UE SPM-PHY-S07-101 Polarization Optics N. Fressengeas Laboratoire Matériaux Optiques, Photonique et Systèmes Unité de Recherche commune à l Université Paul Verlaine Metz et à Supélec Document à télécharger sur http://moodle.univ-metz.fr/ N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 1
Further reading [Hua94, K 85] The physics of polarization optics S. Huard. Polarisation de la lumière. Masson, 1994. G. P. Können. Polarized light in Nature. Cambridge University Press, 1985. N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 2
Course Outline The physics of polarization optics 1 The physics of polarization optics Polarization states Jones Calculus Stokes parameters and the Poincare Sphere 2 Jones Matrices Polarizers Linear and Circular Anisotropy Jones Matrices Composition 3 Formalisms used Propagation through optical devices N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 3
The vector nature of light Optical wave can be polarized, sound waves cannot Polarization states Jones Calculus Stokes parameters and the Poincare Sphere The scalar monochromatic plane wave The electric field reads: A cos(ωt kz ϕ) A vector monochromatic plane wave Electric field is orthogonal to wave and Poynting vectors Lies in the wave vector normal plane Needs 2 components E x = A x cos(ωt kz ϕ x ) E y = A y cos(ωt kz ϕ y ) N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 4
Linear and circular polarization states Polarization states Jones Calculus Stokes parameters and the Poincare Sphere In phase components ϕ y = ϕ x π/2 shift ϕ y = ϕ x ± π/2 1 0.5 0.4-1 -0.5 0.5 1-0.5 0.2-1 π shift ϕ y = ϕ x + π -1-0.5 0.5 1 0.4-0.2 0.2-0.4-1 -0.5 0.5 1-0.2-0.4 Left or Right N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 5
Polarization states Jones Calculus Stokes parameters and the Poincare Sphere The elliptic polarization state The polarization state of ANY monochromatic wave ϕ y ϕ x = ±π/4 0.4 0.2 Electric field E x = A x cos(ωt kz ϕ x ) E y = A y cos(ωt kz ϕ y ) 4 real numbers A x,ϕ x -1-0.5 0.5 1-0.2-0.4 A y,ϕ y 2 complex numbers A x exp( ıϕ x ) A y exp( ıϕ y ) N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 6
Polarization states Jones Calculus Stokes parameters and the Poincare Sphere Polarization states are vectors Monochromatic polarizations belong to a 2D vector space based on the Complex Ring ANY elliptic polarization state Two complex numbers A set of two ordered complex numbers is one 2D complex vector Canonical Basis ([ ] [ ]) 1 0, 0 1 Link with optics? These two vectors represent two polarization states We must decide which ones! Polarization Basis Two independent polarizations : Crossed Linear Reversed circular... YOUR choice N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 7
0.4 0.2-1 -0.5 0.5 1-0.2-0.4 The physics of polarization optics Examples : Linear Polarizations Polarization states Jones Calculus Stokes parameters and the Poincare Sphere Canonical Basis Choice [ ] 1 : horizontal linear polarization 0 [ ] 0 : vertical linear polarization 1 Tilt [ ] cos(θ) sin(θ) θ Linear polarization Jones vector Linear Polarization : two in phase components Two real numbers In a linear polarization basis N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 8
Polarization states Jones Calculus Stokes parameters and the Poincare Sphere Examples : Circular Polarizations In the same canonical basis choice : linear polarizations ϕ y ϕ x = ±π/2 0.4 Electric field E x = A x cos(ωt kz ϕ x ) 0.2 E y = A y cos(ωt kz ϕ y ) -1-0.5 0.5 1-0.2 Jones vector [ ] 1 ± ı -0.4 N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 9
About changing basis A polarization state Jones vector is basis dependent Polarization states Jones Calculus Stokes parameters and the Poincare Sphere Some elementary algebra The polarization vector space dimension is 2 Therefore : two non colinear vectors form a basis Any polarization state can be expressed as the sum of two non colinear other states Remark : two colinear polarization states are identical Homework Find the transformation matrix between between the two following bases : Horizontal and Vertical Linear Polarizations Right and Left Circular Polarizations N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 10
Polarization states Jones Calculus Stokes parameters and the Poincare Sphere Relationship between Jones and Poynting vectors Jones vectors also provide information about intensity Choose an orthonormal basis (J 1, J 2 ) Hermitian product is null : J 1 J 2 = 0 Each vector norm is unity : J 1 J 1 = J 2 J 2 = 0 Hermitian Norm is Intensity Simple calculations show that : If each Jones component is one complex electric field component The Hermitian product is proportional to beam intensity N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 11
Polarization states Jones Calculus Stokes parameters and the Poincare Sphere Polarization as a unique complex number If the intensity information disappears, polarization is summed up in one complex number Rule out the intensity Put 1 as first component Norm the Jones vector to unity Multiplying Jones vector by a complex number does not change the polarization state [ ] 1 Norm the first component to 1 : ξ The sole ξ describes the polarization state Choose between the two Either you norm the vector, or its first component. Not both! N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 12
The Stokes parameters A set of 4 dependent real parameters that can be measured Sample Jones Vector [ ] Ax exp(+ ıϕ/2) A y exp( ıϕ/2) Polarization states Jones Calculus Stokes parameters and the Poincare Sphere P 0 P 2 P 0 = A 2 x + A 2 y = I Overall Intensity π/4 Tilted Basis [ J π/4 = 1 Ax e ıϕ/2 + A y e + ıϕ/2 ] 2 A x e ıϕ/2 A y e + ıϕ/2 P 2 = I π/4 I π/4 = 2A x A y cos(ϕ) P 1 P 3 Intensity Différence P 1 = A 2 x a 2 y = I x I y Circular Basis [ J cir = 1 Ax e ıϕ/2 ıa y e + ıϕ/2 ] 2 A x e ıϕ/2 + ıa y e + ıϕ/2 P 3 = I L I R = 2A x A y sin(ϕ) 4 dependent parameters P 2 0 = P2 1 + P2 2 + P2 3 N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 13
Polarization states Jones Calculus Stokes parameters and the Poincare Sphere The Poincare Sphere Polarization states can be described geometrically on a sphere Normalized Stokes parameters S i = P i /P 0 (S 1,S 2,S 3 ) on a unit radius sphere Unit Radius Sphere 3 i=1 S i = 1 General Polarisation Figures from [Hua94] N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 14
Jones Matrices Polarizers Linear and Circular Anisotropy Jones Matrices Composition Eigen Polarization states Polarization states that do not change after propagation in an anisotropic medium Eigen Polarization states Do not change Except for Intensity Orthogonality 2 eigen polarization states are orthonormal Linear Anisotropy Eigen Polarizations Quarter and half wave plates and Birefringent materials Eigen Polarizations are linear along the eigen axes Circular Anisotropy Also called optical activity e.g in Faraday rotators and in gyratory non linear crystals Linear polarization is rotated by an angle proportional to propagation distance Eigen polarizations are the circular polarizations N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 15
Jones Matrices Polarizers Linear and Circular Anisotropy Jones Matrices Composition Jones Matrices 2D Linear Algebra to compute polarization propagation through devices Jones matrices in the eigen basis Let λ 1 and λ 2 be the two eigenvalues of a given device e.g. for linear anisotropy : λ i = e n ik 0 z [ ] λ1 0 Jones Matrix is 0 λ 2 In another basis Let J 1 = [ u v ] and J 2 = vectors { M J 1 = λ 1 J1 M J 2 = λ 2 J2 M = use Transformation Matrix [ ] v be the orthonormal eigen u [ λ1 uu + λ 2 vv ] (λ 1 λ 2 )uv (λ 1 λ 2 ) vu λ 2 vu + λ 1 vv N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 16
Jones Matrices Polarizers Linear and Circular Anisotropy Jones Matrices Composition The particular case of non absorbing devices Jones matrix is a unitary operator when λ 1 = λ 2 = 1 Nor absorbing neither amplifying devices λ 1 = λ 2 = 1 Det(M) = 1 M M t = M t M = I M is a unitary operator Unitary operator properties Norm is conserved : Intensity is unchanged after propagation Orthogonality is conserved : two initially orthogonal states will remain so after propagation N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 17
Jones Matrix of a polarizer Jones Matrices Polarizers Linear and Circular Anisotropy Jones Matrices Composition In its eigen basis A polarized is designed for : Full transmission of one linear polarization Zero transmission of its orthogonal counterpart [ ] 1 0 Eigen basis Jones matrix : P x = or P 0 0 y = When transmitted polarization is θ tilted [ ] 0 0 0 1 Change base through θ rotation Transformation Matrix [ ] cos(θ) sin(θ) R( θ) = sin(θ) cos(θ) P(θ) = R(θ) [ ] [ 1 0 cos R( θ) = 2 (θ) 0 0 sin(θ) cos (θ) ] sin(θ) cos (θ) sin 2 (θ) N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 18
Jones Matrices Polarizers Linear and Circular Anisotropy Jones Matrices Composition Intensity transmitted through a polarizer From natural or non polarized light From linearly polarized light Half the intensity is transmitted Transmitted Jones vector in polarizer eigen basis: [ ] [ ] [ ] 1 0 cos(θ) cos(θ) = 0 0 sin(θ) 0 Transmitted Intensity : cos 2 (θ) From circularly polarized light MALUS law Show that whatever the polarizer orientation, the transmitted intensity is half the incident intensity. N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 19
Jones Matrices Polarizers Linear and Circular Anisotropy Jones Matrices Composition Linear anisotropy eigen polarization vectors Two orthogonal polarization directions Two different refraction indexes n 1 and n 2 Two linear eigen modes along the eigen directions Jones Matrix in the eigen basis [ e ın 1 k z 0 ] 0 e ın 2k z Quarter and Half wave plates = e ıψ [ e ıφ/2 0 0 e ıφ/2 Express phase delay only ] [ ] e ıφ/2 0 0 e ıφ/2 Find the Jones Matrices of Quarter and Half wave plates Homework Find their action on tilted linear polarization (special case dor π/4 tilt) Find their action on circular polarization N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 20
Jones Matrices Polarizers Linear and Circular Anisotropy Jones Matrices Composition What is circular anisotropy? Two orthogonal circular eigen polarization states Two different refraction indexes n L and n R Jones Matrix in the circular eigen basis Express phase delay only [ ] [ ] [ ] e ın L k z 0 e ıφ/2 0 e ın = Rk z e ıψ 0 e ıφ/2 0 0 e ıφ/2 0 e ıφ/2 Jones Matrix in a linear polarization basis Rotation matrix [ ] use P Cir = 1 1 1 2 transformation matrix i i [ ] P Cir MP 1 cos(φ/2) sin(φ/2) Cir = e ıψ sin(φ/2) cos(φ/2) N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 21
Jones Matrices Polarizers Linear and Circular Anisotropy Jones Matrices Composition Jones Matrices Composition The Jones matrices of cascaded optical elements can be composed through Matrix multiplication Matrix composition If a J 0 incident light passes through M 1 and M 2 in that order First transmission: M 1J0 Second transmission: M 2 M 1J0 Composed Jones Matrix : M 2 M 1 Reversed order Beware of non commutativity Matrix product does not commute in general Think of the case of a linear anisotropy followed by optical activity in that order in the reverse order N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 22
Formalisms used Propagation through optical devices Stokes parameters for partially polarized light Generalize the coherent definition using the statistical average intensity Stokes Vector P 0 S = P 1 P 2 = P 3 I x + I y I x I y I π/4 I π/4 I L I R Polarization degree 0 p 1 P1 2 p = + P2 2 + P2 3 P 0 Stokes decomposition Polarized and depolarized sum P 0 pp 0 (1 p) P 0 S = P 1 P 2 = P 1 P 2 + 0 0 = S P + S NP P 3 P 3 0 N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 23
The Jones Coherence Matrix Formalisms used Propagation through optical devices Jones Vectors are out They describe phase differences Meaningless when not monochromatic Jones Coherence Matrix If [ Ax J = (t)e ıϕx(t) ] A (t)e ıϕy(t) y Γ ij = J i (t) J j (t) Γ = J (t) J (t) Coherence Matrix: explicit formulation [ A Γ = x (t) 2 A x (t)a y (t)e ı(ϕx ϕy) ] A x (t)a y (t)e ı(ϕx ϕy) A y (t) 2 N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 24
Jones Coherence Matrix: properties Trace is Intensity Tr(Γ) = I Formalisms used Propagation through optical devices Base change Relationship with Stokes parameters P 0 1 1 0 0 Γ xx P 1 P 2 = 1 1 0 0 Γ yy 0 0 1 1 Γ xy P 3 0 0 ı ı Γ yx P 1 ΓP Inverse relationship Γ xx 1 1 0 0 P 0 Γ yy Γ xy = 1 1 1 0 0 P 1 2 0 0 1 ı P 2 Γ yx 0 0 1 ı P 3 Transformation P from definition N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 25
Coherence Matrix: further properties Formalisms used Propagation through optical devices Polarization degree p = P 2 1 +P2 2 +P2 3 P 2 0 = 4(ΓxxΓyy ΓxyΓyx) 1 = 1 4Det(Γ) (Γ xx+γ yy) 2 Tr(Γ) 2 Γ Decomposition in polarized and depolarized components Γ = Γ P + Γ NP Find Γ P and Γ NP using the relationship with the Stokes parameters N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 26
Propagation of the Coherence Matrix Formalisms used Propagation through optical devices Jones Calculus If incoming polarization is J (t) Output one is J (t) = M J (t) Coherence Matrix if M is unitary M unitary means : linear and/or circular anisotropy only Γ = J (t) J (t) Γ = M J (t) J (t) M 1 Polarization degree Unaltered for unitary operators Tr and Det are unaltered Not the case if a polarizer is present : p becomes 1 N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 27
Formalisms used Propagation through optical devices Mueller Calculus Propagating the Jones coherence matrix is difficult if the operator is not unitary Jones Calculus raises some difficulties Coherence matrix OK for partially polarized light Propagation through unitary optical devices Hard Times if Polarizers are present The Stokes parameters may be an alternative (linear or circular anisotropy only) Describing intensity, they can be readily measurered We will show they can be propagated using 4 4 real matrices They are the Mueller matrices N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 28
The projection on a polarization state Matrix of the polariazer with axis parallel to V Formalisms used Propagation through optical devices V Projection on V in Jones Basis Orthogonal Linear Polarizations Basis: X and Y Normed Projection Base Vector : V = Ax e ı ϕ 2 X + Ay e ı ϕ 2 Y P V V t V = 1 P V = V V t a a Easy to check in the projection eigen basis : V, V t «N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 29
The Pauli Matrices The physics of polarization optics Formalisms used Propagation through optical devices A base for the 4D 2 2 matrix vector space [ ] [ ] [ ] 1 0 1 0 0 1 σ 1 =,σ 0 1 2 =,σ 0 1 3 =,σ 1 0 1 = P V decomposition P V = 1 2 (p 0σ 0 + p 1 σ 1 + p 2 σ 2 + p 3 σ 3 ) [ ] 0 ı ı 0 N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 30
P V composition and Trace property Formalisms used Propagation through optical devices Projection property V t σj V = ( V t V ) V t σj V = V t ( V V t ) σ j V = V t PV σ j V Projection Trace in its eigen basis (V,V t ) P V eigenvalues : 0 & 1 Tr(P V ) = 1 P V σ j eigenvalues : 0 & α α 1 Tr(P V σ j ) = α t V PV σ jv = α = Tr(PV σ j ) N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 31
Formalisms used Propagation through optical devices P V Pauli composition V t σj V = Tr(PV σ j ) Tr(σ i σ j ) = 2δ ij t V σjv = Tr(PV σ j ) = 1 2 i Tr(σ iσ j )p i = 1 2 i 2δ ijp i = p j Projection P V of the polarization base vectors Using P V = V V t P V X = A 2 x X + Ax A y e ıϕ Y P V Y = A 2 y Y + Ax A y e ıϕ X Using the Pauli Basis P V X = 1 2 (p 0 + p 1 ) X + 1 2 (p 2 + ıp 3 ) Y P V Y = 1 2 (p 0 p 1 ) Y + 1 2 (p 2 ıp 3 ) X N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 32
Formalisms used Propagation through optical devices P V Pauli composition and Stokes parameters Stokes parameters as P V decomposition on the Pauli base p 0 = P 0 = A 2 x ay 2 = I x I y p 1 = P 1 = A 2 x ay 2 = I x I y p 2 = P 2 = 2A x A y cos(ϕ) = I π/4 I π/4 p 3 = P 3 = 2A x A y sin(ϕ) = I L I R N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 33
Formalisms used Propagation through optical devices Propagating through devices: Mueller matrices V = M J V Projection on V P V = V V t = M J V V t MJ t = M J P V M J t Trace relationship ) P i = Tr(P V σ i ) = Tr (M J P V M t J σ i = ) 3 j=0 (M Tr J σ j M t J σ i P j 1 2 Mueller matrix (M M ) ij = 1 2 Tr ( M J σ j M J t σ i ) S = M M S N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 34
Formalisms used Propagation through optical devices Mueller matrices and partially polarized light Time average of the previous study Mueller matrices are time independent S = M M S Mueller calculus can be extended to... Partially coherent light Cascaded optical devices Final homework Find the Mueller matrix of each : Polarizers along eigen axis or θ tilted half and quarter wave plates linearly and circularly birefringent crystal N. Fressengeas UE SPM-PHY-S07-109, version 1.0, frame 35