14.3.RAYTRACING OBSERVATORY TELESCOPES

Arguably one of the best corrected simple systems is the three-mirror system known as Paul, or Paul-Baker telescope. So when a group of astronomers from the University of Arizona's Steward Observatory were looking for a very large, ultra-fast widefield design for the Dark Matter Telescope project, it led them to the Paul-Baker, since no two-mirror system, or a Schmidt-like telescope, could achieve the needed level of correction. The telescope would be limited only by atmospheric seeing and sky background photon noise, which means it has to produce star images smaller than 0.5 arc seconds, achievable occasionally with large telescopes on best sites. The goal was the largest telescope possible with 10m focal length, needed for sufficient sampling with a 15-micron pixel (1 arc second=51 micron).

Their first design, a 6.5m f/1 Paul-Baker telescope with all three mirrors aspherized in Zemax, was capable of producing required image, but had suboptimal plate scale, and unacceptable chromatism from the dewar window. That led to the revised design, a 8.4m f/1.25 telescope with a curved dewar window and added correcting lens. This design was the starting point for the 8.4m f/1.234 Rubin Observatory Simonyi Survey Telescope (El Penon, Cerro Pachon, Chile), with even wider, 3.5-degree field, and a bit better correction (less than 0.3 vs. less than 1/3 arc seconds design image). Since there is no prescription for the Rubin telescope, the University of Arizona's design will be used to illustrate this, we could say unique kind of a telescope. Unfortunately, prescription given in the paper isn't working, but a system with comparable performance can be reconstructed from it.

The main parameter of optical performance is ensquared energy. It is given on the bottom of every 1-arcsec square containing ray spot plots for the 0.80 energy square all values are in meters, so the 7.77^-6 for the axial e-line 0.80 energy square is 7.77 microns). At 1.5 degrees off axis the polychromatic 0.80 energy square side is 24.9 microns, or nearly 0.5 arc seconds. It would probably be the subject of final optimization, generally by near-equalizing error over the field, by allowing more of an error in the inner field. At 1-degree off, the 0.80 energy square is 1/3 of arc second, becoming significantly better toward axis. Astigmatism is minimized, near zero at 1.5 degrees off, with the dominant aberration there being trefoil.

The small dot in the center of each box is the Airy disc; entirely irrelevant here, but illustrates the magnitude of aberrations in this highly corrected large telescope.

More complete picture of the aberrations is given by the Zernike terms. The main contributors are in red (piston is not an aberration in the single-aperture system, just a nominal artifact, and the defocus value - which translates to 0.33 wave RMS, i.e. 1.16 wave P-V - is evident on the astigmatism plot (best focus is not necessarily where astigmatism is at its minimum; in the presence of other aberrations it can shift to another location).

The largest single contributor is trefoil (#9). The next highest one, quadrafoil (#16) is already near negligible, adding only 15% to the trefoil (because Zernike terms add up as square root of their squared values), and even less considering secondary trefoil (#18). Most of the terms are negligible individually, but do have significance as a total, creating a higher-order aberration "noise" in this large and fact telescope.

At 1 degree off axis dominant aberration is astigmatism, but it is significantly smaller than what the longitudinal aberration graph indicates (nearly 6 waves P-V, from W=L/8F², L being the longitudinal aberration, and F the system focal ratio). The reason is that a mix of lower- and higher-order astigmatism of opposite signs has up to four times lower error at the best focus for given longitudinal aberration, and that large central obstruction also lowers the error.

2 - SDSS TELESCOPE: 2.5m f/5 MODIFIED RITCHEY-CHRETIEN

Located at Apache Point Observatory in New Mexico, the main tool of the Sloan Digital Sky Survay project, this telescope uses a two aspheric lens field correctors to achieve a well-corrected 3-degree field. As described in paper by Gunn et al. its second corrector lens is interchangeable, allowing it to be optimized for camera and spectroscopic mode. It is used to map the depths of our Universe.

The two corrector lenses have their radii given indirectly, through the value of a2 coefficient, whose full expression is a₂=d²/R[1+(1-K)(d/R)²], with d being the the surface radial height, R the radius of curvature and K the surface conic. It is not clear why, because the surfaces are spherical (K=0), and (d/R)² is negligibly small, especially for the first corrector, hence for all practical purposes a₂=d²/2R, and R could be expressed directly (for the second corrector, it would give about 3% longer radius, but it would have little consequence).

Raytracing the prescription for camera mode shows, perhaps, less than expected correction-wise. The field is, as described in paper, nearly flat up to about 80% of the 1.5-degree radius, curving inward rather strongly after that. Color correction is good overall, but widely separated lines show significantly different forms of aberrations toward field edge, i.e. their best foci do not coincide. One of the requirements was near-zero distortion; according to OSLO, it just exceeds 0.1% at the field edge.

The outer portion of the field required adjustment of the detector shape. It cannot be fitted with a near-spherical curve, and requires higher order aspherics. Here, the asherics are combined with 67m curvature, which resulted in a good, but still someone uneven fit (the edge is for the best green focus; the red becomes roughly round at 99m curvature, but green out of best focus).

The paper gives no specific performance criteria for the camera mode. Final images were calculated as convolved with 0.8 arc seconds seeing FWHM, which proved to be too optimistic. For the system focal length, one arsecond is 60 microns. On the fitted field, the green line 80% energy square is 18.7 microns on axis, 15.5 microns at 1-degree, and 35.8 microns at 1.5. It indicates that the seeing is limiting factor, rather than optics.

Full spectral range of the telescope is 0.3-1 micron, but the familiar visual 0.43-0.70 range well illustrates its chromatic correction.

3 - EELT: THE 39m, LARGEST-EVER-TO-BE OPTICAL TELESCOPE

European Extremely Large Telescope is designed as a three-mirror anastigmtic aplanat. As described in a paper by Cayrel, the 39-meter f/0.93 ellipsoidal primary consists of 798 hexagonal segments, each 1.45m across. The 4.2m convex secondary and 4m concave tertiary are each a thin meniscus. All three mirrors are active: the primary to make possible for it to conform to the proper shape, and the other two - particularly tertiary - to correct wavefront errors, mainly those caused by atmospheric turbulence, but also those caused by mechanical deformations.

Two folding flat mirrors direct light beam to the side (Nasmyt focus). The first, 2.5m in diameter, is also active, "specified to deliver near infrared diffraction limited images with over 70% Strehl ratio in median atmospheric conditions (0.85 arcsecond seeing, t0 of 2.5ms)". The second flat, 2.2 by 2.7m, corrects for tip-tilt errors. 

Field radius is nearly 5 arc minutes, limited by the central hole in the fourth mirror (even such a small angular radius produces almost 1m image radius). The prescription gives a final f/16.6 system, somewhat faster than f/17.5 cited in the paper; not sure what is causing the difference, but the design correction level is very high.

Even at the field edge, correction is still at the level of 1/12 wave P-V of lower order spherical aberration. With the ~660m focal length, one arc second at the detector is whooping 320 microns - about 1/3 mm. The only significant aberration is field curvature (ray spot plots shown are for the best image surface, of 11.2m radius; on flat field, error at the 5 arc minutes field angle - i.e. 0.94m off center - is as much as 30 waves P-V of defocus).

4 - THE 40-inch YERKES AND 36-inch LICK REFRACTOR

Keeping in the company of the largest - ever wondered how much chromatism there is in the largest refractor ever built: the 40-inch Clark refractor at the Yerkes observatory in Wisconsin? It is a common crown/flint achromat; according to the most relaxed criterion (Sidgwick), its focal ratio for acceptable (roughly 1/4 wave level) color correction should be f/120 - and it is f/19. I was always curious, just how much of color this quarter tone of achromat glass generates. Using general data on the Clark doublet, and Barnard's measurements, gives following picture.

On axis, Airy disc is a barely visible dot in the center of defocused C (656nm), F(486nm), r (706nm) and g (436nm) lines. The F/C error is nearly 7 waves P-V, which puts it at the level of a 100mm f/1.8 achromat. At the field edge, coma is nearly three times the Airy disc - or 0.8 waves P-V - visually unnoticeable (about f/16 paraboloid level). As a side note, Yerkes refractor actually is not the largest ever built. It was the 48-inch (122cm) refractor built for the Great Exhibition in Paris in 1900. It had 57m focal length (f/46.7), so it used siderostat mirror to reflect light into immovable objective lens, with the focuser in a form of a carriage on rails. With thermal issues and bad location on top of that, it performed poorly and did not attract buyers. The Yerkes refractor - the idea of George Ellery Hale, paid for by Charles T. Yerkes and becoming a telescope in October 1897. - is the largest ever used as a full-capacity astronomical telescope. Now, as it is not in use anymore, the largest used refractor is its 36-inch f/19.3 cousin at the Lick observatory. So let's take a closer look at this second largest refractor built by Alvan Clark & Sons.

The James Lick Telescope was built nearly a decade before the 40-inch, in 1888. According to a brief description by its maker, it consisted of a biconvex crown 1734 front lens (n=1.52), and a negative meniscus of flint 1588 (n=1.64) as the rear element (Note on the loss of light in the 36-inch Lick objective, J.H. Moore, 1904.). Lens radii are specified, but the separation is not, so it will be assumed as nearly proportional to that in the Yerkes refractor (comes to ~195mm; some sources state as little as 25mm, but it is highly unlikely, since the main purpose of the gap was to make possible lens maitenance w/o having to take front lens out). The lenses were exceedingly thin - 1.98 and 0.93 inch center - for a few good reasons (light transmission, weight, glass homogeneity) - but it certainly didn't help figuring accuracy. Raytrace below uses less thickness comparable to those used in the Yerkes refractor raytracing, but it is of little significance for the raytracing output. First is presented such design with the standard achromat glasses of this era, with balanced F/C aberration, corrected for coma (top), and then the actual objective - i.e. nearly as close to it as reasonably can come - below (O-ZSL4 is an obsolete Ohara krown; very similar output is produced by the Schott K3 crown).

The first, contemporary correction mode results in a defocus ranging from some 22mm at the 0.7 micron wavelength to zero in proximity of e-line and to 86mm at 0.4 micron wavelength. In the blue (F-line) and red (C-line), defocus is evened up at about 6 waves P-V. Monochromatic aberrations are negligible across the field 1/3 degrees in diameter, over best image surface of 6m radius (it is still well within "diffraction limited" at 0.165° off flat field).

However, according to measurements, the actual C focus of the Lick refractor is midway between its "minimum" (i.e. optimized, 0.565 micron wavelength) and blue (F-line) focus (On the chromatic aberration of the 36-inch refractor of the Lick observatory, J.E. Keeler). If so, the error on the blue spectral end becomes significantly larger, while smaller on the red end. An MTF graph (photopic, for the e-line focus) showing the respective contrast levels is surprising at first: it shows significantly better contrast for the actual correction mode, favoring the red end (nominal cutoff, probably due to the short wavelengths lingering just above zero, is significantly higher than the practical cutoff, which is somewhere around 100cpm). However, it is not too hard to see that it results from the significantly lower photopic sensitivity to the blue/violet end. Also, with achromats generating significant secondary spectrum, best diffraction focus is always shifted from the optimized focus. In this case, the shift is much more significant with the first, balanced correction mode, but even at the best photopic focus its polychromatic Strehl is still somewhat lower than for the Clark's lens: 0.455 vs. 0.490 (focus shift in mm is under "Z"; zero focus shift represents the e-line focus).

For the complete picture, however, both lenses also have to be measured against mesopic sensitivity, since during night time observing eye sensitivity shifts toward that mode. While there is no accurate data on the actual mesopic mode sensitivity (the official version merely takes the average between photopic and scotopic), it can be approximated from experimental studies. The sensitivity is, in general, higher on both ends of visual spectrum (and lower for the photopic peak), but more so on the blue/violet end. The result is that the balanced mode now have higher mesopic poly-Strehl than the Clark's lens: 0.374 vs. 0.317. It indicates that the actual Lick refractor lens performs better with eye in photopic mode, but becomes inferior in the mesopic mode. This level of chromatism is roughly comparable to that in a 100mm f/3 achromat.

With respect to monochromatic aberrations, the original lens had sigificant spherical aberration residual (reportedly, about two waves P-V), and had its front surface aspherized at a later time in order to have it corrected. This arrangement (Clark's radii prescription with the glasses closely matched) has, if all-spherical, 1.5 waves P-V of overcorrection. It could be corrected by changing R3 to -6295mm, but it would throw color correction out of balance (blue F-line focusing about 2mm before e-line, and the red C-line 19mm beyond e-line focus), so it would require regrinding/repolishing of at least one more surface. Aspherization of the front surface is much simpler, in this case requiring 0.32 conic (oblate ellipsoid).

5 - 1m KLEVTSOV-CASSEGRAIN VARIETY (Vihorlatska Observatory, Slovakia)

In his book "New serial telescopes and accessories" (2014), Klevtsov gives prescription obtained from the Slovakian observatory, as a big-scale example of a telescope of the Klevtsov type.

The telescope was made in Odessa, SSSR. Original prescription is with BK7 glass (i.e. K8, its LZOS analog), but K7 glass reduces longitudinal error in the blue/violet to less than half, so that this 1-m telescope easily passes the "true apo" requirement. As with every system of this type, dominant aberration is off axis astigmatism.

6 - 0.7m 5° HOUGHTON-TEREBIZH SURVEY TELESCOPE

One of many designs of Valery Terebizh, the Russian "master designer" as referred to by M.R. Ackerman, is a simple Richter-Slevogt (known as Houghton on the Western side) modification with widely separated corrector lenses, making possible significantly wider fields. Survey telescopes are usually of relatively small apertures, with large corrected field being the primary concern, but larger apertures can be needed for detecting fainter targets.

The Houghton-Terebizh offers 5-degree field at f/3.2. The system shown is slightly tweaked to have the axial error minimized (at no expense to off axis performance).

 Color correction even at this aperture size and relative aperture passes the "true apo" criterion. The 80% energy square is 6.6 microns on axis, and 8 microns at 2.5 degrees off (diffraction images are for the 5 wavelengths with even sensitivity).

This design can also be used as a typical survey telescope, smaller in size and faster. If rescaled to 350mm aperture and f/2.4 focal ratio - parameters of a telescope of this design used in the gamma-ray burst system observatory near Moscow, Russia (The Mobile Astronomical System of Telescope-Robot) - it still preserves a very satisfactory performance.

Note that the actual telescope is somewhat shorter, with the corrector lenses more widely separated and with the final image that would form behind the rear corrector lens, if not directed to the side with a diagonal flat).

7 - 3.8m f/2.5 UK INFRARED TELESCOPE

While designed for work outside the visual spectrum, the UKIRT is based on unusual relay lens design which, with minor adjustments, could be used in the visual range as well. Here is presented its Wide Field Camera mode, as described in this online paper.

System is simplified by omitting optical window and filters, which has a minor influence on its output. The entire field covers 0.93-degree diameter, but the actually used are only four rectangular portions of it, as illustrated in the paper. The telescope is optimized to operate in four IR ranges: Y (0.97-1.07 microns), J (1.17-1.33), H(1.49-1.78) and K(2.03-2.37). As given here, the is somewhat biased toward the lower three, possibly the consequence of omitting mentioned elements.

Sub-aperture aspheric plate helps correct not only spherical aberration, but also coma and astigmatism.

The ray spot plots are given for the central line of each of the four wavebands, when refocused to their respective best focus. Above right are plots for this same system in the visual wavelengths (focused on e-line). Only minor optimization is needed to have it perform satisfactory as a visual telescope.

8 - 8.4m f/1.24 SIMONYI SURVEY TELESCOPE

The Large Synoptic Survey Telescope (LSST), a compact version of the 8.4m telescope on the top of this page, is one of a kind, in that it sports etendue - as a product of its clear aperture and field area - of 318m²deg², over 50 times more than the first next contender. Located on the Cerro Pachon, Chile (60 miles from La Serena), this modified Paul-Baker, or Laux telescope uses three mirrors and a 3-lens corrector to produce a 3.5-degree field diameter with less than 0.2 arc seconds (0.01mm) FWHM star images over its 0.63m diameter detector, with over 3 billion 10-micron pixels. In 5 spectral bands from 400-1030nm it will be used to create by far the most complete picture of the solar system, Milky way and transient optical sky, as well as for exploring dark energy and dark matter.

Based on the published prescription (LSST Camera Optics, Olivier et al, 2006), I raytraced design with SYNOPSYS (free edition). Despite the prescription being unclear in some istances - namely, not specifying front radius of the second lens, what is the filter substrate, and to which surface of the second and third lens were applied given conics and higher order aspherics, the assumed choices - flat 2nd lens front surface, aspherics on the front radius of the 2nd and 3rd lens, filter made of BK7 glass - produced performance level sufficiently close to the description (the only change was in the value of the 6th order aspheric coefficient on the second lens front radius, to 1e-19). It is possible that some other choices would work better.

The field is limited to 1.2-degree radius, when definition begins to deteriorate; but even at 1.75° the condensed core of the roughly four times longer blur (the width is about 0.025mm) is about 0.02mm long by 0.007mm wide, or 0.4 by 0.14 arc seconds. No effort has been made to optimize either axial or edge performance, but there is certainly room for it. Should be noted that this particular setup is optimized for R-bend (red); the blue line is given only to illustrate the overall correction in this particular mode; in its optimized setup, it is further corrected by filter shape and small changes in lens spacing, and should be at the similar level as the red. Despite the prescription being optimized for the red, this take on it still has better correction in the d-line (despite the red blur appearing somewhat smaller, it is more compact, and significantly larger than the dense core of the d-line ray spot, as better show spots above with twice as many rays; note that the ray spot size for highly obstructed aperture is significantly larger than for unobstructed one, for any given level of spherical aberration).

This version of SYNOPSYS doesn't give encircled energy, which would be the best measure for verifying the actual energy spread at any field point, but the size of ray spot plots does confirm, qualitatively, that the field is corrected to FWHM better than 0.2 arc seconds (0.01mm). Another useful indicator of performance level, MTF, shows that better part of the contrast loss comes from the aberrations in the low frequency range, and from the 0.60D central obstruction in the mid-frequency range. In the high frequency range, the telescope performs

slightly better than perfect aperture (MTF plot for 1° off axis roughly coincides with the axial plot). The actual MTF, however, is obtained by multiplying the system MTF with the pixel MTF. Since the 10-micron pixel here is somewhat larger than the FWHM in absence of seeing error, the actual MTF would be lower approximately by a sin(νπ)/νπ factor, with "ν" being the MTF frequency. Since this large aperture even on the best sites will have a substantial seeing-induced error, the pixel MTF degradation factor will likely be superseded by seeing (i.e. the seeing FWHM will be significantly larger than the pixel, reducing pixel MTF degradation to negligible).

9 - HUBBLE SPACE TELESCOPE

As NASA brings our space eye back to life, it warrants taking another look at it. This 2.4m f/24 Ritchey-Chretien system could, with only two perfect conic surfaces, achieve perfect axial corection. The real instrument, after accounting for non-figure errors (i.e. smaller than 1/10 of the mirror diameter), as well as <0.01 arc second pointing error, expected to deliver 1/20 wave axial RMS, or better. Off axis, limiting aberrations are 0.63m field curvature and astigmatism, with the latter limiting "diffraction-limited" (0.80 Strehl) field radius to 4.8 arc minutes (81mm).

Due to testing errors, HST was sent to space with incorrectly figured primary mirror: it was a hyperboloid 2200 nanometers shallower than what it was supposed to be (in terms of mirror conic, -1.0137 instead of -1.0023). The four-wave surface error translates into twice larger error at the paraxial focus, but at the best focus location it diminishes fourfold, to "only" two waves P-V wavefront error of spherical aberration (picture below; note different scales for the flawed and design ray plot spots). While not making the telescope useless - even in 0.5 arc second seeing the seeing-induced long exposure error would have been larger - but it was taking away most of the atmosphere-free environment advantage. Lackily, the error was correctable: all it took was a pair of small, coin-size mirrors placed close to the focal plane of each of the instruments (the first is a plain tilted sphere, reflecting light to the actual corrective mirror with appropriate aspheric, as well as apropriate shape to correct for tilt-induced astigmatism, reflecting light back toward instrument).

While the secondary size needed to fully illuminate the field is only 1/8 of the aperture diameter (D), the effective central obstruction is around 0.31D, needed for proper baffling. MTF graphs show the effect of field astigmatism (left) and figuring error (right) for the base wavelength.

10 - 1.24m f/2.5 U.K. Schmidt Telescope (UKST)

This headline is a bit misleading, because this instrument is not in the U.K. - it's at the Siding Spring Observatory, New South Wales, Australia - nor it is a telescope. While meter-classs professional telescopes are never used for visual observations, they do have accessible image and could, technically, use eyepieces. The UKST, as all Schmidt "telescopes", can't - it is a camera. It is a younger cousin to the Oschin Schmidt "telescope" at the Palomar Observatory, U.S. They are nearly identical in all respects, except that the Oschin Schmidt started out (1948) with a single-glass corrector, replaced with achromatized one in the mid-80s, and its original photographic plate detector was replaced with CCD. On the other hand, UKST had achromatized Schmidt corrector from the get go in 1973. and kept ist photographic plate detectors; from the 2001th on, it was used mainly for multi-object spectroscopy and radial velocity measurements.

The UKST covers field of 6.5x6.5 degrees, which requires 1.9m mirror for zero vignetting (the actual mirror is 1.83m in diameter). Below is what its performance looks like in raytracing, for the single-glass corrector (top) and achromatized one. I didn't find what glass combination was used for the latter, and relied on the achromatic Schmidt coverage in Schroeder's "Astronomical Optics". Given is the one that worked the best, even if KF1 (Schott) is obsolete now, because the corrector is not fully optimized, and the other possible combinations - for instance BALF4/BK7 or NSL36/BSL7 - are close behind. Also, based on the size of the field, central obstruction is approximated at 0.6m in diameter; the actual one shouldn't be significantly different.

At 3° off axis, Zernike analysis for the single-glass system (top) shows only three significant aberrations: primary astigmatism (4), primary spherical aberration (8) and secondary astigmatism (11). However, the first two are not the actual aberrations. This configuration is free of primary aberrations, except field curvature. The "primary astigmatism" is actually a secondary aberration that has identical form, but increases with the 4th power of the field height (lateral astigmatism). Likewise, the "primary spherical" is a secondary aberration, identical in form, that increases with the square of field height (lateral spherical). Obviously, if we have less than 0.01 wave RMS of primary spherical on axis, we can't have 0.081 wave RMS of it at 3° off (as determined by the Zernike term value, 0.181, divided for primary spherical by 5^0.5).

Achromitezed corrector (bottom) significantly improves performance level. Polychromatic Strehl jumps from 0.39 to 0.92 (400-1000nm, even sensitivity), and the square with 80% energy at 3° off axis drops from 0.02mm to 0.011mm (0.74 arcsec; 0.5 and 1.3 arcsec on axis, for the achromatized and single-glass, respectively). This directly determines both, limiting resolution and contrast transfer efficiency. However, achromatized corrector requires significantly deeper curves: 0.281mm and 0.365mm vs. 0.076mm for the single-glass curve (all three have point of inversion, i.e. maximum deviation at ~0.71 zone).

Of course, these figures are valid only for zero atmospheric error, tube currents and misalignment. The first factor is the most significant: even in 1 arcsec seeing, this aperture is subjected to a D/r₀~9 turbulence, with its diffraction pattern broken into a speckle structure, bloated into a blur several times the Airy disc size. In order to have its atmosphere-free PSF maxima still intact, the camera would need 0.2 arcsec, or better seeing. Image below shows the system PSF for achromatized corrector; due to the 0.48D central obstruction, the central maxima diameter is about 20% reduced, and the FWHM for the e-line little more than 7%, at less than 0.0013mm, or below 0.09 arc second (PSF simulations are in different proportions to the graphs, and among themselves for center vs. off axis).

In 1 arcsec seeing, the FWHM would be enlarged to roughly 0.5 arcsec - little better than FWHM of the 8-inch aperture. PSF simulations are normalized to 0.5 intensity, thus the bright central disc approximates the FWHM. The polychromatic FWHM is smaller than the e-line FWHM due to wave interference; neither changes appreciably in size at 3° off axis, although some elongation - mainly due to lateral astigmatism - is noticeable. Either FWHM encircles less than 50% of the energy. Atmospheric enlargement of the FWHM would significantly lower the sytem's design contrast transfer efficiency, shown below.

The MTF shows that the effect of obstruction - determining the contrast transfer limit (green plot) - is much more of a factor than system aberrations with achromatized corrector (top). Transfer efficiency decreases toward outer field, but remains effective, except for the cutoff frequency reduction in tangential plane, due to the aforementioned FWHM elongation (bottom, for achromatized corrector).

11 - The Vatican Advanced Technology Telescope (VATT)

A part of the Mount Graham International Observatory, Arizona, this 1.83m with f/1 primary is rare exception in that it employs aplanatic Gregorian two-mirror system, instead of the usual aplanatic Cassegrain, also known as Ritchey-Chretien. Is there something in the optical properties of its image that makes it the favorable choice, since the compactness obviously is not its advantage? Here's what raytrace shows.

Compared with aplanatic Cassegrain with the primary of the same focal ratio, and with nearly identical focal length (middle), the Gregorian has slightly less curved best image surface, and slightly less astigmatism. The Cassegrain, on the other hand, has slightly smaller central obstruction (0.4m vs. 0.43m, as the secondary diameter needed to fully cover 0.25° field radius, enlarged 10% as the minimum needed for the secondary housing), and 18° shorter secondary-to-final-image separation.

But comparing the VATT with a Cassegrain of nearly identical secondary-to-final-image separation seems more appropriate. In that case, the Cassegrain sports an f/1.5 primary, significantly more relaxed field curvature, nearly 20% lower astigmatism and identical central obstruction (bottom). Hence, the Cassegrain offers better overall performance level, although the difference is still small.

Interesting detail is that all-reflecting systems do have non-zero chromatism (other than that caused by different diffraction pattern size at different wavelengths). In presence of spherical aberration, the magnitude of aberration is inversely proportional to the wavelength.

12 - James Webb Space Telescope

JWST came as a replacement for the Hubble Space Telescope. It is a 6.6m f/20.1 three-mirror Korsch anastigmatic aplanat. The primary is made of 18 hexagonal mirrors forming near-paraboloidal f/1.2 ellipsoidal surface (gray area around it fills in to the simplified shape used for the raytrace). The secondary is a hyperboloid, and tertiary prolate ellipsoid. In order to make it more practical for use with instrumenation chamber, the tertiary is tilted to reflect converging light onto the flat steering mirror placing the final image behind the primary. It will operate in a wide infrared spectrum (0.6-27 micron), with one of primary purposes being collecting information about first light sources in our Universe, within the first galaxies, red-shifted to near-infrared and centered around 2 micron wavelength, at which it should be "diffraction limited" (0.80 Strehl). Apparently, JWST already discovered what should be the oldest known galaxy - GLASS-z13 in Ursa Major - 13.4 billion years old (at the time it formed, it was 3 billion light years from where we are now; light traveled 13.4 billion years to reach us, but the distance between us today is, due to expanding Universe, over 33 billion years). At the time it formed, Sun didn't exist, and the galaxy probably doesn't exist now.

Image below shows the optical system, performance of the basic design (top) and the actual design with tilted tertiary and added flat mirror (bottom). Design data is from "James Webb Space Telescope: large deployable cryogenic telescope in space", P.A. Lightsay et al. (prescription given there does not contain tilt and decenter data, so the design is in that respect approximation). Field angle shown with the basic design is somewhat larger than in the actual telescope (up to 0.1 degree radius, judging on the size of tertiary mirror on a system drawing) in order to make ray paths discernible. Design correction is excellent over the best image radius (2600mm, concave toward secondary). Over flat field though, the edge spot bloats to 3/4 of a milimeter.

Tilting the tertiary induces strong astigmatism, much less of coma, as well as image tilt. Here, field radius is 0.05° (3 arc minutes), which is probably closer to the actual telescope (also, it is about the maximum for the 3° tertiary tilt). After adjusting for image tilt and curvature, correction level is excelent, from better than 1/8 wave P-V equivalent of lower order spherical aberration on axis, to a little over 1/2 wave at 1/20 degree off (minimizing spherical aberration requires slightly lower primary conic, -0.9967). The field could be still slightly corrected for tilt, but it wouldn't produce appreciable gains. Best image surface radius is now -3000mm, concave toward flat mirror. The only point image aberration is low-magnitude primary astigmatism. Ray spot plots bottom left show what the field looks like without corrections for tilt and field curvature. There are big discrepancies between decenter stated in the article (0.19mm for the tertiary) and workable values. If the tertiary is tilted, it induces very strong astigmatism and coma; astigmatism can only be offset by tilting the secondary (which is evident on both published drawings and raytrace presentations), while decentering it induces coma. To have it all balanced requires also decentering the tertiary, which also induces coma, but primarily astigmatism. One peculiarity is that tertiary can't be centered around axial cone; it is shifted up with respect to it (small arow shows bottom of the blue diverging cone at the tertiary). In effect, light falling onto it is using an off-axis section of that ellipsoid. When the tertiary is positioned axially, minimizing astigmatism requires decreasing its tilt angle, and that cannot be done since it is necessary for placing the flat out of the light cone.

Note that the Airy disc shown is for 0.5876 micron wavelength, just below the lower end of the telescope's operational spectrum. Airy disc size changes in proportion to wavelength, so at the 2 micron wavelength it is 3.4 times larger - with the aberration proportionally smaller - and at 27 micron as much as 46 times larger. With the effective focal length of nearly 133m, one arc second spans 0.64mm. Resolving power of the aperture (neglecting central obstruction effect), λ/D ranges from 0.0188 arc seconds at 0.6 micron wavelength (0.0006/6600 times 57.3x60x60, to convert from radians to arc seconds), to 0.84 arc seconds at λ=27 micron. This implies that quality field varies significantly with the wavelength.

Simulated PSF (bottom right) show that the axial diffraction pattern, due to near-hexagonal shape of the primary, has also hexagonal 1st bright ring; central maxima appears slightly non-circular, but larger images show it perfectly round. With intensity normalized to 0.01 - note that scale for these two patterns is different - the pattern shows low-energy regions forming hexagonal wide-spike pattern (intensity points equal and higher than 0.01, or 1% of the central intensity are white). Energy spread by the spider vanes, shown as an inverse aperture, has most of the concentration in the central maxima, roughly 1/30 mm long, two pairs of V-shaped spikes and two elliptical spots above and below the maxima. The approximate width of the vanes is 6 inches. Central obstruction, determined by the missing central segment, is approximately 20% by diameter (23% when extracted from the area).

As already mentioned, this design is approximation, since no complete prescription was available. In the final optimization astigmatism can be even further reduced, but the correction is already very satisfactory. Axial correction error could be further reduced by making the primary figure accurate to yet smaller decimals, but it is neither necessary nor realistic (at the optimum primary conic, -0.996754, coupled with -1.66 secondary conic to keep the coma minimized, axial correction is 1/74 wave P-V of mainly primary astigmatism and some secondary coma, with the error at 0.05° off axis reduced by less than 10%). This design shows that the design limit for JWST is significantly better than 0.80 Strehl in 2 micron wavelength, on and off axis. But fabrication is always less than perfect, and collimation, pointing and thermal errors cannot be entirely eliminated. Below is illustration of the sensitivity of this system to some of the basic possible errors. For clarity, aberrations at the system limit are further reduced (left). It is achieved by tilting the tertiary somewhat more, to 3.15°, with slight changes in the decenter of the secondary (13.705mm) and tertiary (-19.7mm; putting -19.69mm brings zero astigmatism point to the field center, but has no practical significance); optimal primary conic is -0.996755, and -1.66 on the secondary.

Inside the box, effect of very small changes in the figure (primary) and position of the mirrors on the ray spot plots (note that the scale is 2.5 times larger in the box; Airy disc is, as before, for 588nm wavelength). From left, change in the primary conic, primary-to-secondary separation, secondary tilt and decenter, and tertiary tilt and decenter (all changes are smaller in their absolute value than the optimum). In general, induced aberrations change in proportion to the deviation: doubling the deviation doubles the (induced) aberration. As little as 0.0001 deviation in the primary conic - and that is an f/1.2 18-segment surface - induces 0.075 wave RMS (slightly below 0.80 Strehl) of primary spherical aberration in the 588nm wavelength; at 2 microns, it will be smaller in inverse proportion to the wavelength, i.e. 0.022 wave RMS. The sensitivity to the secondary tilt is not a typo: as little as 1/1000 of a degree induces about 0.045 wave RMS (@588nm) of all-field coma.

ACTUAL DESIGN - After I've put here on what appeared to be approximation of the JWST optical design, I had a whisper to my ear (Mike Jones) telling me what the actual design should look like. I'll keep the above system here as a tilted-mirror alternative, but the actual design keeps them orthogonal to the axis. Except above mentioned slight tertiary decenter - purpose of which is to tilt the image so that the used portion becomes nearly perpendicular to the optical axis - there is no other perturbations in the mirrors' rotational symetry vs. optical axis. As image below shows, the flat is centered around optical axis, but it is not in the light path because JWST uses only off-axis section of the image field (that's why it appears that both secondary and tertiary are tilted). In this case, the green cone forms image point at 0.1° off axis, and the blue cone point at 0.2°. Usable image is between 0.1 and 0.2 degree off axis.

Due to this geometry, light forming the image uses only part of the upper half of the tertiary, with the rest being removed to allow light reflected from the flat to form image beyond the tertiary. Beam footprint for the 0.1° field point at the primary (aperture stop, 1), secondary (2) and tertiary (3) is shown below. Since aperture stop is on the primary, any field angle has identical footprint on it. It is differs only slightly on the secondary, while on the tertiary all footprints fit on the upper half of it (for 0.1° field radius on the primary and secondary, for both on the tertiary; needed tertiary radius for zero vignetting is 398mm, and 345mm for the secondary). The dashed circle is the off-axis section on the tertiary containing all beams for a circular field.

Tilt and decenter of the image are only to show its characteristics over the used portion, when positioned optimally vs. cameras. Correction - as the design limit - is still exsquisite (note that this is the same astigmatic field shown above with the basic design, only expanded to 0.2° radius; in the outer field, higher-order astigmatism kicks in causing tangential surface to cross over the sagittal). The field could be extended outward by manipulating astigmatism, without significant effect on the correction level, but the limit to its size is set by the need to keep it nearly flat. In this case, the diameter is 0.1°, or six arc minutes. The field could also be extended toward axial cone, but no more than one arc minut, or so (w/o vignetting).

13 - EXTREMELY ACHROMATIC 19-INCH f/1* CAMERA

"Extremely achromatic" is how Epps and Vogt described the camera they designed for the Keck telescope spectrometer in 1993 (Extremely achromatic f/1 all-spherical camera constructed for the high-resolution echelle spectrometer of the Keck telescope; drawing by Epps with the prescription data published online is from 1990). It is probably as simple as such a camera can be: all-spherical, consisting of a two-singlet full aperture corrector, mirror, and a single field lens made of the same glass as the front end corrector. Unlike the old-fashioned Schmidt/Houghton cameras, it comes with flat field (a comparison could be interesting). Below is how it raytraces with OSLO Edu. As both, LA and OPD plots show, there is no longitudinal chromatism to speak of, although the colors could be made yet little tighter (camera was intended for 0.3-1.1 micron range - and beyond - but here it is raytraced only for the conventional visual 0.43-0.67 micron range; obviously, there is enough room to go toward longer wavelengths, since the red end here is beter corrected than the central line).

The spherical aberration leftover on axis could be minimized to better than "diffraction limited" (0.80 Strehl), but may have been left in on purpose, in order to make the point image energy spread over the field more even. Field seems to be limited to less than 4° radius by departure from flatness and higher order aberrations (in the sense that star images remain similar in size). At 4°, trefoil becomes the dominant aberration form, and quadrafoil becomes the third, after primary astigmatism (which probably includes significant portion of Schwarzschild's lateral astigmatism). Paper cites 1.8 inch (3.4°) field radius, but here it is 3.2°, which gives more even field quality. Diffraction simulations show fairly even blurring accross the field, with the dense blur portion well within 0.001 inch (0.0254mm). 80% energy radius is 0.0072mm at the 70% radius, and 0.0089mm at 4 degrees (2 and 2.4 arc seconds, respectively).

There is a discrepancy between the claimed f/1 focal ratio, and the actual f/1.58. Aperture stop radius on the drawing is given as 9.5 inch, and the focal length given in the abstract is 30 inch; that produces f/1.58, not f/1. With a bit larger field, the front lens radius needed to accept all incoming light goes to 15 inches, but it is not the aperture stop, and the focal ratio remains f/1.58. Pulling the stop all the way back to the font lens changes little in the magnitude of aberrations. If the stop at this location would increase to 15 inch in radius, it would produce a f/1 system - but it would be a different configuration, with significantly inferior performance (although better than with the stop expanded to a 15-inch radius at its original location).

Below are ploted encircled energy (top) and RMS spot size/OPD for the field (bottom). So far, central obstruction effect was omitted, but it obviously cannot be avoided, and has to be relatively significant. For 3.2° field, minimum central obstruction size, determined by the size of field lens (for full field illumination) is about 40% linear (image itself is about half as large). Encircled energy plots for the five wavelengths, even sensitivity, (top left) indicate that obstructed aperture has smaller 80% encircled energy radius up to 70% of the field radius, but keep in mind that the starting point, i.e. unit energy in the obstructed aperture is over 10% smaller PSF (linearly), whose central maxima contains less than 71% of the energy of the unobstructed central maxima (the unmarked plots are for unobstructed aperture, with identical color code field-height wise).

Ray spot plots for obstructed and unobstructed aperture are generally similar. Diffraction simulations (for the actual, obstructed aperture) are about twice larger, for clarity. Patterns are similar to those of unobstructed aperture over most of the field, but closer to the edge they spread wider. The RMS spot size (radius) is generally smaller for unobstructed aperture, even on axis, due to the smaller obstructed spot here being of even density, while the unobstructed spot has two denser inner areas (most dense around the center), not clearly discernible at this scale. The RMS spot size (given as radius) is nearly constant up to 90% of field radius, after which quickly increases, mainly due to trefoil-like deformation. Over about 80% of field radius the RMS spot is little over 0.0002" (0.005mm, e-line) vs. 0.0126mm (diameter) cited in the paper as the whole (3.4°) field average, making it suitable for 7-15 micron pixel chips. Diffraction images above match this averaged RMS spot size fairly well.

This optical arrangement is simple enough to be within reach of the advanced ATM. Scaling it down to, say, 6 inch aperture, would have the blur size reduced by a factor of 0.4, to some 4-5 microns. Below is described system of this kind, using BK7 glass and with the stop at the fron lens, suitable to be used as a stand-alone camera. Still, at this fast focal ratio tolerances are very tight. For instance, just 1mm shorter front field lens radius would negatively affect lateral color correction. As mentioned before, significantly reducing stop separation, even placing it at the front lens, would have relatively minor consequences. As mentioned, minimum central obstruction, due to the assembly that would house the field lens and detector is about 40% linearly at this field size (image itself is always significantly smaller).

To have better idea of the degree of chromatic and overall correction of this arrangement, we'll compare to a known standard, the Schmidt camera of the same aperture and focal ratio. Linear central obstruction is 40% in the Ebbs-Vogt systems, and 20% Schmidts. In addition to the above system (top left), standard Schmidt (bottom left) and flat-field Schmidt (bottom right), included is the above system with silica replaced with the Schott BK7, with the stop at the front lens (stand-alone camera, top right). It compares favorably to the system with a distant stop both, size wise, and with respect to overall correction. In order to make chromatic correction easier to compare, both Schmidt systems and the "short" Epps-Vogt have minimized error on axis; that influences somewhat field correction. System drawings are nearly on the same scale, thus directly comparable size-wise (note that the Schmidts and "short" Epps-Vogt prescrptions are in mm).

The two Epps-Vogt systems have very similar level of chromatic correction, but the "short" version is easier to compare to the Schmidts. Its superiority in the axial chromatic correction is obvious immediately, in all four non-optimized wavelengths (note that the nominal P-V error is not representative in obstructed systems, since measured from the non-existing vertex, while the RMS is measured only over the annulus area). Better overall axial color correction for 0.3-1.1 micron range would have the blue/violet focusing longer - as in the original design, or even more - because yet shorter wavelengths focus shorter, and the error on the violet end is much larger than on the infra-red.

Epps-Vogt systems have about half the lateral color of the flat-field Schmidt, while that of the standard Schmidt is practically non-exsistent on the given scale. Polychromatic encircled energy (top left), however, shows that there is not so much difference in the 80% encircled energy radius (the five wavelengths, even sensitivity). Better chromatic correction of the Epps-Vogt is mainly offset by the better field correction of the Schmidt. Diffraction simulations for 3.2° field radius (polychromatic, the five wavelengths, even sensitivity) show significantly more difference in the intensity distribution pattern, than in encircled energy. Note that placing stop at the front surface produces similar type of field edge pattern in the original Epps-Vogt; for some reason, they opted for somewhat wider, but more evenly mixed color-wise spread toward field edge, even at the price of a widely separated stop (some stop separation is inevitable with spectrographs since the camera has to be preceded by the collimator and dispersive element). The coma-like ray spot plot is actually produced by a wavefront deformation closer to astigmatism; as the wavefront maps show, it affects a narrow outside wavefront side strips, mainly on the bottom half, with the edge wrinkle spreading to the bottom, and becoming nominally larger toward shorter wavelengths. The relative area of deformation is quite small, and so is the amount of energy spread out.

14 - The 500mm f/9 Solar Optical Telescope

Described as "the largest state-of-the-art solar telescope (...) ever completed and flown in space" (The Solar Optical Telescope of Solar-B (Hinode): The Optical Telescope Assembly, Suematsu et al. 2008), it was designed for high-precision photometric and polarimetric observations of the Sun in the visible spectrum (388-668nm). Placed out of the atmosphere - Solar-B satellite, later renamed to Hinode - it is capable of resolving magnetized structures down to 0.2 arc seconds, out of reach of ground based telescopes. This requirement was the main factor determining the aperture size. The system can be separated into two components: (1) Optical Telescope Assembly, consisting of a two-mirror Gregorian, collimating lens unit (CLU), polarization modulator, tip-tilt mirror and astigmatism-correcting lens, added to correct axial astigmatism generated at the primary most likely due to a mounting stress, and (2) Focal Plane Package with CCD detector, where the final image is formed. The collimating lens unit, consisting of six singlets, is in effect a double apochromat, each having opposite sign of low-temperature sensitivity. The tip-tilt mirror stabilizes image and folds collimated beam to the side, toward Focal Plane Package, where it passes through re-imaging lens unit before falling onto a 4000x2000 pixel detector.

The two-mirror optical system is aplanatic Gregorian, with 0.344D central obstruction by the secondary. Secondary is supported on a 120° 3-vane spider, with 40mm wide vanes. Entrance aperture is 200mm ahead of secondary's surface, i.e. 1700mm from the primary. Small diagonal mirror placed at the prime focus is a heat dump mirror (HDM, with an opening about twice the diameter of the solar image). At the final focus, there is a secondary field stop - 361.3x197.4 arc seconds rectangular hole (about 6x3.3 arc minutes, or 8x4.4mm) in a conical mirror of 65mm outer diameter. It determines the usable field.

Gregorian offers the advantage of having two field stops between the primary and secondary mirror. Also, in a similar configuration with f/9 focal ratio, a Cassegrain would require a small, strongly curved secondary, resulting in a twice larger wavefront error at 0.1° on flat field. Ray spot plots on the left are for a perfectly executed design. With a very minor fabrication imperfections in the mirror conics and radii, the error - mainly spherical aberration - exceeds 1/4 wave P-V, but it is practically entirely cancelled increasing the mirror separation by 1.6mm (right). At 0.1° flat field error is still less than "diffraction limited" (0.80 Strehl). Error budget for the system is 0.80 Strehl (36.5nm RMS WFE), or better, with 0.9 Strehl limit (25.8nm RMS) alocated to the optical tube, and as much for the focal plane package. Actual measurements came out better, at the level of 12nm on individual mirrors (about 17nm combined, for the 0.034 wave RMS WFE at 500nm wavelength). The mirrors use protected silver coating with 96% reflectivity and 6.5% solar absorbance.

The vanes are fairly massive, but the good side is that it creates short spikes. Still, MTF shows that they do significantly lower contrast transfer over the entire range of frequencies (there is also relatively small dependance of the contrast transfer on image orientation). They are more pronounced in polychromatic light, along with a loss of inner structure of the diffraction image (peak normalized to 0.005 means that every point with a higher intenity shows white). Polychromatic encircled energy (even sensitivity 430-670nm) gives 80% radius of 12 microns (on axis), wich at this image scale corresponds to 0.56 arc seconds. This is mainly due to loss of energy to the first bright ring, caused by the central obstruction and vanes. Resolution limit for high-contrast details is determined by the size of the central diffraction disk, nearly three times smaller.

15 - 3.6m f/200 AEOS telescope

The U.S. Air Force Advanced Electro-Optical System (AEOS) telescope, located on the Hawaiian volcano Haleakala, uses 3.67m (3.63m clear diameter) f/1.5 primary and two interchangeable secondaries: one with effective diameter less than 5% of the primary diameter, for f/200 and 62"x62" (0.016°x0.016°) field of view, and the other about three times larger, for 202"x202" field of view (in either case about 8x8 inches image area). Primary's shape is controlled with 132 actuators, and the adaptive optics system incorporates CCD wavefront sensor coupled with a 2mm-thick 288mm diameter deformable mirror with 941 actuators. While constructed for space surveillance - primarily satellite tracking - the telescope is also used for astronomical purposes. It operates in spectral range from 540nm into infrared.

System below is a raytrace of similar configuration. Since no prescription is given, it is only a close approximation of the actual system. It is assumed the system is Classical Cassegrain, since there is no benefit from making it aplanatic. The length of converging beam from the secondary is over 30m, with a series of flat mirrors bending it through a couder path around the primary (through the altitue axis, and than through the azimuth axis) to the basement where are located instruments.

Shown is a system with the smaller secondary (the actual secondary is probably a bit smaller, with correspondingly shorter back focal length). Despite the fairly strong field curvature, due to the very small field of view there is no appreciable effect on off-axis performance. Ray spot plots top left are the actual shapes over best field, too small to be recognized within the Airy disc. With the larger secondary and three times larger field, off axis performance is similar over the best field, but over flat field the field edge at 0.024° has 0.32 waves P-V (0.09 waves RMS) of defocus - about three times more than with the f/200 secondary (approx. three times weaker field curvature results in about 1/3 as much of linear defocus, but the wavefront error is inversely proportional to the square of focal ratio). Average measured 200ms FWHM with 850nm filter is 0.13 arc seconds (6.5 pixels), which is 2.7 times the theoretical limit (High resolution imaging with AEOS, Patience et al. 2001). Scaled linearly down to 550nm wavelength, it would come to less than 0.09 arc seconds.

16 - 150mm f/15 Zeiss-Coude-refractor

This rather a small aperture refractor for observatory setting was not intended for professional-level use. It was designed by Zeiss as an observatory/educational instrument for public use, i.e. for popularization of astronomy. Its Coude-style mount makes the final image stationary, independent of the movements of optical tube. The mount is very massive, requiring a permanent, observatory-like setup. Both axes are supplied by an electrical motor, and the tube can also be moved manually.

The standard version uses Zeiss AS objective, doublet achromat using Schott KzF2 "short-flint" as the front element and BK7. It is in reverse to the standard achromat, having positive element (crown) in front, and negative (flint) behind. There is no gain in chromatic correction, but the reverse arrangement - known as the Steinheil configuration - requires more strongly curved lens surfaces, thus all else equal produces more of higher-order spherical residual, as well as more spherochromatism. There is no formal eplanation known to me of why Zeiss opted for the Steinheil arrangment for its AS-objective.

The KzF2 seems to be beyond obsolete nowadays, with hard to find refractive properties data, and was raplaced for raytracing purposes with the Schott KzFN2, its more environmentaly friendly, now obsolete near-equivalent.

While it is commonly cited that this objective has significantly - three-fold, or so - reduced chromatism vs. standard achromat, raytracing shows it is much more modest: about 22% in the F and C lines (when nearly balanced defocus-wise), and in the violet g line. Chromatic focal shift graph shows about half as much of defocus between paraxial F and e focus for the AS objective (and no significant difference vs. achromat in the position of paraxial C-line focus), but that is not relevant in the presence of significant spherochromatism. What matters is where the best focus is, which is displaced from paraxial focus due to spherical aberration, generaly stronger in the reverse, Steinheil configuration. Note that this particular objective has correction set for nearly balanced error in F and C; the actual objective could have somewhat different correction mode, closer to the one with the paraxial F and C foci coinciding (i.e. favoring the red end).

Bottom graphs show the imbalance in the F vs. C defocus error when their paraxial foci coincide: it is significantly larger in the Steinheil configuration, favoring the red line correction. When the two lines are balanced, this AS objective has chromatic correction at the level of the f/19.2 standard achromat.

The second flat can be rotated, allowing alternating between the lower and upper focus position. According to Zeiss, standard accessories include five eyepieces (one of them for Sun image projection), one finder eyepiece, one reticle with illuminating device, one pointing eyepiece, one neutral density glass filter, color filters, one ring micrometer, eyepiece spectroscope, and two 2-fold eyepiece revolvers (one for stelar observtions, the other with light-attenuating device for visual and photographic solar observations). On request, the standard AS achromat can be replaced by a 150mm f/15 or f/11 triplet apochromat (F-objective).

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