Edwin Jan Klein - Universiteit Twente

Chapter 2
∆ =
⎛ β r − β g ⎞
⎜
2 ⎟
⎝ ⎠
16
2
− κ
2
c
(2.5)
where κc is the coupling constant of the port and resonator waveguides and βr and βg
are the propagation constants. The propagation constants are related to the effective
refractive index Neffr of the ring resonator and that of the port waveguide Neffg via:
= 2π ⋅ Neff / λ
(2.6)
β i
i
Examination of Equations (2.2-2.6) shows that the effective refractive indices Neffr
and Neffg are very important parameters in the overall operation of the microringresonator.
The difference between Neffr and Neffg, called the phase mismatch,
determines the maximum of the field coupling between the port and ring waveguides.
In the case that strong coupling is required the mismatch should therefore be carefully
controlled as it may otherwise limit the coupling to a range lower than what is
required for a certain device implementation.
If the effective refractive indices are equal then the coupling regions of the resonator
can be described using the simplified transfer matrix for symmetric couplers:
M
S
=
0
⎡ cos( κ . ) − sin( . )
2
c Leff
j κ c Leff
⎤
1−
χ c ⎢
⎥
⎣−
j sin( κ c.
Leff
) cos( κ c.
Leff
)
(2.7)
⎦
which is found by setting Neffr = Neffg, in Equations (2.2-2.6). In this case complete
coupling between the waveguides is possible.
Equations (2.2) and (2.7) can also be written as:
=
M c
2 ⎡ µ − jκ
⎤
1 − χ ⎢ ⎥
⎣−
jκ
µ
(2.8)
⎦
In this equation the field coupling coefficient κ (0