Ž .Sensors and Actuators 78 1999 28–35www.elsevier.nlrlocatersna

Bistable micromechanical fiber-optic switches on silicon with thermalactuators

Martin Hoffmann ), Peter Kopka, Edgar VogesUniÕersitat Dortmund-Lehrstuhl fur Hochfrequenztechnik, Otto-Hahn-Str. 6, D-44227 Dortmund, Germany¨ ¨

Received 3 November 1998; received in revised form 26 January 1999; accepted 3 February 1999

Abstract

A new bistable moving-fiber switch for applications in optical communication systems is developed by employing advanced bulksilicon micromachining. The bistable switch utilizes actuators based on thermal expansion and the bimetal effect. Switches and switcharrays are successfully fabricated and tested. Due to the low number of process steps and the high yield and reliability, the processtechnology is suitable for low-cost single- and multi-mode fiber optic switches. A typical fiber-to-fiber insertion loss of 2"0.5 dBŽ .single-mode fibers at 1300 nm wavelength, including MT-connector losses of typically 0.5–1.5 dB is achieved in an array of 12

Ž .switches. The cross talk is below the detection limit -y60 dB . The switching power of approx. 700 mW is needed during switchingonly. q 1999 Elsevier Science S.A. All rights reserved.

Keywords: Protection switch; Optical fiber switch; Thermal actuators; Bulk Si-micromachining

1. Introduction

The growing complexity of fiber optical networks ne-cessitates protection mechanisms and rerouting strategies.Optical switches with low insertion loss and high long-termstability are a key element for network protection ap-proaches in the optical domain.

A large number of switch designs and switch technolo-gies has recently been developed. Here, only a very smallselection can be mentioned. Well-known solutions areprecision mechanic switches using microoptics or moving

Ž w x.fibers e.g., Refs. 1,2 as well as integrated optical de-w xvices such as Mach–Zehnder switches 3 and digital

Ž . Ž w x.thermo- optical switches e.g., Ref. 4 . Also, integratedw xopticalrmicromechanical solutions 5,6 , mirror-based de-

Ž w x.vices e.g., Refs. 7,8 or surface micromachined switchesw x9 have been proposed.

Optical cross-connects require for example switcheswith a cross talk below y40 dB. This value is difficult toachieve with integrated optical devices, but it is fulfilledby most mechanical switches. Fiber switches that work

) Corresponding author. Tel.: q49-231-755-4619; Fax: q49-231-755-4631 E-mail: hoffmann@hft.e-technik.uni-dortmund.de

w xlike a relay are commercially available 1,2 . But due tothe small tolerances that are required for single-modefibers, the precision machined moving-fiber switches arestill expensive. A big advantage for the assembly of theseswitches is that no additional optical elements like lenseshave to be aligned and mounted. The cost for moving-fiberswitches is reduced if well-known microelectronics tech-nologies can be used for the actuators and the alignmentstructures. Here, bulk silicon micromachining is used forthe fabrication of 1=2 moving fiber switches with abistable characteristic. Especially, the small size of eachswitch allows the fabrication of compact arrays of switchesfor applications in low-cost space division networks. Thesedevelopments also aim at a fabrication of multiple protec-tion switches for, e.g., fiber ribbons or—most demanding—for switch arrays in flexible optical add-drop-multi-plexers or optical cross-connects.

An example of a low-cost optical backbone networkwith a medium number of nodes and with redundant fibers

Žis developed within the ACTS-project COBNET Corpo-.rate Optical Backbone NETwork . Both a WDM- and a

w xSDM double ring network are field tested 10 . The highestreliability in a network would be reached with a completeduplication of the optical rings. For low-cost private back-

0924-4247r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved.Ž .PII: S0924-4247 99 00200-9

( )M. Hoffmann et al.rSensors and Actuators 78 1999 28–35 29

bone networks, the highest risk for a network is still a fiberbreak. For this reason, COBNET uses duplicated fibersonly. After a fiber break in the ring is detected and located,the transmitters and receivers in the nodes behind thefailure are switched, e.g., from the clockwise operatingring to the counter-clockwise ring.

The necessary long-term stability of these switchesrequires bistable switches. Here, bistability mean that theswitches are driven during the switching cycle only. Other-wise, they are true passive devices such as a splice.

For a bistable operation, a kind of holding mechanismis necessary that fixes the fiber after switching in itsposition. Here, a V-groove clamp is used that clamps theinput fiber in front of one of the two output fibers. Twodifferent mechanisms for the clamp have been investi-gated: a bimetal actuator and a vertical platform actuator.The switching itself is achieved with a bidirectional ther-mal expansion actuator that moves the fiber between itstwo stable states. The complete switch is designed usingadvanced bulk silicon micromachining.

2. Switch principle and design

The design is based on an advanced bulk micromachin-ing process that allows the fabrication of moving-fiberswitches with suitable dimensions for standard single- andmulti-mode fibers with a diameter of 125 mm. For COB-NET, the switches are equipped with single-mode fibers

Ž .which require very high accuracy misalignment -1 mmfor the guiding grooves in the clamping mechanism.Multi-mode fibers could also be used with relaxed de-mands in the precision of the alignment.

A switching cycle is shown in Fig. 1. In stable state IŽ .a , the single input fiber is fixed in front of the left output

Ž .fiber by the fiber clamp. During switching b , the clamp isthermally opened by a bimetal actuator. The free fiber isnow moved by a lateral actuator to the other switchingposition and the fiber clamp is closed again. Now allactuators can be switched off. The fiber remains accurately

Ž .aligned in a V-groove in front of output fiber II c . Stablestate II is reached. In its stable states the switch behaveslike a mechanical fiber splice. Due to the lack of otheroptical elements like lenses or mirrors, the switches shows

Fig. 1. Switching cycle of a 1=2 moving-fiber switch with a bimetalŽ .fiber clamp upper part of the fiber clamp not shown .

Fig. 2. 1=2 switch with a platform clamp during switching.

an insertion loss characteristic comparable to that of asplice. In Fig. 2, a realization with another type of clamp-ing mechanism using buckling of heated beams is shownduring switching.

Key elements for the switch are the fiber-guidinggrooves for the precise alignment in a clamp and theactuators for switching.

For a precise alignment of optical fibers in integrated-Ž .optical and micro-optic applications, V-grooves in 100 -

silicon are well-known structures. They can easily befabricated by anisotropic wet etching in alkaline solutionsŽ .e.g., KOH, potassium hydroxide where etching stops atŽ .111 -plains. The V-groove depth is well defined by thewidth of the mask.

To produce cantilevers with dimensions close to that ofŽ . Ž .standard silica fibers 125 mm diameter in 100 -silicon

which are necessary for this switch the wafer has to beŽ .structured from both sides Fig. 3 . All rectangular areas

which are exposed to KOH are at first etched like V-grooves. If front- and back-side grooves meet each other,

Ž .fast etching 110 -plains are exposed to the KOH at the tipŽ Ž . Ž . .etch rate ratio: 110 r 100 (3r2 . The cantilever now isalso thinned but the triangular shapes are etched muchfaster. Time controlling this etching step results in verticalside walls. A detailed description of the used advanced

w xbulk micromachining process is given in Ref. 11 .While V-grooves in silicon solve the alignment problem

in a simple way, more expense has to be spent in thedesign of the actuators.

3. Actuator concepts

3.1. Thermal actuators

Simple and effective actuators can be designed usingthermal effects in silicon. In contrast to the thermal actua-

ŽFig. 3. Advanced bulk Si-micromachining process dashed line: interme-.diate etching planes .

( )M. Hoffmann et al.rSensors and Actuators 78 1999 28–3530

w xtors used or proposed in Ref. 6 , here bulk silicon mi-crostructures are part of the actuator. The heat needed foractuation is generated in thin film heaters that are fabri-cated in a common lift-off process.

Mainly two different actuator principles can be used forthis application: thermal material expansion and the bimetaleffect. The mechanism of thermal silicon expansion andthe bimetal effect in heater-on-silicon actuators cannot beclearly distinguished. The choice of the heater material andits thickness determine the actuator principle. While a thinisolated metallic layer on bulk silicon mainly works as aheat source for thermal expansion of the material below, itbecomes part of a bimetal actuator when its thickness isincreased and a proper material is chosen.

3.2. In-plane bidirectional actuator

The most important actuator in the micromechanicalfiber switch is the in-plane lateral actuator that moves thefiber between the switching states. For a 1=2 switch withstandard 125 mm fibers, the fiber has to be moved over adistance of approx. "65 mm. Thermal in-plane actuatorsfor moving-fiber switches have been proposed based on

w xelectroplating 6,12 . They show excellent mechanicalcharacteristics. A disadvantage of directly heated Ni bend-ing actuators is that they can only operate in one directiondue to the necessarily asymmetric design.

A symmetric actuator that can realize a bidirectionalin-plane movement is a U-shaped silicon cantilever with

Ž .heat generating elements on both arms see Fig. 4 . If onlyŽ .one arm length L at TsT is heated to TsT qDT , it0 0

expands according to D L s aDTL with a s 2.6 =

10y6 Ky1 for silicon at T(300 K. Due to the stiffcoupling of both arms at the free end, the U-shapedcantilever system bends to the unheated side. The absolutelateral displacement depends on the cantilever length L,the length of the connecting bar b between the two armsand, of course, the temperature increase DT. Due to theidentical shapes of both arms and the boundary conditionsat the fixed side and at the free end the actuator follows acircular in-plane bending. The average bending radius R

Fig. 4. Bidirectional actuator based on thermal expansion.

ŽFig. 5. Calculated displacement of the bidirectional actuator Ls20 mm,.bs350 mm .

of the U-shaped structure assuming LrD L41r2 is givenby

L 1 1Rsb q (b 1Ž .ž / ž /DL 2 aDT

The angle of the circle segment is given by

DL aLDTbs s 2Ž .

b b

For a circular bending, the absolute lateral displacement atthe cantilever tip can easily be derived from

b aLDTdsR 1ycosb s 1ycos 3Ž . Ž .ž /aDT b

Ž . Ž < <. 2Eq. 3 can be simplified with cos x (1yx r2 for< <x <1 resulting in:

dsaDTL2r2 b 4Ž .The main result is that the lateral displacement dependsalmost linearly on the temperature which itself depends onthe heating power. In Fig. 5 the lateral displacement d as

Ž .given in Eq. 3 for DTs0 . . . 300 K is shown for Ls20 mm and bs350 mm.

Ž .If thin metal layers thickness-1 mm are used as thinfilm heaters, the stress induced out-of-plane bending due toa parasitic bimetal effect is negligible for the actuatoritself.

Fig. 6 shows the measured bidirectional displacement ofŽ . Žpoint P Fig. 4 as a function of the heater power which is

.nearly proportional to the cantilever temperature . Thecantilever has a length of 20 mm and the connecting bar alength of 350 mm. It can be seen that the displacement isnearly linear with heater power as described above.

Designing a proper actuator has to take into accountthat the angle b must be kept small. It causes an angular

Fig. 6. Measured bidirectional displacement for a 20 mm long cantilever.

( )M. Hoffmann et al.rSensors and Actuators 78 1999 28–35 31

Fig. 7. 1=4 moving fiber switch with 2 combined bidirectional actuators.

displacement of the fiber. This is not advantageous for astable fixing in the V-groove clamp for 1=4 switches.Otherwise the length L has to be increased which makesthe device impractical and fragile. The problem can beovercome by a platform-like actuator that is driven by 2U-shaped cantilevers and additional straight cantilevers.The basic design is shown in Fig. 7 for a 1=4 switch. Itallows a parallel movement of the fibers. 1=4 switchesusing this actuator concept are under development. Theplatform at the end of two U-shaped actuators allows alsothe integration of four 1=2 switches in a single device for

Ž .a 2=2 switch Fig. 8 . In this case, two input and twooutput fibers are fixed on the platform while the intercon-necting fibers are placed in the clamp.

3.3. Bimetal fiber clamp actuator

For the fiber clamp a movement in the vertical directionhas to be achieved. It should open the V-groove side of thefiber clamp so that the fiber can move laterally. Thevertical movement depends on the depth of the fiber in theV-groove which is typically 30–40 mm. The resultingminimum thickness of the cantilever for such V-grooves is

Ž .85 mm. This minimum thickness which is important forthe bimetal actuator is given by the advanced bulk micro-machining process used here.

The opening of a fiber clamp can be achieved by thebending of the V-groove cantilever by the bimetal effectŽ .Fig. 9 . The low thermal expansion coefficient of silicon

Fig. 8. 2=2 switch based on four 1=2 switches driven by a combinedbidirectional actuator.

Fig. 9. Bimetal fiber clamp actuator.

is advantageous in this case because it is easy to findefficient materials for this actuator. Due to the fabricationprocess used here, the material has to be stable in hot

ŽKOH. A suitable and efficient material is nickel a sNiy6 y1.13.4=10 K . Using standard formulas for this type

Ž . w xof actuator leads to Eq. 5 13

12hsDT a ya t q t LŽ . Ž .Si me Si me2

=14 2t q t y2 t tŽ .Si me Si me12

y13 3E w t E w tSi Si Si me me meq q 5Ž .

E w t E w tme me me Si Si Si

for a cantilever length L much smaller than the bendingradius R.

It describes the vertical displacement of a bimetal actua-tor made from two rectangular layers of width w, thicknesst, length L and Young’s modulus E. The index ‘Si’ standsfor silicon and ‘me’ for the metal coating.

The bidirectional lateral actuator described before is akind of bimetal actuator with separated bimetals also. If allparameters for the metal layer are replaced by the appro-

Ž . Ž .priate silicon parameters and a ya by a , Eq. 5Si me SiŽ .will be reduced to Eq. 4 .

Ž .Eq. 5 can be rewritten as hsDTg . The coefficient g

represents a conversion factor with the dimension mrKand depends on both the geometrical dimensions and thematerial coefficients a and E. It is a measure for theefficiency of the actuator. Due to the use of sputtering for

Table 1Conversion factor for a bimetal actuator

Material Thermal expansion Young’s modulus Conversion factory6 11 2 y8Ž . Ž . Ž .10 rK 10 Nrm 10 mrK

Aluminum 23.0 0.7 3.13Copper 16.7 1.2 3.75Chromium 4.9 2.8 1.42Nickel 13.4 2.1 4.73

Ž .Silicon 2.33 19.0 reference

( )M. Hoffmann et al.rSensors and Actuators 78 1999 28–3532

the metal film the thickness is limited to t -2 mmmeŽ .1.6 mm for Table 1 while t s85 mm. w is set to 500Si Si

mm and the metallic heaters have a width of 160 mm. Thelength of the actuator is 10 mm. It has been calculated forthe given geometry depending on the material coefficients

Ž .of various metals Table 1 . It can be seen that the thermalexpansion coefficient and the Young’s modulus of themetal should both be high. The highest conversion factorof practical materials was found for nickel which alsoexhibits excellent stability in alkaline solutions. Copperand aluminum show also good efficiency but they are notstable in KOH. Due to the small thickness and the re-stricted width of the metal layer relative to the siliconcantilever, this actuator is not very efficient. High tempera-tures have to be achieved for a sufficient bending of thecantilever. The measured displacement of a 20 mm long

Žcantilever other parameters as used for the calculations.above is shown in Fig. 10. Nevertheless, first switches

have successfully been fabricated using this type of actua-w xtor 14 .A non-negligible problem is the stress between the two

materials due to the high temperatures during operation. Itcauses an annealing effect in the sputtered metal layer thatleads to tensile stresses in the metal and a persistingbending of the actuator in the off-state. It can be compen-sated during the assembly of the upper part of the clampbut it is difficult to handle in large arrays of switchesbecause the stress depends on the individual annealingconditions as well as on local deposition inhomogeneities.

3.4. Vertical platform actuator

This basic problem leads to another type of verticalactuator that allows large displacements. A cantilever whichis fixed at both ends and expanded by heating will bendafter a critical load is reached. This effect is well known aspostbuckling of under-etched structures in thin film deposi-tion. A thin layer of, e.g., oxide that is deposited at T4T0

on a silicon wafer under stress free conditions and thancooled down to T shows significant compressive stress0

due to the different thermal expansion coefficients. Beingpartially released it buckles upwards or downwards tocompensate the stress. A detailed analysis of this postbuck-

w xling behavior can be found in Ref. 15 for the SiO rSi-2

Fig. 10. Measured vertical displacement h of a bimetal actuator.

Ž . Ž .Fig. 11. Vertical platform actuator a prebend cantilever at DT s0 bŽ .vertical shifting due to cantilever eleongation at D t)0 c top view of

the H-type platform used in the switches.

system. In our case, the difference Da in expansioncoefficients must be replaced by a , and DT is now theSi

temperature increase of the cantilever due to the heating.This principle is used here to open the fiber clamp. The

stress is caused by selective heating of small silicon can-tilevers which are fixed at both ends in the base plate. Thistype of actuator shows a nonlinear characteristic. For aperfect cantilever two characteristic regions can be distin-guished. Up to the critical load the stress is compensatedwithin the material and therefore no buckling is found. Ifthe stress exceeds the critical load, the bridge can reach 2stable states. Even the smallest imperfection will cause animmediate buckling. For the use in the fiber switch, adefined imperfection is mandatory. Otherwise, the actuatormoves randomly up or down. The imperfection for aselective movement can be caused by slightly pushing

Ž .down the actuator Fig. 11a . Besides the desired imperfec-tion, it also fixes the fibers in a well-defined position in theV-groove. When the cantilever is heated, it always moves

Ž .down as it is necessary for the switch Fig. 11b .For this type of actuator, the nonlinear model with

w ximperfections described in Ref. 15 applies best. The exactcalculation of the deflection d as a function of the tempera-ture increase DT is indicated in Appendix A. Fig. 12shows the calculated deflection for the nonlinear modelwith imperfections.

Fig. 12. Simulated vertical displacement h for a prebend actuator.

( )M. Hoffmann et al.rSensors and Actuators 78 1999 28–35 33

Ž . ŽFig. 13. Measured vertical displacement h Ls20 mm, ts85 mm total.power for an actuator according to Fig. 11c .

The micromechanical switch makes use of a H-typestructure that consists of a platform with the V-groovesand 4 supporting cantilevers that are heated by thin film

Ž .elements Fig. 11c . Due to the boundary conditions in themiddle of the cantilever, a stiff platform can be introducedwithout changing its characteristics.

In Fig. 13, the measured behavior of an unstressedactuator is shown. The length of the supporting cantileverswhich are heated is 10 mm each. The power represents thetotal electrical power applied to all four heaters. Thecharacteristic is close to the theoretically expected behav-ior. In the postbuckling region, the root function can berecognized, as well as the presence of a small imperfectionthat causes a small elongation below the critical load.

A comparable actuator based on surface micromachin-w xing has recently been published 16 . In contrast to that

actuator, the H-type switch actuator requires much lowertemperatures for deflections of up to 100 mm due to therigid clamp within the wafer.

Switch arrays with a bimetal fiber clamp and a bucklingclamp have been fabricated. Taking the requirements ofthe used advanced bulk silicon micromachining processinto account the width of a single switch is less than 2.5mm including a stabilizing connection bar between theinput and output side. The design of the mask for theseswitches becomes more complex than shown because ofthe need of corner compensation structures which avoidthe under-etching at convex corners. Nevertheless, a care-ful design guarantees a high yield of these fragile lookingstructures which show excellent mechanical stability andaccuracy.

4. Fabrication

4.1. Micromechanical base plate

Low-cost components should make use of well-knownand simple technologies. Surface micromachining has suc-cessfully been used to fabricate many new optical mi-crosystems, but the fabrication requires nearly the fullrange of IC-technology, and the devices are rather com-plex. Bulk silicon micromachining can fulfil all require-

ments of these moving-fiber switches also. The number ofprocesses is limited to common UV-lithography, reactive

Žion etching, deposition of metallic layers sputtering or.evaporation and KOH-etching. All relevant mechanical

parts can be fabricated by anisotropic etching of standardsilicon wafers.

As this etching is done in KOH at high temperatures,Ž .silicon nitride Si N is preferred as masking material due3 4

to its excellent stability.Ž .At first the back side layout large square grooves is

defined and anisotropically etched in a closed vessel with20% KOH at 70"0.028C. This step is important for thefinal cantilever thickness and allows to compensate thick-ness variations from wafer to wafer.

In a further lithographic step, the complete switch lay-out including the V-grooves for the precise fiber alignmentis transferred on the front side and the etching windowsare opened using reactive ion etching.

Ž .Due to the low height of the nitride -100 nm , thewafer is still flat for the third lithographic step that definesthe thin-film heaters for the actuators in a lift-off process.While a 1.6 mm thick nickel layer is used for the bimetalactuated switch type, the thickness is reduced to about1 mm for the thermal expansion type switch. In both cases,the nickel is coated with a thin gold or chromium layerthat provides a suitable surface for wire bonding. SputteredNickel exhibits slight compressive stress which can bereduced and even converted into tensile stress by a thermalannealing step. It was found that Cr- or Au-coated or purenickel thin film heaters are neither etched nor detached inthe following etching process in KOH.

The final etching is time controlled to adjust the can-tilever thickness and to achieve nearly vertical sidewalls atthe actuators. This process is not very critical because theetch rate can be carefully controlled by the temperature of

Ž .the closed-cover etching bath "0.02 K . Cantilevers withŽa thickness of about 85 mm lower limit due to the used

.depth of the V-grooves show high mechanical stabilityand high yield.

4.2. Assembly

The switch fabrication is completed by gluing the fibersinto the V-grooves using UV-curing resin. The upper partof the V-groove clamp, a flat glass plate, is put upside-downonto the output cantilevers. The distance between the baseplate and the glass plate is adjusted by short fiber segmentsin special V-grooves outside the structure which are a littlebit broader than that in the clamp. Due to this the coverplate defines the vertical position of the optical fibers andintroduces the necessary imperfection for the platformactuator. For the fixation also UV-curing resin is used. Thecover plate can be extended to allow a hermetical sealingof the active fiber switch which is necessary for a long-termstable operation. It has to be avoided that dust disables theoperation or that water destroys the fibers.

( )M. Hoffmann et al.rSensors and Actuators 78 1999 28–3534

Fig. 14. Photograph of an array of 12 1=2 switches for use in the COBNET demonstrator.

A single 1=2 switch has a width of 3 mm and a lengthof 50 mm. The layout allows the fabrication of arrays. Ourtest array contains 10 switches with different cantileverlayouts and grooves for fiber positioning and has overalldimensions of 39 mm=50 mm.

Fig. 14 shows an assembled switch array connected tofiber ribbons as it is used for the COBNET demonstrator.The most critical coupling region of the fibers is below theglass plate on top of the switch.

5. Experimental results and discussion

1=2 bistable micromechanical arrays of 12 switchesŽ .have successfully been fabricated Fig. 14 . They show an

excellent mechanical operation of the actuators. The powerfor the horizontal shift is about 200 mW. The switchingtime is about 100 ms. The power consumption is slightlyhigher than for the bare actuator due to the additional forceintroduced by the mounted fiber. Both types of fiberclamps allow a sufficient opening of the clamp. While thebimetal clamp has a switching time of 400 ms and a powerconsumption of 700 mW, the platform clamp only requires500 mW for a switching time of 100 ms. Single switcheswith bimetal clamps have successfully been fabricated. Inarrays, the alignment of the upper part of the clamp causessome problems which make this type of actuator unsuit-able for arrays. The platform actuator does not show thisproblem. As the fiber switch is bistable, the switchingpower is needed for 100 ms only.

Optical measurements at 1300 nm and 1550 nm wave-Žlengths exhibit typical insertion losses including MT-con-

nector losses at both sides which add up to typically.0.5–1.5 dB below 2.5 dB, for some switches below 1 dB.

These varying results are mainly due to tolerances of theMT-connectors. The cross-talk is below the limits of our

Ž .measurement equipment y60 dB . The switching time is

estimated to be 400 ms with the present design. Thedifference between the heater-on time and the switchingtime is caused by the relaxation time of the fiber in thegrooves. It may be reduced by using a bidirectional actua-tor as it is proposed in Fig. 7. Return loss has not beenmeasured so far, but it can be reduced to very low valuesby an antireflective coating of the fiber end faces. Anindex matching fluid is not appropriate due to the highsurface tension that would increase the switching time andthe power consumption.

The necessary timing of the heater currents duringswitching is achieved with a simple integrated microcon-troller.

6. Conclusion

Novel bistable moving-fiber switches utilizing thermalactuators on micromachined silicon have successfully beenfabricated and tested. Advanced bulk micromachining pro-cesses allow a low-cost fabrication of arrays for, e.g.,multiple protection switching. Especially, the three differ-ent actuator concepts have been described and tested. Also,the suitability of this concept for arrays and for a batchfabrication was confirmed.

This switch concept can easily be extended to 2=2,1=3 and 1=4 switches or adapted for micromechanicalattenuators which are also investigated.

Further work will investigate a magnetic driving mecha-nism for similar bulk micromachined actuators. This wouldallow higher switching speed.

Acknowledgements

The development of the fiber switch arrays was sup-ported by the European Commission within the ACTS-pro-

Ž .ject COBNET Corporate Optical Backbone NETworks .

( )M. Hoffmann et al.rSensors and Actuators 78 1999 28–35 35

Appendix A. Theoretical description of the platformactuator

The main formulas for the nonlinear model with imper-fections including an adaptation to the actuator used here

w xare given. The equations in Ref. 15 are adopted to ourproblem by replacing Dasa ya with a . TheSi oxide Si

temperature difference DT reflects the temperature in-crease of the cantilevers due to thin film heaters.

Necessary parameters for the calculations are the lengthof the cantilever L, its thickness t and the prebend parame-

Ž .ter gsh DTs0 rL. The vertical movement h is givenhere as a function of the temperature increase DT.

The solution is split into two branches separated by acritical load.

Ž .For loads below the critical load r)0 the displace-ment is given by

3 3' '( (h r)0 s ypr2q r y pr2q r . 6Ž . Ž .Ž .If the load exceeds the critical load r-0 the displace-

ment is given by

'h r-0 s2 yqr3 cos sr3q2 ny1 pr3 . 7Ž . Ž . Ž .Ž .with:

psy16g Lt 2 8Ž .qs16 t 2 y4L2a DTrp 2 yg 2L2 9Ž .Si

rsp2r4 qq3r27 10Ž .y1 'sscos 3 pr 2 q yqr3 11Ž .Ž .Ž .

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Martin Hoffmann was born in Hemer, Germany, in 1966. He received theDipl.-Ing. Degree in January 1992 in electrotechniques and the Dr.-Ing.Degree in 1997 from Universitat Dortmund, Germany. Since 1992, he is¨with the Chair for High Frequency Techniques at Universitat Dortmund¨where he is working on integrated optics on silicon for optical communi-cations. His current interests are also in microoptics and micromechanicson silicon.

Peter Kopka was born in Munster, Germany, in 1969. Received the¨Dipl.-Ing. Degree in August 1995 in electrotechniques from Universitat¨Dortmund, Germany. Since 1996, he is with the Chair for High Fre-quency Techniques at Universitat Dortmund where he is currently work-¨ing in the field of integrated optical and micromechanical switch arrayson silicon for optical communication networks.

Edgar Voges was born in Braunschweig, Germany, in 1941. He receivedhis diploma in physics 1967, his Dr.-Ing. Degree 1970 and his venialegendi for High Frequency Techniques 1973 from the Technische Uni-versitat Braunschweig. Since 1974 he is Professor at the Universitat¨ ¨Dortmund, since 1982 he has the Chair for High Frequency Techniques.His research work concentrates on integrated optic devices for opticalcommunications and optical sensors, and on silicon-based opticalrmicro-mechanical systems.