a curvature‐compensated corrector for drift‐scan observations

A Curvature‐compensated Corrector for Drift‐Scan ObservationsAuthor(s): Paul Hickson and E. Harvey RichardsonSource: Publications of the Astronomical Society of the Pacific, Vol. 110, No. 751 (September1998), pp. 1081-1086Published by: The University of Chicago Press on behalf of the Astronomical Society of the PacificStable URL: http://www.jstor.org/stable/10.1086/316230 .

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1081

Publications of the Astronomical Society of the Pacific, 110:1081–1086, 1998 Septemberq 1998. The Astronomical Society of the Pacific. All rights reserved. Printed in U.S.A.

A Curvature-compensated Corrector for Drift-Scan Observations

Paul Hickson

Department of Physics and Astronomy, University of British Columbia, 2219 Main Mall, Vancouver, BC V6T 1Z4, Canada; [email protected]

andE. Harvey Richardson

Department of Mechanical Engineering, University of Victoria, Victoria, BC V8W 3P6, Canada; [email protected]

Received 1998 April 12; accepted 1998 June 15

ABSTRACT. Images obtained by drift-scanning with a stationary telescope are affected by the declination-dependent curvature of star trails. The image displacement due to curvature and drift-rate variation increaseswith the angular field of view and can lead to significant loss of resolution with modern large-format CCD arrays.We show that these effects can be essentially eliminated by means of an optical corrector design in whichindividual lenses are tilted and decentered. A specific example is presented of a four-element corrector designedfor the Large-Zenith Telescope. The design reduces curvature errors to less than 00.074 over a field of′ ′10 # 20view centered at 497 declination. By changing the positions and tilts of the lenses, the same design can also beused for any field centers between 07 and 5497.

1. INTRODUCTION

In recent years, the technique of time delay integration (TDI,also known as drift-scanning) has become well established inastronomy (McGraw, Angel, & Sargent 1980; Hall & Mackay1984). For this technique, the telescope remains stationary ortracks at a nonsidereal rate, resulting in a steady drift of imagesacross a CCD detector (which is aligned so that the drift di-rection coincides with the orientation of the CCD columns).The CCD is scanned continuously at a rate that moves thephoton-produced charge along the columns at the same speedas the optical image. This prevents charge spread and resultsin sharp images with an integration time equal to the drift timeof the image across the CCD.

TDI offers several benefits compared with conventional im-aging. A chief advantage is that variations in pixel-to-pixelsensitivity are greatly reduced because the response is averagedover all CCD pixels in each column. The required flat-fieldcorrection is essentially one-dimensional. This results in asmuch as a factor of 45 reduction in background variations, afterflat-field correction, for a pixel CCD. Because2048 # 2048background variations are often a limiting factor in photometryof faint or low surface brightness objects, even conventionaltelescopes are sometimes operated in drift-scan mode in orderto take advantage of this and other benefits (Shectman et al.1992; Schneider, Schmidt, & Gunn 1994).

The TDI technique is particularly important for telescopesthat have no tracking capability. Liquid-mirror telescopes(LMTs) employ rotating primary mirrors that are surfaced witha liquid metal such as mercury. Because the rotation axis mustbe vertical, these telescopes are best suited for observations

near the zenith, although it is possible in principle to observeat large-zenith angles using specially designed correctors (Rich-ardson & Morbey 1987; Borra, Moretto, & Wang 1995).By operating their CCD cameras in TDI mode, such zenith-pointing telescopes can survey large areas of sky as the Earthrotates. For example, the UBC-NASA Multiband Survey(Hickson & Mulrooney 1998) employs a pixel2048 # 2048CCD camera at the prime focus of the 3 m LMT of the NASAOrbital Debris Observatory (NODO) (Potter & Mulrooney1997). This survey covers 20 deg2 and reaches inm . 20.4AB

a single TDI scan through medium-band filters.For TDI observations, it is essential that the images of all

objects drift at the same rate, on parallel linear tracks, or imagespread will result. For this reason, it is important that the tele-scopes used for TDI observations be equipped with correctorsthat reduce the pincushion or barrel distortion that is usuallypresent. For a prime-focus configuration, this normally requiresa corrector having at least four optical elements.

After the removal of telescope distortion, however, thereremains a more fundamental difficulty. With the exception ofobjects on the celestial equator, the image tracks are not straightbut concave, in the direction of the nearest celestial pole. Fur-thermore, the linear rate of image motion is a function of dec-lination, so there is no universal CCD scan rate that will preventimage spread. These effects can lead to significant image deg-radation for telescopes at moderate latitudes employing large-format CCD cameras (Gibson & Hickson 1992).

An elegant solution to this problem has been realized withthe great-circle camera (Zaritsky, Shectman, & Bredthauer1996), which uses a combination of rotation and motion in



1082 HICKSON & RICHARDSON

1998 PASP, 110:1081–1086

Fig. 1.—Geometry of star-trail curvature

declination to trace a great circle on the sky. As a result, thenonlinear effects are minimized. While the great-circle cameradoes reduce the curvature effects, it has a fundamental limi-tation. Because of the physical motion required of the camera,the length of the great-circle track is limited. This restrictionis particularly severe for zenith-pointing telescopes that cannotmove to follow a great-circle track. TDI observations wouldneed to be interrupted frequently while the camera position isreset to follow different great-circle arcs.

In this paper, we propose a new solution to the problems ofstar-trail curvature. By offsetting and tilting the lenses of asuitably designed corrector, one can introduce an asymmetricfield distortion that compensates for the nonlinear motion ofthe images. As a result, the images track in parallel straightlines at a common constant rate with high accuracy. This effectcan be achieved without introducing any additional optics inthe light path. Moreover, the same corrector can serve tele-scopes at different latitudes just by repositioning the opticalelements.

The paper is organized as follows. In § 2, we analyze thegeometry of the star-trail curvature and derive formulae for theresulting image displacements. In § 3, we describe an exampleof a prime-focus corrector that compensates for these effects.The performance of the corrector is examined in § 4, followedby our conclusions.

2. ANALYSIS OF STAR-TRAIL CURVATURE

Star-trail curvature effects have been discussed previouslyby Gibson & Hickson (1992), Stone et al. (1996), and Cabanac(1998), who give approximate formulae for the distortions. Ouraim here is to present a simple, exact derivation or the relevantrelations as well as useful approximate formulae. The geo-metrical picture is illustrated in Figure 1. The circle representsthe celestial sphere, which, for convenience, we take to haveunit radius. Its center is the origin of a three-dimensional Car-tesian coordinate system (x, y, z). The z-axis is aligned withthe North Celestial Pole (NCP); hence, the x-y plane intersectsthe sphere at the celestial equator. For clarity, the y-axis, whichis perpendicular to the page, is not shown in the figure. Nowlet us consider a telescope whose axis lies in the x-z plane atan angle to the celestial equator. Therefore, is the dec-d d0 0

lination of the center of the telescope field of view. For a zenith-pointing telescope, is equal to the latitude of the observatory.d0

In the telescope’s focal plane, perpendicular to the optical axis,we set up a two-dimensional Cartesian coordinate system(X, Y), oriented so that the Y-axis is parallel to the y-axis. Theoptical axis passes through the origin of the X-Y coordinatesystem, so that X and Y give the north and east position of theimage with respect to the field center. It is convenient to workwith the projection of the focal plane through the telescopeoptics and back onto the celestial sphere. Figure 1 shows theprojected focal plane that is tangent to the celestial sphere atthe point where it is intersected by the optical axis. Since the

sphere has unit radius, the values of X and Y can be convertedto physical distances in the focal plane by multiplying by thetelescope’s effective focal length F.

Now let us consider the sidereal motion of a star at arbitrarydeclination d. As the Earth rotates, the line of sight to the starsweeps out a cone, which is indicated in Figure 1 by the as-terisks. The intersection of this cone with the tangent planedefines the track of the star’s image on the focal plane. Thistrack is by definition a conic section, which is an ellipse, pa-rabola, or hyperbola depending on whether the declination d

is greater than, equal to, or less than the co-angle (forp 2 d 0

a zenith telescope, the colatitude).The equation of this track is easily derived. The equation of

the cone is

2 2 2 2x 1 y 5 z cot d, (1)

while the plane is defined by

x 5 2X sin d 1 cos d ,0 0

z 5 X cos d 1 sin d . (2)0 0

Eliminating x and z, and setting , we obtainy 5 Y

2 2 2(sin d 2 cos d )X 2 2 sin d cos d X0 0 0

2 2 21 (Y 1 1) sin d 2 sin d 5 0. (3)0

The northerly displacement X of the image as a function of



CURVATURE-COMPENSATED CORRECTOR 1083

1998 PASP, 110:1081–1086

east-west position Y is given by the solution

sin d cos d0 0X 5 2 2sin d 2 cos d0

1/2

2 2 2 2 2(sin d 2 cos d )[(1 1 Y ) sin d 2 sin d ]0 0# 1 2 1 2 .2 2( ){ }sin d cos d0 0

(4)

This equation has a simple Taylor series expansion:

2Y )X 5 tan (d 2 d ) 1 1 . (5)0 2 cot d

From this, we see that the local radius of curvature R, mea-sured at the center of the track, is

2 21d XR { 5 cot d, (6)2F FdY Y50

which, in physical units, is . Note that this curvatureF cot d

depends only on the declination of the object and is independentof the pointing direction or location of the telescope. Equation(5) may be written

X 5 X 1 DX, (7)0

where denotes the X-coordinate of the centerX 5 tan (d 2 d )0 0

of the track ( ) and measures the departure of the trackY 5 0 DXfrom linearity, in the north-south direction. Except at very largedeclinations, the radius of curvature of the trails is nearly con-stant across the field, and we may use the value at the fieldcenter:

1 2DX . Y tan d . (8)02

Now let us consider the rate at which the images move. Letf be the azimuth angle of the object in the (x, y, z) coordinatesystem, referenced to the x-axis. As the Earth rotates, f in-creases at a constant rate. From the definition of f,

Y 5 y 5 x tan f. (9)

Substituting this in equation (1) and using equation (2) to elim-inate z, we obtain

sin fY 5 . (10)

sin d tan d 1 cos d cos f0 0

From this, we see that the rate of image motion is a functionof both the declination of the object and the declination of thefield center. For a given azimuth angle f, objects that crossthe field north of the field center ( ) will have smaller Y-X 1 00

values than those passing south of the field center (in theNorthern Hemisphere; the opposite is true in the SouthernHemisphere).

Let , where denotes the Y-coordinate of aY 5 Y 1 DY Y0 0

star, having the same azimuth angle, whose track passes throughthe field center ( ). From equations (5), (7), and (10), weX 5 00

obtain

1 1 tan d tan d0DY 5 2X Y . (11)0 0 ( )tan d 1 cot d cos f0

The factor in parentheses typically varies very slowly acrossthe field and may be replaced by its value at the field center.This gives

DY . 2X Y tan d . (12)0 0 0

3. CORRECTING CURVATURE EFFECTS

In the previous section, exact and approximate formulae weredeveloped to describe the distortions produced by the star-trailcurvature. For most practical situations, the approximate for-mula of equations (8) and (12) give more than enough accuracy.The maximum distortion can be found by setting andX Y0 0

equal to half the north-south and east-west extent of the de-tector, respectively. The two distortions are closely related. Fora square detector, the east-west image smear, which is due torate variations, is 4 times as large as the north-south smear,which is due to curvature. (Because the distortion changes signas the star crosses the field, the east-west image smear is twice

.)DYAs an example, let us consider observations of a star at

using a CCD camera with a square field of view ofCd 5 45209 ( rad). Equations (8) and (12) giveX 5 Y 5 0.0029090 0

maximum distortions of 0.000004231 and 0.000008462 rad(00.873 and 10.745), respectively, for and . These areDX DYsignificant distortions that would cause unacceptable imagedegradation for TDI observations.

Fortunately, it is possible to eliminate to a large extent theseeffects for zenith telescopes, located even at moderately highlatitudes, by means of an asymmetric corrector. The idea is touse a combination of decenters and tilts of the optical elementsin order to introduce distortion that closely matches that de-scribed by equations (8) and (12), but with an opposite sign.Such a design has been recently developed for the Large-ZenithTelescope (LZT), which employs a 6 m f/1.5 liquid-mercuryprimary mirror (Hickson et al. 1998). The corrector was de-




1998 PASP, 110:1081–1086

Fig. 2.—The LZT corrector configured for observations at 1497179

declination.Fig. 3.—The LZT corrector configured for observations at 1327599

declination.

TABLE 1Corrector Specifications

Surface(1)

Type(2)

R(mm)

(3)A

(4)

07

t(mm)

(5)

1327599 1497179

Dt(mm)

(6)

XDE(mm)

(7)

BDE(deg)(8)

Dt(mm)

(9)

XDE(mm)(10)

BDE(deg)(11)

0 . . . . . . . Object ` ) ` 0.000 ) ) 0.000 ) )1 . . . . . . . Mirror 218000.000 K 5 21.000 28535.429 0.870 ) ) 0.000 ) )2 . . . . . . . BK7 2201.785 ) 230.000 0.000 28.193 3.584 0.000 214.160 6.2153 . . . . . . . Air 2200.549 ) 2253.805 20.660 ) ) 0.000 ) )4 . . . . . . . BK7 2799.735 ) 212.000 0.000 211.002 25.250 0.000 219.215 29.1505 . . . . . . . air 2105.956 0.686E208 256.097 21.083 ) ) 0.646 ) )6 . . . . . . . BK7 594.610 ) 220.000 0.000 11.817 2.839 0.000 21.177 4.9417 . . . . . . . Air 236.851 20.212E207 240.216 0.782 ) ) 0.980 ) )8 . . . . . . . BK7 2131.008 ) 250.000 0.000 211.998 0.822 0.000 221.443 1.5109 . . . . . . . Air 2540.262 0.227E206 215.193 0.260 ) ) 0.358 ) )10 . . . . . . BK7 ` ) 27.000 0.000 0.169 21.023 0.000 0.444 21.83511 . . . . . . Air ` ) 26.000 0.000 ) ) 0.000 ) )12 . . . . . . Quartz ` ) 25.000 0.000 ) ) 0.000 ) )13 . . . . . . Air ` ) 211.000 0.000 ) ) 0.000 ) )14 . . . . . . Image ` ) ) ) ) ) ) ) )

signed by E. H. Richardson to specifications provided by P.Hickson. The required image quality was a 50% encircled en-ergy diameter (EED) of 00.4 or less over a 109.5 # 219 fieldof view (in order to match a pixel CCD). The2048 # 4096procedure was to aim first for a corrector with zero distortion.Global optimizations were employed to keep the maximumclear aperture relatively small, in order to reduce cost, whilemeeting image-quality specifications over the required field ofview. The corrector was then reoptimized to introduce the re-quired sidereal distortion. In this process, curvatures, tilts, de-centers, and spacings were all allowed to change. At first, awedge was allowed for one element, but it was found that thiscould be eliminated, allowing the same lenses to be used atdifferent latitudes. After optimization for the 1497179 latitudeof the LZT, the lens curvatures and thicknesses were frozen.

In order to illustrate the flexibility of the corrector, the designwas then reoptimized for the 1327599 latitude of the NODOLMT and the equator (07), allowing only the tilts, decenters,and locations of the lenses to change.

The optical configurations, for latitudes 1497179 and1327599, are illustrated in Figures 2 and 3. The corrector em-ploys four lenses that are all fabricated from the same opticalmaterial. The details of the corrector design are provided inTable 1, which gives parameters for the two asymmetric con-figurations and for the symmetric case. Each row of Table 1describes an optical surface, in the order in which light reachesit. Light enters parallel to the z-axis of a Cartesian coordinatesystem aligned with surface 1 (the parabolic reflector). Columns(7), (8), (10), and (11) describe translations and rotations ofthe coordinate system in which subsequent surfaces are defined.



CURVATURE-COMPENSATED CORRECTOR 1085

1998 PASP, 110:1081–1086

TABLE 2Encircled Energy Diameters for 1497179

X(deg)(1)

Y(deg)(2)

400 nm 500 nm 600–800 nm

50%(arcsec)

(3)

80%(arcsec)

(4)

100%(arcsec)

(5)

50%(arcsec)

(6)

80%(arcsec)

(7)

100%(arcsec)

(8)

50%(arcsec)

(9)

80%(arcsec)

(10)

100%(arcsec)

(11)

20.088 0.000 0.353 0.502 0.698 0.162 0.265 0.351 0.172 0.244 0.6490.000 0.000 0.154 0.252 0.524 0.072 0.107 0.132 0.176 0.259 0.6760.088 0.000 0.211 0.514 1.046 0.134 0.179 0.360 0.199 0.346 0.857

20.088 0.088 0.371 0.509 0.780 0.153 0.291 0.576 0.203 0.320 0.9180.000 0.088 0.220 0.313 0.853 0.097 0.154 0.357 0.158 0.264 0.7140.088 0.088 0.238 0.631 1.453 0.162 0.222 0.584 0.218 0.351 0.787

20.088 0.176 0.396 0.568 1.297 0.196 0.357 1.107 0.285 0.495 1.1840.000 0.176 0.370 0.569 1.826 0.353 0.533 1.478 0.250 0.417 1.0190.088 0.176 0.370 0.569 1.826 0.353 0.533 1.478 0.397 0.661 1.380

TABLE 3Image Displacements for 1497179

X(deg)(1)

Y(deg)(2)

l

(nm)(3)

DX DY

Error(arcsec)

(8)

Target(arcsec)

(4)

Actual(arcsec)

(5)

Target(arcsec)

(6)

Actual(arcsec)

(7)

20.088 0.176 400 1.131 1.162 1.130 1.123 0.032600 1.131 1.123 1.130 1.091 0.040800 1.131 1.113 1.130 1.068 0.065

0.000 0.176 400 1.131 1.114 0.000 20.014 0.022600 1.131 1.110 0.000 0.022 0.030800 1.131 1.109 0.000 0.032 0.037

0.088 0.176 400 1.131 1.132 21.130 21.188 0.058600 1.131 1.157 21.130 21.103 0.038800 1.131 1.166 21.130 21.065 0.074

0.000 0.132 400 0.636 0.614 0.000 20.035 0.041600 0.636 0.575 0.000 20.025 0.066800 0.636 0.611 0.000 20.022 0.033

20.044 0.088 400 0.283 0.264 0.565 0.554 0.023600 0.283 0.283 0.565 0.526 0.039800 0.283 0.240 0.565 0.513 0.066

0.044 0.088 400 0.283 0.259 20.565 20.605 0.047600 0.283 0.259 20.565 20.569 0.024800 0.283 0.259 20.565 20.553 0.027

Surfaces following such a transformation are aligned on thelocal mechanical axis (z-axis) of the new coordinate system.The new mechanical axis remains in use until changed byanother transformation. In these transformations, the translationis applied before the rotation.

The column headings for Table 1 are as follows: (1) surfacenumber, (2) material that the light enters when crossing thesurface, (3) radius of curvature of the surface, (4) the asphericconstant A (except for surface 1, which is a pure conic section),defined by the equation

2r 4Dz 5 1 Ar , (13)2 2 1/2( )1 1 1 2 1 1 K y /R[ ]

where r is the distance from the z-axis to a point on the surfaceand is the z-displacement of the surface with respect to aDzplane, (5) distance along the z-axis from this surface to thenext for the 07 configuration, (6) change in the z-axis distancerequired for the 1327599 configuration, (7) displacement of theorigin of the coordinate system at the surface, and (8) rotationangle of the coordinate system at the surface. Columns (9),(10), and (11) are the same as columns (6), (7), and (8), butfor the 1497179 configuration.

4. PERFORMANCE OF THE CORRECTOR

The corrector provides a 249 diameter unvignetted field ofview. The effective focal length is 10.000 m, which gives animage scale of 200.63 mm21. Table 2 gives the 50%, 80%, and100% EEDs for various field angles and wavelengths. Themedian values of the 50% EED at 400, 500, and 600–800 nmare 00.353, 00.162, and 00.203, respectively. These values arewell below typical ground-based seeing disk diameters at thesewavelengths, which are typically 00.6 or more at the best as-tronomical sites, so the corrector will not appreciably degradethe image. There is a small, but nonnegligible, focus shift be-tween the three wavelength regions. Since the corrector is in-tended to be used with common broadband (or narrower) filters,this is not an issue for image quality.

Table 3 summarizes the distortion characteristics of the cor-rector. For the field angles specified in columns (1) and (2),columns (4), (5), (6), and (7) give the target and achieved imagedisplacements. The difference in position, which correspondsto the distortion error, is listed in column (8). It can be seenfrom Table 3 that the distortion errors are typically a few tensof milliarcseconds, with the maximum error being 00.074. Sincethese residuals are much less than the seeing diameter, thecorrector effectively eliminates star-trail curvature and rate dif-ferentials as a source of image degradation.

Considering the 6 m diameter and fast focal ratio of theprimary mirror, the corrector is very compact, having an overalllength, from first element to focal plane, of cm. The larg-50.5




1998 PASP, 110:1081–1086

est lens, which has spherical surfaces, has a diameter of only34 cm. This is quite small for a 6 m f/1.5 telescope, whichhelps reduce the cost of the corrector. The three smaller lenseshave diameters less than 15 cm. The rear side of each is a low-order aspheric surface.

5. SUMMARY AND DISCUSSION

We have presented an analysis of star-trail curvature effects.If uncorrected, these effects can result in substantial imagedegradation in drift-scan images obtained with large-formatCCDs. A new technique has been described in which the cur-vature effects are compensated by means of a corrector lensemploying decentered and tilted elements. As an example, wehave discussed a compensated prime-focus corrector designedfor the 6 m LZT.

The corrector requires no additional optical componentsother than the four elements normally needed to remove tele-scope aberrations, including distortion. By varying the posi-tions of the elements, the corrector can be used with a zenith-pointing telescope at any latitude up to at least 5507. It pro-vides a 249 unvignetted field with a median 50% EED of 00.2and a maximum distortion error on the of order 00.07.

An interesting question is how much image quality is sac-rificed in order to achieve the distortion correction. The in-stantaneous image quality of the 497 configuration is slightlyworse than that of a similar corrector optimized for zero dis-tortion. However, the difference is much less than the improve-ment in integrated image quality provided by the distortioncorrection. For example, reoptimizing the corrector for zerodistortion, thus allowing the lens shapes to change as well asthe separations, results in only a 25% improvement in the worst-case rms image spot diameters.

Can this design be extended to larger fields of view? Howwould the result compare with that of the great-circle camera?While we have not explicitly investigated this question, duringthe course of our work, an earlier design was made for a cor-

rector that has a 309 diameter field of view (in order to ac-commodate a pixel CCD) and that assumes a4096 # 40965 m f/1.8 primary mirror. The image quality at 497 is com-parable to that of the present design. Because of the smallerprimary mirror, a strict comparison with the present design isnot possible, but it does show that wider field designs arepossible.

In this context, it should be pointed out that even the great-circle camera is not entirely distortion free. The image dis-placements can be obtained from equations (5) and (11) bysetting and taking d to be the field angle X. This givesd 5 00

and . So while the great-2 2DX 5 Y tan d/2 . XY /2 DY 5 0circle camera is free from rate variation, there is a small amountof star-trail curvature. This results in an image smear that in-creases in proportion to the cube of the field diameter and thatbecomes substantial for field sizes on the order of 17 (the imagedisplacement is 00.548 for a field).C C1 # 1

While the corrector design presented here was specificallydeveloped for a zenith-pointing telescope, the same optical de-sign could also be used with a conventional telescope. Becausethe curvature effects depend only on declination, compensationcould be introduced for any field center by means of mechanicalactuators that would adjust the decenters and tilts of the in-dividual elements in the corrector. The corrector would be re-configured for each pointing of the telescope and then heldfixed during the integration. This would be the case even ifthe scanning is done at a nonsidereal rate, so long as the dec-lination of the field center remains constant during theexposure.

We thank the referee, Stephen Shectman, for helpful com-ments. Research support for P. H. is provided by grants fromthe Natural Sciences and Engineering Research Council of Can-ada (NSERC). The LZT is funded by NSERC CollaborativeProject grant CPG0163307 and the University of BritishColumbia.

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Richardson, E. H., & Morbey, C. L. 1987, in Santa Cruz Workshopon Instrumentation for Ground-based Optical Astronomy, ed. L. B.Robinson (Berlin: Springer), 720

Schneider, D. P., Schmidt, M., & Gunn, J. E. 1994, AJ, 107, 1245Shectman, S. A., Schechter, P. L., Oemler, A. A., Tucker, D., Kirshner,

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a curvature‐compensated corrector for drift‐scan observations

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