Try this first
EXPERIMENT 1 · CLOSE THE APERTURE
The slide is a ruled grating, lines 0.50 µm apart, lit from straight below with green light (550 nm). Drag the aperture slider slowly down from 1.20. Just below 1.10 the two side dots leave the gold ring, and the image goes flat. The lens has not got any worse: it is a perfect lens at every setting. It has stopped catching the light that carries the lines.
What the dots are
A regular structure does to light what a ruled grating does. It sends the light on as a set of separate beams at fixed angles: one straight through (the zeroth order), and on either side, at an angle whose sine is the wavelength divided by the spacing, the first orders; twice that, the second orders; and so on. A lens brings each beam to a single point in its back focal plane, which is panel 2. Every dot there is one beam. The objective accepts beams only up to a certain angle, and that angle, written as n sin θ (the refractive index times the sine), is its numerical aperture. It is the gold ring.
The image is what the caught beams make when they meet again and interfere. One beam alone is an even wash of light. Two beams make fringes, and the fringes have the spacing of the lines on the slide. More beams sharpen the fringes toward the shape of the lines. So the image is not a copy of the slide with some blur added. It is rebuilt from a sample of the slide's diffraction, and the aperture decides which sample.
Ernst Abbe put the consequence in one sentence in 1873:
“Durch kein Mikroskop können Theile getrennt (oder die Merkmale einer real vorhandenen Structur wahrgenommen) werden, wenn dieselben einander so nahe stehen, dass auch der erste durch Beugung erzeugte Lichtbüschel nicht mehr gleichzeitig mit dem ungebeugten Lichtkegel in das Objectiv eintreten kann.”
“No microscope can separate parts (or perceive the marks of a structure that is really there) if they stand so close together that even the first beam produced by diffraction can no longer enter the objective together with the undiffracted cone of light.” (Abbe 1873, p. 455; translation ours.)
Tilt the light and the limit halves
EXPERIMENT 2 · OBLIQUE LIGHT
Same grating, aperture 0.80. With the light straight down the first orders sit at 1.10, outside the ring, and the image is flat. Now press Tilted. The undiffracted beam moves out toward the edge of the ring, and one first order swings inside it on the other side. Two beams: the lines come back, through the same lens.
Abbe gave both numbers. For light straight down, the finest spacing is the wavelength divided by the sine of half the aperture angle; for the most oblique light, “bei jedem Oeffnungswinkel genau halb so gross”, exactly half as large at every aperture (p. 456). In modern notation:
d = λ / NA · d = λ / 2NA
For green light and the widest oil objective here (NA 1.40), that is 0.39 µm with the light straight down and 0.20 µm with it tilted to the rim. The famous version, with the 2 in it, silently assumes the second case, or a condenser whose cone of light fills the aperture. The third button, A cone, is that ordinary condenser: each point of it throws its own copy of the pattern into the back focal plane (the faint discs), each copy makes its own image, and because light from different points of a lamp does not interfere, the images add as brightness. That sum is Abbe's too, and it is the whole of what this page computes.
Stops: pictures of things that are not there
If the image is rebuilt from the beams, then choosing the beams chooses the picture. In 1906 Albert Porter showed this with nothing but an arc lamp, a lens, a card with holes in it and “a piece of wire gauze having about 30 wires to the centimetre”, looked at with the naked eye. The gauze is here too, at 1.6 µm.
EXPERIMENT 3 · PORTER'S GAUZE
A horizontal slit behind the lens passes only the beams spread out sideways, which are the ones the vertical wires make. The horizontal wires vanish from the image. Then turn the slit to 45°. Porter: “neither the vertical nor horizontal wires are seen, but a very real looking set of wires appears running diagonally”. There are no diagonal wires in the gauze.
EXPERIMENT 4 · THE LINES THAT DOUBLE
A grating 1.00 µm apart, with stops placed over its two first orders only. The zeroth and second orders still get through, and the image shows crisp lines 0.50 µm apart: twice as many as the slide has. The readout says so. You can click the stops to remove them, or click other dots to stop them.
Abbe had done this with a real microscope and a ruled plate, and said the false image (his word is Scheinbild) could not be told from the real thing:
“Die so erzeugten Scheinbilder sind aber in Hinsicht auf ihre Schärfe und die Constanz ihres Auftretens bei keiner Vergrösserung von dem normalen Bilde einer wirklich doppelt, dreifach, ... so feinen Streifung gleicher Art zu unterscheiden”
“The false images produced in this way cannot, at any magnification, be told apart in sharpness or constancy from the normal image of a striation of the same kind that really is two, three ... times as fine” (p. 447). He demonstrated it with a doubled ruling shown in the same field of view beside a real ruling of twice the fineness.
Two slides, one image
EXPERIMENT 5 · THE MESSAGE IN THE FINE DETAIL
Slide A is a coarse grating. Slide B is the same grating with two words written into it, using only detail finer than 1.2 cycles per micrometre (nothing in the message repeats more slowly than every 0.83 µm). At aperture 0.60, switch between Slide A and Slide B. The left panel changes. The image does not change by one digit of the arithmetic, and the readout reports the largest difference as zero. Now open the aperture wide, or tilt the light, and read what slide B says. (Straight down, the first of the message's detail gets in once the aperture passes about 0.66, though it takes a wider one to make the words legible.)
This is the conclusion Abbe drew from the stop experiments, and it is the strongest thing in his paper:
“dass verschiedene Structuren stets das nämliche mikroskopische Bild liefern, sobald die Verschiedenheit des an sie geknüpften Beugungseffectes für das Mikroskop künstlich beseitigt wird”
“that different structures always give the same microscopic image as soon as the difference in their diffraction, as far as the microscope is concerned, is artificially removed” (p. 451). An aperture is just such a removal. Two slides whose beams agree everywhere inside the ring are, to that microscope, the same slide. Porter restated the whole theory the same way in 1906: when only part of the diffraction pattern gets through, the image “will correspond to another (virtual) object whose whole diffraction pattern is identical with that portion which passes through the lens”.
So Abbe told microscopists to read fine structure “nicht morphologisch, d. h. als Bilder körperlicher Formen, sondern nur physikalisch”: not as pictures of bodily forms, only as physical marks, “nicht als Abbilder”, not as likenesses (p. 452). From the image one may conclude that the slide diffracts in a certain way, and nothing more.
The diatom everyone argued about
A favourite test of a good objective in the 1870s was a diatom, Pleurosigma angulatum, whose glassy shell carries rows of tiny marks too fine for ordinary lenses. People disagreed about what the marks were, and whether there were two sets of lines on it or three. Abbe's answer was that no microscope could settle it: whether the marks are raised or sunk, how many systems of lines there are, “darüber kann kein noch so vollkommenes Mikroskop und keine noch so hohe Vergrösserung Aufschluss geben”, no microscope however perfect and no magnification however high can tell you (p. 453). What any microscope could ever receive from it, he wrote, was six spectra, symmetrically placed, bent about 65° from the straight beam in blue light, which any structure arranged on “a system of equilateral triangles of 0.484 height” would produce.
EXPERIMENT 6 · PLEUROSIGMA AT ABBE'S SPACING
The diatom slide here is an idealised lattice of pores on Abbe's triangles: rows 0.484 µm apart, so pores 0.56 µm apart. With a good dry lens (NA 0.95) in green light the six first orders sit at 1.13, outside the ring, and the valve is blank. Three ways in, all of them 1870s practice: slide the wavelength down to blue (below about 460 nm, the six dots cross the rim); tilt the light; or use oil. Zeiss made the first oil-immersion objectives in about 1878, to Abbe's calculations and on a suggestion of the English microscopist John Ware Stephenson, at an aperture near 1.26.
The third button blocks the undiffracted light, leaving only the six beams: a dark-field image. Abbe noted (p. 447) that any two or more beams draw sharp detail whether or not the straight-through light is among them, but that different beams draw different detail, which need not match the real structure. Compare the dark-field pattern's spacing with the slide's.
Abbe listed the spacings of his own test objects in a footnote (p. 449). The table puts his numbers through his rule. The blue is his too: a footnote on p. 448 gives dark blue as 0.43 µ.
| Test object (Abbe's spacing) | NA needed, light straight down, 550 nm | tilted, 550 nm | tilted, 430 nm |
|---|---|---|---|
| Butterfly scale, Hipparchia janira, long stripes: 2 µ | 0.28 | 0.14 | 0.11 |
| the same scale, cross stripes: 0.7 µ | 0.79 | 0.39 | 0.31 |
| Pleurosigma angulatum: 0.48 µ | 1.15 (oil) | 0.57 | 0.45 |
| Surirella gemma: 0.34 µ | 1.62 (none) | 0.81 | 0.63 |
| Frustulia saxonica: 0.25 µ | 2.20 (none) | 1.10 (oil) | 0.86 |
“Oil” means more than any dry lens can reach (0.95 is about the practical limit in air); “none” means more than even oil allows (1.515). Read down the first column and you can see why the diatomists of the 1870s were obsessed with oblique light: the finest of these shells could not be resolved by any lens with the light straight down, and could be with the light tilted.
Magnification is not resolution
In the stop experiments Abbe found that with the diffracted light cut off, fine rulings turn into “eine gleichförmige Fläche”, a uniform surface, “welche Vergrösserung auch angewandt wird”, whatever magnification is used (p. 446). The detail is not too small to see; it is not in the image to be enlarged. Microscopists still teach the rule of thumb that follows: total magnification is useful between about 500 and 1,000 times the numerical aperture, and anything above that is “empty magnification”, a larger picture with no more in it.
The equation that is not in the paper
Abbe's 1873 paper runs from page 413 to page 468 of the Archiv für mikroskopische Anatomie, and it states all of this in words. We searched the text of the scanned volume for the whole paper: the only equals signs in it are in one footnote giving the wavelengths of red and dark blue light. No figure is referred to. Hermann von Helmholtz published a derivation of a limit of his own in 1874, saying in a postscript that he had seen Abbe's paper only at the last moment and that its results largely coincided with his. Thirty-two years later Porter could still write that “the complete mathematical development has never been published”.
The formula did end up in stone. A memorial at the corner of Fürstengraben and Weigelstraße in Jena, in front of the university's old main building, is a stone sphere on a pedestal, and the sphere carries d = λ / 2n sin α: the tilted-light version, with the 2.
The same sum, a century and a half on
A chip is printed by a microscope run backwards: a lens images a patterned mask, shrunk, onto a light-sensitive wafer. The software that predicts what will print adds up the images made by each point of the light source, and a standard name for that calculation in the lithography literature is Abbe's method. The limit is the same arithmetic too. The industry writes the printed line width as k1 λ / NA, and for a pattern of equal lines and gaps, Abbe's tilted-light limit on the period, λ / 2NA, is a line width of λ / 4NA: k1 = 0.25. ASML's own figures for its machines:
| Scanner (ASML's figures) | λ | NA | stated resolution | k1 = res × NA / λ |
|---|---|---|---|---|
| TWINSCAN NXT:2000i, water immersion, dipole light | 193 nm | 1.35 | 38 nm | 0.266 |
| NXE (EUV) | 13.5 nm | 0.33 | 13 nm | 0.318 |
| EXE (High-NA EUV) | 13.5 nm | 0.55 | 8 nm | 0.326 |
“Dipole” illumination is two beams of light tilted to opposite sides: Abbe's oblique light, twice. With it, ASML's NXT:2000i prints lines within 7% of the wall Abbe described in words in 1873. (ASML says “resolution” and does not say how it is measured; reading it as the width of one line in a pattern of equal lines and gaps is our assumption, and it is the reading that makes k1 meaningful.) Water raised the NA past 1; the rest of the gain has come from shorter light, which Abbe already saw coming on p. 456, where he notes that photography, working in shorter “chemically active” light, could see structure finer than the eye in the ratio 3 : 2.
What the limit is not
It is not Rayleigh's number. Books give 0.61 λ / NA as “the” resolution, and that is a different criterion for a different question: when two separate points that shine on their own (not lit together, so not interfering) can be told apart. Lord Rayleigh set it out for telescopes and spectroscopes in 1879 and for the microscope in 1896. Abbe's limit is about a lit, regular structure and whether its beams get in at all, and it has no fudge factor: below it the contrast is exactly zero, as experiment 1 shows.
It is not a wall around all seeing. It binds one lens gathering, in the far field, the light scattered by a fixed structure lit all at once. Change any of those and it moves. Electron microscopes use a far shorter wavelength; the question Abbe said no light microscope could settle was taken to one by 1942 (Hamly and Watson, “Electron and Optical Microscope Interpretation of the Wall of Pleurosigma angulatum”). The 2014 Nobel Prize in Chemistry went to Eric Betzig, Stefan Hell and W. E. Moerner “for the development of super-resolved fluorescence microscopy”: methods that arrange for neighbouring fluorescent molecules not to shine at the same time, so the premise of the argument no longer holds.
How this microscope is computed, and what it leaves out
Each slide is held as its Fourier series on a 256 × 256 grid, which is the same as holding the exact set of beams it sends out. The grid keeps every beam that any setting of the instrument can use (the widest reach is 2 × 1.40 / 0.40 µm = 7.0 cycles per µm; the grid holds 7.9, or 7.5 for the coarsest diatom). Tilting the light shifts the beams; the aperture is a sharp-edged disc; stops remove what they cover; the survivors are transformed back and squared. A cone of light is sampled at no more than 60 points.
What it leaves out: the lens is perfect (no aberrations, no defocus); light is treated as a scalar wave, which is not quite right at the highest apertures, where polarisation changes how strongly beams interfere; and the slides are thin patterns that absorb, where a real diatom shell is a transparent, three-dimensional object that mostly shifts the phase of light. None of these moves where the beams go, which is set by the spacing alone, and so none moves the limits; they would change how the images look near them. The colours are an approximation of each wavelength's hue.
The check
Every picture on this page is computed in your browser by engine.mjs. The same engine is held by research/what-the-lens-caught/verify.mjs against things computed another way: its Fourier transform against a plain sum; the image of a grating against the Fourier series written out by hand; its sum over the condenser against Hopkins' cross-coefficients integrated over a continuous disc of light, a route with no transform and no sampled source at all; and each experiment against what the page says it shows. Its companion mutate.mjs breaks the engine in small plausible ways (adding the condenser's images as amplitudes instead of brightness, shifting the beams the wrong way, measuring the aperture in the wrong units) and confirms the verifier fails every time.
-- from verify.mjs (49 passed, 0 failed) PASS 1-D FFT equals the DFT sum (n = 64, random complex input) : max error 1.5e-13 PASS d = 1 um, NA 1.2, 550 nm: orders |m| <= 2 pass and the image is their sum, squared : max error 1.2e-14 PASS axial: contrast is exactly zero 2% below lambda/d and present 2% above, every period tried : 4/4 PASS oblique: the threshold halves (NA ~ lambda / 2d), every period tried : 5/5 PASS every condenser point sampled: Abbe's sum agrees with Hopkins to 1% of the brightest point : max error 0.339% PASS the page's 60-point sampling stays within 5% of it : max error 0.34% PASS stop the first orders of a 1 um grating: the image repeats every 0.5 um (twice as many lines) : strongest line frequency 2 per um; power at 1 per um 0.0e+0 PASS Porter: a 45 degree slit turns the gauze into diagonal lines (constant along one diagonal, varying along the other) : along 1.7e-16, across 0.070 PASS experiment 6: pores 0.56 um (rows 0.485), dry NA 0.95, green: blank PASS ... blue 455 nm: the pattern appears; 465 nm: still blank (threshold 0.95 x 0.485 = 0.46 um) : 0.982 / 0.000 PASS Abbe's six spectra "about 65 degrees for blue": 0.484 um rows bend 439 nm light by 65 degrees, and his own dark blue (0.43 um) by 63 : 65.0 deg PASS axial, NA 0.8: light reaches 1.45 < 1.5 per um, and the images are identical : 0.0e+0 PASS open the aperture (NA 1.3, condenser 0.8): the images part : largest difference 0.095 PASS the test-object table: all 15 apertures recomputed from Abbe's spacings : 5/5 rows PASS the lithography table: k1 = resolution x NA / lambda, recomputed : 0.266, 0.318, 0.326 -- mutate.mjs: 9 of 9 mutations caught
The quotations are from the scanned volume on the Internet Archive; the page numbers are the journal's own. The translations are ours.
Sources
- E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung”, Archiv für mikroskopische Anatomie 9 (1873) 413–468. Scan: archive.org/details/archivfrmikros09berl.
- H. Helmholtz, “Die theoretische Grenze für die Leistungsfähigkeit der Mikroskope”, Annalen der Physik und Chemie, Jubelband (1874) 557–584. Scan: archive.org.
- A. B. Porter, “On the diffraction theory of microscopic vision”, Philosophical Magazine (6) 11 (1906) 154–166. doi:10.1080/14786440609463433.
- E. Abbe, “On Stephenson's system of homogeneous immersion for microscope objectives”, Journal of the Royal Microscopical Society 2 (1879) 256–265.
- Lord Rayleigh, “Investigations in optics, with special reference to the spectroscope”, Phil. Mag. (5) 8 (1879) 261–274; “On the theory of optical images, with special reference to the microscope”, Phil. Mag. (5) 42 (1896) 167–195.
- W. Volk, “Monument of Ernst Abbe in Jena”, Monuments on Mathematicians (the sphere and its formula).
- Nikon MicroscopyU, “Useful Magnification Range”.
- A. Evanschitzky, T. Fühner, A. Erdmann (Fraunhofer IISB), “Image simulation of projection systems in photolithography” (the Abbe and Hopkins formulations).
- ASML, EUV lithography systems and TWINSCAN NXT:2000i (figures as read 2026-09-25).
- D. H. Hamly and J. H. L. Watson, “Electron and Optical Microscope Interpretation of the Wall of Pleurosigma angulatum”, JOSA 32 (1942) 433. doi:10.1364/JOSA.32.000433.
- The Nobel Prize in Chemistry 2014.