W9_parallel_resonance.eps

Week 13: Interference and Diffraction 449

1.0

0

−1.0 sinθ

4I

Figure 191: The graph of combined diffraction and interference, fora= 4λ(same as before) and
d= 8λ.

13.10: Diffraction Through Circular Apertures – Lim-

itations on Optical Instruments

Finally we are ready to understand how the use of waves with a finite (non-zero) wave-

length affects things like vision and optical instrumentation. To start with, I have to

give you a “true fact” concerning diffraction through acircular aperture of radiusD–

something thatcanbe derived but thatI won’t derive just now in this work for you.

It’s not that the derivation is incredibly difficult or exotic – it proceedsmore or less

along the lines we’ve just used for single slit diffraction – it just is easiest to obtain using

integration (which we avoided) and complex variables instead of phasors per se (which

we have also mostly avoided).

In a nutshell, to obtain the result one has to do an integration in a sensible coordinate

system (e.g. cylindrical coordinates) that sums up the differentialelectric field radiated

from every point on the “disk” of Huygens radiators in the circular aperture, including

their phase difference due to the path difference to an arbitrary point on the screen a

distanceZaway from the center of the aperture. To some people^129 this sounds like

a really good time, but I’m guessing that formoststudents using this text it sounds

like a still better time tonotactually do it and hence you’re inclined to forgive me for

presenting something you actually have to just memorize/learn.

That true fact is this. The diffraction pattern produced on the screen by a circular

aperture is itself a cylindrically symmetric “circle” of light, surrounded by alternating,

ever fainter, rings of darkness (where destructive interference causes the total wave to

cancel) and light (where partially constructive interference causes the total wave to peak,

although never at the intensity seen in the central maximum). In fact, thegenericshape

of the diffraction pattern is much the same as that for a slit, only it is cylindrically

symmetric instead of itself being a slit shaped bar with alternating bars of light and dark

on the side. In this diffraction pattern thefirst minimum(the dark ring surrounding the

bright(est) central maximum occurs at the angle given by:

Dsin(θmin) = 1. 22 λ (1087)

Note that this isalmostlike the rule for the slit,asin(θmin) =λ, except that we no longer

get a pretty integer on the right and on the left we have thediameterof the aperture,

not its short-direction width. It certainly makes dimensional sense.

Now consider viewing very distant, point-like objects through a circular aperture. I

prefer to think of viewing stars, for example, as they are very distant indeed and appear

to the eye as mere points of light in the sky, through the aperture of your pupil, or the

(^129) Mostly physics or math majors or other mathochists, granted...