Illustrated Physics
physics, drawn
Series

Imaging — the series

15 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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×.

    What a lens is doing, and why three rays are enough

    A lens bends every ray that reaches it. The construction uses three, because three are all that can be drawn without calculation — and any three that meet prove all the rest do.

    part 1 · optics
  2. A sphere does not have a focus. Parallel rays reflected off a spherical mirror. Rays striking further from the axis cross it nearer the mirror, so there is no single point where all of them meet — the blur is spherical aberration, and a perfect sphere has it inherently.

    The mirror that cannot focus, and the shape that can

    A perfect sphere does not bring parallel light to a point. The blur is not a manufacturing defect — it is what the shape does, and the shape is used anyway, for a reason worth knowing.

    part 2 · optics
  3. What arrives in the back focal plane. A grating of pitch 1.2 µm illuminated at 550 nm, and the spectrum that appears in the objective's back focal plane. Each spatial frequency in the object leaves at its own angle, sinθ = mλ/d, and the bar heights are the Fourier coefficients of the object's transmittance. The aperture admits everything inside sinθ = 0.65, which here is orders -1, 0, 1 — 2 of them carrying information about the pitch. Orders outside it are drawn faint and are simply lost: they never reach the image plane, and no amount of magnification afterwards recovers them. This is where a microscope's resolution is decided — not at the image, not by the eyepiece, but by which of these bars the front of the objective is wide enough to catch.

    The image that is a diffraction pattern twice

    A lens does not project an object onto a screen. It takes the object's spatial frequencies apart, spreading them across its own back focal plane at an angle each, and then puts them back together — so an image is the object's spectrum, filtered by whatever the aperture admits, transformed back. With only the zeroth order through, the image is a uniform grey with no information in it at all.

    part 3 · optics
  4. Abbe's ratio, measured on the traced rays. The quantity h divided by the sine of the angle at which the ray converges on the focus, in units of the paraxial focal length, against how far up the aperture the ray entered. Abbe's sine condition says that a system already free of spherical aberration images a small region round the axis faithfully only if this ratio is the same for every ray. A horizontal line means the condition is met. The parabola departs by 12.96 per cent across the aperture; The sphere departs by 7.18 per cent across the aperture. The paraboloid is the interesting case, because it is exactly stigmatic on axis — every ray from infinity crosses at one point, which is the definition of the shape — and it still fails this test. Perfection at one point buys nothing at the next one along. What the departure predicts is coma, a blur that grows linearly with the distance off axis and quadratically with the aperture, and the offaxis figure measures exactly that blur on the same surfaces. The condition is not a design rule invented for telescopes: it follows from requiring that the same optical path length join object and image for every route, and any instrument that images a field rather than a point has to meet it.

    The condition a lens must meet

    A paraboloid brings every parallel ray to exactly one point. Move the source a fifth of a degree off axis and the image is a fan rather than a point, and the reason is a condition Abbe wrote down that has nothing to do with the axis — perfection at one point buys nothing at the next one along.

    part 4 · optics
  5. Image distance against object distance. Image distance in focal lengths against object distance in focal lengths. At exactly one focal length the image runs off to infinity; inside it the image distance goes negative, which means virtual.

    The focus that is a slab, not a plane

    A lens images one plane and no other, which would make every photograph and every micrograph almost entirely out of focus. What rescues them is a tolerance — and there are two of them, one from rays and one from waves, which give different answers and stop being interchangeable exactly where microscopes work.

    part 5 · optics
  6. The bowl a lens actually focuses onto. Where a lens of 50 mm focal length brings each part of a flat scene to a focus, against distance from the centre of the field, out to 21.6 mm — the corner of a 35 mm frame. The surface of best focus is a sphere of radius 76 mm curving towards the lens, and the corner of the frame sits 3.13 mm in front of the plane the centre is focused on. The shaded band is the depth of focus at f/8, which is 0.070 mm — 45 times smaller than the sag it has to cover. The curvature is not an error in the lens. It is what Σ1/nf comes to for this stack, and it depends on the powers and the glasses and on nothing else: bending the surfaces, moving the stop or stopping down changes every other aberration and leaves this one exactly where it was.

    The flat scene that comes back curved

    A lens does not image a plane onto a plane. It images it onto a bowl, and the curvature of that bowl is fixed by the powers and the glasses alone — not by the shapes of the surfaces, not by where the stop is, and not by stopping down. Everything a designer usually plays with leaves it exactly where it was.

    part 6 · optics
  7. Stopping down removes one colour error and not the other. The two chromatic errors of a single 50 mm N-BK7 lens with its stop 20 mm in front, at an image point 10° off axis, against the f-number. Axial colour — blue and red focusing at different distances, seen as a coloured blur at the yellow focus — is about 775/N µm across: 387 µm at f/2, 97 at f/8, 35 at f/22. Lateral colour — blue and red images of the point centred at different heights — is 55.0 µm at every f-number, because it is carried by the ray through the centre of the stop. The two are equal near f/14, and past that stopping down leaves only the error it cannot touch.

    The colour fringe no aperture can close

    A lens bends blue light more than red, and that one fact produces two different faults. One puts the blue focus nearer the lens than the red, and closing the aperture shrinks it. The other puts the blue image of an off-axis point nearer the centre of the frame than the red one, and closing the aperture does nothing at all, because it is carried by the one ray every aperture keeps. It is why photographs have coloured fringes in their corners and not in their middles, and why a lens designer who cannot remove it with glass can sometimes remove it with symmetry, or with a distance.

    part 7 · optics
  8. Too small blurs by diffraction, too large by shadow. The size of the image of a distant point through a pinhole 100 mm from the screen, in 550 nm light, against the hole's diameter, both on logarithmic axes. The dashed lines are the two simple estimates: the hole's own shadow, as wide as the hole, and the diffraction spread, 2.44λf/d. The dots are the diameter holding half the light, computed from the Fresnel diffraction integral. They follow the shadow for large holes and the diffraction for small ones, and are least — 0.207 mm — for a hole 0.336 mm across, where the simple sum of the two estimates puts its minimum at 0.366 mm. At the best hole the Fresnel number d²/4λf is 0.51: the sharpest hole, judged by where half the light lands, lets through about half of the first Fresnel zone.

    The hole that makes the sharpest picture

    A pinhole camera makes a picture without bending any light: a small hole lets through, from each point of the scene, only the rays that reach one small patch of the screen. A smaller hole makes a sharper picture — until it does not. Below a certain size the hole spreads light by diffraction faster than it narrows the shadow, and the picture blurs again. The best hole balances the two, and at that balance it turns out to pass about half of the first Fresnel zone of the incoming wave: the sharpest pinhole is already beginning to behave like a lens.

    part 8 · optics
  9. Four cosines between a lens and the edge of its picture. Light from a direction 30° off the axis passing through a lens and landing at the edge of the image, with the four factors that dim it relative to the centre. The aperture, seen from that direction, is foreshortened to cos θ of its area. The image point is further from the lens, f/cos θ instead of f, and the light spreads as the square of the distance: cos²θ. The light arrives at the image plane at a slant, spreading over 1/cos θ more area: one more cos θ. Together, cos⁴θ: at 30° the edge receives 0.563 of the light at the centre, 0.83 photographic stops darker. Nothing here is a fault of the glass. A perfect thin lens, or a pinhole, does it, because it is geometry.

    The four cosines at the edge of a photograph

    Photograph a plain, evenly lit wall with a wide-angle lens and the corners of the picture come out darker than the middle — by more than three photographic stops with a 14 mm lens on a full-frame camera. It is usually blamed on the lens. A perfect lens does it too, and so does a pinhole, because the light reaching the edge of an image is reduced by four separate cosines of the angle it came from: three of them pure geometry, and one the slant at which it lands. Fisheye lenses avoid most of the loss by refusing to keep straight lines straight. Every wide lens is a choice between the two.

    part 9 · optics
  10. Three planes that meet in one line. A thin lens of focal length 50, drawn as a vertical line, and an object plane tilted at 45° to the axis, crossing it 150 units in front of the lens. Four points on the object plane are traced to their images through the lens's centre and along the axis-parallel ray. The images lie on a straight line too — a thin lens maps planes to planes — but one tilted far less steeply, and when the object plane, the lens plane and the image plane are extended they meet in a single line, here 150 units below the axis. That is the Scheimpflug rule. A sensor laid along the tilted image plane records the whole tilted object plane sharp, which no sensor square to the axis can do.

    The plane of focus a tilted lens tilts

    An ordinary camera focuses on a plane square to its axis, so a road receding from it, a table top or a field of flowers can be sharp only over the depth an aperture allows. Tilt the lens, or the sensor, and the plane of sharp focus tilts too, by a rule a surveying officer patented in 1904: the plane of the subject, the plane of the lens and the plane of the image always meet in a single line. Turn the focus and the plane of focus does not slide back and forth; it swings about a fixed hinge. The same geometry lets an eye camera see the whole front of the eye in one exposure and a lidar keep a laser beam kilometres long in focus.

    part 10 · optics
  11. The cavities that hold a beam. Every two-mirror cavity as a point (g1, g2), with g = 1 − L/R for each mirror. The shaded region, between the axes and the hyperbolas g1 g2 = 1, holds a ray for ever; everywhere else a ray is walked out. The commonest designs sit on the boundary: two flat mirrors (1, 1), the confocal cavity with the mirrors' foci together (0, 0), the concentric cavity with their centres together (−1, −1), and a flat mirror facing a curved one at its own radius (1, 0). A boundary cavity is stable in principle and marginal in practice, which is why the confocal one, sitting at the corner where the two shaded regions touch, is usually built a little inside it.

    The two mirrors that hold a beam

    A laser is two mirrors facing each other, and the light in it makes thousands of round trips before it leaves. Whether a ray stays between the mirrors for ever or is walked out after a few bounces depends on a single number, built from their spacing and their curvatures, and the answer has a sharp edge: two flat mirrors sit exactly on it, which is why the first lasers were so hard to align. The criterion is the one that decides whether a pumped swing grows, whether a wave crosses a crystal, and whether a particle beam stays in an accelerator.

    part 11 · optics
  12. Sturm's conoid: a line, a circle, a line. The cross-section of a parallel beam after a lens whose focal length is 100 mm in its vertical meridian and 125 mm in its horizontal one, traced ray by ray through a round aperture, at seven distances behind the lens: 60.0, 100.0, 105.6, 111.1, 118.1, 125.0, 135.0 mm. There is no point focus anywhere. At 100 mm the vertical meridian has focused and the beam is a horizontal line; at 125 mm the horizontal meridian has, and it is a vertical line. Between them the beam is an ellipse that turns from wide to tall, and at 111.1 mm — the harmonic mean of the two focal lengths — it is a circle, the circle of least confusion, of diameter 22.2 per cent of the aperture's.

    The focus that is two lines

    A lens curved more steeply in one direction than the other has no focus. It brings light to a short line at one distance, to a line at right angles at another, and in between to an ellipse that turns from wide to tall and is round at exactly one place. Most eyes are built this way to some degree, so do mirrors used at an angle, and the best any ordinary lens can do for them is to put that round patch on the retina. A cylinder, curved in one direction only, can do what no sphere can: move one line onto the other.

    part 12 · optics
  13. A transparent cell, seen four ways. Brightness across the image of a transparent cell 20 μm across, which delays the light passing through its middle by 0.3 radians (and its nucleus by a third as much again) and absorbs none of it, imaged with green light (0.55 μm) through an objective of numerical aperture 0.25; brightness relative to the background. In focus with a plain objective (dashed) the cell is all but invisible: the brightness varies between 0.998 and 1.001. Defocused by 8 μm it shows, 0.96 to 1.06, but only at its edges and nucleus, as bright and dark fringes that do not follow its thickness. With a quarter-wave plate on the undiffracted light the cell appears in focus, its brightness following its thickness: with the plate one way it is brighter where it is thicker, up to 2.02 times the background; with the plate the other way, darker, down to 0.38.

    The specimen that is invisible until its light is delayed

    A living cell in water is nearly transparent: it delays the light passing through it by a fraction of a wavelength and absorbs almost none, and a perfect microscope shows it as nothing. The delay is there in the light, and the eye cannot see a delay. Frits Zernike's remedy was to delay a different part of the light — the part the cell did not scatter — by a further quarter of a cycle, at the one place in a microscope where it is separated from the rest. The invisible phase becomes brightness, the cell appears, and the price is a bright halo round everything large.

    part 13 · optics
  14. Three shapes of one cell, delaying light by the same total. 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.

    The cell weighed by the light it delays

    A living cell is nearly transparent, and the light that passes through it comes out almost as bright as it went in. What changes is its timing: the cell's proteins and nucleic acids raise its refractive index a little above the water around it, and the light is delayed by a fraction of a wave. Measure that delay at every point of the image and add it up, and the sum is the cell's dry mass — the weight of everything in it except water — whatever shape the cell takes and however much water it holds. The water drops out because the medium is water too; the shape drops out because the delay is an integral of concentration along the path. Light becomes a balance that weighs a living cell to a part in a thousand without touching it.

    part 14 · optics
  15. The focus that moves out of the eye. Parallel rays from a distant object entering a simplified eye 24 mm long, in millimetres along its axis: in air the eye's 61 dioptres bring them to a focus 21.9 mm behind its principal plane, on the retina; under water its 18 dioptres would bring them to a focus 73 mm behind, about three times the length of the eye. On the retina itself the underwater rays have converged only a third of the way, and every point of the scene is spread into a disc the size of the pupil's shadow. The eye is not out of focus by a little, as a short-sighted eye is; it is missing most of a lens.

    The eye that loses its lens in water

    Eyes opened under water see only a blur, and the reason is not water in the eye or salt or murk. The human eye does two-thirds of its focusing not with its lens but with the curved front of its cornea, whose power comes from the jump in refractive index between air and tissue. Water is almost as dense optically as the cornea, and the jump all but disappears: about forty-three of the eye's sixty dioptres vanish, and the focus moves to three times the eye's own length behind the retina. A mask gives the cornea back its air; a small pupil and a hard squint buy back a little; animals that hunt in both worlds have flattened their corneas and given the work to the lens.

    part 15 · optics

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