
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. 


