Polarization Optics. N. Fressengeas

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1 Polarization Optics N. Fressengeas Laboratoire Matériaux Optiques, Photonique et Systèmes Unité de Recherche commune à l Université de Lorraine et à Supélec Download this document from N. Fressengeas Polarization Optics, version 2.0, frame 1

2 Further reading [Hua94, GB94] A. Gerrard and J.M. Burch. Introduction to matrix methods in optics. Dover, S. Huard. Polarisation de la lumière. Masson, N. Fressengeas Polarization Optics, version 2.0, frame 2

3 Course Outline 1 The physics of polarization optics Polarization states Jones Calculus Stokes parameters and the Poincare Sphere 2 Jones Matrices Examples Matrix, basis & eigen polarizations Jones Matrices Composition 3 Formalisms used Propagation through optical devices N. Fressengeas Polarization Optics, version 2.0, frame 3

4 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: Acos(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 Polarization Optics, version 2.0, frame 4

5 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/ π shift ϕ y = ϕ x +π Left or Right N. Fressengeas Polarization Optics, version 2.0, frame 5

6 The elliptic polarization state The polarization state of ANY monochromatic wave Polarization states Jones Calculus Stokes parameters and the Poincare Sphere ϕ y ϕ x = ±π/4 1 Electric 0.5 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 A y,ϕ y 2 complex numbers -0.5 A x exp( ıϕ x ) A y exp( ıϕ y ) -1 N. Fressengeas Polarization Optics, version 2.0, frame 6

7 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 Polarization Optics, version 2.0, frame 7

8 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 in a linear polarization basis Linear Polarization : two in phase components N. Fressengeas Polarization Optics, version 2.0, frame 8

9 Examples : Circular Polarizations In the same canonical basis choice : linear polarizations Polarization states Jones Calculus Stokes parameters and the Poincare Sphere ϕ y ϕ x = ±π/ Electric field E x = A x cos(ωt kz ϕ x ) E y = A y cos(ωt kz ϕ y ) Jones vector [ ] ± ı N. Fressengeas Polarization Optics, version 2.0, frame 9

10 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 Polarization Optics, version 2.0, frame 10

11 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 = 1 Hermitian Norm is Intensity Simple calculations show that : If each Jones component is one complex electric field component The Hermitian norm is proportional to beam intensity N. Fressengeas Polarization Optics, version 2.0, frame 11

12 The Stokes parameters A set of 4 dependent real parameters that can be measured Polarization states Jones Calculus Stokes parameters and the Poincare Sphere P 0 Overall Intensity P 1 Intensity Différence P 0 = I P 1 = I x I y P 2 in a π/4 Tilted Basis P 3 in a Circular Basis P 2 = I π/4 I π/4 P 3 = I L I R N. Fressengeas Polarization Optics, version 2.0, frame 12

13 Polarization states Jones Calculus Stokes parameters and the Poincare Sphere Relationship between Jones and Stockes Sample Jones Vector [ ] 4 dependent parameters Ax exp(+ ıϕ/2) J = P A y exp( ıϕ/2) 0 2 = P2 1 +P2 2 +P2 3 P 0 P 2 P 0 = I = A 2 x +A 2 y Overall Intensity in a π/4 Tilted Basis J π/4 = 2 [ Ax e + ıϕ/2 +A y e ıϕ/2 A x e + ıϕ/2 +A y e ıϕ/2 P 2 = J x π/4 Jx π/4 Jy π/4 Jy π/4 = 2A x A y cos(ϕ) ] P 1 P 3 Intensity Difference P 1 = I x I y = A 2 x A 2 y in a 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 = J x Cir Jx Cir Jy Cir Jy Cir = 2A x A y sin(ϕ) N. Fressengeas Polarization Optics, version 2.0, frame 13

14 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 S2 i = 1 General Polarisation Figures from [Hua94] N. Fressengeas Polarization Optics, version 2.0, frame 14

15 Jones Matrices Examples Matrix, basis & eigen polarizations Jones Matrices Composition A polarizer lets one component through Polarizer aligned with x : its action on two orthogonal polarizations [ ] [ ] 1 1 Lets through the linear polarization along x: 0 0 Blocks the linear polarization along y : [ ] 0 1 [ ] 0 0 x polarizer Jones matrix [ ] in this basis N. Fressengeas Polarization Optics, version 2.0, frame 15

16 Jones Matrices Examples Matrix, basis & eigen polarizations Jones Matrices Composition A quarter wave plate adds a π/2 phase shift Birefringent material: n 1 along x and n 2 along y Linear polarization along x: phase shift is ke = k 0 n 1 e Linear polarization along y: phase shift is ke = k 0 n 2 e Jones matrix [ ] e ık 0 n 1 e 0 0 e ık 0n 2 e = e ık 0n 1 e [ ] ± ı thickness e in this basis [ ] ± ı N. Fressengeas Polarization Optics, version 2.0, frame 16

17 Jones Matrices Examples Matrix, basis & eigen polarizations Jones Matrices Composition Eigen Polarizations Eigen polarization are polarizations that do not change upon propagation Eigen Vectors λ C M v = λv v isaneigenvector λisitseigenvalue Polarization unchanged J and λj describe the same polarization Intensity changes Handy basis A matrix is diagonal in its eigen basis Polarizer eigen basis is along its axes Bi-refringent plate eigen basis is along its axes Homework Find the eigen polarizations for an optically active material that rotates any linear polarisation by an angle φ N. Fressengeas Polarization Optics, version 2.0, frame 17

18 A polarizer in a rotated basis Jones Matrices Examples Matrix, basis & eigen polarizations Jones Matrices Composition In its eigen basis Eigen basis Jones matrix : P x = [ ] When transmitted polarization is θ tilted 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 Polarization Optics, version 2.0, frame 18

19 Changing basis in the general case Jones Matrices Examples Matrix, basis & eigen polarizations Jones Matrices Composition Using the Transformation Matrix If basis B 1 is deduded from basis B 0 by transformation P : B 1 = PB 0 Jones Matrix is transformed using J 1 = P 1 J 0 P From linear to circular Optically [ Active media in ] a linear basis : cos(φ) sin(φ) J = sin(φ) cos(φ) Transformation Matrix to a circular basis P = [ ] e ıφ P 1 0 MP = 0 e ıφ [ ] 1 1 I ı example N. Fressengeas Polarization Optics, version 2.0, frame 19

20 Anisotropy can be linear and circular Jones Matrices Examples Matrix, basis & eigen polarizations Jones Matrices Composition Linear Anisotropy Orthogonal eigen linear polarizations Different index n 1 & n 2 Eigen Jones Matrix [ e ıθ Orthogonal linear polarisations basis ] Circular Anisotropy Orthogonal eigen Circular polarizations Different index n 1 & n 2 Eigen Jones Matrix [ e ıθ Orthogonal Circular basis Back to linear basis [ ( cos θ 2) sin ( )] θ 2 sin ( ( θ 2) cos θ ) 2 Optically Active media ] N. Fressengeas Polarization Optics, version 2.0, frame 20

21 Jones Matrices Examples Matrix, basis & eigen polarizations 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 Polarization Optics, version 2.0, frame 21

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 = S P + S NP P 3 P 3 0 N. Fressengeas Polarization Optics, version 2.0, frame 22

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) t Coherence Matrix: explicit formulation [ A Γ = x (t) 2 A x (t)a (t)e ı(ϕx ϕy) ] y A x (t)a y (t)e ı(ϕx ϕy) A y (t) 2 N. Fressengeas Polarization Optics, version 2.0, frame 23

24 Jones Coherence Matrix: properties Formalisms used Propagation through optical devices Trace is Intensity Tr(Γ) = I Base change Relationship with Stokes parameters P Γ xx P 1 P 2 = Γ yy Γ xy P ı ı Γ yx Inverse relationship Γ xx P 0 Γ yy Γ xy = P ı P 2 Γ yx ı P 3 P 1 ΓP Transformation P from definition N. Fressengeas Polarization Optics, version 2.0, frame 24

25 Coherence Matrix: further properties Formalisms used Propagation through optical devices Polarization degree p = P 2 1 +P2 2 +P2 3 P 2 0 = 1 4(ΓxxΓyy ΓxyΓyx) = 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 Polarization Optics, version 2.0, frame 25

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) t Γ = M J(t) J(t) t M 1 Basis change Polarization degree Unaltered for unitary operators Tr and Det are unaltered Not the case if a polarizer is present : p becomes 1 N. Fressengeas Polarization Optics, version 2.0, frame 26

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 Polarization Optics, version 2.0, frame 27

28 The projection on a polarization state Matrix of the polarizer 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 ıϕ 2 X +Ay e ıϕ 2 Y P V V t V = 1 P V = V V t a a Easy to check in the projection eigen basis N. Fressengeas Polarization Optics, version 2.0, frame 28

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 [ ] [ ] [ ] σ 0 =,σ =,σ =,σ = 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 Polarization Optics, version 2.0, frame 29

30 P V composition and Trace property Trace is the eigen values sum 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 P V eigenvalues : 0 & 1 Tr(P V ) = 1 P V σ j eigenvalues : 0 & α α 1 Tr(P V σ j ) = α P V σ j eigenvectors are the same as P V : V associated to eigenvalue α Project the projection V t PV σ j V = α = Tr(PV σ j ) = V t σ j V N. Fressengeas Polarization Optics, version 2.0, frame 30

31 Formalisms used Propagation through optical devices P V Pauli components and physical meaning Express p i as a function of V and the Pauli matrices, then find their signification 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 Project the base vectors on V Using V = A x e ıϕ 2 X +Ay e ıϕ 2 Y 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 P V decomposition on the Pauli Basis P V X = 1 2 (p 0 +p 1 ) X (p 2 + ıp 3 ) Y P V Y = 1 2 (p 0 p 1 ) Y (p 2 ıp 3 ) X Identify N. Fressengeas Polarization Optics, version 2.0, frame 31

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 A 2 y = I x I y p 1 = P 1 = A 2 x A 2 y = 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 Polarization Optics, version 2.0, frame 32

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 Polarization Optics, version 2.0, frame 33

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 Polarization Optics, version 2.0, frame 34