A converging lens making a real image

At its defaults it draws a converging lens making a real image. An object 2.44 focal lengths from a thin converging lens. The image sits where the construction rays cross, at 1.69 focal lengths, magnified -0.69×.

thin-lens is one function in lib/figures/optics.js — rays, lenses, mirrors and what light does to a surface. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

An object 2.44 focal lengths from a thin converging lens. The image sits where the construction rays cross, at 1.69 focal lengths, magnified -0.69×.

The phase delay of 633 nm light across the middle of a model cell holding 300 pg of dry matter, in three shapes — rounded up, spreading and flattened — each with the same contents dissolved in volumes of 942, 965, 982 μm³. The rounded cell delays the light by 3.41 rad at its centre, the flat one by only 0.55. But the delay summed over the whole image is the same for all three, 536 rad·μm², and it gives back 300.0 pg through the refractive increment α = 0.18 mL/g: a cell is weighed by the light it delays, whatever shape it takes.

A model cell of 300 pg dry mass moved into a dilute medium, taking up water and swelling to twice its volume over a few minutes, each quantity relative to its starting value: the cell's volume (dashed), the excess of its refractive index over the medium's (dotted), which falls from 0.0796 to 0.0398 as the contents are diluted, and the phase delay integrated over the cell's image (solid). The volume doubles and the index excess halves, and the light passes through twice the path at half the excess: the integral, and the dry mass it reports, stay at 300.0 pg throughout. Water itself is invisible to the measurement, because the medium is water too.

The refractive increment — the rise in refractive index per gram of dissolved substance per millilitre — for four classes of biological molecule, typical values: proteins, typical, 0.185 mL/g; nucleic acids, 0.170 mL/g; polysaccharides, 0.145 mL/g; lipids, 0.150 mL/g, against the single value of 0.18 mL/g usually assumed (dashed). Proteins, which make up most of a cell's dry mass, sit close to it, and the narrowness of the spread is what makes phase imaging a balance: assuming 0.18 for a cell made entirely of nucleic acid would understate its mass by 6 per cent, and for one made entirely of carbohydrate by 19. Real cells are mixtures dominated by protein, and the uncertainty in their dry mass from this source is a few per cent — larger than the measurement's own noise, and the limit on its absolute accuracy.

The uncertainty in a dry-mass measurement from noise in the phase image, in picograms, against the area of the cell, for phase noise of 1, 10, 100 mrad in each 0.5 μm pixel, on logarithmic axes: σ = (λ/2πα) × the noise × the pixel area × the square root of the number of pixels the cell covers. A bacterium a few square micrometres in area, weighing about a third of a picogram, is weighed to 4.8 fg at 10 mrad; a mammalian cell of 1000 μm², weighing hundreds of picograms, to 0.09 pg — a part in a thousand of its mass. The noise grows only as the square root of the area because independent errors in different pixels partly cancel, which is why the method resolves a growing cell's mass change within minutes.

The dry mass of a model cell doubling from 150 to 300 pg over a 20-hour cycle, if it grows at a constant rate (dashed) or at a rate proportional to its mass (solid), with measurements every half hour from the exponential cell carrying 0.5 pg of noise (dots, seeded). The two laws differ by at most 12.9 pg, 5.6 per cent of the mass, about 11 hours into the cycle. Weighings this precise — by phase, and by vibrating cantilevers that cells flow through — found in the 2010s that many animal cells grow at rates that rise with their size, and adjust them during the cycle: a question about how cells control their size that had been argued for decades with volume, which water makes unreliable, standing in for mass.

physicscheck asserts something about thin-lens that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.