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 *con*structive interference
(what we will now refer to as the PRINCIPAL MAXIMA)
but
we no longer have a simple
criterion for *de*structive 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 wave can be expressed as where is the

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!

For example, two waves of equal amplitude but different phases can be added together algebraically as for BEATS:

where

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.

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
angle
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.

After adding all 7 equal-length phasors in Fig. 4
"tip-to-tail",
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

as can be seen from the blowup in Fig. 5; this can be combined with the analogous

to give the net amplitude

(6) |

From Eq. (3) we know that , and in general , so

where

Although the drawing shows *N*=7 phasors, this result is valid for
an arbitrary number *N* of equally spaced and evenly illuminated slits.

Figure 6
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
physical angles
where
.
That is, where

which is satisfied "trivially" at the CENTRAL MAXIMUM ( ) and otherwise where

The "almost trivial" solution is when is a multiple of (or is a multiple of ), making both tangents zero. This corresponds to the PRINCIPAL MAXIMA where, as expected from Eq. (8),

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

For large

2002-03-26