Wave Optics - IOGS

Wave Optics

Nathalie Westbrook & Arnaud Dubois 5 lectures on Polarization

1 lecture on Interferometry (coherence) 6 lectures on Fourier Optics

NW wave optics( polarization) 1


What do we mean by « Polarization »? Polarized light and its modifications in a functional

description Polarization of light is important when Light is transmitted or reflected under non normal incidence (folded

optical systems, optical fibers, thin films, ellipsometry,…) Light propagates through non isotropic media, Light is diffused, Your light source emits polarized light (ie most lasers) So in most optical systems!

Light propagation in anisotropic media Application to optical devices based on polarization either passive or

actively controlled (electro- or magneto-optical effects,…) Non linear optics: phase matching condition, …

Interferences in polarized light Phase measurement by interference, interference filters (Lyot, Solc)


Outline of the polarization course Lecture 1 : Polarization states / Basic functions Description of polarization states: complex amplitude – Geometrical

representation - Jones vectors and matrices – Poincaré sphere Basic functions and devices: polarizers and retardation plates

Lecture 2 : Propagation in an anisotropic medium Linear birefringence: index ellipsoïd, orthogonal eigenpolarizations Circular birefringence: optical activity, Faraday effect Induced optical effects: Pockels cells

Lecture 3 : Interference with polarized light basic principles, interference colors, Babinet compensator

2 more lectures (end of november): More advanced subjects

Partially polarized light, reversibility theorem Ray constructions in both refraction and reflection Optical path variations at non normal incidence, coherence


Textbooks for this course (available at the IO library)

Optics (E. Hecht) lecture 8 about polarization: physical approach, light on maths

Optical waves in crystals (Yariv) more detailed formalism esp on ray propagation in birefringent media,

also useful for guided and coupled wave course

Other useful references Polarization interferometers (Françon & Mallick) very detailed on this subject, but out of print (available at the IO library)

Principles of Optics (Born&Wolf) very detailed, lots of maths, general optics reference

In French

Atlas des phénomènes optiques et supplément beautiful pictures of optical phenomena (available at the IO library)

Polarisation (S. Huard) french textbook – now available in English translation


Polarized Light

I. Plane Wave in an isotropic media

B,H

Electric Field E

wave vector k Poynting Vector S=ExH (light ray)

E,H wavefront

Polarization = direction of E


Monochromatic plane wave (totally polarized light)

Complex amplitude Ecomplex=E0complexexp-i(ωt-kr)

E0=E0xux + E0ye+iϕuy

E0x , E0y real numbers ϕ >0 phase lag of y with respect to x

Graphical representation: Endpoint of the real electric field Ereal=Re(Ecomplex)

Jones-Vector representation Column vector, usually normalized, defined within a global phase

II. Polarization states and Jones-Vector representation


Linear Polarization

Linear Polarizations are the only ones with a real-number Jones Vector: ε=ε*

Circular polarization

+ left-circular (y lags x by π/2) - right-circular

Elliptical polarization (most general state of totally polarized light)

Parameters for the elliptical polarization:

• angle α where tanα=Eoy/Eox

• phase ϕ


Jones vectors: A few examples

€

εL =121+i⎛

⎝ ⎜

⎞

⎠ ⎟

€

εR =121−i⎛

⎝ ⎜

⎞

⎠ ⎟

a b

Linear Circular Elliptical

a b


• In general the ellipse axes are neither x nor y, nor the diagonal of the rectangle

• Intensity proportional to the square of the diagonal length (E0x2 + E0y

2) All rectangles which include the same ellipse have the same diagonal length

β

θ α

β : ellipticity (0 for linear, ±45° for circular)

θ : main axis orientation

Elliptical polarization:connexion between the electric field expression using (α,ϕ) and the shape of the ellipse using angles (θ,β)


α=20°

0° 15° 30°

90°

45°

165°

α=70° For 2 different values of α

ϕ varies from 0 to π by 15° steps

€

Ecomplex = (E0xux + iE0yuy ) exp(−iωt)

1st specific case: ϕ=90°, θ=0 (or π/2)

Ellipse with axes x and y:


2nd special case: Ellipse set in a square (E0x = E0y): α=45°, θ=±45°

The axes of the ellipse are fixed at 45°

Ellipticity β equals ±ϕ/2 (modulo π/2)


€

εx ⋅εy∗ = 0

€

εL ⋅εR∗ = 0

€

εx =12εL +εR( )

• Orthogonal Polarizations

• Arbitrary choice of basis: any polarization state can be written as a linear combination of two orthogonal polarization states Linear =right-circular+left-circular

linear with angle α= right-circ + left-circ with 2α phase lag

General properties of Jones Vectors

€

cosαsinα⎛

⎝ ⎜

⎞

⎠ ⎟ =

12

− iαe εL +i 2αe εR( )


III. Modification of polarisation states - Jones Matrices

Examples of phenomena modifying polarization states

• Transmission and Reflection Coefficients at non normal incidence on a dielectric or metallic surface in s or p polarization

• Total Internal Reflection: phase shift during reflection

• Propagation through a birefringent medium (refraction index depends on polarization, as in half and quarter wave plates,…) and/or a dichroic medium (absorption depends on polarization, as in polaroïd polarizers)

Main Functions

• Polarizers • Half Wave plate • Quarter Wave plate • Optical rotator


Polarizer

Projects on the polarization state transmitted by the polarizer

P P

Ein Ein

Eout Eout

Iout=Iin cos2 α

P

Ein

Eout

α

Iout=Iin /2


Different types of polarizers

• dichroic polarizers (« polaroid »): absorption depending on polarization

• stack of Brewster plates: Fresnel transmission coefficients at oblique incidence depending on polarization

• grid polarizers: array of metallic wires with a period smaller than λ (see for example the Moxtek company website)

• polarizing beamsplitters cubes using thin films: multiple wave interference

• birefringent polarizer or beamsplitters: angular separation (Rochon, Wollaston prisms) ou total internal reflection (Nicol, Glan Taylor, Glan Thompson) details of those polarizers will be studied in later lectures


Polarizer using Brewster’s angle at an air to glass interface

Stack of 10 Brewster plates: T⊥/ T// =(1-0.15)20 =4%

Reflected polarization: ⊥ to the incidence plane (s or TE)

Transmitted polarization: // to the incidence place (p or TM)


Retardation plates (or wave plates)

Birefringent plate with plane parallel sides cut perpendicular to the optical axis x Phase shift induced at normal incidence:ϕ= 2π/λ (ne – no)e

Positive uniaxial (quartz in general)

Ordinary polarization //y: index no <ne fast axis Extraordinary polarization //x: index ne slow axis The two axes are called the neutral axis of the wave plate

Standard components: half wave plate (ϕ=π) and quarter wave plate (ϕ=π/2)


Half Wave plate

Induces a phase shift between the two components parallel to its neutral axes of ϕ= π , modulo 2π δ= λ/2, modulo λ

Symmetry of the incident polarization with respect to the axes of the half wave plate

Neutral axes of λ/2 plate

Ein

Eout

Ein

Eout

λ/2 plate

Zero order wave plate 0: ϕ= π Higher order wave plate : ϕ= π+2kπ


Quarter Wave plate

ϕ= π/2 , modulo π or 2π δ= λ/4, modulo λ/2 or λ Turns linear at 45° into circular Turns any linear into an ellipse with its axes along the neutral axes of the wave plate (and vice-versa to within a symmetry with respect to the axes)

λ/4 plate (zero order, slow axis)

Ein

Eout

Ein

Eout

Ein

Eout

Note: if the incoming ellipse has axes that are NOT the neutral axes of the wave plate, the outcoming polarization is elliptical with all possible axes and ellipticity (no simple relation in this case)


Optical Rotator

Induces a rotation of an angle α on a linear polarization

Ein Eout

α

Light coming towards the observer

• Components based either on optical activity (quartz used along optical axis) or on Faraday Effect (rotation induced by magnetic field)

• What is the effect of an optical rotator on a circular polarization? On an elliptical polarization? Interpretation as phase shift between left and right circular polarizations (circular birefringence)


Jones matrix for polarization modification

2x2 matrices with complex coefficients, representing the transformation of a Jones Vector in an orthogonal basis of polarizations (linear in general)

Defined to within an overall phase,

Unitary if intensity is conserved (half and quarter wave plates, optical rotators)

Eigenvectors, eigenvalues: polarization states unchanged after propagation, possibly attenuated or dephased


Examples of Jones Matrices

Polarizer //x

Polarizer //y

Halfwave plate axes x and y Quarterwave plate slow axis x Rotator by angle α

€

Rα =cosα − sinαsinα cosα⎛

⎝ ⎜

⎞

⎠ ⎟

€

Px =1 00 0

⎛

⎝ ⎜

⎞

⎠ ⎟

€

Lλ / 2 =1 00 -1

⎛

⎝ ⎜

⎞

⎠ ⎟

€

Py =0 00 1

⎛

⎝ ⎜

⎞

⎠ ⎟

€

Lλ / 4 =1 00 - i

⎛

⎝ ⎜

⎞

⎠ ⎟

Expression of the Jones matrix in the (X,Y) coordinates rotated by an angle θ with respect to (x,y):

€

MXY = R−θ . Mxy . Rθ

Thus the matrix of a polarizer with its axis along X, at 45° from x, can be deduced from the following change of coordinates of the polarizer with transmission axis X:

€

PX = R−45° . Pxy . R45°

€

Pxy = R45° . PX . R−45°

€

Pxy45° =

12

1 −11 1

⎛

⎝ ⎜

⎞

⎠ ⎟

1 00 0

⎛

⎝ ⎜

⎞

⎠ ⎟

12

1 1−1 1

⎛

⎝ ⎜

⎞

⎠ ⎟

=12

1 11 1

⎛

⎝ ⎜

⎞

⎠ ⎟


Elliptical polarizations represented on Poincaré sphere

• Point P: elliptical polarization with main axis θ and ellipticity β

• Orthogonal Polarizations : opposite side of the same diameter

Ellipticity= Constant

Direction of axis=cst

2θ

2β

P

IV. Another representation for polarized light: the Poincaré Sphere


Positions of linear and circular polarizations on the Poincaré Sphere

Left Circular

Right Circular

All Linear Lin//x

Lin//y Lin at 45°

S1

S2

S3


Effect of a phase plate on Poincaré sphere = rotation by an angle ϕ around a direction connecting the two eigenpolarizations

Examples:

• Quarter wave plate with axes x and y = rotation of an angle ϕ=90° around points Lx andt Ly ⇒ L45 (lin at 45°) turns into Cg(left circ)

• Rotator by an angle α=45° = rotation by angle ϕ=2α=90° around points CL et CR ⇒ L45 turns into Ly

Lx L45

CL

CR

Ly


V. Partially polarized light – Stokes parameters

Natural light= randomly polarized, no specific direction of polarization

⇒ direction of E fluctuates rapidly with respect to the response time of the system (detector, physical phenomenon observed) ⇒ Can be represented by a scalar electric field

What is the effect of a polarizer on natural light, both in polarization and in intensity?

Partially polarized light (or partial coherence of polarization)= superposition of natural light and polarized light

⇒ Difficult to describe using an electric field (requires coherence matrices instead of Jones Vectors)

⇒ More convenient using Stokes parameters, description based on intensities


Stokes parameters • Definition based on average values of the electric field

• Physical interpretation in terms of measured intensities

• Degree of polarization

€

S0S1S2S3

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

=

E0x2 + E0y

2

E0x2 − E0y

2

2E0xE0y cosϕ2E0xE0y sinϕ

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥

€

S0S1S2S3

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

=

I totalI x − Iy

I45° − I−45°

I leftcirc − I rightcirc

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

€

V =S12 + S2

2 + S32

S0

V=0 natural light V=1 totally polarized light

Important: not a vector space

IV. Partially polarized light - Stokes parameters and Poincaré sphere

€

S1S2S3

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

For totally polarized light:

are the coordinates on the Poincare

sphere


Stokes parameters: A few examples

€

εL =121+i⎛

⎝ ⎜

⎞

⎠ ⎟

€

εR =121−i⎛

⎝ ⎜

⎞

⎠ ⎟

a b

Linear Circular Elliptical Natural

IV. Partially polarized light - Stokes parameters and Poincaré sphere

Wave Optics - IOGS

Documents

Transcript of Wave Optics - IOGS