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2.2 Photometric image formation 63

f’ = 101mm

c P d

z i’=103mm z o=5m

Figure 2.21 In a lens subject to chromatic aberration, light at different wavelengths (e.g., the red and blur
arrows) is focused with a different focal length f and hence a different depth z , resulting in both a geometric

i
(in-plane) displacement and a loss of focus.
√
Notice that the usual progression for f-numbers is in full stops, which are multiples of 2,
since this corresponds to doubling the area of the entrance pupil each time a smaller f-number
is selected. (This doubling is also called changing the exposure by one exposure value or EV.
It has the same effect on the amount of light reaching the sensor as doubling the exposure
1
1
duration, e.g., from / 125 to / 250, see Exercise 2.5.)
Now that you know how to convert between f-numbers and aperture diameters, you can
construct your own plots for the depth of field as a function of focal length f, circle of
confusion c, and focus distance z o , as explained in Exercise 2.4 and see how well these match
what you observe on actual lenses, such as those shown in Figure 2.20.
Of course, real lenses are not infinitely thin and therefore suffer from geometric aber-
rations, unless compound elements are used to correct for them. The classic five Seidel
aberrations, which arise when using third-order optics, include spherical aberration, coma,
astigmatism, curvature of field, and distortion (M¨ oller 1988; Hecht 2001; Ray 2002).

Chromatic aberration
Because the index of refraction for glass varies slightly as a function of wavelength, sim-
ple lenses suffer from chromatic aberration, which is the tendency for light of different
colors to focus at slightly different distances (and hence also with slightly different mag-
nification factors), as shown in Figure 2.21. The wavelength-dependent magnification fac-
tor, i.e., the transverse chromatic aberration, can be modeled as a per-color radial distortion
(Section 2.1.6) and, hence, calibrated using the techniques described in Section 6.3.5. The
wavelength-dependent blur caused by longitudinal chromatic aberration can be calibrated
using techniques described in Section 10.1.4. Unfortunately, the blur induced by longitudinal
aberration can be harder to undo, as higher frequencies can get strongly attenuated and hence
hard to recover.
In order to reduce chromatic and other kinds of aberrations, most photographic lenses
today are compound lenses made of different glass elements (with different coatings). Such
lenses can no longer be modeled as having a single nodal point P through which all of the
rays must pass (when approximating the lens with a pinhole model). Instead, these lenses
have both a front nodal point, through which the rays enter the lens, and a rear nodal point,
through which they leave on their way to the sensor. In practice, only the location of the front