(Opaque screen / Complementary screens / “Small hole” screen)
This section is about
the diffraction of light by an opaque screen, complementary screens, and “small
hole” screen that are related to Babinet’s principle of diffraction. Another
possible title for this section is “Babinet’s
principle of diffraction.”
1. Opaque screen:
“‘What is an opaque screen?’
Suppose we have a completely opaque screen between a source S and an observer at P, as in Fig. 31–6(a). If the screen is ‘opaque’
there is no field at P (Feynman et al., 1963, p. 31–10).”
According to
Feynman, an opaque screen would have charges moving within the screen due to
an electric field Es and it would generate a new field to exactly cancel the field Es
on the back side of the screen. In addition, the opaque screen has
a large and imaginary index such that the electric field is absorbed
exponentially as it goes deeper into the material of the screen based on the
theory of refractive index. Alternatively, opaque screen can be defined as a
screen that is sufficiently large with a thickness and consists of uniformly
distributed electron-oscillators. It should be good to clarify that this
section is related to Babinet’s
principle of diffraction. This principle shows the equivalence of a hole (or opening)
to an obstacle with respect to the diffraction patterns.
“So if we make the screen
thick enough, there is no residual field, because there is enough opportunity
to finally get the thing quieted down… You know, of course, that a thin
enough sheet of the most opaque material, even gold, is transparent (Feynman et al., 1963, p. 31–10).”
Idealization:
Perhaps Feynman would state Babinet’s principle in terms of a “thick screen”
because he explains that there is no residual electric field if the screen
is thick enough and the electric wave is absorbed exponentially (as it
goes through). However, we can idealize Babinet’s principle by conceptualizing an infinitesimally
thin and perfectly conducting plane screen. In Jackson’s (1998) words, “[a] rigorous statement of Babinet’s principle
for electromagnetic fields can be made for a thin, perfectly conducting plane
screen and its complement (p. 489).” That is, the very thin and perfectly
conducting (metallic) screen would absorb the electric wave such that there is no residual electric field for all
frequencies. Although Feynman adds that gold is
transparent provided it is thin enough, one may clarify that it can reflect red and yellow lights and allow greenish-blue
light to pass through it (Hecht, 2002).
2. Complementary screens:
“We have the result that the field at P when there are holes
in a screen (case b) is the same (except for sign) as the field that
is produced by that part of a complete opaque wall which is located where the holes are! (Feynman et al., 1963, p. 31–11).”
We may define
complementary screens as two screens that
are “complementary” in the sense that the first screen that has transparent
regions can be replaced by the second screen whereby the transparent regions become opaque regions,
or vice versa. In short, “Babinet’s principle states that
the diffracted fields from complementary screens are the negative of each other
(Carcione & Gangi, 1999, p. 1485).” In other words, complementary screen means that we
can replace a hole (or opening) with
an obstacle or replace an obstacle with a hole and still have the same diffraction
pattern. Intuitively, it may not be clear to many (e.g., Poisson) why the Arago spot can be
observed in the shadow of an obstacle. A beauty of Babinet’s principle is that the
Arago spot can be simply explained by the negative electric field, E1 = -E2.
“The field at P is certainly zero in case (c),
but it is also equal to the field from the
source plus the field due to all the motions of the atoms in the walls and in the
plugs. We can write the following equations: Case (b): Eat P
= Es+Ewall, Case (c): E′at P
= 0 = Es+E′wall+E′plug (Feynman et al., 1963, p. 31–11).”
In Landau’s (1971) words, “Let us consider the Fraunhofer
diffraction from two screens which are ‘complementary’: the first screen has holes
where the second is opaque and conversely (p. 155).” That is, a limitation of
Babinet’s principle is that the diffraction patterns of the two complementary
screens are identical provided the condition of Fraunhofer diffraction is met. (In the Audio Recordings* [50 min: 20
sec] of this lecture, Feynman says: “The point is all the fields from the infinity
impinges the opaque screen……,” but this is missing in the edited Feynman’s Lectures.)
Although the Fraunhofer diffraction requirement means the source
should be located at infinity, we may expect a similar diffraction pattern if the
source is relatively far compared to the width of the hole. The assumption
of infinite source distance allows the incident wave to appear like a plane
wave such that all “fields” (or secondary sources) on the same wavefront
near the hole are in phase.
*The
Feynman Lectures Audio Collection: https://www.feynmanlectures.caltech.edu/flptapes.html
3. “Small hole” screen:
“Now if the holes are not too small (say
many wavelengths across), we would not expect the presence of the plugs to change
the fields which arrive at the walls except possibly for a little bit around the
edges of the holes. Neglecting this small effect, we can set Ewall = E′wall
and obtain that Eat P = −E′plug
(Feynman et al., 1963, p. 31–11).”
Feynman briefly mentions that the theory of diffraction is approximately
correct provided the holes are not very small. He explains that the E′plug term will be
small and then the difference between E′wall and Ewall (it is taken to be zero) may be comparable to or larger than the
small E′plug term, and our approximation will no longer be valid. One may
clarify that the approximation is based on the assumption that the interactions
between the electron oscillators around the edges of the holes are negligible.
In the real world, the edge effect is minimal if the screen, hole, or obstacle is
sufficiently thin. This is why physicists idealize an infinitesimally thin and perfectly conducting plane
screen. In short, size matters (i.e., the diffraction pattern is dependent on
the width and thickness of the obstacles or openings).
“We remark again that this theory of diffraction is
only approximate, and will be good only if the holes are not too small (Feynman et al., 1963, p. 31–11).”
In general, one may apply Keller’s (1962) geometrical theory of diffraction
to deduce the edge effects of very small holes. From the perspective of
experiment, it is a challenge to investigate the darker diffraction pattern of a very
small hole because only very
little light passes through the hole. However, it could be apt to end the
chapter with the beauty of Arago spot (Poisson’s spot) or Babinet’s principle. The diffraction pattern
of a hole is almost identical to the diffraction pattern of an obstacle that is
sufficiently thin. For practical
situations, Babinet’s principle is also approximately valid and applicable to curved
screens whose radii of curvature are large compared to the size of the hole
(Jackson, 1998).
Review Questions:
1. How would you idealize an opaque screen (infinitely
thin or perfectly conducting)?
2. How would you state Babinet’s principle
of diffraction (e.g., by including the limitation)?
3. Is Babinet’s principle of diffraction applicable
to very small holes?
The moral of the lesson: Babinet’s principle of diffraction states the equivalence
of a hole (opening) to an obstacle from the perspective of diffraction pattern,
but it is approximately valid depending on the size (or thickness) of the hole and
obstacle.
References:
1. Carcione, J. M., & Gangi, A. F. (1999). Babinet’s principle
for elastic waves: A numerical test. The Journal of the Acoustical Society
of America, 105(3), 1485-1492.
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. Hecht, E. (2002). Optics (4th edition). San Francisco:
Addison Wesley.
4. Jackson, J. D. (1998). Classical
Electrodynamics (3rd ed.). John Wiley & Sons, New York.
5. Keller, J. B. (1962). Geometrical theory of diffraction. Journal of the Optical Society
of America, 52(2), 116-130.
6. Landau, L. D., & Lifshitz, E. M. (1971). The classical theory of fields (Vol. 2). Oxford: Pergamon.
