We all learned that the focal length of plane mirrors is considered to be infinite, as the radius of curvature is infinite. However, imagine this scenario: You have a perfectly flat (no one get mad, this is theoretical)plane mirror, and the earth is perfectly flat as well(theoretical). If you looked into this mirror with a telescope (imagine it is powerful enough), would you be able to see as far away as you wanted to, just as if you were looking into reality? I know that spherical apertures are limited by Lord Rayleigh's criterion (after all, only so many photons can be recorded onto a surface), so are plane mirrors limited by this as well? If so, then we should call the focal length of a plane mirror something besides infinite, right?
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1$\begingroup$ Isn't the focal point defined as the point where reflected parallel rays converge? The reflected rays of a 'flat' mirror never converge, so the best answer for the focal distance is infinite. $\endgroup$– LDC3Jun 29, 2014 at 14:27
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The focal length is a concept in geometric optics which is well-defined only to the extent that the light (or a different radiation) may be approximated by light rays. That usually requires the wavelength to be much shorter than the typical geometric dimensions of the experiment.
Moreover, for the focal length to be well-defined, the light rays going along different paths must converge to one point, the focus. The parallel light rays may be thought of as a limit of light rays converging at distant point, $L\to \infty$, which is why the focal length associated with parallel light rays (even after they get through the lenses or mirrors) is infinite.
Lord Rayleigh's criterion is an inequality indicating when the wave phenomena such as dispersion become important. But when they become important, it's exactly the moment when the light rays cease to converge to a well-defined point or they cease to exist altogether because they must be replaced by a full-fledged wave description of the wave. That's why the focus is no longer uniquely well-defined or geometric optics breaks down completely. At any rate, the images get unavoidably blurry and the focal length becomes ill-defined.
The focal length of some mirrors is computed with the assumption that the relevant light's wavelength is sufficiently short (e.g. much shorter than the aperture etc.) and the issues with dispersion don't arise at all. One may imagine some super-short-wavelength light in every optical device (recall that the wavelength of the visible light is about half a micron, short enough relatively to most telescopes, apertures etc.). When this condition ceases to be valid, one must abandon all these concepts such as "the focus" and "the focal length". They become ill-defined rather than acquiring some particular different value. The focal length quantifies where the sharp image (intersection) of the rays from a very distant object is; it doesn't quantify where the image becomes blurry or how far one may reliably see something.
answered Jun 29, 2014 at 14:51