(a) In a free space optical MEMS system, the Gaussian beam coming from the fibre propagates in free space and its beam size is given as w=w0[1+(lambda /(pi2/0)2]1/2where wo 50

is the original beam waist (wo = 50 um), A is the wavelength of the light (lamda=1.5 um), and z is the distance the beam travels starting from the facet of the fibre. A circular shape reflecting micromirror is placed to spatially alter the direction of the light, and the distance z between the fibrefacet and the mirror is 50 mm. Calculate the minimum radius of the mirror.At the separation z between the fibre facet and the mirror, the longitude loss is given as L = -10 log[4/(M² + 4)], M = lamdaz/(anw3), where n is the refractive index (n=1 in free space). Calculate the longitude loss at 50 mm. (b) To design a cantilever type piezoelectric energy harvester, calculate one set of dimensions to match with the mechanical vibration of 3500 Hz. (UsePZT5A as the piezoelectric material, its density p is 7750 kg/m³) The stiffness of a cantilever is given as k = wt3E/(4L³), where w=30 µm, t, Lare width, thickness, and length of the cantilever, E is the Young's modulus of the material (1.96×1011 Nm-2).[4 marks] (c) Derive the state-space representation (or system matrix) for the two microresonators' dynamic motion equations shown below. Write MATLAB functions for these two equations and solve them with boundary conditions provided. (MATLAB codes need to be included in the answer sheet). Plot xvs. t, i vs. t, and i vs. x, respectively. 13 \ddot{x}+9 \dot{x}+x=\sin (0.28 t) ; \quad x_{0}=0 ; \dot{x}_{0}=0 ; \quad t \in[0200] \check{5 x}+3 x+70 x=4 \quad x_{0}=1 ; \dot{x}_{0}=0 ; \quad t \in[020] Sketch out a process flow for fabricating a MEMS resonator using the bulk micromachining process.