Waves, Light & QuantaTim Freegarde

Web Gallery of Art; National Gallery, London

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Categories of optical polarization

• linear (plane) polarization• non-equal components in phase

• circular polarization• equal components 90° out of

phase• elliptical polarization

• all other cases

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Polarizing components

POLARIZER(filter/

separator)

LINEAR CIRCULAR

WAVEPLATE

(retarder)yx

yx TT

RL

RL TT

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Polarization notation

• circular polarization• right- or left-handed rotation

when looking towards source

• linear (plane) polarization• parallel or perpendicular to

plane of incidence

RCP plane of incidence

perpendicular

parallel

• traces out opposite (right- or left-) handed thread

• plane of incidence contains wavevector and normal to surface

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sin,cos iea

Characterizing the optical polarization

• wavevector insufficient to define electromagnetic wave• we must additionally define the polarization vector

k

yx aa ,ax

yz

sin,cosa• e.g. linear polarization at

angle

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Jones vector calculus• if the polarization state may be

represented by a Jones vector

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

yx aa ,a

2221

1211

aaaa

A

y

x

y

x

aa

aaaa

aa

2221

1211

JONES MATRIX

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Jones vector calculus

y

x

y

x

aa

aaaa

aa

2221

1211

JONES MATRIX

0001

1A transmission by horizontal polarizer

y

x

ii

A

exp00exp

2retardation by

waveplate

cossinsincos

3A projection onto rotated axes

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

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

2221

1211

aaaa

A

yx aa ,a

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Birefringence• asymmetry in crystal

structure causes two different refractive indices

• opposite polarizations follow different paths through crystal

• birefringence, double refraction

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38.5º

Linear polarizers (analyzers)

e-ray

o-ray

e-ray

o-ray

s-ray

p-ray

• birefringence results in different angles of refraction and total internal reflection• many different designs, offering different geometries and acceptance angles

• a similar function results from multiple reflection

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Waveplates (retarders)

WAVEPLATE

• at normal incidence, a birefringent material retards one polarization relative to the other• linearly polarized light becomes elliptically polarized

le 02

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Compensators

adjust

fixed

variable

• 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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Unpolarized light

22yx EE 22

yxyx dEcEbEaE

y

x

y

x

EE

dcba

EE

yxyx EEcdabEdbEca 2222222

• if no correlation between and , xE yE

222222yx EdbEca

• if ,202122 EEE xx

2yx TT

• intensity

• for any system

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Electromagnetic waves• light is a transverse wave: perpendicular to E k

zxE

yBx

y

z

x

y

• Faraday

• Ampère

SBsE d.d.t

SEJsB d.d. 00 t yB

SEsB d.d. 00 t

xE

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Dielectrics

z

• atomic electrons move in response to electric field• resulting atomic dipole radiates field which adds to original

• Faraday

• Ampère

SBsE d.d.t

SEJsB d.d. 00 t

SEJsB d.d. 00 tr

Waves, Light & QuantaTim Freegarde

Web Gallery of Art; National Gallery, London

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Diffraction

S Yoshioka & S Kinoshita, Forma 17 169 (2002)

• irridescence of feathers (Grimaldi, 1665)

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Diffraction

x

d

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Diffraction

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Diffraction

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Huygens’ wave construction

• propagation from a point sourceChristiaan Huygens (1629-1695)

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Huygens’ wave construction

• reflection at a plane surfaceChristiaan Huygens (1629-1695)

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Huygens’ wave construction

Christiaan Huygens (1629-1695)• refraction at a plane surface

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Huygens’ wave construction

• mirages by refraction in the atmosphereChristiaan Huygens (1629-1695)

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Huygens’ wave construction

• Fresnel integral• phasors shorter / rotate more

quickly at distance to give spiral

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Arago’s bright spot• M A Fresnel, La diffraction de la lumière

(1818)Let parallel light impinge on an opaque disk, the surrounding being perfectly transparent. The disk casts a shadow - of course - but the very centre of the shadow will be bright. Succinctly, there is no darkness anywhere along the central perpendicular behind an opaque disk (except immediately behind the disk).

• S D Poisson:

• F Arago: One of your commissioners, M Poisson, had deduced from the integrals reported by [Fresnel] the singular result that the centre of the shadow of an opaque circular screen must, when the rays penetrate there at incidences which are only a little more oblique, be just as illuminated as if the screen did not exist. The consequence has been submitted to the test of direct experiment, and observation has perfectly confirmed the calculation.