Magnification

The number engraved on the barrel is not a property of the objective. It is a property of the objective and the tube lens it was designed for, and those differ by manufacturer: Zeiss builds for a 164.5 mm tube lens, Olympus for 180 mm, Nikon and Leica for 200 mm. Actual magnification is the engraved figure scaled by the ratio of the tube lens you have to the one it expects.

So a Zeiss 40x on a Nikon stand is 40 × 200/164.5 = 48.6x. Not a rounding error — 22% off, silently, on every dimension you derive from it. Every measurement in micrometers, every pixel footprint, every Nyquist calculation built on that number inherits the same 22%.

The error is invisible because nothing looks wrong. The image is sharp, the field is a plausible size, and the scale bar your software draws is confidently incorrect. If your system mixes brands anywhere in the path — and a surprising number do, through adapters and third-party tube lenses — calibrate against a stage micrometer and believe the measurement, not the engraving.

Magnification does not create detail; it distributes detail across pixels. The optics fix what can be resolved, and magnification determines how much sensor area that resolved detail lands on. That makes it the lever that sets your pixel footprint — pixel pitch divided by total magnification — and therefore the lever that decides whether you satisfy Nyquist.

This is why the same camera behaves completely differently on two objectives. A 6.5 µm sensor gives a 325 nm footprint at 20x, 162.5 nm at 40x, and 108 nm at 60x. Against a 273 nm resolution limit, the first is badly undersampled, the second marginal, the third comfortable. The camera did not change. The only thing that changed was the divisor.

Push magnification past the point where the sampling already captures everything the objective transmits and you get a larger image containing nothing new. The classical name is empty magnification, and the classical useful range is between roughly 500 and 1000 times the numerical aperture. Below 500·NA you are throwing away resolution the lens delivered; above 1000·NA you are enlarging blur.

For a 1.25 NA objective that puts the useful window at about 625x to 1250x total. A 100x eyepiece arrangement reaching 1000x sits inside it; pushing to 2000x with a projection lens sits well outside, and produces an image that is bigger, dimmer, and no more informative. Dimmer matters: spreading the same photons over four times the area quarters the brightness for nothing in return.

Total magnification is the objective figure, corrected for tube lens mismatch, times any additional relay or camera adapter. A 40x Zeiss objective on a Nikon stand with a 0.7x camera adapter gives 40 × (200/164.5) × 0.7 = 34.0x — not 40, and not 48.6 either.

Then the footprint: a 6.5 µm pixel divided by 34.0 is 191 nm at the specimen. Against a 273 nm resolution limit, Nyquist wanted 137 nm or finer, so this configuration is undersampled by a factor of 1.4 — and would have looked correctly sampled if you had trusted the engraving and the adapter's nominal figure without multiplying them out.