Download Chapter 8: Polarization • Introduction – Light is a transverse

Chapter 8: Polarization
• Introduction
– Light is a transverse electromagnetic wave.
– Polarization is when the electric field component of an EM wave is
constrained to lie in a certain direction..
– Considered linearly polarized or plane-polarized light: light for which
the orientation of the electric field is constant (though its magnitude
and sign may vary in time).
– What happens when two light waves combine?
• Linear Polarization
– Consider two orthogonal optical disturbances as
E x (z, t) = îE0x cos(kz − ωt),
E y (z, t) = ĵE0y cos(kz − ωt + ).
E(z, t) = E y (z, t) + E x (z, t).
– If ≥ 0, Ey leads Ex . If ≤ 0, Ey lags Ex .
– = 0 or an integral multiple of 2π, then
E(z, t) = (îE0x + ĵE0y )cos(kz − ωt).
– This is linearly polarized. Can resolve any plane polarized wave into
two orthogonal components.
– is an odd integer multiple of ±π, then
E(z, t) = (îE0x − ĵE0y )cos(kz − ωt),
that is linearly polarized but in a different plane to before.
• Circular Polarization
– Assume both constituent waves have equal amplitudes: E0x = E0y =
E0 .
– If = −π/2 + 2mπ, m = 0, ±1, ±2, ..., then
E x (z, t) = îE0 cos(kz − ωt),
E y (z, t) = ĵE0 sin(kz − ωt),
and
E(z, t) = E0 [îcos(kz − ωt) + ĵsin(kz − ωt)].
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– Amplitude of E is constant, E0 , but direction of E is time-varying
and is rotating clockwise at an angular frequency of ω: right circularly
polarized light.
– E vector makes one complete rotation as the wave advances one wavelength.
– If = π/2 + 2mπ, m = 0, ±1, ±2, then
E = E0 [îcos(kz − ωt) − ĵsin(kz − ωt)],
that is E rotates counter-clockwise: left circuularly polarized.
• Elliptical Polarization: E vector rotates and changes its magnitude and
thus traces out an ellipse.
• State of Polarization: Linearly or Plane Polarize: P state, Right or Left
circular polarization: R or L. Elliptical is E.
• Natural light is randomly polarized
• Polarizers: Things which polarize light. Natural light incident on a ideal
linear polarizer - only light in a P state transmitted. The transmitted
P state will have an orientation parallel to the transmission axis of the
linear polarizer. Only the component of the optical field parallel to the
transmission axis of the polarizer will be transmitted.
• Malus’ Law
– Natural light through an ideal polarizer will only allow a P state field
through: this P state field has an orientation parallel to the transmission axis of the polarizer.
– Assume this transmission axis is at an angle θ to the vertical.
– Now put an another linear polarizer whose transmission axis is vertical
and make the beam go through this.
– If the amplitude of the electric field transmitted by the first polarizer
is E01 , only its component, E01 cosθ parallel to the transmission axis
of the second polarizer will be transmitted.
– The irradiance reaching the detector is then given by
I(θ) =
c0 2
E cos2 θ.
2 01
2
– The maximum irradiance is I(0) = c0 E01
/2 and thus, Malus’ Law
states that
Iθ = I(0)cos2 (θ).
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• Dichroism: is the selective absorption of one of the two orthogonal P state
components of an incident beam.
• Wire-Grid Polarizer
– Grid of Parallel conducting wires, fig. 8.12, p. 333.
– Imagine an unpolarized EM wave impinging on the grid from the right.
Resolve this into two components, one parallel and one perpendicular
to the grid. The y-component of the electric field drives the grid
electrons into oscillation along the length of the grid. These oscillating
electrons loose energy through collisions and re-radiate forwards and
backwards. Thus very little of the electric field parallel to the grid
wires is transmitted.
– The x-component of the incident E field tries to oscillate the electrons
in the x-direction but these electrons cannot do this a lot.
– Hence the component of the E field perpendicular to the grid wires is
transmitted.
– Dichroic crystals achieve similar effects through the regular structure
of their crystal lattices - there is some inherent anisotropy in these
structures which leads to anisotropic restoring forces.
– Polaroids contain a molecular analogue of the wire grid.
• Birefringence
– Crystalline substances sometimes can have some anisotropy in the way
the atoms are packed: Binding forces may be stronger/weaker in a
certain direction than other directions.
– Light propogates through a substance by exciting atoms within the
medium. The electrons are driven by the E field and they re-radiate.
These secondary wavelets recombine, and the resultant refracted wave
moves on.
– The speed of the wave, and therefore the index of refraction, depends
on the difference between the frequency of the E field and the natural
frequency of the atoms.
– An anisotropy in the binding force will be manifest in an anisotropy
of the refractive index.
– A material which displays two different indices of refraction in two
different directions is birefringent.
• Scattering and Polarization
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– Examples of scattering - blue sky, exhaled smoke appears white but
smoke rising from a lighted cigarette is blue.
– A linearly polarized incident on an air molecule. The orientation of
the electric field E s follows a dipole pattern such that E s , the Poynting
vector, S and the oscillating dipole are all coplanar.
– The vibrations of induced in the atom are parallel to the E field of
the incoming light wave and so are perpendicular to the direction of
propogation. The dipole does not radiate in the direction of its axis.
– The scattered light in the forward direction is completely unpolarized;
off that axis it is partially polarized, becoming increasingly polarized
as the angle increases. In the normal to the primary beam, the light
is completely linearly polarized.
• Polarization by Reflection
– Reflection from dielectric media can polarize light.
– Consider an incoming plane wave linearly polarized so that its E is
perpendicular to the plane of incidence.
– The wave is refracted at the interface, entering the medium at some
transmission angle θt . The electric dipoles are made to vibrate by the
incident beam. The dipoles oscillate in the direction of the polarization
of the transmitted or refracted light.
– The oscillating dipoles radiate creates the reflected light. Dipoles do
not radiate any energy in the direction they oscillate along.
– Thus if θr + θt = 90, then the dipoles are oscillating along the direction
of what would have been the reflected ray.
– This cant be so there is no reflected ray.
– The particular angle of incidence for which this occurs is Brewster’s
angle.
– From Snell’s law:
ni sinθp = nt sinθt ,
and because θt = 90 − θp ,
tanθp = nt /ni .
– This is Brewster’s angle.
– Thus an incoming, unpolarized wave made up of two incoherent orthogonal P states, only the component polarized normal to the incident
plane and therefore parallel to the surface will be reflected.
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• Stokes Parameters
– Stokes Parameters are functions only of the observables of an EM wave
and describe the polarization state of the wave.
– 4 parameters. I: measure of th epower of polarized radiation. Q, U are
measures of linear polarization. V is a measure of circular polarization.
– Measurements of the polarization of the CMBR can be used to estimate some cosmological model parameters.
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