An optical system is made by two two concave mirrors having the same focal distance $f$, positioned face to face and separated by a distance $d$. What is the relation between $d$ and $f$ for the final image of a luminous point on the main optical axis to coincide with the initial object?
The figure is below:
Let $x1$ and $x2$ be the position of the object, respectively of the image with respect to first mirror.
Then we can write from the mirror equation:
$(1/x1) +(1/x2) =(1/f)$ that is $x2 =(x1*f)/(x1-f)$
The image from the first mirror is an object for the second mirror (see the figure) so that
$x1′ =d-x2= (x1*d-d*f-x1*f)/(x1-f)$
$x2′ =f(x1*d -d*f-x1*f)/(x1*d-d*f-2×1*f+f^2)^2$
Since from the figure we have
$x2′ =d-x1$ then