NATIONAL RADIO ASTRONOMY OBSERVATORY ENGINEERING REPORT NO. 121 ARECIBO THREE-MIRROR SYSTEMS, V: APERTURE ILLUMINATION AND SHAPED SURFACES

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1 NATONAL RADO ASTRONOMY OBSERVATORY ENGNEERNG REPORT NO. 2 ARECBO THREE-MRROR SYSTEMS V: APERTURE LLUMNATON AND SHAPED SURFACES S. VON HOERNER MARCH COPES

2 ARECBO THREE-MRROR SYSTEMS V: APERTURE LLUMNATON AND SHAPNG PROBLEMS S. von Hoerner Summary: The frs sudy phase has resuled n decsons abou many desgn parameers. The nex ask wll be o develop a shapng procedure whch calculaes he fnal surface shapes of he wo upper mrrors for yeldng equal pahlengh wh opmum aperure llumnaon and whou cross polarzaon. The presen repor asks how dffcul he problem s whch he shapng has o solve and how hs may be eased. We show he llumnaon properes of he presen unshaped sysems and we nvesgae he dependence of hese properes on he desgn parameers by calculang wo seres of sysems wh varable parameers. We fnd ha only one parameer s of drec mporance (for llumnaon as well as polarzaon): he angular offse beween he feed axs and he ncomng ray from he aperure cener. Ths sudy shows a general problem: we have eher very odd llumnaons (dffcul for he shapng procedure) or very large erary mrrors (large expensve domes). Some compromses are suggesed. Bu for he selecon of a fnal sysem we frs mus have a shapng program o learn how odd an llumnaon can handle.

3 2. Seled and Open Quesons A he meeng January 985 several decsons were aken ermnang a frs phase of general and prelmnary nvesgaons. Snce he cos of he secondary wll be he man em abou wce he cos of he suppor cables (P Seson) we shall no ncrease he aperure dameer; and snce he ground screen was decded upon we need no dmnsh he dameer or he offse. Thus 700 f dameer wh 50 f offse were frozen. Axsymmerc sysems shall no be furher nvesgaed. Bu he ellpcal aperure of P.S.Kldal was agreed on maybe wh an addonal par "fllng-n beween he ears' of he Gregoran. The radome suggeson of P.Seson was acceped. The longes wavelengh shall be A-- m for sure and A. = m hopefully (Kldal). The full range needs abou 0 recevers and he larges horn shall be a permanen fxure bu acng only as scaer sheld for shor wavelenghs. For changng recevers he 45"-mrror was abandoned n favor of he roang urre. The feed range shall be 90 (sze of erary as seen by feed). - - The mos mporan sll open quesons call for he developmen of a shapng procedure and s applcaon o varous desgns. My su g ges- have a vercal feed axs for avodng cross polarzaon was only ono n analogy o my old wo-mrror calculaons; hs may be acually very dfferen above a sphercal prmary. And how odd an unshaped aperure llumnaon can be urned no he desred one by he shapng we sll do no know. Bu boh quesons mus be answered before we can proceed oward a fnal desgn geomery.

4 2. Aperure llumnaons For he presen purpose we rea only he one-dmensonal case along he aperure dameer. The calculaon of he secondary Gregoran was ndcaed n Repor 2 and descrbed n deal n he Appendx of Repor (where )--. (2 for he presen case). The coordnaes of he erary are hen found as follows where we consder only Gregoran erans all beng ellpses. We call (x 2 z 2 ) he coordnaes of focus F 9 call ) hose of ' ' and R he dsance beween F 2 and F. The erary has one more free parameer s sze whch we fx by he choce of m 5 he smalles dsance of F from he full ellpse of whch he erary s a par see Fg.2. The sum of he wo dsances o F. and F from any pon of he ellpc erary hen s L = R + 2m. () We call c=x -x 2 and g=z -z 2' secondary has he angle B s erary s a dsance k from F2 Any ray afer reflecon a he from he z-axs; and along hs ray he We om he dervaon and gve he resul. The dsance k s L 2 - R2 2(L + g case + c snbs) (2) and he coordnaes of he erary are x = x2 - k snb () s z = z 2 - k cos - s' (4) A he feed he angle beeen he ncomng ray and he z-axs s

5 4 found from an B f = (x - x )/(z - z ). (5) Frs we calculae B f only for he wo exreme rays -400 f and a=+00 f and for he aperure cener a=-50 f. We oban he feed range B (-400)-B (00) and he drecon of he feed axs B = f f a ( (-400)+B f (00))/2. The dfference AB Bf (-50)-B a beween he feed ' axs and he ncomng ray from he aperure cener we call he feed offse and he relave feed offse we call = 4 B/B. (6) Ths we regard as our frs measure of he dffculy o be faced by he shapng procedure. f he demand of zero cross polarzaon means ha an orhogonal polar aperure grd mus be mapped ono he feed paern agan as an orhogonal polar grd hen he aperure cener mus be mapped ono he feed axs or B = 0 afer shapng. (7) Second we call for any ray s cenered feed angle B c B f We assume a Gaussan feed paern a ( 8) exp-(b /B )2/2) c and we adop a 20 ab edge aper whch gves he sandard devaon of he feed paern as (9) = (4 Vln 0). (0) Thrd we go along he aperure dameer n seps of 4a. Afer each sep we call AB s and AB he resulng angular seps a F c) and F ' and now we defne he aperure llumnaons as

6 2 = (AB /4a) from omndreconal feed a F 2' = (AB la) from omndreconal feed a F f ' () (2) P ' from apered feed a F. () The normalzaon of (r/2)(2t/60 ) wh r=870 f serves o le l = a a=0 from omndreconal feed a Fl. Afer compleon of he full aperure dameer we look for he maxma and mnma of 2' and ' and we call k = max( k )mn( k ) = llumnaon rao. (4) We regard as anoher measure of he dffculy faced by he shapng procedure (mosly underesmaed snce we have used large seps of 4 a=5o f excep for dealed drawngs). No only he magnude of ' bu also any srong asymmery of (a) ndcaes a dffculy snce we mus have ( a) = consan and = afer shapng (5) f we wan unform aperure llumnaon for maxmum gan; or we mus have a slghly apered bu sll symmercal (a) f we wan smaller sdelobes.. Our Prevous Sysems As a ypcal example we show n Fg. our sysem 5 of Repor 20. was defned by a feed range of =90 a vercal feed axs and he consran ha he erary does no come oo close o F 2' We see n Fg.lb ha has he well-known nverse aper of a Gregoran over a 2 sphercal prmary. The erary does gve some bu no much dsoron: s smlar o 2 bu somewha more asymmerc. However he shape

7 6 of (a) s very odd and asymmerc and wll be very neresng o fnd ou wheher or no a shapng procedure can handle a case lke ha Also he feed offse s very large A=.6 or B=.5; whch can be seen n Fg lb along he aperure dameer as he large offse of he feed axs ray from he aperure cener. Table. Several prevous sysems. Ordered wh ncreasng llumnaon rao ' (*5 o 8: Repor 20; 4P22: s Addendum; L.Baker: drawng 2--84) name / AB B R m mr 2 # Table summarzes for sx prevous sysems her desgn parameers and he resulng llumnaon raos wh Y 9 = angle beween he x-axs and he drecon from F 2 o F. The able s ordered wh ncreasng and we look for correlaons. There are only wo sgnfcan ones. Frs s well correlaed wh (bu no wh ). Ths means 2 hopefully whaever sysem geomery we selec as a good one for hs feed wll also be good for oher feed paerns. Second he only desgn parameer of drec sgnfcance for o be he relave feed offse B=4B/B ' seems

8 . Two Seres of Sysems We found earler ha B should no be oo large snce mus be zero afer shapng for avodng cross polarzaon. And we jus found ha agan should be small because (and only ) s srongly correlaed wh whch mus be uny afer shapng. We now wan o nvesgae wheher really depends only on B and how o keep B small. We reduce he number of varables. Snce we see no correlaon beeen and 2 seems ha he locaon of F 2 has no drec nfluence so we keep fxed. We also fx R snce seems of no drec m p orance- F henlesonacrcleofraclusrabouf 2 as shown n F g.2. We consder angle y 9 as our ndependen varable of a seres of sysems and hen we have wo choces and use boh: eer we choose fxed-sze erary ellpse (allowng only convenenly small erary mrrors). Or we choose a fxed range B. (of 90 as seleced bes for long wavelenghs). n he frs case range B s a dependen varable and n he second case s he sze of he erary ellpse. For he frs seres Fg. shows he llumnaon raos and he feed range. We have chosen R-6 f and m-6 f for he erary whch gves R+2m=28 f for he long dameer of he ellpse and hs hen s he upper lm for he sze of any erary. Angle f s an mporan desgn parameer wh srong nfluence on and B. For a convenen locaon of he feed cabn a Arecbo we should consder only a lmed range of abou "50. Fg. hen demonsraes a problem: angle ) a should be small for obanng a small ' bu hen he range B s oo small. And a range of 90 0 gves =00 whch s larger han waned (bu hopefully sll managable?) regardng he shapng.

9 8 The second seres s shown n Fg.4 for a fxed feed range of =90. We see a smlar problem l and B are nce and small bu only for small r' where he erary wll be large and vce versa. As seems he man problem s no so much he llumnaon he polarzaon. ' bu B self for Fnally Fg.5 confrms ha he llumnaon rao does depend on he offse B bu on no much else. We added a hrd seres of he second ype wh fxed =60 wch also agreed wh he wo oher seres. Also he sx prevous sysems from Table f n alhough hey have dfferen R F 2 and secondary mrrors. Thus for he choce of a fuure fnal sysem we may jus concenrae on geng a small B ( whn he necessary consrans). 5. Four Examples of he Second Seres Snce he feed range of =90 has been agreed on (lmed horn sze for longes wavelengh) we show some deals of he second seres selecng four nsrucve cases. Fg.6 wh T-40 0 s he case of a small erary dameer d =20 f only bu wh farly large =9 and raher large B=.288 showng n Fg 6b as he very large offse of he feed axs ray from he aperure cener probably dffcul o handle for he shapng. Fg. 7 wh f=20' looks much beer wh only =22. and B=.54 bu now he erary s raher large wh 8 f dameer. seems hen ha a compromse would le beween Fgs.6 and 7 a abou r=00. Fg.8 wh P '=90' serves o show he smalles llumnaon rao =9.24 wh B=.074 only bu havng a rdculously large erary. Fnally Fg.9 wh r=-0' s he case wh B=0 before shapng bu wh an mpossble erary cung even no he causc.

10 v. - ~O"V -M> - DJ 0 /D-) V'D Jl}z' 0 '-": --.--~ F---- ~-~ 7- ~ ~ "'l/ : ; : -~ : \ ~... : " j ~ l..a ~ of \ ';'-' :~ ~ ~ -.-..H- r -rr-' ~. ~ '. ' ' \ ~~ ' -- ~- ~~. f-- ---;-- : -+- \ ~ \ N~u' w.~ ~ \.ll ~.~ ~ ~ : : ~ " ' : J..~.olo'" :'- "'l ; : : ' ; " \ 'or.j. '" J.j. ~ ' ~ r-. 'No. ~~ ~ "' ~ l ~.F " ' 7r~ '" ~ L~~ '~ m ~~ -- : : r- - : ' l ; : 4 ~ - ~ j '~ f; " ~ ~Nl : ' ; ---- : 'c--- " : :..-. ; : : : : ; ' TT : : T U : '".. : ; \ Fg.. Sysem 4f 5 of Repor 20. a) s geomery defned by feed range ~/ and a verval feed axs. b) The llumnaon across he aperure dameer: 2 from omndreconal feed a F 2 ; J same from F J ; from feed wh 20 db edge aper a F J Normalsaon: = from ornnd. feed a F on axs.

11 \ \ \.. 7;- ~-~ >- Fg.2. Two seres of sysems o fnd he dependence of he llumnaon onhe desgn parameers. All sysems have he same secondary mrror defned by he locaon of F2 and he hegh of v. All sysems have F (he feed) on a crcle abou F2 wh radus R 2 6 f bu a varous angles ~ from he x-axs. The frs Geres has a fxed-sze erary ellpse wh m = 6 f. The second seres has a fxed fe~d range of ~ ~ 90.

12 .~. ':== < r ~ r '--- j -" ~ r':j..-~-- ~/_. T _ "c.r --:: _ --.--:c- '\. : -:Lc:~;._=:::'_""'\ 'C.c 8d'.lfJ.-. ~:: =-=~. Fg.. The frs seres: small erares wh m 26f. For each angle <p of Fg.2 he llumnaons ~(a) were calculaed as defned n Fg. and her maxma and mnma were noed. The llumnaon rao hen s defned as k = max(~)/mn(~). These raos show a very srong dependence on <p and so does he feed range ~ Regardng he applcaon o Arecbo especally he locaon of he feed cabn only a small range of <p would be possble abou 60 0 ~ <p ~ 50 0 We may have very nce and small bu h~p s oo narrow. And choosng would gve a large :» 00.

13 o "0 SD ~; :------r..-= -. f-r----" '~ -""";J.-:- ' ~-_~~~-=f--=. ;:=:::-===( ==== C:""'.:;Cc C:-'.' ' " :r- ~ f-~j/ ~-+----~~---- ' -': ~ ':===-f f-"-- ~ ~ ' -- ~j L ' '..: '::-.=:_.:.--~;--=.--=-~ o Fg.4. The second seres: large feed range wh ~ = 90 o a) Values of m as needed for ~ = 90; and resulng relave feed offse B. b) llumnaon raos k as funcons of ~. Feed offse B and llumnaon can be made small bu hen he erary mrror ges large (large m).

14 f l-h+ '" ':l. ==~ ~=::._':=L ~" =. < --~.:~~~.---~~- ~- -~ E ~---- <--+- ==;=- /'r"l ::~... ~.~ - 'r-" -~.j-::..:::~~--"- "r ~ _:- ~.~-=~~ ~=f+=r '- c_=-= :.._:-~-:..._._. ==== ==~-::=_:--' ~- --- ~EEEf;;:: :l_-~-=--~_~_;---::;_ ~-~ = -' -:2 "c..."" _' -'" _= ~.==-~: ~ c_ -.:;.~-~-.~ 2.. '"... Fg.5. The llumnaon rao seems o depend only on he relave feed offse B. A hrd seres wh ~ was added and also agreed. The sx prevous sysems (see Table ) even have dfferen secondary mrrors seres 8 m/r 0 frs second 90 ~ X hrd

15 _~v -yo -;0 -l -[ D JO 'f) so ~ -Vf() )..0 ~- ~- ~ l~ f--: -;--:--: ;) ~-.c:.- ~ ---- ;---;; ~ ;' -'Fl+. ~'_:_; : J. : l -~; : ~ ~ " :s' ( ; :~= J...--.~~ : lr ll "--~ ~ -;--;~--... '-: ':- : ' j--.---'--- : '). ' '. ~ : : J" Tll "' ) ( l r$\ ~ 4 ' : \ LU- " c- V.- ~- -- -~ L ~ ~~ ' ~ " '. : " '-" ' "'. '. : :.. './ ~ : : ~ "l... ' -r'.~ ; ' r : -~ o 0) ~ ~ ': == N l'... j of ')0- j (V~ : ' '.....A ~ 'n ~ ~ ~r'.~ ~fr N.. "" - ' ~ ' r--:----.a : : : "~ : :... l... ' ~... f'' u.-- : '. 'r-- K-~ N \ \ ' \ Fg.5o Second seres wh 'f= 40. a) Geomery: small erary bu large ll\. b) Aperure llumnaon: vares very srong and asymmerc.

16 _~v -yo -;0 -v -o 0 f) 2fJ Jr) 'f) ro ~ -ln - 'O'V _2.D"O _ "J 0 CKJ '-D"V o~:fee' sa~~'s'~~~~fees~'~~'~ ~~~~~EE' ~ ~~~~~7~~:~~~~ : j : ~J[ \.:) : '~l '': ~ * ~ * ~l +4+. bl ~ ~ \-: l ~ '~ ; jl ].l'l "j.~nlj.. lj.: :n \ ~ r~~; ~"" l ~ ; ~~'~: ~+~++~~~~~'~~~~'~~~"~'~:~' T:~'r+++++'+'+'~'~'~~~~'Hr~:~~'~' ~~~~~_ : J N".' ~~~~~+++\~++++\+~4\~~~~\~'~ ~N~~\~ ~~~~~\++.++~++~~~~\~\~~jr;~ ~~~;~:~' ~-r:--' ~~"~'~.. ~ : '"./. ~-4 la ' "NJN:'"h~ ll llj~k+llj : : 4\ "':'+\-+-+\ \ ++\-rl\ \+\ f-l-l +-+= ;:h 'z" _ ~\.++~ L- \ ~\ ~-+- ~~'~N~ H++ "-+j~..;r-+. :rf~----:--a~"7"l Rfh- ~---'-;-\ ~: -~-=:~ ~' ~ 'L-- ~:--'- '\ ' -----r-h-~'.:.-----r--;"'+-+-r-~-h-j---.--j-~--l-~-~ ( l : j : \ : "\\-.- " ' ' ~ \ -+-l-l- ~LJ~~-l-+~ \++ l H-- ---l f--\ H. l--ī -l-h'-'-t't----:---'--;---- \ +-- :: ; 0 L\~\ \-LJ\ L ~\.:..J. ~ \ J...l.\..L.l.\...:...:\ \...:... L:- LL ~\..:...:.. \.l-.\..l..:_~\ L...LL\~\ ~l :...: ~l "--_: -..J.:_'_ '-J...:_. _ \...-_ \-- Fg. 7. Same wh ''= 20 0 mproved ; may be useful afer some changes (smaller erary).

17 -0 -o -'V -'0... "v -yo <).0 -v o J() 'f) ~Q c~ ~:=U.~o-r~~f~~~~~:~~.~~ ~ B~~;~ ~ ~~l~r~~~~ ['. :- " -.".-----=.. -.~~~~ --~T--' -- T ~:u~--"_:.;:~~ l /0 4m.... j~]~c.~.~~s~~ ~~~~~~~'~.~ ~~~~. ~;;= ~ -7 --j -:-::- :-:--::~;-::::=-r=:~_-:::- "l7e----pj c. - :r. ~ 'L~ cl fs~\ ~ -\;- l=s2f~ -~OO ~<\~~:0~~==e~~:~e~~f~~~~ -'f ~-:-::::j..--r-/n//'-c - -d--±=:;: ~A'..o=~~~ ~~ l~ ':. o~~~h'~/rl-~.;l;~~~ ~~c=~~~ ~ ~:~~~\ ~ ~f=;. \..;.~_~;...-.;... L.. ~:.. ~.:._...:_..'.;- :':"'~:::l.;'-~'-' _-r"' - W ~~~ 2. ~'_~-?~+-.. ::=~'~::'.' ~=~ ~.:l~~-~l~" -=~~~~d::~:~.::e:::.:jrr: ~{ '.-_ '.~ ~.~-:~~;: 2~ ~ '.-r ~ L~.'. '''.'.Jl.::::.:'''':.~'~'<~:~;::.~===-- --=:::=~/- '-' ' - ~h- : ~:_:;~::::=:~~::~~'=:~~ ~'~L~:F~ 7~~~ :~~C '~~~~~kl~~~ J-~ 0. J ~&=)-:' ~~~d-~:~t~~:~j7~... '.. -j A~'l-'~ ':':~ F~ l- -;':::;=.:... :'" ~/: ::~:-= ~_Z.:-:: " "--..;...~:-;.:::/-.-- -c- +~"~:...:.;."... -C7 -l~ o.. --~ -- r az:::.~--- ~ f ~ ~- :.: -~- ~- ( ~ ~L.. '" 'r : - -- f- _ -)--..-{ - --;--- : --L : ~... " "\ \ [ \ oj"'\' '''' ~~- j ~..:;. )' j _-==.L l j ff ~ ~. ~ ~ :.- L- '- " : : ' ~-L \ [f ~ll \ ~ l"r f l J ~~ \ """ \ r..+ ~ \ : '... : ' : ".~ ~. ~~ '- ~-+-++ " ~-+- --'--' -- ~ L ' ""'... : );' ".'7'T'..-' : L ]/. L '-.' L. L ~.. [ - /" :r;...-r'"... "C' l" ~ J..' y; '~ \ : l \ V J ~~ ~')-- ~ \ \... ~ rul -"~f j - '. T \ ~ l \ / l..l. : j\ N f \ Fg.a. Second seres wh }'= 90. Terary much oo large. Smalles. \

18 -''0 o 0 r:. \ " : -v -Q "'" l \ l l _o. ~ : ' : ; l~ l : ; :..'..Jf'"'T ) -...f4 \ \ \ -~l~ ~ J.) ~ '~ '' r~ ~ l -. r""r'"' rff ~N..~ ~UJ \. \ c~ l~ l \~ \ ; ~ff:' ~~ l r.{ l Uj " : : ; : - _' "... r--. --;-- l ;:J ~- -~ "'... :.;.~ ~ : : l '77 ; ' V" :+ : ""-~ ~ l l '(" : l --;-:-;- '. V ~ "'" ~' " \ l l JJ Ll ~... ~ l ' :\ :. ; -_. ~ \ ~~ \ 00 \ ' -" \ - l \ \ ) : l -- f-;-- r l '/ l --;--- : ; ' l :. l f " - l l ; : r. l \ \ l " j l l l ; l l \ \ l f \ : ; : f L... _ 7-r l \ r llu \ \ \ \ \ \ Mappng of aperure cener ono feed axs would need N -~