Wavefront error

A perfect lens turns a plane wave into a sphere converging on the focus; a perfect flat mirror reflects a plane wave as a plane wave. Any real optic departs from that ideal by a small path difference that varies across the aperture, the wavefront error $W(x, y)$, usually expressed in waves at a stated wavelength or in nanometres. It is the common currency of optical tolerancing: surface figure, lens aberrations, mounting stress and thermal distortion all add to it, and it determines the Strehl ratio and the point spread function.

Peak-to-valley and rms

The peak-to-valley (P-V) value is the difference between the highest and lowest points of $W$; the rms value is its standard deviation over the pupil. The rms relates directly to image quality, and the P-V is sensitive to a single bad point, so modern specifications favour rms or a robust P-V that discounts isolated spikes. The ratio of the two depends on the shape of the error: 3.46 for pure defocus, 3.35 for balanced primary spherical aberration and 4.90 for astigmatism, and often larger for surfaces with local defects. Rayleigh's quarter-wave criterion (λ/4 P-V) applied to defocus therefore corresponds to λ/13.9 rms, close to the Maréchal criterion of λ/14 rms, and both give a Strehl ratio of about 0.8. Decomposing $W$ into Zernike polynomials gives an rms per aberration term.

Surface and wavefront

A surface error of height $h$ adds a reflected wavefront error of $2h$ at normal incidence and a transmitted error of $(n-1)h$. A mirror with a surface figure of λ/10 P-V at 632.8 nm, 63 nm, reflects a wavefront with λ/5 P-V error, while a glass surface of the same figure adds only about a quarter as much to a transmitted wavefront, since $n - 1 \approx 0.5$ for common glasses. Catalogue flats are commonly graded at λ/4, λ/10 and λ/20 surface flatness at 632.8 nm, and the specification should state whether it refers to surface or wavefront and to P-V or rms.

Measurement

The standard instrument is a Fizeau interferometer at 632.8 nm, which compares the test surface or transmitted wavefront with a reference flat or sphere and reports the error map, P-V, rms and Zernike fit, typically with a repeatability of a few nanometres rms. A Shack-Hartmann sensor, a lenslet array in front of a camera, measures the wavefront slope from spot displacements and is used for laser beams and in adaptive optics. Measurements at 632.8 nm are scaled to the working wavelength by the ratio of wavelengths, which assumes the error is geometric rather than dispersive.

References: M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 9; D. Malacara (ed.), Optical Shop Testing, 3rd ed. (Wiley, 2007); ISO 10110-5, Optics and photonics: Preparation of drawings for optical elements and systems: Surface form tolerances.