Gain Ripple in Small Offset Gregorian Antennas

Dirk I.L. de Villiers

Department of Electrical and Electronic Engineering

Stellenbosch University

Stellenbosch, 7600, South Africa

Email: ddv@sun.ac.za

Abstract—This paper investigates the effect of sub-reflectordiffraction on gain ripple performance in small clear apertureoffset Gregorian reflector antennas. A spherical mode expansionof an analytically specified feed pattern is used to illuminatea given sub-reflector, and method of moment simulations areperformed to find the resulting radiation pattern. It is foundthat the diffracted rays from the sub-reflector edge can interferewith the main beam to produce a sinusoidal form gain ripple forthe dual reflector system. The relative amplitude of the ripple isfrequency dependent, and can be predicted from the sub-reflectordiffraction results. The ripple frequency can be found from theantenna geometry. The effect on the ripple of changing some ofthe important parameters is investigated, and it is shown thataccurate near field simulations of the feed pattern illuminatingthe sub-reflector is required to produce accurate results.

I. INTRODUCTION

In wide band radio telescope systems, such as the South

African Square Kilometer Array (SKA) [1] pathfinder project

MeerKAT [2], special care must be taken in the design of the

reflector and feed systems to limit gain ripple over frequency.

This effect is known as chromatic aberration which, in optics,

equates to differential diffraction of light of different wave-

lengths. Chromatic aberration becomes a limiting factor when

doing radio frequency spectroscopy of radio sources with radio

telescopes exhibiting gain ripple over frequency. Such gain

ripple is usually caused by feed and receiver mismatches, as

well as effects relating to the feed and reflector configuration.

A thorough treatment of some of these effects is given in

[3], where predictions of gain ripple due to blocking and feed

scattering in prime focus as well as symmetrical secondary

focus fed radio telescopes are presented. Typical techniques for

the reduction of gain ripple in symmetrical reflector systems

include apex ’splash’ plates or cones and the use of support

struts with sharp edges toward the reflector to reduce the

backscattered energy into the feed.

When a clear aperture offset reflector configuration, such

as an offset Gregorian, is used the gain ripple caused by the

multipath backscattered energy from the reflector and struts

can be theoretically completely eliminated. However, since

the sub-reflector in such offset designs is normally relatively

small, the diffraction from the sub-reflector edge can still cause

a significant gain ripple in the reflector system. This paper

investigates this diffraction gain ripple in clear aperture offset

Gregorian reflector systems through full wave simulations of

the sub-reflector as well as the full dual reflector system. It

is shown that increasing the projected spacing between the

reflectors will not lead to a reduced ripple amplitude. Care

must be taken when doing simulations of the feed and reflector

system to include the feed near field effects, as it is shown that

by neglecting these erroneous sub-reflector diffraction results

are obtained which can significantly affect the gain ripple

predictions.

II. GAIN RIPPLE DUE TO SUB-REFLECTOR DIFFRACTION

A symmetry plane cut of the general configuration of a

clear aperture offset Gregorian reflector system is shown in

Fig. 1. Also shown in the figure are two geometric optic (GO)

and diffracted ray paths. The path length from the feed to the

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Fig. 1. Typical clear aperture offset Gregorian reflector geometry showingsome important dimensions as well as the geometric optics and main beamdiffracted ray paths at the edges of the sub-reflector.

aperture plane of the GO ray is GOn = ρSn + ρMn + ρAn,

and that of the diffracted ray is Dn = ρSn + ρDn. The path

length difference between a GO ray and a diffracted ray in the

broadside direction (θ = 0◦) is therefore

Δn = GOn −Dn = ρMn + ρAn − ρDn, (1)

with n representing any point on the edge of the GO path,

and n = 1, 2 as indicated in Fig. 1. It is therefore expected

that the interference between the GO and diffracted rays in

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the broadside direction will give rise to a gain ripple with

the frequency dependent on the average path length difference

Δn. For most configurations the path length differences Δ1

and Δ2 are within a few percent and the average path length

can be approximated as Δn ≈ (Δ1 + Δ2)/2. The gain ripple

frequency can be approximately predicted as

fr ≈ c/Δn, (2)

with c the speed of light.

The relative amplitude of the ripple depends on the actual

diffraction pattern of the feed and sub-reflector combination

which is strongly frequency dependent. Full wave Method

of Moment (MoM) simulations of the sub-reflector were

performed in FEKO (using the Multilevel Fast Multipole

Method), with the feed given by a cosn θ radiation pattern

specified by its spherical mode expansion [4] around the

secondary focus to accurately simulate the near field behavior

of the pattern. The effect of ignoring the near field pattern of

the feed is discussed later in the paper. The radiation pattern

results of a DS = 10λ (λ indicating wavelength) sub-reflector

with the feed pattern linearly polarized in the symmetry plane

of the reflector (x-direction in Fig. 1) and with a -10 dB edge

taper is shown in Fig. 2. The θ = 0◦ angle corresponds to the

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Fig. 2. Symmetry plane radiation pattern of a 10λ sub-reflector illuminatedby a -10 dB edge taper pattern.

z-axis direction of the dual reflector system shown in Fig. 1.

The expected diffraction pattern is clearly visible in Fig. 2 in

the area behind the sub-reflector. A plot showing the frequency

dependence of the directivity in the θ = 0◦ direction of the

same sub-reflector and feed combination is shown in Fig. 3.

The frequency dependence of the z-axis sub-reflector

diffraction pattern directivity suggests that the amplitude of

the gain ripple of the dual reflector system will also be

frequency dependent. This is confirmed in Fig. 4, where the

total efficiency [5] of the dual reflector system (DS = 10λ,

dg = 0, D = 27λ and DM = 30λ) as well as the θ = 0◦

directivity of only the sub-reflector and feed combination are

shown on the same graph. Results are shown for different edge

illuminations, showing that a softer edge illumination will have

less sub-reflector diffraction and therefore a correspondingly

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Fig. 3. Directivity of a sub-reflector illuminated by a -10 dB edge taperpattern in the θ = 0◦ direction. The longest dimension of the sub-reflector is10λ at the lowest frequency.

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-10 dB edge taper

-15 dB edge taper

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Fig. 4. Offset Gregorian system illuminated by -10 dB and -15 dB sub-reflector edge taper patterns. The total efficiency of the system and thebroadside directivity of the sub-reflector pattern is shown to illustrate thedependence of the efficiency ripple on the sub-reflector diffraction patternlevel. The longest dimension of the sub-reflector is 10λ at the lowestfrequency, and the path length difference between the broadside GO anddiffracted rays is 38.1λ.

smaller relative ripple amplitude. The total directivity of the

-15 dB edge illuminated system is, however, lower than that

of the -10 dB case since the reflector is no longer optimally

illuminated. Fig. 4 clearly shows the correspondence between

the ripple amplitude and the sub-reflector diffraction pattern

amplitude in the θ = 0◦ direction. The normalized path length

difference for this configuration is Δn ≈ 38.1188λ which,

using (2), corresponds to a normalized ripple frequency of

0.0262, which is very close to the ripple frequency found with

the full wave simulation.

It should be noted here that all the results shown in this

paper are for the symmetry plane polarization case. Similar

results are also found if the feed is polarized in the plane of

asymmetry, but with smaller sub-reflector diffraction resulting

in a smaller relative ripple.

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III. EFFECT OF GAP BETWEEN REFLECTORS

The dimension dg in Fig. 1 is often a free parameter

when designing offset Gregorian/Cassegrain reflector systems.

It represents the projected space between the main- and sub-

reflectors, and is typically greater than or equal to zero in a

clear aperture configuration. In such a configuration changing

dg has no effect on the theoretical GO efficiency of the

antenna since there is no aperture blockage. A larger dg can,

however, have negative mechanical effects on the antenna

structure, since, as can be seen in Fig. 5, a larger dg requires

a larger separation between the main- and sub-reflectors and

correspondingly stiffer arms to support the feed and sub-

reflector assemblies. Close investigation of Fig. 5 also shows

Fig. 5. Three offset Gregorian geometries showing the effect of changingdg . All other dimensions are kept constant (DM = 30λ and DS = 10λ)with dg = [4, 2, 0]λ corresponding, respectively, to the outer, middle andinner geometries.

that changing dg can have a notable effect on the electromag-

netic performance of the antenna system. If all the parameters

are kept constant (the Mizugutch condition is satisfied in all

configurations in this paper [6]) and only dg is varied, the

effect on the sub-reflector in Fig. 5 can be approximated as

a small rotation (change in feed angle γ) of the sub-reflector.

This rotation corresponds to a simple shift in the sub-reflector

diffraction pattern with θ, as is shown in Fig. 6 where the

directivities of the sub-reflector feed combination for different

values of dg are plotted on the same graph. The change in the

angle γ between dg = 0 and dg = 2λ is 1.293◦, and between

dg = 2λ and dg = 4λ is 1.170◦. This is very close to the shift

in diffraction patterns observed in Fig. 6, and therefore gives

a good initial estimate of the effect of changing dg .

Fig. 7 shows the comparison between the directivities of two

cases with different dg values. Note how the ripple amplitude

is similar for both cases, but the position of the peaks and

dips change. If a system is required to have an especially

flat frequency response in a certain narrow important band,

the parameter dg can be adjusted in the design to give a

sub-reflector diffraction pattern which has a dip around the

required frequency. In such a design only the sub-reflector

needs to be analyzed at the required frequency to find the

diffraction pattern with dg = 0, and then dg can be adjusted to

yield a dip in the diffraction pattern in the θ = 0◦ direction by

using the feed angle rotation approximation mentioned earlier.

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dg = 2λ

dg = 4λ

Fig. 6. Sub-reflector diffraction pattern for varying dg . DS = 10λ and anedge illumination of -10 dB is used in all cases.

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dg = 0λ

dg = 4λ

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Fig. 7. Offset Gregorian systems with different dg values illuminatedby a -10 dB sub-reflector edge taper pattern. The total efficiency of thesystem and the broadside directivity of the sub-reflector pattern is shownto illustrate the dependence of the efficiency ripple on the sub-reflectordiffraction pattern level. The longest dimension of the sub-reflector is 10λ

at the lowest frequency. The path length difference between the broadsideGO and diffracted rays is approximately 40λ for dg = 0λ and approximately48λ for dg = 4λ.

IV. IMPORTANCE OF NEARFIELD FEED PATTERNS IN THE

ANALYSIS

As previously stated, all the simulations were performed

with a spherical mode expansion of the feed pattern as source.

This allows the near field effects of the feed to be included

in the model, since the feed is often required to be placed

relatively close to the sub-reflector. Fig. 8 shows a comparison

between the spherical mode source and a radiation pattern

point source in the directivity of the illuminated sub-reflector.

Note that in the reflected part of the pattern the results agree

closely, but in the diffracted part of the field, behind the

sub-reflector, the results differ substantially. Therefore, care

should be taken to include the feed near field effects in the

simulations of the feed sub-reflector combination to accurately

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Radiation Pattern Source

Fig. 8. Comparison of the spherical mode and radiation pattern sourcedirectivities when used to illuminate a sub-reflector of size 10λ.

model and predict the expected gain ripple of the full system.

A comparison of the full simulation results using spherical

mode and radiation pattern sources is shown in Fig. 9, where

the difference in gain ripple responses is obvious.

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Spherical Mode Source

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Fig. 9. Offset Gregorian system illuminated by a -10 dB sub-reflectoredge taper spherical mode and radiation pattern sources. The total efficiencyof the system and the broadside directivity of the sub-reflector pattern isshown to illustrate the dependence of the efficiency ripple on the sub-reflectordiffraction pattern level. The longest dimension of the sub-reflector is 10λ atthe lowest frequency. The path length difference between the broadside GOand diffracted rays is approximately 40λ, with dg = 0λ, D = 27λ andDM = 30λ.

V. CONCLUSION

Chromatic aberration in radio telescopes can be a limiting

factor when doing radio frequency spectroscopy. A reduction

in gain ripple of the reflector antenna system is required to

alleviate this problem, and clear aperture offset Gregorian an-

tennas can provide this due to the elimination of GO multipath

interference. Sub-reflector diffraction, however, can still cause

significant gain ripple in electrically smaller antennas, and this

effect was investigated in this paper. It was shown that softer

edge illumination will reduce this ripple, but possibly at a

cost of efficiency loss. This might be alleviated by shaping

the reflectors to improve the main dish illumination while still

maintaining soft sub-reflector edge illumination. The effect of

changing the projected spacing between the reflectors can be

simply approximated as a rotation of the sub-reflector, and

will not cause significant improvement of the relative ripple

amplitude. Instead, the frequency response of the ripple will

shift, and this result can possibly be used to flatted the gain

response in certain very sensitive frequency bands. Also, in

order to accurately predict gain ripple, care must be taken

to accurately simulate or measure the sub-reflector and feed

combination response - including all the near field effects.

ACKNOWLEDGMENT

The author would like to thank the South African SKA

project and EMSS Antennas in Stellenbosch, South Africa for

financial support of this work.

REFERENCES

[1] P. E. Dewdney, P. J. Hall, R. T. Schilizzi, and T. J. L. W. Lazio, “TheSquare Kilometer Array,” Proceedings of the IEEE, vol. 97, no. 8, pp.1482 – 1496, Aug. 2009.

[2] J. L. Jonas, “MeerKAT - The South African array with composite dishesand wide-band single pixel feeds,” Proceedings of the IEEE, vol. 97,no. 8, pp. 1522–1530, Aug. 2009.

[3] D. Morris, “Chromatism in radio telescopes due to blocking and feedscattering,” Astron. Astrophysics, vol. 67, pp. 221–228, 1978.

[4] J. E. Hansen, Ed., Spherical Near-Field Measurements, ser. IEE Electro-magnetic Wave Series 26. London, UK: Peter Peregrinus Ltd., 1988.

[5] P.-S. Kildal, “Factorization of the feed efficiency of paraboloids andcassegrain antennas,” IEEE Trans. Antennas and Propag., vol. AP-33,no. 8, pp. 903–908, Aug. 1985.

[6] Y. Mizugutch, M. Akagawa, and H. Yokoi, “Offset dual reflector anten-nas,” in IEEE Antennas Propag. Soc. Symp. Dig., Amherst, MA, Oct.1976, pp. 2–5.

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