Page 211 - Optofluidics Fundamentals, Devices, and Applications

186 Cha pte r Ei g h t

A more general result without the assumption of strong initial strain was
provided and the tension per unit length was calculated to be [60]:
⎛ ν − 1⎞
ν +
2
2 2
S 3 ⎜ ⎝ h ⎠ ⎟ + SP CP r = 0 (8-5)
0
where h = thickness of the film
ν = Poisson’s ratio of the film
.
C = 0 04167, a constant.
Solving this equation for S and substituting S into Eqs. (8-3) and
(8-4) allows accurate evaluation of the optical properties of the lens.
Membrane Deflection
A circular membrane with axisymmetric uniform load P with a
clamped edge exhibits deflection profile given by [52,61]:

ur() = P r − ) 2 (8-6)
2
2
r
64 D ( 0
where D = Eh 1( − ν 2 ) / 12 is the stiffness (flexural rigidity) of the
−1
3
plate
E = modulus of elasticity
ν = Poisson’s ratio
h = thickness of the plate.
When no changes in the thickness of the membrane are assumed,
the rigidity is constant across the membrane, and the bending profile
is explicitly given by Eq. (8-6).
Such shells can be used to construct a liquid-filled or a gas-filled
pneumatically actuated lens. For liquid-filled lens, the refractive
properties of the filling differ significantly from those of the outer
environment. So the optical properties of the lens are primarily
defined by the optical path in the liquid, given by ur(). For pneumati-
cally actuated lens, the filling and the outer medium have the same
refractive index, so it is solely the optical path lr() in the membrane
that defines the focal length (see Fig. 8-3a). Assuming constant thick-
ness of the membrane and for small deflections, the optical path
length is approximately given by:

⎡ ⎛ ∂ u⎞ 2 ⎤
lr () ≈ 1 h⎢1 + ⎜ ⎟ ⎥Δ n (8-7)
2 ⎢ ⎣ ⎝ r ∂ ⎠ ⎥ ⎦

Here Δn is the difference in the refractive indices between the
plate (polymer) and the outer medium. Using the best spherical fit for
the function lr(), the radius of curvature of the optical phase and con-
sequently the effective focal length of the lens can be obtained.