Concept 07 / 07Wave Optics
Polarization: Malus's Law, Brewster's Angle and Polarization by Scattering
Polarization proves light is a transverse wave: a polaroid halves unpolarized light, Malus's law gives I = I0 cos-squared theta, and reflection at tan(theta) = mu produces completely polarized light.
Interference and diffraction establish that light is a wave, but they work just as well for sound, so they cannot tell us whether the oscillation is along the direction of travel or across it. Polarization settles that question: light is a transverse wave, with (and ) perpendicular to the direction of propagation. Longitudinal waves such as sound can never be polarized, because there is nothing to orient.
Unpolarized, polarized and partially polarized light. An ordinary source is a jumble of independent atomic emitters, each radiating a short wave train with its own orientation of in the plane perpendicular to the ray. Averaged over even a microsecond, all orientations occur equally often: this is unpolarized (natural) light. If stays along one fixed direction, the light is plane (linearly) polarized; the plane containing and the ray is called the plane of vibration. If one direction is favoured but not exclusively, the light is partially polarized, and its quality is measured by the degree of polarization obtained by rotating an analyser. In diagrams, dots on a ray mean perpendicular to the page and short bars mean in the plane of the page, as in the diagram above.
Polaroids and the factor of one half. A polaroid is a sheet of long aligned molecules that absorbs the component of along the molecular chains and transmits the perpendicular component; the transmitted direction is the transmission axis. Send unpolarized light of intensity into it. A wave whose field makes an angle with the axis contributes , and since all are equally likely we must average: . So the emerging intensity is , whatever the orientation of the polaroid, and the emerging light is plane polarized along the axis. This is why rotating a single polaroid in front of an ordinary lamp produces no change in brightness at all, a fact worth remembering because it is the standard test for unpolarized light.
Malus's law. Now let already-polarized light of intensity , with its field along a direction making an angle with the transmission axis of a second polaroid (the analyser), fall on it. Resolve the amplitude into a component along the axis, which is transmitted, and perpendicular to it, which is absorbed. Since ,
.
That is Malus's law. Two important corollaries: with crossed polaroids () the output is zero, and for unpolarized light entering a polarizer-analyser pair the final intensity is . The half applies only once, at the first polaroid; after that the light is already polarized and Malus's law takes over.
Identifying a beam with a rotating analyser. Rotate a polaroid through a full turn in the beam. If the intensity drops to zero twice, the light is plane polarized. If it varies between a maximum and a non-zero minimum, it is partially polarized. If it does not vary at all, the light is either unpolarized or circularly polarized, and those two are distinguished by first passing it through a quarter-wave plate, which converts circular into plane polarized light but leaves unpolarized light unpolarized.
Polarization by reflection and Brewster's angle. When unpolarized light strikes a transparent surface, the reflected beam is always partially polarized, with the component perpendicular to the plane of incidence reflected more strongly. At one particular angle of incidence the reflected beam becomes plane polarized perpendicular to the plane of incidence. Brewster found experimentally that at this angle the reflected and refracted rays are exactly perpendicular to each other, and that fact gives the law immediately.
Let the polarizing angle be and the refraction angle . Since the reflected ray also leaves at and the reflected and refracted rays are perpendicular, the three angles at the point of incidence satisfy , so
.
Substituting into Snell's law, , hence
: Brewster's law.
For crown glass gives ; for water gives . Since depends on wavelength, so does , slightly.
Why perpendicularity produces complete polarization is worth understanding rather than memorising. The reflected wave is not "bounced off" the surface; it is radiated by the electrons in the medium that the refracted wave sets oscillating. Those electrons oscillate along of the refracted wave, and an oscillating dipole radiates nothing along its own axis. For the component polarized in the plane of incidence, the electrons oscillate in a direction that, when the reflected and refracted rays are perpendicular, points exactly along the reflected ray. That component therefore cannot be radiated into the reflected beam at all, leaving only the perpendicular component: the reflected light is completely polarized perpendicular to the plane of incidence.
A neat consequence: if light enters a glass slab at the Brewster angle, it meets the lower surface at exactly the Brewster angle for the glass-to-air transition, since . Both faces therefore polarize, which is the basis of a pile-of-plates polarizer. Note also that the transmitted beam at is only partially polarized, never completely: the surface removes some of one component, not all of the other.
Polarization by scattering. When sunlight passes through the atmosphere, molecules absorb and re-radiate it. The scattered field must be perpendicular to the scattered ray, so if you look at right angles to the incident beam, only the component of the molecular oscillation perpendicular to both directions can radiate towards you, and the light you receive is completely plane polarized. Scattered light at from the sun is therefore strongly polarized, as a polaroid held up to a patch of clear blue sky shows dramatically. The same Rayleigh scattering, with its dependence, scatters blue about times more strongly than red, making the sky blue and the setting sun red.
Practical uses. Glare reflected from a horizontal surface such as a road or a lake is polarized horizontally, so polaroid sunglasses are made with a vertical transmission axis to block it. Photographic polarizing filters darken skies and cut reflections. Stress patterns in transparent models show up in polarized light (photoelasticity), liquid crystal displays switch polarized light on and off, and sugar solutions rotate the plane of polarization in proportion to concentration (optical activity), which is how sugar content is measured commercially.
Key Takeaways and Formulae:
Polarization proves light is transverse; unpolarized light contains all orientations of equally.
Unpolarized light through one polaroid: , output plane polarized, independent of the polaroid's orientation.
Malus's law for already polarized light: , measured between the vibration direction and the transmission axis.
Brewster: , with ; reflected light is completely polarized perpendicular to the plane of incidence; refracted light is only partially polarized.
Brewster angle from denser to rarer is the complement of the one from rarer to denser.
Degree of polarization .
Scattered light at to the beam is completely plane polarized; Rayleigh scattering goes as .
Common mistake: applying the factor more than once, or applying Malus's law to unpolarized light.
Common mistake: saying the reflected light at is polarized in the plane of incidence. It is polarized perpendicular to it.
Brain Teaser 1: Unpolarized light of intensity falls on two crossed polaroids, so that no light emerges. A third polaroid is now inserted between them with its axis at to the first. Show that light reappears, find its intensity, and find the angle of the middle polaroid that maximises the output.
Solution: Step 1. Unpolarized light through the first polaroid: , polarized along the first axis.
Step 2. Through the middle polaroid at : by Malus's law, , and, crucially, the light emerging is now polarized along the middle polaroid's axis, not the first one's.
Step 3. Through the third polaroid, which is at to the first and therefore at to the middle one: .
So inserting an extra absorbing sheet increases the transmitted light from zero to . The paradox dissolves once you see that a polaroid does not merely filter, it re-orients: the middle sheet rotates the plane of vibration part way round, so the light arriving at the last sheet is no longer perpendicular to its axis.
General angle: with the middle polaroid at to the first, the output is .
This is maximum when , that is , giving the maximum possible output , confirming the first calculation. It vanishes at and , as it must, since then the middle sheet is parallel to one of the crossed pair and changes nothing.
Brain Teaser 2: Light reflected from the surface of a still lake is found to be completely plane polarized. Take . (a) Find the angle of incidence and the angle of refraction. (b) How high is the sun above the horizon? (c) Should the transmission axis of polaroid sunglasses be vertical or horizontal, and why?
Solution: (a) By Brewster's law, , so .
Since the reflected and refracted rays are perpendicular at the Brewster angle, . Check with Snell: . Correct.
(b) The angle of incidence is measured from the normal, which is vertical for a horizontal water surface. The sunlight arrives at from the vertical, hence at from the horizontal. The sun is therefore about above the horizon.
(c) The reflected light is polarized perpendicular to the plane of incidence. The plane of incidence here is vertical (it contains the vertical normal and the incoming ray), so "perpendicular to it" means the electric field of the glare oscillates horizontally. To block the glare, the sunglasses must absorb horizontal vibrations, which means their transmission axis must be vertical. That is exactly how polaroid sunglasses are made, and it is why tilting your head sideways while wearing them makes the glare from a wet road reappear.
Brain Teaser 3: A beam of partially polarized light is examined with a rotating analyser. The transmitted intensity varies between a maximum of and a minimum of . Find the degree of polarization, and find what fraction of the incident power is carried by the polarized part of the beam.
Solution: Model the beam as a completely unpolarized part of intensity plus a completely plane polarized part of intensity , so the total incident intensity is .
The unpolarized part always contributes through the analyser, whatever its orientation. The polarized part contributes , which runs from down to as the analyser turns. Therefore
and .
From the second equation ; substituting into the first, .
Total intensity , so the polarized fraction is , that is of the beam is polarized.
Degree of polarization: .
The two numbers agree, and that is not a coincidence: substituting the general expressions gives identically, so for this simple model the measured degree of polarization is precisely the polarized fraction of the power. Two useful limits: gives (fully plane polarized), and gives (unpolarized).
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