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.