(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+ϕn) 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 Imax. (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 Imax. 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 0o to 360o (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.
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