Length and beamfor boat

This ratio can be assured if the rays entering a lens originate at infinity (they are parallel).  This can be done using a combination of lenses (figure 4).

Shipbeam anddraft

Physically, this means the ratio of the angle at which the ray enters the lens to the angle at which the ray leaves the lens should be constant across the lens to avoid coma.

Length and beamformula

A stop can also be placed in the system.  The position of the stop affects the light rays getting through.  A positive or negative coma can be created by selective positioning.  This coma created by the stop can be used to cancel out coma present in the system independent of the stop.

There are two types of coma: positive and negative.  Negative coma occurs when rays hitting the lens further from the paraxial region focus closer to the axis than rays closer to the paraxial region. Positive coma is just the opposite, with the further rays focusing farther from the axis (figure 1).

Coma is easily seen by holding a lens in sunlight and then tilting it, creating an image that tails off to one side (figure 3).

Length and beamcalculator

Since coma results from a change in magnification with height, for there to be no coma, the magnification must be constant everywhere:

where no, yo, αo and are the index of refraction, height, and slope angle of a ray in object space and ni, yi, αi are the index of refraction, height, and slope angle of the ray in image space.

Coma result from the use of principle planes when considering the propagation of image through a system.  The principle planes are only really planes in the paraxial region.  As you move further from the paraxial region, the surface the rays strike is curved, affecting the focal length for these rays impacting where they focus on the object plane.  The focusing of a ray is therefore dependent on the height off the axis at which it strikes the lens.