Proof of image formation property of spherical mirrors

[Gabriel Maia]

Hi. I'm trying to proof the image formation property of a concave spherical mirror. I know you can do this easily with a particular choice of rays (namely one that hits the vertex and one that passes through the center of the sphere) but I would like to show that a generic ray yields the same result because

1) Any two non-parallel rays will cross paths sonner or later so to use only two rays gives you a result but not a satisfactory proof.

2) The rays chosen for this easy proof pass through special points of the problem's geometry and could be a particular result.

I know there is a purely geometrical approach to this general proof I'm seeking but since I want to keep consistency with the work I've already done, I'm doing it with Analytical Geometry.

The Mirror is given by the equation

$y=R-R\sqrt{1-\left(\frac{\displaystyle x}{\displaystyle R}\right)^2}$,

where $R$ is the radius of the mirror and its axis is parallel to the y-axis. The object in front of the mirror has coordinates $(x_{1},y_{1})$.

The ray reflected at the mirror's vertex has a line equation given by

$y=-\frac{\displaystyle y_{1}}{\displaystyle x_{1}}\,x$

and the one passing through the mirror's center has a line equation

$y=R+\frac{\displaystyle y_{1}-R}{\displaystyle x_{1}}\,x{\displaystyle }$.

These rays cross paths at $x_{2}=\frac{\displaystyle R}{\displaystyle R-2y_{1}}\,x_{1}$ and showing that

$y_{2}=-\frac{\displaystyle R}{\displaystyle R-2y_{1}}\,y_{1}$,

we obtain the known amplification factor $x_{2}=-\frac{\displaystyle y_{2}}{\displaystyle y_{1}}\,x_{1}$.

Now... let's consider a ray that hits the mirror at a point $(x_{0},y_{0})$. When this ray is reflected it will have a line equation

$y=y_{0}+\tan\left(2\arctan\left(-\frac{\displaystyle R}{\displaystyle x_{0}}\sqrt{1-\left(\frac{\displaystyle x_{0}}{\displaystyle R}\right)^2}\right)-\arctan\left(\frac{\displaystyle y_{1}-y_{0}}{\displaystyle x_{1}-x_{0}}\right)\right)(x-x_{0})$.

Under the paraxial approximation (which allows us to consider $x_{0}\ll R$) and using the property of arctan addition we can rewrite it as

$y=y_{0}+\tan\left(\arctan\left(\frac{\displaystyle 2x_{0}R}{\displaystyle R^2-x^2_{0}}\right)-\arctan\left(\frac{\displaystyle y_{1}-y_{0}}{\displaystyle x_{1}-x_{0}}\right)\right)(x-x_{0})$.

I know we can simplify it a bit more by using the $\tan(a+b)$ formula but the way I kept it above is easier to contemplate. I have tested all these rays with mathematica and they converge to the same point so I know everything up to here is ok. My problem is:

If I equal this last ray to one of the others (the one that reflects at the mirror's vertex, for instance) I'll obtain the x at which they cross paths as a function of $x_{0}$ and $y_{0}$. Since all the rays converge to the same point (given the paraxial approximation) I would not expect them to depend upon the point at which they touch the mirror.

What am I missing here? There is some simplification I did not do? Could you lend me a hand?

Thank you very much.

 

[pixel]

I haven't attempted to follow all of your equations, but since no one has responded yet I'll ask one question - you derived a general equation and then applied the paraxial approximation. So why do you still have trig functions in that equation? For instance, if x₀ ≪ R then R² - x₀² ≈ R² so the argument of the first arctan can be written 2x₀/R which << 1 in the paraxial approximation. Etc.

What I'm guessing is that if you rewrite the equation fully making use of the paraxial approximation then either x₀ and y₀ won't appear, or if they do they would be with other expressions that they are much less than.