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In the discussions of ray optics here, and in discussions of optics elsewhere, you will frequently encounter the terms "smooth" and "flat" ( or "optical flat"). What do these words mean?
Rays and Wavelength
So far we have discussed light as rays, abstract objects that travel in straight lines unless the index of refraction of the material they are traveling in changes. In the next section, you'll see that rays are a special case of electromagnetic waves. Each electromagnetic wave has a few different properties. Among them is the wavelength, equal to the distance between successive peaks of the electromagnetic field that makes up the wave.
The wavelength of light varies dramatically, from fractions of a nanometer for gamma ray light to hundreds of nanometers for light that is visible to human eyes to centimeters or meters or kilometers for radio light.
Smooth Surfaces
When a surface is described as "smooth" in the context of optics, we mean that the deviations in the surface from the mathematically smooth and continuous curve that we are using to describe it occur only on length scales much smaller than a wavelength - usually by a factor of 10 or more. This means that a surface that is smooth and produces mirror reflections for some wavelengths of light may not produce mirror reflections for other wavelengths.
For example:
A telescope has a parabolic reflector that has surface irregularities of no more than 100 nanometers. Which kind of light would it be an effective mirror for?
Flat Surfaces
While a reflective surface is smooth if irregularities in its surface are very small compared to the wavelength we care about, a surface is only flat if its entire shape occupies very nearly the same plane - to within a small fraction of a wavelength.
Any mirror that is not so nearly planar will not produce an exact mirror image. Instead, reflections will be distorted. As discussed throughout the rest of this section, that can be extremely useful.