Page 274 - Optofluidics Fundamentals, Devices, and Applications

248 Cha pte r T e n

(i.e., waveguide core) is as large as 2 μm × 5 μm. Truly single-mode
laser oscillation is obtained by inserting a longitudinal optical DFB
resonator of sufficiently large mode spacing, that only one mode fre-
quency falls within the gain spectrum of the laser dye. The mode
spacing, or free spectral range of the DFB resonator is given by

λ
FSR = λ − λ = N , N = 1, 2, 3,… (Bragg) (10-5)
N −1 N N −1

With a gain spectrum width of around 100 nm, this calls for DFB
orders of N ≤ 15.
The light confinement strongly reduces the losses in the laser
resonator, and thereby implicitly reduces the pumping threshold
for lasing. The light confinement in a Bragg grating also depends
on the reflection order, N. For high-order modes the optical field
samples high- and low-refractive index regions equally, whereas
lower reflection order occupies the high-index regions to a larger
extend. This implies that higher-order modes in general are less
localized in the plane of the device, and thereby more lossy. For
this reason it is more advantageous to employ low-order DFB
modes [11]. Considering the Bragg condition, Eq. (10-4), which
implies λ = 2(n L + n L )/N, reduction of Bragg order N requires
N 1 1 2 2
smaller dimensions of the Bragg structure. Current nanofabrica-
tion methods have allowed realization of third-order nanofluidic
DFB lasers with a period of L 600 nm defined by electron beam
lithography in a 300-nm-thick polymer layer [11,12].
Closed-loop waveguiding structures can form optical ring reso-
nators, where resonant modes are determined by the condition of
constructive interference

κ(k )R = N, N = 1, 2, 3,… (Ring) (10-6)
N
Here, κ(ω) is waveguide wave vector [the inverse relation ω(κ) = ck(κ)
is often referred to as the waveguide dispersion relation], R is the radius
of the ring resonator, and N is again a positive integer. Figure 10-4
shows an example of an optofluidic laser where the walls of the glass
capillary are used to form a ring resonator. The light guided in the cap-
illary walls has a small evanescent tail into the hollow part of the capil-
lary. Infiltrating the capillary with liquid dye ensures a small but
sufficient overlap of the modes with the gain material. The design is
thus conceptually different from the previously mentioned designs
where the gain medium to a larger degree occupies the cavity rather
than being situated in the close vicinity of the cavity.
Microdroplet-based optofluidic lasers have close similarities with
the ring resonator–based lasers. Here, modes are only confined at the
exterior boundary of the droplet, while there is no interior boundary