Dynamc Smulaton of Optcal MEM Swtches Tmothy P. Kurzweg, Jose A. Martnez, Steven P. Levtan Department of Electrcal Engneerng, Unversty of Pttsburgh, 348 Benedum Hall, Pttsburgh, PA 15261 tm@ee.ptt.edu, jmart@ee.ptt.edu, steve@ee.ptt.edu Phlppe J. Marchand Optcal Mcro Machnes, 9645 Scranton Rd, Sute 14, San Dego, CA 92121 marchand@ommnc.com Donald M. Charull Department of Computer Scence, Unversty of Pttsburgh, 212 Mneral Industres Buldng, Pttsburgh, PA 1526 don@cs.ptt.edu Abstract: Mcro-optcal-electrcal-mechancal systems (MOEMS) present a new set of challenges for mxed technology systems on a chp (SoC) desgners ncludng the need for mxed-sgnal mult-doman smulaton. We present new modelng technques for optcal and mechancal MEM components and apply these models for mproved system-level smulaton accuracy of MOEMS swtches. 1. Introducton New fabrcaton technques permt mcro- electro-mechancal systems (MEMS) to use the same fabrcaton process as CMOS ntegrated crcuts. In fact, they can be desgned usng many of the standard CAD lay-out packages. Analyss and even synthess of both electrcal and mechancal components have been the topc of recent research. However, one defcency n the MEMS desgn process s the need for a system level smulaton tool that crosses the domans of electronc, mechancal and optcal mcro-systems. In order for a system smulaton tool to be useful, t must be capable of modelng not only frst order but also second order effects n each of the domans. Ths means that for the electrcal doman, we must model both dgtal and analog components, for the mechancal doman we need to capture both lnear and non-lnear stress/stran relatonshps, and for the optcal doman we must support dffractve (scalar) models of optcal propagaton. One applcaton area where the need for ths level of modelng and smulaton s apparent s n the desgn of all-optcal cross-connect swtches for fber based telecommuncatons networks. All-optcal swtches are becomng ncreasngly popular due to the many advantages that they possess over typcal fber optc swtches. No longer s a costly converson between the optcal and electrcal doman needed to swtch data sgnals, as alloptcal swtches also reduce the nserton loss and crosstalk found n conventonal fber swtches. The swtchng s acheved through the mechancal movement of mrrors steerng the data path to the desred output. Wth the advancement of mcro-electro-mechancal systems (MEMS) technology, theses swtches have become a realty, as the swtches are small, fast, relable, and eventually, wll be very cheap to produce. In earler work, we ntroduced Chatoyant [1]. In ths paper, we expand ths work wth new modelng technques for ncreased smulaton accuracy of MOEM systems. We frst dscuss the modelng of non-deal component models, such as the modelng of the surface roughness and nherent curvature of mcro-mrrors. We expand our mechancal lbrary by ncludng models for stress-nduced cantlever beams. Usng these mproved component models, we smulate optcal MEM systems wth ncreased accuracy. We conclude wth a summary and a plan for future work. 2. Models In ths secton, we ntroduce the non-deal component models for both mcro-mrrors and cantlever beams. The mcro-mrror models are enhanced wth the addton of modelng the surface roughness, leadng to lght scatterng, and the nherent curvature of the mrror, resultng n the focusng or defocusng of the lght. The mechancal smulatons are mproved by the modelng of beam curvature due to nduced stress on the beam. 2.1. Mcro-Mrror: Surface Roughness Scatterng s a common optcal effect when lght strkes non-smooth surfaces. In optcal MEM systems, the smoothness of the surface s determned by the materal and technques used to fabrcate the component. Snce we are typcally nterested n lght reflecton from a mrrored surface, n ths dscusson, we examne the surface roughness on an optcal MEM mrror. Commonly, surface roughness measurements are gven by a root mean squared (rms) value, σ, whch s the standard devaton of the heght of the surface. In our modelng, we use the square of ths value as the varance
n a normal or Gaussan dstrbuton wth a mean of zero. Ths s used to smulate the roughness on the surface. Therefore, for each mesh pont of the complex wavefront n the smulaton, there s a varance on the dstance, r, that each wavefront propagates to strke the surface. Our optcal propagaton modelng technque s based on the Raylegh-Sommerfeld formulaton for scalar dffractve propagaton, whch s necessary for near feld effects [2]. As an example of ths technque appled to a rough surface, we smulate a plane wave strkng a 1x1 µm MEM mrror and reflectng straght back towards the source. The lght s detected on a 25x25 µm observaton plane at a dstance of 1 µm from the mrror. In Fgure 1, surface ntensty contours are shown from these smulatons. At the left sde of the fgure (contour A), we show the result of reflecton from a smooth, deal surface. The ntensty dstrbuton of the reflecton off the deal mrror s gven below the contour. As expected wth such a small propagaton dstance, the ntensty dstrbuton shadows the sze of the mrror, that s the small square shape wth peaks at the corners of the aperture. Fgure 1 also presents the results of the same smulaton wth surface roughness rms values of 1 nm, 25 nm, 5 nm, and 1 nm, shown as contours B, C, D, and E, respectvely. When compared to the flat case, (A), t s very notceable how the shape of the beam deterorates as the level of surface roughness ncreases. Of course, ths also leads to lght beng scattered off of the optcal path, resultng n a loss of power at the 25x25 um observaton plane. A) nm rms - % Loss B) 1 nm rms -.5% Loss C) 25 nm rms - 3.3% Loss D) 5 nm rms - 12.6% Loss E) 1 nm rms - 41.5% Loss Fgure 1: Intensty contours of propagaton off a rough mrror 2.2. Mcro-Mrror: Curvature Curved surfaces are common n optcal MEM systems, snce many thn components experence an nherent curvature due to factors ncludng the fabrcaton materal, the fabrcaton processes, and nternal resdual stress. Optcal MEM mrrors are usually covered wth layers of metal to ncrease the reflectveness of the mrror as well as strengthen the mrror, reducng the fnal curvature. We present smulatons that show the effect of usng curved optcal components n a mcro-optcal system. In these smulatons, we agan model reflecton off of a 1x1 µm mrror, however, we now smulate a slght curvature (R=1mm) n the mrror. For these examples, we assume a surface roughness of zero. In Fgure 2, we show ntensty surface and contour dstrbutons of lght reflectng from the mrror and beng detected on a 15x15 µm observaton plane. The smulaton result at the left of the Fgure 2, (A), shows the result of reflecton from an deally flat mrror, as a bass for comparson. In the ntensty dstrbuton n the center, (B), the curvature s n a concave, so the beam starts to converge towards the focus pont of the mrror (f=r/2). In the thrd dstrbuton, (C), the lght strkes a convex mrror, wth the same degree of curvature, resultng n the dvergence or spreadng of the lght. The smulated power detected on each of these observaton planes s essentally the same, although the shape and drecton (.e., convergent or dvergent) of the reflected beam s dfferent. These effects can cause nserton loss and crosstalk as the beam propagates further down the optcal path.
A) Flat B) Concave C) Convex Fgure 2: Intensty dstrbutons of propagaton off a curved mrror 2.3. Stress-Induced Curvature of a Cantlever Beam We have reported prevously the use of nodal analyss and pecewse lnear smulaton for the modelng of mechancal structures [3]. In Chatoyant, we have mplemented ths methodology for the ntegraton of mechancal elements to mcrosystems. In ths methodology, the MEM devce s modeled usng a transformaton of ts general equaton of moton nto a set of frst order dfferental equatons: U, where the state varable vector X = U [ Mb ] X + [ Mk] X = [ E]F U s the dsplacement vector, and U s the velocty vector. The expresson represents a set of lnear ODEs assumng the characterstc matrces are statc and ndependent of the dynamcs n the body. Each mechancal element (beam, plate, etc.) s characterzed by a template composed of characterstc stffness, dampng and mass matrces. The assembly of every element nto the mechancal structure s acheved by mergng together ther ndvdual templates, to create the general matrx representaton (Mb, Mk, E). We now expand our prevous method to model one of the more promsng optcal swtchng MEM devces; a stress-nduced curved cantlever used to move a mrror n and out of an optcal path, shown n Fgure 3. Our system smulaton s based on the fabrcaton and testng of ths swtch at UCLA [4]. When the mrror s n the optcal path, the bar state for the swtch s enabled, when t s out, the swtch s n the cross state. Because of the curvature, the ar gap between the substrate and beam surface s progressvely decreased as the mrror moves from ts ntal poston towards the fxed end of the structure. Ths ntroduces a strong electrostatc force over a secton of the beam near the anchor, whch n turn reduces the amount of appled voltage requred to produce a specfc degree of deflecton on the entre structure. A curved structure as descrbed presents nterestng modelng challenges. Because of the electrcal feld appled over the length of the beam, there s an effectve secton of the beam makng contact wth the substrate (oxde nsulator) as measured from the anchor reference. Ths contact s mechancally a collson between both surfaces. The contact area ncreases as a functon of the level of voltage appled. Collsons, n general, are dffcult to characterze because of the necessty to quantfy the nterchange of energy, the momentum transformaton, and the losses. An addtonal problem for ths model s that the locatons of nodes n ths structure do not correspond to a smple lnear arrangement. Consequently, n order to use our nodal modelng technque, the spatal locaton transformaton for a curved structure must be defned. OUT path Mrror Bar State Curved Cantlever IN path Input + V Oxde layer Cross State Surface Collson: Substrate Fgure 3: Stress-nduced cantlever and Swtchng System The structure s modeled as composed by a seres of basc two-node beams. Ths has the advantage of offerng a characterzaton for hgher modes n the structure. When an electrc feld s appled between the substrate and cantlever surface, all the consttutng elemental beams begn to dsplace toward the substrate.
However, the frst element, measured from the fxed end, wll collde wth the substrate at some tme before any other n the structure, as shown n Fgure 4. At ths tme, any further lnear analyss wll be erroneous because of the necessty to characterze the collson. Instead of characterzng the collson, we smplfy the smulaton by performng the smulaton n dscrete steps. After the collson of each secton wth the substrate, that secton s consdered vrtually anchored to the substrate, resultng n ts nodes unable to move. Consequently, further characterzatons are carred on as a new smulaton problem where the structure s composed of only the remanng sectons wth ntal condtons equal to the ones at an nstant prevous to the tme of the collson. New Smulaton Case Colldng secton Curved Cantlever Vrtual Anchor Nodes Anchor Oxde layer Substrate Curvature: Fgure 4: Step smulaton n a colldng nodal characterzaton In our methodology, a common coordnate reference frame s used for the characterzaton of mechancal structures, snce every template, or element, s characterzed n a local reference system. The process of translaton of these local templates to the global reference system can be descrbed by: S A SA T =, where, A represents the translaton matrx from local dsplacements to global dsplacements (a functon of the structure s geometry), S represents the local template, and S s the correspondng global representaton. For an element whose coordnate system s tlted n relaton to the global reference system as n Fgure 5 (a), the translaton matrx wll be accordng to the drecton cosne of angles between the axs: A = snα snα 1 snα snα 1 In a curved structure, composed of an assembled group of elemental beams, the ndvdual drecton cosnes wll be gven by: s ( + 1) s sn( α ) = cos( ) cos( ) s s ( + 1) s cos( α ) = sn( ) sn( ) s where, n s the number of elemental beams, s s the curved length of the cantlever, ρ s the radus of curvature and s the th elemental beam as shown n Fgure 5 (b). Wth ths transformaton, the curved characterstc of the structure can be ncluded n the nodal characterzaton and can be used n our exstng pecewse lnear smulator.
y y x ρ =, n α Fgure 5: (a), Elemental tlted beam x Fgure 5 (b), Curved beam geometry s 3. Smulatons We have prevously performed dynamc smulatons of a 2x2 optcal MEM swtch [5]. Ths archtecture conssts of a set of four optcal fbers n the shape of a + sgn, wth the nput and output fbers facng each other through a free-space gap. The swtchng system s n the cross state when the lght s passed straght across the free-space gap. However, to swtch to the bar state, a mcro-mrror s nserted between the fbers at a 45-degree angle, and the lght s reflected to the alternate output. The mcro-mrror swtch smulaton mentoned above was modeled as a straght beam wth a unform ar gap. Instead, t s common to have a stressed nduced beam wth a mrror on the end act as the swtchng element of a system [4]. The effectve progressve small ar gap present n a curved beam swtch systems offers a notceable reducton n the requred electrcal voltage for operaton. Wth the advanced models that we have presented n ths paper, our new smulatons represent a more realstc system, wth the ncluson of a stressnduced beam and mrrors wth curvature and surface roughness. System-level results from these smulatons wll be ncluded n our paper and were removed for the space consderatons of ths abstract. 4. Summary Wth the modelng of non-deal components, we are able to perform smulatons of systems that are more realstc to those currently beng fabrcated. We have presented the detals of our modelng technques and have shown the ablty for these models to be easly placed n our smulaton framework Chatoyant. Wth these new detaled models, the CAD tool can gve more detaled and useful results to the user. 4. References [1] Kurzweg, T. P., Levtan, S. P., Marchand, P. J., Martnez, J. A., Prough, K. R., Charull, D. M., CAD for Optcal MEMS, Proceedngs of the 36th IEEE/ACM Desgn Automaton Conference (DAC 99), New Orleans, LA, June 2-25, 1999, pp. 879-884. [2] Kurzweg, T.P., Levtan, S.P., Martnez, J.A., Marchand, P.J., Charull, D.M., Dffractve Optcal Propagaton Technques for a Mxed- Sgnal CAD Tool, Optcs n Computng (OC2), Quebec Cty, CA, June 18-23, 2. [3] Levtan, S.P., Martnez, J.A., Kurzweg, T.P., Marchand, P.J., Charull, D.M., Mxed-Technology System-Level Smulaton, Desgn, Test, Integraton, and Packagng of MEMS/MOEMS, Pars, France, May 9-11, 2. [4] Chen, R.T., Nguyen, H., Wu, M.C., A Hgh-Speed Low-Voltage Stress-Induced Mcromachned 2x2 Optcal Swtch, IEEE Photoncs Technology Letters, Vol.11, No. 11, Nov. 1999, pp.1396-1398. [5] Kurzweg, T.P., Levtan, S.P., Martnez, J.A., Marchand, P.J., Charull, D.M., Modelng and Smulatng Optcal MEM Swtches, Optcal MEMS, Kaua, HI, 21-24 August 2.