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

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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. Fressengeas Polarization Optics, version 2.0, frame 19 The physics of polarization optics Jones Matrices Examples Polarized light propagation Matrix, basis & eigen polarizations Partially polarized light Jones Matrices Composition Anisotropy can be linear and circular

Linear Anisotropy Circular Anisotropy Orthogonal eigen linear Orthogonal eigen Circular polarizations polarizations

Different index n1 & n2 Different index n1 & n2 Eigen Jones Matrix Eigen Jones Matrix 1 0 1 0 0 eıθ˙  0 eıθ˙  Orthogonal linear polarisations basis Orthogonal Circular basis Back to linear basis θ θ cos 2 sin 2  sin θ
cos θ
 − 2 2 Optically Active
media

N. Fressengeas Polarization Optics, version 2.0, frame 20 The physics of polarization optics Jones Matrices Examples Polarized light propagation Matrix, basis & eigen polarizations Partially polarized light Jones Matrices Composition Jones Matrices Composition The Jones matrices of cascaded optical elements can be composed through Matrix multiplication

Matrix composition

If a −→J0 incident light passes through M1 and M2 in that order

First transmission: M1−→J0

Second transmission: M2M1−→J0

Composed Jones Matrix : M2M1 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 The physics of polarization optics Formalisms used Polarized light propagation Propagation through optical devices Partially polarized light Stokes parameters for partially polarized light Generalize the coherent definition using the statistical average intensity

Stokes Vector Polarization degree 0 p 1 ≤ ≤ P0 Ix + Iy h i P2 + P2 + P2 P1  Ix Iy  1 2 3 −→S = = h − i p = q P2 Iπ/4 I π/4 P0   h − − i P   IL IR   3  h − i  Stokes decomposition Polarized and depolarized sum P pP (1 p) P 0 0 − 0 P1  P1   0  −→S = = + = −→S + −−→S P P 0 P NP  2  2    P   P   0   3  3   

N. Fressengeas Polarization Optics, version 2.0, frame 22 The physics of polarization optics Formalisms used Polarized light propagation Propagation through optical devices Partially polarized light The Jones Coherence Matrix

Jones Coherence Matrix Jones Vectors are out A (t) eıϕ˙ x (t) J x If −→ = ıϕ˙ y (t) They describe phase differences Ay (t) e  Meaningless when not Γij = −→J i (t) −→J j (t) h i monochromatic t Γ= −−→J (t)−−→J (t) h i Coherence Matrix: explicit formulation A (t) 2 A (t) A (t)eı˙(ϕx ϕy ) Γ= x x y − h| | ı˙(iϕx ϕy ) h 2 i  Ax (t)Ay (t)e Ay (t)  h − − i h| | i

N. Fressengeas Polarization Optics, version 2.0, frame 23 The physics of polarization optics Formalisms used Polarized light propagation Propagation through optical devices Partially polarized light Jones Coherence Matrix: properties

Trace is Intensity Base change Transformation P 1 Tr (Γ) = I P− ΓP Relationship with Stokes parameters from definition

P0 1 1 0 0 Γxx P1 1 1 0 0 Γyy  = − P 0 0 1 1 Γ  2    xy  P  0 0 ı˙ ı˙ Γyx   3  −    Inverse relationship

Γxx 1 1 0 0 P0 Γyy  1 1 0 0  P1 = 1 − Γ 2 0 0 1ı ˙ P  xy     2 Γyx  0 0 1 ı˙ P     −   3

N. Fressengeas Polarization Optics, version 2.0, frame 24 The physics of polarization optics Formalisms used Polarized light propagation Propagation through optical devices Partially polarized light Coherence Matrix: further properties

Polarization degree

2 2 2 P +P +P 4(Γxx Γyy Γxy Γyx ) 4Det(Γ) 1 2 3 − p = P2 = 1 2 = 1 Tr 2 r 0 r − (Γxx +Γyy ) r − (Γ) Γ Decomposition in polarized and depolarized components

Γ=ΓP +ΓNP

Find ΓP and ΓNP using the relationship with the Stokes parameters

N. Fressengeas Polarization Optics, version 2.0, frame 25 The physics of polarization optics Formalisms used Polarized light propagation Propagation through optical devices Partially polarized light Propagation of the Coherence Matrix

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 t Γ′ = J−−−→′ (t)J−−−→′ (t) h it Γ = M −−→J (t)−−→J (t) M 1 Basis change ′ h i − 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 The physics of polarization optics Formalisms used Polarized light propagation Propagation through optical devices Partially polarized light 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 (linear or circular anisotropy only) Hard Times if Polarizers are present The Stokes parameters may be an alternative 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 The physics of polarization optics Formalisms used Polarized light propagation Propagation through optical devices Partially polarized light V The projection on a polarization−→ state −→ Matrix of the polarizer with axis parallel to V

Projection on −→V in Jones Basis PV Orthogonal Linear Polarizations Basis: −→X and −→Y Normed Projection Base Vector : −ı˙ ϕ ı˙ ϕ −→V = Ax e 2 −→X + Ay e 2 −→Y t −→V −→V = 1 t a PV = −→V −→V

aEasy to check in the projection eigen basis

N. Fressengeas Polarization Optics, version 2.0, frame 28 The physics of polarization optics Formalisms used Polarized light propagation Propagation through optical devices Partially polarized light The Pauli Matrices

A base for the 4D 2 2 matrix vector space × 1 0 1 0 0 1 0 ı˙ σ0 = ,σ1 = ,σ2 = ,σ3 = − 0 1 0 1 1 0 ı˙ 0  − PV decomposition 1 PV = 2 (p0σ0 + p1σ1 + p2σ2 + p3σ3)

N. Fressengeas Polarization Optics, version 2.0, frame 29 The physics of polarization optics Formalisms used Polarized light propagation Propagation through optical devices Partially polarized light

PV composition and Trace property Trace is the eigen values sum

Projection property t t t t t t −→V σj−→V = −→V −→V −→V σj−→V = −→V −→V −→V σj −→V = −→V PVσj−→V ·   ·   · Projection Trace in its eigen basis

PV eigenvalues : 0 & 1 Tr (PV ) = 1

PVσj eigenvalues : 0 & α α 1 Tr (PV σj )= α ≤ PVσj eigenvectors are the same as PV: −→V associated to eigenvalue α Project the projection t t −→V PVσj−→V = α = Tr (PVσj) = −→V σj−→V · ·

N. Fressengeas Polarization Optics, version 2.0, frame 30 The physics of polarization optics Formalisms used Polarized light propagation Propagation through optical devices Partially polarized light

PV Pauli components−→ and physical meaning Express pi as a function of V and the Pauli matrices, then find their signification

t −→V σj−→V = Tr (PV σj ) Tr (σi σj ) = 2δij · t 1 1 −→V σj−→V = Tr (PV σj )= Tr (σi σj ) pi = 2δij pi = pj · 2 i 2 i P P Project the base vectors on −→V ı˙ ϕ ı˙ ϕ Using −→V = Ax e− 2 −→X + Ay e 2 −→Y 2 ıϕ˙ PV−→X = Ax −→X + Ax Ay e −→Y 2 −ıϕ˙ PV−→Y = Ay −→Y + Ax Ay e −→X Using the PV decomposition on the Pauli Basis 1 1 PV−→X = 2 (p0 + p1) −→X + 2 (p2 +˙ıp3) −→Y 1 1 PV−→Y = (p p ) −→Y + (p ı˙p ) −→X 2 0 − 1 2 2 − 3 Identify

N. Fressengeas Polarization Optics, version 2.0, frame 31 The physics of polarization optics Formalisms used Polarized light propagation Propagation through optical devices Partially polarized light

PV Pauli composition and Stokes parameters

Stokes parameters as PV decomposition on the Pauli base 2 2 p = P = A A = Ix Iy 0 0 x − y − 2 2 p = P = A A = Ix Iy 1 1 x − y − p2 = P2 = 2Ax Ay cos (ϕ)= Iπ/4 I π/4 − − p = P = 2Ax Ay sin(ϕ)= IL IR 3 3 −

N. Fressengeas Polarization Optics, version 2.0, frame 32 The physics of polarization optics Formalisms used Polarized light propagation Propagation through optical devices Partially polarized light

−→Propagating through devices: Mueller matrices ′ −→ V = MJ V

Projection on V−→′ t t t t PV′ = V−→′V−→′ = MJ−→V −→V MJ = MJPVMJ Trace relationship t ′ Pi′ = Tr (PV σi ) = Tr MJPVMJ σi =   1 3 t 2 j=0 Tr MJσjMJ σi Pj P   Mueller matrix −→S′ = MM−→S 1 t (MM )ij = Tr MJσjMJ σi 2  

N. Fressengeas Polarization Optics, version 2.0, frame 33 The physics of polarization optics Formalisms used Polarized light propagation Propagation through optical devices Partially polarized light Mueller matrices and partially polarized light Time average of the previous study

Mueller matrices are time independent

−→S = MM −→S h ′i h i 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