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Circular and Elliptical Polarisation

• Circular polarisation is a state where wave components have the same amplitude and frequency
• Circular polarisation happens when orthogonal linear polarisation and a phase difference of π/2 is present
• Two polarisation states are aligned with the z-axis (blue) and the y-axis (green)
• The two waves possess the same wavelength and amplitude, starting with the same phase
• The combined E field vector can be shown as either in the positive or negative direction at 45° to the y,z axis
• At certain points along the propagation, the E field vector is zero
• In circular polarisation, the E field vector is not confined to a single line, completing a circle in one wavelength (2π phase)
• The E field vector has constant magnitude and rotates in a circle in circularly polarized light
• Two forms of circularly polarised light: right hand and left hand
• The difference between right hand and left hand circular polarisation depends on whether the polarisation vector rotates clockwise or anti-clockwise when looking down the z axis toward the origin
• Elliptically polarized light happens if the phase difference between components is not exactly π/2, or the amplitudes of the two components are not the same
• Birefringent material can convert linearly polarised light to circularly polarised light by introducing a specific phase shift between the two polarisation components
• Birefringent materials have have two different refractive indices, one for each polarisation axis
• The two axes of birefringent material are called the fast and slow axis
• A quarter wave plate acts as a filter, cut to have an integer number of wavelengths for one axis and a quarter for the other
• With the correct orientation, a quarter wave plate can convert circular polarisation to linear polarisation, and vice versa
• A half wave plate introduces ½ of wavelength difference and it rotates the angle of both polarisation states
• This is used for linear polarisation and has no effect for circular polarisation

Photoelasticity

• Truly birefringent materials are uncommon
• Several materials exhibit effects where the refractive index varies with a physical stimulus such as temperature known as the thermo-optic effect
• The stress-optic effect is when the change in refractive index is different for polarisations aligned with and against the stress in the material
• Photoelasticity, is common in polymers (plastics)
• Photoelasticity is used to convert stress into a visually resolvable effect
• The material is placed between two crossed polarisers
• The first polariser polarises light; without affecting the angle of polarisation the light is filtered out by the second polariser
• If the material is birefringent, it rotates the polarisation, allowing some light to pass through the second polariser
• The amount and colour of light passed gauge the amount and direction of stress in the object

Scattering

• Scattering occurs when light is absorbed by an atom and re-radiated
• Scattering depends on the frequency, and the radiated light can be polarised
• The intensity of scattering is proportional to f⁴, hence higher frequency light (blue) is more likely to be scattered than low frequency light (red)
• Since the light from the sun is approximately white it means that un-scattered light will appear mostly red, and scattered light is mostly blue
• The light visible looking at the sky is scattered by the atmosphere
• Light is blue because it does not take a direct route from the sun
• Sunrises are red, because the light is un-scattered, passing through the atmosphere
• Clouds scatter light in a relatively high concentration of water vapour and all frequencies end up randomly scattered to more or less the same degree, hence they are white
• Un-scattered light is unaffected and unpolarised
• Scattered light is polarised in a specific direction orthogonal to the direction of the sun
• Many air molecules are able to be polarised and act as a dipole with a specific orientation which re-radiates the light out in a direction with a polarisation determined by the orientation of the dipole.
• Oscillation components radiate light toward the ground, therefore the light seen on the ground will be polarised

Interference in 2 and 3 Dimensions

• Interference happens when two or more waves overlap and is governed by superposition
• Provided that the polarisation is not orthogonal we can get constructive and destructive interference based on the relative phase of the two waves
• Constructive and destructive interference also applies to EM waves, sound wave or waves on a string
• Two sources of interference are
• Two identical waves counter propagating making a standing wave
• Two non-identical waves propagating making a beat frequency
• Constructive or destructive interference depends on the path length difference
• Sinusoidal waves emitted from a point source create wavefronts representing maximum intensity
• Each wavefront is separated by one wavelength
• If radiating an EM wave, the refractive index is constant giving constant wave speed
• When adding a second point source it is generated from the same amplitude, frequency and is in phase as the first source
• With the same frequency, two sources are coherent.
• Both sources have the same polarisation, and therefore interfere

Constructive and Destructive Interference

• Expanding wavefronts come from both sources
• A constant phase relation exists between sources with same frequency and have constant radial spacing
• If this was not the case the two waves would be incoherent
• Incoherence leads to a randomised interference pattern, to the point the waves would not interfere
• On a line bisecting the two sources, the distance between the two sources is the same, therefore constructive interference would be seen
• At a point where wavefronts cross, there is an exact number of wavelengths from S₁ → b = r₁, and S₂ → b = r2
• Waves will arrive in phase and constructively interfere
• Conversely if a position lies on one the wavefronts but halfway the other means the waves are out of phase and will destructively interfere
• Constructive is expressed as: r2 - r₁ = mλ (m = 0, ±1,±2, ±3 ...)
• Destructive is expressed as: r2 - r₁ = (m+ ½ ) λ (m = 0, ±1, ±2, ±3 ...)
• In both cases m is an integer number
• For interference to occur, sources must have the same frequency, and also have the same phase relation

Antinodal and Nodal Curves

• For constructive interference, these are antinodal curves
• Terminology is same for standing waves: nodes never see amplitude, so are points of destructive interference and antinodes sit between
• Higher order lines connect where wavefronts cross
• Creating two coherent sources of sound is achievable, for light is an impossibility due to keeping them in phase due to quantum mechanical effects
• It is easier to take a coherent light source, that has a constant frequency and phase, and split in into two

Setting up coherent light sources

• The secondary sources will maintain coherence over time/distance
• The distance at which coherence remains is known as the coherence length
• Interference occurs using a pair of secondary coherent light sources
• A single slit creates a long, thin light source through diffraction
• A secondary pair of slits creating to secondary line sources occur
• The two slits are placed the same distance from the first slit in order to keep the light hitting and emitting in phase
• As light propagates from these two slit sources bands of interference overlap on a screen
• Light and dark bands are indicative of different orders of both constructive and destructive interference
• This is known as the Young's Slits which was discovered in 1800

Constructive and Destructive Interference

• Constructive occurs when path length difference from the slits fits the equation: r2 - r₁ = d Sin[0]
• The distance between slits, d, are small when compared to the distance between slits and screen R
• The path length is at the slit end, formed from triangle where θ is an angle of rays and is combined to work associated constructive/destructive interference

Measuring Interference

• It is most useful to measure distance on the screen and not the ray's angles
• A reference point for O distance which will be the m=o constructive band is needed
• Apply the equation Ут = R Tan[θm]
• Ут = distance from the 0 order and is related to the mth order band with R
• d is inversely proportional, so if slits are closer interference will spread further
• Destructive interference is found by repalcing by m with (m + ½ )
• Interference is defined into two divisions of amplitude, being thin film and wedges
• The section has a deviation from textbook which as longer explanations

Optical and Refractive Index effect on Interference

• The optical parth length determines phase shift where the whole number of wavelength gives a shift
• Δρ = P1 - P2 shows path length difference
• The process can be affected by refractive index with two optical paths that use the refractive index
• This index will alter wavelength from vacuum state, with equation: λn = λo / n
• Then, p = nL / λO

Thin Film Interference

• Thin film is seen from soap bubbles with bright bands and colour
• Two reflections on the surface causes:
• Generated light reflects directly into the eye from path a->b->c->p
• Generated light is refracted into path a-> b-> d -> e -> f -> p
• With ray 2 longer and material with n > air
• Different colour bands come from the film and can be seen based on the refractive index that appears

Thin Film and Air Wedge

• Using air wedge light interference approximation means rays are at normal and plates are small
• Two reflections are the bottom slides, so doesn't share with the top
• Glass sides are vertical in the diagram and generated so they can be identified
• Use this equation: 2t=(m+1)A

Maxwells Equations

• Wave interference comes from two reflections
• Use equation Er = па-пь / па+пь Ei with the usual meanings of a/b and the reflected amplitude
• (Na > N), Er is positive and (na < n₁), Er is negative
• Boundry conditions in mechanical waves is where speed changes

Amplitude Changes in phases

• Change in amplitude is shown in term and it corresponds to a 180° phase shift
• phase shift = 0 then Na > Nb
• phase shift = ½λ then NaNb

Coherence length conditions

• two rays involved where the same frequency needs to be constant
• bursts are constant and coherent, and when combined become minor
• to find interference seen is less than burst

Film and Refractive Index Layers

• thinner or less dense thickness causes more refractive
• Film needs be "thin" of allow
• glass windows
• longer source allows thicker lengths

Newton Rings and coatings

• Effective at metrology
• Compare two surfaces with optical roller
• Refractive in air between needs to occur
• 1/4 with half reflection