Invisible Exactly in Focus

The stripes that vanished in focus

Zernike did not discover phase contrast at a microscope. He was studying diffraction gratings, and one of their faults. A ruling engine that cuts grooves with a slightly imperfect screw leaves a periodic error in their spacing, and a periodic error acts like a second, much coarser grating laid over the first. It throws faint copies of every spectral line to either side of it, the “Rowland ghosts”. Put your eye where a line falls and look back at the grating, and its surface looks striped. In 1902, H. S. Allen had written that the stripes were nothing real, only the line interfering with its ghosts. Zernike remembered “strongly objecting against his conclusion of unreality”. Around 1930 his laboratory set up a large concave grating, six metres from the eye, and he looked at it through a small telescope:

“Then the unexpected happened. The stripes were seen very clearly, but disappeared as the telescope was exactly focussed on the surface of the grating!”

(Zernike, Nobel Lecture, 11 December 1953, pp. 239–240.) The grating's error changes the path the light travels, not how much of it comes back. It is a phase grating, and that is the whole story in one experiment.

“In the ideal case”

The reason is exact, and short. Glass that is thicker in one place delays the light there. It does not dim it. A perfect lens in perfect focus puts every ray from a point of the slide back together at the matching point of the image with the same delay it had at the slide, and a delay you cannot see at the slide you cannot see in the image. Zernike quotes the physicist Otto Lummer's treatise on Abbe's theory (written with Fritz Reiche and published in 1910 as Die Lehre von der Bildentstehung im Mikroskop von Ernst Abbe), that “in the ideal case the microscope image is exactly similar to the object in structure and phase”, and draws the conclusion Lummer did not: then “the phase object is absolutely invisible « in the ideal case »” (p. 242).

The checker behind this page tests the sentence literally. It gives the engine the drawn cells below, which delay the light by up to 0.19 of a period, and a lens with no aperture limit at all, and asks for the image. All 65,536 pixels come back at the brightness of bare glass, to within a trillionth.

A real objective does have a limit, and what it throws away is what makes a transparent object faintly visible: the image is no longer quite a copy of the slide, so it is no longer quite blank. Which gives a strange result. The better the lens, the more completely it hides a transparent cell:

The fine adjustment

Microscopists had always seen unstained cells, of course. Zernike's account of how is one of the best sentences in the lecture:

“Of course the practical microscopist has never been content with this; as a matter of fact, he never found it out! Without realizing it, he had always turned the fine adjustment, that is, put the object a little out of focus, in order to see the tricky transparent details. Only a somewhat diffuse and watery image is obtained in this way.”

(p. 242.) Defocus a little and the brightness at each point changes by an amount proportional to the curvature of the delay there: light bends toward the thicker parts, as through a weak lens, and piles up or thins out. So a cell appears as a drawing of where its thickness bends, bright on one side of focus and dark on the other, and blurred, because it is out of focus. The checker compares a slightly defocused image with that curvature rule (the transport-of-intensity equation) and finds them within 0.07% of each other.

That second button shows something defocus cannot do. The slide is glass with a raised square on the left and an etched pit of the same depth on the right. Two micrometres out, one is light and the other dark. But which is which depends on which way you turned the knob. The checker proves the identity exactly: a bump 2 µm above focus gives the same image as a dent 2 µm below it, pixel for pixel, to within a trillionth (what difference there is, is rounding in the arithmetic). Unless you know which way you turned, you cannot tell a thicker region from a thinner one.

Dark field sees it, and squares it

The other old trick was to block the direct light altogether: a dark stop in the back focal plane. Then only light the specimen scattered reaches the eye, and the transparent cell shines on a black ground. It works, and it is used to this day, but it loses the sign completely. With the direct light gone, the image is the square of what the specimen scattered, and the light from a slide and the light from its exact opposite are complex conjugates of each other, mirror images in phase, and squaring erases the difference.

Bernard Lyot used exactly this from 1930 to 1938 to study the polish of his coronagraph lenses: a very small opaque screen where the direct light comes to a point, so that only the scattered wave is left to form the image. When he published in 1946 he listed its faults, and the last of them is this theorem, found at the bench: “de plus elle n'indique pas si les défauts sont en creux ou en bosse”, and besides, it does not tell you whether the defects are hollows or bumps (Comptes rendus 222, p. 766). In 1941 he replaced the screen with a plate that shifted the direct light by a quarter period instead.

A quarter of a period

Here is what Zernike saw that nobody had put to use. Split the light leaving any small detail into two parts: the direct light, which would be there if the detail were not, and the diffracted light, the difference the detail makes. For a detail that absorbs, the difference points straight against the direct light, and adding them gives something weaker: dark. For a detail that only delays the light, the difference points at right angles to it, a quarter of a period out of step, and adding something at right angles to a vector turns it without shortening it. Same brightness. Invisible. He had found exactly this quarter period in his gratings: “In the case of the Rowland ghosts the result was: their phases differ by ninety degrees from the principal line” (p. 241).

And the two parts are in different places. In the back focal plane of the objective the direct light is concentrated where the lamp's image falls, and the diffracted light is spread over the whole aperture. So a plate there can treat them differently. Zernike etched a groove into glass with very dilute hydrofluoric acid (a method of Lord Rayleigh's), “of a uniform depth of half a wavelength” (p. 241), which in glass of index about 1.5 changes the light's path by about half of that, a quarter of a wavelength. He laid it where the direct light falls, and shifted that light by a quarter period, bringing it into line with the diffracted light. Then adding them does shorten it. The transparent detail comes out dark, “as if the object had been stained”:

“The direct light is thus advanced by 90°, being then represented by MPh. This causes the detail to be represented by the vector-sum of MPh and MD’, making it darker than the background. Clearly the relations are about the same as in Fig. 1b, the transparent detail may be said to be « optically stained ».”

(p. 245.) For a thin detail that delays the light by a small angle φ (in radians), a plate that shifts the direct light by θ and lets through a fraction a of its amplitude gives, relative to the plate's own background:

contrast = (−2 a φ sin θ + φ²) / a²

At θ = 0 the first term vanishes and you are back to next to nothing. At 90° the detail is dark and its darkness is proportional to how much it delays the light, so the image is a map of thickness. At −90° (a plate that holds the direct light back instead) it is bright. The checker tests the engine against the exact form of this (with Bessel functions, for a grating) at three transmissions, five plate angles and four points, and they agree to within a trillionth.

Dimming the direct light

The formula has a on the bottom. A very thin detail scatters very little, and there is no way to make it scatter more; but you can make the direct light weaker, which makes what the detail scattered a larger share of the result. Zernike put a thin metal film on his strip:

“An absorption of 75 % is often used; the strip then transmits 25 % of the energy, or one half of the amplitude of the direct light. The contrast is thus doubled, a quite marked effect. In my own experiments I could go down to 4% transmission, i.e. a five times enhanced contrast, the limit being set by the unavoidable stray light.”

(p. 245.) The engine, on a grating that delays the light by 0.01 radian: 25% transmission gives 2.00 times the contrast of a clear strip, 4% gives 5.00 times. He adds that the French astronomer Bernard Lyot, who had found the method again independently for inspecting polished lens surfaces, “could use strips that diminished the amplitude to one thirtieth, so that ripples only one thousandth of a wavelength high showed in good contrast”. One thirtieth of the amplitude gives 30.0 times.

Lyot's own account is a four-page note in the Comptes rendus of the Académie des sciences for the sitting of 1 April 1946 (with R. A. Fisher and Wolfgang Pauli in the audience, the minutes record), and it agrees with Zernike's summary more closely than summaries usually do. His lenses were for a coronagraph, where every scratch scatters sunlight over the faint corona, and he looked at them in transmission. His first plates were taken “en affaiblissant un millier de fois l'onde directe”, weakening the direct light a thousandfold (in energy; 31.6 times in amplitude, close to Zernike's thirty), and showed the lens “couverte de sillons et de bosses, avec des contrastes très accusés, bien que la profondeur moyenne de ces défauts ne soit que 6 Å environ”: covered in furrows and bumps, in very marked contrast, though the defects averaged about 6 ångströms deep (p. 768). He also works one example with no dimming at all: a mirror defect two ångströms deep moves the reflected wave four, “et apparaît avec un contraste de 4π·4/5000 = 1 %” (p. 767). The engine, given a four-ångström step of wave in 5,000-ångström light, gives 1.005%. A footnote adds that he learned of Zernike's work only as he was publishing.

On a surface drawn with ripples of a thousandth of a wavelength of optical path, an in-focus brightfield image changes no pixel by even one part in ten thousand; a clear quarter-wave plate spreads it over 2.4% (brightest minus darkest, relative to the background); a plate at one thirtieth of the amplitude spreads it over 72%.

The halo, and why the plates are rings

Every phase-contrast picture has a signature: a bright glow around dark things, and big uniform regions that fade toward the background in their middles. Zernike explained the halo in 1953. The plate is meant for the direct light, but some of the light the detail scattered falls on it too, and gets the plate's treatment; that part, “because of the narrow strip”, would on its own form an image of very poor resolution, and taking it away from the proper image leaves “a very diffuse and weak negative image, appearing as a bright halo round dark details” (pp. 245–246). In this microscope, on the raised square, with a ring plate passing 25% (at the page's default aperture), the glass just outside the square's edge comes out 58% brighter than bare glass.

His first strips were straight, with a line of light to match, and he describes what that did to small bright spots: they looked “as if marked by short crossing pencil streaks”, because the strip's poor image is spread “only in one direction, namely perpendicular to the strip”. So he bent the strip into a ring and the light into a ring to match, and the halo spread in every direction instead, “so that it is much fainter and indeed quite harmless” (p. 246).

Where “optically stained” stops being true

The quarter-period argument assumes the detail delays the light by a small angle, so that its effect is a short arrow at right angles to the direct light. A thick cell is not a small angle. For a glass grating whose delay swings by ±ε, the stripe contrast under a clear quarter-wave plate is exactly 4 J₀(ε) J₁(ε), where J₀ and J₁ are Bessel functions. That grows with ε at first, stops growing at ε = 1.08 radians (about a sixth of a period), and past ε = 2.405, where J₀ crosses zero, the ridges that should be dark come out bright. In the drawn cells, the rounded cell gets darker as the drawing is thickened from a quarter of its thickness to twice it, then at three times comes out 9% brighter than the glass around it (at twice, it was -44%).

This is not a corner case. Quantitative phase measurements of live HeLa cells at 532 nm found an average peak delay of 2.89 radians across eighteen cells (Wang et al. 2013), well past the 1.08 where contrast stops growing, and a 2017 study of stem cells put it plainly: “the phase range of cells is over 3 rad. Thus, they cannot be considered as weak phase object” (Zuo et al. 2017). The drawn rounded cell reaches that at about 2.5x on the thickness slider. Phase contrast still shows such cells, vividly; what it shows is no longer a map of how much material is where.

“We would ourselves have invented it long ago”

In 1932 Zernike took the method, “still in the first somewhat primitive stage”, to the Zeiss works in Jena, the firm Ernst Abbe had built. “It was not received with such enthusiasm as I had expected. Worst of all was one of the oldest scientific associates, who said: « If this had any practical value, we would ourselves have invented it long ago »” (p. 242). Zernike's diagnosis was unkind and specific: Abbe's great achievements in microscopy all dated from before 1890, and his staff had formed “the tradition that everything worth knowing or trying in microscopy had been achieved already”. Abbe's own theory of the image, he adds, had “only applied to the oversimplified cases of a point source of light and an object of regular structure” and “did not explain the peculiarities in the imaging of transparent objects” (p. 243). Zeiss, “who had started with so little enthusiasm, slowly continued with the method”, and after “some years of developing too complicated instruments and after further delay by the War, they brought out phase-contrast objectives and accessories in 1941” (p. 246). The Nobel Prize in Physics followed in 1953, “for his demonstration of the phase contrast method, especially for his invention of the phase contrast microscope”.

Abbe's theory, the one Zernike found incomplete, is the engine under this page, unchanged. Its sibling, What the Lens Caught, is Abbe's microscope with stops you can put over single beams. Zernike's contribution was to let a stop be a complex number instead of a hole: one line of code in the engine, a × exp(−iθ), sixty years after Abbe.

About the drawn cells

The cells are drawn, not photographed. The drawing is ours; its dimensions are chosen inside published ranges, from different cell types, so no real cell is exactly this one.

• How much a cell delays light depends on how much denser than its medium it is. Live HeLa cytoplasm measured 1.3538 against a medium of 1.3370 at 532 nm by optical diffraction tomography (Kim & Guck, bioRxiv 2020, 1,565 cells), a difference of 0.017, which the drawing uses for the whole of both cells. The same study finds the nucleus slightly less dense than the cytoplasm and the nucleoli denser than both; the drawing keeps one index for cytoplasm and nucleus and draws the nucleoli denser (by 0.01) and a scatter of small granules denser again (by 0.03). Other reported values run higher (a 2017 review of the literature gives ranges of 1.360–1.380 for cytoplasm and 1.360–1.391 for nuclei).
• How thick. Living NIH 3T3 fibroblasts measured 4.2 ± 1.1 µm high by atomic force microscopy, and MDA-MB-231 cancer cells, which spread less, 7.4 ± 2.5 µm (Efremov et al. 2019). A lamellipodium, the thin leading sheet, is “thin (0.1–0.3 μm)” (Innocenti 2018). The adherent cell here is 0.2 µm at its edge, 1 µm in its body and 4 µm over its nucleus; the rounded cell is a dome 12 µm across and 6 µm high.
• The rods use the rule of thumb for E. coli, “a diameter of about ≈1µm, a length of ≈2µm” (BioNumbers), and a whole-cell index of 1.384 at 589 nm (Balaev et al. 2002) against water at 1.333: a difference of 0.051.
• So the drawing is on the thin side. Its thickest point, the top of the rounded cell, delays the light by 1.16 radians at 550 nm. The measured HeLa peaks above average 2.89. Use the thickness slider.

curl -L --create-dirs -o research/invisible-exactly-in-focus/verify.mjs https://artwaste.land/checks/research/invisible-exactly-in-focus/verify.mjs
curl -L --create-dirs -o public/strata/invisible-exactly-in-focus/engine.mjs https://artwaste.land/strata/invisible-exactly-in-focus/engine.mjs
curl -L --create-dirs -o public/strata/invisible-exactly-in-focus/index.html https://artwaste.land/strata/invisible-exactly-in-focus/
node research/invisible-exactly-in-focus/verify.mjs

verify.mjs: /strata/invisible-exactly-in-focus/

-- 1. The transform
PASS  1-D FFT equals the DFT sum (n = 64, random complex input)  :  7.9e-14
PASS  2-D forward then inverse returns the input (256 x 256)  :  6.7e-16

-- 2. Lummer's ideal case: a perfect, unlimited lens shows a transparent object as nothing
PASS  every one of the 65,536 pixels of the drawn cells comes out at exactly the background  :  65536 beams kept, largest deviation 1.8e-15
PASS  ... although the light through the thickest part is delayed by more than a tenth of a period  :  1.16 rad = 0.19 of a period

-- 3. A real lens: what little shows is what the aperture threw away, and a better lens shows less
PASS  rms contrast of the in-focus brightfield image falls every time the aperture opens (point light)  :  NA 0.1: 2.08%, NA 0.25: 0.84%, NA 0.4: 0.52%, NA 0.65: 0.31%, NA 0.95: 0.18%
PASS  at NA 0.95 the whole field varies by less than 1% rms  :  0.18%
PASS  closing the condenser diaphragm (sigma 0.7 to 0.2) raises it, as microscopists were taught  :  0.22% to 0.27%

-- 4. Zernike's stripes: a phase grating is invisible exactly in focus, and appears either side
PASS  in exact focus the grating's image has no stripes at all  :  fringe amplitude 0.0e+0
PASS  defocused, the stripes match the Bessel-function closed form at seven focus settings  :  max error 1.4e-16
PASS  they vanish again every 16.22 um of focus  :  0.0e+0, 2.2e-16
PASS  ... which is d^2 / lambda to within 1% (the paraxial Talbot spacing)  :  16.225 vs 16.364
PASS  half way between they are strongest, and opposite in sign either side of focus  :  0.0999 and -0.0999 (2 eps = 0.1)

-- 5. The fine adjustment: a little defocus shows the curvature of the phase (transport of intensity)
PASS  I - 1 = -(lambda z / 2 pi) laplacian(phi), to within 5% rms  :  relative rms error 0.07%

-- 6. Three images that cannot tell a bump from a dent, and one that can
PASS  in-focus brightfield: the slide and its exact opposite give the same image  :  1.6e-15
PASS  dark field: the same, to the last digit  :  2.1e-17
PASS  defocused: a bump 2 um above focus looks exactly like a dent 2 um below it  :  1.3e-15
PASS  ... but not like a dent at the same focus (so focus direction carries the sign)  :  0.139
PASS  phase contrast: the plateau comes out darker than the glass, the pit brighter  :  plateau -19.2%, pit 21.8%
PASS  in dark field the two squares carry the same light  :  5.784e-3 and 5.784e-3

-- 7. The quarter period: Zernike's strip against the closed form
PASS  engine = closed form for 3 plate transmissions x 5 plate phases x 4 points  :  max error 3.3e-16
PASS  the retarded crest under a strip that advances the direct light by 90 degrees is darker ("darker than the background")  :  0.6257
PASS  retard the direct light instead (theta = -90) and the same crest is brighter  :  1.4138
PASS  with no quarter-period shift (0 or 180 degrees) the crest differs from the background only in second order  :  0.0198, 0.0198 vs -0.3743

-- 8. The absorbing strip: Zernike's "doubled", "five times", and Lyot's one-thirtieth
PASS  a strip transmitting 25% of the energy (amplitude 1/2) doubles the contrast of a thin detail  :  2.000x
PASS  4% transmission (amplitude 1/5) gives five times  :  5.000x
PASS  Lyot's amplitude of one thirtieth gives about thirty times  :  30.00x
PASS  Lyot's own arithmetic: 4 pi x 4 / 5000 = 1.005%, and the engine gives a 4-angstrom wave step that contrast  :  1.005% stated, 1.005% computed
PASS  his first plates weakened the direct light "un millier de fois": sqrt(1000) = 31.6 times the contrast  :  31.62x
PASS  ripples of a thousandth of a wavelength: brightfield in focus moves no pixel by even 0.01%  :  largest change 0.00000%
PASS  ... a plain quarter-wave strip shows them at about 1%  :  range 0.9879 to 1.0120
PASS  ... and a strip that also cuts the direct light to 1/30 shows them at tens of per cent ("in good contrast")  :  range 0.670 to 1.390 of the background

-- 9. The halo, and why Zernike made his strips round
PASS  outside the dark plateau, the glass comes out brighter than bare glass: the halo  :  peak just outside the edge 57.8% above background
PASS  inside, the middle of the plateau is paler than its rim: the shade-off  :  middle 8.9%, just inside the edge -35.0%
PASS  with a straight strip (along y) the halo of a small particle is drawn out across it, along x ("short crossing pencil streaks")  :  x-spread / y-spread = 19.29
PASS  with an annular strip it spreads evenly  :  x-spread / y-spread = 0.93

-- 10. Where "optically stained" stops being true: thick specimens
PASS  the engine agrees: stripe contrast is 4 J0 J1 at eps = 0.5, 1.08, 2.0, 3.0
PASS  contrast stops growing at a swing of 1.08 rad  :  1.082 rad = 0.172 of a period
PASS  and reverses (crests turn bright) past 2.405 rad, where J0 = 0  :  amp(2) = 0.516, amp(3) = -0.353
PASS  the rounded cell does not get steadily darker as it is drawn thicker  :  -11.3%, -21.9%, -39.3%, -43.9%, 9.1%

-- 11. The drawing's stated dimensions
PASS  adherent cell: at most 4 um high (lamella 1 um + nucleus dome 3 um)  :  3.99 um
PASS  rods: 1 um thick  :  1.00 um
PASS  rounded cell: 6 um high  :  6.00 um

-- 12. The page's printed numbers, read back out of its HTML
PASS  all 23 numbers the page prints (marked data-v) equal the recomputation
PASS  and the page prints no data-v number that this file does not check

44 passed, 0 failed

mutate.mjs: /strata/invisible-exactly-in-focus/
CAUGHT  plate shifts the wrong way (exp(+i theta))  (11 checks failed)
CAUGHT  plate dims energy, not amplitude (a^2 used as amplitude)  (6 checks failed)
CAUGHT  defocus uses the paraxial parabola  (3 checks failed)
CAUGHT  defocus sign flipped  (4 checks failed)
CAUGHT  specimen phase taken as advance, not delay  (14 checks failed)
CAUGHT  intensity taken as (re + im)^2 instead of re^2 + im^2  (16 checks failed)
CAUGHT  aperture edge off by a bin  (1 check failed)
CAUGHT  plate ring covers only its inner half  (2 checks failed)
CAUGHT  cells drawn with twice the index difference  (1 check failed)
CAUGHT  ring of light loses its symmetry (plain rounding)  (4 checks failed)

10 of 10 caught

What this microscope leaves out

The specimen is thin: all it does is delay and dim the light passing straight through it, where a real cell 6 µm thick also bends light inside itself. The light is scalar (no polarisation) and one colour. The lens is perfect. The ring of light is 32 points, the plate is a hard-edged ring whose radius follows the aperture (0.55 of it) where a real objective's ring has a fixed size and a manufactured profile, and the display is linear. None of these changes the quarter-period argument, the sign theorems, or Zernike's factors of two and five; all of them change the exact shape of a halo, which is why the halo's numbers here describe this microscope and not yours.

Sources

1. F. Zernike, “How I discovered phase contrast”, Nobel Lecture, 11 December 1953; in Nobel Lectures, Physics 1942–1962 (Elsevier, 1964), pp. 239–246. Read in full from the Nobel Foundation's PDF (nobelprize.org). Every quotation on this page is from it and located by its printed page.
2. The Nobel Prize in Physics 1953, prize motivation: nobelprize.org.
3. B. Lyot, “Procédés permettant d'étudier les irrégularités d'une surface optique bien polie”, Comptes rendus de l'Académie des sciences 222 (séance du 1^er avril 1946), pp. [765]–768 (the first page carries no number; it precedes p. 766). Read in a scan of the printed pages (astrosurf.com); the quotations are from pp. 766–768.
4. F. Zernike, “Diffraction theory of the knife-edge test and its improved form, the phase-contrast method”, Monthly Notices of the Royal Astronomical Society 94 (1934) 377–384, doi:10.1093/mnras/94.5.377; and “Phase contrast, a new method for the microscopic observation of transparent objects”, Physica 9 (1942) 686–698, doi:10.1016/S0031-8914(42)80035-X. Listed for the reader; not re-read for this page.
5. K. Kim & J. Guck, “The relative densities of cell cytoplasm, nucleoplasm, and nucleoli are robustly conserved during cell cycle and drug perturbations”, bioRxiv 2020, 10.1101/2020.04.14.040774 (a preprint).
6. Y. M. Efremov et al., Scientific Reports 9 (2019), s41598-019-42077-1. M. Innocenti, Cell Adhesion & Migration (2018), PMC6363039.
7. “How big is an E. coli cell and what is its mass?”, Cell Biology by the Numbers. A. E. Balaev, K. N. Dvoretski & V. A. Doubrovski, Proc. SPIE 4707 (2002) 253–260, doi:10.1117/12.475627 (read in its abstract).
8. Wang et al., Computational and Mathematical Methods in Medicine (2013), PMC3568911 (peak phase of live HeLa cells, 532 nm, 18 cells). C. Zuo et al., Scientific Reports 7 (2017), PMC5550517. The 1.360–1.391 ranges: PMC5451484 (Sci. Rep. 2017).
9. O. Lummer & F. Reiche, Die Lehre von der Bildentstehung im Mikroskop von Ernst Abbe (Vieweg, Braunschweig, 1910); reviewed in Nature 87 (1911), 087141a0. Not read for this page; named because Zernike quotes it.
10. The sibling layer, What the Lens Caught, for Abbe 1873 and the method of computing the image.
11. The Talbot effect: H. F. Talbot, “Facts relating to optical science. No. IV”, Philosophical Magazine 9 (1836) 401–407, doi:10.1080/14786443608649032, cited for the name only; the distance on this page is computed, not quoted.