Physics for Geologists, Second edition

48 Optics
only the light oscillating normal to the plane of incidence is reflected; the
rest is refracted and partly polarized in the plane of incidence. If the incident
light is polarized in the plane of incidence, none is reflected and the refracted
light is polarized in the plane of incidence. If it is polarized normal to the
plane of incidence, it is partly reflected at all angles of incidence and the
refracted light remains polarized normal to the plane of incidence.
Fresnel's reflection formulae for the proportion of light reflected at various
angles of incidence are, - tan (i - r)/ tan (i +Y) for light polarized in the plane
of incidence, and - sin (i - r)/ sin (i + Y) for light polarized normal to the
plane of incidence. These proportions are in terms of amplitudes of the light.
When the incident light is polarized in the plane of incidence and the angle of
incidence is Brewster's angle (i+r = 90°), no light is reflected. The coefficient
of reflection is the ratio of the intensity of the reflected light to the intensity
of the incident light, and the intensity is proportional to the square of the
amplitude. So the coefficients of reflection are, for light polarized in the plane
of incidence, tan2 (i - r)/ tan2 (i + r), and for light polarized normal to the
plane of incidence, sin2 (i - r)/ sin2 (i + r). For unpolarized light, there is
another formula due to Fresnel:
The coefficient of reflection for normal incident light on a transparent surface
can be derived from these formulae and is given by
where n is the refractive index. The derivation uses the fact that when the
angles i and r or (i - r) and (i + r) are very small, sin2 (i - r)/ sin2 (i + r) =
tan2 (i - r)/ tan2 (i + r) x (i - ~ )~/(i + Y ) = ~ (n - ~ )~/(n + I )~.
The blue sky is partly polarized, most strongly polarized at right angles
to the direction of the Sun. This can be verified by looking at the blue sky
through a polaroidTM filter (or one lens of polaroidTM sunglasses) and rotating
it. The amount of light passing through the lens will vary in different
parts of the sky and with different orientations of the polaroidTM filter. The
Vikings knew this and used the polarizing mineral cordierite for navigation
(even on a cloudy day, the position of the Sun can be estimated to 13", it is
said). Bees can also detect polarized light and use it for navigation.
We cannot distinguish the directions of vibration of light without assistance.
If we pass ordinary light through a crystal of tourmaline that has been
cut parallel to its principal axis, and then let it pass through another similar
crystal, we find that the light emerging from the first crystal is polarized
parallel to its axis. We use the second crystal to analyse the light emerging
from the first, which we call the polarizer.
Copyright 2002 by Richard E. Chapman