Gap Excitations and Series Loads in Microstrip Lines: Equivalent Network Characterization with Application to THz Circuits

Transcription

1 Gap Ecitations and Series Loads in Microstrip Lines: Equivalent Network Characterization with Application to THz Circuits Andrea Neto and Peter H. Siegel Jet Propulsion Laboratory, Caliornia Institute o Technology Abstract At submillimeter wavelengths typical gap discontinuities in microstrip, CPW lines or at antenna terminals, which might contain diodes or active elements, cannot be viewed as simple quasi statically evaluated lumped elements. Planar Schotky diodes at 2.5 THz or eample have a ootprint that is comparable to a wavelength. Thus, apart rom modelling the diodes themselves, the connection with their eciting elements (antennas or microstrip) gives rise to parasitics. Full wave or strictly numeric approaches can be used to account or these parasitics but at the epense o generality o the solution and the CPU time o the calculation. In this paper an equivalent network is derived that accurately accounts or large gap discontinuities (with respect to a wavelength) without suering rom the limitations o available numeric techniques. I. INTRODUCTION Planar circuits utilizing microstrip or coplanar waveguide are oten coupled to active devices or antennas through electrically small gaps. However, at THz requencies, practical abrication scales limit devices, such as planar Schottky diodes [1], to dimensions which are comparable to a wavelength. As a consequence, the eect o the gap dimensions on the coupling coeicient or resonant requency o the associated transmission line or antenna cannot be ignored. Nor can it be modelled as a quasi statically derived [2 and reerences there cited] single lumped element. In this paper the authors present a new ormulation or analyzing the eect o inite gap ecitations on microstrip lines. The technique involves separating the ecited modes into reactive elements which are conined near to the gap and propagating quasi TEM components, which are coupled to the surrounding transmission line and load (antenna). The ormulation is very general and can be applied to a large class o problems involving discontinuities in transmisssion line circuits. Veriication o the new method has been obtained using traditional but much more time consuming method o moments (MoM) techniques [3, 4 or eample]. Recent investigations [5, 6, 7] present a direct integration method to derive the spectral Green s unction or microstrip and strip-line. Most notably they urnish a detailed analysis o the singularities o this Green s unction, while enphasizing on the ecitation mechanisms o the propagating and leaky modes compatible with strip line and microstrip. This technique can be used to great advantage in the modeling o sub-mm wave circuits. The same ormalism was adopted by one o the authors, [8, 9] to derive, analytically in that case, the Green s unction o a slot printed at the interace between two dierent homogeneous dielectrics, and in [10] to derive ad hoc basis unctions in the MoM analysis o coplanar waveguide ecited slot antennas. As another application o this methodology, we investigate in this paper the input impedance o gap ecited microstrips coupled to planar antennas. A rigorous equivalent network is derived that accurately accounts or the ield in-and-near large-or-small (with

2 respect to a wavelength) gap discontinuities, without suering rom the lack o physical insight o purely numerically derived solutions. On the contrary the network approach will provide the guidelines on the one hand or the eicient coupling to and rom the gap, and on the other hand or the separate characterization o the reactive energy associated with the eed. Full wave tools, such as Finite Dierence in Time Domain or Finite Element Method may be necessary to characterize the minute details o a complicated diode structure but are typically not needed or the remaining part o the circuit. In our technique the spectral kernel o the Greens unction is irst used or the determination o the bound mode wave number. Then the evaluation o the residue contributions associated with this pole allows the derivation o an equivalent transormer that accounts or the gap launching coeicient on the equivalent transmission line o the bound TEM mode in the microstrip. By subtracting the current associated with this main mode on the ininite line rom the total current, a source attached current is deined. This current, or the circuit substrates used in many sub millimeter planar devices, is mostly conined in a region very close to the gap. From this current a parallel gap impedance is derived that accurately represents the reactive eects o the gap. The same ormalism adopted here or the microstrip coniguration, can be used or characterizing a wide variety o other gap ecited planar transmission lines and active antennas. II. FORMULATION: ELECTRIC CURRENTS ON THE MICROSTRIP The problem under investigation is shown in Fig. 1. Fig.1 Ininitely etended microstrip ecited with by an impressed electric ield distributed over a gap o dimensions ( $,w). An ininitely etended (along ) microstrip, printed on a grounded dielectric slab (% r1, % r2 =1) is considered. The microstrip cross section ( w, along y) is uniorm and much smaller than the wavelength. The structure is assumed to be ecited by an impressed electric ield e (,y)=e (,y) i. The electric current distribution is assumed to be characterized by a separable space-dependence with respect to and y; i.e., 2 1 w1 #y " ç w ü j(,y)=i() (1) where the transverse y-dependence is chosen to veriy the quasi-static edge-singularities 2 and the the normalization constant, w1, has been chosen in such a way that i() represents a net current lowing in the microstrip at any cross section o the slot line. The method adopted to derive the current o a gap ecited microstrip has been presented in [5]-[7]. A #

3 brie review with the minor variations introduced in this paper is presented in the appendi. As a result the electric current in the microstrip is epressed as: _ " -j5 E (k ) e # - _ i()= 1 ' D(k ) dk (2) where E ( k ) is the Fourier Transorm (FT) o the impressed electric ield, with respect to the variable, and _ _ y! " # s y y D(k )= G (k,k )J ( w k )dk (3) with J! is the Bessel unction o zero order and G (k, ky) the FT, with respect to the variables and y o the electric ield radiated by a short electric dipole lying at the interace between the slab and the homogeneous hal space or z<0, in the absence o the metallic strip. An interesting aspect o (2) is that the impact o the eed is all contained in the numerator and in most practical cases it is a simple anayitic unction. When the distribution is uniorm over the eed region e ()=-1/ $ or (- $ /2, $ /2) the ecitation is said to be a $ -gap and E (k )= sinc(k $ /2). This case will be considered speciically in the ollowing. The 1 denominator, D(k ), represents the microstrip spectral Green s unction and or a detailed discussion on the nature o the branch singularities o D(k ) the reader is reerred to [5, 6]. Fig. 2 Comple k plane topology or thin dielectric substrates. Only the TM0 pole o the grounded slab is assumed to be above cut o. The outer integral in (2) can be perormed by adopting a deormation in the comple plane below the real ais or Re ak b<0 and above the real ais or Re ak b>0 ( original path in Fig. 2), in order to avoid the branch cuts arising rom the branch points implicit in D(k ). Here it is suicient to know that these branches are square root type or logarithmic when they originate rom poles or rom the branch (k 0) o the layered Green s unction je G (k,k) y respectively. For a thin dielectric layer coniguration, (that is only the TM0 pole

4 is above cut o in the grounded slab) the comple plane topology o the microstrip s 1 Green s unction, D(k ), is well represented in Fig. 2 which shows the branch cuts arising rom k, k and k. TM0 TE1 0 II.1 Case study It is instructive to perorm a parametric study o the currents as a unction o the gap dimensions. The inormation derived will lead to the current representation and equivalent network o the ollowing sections. Fig. 3. shows the real part (3a) and imaginary part (3b) o the current due to a 1 volt uniorm ecitation or dierent gap dimensions ( $ ) rom 2 to 18. m. The width o the microstrip (w) and the height o the slab (h) are 8. m and 3. m respectively ( h= - 0 /40 at 2.5 THz). The relative dielectric constant considered is % r1 = These parameters and the maimum gap dimension are typical o thin GaAs membrane diode technology [1] or which this analysis is derived. a) b) Fig. 3 Current on an eample microstrip structure at submillimeter wave requencies (2.5 THz) obtained using the present direct integration method (equation 2). Real (a) and Imaginary(b) parts are presented or dierent dimensions o the ecitation gap ( $ varying rom 2 to 18. m). Two aspects o the problem should be noted i) A launching eiciency issue stands out, since the amplitude o the oscillation o the real and imaginary part o the current at a certain distance rom the source is regulated by the gap dimensions. As $ tends to zero, the amplitude o the current is a maimum and tends to one hal o the characteristic admittance o the microstrip line. As $ grows the value o the current diminishes. ii) The reactive energy in the eed zone is strongly aected by the width o the source. When the gap dimensions shrink to zero, the imaginary part o the current tends to grow logarithmically (Fig. 3b) indicating a capacitive reactance. When the dimensions o the gap are large, the imaginary part o the current can be less then zero, indicating an inductive reactance. II.2 Modal and source attached currents The dispersion equation, D(k )=0, deines the position o the pole o the guiding structure. The localization o the zeroes o D(k ) as a unction o the parameters o the structure ( w, % r1 ), can be perormed numerically by means o a descent along the gradient algorithm. As

5 a starting point one can use the analytical ormulas [1], knowing that the actual propagation constant is not drastically dierent. Only one pole, indicated here ater as - ök Bp, is ound or the thin dielectric layers we are interested in ( 3. m= 0 40 at 2.5 THz). In the absence o losses, the pole o the structure is ound on the real ais o the k plane (see Fig. 2 ). Knowingn its location, the integral in (2) can be approimated asymptotically or > $ /2, by the residues o the poles in k æ ökp. These residue contributions are epressed as $ jk 1 res 0 2 D(k ) i ()= 21j í sinc (k ) û p R e R=Res ç ü (4) It is apparent that this current is the one associated with the quasi TEM mode that propagates in the microstrip. A comparison between the residue evaluation (4) o the integral in (2) and the original numerical integration is presented in Fig.4a and Fig. 4b, or two unit $ -gap ecitation conigurations. Outside the source region (gray zone in Fig. 4) the ull integration o (2) and the sole residue contribution are almost superimposed due to the drawing scale. A source attached current can then be deined and will be indicated by s i ()=i()-i () (5) The source attached currents represent an overall current that contains all the separate branch conntributions, without calculating them separately as in [7]. In this respect it is s useul to observe that i can be calculated or > $ /2, deorming the original integration path on a path Pd (Fig. 2 where only the spectrum or Re( k )>0 is shown ) that surrounds all the branch cuts on the top Rieman sheet. This is possible because in thin dielectric cases, the only signiicant singularity encountered in the deormation is the pole in ö k p. res k æk p a) b) Fig. 4 Comparison between asymptotic evaluation (residue only) and direct numerical integration o the current or a sample microstrip structure at submillimeter wave requencies (2.5 THz). Also plotted is the source attached ield (real and imaginary parts in 4.b and only the imaginary part in 4.a since the real part is almost zero and cannot be distinguished in the drawing scale). The width o the microstrip and the dimensions o the ecitation gap ( $ ) are - 0/20 (6. m), - 0/60 (2. m) respectively. The height o the slab is /40 (3 m) in Fig. 4a and /20 (6 m) in Fig. 4b. The relative dielectric constant considered is ( =12.85). %r1

6 The integrand decays eponentially in the deormed path P. d Thus the numerical calculation o i s is much more eicient than along the original path. The deinition o the source attached current is also analytically etended to the source region, < $ /2 (but not this way o calculating it, which must be done in accordance with the deinition in (5)). The source attached ield is also plotted in Fig. 4. In [7] it is demonstrated that each 3/2 s o the branch contributions to the current decays as, thus also i() decays this way or aster. It s value in the source zone depends on the thickness o the dielectric considered. In the very thin dielectric slab ( h=-0/40) considered in Fig. 4a the real part o s i() is not noticeable on the drawing scale and thus is not reported. Also the imaginary s part o i() is very concentrated in the surrounding o the source. In Fig. 4b the same s curves are reported or a thicker dielectric case (h=-0/20). In this case the real part o i () is noticeable, and signiicant or larger distances rom the source. It is worth repeating at this point that ( h=-0/40) corresponds to a 3. m thick dielectric at 2.5 THz, the design target or this work. III. INPUT ADMITTANCE AND EQUIVALENT NETWORK The current representation in terms o source attached and mode contributions is particularly signiicant when only one bound mode is propagating in the microstrip. In this case it is possible to derive a rigorous equivalent network, based on a single transmission line, that characterizes the $ -gap ecited ininite microstrip input impedance. We will see in section IV, that inite (loaded) microstrips can also be studied by resorting to the same simple equivalent network. The input admittance o a $ -gap ecited microstrip can be epressed in the spectral domain as: y = '.5 (6) in _ 2 sinc (k B$ /2) D(k ) -_ Accordingly, the total comple power radiated by the source can be epressed as: B " 2 tot # g in P = V y (7) The knowledge o the microstrip characteristic impedance z 0 [3], together with the amplitude o the traveling wave in the microstrip rom (4), consents to write the power launched in the microstrip quasi TEM mode as: + $ m g 0 P = V ar b z (8) where R = 21j í sinc (k 0 ) û 2 R. This power actually leaves the ecitation zone in orm o a couple o outgoing travelling waves ( ö k p ). As we will see later on, this power is the only one that might give rise to interactions with components o the circuit which are located at a signiicant distance, in terms o a wavelength, rom the eeding zone. In order to derive an equivalent network that rigorously takes into account the gap eects one may associate all the power that is not launched into the the quasi TEM mode as being assigned to a lumped load indicated as. The power conservation then imposes: z gap Pgap= Ptot P m (9)

7 This gap impedance is imagined as in parallel to the source and to the modal loading as in ig. 5a. Accordingly both P and P can also be epressed as: m gap " 2 " 2 m # g m gap # g gap P = V y ; P = V y (10) thus givin the eplicit relation between the various admittance components o the network in ig.5a y gap=y in y m (11) The epression or y m according to this deinition equating the two epression or P m in (8) and (10) y =2aR + b 2 z (12) m 0 Fig. 5 Equivalent network or $ gap ecited microstrip structure. a) admittance decomposition (y in=y gap+y m); b)interpretation o the modal impedance interms o transormer connected equivalent transmission line. So that, given (12), (11) becomes an operative deinition. It is useul to perorm a urther algebraic manipulation epressing the modal admittance as 2z0 R y m= + 2z0R + 2 y0 2z = n 2 (13) 0 where n=2z R =2z 2 jr sinc (k $ a 1 b ) (14) The introduction o the coeicient n consents to represent the modal part o the impedance associated to the series o two transmission lines representing the quasi TEM mode. These transmission line are series ed, via a transormer o turn ratio n, by the parallel combination o the source voltage and the gap impedance. The ull circuit is shown in Fig. 5b. The use o the transormer is a convenient wayo separating the dierent ield behavior associated with the eeding and the propagation o the transmission lines. The electric currents in the source are oriented parallel those in the microstrip, both along. The electric ields in the source and in the transmission line are orthogonal, (along and along

8 z respectively). The value o the transormer n or the same microstrip o Fig.3 (operated again at 2.5 THz) is plotted in Fig. 6. as a unction o the gap s width, $.. Fig. 6 Transormer or $ gap ecited microstrip structure o Fig. 3 Fig. 7 Total and gap admittance (gap and total susceptances are equal) The gap and total admittance or to the same case are presented in ig.7. It should be noted that the gap conductance is virtually zero or the case o the thin dielectric slab, and the gap susceptance, coincident by deinition with the total one, goes rom being capacitive to inductive or the investigated range o $. IV. NUMERICAL EXAMPLES, APPLICABILITY AND CONSIDERATIONS The purpose o deriving the equivalent network is to characterize the input impedance o inite, as opposed to ininite, microstrips. When the bound mode currents are much more intense than the source attached currents, at a certain distance rom the source one may assume that the source attached currents intrinsically satisy Mawell s equations or the inite structure. The loading o the end-points o the microstrip will generate a relection that can be characterized by a load impedance. Thus the interaction between the two outer loads and the source can be accounted or via a complete equivalent network with equivalent lumped elements z z ( Fig. 8) located at the endpoints o the microstrip. l1, l2

9 IV.1 Numerical eamples Two inite length microstrips printed on a GaAs membrane (% r1 =12.85) have been investigated using the equivalent network in ig. 8. Fig.8 Equivalent network or a $ -gap ecited and loaded microstrip. The dielectric thicness and the width o the conductor are 3. m and 2.5. m respectively. The pertinent open end impedance has been derived as in [3]. Fig.9a shows the imaginary part o the input impedance or a 60. m long microstrip as a unction o the requency or three dierent gap dimensions (0.5, 1.15, and 15. m). It is ound that in the three cases the resonant requency is the same and signiicant dierences are observed at larger values o the impedance. Also the pertinent curves obtained via a ull wave Method o Moments are shown or the m and the 15. m cases. The MoM tool uses piece wise sinusoidal sub-domain basis unctions and rectangular weighting unctions in the longitidinal dimension and assumes a rectangular distribution in the transverse dimension. The agreement between the network solution and the MoM solution is ecellent apart or a minimal (0.5 %) resonant requency shit, probably due to the dierent electric current transverse distribution assumed in the two models. a) b) Fig. 9 Impedance o a delta gap ecited microstrip as unction o the requency. a) (l=60. m); b) l=40. m). Also a parametric study as a unction o $ and comparison with MoM are shown. The width o the microstrip and the thickness o the dielectric are 2.5. m and 3. m respectively. The grounded dielectric is made o Ga-As. The case in Fig. 9a ( l=60. m) is geometry corresponds to a case in which the load impedance ( z li, i=1,2), when relected the ecitation zone, is nearly zero. In view o this the gap impedance eect is relatively minimal since it is in parallel to a very low impedance load. The case in Fig. 9b ( l=40. m) corresponds to a very high load impedance

10 at the ecitation zone, and as seen the impact o the gap becomes dramatic. The resonant requency as a unction o the gap width shits very signiicantly or $ varying rom 2.3. m to 6.9. m and the curves can easily be distinguished. For even smaller gaps the resonance requency shits out o the igure since the source attached currents tend to be singular as $ tends to zero. In Fig.9a is also plotted a MoM result, or $ =3.84. m, which again shows ecellent agreement with out model. Finally in Fig. 9b the result that would be achieved by using the equivalent network without the transormer and the gap impedance introduced in this paper is plotted. Not only is the resonance requency poorly modelled but also the absolute values o the impedance vary widely orm those o the more accurate equivalent network analysis. IV.2 Applicability i res(=l/2) 1 We will assume than when i s(=l/2) < 100 the accuracy, A (=l/2) > 100, in the evaluation o input admittance is satisactory. In ig. 10 i res / is is plotted as a unction o the requency at an observation point located at - 0 /10 rom the center o the eed. The microstrip s nominal characteristic impedance is 50 ohms, the dielectric thickness and the width o the conductor strip are 3. m and 2.5. m respectively. The dimension o the ecited gap is -0/40 and the dielectric is composed o silicon (% r1 =11.7). It is ound that i res / is is always larger than 100 or requencies up to 3 THz. Thus, 3/2 since i s() decays algebraically,, any load at a distance larger than -0/10 can be characterized accurately with the present equivalent network. The same is not true or thicker dielectrics. Also the investigated microstrip is only weakly dispersive, as the normalized propagation constant k p/k 0 calculated as eplained in sect. III varies rom 2.8 to 2.9 in the requency range considered in Fig. 10. IV.3 Considerations The loads, Fig. (8), typically represent active impedances. For eample i they represented slot antennas their impedance should be calculated when both slots are radiating in phase. In this case the load connected at the end points o the microstrip will present a larger real part. Also in this case requency shits as signiicant as in Fig. 9b, can be epected. This has already been eperimentally veriied in a similar slot circuit [11]. I the actual source under investigation is not well represented by a $ -gap generator, a correction in the equivalent network parameters can be made by simply substituting the sinc unction in (6), (14) with the relevant FT E (k ), provided the more realistic unction e () is still an even unction, with respect to. I e () also presents some odd components, then a single impedance should not be used, and an even and odd impedance characterization should be adopted. The etra degree o reedom (odd mode) can still beneit rom an equivalent network, that can be derived with the same methodology used here or the even mode. At this point e() could or eample the result o the investigation o complicated source in isolation. Indeed the procedures presented in this paper are ound to be particularly useul in connecting an isolated load or source to the remaining part o the circuit. Full wave tools such as Finite Dierence Time Domain or Finite Element Method may be necessary to caharacterize the tiny details o a complicated source (like a diode) but are oten not necessary or analyzing the remaining part o the circuit.

11 Fig. 10 Ratio between the amplitudes o the residue contribution to the currents and the source attached currents. The graph can be read as 1/ aa(l=0.1-0) b. The microstrip impedance is nominally 50 ohm. The width o the microstrip and the thickness o the dielectric are 2.5. m and 3. m respectively. The grounded dielectric is made o silicon (% r1 = 11.7). The width o the gap is -0/40. V. CONCLUSIONS An equivalent network has been presented or the wide band characterization o the gap in a series ed and loaded microstrip line. The network consists o a transormer and a gap impedance. The transormer accounts or the launching coeicient o the mode on the equivalent transmission line representing the microstrip and is evaluated analytically by simple interpretation o the spectral epression o the microstrip Green s unction. The gap impedance accounts or the current distribution in the region surrounding the eed, and is calculated by simple spectral domain numeric integration (less then 3 minutes on a 300 MHz Pentium II). The procedure, validated via MoM, will work or any kind o dielectric stratiication, but is meaningul when one bound mode is above cut o in the microstrip. Submillimeter s wave problems in which this equivalent network can be applied include the design o receivers (i.e. diodes as miers or requency multipliers) and generators (i.e. photomier based local oscillators). APPENDIX: SOLUTION OF THE INTEGRAL EQUATION According to the equivalence principle an Electric Field Integral Equation (EFIE) is set up and enorced, via a point matching like procedure only on the longitudinal ais o the micro strip (y=0). '' D 0 g (-, 0-y ) j (, y )d dy = - e (, 0) (A0) where D is the entire microstrip domain. The point matching (or razor blade) as testing procedure is selected in order to mantain the same ormalism used [8-10] or the characterization o slot antennas printed between two dierent homogeneous dielectrics. All the quantities are scalar, because o the -polarisation o all the scattered and impressed electric ields involved in the equation, and neglecting cross polarized electric

12 currents (in view o the small cross section o the strip). The let hand side o (A0) represents the electric ield at z=0 radiated by the electric current density j. The Green s unction, g, that appears in the kernel o this integral equation, is the electric ield radiated by a short electric dipole lying at the interace between the slab and the homogeneous hal space or z<0, in the absence o the metallic strip. The electric current distribution is assumed to be characterized by a separable spacedependence with respect to and y; i.e., j(,y )=i( ) j (y ). > (A1) The transverse y-dependence is chosen to veriy the quasi-static edge-singularities j (y)=. (A2) > w1 2 1 #y " ç ü w j(y) possesses a closed-orm Fourier transorm, > # J (k )=FT{j (y)}=j ( wk ), > y >! " # y (A3) where J 9 is the Bessel unction o zero order. The IE (1) is now transormed into the k- Fourier domain as š _ š I( k ) ' G (k, y ) j t(y ) dy = E (k ), (A4) -_ where I( k ) and G (k, -y ) are the FT, with respect to the variable, o i() and g (, y), respectively. Equating the two spectra in k it is apparent that where I(k = ; (A5) E (k ) D(k ) š 1 t 21 y! " # s y y -_ -_ D(k)= ' G(k, y ) j(y )dy = ' G(k,k)J( wk )dk (A6) represent the spectral domain Green s unction o the microstrip line and G(k,k) y is the FT o g(, y) with respect to both the variables and y. The electric current on the microstrip can then be epressed as: _ " -j5 E (k ) e # - _ i()= 1 ' D(k ) dk (A7)

13 Reerences [1] P.H. Siegel, R. P. Smith, M. C. Gaidis, S. Martin "2.5 THz GaAs Monolithic Membrane-Diode Mier," IEEE Trans. Microwave Theory and Techniques, vol. MTT 47, pp , May [2] P. Benedek, P. Sylvester, " Equivalent Capacitance or Microstrip Gap and Steps," IEEE Trans. Microwave Theory and Techniques, vol. MTT 20, pp , Nov [3] P.B. Katehi, N.G. Aleopoulos, "Frequency Dependent Characteristics o Microstrip Discontinuities in mm-wave Integrated Circuits," IEEE Trans. Microwave Theory & Techniques., vol. MTT 33, no. 10, pp , Oct [4] N.G. Aleopoulos, Shih-Chang Wu "Frequency Independent Equivalent Circuit Model or Microstrip Open-End and Gap Discontinuities," IEEE Trans. Microwave Theory & Techniques., vol. MTT 42, no. 7, pp , July [5] C. Di Nallo, F. Mesa, D.R. Jackson. "Ecitation o Leaky Modes on Multilayer Stripline Structure" on IEEE Transactions on Microwave Theory and Techniques, Vol. 46, no. 8, pp , August [6] F. Mesa, C. Di Nallo, D.R. Jackson "The theory o surace-wave and space-wave leaky-mode ecitation on microstrip lines" on IEEE Transactions on Microwave Theory and Techniques, Vol. 47, no. 2, pp , Feb [7] D.R. Jackson, F. Mesa, M.J. Freire, D.P. Nyquist, C. Di Nallo, "An Ecitation Theory or Bound Modes and Residual-Wave Currents on Stripline Structures" Radio Science, Vol. 35, no. 2, pp , March-April [8] A. Neto, S. Maci, "Analytical Solution or Gap-Ecited, Ininite Printed Slot Lines", to be presented at IEEE Antennas and Propagation Symposium, Boston July, [9] A. Neto, S. Maci, "Green s Function o Gap-Ecited, Ininite Printed Slot Lines", NASA New Technology Report (April 2001). [10] A. Neto, P.J.I. De Maagt, S. Maci "Optimized Basis Functions or Slot Anetnnas Ecited by Coplanar Waveguides", submitted to IEEE Trans. Antennas and Propagation (March 2001) [11] R.A. Wyss, A. Neto, W.R. McGrath, B. Bumble, and H. LeDuc, "Submillimeterwave spectral response o twin-slot antennas coupled to hot electron bolometers," Proceedings o the 11-th Int. Symp. on Space THz Tech., Ann Arbor, MI, May 1-3, 2000.