Using the two diagrams on the right, we will derive the intensity pattern for a narrow slit. The upper diagram shows a slit of width a a distance d from a screen, with d>>a. We have divided in the slit into n segment (3 in the figure) and drawn a ray from the top of each segment. The lower diagram shows that the ray from each segment travels a distance dy = a sin q further than the ray from the next higher segment. Hence the field at the screen coming from the n-th seqment is given by
En = (E0 / N) e i w t + 2 p i [a n sin q / (N l)]
where the top segment is numbered n = 0. The factor of 1/N is included to ensure that after we sum over all N contributions, the wave at q = 0 will be a properly-normalized plane wave E = E0 e i w t .
Define a = p a sin q / l . The total field at the screen will then be
E = Sn=0N-1
(E0 / N)
e 2 i n a / N
+ i w t
We can actually do the sum, using
(1 - x) Sn=0N-1 xn = 1 - xN .
to get
E = [E0 / N ] ei w t [1 - e2 i a ] / [1 - e2 i a / N]
Factoring ei a
from the numerator and ei a / N from the denominator gives
E = [E0 / N ] [ei w t + i a (1 - 1/N) ] [e - i a - e i a ] / [e - i a / N - ei a / N]
= [E0 / N ] [ei w t + i a (1 - 1/N) ] [ sin a / sin ( a / N )]
Since N is a large number, sin (a / N ) = a / N
so that E = [E0 / N ] [ei w t + i a (1 - 1/N) ] [ N sin a / a ]
Re (E) = E0 cos [w t + a (1 - 1/N)] [ sin a / a ]
The intensity is the square of the coefficient of the cosine, so
I = I0 [ sin2 a / a2 ]