Polarization Optics Polarized Light Propagation Partially Polarized Light

<a href="#4_0">The physics of polarization optics </a> <a href="#15_0">Polarized light propagation </a> <a href="#22_0">Partially polarized light </a>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 <a href="/goto?url=http://arche.univ-lorraine.fr/" target="_blank">http://arche.univ-lorraine.fr/ </a>N. Fressengeas <a href="#0_0">Polarization Optics, version 2.0, frame 1 </a><a href="#4_0">The physics of polarization optics </a> <a href="#15_0">Polarized light propagation </a> <a href="#22_0">Partially polarized light </a>Further reading [<a href="#2_0">Hua94</a>, <a href="#2_1">GB94] </a>A. Gerrard and J.M. Burch. Introduction to matrix methods in optics. Dover, 1994. S. Huard. Polarisation de la lumière. Masson, 1994. N. Fressengeas <a href="#1_0">Polarization Optics, version 2.0, frame 2 </a><a href="#4_0">The physics of polarization optics </a> <a href="#15_0">Polarized light propagation </a> <a href="#22_0">Partially polarized light </a>Course Outline 123<a href="#4_0">The physics of polarization optics </a> <a href="#4_0">Polarization states </a><a href="#7_0">Jones Calculus </a><a href="#12_0">Stokes parameters and the Poincare Sphere </a><a href="#15_0">Polarized light propagation </a> <a href="#15_0">Jones Matrices Examples </a><a href="#17_0">Matrix, basis & eigen polarizations </a><a href="#21_0">Jones Matrices Composition </a><a href="#22_0">Partially polarized light </a> <a href="#22_0">Formalisms used </a><a href="#26_0">Propagation through optical devices </a>N. Fressengeas <a href="#1_0">Polarization Optics, version 2.0, frame 3 </a><a href="#4_0">The physics of polarization optics </a><a href="#15_0">Polarized light propagation </a> <a href="#22_0">Partially polarized light </a><a href="#4_0">Polarization states </a><a href="#7_0">Jones Calculus </a><a href="#12_0">Stokes parameters and the Poincare Sphere </a>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 <a href="#1_0">Polarization Optics, version 2.0, frame 4 </a><a href="#4_0">The physics of polarization optics </a><a href="#15_0">Polarized light propagation </a> <a href="#22_0">Partially polarized light </a><a href="#4_0">Polarization states </a><a href="#7_0">Jones Calculus </a><a href="#12_0">Stokes parameters and the Poincare Sphere </a>Linear and circular polarization states In phase components ϕy = ϕx π/2 shift ϕy = ϕx π/2 π shift ϕy = ϕx + π Left or Right N. Fressengeas <a href="#1_0">Polarization Optics, version 2.0, frame 5 </a><a href="#4_0">The physics of polarization optics </a><a href="#15_0">Polarized light propagation </a> <a href="#22_0">Partially polarized light </a><a href="#4_0">Polarization states </a><a href="#7_0">Jones Calculus </a><a href="#12_0">Stokes parameters and the Poincare Sphere </a>The elliptic polarization state The polarization state of ANY monochromatic wave ϕy − ϕx = π/4 Electric field Ex = Ax cos (ωt − kz − ϕx ) Ey = Ay cos (ωt − kz − ϕy ) 4 real numbers Ax ,ϕx Ay ,ϕy 2 complex numbers Ax exp (ı˙ϕx ) Ay exp (ı˙ϕy ) N. Fressengeas <a href="#1_0">Polarization Optics, version 2.0, frame 6 </a><a href="#4_0">The physics of polarization optics </a><a href="#15_0">Polarized light propagation </a> <a href="#22_0">Partially polarized light </a><a href="#4_0">Polarization states </a><a href="#7_0">Jones Calculus </a><a href="#12_0">Stokes parameters and the Poincare Sphere </a>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 ꢀꢁ ꢂ ꢁ ꢂꢃ Polarization Basis Two independent polarizations : Crossed Linear 10 01,Reversed circular . . . Link with optics ? These two vectors represent two polarization states YOUR choice We must decide which ones ! N. Fressengeas <a href="#1_0">Polarization Optics, version 2.0, frame 7 </a><a href="#4_0">The physics of polarization optics </a><a href="#15_0">Polarized light propagation </a> <a href="#22_0">Partially polarized light </a><a href="#4_0">Polarization states </a><a href="#7_0">Jones Calculus </a><a href="#12_0">Stokes parameters and the Poincare Sphere </a>Examples : Linear Polarizations Canonical Basis Choice ꢁ ꢂ 10 : horizontal linear polarization ꢁ ꢂ 0 : vertical linear polarization 1Tilt θ<ul style="display: flex;"><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li></ul>cos (θ) sin (θ) <ul style="display: flex;"><li style="flex:1">Linear polarization Jones vector </li><li style="flex:1">in a linear polarization basis </li></ul>Linear Polarization : two in phase components N. Fressengeas <a href="#1_0">Polarization Optics, version 2.0, frame 8 </a><a href="#4_0">The physics of polarization optics </a><a href="#15_0">Polarized light propagation </a> <a href="#22_0">Partially polarized light </a><a href="#4_0">Polarization states </a><a href="#7_0">Jones Calculus </a><a href="#12_0">Stokes parameters and the Poincare Sphere </a>Examples : Circular Polarizations In the same canonical basis choice : linear polarizations ϕy − ϕx = π/2 Electric field Ex = Ax cos (ωt − kz − ϕx ) Ey = Ay cos (ωt − kz − ϕy ) Jones vector <ul style="display: flex;"><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li></ul>1ı˙ 1 2√N. Fressengeas <a href="#1_0">Polarization Optics, version 2.0, frame 9 </a><a href="#4_0">The physics of polarization optics </a><a href="#15_0">Polarized light propagation </a> <a href="#22_0">Partially polarized light </a><a href="#4_0">Polarization states </a><a href="#7_0">Jones Calculus </a><a href="#12_0">Stokes parameters and the Poincare Sphere </a>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 <a href="#1_0">Polarization Optics, version 2.0, frame 10 </a><a href="#4_0">The physics of polarization optics </a><a href="#15_0">Polarized light propagation </a> <a href="#22_0">Partially polarized light </a><a href="#4_0">Polarization states </a><a href="#7_0">Jones Calculus </a><a href="#12_0">Stokes parameters and the Poincare Sphere </a>Relationship between Jones and Poynting vectors Jones vectors also provide information about intensity <ul style="display: flex;"><li style="flex:1">Choose an orthonormal basis </li><li style="flex:1">(J1, J2) </li></ul>Hermitian product is null : J1 · J2 = 0 Each vector norm is unity : J1 · J1 = J2 · J2 = 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 <a href="#1_0">Polarization Optics, version 2.0, frame 11 </a><a href="#4_0">The physics of polarization optics </a><a href="#15_0">Polarized light propagation </a> <a href="#22_0">Partially polarized light </a><a href="#4_0">Polarization states </a><a href="#7_0">Jones Calculus </a><a href="#12_0">Stokes parameters and the Poincare Sphere </a>The Stokes parameters A set of 4 dependent real parameters that can be measured P0 P2 Overall Intensity P0 = I in a π/4 Tilted Basis P2 = Iπ/4 − I P1 P3 Intensity Différence P1 = Ix − Iy in a Circular Basis P3 = IL − IR −π/4 N. Fressengeas <a href="#1_0">Polarization Optics, version 2.0, frame 12 </a><a href="#4_0">The physics of polarization optics </a><a href="#15_0">Polarized light propagation </a> <a href="#22_0">Partially polarized light </a><a href="#4_0">Polarization states </a><a href="#7_0">Jones Calculus </a><a href="#12_0">Stokes parameters and the Poincare Sphere </a>Relationship between Jones and Stockes Sample Jones Vector ꢁ4 dependent parameters P02 = P12 + P22 + P32 ꢂAx exp (+ı˙ϕ/2) Ay exp (−ı˙ϕ/2) J = P0 P2 Overall Intensity P0 = I = A2x + A2y P1 Intensity Difference P1 = Ix − Iy = A2x − A2y in a π/4 Tilted Basis P3 in a Circular Basis ꢂJπ/4 =ꢁAx e+ı˙ϕ/2 − ı˙Ay e−ı˙ϕ/2 <ul style="display: flex;"><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li></ul>Ax e+ı˙ϕ/2 + Ay e−ı˙ϕ/2 −Ax e+ı˙ϕ/2 + Ay e−ı˙ϕ/2 π/4 1 2√√JCir =Ax e+ı˙ϕ/2 + ı˙Ay e−ı˙ϕ/2 2P3 = Jx · JCx ir − Jy · JCy ir =P2 = Jx · Jπx/4 − Jπy/4 · Jπy/4 =<ul style="display: flex;"><li style="flex:1">Cir </li><li style="flex:1">Cir </li></ul>2Ax Ay sin (ϕ) 2Ax Ay cos (ϕ) N. Fressengeas <a href="#1_0">Polarization Optics, version 2.0, frame 13 </a><a href="#4_0">The physics of polarization optics </a><a href="#15_0">Polarized light propagation </a> <a href="#22_0">Partially polarized light </a><a href="#4_0">Polarization states </a><a href="#7_0">Jones Calculus </a><a href="#12_0">Stokes parameters and the Poincare Sphere </a>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 PGeneral Polarisation 3i=1 Si2 = 1 Figures from [<a href="#2_0">Hua94] </a>N. Fressengeas <a href="#1_0">Polarization Optics, version 2.0, frame 14 </a><a href="#4_0">The physics of polarization optics </a><a href="#15_0">Polarized light propagation </a><a href="#22_0">Partially polarized light </a><a href="#15_0">Jones Matrices Examples </a><a href="#17_0">Matrix, basis & eigen polarizations </a><a href="#21_0">Jones Matrices Composition </a>A polarizer lets one component through Polarizer aligned with x : its action on two orthogonal polarizations <ul style="display: flex;"><li style="flex:1">ꢁ ꢂ </li><li style="flex:1">ꢁ ꢂ </li></ul>10 10 Lets through the linear polarization along x: −→ <ul style="display: flex;"><li style="flex:1">ꢁ ꢂ </li><li style="flex:1">ꢁ ꢂ </li></ul>01 00 Blocks the linear polarization along y : −→ <ul style="display: flex;"><li style="flex:1">x polarizer Jones matrix </li><li style="flex:1">in this basis </li></ul><ul style="display: flex;"><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li></ul>1 0 0 0 N. Fressengeas <a href="#1_0">Polarization Optics, version 2.0, frame 15 </a><a href="#4_0">The physics of polarization optics </a><a href="#15_0">Polarized light propagation </a><a href="#22_0">Partially polarized light </a><a href="#15_0">Jones Matrices Examples </a><a href="#17_0">Matrix, basis & eigen polarizations </a><a href="#21_0">Jones Matrices Composition </a>A quarter wave plate adds a π/2 phase shift <ul style="display: flex;"><li style="flex:1">Birefringent material: n1 along x and n2 along y </li><li style="flex:1">thickness e </li></ul>Linear polarization along x: phase shift is ke = k0n1e Linear polarization along y: phase shift is ke = k0n2e Jones matrix ꢁin this basis <ul style="display: flex;"><li style="flex:1">ꢂ</li><li style="flex:1">ꢂ</li><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li><li style="flex:1">ꢁ</li></ul>eı˙k n e 0010 0ı˙ 10 0ı˙ <ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">1</li></ul>= eı˙k n e ≈<ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">1</li></ul>eı˙k n e 20N. Fressengeas <a href="#1_0">Polarization Optics, version 2.0, frame 16 </a><a href="#4_0">The physics of polarization optics </a><a href="#15_0">Polarized light propagation </a><a href="#22_0">Partially polarized light </a><a href="#15_0">Jones Matrices Examples </a><a href="#17_0">Matrix, basis & eigen polarizations </a><a href="#21_0">Jones Matrices Composition </a>Eigen Polarizations Eigen polarization are polarizations that do not change upon propagation Polarization unchanged Eigen Vectors λ ∈ C v is an eigen vector J and λJ describe the same 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 <a href="#1_0">Polarization Optics, version 2.0, frame 17 </a><a href="#4_0">The physics of polarization optics </a><a href="#15_0">Polarized light propagation </a><a href="#22_0">Partially polarized light </a><a href="#15_0">Jones Matrices Examples </a><a href="#17_0">Matrix, basis & eigen polarizations </a><a href="#21_0">Jones Matrices Composition </a>A polarizer in a rotated basis In its eigen basis <ul style="display: flex;"><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li></ul>1 0 0 0 Eigen basis Jones matrix : Px = When transmitted polarization is θ tilted Change base through −θ rotation Transformation Matrix <ul style="display: flex;"><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li></ul>cos (θ) − sin (θ) R (θ) = sin (θ) cos (θ) <ul style="display: flex;"><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li></ul>1 0 0 0 cos2 (θ) sin (θ) cos (θ) sin (θ) cos (θ) sin2 (θ) <ul style="display: flex;"><li style="flex:1">P (θ) = R (θ) </li><li style="flex:1">R (−θ) = </li></ul>N. Fressengeas <a href="#1_0">Polarization Optics, version 2.0, frame 18 </a><a href="#4_0">The physics of polarization optics </a><a href="#15_0">Polarized light propagation </a><a href="#22_0">Partially polarized light </a><a href="#15_0">Jones Matrices Examples </a><a href="#17_0">Matrix, basis & eigen polarizations </a><a href="#21_0">Jones Matrices Composition </a>Changing basis in the general case Using the Transformation Matrix If basis B1 is deduded from basis B0 by transformation P : B1 = P B0 Jones Matrix is transformed using J1 = P−1 J0 P <ul style="display: flex;"><li style="flex:1">From linear to circular </li><li style="flex:1">example </li></ul>Optically Active media in a linear basis : cos (φ) sin (φ) <ul style="display: flex;"><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li></ul>J = − sin (φ) cos (φ) <ul style="display: flex;"><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li></ul>1I1 −ı˙ Transformation Matrix to a circular basis P = <ul style="display: flex;"><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li></ul>eı˙φ 0e−ı˙φ P −1MP = 0N. Fressengeas <a href="#1_0">Polarization Optics, version 2.0, frame 19 </a><a href="#4_0">The physics of polarization optics </a><a href="#15_0">Polarized light propagation </a><a href="#22_0">Partially polarized light </a><a href="#15_0">Jones Matrices Examples </a><a href="#17_0">Matrix, basis & eigen polarizations </a><a href="#21_0">Jones Matrices Composition </a>Anisotropy can be linear and circular Linear Anisotropy Orthogonal eigen linear polarizations Circular Anisotropy Orthogonal eigen Circular polarizations <ul style="display: flex;"><li style="flex:1">Different index n1 & n2 </li><li style="flex:1">Different index n1 & n2 </li></ul>Eigen Jones Matrix ꢁEigen Jones Matrix <ul style="display: flex;"><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li><li style="flex:1">ꢂ</li></ul><ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">0</li><li style="flex:1">1</li><li style="flex:1">0</li></ul><ul style="display: flex;"><li style="flex:1">0 eı˙θ </li><li style="flex:1">0 eı˙θ </li></ul>

Polarization Optics Polarized Light Propagation Partially Polarized Light

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