Ieee transactions on antennas and propagation, vol. 46, No. 12, December 1998
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- Tilting Aplanat RF Telescope
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 46, NO. 12, DECEMBER 1998
achieved this limit. In essence, the continuously inhomogeneous
medium is most similar to a boundary diffraction problem for which
the smooth boundary is permitted to evolve toward an edge, for
example. Finally, we have avoided the issue of turning points since
our asymptotic development is not valid in this situation and the
turning point phenomenon is well-understood , .
We are indebted to the late James R. Wait for his important
comments on this work.
 P. M. Morse and H. Feshbach, Methods of Theoretical Physics—Part II.
New York: McGraw-Hill, 1953, pp. 1092–1106.
 J. R. Wait, Electromagnetic Waves in Stratefied Media.
IEEE Press, 1996, pp. 85–105.
 A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scat-
Englewood Cliffs, NJ: Prentice-Hall, 1991, pp. 62–68.
 C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for
New York: McGraw-Hill, 1978, pp. 14–20.
 D. Bouche, f. Molinet, and R. Mittra, Asymptotic Methods in Electro-
Berlin, Germany: Springer-Verlag, 1994, pp. 91–120.
Tilting Aplanat RF Telescope
R. L. Lucke
Abstract—Beam steering by tilting the secondary is common in two-
mirror RF telescopes, but it is not generally recognized that optical quality
need not suffer as the beam is steered: a tilting aplanat can maintain
diffraction-limited performance over fields of a few degrees.
Index Terms—Beam steering, mirror, radio astronomy.
Ordinarily, RF remote sensors must cover their fields of regard
by moving the entire telescope assembly. But for small fields and
the proper choice of mirror shapes beam steering without loss
of resolution can be accomplished in a two-mirror telescope by
tilting only the secondary mirror. Weight savings can be realized
in mechanical structures and in drive motors and the platform will
be subjected to lower reaction torques—features that are especially
desirable for space-borne instruments. These advantages must be
balanced against the disadvantage of the design: it requires a larger
primary mirror to accommodate beam walk as the secondary is tilted.
Manuscript received March 30, 1998; revised August 3, 1998.
The author is with the Naval Research Laboratory, Code 7227, Washington,
DC 20375 USA.
Publisher Item Identifier S 0018-926X(98)09721-X.
Telescope schematic. The secondary is tilted by angle
the chief ray, which strikes the primary a distance
x from its vertex, through
2 and achieve observation angle with respect to the z axis. h is the
off-axis distance in the focal plane of the primary of the point at which rays
from field angle
are directed. Other symbols are defined in the text.
The tilting aplanat prescription for two-mirror telescopes using
conic-section mirror shapes was given by Bottema and Woodruff 
who applied it to optical systems with modest
f numbers and much
smaller tilt angles than are contemplated here. This paper will apply
it to RF telescopes, which, in order to prevent obscuration of the
primary by the secondary, generally use reflecting surfaces that are
off-axis segments of larger figures with radical
f numbers. It will
be seen that RF tilting aplanats work well if a small departure from
conic shape is applied to one of the mirrors. Depending on the size
and shape of the segments used, further refinements in the optical
prescription such as nonaxisymmetric shapes can be made to improve
performance, but will not be considered here.
Notation will follow that used by Schroeder . The telescope
is shown schematically in Fig. 1, adapted from [2, Fig. 2.7]. The
primary mirror has a vertex radius of curvature
and a focal length
=2 (the sign convention is discussed by [2, pp. 8;
11–12]; since the primary is convex, its focal length is always taken
as positive). The secondary’s vertex radius of curvature is
nominal length of the telescope
d is the distance between the vertices
of the primary and secondary mirrors. The design parameters defined
by [2, pp. 16–17] are
m; ; k; ; where m = (d + f
0 d) is
the magnification of the secondary,
; k = (f
is the ray-height ratio (the vertical distance from the vertex of the
secondary to the point at which the ray strikes it, divided by the
same distance for the primary), and
determines the back focal
length (BFL), measured from the vertex of the primary as a fraction
BFL = f
. Note that
k = (1 + )=(m + 1). As shown in
Fig. 1, by Schroeder’s sign convention,
h are negative and
d, x, m, , k, are positive. A convex secondary is assumed
because it gives the most compact design for a desired final focal
f number, but the formulas given below apply without
change if the secondary is concave (for a concave secondary,
m, , k are negative).
The chief ray is that ray which, after reflection from the secondary,
lies along the
z axis, i.e., along the centerline of the feedhorn,
U.S. Government work not protected by U.S. copyright.
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 46, NO. 12, DECEMBER 1998
which is located at the final focal point. The secondary is tilted
to achieve observation angle , so the chief ray, before
reflection from the secondary, makes an angle
2 with the z axis.
As shown in Fig. 1,
and , incidence angles on the primary and
secondary, respectively, are negative. The chief ray, before it strikes
the secondary, is directed toward a focus in the focal plane of the
primary mirror, a distance
h off axis. h is given to first order by
h = f
h = 2 (f
0 d). Therefore, = =(2k), where k
is the telescope parameter introduced above. The height at which the
chief ray strikes the primary is
x = 02 d = 0f
(m 0 )=(1 + ).
This means that to accommodate a scan range of
6, the vertical
dimension of the primary must be oversized by
2x compared to what
it would be if only on-axis use were required.
In the most basic form of the two-mirror telescope, the classical
Cassegrain, the shape of the primary mirror is paraboloidal—that
of the secondary hyperboloidal. This gives aberration-free imaging
of on-axis sources, but the images of off-axis sources suffer from
substantial coma. In the Ritchey–Cretien telescope design, the conic
constants of the two mirrors are chosen to eliminate third-order spher-
ical aberration and coma. This property—aplanatism—results in a
fairly wide field with minimal aberrations but, for the Ritchey–Cretien
design, the aplanatic condition is not fulfilled if the secondary is
tilted. The primary and secondary mirror conic shapes that maintain
the aplanatic condition at the focal point when the secondary is tilted
were given by Bottema and Woodruff . They can also be derived
using the formalism of Schroeder : inspection of [2, eq. (6.3.2)]
shows that tilting the secondary about any axis perpendicular to and
intersecting the optical axis merely adds another term to the coma
coefficient of a two-mirror telescope, which is given in [2, Table
6.4]. The result can be set equal to zero and solved for the conic
constants of the mirrors. It was assumed by Bottema and Woodruff
 and will be assumed here that the secondary is tilted about its
, the conic constants of the primary and
secondary, respectively, are
(m 0 )
= 0 m + 1
m 0 1
0 m(m + 1)(m
(m 0 )(m 0 1)
Equation (1) can be compared to [2, eqs. (6.2.4), (6.2.5)] for
the Ritchey–Cretien prescription. The difference is that the quantity
+ 1 appears in the above expressions where the quantity two
appears in the Ritchey–Cretien prescription. Since
m is almost never
less than two and may be four or greater,
larger than two, which means that the tilting aplanat prescription
is even farther from the paraboloidal primary condition (that is,
is the Ritchey–Cretien prescription, hence, should be expected to
have higher fifth-order spherical aberration. Test computations with
the BEAM4  optical ray-tracing code confirm this: fifth-order
spherical aberration can easily be an order of magnitude larger than
for the Ritchey–Cretien prescription. The most effective means of
solving this problem is to allow the mirror shapes to have nonconic
components: fifth-order spherical aberration can be eliminated with
a fifth-order deformation parameter (a shape term proportional to the
sixth power of the mirror’s radius) added to either mirror.
A few design examples were evaluated with BEAM4  to
verify that the tilting aplanat design gives diffraction-limited perfor-
mance over few-degree observation angles. With third- and fifth-order
spherical aberration and third-order coma eliminated, the dominant
aberrations are astigmatism and field curvature, the effects of which
depend on the observational problem under consideration. The prob-
lem that led to this work concerned studies of the atmosphere at the
earth’s limb as seen from space, where it is customary to require high
resolution only in the vertical direction, which means that astigmatism
is unimportant and only tangential field curvature remains. For this
case (even at 600 Ghz) it is not difficult to achieve diffraction-limited
performance over a
field, adequate to cover the atmosphere in
a compact telescope design having both nominal length
d and the
primary mirror’s diameter equal to 1 m.
The effect of tangential field curvature is (for most mirror confi-
grations) to move the focal point toward the secondary. If tilt angles
large enough for this effect to be important are needed, it can be
compensated by translating the secondary toward the feedhorn. For
the 1-D application considered here, a simple mechanical design can
effect both the tilting and translating motions with a single actuator.
The disadvantage of reducing the distance between the secondary and
the feedhorn is, of course, that it changes the amount of spillover of
 M. Bottema and R. A. Woodruff, “Third-order aberrations in Cassegrain-
type telescopes and coma correction in servo-stabilized images,” Appl.
Opt., vol. 10, p. 300, 1971.
 D. J. Schroeder, Astronomical Optics.
New York: Academic, 198, chs.
 M. Lampton, BEAM4 Optical Raytracer, Stellar Software, Berkeley,
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