W9_parallel_resonance.eps

418 Week 13: Interference and Diffraction

difference contains a half integral number of wavelengths, so the light arrives at the

screen exactlyout of phase.

By Inspection or Calculus: By inspection, the maxima in the expression for

I(θ) above occur when cos(δ/2) =±1 and the minima occur when cos(δ/2) = 0.

Alternatively, one can differentiate it with respect toδand set the derivative equal

to zero and solve forδfor max’s or min’s that way.

Either path leads one to:

dsin(θ) =mλ Maxima

dsin(θ) = (m+

1

2

)λ Minima

withm= 0,± 1 ,± 2 ,± 3 ....

• N-slit Interference:When there are multiple slits, they will all arrive in-phase at
  the screen when:

δ=kdsin(θ) =

2 π

λ

dsin(θ) =m(2π)

or

dsin(θ) =mλ

form= 0, 1 , 2 .... At theseprinciple intensity maximathe field amplitude isN

times the amplitude of a single slit, so that the intensity is:

I=N^2 I 0

whereI 0 is the intensity produced by a single slit.

• If we use phasors to search for heuristic minima andsecondary maxima, we find
  that we get (zero) minima when the phasors form a closedN-gon. This occurs
  when:

δ=n

2 π

N

for (note well!)n=✁❆ 0 , 1 , 2 , ..N− 1 ,✚✚❩❩N , N+ 1, N+ 2, .. 2 N− 1 ,✟ 2 ❍✟❍N , 2 N+ 1...The

crossed out numbers represent places whereδ is an integer multiple of 2π, but

those are where theprinciple maximaoccur, not another minimum! Secondary

maxima will occurapproximatelyhalf way in between these minima, when:

δ= (n+

1

2

)

2 π

N

for (note well!)n=✁❆ 0 , 1 , 2 , ..N− 1 ,✚✚❩❩N , N+1, N+2, .. 2 N− 1 ,✟ (^2) ❍✟❍N , 2 N+1...Finding
the exact angles for the maxima, however, requires solving a transcendental formula
as there is a small trade off between unwinding the phasors a bit and the resultant
length.

• Rayleigh’s Criterion for Resolution:
  Two (principle) maxima produced by diffraction (or interference using a
  misnamedN-slitdiffraction grating) are considered resolved if the angle
  for the maximum of either one is separated from maximum the other by at
  least the angle to the other’s first minimum.