We have previously considered the problem of the superposition of two scalar waves, and these results will again be applicable here. However, light is a vector phenomenon; both the electric and magnetic field are vector quantities. Understanding this added level of complexity is crucial to understanding many optical phenomena.
Starting with the principle of superposition, the electric field intensity at a particular point in space is generated by the various fields, E1, E2, ..., of the constituent sources,
(13.1)
For the sake of simplicity, consider two point
sources, S1 and S2, emitting monochromatic
waves of the same frequency in a homogenous medium. Let their separation
a be much greater than 
.
Locate the point of observation, P, far enough away from the sources
so that at P the wavefronts will be planes. For now, consider only
linearly polarized waves of the form
(13.2)
and
.
(13.3)
The irradiance at P is given by
.
(13.4)
Recognizing that 
is the time average of the square of the magnitude of the electric field
intensity, we see that
.
(13.5)
The last term is known as the interference
term. For the waves described by (13.2) and (13.3), this can be evaluated
as follows. First, consider the effect of 
:
.
(13.6)
Recall that the time average of a function f(t), taken over an interval T, is
.The period 
of a harmonic function is 
;
for this problem T >> .
After multiplying out and averaging equation (13.6) we have
,
(13.7)
where we used the fact that 
, 
,
and 
.
The interference term is then
,
(13.8)
where 
is the phase difference. It comes from the combined path length
and the initial phase angle difference.
.
In this case, equation (13.8) reduces to
.
(13.9)
Using the fact that
and
,this can be rewritten as
,
(13.10)
so that the total irradiance becomes
.
(13.11)
This reaches a maximum value of
(13.12)
when 
,
where 
.
In this case the distributions are said to be in phase. This is
known as total constructive interference. When 0 < cos 
< 1, the waves are out of phase, I1 + I2
< I < Imax, and the condition is known as
constructive interference. At 
,
cos
= 0, the optical disturbances are said to be 90
out of phase. For 0 > cos
> -1 we get destructive interference, I1 + I2
> I > Imin. The minimum occurs when
(13.13)
when 
,
where .
This is known as total destructive interference.
Another special case is when the amplitudes of both waves are equal. In this case the irradiances from both sources are equal, so let I1 = I2 = I0. Equation (13.11) can then be written as
(13.14)
from which it follows that Imin = 0 and Imax = 4I0.
In a similar way, if we observe the light wave
from a fixed point in space, we see that it appears to be fairly sinusoidal
for some number of oscillations between abrupt phase changes. The corresponding
spatial extent over which the light wave oscillates in a regular, predictable
way has already been identified as the coherence length. If we view
the light beam as a progression of well defined sinusoidal wavegroups of
average length 
xC,
whose phases are uncorrelated to one another, then we find that normal
coherence lengths range from several millimeters for standard laboratory
discharge tubes up to tens of kilometers for some lasers.
Two ordinary sources will normally maintain a
constant relative phase for a time no greater than tC,
so the interference pattern that they produce will randomly shift around
in space at an extremely rapid rate, averaging out and making it impractical
to observe. Until the advent of the laser, it was generally accepted that
no two individual sources would ever produce an observable interference
pattern. The coherence time of lasers, however, is long enough so that
interference of two independent lasers has been detected electronically.
If two beams are to interfere to produce a stable pattern, they must have nearly the same frequency. A significant frequency difference would result in a rapidly varying time dependent phase difference, which would cause I12 to average out to zero during the detection interval. If the sources both emit white light, the component reds will interfere with the reds, and the blues with the blues. A great many overlapping monochromatic patterns will be produces which combine to create a total white light pattern. This final pattern will not be as sharp or extensive as a monochromatic pattern, but white light will produce observable interference.
Assume that the rays pass through a lens (not shown in the diagram), and focus at points P' on the left side of the plate and at P on the right side of the plate. The optical fields at P are given by
(13.15)
where 
is the incident wave.
The terms 
are the contributions to the phase arising from the optical path length
difference between adjacent rays, 
.
There is an additional phase contribution arising from the optical distance
traversed in reaching point P, but this is common to each ray and
thus can be ignored. The resultant reflected scalar wave is then
.
(13.16)
If 
,
and if the number of terms in the series approaches infinity, the series
converges to
.
(13.17)
The reflected flux density at P is
.
(13.18)
The amplitude of the transmitted waves are given by
.
(13.19)
These can be added to yield
.
(13.20)
In turn, this leads to the irradiance of the transmitted beam being given by
.
(13.21)
Using the identity 
,
and setting 
,
(13.18) and (13.21) can be rewritten as
and
or, defining the coefficient of finesse as
,
(13.22)
then
(13.23)
and
.
(13.24)
The term 
is known as the Airy function, and is denoted A(
).
It represents the transmitted flux density distribution. The complementary
function, [1 - A()]
represents the reflected flux density distribution.
Last updated: July 24, 1997
Comments to: D-Suson@tamuk.edu