The Hydrogen Epoch of Reionization Array Dish III: Measuring Chromaticity of Prototype Element with Reflectometry

5.3 MB·

by  Nipanjana Patra

Draft version March 16, 2017
PreprinttypesetusingLATEXstyleemulateapjv.12/16/11
THE HYDROGEN EPOCH OF REIONIZATION ARRAY DISH III : MEASURING CHROMATICITY OF
PROTOTYPE ELEMENT WITH REFLECTOMETRY.
Nipanjana Patra1†, Aaron R. Parsons1, David R. DeBoer1, Nithyanandan Thyagarajan2, Aaron Ewall-Wice7,
Gilbert Hsyu1, T. K. Daisy Leung1, Cherie K. Day1, James E. Aguirre10, Paul Alexander5, Zaki S. Ali1, Adam
P. Beardsley2, Judd D. Bowman2, Richard F. Bradley8, Chris L. Carilli9, Carina Cheng1, Eloy
de Lera Acedo5, Joshua S. Dillon1, Gcobisa Fadana13, Nicolas Fagnoni5, Randall Fritz13, Steve R.
Furlanetto4, Brian Glendenning9, Bradley Greig12, Jasper Grobbelaar13, Bryna J. Hazelton14,15, Daniel C.
Jacobs2, Austin Julius13, MacCalvin Kariseb13, Saul A. Kohn10, Anna Lebedeva1, Telalo Lekalake13, Adrian
Liu1,16, Anita Loots13, David MacMahon1, Lourence Malan13, Cresshim Malgas13, Matthys Maree13, Zachary
7 Martinot10, Nathan Mathison13, Eunice Matsetela13, Andrei Mesinger12, Miguel F. Morales14, Abraham R.
1 Neben7, Samantha Pieterse13, Jonathan C. Pober3, Nima Razavi-Ghods5, Jon Ringuette14, James Robnett9,
0 Kathryn Rosie13, Raddwine Sell13, Craig Smith13, Angelo Syce13, Max Tegmark7, Peter K. G. Williams6,
2 Haoxuan Zheng7

r Draft version March 16, 2017
a
M ABSTRACT
The experimental efforts to detect the redshifted 21 cm signal from the Epoch of Reionization
4
(EoR)arelimitedpredominantlybythechromaticinstrumentalsystematiceffect. Thedelayspectrum
1
methodologyfor21cmpowerspectrummeasurementsbroughtnewattentiontothecriticalimpactof

anantenna?schromaticityontheviabilityofmakingthismeasurement. Thismethodologyestablished
]
M astraightforwardrelationshipbetweentime-domainresponseofaninstrumentandthepowerspectrum
modes accessible to a 21 cm EoR experiment. We examine the performance of a prototype of the
I Hydrogen Epoch of Reionization Array (HERA) array element that is currently observing in Karoo
.
h desert, South Africa. We present a mathematical framework to derive the beam integrated frequency
p response of a HERA prototype element in reception from the return loss measurements between
- 100-200 MHz and determined the extent of additional foreground contamination in the delay space.
o
The measurement reveals excess spectral structures in comparison to the simulation studies of the
r
t HERAelement. CombinedwiththeHERAdataanalysispipelinethatincorporatesinversecovariance
s
weighting in optimal quadratic estimation of power spectrum, we find that in spite of its departure
a
[ from the simulated response, HERA prototype element satisfies the necessary criteria posed by the
foreground attenuation limits and potentially can measure the power spectrum at spatial modes as
2 low as k > 0.1h Mpc 1. The work highlights a straightforward method for directly measuring an
−
v instrume(cid:107)nt response and assessing its impacton 21 cm EoR power spectrum measurements for future
9
experiments that will use reflector-type antenna.
0
2
3 1. INTRODUCTION emissionfromtheneutralhydrogenintheearlyUniverse
0
has gained attention as a powerful probe of both cos-
. Since it was first proposed in Madau et al. (1997);
1 Shaveretal.(1999),measurementoftheredshifted21cm mology and astrophysics. The science case for 21 cm
0 cosmology, particularly during the Epoch of Reioniza-
7 tion, is well established (see, e.g., Furlanetto et al.
1 1Department of Astronomy, University of California, Berke- (2006); Morales & Wyithe (2010); Pritchard & Loeb
ley,CA
v: 2School of Earth and Space Exploration, Arizona State Uni- (2012)). However, the technical path toward measuring
i versity,Tempe,AZ this signal has been extremely challenging. The weak-
X 3PhysicsDepartment,BrownUniversity,Providence,RI ness of the hyperfine line keeps the 21 cm signal below
4Department of Physics and Astronomy, University of Cali- the foreground throughout cosmological history and cre-
r fornia,LosAngeles,CA
a 5Cavendish Astrophysics, University of Cambridge, Cam- ates sensitivity and calibration challenges that are yet
bridge,UK to be fully solved. With the system noise tempera-
6Harvard-Smithsonian Center for Astrophysics, Cambridge, tures dominated by sky noise and foregrounds that are
MA
four to five orders of magnitude brighter than the sig-
7DepartmentofPhysics,MassachusettsInstituteofTechnol-
ogy,Cambridge,MA nal even in the coldest patches of the sky, sky-averaged
8NationalRadioAstronomyObservatory,Charlottesville,VA 21 cm monopole experiments such as EDGES (Bow-
9NationalRadioAstronomyObservatory,Socorro,NM man&Rogers2010);SARAS (Patraetal.2013,2015b);
10DepartmentofPhysicsandAstronomy,UniversityofPenn-
SCIHI (Voytek2015);HYPERION (Presleyetal.2015);
sylvania,Philadelphia,PA
11DepartmentofPhysicsandElectronics,RhodesUniversity, and experiments attempting to detect the 21 cm power
POBox94,Grahamstown,6140,SouthAfrica spectrum such as the LOw Frequency ARray (LOFAR;
12ScuolaNormaleSuperiore,Pisa,Italy van Haarlem et al. 2013), the Murchison Widefield Ar-
13SKA-SA,CapeTown,SouthAfrica
14Department of Physics, University of Washington, Seattle, ray (MWA; ), the Giant Metre-wave Radio Telescope
WA (GMRT; Pacigaetal.2011),theDonaldC.BackerPreci-
15eScienceInstitute,UniversityofWashington,Seattle,WA sionArraytoProbetheEpochofReionization(PAPER;
16HubbleFellow
2 Patra et al.
Parsonsetal.2010),theHydrogenEpochofReionization 2015), these modes may be targeted as the lowest risk
Array (HERA; (DeBoer et al. 2016a)), and the future pathforconstraining21cmreionizationinthenearterm.
Square Kilometre Array (SKA; 17 Mellema et al. 2013) The efficacy of foreground removal from the measured
must furnish their foreground suppression limits given data critically limits the 21 cm power spectrum mea-
their system performance at levels significantly beyond surements. The delay transformation technique of fore-
anything previously achieved in radio telescopes operat- groundremoval,introducedby (Parsons&Backer2009;
ing below 1 GHz. Parsons et al. 2012b), computes the Fourier transform
One of the most problematic effects these experiments of the visibility measured by an interferometer and pro-
face is the chromaticity that are instrument specific and ducesaspectrum,referredtoasthedelayspectrumhere-
have unique manifestation in the measured data. The after, which is a function of the geometric time delay
spectral dimension of 21 cm reionization experiments is corresponding to the physical length of the baseline be-
of vital importance; for line emission, this coordinate tween two antennas. For the visibilities measured over
translates to a line-of-sight distance that can be used a wide field across a wide frequency bandwidth, the de-
to construct three-dimensional maps (and power spec- lay spectrum consists of the convolution of the instru-
tra) of emission, as well as probe the evolution of 21 cm ment response with the foreground signal, and the 21cm
emissionovercosmologicaltimescales. Theresponseofa power spectrum, hereafter referred to as the EoR sig-
radio telescope — either a single dish or an interferom- nal. The technique exploits the smooth spectral charac-
eter — versus spectral frequency, modulates spectrally teristics of the foreground and for a visibility measure-
smooth foregrounds, contaminating spectral modes that mentbyaninterferometerwithabaselineb,confinesthe
might otherwise be used to measure reionization (Bow- foregroundcontributiontothecomputeddelayspectrum
man et al. 2009; Datta et al. 2010; Trott et al. 2012; within the largest possible time delay, τg = b/c, corre-
Vedantham et al. 2012; Liu et al. 2014a; Morales et al. sponding to the given baseline length. In reality, how-
2012). Moreover, measurement sensitivity could be in- ever, foregrounds show a response beyond the maximum
creased by increasing the collecting area of the element geometric delay due to broadband spectrum of the fore-
using reflector structures that results in increased ele- ground sources and chromatic instrument response. For
mentsize. However, the chromaticity of a reflector dish an ideal system, this is a rapidly diminishing response
scales with its diameter, putting the needs of foreground beyond τg spanning the delay range corresponding to
suppression and signal sensitivity in direct tension with the inverse of the measurement bandwidth. Beyond this
one another. limit, for an ideal system performance, the delay spec-
A major step forward for the field of 21 cm cosmology trum would be dominated by any signal with spectral
has been the development of a mathematical description and spatial fluctuations over small scales, such as the
oftelescopechromaticity,howitvarieswithelementsep- EoR signal. The interaction between the sky signal and
aration (or telescope diameter), and what it implies for instrumentresponsecanaltertherelativecontributionof
distinguishing the foreground emission from the cosmo- the foreground, and the EoR signal at a given delay and
logical 21 cm signal (Parsons et al. 2012b; Vedantham influencethedetectabilityoftheEoRsignal. Suchinter-
et al. 2012; Trott et al. 2012; Dillon et al. 2015; Liu actionsmaycausetheforegroundtospilloverintohigher
et al. 2014a,b; Thyagarajan et al. 2013, 2015). The delays and push the upper limit of the foreground and
“wedge”, as it is colloquially known, describes a linear systematics contaminated delay modes to much higher
relationship between the separation between elements in delays affecting the EoR window.
an interferometric baseline and the maximum line-of- In this paper, we use reflectometry measurements to
sight Fourier mode18 that may be occupied by smooth investigatethedelay-domainperformanceofaprototype
spectrum foreground emission. At low-order k modes 14-m parabolic reflector dish and a crossed dipole feed
within the limits of the wedge, foreground con(cid:107)tamina- (figure??)thatareproposedtobeusedinHERA,oper-
tion may be suppressed through a combination of cal- atingfrom100to200MHz. Forclarity,thereturnlossof
ibration and model subtraction. However, calibration thefeedwhenmeasuredoffofthedishwillbereferredto
or modeling errors rapidly re-establish the characteris- as “feed return loss” hereafter. The dish-feed assembly,
tic wedge pattern (Barry et al. 2016). Outside of the when the feed is placed at the focal point of the dish is
wedge, foreground contamination drops rapidly (Pober referred to as “HERA element” hereafter.
et al. 2013; Dillon et al. 2014; Thyagarajan et al. 2015; This paper is one in a series of five papers that studies
Kohnetal.2016),providedthatthespectralresponsesof the time and frequency domain simulation of HERA el-
antenna elements and analog electronics are sufficiently ement (Ewall-Wice et al. 2016), (DeBoer et al. 2016b),
smooth19. In the avoidance-based foreground strategy beam pattern measurements (Neben et al. 2016), and
employed by PAPER (Parsons et al. 2014; Ali et al. foreground limits (Thyagarajan et al. 2016). We com-
parereflectometrymeasurementstothesimulatedspeci-
17 www.skatelescope.org ficationforthespectralperformanceofaHERAelement
18 Assuming a flat sky and using appropriate cosmological (DeBoer et al. 2016b) and to the limits of EoR to fore-
scalars,thespectralaxis,ν,andangleonthesky,θ(cid:126),translatetoco- ground power ratio established by (Thyagarajan et al.
ordinatesinathree-dimensionalvolumeatcosmologicaldistances.
2016). In this paper, Section 2 briefly describes the de-
In describing the spatial power spectrum of emission in this vol-
umeP((cid:126)k),weusethethree-dimensionalwavevector(cid:126)k≡(k(cid:107),(cid:126)k⊥), lay spectrum for the ideal and non-ideal performance of
where k(cid:107) is aligned with the spectral axis, ν, and (cid:126)k⊥ lies in the arefltwecotoemleemtreyntmienatseurrfeermoemnettsear.ndSeescttaiobnlis3hedsetshceribcoensntehce-
planeofthesky.
19 Here, we distinguish the chromaticity of the elements in iso- tion between visibility measurements and reflectometry.
lation from the chromaticity inherent to element separation in an ReflectometryresultsaredescribedinSection5. Section
interferometricbaseline. 6 evaluates the performance of the HERA element for
HERA dish reflectometry 3
HERAdish reflectometry 3
FFigi.g1..—1.—LeLfte:fTt:heTHheERHAERfeAedfceoendsicstosnosfisatspaoifraofpcariorssoefdc-droipsosleeds-odviepro1le.7s2omvedri1am.7e2temrbdaicakmpleatneermbaadckepolfawnieremmaedseh.ofTwheirbeamckepslahn.eTishebackplaneis
susrurroruonudneddebdyb0y.306.3m6wmidewwidierewmireeshmaeroshunadrothueneddgtheereesduglteinrgesinuletnincagsiinngetnhceacsrionsgsetdh-deipcroolessseind-adicpyolilnedsriicnalacacgyeli.n.dRriigchatl:cAagper.o.toRtyigphet: Aprototype
HHEERRAAeleemleemntenatttahtetGhereGenreBeannkBNanRkANORsiAteOcosnitsiestcinognsoifstaincgroossfeaddcripooslseedfeeddipaonldeafepeadraabnodlicarpeflaercatboorldicishre.flectordish.
gdroeutnecdtipoonweorfrtahteio2e1stcamblipshoewdebrysp(eTcthryuamga.rajan et al. The convolutioHn(izs)c=arrHied[Ωout(a1lo+ngzt)h3e+⌧Ωaxi(s1w+hizch)2is+Ω ]1/2
0 M R Λ
2016). In this paper, Section 2 briefly describes the de- Fourierconj•ugiastetohfethHeufrbebquleenccoyn⌫s.taQnutalaitsataivefulyn,cdteiloayn of redshift,
lay sp2e.ctVruImSIBfoIrLtIhTeYideMalEaAnSdUnRonE-iMdeEalNpTerSfoBrmYanAceToWf O spectrumofthHevis=ibil1it0y0rhesuklmtsfrsom1 tMhepccon1v,oluΩtion=of 0.27,Ω =
a two eEleLmEeMntEiNntTerIfeNroTmEeRteFr.ERSOecMtioEnT3ERde:sTcrHibEesDthEeLAYthe instrument d0elay response with−the del−ay specMtrum R
0.73,Ω =1 Ω Ω are the matter, radiation
reflectometry measurementSsPaEndCTesRtaUblMishes the connec- of the sky. It is, therΛefore, i−mpoMrta−nt tRo determine the
and dark energy density parameters respectively.
tionFobretawetewnoveisleibmileitnytminetaesrufreermomenetstearnwdirtehfleactboamseeltirnye. (cid:126)banidnstrumentdelayresponseinordertodetermineskyde-
ReflectometryresultsaredescribedinSection5. Section lay response at a particular delay by de-convolving the
antenna voltage gain pattern a , a , the voltage at the X,Y arecosmologicalscalarsrelateingtheangular
6oeuvtapluuattoesf ethaechpearnfotermnnaanceeleomf ethnet1cHaEn2RbAeewlermitetnetnfaosr, delay transf•ormdeepdevnidsiebnilciteywbiyththsepiencsttrruamlfernetqureesnpcoynsteo. correspond-
detectionof the21cm powerspectrum. Thedelay-transformedvisibilityisrelatedtothepower
spectrumofreidnsghicftoemdo2v1incmg deimstisasniocne.bythe relation,
v (θˆ,ν)=a (θˆ,ν) v (θˆ,ν)
2. VISIBILITY1 MEASUR1EMENTSskByYATWO
ELEMENTINTERFEROMETER:THEDELAY 3. EFFECTS OF MUL�T2IPL2EXR2YEFLECTIONS ON
Considerva2t(wθˆo,νe)le=meaSn2Pt(EiθˆnC,tνeTr)RfevUrsoMkmy(eθˆte,rν)wiet2hπaiν(cid:126)bb·aθˆs/ecline P2H1(EkR?,AkkVc)oI⇡nSIs|(BisV˜It(Ls~bI,oT⌧f)Y|p2aA✓rNa2kbDBol◆DicE⌦rLe�AflBYectSoPrEa(Cn5Tt)eRnUnaMs which
~b. Iwf hweerde,envostkey(tθ(cid:126)h,eνe)leicstrtichefieslkdyskvyolstigangaeliinn tthheeddirireec-ctionwθˆhere provide increased collecting area per array element com-
tiaotna✓ˆfarteqauefrnecqyuνe.ncTyh⌫e bmyevasskuyr(e✓~d,⌫v),istihbeinlittyheisv,oltage paredtoitspredecessorexperimentPAPER.Planewaves
output of eachantenna maybewritten as, inckide=nt2⇡onf abpri,mke =foc2u⇡s⌧pf2a1rHab(zo)li,c dish (a6r)e focussed at
V((cid:126)b,νv)(=✓ˆ,⌫)v=1(aθˆ,(ν✓ˆ,)⌫v)2∗v(θˆ,ν(✓ˆ),⌫e)−2πiν(cid:126)b·θˆ/cdΩ the?feedwDh ichc!isatkthefco(c1a+lpzl)a2neofthedish. Themis-
1 (cid:90) 1 sky match between the impedance of the free space with the
v2(✓ˆ,⌫)==a2A(✓ˆ(,θˆ⌫,)νv)skIys(k✓ˆy,(⌫θˆ),eν2)⇡ie⌫−~b·2✓ˆ/πciν(cid:126)b·θˆ/cdΩ (1) • f2ft1eh:reeedsstakfnyradmstigeranfnraeslqmuweihsnsicliyeonotfhlitenhereer2se1tsucismltsrreaiflndeiacattpieodanr.toiffalocfouthpelinfegedof.
where a , a are(cid:90)the voltage gain pattern of the two f:Tohbissersvigatniaonliflrlueqmuiennactye.sthedishandmostofitisreflected
antewnhnearee.1TAh(eθ2ˆ,vνis)ibi=lityam1(eθˆa,sνu)reda∗2b(yθˆ,tνh)e iinstetrhfeerocmroetsesrpower • back into the space. However, a part of it reflects back
isp,attern and I (θˆ,ν) = v (θˆ,ν) v (θˆ,ν) is the sky z:acnodsmfoorlothgicsaelvreerdaslhtifitmceosrrbeesptwonedeningthteofteheedfraen-d the vertex
sky sky s∗ky • quencyof observation.
ianlotenngsVitth(y~be.,⌫fHr)ee=qnuceen,vcT1y(h✓ˆa,e⌫xF)isovui2⇤.r(e✓iˆ,e,rt⌫h)terea�dn2es⇡lfiao⌫y~br·m✓ˆs/pcdeoc⌦fttrhuem(v1ii)ss,ibility otifonthsegdenisehrawtehimchulitsipslheacdoopwieesdobfyththeeinfeceidde.nStusckhyrseigflneca-l
Z �B: bandwidth centered at the observation fre-
• ofreducedstrengthatvariousdelaysandphasesproduc-
We definVe˜((cid:126)tbh,eτ)=anAte(n(cid:126)bn,aτ)croIss (p(cid:126)bo,wτe)r pattern as quienngcya.dditional correlations in the visibilities data. Iden-
A(✓ˆ,⌫) = a1(✓ˆ,⌫) a⇤2(✓ˆ,⌫) and∗desnkyote the sky intensity tical response can result from reflections internal to the
asIsky(✓ˆ,⌫)=vsky(✓ˆ=,⌫) v∞s⇤kyA(✓ˆ(,τ⌫). ∆Heτn)cIes,ky(τ)d∆τ (2) • kBs:yBstoelmtz,mfaonrnecxoanmstpanlet.between antenna output and back-
−
forVa(~bg,⌫iv)e=n baAse(✓ˆl(cid:90)i,n−⌫e)∞(cid:126)Ibsky((✓Pˆ,a⌫r)soe�n2s⇡ie⌫t~b·✓ˆa/cl.d⌦2012(a2)). The • Dei⌘nnddDer(lzeac)yecisovpmearocveiin.npgWudteisctmaanuecaseisnuaglroenstigmhtehileasyrlisnsteiegomnfasrilgecshpotnotnasme itnoaetsiotin-
Z �Dmattheetchoemseovrineflgedcitsitoannsceaanldontghethreelsiunletionfgsiegffhetcts in power
dTelhaey-Ftoruarniesrfotrrmanesdfovrmisiboiflitthyeivsirsieblailtiteydatloontghethpeofwree-r spec- • cosrpreescptornudmingmteoabsuanredmwiedntths.of observation�B.
trum of redshifted 21 cm emission by the relation,
quencyaxis i.e,thedelay spectrumcanbewritten as, We assume that the reflection coefficient of the
P21(V˜k(~b,,⌧k)=) A(~bV,˜⌧()(cid:126)b⇤,τI)sk2y(~b,⌧λ)2 2 X2Y (3) • Hwhpi(.ezaer),rea=uHbpHo(olz0ni)[c⌦cidMsoi(mst1hhpe+ilseHztp)ue3rbo+iblpllue⌦omRrcto(ii1nnosan+ttaaiznlo)tnt2oa+osift⌦ast⇤hfa]ue1r/ne2dcatiisoohfni1l0lu0m%inoafttiohne
⊥ (cid:107) ≈| | (cid:18)2kB(cid:19) Ω∆B(3) ofinrecdidsheinftt,rHad0ia=tio1n00ishrkeflmecst�io1nMopffc�o1f,th⌦eMdi=sh and reflec-
(wPhaersroens et al. 2012a). For a given baseline~b this can 0te.2rt,i7o,rn⌦adRciao=teiffio0n.c7ia3en,nd⌦t⇤doaf=rkth1ee�nedr⌦igsMyh�diesn⌦usRintyiatrype.atrThameheemteasrkts-y voltage in-
cident on the dish illuminates the part of the dish that
bewrittenas, respectively.
2πf b 2πτf H(z) is not shadowed by the feed. The corresponding dish
21
k = , k = , (4)
V˜(⊥⌧)= D1 A(cid:32)(c⌧(cid:33)��⌧(cid:107))Isky(⌧c)(d1�+⌧ z)2 • Xrp,eYleflxea.crteiTocnohsemcoodelioffisghciciareelnfltseciacsltadioresnntohctaoetedffiraeclsaiet(en1st−tohΓfedta)hnwe-hpiachrtisofcotmhe-
f ,f,z,∆ZB�:1rest frame, observation freq(u4e)ncy of gspudolainsrdhdinesgpheacnoddmoewonvceiednwgbidythisttshapeneccfeet.readlfirseqdueennoctyedtobcyoΓrrde.-The voltage
• th21e21cmradiationandredshiftandbandwidthof reflection coefficient of the feed is Γf and correspond-
ing transmission coefficient is (1+Γ ) (see, e.g. Pozar).
observation. f
Both Γ and Γ are complex numbers that are functions
d f
k : Boltzmann constant. of frequencies. For a given incidence of the sky voltage
B
• v , Γ (1+Γ ) fraction of it is coupled into the cable
sky d f
D D(z)comovingdistancealongthelineofsight leadingtothereceiverbackendfromthefeedwhereasΓ
f
• ≡
4 Patra et al.
fraction of it is reflected off. This subsequently under- prototype HERA element consists of a 14.5 m diam-
goes Γ reflection off the dish and (1+Γ ) of it reenters eter parabolic reflector and a crossed-dipole pair as a
d f
thefeedwitharoundtriptimedelay∆τ =2F/cwhereF feed. The cross dipole antenna pair is identical to the
is the focal length of the dish. Hence, if v is reflected feed of the PAPER antenna and suspended at the fo-
sky
ntimesinbetweenthefeedandthedish,thenetvoltage cal plane of the dish with the support of three vertical
entering the feed after the nth reflection can be written poles. HERA elements will be closely spaced with cen-
as: tre to centre distance between the two adjacent dishes
slightlylargerthanthedishdiameter. Thecrosseddipole
vrec=(1 Γd)(1+Γf)vsky[1+ΓfΓde2πiν∆τ feed is encased in a cylindrical cage with the back plane
−
+(Γ Γ )2(e2πiν∆τ)2+ toreducethecouplingbetweentheadjacentdishes. The
f d
feed is raised and lowered by a pulley system mounted
....+(Γ Γ )n(e2πiν∆τ)n] (5)
f d onthethreepoles. Thefocalheightofthedishis 5m.
≈
Thedetailgeometryandelectromagneticdesignofofthe
feed is presented in DeBoer et al. (2016b).
Duringourmeasurements,eachHERAelementiscon-
nected to an active balun similar to the ones used in the
d PAPER antenna which provides greater than 10dB re-
vsky vin f v tisurrneplolascseodfbpyowaerpaaststivhee ofeneedwouitthpu4t:.1Timhepeadcatinvceebraaltuino
rec
to match the impedance of the 50 Ω transmission line.
vrad v These measurements, therefore, provide a conservative
v trans estimate for Γf and S11.
out
F 4.1. Measurement equations
The measurement presented here is done using a vec-
tor network analyzer (VNA) that is connected to the
antenna using a 15 m long LMR 400 cable. This length
correspond to a 120 ns roundtrip delay. A VNA is con-
Fig. 2.— Schematic diagram of signal propagation through the
dishandthefeedofduringbothtransmissionandreception. Vsky, nected to the HERA element via a ≈ 15 m cable that
VinandVrecrepresentstheskyvoltage,totalreflectedvoltagefrom transmits a broadband noise voltage, vtrans, and a fac-
the dish that is incident on the feed and the net received voltage tor (1+Γ ) of this voltage is radiated by the feed while
atthereceiverinput. Intransmissionmode,Vtrans,VoutandVrad thefractiofnΓ returnstotheVNA.Thetransmittedsig-
are the voltage transmitted by the network analyser, the voltage f
outputofthefeedandthethenetradiatedvoltageafterreflection nal illuminates the dish and most of it is radiated into
onthedish. Γd,Γf arethevoltagereturnlossofthedishandthe thefreespace,afractionΓdofthissignalisreflectedback
feed. towards the feed, and (1+Γ ) fraction of it is received
f
bytheVNA.Thereceivedvoltage,afternreflectionsbe-
or,
tween the dish and the feed (where again n = 0 is the
firstreflectedsignalatthereferenceplane),istherefore:
N
v
vrec =(1−Γd)(1+Γf) [ΓfΓde2πiν∆τ]n vrec=Γfvtrans
sky
n(cid:88)=0 +v (1+Γ )2Γ e2πiν∆τ
(6) trans f d
+v (1+Γ )2Γ e2πiν∆τΓ Γ e2πiν∆τ
trans f d d f
Notethatn=0iswhentheinitialwaveentersthecable
at the feed (where we assume we have set our reference +vtrans(1+Γf)2Γde2πiν∆τ(ΓdΓfe2πiν∆τ)2
plane). ....+v (1+Γ )2Γ e2πiν∆τ(Γ Γ e2πiν∆τ)n
trans f d d f
An accurate measurement of this quantity requires
(7)
receiving v from a well calibrated, wideband source
sky
in the sky in the far field of the HERA antenna element or,
with significant isotropic emission. While this condition
is hard to achieve in practice, reciprocity of the antenna v (1+Γ )2 N
rec =Γ + f [Γ Γ e2πiν∆τ]n (8)
performance in the transmission and reception mode f f d
v Γ
trans f
implies the right hand side of equation 6 could be n=1
(cid:88)
measured using the return loss measurement technique
TheVNAmeasuresthequantityv /v =s which
rec trans 11
with a vector network analyzer (VNA).
isthevoltagereflectioncoefficientoftheHERAelement.
Thefundamentaldifferencebetweentheequations6rep-
resents the system response in reception mode while our
4. REFLECTOMETRY MEASUREMENTS measurementsaredoneintransmissionmode(equations
Reflectometry measurements determine the multiple 6). The n = 0 term in equations 6 represents the trans-
reflection of any signal within a system in either time mission of the beam integrated sky signal from the an-
(Patraetal.2015a)orfrequencydomain. Wecarriedout tenna to the transmission line while the same in equa-
frequency domain reflectometry on a prototype HERA tions 8 represents the reflection at the antenna terminal
elementinGreenBank,WV(figure??)inordertomea- and the transmission line. The zero delay response is
sure the response of the feed and dish assembly. The identical whether or not the feed is suspended on the
HERA dish reflectometry 5
dish. Therefore, we measure the feed return loss alone
TABLE 1
while the feed is kept on the ground facing the sky. This System delay scale
measurements include the reflections from the surround-
ing cage structure but exclude the dish response. Using RoundtripDelay(ns) Lengthscale(m) Source
the return loss measurement of the feed (Γ ), the return
f 30 5 Dishapex
loss measurement s of the dish and feed assembly is
11 2.86 1.72 FeedCavity
corrected via equation 10. Writing v /v = s ,
rec trans 11 120 15 VNAInput
equation 8 can be written as,
of (Thyagarajan et al. 2013, 2016) to estimate the delay
(1+Γ )2 (1+Γ )2 N transform.
s + f Γ = f [Γ Γ e2πiν∆τ]n
11 f f d
Γ − Γ
f f n=0 4.2. Results
(cid:88)
(9)
ReturnlossmeasurementsofthefeedaswellasHERA
Finally,
element is shown in figure 3. Corresponding delay spec-
v Γ tra are shown in figure 4. The delay spectra of the feed
rec f
=(1 Γd) (1+Γf)+ (s11 Γf) and the HERA element closely follow each other up to
v − (1+Γ ) −
sky (cid:20) f (cid:21) delays < 50 ns. Figure 5 shows the delay spectra esti-
(10) matedusingasquarewindowaswellasBlackman-Harris
window. Inthiscase,thedelayspectrafolloweachother
From this, for a two element interferometer with iden-
upto50nsandstartdeviatingafterthat. Thisisbecause
tical antenna elements, the ratio of the received sky in-
at small delays the spectrum is dominated by the small
tensity to true sky intensity will be,
scale reflections associated with the feed cage. Above
2 50 ns, the windowing effect begins to manifest. The rel-
I v
rec = rec = 1 Γ 2 evant length scales inside the system from where the re-
d
Isky (cid:12)vsky(cid:12) | − | × flections are expected and the corresponding delays are
(cid:12) (cid:12) summarized in the table 1. The delay spectrum of the
(cid:12) (cid:12) Γ
[(cid:12)(cid:12)1+Γ(cid:12)(cid:12)f 2+2Re f (s11 Γf) feed exclude the dish reflections. However, the VNA re-
| | (cid:18)(1+Γf) − (cid:19) flectionsattheVNAinputarepresentinboththecases.
Γ 2 We measure the VNA input reflections by connecting an
+ | f| s Γ 2] open load at the feed input of the cable that results in
1+Γ 2| 11− f|
f 100% signal reflection. The combined effect of the VNA
| |
(11) input return loss along with the cable resistive loss is
shown in figure (figure 4). Since our measurement plane
In terms of visibility, the same could be written as,
isattheopenendofthecable, firstreflectionappearsat
zero delay while the second one appears around 120 ns.
Vmul((cid:126)b,ν)= 1 Γ 2 which is 22 dB. Third and consecutive reflections
d
| − | × ≈ −
(cid:90) areburiedinthenoisefloorofthismeasurement. There-
|1+Γf|2+2Re (1+ΓfΓ )(s11−Γf) + ftohree,dieflathyesrpeetcutrrnumloscsoanttathmeinbaatcikoenndduineptuottihse>re−fl2e2ctdioBn,
(cid:20) (cid:18) f (cid:19)(cid:21) at the receiver input will be negligible. Moreover, if the
Γ 2
(cid:20)|1|+fΓ|f|2|s11−Γf|2(cid:21)Iskye−2πiν(cid:126)b·θˆ/cdΩ(12) csiagbnlael,letnhgetsheriesflseucffitiocinesntcloyulidncbreeamseaddewtitohoocuctutrhaetldoeslsayosf
which are not of interest for EoR measurements.
Comparingequations12and1,thecrosspowergenerated
by v1 and v2 has a spurious visibility response due to 4.3. Comparison with simulations
mutual correlation between multiply reflected voltages
4.3.1. HFSS Electromagnetic simulation
representedbythesecondandthethirdtermoftheright
hand side of equation 12. In the ideal case, Γ = 0, in Performance of the HERA feed and the HERA ele-
f
whichcaseequation12resultsinequation1. TheFourier ment has been simulated by using the EM solver ”High
transform of this visibility spectrum along the frequency FrequencyStructuralSimulator(HFSS)”usingthefinite
axis results in the delay spectrum. In the delay domain, element method. These simulations provide a ballpark
visibilities contributed by any two voltage components estimate of the system response under various ideal con-
from two antennas with no mutual delay will be located ditions. The measurement is in agreement with the sim-
at the delay τ =0 whereas any two voltage components ulation in some aspects and differ in some. The simu-
from two antennas having a mutual delay of n∆τ will lated return loss of the feed by HFSS, both magnitude
be centred at n∆τ. These will result in leakage of the andphaseareshownintheupperpaneloffigure3along
foreground to delays where EoR power spectrum could with the measurements. Bottom panel of figure 3 shows
be measured. the same when feed is suspended at the focal plane of
The delay transform of the measured return loss is theparabolicdishwhichisat4.5mfromthedishvertex.
sensitive to the bandwidth of measurements. Wideband The measured return loss is much more complex than
measurementsoverfinitebandwidthisanalogoustowin- what is found from simulation. While the simulated re-
dowing the frequency domain data by a square window turn loss of the feed structure shows two resonant peaks
function that results in multiple side lobes at higher de- around 120 and 160 MHz, the feed resonance occur at
lays. We apply the Blackman-Harris window function slightly higher frequencies. Notable are the difference in
6 Patra et al.
Fig. 3.—Upperpanel: Magnitudeandphaseofthereturnlossofthefeedassimulatedandmeasured. Bothmeasurementandsimulation
shows similar level of return loss across the band with marginally better return loss at the high frequency end of the band. Lower panel:
Magnitudeandphaseofreturnlosswhenthefeedissuspendedatthefocalpointofthedishwhich5mabovethedishvertex.
theshapeoftheresonantpeaks. Whilethesimulatedre- dipole terminal. Therefore, delay spectrum estimated
sponseshowsverywideresonantpeaks,asexpectedfrom from this simulation only has the effects of chromaticity
these dipole structures, the measured return loss has a introduced due to the structural reflections from the
narrow peak at its low frequency resonance. The domi- antenna and did not include the mode antenna mode
nant term in the expression of return loss of the HERA scattering which is a function of frequency if the ter-
elementinreceivingmode(equation10)isthezerodelay minal impedance is frequency dependent. In practice,
term (1+Γ ). Hence the feed return loss and especially dipole impedance is function of frequency that results
f
the shape of the low frequency resonant peak dominates in variation of return loss with frequency. This results
the overall shape of the delay spectrum in figure 6. The in deviation between the simulated delay response and
simulated return loss of the feed shows smooth variation the delay spectrum estimate of our measurements. The
across the band and wide resonant peaks resulting in a simulation and measurements both agree in one aspect.
narrow delay spectrum. Our measurement also confirms that reflections from
the feed structure dominates at lower delays and this is
4.3.2. CST Time domain simulation neither an attribute of the reflector dish response nor it
We also compare our measurements with the time is a computational artifact such as windowing.
domain simulation presented in Ewall-Wice et al.
(2016) in figure 6. In this simulation, the transient
4.3.3. Foreground simulation
electromagnetic response of the HERA element is deter-
mined as a function of time when it is subjected to a Wealsocompareourmeasurementwiththeforeground
plane wavefront. The commercial numerical simulation simulation of Thyagarajan et al. (2016) as shown by
software Microwave Studio, developed by Computer the shaded regions in the first panel of figure 7. The
Simulation Technology is used for the simulation. The shaded region of each panel shows the minimum EoR
simulation assumed a constant 125 Ω impedance at the to foreground power ratio required for detection of the
HERA dish reflectometry 7
Fig. 6.—DelayspectrumofHERAelementestimatedfromthe
reflectometry measurements by a vector network analyser com-
pared to the delay spectrum estimated from the EM simulation
Fig. 4.—DelayspectrumoftheHERAfeed(brown)estimatedby
usingHFSSandtimedomainsimulationusingCST.
takingtheFouriertransformofthemeasuredreturnlossofpower
at the feed output (upper panel in figure 3). The feed, when sus-
pendedonthedishresultsinamorecomplexreturnlossasshown modelparametersusedfortheEoRsignalsimulationare
in the lower panel of figure 3 and corresponding delay response is : Virial temperature of minimum mass of dark matter
shown here in black. A balun with impedance transformation ra- halos that host ionizing sources, Tmin =2 104 K, Ion-
tio 4 : 1 and input return loss of -22 dB across the band is used vir ×
forconvertingtheantennabalancedoutputtounbalancedoutput ising efficiency η = 20%, mean free path of UV photons
voltage. Balunresponseisembeddedinthemeasurements. Delay R = 15 Mpc For these model parameter values, the
mfp
spectrumoftheopenload(gray)showsthereflectionsattheVNA redshift of 50% reionization is predicted to be at z =8.5
input.
at 150 MHz.
The EoR and foreground power spectra are shown by
the blue and the orange curve in figure 8. The details
of the power spectrum computation are given in section
5. The computed power spectra are associated with the
shaded gray regions in following way. For a given sep-
aration in delay, for example, for τ = 200 ns, at delay
t = 200 ns the foreground power spectrum amplitude
should reduce from 1015 (at delay t=0) to 104 mk2 (at
delayt=200)i.e,110dBattenuationofpowerspectrum
amplitude or roughly 55 dB attenuation of power (visi-
bility). Forthesameτ, betweendelayt=100to300ns,
the foreground power spectrum amplitude should be at-
tenuated from 1014 to 104 mk2 which is roughly 100 dB
i.e50dBattenuationoftheforegroundpowerrelativeto
the EoR. At t=400 ns, the foreground should be atten-
uated from 1011 to 103 mk2 between t =200 to 400 ns
which is roughly 80 dB attenuation of the foreground
power spectrum amplitude or or 40 dB attenuation of
the foreground power relative to the EoR. For the sepa-
rationindelayτ =200ns, anytwopointsindelayspace
Fig. 5.—Effectoffinitebandwidthonestimationofdelayspec-
will need different attenuation of foreground power with
trum: Blue line shows the delay spectrum of the HERA element
computedfromthemeasureddatawhichisbandlimitedbetween respect to the EoR power and the maximum foreground
100 to 200 MHz. Black line shows the delay spectrum estimated power attenuation required will be 55 dB for the delays
fromthesamedatasetaftermultiplyingthedatabyaBlackman
where foreground has the highest magnitude. The max-
Harris window. The delay spectrum of the windowed data set
shows significant reduction in the instrument response at higher imum required attenuation of the foreground power at
delays. various values of τ is plotted in the grey shaded region
for different values of τ.
21 cm power spectrum at those individual delays (or in The EoR to foreground power ratio shows that for a
k modes). This ratio is computed by using a achro- givendelaymode,foraparticularfrequency,asuccessful
m(cid:107)atic antenna beam and the sky models. Diffused fore- power spectrum detection at a lower k mode requires a
groundskymodelisincorporatedfromdeOliveira-Costa higherforegroundpowerattenuationre(cid:107)lativetotheEoR
et al. (2008) whereas the point source contribution is power compared to a higher k mode. This simulation
estimated by combining the NRAO VLA Sky Survey provides a baseline for the sy(cid:107)stem performance evalu-
(NVSS) at 1.42 GHz (Condon et al. 1998), Sydney Uni- ation with a conservative strategy of pure foreground
versityMolongloskySurvey(SUMSS) (Bocketal.1999; avoidance without any foreground mitigation strategy.
Mauch et al. 2003) at 843 MHz. The EoR signal is sim- The measurement differs from the simulation in one as-
ulated by using 21cmFAST (Mesinger et al. 2011). The pect. Simulation includes the achromatic antenna gain
8 Patra et al.
excluding the return loss of of power at the antenna ter- ement which convolves with the delay spectrum of the
minal due to frequency dependent antenna impedance. sky signal and produce the delay spectrum. In the delay
This effect is included in the measurements. If included domain, after convolution with the instrument response,
in the model, these limits would provide more conserva- the foreground response must fall below the amplitude
tiveestimatesoftherequiredinstrumentresponsewhich of the 21cm EoR signal to avoid a systematic bias. This
will be even harder to achieve. simple analysis omits the suppression of foreground sys-
tematics that are a feature of more sophisticated power
4.3.4. Delay spectrum of sub bands spectrum estimation techniques. In this section, we re-
In the HERA analysis, the 21 cm power spectrum is evaluate the specification for the delay-domain perfor-
estimated by discretizing the observed data along the manceoftheHERAelementinlightoftheHERApower
line of sight distance i.e in frequency or redshift and in spectrumestimationmethodusingtheoptimalquadratic
the plane of the sky k (Liu et al. 2014a). At particu- estimator(OQE).TheOQEformalismhasbeenoutlined
lar redshift, the bandwidth of observation is so chosen in extensive detail in Dillon et al. (2013), Liu et al.
that over the corresponding ∆z, the 21 cm signal does (2014a), Liuetal.(2014b), Trottetal.(2012), Alietal.
not significantly evolve. Typically, the 21 cm bright- (2015). In order to determine the effect of covariance in
ness temperature fluctuation evolves over redshift scale the measured data, we notationally describe the OQE
∆z >0.5 (Loeb & Zaldarriaga 2004), (Lidz et al. 2008). formalism here. The 21 cm power spectrum P21(k ,k )
Therefore, the delay spectrum is estimated over smaller integratedoverarangeofk ,k ,isestimatedasthe⊥ban(cid:107)d
bandwidths. powerpˆα,whereαrepresen(cid:107)tsa⊥rangeofk ,k . Thema-
We estimate the instrument delay spectrum over var- jor steps of the OQE formalism are, (cid:107) ⊥
ious sub bands between 100 to 200 MHz as shown in The unnormalized band power qˆα, estimated from the
figure 7. Each panel of figure 7 shows the delay spec- data vector x of measured visibilities is
trum estimated from the return loss measurement of the
qˆ =xTEαx, (13)
HERA element between 100-200, 100-140, 130-170, 160 α
to 200 MHz respectively. Each panel also compares the Eα isasymmetricmatrixoperationdenotingtheFourier
delay spectrum estimate obtained EM simulation of De- transform of the data, binning, and foreground reduc-
Boer et al. (2016a) at various subband. Subband delay tion. The normalized estimate of the power spectrum
spectraarewidercomparedtofullbanddelayspectrum. is,
The shape of the resonant peaks determines the roll off pˆ =Mqˆ (14)
α α
rate of the delay spectrum at all frequency ranges. At
higher frequency, our measurement closely follows the where M is the normalization matrix. The true power
EM simulations except around 80-120 ns where the ca- spectrum pα is
blereflectionbetweentheantennaandthebackendman- pˆ =Wp (15)
α α
ifests itself.
where W is the window function matrix,
The shaded regions represent the EoR to Foreground
Various binning and foreground reduction techniques
powerratiosimulatedacrosstheentireband. TheHFSS
result in different forms of Eα resulting in estimates of
simulation of chromatic antenna beam varies smoothly
pˆ with different statistics. A possible choice of Eα is
across frequency resulting in no significant changes in α
thissimulationatvarioussubbands. HERAsystemper- 1
formanceconformsdirectlytothelimitsasposedbysim- Eα = C−1QαC−1 (16)
2
ulations, at frequencies >160 MHz. With the measured
system response including the multiple reflections, the where C = xxt is the covariance matrix of the data
(cid:104) (cid:105)
instrument would be able to probe the spatial modes of vectorx. QαisamatrixoperatorthatFouriertransforms
interest k > 0.2h Mpc 1 with very little foreground the visibilities along the frequency axis and maps them
−
bleedinga(cid:107)roundk 0.2hMpc 1. Atotherfrequencies, intothekspace. ThecriticalstepintheOQEformalism
−
it is possible to do(cid:107)a≈successful measurement by weight- for power spectrum estimation is the weighting of the
ing down the foreground contribution in the measured data by the inverse of its covariance. This can result in
data. This is discussed in the following section. orders-of-magnitude reduction in the foreground power
The HERA analysis pipeline for power spectrum esti- relative to the EoR signal. To derive a delay-spectrum
mation exploits the inverse covariance weighting to re- specification for a prototype HERA element using the
duce the foreground contribution to the measured visi- inverse covariance weighting, we begin with the simula-
bilitydata. Inthelightofthisanalysistechniquewefur- tions of Thyagarajan et al. (2016) and use OQE formal-
ther investigate the limits of EoR to foreground power ism to estimate the amplitude of the power spectrum
ratio from what is established by conservative estimates from a simulated data vector x which contains contribu-
of Thyagarajan et al. (2016). In the following section we tions from the EoR, the foreground and the instrument
brieflydescribethepowerspectrumestimationtechnique noise. The covariance matrix is estimated from 80 dif-
usedinHERAanalysisandinversecovarianceweighting. ferent realizations of the data vector x between 0 24 h
−
Weevaluatetheinstrumentdelayresponseinthecontext of LST. The power spectrum, estimated using one real-
of covariance weighting. ization of x, with and without covariance weighting are
showninfigure8. Theorangecurverepresentsthepower
5. REVISED DELAY SPECTRUM SPECIFICATION spectrum computed from the EoR model for two adja-
USING INVERSE COVARIANCE WEIGHTING cent HERA elements with a given baseline orientation.
Results presented in previous sections focused on the Thebluecurveshowsthepowerspectrumoftheobserved
delay-domain performance of the prototype HERA el- sky signal including both foreground and the EoR using
HERA dish reflectometry 9
Fig. 7.—Delayresponseoftheinstrument(black)estimatedfrommeasurementsofthecomplexreturnlossoftheHERAelement. The
blue curve shows the same estimated from the EM simulation of the HERA element corrected for the zero delay response. From right
to left: the shaded gray regions show the minimum foreground attenuation relative to the EoR power required to detect the EoR power
at a given delay (or in corresponding k(cid:107) mode). At a given delay, detection of EoR power spectrum at smaller k(cid:107) mode requires higher
attenuation of the foreground power. This is shown by different shaded region each represening a minimum k(cid:107) mode that can be probed
for that particular foreground attenuation profile in the delay domain. The instrument delay response estimated over the full band is
dominatedbythesharpestfeaturepresentintheobservationband. At120MHz,thefeedimpedanceisbestmatchedtothe50Ωreference
impedance providing a very low value of return loss. This feature, when Fourier transformed to delay domain results in a wide response.
Athigherfrequencies,themeasuredreturnlossvariessmoothlywithfrequencyresultinginnarrowdelayresponseoftheinstrument.
thedelaytransformtechniqueafterappropriatewindow- regions in Figure 9. This relationship is not so straight-
ing. Here, the contribution from the bright foregrounds forward when using optimal quadratic power-spectrum
dominate the lower delay modes, making separation of estimation. For example, a direction-independent band-
the EoR from the foreground impossible up to delays passshapethatmultipliestheforegroundsignalfallsinto
300 ns. The power spectrum estimated after weight- a single delay mode that can be downweighted essen-
≈
ing the data by the inverse of the covariance matrix is tially to zero. This makes it problematic to interpret
illustrated by the purple curve in the plot. Weighting the output power spectrum in the light of reflectometry
the simulated data by the covariance matrix which is constraintswhichcomesfromthebeamintegratedreflec-
alsoestimatedfromthesimulatedvisibilitydramatically tometry measurements. Without knowing in detail the
reduces the foreground power relative to the EoR signal direction dependence of the HERA element chromatic-
at low delays, opening up the possibility of estimating ity, it is impossible to estimate exactly how many Eigen
the EoR power spectrum at those delays. modes will be occupied by the systematics arising from
In a simple delay-transform power spectrum estima- the foreground-HERA element interaction.
tion, the chromatic antenna response is convolved with Inarelativelyconservativeestimate,wesimplyusethe
the foreground signal in delay domain. As described in inverse covariance weighted foreground power spectrum
Thyagarajan et al. (2016), this enables us to set a spec- from simulations as an effective input foreground ampli-
ification for the reflectometry of the HERA element as tude and repeat the translation to a reflectometry spec-
a function of delay, as illustrated by the grey shaded ification as before. The result is illustrated by the col-
10 Patra et al.
ple reflections in the system. The prototype HERA el-
ement delay response, in conjunction with the inverse
covariance method of foreground suppression, indicates
thattheHERAprototypeelementsatisfiesthenecessary
conditiontomakeasuccessfulpowerspectrummeasure-
ment for the spatial modes as low as k >0.1h Mpc 1.
−
(cid:107)
6. CONCLUSION
The interplay between the extremely bright sky sig-
nal and the system response has remained somewhat
undetermined for the first generation 21cm experiments
suchasMWA,LOFAR,PAPER.Withthetheoryofred-
shifted 21cm experiments very well evolved, it is now
absolutely necessary to quantify the instrumental lim-
its of these measurements in order to produce accurate
power spectrum estimates. The delay-domain perfor-
mance of the HERA dish is central to HERA’s function
asapowerspectruminstrument. Inthispaper, westud-
Fig. 8.—Blue: Powerspectrumoftheskysignalestimatedfrom
ied the performance of a prototype HERA element in
thesimulatedvisibilitywithtwoadjacentHERAelementsandthe
foreground,EoRmodeldescribedinsection4.3.3. Orange: Power both frequency as well as delay domain. We introduced
spectrumofonlytheEoRsignal. Purple: Powerspectrumestimate a mathematical formalism that explicitly relate the de-
of the sky signal after weighting the simulated visibility by the layresponseoftheHERAelementtoEoRtoforeground
inverseofthecovariancematrixcomputedfromthesimulateddata.
power ratio in the delay domain. The effects of multiple
reflections in a HERA element is investigated in detail
and their effects on the measured visibility is estimated.
ReflectometrymeasurementscharacterizedHERA’sper-
formance in delay domain domain and, as equation 9
shows,thesemeasurementsmustbeadjustedforadiffer-
enceintransmission/receptionatthefirstfeedencounter
in order to be interpreted as the delay response of a
HERA element relative to an incident plane wave from
the sky. It is also shown that the windowing the mea-
sured data by the Blackman-Harris window, shown in
Thyagarajan et al. (2016) is critical for accurately mea-
suringtheantennadelayresponseathigherdelays,where
sidelobesfrommuchhigheramplituderesponsesatsmall
delays can easily dominate. Windowing the measured
data conclusively shows that the lower delay response
results from the structural reflections in the feed. Given
Fig. 9.—DelayspectrumofHERAelementestimatedfromthe
reflectometrymeasurementsbyavectornetworkanalyzer(black). the critical nature of the windowing function, it is rec-
ThecoloredregionshowstherevisedEoRtoforegroundpowerra- ommended that all reflectometry measurements be per-
tioasafunctionoftheseparationindelayspaceafterweightingthe
formed in the frequency domain, so that the data could
simulatedvisibilitybytheinverseofitscovariance. Duetoinverse
covariance weighting of the visibility, the foreground contribution be Fourier transformed with the appropriate window.
at each delay is reduced relaxing the limits on the required fore- The delay spectrum estimates are compared with the
groundattenuationrelativetotheEoRbyabout30dBateachde- electromagnetic simulation of the HERA element (De-
lay. Incomparisontothisrevisedspecificationderivedfromthesky
Boer et al. 2016b; Ewall-Wice et al. 2016). The same
powersimulation,theHERAelementdelayresponsedemonstrates
itscapacityforasuccessfulpowerspectrumdetectionwithoutany is then compared with the estimates derived from the
additionaldesignimprovement. foreground simulation of Thyagarajan et al. (2016). The
performanceisalsoevaluatedinthelightofHERApower
ored shaded regions in Figure 9. This approach ignores spectrum estimation technique using inverse covariance
the ability of OQE to identify and invert instrumen- weighting formalism. Taken all together, we conclude
tal covariances, making it a relatively conservative stan- that the HERA antenna element, with a PAPER-style
dard. Ontheotherhand,shouldthenumberofdirection- crossed dipole feed and cylindrical cage, satisfies the cri-
dependentspectraleigenmodesofthedishbecomelarge, teria necessary to meet its science goal.
thisapproachcouldpotentiallyunderestimatetheimpact Acknowledgements: This work was supported by
ofdishchromaticity. Therefore,wepresentitasademon- the U.S. National Science Foundation (NSF) through
stration that our reflectometry specifications could be awards AST-1440343 & AST-1410719. ARP acknowl-
substantially less stringent for HERA’s OQE pipeline, edges support from NSF CAREER award 13 52519. AL
but suggest not over-interpreting the exact level of the acknowledges support by NASA through Hubble Fel-
implied specification without a detailed analysis of the lowship grant #HST-HF2-51363.001-A awarded by the
direction dependence of the reflectometry results. SpaceTelescopeScienceInstitute,operatedbytheAsso-
Thereflectometrymeasurementsdemonstratetheper- ciation of Universities for Research in Astronomy, Inc.,
formance of the HERA element including it intrinsic for NASA, under contract NAS5-26555. This work is
chromaticity and chromaticity generated due to multi- completedaspartoftheUniversityofCaliforniaCosmic