W9_parallel_resonance.eps

430 Week 13: Interference and Diffraction

or the actual anglesθwhere:

dsin(θ) =±mλ (1032)

The intensity is 4I 0 at the maxima.

The minima and maxima occur at precisely the angles that agree with our heuristic rule

from above. We heuristically expect a constructive interference maximum when the path

differencedsin(θ) contains an integer number of wavelengths, and this is exactly what

we get. We heuristically expect a minimum of light from the lower slit travels half a

wavelength farther than light from the upper one, or three half wavelengths farther, or

five half wavelengths farther, and that’s exactly what we get. It’salways nice when our

intuitive, heuristic expectations are confirmed by the actual algebra of the solution. It

gives us confidence that the latter is correct.

13.3: Interference from Three Narrow Slits

d

d

D

P

λ

r + d sinθ

r + 2d sinθ

r

2d sinθ

d sinθ

θ

θ

θ

θ

c

Figure 180: Three narrow slits, equally spaced a distanced > λapart, are illuminated by monochro-
matic light that is coherent over distances long with respect to bothdandλto produce an inter-
ference pattern on a distant screen. Note well that the path difference between any adjacent pair of
slits isdsin(θ).

In the case of three narrow slits, each separated by the same distanced(illustrated in

figure 180, we can follow a more or less identical procedure to find the overall amplitude

from a phasor diagram and square it to find the intensity on the screen in terms of the

intensity produced by a single slit. We can alsobeginthe process of identifying general

rules for finding the angle and amplitude (at least approximately) of important features

of the interference pattern produced, rules that will work for four, five, or indefinitely

many slits. As before we will assume that Fraunhofer conditions hold: the screen is “far”

(compared todandλ) from the slits, and either we will confine our attentions only to

angles that are near the center of the screen or we will consider the screen to “wrap

around” the slits in the shape of a cylinder so that it is all an equal distance from the

central slit^119.

(^119) Not that we couldn’t explicitly include the effect ofr’s gross variation with angle, especially if we programmed