Page 280 - Optofluidics Fundamentals, Devices, and Applications

254 Cha pte r T e n

Pump light

Mask W

Fluorescence

L

FIGURE 10-7 Experimental setup for the one-dimensional diffusion
experiments in (Reprinted with permission from M. Gersborg-Hansen,
S. Balslev, N. A. Mortensen, and A. Kristensen, “Bleaching and diffusion
dynamics in optofluidic dye lasers,” Appl. Phys. Lett. 90(14), 143,501
(2007). Copyright 2007, American Institute of Physics). Closed channel
containing a liquid solution of dye molecules. The dye molecules are optically
pumped by a pulsed, frequency-doubled Nd:YAG laser through a slit of width
w covering the sample.

Consider the device illustrated in Fig. 10-7, which for simplicity
can be modeled as a one-dimensional problem with the concentration
of unbleached dye molecules C being a function of both the spatial
coordinate x as well as the time t. The diffusion-convection dynamics
is governed by the one-dimensional diffusion-convection equation:
∂ ∂ 2 ∂
Cx t) = D Cx t) − v Cx t) − Γ b x (, ttC x t)( , ) (10-12)
(,
(,
(,
∂t ∂x 2 ∂x
The first term on the right-hand side is the diffusion term with D
being the diffusion constant. The second term is the convection term
with v being the liquid flow velocity in the x direction. Finally, the
third term is the source term, which here accounts for the dye bleach-
ing, with Γ being the bleaching rate.
b
In Fig. 10-7, the dye molecules are optically pumped, and thus
bleached, in a narrow segment of width w. In the context of Eq. (10-12)
we thus assume that Γ takes a constant time-averaged value in the
b
pumped region, while it is zero outside the pumped region. Ideally
the concentration of unbleached dye molecules in the optically
pumped region C(x = 0, t) should be kept constant at a high level,