Chromatic aberration
The change of a lens's focal length and magnification with wavelength, caused by the dispersion of its glass. A singlet's focus moves by about f/V between the blue F and red C lines: about 0.78 mm for a 50 mm N-BK7 lens. Achromatic doublets cancel it at two wavelengths; mirrors have none.
Chromatic aberration is the failure of a lens to bring all colors to the same focus. 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 aberration takes 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 color surrounded by out-of-focus halos of the others. Lateral chromatic aberration, or lateral color, is the change of magnification with wavelength, which shows as colored fringes at the edges of the field even when each color is sharp.
Size
The dispersion of a glass in the visible is summarized by its Abbe number, , 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 . 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 micrometers.
Whether the shift is visible depends on the depth of focus. At f/5 the diffraction-limited depth of focus, at 550 nm, is 55 µm, so the singlet's 0.77 mm between F and C spans fourteen depths of focus and a white-light image is soft in every color but one. At f/20 the depth grows to 0.88 mm, larger than the shift, and the same lens gives an acceptably sharp white-light image.
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 , the element powers are and ; for a 50 mm doublet of N-BK7 ( = 64.2) and N-F2 ( = 36.4), that is a 21.7 mm positive element and a −38.2 mm negative one. Computed from the SCHOTT Sellmeier coefficients for thin lenses, both lenses set to 50.00 mm at 587.6 nm:
| Wavelength | N-BK7 singlet | N-BK7 and N-F2 achromat |
|---|---|---|
| 486.1 nm (F) | 49.47 mm | 50.03 mm |
| 555 nm | 49.86 mm | 50.00 mm (shortest) |
| 587.6 nm (d) | 50.00 mm | 50.00 mm |
| 656.3 nm (C) | 50.24 mm | 50.03 mm |
| 850 nm | 50.68 mm | 50.15 mm |
| 1064 nm | 51.00 mm | 50.32 mm |
| 1550 nm | 51.61 mm | 50.85 mm |
The doublet's residual, 0.026 mm between the d focus and the common F and C focus, is the secondary spectrum, about for ordinary crown and flint pairs; it comes from the different shapes of the two glasses' dispersion curves and is smaller than the 55 µm depth of focus at f/5. Apochromats reduce it further with anomalous-dispersion glasses or fluorite, bringing three wavelengths to one focus. Doublets are designed for a band: outside it the correction fails, and the visible achromat above moves its focus by 0.85 mm between 587.6 nm and 1550 nm, more than the singlet does between F and C, which is why 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.
Lateral color
Lateral color is the difference in image height between wavelengths, and it grows in proportion to the distance from the center of the field. A thin lens with the aperture stop at the lens has none, because the chief ray passes through the lens center undeviated at every wavelength; it appears when the stop is away from the lens and in multi-element systems, and designs that are symmetric about the stop cancel much of it. In photographs it shows as red and cyan or blue and yellow fringes along high-contrast edges near the corners of the frame. Because each color is sharp but slightly scaled, cameras and raw converters can correct it by rescaling the color channels. Axial color cannot be removed that way: in fast lenses it shows as purple and green fringes in front of and behind the plane of focus.
Measurement
Axial color 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 color is measured from the difference in image height at the edge of the field between wavelengths.
Common questions
What is chromatic aberration?
The variation of a lens's focus and magnification with wavelength, caused by the dispersion of its glass. A simple lens brings blue light to a focus closer than red, so with white light the image of a point is a colored blur and high-contrast edges show colored fringes.
What causes chromatic aberration?
Dispersion: glass has a higher refractive index for blue light than for red, so a lens bends blue light more. For a singlet the focal length changes by about between the F and C lines, where is the glass's Abbe number: 0.78 mm for a 50 mm lens of N-BK7 ( = 64.2).
What is the difference between axial and lateral chromatic aberration?
Axial (longitudinal) chromatic aberration is a shift of focus along the axis with wavelength, and it blurs the whole image, the center included. Lateral (transverse) chromatic aberration is a change of magnification with wavelength: each color can be sharp, but the images differ in size, so colored fringes grow toward the edge of the field.
How is chromatic aberration corrected?
By combining glasses of different dispersion. An achromatic doublet brings two wavelengths to one focus and leaves a secondary spectrum of about ; apochromats add anomalous-dispersion glass or fluorite to bring three together. Mirrors avoid the problem altogether, and digital cameras remove lateral color in software by rescaling the color channels.
Do mirrors have chromatic aberration?
No. The law of reflection does not depend on wavelength, so a mirror focuses every color at the same point; only its coating's reflectance changes with wavelength. Reflective objectives and off-axis parabolic mirrors are used with broadband light and ultrashort pulses for this reason.
References: W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008); R. Kingslake and R. B. Johnson, Lens Design Fundamentals, 2nd ed. (Academic Press, 2010); SCHOTT Optical Glass datasheets (N-BK7, N-F2); M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 4.