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.

Section 30–1 The resultant amplitude due to n equal oscillators

 (Diffraction / Maximum intensity / Minimum intensity)

 

In this section, Feynman discusses the concept of diffraction and the condition for maximum intensity and minimum intensity.

 

1. Diffraction:

“…although the name has been changed from Interference to Diffraction. No one has ever been able to define the difference between interference and diffraction satisfactorily (Feynman et al., 1963, p. 30–1).”

 

Feynman explains that when there are two sources, then the “result” is usually called interference, but if there is a large number of them, the word diffraction is often used. We should not be discouraged to provide a definition of diffraction or interference. Firstly, interference can be distinguished as constructive interference and destructive interference, whereas diffraction can be distinguished as near-field diffraction and far-field diffraction. To be specific, diffraction of light is a spreading of light waves through a slit or an obstacle whose size is comparable to the wavelength of the light and this results in fringes through interference. On the other hand, interference of light is a superposition of light waves from two or more sources that results in a redistribution of energy as diffraction patterns or interference patterns.

 

In a sense, it may appear difficult to define the difference between interference patterns and diffraction patterns. However, the intensity and location of interference patterns or diffraction patterns can be calculated using complex numbers and path difference between two waves. Mathematically, the diffraction/interference pattern of a diffraction grating (see next section) can be determined by the location and intensity of multiple-slit interference and single-slit diffraction. Interestingly, it is possible to define the degree of diffraction as a parameter to describe the diffractive spreading of a monochromatic light beam (Wu, Yang, & Li, 2015). In essence, the degree of diffraction of light can be related to the degree of paraxiality (El Gawhary & Severini, 2008) from the perspective of energy flow of light.

 

2. Maximum intensity:

“… if ϕ is exactly 0, we have 0/0, but if ϕ is infinitesimal, the ratio of the two sines squared is simply n², since the sine and the angle are approximately equal. Thus the intensity of the maximum of the curve is equal to n² times the intensity of one oscillator (Feynman et al., 1963, p. 30–2).”

 

Feynman mentions that we have to add something like this: R = A[cos ωt + cos (ωt+ϕ) + cos(ωt+2ϕ) + … + cos(ωt+(n−1)ϕ)], (30.1) where ϕ is the phase difference between one oscillator. Alternatively, it could be first written as R = Acos (ωt+ϕ₁) + Acos (ωt+ϕ₂) + Acos (ωt+ϕ₃) … + Acos (ωt+ϕ_n) that is more general, and we can set ϕ₁ = 0, ϕ₂ = ϕ, ϕ₃ = 2ϕ…… Furthermore, Feynman explains that if ϕ is infinitesimal, the ratio of the sines squared is simply n². Some mathematical physicists may disagree with his use of words: “is simply n².” They would prefer to say that the ratio approaches n² and it is not exactly equal to n² because 0/0 does not exist. In short, the n arrows or vectors are effectively in parallel if ϕ is infinitesimal.

 

If n is sufficiently large, then 3π/2n is very small and we can assume sin 3π/2n = 3π/2n (sin q » q). Thus, the intensity at the first maximum is I = I₀(4n²/9π²), whereas n²I₀ is the maximum intensity and so we have I = n²I₀(4/9π²) = 0.045 I_max. (It was 0.047 in the First Edition.) Some may be confused by the multiple “´10” in Fig. 30–2, but they should realize that 0.045 ´10 would result in the maximum of the dotted curve that is close to 0.5 I_max. Next, Feynman elaborates that we have a very sharp central maximum with very weak subsidiary maxima (including the first maximum) on the sides. However, the graph is not drawn to scale because the width of the central maximum should be sharper based on the law of conservation of energy and the factor n² in the maximum intensity.

 

3. Minimum intensity:

“As the phase ϕ increases, the ratio of the two sines begins to fall off, and the first time it reaches zero is when nϕ/2 = π, because sin π = 0. In other words, ϕ = 2π/n corresponds to the first minimum in the curve (Feynman et al., 1963, p. 30–2).”

 

Feynman suggests using arrows (complex numbers or phasors) as shown in Fig. 30–1 to show how to achieve first minimum whereby all the arrows come back to the starting point. In other words, the arrows should form a regular polygon that is equiangular (all angles are equal) and equilateral (all sides have the same length). Similarly, in Fig. 25 of Feynman’s (1986) QED, he states: “[w]hen all the arrows are added, they get nowhere: they go in a circle and add up to nearly nothing (p. 46).” If the oscillators are light sources, it also means that the probability of light to reach there is zero. If there are only two waves or two arrows, they cannot form a polygon, but it could be explained as destructive interference due to “crest meets trough” or the two opposite arrows have the same magnitude.

 

For the condition of minima, Feynman uses the formula (30.6) ndsinθ = λ, but some may prefer dsinθ = λ/n to provide a good contrast to dsinθ = mλ for maxima. To understand physically why we get a minimum at that location, he adds that Nd is the total length L of the array and the contributions of the various oscillators are then uniformly distributed in phase from 0^o to 360^o (thus, the arrows form a closed polygon). Alternatively, one may elaborate that the sum of components of all arrows in any direction such as vertical is also zero. In addition, Feynman could have clarified that the path difference between the 1st oscillator and (N/2)+1 oscillator (including 2nd oscillator and (N/2)+2 oscillator, and so on) are all λ/2 (i.e., dsinθ = λ/2); thus, they all cancel each other and we get the first minimum.

 

Review Questions:

1. Do you agree with Feynman that we are unable to define the difference between interference and diffraction (or interference patterns and diffraction patterns)?

2. How would you explain the condition for the central maximum?

3. How would you explain the condition for the first minimum?

 

The moral of the lesson: we may distinguish diffraction patterns or interference patterns from the viewpoint of their intensity and locations, but the mathematical formulas are the same (complex numbers or phasors plus path differences).

 

Reference:

1. El Gawhary, O., & Severini, S. (2010). Localization and paraxiality of pseudo-nondiffracting fields. Optics communications, 283(12), 2481-2487.

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.

4. Wu, J., Yang, S. Y., & Li, C. F. (2015). Degree of diffraction for monochromatic light beams. Acta Photonica Sinica, 44(1), 126004-0126004.

Section 29–5 The mathematics of interference

 (Trigonometric method / Geometrical method / Analytical method)

 

In this section, Feynman discusses three ways of solving problems pertaining to interference, namely, trigonometric method, geometrical method, and analytical method. Specifically, we may represent the mathematics of interference using trigonometric functions, phasors, and complex functions.

 

1. Trigonometric method:

“In those circumstances, for example (we could call this the trigonometric method of solving the problem), we have (29.9) R = A[cos(ωt+ϕ₁)+cos(ωt+ϕ₂)] (Feynman et al., 1963, p. 29–6).”

 

Using trigonometric method, Feynman explains that the interference of two waves can result in an oscillatory wave having the same frequency with a new amplitude A_R, and a resultant phase ϕ_R. If the amplitude of both waves is the same (A₁ = A₂ = A), the new amplitude is A_R = 2Acos ½(ϕ₁−ϕ₂) and the resultant phase is the average of the two phases, ½(ϕ₁+ϕ₂). Note that it is not difficult to derive the general amplitude: A_R² = A₁² + A₂² + 2A₁A₂cos(ϕ₂−ϕ₁) using the identity sin q² + cos q² = 1. Essentially, the new amplitude can be related to the sum of the intensity A₁² and the intensity A₂² plus the interference effect, 2A₁A₂cos(ϕ₂−ϕ₁). In a sense, the effect of interference is a manifestation of the law of cosines or Schwartz inequality (A₁ – A₂)² £ (A₁² + A₂²) £ (A₁ + A₂)².

 

Idealization: According to Feynman, interference in ordinary language suggests opposition or hindrance, but in physics we often do not use language in the way it was originally designed. However, Michelson opins that the word interference is a misnomer because the waves do not really interfere with each other but rather move independently. Thus, he writes: “… [t]he principle of which these two cases are illustrations is miscalled interference; in reality the result is that each wave motion occurs exactly as if the other were not there to interfere (Michelson, 1902, p. 8).” In other words, interference is an idealization because we assume the real waves are perfect sinusoidal waves that add up linearly at a particular point in accordance with the principle of superposition. Furthermore, we have idealized the three-dimensional waves as one-dimensional for the sake of simplicity.

 

2. Geometrical method:

“Now suppose that we cannot remember that the sum of two cosines is twice the cosine of half the sum times the cosine of half the difference. Then we may use another method of analysis which is more geometrical (Feynman et al., 1963, p. 29–6).”

 

Feynman suggests a second method of analysis which is more geometrical: any cosine function of ωt can be visualized as the horizontal projection of a rotating vector. Interestingly, in a public lecture on QED, Feynman describes wavefunctions of photons using arrows and imaginary stopwatch hands instead of complex numbers. In Feynman’s (1985) words, “[a]lthough it may sound more impressive that way, I have not said any more than I did before - I just used a different language (p. 63).” However, we may use complex number or phasor method that can provide the same geometrical interpretation on the rotating vector. Historically, Steinmetz formalizes the concept of phasor as a rotating vector to represent a sinusoidal signal, but it is equivalent to a complex number that has the geometric significance of √−1.

 

Limitation: In the public lecture on QED, Feynman (1985) adds: “if we put instruments in to find out which way the light goes, we can find out, all right, but the wonderful interference effects disappear. But if we don’t have instruments that can tell which way the light goes, the interference effects come back! (p. 81)” Simply put, if the reliability of the detectors increases, lesser interference is expected. To be specific, interference can be distinguished as classical interference and quantum interference. For example, Taylor’s (1909) experiment of interference patterns using feeble light suggests that a photon can interfere with itself or interact with one of the two slits. Thus, interference may be considered as a manifestation of the difference in phase between the two possible paths (or histories) of the photon.

 

3. Analytical method:

“There is still another way of solving the problem, and that is the analytical way. That is, instead of having actually to draw a picture like Fig. 29–9, we can write something down which says the same thing as the picture (Feynman et al., 1963, p. 29–6).”

 

Feynman suggests a third method of solving the problem and he describes it as the analytical way. That is, instead of drawing the rotating vectors, we can use a complex number or complex function to represent each of the vectors. However, the word analytical may be misunderstood because of its technical meaning in different contexts. For example, one may specify that the exponential function is analytic provided any Taylor series for this function converges not only for x close enough to x₀ but also for all values of x that are real or complex. More important, complex number has the interesting properties such that certain calculations, particularly multiplication and division of complex numbers, can be simplified when expressed in exponential form.

 

Approximation: Feynman explains that the intensity is 2 at 30° in Fig. 29–5 because the two oscillators are ½λ apart and the path difference (dsin θ = λ/2 ´ sin 30 ^o = λ/4) is equivalent to the phase difference, ϕ₂−ϕ₁ = 2πλ/4λ = π/2, and so the interference term is zero. By adding the two rotating vectors (or phasors) at 90^o, the resultant vector is the hypotenuse of a 45° right-angle triangle, which is √2 times the unit amplitude; squaring it, we get twice the intensity of one oscillator alone. However, the formula dsin θ = λ/4 involves an approximation because we have assumed the interference pattern occurs at a distant point from the two oscillators. Technically speaking, this is also known as the Fraunhofer diffraction whereby the light rays emerge from the slit are approximately parallel to each other.

 

Review Questions:

1. How would you explain the meaning of interference using the trigonometric method?

2. Would you use complex number or phasor method to provide the geometrical interpretation on the rotating vector?

3. Would you consider Feynman’s third way of solving the problem pertaining to interference to be an analytical way?

 

The moral of the lesson: we may use trigonometric functions, phasors, or complex functions to solve problems related to interference.

 

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. Michelson, A. A. (1902). Light waves and their uses. Chicago: University of Chicago Press.

4. Taylor, G. I. (1909). Interference fringes with feeble light. Proceedings - Cambridge Philosophical Society, 15,114-5.