Week 13: Interference and Diffraction 451
light source in the sky. Is that blob the image ofoneobject, ortwo? That is, is the source
made up of the light fromtwoobjects (e.g. stars) or is it a slightly asymmetric single
object (e.g. a lenticular galaxy)? Time to return toRayleigh’s Criterion for Resolution!We can easily compute the capability of our telescope to resolve two objects that have a
very small angle in between them using this criterion. Basically, if the peak produced by
one object (center of the illuminated area on the film or charge-coupled device (CCD)^131
is separated from the other by at least the angle of the first diffraction minimum of the
other, we can consider the two objects marginally resolved. This criterion depends on
wavelength, and we intuitively expect our resolution to be better with e.g. blue or violet
light than with red light^132
The critical angle – which is certain to be avery small anglefor any macroscopic aper-
ture and optical frequency light – defining the diffraction resolutionlimit of an optical
instrument is thus:
θc≈sin(θc) =^1.^22 λ
D(1088)
Two stars with an angular separation greater than this critical angle will be clearly
resolved on the film (assuming that the image is otherwise focussed on the film or CCD).
The same is true for two tiny features inside a bacteria or almost anytwo source objects
imaged through a circular aperture. The central rays from object to image must be
separated by more than 1. 22 λ/Dor the two images will blur into one.Imaging nearlyanythinggets dicey when the objects themselves are the order of a wave-
length in size or smaller. If you have ever seen water waves striking apier support that is
much smaller than a wavelength you know that they swirl right around it and recombine
on the far side. A short distance away from the pier there is little signin the shape of
the wavefronts that there was a pier there at all. In order to reflect a wave or obstruct
a wave, an object needs to be (ideally much) bigger than the wavelength of the wave.Practically speaking, it is very difficult to create viewable images of objects much smaller
than a half a micron using visible light. Bacteria are thus visible througha visible light
microscope, butstructuresin or on the bacteria are not. Only the largest of viruses are
visible with visible light.To see objects smaller than the wavelength of visible light, one needsa wave with a
smaller wavelength. Electron microscopes use electron “waves” tosee objects as small
as 5 nm – small enough to see most viruses in considerable (beautiful)detail^133We can see that physicians and physicists alike need to have a fairly clear idea of the role
that waves play in the formation of the magnified images that permit us to see the very
small or the very far away. It is quite easy to build microscopes and telescopes for which
diffraction,wave interferenceand things likechromatic distortionare the limiting factors
that prevent us from being able to see further, smaller, better. Even if you will never
actively design a microscope or telescope, understanding their limitations will make you
a better consumer of the information that they can provide.(^131) Wikipedia: http://www.wikipedia.org/wiki/Charge Coupled Device. A CCD is basically the “electronic film”
used in digital cameras, consisting of a fine-mesh grid of photosensitive electrical units
(^132) This same intuition has driven the invention of e.g. “blue ray” DVD formats that hold more information. Blue
light has roughly half the wavelength of red light, so one canstore roughly 4x as much information at the diffraction
limit of resolution of blue light on disks compared to red. DVDs based on hard ultraviolet (λ∼ 100 −200 nm) would
hold a factor of 4 to 16 more data, and I’m quite certain that the minute I finish buying lots of blue-based movies
UV DVD will be trotted out to replace it all yet again, this time on tiny DVDs...
(^133) Wikipedia: http://www.wikipedia.org/wiki/Virus. This article has some lovely transmission electron micrographs
of viruses, revealing detail that would be completely invisible to the eye even with the aid of a powerful visible light
microscope.