Page 300 - Optofluidics Fundamentals, Devices, and Applications

274 Cha pte r T w e l v e

where α is the loss coefficient of the ring (zero loss: α= 1) and θ =
ωL/c, L being the circumference of the ring, which is given by L = 2πr,
r being the radius of the ring measured from the center of the ring to
the center of the waveguide, c the phase velocity of the ring mode (c =
c /n ), and the fixed angular frequency ω= kc ; c refers to the vacuum
0 eff 0 0
speed of light. The vacuum wavenumber k is related to the wavelength
λ through k = 2π/λ. Using the vacuum wavenumber, the effective
refractive index n can be introduced easily into the ring coupling
eff
relations by
2 π ⋅n
⋅
β= kn eff = λ eff (12-7)

where β is the propagation constant. This leads to

θ= ωL = kc L = ⋅kn eff ⋅2 π =r 2 π ⋅n eff ⋅2 πr = 4 π n eff r r (12-8)
2
0
c c λ λ
From Eqs. (12-4) and (12-6) we obtain
α
j
E = −+ ⋅ te − θ (12-9)
− θ
t1
−αt ∗ + e j
−ακ ∗
E = (12-10)
i2 −αt ∗ + e − θ j
−κ ∗
E = (12-11)
∗ j
t2 1 − αte θ
This leads to the transmission power P in the output waveguide,
t1
which is
2 α 2 + t || 2 −2α||cos( θ + ϕ )
t
P = E t (12-12)
t1 t1 1+ α 2 t || 2 −2α|||cos(θϕ+ )
t
t
t
where t =||exp( j ),||ϕ t representing the coupling losses and ϕ the
t t
phase of the coupler.
The circulating power P in the ring is given by
i2
α 2 ( 1−|| 2
t )
E
P =|| 2 = (12-13)
t −
+
i2 i2 1+ α 2 || 2 2α t ||cos(θ ϕ )
t t
On resonance, (θ + ϕ ) = 2πm, where m is an integer, the following is
t
obtained:
(α 2 −||) 2
t
E
P =|| 2 = (12-14)
t1 t1 ( 1− α t ||) 2