What happens when coherent light comes through more than two slits,
all equally spaced a distance d apart, in a line parallel to the
incoming wave fronts? The same criterion
still holds for completely constructive interference
(what we will now refer to as the PRINCIPAL MAXIMA)
we no longer have a simple
criterion for destructive interference: each successive slit's
contribution cancels out that of the adjacent slit,
but if there are an odd number of slits, there is still
one left over and the combined amplitude is not zero.
Does this mean there are no angles where the intensity goes to zero?
Not at all; but it is not quite so simple to locate them.
One way of making this calculation easier to visualize
(albeit in a rather abstract way) is with the geometrical
aid of PHASORS:
A single " PHASOR" of length
(the wave amplitude)
precessing at a frequency
in the complex plane.
A single wave can be expressed as
is the phase
of the wave at a fixed position x at a given time t.
is the "initial" phase at x=0 and t=0.
At this stage it is usually ignored;
I just retained it one last time for completeness.)
If we focus our attention on one particular location in space,
this single wave's "displacement"
at that location
can be represented geometrically as
a vector of length
(the wave amplitude)
in the complex plane called a " PHASOR"
As time passes, the "direction" of the phasor rotates
at an angular frequency
in that abstract plane.
There is not much advantage to this geometrical description
for a single wave
(except perhaps that it engages the right hemisphere of the brain
a little more than the algebraic expression)
but when one goes to "add together" two or more waves
with different phases, it helps a lot!
Two waves of equal amplitude
but different phases
are represented as PHASORS
in the complex plane. Their vector sum has the resultant
and the average phase
For example, two waves of equal amplitude but different phases
can be added together algebraically as for BEATS:
That is, the combined amplitude
can be obtained
by adding the phasors "tip-to-tail" like ordinary vectors.
Like the original components, the whole thing continues to
precess in the complex plane at the common frequency .
We are now ready to use PHASORS to find the amplitude of
an arbitrary number of waves of arbitrary amplitudes and phases
but a common frequency and wavelength
interfering at a given position.
This is illustrated in Fig. 3
for 5 phasors.
The net amplitude of a wave produced by the interference of
an arbitrary number of other waves of the same frequency
of arbitrary amplitudes
can in principle be calculated geometrically by
"tip-to-tail" vector addition of the individual
PHASORS in the complex plane.
In practice, we rarely attempt such an arbitrary calculation,
since it cannot be simplified algebraically.
Instead, we concentrate on simple combinations of waves of
equal amplitude with well defined phase differences, such as
those produced by a regular array of parallel slits with an
equal spacing between adjacent slits.
It will be conceptually helpful to show a geometrical explanation
of the 6-slit interference pattern in Fig. 6
in terms of phasor diagrams, but clearly the smooth curve shown there
is not the result of an infinte number of geometrical constructions.
It comes from an algebraic formula that we can derive for an arbitrary
and a corresponding phase difference
between rays from adjacent slits.
The formula itself is obtained by analysis of a geometrical construction
like that illustrated in Fig. 4 for 7 slits,
each of which contributes a wave of amplitude a,
with a phase difference of
between adjacent slits.
PHASOR DIAGRAM for calculating the intensity pattern
produced by the interference of coherent light
passing through 7 parallel, equally spaced slits.
Blowup of one of the isosceles triangles formed by a single
phasor and two radii from the center of the circumscribed circle
to the tip and tail of the phasor.
After adding all 7 equal-length phasors in Fig. 4
we can draw a vector from the starting point to the tip of
the final phasor. This vector has a length A (the net amplitude)
and makes a chord of the circumscribed circle, intercepting an angle
where in this case N=7.
The radius r of the circumscribed circle is given by
as can be seen from the blowup in Fig. 5;
this can be combined with the analogous
to give the net amplitude
From Eq. (3) we know that
and in general
Although the drawing shows N=7 phasors, this result is valid for
an arbitrary number N of equally spaced and evenly illuminated slits.
shows an example using 6 identical slits with a spacing
The angular width of the interference pattern
from such widely spaced slits is quite narrow, only 10 mrad
(10-2 radians) between principal maxima where all 6
rays are in phase. In between the principal maxima there are
5 minima and 4 secondary maxima; this can be generalized:
We can understand this analytically from examining Eq. (7)
with calculus: the extrema (maxima and minima) of A occur at the
That is, where
which is satisfied "trivially" at the CENTRAL MAXIMUM (
and otherwise where
The "almost trivial" solution is when
is a multiple of
is a multiple of ), making both tangents zero.
This corresponds to the PRINCIPAL MAXIMA where, as expected from
What about the minima? From Fig. 6 we can see that
the first minimum occurs when
that is, when the
phasor diagram closes on itself. For large N this means that
is a rather small angle and we can use
the small-angle approximation
to satisfy Eq. (9).
Combined with Eq. (8) this predicts that the first minimum
occurs at a laboratory angle
For large N and/or
The intensity pattern produced by the interference of coherent light
passing through six parallel slits 100 wavelengths apart.
PHASOR DIAGRAMS are shown for selected angles.
Note that, while the phase angle difference
between rays from adjacent slits is a monotonically increasing
function of the angle
that the rays make with the "forward" direction,
the latter is a real geometrical angle in space while the former is
a pure abstraction in "phase space".
The exact relationship is
for very small .
Note the symmetry about the
mrad the intensity is back up to the
same value it had in the central maximum at
this is called the first PRINCIPAL MAXIMUM.
Then the whole pattern repeats . . . .
Jess H. Brewer