Section 27–7 Resolving power

(Optical resolution / Rayleigh’s criterion / Limitations of geometrical optics)

 

The three interesting concepts discussed in this section are optical resolution, Rayleigh’s criterion, and the limitations of geometrical optics.

 

1. Optical resolution:

“The general rule for the resolution of any optical instrument is this: two different point sources can be resolved only if one source is focused at such a point that the times for the maximal rays… (Feynman et al., 1963, section 27–7 Resolving power).”

 

Feynman explains resolving power by providing the general rule for optical resolution that is related to resolving two different point sources such as looking at a bacterium. However, there could be more discussions on the definition of resolving power and the resolution of an optical instrument. Firstly, the resolving power of a microscope or a telescope is its ability to separate the images of two objects and it can be expressed in terms of angular resolution as q = 1.22 l/D in which D is the diameter of the aperture. Next, the resolving power of a spectroscope or diffraction grating is its ability to separate the wavelengths and it can be expressed as l/Dl. In general, the better the resolving power (smaller resolving power) implies the better the optical resolution (or the smaller size the instrument can resolve).

 

Feynman was aware of the resolution or resolving power of electron microscopes. In his lecture titled There’s plenty of room at the bottom, Feynman (1959) says: “I would like to try and impress upon you while I am talking about all of these things on a small scale, the importance of improving the electron microscope by a hundred times. It is not impossible; it is not against the laws of diffraction of the electron (p. 124).” Interestingly, Feynman poses the challenge of a more powerful electron microscope: “there are theorems which prove that it is impossible, with axially symmetrical stationary field lenses, to produce an f-value any bigger than so and so; and therefore the resolving power at the present time is at its theoretical maximum. But in every theorem there are assumptions. Why must the field be symmetrical? (p. 126).”

2. Rayleigh criterion:

“A corresponding formula exists for telescopes, which tells us the smallest difference in angle between two stars that can just be distinguished (Feynman et al., 1963, section 27–7 Resolving power).”

 

According to Feynman, a corresponding formula exists for telescopes, which tells us the smallest difference in angle between two stars that can just be distinguished. In section 30-4, Feynman states the resolving power of a telescope as θ = 1.22λ/L, where L is the diameter of the telescope. Essentially, he considers Rayleigh criterion to be the limit of resolving power whereby two point-sources are just resolved when the central maximum of one image coincides with the first minimum of the other. Currently, Rayleigh’s criterion is no longer considered to set the limit of resolving power. For example, Born and Wolf (1980) write: “[w]ith other methods of detection (e.g. photometric) the presence of two objects of much smaller angular separation than indicated by Rayleigh’s criterion may often be revealed (p. 418).”

 

Feynman mentions that if the distance of separation of the 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 and suggests the best resolution is approximately the wavelength of light. It seems that Feynman would describe the criterion to be a rough idea. (In Chapter 30, a footnote is stated: This is because Rayleigh’s criterion is a rough idea in the first place.) However, it is not obvious nor practical to operationalize the limit of resolving power (t₂ − t₁ > 1/ν) as stated by Feynman. For instance, it is more practical to resolve binary stars using Dawes’ limit that depends on the difference in brightness between the binary star components and the observer’s visual acuity instead of simply the optical resolving power of the telescope.

 

3. Limitations of geometrical optics:

“…we still could not see two points that are too close together because of the limitations of geometrical optics, because of the fact that least time is not precise (Feynman et al., 1963, section 27–7 Resolving power).”

 

Feynman explains that we cannot keep on magnifying the image because of the limitations of a microscope. He adds that this is due to the limitations of geometrical optics because of the fact that least time is not precise. However, many may expect Feynman to explain the limitations of geometrical optics that are related to the diffraction and interference of light waves. That is, wave properties of light cause difficulties to see two objects or light sources that are very close together. It implies that even we can compensate for aberrations, we should not expect to achieve perfectly sharp images because of the diffraction limit.

 

Feynman seems pessimistic to suggest that if “the difference in time is less than about the period that corresponds to one oscillation of the light, then there is no use improving it any further” (the end of the previous section). Currently, there are many ways to achieve a better resolution that is not limited by the diffraction effects (Tsang, Nair, & Lu, 2016). Physicists can use quantum optics, quantum metrology, and statistical analysis to provide a better estimate of the separation of two light sources. Historically, Rayleigh’s criterion was not rigorously proved and it was based on Huygen’s wave theory. In Sparrow’s (1916) words, “[a]s originally proposed, the Rayleigh criterion was not intended as a measure of the actual limit of resolution, but rather as an index of the relative merit of different instruments (p. 76).”

 

Review Questions:

1. Is the best resolution approximately the wavelength of light or the size of a molecule (See Hell’s Nobel lecture)?

2. Would you consider Rayleigh’s criterion of resolution to be a rough idea?

3. What are the limitations of geometrical optics?

 

The moral of the lesson: we may not be able to resolve two light sources that are close together because of the diffraction and interference of light waves.

 

References:

1. Born, M. & Wolf, E. (1980). Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (6th ed.). Oxford: Pergamon.

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. Hell, S. W. (2015). Nanoscopy with focused light (Nobel Lecture). Angewandte Chemie International Edition, 54(28), 8054-8066.

4. Sparrow, C. M. (1916). On spectroscopic resolving power. The Astrophysical Journal, 44, 76-86.

5. Tsang, M., Nair, R., & Lu, X. M. (2016). Quantum theory of superresolution for two incoherent optical point sources. Physical Review X, 6(3), 031033.