yyrxxryyrxxr                     The time average Poynting vector is
48 POLARIZATION SOLO ( ) ( ) ( )[ ] zyP jj yPxPxPUP AeeAAAAcnkS 1 22222 sincossincos2/ ∧ −−− ++++= ββββ ϕδϕδ  2 2sin cossin 2 2cos1 sin 2 2cos1 cos 2 2 β ββ β β β β = − = + = ( ) ( ) ( ) ( )[ ]{ } ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] [ ]βϕβϕβ βϕδβϕδβ βϕβϕβ βϕϕϕϕβ δδδδ δδ 2sinsin2sincos2cos 2 2sinsinsin22sincossin22cos 2 2sinsin2sincos2cos 2 2sinsincossincos2cos 2 ******** 22222 22222 22222 xyyxxyyxyyxxyyxx yPxPyPxPyPxPyPxPUP jj yPxP jj yPxPyPxPyPxPUP jj yPxPyPxPyPxPUP EEEEjEEEEEEEEEEEE cnk AAjAAAAAAA cnk eeAAjeeAAAAAAA cnk jejeAAAAAAA cnk S −+++−++= ++−+++= −−++−+++= ++−+−+++= −− −  Measuring the Stokes Parameters (continue – 2) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] zz yyrxxryyrxxr j yP j UP j xP j UP j yP j UP j xP j UP eAeAeAeAeAeAeAeAcnkEEcnkS 11 sincossincos 2/2/2/2/ ∧ −−−+−−−+ ∧ ∗ +++⋅+++=⋅= ββββ ϕδδϕδδϕδδϕδδ  [ ]βϕβϕβ 2sinsin2sincos2cos 2 3210 SjSSS cnk S +++=  ( ) ( ) ( ) ( )xyyPxPxyyx xyyPxPxyyx yPxPyyxx yPxPUPyyxx AAEEEEjS AAEEEES AAEEEES AAAEEEES δδ δδ −=−= −=+= −=−= ++=+= ∗∗ ∗∗ ∗∗ ∗∗ sin2 cos2 3 2 22 1 222 0
49 POLARIZATION SOLO Measuring the Stokes Parameters (continue – 3) ( ) [ ]βϕβϕβϕβ 2sinsin2sincos2cos 2 , 3210 SjSSS cnk S +++=  ( ) ( ) ( ) ( )xyyPxPxyyx xyyPxPxyyx yPxPyyxx yPxPUPyyxx AAEEEEjS AAEEEES AAEEEES AAAEEEES δδ δδ −=−= −=+= −=−= ++=+= ∗∗ ∗∗ ∗∗ ∗∗ sin2 cos2 3 2 22 1 222 0 The Stokes Parameters are measured by first removing the waveplate φ = 0 ( ) [ ]ββϕβ 2sin2cos 2 0, 210 SSS cnk S ++==  Now the polarizer is sequentially rotate to β = 0, π/4 and π/2 ( ) [ ]10 2 0,0 SS cnk S +=== ϕβ  ( ) [ ]20 2 0,4/ SS cnk S +=== ϕπβ  ( ) [ ]10 2 0,2/ SS cnk S −=== ϕπβ  For the final measurement we add the waveplate with φ = π/2 and polarizer at β = π/4 ( ) [ ]30 2 2/,4/ SS cnk S −=== πϕπβ  ( ) ( )[ ]0,2/0,0 2 0 ==+=== ϕπβϕβ SS cnk S  ( ) ( )[ ]0,2/0,0 2 1 ==−=== ϕπβϕβ SS cnk S  ( ) 02 0,4/2 2 SS cnk S −=== ϕπβ  ( )2/,4/2 2 03 πϕπβ ==−= S cnk SS 
50 POLARIZATION SOLO Measuring the Stokes Parameters (continue – 4) ( ) [ ]βϕβϕβϕβ 2sinsin2sincos2cos 2 , 3210 SjSSS cnk S +++=  ( ) ( ) ( ) ( )xyyPxPxyyx xyyPxPxyyx yPxPyyxx yPxPUPyyxx AAEEEEjS AAEEEES AAEEEES AAAEEEES δδ δδ −=−= −=+= −=−= ++=+= ∗∗ ∗∗ ∗∗ ∗∗ sin2 cos2 3 2 22 1 222 0 The Stokes Parameters can measure the degree of polarization of a beam. We can see that a beam is unpolarized iff: 00 321 2 0 ===>=+= ∗∗ SSSAEEEES UPyyxx The Degree of Polarization is defined as: 10 0 2 3 2 2 2 1 222 22 ≤≤ ++ = ++ + = P S SSS AAA AA P yPxPUP yPxP If the beam is completely polarized then AUP = 0: 0 2 3 2 2 2 1 2 0 >++= SSSS Return to Table of Content
51 POLARIZATION SOLO H. Poincaré, “Théorie Mathématique de la Lumiere”, Gauthiers-Villars, Paris, 1892, Vol.2, Ch.12 Poincaré Sphere Jules Henri Poincaré 1854-1912 We found 03 02 01 /2sin /2sin2cos /2cos2cos SS SS SS = =⋅ =⋅ χ ψχ ψχ We see that the normalized Stokes Parameters can be represented as a unit vector on a sphere (called Poincaré Sphere) using the Polarized Ellipse parameters χ and ψ.
52 POLARIZATION SOLO Poincaré Sphere J.D. Kraus, “Electromagnetics”, 4th Ed., McGraw Hill, 1992, p.607 Return to Table of Content
53 POLARIZATION SOLO The Mueller Matrices for Polarizing Components In 1948 Hans Mueller, then a professor of physics at MIT devised a matrix method that deals with the relation of Stokes vector at the output of an optical device to the Stokes vector at the input of the optical device.                             =               3 2 1 0 33323130 23222120 13121110 03020100 3 2 1 0 ' ' ' ' S S S S mmmm mmmm mmmm mmmm S S S S inputoutput SMS  = or Hans Mueller, “The Foundation of Optics”, J. Opt. Soc. Am., 37, pg. 110 (1947) 38, pg. 661(1948)
54 POLARIZATION SOLO The Mueller Matrices for Polarizing Components Suppose that we have a few optical elements through which the ray passes . ( ) inputinputoutput SMSMMMSMMSMS  ==== 12312323 or 123 MMMM =
55 POLARIZATION SOLO The Mueller Matrices for Polarizing Components Polarizer 10' 10' ≤≤= ≤≤= yyyy xxxx pEpE pEpE The Polarizer is described by two orthogonal transmission axes that are characterized, respectively, by transmission factors px and py (0 ≤ px,py ≤ 1). ( )∗∗ ∗∗ ∗∗ ∗∗ −= += −= += xyyx xyyx yyxx yyxx EEEEjS EEEES EEEES EEEES 3 2 1 0 ( )∗∗ ∗∗ ∗∗ ∗∗ −= += −= += ''''' ''''' ''''' ''''' 3 2 1 0 xyyx xyyx yyxx yyxx EEEEjS EEEES EEEES EEEES                             +− −+ =               3 2 1 0 2222 2222 3 2 1 0 2000 0200 00 00 2 1 ' ' ' ' S S S S pp pp pppp pppp S S S S yx yx yxyx yxyx ( )               +− −+ = yx yx yxyx yxyx yxPOL pp pp pppp pppp ppM 2000 0200 00 00 2 1 , 2222 2222
56 POLARIZATION SOLO The Mueller Matrices for Polarizing Components Polarizer ( ) 22 1 : /tan: yx xy ppp pp += = − βLet define                             =               3 2 1 0 2 2 22 22 3 2 1 0 2sin000 02sin00 002cos 002cos 2 1 ' ' ' ' S S S S p p pp pp S S S S β β β β ( )               = β β β β β 2sin000 02sin00 0012cos 002cos1 2 , 2 p pMPOL β β sin cos pp pp y x = =
57 POLARIZATION SOLO The Mueller Matrices for Polarizing Components Polarizer (continue -1) 0' 10' = ≤≤= y xxxx E pEpE Ideal Linear Polarizer py = 0 and 0 ≤ px ≤ 1. ( ) ( )               +=                             =               0 0 1 1 2 0000 0000 0011 0011 2 ' ' ' ' 10 2 3 2 1 0 || 2 3 2 1 0 SS p S S S S p S S S S x M x POL    Regardless the state of polarization of the input beam the output beam is Linear Horizontally Polarized (LHP) 10' 0' ≤≤= = yyyy x pEpE E Ideal Linear Polarizer px = 0 and 0 ≤ py ≤ 1. ( ) ( )               − −=                             − − =               ⊥ 0 0 1 1 2 0000 0000 0011 0011 2 ' ' ' ' 10 2 3 2 1 0 2 3 2 1 0 SS p S S S S p S S S S y M y POL    Regardless the state of polarization of the input beam the output beam is Linear Vertically Polarized (LVP)
58 POLARIZATION SOLO The Mueller Matrices for Polarizing Components Polarizer (continue -2) Crossed Polarizers ( ) ( )               =                             − −               =               ⊥ 0 0 0 0 22 0000 0000 0011 0011 2 0000 0000 0011 0011 2 ' ' ' ' 22 3 2 1 0 2 || 2 3 2 1 0 yx M y M x pp S S S S pp S S S S POLPOL      Regardless the state of polarization of the input beam the output beam is completely blocked Crossed Polarizers are a combination of a Linear Polarizer with the transmission axes normal to the x-axis, followed by a Linear Polarizer with the transmission axes parallel to the x-axis (or vice-versa).
59 POLARIZATION SOLO The Mueller Matrices for Polarizing Components Polarizer (continue -3) Eigenvalues and Eigenvectors of the Linear Polarizer Mueller Matrix Let find the Input Stokes Vectors that are unaffected by the Polarizer; i.e.:               =                             +− −+ =               3 2 1 0 3 2 1 0 2222 2222 3 2 1 0 2000 0200 00 00 2 1 ' ' ' ' S S S S S S S S pp pp pppp pppp S S S S POLM yx yx yxyx yxyx λ    ( ) 0 22000 02200 002 002 2 1 , 2222 2222 44 = − − −+− −−+ =− λ λ λ λ λ yx yx yxyx yxyx xyxPOL pp pp pppp pppp IppM               == 0 0 1 1 & 1 2 1 Spx  λ               − == 0 0 1 1 & 2 2 2 Spy  λ 1st eigenvalue- eigenvector 2nd eigenvalue- eigenvector We can see that only LHP and LVP are unaffected by the Linear Polarizer.
60 POLARIZATION SOLO The Mueller Matrices for Polarizing Components Waveplate y j y x j x EeE EeE 2/ 2/ ' ' ϕ ϕ − = = The Waveplate is a polarizing element that introduced a phase shift φ between the orthogonal components of an optical beam. A Waveplate is a phase-shifter but is also called a retarder or a compensator. ( )∗∗ ∗∗ ∗∗ ∗∗ −= += −= += xyyx xyyx yyxx yyxx EEEEjS EEEES EEEES EEEES 3 2 1 0 ( )∗∗ ∗∗ ∗∗ ∗∗ −= += −= += ''''' ''''' ''''' ''''' 3 2 1 0 xyyx xyyx yyxx yyxx EEEEjS EEEES EEEES EEEES                             − =               3 2 1 0 3 2 1 0 cossin00 sincos00 0010 0001 ' ' ' ' S S S S S S S S ϕϕ ϕϕ ( )               − = ϕϕ ϕϕ ϕ cossin00 sincos00 0010 0001 WP M
61 POLARIZATION SOLO The Mueller Matrices for Polarizing Components Waveplate (continue – 1) Quarter-Waveplate φ = π/2 The Quarter-Waveplate transforms Linearly Polarized (L+45P or L-45P) to Right or Left Circularly Polarized (RCP or LCP) or vice-versa. ( )               − == 0100 1000 0010 0001 2/πϕWPM ( )    RCPPLPL WPM               =                             − =               = ++ 1 0 0 1 0 1 0 1 0100 1000 0010 0001 0 1 0 1 2/ 4545 πϕ ( )    LCPPLPL WPM               − =               −               − =               − = −− 1 0 0 1 0 1 0 1 0100 1000 0010 0001 0 1 0 1 2/ 4545 πϕ ( )    PLRCPRCP WPM 45 0 1 0 1 1 0 0 1 0100 1000 0010 0001 1 0 0 1 2/ −               − =                             − =               = πϕ ( )     LCP PLLCPLCP WPM 45 0 1 0 1 1 0 0 1 0100 1000 0010 0001 1 0 0 1 2/ +               =               −              − =               − = πϕ
62 POLARIZATION SOLO The Mueller Matrices for Polarizing Components Waveplate (continue – 2) Half-Waveplate φ = π ( )               − − == 1000 0100 0010 0001 πϕWPM               − − =                             − − =               3 2 1 0 3 2 1 0 3 2 1 0 1000 0100 0010 0001 ' ' ' ' S S S S S S S S S S S S χ ψχ ψχ 2sin 2sin2cos 2cos2cos 0 3 0 2 0 1 = ⋅= ⋅= S S S S S S       =       = − − 0 31 1 21 sin 2 1 tan 2 1 S S S S χ ψ 2 ' 2 ' π χχ ψ π ψ −= −= The Half-Waveplate reverses the orientation and ellipticity of the polarization ellipse (polarization state).
63 POLARIZATION SOLO The Mueller Matrices for Polarizing Components Waveplate (continue – 3) The phases of two adjacent Half-Waveplates add. ( ) ( ) ( ) ( )                             ++ +−+ =                             −               − =               3 2 1 0 2121 2121 3 2 1 0 22 22 11 11 3 2 1 0 cossin00 sincos00 0010 0001 cossin00 sincos00 0010 0001 cossin00 sincos00 0010 0001 ' ' ' ' S S S S S S S S S S S S ϕϕϕϕ ϕϕϕϕ ϕϕ ϕϕ ϕϕ ϕϕ Eigenvalues and Eigenvectors of the Waveplate Mueller Matrix ( ) 0 cossin00 sincos00 0010 0001 4 = − −− − − =− λϕϕ ϕλϕ λ λ λϕ IMWP  LHP S               == 0 0 1 1 &1 11  λ  LVP S               − =−= 0 0 1 1 &1 22  λ 1st eigenvalue- eigenvector 2nd eigenvalue- eigenvector We can see that only LHP and LVP are unaffected by the Waveplate Polarizer.
64 POLARIZATION SOLO The Mueller Matrices for Polarizing Components The Mueller Matrix of a Rotator (Coordinate Rotation) ( ) ( ) θθθγ θθθγ cossinsin' sincoscos' yxy yxx EEEE EEEE +−=−= +=−= Assume that the polarizing device rotates its orthogonal axes along the ray propagation direction by an angle θ. The orthogonal axes are defined as (‘). ( )∗∗ ∗∗ ∗∗ ∗∗ −= += −= += xyyx xyyx yyxx yyxx EEEEjS EEEES EEEES EEEES 3 2 1 0 ( )∗∗ ∗∗ ∗∗ ∗∗ −= += −= += ''''' ''''' ''''' ''''' 3 2 1 0 xyyx xyyx yyxx yyxx EEEEjS EEEES EEEES EEEES                             − =               3 2 1 0 3 2 1 0 1000 02cos2sin0 02sin2cos0 0001 ' ' ' ' S S S S S S S S θθ θθ ( )               − = 1000 02cos2sin0 02sin2cos0 0001 θθ θθ θROTM '1'1 1111 '' sincos yx yxyx yx yx EE EEEE ∧∧ ∧∧∧∧ +=         +=+= γγ
65 POLARIZATION SOLO The Mueller Matrices for Polarizing Components Assume that the polarizing device rotates its orthogonal axes along the ray propagation direction by an angle θ. The Mueller Rotation Matrix from x,y to x’,y’ is ( )               − = 1000 02cos2sin0 02sin2cos0 0001 θθ θθ θROTM The Mueller Polarizator Matrix is ( )               = β β β β β 2sin000 02sin00 0012cos 002cos1 2 , 2 p pMPOL ( ) SMS ROT  θ=' ( ) ( ) ( ) SMpMSpMS ROTPOLPOL  θββ ,'," == The Mueller Rotation Matrix from x’,y’ to x,y is ( )               − =− 1000 02cos2sin0 02sin2cos0 0001 θθ θθ θROTM ( ) ( ) ( ) ( ) SMpMMSMS ROTPOLROTROT  θβθθ ,"'" −=−= ( ) ( ) ( ) ( )θβθθβ ROTPOLROTPOL MpMMpM ,,, −= The Mueller Matrix of a Rotated Polarizer
66 POLARIZATION SOLO The Mueller Matrices for Polarizing Components ( ) ( ) ( )               +− −+ = β θβθθθβθβ θθβθβθθβ θβθβ θβ 2sin000 02cos2sin2sin2cos2sin2sin12sin2cos 02cos2sin2sin12sin2sin2cos2cos2cos 02sin2cos2cos2cos1 2 ,, 22 222 p pMPOL The Mueller Matrix for Rotated Polarizer is ( ) ( ) ( ) ( )               −                             − = −= 1000 02cos2sin0 02sin2cos0 0001 2sin000 02sin00 0012cos 002cos1 1000 02cos2sin0 02sin2cos0 0001 2 ,,, 2 θθ θθ β β β β θθ θθ θβθθβ p MpMMpM ROTPOLROTPOL The Mueller Matrix of a Rotated Polarizer (continue – 1)
67 POLARIZATION SOLO The Mueller Matrices for Polarizing Components ( ) ( ) ( )               − +− −−+ = ϕϕβϕβ ϕβϕββϕββ ϕβϕββϕββ ϕβ cossin2cossin2sin0 sin2coscos2cos2sincos12cos2sin0 sin2sincos12cos2sincos2sin2cos0 0001 , 22 22 WPM The Mueller Matrix for Rotated Waveplate is ( ) ( ) ( ) ( )               −               −               − = −= 1000 02cos2sin0 02sin2cos0 0001 cossin00 sincos00 0010 0001 1000 02cos2sin0 02sin2cos0 0001 ,, θθ θθ ϕϕ ϕϕθθ θθ θϕθθβ ROTWPROTPOL MMMpM The Mueller Matrix of a Rotated Waveplate Return to Table of Content
68 POLARIZATION SOLO The Jones Polarization Parameters R. Clark Jones, “A New Calculus for the Treatment of Optical Systems”, J. Opt. Soc. Am., Vol.31, July 1941, pp.500-503 J. Opt. Soc. Am., Vol.32, Aug. 1942, pp.486-493 J. Opt. Soc. Am., Vol.37, Feb. 1947, pp.107-110 J. Opt. Soc. Am., Vol.37, Feb. 1947, pp.110-112 J. Opt. Soc. Am., Vol.38, Aug. 1948, pp.671-585 R. Clark Jones 1916-2004 Mueller matrices deal with the intensity of the beam. If the phase information is important we must use Jones formalism. Jones calculus was developed in the same time with the Mueller calculus by R. Clark Jones who introduced Jones vectors and Jones matrices: Jones vectors describe the polarization of light: Jones matrices describe the optical component:         + = y x yx E E EE J 22 1 [ ]       = 2221 1211 jj jj J Jones calculus deals only with polarized light.
69 POLARIZATION SOLO Stokes and Jones Vector for Different Polarization Types LHP       =               = 0 1 0 0 1 1 0 JIS  2 0 0 x y AI A = = LVP 2 0 0 y x AI A = =       =               − = 1 0 0 0 1 1 0 JIS        =               = 1 1 2 1 0 1 0 1 0 JIS  L+45P 2 0 2 0 x yx AI AA = = =δ       − =               − = 1 1 2 1 0 1 0 1 0 JIS  L-45P 2 0 2 x yx AI AA = = = πδ       =               = i JIS 1 2 1 1 0 0 1 0  RCP 2 0 2 2 x yx AI AA = = = π δ       − =               − = i JIS 1 2 1 1 0 0 1 0  2 0 2 2 3 x yx AI AA = = = π δ LCP δ δ sin2 cos2 3 2 22 1 22 0 yx yx yx yx AAS AAS AAS AAS = = −= += 2 3 2 2 2 1 2 0 SSSS ++= ( ) ( ) yxyx yx zktj y zktj xyx eAeAEEE 1111 ∧ +− ∧ +− ∧∧ +=+= δωδω Return to Table of Content
70 POLARIZATION SOLO Faraday Effect (Hecht p.261) Michael Faraday (England) 1845 described the rotation of the plane of polarized light that passed through glass in a magnetic field. Return to Table of Content
71 POLARIZATION SOLO Pockels Effect (Hecht p.263, Chuang p.509, Meyer-Arent p.318) Vnr nretardatio Er n n n Ln Ln yx z yy xx 3 0'' 32 '' '' 2 : 21 2 2 λ π λ π λ π =Φ−Φ=Φ ⇒ = ∆ −=      ∆ =Φ =Φ Pockels Effect 1893 Electro – Optical Effects The electro – optical effects are called Pockels or Kerr where the refractive index changes linearly or quadraticly, respectively.    = = =      ∆ Kerrk Pockelsk EKn n k 2 11 02 Frederich Carl Alwin Pockels (1865-1913) http://www.physi.uni-heidelberg.de/~schmiedm/Vorlesung/LasPhys02/LectureNotes/OpticsCrystals.pdf#search='Pockels%20Effect' http://en.wikipedia.org/wiki/Pockels_effect
72 POLARIZATION SOLO Electro – Optical Effects Kerr Effect (Hecht p.263, Chuang p.509, Meyer-Arent p.318) The electro – optical effects are Kerr or Pockels 2 0 EKn λ=∆ 2 2 0 2 d VKλπ =Φ∆ In the Kerr Electro-optic Effect 1875 is the electric field that causes the substance to become birefringent. Return to Table of Content
73 POLARIZATIONSOLO Plane Polarized Wave in an Absorbing Medium Circular Polarized Wave in an Absorbing Medium
74 POLARIZATIONSOLO Plane Polarized Wave in an Refracting Medium Circular Polarized Wave in an Refracting Medium
75 POLARIZATIONSOLO Plane Polarized Wave in a Medium Showing Circular Dichroism Plane Polarized Wave in a Medium Showing Circular Birefrigens
76 POLARIZATIONSOLO Plane Polarized Wave in a Medium Showing Both Circular Dichroism and Circular Birefrigens
77 OPTICSSOLO http://microscopy.fsu.edu/ Return to Table of Content
78 OPTICSSOLO References Optics Polarization A. Yariv, P. Yeh, “Optical Waves in Crystals”, John Wiley & Sons, 1984 M. Born, E. Wolf, “Principles of Optics”, Pergamon Press,6th Ed., 1980 E. Hecht, A. Zajac, “Optics”, Addison-Wesley, 1979, Ch.8 C.C. Davis, “Lasers and Electro-Optics”, Cambridge University Press, 1996 G.R. Fowles, “Introduction to Modern Optics”,2nd Ed., Dover, 1975, Ch.2 M.V.Klein, T.E. Furtak, “Optics”, 2nd Ed., John Wiley & Sons, 1986 http://en.wikipedia.org/wiki/Polarization W.C.Elmore, M.A. Heald, “Physics of Waves”, Dover Publications, 1969 E. Collett, “Polarization Light in Fiber Optics”, PolaWave Group, 2003 W. Swindell, Ed., “Polarization Light”, Benchmark Papers in Optics, V.1, Dowden, Hutchinson & Ross, Inc., 1975 http://www.enzim.hu/~szia/cddemo/edemo0.htm (Andras Szilagyi)
79 ELECTROMAGNETICSSOLO References Electromagnetics J.D. Jackson, “Classical Electrodynamics”, 3rd Ed., John Wiley & Sons, 1999 R. S. Elliott, “Electromagnetics”, McGraw-Hill, 1966 J.A. Stratton, “Electromagnetic Theory”, McGraw-Hill, 1941 W.K.H. Panofsky, M. Phillips, “Classical Electricity and Magnetism”, Addison- Wesley, 1962 F.T. Ulaby, R.K. More, A.K. Fung, “Microwave Remote Sensors Active and Passive”, Addison-Wesley, 1981 A.L. Maffett, “Topics for a Statistical Description of Radar Cross Section”, John Wiley & Sons, 1988
80 ELECTROMAGNETICSSOLO References 1.W.K.H. Panofsky & M. Phillips, “Classical Electricity and Magnetism”, 2.J.D. Jackson, “Classical Electrodynamics”, 3.R.S. Elliott, “Electromagnetics”, 4.A.L. Maffett, “Topics for a Statistical Description of Radar Cross Section”, Return to Table of Content
January 4, 2015 81 SOLO POLARIZATION Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 –2013 Stanford University 1983 – 1986 PhD AA

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Polarization

  • 1.
  • 2.
    POLARIZATION History SOLO TABLE OF CONTENT Naturalor Unpolarized Light Monochromatic Planar Wave Equations Linear Polarization or Plane-Polarization Circular Polarization Elliptically Polarization Methods of Achieving Polarization Polarization Ellipse Degenerated States of Polarization Ellipse The Stokes Polarization Parameters Measuring the Stokes Parameters Poincaré Sphere The Mueller Matrices for Polarizing Components The Jones Polarization Parameters
  • 3.
    3 POLARIZATIONSOLO TABLE OF CONTENT(Continue – 1) Faraday Effect Electro – Optical Effects ( Pockels, Kerr) References Optics Polarization
  • 4.
    4 POLARIZATION Erasmus Bartholinus, doctorof medicine and professor of mathematics at the University of Copenhagen, showed in 1669 that crystals of “Iceland spar” (which we now call calcite, CaCO3) produced two refracted rays from a single incident beam. One ray, the “ordinary ray”, followed Snell’s law, while the other, the “extraordinary ray”, was not always even in the plan of incidence. SOLO History Erasmus Bartholinus 1625-1698 http://www.polarization.com/history/history.html
  • 5.
    5 POLARIZATIONSOLO History Étienne Louis Malus 1775-1812 EtienneLouis Malus, military engineer and captain in the army of Napoleon, published in 1809 the Malus Law of irradiance through a Linear polarizer: I(θ)=I(0) cos2 θ. In 1810 he won the French Academy Prize with the discovery that reflected and scattered light also possessed “sidedness” which he called “polarization”.
  • 6.
    6 POLARIZATION Arago and Fresnelinvestigated the interference of polarized rays of light and found in 1816 that two rays polarized at right angles to each other never interface. SOLO History (continue) Dominique François Jean Arago 1786-1853 Augustin Jean Fresnel 1788-1827 Arago relayed to Thomas Young in London the results of the experiment he had performed with Fresnel. This stimulate Young to propose in 1817 that the oscillations in the optical wave where transverse, or perpendicular to the direction of propagation, and not longitudinal as every proponent of wave theory believed. Thomas Young 1773-1829 Return to Table of Content
  • 7.
    7 POLARIZATIONSOLO The Natural Lightis emitted by excitation of material atoms. Each excited atom radiates a wave train for roughly 10-8 sec. The Natural light is the result of radiation of a large collection of such atoms. New waves are emitted in a completely unpredictable fashion and the light is actually composed of a succession of short living polarization states. Natural or Unpolarized Light A light source consists of a very large number of randomly oriented atoms emitters. We say that the Natural light is compose of a collection of monochromatic (polichromatic) unpolarized rays. One mathematical description of a monochromatic unpolarized ray moving in z direction, at a certain location in space, is by a Electric Intensity phasor of constant amplitude and a random phase: ( ) ( )( ) ( )( ) yx tzktjtzktj yx eAeAtE 11 ∧ +− ∧ +− += δωδω zyx 111 ,, ∧∧∧ are orthogonal unit vectors and δx (t) are δy (t) are randomly phase angles.
  • 8.
    8 POLARIZATIONSOLO Light is atransverse electromagnetic wave; i.e. the Electric and Magnetic Intensities are perpendicular to each other and oscillate perpendicular to the direction of propagation. For the natural light the direction of the Electric Intensity vector changes randomly from time to time. We say that the natural light is Unpolarized. A Planar wave (in which the Electric Intensity propagates remaining in a plane – containing the propagation direction) is said to be Linearly Polarized or Plane-Polarized. If light is composed of two plane waves of equal amplitude but differing in phase by 90° then the light is aid to be Circular Polarized. If light is composed of two plane waves of different amplitudes and/or the difference in phase is different than 0,90,180,270° then the light is said to be Elliptically Polarized. E  Return to Table of Content
  • 9.
    9 ELECTROMAGNETICSSOLO To satisfy theMaxwell equations for a source free media we must have: Monochromatic Planar Wave Equations we haveUsing: 1ˆˆ&ˆˆ 0 =⋅== kkknkkk εµω         =⋅∇ =⋅∇ −=×∇ =×∇ 0 0 H E HjE EjH ωµ ωε            =⋅ =⋅ =× −=× 0ˆ 0ˆ ˆ ˆ 0 0 00 00 Hk Ek HEk EHk ε µ µ ε        =⋅− =⋅− −=×− =×− ⇒ ⋅− ⋅− ⋅−⋅− ⋅−⋅− −=∇ ⋅−⋅− 0 0 0 0 00 00 rkj rkj rkjrkj rkjrkj ekje eHkj eEkj eHjeEkj eEjeHkj rkjrkj           ωµ ωε ( ) * 2 * 2 & 2 ˆ 2 ˆ HHwEEwwcn k wwcn k S meme µε ====+=  Time Average Poynting Vector of the Planar Wave ( ) ( )rktjrktj eHHeEE  ⋅−⋅− == ωω 00 & Return to Table of Content
  • 10.
    10 POLARIZATIONSOLO A Planar wave(in which the Electric Intensity propagates remaining in a plane – containing the propagation direction) is said to be Linearly Polarized or Plane-Polarized. http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html http://www.enzim.hu/~szia/cddemo/edemo0.htm)Andras Szilagyi( Linear Polarization or Plane-Polarization ( ) yyzktj y eAE 1 ∧ +− = δω Return to Table of Content
  • 11.
    11 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html If light iscomposed of two plane waves of equal amplitude but differing in phase by 90° then the light is said to be Circular Polarized. http://www.optics.arizona.edu/jcwyant/JoseDiaz/Polarization-Circular.htm ( ) ( ) yx xx zktjzktj eAeAE 11 2/ ∧ ++− ∧ +− += πδωδω Circular Polarization Return to Table of Content
  • 12.
    12 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html If light iscomposed of two plane waves of different amplitudes and/or the difference in phase is different than 0,90,180,270° then the light is said to be Elliptically Polarized. ( ) ( ) yx yx zktj y zktj x eAeAE 11 ∧ +− ∧ +− += δωδω Elliptically Polarization Return to Table of Content
  • 13.
    13 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Methods of AchievingPolarization Polarization is based on one of four fundamental physical mechanisms: 1. Dichroism or selective absorbtion 3. Reflection 2. Scattering 4. Birefrigerence To obtain Polarization we must have some asymmetry in the optical process.
  • 14.
    14 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Methods of AchievingPolarization (continue – 1) Polarization by Dichroism A dichroic material has different absorption properties for perpendicular incident planes. An example of a dichroic material is the tourmaline that is a class of borone silicate. The tourmaline has a unique optic axis, and any electronic field normal to it is strongly absorbed.
  • 15.
    15 POLARIZATION SOLO Polaroids (Dichroism) http://en.wikipedia.org/wiki/Edwin_H._Land Edwin H.Land 1909-1991 In 1928 Edwin H. Land undergraduate at Harvard College invented the Polaroid J-sheet. It consists of many microscopic Crystals of iodoquinine sulphate embeded in a transparent Nitrocellulose polymer film. The sunglasses use polaroid material that uses dichroism to achieve absorption..
  • 16.
    16 Methods of AchievingPolarization (continue – 1) Wire-Grid Polarizer (Dichroism) POLARIZATIONSOLO Grid of parallel conducting wires with a spacing comparable to the wavelength of the electromagnetic wave. The Electric Field vector parallel to the wires is attenuated because of the currents induced in the wires.
  • 17.
  • 18.
    18 POLARIZATIONSOLO Methods of AchievingPolarization (continue – 3) Polarization by Scattering
  • 19.
  • 20.
    20 POLARIZATIONSOLO Methods of AchievingPolarization (continue – 5) Polarization by Reflection The Pile-of-plate Polarizer The problem encounter using the Brewster Effect is that the reflected beam although completely polarized is weak and the refracted beam is only partially polarized. The solution is to use a pile-of- plates polarizer as in Figure. This was invented by F.J. Arago in 1812. Dominique François Jean Arago 1786-1853
  • 21.
    21 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Polarization by Birefrigerence(continue – 4) Polarization can be achieved with crystalline materials which have a different index of refraction in different planes. Such materials are said to be birefringent or doubly refracting. Nicol Prism The Nicol Prism (1828) is made up from two prisms of calcite cemented with Canada balsam. The ordinary ray can be made to totally reflect off the prism boundary, leving only the extraordinary ray. William Nicol(1768 ?– 1851) Scottish physicist Methods of Achieving Polarization (continue – 6)
  • 22.
    22 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Polarization by Birefrigerence(continue – 4) Polarization can be achieved with crystalline materials which have a different index of refraction in different planes. Such materials are said to be birefringent or doubly refracting. Wollaston Prism William Hyde Wollaston 1766-1828 Methods of Achieving Polarization (continue – 7)
  • 23.
    23 POLARIZATIONSOLO http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Polarization by Birefrigerence(continue – 4) Polarization can be achieved with crystalline materials which have a different index of refraction in different planes. Such materials are said to be birefringent or doubly refracting. Glan-Foucault Polarizer Methods of Achieving Polarization (continue – 8)
  • 24.
    24 POLARIZATION Methods of AchievingPolarization (continue – 9)
  • 25.
    25 POLARIZATION Methods of AchievingPolarization (continue – 10)
  • 26.
  • 27.
    27 POLARIZATION SOLO Polarizations Prisms Overview http://www.unitedcrystals.com/POverview.html Methodsof Achieving Polarization (continue – 12) Return to Table of Content
  • 28.
    28 POLARIZATIONSOLO The Polarization ofa monochromatic planar wave is defined in terms of the behavior of the tip of the phasor vector as a function of timeE ( ) ( ) yx yx zktj y zktj x eAeAE 11 ∧ +− ∧ +− += δωδω Tacking the real part we obtain: ( )x x x zkt A E δω +−= cos ( ) ( ) δδωδδωδδδω δ sinsincoscoscos xxxyx y y zktzktzkt A E +−−+−=         −++−=  δδ sin1cos 2         −= x x x x y y A E A E A E  δδ 2 22 sin1cos                 −=         − x x x x y y A E A E A E δδ 2 22 sincos2 =         +−        y y y y x x x x A E A E A E A E 1 sinsin cos 2 sin 2 2 2 =         +−        δδ δ δ y y y y x x x x A E A E A E A E ellipse ( )y y y zkt A E δω +−= cos Polarization Ellipse
  • 29.
    29 POLARIZATION SOLO Let transform theellipse equation to canonical form by using a linear transformation through an angle ψ (to be defined) 1 sinsin cos 2 sin 2 2 2 =         +−        δδ δ δ y y y y x x x x A E A E A E A E               − =        ⇔              − =        η ξ η ξ ψψ ψψ ψψ ψψ E E E E E E E E y x y x cossin sincos cossin sincos 1 sin cossin cos sin cossin sin sincos 2 sin sincos 22 =         + +         +         − −        − δ ψψ δ δ ψψ δ ψψ δ ψψ ηξηξηξηξ yyxx A EE A EE A EE A EE ( ) 1sincos sin cos2 sin cossin2 sin cossin2 cos sin cos sin sin 2 sin cos sin sin cos sin sin sin cos 2 sin sin sin cos 0 22 22222 22 2 22 2 2 22 2 22 2 2 =         −−+−+         +++         −+    ψ ηξ ηξ ψψ δ δ δ ψψ δ ψψ δ δ ψ δ ψ δ ψ δ ψ δ δ ψ δ ψ δ ψ δ ψ definingby yxyx yxyxyxyx AAAA EE AAAA E AAAA E Choose ψ such that the last term is zero, or 02cos cos211 2sin 22 =−         − ψ δ ψ yxxy AAAA ( ) δαδδψ α cos2tancos /1 / cos 2 2tan /:tan 222 xy AA xy xy yx yx AA AA AA AA = = − = − = Polarization Ellipse (continue – 1)
  • 30.
    30 POLARIZATION SOLO 1 22 =         +         η η ξ ξ A E A E Define (see Figure) δ δ ψ δ ψ δ ψ δ ψ δ δ ψ δ ψ δ ψ δ ψ χ ξ η cos sin cos sin sin 2 sin cos sin sin cos sin sin sin cos 2 sin sin sin cos :tan 22 2 22 2 22 2 22 2 2 2 yxyx yxyx AAAA AAAA A A ++ −+ =         = () ( ) ( ) ( ) δψψαψψα δψψαψψα δψψψψ δψψψψ χ α coscossintan2cossintan coscossintan2sincostan coscossin/2cossin/ coscossin/2sincos/ tan 222 222tan/ 222 222 2 ++ −+ ++ −+ = = =xy EE xyxy xyxy EEEE EEEE ( )( ) ( ) α δψααψ α δψψααψψ χ χ χ 2 2 2 222 2 2 tan1 cos2sintan2tan12cos tan1 coscossintan4tan1sincos tan1 tan1 2cos + +− = + +−− = + − = From δαψ cos2tan2tan = α ψ δ 2tan 2tan cos = and α α α α α α 2 2 2 tan1 tan1 2cos& tan1 tan2 2tan + − = − = ( ) ψ α ψ ψ ψ α α α α ψ ψααψ χ ψ α 2cos 2cos 2cos 2sin 2cos tan1 tan1 tan1 2tan 2tan 2sintan2tan12cos 2cos 2cos/1 2 2cos 2 2 2 2 =      + + − = + +− =    δαψ cos2tan2tan = Polarization Ellipse (continue – 2)
  • 31.
    31 POLARIZATIONSOLO 1 22 =         +         η η ξ ξ A E A E χ α ψ 2cos 2cos 2cos = δαψcos2tan2tan = χ δ αδα χ α ψψψ 2cos cos 2sincos2tan 2cos 2cos 2tan2cos2sin === Therefore 1cos sin cos sin sin 2 sin cos sin sin cos sin sin sin cos 2 sin sin sin cos 22 2 22 2 2 22 2 22 2 2 =         +++         −+ δ δ ψ δ ψ δ ψ δ ψ δ δ ψ δ ψ δ ψ δ ψ ηξ yxyxyxyx AAAA E AAAA E Also δ δ ψ δ ψ δ ψ δ ψ δ δ ψ δ ψ δ ψ δ ψ η ξ cos sin cos sin sin 2 sin cos sin sin1 cos sin sin sin cos 2 sin sin sin cos1 22 2 22 2 2 22 2 22 2 2 yxyx yxyx AAAAA AAAAA ++= −+= δηξ 22222 sin 11111         +=+ yx AAAA Polarization Ellipse (continue – 3) To Stokes Parameters
  • 32.
    32 POLARIZATION SOLO Let compute thearea of the Polarization Ellipse ( ) ( )yyyyy xxxxx AzktAEy AzktAEx δτδω δτδω τ τ +=         +−== +=         +−== coscos: coscos:   ( ) ( ) ( ) ( ) δπτδδτδδ τδτδτ π δ π sin2sinsin 2 1 sincos 2 0 2 0 yxyxxyyx xxyy AAdAA dAAdxyArea =         ++−−= ++−== ∫ ∫∫  But the area of the Polarization Ellipse is1 22 =         +         η η ξ ξ A E A E ηξπ AAArea = Therefore δηξ sinyx AAAA = Using δηξ 22222 sin 11111         +=+ yx AAAA δηξ ηξ 222 22 22 22 sin 1 yx yx AA AA AA AA + = + 2222 yx AAAA +=+ ηξ Energy Equation Polarization Ellipse (continue – 4)
  • 33.
    33 POLARIZATION SOLO δηξ sinyx AAAA = 2222 yxAAAA +=+ ηξ Energy Equation ξ η χ A A =:tanwe defined Therefore δαδ δ χ χ χ ηξ ηξ ξ η ξ η sin2sinsin 1 2 sin22 1 2 tan1 tan2 2sin 2222222 =         + = + = + =         + = + = y x y x yx yx A A A A AA AA AA AA A A A A δαχ sin2sin2sin = Polarization Ellipse (continue – 5) δαψ cos2tan2tan =We also found that ψ χ ψχψ χ αψ χ αψ αχ δ δ δ 2sin 2tan 2cos2cos2tan 2sin 2cos2tan 2sin 2tan/2tan 2sin/2sin cos sin tan = ⋅⋅ = ⋅ === ψ χ δ 2sin 2tan tan = ψχα 2cos2cos2cos ⋅= To Stokes Parameters
  • 34.
    34 POLARIZATION SOLO Summary Polarization Ellipse (continue– 6) δηξ sinyx AAAA = 2222 yx AAAA +=+ ηξ    = = δαχ δαψ sin2sin2sin cos2tan2tan      = ⋅= ψ χ δ ψχα 2sin 2tan tan 2cos2cos2cos       ⇐       χ ψ δ α       ⇐       δ α χ ψ ξ η χ A A =:tan x y A A =:tanα 1 sinsin cos 2 sin 2 2 2 =         +−        δδ δ δ y y y y x x x x A E A E A E A E 1 22 =         +         η η ξ ξ A E A E               − =        η ξ ψψ ψψ E E E E y x cossin sincos xy δδδ −=( ) ( ) yxyx yx zktj y zktj xyx eAeAEEE 1111 ∧ +− ∧ +− ∧∧ +=+= δωδω Return to Table of Content
  • 35.
    35 POLARIZATIONSOLO ( ) () yx yx zktj y zktj x eAeAE 11 ∧ +− ∧ +− += δωδω Linearly Horizontally Polarized (LHP): ( )x x x zkt A E δω +−= cos Degenerated States of Polarization Ellipse ( ) 01 == ∧ +− y zktj x AeAE xxδω http://www.enzim.hu/~szia/cddemo/edemo0.htm (Andras Szilagyi) Linearly Vertically Polarized (LVP): ( ) 01 == ∧ +− x zktj y AeAE yyδω ( )y y y zkt A E δω +−= cos
  • 36.
    36 POLARIZATIONSOLO Linear + 45°Polarized (L+45P) Degenerated States of Polarization Ellipse ( )tzkj yx eAAE yx ω−− ∧∧       += 11 ( )tzkj yx eAAE yx ω−− ∧∧       −= 11 Plane Polarization Mixed 0=−= xy δδδ πδδδ =−= xy Linear - 45 ° Polarized (L-45P) http://www.enzim.hu/~szia/cddemo/edemo0.htm (Andras Szilagyi) ( ) ( ) yx yx zktj y zktj x eAeAE 11 ∧ +− ∧ +− += δωδω
  • 37.
    37 POLARIZATIONSOLO ( )x x zkt A E δω +−=cos ( )x y zkt A E δω +−−= sin Degenerated States of Polarization Ellipse Right Circular Polarization (RCP) AAA yxxy ===−= &2/πδδδ 1 22 =      +      A E A E yx http://www.enzim.hu/~szia/cddemo/edemo0.htm (Andras Szilagyi) ( ) ( ) yx yx zktj y zktj x eAeAE 11 ∧ +− ∧ +− += δωδω ( )x x zkt A E δω +−= cos ( )x y zkt A E δω +−= sin Left Circular Polarization (LCP) AAA yxxy ===−= & 2 3π δδδ
  • 38.
    38 POLARIZATIONSOLO ( )x x zkt A E δω +−=cos ( )x y zkt A E δω +−= sin Degenerated States of Polarization Ellipse Left Circular Polarization (LCP) AAA yxxy ===−= & 2 3π δδδ 1 22 =      +      A E A E yx http://www.enzim.hu/~szia/cddemo/edemo0.htm (Andras Szilagyi) Superposition of Two Circular Polarizations ( ) ( ) yx yx zktj y zktj x eAeAE 11 ∧ +− ∧ +− += δωδω Return to Table of Content
  • 39.
    39 POLARIZATION SOLO The Stokes PolarizationParameters George Gabriel Stokes 1819-1903 G.G. Stokes, “On the Composition and Resolution of Streams of Polarized Light from different Sources”, Trans. Cambridge Phil. Soc., Vol.9, 1852, pp.399-416 ( ) ( ) yx yyxx zktAzktAE 11 coscos ∧∧ +−++−= δωδω  where ( ) ( ) ( ) ( ) δδ 2 22 sin , cos ,, 2 , =         +−        y y y y x x x x A tzE A tzE A tzE A tzE The Polarization Ellipse is All the information of polarization is contained in this equation. In order to observe the quantities involved let take the time average <… > of the time dependent quantities in the Polarization Ellipse equation. ( ) ( ) ( ) ( ) δδ 2 2 2 2 2 sin , cos ,, 2 , =+− y y yx yx x x A tzE AA tzEtzE A tzE ( ) ( ) ( ) ( ) yxjidttzEtzE T tzEtzE T ji T ji ,,,, 1 :,, 0 lim == ∫→∞ ( ) ( ) ( ) ( ) ( )22222 sin2,4cos,,8,4 δδ yxyxyxyxxy AAtzEAtzEtzEAAtzEA =+−
  • 40.
    40 POLARIZATION SOLO The Stokes PolarizationParameters (continue – 1) We obtain ( ) ( ) ( ) ( ) ( )22222 sin2,4cos,,8,4 δδ yxyxyxyxxy AAtzEAtzEtzEAAtzEA =+− ( ) ( ) ( ) ( ) ( ) ( ) δδδωδδ δωδω δ cos 2 22coscos 1 2 coscos 1 ,, 0 0 lim lim yx T xyyx T yx T yxyx T zx AA dtzkt T AA dtzktzktAA T tzEtzE =         ++−−−= +−+−= ∫ ∫ →∞ →∞  ( ) ( ) ( )[ ] 2 2cos1 1 2 cos 1 , 2 0 2 0 222 limlim x T x T x T xx T x A dtzkt T A dtzktA T tzE =+−−=+−= ∫∫ →∞→∞ δωδω ( ) ( ) 2 cos 1 , 2 0 222 lim y T yy T y A dtzktA T tzE =+−= ∫→∞ δω ( ) ( )222222 sin22cos22 δδ yxyxyxyx AAAAAAAA =+− By adding and subtracting on the left side of this equation we obtain ( ) ( ) ( ) ( )22222222 sin2cos2 δδ yxyxyxyx AAAAAAAA =−−−+ 44 2 yx AA +
  • 41.
    41 POLARIZATION SOLO The Stokes PolarizationParameters (continue – 2) The Stokes Parameters are defined as ( ) ( ) ( ) ( )22222222 sin2cos2 δδ yxyxyxyx AAAAAAAA ++−=+ δ δ sin2 cos2 3 2 22 1 22 0 yx yx yx yx AAVS AAUS AAQS AAIS == == −== +== 2 3 2 2 2 1 2 0 SSSS ++= The Stokes Vector is defined as                 − + =               =               = δ δ sin2 cos2 22 22 3 2 1 0 yx yx yx yx AA AA AA AA V U Q I S S S S S  The Stokes Parameters are observable using a proper experiment.
  • 42.
    42 POLARIZATION SOLO The Stokes PolarizationParameters (continue – 3) Consider a quasi-monochromatic wave of mean frequency ω propagating in z direction ( ) ( )( ) ( ) ( )( ) ( ) ( ) yxyx ztEztEetAetAE yx ttzkj y ttzkj x xy 1111 ,, ∧∧∧ +−− ∧ +−− +=+= δωδω For monochromatic waves Ax, Ay, δx, δy, ω are constant. Quasi-monochromatic Light For quasi-monochromatic waves Ax, Ay, δx, δy , ω are slowly changing with time. We have ( ) ( ) ( ) ( ) yxjidttzEtzE T tzEtzE T ji T ji ,,,, 1 :,, 0 lim == ∫ ∗ →∞ ∗ ( ) ( ) 2 0 21 :,, lim x T x T xx AdtA T tzEtzE == ∫→∞ ∗ ( ) ( ) 2 0 21 :,, lim y T y T yy AdtA T tzEtzE == ∫→∞ ∗ ( ) ( ) ( )yxyx j yx T j y j x T yx eAAdteAeA T tzEtzE δδδδ −− →∞ ∗ == ∫ 0 1 :,, lim ( ) ( ) ( )yxxy j yx T j x j y T xy eAAdteAeA T tzEtzE δδδδ −−− →∞ ∗ == ∫ 0 1 :,, lim
  • 43.
    43 POLARIZATION SOLO The Stokes PolarizationParameters (continue – 3) ( ) ( )( ) ( ) ( )( ) ( ) ( ) yxyx ztEztEetAetAE yx ttzkj y ttzkj x xy 1111 ,, ∧∧∧ +−− ∧ +−− +=+= δωδω Quasi-monochromatic Light The Stokes Parameters are defined as ( ) ( ) ( ) ( ) ∗∗ ∗∗ ∗∗ ∗∗ −=−= +=−= −=−= +=+= xyyxxyyx xyyxxyyx yyxxyx yyxxyx EEEEjAAS EEEEAAS EEEEAAS EEEEAAS δδ δδ sin2 cos2 3 2 22 1 22 0 ( ) ( ) 2 0 21 :,, lim x T x T xx AdtA T tzEtzE == ∫→∞ ∗ ( ) ( ) 2 0 21 :,, lim y T y T yy AdtA T tzEtzE == ∫→∞ ∗ ( ) ( ) ( )yxyx j yx T j y j x T yx eAAdteAeA T tzEtzE δδδδ −− →∞ ∗ == ∫ 0 1 :,, lim ( ) ( ) ( )yxxy j yx T j x j y T xy eAAdteAeA T tzEtzE δδδδ −−− →∞ ∗ == ∫ 0 1 :,, lim
  • 44.
    44 POLARIZATION SOLO The Stokes PolarizationParameters (continue – 4) Stokes Vector for Different Polarization Types LHP               = 0 0 1 1 0IS  2 0 0 x y AI A = = LVP 2 0 0 y x AI A = =               − = 0 0 1 1 0IS                = 0 1 0 1 0IS  L+45P 2 0 2 0 x yx AI AA = = =δ               − = 0 1 0 1 0IS  L-45P 2 0 2 x yx AI AA = = = πδ               = 1 0 0 1 0IS  RCP 2 0 2 2 x yx AI AA = = = π δ               − = 1 0 0 1 0IS  2 0 2 2 3 x yx AI AA = = = π δ LCP δ δ sin2 cos2 3 2 22 1 22 0 yx yx yx yx AAS AAS AAS AAS = = −= += 2 3 2 2 2 1 2 0 SSSS ++= ( ) ( ) yxyx yx zktj y zktj xyx eAeAEEE 1111 ∧ +− ∧ +− ∧∧ +=+= δωδω
  • 45.
    45 POLARIZATION SOLO The Stokes PolarizationParameters (continue – 5) The Stokes Parameters are defined as δ δ sin2 cos2 3 2 22 1 22 0 yx yx yx yx AAS AAS AAS AAS = = −= += 2 3 2 2 2 1 2 0 SSSS ++= δαχ sin2sin2sin = αψχ 2cos2cos2cos =⋅ δαψχ cos2sin2sin2cos ⋅=⋅ We found 22 22 2 2 222 tan1 tan1 2cos& tan1 tan 2sin yx yx yx yx AA AA AA AA + − = + − = + = + = α α α α α α x y A A =:tanα 0 1 22 22 2cos2cos S S AA AA yx yx = + − =⋅ ψχ 0 2 22 cos cos2sin2sin2cos S S AA AA yx yx = + ⋅ =⋅=⋅ δ δαψχ 0 3 22 sin sin2sin2sin S S AA AA yx yx = + ⋅ == δ δαχ       =       = − − 0 31 1 21 sin 2 1 tan 2 1 S S S S χ ψ Return to Table of Content
  • 46.
    46 POLARIZATION SOLO Consider a quasi-monochromaticwave of mean frequency ω propagating in z direction composed of a Unpolarized component AUP with random phases δrx and δry and a Polarized component Ax, δx , Ay, δy ( ) ( ) ( )tzkjj yP j UP j xP j UPyx eeAeAeAeAEEE yxyx yyrxxr ωδδδδ −− ∧∧∧∧     +++=+= 1111 Measuring the Stokes Parameters Pass the beam through a waveplate that induces a wave retardation of φ and a polarizer with a transmission axis at an angle β relative to x axis ( ) ( ) ( ) ( ) ( )tzkjj yP j UP j xP j UP eeAeAeAeAE yx yyrxxr ωϕδδϕδδ −− ∧ − ∧ +     +++= 11 2/2/ ' ( ) ( ) ( ) ( )[ ] ( )tzkjj yP j UP j xP j UP eeAeAeAeAE yyrxxr ωϕδδϕδδ ββ −−−+ +++= sincos" 2/2/ The waveplate that induces a wave retardation of φ between the phases of x and y components of the polarized light but will not affect the random phase of the unpolarized light. The polarizer will transmit only the component along the transmission axis
  • 47.
    47 POLARIZATION SOLO Measuring the StokesParameters (continue – 1) ( ) ( ) ( ) ( )[ ] ( )tzkjj yP j UP j xP j UP eeAeAeAeAE yyrxxr ωϕδδϕδδ ββ −−−+ +++= sincos" 2/2/ ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] zz yyrxxryyrxxr j yP j UP j xP j UP j yP j UP j xP j UP eAeAeAeAeAeAeAeAcnkEEcnkS 11 sincossincos"", 2/2/2/2/ ∧ −−−+−−−+ ∧ ∗ +++⋅+++=⋅= ββββϕβ ϕδδϕδδϕδδϕδδ  ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) βββ βββ βββ βββ ββββ ϕδδϕδδϕδδ δϕδδϕδδδ ϕδδϕδδδϕδ δδδϕδδϕδ ϕδδϕδδϕδδϕδδ 22 0 2/ 0 2/ 2 0 2/2 0 2/ 0 2 0 2/22 0 2/ 00 2/22 0 2/2 2/2/2/2/ sincossin sin2/cossin cossincos cossincos2/ sincossincos         ++         ++         ++         ++         ++         ++         ++         += +++⋅+++ −−−−−−− −−−+− −−+−−+ −−−−+ −−−+−−−+ yP jj yPUP j yPxP jj yPUP jj yPUPUP jj UPxP jj UP j xPyP jj xPUPxP jj xPUP jj UPyP jj UP jj xPUPUP j yP j UP j xP j UP j yP j UP j xP j UP AeeAAeAAeeAA eeAAAeeAAeeA eAAeeAAAeeAA eeAAeeAeeAAA eAeAeAeAeAeAeAeA yxpxyyxr yyyrxyrrr xyxyrxx xryxryxrx yyrxxryyrxxr                     The time average Poynting vector is
  • 48.
    48 POLARIZATION SOLO ( ) () ( )[ ] zyP jj yPxPxPUP AeeAAAAcnkS 1 22222 sincossincos2/ ∧ −−− ++++= ββββ ϕδϕδ  2 2sin cossin 2 2cos1 sin 2 2cos1 cos 2 2 β ββ β β β β = − = + = ( ) ( ) ( ) ( )[ ]{ } ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] [ ]βϕβϕβ βϕδβϕδβ βϕβϕβ βϕϕϕϕβ δδδδ δδ 2sinsin2sincos2cos 2 2sinsinsin22sincossin22cos 2 2sinsin2sincos2cos 2 2sinsincossincos2cos 2 ******** 22222 22222 22222 xyyxxyyxyyxxyyxx yPxPyPxPyPxPyPxPUP jj yPxP jj yPxPyPxPyPxPUP jj yPxPyPxPyPxPUP EEEEjEEEEEEEEEEEE cnk AAjAAAAAAA cnk eeAAjeeAAAAAAA cnk jejeAAAAAAA cnk S −+++−++= ++−+++= −−++−+++= ++−+−+++= −− −  Measuring the Stokes Parameters (continue – 2) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] zz yyrxxryyrxxr j yP j UP j xP j UP j yP j UP j xP j UP eAeAeAeAeAeAeAeAcnkEEcnkS 11 sincossincos 2/2/2/2/ ∧ −−−+−−−+ ∧ ∗ +++⋅+++=⋅= ββββ ϕδδϕδδϕδδϕδδ  [ ]βϕβϕβ 2sinsin2sincos2cos 2 3210 SjSSS cnk S +++=  ( ) ( ) ( ) ( )xyyPxPxyyx xyyPxPxyyx yPxPyyxx yPxPUPyyxx AAEEEEjS AAEEEES AAEEEES AAAEEEES δδ δδ −=−= −=+= −=−= ++=+= ∗∗ ∗∗ ∗∗ ∗∗ sin2 cos2 3 2 22 1 222 0
  • 49.
    49 POLARIZATION SOLO Measuring the StokesParameters (continue – 3) ( ) [ ]βϕβϕβϕβ 2sinsin2sincos2cos 2 , 3210 SjSSS cnk S +++=  ( ) ( ) ( ) ( )xyyPxPxyyx xyyPxPxyyx yPxPyyxx yPxPUPyyxx AAEEEEjS AAEEEES AAEEEES AAAEEEES δδ δδ −=−= −=+= −=−= ++=+= ∗∗ ∗∗ ∗∗ ∗∗ sin2 cos2 3 2 22 1 222 0 The Stokes Parameters are measured by first removing the waveplate φ = 0 ( ) [ ]ββϕβ 2sin2cos 2 0, 210 SSS cnk S ++==  Now the polarizer is sequentially rotate to β = 0, π/4 and π/2 ( ) [ ]10 2 0,0 SS cnk S +=== ϕβ  ( ) [ ]20 2 0,4/ SS cnk S +=== ϕπβ  ( ) [ ]10 2 0,2/ SS cnk S −=== ϕπβ  For the final measurement we add the waveplate with φ = π/2 and polarizer at β = π/4 ( ) [ ]30 2 2/,4/ SS cnk S −=== πϕπβ  ( ) ( )[ ]0,2/0,0 2 0 ==+=== ϕπβϕβ SS cnk S  ( ) ( )[ ]0,2/0,0 2 1 ==−=== ϕπβϕβ SS cnk S  ( ) 02 0,4/2 2 SS cnk S −=== ϕπβ  ( )2/,4/2 2 03 πϕπβ ==−= S cnk SS 
  • 50.
    50 POLARIZATION SOLO Measuring the StokesParameters (continue – 4) ( ) [ ]βϕβϕβϕβ 2sinsin2sincos2cos 2 , 3210 SjSSS cnk S +++=  ( ) ( ) ( ) ( )xyyPxPxyyx xyyPxPxyyx yPxPyyxx yPxPUPyyxx AAEEEEjS AAEEEES AAEEEES AAAEEEES δδ δδ −=−= −=+= −=−= ++=+= ∗∗ ∗∗ ∗∗ ∗∗ sin2 cos2 3 2 22 1 222 0 The Stokes Parameters can measure the degree of polarization of a beam. We can see that a beam is unpolarized iff: 00 321 2 0 ===>=+= ∗∗ SSSAEEEES UPyyxx The Degree of Polarization is defined as: 10 0 2 3 2 2 2 1 222 22 ≤≤ ++ = ++ + = P S SSS AAA AA P yPxPUP yPxP If the beam is completely polarized then AUP = 0: 0 2 3 2 2 2 1 2 0 >++= SSSS Return to Table of Content
  • 51.
    51 POLARIZATION SOLO H. Poincaré, “ThéorieMathématique de la Lumiere”, Gauthiers-Villars, Paris, 1892, Vol.2, Ch.12 Poincaré Sphere Jules Henri Poincaré 1854-1912 We found 03 02 01 /2sin /2sin2cos /2cos2cos SS SS SS = =⋅ =⋅ χ ψχ ψχ We see that the normalized Stokes Parameters can be represented as a unit vector on a sphere (called Poincaré Sphere) using the Polarized Ellipse parameters χ and ψ.
  • 52.
    52 POLARIZATION SOLO Poincaré Sphere J.D. Kraus,“Electromagnetics”, 4th Ed., McGraw Hill, 1992, p.607 Return to Table of Content
  • 53.
    53 POLARIZATION SOLO The Mueller Matricesfor Polarizing Components In 1948 Hans Mueller, then a professor of physics at MIT devised a matrix method that deals with the relation of Stokes vector at the output of an optical device to the Stokes vector at the input of the optical device.                             =               3 2 1 0 33323130 23222120 13121110 03020100 3 2 1 0 ' ' ' ' S S S S mmmm mmmm mmmm mmmm S S S S inputoutput SMS  = or Hans Mueller, “The Foundation of Optics”, J. Opt. Soc. Am., 37, pg. 110 (1947) 38, pg. 661(1948)
  • 54.
    54 POLARIZATION SOLO The Mueller Matricesfor Polarizing Components Suppose that we have a few optical elements through which the ray passes . ( ) inputinputoutput SMSMMMSMMSMS  ==== 12312323 or 123 MMMM =
  • 55.
    55 POLARIZATION SOLO The Mueller Matricesfor Polarizing Components Polarizer 10' 10' ≤≤= ≤≤= yyyy xxxx pEpE pEpE The Polarizer is described by two orthogonal transmission axes that are characterized, respectively, by transmission factors px and py (0 ≤ px,py ≤ 1). ( )∗∗ ∗∗ ∗∗ ∗∗ −= += −= += xyyx xyyx yyxx yyxx EEEEjS EEEES EEEES EEEES 3 2 1 0 ( )∗∗ ∗∗ ∗∗ ∗∗ −= += −= += ''''' ''''' ''''' ''''' 3 2 1 0 xyyx xyyx yyxx yyxx EEEEjS EEEES EEEES EEEES                             +− −+ =               3 2 1 0 2222 2222 3 2 1 0 2000 0200 00 00 2 1 ' ' ' ' S S S S pp pp pppp pppp S S S S yx yx yxyx yxyx ( )               +− −+ = yx yx yxyx yxyx yxPOL pp pp pppp pppp ppM 2000 0200 00 00 2 1 , 2222 2222
  • 56.
    56 POLARIZATION SOLO The Mueller Matricesfor Polarizing Components Polarizer ( ) 22 1 : /tan: yx xy ppp pp += = − βLet define                             =               3 2 1 0 2 2 22 22 3 2 1 0 2sin000 02sin00 002cos 002cos 2 1 ' ' ' ' S S S S p p pp pp S S S S β β β β ( )               = β β β β β 2sin000 02sin00 0012cos 002cos1 2 , 2 p pMPOL β β sin cos pp pp y x = =
  • 57.
    57 POLARIZATION SOLO The Mueller Matricesfor Polarizing Components Polarizer (continue -1) 0' 10' = ≤≤= y xxxx E pEpE Ideal Linear Polarizer py = 0 and 0 ≤ px ≤ 1. ( ) ( )               +=                             =               0 0 1 1 2 0000 0000 0011 0011 2 ' ' ' ' 10 2 3 2 1 0 || 2 3 2 1 0 SS p S S S S p S S S S x M x POL    Regardless the state of polarization of the input beam the output beam is Linear Horizontally Polarized (LHP) 10' 0' ≤≤= = yyyy x pEpE E Ideal Linear Polarizer px = 0 and 0 ≤ py ≤ 1. ( ) ( )               − −=                             − − =               ⊥ 0 0 1 1 2 0000 0000 0011 0011 2 ' ' ' ' 10 2 3 2 1 0 2 3 2 1 0 SS p S S S S p S S S S y M y POL    Regardless the state of polarization of the input beam the output beam is Linear Vertically Polarized (LVP)
  • 58.
    58 POLARIZATION SOLO The Mueller Matricesfor Polarizing Components Polarizer (continue -2) Crossed Polarizers ( ) ( )               =                             − −               =               ⊥ 0 0 0 0 22 0000 0000 0011 0011 2 0000 0000 0011 0011 2 ' ' ' ' 22 3 2 1 0 2 || 2 3 2 1 0 yx M y M x pp S S S S pp S S S S POLPOL      Regardless the state of polarization of the input beam the output beam is completely blocked Crossed Polarizers are a combination of a Linear Polarizer with the transmission axes normal to the x-axis, followed by a Linear Polarizer with the transmission axes parallel to the x-axis (or vice-versa).
  • 59.
    59 POLARIZATION SOLO The Mueller Matricesfor Polarizing Components Polarizer (continue -3) Eigenvalues and Eigenvectors of the Linear Polarizer Mueller Matrix Let find the Input Stokes Vectors that are unaffected by the Polarizer; i.e.:               =                             +− −+ =               3 2 1 0 3 2 1 0 2222 2222 3 2 1 0 2000 0200 00 00 2 1 ' ' ' ' S S S S S S S S pp pp pppp pppp S S S S POLM yx yx yxyx yxyx λ    ( ) 0 22000 02200 002 002 2 1 , 2222 2222 44 = − − −+− −−+ =− λ λ λ λ λ yx yx yxyx yxyx xyxPOL pp pp pppp pppp IppM               == 0 0 1 1 & 1 2 1 Spx  λ               − == 0 0 1 1 & 2 2 2 Spy  λ 1st eigenvalue- eigenvector 2nd eigenvalue- eigenvector We can see that only LHP and LVP are unaffected by the Linear Polarizer.
  • 60.
    60 POLARIZATION SOLO The Mueller Matricesfor Polarizing Components Waveplate y j y x j x EeE EeE 2/ 2/ ' ' ϕ ϕ − = = The Waveplate is a polarizing element that introduced a phase shift φ between the orthogonal components of an optical beam. A Waveplate is a phase-shifter but is also called a retarder or a compensator. ( )∗∗ ∗∗ ∗∗ ∗∗ −= += −= += xyyx xyyx yyxx yyxx EEEEjS EEEES EEEES EEEES 3 2 1 0 ( )∗∗ ∗∗ ∗∗ ∗∗ −= += −= += ''''' ''''' ''''' ''''' 3 2 1 0 xyyx xyyx yyxx yyxx EEEEjS EEEES EEEES EEEES                             − =               3 2 1 0 3 2 1 0 cossin00 sincos00 0010 0001 ' ' ' ' S S S S S S S S ϕϕ ϕϕ ( )               − = ϕϕ ϕϕ ϕ cossin00 sincos00 0010 0001 WP M
  • 61.
    61 POLARIZATION SOLO The Mueller Matricesfor Polarizing Components Waveplate (continue – 1) Quarter-Waveplate φ = π/2 The Quarter-Waveplate transforms Linearly Polarized (L+45P or L-45P) to Right or Left Circularly Polarized (RCP or LCP) or vice-versa. ( )               − == 0100 1000 0010 0001 2/πϕWPM ( )    RCPPLPL WPM               =                             − =               = ++ 1 0 0 1 0 1 0 1 0100 1000 0010 0001 0 1 0 1 2/ 4545 πϕ ( )    LCPPLPL WPM               − =               −               − =               − = −− 1 0 0 1 0 1 0 1 0100 1000 0010 0001 0 1 0 1 2/ 4545 πϕ ( )    PLRCPRCP WPM 45 0 1 0 1 1 0 0 1 0100 1000 0010 0001 1 0 0 1 2/ −               − =                             − =               = πϕ ( )     LCP PLLCPLCP WPM 45 0 1 0 1 1 0 0 1 0100 1000 0010 0001 1 0 0 1 2/ +               =               −              − =               − = πϕ
  • 62.
    62 POLARIZATION SOLO The Mueller Matricesfor Polarizing Components Waveplate (continue – 2) Half-Waveplate φ = π ( )               − − == 1000 0100 0010 0001 πϕWPM               − − =                             − − =               3 2 1 0 3 2 1 0 3 2 1 0 1000 0100 0010 0001 ' ' ' ' S S S S S S S S S S S S χ ψχ ψχ 2sin 2sin2cos 2cos2cos 0 3 0 2 0 1 = ⋅= ⋅= S S S S S S       =       = − − 0 31 1 21 sin 2 1 tan 2 1 S S S S χ ψ 2 ' 2 ' π χχ ψ π ψ −= −= The Half-Waveplate reverses the orientation and ellipticity of the polarization ellipse (polarization state).
  • 63.
    63 POLARIZATION SOLO The Mueller Matricesfor Polarizing Components Waveplate (continue – 3) The phases of two adjacent Half-Waveplates add. ( ) ( ) ( ) ( )                             ++ +−+ =                             −               − =               3 2 1 0 2121 2121 3 2 1 0 22 22 11 11 3 2 1 0 cossin00 sincos00 0010 0001 cossin00 sincos00 0010 0001 cossin00 sincos00 0010 0001 ' ' ' ' S S S S S S S S S S S S ϕϕϕϕ ϕϕϕϕ ϕϕ ϕϕ ϕϕ ϕϕ Eigenvalues and Eigenvectors of the Waveplate Mueller Matrix ( ) 0 cossin00 sincos00 0010 0001 4 = − −− − − =− λϕϕ ϕλϕ λ λ λϕ IMWP  LHP S               == 0 0 1 1 &1 11  λ  LVP S               − =−= 0 0 1 1 &1 22  λ 1st eigenvalue- eigenvector 2nd eigenvalue- eigenvector We can see that only LHP and LVP are unaffected by the Waveplate Polarizer.
  • 64.
    64 POLARIZATION SOLO The Mueller Matricesfor Polarizing Components The Mueller Matrix of a Rotator (Coordinate Rotation) ( ) ( ) θθθγ θθθγ cossinsin' sincoscos' yxy yxx EEEE EEEE +−=−= +=−= Assume that the polarizing device rotates its orthogonal axes along the ray propagation direction by an angle θ. The orthogonal axes are defined as (‘). ( )∗∗ ∗∗ ∗∗ ∗∗ −= += −= += xyyx xyyx yyxx yyxx EEEEjS EEEES EEEES EEEES 3 2 1 0 ( )∗∗ ∗∗ ∗∗ ∗∗ −= += −= += ''''' ''''' ''''' ''''' 3 2 1 0 xyyx xyyx yyxx yyxx EEEEjS EEEES EEEES EEEES                             − =               3 2 1 0 3 2 1 0 1000 02cos2sin0 02sin2cos0 0001 ' ' ' ' S S S S S S S S θθ θθ ( )               − = 1000 02cos2sin0 02sin2cos0 0001 θθ θθ θROTM '1'1 1111 '' sincos yx yxyx yx yx EE EEEE ∧∧ ∧∧∧∧ +=         +=+= γγ
  • 65.
    65 POLARIZATION SOLO The Mueller Matricesfor Polarizing Components Assume that the polarizing device rotates its orthogonal axes along the ray propagation direction by an angle θ. The Mueller Rotation Matrix from x,y to x’,y’ is ( )               − = 1000 02cos2sin0 02sin2cos0 0001 θθ θθ θROTM The Mueller Polarizator Matrix is ( )               = β β β β β 2sin000 02sin00 0012cos 002cos1 2 , 2 p pMPOL ( ) SMS ROT  θ=' ( ) ( ) ( ) SMpMSpMS ROTPOLPOL  θββ ,'," == The Mueller Rotation Matrix from x’,y’ to x,y is ( )               − =− 1000 02cos2sin0 02sin2cos0 0001 θθ θθ θROTM ( ) ( ) ( ) ( ) SMpMMSMS ROTPOLROTROT  θβθθ ,"'" −=−= ( ) ( ) ( ) ( )θβθθβ ROTPOLROTPOL MpMMpM ,,, −= The Mueller Matrix of a Rotated Polarizer
  • 66.
    66 POLARIZATION SOLO The Mueller Matricesfor Polarizing Components ( ) ( ) ( )               +− −+ = β θβθθθβθβ θθβθβθθβ θβθβ θβ 2sin000 02cos2sin2sin2cos2sin2sin12sin2cos 02cos2sin2sin12sin2sin2cos2cos2cos 02sin2cos2cos2cos1 2 ,, 22 222 p pMPOL The Mueller Matrix for Rotated Polarizer is ( ) ( ) ( ) ( )               −                             − = −= 1000 02cos2sin0 02sin2cos0 0001 2sin000 02sin00 0012cos 002cos1 1000 02cos2sin0 02sin2cos0 0001 2 ,,, 2 θθ θθ β β β β θθ θθ θβθθβ p MpMMpM ROTPOLROTPOL The Mueller Matrix of a Rotated Polarizer (continue – 1)
  • 67.
    67 POLARIZATION SOLO The Mueller Matricesfor Polarizing Components ( ) ( ) ( )               − +− −−+ = ϕϕβϕβ ϕβϕββϕββ ϕβϕββϕββ ϕβ cossin2cossin2sin0 sin2coscos2cos2sincos12cos2sin0 sin2sincos12cos2sincos2sin2cos0 0001 , 22 22 WPM The Mueller Matrix for Rotated Waveplate is ( ) ( ) ( ) ( )               −               −               − = −= 1000 02cos2sin0 02sin2cos0 0001 cossin00 sincos00 0010 0001 1000 02cos2sin0 02sin2cos0 0001 ,, θθ θθ ϕϕ ϕϕθθ θθ θϕθθβ ROTWPROTPOL MMMpM The Mueller Matrix of a Rotated Waveplate Return to Table of Content
  • 68.
    68 POLARIZATION SOLO The Jones PolarizationParameters R. Clark Jones, “A New Calculus for the Treatment of Optical Systems”, J. Opt. Soc. Am., Vol.31, July 1941, pp.500-503 J. Opt. Soc. Am., Vol.32, Aug. 1942, pp.486-493 J. Opt. Soc. Am., Vol.37, Feb. 1947, pp.107-110 J. Opt. Soc. Am., Vol.37, Feb. 1947, pp.110-112 J. Opt. Soc. Am., Vol.38, Aug. 1948, pp.671-585 R. Clark Jones 1916-2004 Mueller matrices deal with the intensity of the beam. If the phase information is important we must use Jones formalism. Jones calculus was developed in the same time with the Mueller calculus by R. Clark Jones who introduced Jones vectors and Jones matrices: Jones vectors describe the polarization of light: Jones matrices describe the optical component:         + = y x yx E E EE J 22 1 [ ]       = 2221 1211 jj jj J Jones calculus deals only with polarized light.
  • 69.
    69 POLARIZATION SOLO Stokes and JonesVector for Different Polarization Types LHP       =               = 0 1 0 0 1 1 0 JIS  2 0 0 x y AI A = = LVP 2 0 0 y x AI A = =       =               − = 1 0 0 0 1 1 0 JIS        =               = 1 1 2 1 0 1 0 1 0 JIS  L+45P 2 0 2 0 x yx AI AA = = =δ       − =               − = 1 1 2 1 0 1 0 1 0 JIS  L-45P 2 0 2 x yx AI AA = = = πδ       =               = i JIS 1 2 1 1 0 0 1 0  RCP 2 0 2 2 x yx AI AA = = = π δ       − =               − = i JIS 1 2 1 1 0 0 1 0  2 0 2 2 3 x yx AI AA = = = π δ LCP δ δ sin2 cos2 3 2 22 1 22 0 yx yx yx yx AAS AAS AAS AAS = = −= += 2 3 2 2 2 1 2 0 SSSS ++= ( ) ( ) yxyx yx zktj y zktj xyx eAeAEEE 1111 ∧ +− ∧ +− ∧∧ +=+= δωδω Return to Table of Content
  • 70.
    70 POLARIZATION SOLO Faraday Effect (Hechtp.261) Michael Faraday (England) 1845 described the rotation of the plane of polarized light that passed through glass in a magnetic field. Return to Table of Content
  • 71.
    71 POLARIZATION SOLO Pockels Effect (Hechtp.263, Chuang p.509, Meyer-Arent p.318) Vnr nretardatio Er n n n Ln Ln yx z yy xx 3 0'' 32 '' '' 2 : 21 2 2 λ π λ π λ π =Φ−Φ=Φ ⇒ = ∆ −=      ∆ =Φ =Φ Pockels Effect 1893 Electro – Optical Effects The electro – optical effects are called Pockels or Kerr where the refractive index changes linearly or quadraticly, respectively.    = = =      ∆ Kerrk Pockelsk EKn n k 2 11 02 Frederich Carl Alwin Pockels (1865-1913) http://www.physi.uni-heidelberg.de/~schmiedm/Vorlesung/LasPhys02/LectureNotes/OpticsCrystals.pdf#search='Pockels%20Effect' http://en.wikipedia.org/wiki/Pockels_effect
  • 72.
    72 POLARIZATION SOLO Electro – OpticalEffects Kerr Effect (Hecht p.263, Chuang p.509, Meyer-Arent p.318) The electro – optical effects are Kerr or Pockels 2 0 EKn λ=∆ 2 2 0 2 d VKλπ =Φ∆ In the Kerr Electro-optic Effect 1875 is the electric field that causes the substance to become birefringent. Return to Table of Content
  • 73.
    73 POLARIZATIONSOLO Plane Polarized Wavein an Absorbing Medium Circular Polarized Wave in an Absorbing Medium
  • 74.
    74 POLARIZATIONSOLO Plane Polarized Wavein an Refracting Medium Circular Polarized Wave in an Refracting Medium
  • 75.
    75 POLARIZATIONSOLO Plane Polarized Wavein a Medium Showing Circular Dichroism Plane Polarized Wave in a Medium Showing Circular Birefrigens
  • 76.
    76 POLARIZATIONSOLO Plane Polarized Wavein a Medium Showing Both Circular Dichroism and Circular Birefrigens
  • 77.
  • 78.
    78 OPTICSSOLO References Optics Polarization A.Yariv, P. Yeh, “Optical Waves in Crystals”, John Wiley & Sons, 1984 M. Born, E. Wolf, “Principles of Optics”, Pergamon Press,6th Ed., 1980 E. Hecht, A. Zajac, “Optics”, Addison-Wesley, 1979, Ch.8 C.C. Davis, “Lasers and Electro-Optics”, Cambridge University Press, 1996 G.R. Fowles, “Introduction to Modern Optics”,2nd Ed., Dover, 1975, Ch.2 M.V.Klein, T.E. Furtak, “Optics”, 2nd Ed., John Wiley & Sons, 1986 http://en.wikipedia.org/wiki/Polarization W.C.Elmore, M.A. Heald, “Physics of Waves”, Dover Publications, 1969 E. Collett, “Polarization Light in Fiber Optics”, PolaWave Group, 2003 W. Swindell, Ed., “Polarization Light”, Benchmark Papers in Optics, V.1, Dowden, Hutchinson & Ross, Inc., 1975 http://www.enzim.hu/~szia/cddemo/edemo0.htm (Andras Szilagyi)
  • 79.
    79 ELECTROMAGNETICSSOLO References Electromagnetics J.D. Jackson,“Classical Electrodynamics”, 3rd Ed., John Wiley & Sons, 1999 R. S. Elliott, “Electromagnetics”, McGraw-Hill, 1966 J.A. Stratton, “Electromagnetic Theory”, McGraw-Hill, 1941 W.K.H. Panofsky, M. Phillips, “Classical Electricity and Magnetism”, Addison- Wesley, 1962 F.T. Ulaby, R.K. More, A.K. Fung, “Microwave Remote Sensors Active and Passive”, Addison-Wesley, 1981 A.L. Maffett, “Topics for a Statistical Description of Radar Cross Section”, John Wiley & Sons, 1988
  • 80.
    80 ELECTROMAGNETICSSOLO References 1.W.K.H. Panofsky &M. Phillips, “Classical Electricity and Magnetism”, 2.J.D. Jackson, “Classical Electrodynamics”, 3.R.S. Elliott, “Electromagnetics”, 4.A.L. Maffett, “Topics for a Statistical Description of Radar Cross Section”, Return to Table of Content
  • 81.
    January 4, 201581 SOLO POLARIZATION Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 –2013 Stanford University 1983 – 1986 PhD AA