Edwin Jan Klein - Universiteit Twente

2.4.1 The drop response
19
The Micro-Resonator
In the model of Figure 2.5 only two paths can be identified between the in and the
drop port. This leads to the simplified model of Figure 2.6 in which the direct path
from the in to the drop port as well as the feedback loop caused by the resonator can
be discerned.
In
Figure 2.6. Simplified resonator model used to obtain the drop response.
By applying Mason’s rule to this simplified model transfer function (2.13) is obtained.
E
E
Drop
In
− jϕ
r
2
− κ1κ
2 ⋅ e
=
1−
µ µ ⋅ e
1
2
− jϕ
r
χ
χ
r
r
(2.13)
In order to find the power dropped by the ring resonator the equation is squared and
rearranged as:
P
P
Drop
In
1
-jκ1
2
− jϕ
r
2
− κ1κ
2 ⋅ e
=
1−
µ µ ⋅ e
− jϕ
r
χ
χ
r
2
2
r
2 2
k1
κ2
⋅ χr
=
2
( 1−
µ µ ⋅ χ ) + 4µ
µ ⋅ e
This can be simplified further and is most often written as:
where H is given by:
P
P
Drop
In
H =
1
2
H
=
2
1+
F sin ( ϕ / 2)
C
κ κ ⋅ χ
2 2
1 2 r
( 1−
µ 1µ
2 ⋅ χr
r
r
)
2
1
2
−α
r
sin
2
( ϕ / 2)
Also, H is also the maximum power available in the drop port for a resonator in
resonance:
P
Drop _ Max
H
= H =
2
1+ FC
sin (( ϕ r = 2
Fc is the so-called finesse factor defined as:
+
µ1
− jϕr 2
e χ
− jϕr 2
e χ
r
r
π
µ2
) / 2)
-jκ2
r
Drop
(2.14)
(2.15)
(2.16)
(2.17)