The Grating
This is a large number of slits, equally
spaced, distance d from center to center of adjacent slits. Then if we shine
light normally on the grating toward a screen a distance D away, dsinq = ml is valid for
the maxima when D is huge compared to d.
If the above is not clear, you need to go to
the double slit
first. The reasoning is the same. The rest of what we learn about the grating
is by using phasors, which are rotating vectors.
If we analyze the behavior of a small number
of slits, we will know how to handle a large number. Consider 6 slits. The
phasor diagram for the central maximum is 
Oops, that's five. Oh well…. Central max is zero order.
So if Eo
is the electric field amplitude of the wave from one slit, the amplitude at the
central maximum is 6Eo. For N slits, NEo. Then if we move a little away from
center, the phasor diagram could be
and
the amplitude is a little smaller. The angle f is the phase angle between
adjacent rays, and we find since one traveled a distance dsinq farther
than the other, f in radians is
2p
dsinq /l .
(This is because dsinq /l is the fractional part of
the wavelength, and one l is 2p radians.) Move a little farther and we come to
f
= 2p /6 (2p /N in general), and we find the first zero: Equate
this to
2p
dsinq /l , and
we have
dsinq = l /N for the first zero. Keep increasing f , and you get a minor
relative maximum, then another zero when
f
= 2(2p /N): 
(It looks like half the number of vectors, but they are piled on top of each
other.)
People are generally not interested in these
minor maxima and zeros, just the first zero. But it is worth looking into if
you want to understand the single slit. These minor ups and downs are
essentially the same as the single slit pattern. The pattern continues until f = 2p , where we have a maximum
equal to the central maximum, and we call this the first order maximum. The m is 1 in dsinq = ml .
Then the pattern repeats until m = 2, 3,… until q gets to 90o (sinq = 1), so
the maximum m is d/l (or the largest whole number below d/l ). If d<l, we would just have the little ups and downs beyond
the central maximum: the single slit pattern.
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Comments, questions: fredrick.gram at tri-c.edu (replace at with @)