Bill Shillue
National Radio Astronomy Observatory
Tucson,AZ 85721
The maximimum distance off-axis that a feed can be placed in a symmetrical Cassegrain antenna without significant degradation in efficiency is investigated. The results place limits on the size and position of receivers at the secondary focus. For example, for the proposed MMA geometry, if we use a one percent reduction in gain as a figure-of-merit, then the maximum off-axis distance is about 165 mm or 6.5 in. for a 350 micron wavelength. For longer wavelengths the allowable distance is much greater.
Early discussions of MMA receiver geometries have suggested the use of one or two large dewars at the secondary focus with feeds clustered around the central axis of the antenna. Aberrations are thereby introduced which degrade antenna performance, and this memo attempts to quantify the degradation. One of the receiver geometries that has been suggested is used as an example.
The loss of antenna efficiency due to lateral feed displacement for a symmetric cassegrain antenna, such as is planned for the MMA, is investigated. The assumption is made that the feed is repointed towards the center of the subreflector so that the field amplitude aperture illumination is symmetric to first-order. This is a necessary condition for many applications where the offset of the feed is so large that the illumination of the subreflector would be vignetted were the feed not repointed.
Classical optical theory is used and the system is treated as an equivalent
prime-focus telescope with a focal length of 
, where m is the
Cassegrain magnification parameter and F is the focal length of the
primary mirror. The expression for the Seidel aberrations, which are
classical optical aberration
functions that give wavelength dependent phase-shifts (and which excludes
the linear phase-shift which leads to a pure beam squint) is [1]:
where the coefficients (C-F) are given by:
Eq.(1) is an expression of the path length errors developed on
the primary aperture due to the off-axis feed. The variable r is
radius on the primary, and 
where h is the off-axis
distance and 
is the equivalent focal length of the Cassegrain.
The variables 
is the distance from the primary to the
secondary focus. The variable d is the distance from the primary
focus to the vertex of the hyperbolic subreflector.
We are interested in a measure of the gain reduction. In general, quadratic and higher order terms in the phase distribution widen the antenna beamwidth. Severe phase errors also bring up the sidelobe level, including the well-known coma lobe. Since we are interested in limiting the gain reduction to one percent or less, the effect should only be a slight beam-broadening.
It is noted that the loss in gain is proportional to the weighted
squared sum of phase deviations. There are two ways to reduce the
aberrations on the telescope: repointing the antenna to the beam peak,
and refocussing the antenna, usually by moving the subreflector.
Re-pointing takes out aberration terms proportional to 
and refocussing takes out terms in 
. This leaves the astigmatism
and coma terms as the most significant.
The astigmatism term, when expressed as
, contains a net focus error; removing this focus
error minimizes the residual wavefront RMS and results in
an astigmatic error of the form 
.
The coma term expressed in the form 
contains a net
telescope pointing error. The magnitude of the best fit pointing
error depends on the illumination taper. Integrating as 
removes the pointing error for a uniform
aperture illumination, and integrating as 
removes the pointing error for a 12-dB tapered parabolic illumination.
There is another detrimental effect of moving the feed off-axis.
The feed illuminates the primary aperture in an asymmetrical way,
which increases the spillover on one side of the primary.
Lugten and Welch discussed this and estimate that it would add
to the system temperature [3]. Alternatively, they
say, a skirt could be added to the primary to direct the spillover
onto the sky.
The aberration function can be used in the expression for the gain of a circular aperture:
Here the function 
expresses the aperture illumination
function, with 
the normalized radial parameter.
In our calculations we used:
This is a parabolic illumination function with a 12-dB edge taper.
Eq (6) can be integrated directly to get the gain degradation due to aperture phase errors. Alternatively, Ruze [4] gives a simplified expression for Eq (6) by expanding the exponential, which is valid for phase errors only on the order of a radian or less. In practice, we are looking for the lateral offset which will give us a one percent gain degradation, which invariably will meet the criterion of a radian or less. The expanded expression is given:
where:
Eq (8) was calculated for the MMA geometry proposed by Lugten and Welch [3]. The relevant parameters used were:
| Antenna Parameters | |
| D=8000 mm | Diameter of Primary Mirror |
| d=2777.27 mm | Primary Vertex to Secondary Vertex |
| F=3040 mm | Primary Focal Length |
 mm | Primary to Secondary Focal Distance |
| m=16.35 | Cassegrain Magnification |
 mm | Effective focal length of Cassegrain |
Classical aberration theory is based on paraxial optics, in which it
is assumed that all geometric rays are nearly parallel to the optical
axis. The theory breaks down when the off-axis angles are too large.
Realizing this, we checked the validity of the aberration theory on
the proposed MMA geometry, in which receivers will be 10.5 arcsec off
of the axis of the equivalent paraboloid (resulting in a beam scan
of about 60 beamwidths on the sky at 0.35 mm wavelengths). To check
the aberration theory, the method of geometrical ray-tracing was used,
in which the path length from the feed to the aperture in the
Cassegrain system was explicitly calculated. After subtracting the
tilt term in , a curve fit was done to the phase
expression of Eq.[1] with the coefficients as unknowns, and the
results were compared to the coefficients defined by Eq. [2], [3], and [5].
The results were as follows:
| Coefficient | Classical Theory | Ray Tracing | Percent Difference |
| C (Astigmatism) | -1.06 e-4 | -1.11 e-4 | 4.4 |
| D (Curvature) | -1.74 e-3 | -1.67 e-3 | 4.0 |
| F (Coma) | -1.01 e-10 | -1.00 e-10 | 1.0 |
The proposed MMA geometry of Lugten, Welch [3] calls for the shortest wavelength receiver (0.35 mm) to be 6 inches off-axis. Fig.1 shows the loss in percent for the receiver at that frequency versus off-axis distance. The curve that shows coma and astigmatism together is the relevant result, and the curves for coma and astigmatism alone are incuded as well. The loss at an off-axis distance of 6 in. (152.4 mm), and at a wavelength of 0.35 mm, which is the highest loss condition for the proposed MMA receiver layout, is 0.75 percent. This is very small, especially considering that at the shorter wavelengths the phase efficiency degradation will be dominated by the loss of efficiency due to the surface accuracy of the dish. It should also be noted that the plot is only showing the shortest wavelength, and that the loss at longer wavelengths is negligible.
The calculation of the integrals shown in Eq [9] and Eq [10] were done in Mathematica.
Discussions with Peter Napier alerted me to the fact that that the classical aberration terms might not be valid for our geometry.
John Lugten corrected my first calculation by pointing out that using the aberration terms from Ref. [1] overstates the loss, and suggested the corrected form that is dicussed in Section 3.
Discussions with James Lamb led me to find an error in my calculation. James had done calculations in parallel to these, with a different formulation but nearly identical result.
Jingquan Cheng made some helpful suggestions concerning the ray-trace program.
REFERENCES
Figure 1: Loss of Gain for Proposed MMA Geometry
due to Off-Axis Feed at 350 Micron Wavelength