Strehl ratio

The Strehl ratio compares a real optical system with a perfect one of the same aperture and wavelength. Both focus a point source into a point spread function; for the perfect circular aperture that pattern is the Airy disk. Aberrations do not change the total energy, but they move some of it out of the central peak into the rings and surrounding halo, so the peak falls. The Strehl ratio is the aberrated peak divided by the ideal peak, from 1 for a perfect system down toward zero.

For small aberrations, the Strehl ratio depends only on the root-mean-square wavefront error $\sigma$, whatever its shape. The Maréchal approximation, $S\approx \exp [-(2\pi \sigma \mathrm{/}\lambda {)}^2]$, gives 0.82 for an rms error of $\lambda \mathrm{/}14$, 0.91 for $\lambda \mathrm{/}20$ and 0.67 for $\lambda \mathrm{/}10$. The simpler form $1-(2\pi \sigma \mathrm{/}\lambda {)}^2$, used in the aberrations entry, agrees closely in the range where either is valid, giving 0.80 at $\lambda \mathrm{/}14$. The Maréchal criterion takes a Strehl ratio of 0.8, an rms wavefront error of about $\lambda \mathrm{/}14$, as the point at which a system counts as diffraction-limited. The exponential form stays within about 0.05 of the exact value down to a Strehl ratio of roughly 0.3, and a system farther from ideal is better described by its full point spread function or its modulation transfer function; the simpler form is reliable only above about 0.8 and reaches zero at $\sigma =\lambda \mathrm{/}2\pi$.

The number is used wherever an optical system must approach the diffraction limit. Lens and telescope designers quote it as a tolerance target; adaptive-optics systems in astronomy report it as the measure of how well atmospheric turbulence has been corrected, often at a few tenths in the near infrared; laser beam delivery and focusing optics are specified by it; and microscope objectives are tested against it. Because it compares peak intensities, it relates directly to what matters in many laser applications, the intensity reached at focus, which is why it is used alongside beam quality factors in high-power systems.

Measuring the Strehl ratio directly means imaging a point source and normalizing the peak to that of a computed ideal pattern with the same total energy, which is sensitive to detector sampling, background and the normalization itself. More commonly the wavefront is measured with an interferometer or a Shack-Hartmann sensor, and the Strehl ratio is computed from the rms error.

Reading the surface and wavefront specifications that feed into the Strehl ratio is covered in How to Read an Optics Spec Sheet.

References: M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 9; V. N. Mahajan, J. Opt. Soc. Am. 72, 1258 (1982).