Chromatic aberration

The refractive index of glass falls with wavelength, so a lens bends blue light more than red and its focal length grows toward the red. The result is chromatic aberration, in two forms. Axial (longitudinal) chromatic aberration is the shift of focus along the axis with wavelength: with white light, a point images as a sharp spot of one colour surrounded by out-of-focus halos of the others. Lateral chromatic aberration, or lateral colour, is the change of magnification with wavelength, which shows as coloured fringes at the edges of the field even when each colour is sharp.

Size

The dispersion of a glass in the visible is summarized by its Abbe number, $V = (n_d - 1)/(n_F - n_C)$, using the indices at the helium d line (587.6 nm) and the hydrogen F (486.1 nm) and C (656.3 nm) lines. A thin singlet's focal length changes between F and C by

N-BK7 has an Abbe number of 64.2, so a 50 mm plano-convex lens focuses the F line 0.77 mm closer than the C line, 49.47 mm against 50.24 mm by the Sellmeier equation, in line with $f/V$. In the near infrared the dispersion is smaller but the effect remains: the same lens has focal lengths of 51.31 mm at 1310 nm and 51.61 mm at 1550 nm, a 0.30 mm shift that matters when one lens couples both wavelengths into a single-mode fiber, where the focal depth is a few tens of micrometres.

Correction

An achromatic doublet combines a positive element of low-dispersion crown glass with a negative element of higher-dispersion flint so that the focal shifts cancel at two wavelengths, usually F and C. For a thin cemented doublet of total power $\phi$, the element powers are $\phi_1 = \phi V_1/(V_1 - V_2)$ and $\phi_2 = -\phi V_2/(V_1 - V_2)$; for a 50 mm doublet of N-BK7 ($V$ = 64.2) and N-F2 ($V$ = 36.4), that is a 21.7 mm positive element and a −38.2 mm negative one. The residual focus variation between the corrected wavelengths is the secondary spectrum, typically about $f/2000$ for ordinary glass pairs, and apochromats reduce it further with anomalous-dispersion glasses or fluorite. Doublets are designed for a band: a visible achromat used at 1550 nm is no longer corrected there, and near-infrared achromats are sold separately.

Reflective optics have no chromatic aberration, which is why off-axis parabolic mirrors are used to collimate and focus broadband sources, supercontinuum light and ultrashort pulses. Diffractive optics and metasurfaces have strong chromatic aberration of the opposite sign to glass, which hybrid refractive-diffractive lenses use for correction.

Measurement

Axial colour is measured by finding the best focus at several wavelengths, using laser lines or a monochromator, with a camera or knife-edge scan; the plot of focus position against wavelength is the focal shift curve that achromat datasheets show. Lateral colour is measured from the difference in image height at the edge of the field between wavelengths.

References: W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008); SCHOTT Optical Glass datasheets (N-BK7, N-F2); M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 4.