Strehl Ratio and Wavefront Error
Strehl ratio from an RMS or peak-to-valley wavefront error: the Maréchal approximation against the exact value for defocus, astigmatism, coma or spherical aberration, with the λ/14 and S = 0.8 tolerances.
PV/RMS is fixed only for a single Zernike term (3.464 for defocus). A measured surface mixes terms, so convert its PV with care; RMS is the number the Strehl ratio depends on.
Results
Maréchal's λ/14 is where the first-order form 1 − (2πσ/λ)² reaches 0.8; the exponential form reaches 0.8 at 0.0752λ. The exponential stays within 2 % of the exact Strehl of each term up to σ = 0.1λ (S ≈ 0.67); the first-order form is already 10 % low there. The exact Strehl is the squared pupil average of exp(2πiσZ/λ) for one balanced Zernike term (Noll normalisation, RMS = coefficient).
Strehl ratio vs RMS wavefront error
Model and references
Exact. The equations are exact within the stated assumptions, or are a standard's tabulated values. The results are as good as the inputs. About the tiers
Exact Strehl ratio for one Noll-normalised Zernike term on a uniform circular pupil (radial integral, numerical), shown with the extended Maréchal approximation exp(−(2πσ/λ)²). PV/RMS is the ratio of that single term.
- Noll R. J. (1976). Zernike polynomials and atmospheric turbulence. J. Opt. Soc. Am. 66, 207–211. Zernike normalisation: the coefficient is the RMS of the term. doi:10.1364/JOSA.66.000207
- Mahajan V. N. (1982). Strehl ratio for primary aberrations: some analytical results for circular and annular pupils. J. Opt. Soc. Am. 72, 1258–1266. Strehl ratio of balanced primary aberrations. doi:10.1364/JOSA.72.001258
- Mahajan V. N. (1983). Strehl ratio for primary aberrations in terms of their aberration variance. J. Opt. Soc. Am. 73, 860–861. The approximation exp(−(2πσ/λ)²). doi:10.1364/JOSA.73.000860