## Abstract

Tunable multi-chamber microfluidic membrane microlenses with achromaticity over a given focal length range are demonstrated. In analogy to a fixed-focus achromatic doublet lens, the multi-lens system is based on a stack of microfluidic cavities filled with optically optimized liquids with precisely defined refractive index and Abbe number, and these are independently pneumatically actuated. The membranes separating the cavities form the refractive optical surfaces, and the curvatures as a function of pressure are calculated using a mechanical model for deformation of flexible plates. The results are combined with optical ray tracing simulations of the multi-lens system to yield chromatic aberration behavior, which is verified experimentally. A focal length tuning range of 5 – 40 mm and reduction in chromatic aberration of over 30% is demonstrated, limited by the availability of optical fluids.

©2011 Optical Society of America

## 1. Introduction

A wide variety of tunable microlenses has been demonstrated [1], including devices based on electrowetting of liquids [2, 3], actuation of liquid crystals [4], deformation of polymers [5], flow rate controlled lenses [6, 7] and pressure actuation of liquid-filled membrane-defined microfluidic cavities [8].

The latter are of particular interest since the behavior of the lens is defined by the optical properties of the liquids employed. By using liquids with differing refractive indices or dispersion attributes, the optical functionality of the lens may be tuned along with the focal length, allowing the realization of a variety of lens types with differing optical characteristics.

We employ this variability here to demonstrate an achromatic multi-chamber membrane-based pressure-actuated liquid microlens which is tunable in focal length. As is seen in the schematic diagram of Fig. 1, the lens consists of three microfluidic cavities, two filled with liquid and the third with air, separated by distensible polymer membranes. The structure is conceptually analogous to a fixed-focus achromatic doublet, but with tunable lens curvature.

By applying pneumatic pressure to the three cavities independently, the membrane curvatures may be varied, resulting in a change in focal length. Previous theoretical work has shown that chromatic aberration in such tunable multi-element lenses may be completely eliminated for the paraxial case [9] and extensive development of replication processes has allowed these relatively complex optical, mechanical and fluidic structures to be fabricated using low-cost manufacturing techniques [10].

We show here theoretically and experimentally that achromatic tunable membrane lenses are feasible, and find that a proper choice of liquids results in a reduction in chromatic aberration of up to 30% for a tuning range of 5 – 40 mm. To allow determination of the required relative pressures between the cavities, we first model the membrane distention as a function of pressure, combining the resulting predicted curvature with a ray tracing model to determine the optical behavior of the lens stack. Using pressure variations stipulated by these simulations, we then employ a rigorous means to measure chromatic aberration and show that its value can be strongly reduced, using two focal lengths as reference values.

## 2. Design and Fabrication

The design of the multi-chamber lenses is shown in Fig. 1(b). The three cavities are separated by 100 μm thick silicone membranes with a diameter of 2 mm and each cavity has a separate fluidic connection using a system of vias from the top and bottom of the lens system. The single lens elements are fabricated through a replication process [11] using an aluminum mold and the polydimethylsiloxane (PDMS) *Sylgard184*, a silicone from DowCorning with a reported Young’s modulus of 1.76 MPa [12]. The thickness of the lens membranes can be adjusted by precision steel spacers in the molding process. Since the curvature of the distended membrane deviates strongly from the spherical close to the edges of the lens, two aperture stops, photolithographically-defined chromium on glass, are integrated into the lens stack as shown in Fig. 1(a). This flexible design and fabrication process enables the membrane thickness to easily be varied between molding steps, and the concept also allows integration of internal pressure actuators [13].

To control the chromatic aberration properties of this lens system, two optical liquids with different refractive indices and dispersion properties are needed. The first chamber is filled with an aqueous ammonium sulfate solution (refractive index *n* = 1.38, Abbe number *V* = 60.8), the second one with water (*n* = 1.33, *V* = 55.8), and the third chamber with air. Pressure is applied individually to each chamber using three external pressure controllers; in the present case, the pressure difference between the first and the second chamber Δ*p*
_{1} determines the curvature of the first membrane, and Δ*p*
_{2}, the pressure difference between the second and third chamber, is responsible for the deformation of the second membrane.

## 3. Membrane Deformation

The refractive properties of the tunable lenses are given by the membrane profiles, which are modified by changing the pressure differences between adjacent chambers. With the knowledge of the exact relation between the pressure and the corresponding mechanical response of the membrane, its profile and thus optical function may be determined. We thus first analyze the mechanical deformation of the membrane (a flexible circular plate) as a function of pressure and then verify these results with profilometer measurements of the structures.

#### 3.1. Modeling

The equilibrium state of a clamped circular membrane under a uniformly distributed load may be described by the following differential equations, relating *w*, the displacement normal to the plane of the unbent membrane, to the radial coordinate *r*, as [14]

*u*is the displacement in the radial direction;

*h*is the membrane thickness;

*ν*the Poisson ratio;

*D*the flexural rigidity of the membrane; and

*p*the pressure applied to the membrane.

A solution to this set of equations is given by

*C*are recursively related to each other [15]. Applying the boundary conditions that, at the edge of the membrane

_{k}*r*=

*a*, both the deflection

*w*(

*r*=

*a*) and its derivative

*dw/dr*(

*r*=

*a*) are zero, the membrane profiles for different pressure values

*p*can be calculated. Examples of these solutions are shown in Fig. 2, which are calculated for a membrane diameter of 2 mm, a thickness of 100 μm, a Young’s modulus of 1.76 MPa and a Poisson ratio of 0.5. The flexural rigidity

*D*correlates with Young’s modulus

*E*as follows:

An axially symmetric aspherical surface, such as this distended membrane, may be described mathematically as

*c*is the vertex curvature of the profile and

*K*is the conic constant. If the higher aspheric coefficients

*a*in Eq. (4) are neglected, the well-known conic sections for different values of K result, defining a sphere (

_{k}*K*= 0), a paraboloid (

*K*= −1), a hyperboloid (

*K*< −1), or an ellipsoid (−1 <

*K*< 0 or

*K*> 0).

Within the accuracy of the third order aberration theory [16], Eq. (2) can be converted into Eq. (4), such that the analytic expression of Eq. (4) can be used to describe the profile of the distended membrane as a function of the applied pressure if the following substitutions are made:

*c*and conic section

*K*, as well as the aspheric coefficients

*a*, can thus be found from the solution of Eq. (2) and related to pressure and the mechanical characteristics of the membrane.

_{k}Using this analysis, Fig. 3(a) shows the calculated curvature with respect to applied pressure for membrane thicknesses varying between 50 and 200 μm. Close examination reveals that there is a linear regime at low pressure, but at a certain point, where the sag height approaches the thickness of the membrane, the slope decreases. This nonlinear behavior is a result of shear stress in the membrane plane, which is only negligible for membrane deflections small compared to its thickness.

The correlation between the applied pressure and the conic constant *K*, which determines spherical aberration, is depicted in Fig. 3(b), again for membrane thicknesses between 50 and 200 μm. For the low pressure regions, the conic constant indicates that the membrane profiles are highly hyperbolic (*K* ≪ −1), whereas for high pressures the profiles become more parabolic (*K* → −1). For very thin membranes (*h* < 60μm), there are even pressure regimes for which the conic constant is positive, yielding ellipsoidal membrane profiles.

In summary, for very thin membranes, the mechanical calculations indicate that the conic constants vary over a wide range, such that the form of the membrane profile (spherical, hyperbolic, parabolic), and the associated spherical aberrations, will vary strongly with pressure. On the other hand, for very thick membranes, very high pressures need to be applied to achieve the required curvatures. As a result, we have chosen an intermediate membrane thickness, 100 μm, which permits operation with a constant type of asphere without the need to apply excessively high pressures.

#### 3.2. Profilometric Measurements

For verifying the calculations of the last section, individual lens membranes with different membrane thicknesses (50, 75 and 100 μm) were fabricated and the membrane deformation was measured for pressure values between 0 and 200 mbar using a tactile profilometer (*KLA-Tencor P-11*). Since the distended membranes are soft, we considered the influence of the tip pressure on the measured profile data. Using two different forces on the tip (0.5 and 1 mg), no significant difference between the measurements was noted.

Figure 4(a) shows measured profiles for 100 μm thick membranes for pressures varying between 5 and 300 mbar. Since the lens membrane can be considered as rotationally symmetric, the deformation curves were fit to the standard aspheric lens equation [Eq. (4)] and the curvature *c*(Δ*p*) is plotted for the three membrane thicknesses with respect to the applied pressure in Fig. 4(b).

The experimentally and theoretically determined profiles are quite similar, and the change in membrane curvature with pressure as well as membrane thickness shows the same variation for the two approaches. Remaining differences in the curves are due to prestress in the membranes, resulting from shrinkage of the silicone during the curing process, an effect which is not considered in the theoretical description. Nevertheless, theory and experiment are comparable and make the prediction of optical parameters during mechanical deformation possible, especially in case of a redesign of the lens dimensions or when switching to materials with different mechanical properties.

## 4. Ray Tracing Simulation

Based on the response of this single membrane (curvature as a function of pressure), ray tracing simulations were then employed to determine the optical response of the entire multi-lens system. We employed *Zemax* using the following geometrical parameters: lens chamber height 1.5 mm and membrane diameter 2 mm. We assume that Eq. (4) adequately describes the aspheric profile and that the lens is axially symmetric.

The curvature *c*(Δ*p _{k}*) and conic constant

*K*(Δ

*p*) only depend on the applied pressure difference Δ

_{k}*p*between the relevant adjacent chambers; the pressure dependence is determined by fitting the profilometric measurements for a 100 μm thick membrane. As noted above, the first lens cavity is filled with a liquid of refractive index

_{k}*n*= 1.38 and an Abbe number of

*V*= 60.8, whereas the parameters of the second chamber are

*n*= 1.33 and

*V*= 55.8; the third chamber is filled with air.

For the imaging, the object distance is chosen to be 20 mm. Simulations for optimizing the chromatic error are carried out in the same way as in the experiments will be: for a given back focal length, here *BFL* = 12mm, the pressure difference Δ*p*
_{2} is varied in the range between 8 and 24 mbar and the pressure difference Δ*p*
_{1} is correspondingly adjusted to yield a sharp focus (minimized RMS (root mean square) spot size) for a wavelength of *λ*
_{1} = 452nm. Keeping the pressures constant, the image plane is then moved to yield a sharp focus *λ*
_{2} = 682nm; the chromatic aberration is then the difference in the two focal lengths calculated by using the two image plane positions. Figure 5 shows a ray tracing result for *λ*
_{1} = 452nm and a back focal length of *BFL* = 12mm.

The simulated *CA* as a function of the applied pressure difference Δ*p*
_{2} for a focal length of 12 mm using this approach is shown in Fig. 8a, below. The blue dotted line indicates the pressure difference Δ*p*
_{1} that needs to be applied for a given Δ*p*
_{2} such that the focal length remains fixed at 12 mm. As we move along the blue curve, then, *f* is constant, but we see that chromatic aberration varies, peaking at about Δ*p*
_{2} ≈ −20mbar. By moving along the blue curve to lower Δ*p*
_{2} values, chromatic aberration is reduced.

To verify that chromatic aberration can be reduced while tuning the focal length, the simulation was repeated for a second focal length, 15 mm, the results for which are shown in Fig. 8b; similar behavior is seen. For both focal lengths, the simulated chromatic aberration exhibits a maximum at Δ*p*
_{1} ≈ −50mbar, indicating a plano-concave shape of the first (left-most) lens, and a bi-convex shaped second lens. To optimize the chromatic aberration, the pressure difference Δ*p*
_{1} must be positive, which yields a plano-convex first lens and a meniscus-shaped second lens.

The simulation suggests that the chromatic aberration can be reduced by more than 30% in both cases, but its value does not reach zero for the refractive index and dispersion properties of the liquids employed. Since liquids available for experiment were used in the model, the dispersive properties are not ideal, but corresponded to what was commercially available; we discuss this point in the Conclusions.

## 5. Optical Measurements

The simulation results show that chromatic aberration is relatively small, with values on the order of several hundred micrometers; therefore, its measurement requires some care. Measurement of chromatic aberration was done by generating an image of highest sharpness for a particular combination of Δ*p*
_{1} and Δ*p*
_{2} yielding a specific focal length *f*. The object to be imaged is a razor blade, homogeneously illuminated by the collimated light from a broadband flat panel LED directly placed in front of the object. The measurement setup is illustrated in Fig. 6; for measurement at the two wavelengths *λ*
_{1} = 452nm and *λ*
_{2} = 682nm, two interference filters were inserted into the optical path. To reduce the influence of coma on the measurand, the object size is taken to be small compared to the lens diameter, spherical aberration is minimized by the small aperture size.

The short focal lengths of the microlens system call for a transfer of the image which is relatively close to the microlens system, realized by afocal image relay optics which maintain a collimated and homogeneous illumination of the intermediate image. A CCD camera is placed on a precision linear stage and the position of the sharpest image is found by evaluating the image contrast by an autofocus algorithm. Figure 7 explains the image processing: three images of different sharpness are taken using the microlens system with the razor blade as the object. From the vertical cross-sections of these three images, as shown in the right of the figure, the sharpest image (green line) is identified by the highest slope, determined by one-dimensional Sobel-filtering.

Experimentally, the same procedure as employed for the simulations of section 4 was used: for two focal lengths, 12 mm and 15 mm, the chromatic aberration is tuned by first varying the pressure difference at the second membrane Δ*p*
_{2}, and then adapting Δ*p*
_{1}, such that the desired focal length for the first wavelength *λ*
_{1} remains constant. Then the position of highest sharpness for the second wavelength *λ*
_{2} is determined for the same settings. The difference in the camera positions leads to the image distance for *λ*
_{2} and hence to the corresponding focal length, yielding chromatic aberration. Figure 8 shows the comparison between simulations and experimentally determined *CA*. We see that the measurement points agree well with the results of the simulation; the relatively large error bars result from the low signal-to-noise ratio of the image contrast and the comparably large depth of focus. As predicted by simulation, chromatic aberration can be reduced by slightly more than 30% for both focal lengths, limited by the optical properties of the liquids employed.

## 6. Conclusion and Outlook

We have seen that multi-chamber membrane lenses may be used to realize tunable achromatic “doublets” with tunable focal length and reduced chromatic aberration. Whereas a reduction of 30% compared to a single lens could be demonstrated, complete achromaticity requires liquids with optimized values for *n* and *V*. Since the liquids employed need not only the specific optical properties, but must also be compatible (in terms of viscosity, vapor pressure, transparency or usability with PDMS) with the fabrication process used, the spectrum of available materials is limited. Nevertheless, we are continuing the search for liquids which will allow complete elimination of chromatic aberration.

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