Page 234 - Optofluidics Fundamentals, Devices, and Applications

Bio-Inspir ed Fluidic Lenses for Imaging and Integrated Optics 209

shape is an oblate ellipsoid. The shape difference has mechanical
and optical implications. Mechanically, when k > 0, more fluid is
needed to create the same curvature than a spherical lens (k = 0).
Conversely, when k < 0, less fluid is needed to achieve the same
curvature. Since curvature, in a first-order approximation, deter-
mines the lens power, the sign and value of conic factor means dif-
ferent requirements for the actuators that have to move a certain
amount of fluid into and out of the lens chamber to vary the focal
length. Optically, different conic factors mean different geometric
aberrations. Hence, the value of conic factor affects how the optical
system is designed, and the ability of controlling the conic factor is
important to achieve optimal system performance. In this sense, the
IOL in human eye has a highly desirable aspherical shape for low-
ered spherical aberration and high tuning efficiency for the actuator
(i.e., ciliary muscles).
To better understand the lens profile, the elastic membrane is
modeled using the COMSOL Multiphysics simulation software.
The analysis is done by solving partial differential equations
(PDEs) using the finite element method (FEM). The most suitable
mechanical model for rubbers where PDMS belongs to is the
hyperelastic model where the stress-strain relation is specified by
a strain-energy density function [40]. The strain-energy function
(W) is a function of the deformation gradient tensor (e.g.,
W = W()F where F is the deformation gradient tensor). For
hyp
isotropic material, the strain-energy function becomes a function
of the principal invariants of the right Cauchy-Green deformation
(,
tensor, for example, W hyp = W I I I , ) [41,42]. The right Cauchy-
3
2
1
Green deformation tensor is defined as C = F F and the principal
T
invariants are
I = trace()C
1
I = 1 { trace(( ))C 2 − trace(C 2 )}
2 2
I = det( )C
3
We use Mooney-Rivlin constitutive equations to model the
incompressible isotropic elastomer (i.e., PDMS). The strain energy
function is defined in COMSOL as follows:

W = C ( I − 3) + C ( I − 3) + 1 K J − 1( ) 2 (9-2)
hyp 10 1 01 2 2 el

where C , C , and K are material properties for PDMS membrane.
01
10
⋅
I = I J − / and I = I ⋅ J − / where I and I are the first and second
23
43
1 1 2 2 1 2