Edwin Jan Klein - Universiteit Twente

Substitution with (2.18) then gives:
SRR
⎛ ( 1+
µ 1µ
2 ⋅ χr
)
10.
log ⎜
⎝ ( 1−
µ 1µ
2 ⋅ χr
)
= 2
23
2
⎞
⎟
⎠
The Micro-Resonator
(2.28)
where SRR is expressed in dB.
The SRR is proportional to the finesse of the resonator. The exact relationship between
the finesse and the SRR can be found by rewriting (2.28) as a function of SRR:
µ µ ⋅ χ =
1
2
r
10
10
Substitution of (2.29) in (2.19) then leads to:
S RR
10
S RR
10
−1
+ 1
S RR
10
π µ 1µ
2 ⋅ χr
π 10
F ≈
=
1−
µ µ ⋅ χ 2
1
2
r
−1
(2.29)
(2.30)
In general filter design there is always a tradeoff to be made between the bandwidth
of a filter and the filter rejection ratio. That a micro-resonator based filter is no
exception to this rule is apparent by substituting (2.30) in (2.22):
∆λ
FWHM
=
2.
FSR
S RR
=
10 2 10
π 10 −1
n ⋅ R ⋅π
10 −1
g
λ
2
0
S RR
(2.31)
This equation can be used to plot the bandwidth ∆λFWHM of the resonator as a function
of the rejection ration for a number of effective resonator radii (REff=R.ng) as is done
in Figure 2.11.
Bandwidth (GHz)
300
275
250
225
200
175
150
125
100
75
50
25
20 µm
50 µm
75 µm
100 µm
200 µm
500 µm
1000 µm
0
5 10 15 20 25 30 35 40
SRR (dB)
Figure 2.11. Maximum achievable bandwidth as a function of SRR for a number of
different effective radii (simulated for λ0=1550nm)