Say you have a convex lens with one of the sides completely coated with a mirror like substance, effectively rendering one side into a mirror. How would this lens work? Would the usual formulas like $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$ or $\frac{1}{f} = (n-1)(\frac{1}{R_1} + \frac{1}{R_2})$ still apply?
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Yes. That is exactly the same as a convex mirror.
Instead of the lens formula $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$, There is a formula called the mirror formula $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$.
The Lens Maker's relation has a counterpart where $f = \frac{R}{2}$.
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$\begingroup$ Thanks, but does that mean that the index of refraction or the original focus of the lens have no impact on the final focus? $\endgroup$ Commented Jan 21 at 8:54
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$\begingroup$ That would require assuming that the ray passes through the glass to reach the reflecting layer which is quite different than how the curved mirrors actually work. There, the index of refraction would definitely have an effect on the focal length. $\endgroup$ Commented Jan 21 at 9:55