BrainFlux

Imaging: Abbe and phase contrast

Object

t = a e^iφ

Paint on the object with Add ink

Fourier plane

Paint on the Fourier plane: Block an order (a numbered dot) and watch the image

Image

intensity |E|²

Profile

the image's brightness along the dashed line (solid) against the object's (dashed); drag on the image to move the line

Experiments

Start here

A microscope, taken apart: the light that leaves the object (left) is sorted by direction in the Fourier plane (middle), where the lens's aperture and whatever you put there decide what goes on to make the image (right). The sketch at the top shows the same from the side.

Paint on the Fourier plane with Block over one of the numbered dots (the grating's orders) and watch the image; slide the NA in panel 3 to open and close the aperture; paint on the object with Add ink; drag on the image to move the profile line.

Look for this: the image is made by the orders that get through, not copied from the object. Take away the first orders and the grating is gone; take away the zero order (the one in the middle) and it has twice as many lines.

1. Abbe's experiment: the image is made of the orders that get in

A grating of period $d=1.28~\mu\mathrm{m}$ in bright field (nothing in the Fourier plane but the aperture, NA 0.5). It sends its light in a few directions only, $\sin\theta=m\lambda/d$: the orders, single points in the Fourier plane, ringed and numbered (white ones get through the aperture, red ones are blocked; the sketch shows the same). The zero order, $m=0$, is the light that went straight through; the orders $\pm1$ get in too, and the image is a sinusoid of period $d$, which the profile shows against the square bars of the object. Slide the NA (panel 3) down past $\lambda/d=0.43$: the $\pm1$ orders turn red, and the image is an even grey, $I=0.250$ everywhere, the square of the grating's average transmission, ½. The light is all there; the grating is not. Slide it up past $3\lambda/d=1.29$ and the $\pm3$ orders come in too (equal bars and gaps have no even orders): the bars get sharper edges. This is Abbe's rule (1873): with the light along the axis, a period shows only if its first orders get in, $d\ge\lambda/\mathrm{NA}$.

2. Block the zero order and the period halves

The aperture (NA 0.6) passes the orders $-1$, $0$ and $+1$, and a small stop in the middle blocks the zero order. What is left is the two first orders, $E\propto\cos(2\pi x/d)$, whose intensity $\cos^2(2\pi x/d)=\frac12\left[1+\cos(4\pi x/d)\right]$ repeats every $d/2$: the image has twice as many lines as the grating (the profile reads 0.640 µm). The image is made by the orders interfering; it is not a copy of the object. Choose Open to compare.

3. Oblique light doubles the resolution

A finer grating, $d=0.8~\mu\mathrm{m}$: its first orders, at $\sin\theta=\lambda/d=0.69$, are outside an NA 0.4 aperture, and light along the axis shows nothing (set the tilt in panel 1 to 0 to see). Here the light comes in tilted, at $\sin\theta=0.35$: every direction leaving the object tilts with it, so the whole pattern in the Fourier plane slides over. The zero order moves to one side of the aperture and the $+1$ order to the other, both inside, and the two make fringes of period $d$. Drag in the Fourier plane with Tilt the light and watch the grating come and go. With the tilt at $\sin\theta=\mathrm{NA}$, the finest period is $d=\lambda/2\mathrm{NA}$, half of Abbe's limit. A condenser lights the object from every side at once: set the tilt to 0, choose Condenser with NA 0.4, and the grating shows again, with less contrast (0.08, against 0.85 here), because only the directions on one side bring an order in with them.

4. A diatom, the classic test object

The silica shell of a diatom is pierced by a lattice of pores whose rows are $h=0.52~\mu\mathrm{m}$ apart; nineteenth-century microscopists judged their objectives by such shells. The dashed rings mark the lattice's six first orders, at $\sin\theta=\lambda/h=1.06$. At NA 0.65 they are outside the aperture: the outline and the midrib show, but the pores are gone. Raise the NA past 1.06 (an oil-immersion objective) and they appear. Or keep NA 0.65 and tilt the light (sin θ 0.53, direction −65°): one order joins the zero order in the aperture, and one family of rows appears, as stripes. Then make the pore spacing smaller and find where the pores disappear again.

5. A transparent cell is invisible

These cells only delay the light, by up to 0.3 rad; $|t|=1$ everywhere. With the aperture wide open (NA 1.4) the image is even to 0.15%: $I=|e^{i\varphi}|^2=1$, nothing to see, though the object's colours (its phase) show the cells plainly. Close the aperture to NA 0.25: the light that the cells' steep edges send out at wide angles is lost, and faint rims appear (contrast 0.006: choose Stretch contrast under the image to see them). The Fourier plane shows why phase objects are hard to see: nearly all their light is in the zero order (99.7%, says the sketch's legend).

6. Defocus makes it faintly visible

The same cells, with the microscope focused $z=1~\mu\mathrm{m}$ beyond them (the pupil gains the phase $e^{-i\pi\lambda zf^2}$: each direction travels a little further). A cell is thicker in the middle, so it delays the light most there, like a weak converging lens: just past it, the light bunches into its middle. Its nucleus is 3% brighter than the background and its rim darker. For small $z$ (the transport-of-intensity equation) $$I\approx1-\frac{\lambda z}{2\pi}\nabla^2\varphi,$$ and $\nabla^2\varphi<0$ where $\varphi$ peaks. Slide $z$ through zero to $-1$ µm and the contrast reverses: the nucleus is 3% darker. Biologists did this for a century: a slightly defocused microscope shows transparent cells.

7. Dark field

A stop of radius 0.08 NA in the middle of the Fourier plane removes the zero order (the light that went straight through) and the lowest frequencies. Only 0.73% of the light gets through, but these letters, written in phase alone, now glow on black at their edges, where the phase changes fastest. (With Zero order only the whole letters shine, as $I\approx(\varphi-\langle\varphi\rangle)^2$.) A real dark-field microscope instead lights the object from beyond the objective's NA, so the direct light misses the aperture altogether: choose Open and set the tilt to 0.6 (more than NA 0.5); the letters glow again, lit from one side.

8. Zernike's phase contrast

A transparent object, $t=e^{i\varphi}\approx1+i\varphi$, is two waves. The 1 is the light that goes straight through, the zero order, bright in the middle of the Fourier plane. $i\varphi$ is the weak light the object diffracts, and the factor $i=e^{i\pi/2}$ puts it a quarter period behind. Waves a quarter period apart don't change each other's brightness: $|1+i\varphi|^2=1+\varphi^2$, nothing to first order. Zernike's dot multiplies the zero order alone by $e^{i\alpha}$: $$I=|e^{i\alpha}+i\varphi|^2\approx1+2\varphi\sin\alpha.$$ With $\alpha=-90°$ the dot advances the direct light a quarter period, so the diffracted light is now half a period behind it and cancels some of it: $I\approx1-2(\varphi-\langle\varphi\rangle)$. The thick parts of the cells, which delay the light most, look dark: this is positive phase contrast, the usual kind. With $+90°$ (a delay) they look bright. Only differences from the average phase $\langle\varphi\rangle$ show, because the zero order carries the average. The profile compares the image with the formula: at NA 0.5 the image is a blurred version of it; at NA 1.4 they agree to within 0.073. Raise $\varphi_0$ and watch the formula fail. A dot that also absorbs, transmitting $a$, gives $I\approx a^2\mp2a(\varphi-\langle\varphi\rangle)$: relative to the dimmer background $a^2$ the contrast is $1/a$ times higher (with $\varphi_0=0.1$, $a=0.5$ takes it from 0.103 to 0.213). Zernike, 1935; Nobel prize, 1953.

9. The halo

A real phase dot has a size (here 0.1 NA), and it shifts not only the zero order but the lowest frequencies of the object too: the ones that make up a large cell's broad, smooth inside. Those are then imaged as in bright field, invisibly, so the middle of a big cell fades back towards the background ("shading off") and a bright halo rings it. Make the dot bigger (0.2) and the cells all but vanish (the contrast falls from 0.23 to 0.09), leaving bright rims; make it smaller and the halo fades. Real microscopes need a ring wide enough to catch all the condenser's direct light, which is why phase-contrast pictures of cells glow at the edges.

10. Schlieren: a knife edge

Foucault's knife edge blocks one half of the Fourier plane and keeps the zero order. For a weak phase grating, $\varphi=\epsilon\cos kx$, $t\approx1+\frac{i\epsilon}{2}\left(e^{ikx}+e^{-ikx}\right)$; the knife removes the $e^{-ikx}$ order and leaves $$I\approx\left|1+\tfrac{i\epsilon}{2}e^{ikx}\right|^2\approx1-\epsilon\sin kx=1+\frac1k\frac{\partial\varphi}{\partial x}:$$ where the phase rises towards the kept side (to the right, with the knife at 0°) the image is brighter, where it falls darker. The cells look lit from one side, in relief (Hoffman's modulation contrast microscope, 1975, refines this for living cells). Turn the knife's angle and the light swings round. With the Condenser on, the knife also cuts the image of the light source, as in the schlieren cameras that photograph shock waves and hot air.

11. The hidden portrait

Fourier's portrait written in phase alone, 0.3 rad at the darkest ink: the image shows almost nothing (a contrast of 1.5%). Choose Phase dot in the Fourier plane and he appears (contrast 0.32). The $-90°$ dot shows him as engraved, dark ink dark; $+90°$ as a negative.

12. Coherent or incoherent light

The bar target in coherent light at NA 0.5. Periods finer than $\lambda/\mathrm{NA}=1.1~\mu\mathrm{m}$ should vanish, and from the 0.79 µm group on they do: grey blocks (the 0.79 and 0.63 µm groups keep 5% and 4% of their bars' contrast). The 1 µm group, just past the limit, keeps 44%: three bars are not an endless grating, and some of their spread of directions gets into the aperture. Edges ring. Beside a single straight edge the light overshoots by 19% (try Edge: $I$ peaks at 1.190); between close bars the ripples add, and the image reaches $I=1.664$ (see the readout), though nothing in the object transmits more than 1. Choose Incoherent: now intensities add, not fields, and the transfer function is the pupil's autocorrelation (the OTF, the optical transfer function: choose show the OTF over the Fourier plane), reaching twice as far, to $\lambda/2\mathrm{NA}=0.55~\mu\mathrm{m}$, but falling off gradually. The 0.79 and 0.63 µm groups come back, with 16.5% and 11.8% of their contrast; the 0.5 µm group, finer than that, stays grey. Nothing overshoots ($I$ peaks at 0.997): an incoherent image is never brighter than the brightest part of the object. A condenser is in between: with NA 0.25 the image still rings a little ($I$ peaks at 1.27), and periods down to $\lambda/(\mathrm{NA}+\mathrm{NA_c})=0.73~\mu\mathrm{m}$ get through.

13. Where resolution runs out: the Siemens star

The star's 18 dark spokes are a grating whose period shrinks towards the centre, as $2\pi r/18$. Inside the dashed circle, $r=\frac{18\lambda}{2\pi\,\mathrm{NA}}=3.15~\mu\mathrm{m}$, it is finer than $\lambda/\mathrm{NA}$ and the star turns grey. In Incoherent light the circle shrinks by half. Then set the defocus to 4 µm: between $r\approx3$ and 6 µm the spokes come back faintly, with black and white swapped (choose Stretch contrast to see them). The OTF view shows why: defocus makes the transfer function negative in a ring (cyan). This is "spurious resolution", and a reason to focus on something with sharp edges.

14. Two points and Rayleigh's criterion

Two pinholes $0.6~\mu\mathrm{m}$ apart at NA 0.56: Rayleigh's limit, $0.61\lambda/\mathrm{NA}=0.599~\mu\mathrm{m}$, where the first dark ring of one point's image falls on the other. In incoherent light (two stars, two fluorescent molecules) the profile dips by 26.6% between them (26.5% for ideal points): just resolved. In coherent light the amplitudes add first: in step, the two make one blob, brightest in the middle; set the phase to 180° and a dark line separates them however close they are. Coherent resolution depends on the object's phase.

How it works

A microscope is two Fourier transforms; the sketch at the top of the page shows it from the side. Light of wavelength $\lambda$ (panel 1) passes through the object (2), whose transmission $t(x,y)=a\,e^{i\varphi}$ can dim the light (the amplitude $a$) or delay it (the phase $\varphi$). The objective lens (3) sorts the light by direction: in its back focal plane, the Fourier plane (4), light that left the object at angle $\theta$ to the axis arrives a distance $F\sin\theta$ from the axis ($F$ is the lens's focal length), whichever point of the object it came from. So each point of that plane is one direction, a spatial frequency $f_x=\sin\theta/\lambda$, and the plane holds the object's Fourier transform $T(f)$. The light that goes straight through, the zero order, is focused to the middle. The lens's aperture is a circle there, of radius $\mathrm{NA}/\lambda$ in frequency (the numerical aperture $\mathrm{NA}=n\sin\theta_\text{max}$, with $n$ the index between the lens and the object): light outside it never reaches the image. The tube lens transforms back: $$E(x,y)=\mathcal{F}^{-1}\left[P(f)\,T(f)\right],\qquad I=|E|^2,$$ where the pupil $P$ (the aperture and whatever is in it) is 1 inside the aperture and 0 outside, times whatever you put in the plane (panel 4) or paint on it.

In the Fourier plane the colour is the phase of $T$ (red real and positive, yellow-green $+i$, cyan negative, violet $-i$) and the brightness its size; what the pupil blocks is dimmed. The object is drawn as its transmitted intensity $|t|^2$ in grey, or as its phase delay $\varphi$ in colour. Waves go as $e^{i(kz-\omega t)}$, so a positive $\varphi$ is a delay: light through a thicker or denser part of a cell arrives later. The field of view is $25.6~\mu\mathrm{m}$ across, on 256 × 256 points.

Things to know

Scalar, thin-object, paraxial optics with perfect lenses: the object is a mask, the light has one polarisation, and the lenses add no aberrations of their own. Defocus is the paraxial phase $e^{-i\pi\lambda zf^2}$; at high NA the true one differs. An NA above 1 means an immersion objective, with the object in oil; the Fourier plane's $\sin\theta$ is then $n\sin\theta$.
In the sketch each lens is a focal length $F$ from the planes on either side of it (a "4f" system). It is not to scale across the axis: up the page is $x$ (or the direction of the tilt), and a direction $n\sin\theta$ is drawn at a height proportional to it, as in the Fourier plane.
The image is drawn upright (a real 4f system turns it over) and at the object's scale.
Everything is periodic on the 256 × 256 grid: light leaving one edge comes back at the other. The grating's period is rounded so that a whole number of periods fits.
The condenser's light is a sum over up to 40 directions inside its NA (Abbe's own method), added a few at a time, so the image settles over a few frames. Incoherent light uses the OTF and is exact.
The image's grey: by default its brightest point is white; Clear glass = white fixes $I=1$ at white, so dimmer images look dimmer; Stretch contrast makes the darkest point black and the brightest white, to see faint contrast. The tag in the corner says which intensity is white, and the readout gives the true intensities.
The profile's dashed curve is the object, $|t|^2$; for a transparent object it is its phase (right-hand axis) or, with a phase dot, the small-phase formula. Drag on the image to move the line.
Painting the Fourier plane: Block makes it opaque, Pass clear (even outside the aperture), $-90°$ and $+90°$ shift the phase a quarter period, and Unpaint goes back to what panel 4 sets. Mirror paints the opposite point too.

New in 2026 for BrainFlux. Source: github.com/gallicchio/brainflux, MIT licence.