
Wave optics starts with Huygens
(1629  1695) and Young (1773  1829); see the link for some details. Wave optics proceeds in essentially two steps. First,
the Huygens principle is applied, second interference
between the resulting waves is added. 


Step 1: A wave hitting some edge or just about anything
will produce a circular wave as shown below. The effect is easily visible when looking
at water waves on some relatively undisturbed water surface with obstacles. 


(Plane) waves hitting
an obstacle as shown below thus are detectable even in "shaded" places. Note that the circular wave would go all
the way around the edge; this is not shown for simplicity. The consequences are clear: 
 
Even sharp edges appear always blurred if one looks closely enough. i.e. on a µm
scale. This happens for any plane wave, be it light, radio waves (allowing you to receive the rhythmic noise of your choice
even in the "shade" of, e.g., buildings), or electron waves (in electron microscopes). Just the lengths scales
are different, going with the basic wave lengths of the wave considered. 



The blue lines show the maxima (or minima; whatever you prefer) of the amplitude of a traveling wave at some point in
time t_{0}. The distance between the lines is thus the wavelength l
Somewhat later, the whole system of lines would have moved somewhat to the right (the propagation direction here). The distance
of one wavelength is covered after a time t_{l} = l/v;
v is the propagation speed. The time t_{l} is the inverse frequency
n Thus we have v = n · l
= c for light. 



The consequences are clear: The resolution d_{min}
of a lens with the numerical aperture
NA, i.e. its capability to image two points at a distance d_{min} separately and not
as some blur, is wavelength limited and given by 
 



It's easy to see in a "handwaving" manner why the numerical aperture comes in.
Imagine some lens and reduce its numerical aperture by putting a real opaque aperture with a hole in front of it. The aperture
edges will induce a blur that get's worse the smaller the hole and therefore NA. Your resolution goes down with decreasing
NA. 


On the other hand, your lens aberrations
become worse with increasing NA. The resulting conflict for optimized optical apparatus is clear and encompasses
a lot of intricate and very advanced topics in optics, e.g. how to make structures on microelectronic chips with lateral
dimensions around 30 nm < l with optical
lithography. 

Now we consider step 2: two waves can
interfere with one another. The principle as shown below is clear. 
 



For a phase difference
= 0 we have constructive interference; the amplitudes are doubled.
For a phase difference = 180^{0} = p destructive
interference "cancels" the wave; the amplitude is zero. 


The apparent paradox of how you can get nothing
from something (where are the two single waves and the energy they carry now?) is not trivial to solve; for details see
the link. 

The paradigmatic experiment for showing interference effects is, of course, the double slit experiment. If you consider that for electron waves, and in particular just for
one electron (or photon), you are smack in the middle of quantum theory. 


In the wave
picture the two spherical (or here cylindrical) waves emanating from the
two slits interfere to give the pattern shown below. There is no problem at all. 
 
In the photon
picture, a photon (or electron) passing through the two slits interferes with itself. This boggles the mind quite a bit but the result is the same: You get
the interference pattern as shown, with pronounced minima and maxima of the intensity, which now correspond to the probability of detecting the particle. 




Whenever we look at nontrivial optics, we need to consider interference effects. In the real
world (as opposed to the ideal world shown in the pictures above), we need to consider the fact that our waves are almost
never monochromatic (all have the same wavelength) and coherent
(all have the same phase) plane waves extending into infinity in every direction. 
 
The exception is, of course, the typical Laser
beam, where we have an (almost) monochromatic and (almost) coherent beam. However, a "Laser beam" is typically
"thin" and doesn't extend in all directions. So it is not a simple
plane wave! 

A first important conclusion can be arrived at. 


If we look at an ensemble of waves with the same wavelength, or better: with the same wave vector
k = 2p/l since it contains in addition to the
wavelength also the direction of propagation (that's why it's a vector),
we note that: 
 
An ensemble of sufficiently many waves with the same ±k
and random phases interferes to exactly zero (plus some noise) 



"Random phases" means that all phases are equally
probable. The proof of the theorem is easy: If one of the many waves has a phase a, there
will be some other wave with the phase –a  the two will cancel. A visual proof
constructed in a different context (that should be familiar) but fits just as well here is shown in the link. 

From this you realize immediately that inside some hollow tube of length L that
reflects waves at either end, only waves with l = 2L/m; m = 1, 2, 3,... can exist.



Waves not meeting this criterion will, upon reflection at the end of the tube, produce a phaseshifted
wave with some new phase, twice this phase upon the second reflection and so on. Pretty soon you have waves with random
phases in there and—see above. 


What that also means is: We now have also all musical
instruments covered, in fact everything where the term "resonator" comes up. This
includes also the free electron gas and to some
extent electrons in a periodic potential as well
as the Bragg condition for diffraction
of waves at crystals. The list goes on. Quantum theory deals with wave functions y
after all, and the big difference to classical physics comes from the simple fact that you let your wave functions interfere
before you take the square, producing in terms of probabilities the classical equivalent of a particle. 
  