SIS MIXER ANALYSIS WITH NON-ZERO INTERMEDIATE FREQUENCIES. S.-K. Pan and A. R. Kerr National Radio Astronomy Observatory* Charlottesville, VA 22903

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1 3- SIS MIXER ANALYSIS WITH NON-ZERO INTERMEDIATE FREQUENCIES S.-K. Pan and A. R. Kerr Natonal Rado Astronomy Observatory* Charlottesvlle VA 2293 ABSTRACT Most desgn and analyss of Superconductor-Insulator-Superconductor (SIS) mxers has been based on Tucker's quantum theory of mxng always wth the assumpton of a zero ntermedate frequency. Ths paper relaxes the zero IF constrant and explores the performance of SIS mxers at ntermedate frequences whch are a sgnfcant fracton of the LO frequency. The complete expresson for the elements of the admttance matrx [Y] of a quantum mxer has been gven by Tucker []. Examnaton of ths expresson shows that for a non-zero IF the symmetry of the admttance matrx elements between opposte sdebands.e. Y Y s broken by the quantzaton of the IF. Therefore the converson gan and nose temperature of opposte sdebands wll be dfferent. Also the output mpedance of the mxer s no longer real and the output reactance vares as a functon of the IF. In ths study a quas fve-frequency approxmaton to Tucker's theory (.e. snusodal LO and fve small-sgnal sdebands wth non-zero IF) s used to smulate the performance of SIS mxers at dfferent ntermedate frequences. It s found that the zero-if approxmaton s approprate for IF/LO <.. The SIS mxer tself should be capable of excellent performance for a very wde IF bandwdth (2% to 3% of the LO frequency) and the desgn of an optmum couplng network between the mxer and IF amplfer should be straghtforward. The Natonal Rado Astronomy Observatory s a faclty of the Natonal Scence Foundaton operated under cooperatve agreement by Assocated Unverstes Inc.

2 I. INTRODUCTION SIS mxer technology has progressed very rapdly n the past few years. SIS recevers are now well establshed as the most senstve recevers over much of the mllmeter and submllmeter spectrum and have been used on almost all mllmeter- and submllmeter -wave rado telesco p es around the world. The development of SIS mxer technology has been guded manly by Tucker's quantum mxng theory whch predcts non-classcal behavor -- quantum-lmted senstvty negatve nput and output mpedance and converson gan -- n a resstve mxer wth a sharp I -V characterstc. Although Tucker's theory provdes a complete framework for understandng the behavor of SIS mxers due to ts complexty the quantum mxng theory has always been appled wth the assum p ton of a zero ntermedate frequency (IF). The output (IF) frequency of most exstng mllmeter-wave astronomy recevers s much smaller than the nput (RF) frequency and the IF photon voltage (hwfe) s small compared wth the voltage scale of juncton's dc nonlnearty. In such cases the ntermedate fre quen c y s suffcently small and the zero-if assumpton s justfed. For applcatons such as contnuum observatons a very wde IF would actually be preferable and the valdty of the zero-if assumpton needs to be reexamned. Mllmeter- and submllmeter -wave recevers wth an extremely wde IF (- several tens of GHz) would have the advantage that a wde RF bandwdth could be covered by a sngle fxed-frequency LO whch substantally smplfes the desgn of the LO system. The am of ths paper s to use Tucker's theory wthout the zero-if assumpton to examne the behavor of SIS mxers wth hgh IFTs. H. SIMULATIONS We have nvestgated the behavor of SIS mxers as functons of IF at two dfferent LO frequences 5 and 345 GHz and wth varous source and load mpedances. In order to compare the results of ths work to the zero-if case studed n our earler paper [2] we use the same I -V curve that of a 4-juncton array of Nb/A -Al 2 3 /Nb junctons fabrcated by Hypres shown n Fg.. (Tucker's theory for sngle juncton can be appled to an N juncton array by scalng the nduced photon step current and voltage scale by a factor of N.) We assumed that: () the array s voltage-based at the center of the frst photon step below the gap voltage;.e. Vo= Vgap NrICOp2e where V gal s the gap voltage of the array N s the number of junctons n the array and w p s the LO fre q uenc y and (2) the pumpng parameter a = ev lnrco p =.2 where V I s the ampltude of the LO voltage at frequency wp. Although the analytcal forms of the complete expressons for the elements of the admttance matrx [Y] and the nose current correlaton matrx [H] of a tunnel juncton mxer have been derved by Tucker [] usng a perturbaton techn q ue certan approxmatons can be made to reduce computatonal effort n calculatng these elements. In an earler paper [2] we showed that for most practcal desgn parameters the behavor of a SIS mxer wth a zero-if could be modeled qute accurately usng Tucker's theory assumng a quas fve-frequency approxmaton n whch fve small-sgnal sdebands are allowed but the LO voltage s assumed snusodal. In ths paper we extend ths approach by ncludng the ntermedate frequency as an ndependent parameter. Usng ths model the 5x5 small-sgnal admttance matrx elements Y mn and current correlaton matrx elements of our hypothetcal mxer can be calculated drectly from the closed-form expressons gven n Tucker and Feldman's paper [3]. As explaned n [2] the quas fve-frequency approxmaton assumes that the embeddng mpedance seen by the juncton s fnte at IF ( ) the upper and

3 lower sdebands C. ± IF) the second harmonc sdebands (2w p ± w d but the second LO harmonc (2w p ) s short-crcuted at the juncton (LO voltage waveform s snusodal). In the present work we further assume that () at the second harmonc sdebands the juncton s termnated by the juncton capactance only (2) the juncton capactance s tuned out at IF by the load susceptance and at both the upper and lower sdebands by the source susceptance (3) both upper and lower sdebands are termnated by the same source conductance 4 e Y - USB = G S = YLSB) and (4) the RF source and IF load conductance are equal. Ths s shown n Fg. 2. Although these assumptons may be dffcult to mplement n a real mxer they provde a convenent smplfcaton for ths ntal study of the effects of hgh ntermedate frequences. Also for smplcty only the case of w p R N C = 4 s examned n ths work (R N s the normal resstance of the juncton (array) and C s the juncton (array) capactance). The accuracy of the quas fve-frequency approxmaton s very good for w p R N C 4 and because the juncton capactance s tuned out at IF and the sgnal and mage frequences the value of w p R N C should have very lttle effect on the mxer performance as a functon of the ntermedate frequency. Three dfferent values of R RF (the recprocal of the source conductance.e. R /G 5 ) are used n the smulatons for each LO frequency: R RF =.2*RN.6*R N and.*r N for 5 GHz and R RF =.6*R N.*R N and.4*r N for 345 GHz. The mxer converson gan nose temperature nput return loss and output admttance are calculated n each case as functons of the normalzed ntermedate frequency (IF/LO). III. RESULTS The converson gan of 5 GHz and 345 GHz mxers s shown n Fgs. 3(a)- (b). As expected the upper sdeband (USB) gan and the lower sdeband (LSB) gan are not equal. At low IF the USB and LSB gans converge to the zero-if results gven n [2]. For IF/LO <.3 the LSB gan ncreases as IF/LO ncreases whle the USB gan decreases. At IF/LO =.3 the dfference n gans s 2-3 db. The gan of both sdebands drops very quckly for IF/LO >.5. Fgs. 4(a)-(b) show the equvalent nput nose temperatures of the mxers. Dfferences n nose temperature between two sdebands are less than 5 K for IF/LO <.3 but ncrease very rapdly for IF/LO >.5. The nput return loss of the same mxers s shown n Fgs. 5(a)-(b). In each case the return loss s reasonable for IF/LO <.3; the worst RF nput match occurs between IF/LO =.4 and.6. The output admttance of the mxer s computed for each case and converted nto the recprocal output conductance ( R mt = /Gut) and the recprocal output susceptance = /But) - These data are then normalzed to the recprocal load conductance ( R f = /G f ) and plotted n Fgs. 6(a) and (b). In general the mxer output mpedance s large but no longer real and the output reactance s capactve. Snce the output of the mxer s capactve t can be represented as shown n Fg. 7 by a resstor R mt whose value s gven n Fgs. 6(a)-(b)) n parallel wth a capactor whose value s gven n Fgs. 8(a)-(b). Also plotted n these two fgures for comparson s the juncton capactance. For IF/L <.3 the equvalent output capactance s almost constant and s small compared to the juncton capactance. It shows a broad peak between IF/LO =.4 and.8.

4 The results presented above can be summarzed as follows: the performance dfference between the upper and lower sdebands s almost neglgble and the zero-if approxmaton s appro p rate for IF/LO <.. As IF/LO approaches.3 the dfference between two sdebands becomes sgnfcant. The overall mxer performance deterorates very rapdly for IF/LO >.5. The output admttance of the mxer s no longer real and the output susceptance s capactve and vares a functon of IF. DISCUSSION It s well-known that n the zero-if case because of the quantzaton of the LO frequency the behavor of an SIS mxer s completely determned by the current at only those voltages equal to the bas voltage plus multples of hco/e (.e. V = Vo + nrw p /e where n = ± ± 2... whch are called photon ponts). The matrx elements between opposte sdebands thus have the followng symmetry: rnn = and H rnn = Furthermore snce the crcut external to the nonlnear mxer element (juncton) does not dstngush between the sgnal and the mage (.e. Y -sgnal = Ymage) the mxer operates n the double sdeband mode and the IF output mpedance s real. In ths paper we have shown that when the ntermedate frequency s not zero even though both sdebands of the mxer are termnated by the same source admttance the mxer converson gan nose temperature and the nput return loss at the two sdeband frequences are dfferent. The behavor of the SIS mxers presented n ths paper can be understood by examnng the expresson for the admttance matrx elements Y mn of the nonzero IF mxer gven n [3]. These expressons consst of complcated Bessel seres summatons nvolvng the juncton's dc current I dc (V) and ts Kramers- Krong transform I KK (V) at not only the "photon Ponts" gven by V = V - + nhco p /e but also at voltage ponts equal to the "photon ponts" plus/mnus one IF photon due to the quantzaton of the IF.e. V = Vo + nhco p /e hcoule where n = ± ± 2 ± 3... Ths breaks the symmetry of the admttance matrx elements between opposte sdebands.e. Y*m...n. For the same reason the nose current correlaton matrx elements also lose ther symmetry (.e. Hun _ rn ). Therefore although the termnatons at the upper and lower sdebands are dentcal the performance of these two sdebands s dfferent. Examnaton of these Bessel seres summatons also reveals that for typcal values of the pumpng parameter a the frst few terms especally the ones nvolvng the photon ponts at V = Vo + nhco p /e rcou/e where n = and domnate the Bessel sum for the Y mn and n. As IF/LO approaches.5 wth the juncton voltage based at the center of the frst photon step below the gap voltage the photon ponts at V = Vo + hw F e and V = Vo + heo p /e hcoule approach the gap voltage. Ths causes the mxer nose temperature to ncrease very rapdly and the gan of the mxer to peak at IF/LO -.5. Ths argument also mples that the mxer's usable IF bandwdth decreases f t s based at voltages other than the center of the photon step. Recent work by S. Padn et al. of OVRO has demonstrated that t s possble to acheve good SIS recever Performance wth moderately broad IF bandwdth (- 4 GHz) usng a smple couplng network between the SIS mxer and HFET IF amplfer [4]. The results presented n ths paper show that the SIS mxer tself should be capable of excellent performance wth a much wder IF bandwdth (2% to 3% of the LO frequency). Furthermore because the equvalent output capactance s almost constant and s small compared to the juncton capactance the desgn of an optmum couplng network between the mxer and IF amplfer should be straghtforward.

5 V. ACKNOWLEDGMENT The authors would lke to thank Dr. Maran W. Pospeszalsk for many helpful dscussons. REFERENCES [] J. R. Tucker "Quantum lmted detecton n tunnel juncton mxers" IEEE J. Quantum Electron. vol. QE-5 no. pp Nov [2] A. R. Kerr S.-K. Pan and Stafford Wthngton "Embeddng mpedance approxmatons n the analyss of SIS mxers" IEEE Trans. Mcrowave Theory Tech. vol. 4 no. 4 pp Aprl 993. [3] J. R. Tucker and M. J. Feldman "Quantum detecton at mllmeter wavelength" Rev. nod. Phys. vol. 57 no. 4 pp Oct [4] S. Padn D. P. Woody J. A. Stern H. G. LeDuc R. Blundell C.-Y. E. Tong and M. W. Pospeszalsk "An ntegrated SIS mxer and HEMT IF amplfer" Proc. of the Sxth Int. Symp. on Space Terahertz Tech. March 995.

6 2 I HA HY43:5K 4 2 K 4 junctons R = 72 ohms NA 5 5 V mv Fg.. I-V curve used n the smulatons. Ths curve s for a seres array of four Hypres Nb/A-Al 2 3 /Nb junctons at 4.2 K as used n [2]. INTRINSIC JUNCTION 2co p + 6.)IF tuned out L. tuned out CJp (USB) L a)(sb tuned out GIF GS 2W P WIF Fg. 2. The mxer's small-sgnal equvalent crcut showng the sdeband termnatons used n the smulaton.

7 . IF/LO Fg. 3(a). Upper and lower sdeband converson gan as functons of IF/L for a 5 GHz SIS mxer. Results are shown for Ra F /R N =.2.6 and. 4 2 LO. 345 GHz Rrf/Rn =.4 X ' _. u. USB L = LSB /. /. V P..: -- I " Rrf/Rn = L- 3 A...(.. : _. Rrf/Rn =.6 ***4"e"t " '''' ;c '.%... N.. l'' "... : : / : IF/LO Fg. 3(b). Upper and lower sdeband converson gan as functons of IF/LO for a 345 GHz SIS mxer. Results are shown for R RF /R N =.6 and.4.

8 6 LO = 5 GHz I u =USB L = LSB /.."' 2..:.-- t.2..." ' _ Fg. for 4(a). Upper and lower sdeband nose a 5 GHz SIS mxer. Results are shown temperature as functons of IF/Lo for RRF/RN..2.6 and. 6 LO = 345 GHz z 4 U = USB L = LSB."... R rf /Rn D ) co (/) c..-. AVA.4.44v.2.4 Fg. 4(b). Upper and lower sdeband nose temperature as functons of IF/LO for a 345 GHz SIS mxer. Results are shown for R RF /R =.6 and.4. IF/LO.6.8

9 Fg. 5(a). Upper and lower sdeband nput match ( Pnput 2) as functons of IF/LO for a 5 GHz SIS mxer. Results are shown for R RF /RN =.2.6 and. LO = 345 GHz.... AO -5 2sy U USB : L = LSB I v Rfan = z -2 :. I IF/LO Fg. 5(b). Upper and lower sdeband nput match ( I Pnput 2 ) as functons of IF/LO for a 345 GHz SIS mxer. Results are shown for R RE IR N =.6 and.4.

10 . 8 6 cc 4 c 2 x -2 - = c LO = 5 GHz Rout-P/Rf I : AI fl.. :::.t s r ;.... :. ''';..._ ' e : Rout-P/Rf :.; F/LO Fg. 6(a). Normalzed recprocal output conductance and susceptance as functons of IF/L for a 5 GHz SIS mxer. Results are shown for RRF/RN.2.6 and. 8 6 LO = 345 GHz z 4 N Rout-P /Rf : I 4... s ' 4" * ---'''' : / v....ṭ... Xout-P /R.f I :.: : '. Nw - -4-'-' ; - - Rf/Rn = 4 '.. ' Rf/R.n.--...r... 2f/R.r..6 f ; :...tt..... : ' ; F/LO Fg. 6(b). Normalzed recprocal output conductance and susceptance as functons of IF/LO for a 345 GHz SIS mxer. Results are shown for R RF /R N =.6 and.4.

11 WIF R ou4 Co t tuned out F = G S INTRINSIC JUNCTION Yout Fg. 7. The equvalent crcut of the mxer's output port. It s represented by a resstor whose value R. ut = /Gut n parallel wth a capactor whose value s shown n Fgs. 8 (a)-(b).

12 . LO= 5 GHz.8 a) C c '.6 ccs C) Juncton Capactance Rrf/Rn = C. Rrf/Rn = IF/L.6.8 Fg. 8(a). Equvalent output capactance as a functon of IF/L for a 5 GHz SIS mxer. Results are shown for R RF /R N =.2.6 and. Also shown s juncton capactance for comparson LO = 345 GHz Immlmple -.2 a) C) a..5 cts '5. j.. Juncton Capactance ' l RrfRn.6 I -- I. s. R.rf/Rn = I. ' z l --tt 4. '... = 'Nt... :...-` ; ' W IF/L. Fg. 8(b). Equvalent output capactance as a functon of IF/L for a 345 GHz SIS mxer. Results are shown for R RF /R =.6 and.4. Also shown s juncton capactance for comparson.