Resolution Limit

Resolution is not limited by manufacturing. Take a lens with no aberration whatever, figure it to atomic tolerance, and a point of light still arrives as a small disc surrounded by faint rings. The aperture itself is the cause: light passing through a finite opening diffracts, and the image of a point is the diffraction pattern of that opening. Two points closer together than the width of their own patterns produce one merged blob, and no amount of lens quality separates them.

Two criteria put a number on it. Rayleigh places the limit where one pattern's peak falls on its neighbor's first dark ring, giving d = 0.61λ/NA. Abbe works from the finest grating whose diffracted orders the lens can still collect, giving λ/2NA. They differ by about 20% and disagree about nothing important: both say resolution improves with shorter wavelength and larger numerical aperture, and with nothing else.

There is a more useful way to hold this. Any image can be described as a sum of spatial frequencies: broad gentle variations at low frequency, fine repeating detail at high frequency. An objective transmits the low ones faithfully, passes the middle ones with progressively less contrast, and beyond a hard cutoff transmits nothing at all. It is a low-pass filter, and the cutoff is the resolution limit expressed as a frequency rather than a distance.

This reframing is what makes the rest of the subject tractable. A lens has a bandwidth in the same sense a telephone line does. Detail above that bandwidth is not faint or noisy in the image — it is absent, having never entered the optical system's output at all. And because the objective has a definite bandwidth, every question about how finely to sample it becomes a question that was already answered, for telegraph wires, in 1924.

Magnification and resolution are independent, and conflating them is the oldest error in microscopy. Magnification makes the image larger. Resolution determines whether there is anything new to see in the larger image. Past the point where the optics have delivered everything they can, additional magnification enlarges the blur along with everything else — the classical name for this is empty magnification, and it is why a specification quoting magnification without numerical aperture is quoting the number that does not matter.

The same logic bounds what processing can do. Deconvolution sharpens by reversing a known blur, and it genuinely helps within the passband. It cannot restore frequencies the objective never transmitted, because there is nothing in the data to restore — only an estimate of what might plausibly have been there.

Take green-yellow emission at 560 nm through a 40x / 1.25 NA objective. Rayleigh gives d = 0.61 × 560 / 1.25 = 273 nm; Abbe gives 560 / 2.5 = 224 nm. So two structures 300 nm apart resolve, two structures 200 nm apart do not, and that verdict is fixed before the light reaches any camera.

Change one variable at a time to feel the dependence. Move to 488 nm excitation-side emission and the limit improves to 238 nm. Drop to a 0.75 NA air objective and it degrades to 455 nm — worse by a factor of 1.7 for the same wavelength, from the aperture alone. Switch from 40x to 100x with the same 1.25 NA and it does not move at all, because magnification is not in the formula. That last comparison is the one worth remembering.