# What are the limitations on the Paraxial Approximation?

What my physics book referred to as the paraxial approximation derived to be thus:

$$\frac{1}{s}+\frac{1}{s'}=\frac{2}{R}$$

as a way of showing that in a concave spherical mirror, all reflected rays hit the a single point P. I think it is fair use to show just this diagram, since the book is more than 1000 pages:

It used these equations, and simply removed the $\delta$ term as a simplification for when $$\tan{\alpha}=\frac{h}{s-\delta}\space\space\tan{\beta}=\frac{h}{s'-\delta}\space\space\tan{\phi}=\frac{h}{R-\delta}$$. Using WolframAlpha, I solved for the original Equation:

$$s'=\frac{\delta(s-2R)+Rs}{\delta+R-2s}$$

But this didn't help me understand why exactly all rays converge to P. The book says: "The equation does not contain a single $\alpha.$. It seems, in fact, that since $\theta$ depends on $\alpha$, that in fact the image is somewhat distorted for anything BUT a point.

Changing any single variable in this equation changes all the others! It continues: "All rays from P that make sufficiently small angles with sufficiently small angles with the axis intersect a P' after they are reflected..." but that doesn't make sense to me either.

I guess my question could be summed up as this:

A.) Why is this approximation useful? Would an actual engineer/scientist use it or is it mostly a learning tool?

B.) What is $$\frac{ds'}{d\alpha}$$? Am I correct to infer this value shrinks with the value of $\alpha$?

The book is University Physics by Sears & Zemansky.

• The image IS distorted even for a single point. Spherical mirrors have pretty poor optical properties and if you want to go past the approximation, then you will have to read entire books on technical optics that contain the methods to design better imaging systems without these errors. Physics textbooks rarely talk about the level of actual understanding needed to design even the most simple useful optics. May 3, 2015 at 6:22
• Just want to point out for completeness, the correct mirror shape is ellipsoidal -- a prolate spheroid to be precise -- where the object point is at one focus and the image point is at the other focus. A sphere is only an approximation to the correct shape of the end-cap of the spheroid, hence the need for a small-angle approximation. May 3, 2015 at 7:55
• @CuriousOne Could you be so kind as to suggest something? I have an astronomy hobby and a VR hobby and I actually am interested in designing or at least understanding simple optics, though I'm far less than fully trained Engineer... highest math I've had is differential equations. Something to do on weekends.... May 3, 2015 at 16:10
• Unfortunately, I don't know enough about technical optics to give good advice about the best way to learn about it. I would suggest you look for a good introductory textbook that satisfies your curiosity on the level you are most comfortable with. That approach usually leads to a new level of insight from which you can go on reading the more technical literature. The level of math that seems to be useful for many optics applications is a combination of linear algebra and multivariate calculus. If you are comfortable with those, you should have no problems with the topic. May 4, 2015 at 5:04