Astronomy Principles and Practice Fourth Edition.pdf

232 The radiation laws
radiation happens to pass. Consequently, a plane wave electromagnetic disturbance can be expressed
simply by using either the electrical or magnetic disturbances alone. In the optical region, it is the
electric vector which usually plays the dominant role in any interaction of the radiation and matter, and
the equation describing the electric disturbance is normally used to express the form of the disturbance.
Thus, provided that the optical properties of the medium are known, a plane wave electromagnetic
disturbance may be summarized by any equation in the form:
(
E x = E x0 cos ωt − 2πz )
λ
+ δ x .
The strength of the radiation associated with such a wave is proportional to the square of the
amplitude of the electric field disturbance and it is related to the brightness of the source.
At any instant in time, the disturbance passing through a point in space is the resultant of the
radiation originating from atoms which happened to be emitting energy at exactly the same time. This
resultant may be described by considering it to have components vibrating in the xz and yz planes,
with amplitudes and phases which are unlikely to be the same. Thus, the resolved components of the
disturbance may be written as
E x = E x0 cos
(
ωt − 2πz )
λ
+ δ x
and
(
E y = E y0 cos ωt − 2πz )
λ
+ δ y . (15.29)
If it were possible to measure the electric disturbance at a point over one cycle of the wave (this
would correspond to a time approximately equal to 10 −15 s for visible radiation), the resultant electric
vector would sweep out an ellipse. This ellipse is known as a polarization ellipse and for the short
time of the measurement, the radiation can be considered to be elliptically polarized. The geometry
of the polarization ellipse can be investigated by combining equations (15.29) and removing the time
dependence. Thus, the equation describing the ellipse can be written as
( ) 2 ( ) 2 Ex Ey
+ − 2E x E y cos(δ x − δ y )
= sin 2 (δ x − δ y ). (15.30)
E x0 E y0 E x0 E y0
A special form of polarization occurs when the phase difference (δ x −δ y ) between the components
is zero. For this condition, equation (15.30) shows that the resultant oscillates along a line and the
disturbance is known as linear polarization. Another special form occurs when the amplitudes are
equal (E x0 = E y0 ) and the phase difference is equal to π/2. For these conditions, the resultant has
a constant magnitude and rotates at the frequency of the radiation and the disturbance is known as
circular polarization.
In order to describe the general polarization ellipse, three parameters are required. They are the
azimuth (i.e. the angle that the major axis makes with some reference frame), the ellipticity and the
sense of rotation of the electric vector or its handedness. These three parameters, together with the
strength of the radiation (i.e. the size of the ellipse), describe the characteristics of the radiation. The
various polarization forms are depicted in figure 15.12.
The general polarization form described here was considered by measuring the electric vector over
one cycle of the wave. However, during this small period of time, some of the wave-trains emitted by
the collection of atoms will have passed through the point of observation and other wave-trains emitted
by different atoms in the collection will have just started to contribute to the resultant disturbance. If
the atoms are radiating in a random way, the amplitudes and phases of the resolved components will be
changing continuously, thus causing the polarization to change. After a period of time equal to a few
cycles of the wave, the ‘instantaneous’ polarization form bears no relation to the initial ‘instantaneous’
polarization form. If the radiation from a collection of atoms, with no special properties, were to be
analysed over a time period required for normal laboratory experiments, no preferential polarization