22.5 A short bestiary of classic designs⧉
The previous chapter left us with a dilemma. The ideal thin-lens model describes only power, $1/f$. A physical singlet has additional choices of bending, thickness, and glass even after that power is fixed, but limited freedom remains to also flatten the field, to bring the marginal rays to the same focus as the paraxial ones, to keep red and blue together, or to straighten the lines. So a lone element fails several aberrations at once, by construction. The way out is structural: add elements. Each new element contributes two more refracting surfaces (two more curvatures to choose), one more glass (a fresh index and a fresh dispersion), and, if it floats on a gap, an air space whose width is another parameter, adding another design variable that can be turned to make the aberrations of one element cancel those of another rather than accumulate. A compound lens is exactly that: a system of elements whose individual flaws are arranged to subtract.
There is a cost, which is why lenses are not built from a hundred elements. Each one adds weight, bulk, and money, and (the concern of the Glare suppression chapter) each uncoated air–glass surface reflects a few percent of the light, with coatings greatly reducing the loss, so more surfaces mean more stray reflections and lower contrast. The art of lens design is therefore to reach the required quality with as few elements as possible. The designs that follow are, in effect, a set of minimal arrangements that each add a new level of correction that recur, with variations, in nearly every lens ever sold. We name them and show the family resemblances; the quantitative argument for why each works was set up in the previous chapter, and the achromat in particular is derived there in full (the achromat condition). (Figure 22.5.1)
The achromatic doublet. The simplest useful compound lens is two elements: a positive lens of low-dispersion crown glass bonded to a negative lens of high-dispersion flint glass, usually cemented face-to-face so the pair acts as one unit. The negative flint reverses the color spread of the positive crown, so that two wavelengths, typically a red and a blue, are brought back to a common focus while the pair keeps net positive power. It is the minimum system that corrects chromatic aberration (the colored halos and fringes of the previous chapter), and the extra surfaces let the designer simultaneously tame the spherical aberration a single element of that power would carry. The doublet appears in many systems, including the front objective of inexpensive telescopes and binoculars, and a building block buried inside larger lenses. Its working principle, pairing opposite optical powers with different amounts of dispersion so the chromatic terms cancel, is the achromat condition from Chapter 22.2, Aberrations correction.
The Cooke triplet. Three elements, positive, negative, and positive, with the diaphragm by the central negative element. This classic three-element form provides enough useful design freedom to balance the primary aberrations over a practical field and aperture: six surface curvatures, thicknesses, air gaps, glass choices, and stop placement. Bending an element changes its curvatures; it is not an extra independent parameter. The triplet does not exactly eliminate every aberration at every field point, but its combination of positive and negative powers gives a much better compromise than a simple singlet or doublet, which is why it is the ancestor of an enormous family of inexpensive lenses. Many simple primes are recognizable triplets or lightly modified ones: splitting an element into a cemented pair, for instance, to buy a little more correction. Designed by H. Dennis Taylor in the 1890s Taylor 1893 (Cooke triplet), it became an influential economical design within an already established tradition of multi-element photographic optics.
The double-Gauss (Planar). When you want a fast normal lens, around $f$/2 or wider, the triplet runs out of freedom, and the dominant solution is the double-Gauss, six or seven elements in two groups with rough mirror symmetry about the aperture stop. The symmetry is the whole point: a perfectly symmetric system, used at unit magnification, automatically cancels the odd aberrations (coma, distortion, and lateral chromatic aberration) because each half introduces them with the opposite sign, so they subtract. Real photographic double-Gauss lenses are only nearly symmetric (they image distant objects, not at 1:1), but enough of the cancellation survives that this layout has dominated fast standard lenses for a century. Zeiss's Planar is the canonical name; most 50 mm $f$/1.4 and $f$/1.8 lenses are its descendants. (Figure 22.5.3)
Telephoto and retrofocus. The two remaining classics solve a different problem: physical length rather than image quality. Naively a lens of focal length $f$ ought to be roughly $f$ long, which is awkward for a 300 mm telephoto (too long to carry) and impossible for a wide-angle on a camera with a mirror in the way (too short to clear it). Both are fixed by ordering a positive and a negative group so as to slide the rear principal plane, defined in the next section, to a more convenient place. (Figure 22.5.4)
- A telephoto lens puts a positive group in front and a negative group behind. The rear negative group pushes the rear principal plane forward, out past the front of the lens, so the whole assembly is physically shorter than its focal length: a 300 mm telephoto in a barrel only 200 mm long. This is what "telephoto" properly means: the length-shortening positive-then-negative construction, not the everyday sense of "long lens."
- A retrofocus lens (also called inverted telephoto) does the reverse: a negative group in front, a positive group behind, placing the rear principal plane forward of the rear vertex sufficiently to lengthen the back-focal distance: the air gap between the last glass and the sensor. A wide-angle lens has a short focal length and would normally sit almost on the sensor, but on a single-lens reflex (SLR) camera the swinging mirror needs room, so the lens must stand off; retrofocus design buys that standoff. Removing the mirror-box clearance constraint can simplify a wide-angle design, though sensor angular response, packaging, and desired correction may still favor a retrofocus-like form.
The zoom. Finally, the zoom lens is a system whose focal length is variable. The following variator/compensator arrangement describes a classical parfocal design; photographic zooms may need focus correction as focal length changes. Internally one group of elements, the variator, slides along the axis to change the magnification and hence $f$, while a second group, the compensator, slides in a coordinated way to keep the image plane fixed so the picture stays in focus as you zoom (mechanically the two motions are coupled by a cam). The price is steep: a zoom must be reasonably corrected at every focal length in its range, not just one, so it carries many elements and many compromises, and it is the hardest kind of lens to optimize well. Almost every aberration trade-off in this part gets harder when the geometry itself is allowed to move. Figure 22.5.5 traces the whole mechanism live: drag the zoom and watch the variator and compensator slide while the image stays pinned to the sensor.
The architectures these classes are drawn after are not guesswork: many lens patents publish numerical example prescriptions (radii, thicknesses, glasses, and zoom configurations), though these need not match the final production lens and Bill Claff's Optical Bench Hub on Photons to Photos has transcribed hundreds of them into an interactive ray-tracer you can pan, zoom, and step through configuration by configuration. It is the best public window onto how real primes, zooms, and mirror lenses are actually built. See photonstophotos.net/GeneralTopics/Lenses/OpticalBench/OpticalBenchHub.htm.
The main pattern in the bestiary is that element count makes correction visible. A two-element doublet fixes color; three elements permit a useful balance of primary aberrations; six or seven buy a fast normal lens; ten to twenty buy a high-performance zoom. Each stage spends glass to cancel one more class of the errors cataloged in the previous chapter. The next section gives the language to describe any of these stacks, however many elements they contain, as a single equivalent system.
22.5.1 The lens as a system: cardinal points, pupils, f-number, T-stop⧉
In most calculations, you almost never need to think about the individual elements. In first-order paraxial optics, a centered compound system behaves, for the purpose of locating conjugate planes, like an equivalent lens, provided you measure distances from the right reference planes and the aperture at the right opening. Those reference planes and openings are the system's cardinal points and pupils, and they are what a lens's published specifications (focal length, f-number, angle of view) actually refer to.
Principal planes and the thick-lens equation. A thin lens has one obvious plane to measure from: the lens itself. A thick, multi-element system has no single such plane, but it has two effective ones, the front and rear principal planes, written $H$ and $H'$. They are the planes from which object and image distances must be measured for the familiar equation to hold: with $u$ measured from $H$ and $v$ measured from $H'$,
holds for a system with the same surrounding medium on both sides and the usual real-conjugate distance convention, with $f$ the system's effective focal length. The whole black box then acts like a single thin lens placed at $H$ for the object side and at $H'$ for the image side. The two planes need not lie inside the glass at all, and indeed, as we just saw, the telephoto and retrofocus designs are precisely tricks for moving $H'$ to where it is wanted. This is also why the effective focal length (measured from $H'$) and the back focal length (measured from the last glass surface to the sensor) are different numbers: the telephoto's effective focal length is large while its back focal length is small, and the retrofocus is the reverse. (Figure 22.5.6)
Nodal points. A second pair of cardinal points, the nodal points, are defined by a different property: a ray aimed at the front nodal point leaves the rear nodal point traveling parallel to its original direction: it crosses the axis undeviated in angle. When the lens has air on both sides (the ordinary camera case) the nodal points coincide with the principal points. Nodal points are often confused with a panorama pivot. The appropriate pivot in the central-camera approximation is the entrance pupil, discussed next, not generally either nodal point.
Entrance and exit pupils. Somewhere inside the lens is the aperture stop: the diaphragm whose blades you open and close. But you rarely see the stop directly; you see its image formed by the glass in front of or behind it. Looking into the front of the lens, the apparent opening is the entrance pupil (EP): the image of the aperture stop projected into object space by the elements ahead of it. Looking into the back, you see the exit pupil, the image of the stop projected into image space by the elements behind it. The entrance pupil matters enormously because it is the opening that actually gathers light: a scene point sends into the lens exactly the cone of rays that fills the entrance pupil. So the entrance pupil (not the physical front element, and not the physical stop) is the effective aperture of the system. It is also the true center of perspective: rotating the camera about it minimizes parallax in the central-camera approximation; a real wide-angle lens can have a field-dependent apparent pupil, so no single pivot is perfect over every direction. This explains why panoramic rotation is usually arranged around the entrance pupil rather than a nodal point.
The f-number, pinned to the entrance pupil. FUNDAMENTALS defined the f-number $N = f/D$ as focal length over aperture diameter, controlling both light-gathering (a stop is a factor of $\sqrt 2$ in diameter, a factor of $2$ in area and exposure) and depth of field. Now we can say which diameter: the entrance-pupil diameter $D_{\text{EP}}$, so that
Using the front-element diameter instead is a genuine mistake. In a retrofocus or telephoto lens, the front element can be much larger or smaller than the entrance pupil, and only $D_{\text{EP}}$ gives the right light-gathering and depth of field. Off-axis, the barrel and the front and rear elements can clip the pupil (the oblique cone of light no longer passes cleanly), which darkens the corners (optical vignetting, distinct from the natural $\cos^4$ falloff of FUNDAMENTALS) and squeezes out-of-focus highlights into "cat's-eye" shapes. We pick that up in Depth of field; here it is enough to note that the pupil, not the stop, is what gets vignetted.
The T-stop. The f-number quietly assumes that all the light entering the entrance pupil reaches the sensor: that the lens transmits 100% of it. Real glass absorbs a little and, more importantly, every air–glass surface reflects a few percent (which we attack with coatings in Glare suppression), so a multi-element lens transmits only a fraction $\tau < 1$ of the light. Two lenses of the same f-number can therefore deliver different actual exposures. Cinematographers, who cannot tolerate a brightness jump when they switch lenses, mark their lenses not in f-stops but in T-stops, which fold in the measured transmittance:
The square root appears because exposure goes as the inverse square of the f-number (it is an area), so a transmittance $\tau$ reduces the effective area by $\tau$ and raises the equivalent f-number by $1/\sqrt{\tau}$. A T-stop of $T$ thus means "this lens gives the exposure that a perfect (lossless) $f$/$T$ lens would," which is exactly the number you need to set exposure reliably. As coatings push $\tau$ toward 1 (with the actual value depending on lens, wavelength, and aperture), the T-stop creeps back toward the f-number.
How many elements, in practice. Putting it together: the visible cost of canceling every aberration in glass is the element count, and modern lenses are extravagant. A high-performance zoom or a fast prime commonly runs 10 to 20-plus elements in several groups, sprinkled with aspheric surfaces and exotic low-dispersion glasses to do more correction per element (both detailed in Aberrations correction). Even a smartphone "lens," for all that it looks like a single dot, is a stack of five to eight tiny molded plastic or glass elements, several strongly aspheric. Every one of those elements is there to cancel something. The central argument of this part is that not everything must be corrected in glass. You can let the optics get close and finish the job in software: correct the residual distortion, vignetting, and color computationally, which is cheaper than another aspheric element. That trade, glass against computation, shapes the chapters ahead, and the cardinal-point picture built here is what lets us reason about it cleanly: it tells us exactly what the optics is promising, so we know precisely what the software must clean up.