Before proceeding to example QRFH designs of the next chapter, a method for calculatingT E and T M mode coefficients required at a circular aperture to achieve given radiation patterns is presented.
This procedure is based on [29] and is used in the next chapter to evaluate aperture mode content of the quad-ridged horns.
The approach relies on the fact that far-field patterns of all hollow circular waveguide modes are expressed in closed form [29, 51]. Neglecting reflections at the aperture, the far-field patterns of transverse-electric modeT Emn are
Eθ,mnT E =−amnjmmJm(X0mn)Jm(kasinθ)
kasinθ cosmφΨ (θ) (3.10)
Eφ,mnT E =amnjmXmn02 Jm(X0mn) J0m(kasinθ)
X0mn2 −(kasinθ)2sinmφΨ (θ) (3.11)
and those of the transverse-magnetic modeT Mmn are
Eθ,mnT M =bmnjmXmnJ0m(Xmn)kasinθJm(kasinθ)
Xmn2 −(kasinθ)2 cosmφΨ (θ) (3.12) where
Ψ (θ) = ka2
2 (1 + cosθ)e−jkR R k= 2π
λ
λ= Wavelength
a= Waveguide aperture radius R= Far-field radius
Xmn=nth zero ofJm(x) Xmn0 =nth zero ofJm0 (x),
m∈[0,∞], n∈[1,∞] ;amnandbmnare complex amplitude coefficients of theT EandT M modes, respectively. These equations underline an important feature. Specifically, φ-directed far-fields are determined only by theT Emodes, and the shape of theθ-directed far-fields of theT Emodes depend onmbut not onn.This implies that, for a givenm,theφ-component of the radiation patterns can be synthesized entirely fromT E modes, and then the θ-component may be synthesized from T M modes without affecting the former.
Consequently, an arbitrary radiation patternf(θ, φ) can be written as a sum of patterns of each mode, i.e.,
f(u, φ) = 1 + q
1− kau
2
2
M,N
X
m=0n=1
Amn
mJm(u) Xmn02 u −Bmn
uJm(u) u2−Xmn2
cosmφθˆ
+
Amn Jm0 (u) u2−Xmn02
sinmφφˆ
(3.13) where
Amn=−amnjmkJm(X0mn)Xmn02 a2
Bmn=bmnjmkJ0m(Xmn)Xmna2 (3.14) u=kasinθ
After some algebra and using orthogonality of sine and cosine, the final expressions for the complex
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized Aperture E−field Magnitude
Normalized Aperture Radius Increasing frequency from 1 to 6 f
lo
(a)
0 30 60 90 120 150 180
−30
−25
−20
−15
−10
−5 0
θ [deg]
Relative Power [dB]
Desired Gaussian cosq, q = 3.322
(b)
Figure 3.7: (a) Aperture distributions as a function of frequency and (b) the resultant far-field patterns (black) in addition to Gaussian (red dashed) and cosq (green dotted) radiation patterns.
10 dB beamwidth is 90 degrees.
mode coefficients are found to be [29]
Apq= 2
1 + r
1−X0 pq
ka
2
2Xpq0 πJp00 Xpq0
2π
Z
0
fφ Xpq0 , φ
sinpφdφ. (3.15)
Bpq= 2
1 + r
1−X
pq
ka
2
2 πJp+1(Xpq)
2π
Z
0
fθ(Xpq, φ) cospφdφ (3.16)
which demonstrates that theT EandT M coefficients are in fact independent, as alluded to earlier.
Before proceeding into calculation of mode coefficients of an “ideal” radiation pattern, it is worth- while to make a few observations. First, the summation in (3.13) is finite so long as the aperture is of finite extent and wavelength is greater than zero, because only modes above cutoff at the aperture can contribute to far-field radiation patterns, and for a given aperture size, number of such modes is always finite. Secondly, fits to the desired radiation pattern obtained through this approach are necessarily approximate, because the fitting is performed at a set of discrete points, namelyu=Xpq
for theθ-component andu=Xpq0 for theφ-component.
Another important observation from these equations is that the radiation patterns of individual modes reach their maxima near the pointsu=Xmn and u=Xmn0 for T Mmn andT Emn modes, respectively. This implies that for a fixed aperture size, adding more modes broadens the radi- ation pattern. Conversely, using more modes necessitates a larger aperture to maintain constant beamwidth [29]. However, phase errors in the horn aperture can significantly degrade performance
2 4 6 8 10 12 0
10 20 30 40 50 60 70 80 90 100
TEmn Coefficients
Frequency [GHz]
Percent of Total Power
1, 1 1, 2 1, 3 1, 4 1, 5 1, 6 1, 7
(a)
2 4 6 8 10 12
0 10 20 30 40 50 60 70 80 90 100
TMmn Coefficients
Frequency [GHz]
Percent of Total Power
1, 1 1, 2 1, 3 1, 4 1, 5 1, 6 1, 7
(b)
Figure 3.8: (a)T Eand (b)T M modes required at a circular aperture of radiusa= 0.6λloto achieve the desired far-field pattern of Figure 3.7 from 1 to 6flo.Mode coefficients are normalized such that total power in all modes at a given frequency sums to 1.
for large apertures. This is the reason it is more difficult to design octave-band corrugated horns with wide beamwidths. The same holds for quadruple-ridged horns as demonstrated in Chapter 4.
Equations (3.15) and (3.16) are now used to calculate the necessary modes at an aperture with radius 0.6λloto achieve circularly symmetric radiation pattern with 10 dB beamwidth of 90 degrees.
Figure 3.7(a) shows the aperture distributions required to achieve far-field patterns with 90 degree beamwidth over 6:1 frequency band which are calculated using the results in [27, 52]. The far-field pattern obtained from these distributions are plotted in part (b) of the same figure in black along with Gaussian and cosq patterns yielding the same beamwidth.
The mode coefficient computations are performed in MATLAB [53] and the results are provided graphically in Figure 3.8. A subset of the results are also listed in Table 3.2. As expected, theT E11
mode is the dominant mode throughout the frequency range; however, its relative power decreases monotonically with frequency. The next three most significant modes areT M11, T E12, andT M12. This observation, as well as the relative power content in each mode, is in good agreement with the power distribution in an optimum four-mode horn [29].
The requirement of circular symmetry in the radiation patterns necessitates far-field patterns exhibiting the following azimuthal dependence1
f(u, φ)∝Vθ(u) cosφθˆ+Vφ(u) sinφφ.ˆ (3.17) As observed from (3.13), such patterns are attained with onlym= 1 modes at the aperture. The results of Figure 3.8, where only m= 1 modes are present, confirm this point. Lack of even-order
1This is the “optimum” far-field pattern of Ludwig [29].
Mode Frequency
1flo 3flo 4.5flo 6flo
T E11 1.0000 1.0000 1.0000 1.0000 T M11 0.0878 0.6027 0.6580 0.6789 T E12 0 0.4996 0.6079 0.6517 T M12 0 0.3605 0.5190 0.5897 T E13 0 0.2556 0.4480 0.5440 T M13 0 0.1595 0.3645 0.4828 T E14 0 0.0949 0.2941 0.4289
T M14 0 0 0.2248 0.3687
T E15 0 0 0.1693 0.3155
T M15 0 0 0.1209 0.2619
T E16 0 0 0 0.2159
T M16 0 0 0 0.1728
T E17 0 0 0 0.1372
T M17 0 0 0 0.1058
Table 3.2: Amplitudes of T E and T M modes, normalized to that ofT E11,required to realize the desired radiation pattern of Figure 3.7 with a circular aperture of radiusa= 0.6λlo.All modes are in phase withT E11.
modes, e.g. m= 0,2,4,· · ·, is due to perfect two-fold symmetry of the aperture field distribution.
Moreover, the curves in this figure reinforce the earlier point on the need to have more modes at the aperture to maintain large beamwidth and circular beam as aperture size grows in terms of wavelength.
We conclude this section by highlighting the inherent assumptions in the above development.
One of these assumptions, neglecting reflections at the aperture, has already been mentioned. More importantly, this development also approximates the guide wavelength at the aperture with the free- space wavelength. This is certainly not true for the example calculation presented in Figure 3.8 at the low end of the frequency band where, for instance, the aperture diameter is only 1.2λlo. On the other hand, it is a fairly good assumption in the upper half of the frequency band for the first three or four modes. It is possible to address both of these assumptions by modifying equations (3.10-3.12) as outlined in [51]. This, however, is not pursued, because the increase in accuracy gained from such an exercise is thought to be small, especially when the aperture mode calculation is applied to circular quadruple-ridged horns as is done in Chapter 4.
Chapter 4
Example Designs
The flexibility to design the QRFH for varying nominal 10 dB beamwidths through proper choice of ridge/wall profiles has made it a very attractive radio telescope feed antenna in the two years since the first QRFH was built. Its appeal is further increased because it is:
1. currently the only ultra-wideband feed that requires one single-ended 50 Ohm LNA per polar- ization;
2. the easiest to scale for operation in different frequency bands;
3. very stable and repeatable structurally. It can be accurately and cheaply machined from aluminum;
4. inherently low loss because of no dielectrics and relatively low current density with no thin metal surfaces.
As a result, there has been ample interest in quad-ridge horns designed at Caltech. Table 4.1 lists status of all the QRFH designs built to date in addition to those that are currently in discussion.
Figure 4.1 displays this list in a more visual format.
Majority of today’s cm-wave radio telescopes havef-numbers (f /D ratios) between 0.3 and 0.5 at primary, because lowf-numbers yield more compact telescope designs. On the other hand, there are some notable secondary focus antennas with highf /Dsuch as the Green Bank Telescope, the 65 meter antenna at Shanghai Astronomical Observatory, and NASA’s Deep Space Network antennas1. In this chapter, simulated—and, where available, measured—performance of five QRFH antennas are presented. The designs are selected to demonstrate the suitability of the quad-ridge horn in telescopes withf-numbers from 0.3 to 2.5. Measured system performance of a radio telescope with one of the quad-ridge designs presented herein is also provided while predicted system performance curves are included for the remaining designs. A square QRFH design is demonstrated which is very
1Deep Space Network antennas are technically not radio telescopes, but they are sometimes used for radio astron-
omy.
Table4.1:ListofQRFHantennasdeliveredtotelescopesaroundtheworldaswellasthosecurrentlyindiscussion.Thedesignsthatareinbold printarepresentedherein. TelescopePrimaryQRFH10dBFrequencyFeedDiam OperatedbyStatusSee Diam.[m]beamwidth[deg]range[GHz]&Length[cm]Section NASAGoddard12902-1218x16.4MITHaystackObservatoryOntelescope4.4 Westford18.31402.3-1414.3x11.9MITHaystackObservatoryOntelescope4.6 Effelsberg1001400.6-2.574.6x35MaxPlanckInstitute Ontelescope forRadioAstronomy Caltech61500.6-372.6x32CaltechOntelescope JapaneseVLBI13.21202.3-1420x13.4GeospatialInformation Shipped4.5 AuthorityofJapan CaltechOVRO271201-6and3-1812.4x5.7(3-18)CaltechOVROInfabrication HaystackVLBIN/A1202.3-1420x13.4MITHaystackObservatoryUndertest ShanghaiVLBIN/A1202.3-1420x13.4ShanghaiAstronomicalObservatoryUndertest DeepSpaceNetwork70300.5-3.5230x401NASA/JPLIndiscussion4.2 Shanghai65304-28?TBDShanghaiAstronomicalObservatoryIndiscussion AustralianVLBIN/A902.3-14TBDCSIROIndiscussion GAVRT34650.7-4.982x73.2LewisCenterforEducational Onhold4.3 Research/Caltech
Figure 4.1: A world map showing locations of the quad-ridge flared horns delivered to date (red place marks) and those that are in discussion phase (yellow place marks). The map was generated using Google Maps.
attractive for low-frequency operation due to relative ease of fabrication. Additionally,T E andT M mode coefficients at the aperture of the first three quad-ridge horns are presented.
The first section describes the antenna far-field measurement setup used for the QRFH measure- ments. The subsequent sections detail the five QRFH designs and are arranged in ascending order of nominal beamwidth. Target applications for these horns can be found in Table 4.1. Figure 4.2 com- pares profiles of the five horns which underscores the relationship between flare angle and nominal beamwidth. All results of this chapter, except measured and predicted system performance curves, are scaled such that the lowest frequency of operation, flo, of all quad-ridge horns is identical and relative bandwidths are used instead of absolute ones.