Section 30–2 The diffraction grating

(Diffraction grating / Reflection grating / Transmission grating)

 

In this section, Feynman discusses the principle of diffraction grating and specifically, reflection grating as well as transmission grating.

 

1. Diffraction grating:

“In one of its forms, a diffraction grating consists of nothing but a plane glass sheet, transparent and colorless, with scratches on it. (Feynman et al., 1963, p. 30–4).”

 

Feynman says that a diffraction grating may consist of a plane glass sheet that is transparent and colorless, with scratches (reduce light transmission) on it. There are often several hundred scratches within a millimeter that are carefully arranged so as to be equally spaced. However, we can define a diffraction grating as an optical device consisting of a system of narrow slits or grooves, which by “diffracting light” to result in an “interference pattern” (due to interfering light beams). This device is able to split electromagnetic radiation into its constituent wavelengths and it is preferred over a prism because it does not absorb much ultraviolet or infrared radiation. Currently, we may use holographic diffraction gratings that are generated by recording an interference pattern in a photo-resist coated substrate.

 

According to Feynman, if the wire spacing is greater than the wavelength, we will get a strong intensity of scattering in the normal direction, and in certain other directions given by dsin θ = mλ (30.6). In short, the principle of diffraction grating is based on path difference dsin θ that may be a multiple of wavelength. However, the principle of diffraction grating involves the following: 1. Idealization: we assume the slits in a diffraction grating are all in phase, as if they are connected to a plane wave (from a light source that is “practically at infinity”) and parallel to the plane of the slits. 2. Approximation: The grating equation dsin θ = mλ is approximately correct because the slits are all very narrow and the screen is very far away from the grating such that the angles of diffracting waves (or interference fringes) are almost the same.

 

2. Reflection grating:

“So not only do we get a beam in the same direction as the incoming beam but also one in another direction, such that the angle of incidence is equal to the angle of scattering. This we call the reflected beam (Feynman et al., 1963, p. 30–5).”

 

Feynman mentions that a grating is often made with little “sawtooth” cuts instead of little symmetrical notches. Then, he adds the basic principle of reflection grating: the incoming light wave generates motions of the atoms in the reflector, and the reflector then regenerates a new light wave. However, a reflection grating may be defined as a system of equally spaced ridges or grooves on a reflective screen (or made from a mirror). One may elaborate that the surface of the grating should be periodic such that the diffracted light waves can constructively interfere in certain special directions. If Feynman were alive today, he might discuss why we see a rainbow of colors that is apparently “reflected through” a compact disc.

 

A simple reflection grating is a mirror that has hundreds or thousands of narrow, parallel grooves on its surface. In QED, Feynman (1986) explains that “[s]uch a mirror is called a diffraction grating, and it works like a charm. Isn’t it wonderful – you can take a piece of mirror where you didn’t expect any reflection, scrape away part of it, and it reflects (pp. 46-47).” In essence, we can modify a mirror so that it behaves like a reflection grating, but a reflection grating does behave like a mirror. Importantly, we should distinguish the “reflected beam” of a reflection grating and a mirror.

 

Reflected beam: Feynman says that there is a reflected beam because we get another beam in the same direction as the incoming beam such that the angle of incidence is equal to the angle of scattering. Mathematically, sin θ_out = sin θ_in may mean that θ_out is the supplement of θ_in because the light comes out in the same direction as the light which was exciting the grating. However, the term reflected beam used by Feynman is potentially misleading. The so-called reflected beam is an effect of diffracting waves and interfering waves that occurs after reflection (or scattering). Instead of saying reflected beam, it may be more appropriately replaced by “interference after scattering” or “diffraction and interference after reflection.”

 

3. Transmission grating:

“Next, we discuss the special case when d→0. That is, we have just a solid piece of material, so to speak, but of finite length. In addition, we want the phase shift from one scatterer to the next to go to zero (Feynman et al., 1963, p. 30–5).”

 

d→0: Feynman explains that when we put more antennas between the other ones, so that each of the phase difference approaches zero, but the number of antennas is increasing such that the total phase difference, between one end of the line and the other, is constant. In addition, if we recognize n²I₀ as I_m, the maximum intensity at the center of the beam, we get (30.8) I = 4I_msin² ½ Φ/Φ² and this limiting case is shown in Fig. 30–2. However, the antennas are analogous to the slits, and the effect of an infinite number of very narrow slits is equivalent to one wide slit. Perhaps Feynman should elaborate on Fig 30.1: it would appear to be a circular arc instead of a polygon. This is because as the number of sides of a polygon increases to infinity, the length of each side and the phase difference between adjacent antennas approach zero.

 

d<λ: Note that m = 0 is the only solution if d is less than λ because sin θ_out = sin θ_in, which means that θ_out is the supplement of θ_in so the light comes out in the same direction as the light which was exciting the grating. (It means the atoms of the grating absorb the light, but new light was exiting the grating.) That is, the light “does not really go right through” the grating because new light is generated by scattering at the grating. Essentially, the wavelength of light is too long (λ > d) to allow constructive inference to occur at a larger angle in accordance to the formula sin θ = λ/d (there is no θ whereby sin θ = λ/d > 1). It should be worthwhile mentioning that the phrase “distance D” used by Feynman does not refer to the path difference dsin θ, but it is Lsin θ as shown in Fig. 30–3.

 

Review Questions:

1. How would you explain the principle of diffraction grating?

2. Would you use the term reflected beam that was suggested by Feynman?

3. How would you define the equatorial plane in Fig. 30–5?

 

The moral of the lesson: the principle of diffraction grating is based on the grating equation dsin θ = mλ, but it also involves diffraction and interference as well as idealization (source at infinity) and approximation (screen at infinity).

 

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.