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6.5. 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 the level of image quality in the 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.

Similar type of indicator is the Struve ratio, which expresses peak diffraction intensity of aberrated vs, aberration-free line spread function (LSF). It requires slightly tighter 0.80 ratio level requirement for primary coma (0.58 vs. 0.63 wave P-V), more relaxed for spherical aberration (0.27 vs. 0.25) and defocus (0.29 vs. 0.26) than the Strehl ratio (the tolerance for astigmatism is nearly identical). However, it has far less universal appeal than the Strehl ratio, expressing vital property of the single most important optical indicator, the PSF, building stone of nearly all intensity distribution forms, including the LSF.

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 tells nothing about its extent over the wavefront area. 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 (note that the RMS error itself is accurately representing the magnitude of wavefront deviation only when it is affecting relatively large wavefront area, which is generally the case with the conic surface aberrations).

For relatively small errors - roughly 0.15 wave RMS, and smaller - the RMS wavefront error, and the resulting Strehl ratio, accurately reflect the effect of overall change in energy distribution, regardless of the type of aberration. With larger errors, the correlation between the RMS error and the Strehl vanishes: larger RMS error can produce higher PSF peak intensity, and better image quality than the lower errors. Mid astigmatic focus, for instance, has identical PSF peak intensity at 2 and 3 waves P-V wavefront error, despite the latter having 50% higher RMS/P-V. Similar RMS-to-Strehl inconsistency above 0.15 wave RMS exist for spherical aberration, and aberrations in general.

As a general rule for aberrations below 0.15 wave RMS, the relative drop in peak diffraction intensity indicates how much of the energy is lost, relatively, from the Airy disc. For instance, 0.90 Strehl indicates about 10%  lower energy within the Airy disc. But the exceptions are possible, and generally larger in magnitude with larger error levels.

For instance, the drop in peak diffraction intensity is nearly identical at 0.0745 wave RMS and 0.15 RMS wavefront error - 20% and 59% respectively - for all three, spherical aberration, coma and astigmatism. At the same time, the accompanying drop in the energy encircled within the Airy disc is 20% and 11% at 0.745 wave RMS, and 61%, 56% and 38% 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 more, and less than the other two in the lower frequency range.

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

(56)

where e is the natural logarithm base (2.72, rounded to two decimals), and ω is usually the RMS wavefront error in units of the wavelength. Note that use of the RMS wavefront error can yield inaccurate result; the actual Strehl value - and the original form of approximation - are phase dependant, thus determined by phase variance φ2 and, more directly, by the phase analog to the OPD-based RMS wavefront error, φ, with φ2=(2πφ)2.

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 does not exceed 10%. It overestimates true Strehl for balanced primary aberrations, and underestimates it for classical aberrations.

This approximation is also known as "extended Maréchal's approximation", as opposed to the original Maréchal's approximation, S~(1-0.5φ2)2~[1-0.5(2πφ)2]2 which, for φ~ω, can be written in terms of the RMS wavefront error as S~[1-2(πω)2]2.

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. 97). For errors larger than ~1/15 wave RMS, and smaller than 1/5 wave RMS, a 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 97: Strehl ratio as a function of RMS wavefront error. LEFT:  Plots for three ratio approximations and the true Strehl value for primary spherical aberration at the best focus (balanced spherical; identical to the Siedel - i.e. Gaussian focus' - spherical aberration) in unobstructed aperture. Strehl ratio approximations, from the top down, Mahajan's (also known as "extended Maréchal approximation"), Maréchal's, and simplified Maréchal's, the latter with the 4th power term in the expansion neglected. The lower two approximations are accurate for RMS errors smaller than ~0.07, while Mahajan's 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. It remains close to the true Strehl for spherical aberration for wavefront errors in excess of 0.25 wave RMS.
For larger RMS errors (~0.1 wave RMS and larger) the true Strehl ratio for best focus coma (not shown) is slightly higher than the ratio for spherical aberration, and the ratio for best focus astigmatism (not shown) is slightly higher then that for best focus coma, with the latter only slightly lower than Mahajan's approximation. Both, coma and astigmatism Strehl ratios become slightly lower than the ratio for spherical aberration as the RMS error exceeds ~0.2. In general, for RMS wavefront errors over ~0.1 wave, Strehl value for given large RMS error varies slightly with the aberration type. This variation becomes more pronounced expanding to other wavefront forms; for instance, true Strehl for Siedel coma of 0.25 wave RMS is over 60% higher than for the three balanced forms (spherical, coma, astigmatism), and true Strehl for Siedel astigmatism more than doubled (that despite the RMS wavefront error keeping the same lower ratio for the balanced aberration forms; similar shift of the PSF peak away from the focus point with minimum wavefront deviation occurs with spherical aberration as well).
RIGHT: Change in the Strehl ratio due to central obstruction, for balanced primary spherical, defocus, coma and astigmatism, all at the best focus, for selected central obstruction sizes. Horizontal scale shows the RMS error for zero obstruction. Change of the Strehl is the consequence of the change in the RMS wavefront error due to obstruction of a portion of the wavefront, with the RMS of the annulus varying somewhat with the aberration type. With balanced spherical and defocus, Strehl ratio of the annulus continually increases with the size of central obstruction in a similar way, only the magnitude of increase is larger for the former. With coma, the change is negligible except for obstructions nearing 0.5D and larger, for which the Strehl increases by nearly 10%, or more (due to the shape of the comatic wavefront, with nearly flat central area and deformation increasing toward the edge, smaller obstructions - up to about 0.3D - cause relatively small increase in the annulus RMS error, but as the obstruction becomes large enough to significantly block out the deformed areas, the RMS and Strehl quickly start improving, becoming better than for the unobstructed wavefront). With astigmatism, the Strehl continually decreases, falling over 50% at 0.5D obstruction size.
The Strehl and RMS error are, of course, only one side of the story. In addition to the effect of optical path difference due to wavefront deviation from perfect sphere, wave interference at the focus is also affected by missing wave contributions due to wavefront obstruction. The PSF peak here is a product of the Strehl and PSF peak degradation due to central obstruction, given by
(1-ο2)2, where o is the relative linear obstruction diameter in units of aperture diameter.
For errors larger than about 0.15 wave RMS, the correlation between the RMS wavefront error and Strehl ratio becomes more loose, and the PSF peak shifts away from the point of minimum wavefront deviation. Since the plots are based on the RMS error at this point, they are accurate only up to about that error level. Thus, part of the plot beyond 0.15 wave RMS is only approximation of the actual change in the Strehl ratio.

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 only concerns wavefront quality; presence of other factors negatively affecting image quality, such as aperture obstruction, or chromatism, would result in further deterioration in quality of the diffraction image. Thus achieving "diffraction-limited" level in such circumstances requires higher wavefront quality, according to the magnitude of additional error.

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.

Strehl and encircled/ensquared energy

Potentially more versatile indicator of the effect of aberrations is the amount of point image energy contained in a circle of given radius, or a square of given side (encircled and ensquared energy, respectively). 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.

Illustration at left shows Point Spread Function (PSF) - with its peak intensity determining the value of Strehl ratio - and encircled energy (EE) of a perfect (aberration-free) and aberrated aperture (0.25 wave P-V of primary 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. It gets more complicated 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 the clear simplicity of the Strehl.

For pixel-based detectors like CCD, more relevant is the information on ensquared energy, although the difference between the two is generally small. The amount of energy contained within a square of given side vary with the form of aberration, and can be significant even if the Strehl number remains similar (FIG. 98).

FIGURE 98: Change in ensquared energy with increase in primary spherical aberration at best focus from zero to λ/4, λ/2 and
λ wave P-V, in comparison to four other aberration forms of comparable RMS wavefront errors (the RMS-to-PV ratio is
11.25, 28, 12, 32 and 24 for primary spherical, balanced secondary spherical, defocus, primary coma and astigmatism, respectively). Included is also the effect of central obstruction for 0, 0.3, 0.4 and 0.5 obstruction ratios. Since the RMS wavefront errors are identical, the corresponding Strehl values - 0.80, 0.39 and 0.14 - are either practically identical, for the RMS errors of about 0.15 and smaller, but are somewhat different for the 0.3 wave RMS level (for large errors, focus location with the lowest RMS does not coincide with the maximum Strehl focus location, and the difference vary somewhat with the form of aberration). Yet the corresponding ensquared energies may be significantly different, and particularly for larger error levels. It is the consequence of a different pattern of energy transfer out of the Airy disc: while for given Strehl all aberrations have similar amounts of energy lost from the central maxima, those with more of the transferred energy ending up closer to the central maxima - like astigmatism and defocus - have more ensquared energy at all significant aberration levels, and more so the larger the aberration magnitude.
In general, balanced secondary spherical spreads the energy most extensively, while astigmatism and defocus do the least amount of damage (encircled energy for coma is somewhat better if its centroid, not Gaussian focus, coincides with the square center). As a measure of the energy spread here is used the energy content in a square with the side equaling Airy disc diameter (87% of the total energy, marked by the horizontal dashed line). Aberrations at the level of λ/4 of primary spherical are already not negligible in this respect, with the 87% square side increasing from about 40% (astigmatism) to 150% (balanced secondary spherical), increasing for larger aberration levels roughly in proportion to this initial level. The effect of central obstruction in its usual range of sizes in imaging systems is comparatively small, roughly at the level of λ/4 wave P-V of primary spherical up to about 0.4D obstruction size, and somewhat worse for larger obstructions.
The blue FWHM (Full-Width-at-Half-Maximum) squares mark the diameter of the corresponding PSF FWHM, and also show how much energy is contained within it. The importance of the FWHM is in it being considered the determinant of point-source resolution, hence it complements the information on EE with this important aspect. There is little change in the size of FWHM in this aberration magnitude range for primary and secondary spherical and defocus, but its energy content drops significantly with the increase in aberration. At the 0.3 wave RMS level of defocus, there is no FWHM in terms of central maxima since the intensity drops to zero in the center, gradually increasing away from it (FIG. 47, 1λ P-V defocus). Half maximum of the decreasing intensity, outlining the ring-like central dot is quite wide, about 5.4λF in diameter. Similarly, balanced secondary spherical at the minimum RMS focus has central maxima of lower intensity than the surrounding rings (this inverse intensity pattern in the PSF center is indicated by the EE plot laying near flat close to the origin). However, the same aberration at the maximum Strehl focus (0.24mm away axially at
ƒ/8.15) does form the central maxima, resulting in more ensquared energy closer to pattern center, but less going farther out (that would result in better contrast transfer in the high MTF frequencies, and worse in the lower ones). Unlike it, primary spherical forms central maxima at both, minimum RMS and maximum Strehl foci (separated by about 0.25mm axially), only the former has somewhat more energy in the first few rings.
With coma, central maxima becomes elongated already at the 0.075 wave RMS level, with the cross section of the maxima with the pattern in sagittal (vertical) orientation being slightly narrower than that in aberration-free aperture, but more than twice wider in the tangential (horizontal) pattern orientation. With further increase in magnitude, central maxima becomes larger and more elongated, also shifting away from the point of Gaussian focus, as shown on the coma peak intensity shift graph.
Astigmatism doesn't change the FWHM appreciably up to about 0.15 wave RMS level, but quickly widens it after that, also developing central depression at 0.3 wave RMS at its minimum RMS focus, although not as deep as with 0.3 wave RMS of defocus. Its maximum Strehl focus for this aberration level is nearly coinciding with either tangential or sagittal focus, with a large discrepancy in the FWHM diameter along the two perpendicular axes.
Of course, both EE and FWHM are significantly affected (enlarged) due to the aberrations induced to the optical train, such as the seeing error, thermal effects on the optics and air within the instrument, miscollimation, and others. It is difficult to measure all those sources of error separately, and it is usually the easiest approach to model performance level with MTF and empirically measured FWHM.
Note that the corresponding square side in microns is given by a product of the wavelength λ in microns and the focal ratio F.

Still, encircled/ensquared energy remains quantitative indicator of image quality. 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. The needed tool is found in the optical transfer function (OTF), a Fourier transform of the PSF.

Strehl and MTF, Hopkins ratio

Being based on the system's PSF, Strehl ratio is directly related to its MTF, with the PSF being the inverse Fourier transform of the MTF. In effect, the Strehl represents the MTF averaged over all frequencies - in other words, it represents the averaged MTF contrast transfer. Thus the quantity 1-S represents the averaged MTF contrast loss due to the aberrations. General consensus for general observing is that contrast loss of up to 5% is inconsequential, and that loss of up to 20% does not significantly degrade performance.

The problem with such generalization is that: (1) contrast loss for most aberrations is not uniform over the range of MTF frequencies, and (2) the effect of contrast loss depend primarily on the inherent object contrast, and it varies widely from one object type to another. Hence 20% loss may not significantly degrade performance with some objects and details - possibly majority of them - but it will with some others, generally those with the lowest inherent contrast. That puts the acceptable contrast loss - depending of the object of observation - anywhere between 20%, or somewhat more, to 5%, or somewhat less.

As for the contrast loss variation over MTF frequencies for a given Strehl (i.e. aberration level), it is evident on the typical MTF. Even at relatively low aberration level, resulting in 0.80 Strehl, it can cause potentially noticeable differences in performance with specific object types. For clarity, it is presented as contrast transfer vs. that in a perfect aperture normalized to 1 for every frequency, i.e. as the MTF relative contrast (FIG. 99; plots generated by Aperture, R. Suiter).

FIGURE 99: MTF contrast variation for 0.80 Strehl. Contrast is normalized to 1 for contrast transfer in a perfect aperture at every frequency (i.e. the contrast transfer of a perfect aperture coincides with the top horizontal scale). All four wavefront deformations result in 0.80 Strehl, but the differences in their contrast transfer over local frequencies - with the Strehl representing the average contrast over all frequencies, the local contrast transfer is effectively a local Strehl - can be very significant. At the resolution limit for planetary details, for example, where less than 5% of contrast differential can produce detectable difference in performance, the "local Strehl" for the four 0.80 Strehl deformations ranges from 0.71 for defocus, to 0.82 for spherical aberration and turned edge.

Even with the all four aberrations being at the "diffraction limited" level, the differences in the contrast transfer are not negligible, and can be substantial. The worst effect has turned edge, which underperforms at both ends of the frequency range. At the low-frequency end, for details of about 10 Airy disc diameters, and larger (since the cutoff frequency is 2.5 times smaller than Airy disc diameter, frequency equaling the Airy disc diameter is 0.4, and 0.04 is ten times larger), it quickly loses nearly 10% of the contrast. While it is still a relatively small loss, generally speaking, it indicates wide spread of energy that can brighten background, and soften - even wash off entirely - faint objects in proximity of bright objects. On the high-frequency end, contrast with turned edge begins its dive to zero as the detail size goes under half the Airy disc diameter, hitting zero at some 96% of the resolution limit. Needles to say, it will noticeably affect not only performance in splitting unequal doubles, of resolving critical lunar details, but also the resolution of near-equal doubles.

Glance at this relative contrast transfer variation over the range of MTF frequencies indicates that the contrast drop tends to be smaller toward either low or high frequency end, and larger over mid frequencies. When that is the case, the aberration tolerance for such sub-range widens. Hopkins found specific aberration tolerances producing 0.8 Hopkins ratio - the contrast drop of 20%, analogous to the Strehl ratio - or better, for MTF frequencies equal to, or lower than 0.1. Shown at left are peak aberration coefficients as a function of spatial frequency ν in this frequency sub-range. The tolerances are significantly larger than in the conventional treatment of aberrations, placing the lower limit at 0.80 Strehl; consequently, their corresponding conventional Strehl values are significantly below 0.80, for which the coefficient value is S=1, C=0.63, A=0.37 and P=0.26 for primary spherical aberration, coma, astigmatism and defocus, respectively (coma with θ=0 is for the blur orientation same as that of MTF bars).

Note that at this large aberration levels the coefficient equals the actual P-V wavefront error only for defocus. For primary spherical aberration, the aberration minimum for Hopkins ratio is at a point defocused by PS=-(1.33-2.2ν+2.8ν2)S, generally more defocused than for the point of minimum wavefront deviation (PS=-S), where the P-V wavefront error equals S/4. This is the consequence of the shift of the PSF peak away from the minimum deviation focus as the P-V wavefront error exceeds 0.6 wave, i.e. for the peak aberration coefficient values of 2.5 and larger. With the Strehl at these aberration levels being up to several times higher for the PSF peak than for the Gaussian (paraxial) focus, the actual error is also significantly smaller, corresponding to roughly 2-3 times smaller P-V wavefront error.

Similarly, for large errors of astigmatism (about 1 wave P-V, which is nominally equal the coefficient, and larger), the PSF peak also shifts away from the point of minimum wavefront deviation (mid focus, i.e. defocused from either tangential or sagittal focus by DA=A/2) toward sagittal and tangential focus (double peak), with the PSF at these peaks for larger aberration being up to several times higher than at the mid focus.

For coma, the P-V wavefront error at the tilt-corrected focus, the one with the minimum RMS wavefront deviation, is 2/3 of the aberration coefficient, with the actual effect on MTF contrast ranging from the maximum for θ=0 (the blur length perpendicular to MTF bars), to the minimum for θ=π/2 (blur length parallel to the bars). The shift of the PSF peak away from the focus of minimum wavefront deviation begins with the P-V error nearing 1λ (i.e. peak aberration coefficient 1.5).

Graphs below show MTF for the aberration level resulting in 0.80 Hopkins ratio at 0.1 frequency (ν=0.1, line pair width little over four Airy disc diameters), with the center of the small square marking the 0.8 contrast drop point for this frequency. Expectedly, all plots are at or near this point at this frequency. Small deviations might be result of the specific MTF algorithms applied by OSLO. For coma and astigmatism MTF shows sagittal (x) and tangential (+), i.e. vertical and horizontal, respectively - blur orientation.

The most convenient general indicator of the magnitude of tolerance change with the frequency in the conventional P-V wavefront error context is defocus, which does not have neither axial nor tilt correction aspect. It shows the tolerance increasing inversely to the spatial frequency, from 1/2 wave P-V at ν=0.1, to 1 wave at ν=0.05, 2 waves at ν=0.025, and so on.

Hopkins ratio confirms the practical experience finding that observation of dim, low contrast details - whose resolving range (inset A, right) does not extend to frequencies significantly higher than 0.1 - has lower requirement with respect to optical quality. But it also shows that the aberration tolerance for this type of objects varies significantly with the type of aberration. Of the four aberrations here, the tolerance is the stringest for primary spherical aberration (0.34λ P-V, or 0.64 Strehl), somewhat more forgiving for defocus and significantly more forgiving for coma and astigmatism. With coma blur aligned with MTF bars (the case to which applies the relation for Hopkins ratio given above), nearly 1 wave P-V of coma is needed to cause 20% contrast loss. For the perpendicular blur orientation, not more than 0.6 wave P-V, and for the average contrast drop midway between the two orientations about 0.7 wave P-V.

The contrast is even more forgiving to astigmatism, with 1 wave P-V needed to cause a 20% drop with the astigmatic cross aligned with MTF bars, and 0.9 wave P-V with it rotated 45° (not shown). The top Strehl given for astigmatism is for the minimum RMS focus, and the bottom is the peak Strehl focus, defocused 0.14mm (at ƒ/8.2) from the former; astigmatism is the only aberration here for which the difference in the Strehl at the point of minimum RMS focus vs. peak Strehl focus is significant.

Similar results, only in the direction of tightening the tolerance, can be expected for the frequencies toward the high end (certain exception being turned edge, which causes unacceptable contrast drop in this sub-range at the 0.80 conventional Strehl already). On the other hand, the conventional Strehl that would secure no more than 20% contrast drop in the mid range, where are the resolution threshold for bright low-contrast details, probably wouldn't be significantly below 0.90. For ensuring no more than 5% contrast drop at mid frequencies, the conventional Strehl would need to be above 0.97 (equivalent of 1/11 wave P-V of spherical aberration, or better). That, however, would strictly apply only to very small apertures with near-perfect correction and negligible induced errors (seeing, thermals, miscollimation...). At the relatively large error levels, the finest details are washed out, and those more coarse that remain are generally less affected by any given contrast drop.