# Full text of "Scientific Papers - Vi"

## See other formats

f it
1917]   REFLECTION OF LIGHT FROM A REGULARLY STRATIFIED MEDIUM       493                                         {#\$
If'
where ^ is the refractive index, T the thickness, and a! the angle of refraction                                 |Mi
within the plate.   Also k = 27T/X, X being the wave-length,    Adding together                                 l||
the various reflections and summing the infinite geometric series, we find
IIP
_
In like manner for the wave transmitted through the plate we get
»-e-™+...=_C'f_iks>..................(3)
the incident and transmitted waves being reckoned as at A,
The quantities &', c', e', f are not independent. The simplest way to find the relations between them is to trace the consequences of supposing 8 = 0 in (2) and (3). For it is evident a priori that, with a plate of vanishing thickness, there must be a vanishing reflection and an undisturbed total transmission*. Accordingly,
b' + e'-Q,   c'f = l~e/2   ........................(4)
1                      '            »/                                  3                                                                         \    J
the first of which embodies Arago's law of the equality of reflections, as well as the famous " loss of half an undulation." Using these, and substituting ^ for e, we find for the reflected vibration,
_^il^!±)j .................................(5)
and for the transmitted vibration
fr^ps..................................(6)
In dealing with a single plate, we are usually concerned only with intensities, represented by the squares of the moduli of these expressions. Thus,
T         •       f    n       i i- i         o Intensity of reflected light = tf .-— J                     °           (1—
n
Intensity of transmitted light = jir^
the sum of the two expressions being unity, as was to be expected.
According to (7), not only does the reflected light vanish completely when 8-0, but also whenever ^kS^srr, s being an integer;  that is, whenever
Returning to (5) and (6), we may remark that, in supposing k real, we are postulating a transparent plate. The effect of absorption might be included by allowing k to be complex.
*  " Wave Theory of Light," Ency. Brit. Vol. xxiv. 1888; Scientific Papers, Vol. in. p. 64.n the understanding that the ratio of the right-hand side of (12) to that of (11) is zero when n = 0, which is not the case when a absolutely = 0. W. F. S.]