WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

The optimization problem in Eq. (4) is based on a Galerkin finite element discretization of Eqs. (1)-
(3) which leads to the system equation consisting of a set of linear complex equations. A single design
variable ̺e is assigned to each finite element within the chosen design domain (Nd elements). This design
variable is then used with a SIMP-like model to control the material properties in the corresponding
elements:
Ae = A1 + ̺e(A2 − A1)
(5)
Be = B1 + ̺e(B2 − B1).
The optimization problem is formulated as a max-min problem aiming to maximize both objectives Φ1
and Φ2. As objective we wish to maximize the wave transmission at the two output ports, so we specify
the two objectives as the time-averaged Poynting vector integrated over these ports, which quantifies
the corresponding energy flux. The point-wise time-averaged Poynting vector is given as:
P(x) = {Px Py} T = 1
2 ωAℜ iu∇u . (6)
For further details regarding the discretization of Eq. (6) see [6]. Analytical sensitivity analysis is facilitated
by the adjoint method ([4], [6]), and the optimization problem stated in Eq. (4) is solved using
the mathematical programming software MMA [5].
5.2. Penalization - PAMPING
We relate the element design variables to the element material properties using Eq. (5). Since we do
not impose a volume constraint, which does not make sense in this problem, we cannot rely on using
a penalization factor as in the usual SIMP model to ensure a well defined binary structure. Thus, we
use a simple linear interpolation (Eq. (5)) and introduce instead a penalization of intermediate design
variables based on artificial dissipation [6].
This is done by adding an extra conduction (or dissipation) term in each element within the design
domain:
σe ∼ α̺e(1 − ̺e), (7)
where α is a scaling factor. Thus, elements with intermediate values of ̺e dissipate energy and consequently
reduce the objective function. In this way the element design variables will be forced towards 0
and 1, if α is sufficiently large.
We now exemplify results of applying the optimization algorithm to the model problem in Figure 4.
5.3. T-splitter - TM polarization
The first example is for TM-polarized waves. We use the setup described in [16], using straight waveguides
of width w = 200nm and a dielectric material with refractive index n = 3.2. A similar system was
recently studied in [17]. As previously reported the reflection in the wavelength range around 1.55µm is
about 25% for a plain simple T-junction.
Figure 5 shows the optimized waveguide for a single frequency corresponding to a wavelength of
1.55µm. In this case we have chosen a very small design domain allowing only for small modifications of
the material distribution and thus giving limited possibilities for improving the performance. In Figure
5 two different designs are shown along with the corresponding wave patterns. The structure at the top
(structure 1) has the better performance and has been obtained by a straightforward implementation
of the optimization algorithm. It is noticeable that the designed waveguide is discontinuous and the
waves has to cross an air bridge. In the 2D loss-free model this leads to a good performance (see Figure
7), but for a real 3D structure large out-of-plane scattering losses can be expected. To avoid this, we
have designed the bottom structure (structure 2) with simulated dissipation in the air. Consequently
discontinuities are less favorable since the waves have to travel a distance in the lossy medium. The
resulting structure performs worse with the loss-free 2D model, but we anticipate that it will perform
better than structure 1 using 3D simulations and also in experiments. This is ongoing work.
We now attempt to improve the performance by increasing the design domain. Figure 6 shows two
examples of optimized structures with larger design domains. Both designs have been obtained with
dissipation in the air. Figure 6 (top) (structure 3) has been obtained by enlarging the design domain in
a straightforward manner with a very good performance as a results (see Figure 7), showing practically
full transmission near the target frequency. In Figure 6 (bottom) (structure 4) the design domain from
5