Page 299 - Optofluidics Fundamentals, Devices, and Applications

Optofluidic Resonators 273

t
E i1 E t1

κ t * –κ ∗
E i2 E i2

α

FIGURE 12-2 Model of a single ring resonator with one waveguide.

One of the emerging and promising new optofluidic devices are
ring resonator configurations. The basic configuration (Fig. 12-2),
which consists of unidirectional coupling between a ring resonator
with radius r and a waveguide, is described briefly in the following
paragraph, which is adapted from Ref. 5.
Defining that a single unidirectional mode of the resonator is
excited, the coupling is lossless, single polarization is considered, none
of the waveguide segments and coupler elements couple waves of dif-
ferent polarization, and the various kinds of losses occurring along the
propagation of light in the ring resonator filter are incorporated in the
attenuation constant, the interaction can be described by the matrix
relation:
⎛ E ⎞ ⎛ t κ ⎞ ⎛ E ⎞
i1
t1
⎟ ⎜
⎜ ⎝ E ⎠ ⎟ = ⎜ ⎝−κ ∗ t ⎠ ⎝ E ⎠ ⎟ (12-4)
∗
t2 i2
The complex mode amplitudes E are normalized, so that their squared
magnitude corresponds to the modal power. The coupler parameters
t and κ depend on the specific coupling mechanism used. The ∗
denotes the conjugated complex value of t and κ, respectively.
The matrix is symmetric because the networks under consider-
ation are reciprocal. Therefore,

2
κ + t 2 = 1 (12-5)

In order to further simplify the model, E is chosen to be equal to 1.
i1
Then the round trip in the ring is given by
E = α ⋅ e E (12-6)
jθ
i2 t2