Encircled Power Study of Focal Plane Field for Estimating Focal Plane Array Size

Douglas B. Hayman* (1-3), Trevor S. Bird(2,3), Karu P. Esselle(3), and Peter Hall(4) (1) CSIRO ATNF, PO Box 76 Epping, NSW, Australia 1710

(2) CSIRO ICT Centre, PO Box 76 Epping, NSW, Australia 1710 (3) Dept. Electronics, ICS, Macquarie University, NSW, Australia 2109

(4) International SKA Project Office, c/o ASTRON, PO Box 2, 7990, AA Dwingeloo, The Netherlands

The relationship between the size of a focal plane array in a prime focus dish and achievable scan angle is studied in order to find a starting point for designing such systems. A physical optics model is used to obtain focal plane fields and the radii in the focal plane required to encircle 3 dB and 1 dB below the incident power is calculated for a range of dish diameters and focal length over dish diameter ratios (F/D). The study found the focal plane fields are largely independent of dish size when the scan angle is expressed in beamwidths and gives guidance for the optimal F/D to achieve maximum field of view for a given focal plane array size.

Introduction In the radio-astronomy community there is considerable attention being given to densely-sampled focal plane arrays (FPA) with each beam being formed from the complex weighted sum of a number of the elements. The applications include retrofitting large dish radio telescopes and design of new radio telescopes such as the Square Kilometre Array (SKA), where access to large areas of sky below 1.5 GHz is of paramount importance. The potential benefits of focal plane arrays include the ability to fully sample the field of view, correct for aberrations [1,2], and mitigate interference [3].

There have been many studies of focal region fields [4,5] and scanning relationships [6] but none have focused on the field-of-view achievable over a wide range of F/D and dish diameters. While semi-empirical beam-deviation factors [6,7] provide a good estimate for the peak field strength in the focal plane, the FPA must collect the energy falling beyond the peak value. An estimate of the overall size required is needed to calculate overall number of elements and blockage. Given a particular size of focal plane array, the number of elements is dependant on the spacing. The element spacing is largely independent of F/D because the wide beamwidth of most candidate elements, such as Vivaldi antennas, requires the suppression of grating lobes to reduce radiation from the ground.

Calculation of Focal Plane Fields A physical optics (PO) model was used to calculate the focal plane fields for an incident plane wave on a reflector. This was repeated for F/D ratios from of 0.3 to 2; with dish sizes from 30λ to 160λ and scan angles from 0 to 10 beamwidths (BW) (one beamwidth is taken as 70λ/D degrees). The Poynting vector (S) was calculated from the electric and magnetic fields at each point in the focal plane and the component normal to the focal plane (Sz) is used in this analysis, giving a measure of the power density available for reception by the focal plane array.

-2 0 2 4 6

-4

-2

0

2

4

λ Fig 1. D=30λ, scan 7° (3 BW),

F/D=0.3.

-2 0 2 4 6

-4

-2

0

2

4

λ Fig 2. D=70λ, scan 3° (3 BW),

F/D=0.3.

-5 0 5 10 15

-10

-5

0

5

10

λ Fig 3. D=30λ, scan 23.3° (10 BW),

F/D=0.3.

-5 0 5 10 15

-10

-5

0

5

10

λ Fig 4. D=70λ, scan 10° (10 BW),

F/D=0.3.

-5 0 5 10 15

-10

-5

0

5

10

λ Fig 5. D=70λ, scan 10° (10 BW), F/D=0.3. Tapered reflector field.

0 5 100

2

4

6

8

Scan in BW

FP

A R

adiu

s/λ

D = 30λD = 70λ

Fig 6. Focal plane radius required to

collect half the incident power for D=30λ and D=70λ.

Plots of Sz are shown in Figs. 1-5. The plots have contour lines at -1, -3, -6, -10, -20 and -30 dB relative to the peak in each plot.

The peak value of Sz was compared with the beam-deviation factor given in [7] and the correspondence is very good over the range of parameters studied, only breaking down for scan angles above about 7 BW.

For F/D ratios of 1 and higher, Sz very closely approaches the Airy pattern: /)(1 UUJ where λπρ cU Ψ= sin2 , ρ is the radius in the focal plane and Ψc is

the semi angle between the rim of the reflector and the focus. As F/D gets smaller, the sidelobes in the focal-plane pattern become higher and the central spot gets smaller. As the scan angle increases, the pattern becomes less symmetrical, with sidelobes increasing beyond the central spot.

By normalizing the scan angle to half power beamwidths for the reflector radiation pattern, the results are found to be largely independent of dish size up to about 7 BW (compare Figs. 1 and 2, and Figs. 3 and 4). The case of 0.3 F/D is shown as it has the most pronounced differences between different dish sizes and the most asymmetric field.

In transmission, the beam pattern will approach that of a uniformly illuminated aperture as the focal plane array size increases. This configuration will give maximum gain (super gain aside) but will result in significant spillover (and hence high noise temperature) and high sidelobes, whereas, particularly for astronomy, gain over system temperature (G/Tsys) is the key figure of merit. In order to model a more desirable pattern, the incident field on the reflector was weighted with a raised parabola taper, having an edge taper of -12 dB. This resulted in a more compact field distribution as expected (compare Figs. 4 and 5).

Encircled Power Calculations Sz was integrated over a disk in the focal plane to calculate the power available to an FPA of corresponding radius. With a number of assumptions, this power is available to the receiver. Firstly, Poynting’s theorem requires the surface of integration to be closed. Secondly, the focal plane array will distort the field at its edges resulting in array size calculations being accurate only to perhaps half a wavelength. Thirdly, the E and H fields in the focal plane can have a significant component in the axial direction (-5 dB for F/D of 0.3). A carefully designed horn feed may be able to closely match the fields in the focal plane (eg. [5]). However, with elements considered for wide band FPAs such as Vivaldis, there is likely to be a significant mismatch, reducing the efficiency.

The power in the focal plane was normalized by dividing it by the power intercepted by the reflector for an on-axis source. The radii required to intercept 3 dB and 1 dB below the available power were calculated. In Figs. 1-5, the solid and dotted heavy lines shown correspond to -3 dB and -1 dB encircled power radii respectively. In the case of Fig. 3, the -1 dB circle is beyond the limits of the calculated field.

By normalizing the scan angle to the beamwidth, the results are largely independent of dish size for scans up to about 7 BW and the -3 dB radii to at least 10 BW (see Fig. 6).

Fig. 7 shows the integration of encircled power as a function of radius. If 50% efficiency is used as the criterion, a low F/D is indicated yielding the smallest FPA. For efficiencies in the range of 70 to 80%, an F/D of around 0.4 results in the smallest focal plane array size. For very high efficiencies, high F/D ratios

capture more of the energy in spite of the central spot becoming larger.

Applying these results to the SKA field of view specification of 200 square degrees at 0.7 GHz and assuming 12 m diameter dishes with F/D=0.4, the -3 dB power circle indicates a minimum focal plane array radius of about 0.7 m.

Conclusion Further work involving optimizing the radiation pattern and G/Tsys from a focal plane array is required to show how these measures relate to achievable results. Nevertheless, the optimal FPA radius is likely to lie between the 3 dB and 1 dB radii for a tapered aperture as this takes into account the spillover and radiation pattern. Also, an F/D ratio of 0.4 is a good starting point for reflector design with close-packed FPAs.

References: [1] Loux, P. and Martin, R., "Efficient aberration correction with a transverse focal plane

array technique," IRE International Convention Record, pp. 125-131, 1964. [2] Galindo-Israel, V., Shung-Wu Lee, and Mittra, R., "Synthesis of a laterally displaced

cluster feed for a reflector antenna with application to multiple beams and contoured patterns", Antennas and Propagation, IEEE Transactions on, vol. 26, no. 2, pp. 220-228, 1978.

[3] Hansen, C., Warnick, K. F., and Jeffs, B. D., "Interference cancellation using an array feed design for radio telescopes," Antennas and Propagation Society Symposium, 2004. IEEE, pp. 539-542 Vol.1, 2004.

[4] Minnett H. C. and Thomas, B. M., “Fields in the image space of symmetrical focussing reflectors,” Proc. IEE, vol. 115, no. 10, pp. 1419-1430, Oct, 1968.

[5] Thomas, B., “Matching focal-region fields with hybrid modes,” Antennas and Propagation, IEEE Transactions on, vol. 18, no. 3, pp. 404-405, 1970.

[6] Lee, S.-W. and Rahmat-Samii, Y., “Simple formulas for designing an offset multibeam parabolic reflector,” Antennas and Propagation, IEEE Transactions on, vol. 29, no. 3, pp. 472-478, 1981.

[7] Lo, Y. T. and Lee, S. W. The antenna handbook, Van Nostrand Reinhold (ITP), 1993.

0 5 10 15 200

0.2

0.4

0.6

0.8

1

Radius in focal plane in wavelengths

Fra

ctio

nal P

ower

F/D 0.3 F/D 0.4 F/D 0.5 F/D 0.6

Fig 7. Encircled power curves for

D=70λ, scan 10° (10 BW) for a range of F/D values.

Encircled Power Study of Focal Field for Estimating

Focal Plane Array Size2005 IEEE AP-S INTERNATIONAL SYMPOSIUM

Washington DC

Douglas B. Hayman* (1-3), Trevor S. Bird(2,3), Karu P. Esselle(3), and Peter J. Hall(4)(1) CSIRO ATNF, PO Box 76 Epping, NSW, Australia 1710

(2) CSIRO ICT Centre, PO Box 76 Epping, NSW, Australia 1710(3) Dept. Electronics, ICS, Macquarie University, NSW, Australia 2109

(4) International SKA Project Office, c/o ASTRON, PO Box 2, 7990, AA Dwingeloo, The Netherlands

2

Outline

Context: Radio astronomy and the SKA“Focal Plane Array” definitionEncircled power formulationResults

Tapered Aperture

Application

Conclusion

3

Focal Plane Arrays for Radio Astronomy

The Square Kilometre Array (SKA) is a proposed radio telescope 100 times larger than the largest imaging interferometer todayFocal Plane Arrays (FPAs) are important to almost all proposals:

Provides wide field of view

Australia’s plans for demonstrators:Two 13.7m dishes with focal plane arrays to be completed in 200720 15m dishes with focal plane array ~2008 – Fast survey radio-telescope

Artist’s impression of SKA: Chris Fluke, Swinburne University

4

Focal Plane Array Definition

Multi-feed system Focal plane array (FPA) a.k.a. Cluster Feed

One beam per element Each beam from a combination (weighted sum) of elements.Elements can be used for more than one beam.

Focal plane undersampled Full sampling (hence sometimes Fully Sampled FPA)

Ideally we want to fully control the match to the field in the focal plane

Amplitudeand phase weighting

AD

LNA

∑

Beam 1

∑

Beam 2

∑

Beam 3

AD

LNA

AD

LNA

AD

LNA

Conceptual beamformer architecture

Overlappingfar fieldbeams

Focal plane array

5

Radio Astronomy Goals

“Wide field of view” SKA goal: ~200 square degrees at 700MHz (~8° half scan angle)

Bandwidth: >2:1High sensitivity (G/T)

Low noise – low spilloverFlexible – the wish list includes:

Contiguous beams for efficient mappingInterference mitigation (Null steering)Electronic fine pointingDish deformation correction…

Digital beamforming to provide flexibility

6

Aim of This Work

Estimate FPA size as a function F/D (focal length/dish diameter), field of view (FoV) and dish diameter D.

FPA size has is a strong driver of system costChoice of F/DStarting point for system design

Prime focus paraboloids only considered in this work

7

Encircled Power Calculations

Physical optics used to calculate E and H fields in the focal region

Reflector currents excited from a plane wave from a given directionPower flux density taken as axial component Poynting Vector

Integral of flux density taken over disk of radius ρ

LimitationsConjugate field match (CFM) assumed:

• FPA will only approximate this –some energy will be reflected

FPA will distort the field so ρ is not precise

zHESz ˆ)Re( *21 ⋅×=

Half scan angleθscan

Incident plane wave

PO currentson reflector

ρ

Focal planefield

Focal plane array

8

Field and Encircled Power Plots –Uniform Dish Aperture

Sz (dB)

Nor

mal

ized

Enc

ircle

d Po

wer

ρ (λ)

F/D 0.4, D70λ, θscan 4°Sz (dB)N

orm

aliz

ed E

ncirc

led

Pow

er

ρ (λ)

F/D 0.4, D70λ, θscan 0°

Power Flux Densityin Focal Plane

9

FPA Size vs. F/D

Optimum F/D depends on quality of edge beamsFor half scan of 4°:

3dB loss 0.35 – 0.41dB loss 0.4 – 0.5

10

Dish Size Independence

Express scan angle in beamwidths, FPA size in wavelengths

Then for moderate θscan and D, ρ is independent of dish size.Not surprising as:

Long F/D: Field approaches Airy patternFor small angles:

Shift by

Both follow this independence

F/D 0.25, D70λ, θ 4.0° = 4×HPBW

F/D 0.25, D30λ, θ 9.3° = 4×HPBW

-3dB -1dB

deg70θλθ D

BW ≈

-1dB radiivs.

dish size

11

Combined Plots

0.5 1 1.5 20

5

10

15

20

25

30

35

40Encircled Power 1dB, D 70

F/D

FPA

Rad

ius

( λ)

BW 0BW 1BW 2BW 4BW 6BW 8BW 10BW 15

0.5 1 1.5 20

5

10

15

20

25

30

35

40Encircled Power 3dB, D 70

F/D

FPA

Rad

ius

( λ)

BW 0BW 1BW 2BW 4BW 6BW 8BW 10BW 15

-3dB (50%) Encircled Power, D 70λ -1dB (79%) Encircled Power, D 70λ

Combined plots provide relationship between F/D, scan angle (or field of view) and FPA radiusF/D of 0.4 to 0.6 looks a reasonable choice

12

Tapered Dish Aperture

Truncation of a the focal plane field from a uniform dish illumination will result in high spillover (and high Tant)Practical reflector illuminations require edge taper

Reduce spillover noiseReduce sidelobes in far field

Tapering the reflector currents:Parabola on a pedestal-15dB edge taperReduces sidelobes in FFReduces extent of field in focal plane

ρ

Focal planefield

Far fieldpattern

Uniform

Parabola on-15dB pedestal

Amplitude ofPO currentson reflector

13

Tapered vs. Uniform Aperture – Focal Fields

Structure of focal field is similar Tapered case has less power outside the central spot

Contours at -3, -6 & -10dB

Encircled power: -3dB line solid-1dB line dottedTapered case is more compact

Uniform Dish Aperture

Tapered Dish Aperture

Sz: θscan 4° = 4×HPBWF/D 0.4, D70λ

14

Tapered vs. Uniform Aperture – Encircled Power

θscan 4° = 4×HPBW F/D 0.4, D70λ F/D 0.4, D70λ

FPA Radius (λ) Half Scan Angle in HPBWFP

A R

adiu

s (λ)

Tapered case normalized to give εt = 0.86 as maximum powerSimilarity in results increases confidence that weighting of theFPA elements can result in practical dish aperture tapers

15

Application to the New Technology Demonstrator (NTD)

ASTRON Vivaldi Array

Bandwidth 800 to 1800 MHz10x9 pol A and 9x10 pol B elements in Vivaldi egg-crate array

Assuming λ/2 spacing at 1400MHz (0.29λ to 0.64λ over band)FPA radius 48 cm

13.7m Dishes, F/D 0.4 (D/λ = 37 to D/λ = 82)Field of view from interpolation:

Constant in high frequency limitFocal spot size increases relative to the offset in the focal plane at low frequencies

16

Focal field overlaid with array

800MHz 1.9° scan 1800MHz 2.6° scan

λ λContours at -6 and -10dBNumber of elements required to form a beam:

About 8 elements in both cases should give a basic beamMore are required to clean up the beam and increase efficiency

17

Conclusion and Acknowledgements

Encircled power shows the general behaviour of the relationship between FPA size, FoV, F/D and dish sizeWith appropriate scaling, the results are independent of dish size.Practical systems will probably correspond to encircled power levels between the -1dB and -3dB levels

Actual levels should become more clear with G/T calculations

Further work:Calculate G/T for ideal and practical array elements with ideal weightingsLook at cross-pol and beam shape

Thanks for helpful discussions with:Dr. Marianna Ivashina from ASTRON, Netherlands Drs. John Kot and Stuart Hay from CSIROProf. Paul Smith from Macquarie University