Considering the effectiveness of 3rd order aberration theory from the view point of sine condition in the presence of spherical aberration

Even though the level of computer simulation for optical designing is becoming powerful, the 3rd order aberration theory is still useful for prospecting optical characteristics and developing optics. Therefore, it is important to confirm its validity even now. Considering the physical meaning of imaging, pupil coordinates should be defined by the direction cosine of the ray. The sine condition in the presence of spherical aberration has been derived by using this coordinate and its validity was later confirmed by practical lens designing. On the other hand, in order to deal with object imaging and pupil imaging equivalently, conventional aberration theory uses the pupil coordinate defined by the slope of the ray. By applying this theory, it had been deduced that no spherical aberration exists when the isoplanatic condition is fulfilled and it was then concluded that the sine condition in the presence of spherical aberration is meaningless. Therefore, one might think that this conventional aberration theory is less useful. However, this time I found that the 3rd order spherical aberration can exist with no 3rd order coma aberration under this aberration theory. As a result, the conventional aberration theory is meaningful at least in the 3rd order region.