*Homemade micrometer for measuring the wavelength of light using shadow fringes.*

In
the spring of 1970 I was playing around with a convex lens at night. For some idle
reason I pointed the lens toward the window and looked through it at the street
lamps in the distance. Of course, if you butt your eye up against a convex lens
and look at a distant lamp it becomes a blurry blob of light. But what caught
my attention was that although I was about 6 feet from the window pane I could
see silhouettes of the marks and imperfections on the window not only in astonishing detail but they were also surrounded with very clear fringes of
wave interference. I was in the sixth form at the time and in physics classes
we had done the usual Young’s slits experiment with its well-known striped interference
pattern. As is well known Young’s slits is an experiment which has tested our
concept of reality sorely and consequently it has been the cause of much angst
in physics. But this experiment requires quite a sophisticated laboratory set
up and yet here quite accidentally I was confronted with a similar observation. I
jotted down some notes (with a fountain pen!) explaining what I thought I was
seeing and these notes can be seen in the scan below:

What’s
happening here is that when the relaxed eye is looking through the lens it is
seeing a magnified image an inch of two in front of the lens; this is the point
where you would place an object if you want to examine it using the lens. But
because in this case there was nothing at the lens’ focal point what I was actually
focusing on were the shadows formed by the marks on the window. I could see the
edges of those shadows in sharp outline albeit fringed by interference
patterns. A bonus was that the
fringes increased in size the further I got from the window. But this magnification
effect was offset somewhat by the fact that the street lamps, although distant,
weren’t true point sources of light and so the further away I got from the
window the more the shadows and the fringes started to blur as umbra effects set
in (Conceivably this effect could be used as method to measure the angular size
of the light source).

An
elementary version of the wave theory behind this effect, which I was
working toward in my jottings, can be worked out as follows:

In
the above diagram the vertical blue line represents the incoming wave front
from the approximate point source. This
wave front meets an obstacle shown in green. The shadow cast by the obstacle is
shown in black on the right of the diagram. If light motion was described purely
by geometric mechanics then for a point source the shadow would be precisely
sharp. But of course light motion is described using wave mechanics and this
explains the interference fringes. In the model above we imagine the top of the
obstacle (which I have labelled as “A”) to become the source of a reflected secondary
wave. When this secondary wave is combined with the incoming plane wave it produces
an interference pattern in the region of B.

At
point B the reflected wave has traveled a distance AB whereas the plane wave
front has travelled a distance equal to AC.
Now, let as put AB =

*h*and AC =*d*and BC =*x.*If the wavelength of light is l then for the reflected wave and the plane wave to form a node of complete cancellation this requires a difference in their travel distance of half a wavelength. That is, for a dark node to be seen at B we require:
1.0

…where

*n*is an integer.
Therefore:

2.0

Since

*d*>>*x*we can use the binomial approximation and 2.0 becomes:
3.0

After
some manipulation this becomes:

4.0

Note
that

*n*increase much faster than**and hence the convergent appearance of the fringe lines surrounding an obstacle. For***x**n*= 1
5.0

Some
years after working out this theory, in 1976 to be precise, I built the micrometer
shown in the picture at the head of this
article. I used this micrometer to measure the distance to the first dark node for
the shadows cast by an approximate point source. In this experiment I set

*d*= 18cm and measured*x*to be 0.029cms. Hence, using equation 5.0 I measured the wavelength of light to be about 4.7 × 10^{-7}m. As I only had a white light source this value is a ball-park figure and this ball-park is the window of the visible spectrum which ranges from 4.00 × 10^{−7}to 7.00 × 10^{−7}m.
***

This
experiment, based as it is in theory that goes back to Huygens, is pretty passé.
But even so there are several profound features here worthy of note:

1. .The micrometer I was using was a homemade
affair which employed a standard 60 threads/inch screw. As I was attempting to measure to the nearest
1/10 of a turn this means I was trying
to measure to 1/600

^{th }of an inch. That works out at 4.2 x 10^{-5}m. And yet the wavelength I was measuring was about 100 times smaller than that. So, this home-brewed experiment was effectively measuring distances of at least ½ millionth of a metre. This accuracy is achieved because in equation 4.0 we see that*d*effectively magnifies the value of*x*, although only by a factor proportional to the square root of*d*.
2. All interference patterns of this
kind pose the well-known conundrum of quantum mechanics: Viz: If the point source was emitting one
photon at a time we would still get the interference pattern when integrated over long enough time, implying that
even a single photon is associated with an extensive wave field and therefore
sensitive to the material geometric layout across a volume of space; photons
are not localised point particles until the wave front “collapses”.

3. There is a peculiar (and profound) feature of
the theoretical model I used to derive equation 5.0. In this model we imagined the
plane wave and the reflected secondary wave to

*travel independently*to point B. But if this is so then to get a predictable interference pattern the precision with which the waves fill the space between the obstacle and the shadow is nothing short of mind-boggling. For let’s say there was a small percentage random variation in the lengths of each of the waves between obstacle and shadow. Given the very high number of waves between these two points then this variation, if modeled as a version of random walk, would likely build up between obstacle and shadow. This would mean that the synchronisation of wave effects required for equation 1.0 to work would be disrupted and either a spurious variable value of*x*observed or the interference pattern too blurred to be seen. Presumably the length*d*could be many miles and yet the interference pattern still observed implying that each wave has the same length to a very high precision indeed. The observed wave fringes may be an indication that space itself is made up of discrete numbers of nodes with exact numbers of these nodes supplying the accuracy needed for the interference phenomena.