# Tunable Optical Filter (38875)

### Tunable Optical Filter (38875)

Dennis Hohlfeld, hohlfeld@imtek.uni-freiburg.de

Tamara Bechtold, bechtold@imtek.uni-freiburg.de

Hans Zappe, zappe@imtek.uni-freiburg.de

The DFG project AFON (funded under grant ZA 276/2-1) aimed at the development of an optical filter, which is tunable by thermal means. The thin-film filter is configured as a membrane in order to improve thermal isolation. Fabrication is based on silicon technology. Wavelength tuning is achieved through thermal modulation of resonator optical thickness, using metal resistor deposited onto the membrane. The devices features low power consumption, high tuning speed and excellent optical performance [1].

The benchmark contains a simplified thermal model of a filter device. It helps designers to consider important thermal issues, such as what electrical power should be applied in order to reach the critical temperature at the membrane or homogeneous temperature distribution over the membrane. The original model is the heat transfer partial differential equation.

There are two different benchmarks, 2D model and 3D model (see Table 1). Due to modeling differences, their simulation results cannot be compared with each other directly.

Code | comment | dimension | nnz(A) | nnz(E) |

filter2D | 2D, linear elements, PLANE55 | 1668 | 6209 | 1668 |

filter3D | 3D, linear elements, SOLID90 | 108373 | 1406808 | 1406791 |

The device solid models have been made, meshed and discretized in ANSYS 6.1 by the finite element method. All material properties are considered as temperature independent. Temperature is assumed to be in Celsius with the initial state of 0 C. The Dirichlet boundary conditions of T = 0 C have been applied at the bottom of the chip.

The output nodes for the models are described in Table 2 and schematically displayed in Fig 2. Output 1 is located at the very center of the membrane. By simulating itÕs temperature one can prove what input power is needed to reach the critical membrane temperature for each wavelength. Furthermore, the output 2 to 5 must be very close to output 1 (homogenous temperature distribution) in order to provide the same optical properties across the complete diameter of the laser beam.

Number | Code | Comment |

1 | Memb1 | Membrane center |

2 | Memb2 | Membrane node with radius 25E-6, theta 90 |

3 | Memb3 | Membrane node with radius 50E-6 theta 90 |

4 | Memb4 | Membrane node with radius 25E-6, theta 135 |

5 | Memb5 | Membrane node with radius 50E-6 theta 135 |

Fig. 2. Schematical position of the chosen output nodes.

The benchmark contains a constant load vector. The input function equal to 1 corresponds to the constant input power of of 1 mW for 2D model and 10 mW for 3D model. The linear ordinary differential equations of the first order are written as:

E dT/dt = A T + B u

y = C T

where E and A are the symmetric sparse system matrices (heat capacity and heat conductivity matrix), B is the load vector, C is the output matrix, and T is the vector of unknown temperatures.

The output of the transient simulation for node 1 over the rise time of the device (0.25 s) for 3D model can be find in Filter3DTransResults. The results can be used to compare the solution of a reduced model with the original one. The time integration has been performed in ANSYS with accuracy of about 0.1 %. The results are given as matrices where the first row is made of times, the second of the temperatures.

Download matrices in the Matrix Market format: File 1, 106502 bytes; File 2, 37417415 bytes. The matrix name is used as an extension of the matrix file. File *.C.names contains a list of ouput names written consecutively. The system matrices have been extracted from ANSYS models by means of mor4fem.

The discussion of electro-thermal modeling related to the benchmark can be found in [2].

## Bibliography

1.
D. Hohlfeld, H. Zappe, *All-dielectric tunable optical filter based on the thermo-optic effect*, Journal of Optics A: Pure and Applied Optics, **6**(6), 504- 511 (2003).

2.
T. Bechtold, *Model Order Reduction of Electro-Thermal MEMS*, PhD thesis, University of Freiburg, Germany (In preparation).