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, axial 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 the 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. 44).

FIGURE 44: Chromatic aberration in a lens, grossly exaggerated. LEFT: 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). 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), also grossly exaggerated in the illustration. RIGHT: The white-light chief ray (the one orthogonal to the center of the incident wavefront at an angle to the optical axis), splits laterally into the component wavelengths chief rays, setting the stage for lateral chromatism. With the stop at the surface, it passes through the center of the lens, which acts as a plane-parallel plate, having the chief rays of different wavelengths exit slightly separated, but at nearly identical angle. Consequently, lateral chromatism approaches zero, unless the lens is exceedingly thick. With the aperture stop displaced from the lens, the chief ray passes through a portion of lens with significant optical power, resulting in chief rays of different wavelengths exiting the lens ad different angles, thus focusing at different heights in the image space. Since lateral color and secondary spectrum have different origins and magnitude, correcting one doesn't necessarily cancel the other.

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:

with n being the glass refractive index, and ι 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 ι 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:

Thus, the lower Abbe number, the higher glass dispersive power, and vice versa.

Lens' lateral color is near-zero with the stop at the surface; for displaced stop, it increases with stop displacement, in proportion to the field angle, depending on lens' dispersive and optical powers, as well as its thickness). However, secondary spectrum remains present regardless of stop location. A single BK7 lens of the focal length ƒ=1000mm, would have the axial F-C separation of over 15mm. 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 of lens objectives, achromats and apochromats.

◄ 4.7. Field curvature ▐ 4.8.1. Secondary spectrum and spherochromatism ►

Home | Comments