Download Lecture 21: Polarisation of light and other waves

Lecture 21: Polarisation of light and other waves
1. Transverse polarisation
Ordinary waves on a string can occur either in the vertical (V) direction or, equally well, the horizontal
(H) direction. Or in any intermediate direction.
We can express this by writing the
of the wave
polarisation
1
0
as a 2-D vector normalised to 1. So
represents vertical polarisation and
represents horizontal
0
1
√ 1/√2
polarisation and
represents waves polarised at 45◦ .
1/ 2
This is especially important for Electromagnetic radiation where the polarisation vector refers to the
direction of the electric field. Radio waves emitted by an aerial are polarised, and the alignment of a receiving
aerial relative to that polarisation affects the strength of the signal received. Which is why they are often
adjustable on FM radio and TV receivers.
Light waves are generally produced in an unpolarised mixture. But light can become polarised by
various phenomena.
1.1 Birefringence
In a previous lecture we considered the effect of EM radiation on atoms, and how the induced charges
in the atoms led to refractive index, dispersion, and absorption. (Please note that the ‘polarisation’ of the
charge clouds in atoms is not the same as the ‘polarisation’ of light: an unfortunate verbal coincidence.)
Now consider molecules. A molecule has a shape which is more complicated than that of the simple
spherically symmetric atom. A molecule will respond to an applied electric field in a way that depends on
the orientation of the molecule in the field. A long thin molecule with a floppy electron cloud will be easily
polarised along the direction of the molecule, but hard to polarise in the transverse directions.
If the molecules are just aligned at random, as they are in a gas, then this all smears out and you just
get an averaging effect. But if the molecules are lined up, either in a regular crystal lattice or in a phase
where there is some order (such as a liquid crystal, in which the directions are the same though the positions
are random) then you can get a situation where the refractive index is different in the H and V directions.
So if you have such a crystal then light polarised in the H and V directions will travel at different speeds
(H and V are now defined with respect to some crystal coordinate axis). If you use it to refract light at
some angle of incidence θi there are two different angles of refraction θr for the H polarisation and the V
polarisation.
If light of some intermediate polarisation is incident on the crystal, you do not get refraction at some
intermediate angle! The incident light is split into its separate H and V components. There are two emerging
rays, with some of the light refracted at one angle and some at the other.
1.2 Polaroid
An extreme example of this is polaroid film. It has long molecules that are all aligned in a particular
direction. Light polarised in that direction is actually absorbed, even in a thin sheet. So light that emerges
on the far side of the sheet is polarised in a particular direction (normal to the direction of the molecules).
If the axis of the polaroid coincides with the direction of polarisation, all the light goes through. If it is
perpendicular, none of the light goes through. If it is intermediate, with some angle φ between the polaroid
axis and the polatrisation direction, then an intensity cos2 φ goes through and the emerging light is polarised
along the polaroid axis.
1.3 Reflection
We have previously dealt with the reflection and transmission of 1-dimensional waves at a boundary,
and the transmission and reflection coefficients T and R. We also know that in 2 (or 3) dimensions when
light falls on a boundary there is a reflected wave and a refracted wave.
What we don’t know is the equivalent of the T and R coefficients for 2 dimensions. How much of the
light incident at some angle θ is reflected and how much is refracted? That lies outside the scope of this
course, as it uses some E & M that you don’t get till next year. But what we do need to know is that the T (θ)
and R(θ) coefficients depend on the polarisation. The Electric field of the wave has a component parallel
to the surface E|| and a component perpendicular to that, E⊥ . In general the perpendicular component is
reflected less and refracted more.
In particular there is one angle - Brewster’s angle - at which there is no reflected E⊥ . This occurs when
the reflected and refracted rays are at 90◦ to each other. In this situation the molecules in the glass are
dipoles oscillating along the direction where the reflected ray would go, and so they don’t produce any effect.
This is why sunglasses are often made of polaroid. They let through only part of the light - though
any semitransparent glass would do that. But the ‘glare’ of reflected sunlight, off road surface puddles or
Lecture 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polarisation of light and other waves
swimming pools, is predominantly polarised in the horizontal (being parallel to the horizontal puddles and
swimming pools), so glasses made from polaroid admitting only vertically polarised light cut out the glare.
1.4 Rayleigh scattering.
The blue sky is due to scattering of the sunlight, which polarises the upper atmosphere and shakes the
electrons around. (And this happens most strongly at high frequencies/short wavelengths, i.e. blue light).
So when looking at a piece of blue sky, the light will be polarised in a direction normal to the line
between the sun and this particular piece of sky. The fact that the blue sky is polarised, and that the
direction of this polarisation varies across the sky, is readily observed.
2. Circular Polarisation
◦
Light polarised at 45 , described by the polarisation vector
√1
2
1
, has equal H and V components
1
oscillating in phase.
◦
Now
suppose that the V component, instead of being in phase with the H component, is 90 ahead.
i
√1
. This describes an electric field for which, at a particular point, the H and V component are both
2
1
oscillating, but in such a way that the total is constant and the direction describes a clockwise circle. This is
Right-Handed (RH) polarised light. If the V component lags by 90◦ this gives Left Handed (LH) polarised
light, with the E field going round anticlockwise.
If 45◦ plane polarised light is incident on a plate which is birefringent, then the H and V phases, which
were initially equal, are out of step when they emerge from the plate. The extent to which they are out of
step depends on the difference in refractive indices and on the thickness of the plate (and on the wavelength
of the light). If the thickness is chosen such that the difference is π/2 then plane polarised light is converted
into circularly polarised light. This is called a Quarter-wave plate.
Many more complicated molecules, particularly the organic molecules involved in biology, exist as
‘stereoisomers’: two molecules with the same formula but which cannot be superimposed on each other.
Like a left- and right-handed glove. These isomers respond differently to LH and RH polarisations. (They
do this independently - they don’t need to be aligned in a crystal.)
So light can be plane polarised in 2 ways (H and V) and circularly polarised in 2 ways (RH and LH).
But there are always only two possibilities, not four.
Consider plane polarised light passing through a solution containing a stereoisomer (sugar is a typical
example). In the solution it is the circular polarisation which matters, so we say that the light is equal parts
of RH and LH light, exactly in phase.
After going through the solution the light is still equal parts of RH and LH, but they are no longer in
phase as the refractive indices are different. They emerge, and suppose the next thing they meet is a sheet
of polaroid. Now it is the plane polarisation which matters. Adding equal amounts of RH and LH will give
plane polarised light, but the direction will depend on the phase difference between the two. As the light
goes through the solution the plane of polarisation changes - and that can be measured by rotating the angle
of the polaroid sheet and finding the maximum. This is used to measure the amount of sugar in a solution.
3. Polarisation in sound
Polarisation is irrelevant for sound in air, as there’s only one longitudinal direction. But sound can also
travel through solids. And then it has not only longitudinal motion (planes of atoms moving up to the next
plane and getting pushed back by the repulsive core of the intermolecular force) but also transverse motion
(planes of atoms displaced sideways and getting pulled back). The speeds of transverse and longitudinal
sound are different, involving the shear modulus and Young’s modulus, which are similar but not identical.
3.1 Seismology
Sound waves in solids are important in seismology. Earthquakes create sound waves in the earth,
detectable by seismographs and other means. Two signals are received, first from the longitudinal and a
second one from the transverse wave. The difference in time indicates how far away the earthquake happened.
The analysis is complicated because the speed of sound depends on the type of rock, and also on the
depth. As the depth increases so does the pressure and so do both speeds of sound. This produces refraction,
and the sound travels along a curved path, bending away from the vertical on the way down and back towards
it again on the way up.
If the seismograph is far (several thousand miles) from the origin then only one signal - the longitudinal
one - is received. Long paths require the sound to go down to a great depth: something happens there which
means that transverse waves are not supported. This is how we know that the Earth has a liquid core.
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