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.