W9_parallel_resonance.eps

Week 13: Interference and Diffraction 419

If this criterion is satisfied, there is a resolvable dip in intensity in between the

two separate maxima. If the two maxima are any closer, there is just one broad

central maximum and one cannot tell that the images of the two source points or

wavelengths are distinct (that is, one cannot tell that there are two source points

thereat allfrom the image).

• The Diffraction Grating: If one illuminatesNslits with the distance between
  adjacent slitsd(such that allNslits are within the coherence length of the light)
  then different wavelengths in the light source have principle maxima atdifferent
  angles for any given order. This can be used to perform experimental spectroscopy
  and invert the observation as ameasurementof the wavelengths of the light in
  the source. From the discussion ofN-slit interference, we know that the principle
  maxima are brightened by a factor ofN^2 relative to the light from a single slit and
  that these maxima occur at the angle(s) where:

dsin(θ) =mλ

form= 0, 1 , 2 ....

• The resolving powerRof a diffraction grating depends on the order of the maximum.
  In the small angle approximation,

R=mN=

λ

∆λmin

where ∆λminis theminimum separation in wavelength that can be resolved

according to the Rayleigh criterion from the wavelengthlambda, at any given order

m. Inverting this:

λmin=

λ

mN

so that resolution improves (closer wavelengths can be resolved) with both the

number of slits and the order of the maxima being resolved.

• Single Slit Diffraction:The intensity of light of wavelengthλpassing through a
  single slit of widthato strike a distant screen is:

I(θ) =I 0

(

sin(φ/2)

φ/ 2

) 2

where the phase angleφ=kasin(θ) and whereθis, as usual, the angle from the

center of the slit to the point on the screen. The phase angleφcan be thought of

as the phase difference between light from the first Huygens radiator on one side of

the slit and light from the last Huygens radiator on the other side of the slit, the

difference accumulated across thewidthof the slit.

• A simple heuristic (described in the text) can be used to show thatminimaoccur
  in this “diffraction pattern” (the intensity function given above) when:

asin(θ) =mλ

for (note well!)m= 1, 2 , 3 .... Note the omission ofm= 0. This is becauseθ= 0

(corresponding tom= 0) is always the position of thecentral maximumof the

diffraction pattern, with peak intensityI 0.

• In between the minima given at theseexactangles are secondary maxima of strictly
  descending intensity at theapproximateangles: