Maria Bayard Dühring - Solid Mechanics

two air holes, is Λ = 3.1 µm, the hole diameter over the pitch ratio is d/Λ = 0.92 and the diameter
is d = 2.852 µm. The radius of the air core in the center is R = Λ. The core region Ωc where the
objective function is optimized has the radius 0.5R, and the design domain Ωd is defined by the area
between the two half circles with radius equal to 0.5R and 1.5R, respectively. The half circle at the
center where the source is applied is equal to 0.05R. The maximum element size is 0.1 µm in Ωd
and 0.8 µm in the rest of the domain. The refractive indexes for the chosen wavelength are na = 1
for air and ns = 1.43791 + 0.0001i for silica, where the imaginary part in ns causes absorption and
is denoted as the absorbing coefficient αs. Before the optimization, a band-gap check is performed
with the commercial program BandSolve, see [31]. The band diagram is calculated for the periodic
structure with k0Λ as function of k3Λ, where k3 is the propagation constant in the x3-direction. The
point (k3Λ, k0Λ) = (9.5127, 9.73894) for the fundamental mode was shown to be in the calculated
band gap. The wave equation (2) is then solved as an eigenvalue problem with the absorption
coefficient αs equal to zero and the propagating constant β for the guided mode is found, which is
used together with the fixed value of k0 = 2π/λ0 to solve the model by employing the forcing term.
3.1 Study of the optimized design
The optimization is performed with the filter radius r = 0.15 µm, a move limit equal to 0.05 for
the maximum change in the design variables in each iteration step and the absolute tolerance 0.01
to terminate the optimization. The continuation method for the damping is applied such that αs
is equal to 1 when the optimization starts and is reduced by a factor 10 after 100 iterations or
if the tolerance is satisfied. This is repeated until αs reaches the value 0.0001. Introducing the
damping lowers and widens the peak at resonance as seen on figure 2 where log(Φ/Φinit) is plotted
as function of λ0 for increasing αs. Φinit is the value of the objective function for the initial design.
The resonance is kept at the original wavelength λ0 = 2 µm for the low values of αs, but is slightly
shifted to higher wavelengths for the two highest values of αs such that the optimization starts
away from resonance. The optimized design is found in 316 iterations and the objective function
Φ is increased 375% compared to the initial design. In figure 2, log(Φ/Φinit) as function of λ0 is
compared for the initial and the optimized design. The wavelength at resonance is the same before
and after the optimization and the peak value has increased for the optimized design. In figure 3(a)
and 3(b) the initial and optimized design are plotted, respectively, and the distribution of the energy
measure from the objective function normalized with the maximal value for the initial design, Φmax,
is indicated with the contour lines. The optimized design is close to a 0-1 design and all parts are
log(Φ/Φ init ) [−]
1
0
−1
−2
−3
−4
−5
−6
optimized design, α s =0.0001
initial design, α s =0.0001
initial design, α s =0.001
initial design, α s =0.01
initial design, α s =0.1
initial design, α s =1.0
−7
1.99 1.995 2
wavelength, λ [μm]
0
2.005 2.01
Figure 2: The logarithm to the normalized objective function Φ/Φinit as function of the optical
wavelength λ0 is indicated both for the initial design with different values of the absorption coefficient
αs and for the optimized design with αs = 0.0001.
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