494
ON THE REFLECTION OF LIGHT
[422
When we pass from a single plate to consider the operation of a number of plates of equal -thicknesses and separated by equal intervals, the question of phase assumes importance. It is convenient to refer the vibrations to points such as 0, 0', bisecting the intervals between the plates; see figure, where for simplicity the incidence is regarded as perpendicular. When we
0
o'
I
B
reckon the incident and reflected waves from 0 instead of A, we must introduce the additional factor e~^iks>, 8' for the interval corresponding to 8 for the plate. Thus (5) becomes
~ fTTT^Js— = r.........'................(9)
So also if we reckon the transmitted wave at $', instead of A, we must introduce the factor e~^lS+&'}, and (6) becomes
The introduction of the new exponential factors does not interfere with
the moduli, so that still
rM + |tf2 =1...............................(11)
Further, we see that
r/yi i I —~. <0"~M*0 \ V'J'M Q1Y1 -i-/"/S
11 \ JL & I Łjvll OJLJLJ. fYl\j\J /~t C\\
j, ~~ /-i 2\ /-,__ii'&S "~" ~~" 1 2 ? ...........*......v"^^*1/
and thus (in the case of transparency) r/t is a pure imaginary. In accordance with (11) and (12) it is permitted to write
T = sin 6. ei?, t = ico$6. e^,.....................(13)
in which 6 and p are real and
/I ^1 SlU 7? A/O /t A\
tan c/ = —-------—............................\*ty
Also from (9), (13)
where s is an integer and
, 7?3 sin AS /.„>
tanv = ~-^—------=-5..........................(16)
A I-7)*coskS ^
The calculation for a set of equal and equidistant plates may follow the lines of Stokes5 work for a pile of plates, where intensities were alone regarded*. * Roy. Soc. Proc. 1862; Math, and Phys. Papers, Vol. iv. p. 145. supposing k real, we are postulating a transparent plate. The effect of absorption might be included by allowing k to be complex.