Edwin Jan Klein - Universiteit Twente

P
Through
P
In
P
Drop =
P
In
= 1+ F5.
F
F5.F4
75
3
Simulation and analysis
(4.11)
(4.12)
The partial terms F1 to F4 can be pre-calculated since they are not dependent on φ.
The calculation of the through and drop response therefore only requires the
recalculation of (4.10-4.12) which, assuming that φ is known, now only requires 1
cosine, 3 multiplications, 1 division, 1 subtraction and 1 addition.
4.3 RFit
The RFit tool, used for the parameter fitting of micro-resonator through and drop
responses, was based on Equations (4.6-4.12). These allowed for a significant
reduction (>10 times) of the fitting time compared Equations (4.4) and (4.5) because
these equations are easily called several million times during the fitting process.
The fitting process uses a simple two-step brute force method that first fits the free
spectral range and then the curve itself. When started the program requires a few
approximate values to be set for αdB, κ, ng and R. The first three of these are not
critical and are easily estimated by the user while the more critical value of the radius
is almost always known from the design.
The free spectral range fit calculates the sum
Σ
λ
m
[ ] 2
P n , ∆λ,
λ ) − P ( λ )
Sim
( g m Meas m
(4.13)
for variations in the group index ng and the response shift ∆λ within a range that has
been defined by the program user. The latter variable is only used as a helper variable
to shift the simulated response with respect to the measured response so that they are
aligned correctly (i.e. resonances at the same wavelengths). It is added to the phase
calculation of the simulated response using:
( 2.
π )
ϕ r = ⋅ R ⋅ n
λ + ∆λ
m
2
g
(4.14)
The sum is only calculated for the wavelengths λm for which measurement data is
available. The best fit is found for those values of ng and ∆λ at which (4.15) is at its
minimum.
The next step in the fitting process is then to scan the variables κ1, κ2 and αdB across a
user defined range and find the values for which the sum
Σ
m
[ ] 2
log( P ( κ , κ , α , λ )) −10
log( P ( λ ))
λ 10 Sim 1 2 dB m
Meas m
(4.15)
is at its minimum. A possible outcome of such a fitting process is given in Figure 4.4.