P-V vs. RMS
Definitions
The RMS (Root Mean Square) deviation of the surface of a mirror is the square root of the area-weighted mean squared deviation from the parabolic surface which minimizes this quantity. That parabola will be called the "best-fit" parabola. A convenient unit for the surface RMS is nanometers; another popular unit is waves, twice the surface RMS divided by the reference wavelength of light (usually 550 nm). The importance of the RMS is that the Strehl ratio of the mirror can be found in terms of the RMS by the Mahajan approximation
with surface RMS s and reference wavelength L. "The general implication of a good-quality image is that the Strehl ratio is 0.8." Solving Mahajan's approximation gives a passing surface RMS of 20.7 nm.
The P-V (Peak-Valley) deviation of the surface of a mirror is the maximum deviation minus the minimum deviation from the best-fit parabola. Parallel to the RMS, a convenient unit for the surface P-V is nanometers; it is also popularly given in terms of waves, twice the surface P-V divided by the reference wavelength of light. The Rayleigh Criterion says that the P-V should be less than ¼ wave, or surface P-V < 68.8 nm.
Examples
These deviation plots are for a 200 mm f/6 mirror. The plot x-axis is zone radius in mm, the y-axis is the surface deviation in nm. The first is the worst ¼-wave mirror I could find:
The surface RMS is 22.9 nm and Strehl ratio 0.760, not quite passing.
The next example, gotten by flattening the outermost deviations from the above "bad" ¼-wave mirror, is a "good" ¼-wave mirror:
With surface RMS 13.3 nm and Strehl ratio 0.912, it is an excellent mirror.
Finally, a 1-wave mirror:
which passes even without a secondary shadow, surface RMS 19.9 nm, 0.813 Strehl ratio.
These three examples show the general P-V vs. RMS situation, which is illustrated by the usual "intersection" diagram:
The yellow region represents mirrors that are both ¼-wave and 0.8 Strehl, green 0.8 Strehl but not ¼-wave, and the red sliver is the small number of ¼-wave mirrors that don't "pass".
