Page 281 - Optofluidics Fundamentals, Devices, and Applications

Optofluidic Dye Lasers 255

preferable the initial concentration level C = C(x = 0, t → 0). This can
0
be achieved by means of a convective flow, that is, a sufficient high
flow velocity. The corresponding convection rate is given by Γ = v/w.
c
The condition that
Γ >> Γ (convective replenishment condition) (10-13)
c b
is amply satisfied with the v 5 m/s jet flow in conventional dye
lasers, and likewise, convective flow is an efficient dye replenishment
mechanism in optofluidic dye laser. Making a similar analysis of the
diffusion term we arrive at a diffusion rate given by

Γ = D (10-14)
d
w 2
In microfluidics it is a key observation that while the diffusion
constant is scale invariant, that is, D does not depend on the size of
the device, the diffusion rate Γ increases as 1/w when w goes to zero
2
d
[17]. Thus, a steady state can equally be achieved by diffusive driven
molecule exchange with a large reservoir, or an ideal reservoir where
∂C/∂t = 0. The condition for this is that
Γ >> Γ (diffusive replenishment condition) (10-15)
d b
Equation (10-12) as well as experimental studies [18] indicate that
diffusion alone may be sufficient to replenish bleached dye in a min-
iaturized dye laser under typical optical pumping levels and repeti-
tion rates.
As another example where dye replenishment is achieved through
a combination of convection and diffusion is shown in Fig. 10-8. The
figure shows a finite element calculation of the laminar flow profile in
the laser device in Fig 10-1c [10]. In this device the laminar flow
profile, and hence also the convective dye replenishment is spatially
very inhomogeneous. The flow simulations in Fig. 10-8 reveal that
convective flow only occurs off-center in the microfluidic channels,
while stagnant fluid volumes (v ~ 0) are present in between the poly-
mer posts in the center of the channel. In the stagnant regions the dye
replenishment must instead rely on dye molecule diffusion between
the stagnant volume and convective flow regions. In this context the
convective flow regions act as ideal reservoirs. Using the previously
estimated diffusion constant for rhodamine 6G in ethylene glycol, D ~
2
1.5 × 10 m /s, and a typical width w ~ 1 μm of the stagnant regions,
−11
−1
we arrive at a characteristic diffusion rate Γ = D/w ~ 15 s . This is
2
d
larger than typical repetition rates of the pulsed pump radiation, thus
ensuring an efficient diffusive dye replenishment in the stagnant
regions, allowing for a steady laser output.