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4.8. Chromatic aberration
Chromatic aberration, as the name implies
(khroma=color in Greek) is the phenomenon of white light wavefront
splitting into its spectral components wavefronts, as light encounters media of different
optical density. It is caused by different wavelengths moving at
different speeds through media denser than air. Denser media slow down
all wavelengths, but shorter wavelengths more than
longer wavelengths.
As illustrated on
Fig. 4, points of
wavefront that simultaneously move through media of different optical
densities, generate wavelength-specific phase difference, due to
different wavelengths now moving at different speeds in new media. This causes change in
a form of the wavefront (in terms of rays, it results in
refraction).
Obviously, chromatic wavefront split
will take place even if the shape of optical surface coincides with the
shape of wavefront, for instance, when flat wavefront passes through an
optical window at zero inclination angle. However, this effect is
negligibly small; in BK7 glass, the green e-line is only 0.241% faster
than blue F-line, and 0.29% slower than red C-line. In, say, 5mm thick
window, the separation between blue and red wavefront at the rear
surface would
be 0.0266mm.
Much more important is the effect of
change in the wavefront form when shapes of the wavefront and optical
surface don't coincide - a common scenario with telescope lens
objectives, made to change flat incoming wavefronts into spherical.
Here, the in-glass light speed varying with the wavelength causes shorter
(faster) wavelengths to
form more strongly curved wavefront - or, in terms of rays, refract more strongly - and
focus closer than longer wavelengths (FIG. 29).
FIGURE 29: Chromatic aberration in a lens.
Left: the white-light wavefront
W splits into its component wavelengths wavefronts, as
shorter wavelengths lag behind in a denser media of the
refractive n', forming more strongly curved wavefronts with
shorter foci. The path of refracted rays (dotted lines) is
orthogonal with respect to the transforming wavefronts.
Longitudinal focus disparity for different wavelengths is called
longitudinal chromatism, or secondary
spectrum; variation of spherical aberration with the
wavelength is spherochromatism
(due to, strictly, only a single wavelength can come out with
perfectly spherical wavefront).
Right: the very center of the lens, where the white-light chief ray (the
one orthogonal to the center of the incoming wavefront
Wi
at an angle to the optical axis), splits laterally into the component
wavelengths chief rays. The longer wavelength
(red),
being slowed down less in the glass, tilts - or refracts - less,
forming the point image higher in the image plane. It is called
lateral chromatism. Note that
the axial separation of the wavefronts themselves is exaggerated; typically,
it is negligible in comparison to
the effect of wavefront radii variation (i.e. the extent of
their focal length variation).
The aberration geometry
is governed by the law of refraction, stating
that the
sines of
the
incident (α) and refracted (α') ray angle
to
the
surface normal relate in inverse proportion to the
refractive indici of the incident and refractive media, or
sinα'/sinα
= n/n', with n and n' being the incident and refractive media
refractive index, respectively.
Chromatism of a single lens is overwhelming:
it smears the wavelengths into a rainbow of colors, both longitudinally
and laterally. While the lens focal
length is directly related to the glass refractive index, degree of
chromatism depends on the dynamics of index change with the wavelength -
called "dispersion". It varies with different glass types. For object at infinity, the two forms of chromatism are
proportional to 1/V, V being the glass dispersive constant, or
Abbe
number. Specifically, this means that a lens of the focal
length ƒ will
have longitudinal chromatism of ƒ/V, and lateral chromatism of h/V at the
height h in the image plane.
General form of the Abbe number is:
V = (n-1)/i
(40)
with n being the
glass refractive index, and i
the index differential for the chosen range. The usual choice for n
is either e (λ=0.5461μ) green, or d
(λ=0.5876μ)
line of yellow light, and for i
the
index differential between the blue F-line (λ=0.4861μ) and red C-line
(λ=.6563μ),
nF-nC.
Taking the e line
and BK7 glass (ne=1.519,
nF=1.522
and nC=1.514),
gives the appropriate Abbe number as: V = (ne-1)/(nF-nC)
= 64.9
(41) So, a single BK7 lens of the
focal length ƒ=1000mm, would have axial F-C separation of over 15mm, and
lateral F-C separation of h/64.9, h being the height in the image
plane. In other words, it would be all but useless
for observing of
pretty much anything but the chromatism itself.
In order to minimize chromatism in a lens
objective, it has to be made of two or more glass elements of different
optical properties. This leads to compounded refracting objectives,
usually consisting of two or three lens elements. Depending on the level
of correction of chromatism, most of them belong to one of the two main
groups, achromats and apochromats.
◄
4.7. Distortion
▐
4.8.1. Secondary
spectrum and spherochromatism ►
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