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22.0 COMPOUND LENSES, AND ABERRATION CORRECTION⧉

The FUNDAMENTALS part introduced an idealized model. The thin lens, a single ideal element of zero thickness for which $1/f = 1/u + 1/v$, is what you draw on the whiteboard, and it is exactly right, as long as you keep only the first term of everything. It is the paraxial approximation: replace $\sin\theta$ by $\theta$ (drop the cubic term and all that follow), pretend there is only one wavelength, and pretend the glass has no thickness. Within this first-order, monochromatic geometric model, rays from a conjugate point meet at its image point. A thick system can also be described paraxially using principal planes; thickness and nonparaxial aberration are separate issues.

Reality keeps the terms the thin lens threw away, and the whole story of this part falls out of that one fact. Snell's law, $n_1\sin\theta_1 = n_2\sin\theta_2$, is non-linear: restore the next term, $\sin\theta \approx \theta - \theta^3/6$, and rays through the edge of the aperture no longer land where the paraxial rays do. The refractive index depends on wavelength ($n(\lambda)$, dispersion), so red and blue focus in different places. And real glass has thickness, real lenses have many surfaces, and real barrels have mounts, dust, and air gaps that scatter stray light. The consequence is stark: no single element can make all the rays, from all the field points, at all the wavelengths, through all of the aperture, converge to the right place. It is not a manufacturing failure; it is geometry.

So the lens you actually own is not one piece of glass. It is a system of six, ten, or twenty elements, and it was found not by a formula but by optimization: vary the curvatures, glasses, thicknesses, and spacings to minimize the leftover blur, exactly the data-fit-versus-trade-off shape the rest of the book uses for inverse problems, here with a lens as the unknown and ray-tracing as the forward model. And whatever the glass cannot fix, computation calibrates and corrects. Software undoes distortion, vignetting, color fringing, even some of the blur, from a measured profile of the lens. That partnership is the spine of the part. The optics must preserve enough information for the algorithm to work with; correction cannot restore arbitrary detail lost to clipping, sampling, or noise. Optics and computation are two ends of one design space, and the most interesting modern cameras are designed across both at once.

The part is organized around one question: where the thin lens breaks, and how we fix it: in glass, in mechanism, or in software.

The opening chapters identify Chapter 22.1, Aberrations and optical challenges, explain Chapter 22.2, Aberrations correction, and establish Chapter 22.3, Measuring lens quality. Chapter 22.4, Lens optimization then treats design as a search over curvatures, glass types, thicknesses, and spacings, using ray tracing and a merit function to compare candidates, the job of commercial design programs such as Zemax OpticStudio and CODE V and, in research, of differentiable ray tracers that let the lens be optimized together with the image-processing that follows it. Chapter 22.5, A short bestiary of classic designs explains the forms that organize this search: achromatic doublets, Cooke triplets, double-Gauss lenses, telephotos, retrofocus lenses, and zooms. Cardinal points and pupils let us describe each compound system without tracing every ray: the f-number is $N=f/D_{\mathrm{EP}}$, using entrance-pupil diameter, while the T-stop also accounts for transmission. Chapter 22.6, Scaling laws in optics asks what changes when these designs are enlarged or miniaturized.

Chapter 22.7, Special optics is a gallery of designs that deliberately break a thin-lens assumption to gain a capability: tilt the plane of focus (the Scheimpflug condition), abandon rectilinear projection for a 180°-plus fisheye, fold the path with mirrors (catadioptric, with its signature doughnut bokeh), squeeze the frame (anamorphic), image life-size (macro), and, as a bridge to the Advanced part, trade the bulk of the glass for a thin structure, from the Fresnel lens through diffractive optics and gradient-index (GRIN) to flat metalenses.

Chapter 22.9, Autofocus treats focusing as choosing the conjugate that lands the subject on the sensor. The mechanics are simple, but deciding which way to move is the real problem, solved by a ladder of methods: contrast-detection (climb the sharpness hill, but blind to direction), phase-detection autofocus (PDAF) (a split-pupil view that is literally stereo inside the lens, giving a signed defocus in one shot), its on-sensor and dual-pixel forms, and learned subject and eye detection on top. The following part, beginning with Chapter 23.1, Bokeh, focus stacking, and depth-of-field control, examines the blur itself, why an out-of-focus highlight takes the shape of the aperture (bokeh), why it clips to cat's-eye lemons toward the corners, and on extending it (focus stacking) or faking it (depth-aware portrait-mode blur), leaning on the FUNDAMENTALS geometry rather than re-deriving it.

Two further chapters treat unwanted light and camera motion. Chapter 22.10, Glare suppression addresses the opposite problem from aberrations: aberrations are wanted rays in the wrong place, glare is unwanted light (inter-surface reflections and scatter) landing as ghosts, flare, and contrast-killing veiling glare. The cures split the familiar way: in hardware (anti-reflection coatings by thin-film interference, hoods, internal baffles) and in software (deflare and glare deconvolution). And Chapter 22.11, Optical stabilization fights the motion blur that hand-shake smears across an exposure: gyroscopes sense the shake, and a floating lens group (optical image stabilization, OIS) or the sensor itself (in-body image stabilization, IBIS) moves to keep the projected image still, buying several stops of shutter speed. The caveat is that it cancels camera motion only and cannot un-blur a moving subject, so subject motion still calls for suitable exposure time, illumination, or a reconstruction with adequate measurements.

Chapter 22.12, The eye as an optical instrument: vision and its correction closes the part with a familiar optical system whose accommodation, pupil, and aberrations connect these ideas to vision. Across the part, the same pattern recurs: a defect that the thin lens hid, met either in glass (more elements, exotic surfaces, coatings) or in computation (calibrate, deconvolve, correct) and, increasingly, met by designing the two together. That last move, end-to-end or computational optics, is forward-referenced throughout to the Advanced part (coded apertures, wavefront coding, metalenses, lensless imaging), where the optics may look "wrong" by classical metrics yet give the better picture. Keep the partnership in view and the part stops looking like a catalog of lens trivia and starts looking like one idea: the optical design determines which measurements reach the sensor, and the processing determines how well the final image uses them.

Contents of this part

▸ full collapsible outline of this part