Strehl ratio

The Strehl ratio, named after the German physicist and mathematician Karl Strehl (1864-1940), is a measure for the optical quality of telescopes and other imaging instruments. It is defined as the ratio of the observed peak intensity at the detection plane from a point source as compared to the theoretical maximum peak intensity of a perfect imaging system working at the diffraction limit.^[1] This is closely related to the sharpness criteria for optics defined by Karl Strehl.^[2][3] It can be computed from Fresnel’s diffraction theory as:

${\displaystyle S=|\langle e^{i2\pi \delta /\lambda }\rangle |^{2}}$

where i is the imaginary unit, δ is the deviation of the wavefront at some point in the exit pupil, λ is the wavelength and ⟨·⟩ means averaging over the entire exit pupil. The Strehl ratio is often estimated using the Mahajan’s approximate formula

${\displaystyle S\approx {e^{-(2\pi \sigma /\lambda )^{2}}}}$

with sigma (σ) being the root mean square deviation of the wavefront (i.e. σ² = ⟨δ²⟩ - ⟨δ⟩²).^[4][5] Unless stated otherwise, the Strehl ratio is usually calculated at the best focus of the imaging system under study.

Usage

The ratio is commonly used to assess the quality of astronomical seeing in the presence of atmospheric turbulence and assess the performance of any adaptive optical correction system. It is also used for the selection of short exposure images in the lucky imaging method.

In industry, the Strehl ratio has become a popular way to summarize the performance of an optical design because it gives the performance of a real system, of finite cost and complexity, relative to a theoretically perfect system, which would be infinitely expensive and complex to build and would still have a finite point spread function. It provides a simple method to decide whether a system with a Strehl ratio of, for example, 0.95 is good enough, or whether twice as much should be spent to try to get a Strehl ratio of perhaps 0.97 or 0.98.

The Airy disk

Computer-generated image of the Airy disk

Graph of the Airy intensity function vs. normalized radius

Due to diffraction, even a focusing system which is perfect according to geometrical optics will have a limited spatial resolution. In the usual case of a uniform circular aperture, the point spread function (PSF) which describes the image formed from an object with no spatial extent (a "point source"), is given by the Airy disk as illustrated here. For a circular aperture, the peak intensity found at the center of the Airy disk defines the point source image intensity required for a Strehl ratio of unity. An imperfect optical system using the same physical aperture will generally produce a broader PSF in which the peak intensity is reduced according to the factor given by the Strehl ratio.

Note that for a given aperture the size of the Airy disk grows linearly with the wavelength ${\displaystyle \lambda }$, and consequently the peak intensity falls according to ${\displaystyle \lambda ^{-2}}$ so that the reference point for unity Strehl ratio is changed. Typically, as wavelength is increased, an imperfect optical system will have a broader PSF with a decreased peak intensity. However the peak intensity of the reference Airy disk would have decreased even more at that longer wavelength, resulting in a better Strehl ratio at longer wavelengths (typically) even though the actual image resolution is poorer.

Limitations

Characterizing the form of the point-spread function by a single number, as the Strehl Ratio does, will be meaningful and sensible only if the point-spread function is little distorted from its ideal (aberration-free) form, which will be true of a well-corrected system that operates close to the diffraction limit. That includes most telescopes and microscopes, but excludes most photographic systems, for example. The Strehl ratio has been linked via the work of Marechal to an aberration tolerancing theory which is very useful to designers of well-corrected optical systems, allowing a meaningful link between the aberrations of geometrical optics and the diffraction theory of physical optics. A significant shortcoming of the Strehl ratio as a method of image assessment is that, although it is relatively easy to calculate for an optical design prescription on paper, it is normally difficult to measure for a real optical system, not least because the theoretical maximum peak intensity is not readily available.

See also

• Circle of confusion
• Fraunhofer diffraction
• Fraunhofer diffraction (mathematics)
• Huygens–Fresnel principle
• Optical transfer function

References

External links

• Discussion page R.F. Royce' explanation of Strehl ratio in lay terms
• Strehl meter W.M. Keck Observatory Strehl calculator page
• Definition page Eric Weisstein's World of Physics
• Strehl ratio Telescope Optics Net practical explanation of Strehl ratio for amateur telescope makers