WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

1196 J. Opt. Soc. Am. B/Vol. 22, No. 6/June 2005 J. S. Jensen and O. Sigmund
Fig. 6. Transmission through the top and bottom output ports.
Results for the standard junction is shown along with three optimized
designs for three different frequencies.
Fig. 7. Illustration of update scheme for target frequencies. The
frequency curve is computed with Padé approximations to obtain
high frequency resolution at low computational cost. Discrete
markers indicate transmission values computed directly.
I 1 = †˜ 1;˜ 2, ..., I n = ˜ N;˜ N+1‡, 20
where N is the number of target frequencies and ˜ N+1
−˜ 1 is the entire frequency range of interest, divided into
N equally sized intervals. The power transmission is normalized
for each frequency with the corresponding transmission
for the straight waveguide J * . In this way a
proper weighting of the different frequencies is ensured.
The implementation of the the active-set strategy is illustrated
in Fig. 7. A number N of target frequencies is
chosen, and the frequency interval is divided into N
equally sized sections. At regular intervals during the optimization,
e.g., every 20 or 30 iterations, the transmission
spectrum is computed, and the frequency with the
lowest transmission is identified in each interval. These
frequencies now become the new target frequencies in
subsequent iterations. It is important that the spectrum
is computed with high frequency resolution to accurately
detect the critical frequencies. Unfortunately, this requires
solving the direct problem [Eq. (4)] for many frequencies,
thus slowing down the optimization procedure
significantly. A way to overcome this is to compute the
spectrum with a fast-frequency-sweep technique with
Padé approximates. 20 We then need to solve Eq. (4) only
once for each frequency interval and can then expand the
solution in the neighboring frequency range with high accuracy
at low computational cost. The transmission spectrum
in Fig. 7 is computed with a fast frequency sweep,
and the solutions computed by our solving the direct problem,
depicted with discrete markers, illustrate the accuracy
of the expansion.
In the first example we use the original design domain
(Fig. 2) and attempt to design the junction with full transmission
in the entire frequency range from ˜ =0.32 to ˜
=0.44. We start out with three target frequencies and increase
this number to 12 as the continuation parameter
is reduced to its final value at =0.00625. Throughout the
optimization we keep the pamping parameter =0.1. Figure
8 shows the final design and the field computed for
˜ =0.38. Figure 10 shows the corresponding transmission
spectrum. However, note that the transmission still drops
near the extremal frequencies.
It appears that the chosen design domain is unable to
provide a full transmission in the entire frequency range,
and therefore the domain is increased by an additional
eight unit cells near the corners of the junction. We repeat
the optimization procedure described above by using the
previous design as the initial design and obtain the structure
displayed in Fig. 9 that shows also the field computed
for ˜ =0.38. As can be seen in the transmission diagram
(Fig. 10), the transmission is now improved and
especially increased near the limits of the frequency
range. By choosing even larger design domains, one can
expect further improvements.
5. CONCLUSIONS
We have developed a design method based on topology optimization
and used it to design a T junction in a photonic
crystal waveguide with high transmission in a large frequency
range.
The optimization algorithm is based on a frequencydomain
finite-element model of 2D plane polarization. Elementwise
constant design variables govern the distribu-
Fig. 8. Design optimized for frequency range ˜ =0.32–0.44. The
field is computed for ˜ =0.38.