WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

power flow is evaluated for several frequencies (up to 6) in the chosen frequency range.
During the optimization procedure we maximize the output for all frequencies, and update
these target frequencies every 10th or 20th iteration in order to eliminate transmission dips in
the frequency range [21].
The optimized design is shown in Fig. 1 (right) where the green areas highlight the
optimized design domains. This design was obtained after approximately 1000 iterations of
the optimization algorithm, however, with the qualitative structure of the design appearing
after about 200 iterations. Clearly, the bends have been smoothened by applying a soft
curvature in the bend region and one hole has been removed on the inner side of the bend.
However, the smoothening is not trivial as the design domain still contains complex
structures. Note that the optimized 60° bend mostly resembles an etched mirror [22], whereas
the topology-optimized 120° bend [12] retained its original crystal structure with deformed
holes. The major strength of the topology optimization method is that the superior type of
structure does not need to be known in advance; it will appear from the optimization
procedure.
Fig. 2. Scanning electron micrographs of fabricated photonic crystal waveguides containing
two consecutive 60° bends. The pitch of the triangular lattice is Λ≈400nm with hole diameter
D≈275nm. Left: Waveguide with generic. Right: Waveguide with topology-optimized bends.
The number, shape and size of the holes at each bend are designed using topology
optimization. The contrast and brightness of the images have been changed for clarity.
Figure 2 shows scanning electron micrographs of the fabricated PhCWs containing two unoptimized
(left) and two topology-optimized (right) 60° bends. The PhC structures have been
fabricated without applying any special proximity corrections to the irregular shaped holes
during the e-beam patterning. Nonetheless, the fabricated topology-optimized structures
nicely resemble those shown in Fig. 1 (right).
3. Simulation and experimental results
Figure 3 shows the steady-state magnetic field distribution for the fundamental PBG mode of
the fabricated PhCWs simulated using 2D FDTD. The left image shows the mode behavior for
light incident from the bottom-left through the PhCW with the generic bends. It is clearly seen
that the generic bend forms a severe discontinuity in the straight PhCW and excite an odd
mode, which is not well guided in the PhCW. Moreover, the lower bend introduces large
reflections and scattering of light to the PhC structure. In contrast, the right image shows that
the topology-optimized bend regions guide the fundamental PBG mode nicely through the
two bends without disturbing the mode profile.
#5520 - $15.00 US Received 19 October 2004; revised 12 November 2004; accepted 15 November 2004
(C) 2004 OSA 29 November 2004 / Vol. 12, No. 24 / OPTICS EXPRESS 5919