W9_parallel_resonance.eps

Week 13: Interference and Diffraction 435

Note well that the series doesn’t continue indefinitely – the largestmthat con-

tributes is one where:

θmprinciple max= sin−^1

(

mλ

d

)

(1046)

exists, somλ/dhas to be less than or equal to 1. This condition constrains all of

the other series (below) as well, just as it did for 2 or 3 slits.

b) Minima occur when theN-gon formed by the amplitudes closes (forming pentagons

or five pointed stars in theN= 5 case). The anglesδwhere these minima occur

clearly form the series:

δmin=^2 πm

5

m=⊗, 1 , 2 , 3 , 4 ,⊗, 6 , 7 , 8 , 9 ,⊗, ... (1047)

where I’ve⊗’d out the valuesm= 0, 5 , 10 , .... Wehaveto skip those in the series

because e.g. 10π/5 = 2π, and we already know thatδ= 2πis a principlemaximum.

Clearly this generalizes to:

δmin=

2 πm

N for

m=⊗, 1 , 2 , ..., N− 1 ,⊗, N+ 1, N+ 2, ..., 2 N− 1 ,⊗, 2 N+ 1, ...(1048)

where we have to skip everyNth value ofm.

Take a moment and verify that this rules works forN= 2 andN= 3 slits, and

derive the related expression fordsin(θm) and henceθminm forNslits.

c) In between any pair of adjacent, isolated minima, a smooth function must have

a maximum. We therefore expect that in between each adjacent pair of minima

enumerated above, there must be a maximum. The principle maxima have already

been enumerated, but there also exist a whole list ofsecondary maxima. These

occur as the “chain” ofE-field vectors twists around in between closedN-gons,

and occurclose to(but not exactly at) where the (N−1)-gon closes, leaving a

single “dangling”E 0 at the end. If one evaluates the maxima more carefully (using

calculus) one finds that they aren’t exactly at the (N−1)-gon angles, and don’t have

the exact lengthE 0 , but they are allclose tothese angles and lengths and we’ll

consider this to be “good enough” to help us draw a semi-quantitatively correct

graph of the intensity.

This was illustrated in the 5-slit example above as:

δsecondary max=

2 πm

4

=

πm

2

m=⊗, 1 , 2 , 3 ,⊗, 5 , 6 , 7 ,⊗... (1049)

where we note that we again have to skip the values ofmthat would lead to aδ

that is an integer multiple of 2π, and generalizes to:

δsecondary max=

2 πm

N− 1

m=⊗, 1 , 2 , ..., N− 2 ,⊗, N, N+ 1... (1050)

and so on.

These rules are more than sufficient to allow us to draw a qualitatively correct graph

both of the intensity produced by 5 slits and a “generic” graph of “N“ slits (where of

course we have to pick some large but finite number to illustrate).

You might wonder why we are spending so much time looking at interference through

multiple slits, when we hardly ever run into problems involving interference through just

twoslits while shopping at the mall. There are two simple reasons. The firstis that

interference from many closely spaced slits is the basis for thediffraction grating, which

in turn is the basis for modern spectrographs. Spectrographs are optical instruments