Coma

When a lens images a point that lies off its axis, each annular zone of the aperture forms a small ring image, and in a system with coma those rings have different sizes and are displaced along the radial direction of the field. Their overlap is a bright point with a flare spreading to one side, the comet shape that gives the aberration its name. Coma is the first of the Seidel aberrations to appear off axis, and it is particularly visible because it is asymmetric: the image has a centre of intensity that is shifted from its peak.

The wavefront error is $W = W_{131}\,H\rho^3\cos\theta$, where $H$ is the normalized field height and $\rho$ and $\theta$ are pupil coordinates. It is therefore linear in the field and cubic in the pupil; the transverse size of the comet grows as the square of the aperture. Balanced with a small tilt, it becomes the Zernike coma term, with an rms of $W_{131}/(6\sqrt{2})$, or $0.118\,W_{131}$, at unit field. In the geometric image the tangential coma is three times the sagittal coma, so the comet is 1.5 times as long as it is wide and fills a 60° wedge.

Where it appears

A lens with coma cannot give sharp images of points away from the axis even when the axial image is perfect, so it limits the usable field of telescopes, microscopes and scanning systems. A system free of both spherical aberration and coma is called aplanatic; this is the condition expressed by the Abbe sine condition, and it is what a microscope objective or a Ritchey-Chrétien telescope is designed to meet. In laser work, coma appears when a lens or a spherical mirror is tilted relative to the beam, when a beam passes a lens off centre, and in off-axis parabolic mirrors that are misaligned: a focused spot with a one-sided tail usually means the beam does not share the lens's axis. Centring the beam and removing the tilt removes the coma; a small deliberate tilt is sometimes used to cancel coma from another element.

Measurement

In an interferometric test coma appears in the Zernike fit as the $Z_3^{\pm1}$ terms, and it can be distinguished from misalignment of the test itself by how it changes when the part is tilted. On a bench, imaging a focused spot onto a camera while tilting the lens slightly shows whether a tail appears or disappears; the orientation of the tail points along the direction of the offending tilt or decentre.

References: W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986); V. N. Mahajan, Optical Imaging and Aberrations, Part I (SPIE Press, 1998); M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 5.