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Look at two car headlights far down a straight road at night. At some distance they merge into one patch of light, however sharp your eyesight. No lens, mirror or eye can show detail finer than a certain limit, and the physicist who turned that limit into a usable rule was Lord Rayleigh. Here is who he was, what he worked out in 1879, how to use his rule with real numbers, and the mistakes to avoid.
Who was Lord Rayleigh?
John William Strutt, third Baron Rayleigh, was born on 12 November 1842 at Langford Grove, Maldon, Essex, and died on 30 June 1919, according to his Nobel Prize facts page. His Nobel biography describes a frail child whose schooling was repeatedly broken by illness, who still graduated from Cambridge in 1865 as Senior Wrangler, the top student in the Mathematical Tripos. In 1879 he followed James Clerk Maxwell as head of the Cavendish Laboratory.
He won the 1904 Nobel Prize in Physics "for his investigations of the densities of the most important gases and for his discovery of argon in connection with these studies". That story is told in our post on William Ramsay and the noble gases. His 446 collected papers range from sound to light scattering; here we stay with how sharp an image can be.
What problem was Rayleigh trying to solve?
In 1879 Rayleigh published a series of papers in the Philosophical Magazine called Investigations in optics, with special reference to the spectroscope. The question: how close can two spectral lines, or two stars, be before an instrument shows them as one?
His starting point was that light is a wave, so even a perfect lens cannot focus a point of light to a point. In the paper, reprinted in volume 1 of his Scientific Papers, he explains that the image of a star seen through a circular opening is "a central disk of light, surrounded by luminous rings of rapidly diminishing brightness". The angular radius of that disk is 1.2197λ/D, where λ is the wavelength and D the diameter of the opening.
What exactly is the Rayleigh criterion?
For two spectral lines of equal strength, Rayleigh took the case where "the central line in the image of one coincides with the first zero of brightness in the image of the other". He calculated that the brightness midway between the two peaks is then 0.8106 of the brightness at the peaks, and called this "about the limit of closeness at which there could be any decided appearance of resolution."
You can check his number. For a slit, the brightness pattern of one line is (sin u / u)², with the first zero at u = π. Halfway between the peaks each line has u = π/2, so each gives (1/(π/2))² = 4/π². Together that is 8/π² = 0.8106. At each peak the other line is dark, so the peak is 1.
He admitted it was a convention: the rule "is convenient on account of its simplicity; and it is sufficiently accurate in view of the necessary uncertainty as to what exactly is meant by resolution."
For a circular lens the first dark ring sits at 1.22λ/D, so two point sources are just resolved when their angular separation is
θ = 1.22 λ / D
Wider lenses and shorter wavelengths give finer detail.
Worked example: your eye and a large telescope
Sample numbers: green light with λ = 550 nm and a pupil 3 mm across.
Angle. θ = 1.22 × 550 × 10⁻⁹ / (3 × 10⁻³) = 2.24 × 10⁻⁴ rad.
In arcminutes. 2.24 × 10⁻⁴ × (180/π) × 60 = 0.77 arcminute.
At reading distance. Two dots 25 cm away must be at least 0.25 × 2.24 × 10⁻⁴ = 5.6 × 10⁻⁵ m apart, about 0.06 mm. This is the limit set by diffraction alone.
Headlights. If the lamps are 1.5 m apart, they merge beyond 1.5 / (2.24 × 10⁻⁴) = 6.7 km.
The NCERT Class 12 Ray Optics chapter mentions the 40-inch (about 1.02 m) lens at Yerkes Observatory. For the same light, θ = 1.22 × 550 × 10⁻⁹ / 1.02 = 6.6 × 10⁻⁷ rad, about 0.14 arcsecond. The lens is 340 times wider than the pupil, so it resolves angles 340 times smaller.
How did Rayleigh test his rule?
For the two yellow sodium lines he calculated that a prism of dense glass needed an effective thickness of 1.02 cm to separate them. He then fitted cardboard screens to a real prism and narrowed the opening until, "in the judgment of the observer, the line was barely resolved." One observer put the point at 1.00 to 1.20 cm, close to his prediction.
What mistakes do students make with resolution?
Using the radius. D is the full diameter of the lens or pupil.
Mixing up slit and circle. A slit of width a has its first dark fringe at θ ≈ λ/a, as in the NCERT Wave Optics chapter. The factor 1.22 belongs to circular openings.
Confusing magnification with resolution. The Ray Optics chapter notes that resolving power depends on the diameter of the objective. Magnifying a blurred image adds no detail.
Degree-radian slips. Forgetting to convert changes the answer by a factor of about 57.3.
Where does this sit in your syllabus?
The JEE (Main) 2026 syllabus lists single-slit diffraction and the magnifying powers of microscopes and telescopes, but not the resolving-power formula, so treat 1.22λ/D as useful background and check the current year's syllabus. Our post on single-slit diffraction and polarisation covers the building block, and microscopes and telescopes covers magnification.
If the overlap of two diffraction patterns is hard to picture from equations, the Fermi Simulation Lab has interactive simulations for ideas that are easier to see than read.
What can you take from how Rayleigh worked?
Make a fuzzy word precise. "Resolved" meant different things to different observers. He picked one clear rule and said openly why.
Check the number yourself. The 0.8106 takes two lines of algebra to rederive.
Test against a real instrument. His cardboard screens on a prism turned a formula into a measurement.
Next step: redo the headlight calculation for a 7 mm night-time pupil and see how far the merging distance moves.
