with Battle–Lemarie scaling functions, 6th- and 10th-order centered differences are very close to the accuracy limit. It agrees with the theoretical evaluation. Therefore, the 6th-order accurate method can sufficiently eliminate the numerical-dispersion error of the ADI-FDTD method, which is caused by approximating the spatial derivatives using the finite-difference method. 6. CONCLUSIONS

In this paper, we have developed a high-order ADI-FDTD method by approximating the spatial derivatives of the ADI-FDTD method with Battle–Lemarie scaling functions and high-order centered differences. It has been proved that the method is unconditionally stable. The numerical dispersion of the high-order ADI-FDTD method with different schemes and the accuracy limit of the ADI-FDTD method for a given time-step size have been derived. It was found that the numerical dispersion is improved. In addition, it was observed that the numerical-dispersion error is very close to the accuracy limit at any mesh density when the 6th-order centered difference is applied.

N-PORT NETWORK FOR THE ELECTROMAGNETIC MODELING OF MEMS SWITCHES E. Perret,1 H. Aubert,1 and R. Plana2 ENSEEIHT 2, rue Charles Camichel 31071 Toulouse, France 2 LAAS-CNRS 7 avenue Colonel Roche 31400 Toulouse, France

1

Received 21 September 2004 ABSTRACT: An efficient fullwave analysis of micro-electromechanical systems (MEMS) is presented. Based on the connection of scale-dependent equivalent networks, the proposed integral-equation method allows a substantial reduction in computational time and memory, as compared to classical numerical techniques using the spatial discretization of the whole circuit. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 45: 46 – 49, 2005; Published onlinePublished online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20718

ACKNOWLEDGMENT

The work described in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China, under project no. PolyU 5131/01E. REFERENCES 1. K.S. Yee, Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans Antennas Propagat AP-14 (1966), 302–307. 2. A. Taflove, Computational electromagnetics: The finite-difference time-domain method, Artech House, Norwood, MA, 1995. 3. T. Namiki, A new FDTD algorithm based on alternating direction implicit method, IEEE Trans Microwave Theory Tech 47 (1999), 2003–2007. 4. T. Namiki, 3D ADI-FDTD method-unconditionally stable time-domain algorithm for solving full vector Maxwell’s equations, IEEE Trans Microwave Theory Tech 48 (2000), 1743–1748. 5. G. Sun and C.W. Trueman, Some fundamental characteristics of the one-dimensional alternate-direction-implicit finite-difference timedomain method, IEEE Trans Microwave Theory Tech 52 (2004), 46 –52. 6. T. Namiki and K. Ito, Investigation of the numerical errors of the two-dimensional ADI-FDTD method, IEEE Trans Microwave Theory Tech 48 (1999), 1950 –1956. 7. M. Krumpholz and L.P.B. Katehi, MRTD: New time-domain schemes based on multiresolution analysis, IEEE Trans Microwave Theory Tech 44 (1996), 555–571. 8. W.Y. Tam, Comments on new prospects for time domain analysis, IEEE Microwave Guided Wave Lett 6 (1996), 422– 423. 9. F. Zheng and Z. Chen, Numerical dispersion analysis of the unconditionally stable 3D ADI-FDTD method, IEEE Trans Microwave Theory Tech 49 (2001), 1006 –1009. 10. F. Zheng, Z. Chen, and J. Zhang, Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method, IEEE Trans Microwave Theory Tech 48 (2000), 1550 – 1580. 11. G. Sun and C.W. Trueman, Analysis and numerical experiments on the numerical dispersion of two-dimensional ADI-FDTD, IEEE Antennas Wireless Propagat Lett 2 (2003), 78 – 81. 12. G.D. Smith, Numerical solution of partial differential equations: Finite-difference methods, 3rd ed., Clarendon Press, Oxford, UK, 1985. 13. X. Zhang, J. Fang, and K.K. Mei, Calculations of the dispersive characteristics of microstrips by the time-domain finite difference method, IEEE Trans Microwave Theory Tech 36 (1988), 263–267. © 2005 Wiley Periodicals, Inc.

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Key words: MEMS switches; N-port network; surface impedance 1. INTRODUCTION

Some authors have recently reported the design of a new class of microelectronics systems named radio-frequency micro-electromechanical systems (RF-MEMS). Based on RF-MEMS switches, this class includes, for example, quasi-optical grid arrays [1], reconfigurable integrated circuits [2], and phase shifters [3]. Efficient numerical techniques are necessary for the accurate electromagnetic modeling of such promising systems. As reported in [4] (and the references therein), this task is very challenging because MEMS exhibit generally complex geometries and critical “aspectratios”. The accurate design of such circuits requires electromagnetic simulations that combines full-wave analysis (for the description of the distributed passive part of the circuits) with equivalentcircuit models (for lumped passive and active devices) [5]. Because their dimensions are much smaller than the wavelength, a MEMS switch is usually modeled by a shunt CLR lumped-element circuit derived from quasi-static analysis. However, the incorporation of such a lumped element in an electromagnetic simulator based on a spatial discretization may create an ambiguity due to the distribution of this model over the mesh [6]. An alternative electromagnetic analysis consists of applying a direct full-wave method to the overall MEMS circuit (see, for example, [7, 8] for modeling of air-bridges in coplanar waveguide). But these methods are based on the spatial discretization of the whole circuit (MEMS switches included) and, consequently, require a large computer storage capability and are time-consuming as the number of switches increases. Moreover, the wide diversity of scales in MEMS circuits may generate ill-conditioned matrices in the computation of the boundary value problem. Combining the advantages of different numerical techniques’ hybrid approaches has provided encouraging results for MEMS electromagnetic modeling [9]. In this paper, we propose an alternative approach named the multi-scale integral-equation (MS-IE) method. The first version of this technique has been reported recently for the modeling of planar active antennas [10, 11]. An extension of this precursor is proposed here for the analysis of RF-MEMS. A coplanar-waveguide shunt MEMS switch-transmission coefficient is computed from the proposed technique and compared to one obtained from a commercial electromagnetic

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box is then obtained [Fig. 2(b)]. The admittance matrix [YMEMS] of this equivalent network is computed from a classical IE technique. For a single eigenmode (N ⫽ 1), the resulting one-port network is equivalent to a surface impedance Z S ⫽ 1/j ␻ C, where the capacitance C is given by

C ⫽ 2␧ 0

wh v

冘 ⬁

n⫽0,1,2,. . .

coth共␥2n⫹1 h兲 ␥2n⫹1 h

冦

冋

␲w 2v ␲w 共2n ⫹ 1兲 2v

sin 共2n ⫹ 1兲

册

冧

2

,

(1) 2 2 2 where ␥ 2n ⫹ 1 ⫽ [(2n ⫹ 1) ␲ / 2v] ⫺ k 0 and k 0 designates the free-space propagation constant. This simple one-port network is useful for a first rapid design of MEMS containing a great number of switches. However, we show below that the number of modes (or ports) must be greater than one if accurate numerical results are required.

Figure 1 (a) Coplanar waveguide containing a MEMS switch and (b) MEMS series switch in a microstrip line

software based on the spatial discretization of the whole MEMS. The numerical results are in very good agreement with a substantial reduction in the computational time in case of the multi-scale integral-equation method. 2. THEORY

MEMS switches are devices in which a thin metallic movable bridge is suspended above a circuit surface. Actuation of the switches is generally obtained through electrostatic-field application. The device under consideration is shown in Figure 1 in coplanar waveguide and microstrip technologies. For the sake of clarity in the presentation of the MS-IE method, consider the microstrip-line series MEMS switch shown in Figure 1(b). As sketched at the top of Figure 2, this circuit is composed of two substructures or two scale levels: a small 3D object [see the cantilever shown Fig. 2(a)] and a large planar circuit [see the microstrip line displayed in Fig. 2(c)]. A small nonmetallic domain ⍀ is particularized below the thin metallic movable membrane. This small rectangular surface belongs to the two following structures: the small MEMS switch and the large distributed part of the circuit. As described below in steps 1 and 2, each scale level is first modeled by an N-port network obtained both from an integralequation (IE) technique. Finally, the connection of the two resultant N-port networks allows the estimation of the S-parameters of the whole circuit (step 3). Thus, the various scale levels combination in the whole structure is simply performed via the connection of equivalent networks. This approach extends to the nonplanar circuits of the MS-IE introduced in [10, 11]. 2.1. Derivation of the N-Port Network at the Scale Level of the MEMS Switch (Step 1) A planar subdomain ⍀ and proper boundary conditions (magnetic or electric walls) are first chosen for enclosing the MEMS switch in an artificial box [Fig. 2(a)]. From the N eigenmodes defined in the subdomain ⍀ (these modes are the propagating and evanescent modes in a rectangular waveguide having the ⍀ domain as a cross section), the equivalent N-port network representation of such a

2.2. Derivation of the Equivalent Network at the Scale Level of the Planar Circuit (Step 2) The distributed part of the circuit is sketched in Figure 2(c) and its equivalent network is shown in Figure 2(d). In addition to the two access voltage sources e 1 and e 2 , each eigenmode defined in the subdomain ⍀ is considered as a voltage excitation port. The admittance matrix [YCIRCUIT] of the equivalent (N ⫹ 2)-port network is then computed from the IE Technique using entire domain trial functions [12]. 2.3. Electromagnetic Modeling of the MEMS Switch-Wave Interaction (Step 3) The two equivalent networks connection obtained from steps 1 and 2 allows the determination of the two-port admittance matrix that relates the currents and voltages of the two access sources. The S-parameters of the whole circuit are then deduced. The number N of ports (that is, the number of eigenmodes defined in the subdomain ⍀) is taken such that the convergence of the numerical results is reached. When a large diversity of scales is present in a MEMS computational domain, space discretization methods such as the finitedifference time-domain method, the transmission-line method, or the finite-element method generate a dense mesh that requires

Figure 2 (a) Small MEMS switch and (b) its equivalent N-port network; (c) the large distributed part of the circuit and (d) its equivalent (N ⫹ 2)-port network

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Figure 3 Transmission coefficient of a membrane-supported coplanar waveguide containing a MEMS switch vs. frequency for various switch heights (h ⫽ 0.6, 0.8, 1.0, 2.0, 3.0 ␮m): (x x x) surface impedance model (one-port network); (——) eighteen-port network; (– – –) MoM electromagnetic simulations. Dimensions are [see Fig. 1(a)]: d ⫽ 40 ␮m, w ⫽ 50 ␮m, s ⫽ 260 ␮m, v ⫽ w ⫹ s/ 2, h sub ⫽ 400 ␮m and ␧ r ⫽ 1

large memory capabilities and computational time [4]. The MS-IE method presented here overcomes these difficulties by segmenting the entire MEMS domain into various scale levels and analyzing, at each scale level, a substructure that does not present critical aspect-ratios. Moreover, when the geometry of the MEMS switch is modified, only a new admittance matrix [YMEMS] has to be derived, while the matrix [YCIRCUIT] remains unchanged. 3. NUMERICAL RESULTS

The transmission coefficient of a membrane-supported coplanar waveguide containing a MEMS switch is shown in Figure 3. Simulations are performed for various heights of the MEMS switch. The minimal (respectively maximal) height value h ⫽ 0.6 ␮m (h ⫽ 3 ␮m) corresponds to the down-state (up-state) of the switch. We have observed (not shown here) that the numerical convergence of the S-parameters is reached when at least eighteen modes are taken in the subdomain ⍀; an 18-port network is then required for the accurate estimation of the transmission coefficient. As displayed in Figure 3, the numerical results obtained are in very good agreement with those given by a method of moments (MoM) software. However, our computer program based on the MS-IE method is three times faster than the MoM-based software for the analysis of this single MEMS switch. This substantial reduction in computer time can be explained by essentially two reasons: (i) when the height of the MEMS switch is modified, only a new admittance matrix [YMEMS] (step 1) has to be derived, while the matrix [YCIRCUIT] remains unchanged; (ii) we have observed that the set of modes in the ⍀ domain acts as a bottleneck for the representation of the electromagnetic field: the number of such modes is relatively low, compared to one required in the large distributed circuit or in the MEMS switch. The difference between the numerical results obtained from the surface-impedance model (N ⫽ 1) and those given when the convergence is reached (N ⫽ 18) is not negligible; it is found to be close to 18% at 10 GHz. We have obtained similar conclusions in the electromagnetic analysis of the series MEMS switch in a microstrip line. The S-parameters of a MEMS series switch is shown in Figure 4. When

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Figure 4 S-parameters (magnitudes) of a MEMS series switch (up state) in a membrane-supported microstrip line: (x x x) surfaceimpedance model (one-port network); (——) equivalent network built with 59 modes; (– – –) MoM electromagnetic simulation. Dimensions are [see Fig. 1(b)]: h ⫽ 3 ␮m, d ⫽ 70 ␮m, w ⫽ 100 ␮m, v ⫽ 220 ␮m, hsub ⫽ 300 ␮m, and ␧r ⫽ 1

the surface-impedance model (or one-port network) is adopted, the difference between our numerical results and those obtained from MoM-based software is close to 17%. However, this difference is very low when 59 modes are used in the ⍀ domain, that is, when the convergence of the S-parameters is reached. The RF currents in a CPW configuration are calculated using this method and are displayed in Figure 5. The results have been provided for an incident RF power of 1 W. The MEMS is in the up-state position, 3 ␮m over the CPW line, and is 140-␮m wide. The substrate is made of SiHR with a BCB layer. The current is concentrated in the region over the CPW slots along the edges (in a 12-␮m strip width). A current peak (more than 2200 A/m) in this region is observed, this one is even more important than maximum current on the CPW slot edges. 4. CONCLUSION

The multiscale integral-equation (MS-IE) method combines the various scale levels in micro-electromechanical systems by performing the connection of scale-dependent equivalent networks. The incorporation of such a network in full-wave electromagnetic

Figure 5 Current distribution (A/m) on a half-part of the CPW line and the MEMS switch in the up-state position. Dimensions are [see Fig. 1(a)]: s ⫽ 240 ␮m, w ⫽ 100 ␮m, d ⫽ 140 ␮m, 2v ⫽ 1240 ␮m, h ⫽ 3 ␮m, h sub ⫽ 170 ␮m (BCB 20 ␮m ⫹ SiHR 150 ␮m)

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simulators based on the integral-equation technique is straightforward and the accuracy of the numerical results is improved by increasing the number of ports. We have obtained a substantial reduction of computation time and memory, as compared to using numerical techniques based on spatial discretization. The proposed technique is very promising for the design of a MEMS circuit having a great number of MEMS switches.

EFFECT OF POLES ON SUBWAVELENGTH FOCUSING BY AN LHM SLAB

ACKNOWLEDGMENT

Jie Lu, Tomasz M. Grzegorczyk, Bae-Ian Wu, Joe Pacheco, Min Chen, and Jin Au Kong Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139

The authors wish to thank the Midi-Pyre´ne´es Regional Council for financial support in the framework of the MEMSCOM Project.

Received 17 September 2004

REFERENCES 1. W.K. Zhang, F. Jiang, C.W. Domier, and N.C. Luhmann, Quasioptical E-Band MEMS Switching Arrays, IEEE MTT-S Int Microwave Symp Dig, Seattle, WA (2002), 1531–1534. 2. E.R. Brown, RF-MEMS switches for reconfigurable integrated circuits, IEEE Trans Microwave Theory Tech MTT-46 (1998), 1868 – 1880. 3. T. Guan-Leng, R.E. Mihailovich, J.B. Hacker, J.F. DeNatale, and G.M. Rebeiz, A 2-bit miniature X-band MEMS phase shifter, IEEE Microwave Wireless Compon Lett 13 (2003), 146 –148. 4. R. Sorrentino and P. Mezzanotte, Electromagnetic modeling of RF MEMS structures, Open Symp MEMS Microwaves Millimeterwaves Optics, URSI GA, Maastricht, Netherlands, 2002. 5. P. Taillardat, H. Aubert, and H. Baudrand, A combination of quasistatic approach with an integral method for the characterization of microwave planar circuits, IEEE MTT-S Int Microwave Symp Dig 1 (1994), 417– 420. 6. G. Emili, F. Alimenti, P. Mezzanotte, L. Roselli, and R. Sorrentino, Rigorous modeling of packaged schottky diodes by nonlinear lumped network (NL2N)-FDTD approach, IEEE Trans Microwave Theory Tech MTT-48 (2000), 2277–2282. 7. M. Rittweger, M. Abdo, and I. Wolff, Full-wave analysis of coplanar discontinuities considering three dimensional bond wires, IEEE MTT-S Int Microwave Symp Dig, Boston, MA (1991), 465– 468. 8. K. Beilenhoff, W. Heinrich, and H.L. Hartnagel, The scattering behavior of air bridges in coplanar MMICs, Euro Microwave Conf, 1991, pp. 1131–1135. 9. G. Pierantoni, M. Farina, T. Rozzi, F. Coccetti, W. Dressel, and P. Russer, Comparison of the efficiency of electromagnetic solvers in the time- and frequency-domain for the accurate modeling of planar circuits and MEMS, IEEE MTT-S Int Microwave Symp Dig 2 (2002), 891– 894. 10. E. Perret and H. Aubert, A multi-scale technique for the electromagnetic modeling of active antennas, IEEE AP-S Int Antennas Propagat Symp, Monterey, CA, 2004. 11. M. Nadarassin, H. Aubert, and H. Baudrand, Analysis of planar structures by an integral multi-scale approach, IEEE MTT-S Int Microwave Symp Dig 2 (1995), 653– 656. 12. M. Nadarassin, H. Aubert, and H. Baudrand, Analysis of planar structures by an integral approach using entire domain trial functions, IEEE Trans Microwave Theory Tech MTT-10 (1995), 2492–2495. © 2005 Wiley Periodicals, Inc.

ABSTRACT: The poles of the electric field inside a slab with negative permittivity and negative permeability are analyzed. The theoretical solution shows that there is a pole on the integral path which moves toward infinity along the imaginary kz axis as the permittivity and permeability both approach the real value of ⫺1. The surface plasmon wave and the resolution ability of the slab are shown to be determined by this pole. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 45: 49 –53, 2005; Published online in Wiley InterScience (www.interscience. wiley.com). DOI 10.1002/mop.20719 Key words: LHM; metamaterial; perfect lens; imaging; resolution 1. INTRODUCTION

Left-handed materials (LHM) having negative permittivity and permeability, proposed in [1] and realized in [2], exhibit extraordinary electromagnetic characteristics which result in many new applications such as the theoretical perfect lens introduced in [3]: by using an LHM slab, the evanescent waves can be amplified and the subwavelength focusing can be realized, that is, the field distribution at the source can be perfectly reproduced at the image point. A transfer function has been used to analyze the resolution abilities of LHM slabs for lossless [4] and lossy [5] cases. The results showed that the resolution is determined by the deviations from the real value of ⫺1 in the relative permeability and permittivity, as well as the thickness of the slab. 2. CONFIGURATIONS OF THE PROBLEM

Here we discuss this problem by directly calculating the electric and magnetic fields inside a slab of LHM. We first show that there is an infinite number of poles in the complex k z plane (the zˆ component of wave vector k), and that a specific pole is located on the integral path of the field in all regions. We analyze the importance of this specific pole, and its physical meaning. In this paper, we consider the standard LHM lensing configuration shown in Figure 1. The permittivity ␧ i and permeability ␮ i for i 僆 {1, 2, 3} shown here represent relative values. Regions 1 and 3 are free-space regions, with ␮1 ⫽ ␮3 ⫽ 1, and ␧1 ⫽ ␧3 ⫽ 1. An LHM slab is located in region 2 between z ⫽ d 1 and z ⫽ d 2 , and is infinite in both xˆ and yˆ directions. In the examples discussed in this paper, d 1 ⫽ ␭ / 2 and d ⫽ d 2 ⫺ d 1 ⫽ ␭ . The relative permeability and permittivity of the slab, ␮2 and ␧2, have real parts that may take negative values. The current and electric field are ៮ (r៮ ), which are related to the time expressed in frequency domain, ⌰ ៮ (r៮ , t) ⫽ ᑬ{⌰ ៮ (r៮ )e ⫺i␻ t }, where ⌰ repredomain quantities by ⌰ sents current J or electric field E. A line source is located at the origin, whose current can be expressed as J៮ 共r៮ 兲 ⫽ xˆ I ␦ 共 y兲 ␦ 共 z兲.

(1)

This work was sponsored by the ONR under Contract N00014-01-1-0713 and DARPA under Contract N00014-03-1-0716.

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