Field curvature
An aberration in which a lens images a flat object onto a curved surface, so the centre and edges of a flat detector cannot be in focus at the same time. For a single thin lens the curved (Petzval) surface has radius n·f: 76 mm for a 50 mm N-BK7 lens.
A positive lens brings off-axis points to focus closer to the lens than on-axis points, so the sharp image of a flat object lies on a surface curved toward the lens. A flat camera sensor or a flat fiber array placed at the axial focus then sees the edges of the field out of focus, and refocusing for the edges blurs the centre. That is field curvature, one of the five Seidel aberrations.
Without astigmatism, the image surface is the Petzval surface. For a set of thin lenses in air its curvature depends only on the focal lengths and refractive indices of the elements, not on their shapes or spacings:
With astigmatism present, the tangential and sagittal images lie on two further surfaces, both displaced from the Petzval surface in the same direction, the tangential by three times as much, and best focus lies between them.
Typical values
A single thin N-BK7 lens of 50 mm focal length has a Petzval radius of = 76 mm. At an image height of 5 mm, the Petzval surface departs from the flat focal plane by = 0.16 mm, while the depth of focus at f/4 and 550 nm is only about ±18 µm, so a flat detector cannot hold the edge and the centre in focus together. Because the sum depends only on power and index, a lens cannot be flattened by bending its surfaces; it needs negative elements, placed where they contribute to the Petzval sum more than to the power.
Correction and practice
Flat-field designs balance positive and negative elements; the Cooke triplet was the classic solution, and microscope objectives labelled "plan" are corrected this way. A field-flattening lens near the image plane cancels the curvature with little effect on focus. Where the field is scanned rather than imaged, as in laser scanning systems, a flat-field scan lens is designed so that the focused spot stays on a plane across the scan. Curved detectors, such as the curved focal planes of some astronomical cameras, accept the curvature instead of correcting it.
Measurement
Field curvature is measured by imaging a flat target and recording the best-focus position at several field heights, with a camera on a translation stage or with through-focus MTF measurements across the field. Plotting focus position against field height, separately for tangential and sagittal lines, gives the field curves that lens datasheets show.
References: W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008); W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986); M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 5.