(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 θout = sin θin may mean that θout is the supplement
of θin 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 Im, the maximum intensity at the center of the beam, we get (30.8) I = 4Imsin2 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
θout = sin θin, which means that θout is the supplement
of θin 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.
