What is abbe's rule in optics? I have read wikipedia but can't really understand what they mean to say. The usual explanations are given in terms of Fourier optics, which I don't yet have the background for. Can anyone explain it with an illustration, or give another high school level explanation? I've a high school physics background.
 A: Departure from Abbe's sine condition is a concept used in lens design to characterize aberrations or lack of sharpness of the image made by a lens, as mentioned in the Wikipedia article.  The traditional use of this condition does not require higher mathematics, but the subject may easily get confusing, because it consists of many different parts of optical theory.
Let us start with paraxial optics which is a simplified version of geometrical optics. Aberrations are defined as departures from the predictions of paraxial optics.   Here all rays from an object point converge to a single point image, real or virtual.   Principal planes is another concept from paraxial optics.  These planes are defined as the specific  object and image planes where the  magnification is unity. This means that if we put a luminous point in the first principal plane at a certain distance from  the axis of the lens,  we will find that the lens will make an image of this point in the second principal plane at the same distance from the axis.  Next, let us select a ray parallel to the axis and compute its path through the lens to the image.  If we extend the starting and final pieces of the ray, we find that they cross at the second principal plane. 
Departing from paraxial optics to geometric optics proper, the principal planes are no longer planes, but curved surfaces, their  shapes are fixed, being given by the design of the lens.   We now want to make a sharp image of an object at a certain distance.  If the principal surfaces are spheres, centered on the object and the image,  Abbe's sine condition is fulfilled for this object distance, and we obtain an image free from the aberration called coma. 
Coma is an aberration that makes the image of a luminous point, (not at the center of the field of view), look like a small  comet,  pointing either towards or away from the center of the field.  Since the image is spread out radially from the center of the field, one can ascribe this to a change of  magnification along the comatic image.  It turns out that there is a connection between the sine condition and magnification, therefore we also have the connection between the sine condition and coma.
Coma is a troublesome aberration for various reasons, and should be corrected if possible.  Freedom from coma is also a condition for the use of Fourier optics.
Sometimes the sine condition makes it possible to predict the presence of coma without any computation.  A flat Fresnel lens has coma because the rays are bent at the flat surface which is not a sphere centered at the image.  This could be remedied by using a spherical substrate for the Fresnel lens.  The same reasoning holds for a flat holographic lens.  A parabolic mirror has coma because the rays are reflected at the parabolic surface which is not a sphere.  A spherical mirror will also have coma because the spherical surface has a radius that is twice the focal length.  
A: This article on abbe sine condition has a full explanation but in short It relates the angle of the input marginal ray with the angle of the output marginal ray.  It is used by optical engineers to reduce aberrations like coma.
