It was discussed in a previous article, Mirrors in Geometrical Optics, Paraxial Approximation that the paraxial approximation is used to consider an apparently imperfect or flawed system as a perfect system.
The paraxial approximation was proposed in response to a normal occurrence in optical systems where the focal point is inconsistent for incident rays of higher incidence angles.The focal point F for a spherical mirror is understood under the paraxial approximation to be half the radius of curvature. Without the paraxial approximation, the system becomes increasingly complicated, as the focal point is a varying trigonometric function of the angle of incidence. The paraxial approximation assumes that all incident angles will be small.
The paraxial approximation can be likened (and when analyzed fully, this is it exactly) to a case of a triangle of base B, hypotenuse H and angle θ. Consider a case where H/B is very close to 1. θ will also be very small. In this case, it is of little harm to consider such a triangle as a triangle with θ=0, virtually to lines on top of each other, H and B, and more explicitly, H=B. This is precisely what is done when using the paraxial approximation.
An interesting question to ask is, what angle should be the limit to which we allow a paraxial approximation? The answer would be, it depends on how accurate, or clear the image must be. When discussing optical systems, an aberration is a case in which rays are not precisely focused at the focal point of a mirror (or another type of optical system involving focusing). An aberration will actually cause the image clarity to be reduced at the output of the system. The following image would be an example of the result of an aberration to an image in an optical system:
Here is an example of a problem that makes clear an example of the issue of an aberration. Two rays appear to be correctly aligned to the focal point, however another ray with angle of incidence of 55 degrees is not focused at the focal point. A system that would allow a ray of incidence of 55 degrees may be acceptable under some circumstances, however one would expect to have an aberration or some level of blurriness to the image.