Mechanical modeling of fluid-driven polymer lenses.
ABSTRACT A finite-element model (FEM) is employed to study the pressure response of deformable elastic membranes used as tunable optical elements. The model is capable of determining in situ both the modulus and the prestrain from a measurement of peak deflection versus pressure. Given accurate values for modulus and prestrain, it is shown that the two parameters of a standard optical shape function (radius of curvature and conic constant) can be accurately predicted. The effects of prestrain in polydimethylsiloxane (PDMS) membranes are investigated in detail. It was found that prestrain reduces the sensitivity of the membrane shape to the details of the edge clamping. It also reduces the variation of the conic constant with changes in curvature. Thus the ability to control the prestrain as well as thickness and modulus is important to developing robust optical designs based on fluid-driven polymer lenses.
- [Show abstract] [Hide abstract]
ABSTRACT: In this paper, the dynamic mechanical stability of the liquid-filled lenses was studied, in which acoustic excitation was used as broad band perturbation sources and the resultant response of the lens was characterized using non-contact laser Doppler vibrometer. To the best of our knowledge, it's the first time that the mechanical stability of liquid-filled lenses was experimentally reported. Both experimental results and theoretical analysis demonstrate that the resonance of the lens will shift to higher frequency while the vibration velocity as well as its magnitude will be reduced accordingly when the pressure in the lens cavity is increased to shorten the focal length. All of these results will provide useful references to help researchers design their own liquid-filled lenses for various applications.Optics Express 10/2012; 20(21):23720-7. · 3.55 Impact Factor - SourceAvailable from: Antonin Miks[Show abstract] [Hide abstract]
ABSTRACT: The theory of third-order aberrations for a system of rotationally symmetric thin tunable-focus fluidic membrane lenses with parabolic surfaces is described. A complex analysis of the third-order design of tunable fluidic lenses is performed considering all types of primary aberrations. Moreover, formulas are derived for the calculation of the change of aberration coefficients of the parabolic tunable fluidic membrane lens with respect to the wavelength. It is shown that spherical aberration of a simple tunable-focus fluidic membrane lens with parabolic surfaces can be corrected, which is not possible with a classical spherical lens. The presented analysis is explained on examples. Derived formulas make possible to calculate parameters of optical systems with fluidic membrane lenses with small residual aberrations.Applied Optics 04/2013; 52(10):2136-44. · 1.69 Impact Factor - SourceAvailable from: A. Santiago-AlvaradoA. Santiago-Alvarado, J. González-García, F. Itubide-Jiménez, M. Campos-García, V.M. Cruz-Martinez, P. Rafferty[Show abstract] [Hide abstract]
ABSTRACT: A variable focal length liquid-filled lens (VFLLFL) is a lens that changes its focal length by modifying the amount of water contained on it. Recent studies show that the use of VFLLFL in micro-optical devices makes them light, simple and compact. The VFLLFL under study is composed of a cylindrical metal mount with a compartment for two elastic membranes and a liquid medium between them. Unlike previous studies that have focused on developing micro-lenses filled with liquid and with thin flat membranes, this paper presents the design, simulation, and analysis of the opto-mechanical behavior of the VFLLFL formed with thick membranes of different profiles and 2 cm diameter. The study considered three lenses with membranes of different profiles. To do so, a preliminary optical design was done of the lenses to reduce the spherical aberration; next, the study describes the modeling, simulation, and analysis of the mechanical behavior of the VFLLFL using FEM. Then a Genetic Algorithm application was developed to obtain the geometrical parameters of the lens when the shape changes due to pressure applied by the liquid medium on the internal surfaces of the membranes. Finally, with the initial geometrical parameters which the lens begins to adjust due to changes of pressure, an analysis and simulation were done of the optical behavior of the lenses using the OSLO commercial ray-trace program. The results obtained are shown.Optik - International Journal for Light and Electron Optics 06/2013; 124(11):1003–1010. · 0.77 Impact Factor
Page 1
Mechanical modeling of fluid-driven polymer lenses
Qingda Yang,1Paul Kobrin,2,* Charles Seabury,2Sridhar Narayanaswamy,2
and William Christian2
1Department of Mechanical and Aerospace Engineering, University of Miami, Coral Gables, Florida 33124, USA
2Teledyne Scientific & Imaging, 1049 Camino Dos Rios, Thousand Oaks, California 91360, USA
*Corresponding author: pkobrin@teledyne.com
Received 13 March 2008; revised 4 June 2008; accepted 10 June 2008;
posted 11 June 2008 (Doc. ID 93759); published 9 July 2008
A finite-element model (FEM) is employed to study the pressure response of deformable elastic mem-
branes used as tunable optical elements. The model is capable of determining in situ both the modulus
and the prestrain from a measurement of peak deflection versus pressure. Given accurate values for
modulus and prestrain, it is shown that the two parameters of a standard optical shape function (radius
of curvature and conic constant) can be accurately predicted. The effects of prestrain in polydimethylsi-
loxane (PDMS) membranes are investigated in detail. It was found that prestrain reduces the sensitivity
of the membrane shape to the details of the edge clamping. It also reduces the variation of the conic
constant with changes in curvature. Thus the ability to control the prestrain as well as thickness
and modulus is important to developing robust optical designs based on fluid-driven polymer lenses.
© 2008 Optical Society of America
OCIS codes:
310.6805, 160.5470.
1.
The concept of an optical lens with a variable–
tunable focal length (FL) has been of considerable
interest for a number of years [1,2]. The ability to
construct auto focus, zoom, or self-steering optical
systems that do not require a change in external phy-
sical dimensions would enable small robust designs
for a number of important applications. As new ma-
terials and alternative electromechanical systems
have become available, the search for practical de-
signs has gained momentum.
For example, the ability of liquid crystals to change
index in response to an applied field stimulated a
number of designs for variable-focus flat lenses [3].
However, the polarization requirement and the lim-
ited dynamic range have restricted the use of this
technique. Electrowetting, the change in surface con-
tact angle in response to an electric field, has also
been exploited to produce tunable optical elements
[4–6]. Electrowetting avoids the polarization restric-
Introduction
tion and may find applications in fine adjustments
such as autofocus, but it has a limited range of refrac-
tive power adjustment. The most straightforward
tactic is to directly adjust the curvature of the pri-
mary refractive interface with air [1,2,7–10]. Ad-
vances in the optical quality of flexible polymer
membrane materials, such as polydimethylsiloxane
(PDMS), developed for ocular implants and contact
lenses, have made this appear a viable technology.
A number of groups have demonstrated a variation
of FL by changing fluid pressure on a fluid-filled lens
with flexible membranes, which will be referred to in
this paper as a fluid-filled polymer lens [2,7–11]. The
large selection of potential fluids, the high index dif-
ference with air, and the wider range of curvatures
achievable give the optical designer considerably
more flexibility. The approach does require a pump-
ing system, but various ingenious mechanical de-
signs have been proposed. [6,8,12].
In most of the systems demonstrated, simple mod-
els assuming spherical surfaces have been used. If
this were an adequate representation for fluid-filled
polymer lenses, the entire optical design history
based on spherical surfaces would be available.
0003-6935/08/203658-11$15.00/0
© 2008 Optical Society of America
3658 APPLIED OPTICS / Vol. 47, No. 20 / 10 July 2008
Page 2
Unfortunately, even a qualitative consideration of
the mechanics suggests that the spherical assump-
tion is not accurate. Edge bonding conditions and fi-
nite film stiffness result in inflection points that do
not occur on a sphere. While to first order this devia-
tion can be ignored in determining the basic FL,
the picture is not adequate to predict the ultimate
aberration-limited performance. In order to design
precise optical systems a higher-order shape function
is needed. In this work, a parametric mechanical
model is first developed for the deformation, based
on the geometric dimensions and film properties in-
cluding thickness, boundary conditions, modulus,
and prestrain. The material constants are then ex-
tracted from a comparison with simple point mea-
surements on real devices at various pressures.
Those values are then inserted into a finite-element
model (FEM model) to predict the precise shape over
a wide range of operating conditions. This form is
then fitted with a standard optical lens shape func-
tion containing a radius of curvature, R, and a conic
constant, K and is used for predictive optical design.
The paper is organized as follows. In Section 2, a
parametricstudybasedonasimpleapproximateana-
lytical formulation for the deflection of a circular
membrane under transverse pressure is discussed
in order to highlight the significance of boundary
conditions and prestrain. In Section 3, specimen fab-
rication and membrane deformation measurement
procedures are described. In Section 4 a FEM-based
self-consistent parameter extraction process that can
accurately determine the in situ values of membrane
modulus and prestrain is established, and its effec-
tiveness is demonstrated. Numerical predictions
based on the determined in situ modulus and
prestrain are compared with experimental measure-
ments. Section 5 discusses the implication of pre-
strain on polymer lens design and some practical
means to quickly assess prestrain from measured
data. Finally, in Section 6 some highlights of this
study are summarized.
2.
To qualitatively understand the influence of possible
membrane prestrain on optical parameters (e.g., the
conic constants and radius of curvature of a deformed
membrane), a simple analytical expression relat-
ing applied pressure and peak deflection of a pres-
sured membrane was derived by using the Galerkin
method (e.g., [13]) The necessary elastic and geo-
metric parameters that have to be considered in a
membrane deformation analysis are illustrated in
Fig. 1. The deformation of such a circular membrane
will approach two limiting cases. At low pressure,
which is of interest to long FL applications, the de-
formation is small and can be described by small-
deformation plate-bending theory. At sufficiently
high pressure the deformation approaches the so-
called membrane limit by biaxial stretching. The op-
erating regime of membranes in fluid-filled polymer
lenses is frequently between these two limits.
Parametric Study with Consideration of Prestrain
First, the influences of Young’s modulus, E, and
prestrain, ε0, on the relationship between peak de-
flection (or, in optical terminology, sag) of the mem-
brane, δ0, and the applied pressure, p, is examined.
Since there is no analytical solution that can cover
the regime of transition from small to large deflec-
tion, an approximate solution is derived by using
an analytical form based on the Galerkin method.
This method assumes that the deflection shape is
of the form ½1 − ðr=aÞ2?2and is proportional to the
peak deflection,
WðrÞ ¼ δ0½1 − ðr=aÞ2?2;
ð1aÞ
where r is the radius of the point in cylindrical coor-
dinates. Equation (1a) automatically satisfies the
displacement boundary conditions and following
the standard Galerkin procedure; an approximate
analytical expression that relates pressure and peak
deflection is found to be
p
E¼
1
21ð1 − ν2Þ
þ 84ε0ð1 þ vÞ
?
112
?h
a
?2
??h
þ ð46 þ 28ν − 18ν2Þ
?
?δ0
a
?2
ð1bÞ
??δ0
a
a
:
It can be easily demonstrated that this solution ap-
proaches both small and large deflection limits cor-
rectly. Also, note that in the limit of ε0→ 0 and
with small δ0=a and h=a, the solution
p
E¼
16
3ð1 − ν2Þ
?h
a
?3δ0
a
is the correct solution for small deformation of the
thin-plate-bending problem.
Equation (1b) reveals that for a membrane with a
given geometry h=a, the normalized peak deflection
δ0=a is dependent on the normalized pressure p=E
and prestrain and is weakly dependent on Poisson’s
ratio v. In this study, the membrane material, PDMS,
is nearly incompressible, and there was no evidence
that it is changed by the various manufacturing
Fig. 1.
and thickness h) bonded to a rigid washer.
Illustration of a thin-film polymer membrane (of radius a
10 July 2008 / Vol. 47, No. 20 / APPLIED OPTICS3659
Page 3
procedures, i.e., spin coating under varying tempera-
ture and electron beam irradiation. Further, as will
be seen later in the numerical analyses (Section 4),
the deformation of PDMS membranes can be
accurately characterized by the optimum choice of
Young’s modulus E and prestrain ε0, which indicates
that any uncertainty associated with a processing-
induced change of v will be compensated by E and
ε0. Therefore a Poisson’s ratio of 0.48 was used in
all analyses hereafter.
It is immediately seen from Eq. (1b) that for any
thin-film, polymer-based membrane (h=a ≪ 1) the
prestrain will have significant influence in the small
deflection regime (small δ0=a). Note that, in this lim-
it, the normalized peak deflection is proportional to
the normalized pressure, and the proportionality
scales linearly with the level of prestrain,
p
E¼
4ε0
ð1 − νÞ
h
a
δ0
a
?δ0
a≪ 1;
h
a≪ 1
?
:
ð2aÞ
In the other limit, when δ0=a is sufficiently large,
the peak deflection is independent of the prestrain,
p
E¼46 þ 28ν − 18ν2
21ð1 − ν2Þ
h
a
?δ0
a
?3
?
largeδ0
a
?
:
ð2bÞ
The effect of 2% prestrain on the normalized de-
flection versus pressure is shown in Fig. 2. Note that
the bottom left corner corresponds to the small de-
flection limit and the upper right corner corresponds
to the membrane limit. It is obvious that the pre-
strain is of significant influence on the membrane de-
flection in the small-deformation limit. For typical
fluid-driven polymer lenses, the operational regime
is indicated by the dashed box in Fig. 2(a), which
is magnified in Fig. 2(b). It is seen that the prestrain
has significant influence on the sag versus pressure:
it affects not only the magnitudes, but also the char-
acteristics of the δ0=a versus p=E curves, especially
the initial slopes. It will be shown below that the pre-
strain can also significantly influence the shape of
polymer membranes.
It is interesting to note that in the small-
deformation limit (large FL), the prestrain can actu-
ally increase the pressure sensitivity of polymer
based membranes, but it has less effect on the pres-
sure sensitivity at large deformations (small FL).
It is emphasized that Eqs. (1a) and (1b) are far
from a rigorous solution but are a convenient first-
order estimate of the influence of prestrain on mem-
brane sag. While the δ0=a versus p=E relation given
by Eq. (1b) is within 5% of our rigorous numerical
(FEM) solutions, the assumed shape function of
Eq. (1a) is generally erroneous. To accurately predict
the deformed shape, detailed numerical models
using FEM must be used, and this will be detailed
in Section 4.
3.
Measurement
To investigate the prestrain effects, specimens with
different levels of prestrain were fabricated, and
their deformation profiles under various fluidic pres-
sures were measured. The membranes were fabri-
cated under different conditions and subjected to
different treatments; so their properties may vary
significantly. Descriptions of the samples and the
measurement procedures are described below.
Specimen Fabrication and Deformation
A.
The membranes in this study were made of Dow
Corning Silgard 185 silicone. The precursor liquid
was spun onto standard semiconductor-grade silicon
wafers. The wafers had previously been coated with a
thin release layer. For the lower temperature curing
the release layer was photoresist. The silicone was
spin coated and cured twice to achieve a total thick-
ness of 120μm.
The support structures consisted of rigid circular
washers with a 10mm inner diameter and a 12mm
or larger outer diameter. The baseline design used
machined aluminum polished to an optical-quality
Specimen Fabrication Procedure
Fig. 2.
ship between normalized peak deflection (δ0=a) and applied pres-
sure(p=E)forvariousmembrane
(b) magnified view of δ0=a ∼ p=E at typical operational regime.
(Color online) (a) Effects of prestrain (ε0) on the relation-
geometries(h=a),and
3660APPLIED OPTICS / Vol. 47, No. 20 / 10 July 2008
Page 4
surface. Because of the concern that the edge round-
ing induced by polishing would affect the edge con-
straints and thus the lens shape, silicon washers
were sometimes used. The silicon washers were cut
to shape with laser etching, which left very sharp cor-
ners. A thin film of SiO2wassputtered onto the rings.
The rings were dry pressure bonded to the supported
membrane after both surface were exposed to an oxy-
gen plasma. Then the membrane was cut around the
ring, and the release layer was dissolved away. Very
good bonding quality was achieved by this process,
and no debonding between washer and membrane
was observed in any of the specimens.
Variation of prestrain and possibly modulus re-
sulted from differences in cure temperature. As de-
tailed in Table 1, the cure temperature was 90°C for
baselinesamplesand150°Cforthehigh-temperature
samples. Two mechanisms may contribute to pre-
strain in the final mounted membranes. One is
that there is some shrinkage as the film cures. The
other is that there is a large mismatch between the
coefficient of thermal expansion of the PDMS (∼6 ×
10−4=°C) and the silicon substrate (∼3 × 10−6=°C). If
the film is assumed to be at zero stress when the
cure is complete at the cure temperature, this coeffi-
cient of thermal expansion difference will result in
atensilestressofafewpercentwhenthefilmiscooled
to room temperature. This prestrain should in gen-
eral increase as the cure temperature is increased,
althoughitisverydifficulttocontrolitquantitatively.
Further, different cure temperatures may also lead
to variations in the Young’s modulus because of the
temperature dependence of polymerization. There-
fore, a method that is capable of accurately determin-
ing in situ modulus and prestrain from an already
mounted membrane was developed and is described
in Section 4.
In addition to the two sets of membranes prepared
at different cure temperatures, several films pre-
pared with the baseline process (90°C cure) were
further subjected to a flux of high-energy electron ir-
radiation. This process is known to increase the cross
linking of the polymer and will increase the resulting
modulus of the membrane [14]. This radiation
process may also alter the initial prestrain, because
volume and temperature variations were observed
during the irradiation process.
B.
Shape versus pressure measurements were made
using a noncontact optical technique at the facilities
of Micro Photonics Inc. (Irvine, California, www
.microphotonics.com). The micromeasure profiler
uses a polychromatic beam focused on the surface
by a highly achromatic lens and a spectrometer to de-
tect the wavelength of a reflection maximum, which
indicates the vertical height (Fig. 3). The sample is
scanned in the X–Y plane under the probe to obtain
a 3D profile. Two different configurations were used,
but in both cases the vertical resolution wasless than
100nm, and the noise in the data is limited by the
surface roughness of ∼500nm rms. The horizontal
step was 10μm, roughly the same as the optical spot
size. The X–Y translation stage reproducibility is
2μm according to the manufacturer and is small en-
ough to have no noticeable effect on this work. The
reflection comes primarily from the air–PDMS inter-
face. The smaller optical index difference between
the second PDMS surface and the pressurized fluid
produces an insignificant reflection. At the edges of
the aperture, however, where the film is bonded to
the washer, the second PDMS surface reflection is
stronger than the first, and some noise and uncer-
tainty result. Also, the collection optics cannot mea-
sure the membrane when the slope exceeds 30°. This
occurs at the outer radius of the membrane at the
highest pressures. The system is nevertheless very
effective for measuring the range of curvatures
and apertures useful for optical devices.
The apparatus did allow for measurements at both
positive and negative pressures. No evidence of peel-
ing at the edges was seen under positive pressure in
thisdata.Withthefilmontopofthewasher,theentire
aperture could be scanned (within the slope limit).
Membrane Deformation Measurement
4.
Deformation
As mentioned in Section 2, the approximate solution
of Eqs. (1a) and (1b) should not be used to evaluate
deformation profiles for designing optical compo-
nents. In this section we use a rigorous, 3D FEM with
large deformation theory to study the effects of mod-
ulus and prestrain on the deformed profile.
FEM Analyses of Prestrained Membrane
Table 1. Process Parameters for Fabrication of the PDMS
Films Used in this Study
ParameterBaseline High Temp
Spin speed
Spin time
No. of spins
Cure temperature
Cure time
Final thickness
1000RPM
60s
2
90°C
4h
120 μm
1000RPM
60s
2
150°C
1h
120 μm
Fig. 3.
surements.
Test configuration for the pressure-dependent shape mea-
10 July 2008 / Vol. 47, No. 20 / APPLIED OPTICS3661
Page 5
While the mechanics would suggest an intuitive
parameterization based on the physical constraints,
the need to assess the effect of the shape on optical
designs requires conformation to the conventions
commonly used in optics. In this study, the standard
surface used in a popular ray tracing program (ZE-
MAX Development Corporation, 3001 112th Avenue
NE, Suite 202, Bellevue, Washington 98004-8017,
USA) was chosen. The normalized two-parameter
equation that characterizes the deformed membrane
surface or conic profile is
δðrÞ
a
¼
ðr=aÞ2ða=RÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ
1 − ð1 þ KÞðr=aÞ2ða=RÞ2
p
;
ð3Þ
where R and K are the radius of curvature (which is
inversely proportional to the FL) and the conic con-
stant, respectively. They are related to the lengths
of the semi-major axis (b1) and the semi-minor axis
(b2) by
R ¼ ?b12=b2;
K ¼ −ðb12− b22Þ=b12:
ð4Þ
In geometrical terms, the conic profile is a hyperbola
when K < −1, a parabola when K ¼ −1, an ellipse
when −1 < K < 0, a sphere when K ¼ 0, and an ellip-
soid when K > 0. Thus, with only two parameters, a
wide range of optically important shapes can be mod-
eled. By fitting the experimentally measured shape
and the model derived shape to this form the effect
on image quality can be readily evaluated.
A.
The 3D axial-symmetric FEM model is shown in
Fig. 4. The numerical model contains 10 layers of
axial-symmetric elements for the thin-film mem-
brane and one layer of axial-symmetric rigid ele-
ments for the washer. The washer deformation is
ignored because its Young’s modulus is several or-
ders of magnitude larger than that of the PDMS poly-
mer. Local details around the bonding edge including
rounded corners and possible contact (touching) of
the deformed membrane to the washer were care-
Finite Element Model (FEM)
fully modeled by using contact elements. Such local
details were found important because they dictate
the bonding edge boundary conditions that have con-
siderable influence on membrane deflection profile,
as shown in Fig. 5(a).
The prestrain was modeled by using a pseudo-
temperature method, that is, by inducing a false tem-
perature change so that the intended level of pre-
strain due to thermal contraction was developed
before applying the mechanical pressure load. Extra
care was taken so that the membrane thickness after
the pseudo-temperature excursion was equal to the
measured thickness of a physical sample.
The FEM-predicted effects of ε0on the deformed
membrane profiles are plotted in Fig. 5(a) for a mem-
brane with h=a ¼ 0:025. The FEM predicted de-
formed shapes at prestrain levels of 0% , 2%, and
Fig. 4.
membrane system used for membrane deformation calculations
(Color online) 3D axial-symmetric FEM of the bonded
Fig. 5.
shape (δðrÞ=a) as a function of normalized position (ðrÞ=a) at var-
ious deformation levels for ε0¼ 0% (red dashed curves with cir-
cles), ε0¼ 2% (black single-dotted–dashed curves), and ε0¼ 4%
(pink solid curves). Also superimposed are spherical shapes at
the corresponding peak deformation levels (blue double-dotted–
dashed curves) for comparison. (b) Microsoft Excel fitted conic con-
stant (K) and normalized radius of curvature (R=a) from the defor-
mation data of (a) plotted as functions of applied pressure (p=E).
The solid curves refer to the left Y axis and dashed curves to the
right Y axis
(Color online) (a) FEM predicted normalized deformation
3662APPLIED OPTICS / Vol. 47, No. 20 / 10 July 2008
Page 6
4% (see legend) are plotted for four different defor-
mation levels (δ0=a ¼ 0:021;0:055;0:10;0:22). For
comparison, the corresponding spherical shapes hav-
ing the same peak deflection are also superimposed.
It is seen that the prestrain can greatly improve the
optical profiles in that they can greatly reduce the
boundary effects and make the deformed profiles clo-
ser to a spherical shape. This is particularly impor-
tant at relatively small deformations. Without the
prestrain, the deformed shapes are significantly dif-
ferent from spherical shapes and are closer to (an un-
wanted) hyperbolic shape. However, some boundary
effects persist (especially at large deformations) even
for a large prestrain. Therefore, for practical pur-
poses, only the central 80% of the membrane (80%
aperture) should be used in an optical design, and
this convention will be followed in the remainder
of this paper.
The numerically computed deformation data was
fitted to the standard conic surface in normalized co-
ordinates using the optimizing solver function in a
Microsoft Excel spread sheet by setting the target
function to be in the form of Eq. (3). The resulting
normalized radius of curvature, R=a, and conic con-
stant, K, are plotted in Fig. 5(b) as functions of nor-
malizedpressure, p=E, with the
referring to the left vertical axis and dashed curves
to the right vertical axis. The effect of prestrain is
clearly evident in the variation at low pressure (long
FL), but the curves merge together at high pressure
(short FL). That is, ε0can dramatically reduce the
membrane deformation (δ=a) at small p=E, and hence
greatly increase R=a. However, the influence di-
minishes as pressure increases. Both of these limits
are consistent with the results of the approximate
analytical solution (Fig. 2).
The existence of prestrain also has important in-
fluences on K (solid curves in Fig. 5(b), referring to
the left vertical axis). With ε0¼ 0, the deformed
membrane is highly hyperbolic (negative K values)
at small pressure (i.e., small deflection) and quickly
becomes ellipsoidal as pressure increases. When
pressure is sufficiently large (i.e., large deformation),
K remains slightly ellipsoidal (K > 0), but asympto-
tically approaches 0 (spherical shape) at high pres-
sure. As the prestrain increases, the change in K
becomes more gradual, as evidenced by the ε0¼
2% and ε0¼ 4% curves. More interestingly, when
ε0is sufficiently large (for example ε0¼ 4%) the conic
constant becomes almost independent of the radius
of curvature within a fairly large range of pressures
(or equivalently, deformation). This is desired by op-
tical designers because a constant K could signifi-
cantly simplify the designing procedure.
solidcurves
B.
Figure 5 suggests that the prestrain significantly
influences membrane shape, and hence the optical
performance, particularly at low pressure. Unfortu-
nately, this is a difficult region to accurately measure
the shape, and therefore one has to make use of the
Determining in situ E and ε0
transition regime as well. In this study, an optimiza-
tion procedure has been used to adjust E and ε0to
match the predicted δ0versus p curve with the mea-
sured one. The linear elastic nature of the system
guarantees that the extracted E and ε0approach
the true solution. The self-consistency of the para-
meter determination procedure can then be vali-
dated by comparing predictions of the membrane
deformation profiles against experimental measure-
ments over the entire range (r=a from 0 to 1).
Figure 6 gives such an example. In this particular
sample, the PDMS membrane had been subjected to
electron irradiation to enhance its elastic modulus.
The experimentally measured δ0versus p data are
depicted in Fig. 6(a) as filled dots. The FEM calcu-
lated δ0versus p curves of three different ðE;ε0Þ com-
binations are also plotted. The numerical δ0versus p
curve is sensitive to both E and ε0. However, it is
found that the best fit to the experimental data can-
not be achieved by adjusting E or ε0individually.
They must be adjusted simultaneously to achieve
the best fit. The optimum combination was found
to be ε0¼ 1:3% and E ¼ 12:9MPa. Additional simu-
lations were also carried out to examine the possible
influence of Poisson’s ratio. It was concluded that
such effects were secondary and they can be fully ac-
counted for by a minor change of E. Therefore, to
keep the optimization process simple, we will use
E and ε0only and keep Poisson’s ratio as a constant
of 0.48.
The determined E and ε0are then used to predict
the deformed profiles of the same membrane under a
wide range of deformation levels, as shown by the so-
lid curves in Figs. 6(b) and 6(c). It can be seen that
the numerically predicted shapes follow the experi-
mental curves closely, at both large [Fig. 6(b)] and
small deformations [Fig. 6(c)]. The excellent agree-
ment between numerically predicted and experimen-
tally measureddeformation
demonstrates that the determined membrane modu-
lus and prestrain are true properties of the mem-
brane and suggests that they can be used for
design purposes.
profiles clearly
C.
For each of the three types of membrane listed in
Table 2, the deformed profiles at different pressure
levels were carefully measured. The peak deflection
versus pressure data was then used to obtain the
Young’s modulus and prestrain in the membrane
through FEM and the parameter determining proce-
dure described in Subsection 4.B. The obtained E and
ε0for each membrane are listed in Table 2, together
with the membrane geometries.
Although there is some scatter in the data, the
mean values for each type of sample do show the ex-
pected differences. The films cured at 90°C have
∼1:9% prestrain, whereas the films cured at 150°C
have ∼2:5%. There is also a small difference in the
modulus, which, though not surprising, is not ex-
plained definitively. The irradiated films did show
FEM Analyses of Experiments
10 July 2008 / Vol. 47, No. 20 / APPLIED OPTICS3663
Page 7
the expected increase in modulus. However they also
show a reduction in prestrain, which was not ex-
pected, but may be explained as a virtual annealing
effect as the polymer chains are broken and re-
bonded. It is clear from these variations that the abil-
itytoextractthe material
fabricated membranes is essential to being able to
eventually engineer optimized devices.
The repeatability of experiments is reasonably
good. It can be seen from Table 2 that FEM deter-
mined E and ε0are fairly consistent among speci-
mens, as long as the local details of the washers are
reasonably reproduced. Therefore in the following
wechoosetouseonlyonetypicaldatumfromeachspe-
cimen group for detailed analysis and discussion.
Three typical δ0versus p curves are shown in
Fig. 7(a), where the symbols are measured data
points and the solid lines are the numerical best fits
using the E and ε0shown in the corresponding boxes.
Figure 7(b) is the same data but is plotted in normal-
ized scales (δ0=a versus p=E). Note that the apparent
convergence of the non-normalized data at low pres-
sures is the result of the compensating effects of
the higher modulus and the lower prestrains in
parametersfrom
Fig. 6.
(δ0) as a function of applied pressure (p) (red solid curve) as com-
pared with the experimentally measured data (squares). Two
other FEM curves of δ0∼ p using different moduli and prestrains
are also plotted to show the influence of these two parameters on
theδ0∼ p data.(b) FEM predicteddeformation profiles (blacksolid
curves) under various high pressure levels, compared with the ex-
perimentally measured deformation (curves with high-frequency
undulations). (c) Comparison of FEM predicted deformations
(black solid curves) with experimental measurements (curves with
undulations) at lower pressure levels.
(Color online) (a) Best fit of FEM result of peak deflection
Fig. 7.
and ε0as indicated for three membranes (solid curves) compared
with experimentally measured data (symbols). (b) Normalized
δ0=a ∼ p=Ecurvesshowingthedifferentinitialslopesthatareinver-
sely proportional to the levelof prestrainsin the three membranes.
(Color online) (a) FEM calculated δ0∼ p using best-fit E
3664 APPLIED OPTICS / Vol. 47, No. 20 / 10 July 2008
Page 8
the PDMS_EB samples. In the normalized plot the
differences in initial slope reflect the differences in
ε0as described by Eq. (2a). A quick calculation using
the normalized initial slopes gives prestrains of 0.4%
for PDMS_EB2, 1.7% for PDMS_2, and 2.3% for
PDMS_HT1, which are in excellent agreement with
the numerically obtained values.
Using the E and ε0values derived from the δ0ver-
sus p data, FEM simulations of the membranes at
different pressure levels were performed. These were
then fitted with the standard conic surface, and
values of radii of curvature and conic constants were
obtained and compared with the experimental
values. Figure 8(a) shows the comparisons of normal-
ized radius, R=a, as a function of normalized pres-
sure p=E. The predicted R=a versus p=E relation is
remarkably successful. The numerical results (solid
curves) are almost identical to the experimental
curves (dotted curve with symbols). Note the rela-
tively large difference between samples PDMS_2
and PDMS_HT1, despite the fact that their δ0versus
p curves are almost identical (Fig. 7(a)).
The predicted K versus p=E curves are shown in
Fig. 8(b). The predictions follow experimental curves
over a large range of p=E, but the agreement in the
small p=E range is plagued by the large oscillation of
experimental data. There are two reasons for the
lack of consistency of experimental fitting. The first
is noise in the data. At small pressures and deforma-
tions, the fractional errors in pressure and height in-
crease (Δperror ¼ ∼0:14KPa), as can be seen from
Fig. 6(c). Also, since K represents the deviation from
a spherical shape, it is a second-order effect, and the
fitting becomes increasingly unstable for a flat
surface.
From Fig. 8(b), the effect of ε0on the K versus R=a
relation is clearly seen. Of the three samples,
PDMS_HT1 has the largest prestrain (ε0¼ 2:4%),
and K remains nearly constant with a value of about
2.5 over a fairly large range of p=E. This is a desirable
condition, since, in developing a multielement vari-
able design, a changing K value can be problematic
for optimization. As the level of prestrain decreases,
the K values, in general, increase with the decrease
of p=E, before they become unstable. Note that, even
at large p=E values, the numerical predictions are
not completely identical to experimental results.
Table 2.Membrane Geometries and Calculated Modulus and Prestrain
Specimen
E (MPa)
ε0(%)
a (mm)
h (mm)
v
PDMS membrane cured at 150°C
2.4 (0.88)
2.5 (0.86)
2.7 (0.90)
PDMS_HT1
PDMS_HT2
PDMS_HT3
2.30 (6.45)
2.25 (6.54)
2.60 (7.89)
PDMS membrane subjected to electron-beam irradiation after curing at 90°C
10.3 (17.6)0.65 (0.45)
11.7 (16.7)0.35 (0.34)
11.5 (16.4) 0.4 (0.34)
PDMS membrane cured at 90°C (baseline)
2.75 (6.06)2.3 (1.12)
2.85 (6.40)1.65 (0.81)
2.90 (5.99) 1.1 (0.57)
5.015
5.015
5.015
0.121
0.121
0.121
0.48
0.48
0.48
PDMS_EB1
PDMS_EB2
PDMS_EB3
5:00 þ 0:250a
5.015
5.015
0.123
0.108
0.109
0.48
0.48
0.48
PDMS_1
PDMS_2
PDMS_3
5:000 þ 0:300a
5.015
5:015 þ 0:235a
0.124
0.124
0.124
0.48
0.48
0.48
aThe numbers after “þ” indicate the local radius of curvature of the rounded hole edge of the rigid washer, which has to be explicitly
modeled as illustrated in Fig. 4. The rest of the specimens have sharp edge.
Fig. 8.
curves) and experimentally measured (symbols) normalized ra-
dius of curvature (R=a) as functions of normalized pressure
(p=E) for the three membranes with different mechanical proper-
ties. (b) Conic constant K predicted (solid) and extracted from ex-
perimental measurements (symbols).
(Color online) (a) Comparison of FEM predicted (solid
10 July 2008 / Vol. 47, No. 20 / APPLIED OPTICS 3665
Page 9
This isbecause the conic constant ismore sensitive to
the boundary conditions than is the radius of curva-
ture. Despite considerable effort, the details of the
bonding contain subtleties that can not be completely
captured in the model (see Fig. 4). Fortunately, for
optical design, the effect of K is greatest for small
FLs (small R), whereas the predictions are most ac-
curate and less important at long FLs (large R near
flat conditions).
5.
It has been clearly demonstrated that the variations
in process parameters can influence the material
modulus and prestrain in spin-cast films. Our nu-
merical procedure to obtain the in situ prestrain
and Young’s modulus of a polymer lens has been pro-
ven to be very effective. More important, numerical
predictions of membrane optical performance para-
meters based on the determined membrane proper-
ties are in excellent agreement with experimental
measurements. Especially, the numerical predictions
on the radius of curvature of a deformed membrane
at various pressure levels are of very high fidelity
(Fig. 8). Using such a high-fidelity numerical model,
a number of issues important for polymer-based lens
design can be investigated.
Discussion
A.
The existence of prestrain is actually beneficial to
the optical performance of thin-film polymer mem-
branes because it helps to reduce the negative in-
fluence of boundary conditions on the deformed
membrane profile (Fig. 5(a)). The deformed lens
profile with prestrain is closer to a spherical shape,
which is typically assumed in optical lens design.
Further, the prestrain helps to maintain a near-
constant value of K within a large pressure range
(Figs. 5(b) and 8(b)), which will greatly facilitate
optical design of polymer membranes.
Anotherimportantbenefitofprestrainisthatitwill
lead to much improved pressure sensitivity (p=δ0) of
polymer membranes in the small-deformation re-
gime. Pressure sensitivity is important for the accu-
rate control for variable-field-of-view applications.
According to Eqs. (2a) and (2b) and as clearly shown
inFig7(b),atsmalldeformationp=Eisproportionalto
ε0. However, it is noted that the prestrain effect fades
away gradually as membrane deformation increases
(δ0=a increases), which means there is no extra bur-
den for pressure increase in the small-field-of-view
operatingregime.Thisispreferable tootherpressure
sensitivity enhancement methods, suchasincreasing
membrane modulus or thickness.
Increasing the modulus of a polymer membrane
will improve the pressure sensitivity in the small-
deformationregime.However,theincreasedmodulus
requires a proportionally increased pressure to
achieve the same deformation for large deformations,
which could be an extra burden for a miniature lens
system. This can be illustrated using the PDMS_HT1
and PDMS_EB2 curves shown in Fig. 7(a). The mod-
Effects of Prestrain
ulusofthePDMS_EB2membrane(EEB2¼ 11:7MPa)
is about 5.1 times the modulus of the PDMS_HT1
membrane (EHT1¼ 2:3MPa). If both lenses are de-
formed to have identical radii (for example, R=a
¼ 20),therequiredpressureforthePDMS_HT1mem-
brane is p ¼ 0:00012EHT1¼ 0:28KPa, which is actu-
ally larger than the pressure required for the
PDMS_EB2membrane
KPa), owing to the much larger ε0in the PDMS_HT1
membrane. However, at the large deformation of
R=a ¼ 3, the required pressure for PDMS_HT1 mem-
brane is p ¼ 0:0116EHT1¼ 26:7KPa, which is less
than half of that required by the PDMS_EB2 mem-
brane, p ¼ 0:005EEB2¼ 58:5KPa. This explains the
apparent contradiction regarding the δ–p curves
shown in Fig. 7(a). Similarly, increasing polymer
membrane thickness (increasing h=a in Fig. 2) can
also enhance the pressure sensitivity at small de-
formations rather dramatically because the pressure
is proportional to the third power of membrane thick-
ness in this limit in the absence of ε0. However,
this method suffers the same drawback as the high-
modulus membrane at large deformation; i.e., the
pressure required is proportional to the thickness.
(p ¼ 0:00002EEB2¼ 0:23
B.
Polymer Membranes
If the Young’s modulus is known, it is possible to
quickly extract the prestrain level in a polymer mem-
brane of a given geometry (i.e., known h=a) from the
measured δ0− p data [e.g., Fig. 7(a)]. This can be
done through measuring the initial slope in the
small-deformation range. If the membrane thickness
is very small (h=a ≪ 1), the initial slope is directly
related to the prestrain by Eq. (2a). With E known,
the prestrain can be quickly assessed by
Quick Estimate of Prestrain in Thin-Film
ε0≅ð1 − νÞ
4
a
h
a
kE;
ð5Þ
where k ¼ δ0=p is the initial slope of the δ0− p curve.
Unfortunately, for most thin polymer membranes the
elastic modulus itself is, strictly speaking, unknown.
Nevertheless, the bulk modulus of the polymer mem-
brane may be used for a quick, back-of-the-envelope
estimate of the prestrain in the membrane.
Also, note that at large deformations (δ0=a > 1) the
δ0− p relation is independent of the prestrain
[Eq. (2b)]. This feature can be used, in theory, to cal-
culate the Young’s modulus (E) of a polymer mem-
brane. However,fewpolymer
survive such a large deformation without bursting.
At the deformation level involved in this study,
δ0=a ∼ 0:2, it is readily seen that the membrane mod-
uli obtained (as listed in the parentheses in Table 2)
are typically overestimated by a factor of about 2–3
(assuming a fixed Poisson’s ratio of 0.48).
membranescan
C.
In almost all of the previously reported fluid-driven
polymer lens designs, it was implicitly assumed that
Optical Design with Considerations of Prestrain
3666APPLIED OPTICS / Vol. 47, No. 20 / 10 July 2008
Page 10
the membranes were deformed into a spherical
shape, i.e., K ¼ 0 [1,2,4–7,9,15,16]. However, both
analyses and experimental measurements in this
study showed that K is nonzero: K is mostly around
2.5 for a wide range of FLs, with rather significant
deviations (from K ¼ 0) in the small-deformation re-
gime (small p=E), depending on the prestrain level
(Fig. 8). It is noted that the conic constant does have
a profound effect on such issues as image quality.
Figure 9 shows a calculation of the spot size for a sim-
ple biconvex lens with two different FLs but varying
conic constant. For negative conic values the focus is
considerably improved over a simple spherical
shape, and for positive values it is degraded. Clearly
ignoring the conic term would result in a disappoint-
ing correlation between the optical model and real
performance.For the long FLcase R ¼ 25mm (R=a ¼
6:25), the effect is much less than for the short FL
case, R ¼ 8mm (R=a ¼ 2:0).
The clear implication from Fig. 9 is that to achieve
better image quality from a single element, a nega-
tive K value is preferred. Unfortunately, both the
model analyses and experiments in this study show
that this can rarely be achieved with uniform mem-
brane geometry and modulus, except at very small
deformation levels (long FL, large R=a or small
p=E). However, increasing prestrain can bring the de-
formation profile closer to K ¼ 0. The analysis model
in this paper shows that optimum lens shapes with
K < 0 may be achieved by systematic change of the
modulus or membrane thickness as a function of r.
Such attempts are currently ongoing.
6.
In this paper, a numerical model is developed for the
pressure response of a deformable elastic membrane
used in a tunable optical element. The model is cap-
able of determining both the modulus and the pre-
strain from a measurement of peak deflection
versus pressure. A simple analytical model can be
used to accurately determine the prestrain if the
modulus is known. Given accurate values for E
and ε0, we have shown that the two parameters of
the standard optical shape function (curvature and
conic constant) can be predicted. The accuracy of
the prediction of the conic constant is best at large
values of curvature and poorest near flat conditions.
Fortunately, the sensitivity of optical designs to the
value of the conic constant is greatest at curvature.
In general it was found that prestrain reduces the
sensitivity of the membrane shape to the details of
the edge clamping (which are often difficult to model
accurately). It also reduces the variation of the conic
constant with changes in curvature.
Thus the ability to control the residual stress as
well as the thickness and modulus is important to
developing robust optical designs based on fluid-
filled polymer lenses.
The authors acknowledge Michael Rubner and his
students for the irradiation of the PDMS mem-
branes. This work was supported by the Defense
Advanced Research Projects Agency (DARPA) under
contract HR0011-04-C-0042.
Conclusions
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3668 APPLIED OPTICS / Vol. 47, No. 20 / 10 July 2008