Light and Optics - 3Propagation of light

Electromagnetic waves (light) in vacuum and matterReflection and refraction of lightHuygens’ principlePolarisation of light

Geometric opticsPlane and curved mirrorsThin lenses

InterferenceDouble slits

DiffractionSingle slitDouble slits

Luke Wilson (Luke.wilson@... Room E17)

Polarisation

Direction of polarisation defined to be in the direction of the electric field vector, E

e.g. above we have ( ) ( )tkxEtx ω−= cos, maxjE

Polarising filters (polariser)

Polarising filters (polariser)

Ratio of transmitted to incident amplitude is cos φRatio of transmitted to incident intensity is cos2 φ

Intensity of light transmitted through the analyser is then

φ2max cosII =

Polarising filters example

e.g. Unpolarised incident light intensity I0 .Find the intensities transmitted by 1st and 2nd polariser for 60° angle between axes of polarisers.

After polariser 1: (avg. value cos2 = ½)

After polariser 2 (analyser):2

cos 0201

III == φ

84cos 012

12IIII === φ

Polarisation by reflection

‘p’ polarisation, E lies in plane of incidence‘s’ polarisation, E perpendicular to plane of incidence

Polarising angle a.k.a. Brewster’s angle

Polarisation by reflection

At the polarising angle θp the reflected and refracted rays are perpendicular to each other

( )

a

b

bb

nn

nnn

nn

=

=−°=

=

p

pppa

bbpa

tan

cos90sinsin

sinsin

θ

θθθ

θθ

Brewster’s Law

Circular polarisation

We can think of both linear and circular polarisation by separating the E-field into orthogonal components.

The only difference is the relative phase of the components.

Circularly polarised light (top) Linearly polarised light (bottom)

The blue and green curves are projections of the red lines on the vertical and horizontal planes, respectively.

Circular polarisation - vectorsEasy to see with vector representation:

( ) ( )( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( ){ }( ) ( ){ }tkztkzE

tkztkzE

tkzEtkzE

tkzEtztkzEtztkzEtz

ωωωω

ωω

ωωω

−−−=−−+−=+

−−=−−+=+

−=−=−=

sincossincos

coscos

sin,cos,

cos,

031

031

021

021

03

02

01

jiEEjiEE

jiEEjiEE

jEjEiE

x

yE1

E2, E3

E1+E2

E1-E2

E1+E3

E1-E3Right circular polarisation for E rotating clockwise and wave coming towards you

Creating circularly polarised lightA phase shift can be introduced by passing light through a birefringent material.

Material that exhibits birefringence has a different refractive index for different directions of polarisation.

Creating circularly polarised light – quarter wave plate

Consider linearly polarized light which strikes the plate to be divided into two components with different indices of refraction.

If the birefringent crystal is just thick enough then a quarter-cycle phase difference can be produced between the two components.

By adjusting the plane of the incident light so that it makes 45° angle with the optic axis then linearly polarised light is converted to circular (and vice versa).

Quarter wave plate thicknessConsider birefringent material with refractive indices n1 and n2 for two orthogonal components of linearly polarised light.

Wavelengths in the material are given by

For circularly polarised light production, the number of wavelengths of each component in the material must differ by ¼.

The number of wavelengths in a thickness t of material is t / λ . The condition for the quarter wave plate is therefore:

2

02

1

01

nnλλλλ ==

( )

( )21

0

0

21

0

2

0

1

0

2

0

1

21

4

41

41

41

41

nnt

nnttntntntn

tt

−=

=−

⇒=−⇒+=

+=

λλλλλλ

λλ(assuming n1 > n2, so λ2 > λ1)

(minimum thickness, typically ~ 1μm or less)

Polarisation summaryConsider only the E-field component of EM light wave

Linearly polarised light can be created by passing unpolarised light through a polaroid film. The initial (I0) and final (I1) light intensities when passing through a linear polariser are related by

Linearly polarised light can also be produced by reflection. Theangle for which the p-polarised reflection is minimum can be found from

Birefringent material can be used to produce circularly polarised light. The important property is different refractive indices for different linear polarisations of light in the material. The minimum thickness for a quarter wave plate was found:

φ201 cosII =

a

bnn

=ptanθ

( )21

04 nn

t−

=λ