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6.4. Diffraction pattern and aberrations   ▐    6.5. Strehl ratio

6.4.1. Star testing telescope quality

As FIG. 96 shows, even slight presence of any wavefront aberration changes the form of diffraction pattern, leaving its unique fingerprint on it. With the knowledge of what those fingerprints look like, with some practical experience, the aberrations can be detected and, most often, quantified with sufficient accuracy. Thus, all that is needed to test telescope optics for quality is a single point-source of light. It can be either a star, or a small, man-made source of light that fits the definition of a point-source. The telescope should be well collimated, and in thermal equilibrium with surrounding air (depending on the telescope size, type and initial differential, it can require anything from minutes to hours).

In order to visually analyze diffraction pattern, it has to be sufficiently magnified. Minimum magnification for conducting star test is considered to be 25x per inch of aperture (or equal to the aperture diameter in mm), but more is better. Optimum magnification level is probably between 40x and 50x per inch, combined with moderate defocus, of between 3 to 5 waves (number of rings in the defocused pattern roughly corresponds to the defocus in waves).

When testing on a star, it shouldn't be too bright, nor too faint; either can hide more intricate, yet potentially important features of the diffraction pattern. Optimum level of brightness for a 6 inch aperture is ~2nd magnitude star, which helps define best star magnitude for the test, in terms of aperture diameter D, as m~5logD-8.9 for the aperture D in mm, or m~5logD-1.9, for D in inches. Test star should be close to zenith, to minimize the effect of seeing, although for telescopes without tracking mechanism Polaris may be the best choice most of the time. Needless to say, less than good seeing quality will compromise reliability of star test results (steady air is also required for artificial star testing; no heat radiating object or areas should be close to the light path).

When star testing with an artificial star, two more requirements have to be met:

(1) angular size of the light source needs to be less than 1/2 of the Airy disc size, and

(2) distance between the source and the telescope needs to be large enough to keep spherical aberration induced by the finite object distance negligible.

The first requirement demands the artificial star to be what it needs to be for the test: effectively a point source of light. Diffraction theory shows (Optical Imaging and Aberrations 2, V. Mahajan, p214), that if the source diameter exceeds 1/4 of the Airy disc's, resulting diffraction pattern begins to change: its full width at half maximum (FWHM), first minima radius and overall ring structure all become larger. At the source diameter of 1/2 the Airy disc's, the FWHM is nearly 15% larger, with the first minima nearly doubling in radius (in effect, the first dark ring vanishes, with the central disc and first bright ring merging). At the source size equaling the Airy disc's, the FWHM is already 60% larger, and the in-focus ring structure diminishes further. There is no specifics on how it affects defocused patterns, but it is a common sense to try to preserve the form of diffraction pattern produced by artificial star closely resembling the one that we are comparing it to. With today's availability of materials and products, this shouldn't be an obstacle.

Obviously, angular size of an artificial star is a function of its diameter and its distance from a telescope. Simple visualization of the Airy disc in the focus of a telescope, with two lines extending from its top and bottom to cross at the aperture stop and diverge outwards, helps define the size of Airy disc projection for any given distance from the telescope. Knowing that the Airy disc diameter in radians is given by 2.44λ/D, which for λ=0.00055mm comes to 1/745D for the aperture D in mm, linear diameter corresponding to angular Airy disc diameter at any distance L is given by =L/745D. Since we want to be as close to an actual point-source pattern as possible, the target size for the artificial star is 1/4 as much, or

~ L/3000D

(note that L needs to be in the same measuring unit as D, here in mm). For instance, for D=100mm telescope, an artificial star placed at 50m distance shouldn't be significantly larger than 0.17mm in diameter. For distance L in feet, aperture D in inches, the maximum source diameter in mm is ~L/250D.

It is a bit more complicated when it comes to determining artificial star distance that will not induce spherical aberration appreciably affecting test results. The complexity arises mainly from the great variety of different telescope systems in use today. While they are all optimized for object at infinity - or at least at a large distance - the amount of sensitivity to reduced object distance can vary significantly from one type to another. Some systems are easier to deal with, in a sense that they come with the basic specifications known and/or relatively constant. Some others have undetermined optical specs, thus cannot be predicted with respect to their specific object distance sensitivity.

The easiest are single mirror systems: Newtonians with either paraboloidal or spherical mirror (diagonal, by definition, has no optical power, and its surface errors do not induce spherical aberration). For object distance in units of mirror's focal length L/, and its reciprocal value ψ=/L, the P-V wavefront error of spherical aberration at the best focus induced by object distance is given by a simple relation,


where K is the mirror conic, D the aperture diameter in mm, and F the focal ratio. The sign of wavefront error indicates the form of aberration; when the aberrated portion of the wavefront is closer to best focus than the reference sphere, it is over-correction, and vice versa. Therefore, the aberration induced is over-correction for K<-(1-2ψ)2. Taking relatively close object, at twenty focal lengths away (ψ=1/20), conics smaller than -0.81 will be inducing over-correction error, those greater than -0.81 under-correction, and the error will be cancelled for K=-0.81.

    With spherical mirror, there is no need to worry about the close object error; being under-corrected for object at infinity, and with closer objects inducing overcorrection, it actually becomes better corrected with reduced object distance. The closer object, the more so - up to the object placed at mirror's very center of curvature, where sphere provides perfect imaging. For testing purposes, however, it is good to know what level of error it should have - if it is really spherical. By substituting 0 for K in the above relation, the wavefront error is given as Ws=D(1-2ψ)2/2048F3. Deviation ratio δ of the actual vs. implied value indicates non-spherical surface, with its conic given as K=(δ-1)(1-2ψ)2. In other words, it is prolate ellipsoid for 0<δ<1 and oblate ellipsoid for δ>1 (negative δ value implies hyperboloid).

    Taking, for instance, δ=0.6, implies K=-0.324 for object at ten focal lengths away (ψ=1/10), and K=-0.361   for object at twenty focal lengths. The positive sign of δ implies that the actual error is under-correction, of the same sign as that of spherical mirror (which induces under-correction with any object farther away than its center of curvature). The same amount of over-correction error - thus resulting in a negative ratio value - would imply the corresponding conics -1.296 and -1.444, respectively (hyperboloids). The actual-to-ideal δ value of 1.4, on the other hand, implies K=0.324 and K=0.361, respectively (oblate ellipsoids).

    For a paraboloid, which is axially perfect for object at infinity, there is a price to pay with closer objects and the currency is over-correction. Substituting K=-1 in Eq. (a) gives Ws=(ψ-1)ψD/512F3, and neglecting the ψ term in the bracket (for any serious test, object needs to be significantly farther away than 10 focal lengths, which makes ψ significantly smaller than 0.1), it reduces to Ws~ψD/512F3. With ψ=/L=FD/L, this gives Ws~D2/512LF2, and the object (i.e. star) distance in terms of the wavefront error as:

(all three parameters, D, W and L are in the same units). Since this value for the light source distance L is effectively based on the P-V wavefront error Ws larger by a factor 1/(1-ψ) than the actual error, it is larger by a factor 1/(1-ψ), or L/(L-), than a distance corresponding to the actual Ws value.

For the wavelength in units of 0.00055mm (and D in mm), test distance is L~3.55D2/WsF2.

For L in feet and D in inches, it is L~7.5D2/WsF2.

Substituting optional maximum tolerable P-V error of spherical aberration for Ws gives the appropriate minimum distance for the artificial star. For instance, by substituting W=λ/20=0.0000275mm for λ=550nm, gives the distance for this error level for D=400mm, /4, as L=710m. As the original relation implies, the actual distance for that error level is smaller by a factor of (1-ψ) than that given by the approximation. That comes to 11% difference at a ten focal length distance, and 5% at twenty focal  lengths. Nearly exact expression for the appropriate distance, for all practical purposes, is given by L=(1-ψ')D2/512WsF2, where ψ' is obtained from ψ'=/L, using the distance L given by the approximation.

Solving Eq. (a) for ψ gives the exact value as ψ=0.5-0.5(2048WF3/D)0.5, which is used to plot required distance for 1/20 wave P-V of spherical aberration as a function of mirror aperture diameter D and focal ratio F, shown below.

Required distance ranges from 5.5m with a 100mm /10 and ψ~0.18 (1/0.18=5.5m), to 2km (1.24 mile) with 500mm /3 paraboloid and ψ=0.00075 (1.5/0.00075=2000m). The distance scales inversely to the wavefront error, so half as large error (1/40 wave P-V) would require doubling the distance.

Obviously, required artificial star distance becomes impractical for larger, fast paraboloids. Is 1/20 wave error tolerance really necessary? It is; moreover, it may be still to much. For instance, if a mirror is 1/6 wave P-V inherently over- or under-corrected, a 0.05 wave of over-correction induced by the object distance would alter test result into -1/4.6 wave or 1/8.6 wave P-V, respectively. For accurate star testing, the error induced by object distance shouldn't significantly exceed 1/50 wave P-V. This pretty much rules out an artificial star test for larger Newtonians.

Catadioptric Newtonians with full-aperture Maksutov or Schmidt corrector are significantly less sensitive to reduced object distance than paraboloidal Newtonians, due to their spherical mirrors. Catadioptric two-mirror systems are not uniform enough in their production types to fall under some type of generalization.

For all-reflecting two-mirror systems, close object error of spherical aberration is given by Eq. 92 (also plots for the three most common systems), and for typical commercial SCT by Eq. 120.3.

Ordinary doublet achromat is very tolerant to the reduction in object distance in focal ratios ~/10 and slower. It is in part due to its relatively small apertures, but even a 200mm achromat will likely generate less than 1/20 wave P-V of under-correction with the object (artificial star) as close as 10 focal lengths away (given relative aperture, the error level is nearly in proportion to the aperture size). On the fast end, however, the sensitivity can be several times, or more, greater. A 4 inch /6 achromat can generate in excess of 1/5 P-V of under-correction with the object at 10 focal lengths away, due in part to the generated lower-order aberration falling out of balance with the higher-order component (the error is nearly inversely proportional to object distance). This aberration duality makes the sensitivity of these instruments to object distance fairly unpredictable, because the level of higher-order aberration and proportion of balanced lower- and higher order spherical aberration vary from one system to another. Similar applies to apochromatic refractors.

Proficient star testing requires the ability to properly interpret diffraction pattern seen in the eyepiece. Familiarity with the characteristic forms associated with particular aberrations is easiest to acquire by way of software-generated visual simulations. While the actual testing commonly involves more complex pattern forms than those caused by a single aberration, knowledge of the single-aberration patterns is always the starting point. Following simulations are generated by Aberrator (Cor Berrevoets), except the zonal error, which is generated by Aperture (Harold Suiter). Shown are all common conic surface aberrations, at the level of 0.80 and 0.95 Strehl, for unobstructed and 0.3D (30% linear) obstructed aperture. Light travels from left to right, thus defocus in the intrafocal direction is numerically negative, and positive for extrafocal direction (patterns are given for defocus values -4, -2, 0, 2 and for waves, since this range of defocus is generally the most sensitive for detecting aberrations). Axial cross section is not commonly presented, but it helps better understand how particular patterns form.

The basic reference pattern is always the one of aberration-free aperture, unobstructed or obstructed. The presence of obstruction changes both, aberration-free and aberrated pattern, but in the absence of aberrations patterns on the two sides of defocus are identical for given defocus value.

The level of primary spherical aberration at best focus resulting in 0.80 Strehl is 0.25λ P-V WFE (0.0745λ wave RMS). Shown is overcorrection, which is readily apparent from the side at which the defocused pattern is larger and with less well defined ring structure (the consequence of the outer portion of the converging cone widening before the focus, since focusing behind it; opposite for undercorrection). The tail tale of primary spherical is that defocused patterns on one side are brighter in the central portion, while brighter in the outer portion on the other. Generally, it is easier to detect in the defocused patterns, with the focused pattern differing from the aberration-free pattern mainly for its brighter first ring. It is easily detectable at λ/4. and fairly easy even at λ/8
P-V wavefront error.

Similarly to the primary aberration, shown is overcorrection, with the outer rays focusing behind paraxial focus. For undercorrection, the patterns are reversed. Mainly due to its somewhat different wavefront deviation form, namely additional "wrinkle" at the edge, throwing light out at a greater angle, the intra- and extrafocal patterns differ more than with the primary spherical. In unobstructed aperture, the tail tale sign can be a pair of equally bright rings in the focused pattern, but the aberration is even easier to detect than the primary form, either at the 0.95 Strehl level (0.20λ P-V), or the "diffraction-limited" 0.80 Strehl (0.40λ P-V, 0.075λ RMS).

The 0.37λ P-V wavefront error of primary astigmatism (0.0745λ RMS), resulting in 0.80 Strehl, is easily detectable in both, focused and defocused patterns. The former by the first bright ring morphed into a cross-like pattern around the central maxima, and the latter by elliptical elongation of defocused patterns, with the orientation perpendicular one to another for the two opposite sides of defocus. At twice lower nominal wavefront error it is, however, harder to detect than spherical aberration; it could give only a hint of the cross-like change in the ring form with bright stars.

Unlike the other three, primary coma has identical patterns on either side of defocus. It is the consequence of its wavefront deviation shape, which has one half of it flatter, and the other more curved to identical degree. Hence the former focuses as much longer, as the other focuses shorter, with the identical angles of conversion, only of opposite sign (in this particular pattern, the top half of the wavefront focuses farther, and the bottom half closer). Luckily, it is the easiest to detect in its focused pattern. Even at the 0.95 Strehl level (0.21λ P-V), one side of the first bright ring is noticeably brighter.

These patterns illustrate the effect of zonal error. The zone generates 0.075λ RMS wavefront error (WFE), thus produces 0.80 Strehl. To some extent, it resembles spherical aberration (it is a raised zone, focusing behind paraxial focus, thus resembling overcorrection), but has a distinctive double ring in addition to the first bright ring in the focused pattern.

As mentioned, actual instrument will most often have a mixture of different aberrations. Characteristic patterns for some other aberration forms are given in FIG. 96.

Harold Suiter's "Star Testing Astronomical Telescopes" discusses in details star testing, connecting it to the underlying optical theory. An interesting new development (relatively speaking) is Roddier's test, (freeware) which uses CCD image of two defocused diffraction patterns of a real star to determine wavefront quality (seeing effect is averaged out through the exposure length, and also relatively insignificant, due to the large size of defocused patterns).

Follows more detailed description of the Strehl ratio and MTF.

6.4. Diffraction pattern and aberrations   ▐    6.5. Strehl ratio

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