Questions: Diffraction Limit and the Rayleigh Criterion

Question 1

A telescope manufacturer builds a perfect, completely aberration-free lens with exquisite optical coatings. Can this lens resolve two stars separated by an angle smaller than 1.22 λ/D?

• A. Yes — eliminating aberrations removes all resolution limits
• B. Yes — better coatings allow more light collection, improving resolution
• C. No — the diffraction limit is a fundamental physical constraint, not an engineering one
• D. No — but a larger magnification eyepiece would overcome this

Answer: (C)

The Rayleigh criterion expresses a fundamental limit set by wave optics, not by imperfections in the instrument. Even a perfect lens diffracts light at its aperture, producing an Airy disk rather than a point image. Eliminating aberrations, improving coatings, or increasing magnification cannot reduce the size of the Airy disk — only a larger aperture D or shorter wavelength λ can do that. This is the key insight: the diffraction limit belongs to physics, not engineering.

Question 2

A radio telescope observing at λ = 21 cm needs to match the angular resolution of an optical telescope with a 10 cm aperture observing at λ = 500 nm. Approximately how large must the radio telescope be?

• A. About 10 cm — resolution depends only on aperture, not wavelength
• B. About 420 cm — wavelength ratio times the optical aperture
• C. About 42,000 cm (420 m) — because radio wavelengths are ~4,200× longer
• D. About 21 m — because radio wavelengths are about 210× the optical aperture

Answer: (C)

The Rayleigh criterion θ ≈ 1.22 λ/D shows resolution depends on the ratio λ/D. To achieve the same θ, if λ increases by a factor of 4,200 (from 500 nm to 21 cm), D must increase by the same factor: 10 cm × 4,200 = 42,000 cm = 420 m. This is why real radio telescopes are enormous — it is not a matter of poor design, but of compensating for wavelengths thousands of times longer than visible light.

Question 3

A perfectly crafted, aberration-free lens will eventually beat the Rayleigh criterion if the optical quality is high enough.

• T. True
• F. False

Answer: False

The diffraction limit is not a consequence of imperfect optics — it arises from the wave nature of light itself. Any aperture, no matter how perfect, diffracts light and produces an Airy disk. The Rayleigh criterion sets the minimum resolvable separation for a given aperture and wavelength regardless of optical quality. Super-resolution microscopy methods that go beyond λ/D do so by exploiting molecular photochemistry, not by improving lens quality.

Question 4

Using shorter-wavelength light to illuminate a specimen in a microscope will improve its angular resolution.

• T. True
• F. False

Answer: True

Since θ_min ≈ 1.22 λ/D, decreasing λ directly decreases the minimum resolvable angle, improving resolution. This is why electron microscopes can image atomic structures — electron de Broglie wavelengths (~pm) are far shorter than visible light (~hundreds of nm). UV microscopy and X-ray crystallography exploit the same principle: shorter wavelength yields finer resolution.

Question 5

Explain why simply building a larger telescope improves its angular resolution, in terms of the physics of diffraction.

Model answer: A larger aperture D reduces the angular size of the Airy disk produced by each point source, because the first dark ring of the diffraction pattern falls at an angle θ ≈ 1.22 λ/D. When D increases, θ_min decreases, so two sources that previously fell within each other's Airy disks now produce distinguishable peaks. The telescope isn't 'seeing more detail' because it collects more light — it resolves finer detail because it diffracts less, spreading each point source's image into a smaller disk.

This answer demonstrates understanding that resolution is about diffraction, not light collection. Many students confuse the two benefits of larger apertures (more light AND better resolution). The resolution gain is entirely due to the wave optics of diffraction at the aperture, captured by θ ≈ 1.22 λ/D.