W9_parallel_resonance.eps

432 Week 13: Interference and Diffraction

2 π

3

4 π

3

4 π

3

2 π

3

π

π

δ =

Principle maxima a^000

E E E

E 0 E 0

E 0

E 0

E 0 E 0

E 0

Secondary maxima

Minima

Figure 182: Phasor diagrams illustrating principle maxima, minima, and secondary maxima in the
interference pattern. Note that we get minima when the three phasorscloseto get a three-sided
polygon or 3-gon (a.k.a. an equilateral triangle in this case). In between the minima we get maxima,
but the secondary maxima are much weaker than the principle maximathat occur when all three
slits arrive in phase becausedsin(θ) =mλ.

If we divide by 2πand multiply byλ, we see that this corresponds to:

dsin(θ) =±mλ (1037)

just as beforefor two slits separated byd, so that the angles for principle maxima are:

θprinciple maxm =±mλ

d

(1038)

This isimportant:The location of the principle maxima ofNslits is determined

by the slit separationd, not byN!The two signs just mean that the pattern obtained

is symmetric, with maxima at the same angles above and below the horizontalθ= 0

line. We will (from now on) ignore this and just present positivemand find positiveθ’s,

and remember that the intensity pattern is symmetric for negativeθ.

Now let’s consider the minima. We immediately note that the intensity cannot be nega-

tive – I don’t know what a negative intensity wouldmeanfor light – the Poynting vector

can have a sign relative to some coordinate frame, but the intensityis just the absolute

power per unit area that flows past any given point in space. The smallest it can possibly

be is zero.

For this problem it willbezero when the phasors for the fieldadd upto zero, which, given

three equal field strengths, occurs when the phasors form aclosed, three sided figure, that

is, aunilateral triangle^121. The two triangles in the figure above thus represent phase

angles that lead tominima.

We observe that we close these triangles when:

δmin=

2 π

3

or

4 π

3

(1039)

(^121) To begin to get ready for the next topic, you might want to think about a unilateral triangle as a3-gon, a polygon
with three sides.