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The physics of polarization optics
Polarized light propagation
Partially polarized light

Polarization Optics

N. Fressengeas

Laboratoire Mat´eriaux Optiques, Photonique et Syst`emes
Unit´e de Recherche commune `a l’Universit´e de Lorraine et `a Sup´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.

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The physics of polarization optics
Polarized light propagation
Partially polarized light

Course Outline

123

The physics of polarization optics
Polarization states Jones Calculus Stokes parameters and the Poincare Sphere

Polarized light propagation
Jones Matrices Examples Matrix, basis & eigen polarizations Jones Matrices Composition

Partially polarized light
Formalisms used Propagation through optical devices

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The physics of polarization optics

Polarized light propagation
Partially polarized light

Polarization states

Jones Calculus 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 )

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Polarization Optics, version 2.0, frame 4

The physics of polarization optics

Polarized light propagation
Partially polarized light

Polarization states

Jones Calculus Stokes parameters and the Poincare Sphere

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

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Polarization Optics, version 2.0, frame 5

The physics of polarization optics

Polarized light propagation
Partially polarized light

Polarization states

Jones Calculus Stokes parameters and the Poincare Sphere

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 )

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Polarization Optics, version 2.0, frame 6

The physics of polarization optics

Polarized light propagation
Partially polarized light

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

ꢀꢁ 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 !

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The physics of polarization optics

Polarized light propagation
Partially polarized light

Polarization states

Jones Calculus

Stokes parameters and the Poincare Sphere

Examples : Linear Polarizations

Canonical Basis Choice

ꢁ ꢂ

10
: horizontal linear polarization

ꢁ ꢂ

0
: vertical linear polarization
1

Tilt

θ

  • ꢁ
  • ꢂ

cos (θ) sin (θ)

  • Linear polarization Jones vector
  • in a linear polarization basis

Linear Polarization : two in phase components

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Polarization Optics, version 2.0, frame 8

The physics of polarization optics

Polarized light propagation
Partially polarized light

Polarization states

Jones Calculus

Stokes parameters and the Poincare Sphere

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

  • ꢁ
  • ꢂ

1ı˙

1
2

√

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Polarization Optics, version 2.0, frame 9

The physics of polarization optics

Polarized light propagation
Partially polarized light

Polarization states

Jones Calculus

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

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The physics of polarization optics

Polarized light propagation
Partially polarized light

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
  • (J1, J2)

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

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The physics of polarization optics

Polarized light propagation
Partially polarized light

Polarization states Jones Calculus

Stokes parameters and the Poincare Sphere

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 Diff´erence

P1 = Ix − Iy

in a Circular Basis

P3 = IL − IR

−π/4

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The physics of polarization optics

Polarized light propagation
Partially polarized light

Polarization states Jones Calculus

Stokes parameters and the Poincare Sphere

Relationship between Jones and Stockes

Sample Jones Vector

ꢁ

4 dependent parameters

P02 = P12 + P22 + P32

ꢂ

Ax 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

ꢂ

Jπ/4

=

ꢁ

Ax e+ı˙ϕ/2 − ı˙Ay e−ı˙ϕ/2

  • ꢁ
  • ꢂ

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

2

P3 = Jx · JCx ir − Jy · JCy ir

=

P2 = Jx · Jπx/4 − Jπy/4 · Jπy/4

=

  • Cir
  • Cir

2Ax Ay sin (ϕ)
2Ax Ay cos (ϕ)

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Polarization Optics, version 2.0, frame 13

The physics of polarization optics

Polarized light propagation
Partially polarized light

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
(S1, S2, S3) on a unit radius sphere

Si = Pi /P0

Unit Radius Sphere

P

General Polarisation

3

i=1 Si2 = 1

Figures from [Hua94]

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The physics of polarization optics

Polarized light propagation

Partially polarized light

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

  • ꢁ ꢂ
  • ꢁ ꢂ

10
10
Lets through the linear polarization along x:

−→

  • ꢁ ꢂ
  • ꢁ ꢂ

01
00
Blocks the linear polarization along y :

−→

  • x polarizer Jones matrix
  • in this basis

  • ꢁ
  • ꢂ

1 0 0 0

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The physics of polarization optics

Polarized light propagation

Partially polarized light

Jones Matrices Examples

Matrix, basis & eigen polarizations 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

  • ꢂ
  • ꢂ
  • ꢁ
  • ꢂ
  • ꢁ

eı˙k n e

0

0

10
0ı˙
10
0ı˙

  • 0
  • 1

= eı˙k n e

≈

  • 0
  • 1

eı˙k n e

2

0

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The physics of polarization optics

Polarized light propagation

Partially polarized light

Jones Matrices Examples

Matrix, basis & eigen polarizations

Jones Matrices Composition

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

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Polarization Optics, version 2.0, frame 17

The physics of polarization optics

Polarized light propagation

Partially polarized light

Jones Matrices Examples

Matrix, basis & eigen polarizations

Jones Matrices Composition

A polarizer in a rotated basis

In its eigen basis

  • ꢁ
  • ꢂ

1 0 0 0
Eigen basis Jones matrix : Px =

When transmitted polarization is θ tilted

Change base through −θ rotation Transformation Matrix

  • ꢁ
  • ꢂ

cos (θ) − sin (θ)
R (θ) = sin (θ) cos (θ)

  • ꢁ
  • ꢂ
  • ꢁ
  • ꢂ

1 0 0 0 cos2 (θ) sin (θ) cos (θ) sin (θ) cos (θ) sin2 (θ)

  • P (θ) = R (θ)
  • R (−θ) =

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Polarization Optics, version 2.0, frame 18

The physics of polarization optics

Polarized light propagation

Partially polarized light

Jones Matrices Examples

Matrix, basis & eigen polarizations

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 = P B0

Jones Matrix is transformed using J1 = P−1 J0 P

  • From linear to circular
  • example

Optically Active media in a linear basis : cos (φ) sin (φ)

  • ꢁ
  • ꢂ

J =

− sin (φ) cos (φ)

  • ꢁ
  • ꢂ

1

I

1
−ı˙
Transformation Matrix to a circular basis P =

  • ꢁ
  • ꢂ

eı˙φ

0

e−ı˙φ

P
−1MP =

0

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Polarization Optics, version 2.0, frame 19

The physics of polarization optics

Polarized light propagation

Partially polarized light

Jones Matrices Examples

Matrix, basis & eigen polarizations

Jones Matrices Composition

Anisotropy can be linear and circular

Linear Anisotropy

Orthogonal eigen linear polarizations

Circular Anisotropy

Orthogonal eigen Circular polarizations

  • Different index n1 & n2
  • Different index n1 & n2

Eigen Jones Matrix

ꢁ

Eigen Jones Matrix

  • ꢁ
  • ꢂ
  • ꢂ

  • 1
  • 0
  • 1
  • 0

  • 0 eı˙θ
  • 0 eı˙θ

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  • Instrumental Optics

    Instrumental Optics

    Instrumental optics lecture 3 13 March 2020 Lecture 2 – summary 1. Optical materials (UV, VIS, IR) 2. Anisotropic media (ctd) • Double refraction 2. Anisotropic media • Refractive index: no, ne() • Permittivity tensor • Birefringent materials – calcite, • Optical axis YVO4, quartz • Ordinary wave Polarization perpendicular to o.a.-푘 plane • Extraordinary wave Polarization in o.a.-푘 plane • Birefringent walk-off 3. Polarization of light • Wave vector ellipsoid • states of polarization Polarization of light linear circular elliptical 휋 휋 Δ휙 = 0, sin Θ = 퐸 /퐸 Δ휙 = , 퐸 = 퐸 Δ휙 ∈ (0, ) 푦 푥 2 푥 푦 2 Plane wave, 푘||푧, 푧 = const: • Fully polarized light 푖(휔푡+휙푥) 퐸푥(푡) 퐸0푥푒 퐸0푥 • Partially polarized light 푖(휔푡+휙 ) i(휔푡+휙푥) iΔ휙 퐸 푡 = 퐸푦(푡) = 퐸 푒 푦 =e 퐸 e • Fully depolarized light 0푦 0푦 0 0 0 Δ휙 = 휙푦 − 휙푥 Note: electric polarization 푃(퐸) vs. polarization of a wave! Linear polarizers Key parameter: Other parameters: 퐼 extinction coefficient, 푦 • Spectral range, 퐼푥 • damage threshold, (assumption: incident light is • angle of incidence, unpolarized!) • beam geometry For polarizing beamsplitters: 2 extinction coefficients, reflection typically worse • clear aperture Polarizer types: • Dichroic • Crystalline • Thin-film Dichroic polarizers Dichroic polarizer: • Made from material with anisotropic absorption (or reflection) coefficient long, „conductive” molecules/structures • Polaroid: polyvinyl alcohol + iodine – one polarization component is absorbed. • Cheap, large aperture, broadband, poor extinction, low damage threshold • Wire grid / nanowire:
  • Lab #6 Polarization

    Lab #6 Polarization

    EELE482 Fall 2014 Lab #6 Lab #6 Polarization Contents: Pre-laboratory exercise 2 Introduction 2 1. Polarization of the HeNe laser 3 2. Polarizer extinction ratio 4 3. Wave Plates 4 4. Pseudo Isolator 5 References 5 Polarization Page 1 EELE482 Fall 2014 Lab #6 Pre-Laboratory Exercise Bring a pair of polarized sunglasses or other polarized optics to measure in the lab. Introduction The purpose of this lab it to gain familiarity with the concept of polarization, and with various polarization components including glass-film polarizers, polarizing beam splitters, and quarter wave and half wave plates. We will also investigate how reflections can change the polarization state of light. Within the paraxial limit, light propagates as an electromagnetic wave with transverse electric and magnetic (TEM) field directions, where the electric field component is orthogonal to the magnetic field component, and to the direction of propagation. We can account for this vector (directional) nature of the light wave without abandoning our scalar wave treatment if we assume that the x^ -directed electric field component and y^ -directed electric field component are independent of each other. This assumption is valid within the paraxial approximation for isotropic linear media. The “polarization” state of the light wave describes the relationship between these x^ -directed and y^ -directed components of the wave. If the light wave is monochromatic, the x and y components must have a fixed phase relationship to each other. If the tip of the electric field vector E = Ex x^ + Ey y^ were observed over time at a particular z plane, one would see that it traces out an ellipse.
  • Development of Instrumentation for Mueller Matrix Ellipsometry

    Development of Instrumentation for Mueller Matrix Ellipsometry

    Author Frantz Stabo-Eeg Title SDevelopmentubtitle? Subtitle? S uofbti tinstrumentationle? Subtitle? Subtitle? Subtitle? Subtitle? Subtitle? for Mueller matrix ellipsometry Thesis for the degree of Philosophiae Doctor Trondheim, February 2009 Thesis for the degree of Philosophiae Doctor Norwegian University of Science and Technology FTrondheim,aculty of FebruaryXXXXXXX 2009XXXXXXXXXXXXXXXXX Department of XXXXXXXXXXXXXXXXXXXXX Norwegian University of Science and Technology Faculty of Natural Sciences and Technology Department of Physics NTNU Norwegian University of Science and Technology Thesis for the degree of Philosophiae Doctor Faculty of Natural Sciences and Technology Department of Physics © Frantz Stabo-Eeg ISBN 978-82-471-1432-2 (printed ver.) ISBN 978-82-471-1434-6 (electronic ver.) ISSN 1503-8181 Doctoral theses at NTNU, 2009:32 Printed by NTNU-trykk Abstract This thesis gives an introduction to the Mueller-Stokes calculus, which is used to describe partially and fully polarized light. It describes how polarized light inter- acts with various sample configurations resulting in a Mueller matrix, and how this can be measured by using appropriate instrumentation. Specifically, two such Mueller matrix ellipsometers have been realized. One system is based on rotating Fresnel bi-prism and the other on Ferro-electric liquid crystals. The systems are unique in different ways. The rotating Fresnel bi-prism Mueller matrix ellipsome- ter is an optimal UV-NIR achromatic suitable for high power angular scattering measurements. The Ferro-electric liquid crystals Mueller matrix ellipsometer is a fast acquisition system capable of measuring Mueller matrices at 50 Hz. The thesis gives an introduction to the topics discussed in the attached six scientific papers. The first three papers report on construction and design of the ellipsometers.