Astigmatism

In an astigmatic system the focusing power differs between two perpendicular planes. For an off-axis point in an otherwise good lens, those are the tangential plane, containing the point and the axis, and the sagittal plane perpendicular to it. Rays in the tangential plane focus first, to a short line perpendicular to that plane; rays in the sagittal plane focus farther on, to a line at right angles to the first. Midway is the circle of least confusion, the round blur usually taken as best focus. The separation of the two line foci measures the aberration.

As a Seidel term the wavefront error is $W = W_{222}\,H^2\rho^2\cos^2\theta$, quadratic in field height $H$ and in pupil radius $\rho$. Balanced with defocus, it becomes the Zernike astigmatism term, with an rms of $W_{222}/(2\sqrt{6})$, or $0.204\,W_{222}$, at unit field. Its peak-to-valley to rms ratio, 4.90, is higher than for defocus or spherical aberration, which matters when a specification is written in peak-to-valley terms.

Sources in practice

Lens astigmatism off axis is corrected in anastigmatic designs such as the Cooke triplet. On the bench, the more common causes are elements used off their design geometry: a tilted lens, a spherical mirror used at an angle (whose tangential and sagittal focal lengths are $f\cos\theta$ and $f/\cos\theta$), and a tilted plane-parallel plate, such as a beamsplitter or window, in a converging beam. Folded laser resonators use the angle of a curved mirror to cancel the astigmatism of a Brewster-angled crystal.

Edge-emitting laser diodes produce astigmatic beams: the fast-axis beam waist sits at the facet while the apparent slow-axis waist lies behind it, by a few micrometres for index-guided devices and tens of micrometres for gain-guided and broad-area ones. Combined with the unequal far-field divergence of the two axes, this is why diode collimation often uses cylindrical optics or anamorphic prism pairs.

The eye's astigmatism, from a cornea with different curvatures in two directions, is corrected with cylindrical spectacle lenses and quoted in dioptres of cylinder; it produces the same two line foci, but on axis, from a toric surface rather than from field angle.

Measurement

Astigmatism shows as two separate line foci: scanning a camera through focus finds the two planes where the spot is narrowest in $x$ and in $y$, and their separation gives the astigmatic difference. For laser beams, ISO 11146 treats this as a waist-position difference between the two principal axes. Interferometers report it as the $Z_2^{\pm2}$ coefficients, and Shack-Hartmann sensors as the same terms of a Zernike fit.

References: W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986); A. E. Siegman, Lasers (University Science Books, 1986), Ch. 17; H. W. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, IEEE J. Quantum Electron. 8, 373 (1972).