6.4. Strehl ratio

One of the most frequently used optical terms in both, professional and amateur circles is the Strehl ratio. It is the simplest meaningful way of expressing the effect of wavefront aberrations on image quality. By definition, Strehl ratio - introduced by the German physicist, mathematician and astronomer Dr. Karl Strehl at the end of 19th century - is the ratio of peak diffraction intensities of an aberrated vs. perfect wavefront. The ratio indicates image quality in presence of wavefront aberrations; often times, it is used to define the maximum acceptable level of wavefront aberration for general observing - so-called diffraction-limited level - conventionally set at 0.80 Strehl.

Wavefront deviations from perfect spherical are directly related to the size of phase errors at all points of wave interference that form diffraction pattern. In other words, it is a nominal wavefront deviation from spherical that determines the change in pattern's intensity distribution. However, it is not the peak-to-valley nominal aberration, which only specifies the peak of deviation, and is entirely meaningless in regard to the extent of the aberration over the wavefront. It is the root-mean-square, or RMS wavefront error, which expresses the deviation averaged over the entire wavefront. This average wavefront deviation determines the peak intensity of diffraction pattern and, hence, numerical value of the Strehl ratio.

For relatively small errors - roughly 1/2π wave RMS (in units of the wavelength), and smaller - the RMS wavefront error, and the resulting Strehl ratio, mainly accurately reflect the effect of overall change in energy distribution, regardless of the type of aberration. This is more so with respect to the effect of change in distribution, than the nominal change itself. For instance, peak diffraction intensity drop is nearly identical at 0.0745 wave RMS and 0.15 RMS wavefront error, 20% and 33% respectively, for all three, spherical aberration, coma and astigmatism. At the same time, drop in energy encircled within the Airy disc is 20% and 12% at 0.745 wave RMS, and 33% and 20% at 0.15 wave RMS, for spherical aberration and coma vs. astigmatism, respectively. However both, nominal Strehl and overall contrast level remain nearly identical for all, due to the energy transferred by astigmatism effectively transforming central disc into a larger, cross-like form, reducing contrast level over the higher range of MTF frequencies.

While the actual Strehl calculation requires complex math, simple empirical expression given by Mahajan gives a very close approximation of the Strehl ratio in terms of the RMS wavefront error:

S ~ e-(2πω)2 ~ 1/e(2πω)2 (56)

where e is the natural logarithm base (2.72, rounded to two decimals), and ω is the RMS wavefront error in units of the wavelength. The approximation is accurate to a couple of percent for RMS errors of ~1/10 wave, with the difference diminishing for smaller errors. The difference vs. exact Strehl value gradually increases with the RMS error, but even at S~0.3 it still doesn't exceed 10%. It overestimates true Strehl for balanced primary aberrations, and underestimates it for classical aberrations.

For small RMS errors (~1/15 wave or less), a simpler approximation, given by S~1-(2πω)2, or S~1-39.5ω2, is also accurate; however, it becomes increasingly inaccurate with larger RMS errors - at 1/10 wave it already underestimates the true Strehl by more than 10%, and drops to zero at ~0.16 wave RMS (FIG. 60). For errors larger than ~1/15 wave RMS, and smaller than 1/5 wave RMS, another simple empirical approximation S~1-10ω1.5 gives slightly less accurate result than Mahajan's approximation for RMS<0.2 (within 2%), but has better overall accuracy than the two alternative approximations.

FIGURE 60: Strehl ratio curve plotted for four ratio approximations and the true Strehl value for primary spherical aberration at best focus (balanced spherical). For larger RMS errors (~0.1 wave RMS and larger) the true Strehl ratio for best focus coma is slightly higher than the ratio for spherical aberration, while the ratio for best focus astigmatism is slightly higher then that for the coma (the astigmatism Strehl curve is only slightly lower than that for Mahajan's approximation, plotted with the dashed-dotted line). Both become lower than the ratio for spherical aberration as the RMS error exceeds ~0.2. This implies that for larger wavefront errors, a given RMS wavefront produces slightly different Strehl value with different aberrations.

The top three approximations are accurate for RMS errors smaller than ~0.05, while the approximation from Mahajan remains reasonably close to the true ratio value even for RMS errors in excess of ~0.2, for classical and balanced (best focus) aberrations in general. For best focus (balanced) spherical aberration, the approximation remains close to the true Strehl for wavefront errors in excess of 0.25 wave RMS. Bottom approximation (red) is close to the true Strehl for RMS errors smaller than 0.18. Optimized to a narrower RMS error range, 0-0.1 wave, S~1-20ω1.77, it is within a fraction of 1%, relatively, from the true Strehl.

Conventional "diffraction-limited" aberration level is set at the Strehl ratio of 0.80 or, in terms of the RMS wavefront error, 0.0745 (or 1/√180), regardless of the type of aberration. This assumes no image quality factors other than wavefront quality present; presence of other factors affecting image quality (aperture obstruction, chromatism) put higher demands on wavefront quality for achieving "diffraction-limited" level.

The RMS wavefront error in terms of Strehl ratio is, from Eq. 56, closely approximated as ω~0.24√-logS. For the range of aberration mentioned, drop in the peak intensity expressed by the Strehl ratio also indicates the relative amount of energy transferred from the central disc to the ring area of the diffraction pattern, given as (1-S). Moreover, this relative number also indicates the average contrast loss over the range of resolvable frequencies. Regardless of the aberration type, these three basic properties of an aberrated pattern - the relative drop in central intensity, relative amount of energy transferred to the rings area, and averaged relative contrast loss - are practically identical for a given RMS wavefront error.

While the Strehl ratio furnishes very useful quantitative information about the effect of an aberrated wavefront, it is of general nature. It doesn't give specific indications on how the contrast varies for details of different angular size, nor how it affects the resolution limit. Also, there are factors affecting intensity distribution within diffraction pattern - such is pupil obstruction or apodization - not originating from wavefront aberrations. Hence, the Strehl figure doesn't include such effects. The effects of change in the pupil transmission factor due to obstructions of various forms still can be expressed through the PSF, as a single number comparable to the Strehl ratio.

Potentially more versatile indicator of the effect of aberrations is encircled energy (FIG. 61). It shows what portion of the energy is contained within a circle of given radius, centered at the intensity peak of the diffraction pattern. If specified for more than a single radius, it gives more detailed picture of intensity distribution.

FIGURE 61: Illustration to the left shows Point Spread Function (PSF), whose peak intensity determines the value of Strehl ratio, and encircled energy (EE) of a perfect (aberration-free) and aberrated aperture (0.25 wave P-V of spherical aberration), as a function of diffraction pattern radius, given in units of λF. In the presence of aberrations, the energy is spread wider, thus the energy encircled within a given pattern radius diminishes. Encircled energy figure can be given not only for the Airy disc, but also for any radius of the diffraction pattern. It can indicate possible change in size of the central disc, or furnish some other information of particular interest. An additional EE value for, say, 2.5λF radius, would indicate how much of the energy lost from the disc ended up in the first bright ring. The difficulty in using the EE in this manner arises with asymmetrical aberrations, since the amount of energy at any radius can vary significantly with the pupil angle. Showing this aspect of energy distribution would require several EE figures for each of various radial angles, or some kind of a graphical (contour) EE presentation - far from clear simplicity of the Strehl.

However, for more specific information on the effect of wavefront aberrations on image quality, as well as the effect of other factors affecting wave interference in the focal zone, the calculation has to expand from the characteristics of a single point-image (PSF), to those of the images of standardized extended objects, covering the entire range of resolution. Needed tool is found in the optical transfer function (OTF), a Fourier transform of the PSF.

◄ 6.3.1. Star testing telescopes ▐ 6.5. Effects of aberrations: MTF ►

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