Section 30–3 Resolving power of a grating

 (Rayleigh’s criterion / Grating’s resolving power / Reciprocal time difference)

 

In this section, Feynman discusses Rayleigh’s criterion of resolution, derives the resolving power of a diffraction grating, and relates it to the reciprocal time difference between extreme paths of light.

 

1. Rayleigh’s criterion:

“In order to be able to just make out the double bump, the following simple criterion, called Rayleigh’s criterion, is usually used. It is that the first minimum from one bump should sit at the maximum of the other (Feynman et al., 1963, p. 30–6).”

 

According to the Rayleigh criterion, two images are just resolved when the maximum intensity of one Airy disk (smallest diffraction-limited spot size) coincides with the first minimum of the other disk. While this provides a useful rule of thumb, it remains an arbitrary convention rather than a fundamental physical limit. Importantly, Rayleigh’s intention was to use this criterion to compare the resolution of different optical instruments. In Investigation in Optics, Rayleigh (1880) writes “[a]ccording to the principles of common optics, there is no limit to resolving-power, nor any reason why an object, sufficiently well lighted, should be better seen with a large telescope than with a small one.” However, his effort to quantify resolving power in practical terms, imply the choice of a 26.3% intensity drop relative to the maximum is a matter of convenience rather than an inherent property of light. The Rayleigh criterion has historical significance, but modern imaging and computational methods surpass its limitations.

In Rayleigh’s (1880) words, “[w]e conclude that a double line cannot be fairly resolved unless its components subtend an angle exceeding that subtended by the wavelength of light at a distance equal to the horizontal aperture.” In essence, Rayleigh initially suggested a criterion of resolution whereby the subtended angle due to a wavelength of light ensure the double line to be “fairly resolved” (or well resolved). Subsequently, Rayleigh (1896) built upon Airy’s work and proposed the following: “[i]t appeared that two neighbors, whether constituting a single pair of points or forming part of an extended series of equidistant points, could not be properly distinguished if the distance were less than half the wavelength of the light employed.” In short, this criterion requires a minimum of half wavelength of the light such that the double line can be “just resolved.”

 

2. Grating’s resolving power:

“That is, we want Δ to be exactly one wavelength λ more than mnλ. That is, Δ=mnλ+λ = mnλ′. Thus if λ′=λ+Δλ, we find (30.9) Δλ/λ=1/mn (Feynman et al., 1963, p. 30–6).”

 

Feynman’s derivation of resolving power of a grating could be “confusing” to some students (e.g., why Δ=mnλ+λ = mnλ′?). Below is an alternative derivation:

The maximum intensity for λ (order m) corresponds to nf/2 = mnp.

It is related to I = I₀sin²(nf/2)/sin²(f/2) = I₀sin²(npdsin q/l)/sin²(pdsin q/l).

Similarly, the first minimum for λ′ (order m) corresponds to nf/2 = mnp-p.

(Longer wavelength λ′=λ+Δλ Þ more deviation and lesser phase mnp-p needed.)

Thus, we have npdsin q/(l+Dl)= mnp-p ---(1) and npdsin q/l = mnp ---(2).

Equation (2)/Equation (1) gives (l+Dl)/l = mnp/(mnp-p) = mn/(mn-1)

Dl/l = (mn – mn + 1)/(mn – 1) = 1/(mn – 1) or approximately 1/mn.

 

Rayleigh is the first person to derive the formula of the resolving power of a grating. In an article titled On the manufacture and theory of diffraction gratings, Rayleigh (1874) provides a derivation by drawing and explaining a diagram: “[s]uppose now that l + dl is the wavelength for which BQ gives the principal maximum, then (mn + 1)l = mn(l + dl); whence dl/l = 1/mn which shows that the resolving power varies directly as m and n.” Rayleigh’s derivation of the formula is essentially the same as Feynman’s derivation, but it is based on his earlier criterion that corresponds to one wavelength. Current textbook authors may prefer to use the equation df = (2pd cos q)dq/l and let the phase difference between a maximum and the first adjacent minimum Df to be equal to 2p/n.

 

3. Reciprocal time difference:

“…this formula is equivalent to the formula that the error in frequency is equal to the reciprocal time difference between extreme paths that are allowed to interfere: Δν = 1/T (Feynman et al., 1963, p. 30–6).”

 

According to Feynman, the formula of resolving power λ/Δλ is equivalent to the formula Δν = 1/T. To prove the equivalence of the two formulas, we can use the hint provided in the footnote “In our case T = Δ/c = mnλ/c, where c is the speed of light. The frequency ν = c/λ, so Δν = cΔλ/λ²”. However, the notation T used in the footnote could be confusing. Let’s recall chapter 27 where Feynman mentions that if the distance of separation of two points is D and if the opening angle of the lens is θ, then the inequality t₂ − t₁ > 1/ν is exactly equivalent to D > λ/nsin θ. As a suggestion, we could replace T = Δ/c = mnλ/c by t₂ − t₁ = mnλ/c. Therefore, Δν = cΔλ/λ² = (c/λ)(Δλ/λ) = (mn/[t₂ − t₁])(1/mn) = 1/(t₂ − t₁).

 

Feynman suggests that we should remember the general formula Δν = 1/T because it works not only for gratings, but for any other instrument, while the formula dl/l = 1/mn is applicable only to gratings. Interestingly, in Investigations in Optics, Rayleigh (1880) concludes that “[i]t is not easy to decide whether the highest resolving-power is more likely to be obtained by gratings or by prisms.” To resolve a double line, he stipulates that the aggregate thickness of the prisms (t) should exceed the value given by the equation t = l/dm in which dm is the change in refractive index. He adds that the resolving-power of a prismatic spectroscope of a dispersive material is proportional to the total thickness used, but it is independent of the number, angles, or setting of the prisms, is perhaps the most important proposition in this subject.

 

Review Questions:

1. How would you state Rayleigh’s criterion of resolution for an optical system?

2. Would you use Rayleigh’s method (or Feynman’s method) that is based on his earlier criterion?

3. How would you show that the formula of resolving power λ/Δλ is equivalent to the formula Δν = 1/T?

 

The moral of the lesson: we may derive the resolving power of a diffraction grating as 1/(mn – 1) » 1/mn by taking the phase difference Df between a maximum and the first adjacent minimum to be equal to 2p/n.

 

References:

1. 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.

2. Strutt, J. W. (Lord Rayleigh) (1874). On the manufacture and theory of diffraction gratings. Philosophical Magazine, 47(310), 81-93.

3. Strutt, J. W. (Lord Rayleigh) (1880). Investigations in optics, with special reference to the spectroscope. Philosophical Magazine, 8(49), 261-274.

4. Strutt, J. W. (Lord Rayleigh) (1896). On the theory of optical images, with special reference to the microscope. Philosophical Magazine, 42(255), 167-195.

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.