## Abstract

A design method for correcting chromatic as well as spherical aberrations of variable-focus, multi-chamber liquid lenses is described. By combining suitable optical liquids with appropriate radii of the liquid’s interfaces, liquid lenses with superior, diffraction-limited resolution over a wide focal tuning range are possible. For an infinite object distance, the analytic thin-lens approximation of an achromatic positive/negative varifocal liquid lens is derived and the obtained results are compared with ray-traced optimized designs which consider finite thicknesses and rigid cover glasses. As a design example, the optical performance of a 4 mm-diameter positive/negative *f*/3.6 achromatic liquid lens is given in detail.

©2007 Optical Society of America

## 1. Introduction

A major trend in current optics research is the broadening of the functionality of optical elements by endowing them with more flexibility. In recent years, the old idea of varifocal lenses [1], modeled on one of the numerous capabilities of the human eye, has been revived in the form of tunable, liquid based micro-lenses. The most popular approaches involve either the use of electrowetting or membrane-covered pressure-variable micro-fluidic cavities, since these allow focal-range tunability due to a simple curvature variation of the refracting interfaces. By changing the shape of the liquid lens, its focal length will change. Electrowetting lenses [2, 3, 4] utilize a driving voltage to vary the contact angle of one of the liquids, and thus the power of the liquid lens. Membrane lenses [5, 6, 7] are formed by a fluid-filled cavity which is covered with at least one, thin elastomeric membrane. The shape variation is introduced by a pressure-induced liquid redistribution. Alternatively, the meniscus of immiscible liquids can also changed by an applied pressure without a separating membrane [8].

In many publications on varifocal lenses, the authors emphasize the wide focal length tunability and support their argumentation with examples of hemispherical lenses. However, it is well known that the optical performance of planoconvex lenses with small *f*-numbers is fairly poor, especially when used in the wrong orientation. To remedy this situation, multi-chamber liquid lens systems can be used either to correct for or to introduce specific monochromatic aberrations [9]. Furthermore, the ability to achromatize liquid lenses is of special interest as already outlined recently [10].

The purpose of this paper is to address both the correction of chromatic as well as the most critical monochromatic (i.e. the primary spherical aberration) of varifocal liquid lenses. We intend to provide analytic pre-design rules which may be helpful in defining the parameter space and the performance range of variable-focus liquid lenses, and moreover, to ease further developments in these promising technologies.

## 2. Achromatic lens design fundamentals

For the sake of completeness, we briefly recapitulate some common design principles for achromatic refractive lenses. In what follows, the paraxial approximation is employed [11].

The power *φ* of a single thin lens surrounded by air is given by the well-known relation *φ* = *c*(*n* - 1), where *n* refers to the refractive index and *c* to the bending of the lens, which is defined as *c* = *c*
_{1} - *c*
_{2} with *c*
_{1},*c*
_{2} as the curvatures (inverse radii) of the spherical boundaries, respectively.

The dispersion of an optical material in the visible spectral range can be specified by the Abbe number

where *n*
_{d} is the refractive index with respect to the yellow Fraunhofer d-line and Δ*n* = *n*
_{F} - *n*
_{C} is the the mean dispersion.

One way to correct a lens for chromatic aberration is to cement two materials with different dispersions and refractive indices together. The total refractive power of such cemented thin lenses is then simply the sum of the powers of all the single elements

Achieving achromatism for such combined lenses requires that

Equations (2) and (3) represent the basic formulas for an achromatic lens design which should now be applied to liquid lenses. Evidently, the achromatism equation (3) states nothing about the ordering of the optical liquids as well as the bending of the lens. Thus, both the order as well as the curvature of elements in a multi-chamber liquid lens system can be utilized as additional degrees of freedom to minimize monochromatic aberrations.

The primary longitudinal spherical aberration of *k* thin lenses in contact, surrounded by air have been given by [12, 11] to

with *s* and *s*′ being the paraxial object and image intersection-lengths, respectively, *y* being the paraxial incident ray height. *LA*
_{p} is the primary spherical aberration of the object; if non-zero, it has to be transferred to the image by the magnification rule. The so-called spherical *G*-sum in Eq. (4) refers to the expression

where *c* is the bending of each lens, *c*
_{1} is the curvature of the first surface of each element, and *v*
_{1} is the reciprocal of the object intersection length of each element in air. The *G*-functions are pure functions of the refractive index, defined by Conrady [12] as follows

$${G}_{4}=\frac{1}{2n}\left(n+2\right)\left(n-1\right)\phantom{\rule{1.2em}{0ex}}{G}_{5}=\frac{2}{n}\left({n}^{2}-1\right)\phantom{\rule{1.2em}{0ex}}{G}_{6}=\frac{1}{2n}\left(3n+2\right)\left(n-1\right).$$

The *G*-sum has to be calculated for each element of a thin lens system, and therefore, the meanings of *n*, *c*
_{1}, and *v*
_{1} in Eq. (5) have to be changed. To give an example, for the second lens of a doublet we set *n* = *n*
_{b}, (*c*
_{1})_{b} = (*c*
_{1})_{a} - *c*
_{a}, and (*v*
_{1})_{b} = (*v*
_{1})_{a} + *c*
_{a} (*n*
_{a} - 1).

Provided that the total bendings (but not the single curvatures) of each lens are given, the *G*-sum of each lens has to be expressed in terms of the selected bending parameter (most often the curvature of the first lens, *c*
_{1}). It should be mentioned that (*v*
_{1})_{a} = constant for a finite object distance and (*v*
_{1})_{a} = 0 for an object at infinite distance. The summation of all *G*-sums yields the total *G*-sum which is an ordinary parabola with a vertical axis of the form (*G* sum)_{t} = *α*
_{t}
*c*
^{2}
_{1} + *β*
_{t}
*c*
_{1} + *γ*
_{t} in which the coefficients *α*,*β*,*γ* have definite numerical values. The zeros of this function will provide the curvature *c*
_{1} and thus all other curvatures for which the primary spherical aberration of the lens system is zero.

For the special case of an infinite object distance, we have *v*
_{1} = *s*
^{-1} = 0 and *s*′ = *f*′ and thus, Eq. (4) simplifies to

In the following, we apply these considerations to all-liquid optical systems comprising *l* thin lenses in contact. The liquid interfaces are separated by either a meniscus between two immiscible liquids (in particular: conductive/insulating liquids) or a negligibly thin elastic membrane to form multi-chamber systems. In the analytic predesign, the influence of the glass cover slides, elastic membranes, and finite thicknesses of the liquid chambers are neglected. Furthermore, we consider the case of infinite object distance and assume ideal spherical interfaces.

## 3. Design of a spherically corrected, variable-focus liquid lens

Let us consider a liquid lens system as schematically depicted in Fig. 1. It comprises six chambers, separated in pairs by rigid transparent plates to ensure an individual adjustment of the liquid interfaces. In an optimal arrangement, the subchambers (b1, b2) and (c1, c2) would each be filled with the same liquid. Considering such a system, we then have for the bending variables ${c}_{a}=-\frac{1}{{R}_{3}}=-{c}_{3}$, ${c}_{b}=\frac{1}{{R}_{3}}-\frac{1}{{R}_{5}}={c}_{3}-{c}_{5}$, ${c}_{c}=\frac{1}{{R}_{5}}-\frac{1}{{R}_{7}}={c}_{5}-{c}_{7}$ and ${c}_{d}=\frac{1}{{R}_{7}}={c}_{7}$. Consequently, the total refractive power is

whereas the achromatism condition yields

Eqn. (8) and (10) represent an under-determined system of linear equations which leads to an infinite number of possible solutions for *c*
_{3},*c*
_{5},*c*
_{7} to achieve achromatization only. Thus, two degrees of freedom can be used to correct chromatic aberration, and one degree of freedom is left for correcting 3rd order spherical aberration. However, the expression for primary spherical aberration as a function of lens bendings or curvatures is not a linear one, which makes an analytic solution more complex. The problem could be solved numerically, but for two reasons explained below we follow here Kingslake’s ideas for analytically predesigning a spherically corrected achromat [11].

Firstly, it is known that the chromatic aberration depends only on the power of the lens, whereas the spherical aberration most easily can be adjusted by lens bending. However, the total bending can only be varied if the front and the back surface of the liquid lens system is variable too – which is not practicable in case of rigid cover glass plates. Secondly, with curved front and back surfaces, a higher refractive power for a given lens system could be achieved since geometrical constraints restrict the maximum curvature of each lens interface.

We address these issues by filling both the first and the last chambers of the lens system with air. We leave the glass cover plates at the extremities of the system to act as protection for the internal components. Thus *n*
_{a} = *n*
_{d} ≈ 1 and Δ*n*
_{a} = Δ*n*
_{d} = 0, whereby Eqn. (7) and (9) simplify to

and

which have to be solved for the lens bendings *c*
_{b} and *c*
_{c}, not for single curvatures. The pre-design procedure for a spherically corrected variable-focus liquid achromatic lens system is then basically as follows [11]:

- Selection of two suitable liquids with complementary dispersive characteristics such as ‘crown-like’ and ‘flint-like’ liquids
- Predefining the focal tuning range
- Determination of the
*c*_{b}and*c*_{c}values for thin-lens chromatic correction - Setting up the
*G*-sum expressions for primary spherical aberration of each lens for a predefined ordering of the liquids (two options: crown-in-front or flint-in-front) - Solving the two total spherical
*G*-sum expressions of the thin-lens system for the bending parameter*c*_{3}, by determining the zeros or minimum values - Picking out the best solution (left-hand side or right-hand side and crown-in-front or flint-in-front) according to which requires the most moderate curvatures (∑|
*c*| → Min) or mathematical minimum spherical aberration, respectively_{n}

Next, we elaborate on this procedure. For predesigning a liquid achromatic lens as configured in Fig.1, we choose the curvature of the first liquid interface *c*
_{3} as the bending variable. Chambers (a) and (d) are filled with air, whereas the chambers (b) and (c) are filled with two different optical liquids.

By solving Eqs. (11) and (12) we get for the (*c*
_{b},*c*
_{c})-formulas:

The *G*-sums of both thin lenses (combined chambers *b*
_{I}, *b*
_{II} and *c*
_{I}, *c*
_{II}, respectively) are set up in terms of *c*
_{3} and added together. For an infinite-corrected achromatic lens the primary spherical aberration according to Eq. (6) then can be written as *LA*
_{p}′ = -*y*
^{2}
*f*′^{2}(*α*
_{t}c^{2}
_{3} + *β*
_{t}
*c*
_{3} + *γ*
_{t}), with the total coefficients *α*
_{t} = *α*
_{b} + *α*
_{c}, *β*
_{t} = *β*
_{b} + *β*
_{c}, and *γ*
_{t} = *γ*
_{b} + *γ*
_{c}. The corresponding lens contributions have been derived from the *G*-sums as:

Since neither *y* nor *f*′ in Eq. (6) will become zero, for *LA*′_{p} = 0 the quadratic function in parentheses has to be zero. Solving for *c*
_{3} yields

where the positive root gives the correct solution. The other curvatures are then simply given by *c*
_{5} = *c*
_{3} - *c*
_{a} and *c*
_{7} = *c*
_{3} - *c*
_{a} - *c*
_{b}.

There are further geometrical constraints which have to be considered. The maximum curvature |*c*
_{max}| of each membrane (hemispherical surface) is limited by the inverse aperture radius. Moreover, suitable edge thicknesses have to be introduced to avoid membrane contact during tuning, which is especially important for high power lenses. The required edge thickness is also determined by |*c*
_{max}|; it can derived from the sagitta of the spherical membrane given by *z* = *ch*
^{2}/[1 + (1 - *c*
^{2}
*h*
^{2})^{1/2}].

For predesigning an example achromatic lens, a crown-like liquid with *n*
_{1} = 1.3250, Δ*n*
_{1} = 0.003788, *V*
_{1} = 85.8 and a flint-like liquid with *n*
_{2} = 1.5000, Δ*n*
_{2} = 0.0120, *V*
_{2} = 41.7 (Cargille Labs, [13]) were chosen to fill the chambers (b) and (c), respectively. The aperture variables are *h* = 2.5 mm for the clear aperture radius and and *y* = 2 mm for height of the marginal ray. With these parameters, Fig. 2 visualizes the primary longitudinal spherical aberration *LA*′_{p} at different focal lengths as a function of the bending variable *c*
_{3} and for both possible orderings. The dashed lines indicate the curvatures *c*
_{3} for which *LA*′_{p} is zero.

An analysis of the four solutions shows that in the case of a crown-in-front achromat [Fig.2(a)] for *f*′ > 0 the right-hand solution is most moderate in its curvatures (|*c*
_{3}| + |*c*
_{5}| + |*c*
_{7}|)_{RH} < (|*c*
_{3}| + |*c*
_{5}| + |*c*
_{7}|)_{LH}, whereas for *f*′ < 0 the left-hand solution gives weaker surfaces. This solution corresponds to the dashed lines in Fig.2(a) which are far from the ordinate at *c*
_{3} = 0. In contrast, for the case of a flint-in-front achromat [Fig.2(b)] the best solution is given by the dashed curve closer to the ordinate. These solutions are preferred since they will have minimum zonal aberration.

The results of the analytic predesign have been analyzed and compared with an optimized design by using ZEMAX^{®} raytracing software. The required radii for the membrane interfaces of a spherical corrected tunable achromatic lens for the crown-in-front (CIF) as well as the flint-in-front (FIF) solution are shown in Fig. 3. The optical system modelled in ZEMAX^{®} considers finite thicknesses of the chambers (edge thickness of 1 mm at a radial height of 2.5 mm) and additionally we introduce four 400 μm thin, low-dispersive silica glass plates for chamber separation as well for front and back covers (cf. layouts in Fig. 5). For optimization, the predesigned radii serve as initial values which are then varied to minimize an rms-wavefront based merit function. Despite the fact that the ‘real’ systems are truly not thin, the analytic predesign delivers results relatively close to the final solution, especially for the first and second radii *R*
_{3} and *R*
_{5}, respectively. Finite thicknesses and an additional spherical aberration introduced by the plane parallel plates give rise to the radius offset of the third membrane radius *R*
_{7}. Furthermore, we observe a non-linearity in *R*
_{7} for a negative, FIF achromatic lens which indicates the occurrence of high-order spherical aberration for which the optimization routine compensates. In any case, the CIF-solution (Fig. 3a) is generally preferred for two reasons: first, it has a slightly improved Strehl ratio (i.e. smaller spherical aberration, cf. Fig. 4) and second, higher numerical apertures can be realized by maintaining the given edge thickness (cf. the gap around zero).

Figure 5 illustrates the excellent optical performance at the two largest possible positive/negative refractive powers for the exemplary crown-in-front spherically corrected, achromatic variable-focus liquid lens. We conclude from this study that it is possible to design an all-liquid achromatic lens which is tunable over a wide focal range and simultaneously corrected for primary spherical aberration. In spite of finite thickness and glass plates, the theoretical optical performance of such liquid lenses is still diffraction limited.

## 4. Concluding Remarks

In summary, we have presented a design procedure for spherically corrected, achromatic variable-focus liquid lens systems. The analytic thin-lens predesign delivers solutions close to the final ray-traced optimized solution and therefore the given design rules may be very useful at an early design stage, i.e. in choosing the right liquids, defining the mechanical housing, and determining the theoretically achievable optical performance.

To realize such a system in practice, the lens can be fabricated by stacking three two-chamber systems one on top of the other. Standard MEMS technology enables cost-effective wafer-based batch processing. The described design scheme may applied for membrane-based as well as for electrowetting-based liquid lenses. Both types of liquid lenses have their own advantages and disadvantages. In particular, the actuation of electrowetting lenses is easier to miniaturize and control, but the liquid selection is more restricted due to special electrical and chemical properties which are required. On the other hand, pneumatic actuation for membrane lenses is more complex but more liquids are available which have only to be chemically compatible with the elastomer material. In this context, for the lens designer, it would be most desirable to have a comprehensive ‘liquid map’ – similar to the glass catalogue – that covers optical, chemical, thermal, and electrical properties of liquids. Since the optical properties of liquids can often be tailored within a certain parameter space, the design freedom is higher than for glasses, thus providing greater flexibility for liquid achromats such as reducing of the secondary spectrum by choosing liquids with different Abbe numbers but nearly identical partial dispersions [14].

## Acknowledgments

This study was supported by the Landesstiftung Baden-Württemberg, program “Forschung Optische Technologien” under the project MEMLENS.

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