Cosine square law
in Polarised light

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© J.M
Frère


description :

This is a simple (and
cheap) experiment/demonstration showing the cosinesquared
dependence of light transmission throught a pair of polarising filters.
It uses cheap equipment, (total cost approx 60 euro), and is easy to
perform in daylight ( it uses daylight)
The accuracy is good (a few percent)

why?

This is an essential
experiment in physics, and students must see it, either as a lab
experiement, or at least as a demonstration.
Indeed it illustrates:
 The VECTOR nature of light (characterized by the
Electric field E)
 The projection of such vectors in various
directions
 The "square of the amplitude" dependence of the
energy, characteristic of wave mechanics (and later of quantum
mechanics)
 It relates the square of the amplitude in wave
formulation to the electrical energy,
Maybe more importantly,
 It
is a crucial link to serious Quantum Mechanics courses.

link to QM

 the polarisation states,
 the projection operation operated by filters
are used as the standard analogy in the DIRAC formulation of quantum
mechanics, (at the basis of quantum field theory),
see for instance the classical ref :
 Dirac, P. A. M., 1930, The Principles of
Quantum Mechanics, Oxford:
Clarendon Press.

apparatus

 lightmeters are now cheaply available (mine is
imported by Velleman, and costs about 50 euros).
 A pair of linear polarisers can be obtained in
rotating mount from (used) photo shops: they no longer work with
autofocus or light measurement in the latest SLRs (where they are
replaced by circular polarisers) : the price is negligeable
they are optically excellent and rather neutral, but
when crossed, a little blue light can still seep through)
Alternatively, cheap linear polarising filters (plastic) can be
obtained from standard physics suppliers.
 A tube is used to adapt the filters on the
lightmeasuring element: this can be anything opaque (like plastic
drain pipe, heavy cardboard tubes, ...)

some work

using a photographic
filter, the only real work consists in attaching a paper tape around
one of the filters, after graduating it in angles.
The zero setting may be chosen arbitrarily, but as a matter of
elegance, it is convenient to fix it using the Brewster angle (see
below)

Brewster angle

light reflecting from
a nonconducting (dielectric) surface (tradition calls for a piece of
black glass) is polarized. At the Brewster incidence angle, reflected
light cannot be polarised with the electric vector in the incidence
plane (it must thus, in case of a flat surface be both
perpendicular to the direction of the light ray, and parallel to the
surface).
This is easily realised on the optical bench, by placing the polariser
in its rotating mount on the light source, and fitting a piece of black
glass on a goniometer.
It is however very easily demonstrated in the class room by looking at
a set of windows from a nearby building (in the horizontal plane). One
window will likely be close to the Brewster angle, and rotating the
filter will kill the reflected light: at this moment, the filter is
only allowing the light polarised in the incidence plane. (the Brewster
angle for glass is about 56 degrees from the normal incidence).

mode of operation

 in principle, the measurement can be made using
only available daylight, inside a room. The lightmeter has a highest
sensitivity scale reaching 199.9 lux. (with scales up to 50 000)
My first measurements in a
northfacing room in fall gave a max of 50 lux, and a background
(polarisers in crossed position) of 1.8 lux;
 but clearly, a desk lamp, possibly with a coloured (not blue)
filter can help.
 stack the 2 filters on top of the tube, and the
tube snuggly on the lightmeter. the filter with a scale could be on the
bottom
 rotating the other filter, check the lightmeter
for the minimum of light; when this is reached, put a mark on the
scaleless filter in front of the 90 degrees notch on the other
 rotate back the top filter to 0 degrees, and
check that you indeed have a maximum
 measure the amount of light for various angles
(for instance, with a spacing of 15 degrees)
 put an opaque screen on top of the assembly, this
allows to measure the background (light leakage), which can be
substacted

further measurements

 in daylight conditions : the same apparatus can
be used to check 1/2 and 1/4 wavelength plates (check that the
polarisation is reversed, or circular, after carefully adjusting the
plates orientation)
 alternatively, precision measurements can be made
in a darkened room, using the optical bench. Since come light is needed
to read the lightmeter, it may still be advisable to use the tube
setup.
 obviously, narrowfiltered light, or
monochromatic (Na) light would be preferred, and it may also be
preferable to adapt the sensor to make it more punctual and directional
(remove the white ambiance filter, at the cost of loosing the absolute
scale)

results

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fullscale graphs





data handling

It is obviously
tempting to use a spreadsheed to plot the results.
However, care should be taken if an automatic line fitting (least
squares) is used. Many programs indeed assume equal errors (and often
neglect the error on x in xy plots). This is certainly not the case
here: the error on the light measurement is typically a fraction of the
measurement + a constant, while, if the cosine squared is used as a
variable, the error on this quantity is proportional to 2 * sin* cos*
Delta(theta) . The error is thus maximal for 45 degrees, and vanishes
(at first order) for 0Â° and 90Â° , which are both
local extrema of cos
squared in terms of the angle.
A precise chisquared treatment is of course possible (see
http://homepages.vub.ac.be/~frere/ , under the heading general physics)
To avoid entering these more elaborate discussions for this elementary
measurement, and more specifically in the framework of a demonstration,
I suggest to present:
 the raw curve, with the angular dependence, and
no substraction of the background (that is: light measured vs
polarisers angle)
 a cossquared vs lightmeasured curve, after
substraction of the parasitic light remaining when polarisers are
crossed (assuming thus it is fully due to leaks) : compare the
intermediary points to the "stable" ones, corresponding to 0 and 90
degrees, by which a straight line is fitted
Obviously, for a really quantitative, opticalbenched
based experiment, a suitable data handling (like the proper chisquared
mentioned above) should be used, while a "blind" use of the fitting routine of
standard spreadsheets could be very misleading in this case.

METER accuracy

the meter (VELLEMAN
DVM1300) is a cheap model, and its accuracy is listed as 5% of reading
+ 10 LSD. This applies to the absolute reading,
the linearity is probably much better, and will be checked in separate
experiment testing the 1 over R squared behaviour.
