Page 141 - Fundamentals of Light Microscopy and Electronic Imaging

124 PROPERTIES OF POLARIZED LIGHT

The amount of light transmitted through two polars crossed at various angles can be
calculated using Malus’law, which states that
2
I I cos ,
o
where I is the intensity of light passed by an analyzer, I is the intensity of an incident
o
beam of linearly polarized light, and is the angle between the azimuth of polarization
of the incident light and the transmission axis of the analyzer. Thus, for two polars
2
crossed at a 90° angle, cos 0, and I 0, so polarized light produced by the first fil-
ter (the polarizer) is completely excluded by the second filter (the analyzer). For polar-
izers partially crossed at 10° and 45°, the light transmitted by the analyzer is reduced by
97% and 50%, respectively.

DOUBLE REFRACTION IN CRYSTALS

Many transparent crystals and minerals such as quartz, calcite, rutile, tourmaline, and
others are optically anisotropic and exhibit a property known as double refraction. When
letters on a printed page are viewed through a calcite crystal, remarkably each letter
appears double (Figure 8-6). Calcite is therefore said to be doubly refracting, or birefrin-
gent. Birefringent materials split an incident ray into two components that traverse dif-
ferent paths through the crystal and emerge as two separate rays. This occurs because
atoms in crystals are ordered in a precise geometric arrangement causing direction-
dependent differences in the refractive index. In contrast, a sheet of glass, such as a
microscope slide, which is an amorphous mixture of silicates, is usually optically
isotropic and does not exhibit double refraction.
When a ray of light is incident on a birefringent crystal, it usually becomes split into
two rays that follow separate paths. One ray, the ordinary ray or O ray, observes the laws
of normal refraction, while the other ray, the extraordinary ray or E ray, travels along a
different path. Thus, for every ray entering the crystal there is a pair of O and E rays that

Figure 8-6
Double refraction in a calcite crystal. A letter viewed through a plane surface of the crystal
appears double, the two images corresponding to the ordinary and extraordinary rays. As the
crystal is rotated, the E ray rotates around the O ray. The O ray obeys the normal laws of
refraction and does not rotate.