# Requirements for rays to converge exactly at a point $P'$

I am currently studying Modern Optical Engineering, fourth edition, by Warren Smith. The textbook says the following:

In Fig. 1.10 several ray paths have been drawn for the case of a converging lens. Note that the rays originate at point $$P$$ and proceed in straight lines (since the media involved are isotropic) to the surface of the lens where they are refracted according to Snell’s law (Eq. 1.3). After refraction at the second surface the rays converge at the image $$P'$$. (In practice the rays will converge exactly at $$P'$$ only if the lens surfaces are suitably chosen surfaces of rotation, usually nonspherical, the axes of which are coincident and pass through $$P$$.) This would lead one to expect that the concentration of light at $$P'$$ would be a perfect point. However, the wave nature of light causes it to be diffracted in passing through the limiting aperture of the lens so that the image, even for a “perfect” lens, is spread out into a small disc of light surrounded by faint rings as discussed in Chap. 9.

This part is unclear to me:

(In practice the rays will converge exactly at $$P'$$ only if the lens surfaces are suitably chosen surfaces of rotation, usually nonspherical, the axes of which are coincident and pass through $$P$$.)

I'm presuming that the "surfaces of rotation" that the author refers to are surfaces of revolution? Why must the surface of rotation be nonspherical? And what is meant by "the axes of which are coincident and pass through $$P$$"? Furthermore, why do we require all of this in order for the rays to converge exactly at $$P'$$?

I would greatly appreciate it if people would please take the time to clarify this.

## 1 Answer

Yes, revolution, meaning the lens has cylindrical symmetry. You want that the rays that refract at different heights from the center converge into a point. But this is by no means guarantee. In particular, a spherical lens will not satisfy this requirement very well, look at spherical aberration .

Ideally you want a lens with a shapes that allows all rays of different heights to converge at the same point. it is difficult to make a perfect lens, see here .

And the focus will be better when the object is in the axis of the lens.

• Thanks for the answer. What are some examples of nonspherical surfaces of revolution? And, just to clarify, "the axes of which are coincident and pass through $P$" means that the object is on the optical axis of the lens? Mar 2, 2020 at 22:44
• yes to the last question. Regarding the first, I do not think those shapes resemble any of the known ones. I do not remember right now the details, but some shapes like parabolic have some advantages over a spherical lens in some specific cases. making good lenses with the right shape is a science in itself.
– user65081
Mar 2, 2020 at 22:50
• Oh, ok, so nonspherical surfaces of revolution are quite atypical shapes? Mar 2, 2020 at 22:53
• it will not look weird to you, it will likely look spherical to you, but the shape details are not the one of any standard geometrical figure we use to learn at college.
– user65081
Mar 2, 2020 at 23:04
• Ok, thank you for clarifying this. Mar 2, 2020 at 23:05