Fundamentals of Radio Interferometry

Transcription

1 A lon tim ao, in a alaxy far, far away Fundamntals of Radio Intrfromtry Fundamntals of Cohrn Thory Gomtris of Intrfromtr Arrays Ral Intrfromtrs An ltron was movd. This ation ausd an ltromanti wav to b launhd, whih thn propaatd away, obyin th wll-known Maxwll s quations. At a latr tim, at anothr loal, this EM wav, and many othrs from all th othr ltrons in th univrs, arrivd at a snsin dvi (a.k.a. antnna ). Th suprposition of all ths filds rats an ltri urrnt in th antnna, whih (thanks to vry lvr ninrs) w an masur, and whih ivs us information about th ltri fild. What an w larn about th radiatin sour from suh masurs? Lt us dnot th oordinats of our ltron by: (R, t), and th vtor ltri fild by: E(R,t). Th loation of th antnna is dnotd by r. An mittin ltron (on of many) It is usful to think of ths filds in trms of thir sptral ontnt. Imain th volta wavform oin into a lar filtr bank, whih domposs th tim-ordrd fild into its mono-hromati omponnts of th ltri fild, E (R). Baus th mono-hromati omponnts of th fild from th farrahs of th univrs add linarly, w an xprss th ltri fild at loation r by: E ( r) P ( R, r) E ( R)dV An obsrvr R R Th lstial sphr Whr P (R,r) is th propaator, and dsribs how th filds at R influn thos at r. r 3 4 At this point w introdu simplifyin assumptions:. Salar filds: W onsidr a sinl salar omponnt of th vtor fild. Th vtor fild E boms a salar omponnt E, and th propaator P (R,r) rdus from a tnsor to a salar.. Th oriin of th mission is at a rat distan, and thr is no hop of rsolvin th dpth. W an thn onsidr th mission to oriinat from a ommon distan, R -- and with an quivalnt ltri fild E (R ) 3. Spa within this lstial sphr is mpty. In this as, th propaator is partiularly simpl: P ( R i, r ) R π R r whih simply says that th phas is rtardd by π R-r / radians, and th amplitud diminishd by a fator / R-r. r / W thn hav, for th monohromati fild omponnt at our samplin point: iπ R r / E ( r ) Ε ( R ) ds R r Not that th intration ovr volum has bn rplad with on of th quivalnt fild ovr th lstial surfa. So what an w do with this? By itslf, it is not partiular usful an amplitud and phas at a point in tim. But a omparison of ths filds at two diffrnt loations miht provid usful information. This omparison an b quantifid by formin th omplx produt of ths filds whn masurd at two plas, and avrain. Dfin th spatial ohrn funtion as: * V E ( r ) E ( r ) 5 6

2 W an now insrt our xprssion for th summd monohromati fild at loations r and r, to obtain a nral xprssion for th quantity V,. Th rsultin xprssion is vry lon -- s Equation 3- in th book. W thn introdu our fourth and vry important assumption: 4. Th filds ar spatially inohrnt. That is, * E ( R ) E ( R) whn R R This mans thr is no lon-trm phas rlationship btwn mission from diffrnt points on th lstial sphr. This ondition an b violatd in som ass (sattrin, illumination of a srn from a ommon sour), so b arful! Usin this ondition, w find (s Chap. of th book): V ( r, r ) E ( R ) R iπ R r / i R r R Now w introdu two important quantitis: Th unit dirtion vtor, s: Th spifi intnsity, I : / π R r / s R R r I R s E ( ) And rpla th surfa lmnt ds with th lmntal solid anl: ds R dω Rmmbrin that R >> r, w find: ds 7 8 V ( r, r ) I ( s) i π s ( r r )/ dω This bautiful rlationship btwn th spifi intnsity, or brihtnss, I (s) (whih is what w sk), and th spatial ohrn funtion V (r,r ) (whih is what w must masur) is th foundation of aprtur synthsis in radio astronomy. It looks lik a Fourir Transform and in th nxt stion w look to s undr what onditions it boms on. A ky point is that th spatial ohrn funtion ( visibility ) is only dpndnt upon th sparation vtor: r - r. W ommonly rfr to this as th baslin: b r r Gomtry th prft, and not-so-prft Cas A: A -dimnsional masurmnt plan. Lt us imain th masurmnts of V (r,r ) to b takn ntirly on a plan. Thn a onsidrabl simplifiation ours if w arran th oordinat systm so on axis is normal to this plan. Lt u, v, w b th rtanular omponnts of th baslin vtor, b, masurd in units of th wavlnth. Orint this rfrn systm so w is normal to th plan on whih th visibilitis ar masurd. Thn, in th sam oordinat systm, th unit dirtion vtor, s, has omponnts (th dirtion osins) as follows: and ( l, m, n ) dω 9 w s W thn t: iπ v α γ β v whih is a -dimnsional Fourir transform btwn th projtd brihtnss: I / and th spatial ohrn funtion (visibility): V (u,. l os( α) m os( β ) n os( γ ) u And w an now rly on a ntury of ffort by mathmatiians on how to invrt this quation, and how muh information w nd to obtain an ima of suffiint quality. Formally, l m iπ v V ( u, With nouh masurs of V, w an driv I. dudv

3 Cas B: A 3-dimnsional masurmnt volum: But what if th intrfromtr dos not masur th ohrn funtion within a plan, but rathr dos it throuh a volum? In this as, w adopt a slihtly diffrnt oordinat systm. First w writ out th full xprssion: v, w) I ( l, (Not that this is not a 3-D Fourir Transfor. Thn, orint th oordinat systm so that th w-axis points to th ntr of th rion of intrst, and mak us of th small anl approximation: n m i π vm+ wn) ( l + )/ θ / v, w) I ( l, iπ w iπ vm wθ / ) Th quadrati trm in th phas an b nltd if it is muh lss than unity: wθ << Or, in othr words, if th maximum anl from th ntr is lss than: λ θ max < ~ ~ θsyn w B thn th rlation btwn th Intnsity and th Visibility aain boms a -dimnsion Fourir transform: ' iπ v 3 4 Whr th modifid visibility is dfind as: ' π iw V V And is, in fat, th tru visibility, projtd onto th w plan, with th appropriat phas shift for th dirtion of th ima ntr. Th Stationary, Radio-Frquny Intrfromtr Th simplst possibl intrfromtr is skthd blow: b s / s s I lav to you th rst of Chaptr in th book. It ontinus with th ffts of disrt samplin, th fft of th antnna powr rption pattrn, som ssntials of sptrosopy, and a disours into polarimtry. W now o on to onsidr a ral intrfromtr, and larn how ths omplx ohrn funtions ar atually masurd. V A os[ ω( t )] b An antnna V A os( ωt) A [os( A ω ) + os(ωt ω )] R A A os( ω ) A A os(πυ b s/ ) 5 6 In this xprssion, w us A to dnot th amplitud of th sinal. In fat, th amplitud is a funtion of th antnna ain and abl losss (whih w inor hr), and th intnsity of th sour of mission. Th sptral intnsity, or brihtnss, is dfind as th powr pr unit ara, pr unit frquny width, pr unit solid anl, from dirtion s, at frquny. Thus, (inorin th antnna s ains and losss), th powr availabl at th volta multiplir boms: dp I (s) dωdad Th rspons from an xtndd sour (or th ntir sky) is obtaind by intratin ovr th solid anl of th sky: R C d da I ( s)os(πb s/ ) dω 7 This xprssion is los to what w ar lookin for. But baus th osin funtion is vn, th intration ovr th sky of th orrlator output will only b snsitiv to th vn part of th brihtnss distribution it is insnsitiv to th odd part. W an onstrut an intrfromtr whih is snsitiv to only th odd part of th brihtnss by buildin a nd multiplir, and insrtin a 9 dr phas shift into on of th sinal paths, prior to th multiplir. Thn, a straihtforward alulation shows th output of this orrlator is: R S d da I ( s)sin(πb s/ ) dω W now hav two, indpndnt numbrs, ah of whih ivs uniqu information about th sky brihtnss. W an thn dfin a omplx quantity th omplx visibility, by: πi b s / R RC + irs I ( s) dωdad 8

4 This is th sam xprssion w found arlir allowin us to idntify this omplx funtion with th spatial ohrn funtion. So th funtion w nd to masur, in ordr to rovr th brihtnss of a distant radio sour (th intnsity) is providd by a omplx orrlator, onsistin of a osin and sin multiplir. What s oin on hr? How an w onvnintly think of this? Th COS orrlator an b thouht of astin a sinuoisidal frin pattrn onto th sky. Th orrlator multiplis th sour brihtnss by this wav pattrn, and intrats (adds) th rsult ovr th sky. In this analysis, w hav usd ral funtions, thn ratd th omplx visibility by ombinin th osin and sin outputs. This orrsponds to what th intrfromtr dos, but is lumsy analytially. A mor powrful thniqu uss th analyti sinal, whih for this as onsists of rplain os(ωt+ϕ) with i( ω t+ ϕ ), thn takin th omplx produt <V V *>. A dmonstration that this lads (mor lanly) to th dsird rsult I lav to th studnt! Frin pattrn ast on th sour. Orintation st by baslin omtry Frin sparation st by baslin lnth Frin Sin Th SIN orrlator pattrn is offst by ¼ wavlnth. 9 Th mor widly sparatd th frins, th mor of th sour is sn in on frin lob. Widly sparatd frins ar nratd by short spains hn th total flux of th sour is visibl only whn th frin sparation is muh ratr than th sour xtnt. Convrsly, th fin dtails of th sour strutur ar only disrnibl whn th frin sparation is omparabl to th fin strutur siz and/or sparation. To fully masur th sour strutur, a wid varity of baslin lnths and orintations is ndd. On an build this up slowly with a sinl intrfromtr, or mor quikly with a multi-tlsop intrfromtr. Th Efft of Bandwidth. Ral intrfromtrs must apt a ran of frqunis (amonst othr thins, thr is no powr in an infinitsimal bandwidth)! So w now onsidr th rspons of our intrfromtr ovr frquny. To do this, w first dfin th frquny rspons funtions, G(), as th amplitud and phas variation of th sinals paths ovr frquny. Insrtin ths, and takin th omplx produt, w t: + * V I ( ) G ( G ( s π i d Whr I hav lft off th intration ovr anl for larity. If th sour intnsity dos not vary ovr frquny width, w t iπ V I (s)sin( ) dω whr I hav assumd th bandpasss ar squar and of width. 3 4

5 Th sin funtion is dfind as: sin( π x) sin( x) π x ( πx) whn πx << 6 This shows that th sour mission is attnuatd by th funtion sin(x), known as th frin-washin funtion. Notin that ~ B/ sin(θ) ~ Bθ/λ (θ/θ rs )/, w s that th attnuation is small whn θ << θ rs In words, this says that th attnuation is small if th frational bandwidth tims th anular offst in rsolution units is lss than unity. If th fild of viw is lar, on must obsrv with narrow bandwidths, in ordr to masur a orrt visibility. 5 6 So far, th analysis has prodd with th impliit assumption that th ntr of th ima is stationary, and loatd straiht up, prpndiular to th plan of th baslin. This is an unnssary rstrition, and I now o on to th mor nral as whr th ntr of intrst is not straiht up, and is movin. Th Stationary, Radio-Frquny Intrfromtr with insrtd tim dlay s s s s b s/ In fat, this is an lmntary addition to what w v alrady don. Sin th fft of bandwidth is to rstrit th rion ovr whih orrt masurs ar mad to a zon ntrd in th dirtion of zro tim dlay, it should b obvious that to obsrv in som othr dirtion, w must add dlay to mov th unattnuatd zon to th dirtion of intrst. That is, w must add tim dlay to th narr sid of th intrfromtr, to shift th unattnuatd rspons to th dirtion of intrst. V A os[ ω( t )] b s / b An antnna V A os[ ω( t )] os[ ω( )] + os[ωt ω( )] A A os[ ω( )] A A os[πυ b ( s s) / ] 7 8 It should b lar from insption that th rsults of th last stion ar rprodud, with th hromati abrration now ourrin about th dirtion dfind by. That is, th ondition boms: θ/θ rs > / Rmmbrin th oordinat systm disussd arlir, whr th w axis points to th rfrn ntr (s ), assumin th introdud dlay is appropriat for this ntr, and that th bandwidth losss ar nliibl, w hav: b s/ ul + vm + wn b s/ w n dω / n 9 Insrtin ths, w obtain: iπ [ ul+ vm+ w( l m )] This is th sam rlationship w drivd in th arlir stion. Th xtnsion to a movin sour (or, mor orrtly, to an intrfromtr loatd on a rotatin objt) is lmntary th dlay trm hans with tim, so as to kp th pak of th frin-washin funtion on th ntr of th rion of intrst. W will now omplt our tour of lmntary intrfromtrs with a disussion of th ffts of frquny downonvrsion. 3

6 Idally, all th intrnal ltronis of an intrfromtr would work at th obsrvin frquny (oftn alld th radio frquny, or RF). Unfortunatly, this annot b don in nral, as hih frquny omponnts ar muh mor xpnsiv, and nrally prform mor poorly, than low frquny omponnts. Thus, narly all radio intrfromtrs us downonvrsion to translat th radio frquny information to a lowr frquny band. For sinals in th radio-frquny part of th sptrum, this an b don with almost no loss of information. But thr is an important sid-fft from this opration, whih w now quikly rviw. os(ω IF t-ω RF ) ω LO φ LO os(ω IF t-ω IF +φ) i( ω RF ω IF + φlo V ) This is idntial to th RF intrfromtr, providd φ LO ω LO os(ω RF t) os(ω IF t+φ LO ) (ω RF ω LO +ω IF ) 3 3 Thus, th frquny-onvrsion intrfromtr (whih is ttin quit los to th ral dal, will provid th orrt masur of th spatial ohrn, providd that th phas of th LO (loal osillator) on on sid is offst by: δφ π f Th rason this is nssary is that th dlay,, has bn addd in th IF portion of th sinal path. Thus, th physial dlay ndd to maintain broad-band ohrn is prsnt, but baus it is addd at th wron (IF) frquny, rathr than at th riht (RF) frqundy, an inorrt phas has bn insrtd. Th nssary adjustmnt is that orrspondin to th diffrn frquny (th LO). LO Som Conludin Rmarks I hav ivn hr an approah whih is basd on th ida of a omplx orrlator two idntial, paralll multiplis with a 9 dr phas shift introdud in on. This lads quit naturally to th formation of a omplx numbr, whih is idntifid with th omplx ohrn funtion. But, a omplx orrlator is not nssary, if on an find anothr way to obtain th two indpndnt quantitis (Cos, Sin, or Ral, Imainary) ndd. A sinl multiplir, on a movin (or rotatin) platform will allow suh a pair of masurs for th frin pattrn will thn mov ovr th rion of intrst, and th sinusoidal output an b dsribd with two paramtrs (.., amplitud and phas) This approah miht sm attrativ (fwr multiplirs) until on onsidrs th rat at whih data must b lod. For an intrfromtr on th arth, th frin frquny an b shown to b: dw F ωu osδ dt Hr, u is th E-W omponnt of th baslin, and ω is th anular rotation rat of th arth: 7.3 x -5 rad/s. For intrfromtrs whos baslins xd thousands of wavlnths, this frin frquny would rquir vry fast (and ompltly unnssary) data loin and analysis. Th purpos of stoppin th frins is to prmit a data loin rat whih is basd on th diffrntial motion of sours about th ntr of th fild of intrst. For th VLA in A onfiuration, this is typially a fw sonds. 35