(Single surface reflection / Thin-film interference
/ Three-dimensional grating)
The three
interesting concepts discussed in this section are a reflection of a light wave
at a surface of a material, thin-film interference (due to front reflection and back reflection), and three-dimensional grating (reflection at
atoms of crystals).
1. Single
surface reflection:
“…when a light wave hits a surface of a material with an index n, let us say at normal incidence, some of the light is reflected. The
reason for the reflection we are not in a position to
understand right now; we shall discuss it later (Feynman et al., 1963, p. 30–7).”
Feynman says that we are not
in a position to understand the reflection of a light wave right now. In Volume
II, he adds that “the amplitude of a surface reflection is not a property of
the material, as is the index of refraction. It is a
‘surface property,’ one that depends precisely on how the surface is made
(Feynman et al., 1964, Chapter 33 Reflection from surfaces).” In short, the
amount of light reflected by the surface is dependent on the smoothness of the
surface or the arrangement of atoms in an object. Essentially, the
free electrons of the atoms oscillate in response to the incident light waves
and they may cause the reflected light waves to be either strong or weak. In
other words, the reflection at the boundary between two media is a process of
scattering and interference of electromagnetic waves.
“But there are a
number of other examples, and even though we do not understand the fundamental
mechanism yet, we will someday, and we can understand even now how the
interference occurs (Feynman et al., 1963, p. 30–7).”
Feynman mentions that we do not understand the
fundamental mechanism of reflection and we can understand how the interference
occurs. Interestingly, physicists have argued whether two photons can be said to
interfere with each other (Glauber, 1995). More important, some may prefer this
explanation of reflection: “[w]hen I talk about the
partial reflection of light by glass, I am going to pretend that the light is
reflected by only the surface of the glass. In reality, a piece of glass is a
terrible monster of complexity - huge numbers of electrons are jiggling about.
When a photon comes down, it interacts with electrons throughout the glass, not
just on the surface. The photon and electrons do some kind of dance, the net
result of which is the same as if the photon hit only the surface (Feynman,
1985, p. 16-17).” That is, the fundamental mechanism of reflection can be
explained by light waves or photons.
2. Thin-film interference:
“Then, if we look at the
reflection of a light source in a thin film, we see the sum of two
waves; if the thicknesses are small enough, these two waves will produce an
interference, either constructive or destructive, depending on the signs of the
phases (Feynman et
al., 1963, p. 30–7).”
According to Feynman, if we look at the reflection
of a light source in a thin film, we see an interference pattern that depends
on the signs of the phases, provided the thickness of the thin film is small
enough. On the other hand, there is also a strong reflection even if the “thin-film”
is not small enough (e.g., single crystal x-ray diffraction). To be specific,
one may clarify that the interference is observable provided the thickness of
the thin film is of the order of about ¼ to 10 wavelengths of visible light. In addition, this is an interference of reflected waves at the
front surface and back surface of the thin film. Thus, we may define thin-film interference as an interference of light
waves that occurs when light interacts with the front and back surface of a thin film of material.
“So we see colors
when we look at thin films and the colors change if we look at
different angles, because we can appreciate that the timings are different
at different angles. Feynman et al., 1963, p. 30–8).”
Feynman explains that we can see colors change at thin films because we can appreciate that the timings are different at
different angles. Some may be surprised that his explanation is in terms of
different timings instead of path difference, and he did not provide a formula.
However, the principle can be based on the optical path difference due
to the front reflection and back reflection of a thin film (d = nl/4 for constructive inference). In a lecture on QED,
Feynman (1985) elaborates that “[t]he ‘front reflection’ arrow
is drawn opposite to that of the stopwatch hand when it stops turning… The ‘back
reflection’ arrow is drawn in the same direction as the stopwatch hand (pp.
28-29).” One may add that the front reflection is drawn opposite to the
stopwatch hand because of a phase shift of 180 degrees when light waves move
from a low refractive index medium to a high refractive index medium.
3.
Three-dimensional grating:
“This principle is used
to discover the positions of the atoms in a crystal. The only complication is that a crystal is three-dimensional;
it is a repeating three-dimensional array of atoms (Feynman et
al., 1963, p. 30–8).”
Feynman discusses the principle for determining the
positions of atoms: Based on the difference in intensity of the various images,
we could find out the shape of the grating scratches, whether the grating was
made of wires, sawtooth notches, and so on. Currently, some physicists prefer
to use the term condition and state Bragg’s two conditions (or Bragg’s
law) and Laue’s condition. Bragg’s first condition is about the regular
reflection of x-rays whereby the angle of incidence equals to angle of
scattering, whereas the second condition requires the path
difference between two scattered waves equals to an integer number of
wavelengths (2d sin q = nl). On
the other hand, Laue’s condition is a relation of an incident wave and
scattering wave from the crystal to the reciprocal lattice vector.
According to Feynman, we must use radiation of a very
short wavelength, i.e., x-rays, whose wavelength is less than the space between
the atoms such that there are diffraction patterns. In a sense, this is not
correct because we can use electrons and neutrons instead of x-rays.
Furthermore, one should elaborate that glass is an amorphous (non-crystalline)
solid in which the atoms are not in regular arrangement (definite lattice
pattern). More important, the symmetry (e.g., fourfold symmetry) in the diffraction
pattern corresponds to the symmetrical axis or periodicity of atoms. However,
the object need not be a crystal, e.g., the “cross shape” diffraction pattern indicates
a helical arrangement of DNA.
As a suggestion, you may
want to read his lecture on QED: “I can’t resist telling you about a
grating that Nature has made: salt crystals are sodium and chlorine atoms
packed in a regular pattern. Their alternating pattern, like our grooved
surface, acts like a grating when light of the right color (X-rays, in this
case) shines on it. By finding the specific locations where a detector picks up
a lot of this special reflection (called diffraction), one can determine
exactly how far apart the grooves are, and thus how far apart the atoms are
(see Fig. 28). It is a beautiful way of determining the structure of all kinds
of crystals as well as confirming that X-rays are the same thing as light. Such
experiments were first done in 1914. It is very exciting to see, in detail, for
the first time how the atoms are packed together in different substances…
(Feynman, 1985, pp. 48-49).”
Review Questions:
1. How would you explain the reflection of light
waves at the boundary between two
media?
2.
Would you explain that colors change at a thin
film if we look at different angles because the timings are different at
different angles (or path differences)?
3. How would you state the principle for determining the
positions of atoms in a crystal?
The moral of the lesson: thin-film interference and
the diffraction of light in a crystal are related to the reflection of light
waves at the surfaces of a material or atoms in the crystal.
References:
1. Feynman, R. P. (1985). QED: The strange theory of light and matter.
Princeton: Princeton University Press.
2. Feynman, R. P., Leighton, R. B., &
Sands, M. (1963). The Feynman
Lectures on Physics, Vol I: Mainly mechanics, radiation, and
heat. Reading, MA: Addison-Wesley.
3. Glauber, R. J. (1995). Dirac’s Famous Dictum on Interference: One Photon or Two?. American Journal of Physics, 63(1), 12.
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