Week 13: Interference and Diffraction 457
wavelengths of light in the visible band and diminishment of others, constantly varying
as the soap swirls around in the film (and the film thickness changes minutely) and as
the angle of incidence and reflection of the light is varied by perspective.
If you blow a nice, big bubble that just hangs there for a time on a stillday, supported
by the slight buoyancy of the warm air of the breath with which you blew it, you will
probably observe the following, although how successful you are may depend on the
particular mix of soap you are using (some soap mixtures ‘pop’ more quickly than others).
As you watch, the color swirl will settle down and become colored not-quite rainbow
likerings concentric around the vertical axis, and concentrated in the bottom half of
the bubble. You may see several sets of rings at some point. What ishappening is that
the bubble soap is sinking under the influence of gravity and “bulging”the film at the
bottom and thinning it out on top. At the same time, of course, the film is evaporating- getting thinner as the water molecules in the film thermally bounce free.
On the top, a curious thing happens. The film stops exhibiting color atall – it becomes
completely transparent! In fact, as the water evaporates, the entire bubble may become
almost completely invisible, revealed only by a hint of distortion at the outside edge
of the sphere and an almost invisible tracing of lines where the soap is ever so slightly
thicker and holding the bubble together.
This transparency is caused, as noted above, but light reflecting off of thefirstsurface
with a phase shift ofπ(functionally, a half of a wavelength) and reflecting off of the second
surface with no phase shift. Once the film is much thinner than a wavelength, light in all
wavelengths thus recombinesdestructively, largely cancelling the reflected wave. Light
that isn’t reflected is transmitted; hence the soap bubble becomestransparent.
This trick is used to advantage to make advanced optical coatings for e.g. binoculars,
telescopes, microscopes, and other optical instruments. By covering the outer surface
of the primary lens with a thin (<100 nm) coating with ahigherindex of refraction
than the glass, destructive interference in all visible wavelengths isassured, resulting in
a lens thatmaximizes light transmission. High quality coated optics deliver 90+% of the
light that is incident on them to the eye of the observer, which makesa big difference
when compared to expected reflection/transmission intensities for the glass-air interface
alone^138.
(^138) In my online bookClassical Electrodynamics III derive thetransmission coefficient
T=(n^4 n^1 n^2
1 +n 2 )^2
for normal reflection. This is the fraction of intensity thatis transmitted at an interface between two otherwise perfectly
transparent media with differing indices of refraction. We omit discussing transmission and reflection coefficients in
this book because they are too difficult to derive or handwave,arising from solving the boundary value problem on
the surface between the two media.
However, for air (na≈1) and glass (ng≈ 3 /2) the expected transmitted fraction of the intensity from each air-glass
surface (in either direction) is thusT= 0.96. For four surfaces (two lenses), this means that only 85% of the light
makes it through to the eye, less if there are additional reflecting surfaces or lenses in the optical path, less still from
filters or absorption by the glass (which is small but not zero). Coating can increase the transmitted fraction to
0.98-0.99 (per surface) and thus transmit an easy 10% more light.