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.