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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. |
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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. |
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(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: |
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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. |
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The blue lines show the maxima (or minima; whatever you prefer) of the
amplitude of a traveling wave at some point in time
t0. 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 tl =
l/v; v is the propagation speed. The time
tl is the inverse frequency
n
Thus we have v = n · l = c for light. |
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The consequences are clear: The
resolution dmin of a lens with the
numerical aperture
NA, i.e. its capability to image two points at a distance
dmin separately and not as some blur, is wave-length
limited and given by |
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It's easy to see in a "hand-waving"
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. |
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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. |
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Now we consider step 2: two waves can interfere with one another. The principle as shown
below is clear. |
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For a phase
difference = 0 we have constructive interference; the
amplitudes are doubled. For a phase difference = 1800 =
p destructive interference
"cancels" the wave; the amplitude is zero. |
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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. |
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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. |
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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. |
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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. |
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Whenever we look at non-trivial 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 mono-chromatic (all have
the same wavelength) and coherent (all have
the same phase) plane waves extending into infinity in every direction. |
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The exception is, of course, the typical
Laser beam, where we have
an (almost) mono-chromatic 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! |
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A first important conclusion can be arrived
at. |
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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:
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An ensemble of sufficiently many waves with the same
±k
and random phases
interferes to exactly zero (plus some noise) |
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"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. |
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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. |
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Waves not meeting this criterion will, upon
reflection at the end of the tube, produce a phase-shifted 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. |
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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. |
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Standing Waves |
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If we look at simple waves
propagating in just one direction, i.e. at
a one-dimensional problem like sound waves inside an organ pipe, we quickly get
the concept of a standing wave, the
superposition of two plane waves with everything equal except the sign of the
wave vector k. |
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First let's look at the pictures and relations
below; they are only meant to refresh your memory |
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| "Running" Plane Wave |
Standing Wave |
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| E(r,t)
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= E0 · exp{i(kr –
wt)} |
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| Re E |
= E0
· cos{2p/l –
wt} |
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| E(r,t)
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= E0 · exp{i(kr
–wt)} ±
E0 · exp–{i(kr –wt)} |
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| Re E |
= 2E0 · cos(2p/l) · cos(wt) |
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| l |
= 2L/m; m = 1, 2, 3,..; (a
"quantum" number) |
| L |
= resonator
length |
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The pictures and equations are true for acoustic
waves, light waves or electron "waves" - just for any wave. You
should know standing waves from acoustics - it's the base of any musical
instrument, after all, and you hear them
all the time. |
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How about standing light waves? You ever
seen some? |
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No? So put two mirrors at some distance
L in the cm range and admit some light. Are you now going
to see a standing light wave between the
mirrors, as you should expect from all of the above? No you don't - for several
reasons:
- The coherence length
lcoh of normal light is too short. Normal light
is not a infinitely extended plane wave but has some finite "length"
that is far below cm. You're essentially missing the extended waves
between the mirrors that are superimposed and thus you can't have a standing
wave. Organ pipes that are 500 m long don't work either.
- Fine, so let's use coherent light,
however made. OK - you will get standing waves now but you won't notice. The
minimal difference in wavelength between two allowed standing waves is Dl = [2L/m – 2L/(m + 1); for large
m this simplifies to Dl
» 2L/m2. Since l is in the µm region, and L
in the cm region, m is around 10.000 and Dl » 10–4 l. In other words, the allowed wavelengths of the
standing waves are so close to each other that pretty much all light
wavelengths can live inside your resonator. You won't notice a difference to an
arbitrary spectrum. Our lecture room here, even so it is a resonator for
acoustic waves in principle, doesn't appear to produce nice musical tones
because it is simply too large for acoustic wave lengths.
- OK, so let's make a resonator - two mirrors once more - but only separated by a distance of a few µm. Now
you did it. You could have distinct standing light waves in there. But what do
you expect to see now? Think!
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Why do you hear the standing acoustic waves in an organ pipe or
in any (classical) musical instrument? Because some of the wave leaks out,
eventually hitting your ear. The tone (= pressure amplitude inside the pipe)
then would soon be gone if one wouldn't keep feeding acoustic waves into the
resonator (by blowing into the organ pipe, for example). Same here. Some light
must leak out so you can see it. If the leaking (and the feeding light into the
resonator) is done in a certain way, we call the resulting instrument a
Laser. We'll get back to this. |
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The long and short is: Yes, interference effects
and standing waves are of supreme importance for modern optics and you should
refresh you memory about the
basics
of that if necessary. |
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The following picture shows standing light waves
very graphically. We are looking at the unexposed section of a photo resist
from microelectronics. The part that was exposed to light has been etched off.
The ripples on the left-over resist (= light sensitive polymer) correspond to
the extrema of the amplitude of a standing light wave. |
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The principle of what is happening. |
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The surface of the resist and the surface of the
substrate were rather flat and only a few wavelengths apart. When light
(monochromatic and rather coherent) was fed to the system, a standing wave
developed inside the resist. While you
wouldn't have seen anything special, the light intensity "seen" by
the resist varied periodically with depth, and that's why the light-induced
chemical changes that allow to etch out the exposed part, leave
"ripples" on the side walls. |
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If this is all gobbledegook to you, you need to
look up "lithography" within the context of
semiconductor
technologies. |
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© H. Föll (Advanced Materials B, part 1 - script)
