Characterization of an 2x2 SCB Optical Switch Integrated with VOA

Characterization of an 2x2 SCB Optical Switch Integrated with VOA

Hen-Wei Huang and Yao-Joe Yang*

Department of Mechanical Engineering, National Taiwan University, Taiwan

(Received 18 September 2011; Accepted 23 March 2012; Published on line 1 June2012)
*Corresponding author: yjy@ntu.edu.tw
DOI: 10.5875/ausmt.v2i2.107

Abstract: This work presents the modeling, simulation and characterization of an innovative micromachined 2x2 optical switch monolithically integrated with variable optical attenuators. The device uses bi-stable mechanisms for optical switching, and can be easily realized by a standard micromachining process. The split-cross-bar design (SCB) is employed as the optical path configuration. A one-dimensional (1-D) heat transfer model is developed for estimating temperature elevation. An analytical solution is also proposed for the thermo-elastic bending and buckling problem of thermal V-beam and curved beam (pre-shaped buckled beam) actuators. The resulting governing equations with external y-directed force and thermal strain force are solved analytically. Results of the analytical solutions and the finite element (FEM) calculations are compared, with prediction accuracy within 10% of the nonlinear FEM solution, which agrees well with the experimental data.

Keywords: split cross bar; optical switch; converter; variable optical attenuators; analytical

I. Introduction

The recent rapid growth of optical fiber communication networks has boosted demand for various optical components, such as optical switches (OS) [1, 2] and variable optical attenuators (VOA) [3]. In an optical network, OS and VOA are usually deployed together. System complexity and cost increase significantly with the simultaneous use of a large number of single-function switches and attenuators. Therefore, an optical device integrated with switching and attenuating capabilities is desirable to lower the cost and the complexity of optical networks. In this work, we present a novel approach to realize a bi-stable 2x2 OS monolithically integrated with VOAs. The proposed dual-function device can be easily fabricated by a very simple process with a single photo-mask. The key to this approach is the use of the split-cross-bar (SCB) configuration [4, 5] in which all sub-components can be accommodated on a small SOI die without blocking optical paths. The heat transfer behavior of similar designs has been solved analytically [6-9], but estimating thermo-elastic deformation is primarily accomplished through finite-element methods (FEM) [10-14]. In addition, the analytical model of the bi-stable curved beam is solved in [15]. In this work, we develop an analytical model of the electro-thermo-elastic problems for the V-beam and curved beam actuators with applied external y-directed force and thermal strain force. The temperature of the entire actuator is calculated to accurately determine the tip displacement of the V-beam and curved beam. The results of the analytical solutions and the finite element (FEM) calculations are also compared.

This paper is organized as follows: The device configuration is reviewed in Section II. The analytical models of thermo-elastic behaviors are described in Section III. FEM modeling is presented in Section IV. Section V compares and discusses the results of the analytical models and the FEM approaches, and provides a VOA performance comparison between the measured and modeling results. Finally, Section VI presents conclusions.

II. Device configuration

Figure 1 shows the schematic of the dual-function device comprised of two bi-stable mechanisms, two small V-beam actuators, four large V-beam actuators, four fiber grooves, and four input/output fibers. Figure 2 shows the operational principle of the proposed device serving as an OS. As shown in Figure 2(a), four mirrors (i.e., M1, M2, M3 and M4) are used in the OS operation. Figures 2(a) and 2(b) show that optical switching is achieved by moving M1 and M3. Figure 3 illustrates the operational principle of the device operating as a VOA. Figure 3(a) shows the schematic serving as a VOA at OS-State-1. Slightly moving M1 (or M3) in a certain direction (indicated by \({{\theta }_{0}}\)) results in misalignment between the reflected light beam and the output channel which, in turn, attenuates optical power. As the displacement of the mirror increases, the optical power decreases. Similarly, Figure 3(b) shows the schematic serving as a VOA at OS-State-2 while Figure 4 shows the SEM pictures of the proposed device realized by a standard DRIE process on an SOI wafer [5].

It is well-known that a relatively large force is required to push the bi-stable mechanism back and forth. One possible method for pushing the mechanism is to use an electro-thermal V-beam actuator [16], which consists of many arched clamped-clamped beams (i.e., the V-structure shown in Figure 5) and undergoes thermal expansion when a current is applied. The arched structure will then deflect toward the arched direction with a large force.

III. Thermomechanical Model

A. Heat Transfer Model

To estimate the thermomechanical behavior of the device, we start with the heat transfer effect on the suspended structure. Assuming that convection and radiation effects are negligible, the 1-D heat transfer governing equation [13] for a suspended microstructure over substrate with a volume heat generation by Joule heating is given by:

\[{{T}_{,xx}}-\frac{s}{{{G}_{u}}t{{k}_{p}}}\left( T-{{T}_{s}} \right)+\frac{{{v}^{2}}}{\rho {{L}^{2}}{{k}_{p}}}=0,\tag{1}\]

where \({{G}_{u}}\) is the thermal conductance of air and silicon, \({{k}_{p}}\) is the thermal conductivity, \(s\) is the shape factor, \(t\) is the thickness, \(L\) is the distance between the two anchors of each curved beam actuator, \(v\) is the driving voltage, \(\rho \) is the specific resistance, and \({{T}_{s}}\) is the substrate temperature (i.e., the anchor temperature). For MEMS devices, the heat flux dissipated toward the substrate (anchor) is significant. The three terms in (1) respectively correspond to the diffusive heat flux along the beam, the heat flux conducted to the substrate, and the volume heating generation (Joule heating). Also, \(s\) is the shape factor accounting for the additional heat flux flowing into the substrate across the side walls of the actuator. The schematic of the solid model used in the FEM simulation is shown in Figure 5. An empirical expression for the shape factor for improving the accuracy was reported in [17] as

\[s=1.685{{\left[ \log \left( \frac{{{t}_{air}}+w}{w} \right) \right]}^{-0.59}}{{\left( \frac{{{t}_{air}}}{t} \right)}^{-0.078}}L,\tag{2}\]

where \({{t}_{air}}\) is the thickness of the air gap between the movable structure and the substrate, and t and w are respectively the thickness and width of the actuator. To simplify the model, thermal conductivity and resistance are assumed to be constant. Since substrate and air layer act as two thermal paths in series, the thermal conductance \({{G}_{u}}\) can be defined as

\[{{G}_{u}}=\frac{{{t}_{air}}}{{{k}_{air}}}+\frac{{{t}_{s}}}{{{k}_{s}}},\tag{3}\]

where \({{t}_{s}}\) and \({{k}_{s}}\) are respectively the thickness and thermal conductivity of the substrate. With the boundary conditions of \(T(0)={{T}_{s}}\) and \({{T}_{,x}}\left( {L}/{2}\; \right)=0\), the solution for (1) is

\[T(x)-{{T}_{s}}=\frac{\gamma }{{{k}^{2}}}\left[ 1-\frac{{{e}^{kx}}-{{e}^{k(L-x)}}}{1+{{e}^{kL}}} \right],\tag{4}\]

where

\[{{k}^{2}}=\frac{s}{{{G}_{u}}t{{k}_{p}}},\tag{5}\]
\[\gamma =\frac{{{v}^{2}}}{\rho {{L}^{2}}{{k}_{p}}}.\tag{6}\]

Figure 6 shows the temperature comparison by the FEM and the analytical solution. The error between FEM and the analytical model is about 10%. Once the temperature distribution is determined, the tip displacement of the thermal actuators can be estimated using thermal expansion of the beam. The details of the FEM analysis are discussed in next section.

B. Curved-beam Actuator

In this work, the curved beam plays two major roles: a bi-stable mechanism for the optical switch, and an in-plane-motion actuator for VOA. When the curved beam serves as a bi-stable mechanism, the governing equation for a curved beam subjected to an axial load \(P\) and a y-directed force \(F\) on the mid-point is as follows:

\[EI\cdot {{y}_{,xxxx}}+P\cdot {{y}_{,xx}}=EI\cdot {{y}_{0,xxxx}}-F\cdot \delta (x),\tag{7}\]

where \(E\) and \(I\) are the Young’s modulus and the inertia moment of the beam, and \(\delta (x)\) is the impulse function. According to the theoretical model [15], the shape of the curved beam is designed as the first buckling mode of a straight beam subjected to an axial load, as shown in Equation (8) and Figure 7(a):

\[{{y}_{0}}=\frac{h}{2}\left( 1+\cos {{k}_{1}}x \right),\tag{8}\]

where \(-L/2\le x\le L/2\), \({{k}_{n}}={2n\pi }/{L}\;\), \(h\) is the central offset of the curved beam. Also, the shape function satisfies the following boundary conditions: \({{y}_{,x}}=0\) at \(x=0\), \(y={{y}_{,x}}=0\) at \(x=-L/2\) and \(L/2\). We assume that the shape of the thermally-deformed beam is described by the similar form of Equation (8):

\[y=\sum\limits_{n=1}^{\infty }{\frac{{{w}_{n}}}{2}}\cdot \left[ 1+\cos ({{k}_{n}}x) \right],\tag{9}\]

where \({{w}_{n}}\) is the amplitude of the nth mode. Applying a y-directed force and voltage to the pre-shaped beam generates strain \(\varepsilon \) and axial force \(P\).

\[\varepsilon ={{\varepsilon }_{T}}+{{\varepsilon }_{x}}=\alpha \cdot \overline{T}-\frac{1}{2L}\int_{-L/2}^{L/2}{\left( y_{,x}^{2}-y_{0,x}^{2} \right)dx},\tag{10}\]

where \({{\varepsilon }_{T}}\) and \({{\varepsilon }_{x}}\) are respectively the thermal strain and the strain due to y-directed force applied to the curved beam. The change of \(\varepsilon \) gives rise to the axial force \(P\):

\[P=EA\varepsilon =EA\cdot \left[ \alpha \cdot \overline{T}-\frac{1}{2L}\int_{-L/2}^{L/2}{\left( y_{,x}^{2}-y_{0,x}^{2} \right)dx} \right],\tag{11}\]

where

\[\overline{T}={\int_{-L/2}^{L/2}{(T(x)-{{T}_{s}})dx}}/{L}\;=\frac{\gamma }{{{k}^{3}}L}\left( kL+2Cosh\left[ \frac{kL}{2} \right]-2Sinh\left[ \frac{kL}{2} \right] \right)\]

is the average temperature increase of the beam. By substituting Equations (8) and (9) into Equation (11), it can be rewritten as

\[P=EA\cdot \left[ \alpha \cdot \overline{T}-\frac{1}{4}\cdot \left[ \sum\limits_{n=1}^{\infty }{{{(\frac{{{w}_{n}}\cdot {{k}_{n}}}{2})}^{2}}-{{(\frac{h\cdot {{k}_{1}}}{2})}^{2}}} \right] \right].\tag{12}\]

After substituting the shape functions into the governing equation, we obtain

\[\begin{align} & \sum\limits_{n=1}^{\infty }{\left[ \left( EI\cdot k_{n}^{4}-Pk_{n}^{2} \right)\cdot \frac{{{w}_{n}}}{2}\cdot \cos \left( {{k}_{n}}x \right) \right]} \\ & =EIk_{1}^{4}\cdot \frac{h}{2}\cdot \cos \left( {{k}_{1}}x \right)-F\cdot \delta (x) \\ \end{align}.\tag{13}\]

By integrating Equation (14) from \(x=-L/2\) to \(x=L/2\) after multiplying it by \(\cos {{k}_{1}}x\), and employing the orthogonality properties of the eigen-functions, one can derive an analytical form:

\[\left( EI\cdot k_{1}^{4}-{{P}_{1}}\cdot k_{1}^{2} \right)\cdot \frac{{{w}_{1}}\cdot L}{4}=EI\cdot k_{1}^{4}\left( \frac{hL}{4} \right)-F,\tag{14}\]

where \({{P}_{1}}\) is the axial force in first mode \({{n=1}}\). Substituting this into Equation (14), we obtain:

\[F=-\frac{k_{1}^{4}EAL}{64}\cdot \left[ w_{1}^{3}+\left( \frac{16I}{A}-\left( \frac{16\alpha \overline{T}}{k_{1}^{2}}+{{h}^{2}} \right) \right){{w}_{1}}-16\frac{Ih}{A} \right].\tag{15}\]

Under a special condition of zero loading (i.e., \(F=0\)), we obtain

\[w_{1}^{3}+\left( \frac{16I}{A}-\left( \frac{16\alpha \overline{T}}{k_{1}^{2}}+{{h}^{2}} \right) \right){{w}_{1}}-16\frac{Ih}{A}=0.\tag{16}\]

The curved beam expands due to Joule heating as shown in Figure 7(a). Solving Equation (16), we obtain two stable positions of the bistable mechanism, which are indicated in Figure 8(a). The first stable position due to thermal expansion can be shown as

\[\begin{align} & {{w}_{fst}}=\sqrt[3]{-b/2+\sqrt{{{b}^{\text{2}}}/4+{{a}^{3}}/27}} \\ & -\sqrt[3]{b/2+\sqrt{{{b}^{\text{2}}}/4+{{a}^{3}}/27}} \end{align},\tag{17}\]

where \(a=\frac{16I}{A}-\left( \frac{16\alpha \overline{T}}{k_{1}^{2}}+{{h}^{2}} \right)\) and \(b=-16\frac{Ih}{A}\), which corresponds with the results in [18]. Figure 8(b) shows the applied external force versus the amplitude of the deformation \(\left( d=h-{{w}_{1}} \right)\) at different average temperature \(\overline{T}\) of the curved beam. Note that the points in A and B indicate the amplitudes due to thermal expansion without applying external force (the displacement of the first stable position) given respective average temperatures of 50 K and 100 K. The amplitude of the first mode (\(n=1\)) at the critical positions can be derived by solving \({dF}/{d{{w}_{1}}}\;=0\) [18]:

\[{{w}_{1,crt}}=\pm \sqrt{\frac{16I}{3A}-\frac{16\alpha \overline{T}+k_{1}^{2}{{h}^{2}}}{3k_{1}^{2}}}.\tag{18}\]

Substituting the two critical positions of Equation (18) into Equation (15), the critical external y-directed forces can be found at the local maximum and local minimum (see Figure 8(a)):

\[\begin{align} & {{F}_{crt,d}}=-\frac{k_{1}^{4}EAL}{64}\cdot \\ & \left[ {{\left( \frac{16I}{3A}-\frac{16\alpha \overline{T}+k_{1}^{2}{{h}^{2}}}{3k_{1}^{2}} \right)}^{3/2}}\cdot \left( \frac{\sqrt{3}+1}{\sqrt{3}} \right)-\frac{16Ih}{A} \right] \end{align} \tag{19}\]

and

\[\begin{align} & {{F}_{crt,u}}=\frac{k_{1}^{4}EAL}{64}\cdot \\ & \left[ {{\left( \frac{16I}{3A}-\frac{16\alpha \overline{T}+k_{1}^{2}{{h}^{2}}}{3k_{1}^{2}} \right)}^{3/2}}\cdot \left( \frac{\sqrt{3}+1}{\sqrt{3}} \right)+\frac{16Ih}{A} \right] \end{align}.\tag{20}\]

In the analytical model, the two stable positions and the critical force are dependent on the average temperature. As shown in Figure 8(b), increases in the average temperature increase the distance between the two stable positions and the critical forces [20].

C. V-beam Actuator

When applying voltage on the anchors of the V-beam actuator, the two arms expand and push the tip along the y-axis, as shown in Figure 7(b). Being symmetric, the two arms can be considered to be two fixed-guided beams. The guided end of each arm moves along the y-axis. The boundary conditions of the guided end, including a reactive force (\({{P}_{x}}\)), a reactive moment (\({{M}_{0}}\)), a virtual force applied on y-directed direction (\({{F}_{v}}\)) and a y-directed applied force (\({{F}_{\ell }}\)), are also shown in Figure 7(b). In addition, the thermal force \({{F}_{t}}\) due to the elongation of the arm by heating can be modeled as:

\[{{F}_{t}}=EA\cdot {{\varepsilon }_{T}}=EA\alpha \overline{T}.\tag{21}\]

The bending moment around the z-axis (\({{M}_{z}}\)) and the axial force (\({{F}_{{{x}^{'}}}}\)) can be written as

\[{{F}_{{{x}^{'}}}}={{F}_{t}}-{{P}_{x}}\cdot \cos \theta +{{F}_{v}}\cdot \sin \theta -{{F}_{\ell }}\cdot \sin \theta ,\tag{23}\]
\[{{M}_{z}}={{M}_{o}}+{{F}_{v}}\cdot {{x}^{'}}\cos \theta -{{F}_{\ell }}\cdot {{x}^{'}}\cdot \cos \theta +{{P}_{x}}\cdot {{x}^{'}}\sin \theta .\tag{23}\]

The strain energy \({{U}_{p}}\) due to the axial force \({{F}_{{{x}^{'}}}}\) can be calculated as

\[{{U}_{p}}=\frac{1}{2EA}\int_{0}^{\ell }{F_{{{x}^{'}}}^{2}d{{x}^{'}}}=\frac{{{\left[ {{F}_{t}}-{{P}_{x}}\cos \theta +({{F}_{v}}-{{F}_{\ell }})\sin \theta \right]}^{2}}\cdot l}{2EA}.\tag{24}\]

The strain energy \({{U}_{b}}\) due to the bending moment can be calculated as

\[\begin{align} & {{U}_{b}}=\frac{1}{2EI}\int_{0}^{\ell }{M_{z}^{2}}d{{x}^{'}} \\ & =\frac{1}{2EI}{{\int_{0}^{\ell }{\left[ {{M}_{o}}+({{F}_{v}}-{{F}_{\ell }}){{x}^{'}}\cdot \cos \theta +{{P}_{x}}\cdot {{x}^{'}}\cdot \sin \theta \right]}}^{2}}d{{x}^{'}} \end{align}.\tag{25}\]

Finally, the total strain energy in the system is:

\[{{U}_{t}}={{U}_{p}}+{{U}_{b}},\tag{26}\]

where \(\theta \) is the incline angle of the arm and \(\ell \) is the half length of the v-beam actuator.

The boundary conditions at the guided end provide the required equations for the analytical solution. The x-displacement \({{u}_{x}}\) and rotation around the z-axial \({{\theta }_{z}}\) are both zero at the guided end. Then, applying Castigliano’s displacement theorem, the following equations are obtained

\[{{u}_{x}}=0\Rightarrow \frac{\partial {{U}_{t}}}{\partial {{P}_{x}}}=0,\tag{27}\]
\[{{\theta }_{z}}=0\Rightarrow \frac{\partial {{U}_{t}}}{\partial {{M}_{o}}}=0.\tag{28}\]

Using the above two equations, \({{P}_{x}}\) and \({{M}_{o}}\) can be expressed in terms of \({{F}_{t}}\), \({{F}_{\ell }}\) and the geometrical parameters of the actuator. Then, the y-displacement of the arm \(\Delta y\) can be calculated by applying a virtual force \({{F}_{v}}\) in y-direction at the free end of the beam.

\[{{P}_{x}}=\frac{12EI\cdot {{F}_{t}}\cos \theta +\frac{1}{2}\left( EA{{l}^{2}}-12EI \right){{F}_{\ell }}\sin 2\theta }{12EI{{\cos }^{2}}\theta +EA{{l}^{2}}{{\sin }^{2}}\theta },\tag{29}\]
\[{{M}_{o}}=\frac{l}{2}\left( {{F}_{\ell }}\cos \theta -{{P}_{x}}\sin \theta \right).\tag{30}\]

Applying Castigliano’s theorem and setting the virtual force to zero (\({{F}_{v}}=0\)) yields

\[\begin{align} & \Delta y=\frac{\partial {{U}_{t}}}{\partial {{F}_{v}}} \\ & =\frac{\left( {{F}_{t}}-{{P}_{x}}\cos \theta -{{F}_{\ell }}\sin \theta \right)l\sin \theta }{EA} \\ & +\frac{{{M}_{o}}{{l}^{2}}\cos \theta +\frac{{{l}^{3}}}{3}\left( {{P}_{x}}\sin 2\theta \right)-2{{F}_{\ell }}{{\cos }^{2}}\theta }{2EI} \end{align}.\tag{31}\]

Substituting \({{P}_{x}}\) and \({{M}_{o}}\) into Equation (29) yields a simple relationship between the thermal strain force, y-directed applied force and tip displacement.

\[\Delta y=\frac{\left( {{F}_{t}}\sin \theta -{{F}_{\ell }} \right){{l}^{3}}}{12EI{{\cos }^{2}}\theta +EA{{l}^{2}}{{\sin }^{2}}\theta }.\tag{32}\]

Note that if the external y-directed force \({{F}_{\ell }}=0\), Equation (33) is the same as in [21].

IV. Results and Discussions

The thermal-mechanical behavior of the V-beam and curved beam actuator were also simulated using the commercial FEM package CoventorWareTM using 3D models. Figure 5 shows the schematic of the simulated model of a V-structure. For electro-thermal analysis, a voltage difference was applied between the anchors of the actuator. The temperature of the substrate bottom was fixed at constant temperature, and a convection boundary condition was applied for all surfaces. For thermo-mechanical analysis, the bottom surfaces of the anchors were fixed on the substrate surfaces. The electro-thermal analysis computed the temperature distributions of the structure. The temperature distributions were then used as the volume conditions for the thermo-mechanical analysis to calculate structural deformations.

Figure 9 shows the simulated results of the thermal actuator with an applied voltage of 5 V. The dimensions of the curved beam and v-beam actuators used in the FEM simulations and analytical models are presented in Table I. All the thermal properties and the material constants can be found in Tables II and III. The calculated tip displacements for different V-beam structures (1000 µm, 3000 µm and 4000 µm in length) are shown in Figures 10(b) and 11, which also show the results by the analytical solution of Equations (17) and (32) without applied external y-directed force. FEM simulations and measured displacements at different input voltages for the V-beam and curved-beam actuators are also shown. Figures 10(a) and (b) show the tip displacement versus input voltage for the curved beam and small V-beam actuator with respective beam lengths of 3000 µm and 1000 µm. Figures 11(a) and (b) show the tip displacement versus input voltage for the large V-beam actuator with respective beam lengths of 4000 µm and 3000 µm. The tip displacement is seen to increase with the beam length. Figure 12 shows the analytical solution of Equation (32) and the semi-analytical solution from [10] with an applied y-directed force of \({{F}_{\ell }}=1000\text{ }\!\!\mu\!\!\text{ N}\). All the parameters used in Figure 12 are the same as those used in [10].

Figure 13 illustrates the measured and analytical attenuation performance versus applied voltages, showing good agreement between the measured results and the analytical results for state 1 and state 2 with different angles. Note that the analytical attenuation is calculated from the displacement of the mirror [25]. For the VOA@OS-State-1, the measured attenuation reaches 33 dB under an applied voltage of 20 V. For the VOA@OS-State-2, an attenuation of 34 dB can be achieved with 15 V. The measured results are in a good agreement with the calculated results.

V. Conclusion

This work presents the modeling, simulation and characterization of an MEMS optical device. The modeled device is fully-monolithically integrated using micromachining techniques and has dual functions: optical switching and variable optical attenuating. A curved beam bi-stable mechanism is adopted for optical switching, and the split-cross-bar design (SCB) is used as the optical path configuration. Analytical solutions of the thermo-elastic bending and buckling of the microstructures in the device are proposed. The results predicted by the analytical solutions are compared with the finite element (FEM) calculated results, with a discrepancy of under 10%. The experimental results are also in a good agreement with theoretical predictions, which suggests that the analytical modeling results presented in this work are useful for applications in MEMS devices with curved beam bi-stable structures.

Acknowledgement

This work was supported in part by the National Science Council, Taiwan, R.O.C. (Contract No: NSC 94-2212-E-002-060). The authors would like to thank Mr. Sun-Chih Shih and Mr. Shen-An Yang for their help on the packaging process.

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