It is very difficult to fabricate tunable optical systems having an aperture below 1000 micrometers with the conventional means on macroscopic scale. Krogmann et al. (J. Opt. A 8, S330-S336, 2006) presented a MEMS-based tunable liquid micro-lens system with an aperture of 300 micrometers. The system exhibited a tuning range of back focal length between 2.3mm and infinity by using the electrowetting effect to change the contact angle of the meniscus shape on silicon with a voltage of 0–45V. However, spherical aberration was found in their lens system. In the present study, a numerical simulation is performed for this same physical configuration by solving the Young-Laplace equation on the interface of the lens liquid and the surrounding liquid. The resulting meniscus shape produces a back focal length that agrees with the experimental observation excellently. To eliminate the spherical aberration, an electric field is applied on the lens. The electric field alters the Young-Laplace equation and thus changes the meniscus shape and the lens quality. The numerical result shows that the spherical aberration of the lens can be essentially eliminated when a proper electric field is applied.
© 2008 Optical Society of America
Tunable optical systems of sub-millimeter size with high optical quality have a fast growing market. Mobile cameras, optical communications, optical sensors, optical pick-up for CD/DVD reading or writing, are some of the examples. However, such a tiny optical system is very difficult to fabricate with the conventional means on macroscopic scale. In 2000, Berge and Peseux  presented a tunable optical lens consisting of an oil drop and surrounding water. Electrowetting was used to change the contact angle of the oil drop on the glass, thereby changing its curvature and thus tuning the focal length. The contact angle hysteresis was negligibly small. Their results opened the possibility of fabricating cheap electrically controllable lenses in the range of 0.1 to 10 mm size. Other than electrowetting, dielectric force can also be used to tune the contact angle for liquid lens systems as demonstrated by Cheng and Yeh . However, the lens system developed by Cheng and Yeh  poses to a serious contact angle hysteresis.
One of the major problems about the liquid lens systems proposed by Berge and Peseux  and Cheng and Yeh  is the center-alignment of the liquid drop. After examining a few possible methods for meniscus center-alignment, Berge  found that the geometrical centering (conical geometry) worked very efficiently. Similar finding was reported by Kuiper and coworkers [4, 5] who packaged two immiscible liquids in a tube to form a self-centered variable-focus lens. Recently, Krogmann et al.  utilized the MEMS technology to fabricate a liquid micro-lens system. The conical hole formed in a standard silicon wafer by KOH etching provided a stable center-alignment for the liquid lens. This is a great progress in the development of liquid micro-lens.
For tunable liquid optical systems of sub-millimeter size, the gravity effect can be neglected so that the lens system is independent of orientation, and rather insensitive to external vibration and shocks, if the density difference between the two liquids is sufficiently small. Under such a situation, however, the meniscus shape becomes spherical that might pose to a significant spherical aberration as reported in Ref. [2, 4–6]. In the present study, a numerical simulation is performed to investigate the MEMS-based liquid lens developed by Krogmann et al. . In addition, an electric field is applied on the liquid lens to yield a non-spherical meniscus shape that is essentially free of spherical aberration.
2. Theoretical analysis
2.1 Electrowetting on dielectrics
In their study, Krogmann et al.  presented a tunable plano-convex liquid micro-lens that exhibited a tuning range of back focal length between 2.3 mm and infinity achieved by applying a voltage of 0-45 V. The system was completely fabricated in MEMS technology. As illustrated in Fig. 1, a conical hole was formed in a standard silicon wafer. A well defined sidewall angle e of α=54.7° was achieved. The hole surface was covered with a dielectric layer consisting of a silicon dioxide layer and a hydrophobic layer. An indium-tin-oxide (ITO) structured Pyrex wafer was bonded to the bottom. The lens liquid as well as an immiscible surrounding liquid was then deposited in the conical hole. Finally, the system was closed by a glass cover.
The lens liquid used in Ref.  was a water-based inorganic salt solution with a refractive index of 1.510 and a density of 2100 kg/m3, while the surrounding liquid was a density-matched perfluorocarbon with a refractive index of 1.293. The volume of the lens liquid was 100 nl with a tolerance of 10%. The aperture of the lens, which is defined by the bottom opening of the conical hole in the silicon wafer, was 300µm. The initial contact angle (i.e. without an applied voltage) was θ0=96° that gave rise to a back focal length of 2.3 mm. By applying a voltage of 45V across the dielectric layer (see the electric connectors in Fig. 1), the system reached a back focal length of infinity (20 mm).
In the present work, this same system configuration is investigated on a cylindrical coordinate system (r,z) which is normalized with the reference length L (a half lens aperture) as depicted in Fig. 1. The curvature of the meniscus between the lens liquid and the surrounding liquid is expressible as the Young-Laplace equation [7–9]
where z=h(r) stands for the profile of the meniscus. The primes represent derivatives with respect to r. The notations µ and ρ are viscosity and density, while the subscript l and s denote the lens liquid and surrounding liquid, respectively. The capillary number Ca is based on a reference velocity Uc. Definition of the flow-induced pressure (or dynamic pressure) on the lens liquid side and the surrounding liquid side of the meniscus (p̂l and p–s) can be found in Refs. [8, 9]. For a static problem considered here, the second term on the right-hand-side of Eq. (1) vanishes such that Eq. (1) reduces to
where κ(0) is the curvature of the meniscus at the apex (r=0).
Equation (4) is a nonlinear second-order ordinary differential equation for h(r) subject to the two associated boundary conditions
where r 0=1+z cot α represents the surface of the conical hole. The contact angle θ can be estimated from the Lippmann equation 
where d and εd are, respectively, the thickness and the relative permittivity of the dielectric layer, while ε0 denotes the vacuum permittivity (ε 0=8.854×10-12 F/m). Equation (6) reveals that the contact angle θ is always less than θ 0 whenever a voltage across the dielectric layer Vd is applied. This is known as the electrowetting effect.
In the present study, the problem (4)–(6) is solved with a simple algorithm for a given electrowetting parameter C and a lens liquid volume. For convenience, the algorithm is described as follows.
(a) Guess h(0).
(b) Guess κ(0).
(d) Adjust the value of κ(0) with a shooting method and then return to Step (c) until the boundary condition (5b) is satisfied.
(e) Evaluate the volume of the lens liquid from the resulting h(r) solution.
(f) Stop the computation if the volume of the lens liquid is satisfied within a prescribed tolerance. Otherwise, adjust the value of h(0) with a shooting method and then return to Step (b) and repeat the numerical procedure.
Once the meniscus shape is determined, the optical performance of the liquid lens is examined by tracing the axial rays (i.e. rays that parallel to the lens axis) through the lens.
Figure 2 shows a schematic axial ray going through air, surrounding liquid, lens liquid, and air again. The incident ray is parallel to the lens axis with distance b. After a few refractions, the ray eventually reaches the lens axis at the point z β(b) which depends on the value of b. The so-called back focal length (BFL) is defined by
For simplicity, the notation β(0) will be referred to as the back focal length. In the present study, the ray refraction is assumed to follow the Snell’s law 
when the ray passes through a boundary between two isotropic media, where n and I are refractive index and incidence angle preceding the boundary, while n′ and I′ are the counterparts following the boundary. There are three boundaries in the present problem, namely, air and surrounding liquid, surrounding liquid and lens liquid, lens liquid and air. Effects of the glass cover and the ITO-structured Pyrex wafer are neglected for their uniform thicknesses.
2.2 Dielectric force
Another means of tuning the back focal length is to apply an electric field on the system. Let the glass cover shown in Fig. 1 be replaced with an ITO layer, and a voltage e V0 be applied on the top of the lens (at z=z0) while the bottom of the lens (at z=0) is grounded to zero. This produces an electrostatic potential ϕ(r,z) that can be determined from the dimensionless axisymmetric problem [12, 13]
where the electrostatic potential ϕ(r,z) has been normalized with V0 while n is a normal coordinate to the boundary. The dimensionless electric conductivity σ is a step function across the meniscus, i.e.
where σl and σs are the electric conductivity of the lens liquid and the surrounding liquid, respectively. Equation (9) is still valid for alternating current if the root-mean-square voltage Vrms is used instead of V 0 .
In the present study, the problem (9) and (10) is solved with the integration scheme [15, 16] on a uniform Cartesian grid system. For a desired grid mesh Δr, the other grid mesh Δz is determined by the mesh ratio
such that the grid points are always located at the surface of the conical hole. This arrangement greatly simplifies the numerical procedure on the inclined boundary. Upon discretizing Eq. (9a) at point P, one gets
where ϕW denotes the electrostatic potential at point W. The subscripts W, E, S, N, represent the properties at the neighboring grid points of P that is in the west, east, south, and north of point P, respectively. The effective electric conductivity σeff is defined by
for an interval if one portion of the interval sΔr (or sΔz) lies in the lens liquid region, while the other portion (1-s)Δr (or (1-s)Δz) in the surrounding liquid region. Note also that σeff reduces to σl/σs as s=1 and becomes unity as s=0. Hence, for the case illustrated in Fig. 3, the weighting factors should be
where ϕE′ and ϕS′ are the result of the previous iteration at points E′ and S′ which are the mirror-reflection of E and S about the inclined boundary, respectively (see Fig. 4).
where εl and εs are the relative permittivity of the lens liquid and the surrounding liquid, while En and Et are the normal and the tangent components of the electric field on the meniscus. As a result, the Young-Laplace equation becomes
Equation (17) indicates that the meniscus becomes non-spherical in the presence of the dielectric force. The numerical algorithm for the h(r) result and the axial ray tracing are similar to that presented in the previous subsection except for the coupling with the electric field (9).
3. Result and discussion
Perfluorocarbon liquid, a well-known material in ophthalmology [17–20], has a relatively low refractive index. Due to such an important optical property, Krogmann et al.  employed a perfluorocarbon liquid as the surrounding liquid in their liquid lens system to achieve a short back focal length. The physical properties of the materials used in the experiment , i.e.
are adopted in the present computation. The gravity is g=9.806 m/s2. The thickness of the dielectric layer on the conical hole surface is d=0.3µm. The refractive index of air is assumed to be unity. The volume of the lens liquid is 117×10-12 m3 while the reference length is L=0.150 mm. The Bond number
is very small as compared to κ(0) even when ρl/ρs≠1. Hence, the gravity effect is neglected in the present study.
After a series of grid tests, the grid mesh Δr=0.01 is found adequate for all of the cases examined in the present study. Figure 5 shows the meniscus shape for various contact angles θ without the body force effects. The resulting back focal length (has been normalized with the reference length L=0.150 mm) is depicted in Fig. 6. Due to the absence of the body forces (Bo=Boe=0), Eq. (17) reduces to
such that the meniscuses in Fig. 5 are all spherical. Figures 5 and 6 show that a decrease in the contact angle θ gives rise to a decrease in the curvature of the meniscus and thus an increase in the back focal length. Influence of the electrowetting parameter C on the back focal length β(0) is revealed in Fig. 7. The experimental measurement  non-dimensionalized with the physical properties (19) is also plotted in Fig. 7 for comparison. Excellent agreement between the present numerical simulation and the experimental data  is found for 0≤C≤0.4 which is equivalent to 0≤Vd≤19.52 V in the present case. The corresponding back focal length is in the range of 2.3–6.0mm. As remarked by Krogmann et al. , the great discrepancy between the numerical simulation and the experimental dada at large electrowetting might be attributed to the contact angle saturation [21, 22] that the Lippmann equation (6) does not take into account. Unfortunately, the mechanism of contact angle saturation is still not well-understood. Onset of contact angle saturation is highly problem-dependent. This reflects the need of a more precise/reliable model for electro-wettabilty.
The numerical result of the electrostatic potential ϕ(r,z) is illustrated in Fig. 8(a) for the case of Boe=0.046 and θ=96°. The corresponding normal and tangent components of the electric field on the meniscus are shown in Fig. 8(b), while the resulting dielectric force function G(r) is revealed in Fig. 8(c). Finally, the influence of the electric Bond number Boe on the meniscus shape is depicted in Fig. 9. From Figs. 8(a) and 8(b), it is seen that the normal component of the electric field on the surrounding liquid side of the meniscus (en)s is larger than that on the lens liquid side (en)l at a great amount because of the large electric conductivity ratio (σl/σs≫1). In addition, the electric field is stronger in the lens axis region than that in the lens rim region due to the convex meniscus shape. As a result, the dielectric force function G(r) monotonically decreases with respect to r. It thus gives rise to an increase in the curvature of the meniscus in the lens axis region as can be seen from Eq. (17) and Fig. 9. This finding is consistent with that reported in the literature (see e.g. Ref. [12, 13]). Note that the zigzag appearing in the numerical result of G(r) is due to the numerical error when the electric field is evaluated from the electrostatic potential on a Cartesian grid system. Fortunately, it poses no significant error when the meniscus shape is computed (see Fig. 9).
Figure 10(a) shows the influence of the electric Bond number Boe on the function β(b) for an incident axial ray at r=b under the contact angle of θ=96°. The corresponding spherical aberration (β(b)-β(0)) of the lens is revealed in Fig. 10(b). According to Fig.10(a), the back focal length β(0) decreases whenever an electric field is applied. Surprisingly, the lens is essentially free of spherical aberration at Boe=0.046 or V 0=110.5 V as observable from Fig. 10(b) and the axial ray tracing revealed in Fig. 11. This implies that the spherical aberration can be efficiently eliminated with a proper electric field. The optimal electric Bond number (Boe)opt and the resulting back focal length β(0) for various contact angles θ are shown in Fig. 12. The back focal length β(0) with Boe=0 is also plotted in Fig. 12 for comparison. The maximum spherical aberration (occurring at the contact angle θ=96°) reduces from 1.5 to 0.0015 (225 to 0.225 µm) when the optimal electric Bond number is applied. Nevertheless, the optimal electric Bond number shows only little influence on the back focal length (see Fig. 12). This implies that tuning the back focal length with electrowetting while eliminating the spherical aberration with dielectric force might be a good strategy for developing a high performance liquid micro-lens. In this connection, one needs a particular design that allows the two parameters, the electrowetting parameter C and the electric Bond number Boe, be handled independently.
A numerical simulation is performed in the present work to investigate an existing MEMSbased liquid lens with and without electric field. The numerical methods employed in the present study simulate the problem successfully. The resulting back focal length agrees with the existing experimental measurement in the range of 2.3–6.0mm. The corresponding voltage is 0–19.52 V. The discrepancy between numerical simulation and experimental dada at large electrowetting might be attributed to contact-angle saturation that the Lippmann equation does not take into account. A precise and reliable model for electro-wettabilty is needed. The numerical results show also that the spherical aberration of the lens can be essentially eliminated when a proper electric field is applied. Finally, it is mentioned that tuning the back focal length with electrowetting while eliminating the spherical aberration with a proper electric field might be a good strategy for developing a high performance liquid micro-lens.
The authors wish to express their appreciation to Dr. Cheng-Huan Chen, an Assistant Professor in their Department, for his providing valuable information about Optics. This work is supported by the National Science Council of Taiwan through the contract NSC 94-2218-E-007-012.
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