W9_parallel_resonance.eps

434 Week 13: Interference and Diffraction

E 0 E 0 E 0 E 0 E 0

Principle Maxima

• 
  
  
  
  • 
    
    
    
    • 
      
      
      
      • = 5E 0
      
      
      
      
    
    
    
    
  
  
  
  

2 π/5

4 π/5

6 π/5

Minima 8 π/5

Secondary Maxima π/2

π

3π/2

Figure 183: Phasor diagrams principle maxima, minima, and secondarymaxima for five slits. The
amplitude of the secondary maxima aren’texactlyE 0 (or equal) and the angles aren’texactlyat
δ= 2π/(N−2) (forN= 5) but this is close enough for an excellent semi-quantitative graphof the
intensities (and our heuristic understanding).

whereδ=kdsin(θ) is the phase angle produced by the path difference between anytwo

adjacent slitsin a set ofNslits.

To see exactly how the results generalize, let’s draw the phasors for one more set of slits,

this one withN= 5, in figure 183. That should be plenty for us to infer a rule and

understand how diffraction gratings (our next subject) and singleslit diffraction (the

one after that) work.

Note the following features, described in terms of thegeneralrules that they represent:

a) Principle maxima have field amplitude ofN E 0 (forN= 5) when the field phasors

“all line up”. They do so whenever the phase angleδis an integer multiple of 2π.

Clearly this result (which held forN= 2 and 3 as well) is general. Thus forallN

we find:

δprinciple max= 2πm m= 0, 1 , 2 , 3 ... (1044)

or:

δprinciple max=kdsin(θ) = 2πm

2 π

λ

dsin(θ) = 2πm

dsin(θ) = mλ (1045)

Principle maxima occur when the light fromallof the slits arrives at the point of

observationin phase, which in turn happens when the path travelled by light from

any two adjacent slits differs by an integer number of wavelengths. This makes

perfect sense.