Section 30–6 Diffraction by opaque screens

(Effective light sources / Cornu’s spiral / Geometrical shadow edge)

 

The three interesting concepts discussed in this section are effective light sources, Cornu’s spiral, and geometrical shadow edge.

 

1. Effective light sources:

“Of course, actually there are no sources at the holes, in fact that is the only place that there are certainly no sources… If we use the theorem that we have not yet proved, then we can replace the actual problem by a set of effective sources uniformly distributed over the open space beyond the object (Feynman et al., 1963, p. 30–8).”

 

Some may enjoy how Feynman explained the diffraction of light waves at an opaque sheet with holes in it: We have assumed that there are sources distributed with uniform density across the open holes, but there are actually no sources at the holes. However, there is a gap in his explanation because he has not proved the theorem that justifies why we can replace the problem with a set of effective light sources uniformly distributed over the open space. For example, one may state the theorem as Huygens-Fresnel Principle and then explain that every point on a wavefront is the source of spherical wavelets... In short, the idealized light sources on the wavefront do not exist as physical objects, but they are useful mathematical objects. Perhaps Feynman did not like this principle partly because he wrote that “[a]ctually Huygens’ principle is not correct in optics… (Feynman, 1942, p. 91)” in his PhD thesis.

 

Nevertheless, we get the correct diffraction patterns by considering the holes to be the only places that there are sources; that is a rather peculiar fact. We shall explain later why this is true, but for now let us just suppose that it is (Feynman et al., 1963, p. 30–8).

 

Feynman says that we can get the correct diffraction patterns by considering the holes to be the only places where there are sources, but he has planned to explain later why this is true. Note that he does not provide the explanation for the diffraction patterns in this chapter. In section 31–6 Diffraction of light by a screen, he mentions: “[w]e 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).” Furthermore, he remarks that this theory of diffraction is only approximate, and it is valid only if the holes are not too small. In the real world, there are real sources, for example, a beam of radiation incident on an atom causes the electrons in the atom to oscillate and thus the electrons can radiate in various directions (Feynman et al, 1963, Chapter 32).

 

2. Cornu’s spiral:

“To construct that curve involves slightly advanced mathematics, but we can always construct it by actually drawing the arrows and measuring the angles. In any case, we get the marvelous curve (called Cornu’s spiral) shown in Fig. 30–8. (Feynman et al., 1963, p. 30–9).”

 

According to Feynman, Fig. 30–8 shows a marvelous curve that is called Cornu’s spiral. Strictly speaking, the figure does not really show a curve but a series of arrows representing the addition of amplitudes for many in-phase oscillators or antennas. Simply phrased, the arrows having the same length means that the idealized antennas (instead of a continuous line source) have the same electric field strength and are equally spaced. The slightly advanced mathematics is essentially Fresnel integrals that is also known as Euler’s identity: ò₀^u e^i(p/2)u² du = ò₀^u cos (p/2)u² du + iò₀^u sin (p/2)u² du. In other words, Cornu’s spiral is a continuous curve in the complex plane of the points Z = C(z) + iS(z) in which C(z) and S(z) are Fresnel Integrals.

 

A property of Cornu’s spiral is its curvature at any point is linearly proportional to its arc length (distance along the spiral) from the origin. The curve spirals towards a point relatively quickly because the diffraction pattern is mainly due to a small region of effective sources. In essence, the intensity of diffraction pattern is mainly contributed by light rays of the shortest and shorter paths. We can find similar spirals for phenomena including reflection and refraction. Similarly, in his lecture on QED, Feynman (1985) wrote: “[b]elow the graph is the direction of each arrow, and at the bottom is the result of adding all the arrows. It is evident that the major contribution to the final arrow’s length is made by arrows E though I, whose directions are nearly the same because the time of their paths is nearly the same (p. 43).”

 

3. Geometrical shadow edge:

“The intensity near the edge of a shadow. The geometrical shadow edge is at x₀ (Feynman et al., 1963, p. 30–9).”

 

Perhaps Feynman could have clarified the meaning of geometrical shadow edge or geometrical shadow. For instance, the geometrical shadow may be explained as the idealized shadow that would have been seen, assuming there are no diffraction effects. That is, the diffraction of light due to a semi-infinite opaque screen causes the edge of geometrical shadow to be fuzzy and thus physicists define the geometrical shadow edge at x₀. Better still, the geometrical shadow edge could be defined as the boundary between the illuminated region and shadow region. Additionally, the intensity at x₀ is ¼I₀ whereby I₀ is the unobstructed intensity if there is no semi-infinite opaque screen. Perhaps Fig. 30–9 could be modified as shown below. 

Some may ponder whether it is appropriate to say “the intensity is ¼ of the incident light.” This is related to the explanation “actually there are no sources at the holes, in fact that is the only place that there are certainly no sources,” but it is good to stress that I₀ is not the maximum intensity. Theoretically, the intensity at the geometrical shadow edge (x₀) is ¼I₀ implies that I₀ would be the intensity of a plane wave that is parallel to the semi-infinite opaque screen. In other words, the intensity of the incident light is I₀ at all points on the same wavefront and the intensity at the screen or anywhere would remain I₀ if there is no obstacle. Alternatively, we can deduce the intensity at x₀ to be ¼I₀ without using Cornu’s spiral by simply explaining the amplitude is halved because of the semi-infinite opaque screen. However, the plane wave does not exist in the real world and “actually there are no sources at the holes.” (It should also be a plane wave because "we have light coming in from infinity" as mentioned by Feynman.) 

 

Review Questions:

1. Would you explain that there are effective sources or no sources at the holes?

2. Would you say that Fig. 30–8 shows a marvelous curve that is called Cornu’s spiral?

3. How would you define the geometrical shadow edge?

 

The moral of the lesson: The intensity of diffraction pattern is dependent mainly on light rays that traveled by the shortest path to the screen followed by those traveled by slightly shorter paths.

 

References:

1. Feynman, R. P. (1942/2005). Feynman’s thesis: A New Approach to Quantum Theory. Singapore: World Scientific.

2. Feynman, R. P. (1985). QED: The strange theory of light and matter. Princeton: Princeton University Press.

3. 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.