W9_parallel_resonance.eps

Week 13: Interference and Diffraction 459

Joe Braggart claims to have really, really good vision. “Why,” he says. “My vision is

sogood I can make out the Galilean moons of Jupiter with my naked eyes on a really

clear night. If I’d been around at the time of Galileo we wouldn’t have had to invent the

telescope in order to confirm the Copernican theory.”

Callisto is the moon with the largest orbit and has a maximum distance from Jupiter of

just under 2× 106 kilometers. At its closest point to the earth, it is around 600× 106

kilometers away. Assuming that he is using visible light, is there a chance that he’s

telling the truth? Note well: This is a problem on resolution, not lenses or the sensitivity

of the retina, so the determine whether or not Jupiter and its moonareresolvedby the

human eye at this distance.

Problem 7.

Derive the intensity as a function ofθfor the single slit problem. Fora= 3λ, find

the angles where the intensity is a minimum. Sketch the diffraction pattern fromθ∈

[−π/ 2 , π/2]. If you prefer, you can solve for the sines of the angles and sketch the

diffraction pattern from sin(theta)∈[− 1 ,1] instead.

Advanced Problem 8.

From your algebraic answer to the previous problem, obtain an expression for the angles

where diffractionmaximaoccur. You might find the following useful:

d f^2

dx

= 2f

df

dx

which has zerosbothwheref= 0 (the minima, except for the one atθ= 0)andwhere

df

dx= 0 independently. Also recall from the footnote in the text above that:

xlim→ 0 sin(x)

x

= 1

and hence is not “undefined”.

Advanced Problem 9.

Derive the expressionR=mN=∆λλfor resolution for a diffraction grating withNslits

of separationd. This proceeds as follows: First use a phasor diagram to determine the

angle(s) where theprinciplemaxima occur. Then use it to find the angles where thefirst

minimumfollowing such a maximum occurs for any given orderm. This tells you the

angular half-widthof the maximum for a givenλ. Use Raleigh’s criterion for resolution

to determine the minimum ∆λthat can be resolved (considerλ′=λ+ ∆λ), and verify

the expression above.