Edwin Jan Klein - Universiteit Twente

2.4.2 The through response
21
The Micro-Resonator
The through response of a micro-resonator can be derived from Figure 2.5 in a way
similar to that of the drop response by identifying all the paths the light may follow
from the In port to the Through port. This leads to the simplified model of Figure
2.8. in which the feedback loop caused by the resonator and a feed-forward path
through the first coupler can be discerned.
In
-jκ1
Figure 2.8. Simplified resonator model used to obtain the through response.
The application of Mason’s rule then leads to the field transfer function of (2.23):
E
E
Through
In
µ 1 − µ 2 ⋅ e
=
1−
µ µ ⋅ e
The power in the through port is then given by:
P
Through
P
In
1
2
1
2
− jϕ
− jϕ
r χr
r χ
2
µ 1 − 2χ
r ⋅ µ 1µ
2 ⋅cos(
ϕ)
+ χ ⋅ µ
=
1−
2χ
⋅ µ µ ⋅cos(
ϕ)
+ µ ⋅ χ
r
r
2
r
2 2
1 µ 2
By substitution of the cosine in the denominator and rearranging expression (2.24),
the through response can be written in a way analogous the drop response of (2.15):
where G is given by:
+
P
P
− jϕr 2
e . χ
Through
In
r
2
2
2
r
(2.23)
(2.24)
G
= 2
1+
F sin ( ϕ / 2)
(2.25)
2
2 2
µ 1 − 2χ
r ⋅ µ 1µ
2 ⋅ cos( ϕ)
+ χr
⋅ µ 2
G =
2
( 1−
µ µ ⋅ χ )
1
C
2
µ2
µ1
µ1
r
Through
(2.26)
The trough response of (2.24) is shown in Figure 2.9 where three fringes are drawn
for R=50 µm, κ1= κ2=0.5, αdB=1 dB/cm and ng=1.5.
− jϕr 2
e . χ
r
-jκ1 +