Light and Matter

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Light and Matter
Controlling light with matter
Tim Freegarde
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Optical polarization

• for any wavevector, there are two field components

• any wave may be written as a superposition of the
  two polarizations

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The Fresnel equation
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The Fresnel equation
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The Fresnel equation

• for isotropic media

• electromagnetic waves are transverse

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Characterizing the optical polarization

• wavevector insufficient to define electromagnetic
  wave

• we must additionally define the polarization
  vector

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Jones vectors

• normalized polarization vector is known as the
  Jones vector

• real field corresponds to superposition of
  exponential form and complex conjugate

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Categories of optical polarization

• linear (plane) polarization

• coefficients differ only by real factor

• circular polarization

• elliptical polarization

• all other cases

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Categories of optical polarization

• linear (plane) polarization

• coefficients differ only by real factor

• circular polarization

• elliptical polarization

• all other cases

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Polarization notation

• circular polarization

RCP
plane of incidence

• right- or left-handed rotation when looking
  towards source

perpendicular

• traces out right- or left-handed thread

parallel

• linear (plane) polarization

• parallel or perpendicular to plane of incidence

• plane of incidence contains wavevector and normal
  to surface

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Categories of optical polarization

• complex electric field given by

• real electric field corresponds to superposition
  with complex conjugate

• for monochromatic fields, Jones vector is constant

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Polarization of time-varying fields

• complex polychromatic electric field given by

• beating between frequencies causes field to vary
  with time

• even stabilized lasers have linewidth in the MHz
  range

• Jones vector may therefore vary on a microsecond
  timescale or faster

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Stokes parameters

• with polychromatic light, the Jones vector varies

• we therefore describe polarization through
  averages and correlations

STOKES PARAMETERS
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Stokes parameters

• with polychromatic light, the Jones vector varies

• we therefore describe polarization through
  averages and correlations

STOKES PARAMETERS
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Stokes parameters

• total intensity, I1

• related to horizontally polarized component, I2

• component polarized at 45º to horizontal, I3

• right circularly polarized component, I4

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Unpolarized (randomly polarized) light

• average horizontal component average vertical
  component

• average 45º component average -45º
  component

• average RCP component average LCP
  component

half total intensity

• orthogonal polarizations are uncorrelated

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Degree of polarization

• for partially polarized light, the quantity

represents the degree of polarization, where
unpolarized (randomly polarized)
completely polarized
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Completely polarized light

• constant Jones vector

• Stokes parameters given by

• when simply defining the polarization state, it
  is common to drop the intensity factor I1

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The Poincaré sphere
(a)
(f)
right elliptically polarized
(a) right circularly polarized
0, 0, 1
(d)
(b) left circularly polarized
0, 0,-1
(c) horizontally polarized
1, 0, 0
(g)
(e)
(c)
(d) vertically polarized
-1, 0, 0
(e) polarized at 45º
0, 1, 0
left elliptically polarized
(f) elliptically polarized
d1,d2,d3/d0
(g) unpolarized
0, 0, 0
(b)
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Polarizers

• many optical elements restrict or modify the
  polarization state of light

• polarization-dependent transmission/reflection

• sheet polarizers (Polaroid)

• Nicol, Wollaston prisms etc

• polarizers, polarizing filters, analyzers

• polarization-dependent refractive index

• waveplates, retarders

• four categories of physical phenomena

• polarization-sensitive absorption (dichroism)

• polarization-sensitive dispersion (birefringence,
  optical activity)

• reflection at interfaces

• scattering

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Polarizers
plane of incidence

• each mechanism may discriminate between either
  linear or circular polarizations

• mechanisms depend upon an asymmetry in the device
  or medium

perpendicular
parallel
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Linear polarization upon reflection

• for normal incidence, no distinction between
  horizontal and vertical polarizations

• if wavevector makes angle with interface normal,
  s- and p-polarizations affected differently

• we consider here the reflection of p-polarized
  light s-polarized beams may be treated similarly

• we resolve the electric field into components
  parallel and normal to the interface

• all magnetic field components are parallel to the
  interface

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Linear polarization upon reflection
combine forward and reflected waves to give total
fields for each region
apply continuity conditions for separate
components
hence derive fractional transmission and
reflection
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Fresnel equations
combine forward and reflected waves to give total
fields for each region

• p-polarization

apply continuity conditions for separate
components

• s-polarization

hence derive fractional transmission and
reflection
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Linear dichroism

• conductivity of wire grid depends upon field
  polarization

• electric fields perpendicular to the wires are
  transmitted

• fields parallel to the wires are absorbed

WIRE GRID POLARIZER
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Linear dichroism

• crystals may similarly show absorption which
  depends upon linear polarization

• absorption also depends upon wavelength

• polarization therefore determines crystal colour

• pleochroism, dichroism, trichroism

TOURMALINE
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Circular dichroism

• absorption may also depend upon circular
  polarization

• the scarab beetle has polarization-sensitive
  vision, which it uses for navigation

• the beetles own colour depends upon the circular
  polarization

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Polarization in nature

• the European cuttlefish also has
  polarization-sensitive vision

• and can change its colour and polarization!

(red horizontal polarization)
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Birefringence

• asymmetry in crystal structure causes
  polarization dependent refractive index

• opposite polarizations follow different paths
  through crystal

• birefringence, double refraction

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Linear polarizers (analyzers)
o-ray

• birefringence results in different angles of
  refraction and total internal reflection

e-ray

• many different designs, offering different
  geometries and acceptance angles

e-ray
o-ray
s-ray

• a similar function results from multiple
  reflection

p-ray
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Waveplates (retarders)

• at normal incidence, a birefringent material
  retards one polarization relative to the other

• linearly polarized light becomes elliptically
  polarized

WAVEPLATE
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Compensators

• a variable waveplate uses two wedges to provide a
  variable thickness of birefringent crystal

• a further crystal, oriented with the fast and
  slow axes interchanged, allows the retardation to
  be adjusted around zero

SOLEIL COMPENSATOR

• with a single, fixed first section, this is a
  single order (or zero order) waveplate for
  small constant retardation

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Optical activity (circular birefringence)

• optical activity is birefringence for circular
  polarizations

• an asymmetry between right and left allows
  opposing circular polarizations to have differing
  refractive indices

l-limonene (orange)
r-limonene (lemon)

• optical activity rotates the polarization plane
  of linearly polarized light

CHIRAL MOLECULES

• may be observed in vapours, liquids and solids

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Jones vector calculus

• if the polarization state may be represented by a
  Jones vector

JONES MATRIX

• then the action of an optical element may be
  described by a matrix

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Jones vector calculus
JONES MATRIX
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Müller calculus

• field averages and correlations following optical
  element depend linearly upon parameters
  describing incident beam

• Müller matrix elements may be written in terms of
  Jones matrix elements, e.g.

MULLER MATRIX
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Müller calculus

• the actions of optical materials can be
  represented by geometrical transformations of the
  Stokes vector in the Poincaré sphere

right elliptically polarized

• optical activity rotation about a vertical axis

left elliptically polarized
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Müller calculus

• the actions of optical materials can be
  represented by geometrical transformations of the
  Stokes vector in the Poincaré sphere

right elliptically polarized

• optical activity rotation about a vertical axis

• birefringence rotation about a horizontal axis

left elliptically polarized