WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

J. S. Jensen and O. Sigmund Vol. 22, No. 6/June 2005/J. Opt. Soc. Am. B 1195
should be emphasized that the CPU time per iteration is
almost independent of the design domain size (for the
given discretization).
We optimize the junction for ˜ =0.32, ˜ =0.38, and ˜
=0.44 ˜ =2c/a. This corresponds to two frequencies
near the extremal values of the guided-mode frequency
range and one center frequency. For an initial design, we
choose the unoptimized structure as depicted in Fig. 2.
Figures 3–5 show three optimized designs together
with the fields for the corresponding target frequencies.
In Figure 6 the transmission spectrum is depicted for the
three optimized designs as well as for the initial structure.
The spectra have been normalized with the power
transmission through a straight waveguide, so that 0.5
corresponds to 50% transmission through both upper and
lower output ports of the symmetrical design, and hence
there is no reflection at the junction.
For all three designs, practically full transmission is
obtained at the specified target frequency. Except for the
design obtained for ˜ =0.32, a good performance is also
seen away from the target, which indicates that the continuation
approach has been effective in eliminating local
maxima that are based on local resonances. These may
display high transmission at single frequencies but are
normally associated with excessive peaks and valleys in
the transmission spectrum. Starting from different initial
designs in most cases, we obtain different optimized designs,
which indicates the strong nonuniqueness of the
optimization problem. However, all designs obtained perform
equally well (full transmission) at the target frequency.
Owing to the long wavelength of the guided mode at
˜ =0.32, the optimized structure for this frequency is not
well suited for higher frequencies at which the wavelength
is significantly shorter. Similarly, the optimized design
for ˜ =0.44 performs poorly at lower frequencies.
However, the design for ˜ =0.38 gives a good transmission
in a large frequency range, and the transmission drops
significantly only for frequencies below ˜ =0.35 and above
˜ =0.41.
4. FREQUENCY-RANGE OPTIMIZATION BY
USE OF ACTIVE SETS
To get a larger bandwidth with high transmission, we
need to optimize the junction for several frequencies in
the specified frequency range simultaneously. In Ref. 4
the sum of the transmission for a number of target frequencies
was considered. With this approach the frequencies
should be chosen carefully, and even then the transmission
may still drop significantly between these
frequencies. This problem could be partially remedied by
use of a large number of frequencies. However, this would
be CPU time expensive.
Instead, we introduce an active-set strategy in which
we no longer keep the target frequencies fixed but let
them vary according to the most critical frequencies, i.e.,
those with minimum transmission.
We now write the objective as
max
x e
1 ,..., N
minJi/J
iIi * i, Fig. 3. Optimized T-junction topology for target frequency ˜
=0.32 and the corresponding field distribution.
Fig. 4. Optimized T-junction topology for target frequency ˜
=0.38 and the corresponding field distribution.
Fig. 5. Optimized T-junction topology for target frequency ˜
=0.44 and the corresponding field distribution.