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Polarization Optics Polarized Light Propagation Partially Polarized Light The physics of polarization optics Polarized light propagation Partially polarized light Polarization Optics N. Fressengeas Laboratoire Mat´eriaux Optiques, Photonique et Syst`emes Unit´ede Recherche commune `al’Universit´ede Lorraine et `aSup´elec Download this document from http://arche.univ-lorraine.fr/ N. Fressengeas Polarization Optics, version 2.0, frame 1 The physics of polarization optics Polarized light propagation Partially polarized light Further reading [Hua94, GB94] A. Gerrard and J.M. Burch. Introduction to matrix methods in optics. Dover, 1994. S. Huard. Polarisation de la lumi`ere. Masson, 1994. N. Fressengeas Polarization Optics, version 2.0, frame 2 The physics of polarization optics Polarized light propagation Partially polarized light Course Outline 1 The physics of polarization optics Polarization states Jones Calculus Stokes parameters and the Poincare Sphere 2 Polarized light propagation Jones Matrices Examples Matrix, basis & eigen polarizations Jones Matrices Composition 3 Partially polarized light Formalisms used Propagation through optical devices N. Fressengeas Polarization Optics, version 2.0, frame 3 The physics of polarization optics Polarization states Polarized light propagation Jones Calculus Partially polarized light Stokes parameters and the Poincare Sphere The vector nature of light Optical wave can be polarized, sound waves cannot 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 Ex = Ax cos (ωt kz ϕx ) − − Ey = Ay cos (ωt kz ϕy ) − − N. Fressengeas Polarization Optics, version 2.0, frame 4 The physics of polarization optics Polarization states Polarized light propagation Jones Calculus Partially polarized light Stokes parameters and the Poincare Sphere Linear and circular polarization states In phase components ϕy = ϕx π/2 shift ϕy = ϕx π/2 ± 0.4 1 0.2 -1 -0.5 0.5 1 0.5 -0.2 -0.4 -1 -0.5 0.5 1 π shift ϕy = ϕx + π 0.4 -0.5 0.2 -1 -0.5 0.5 1 -0.2 -1 -0.4 Left or Right N. Fressengeas Polarization Optics, version 2.0, frame 5 The physics of polarization optics Polarization states Polarized light propagation Jones Calculus Partially polarized light Stokes parameters and the Poincare Sphere The elliptic polarization state The polarization state of ANY monochromatic wave ϕy ϕx = π/4 − ± 1 Electric field Ex = Ax cos (ωt kz ϕx ) − − Ey = Ay cos (ωt kz ϕy ) 0.5 − − 4 real numbers Ax ,ϕx -1 -0.5 0.5 1 Ay ,ϕy 2 complex numbers -0.5 Ax exp(˙ıϕx ) Ay exp(˙ıϕy ) -1 N. Fressengeas Polarization Optics, version 2.0, frame 6 The physics of polarization optics Polarization states Polarized light propagation Jones Calculus Partially polarized light 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 Polarization Basis , 0 1 Two independent polarizations : Crossed Linear Link with optics ? Reversed circular These two vectors represent ... two polarization states YOUR choice We must decide which ones ! N. Fressengeas Polarization Optics, version 2.0, frame 7 The physics of polarization optics Polarization states Polarized light propagation Jones Calculus Partially polarized light Stokes parameters and the Poincare Sphere Examples : Linear Polarizations Canonical Basis Choice 1 : horizontal linear polarization 0 0 : vertical linear polarization 1 Tilt θ 0.4 cos (θ) 0.2 -0.5 0.5 sin(θ) -0.2 -0.4 Linear polarization Jones vector in a linear polarization basis Linear Polarization : two in phase components N. Fressengeas Polarization Optics, version 2.0, frame 8 The physics of polarization optics Polarization states Polarized light propagation Jones Calculus Partially polarized light Stokes parameters and the Poincare Sphere Examples : Circular Polarizations In the same canonical basis choice : linear polarizations ϕy ϕx = π/2 − ± 1 Electric field 0.5 Ex = Ax cos (ωt kz ϕx ) − − Ey = Ay cos (ωt kz ϕy ) − − Jones vector -1 -0.5 0.5 1 1 1 √2 ı˙ -0.5 ± -1 N. Fressengeas Polarization Optics, version 2.0, frame 9 The physics of polarization optics Polarization states Polarized light propagation Jones Calculus Partially polarized light Stokes parameters and the Poincare Sphere About changing basis A polarization state Jones vector is basis dependent 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 The physics of polarization optics Polarization states Polarized light propagation Jones Calculus Partially polarized light Stokes parameters and the Poincare Sphere Relationship between Jones and Poynting vectors Jones vectors also provide information about intensity Choose an orthonormal basis (J1, J2) Hermitian product is null : J J = 0 1 · 2 Each vector norm is unity : J J = J J = 1 1 · 1 2 · 2 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 The physics of polarization optics Polarization states Polarized light propagation Jones Calculus Partially polarized light Stokes parameters and the Poincare Sphere The Stokes parameters A set of 4 dependent real parameters that can be measured P0 Overall Intensity P1 Intensity Diff´erence P0 = I P = Ix Iy 1 − P 2 in a π/4 Tilted Basis P3 in a Circular Basis P2 = Iπ/4 I π/4 P3 = IL IR − − − N. Fressengeas Polarization Optics, version 2.0, frame 12 The physics of polarization optics Polarization states Polarized light propagation Jones Calculus Partially polarized light Stokes parameters and the Poincare Sphere Relationship between Jones and Stockes Sample Jones Vector 4 dependent parameters Ax exp(+ıϕ/˙ 2) 2 2 2 2 J = P0 = P1 + P2 + P3 Ay exp( ıϕ/˙ 2) − P0 Overall Intensity P1 Intensity Difference P I A2 A2 0 = = x + y 2 2 P = Ix Iy = A A 1 − x − y P2 in a π/4 Tilted Basis P3 in a Circular Basis J = π/4 +ıϕ/˙ 2 ıϕ/˙ 2 +ıϕ/˙ 2 ıϕ/˙ 2 1 Ax e ı˙Ay e− A e + A e JCir x y − = √ +ıϕ/˙ 2 − ıϕ/˙ 2 √2 +ıϕ/˙ 2 ıϕ/˙ 2 2 Ax e + ı˙Ay e− Ax e + Ay e− x x y y −x x y y P3 = JCir JCir JCir JCir = P2 = Jπ/4 Jπ/4 Jπ/4 Jπ/4 = · − · · − · 2Ax Ay sin(ϕ) 2Ax Ay cos (ϕ) N. Fressengeas Polarization Optics, version 2.0, frame 13 The physics of polarization optics Polarization states Polarized light propagation Jones Calculus Partially polarized light Stokes parameters and the Poincare Sphere The Poincare Sphere Polarization states can be described geometrically on a sphere Normalized Stokes parameters (S1, S2, S3) on a unit radius sphere Si = Pi /P0 Unit Radius Sphere 3 2 i=1 Si = 1 General PolarisationP Figures from [Hua94] N. Fressengeas Polarization Optics, version 2.0, frame 14 The physics of polarization optics Jones Matrices Examples Polarized light propagation Matrix, basis & eigen polarizations Partially polarized light 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 0 0 Blocks the linear polarization along y : 1 −→ 0 x polarizer Jones matrix in this basis 1 0 0 0 N. Fressengeas Polarization Optics, version 2.0, frame 15 The physics of polarization optics Jones Matrices Examples Polarized light propagation Matrix, basis & eigen polarizations Partially polarized light Jones Matrices Composition A quarter wave plate adds a π/2 phase shift Birefringent material: n1 along x and n2 along y thickness e Linear polarization along x: phase shift is ke = k0n1e Linear polarization along y: phase shift is ke = k0n2e Jones matrix in this basis eı˙k0n1e 0 1 0 1 0 = eı˙k0n1e 0 eı˙k0n2e 0 ı˙ ≈ 0 ı˙ ± ± N. Fressengeas Polarization Optics, version 2.0, frame 16 The physics of polarization optics Jones Matrices Examples Polarized light propagation Matrix, basis & eigen polarizations Partially polarized light Jones Matrices Composition Eigen Polarizations Eigen polarization are polarizations that do not change upon propagation Polarization unchanged Eigen Vectors λ C ∈ J and λJ describe the same v is an eigen vector polarization M v = λv · ⇔ λ is its eigen value 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 The physics of polarization optics Jones Matrices Examples Polarized light propagation Matrix, basis & eigen polarizations Partially polarized light Jones Matrices Composition A polarizer in a rotated basis In its eigen basis 1 0 Eigen basis Jones matrix : Px = 0 0 When transmitted polarization is θ tilted Change base through θ rotation Transformation Matrix − cos (θ) sin(θ) R (θ)= − sin(θ) cos (θ) 1 0 cos2 (θ) sin(θ) cos (θ) P (θ)= R (θ) R ( θ)= 0 0 − sin(θ) cos (θ) sin2 (θ) N. Fressengeas Polarization Optics, version 2.0, frame 18 The physics of polarization optics Jones Matrices Examples Polarized light propagation Matrix, basis & eigen polarizations Partially polarized light Jones Matrices Composition Changing basis in the general case Using the Transformation Matrix If basis B1 is deduded from basis B0 by transformation P : B1 = PB0 1 Jones Matrix is transformed using J1 = P− J0 P From linear to circular example Optically Active media in a linear basis : cos (φ) sin(φ) J = sin(φ) cos (φ) − 1 1 Transformation Matrix to a circular basis P = I ı˙ − ıφ˙ 1 e 0 P− MP = ıφ˙ 0 e− N.
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