## Abstract

A novel design of a zoom lens system without motorized movements is proposed. The lens system consists of a fixed lens and two double-liquid variable-focus lenses. The liquid lenses, made out of two immiscible liquids, are based on the principle of electrowetting: an effect controlling the wetting properties of a liquid on a solid by modifying the applied voltage at the solid-liquid interface. The structure and principle of the lens system are introduced in this paper. Detailed calculations and simulation examples are presented to show that this zoom lens system appears viable as the next-generation zoom lens.

© 2007 Optical Society of America

## 1. Introduction

Zoom lenses in imaging systems such as cameras must satisfy two basic requirements: adjustable focal length and fixed image plane. Traditional zoom lens systems usually comprise several fixed lenses, which satisfy the requirements by motorized movements [1]. As a result, delicate driving motors must be used to provide desired control over the mechanical positions of lenses and precise space cams must be utilized to implement synchronized movements. Such conventional solutions have been considered to be complicated, fragile and thus expensive. In addition they are also not convenient to be implemented in small spaces, such as in mobile phone cameras.

An alternate solution, a kind of refraction-diffraction combined variable-focus lens has been presented [2, 3]. It makes use of LCD Fresnel lenses. Electrically motivated, the LCD Fresnel lenses can vary the focal length by changing the refractive index of LCD without involving any motorized movements. However, disadvantages such as large chromatic aberrations, multi-focuses and difficulty in keeping the imaging plane fixed etc. pose an obstacle to this solution.

Researchers at Philips recently reported a double-liquid variable-focus lens based on electrowetting [4]. Unlike conventional solutions, a double-liquid lens changes its focal length electrically, not mechanically. On the other hand, the disadvantages of the LCD Fresnel lenses mentioned above are also avoided. However, one double-liquid lens cannot keep the image plane fixed while it changes the focal length.

In order to meet the two basic requirements at the same time, two liquid lenses are required, as pointed out recently by Hendricks et al [5]. In this paper, we present a new design by combining two double-liquid variable-focus lenses with a fixed lens. These two liquid lenses under suitable external voltages can not only adjust the whole focal length of the system but also compensate the position changes of the image plane due to each liquid lens. We start by illustrating the configuration and principle of the proposed zoom lens system. A detailed theoretical analysis on the system is then performed. Finally, simulation examples and results are presented to show that the described zoom lens system is a highly viable and promising alternative to the conventional zoom lens system.

## 2. Principle of the zoom lens

The double-liquid lens is the main component of our zoom lens. Figure 1 shows a simplified schematic of the double-liquid lens. For convenience all the lens configurations in this paper are plotted horizontally despite the fact that experimentally they have a vertical setup. There are two immiscible liquids contained in the cylindrical chamber, one conducting (Liquid 1) with refractive index *n*
_{1} and the other insulating (Liquid 2) with refractive index *n*
_{2}. These two liquids have the same density, thus the interface of liquid 1 and liquid 2 is perfectly spherical [6]. A thin transparent electrode and a hydrophobic dielectric layer with thickness *d* and relative permittivity *ε _{r}* are subsequently coated inside the cylindrical chamber. A voltage

*U*applied between the conducting liquid and the electrode results in a variation of the interface curvature. If

*n*

_{1}≠

*n*

_{2}, the double-liquid system can be considered as a variable focus lens. Equation (1) [7] describes how the applied voltage changes the radius of the interface

*r*.

where *a* is the inner radius of the cylindrical chamber, *θ*
_{0} is the initial contact angle when there is no external voltage, *ε*
_{0} is the permittivity of the free space and *γ*
_{12} is the tension coefficient of the interface between liquid 1 and liquid 2.

A schematic of the zoom lens we proposed, which is composed of one fixed lens and two double-liquid variable-focus lenses is shown in Fig. 2. Two liquid lenses are placed between the exit pupil and the back focus plane of the fixed lens.

Obviously, the focal power of the system is mainly determined by the fixed lens. When the first liquid lens acts as a concave lens and the second as a convex lens under applied voltages, the former diverges rays and the latter converges rays. Thus the convergence angle on the image plane of the fixed lens increases and the combined focal length decreases if the image plane remains fixed. Conversely, the combined focal length increases when the first liquid lens acts as a convex lens and the second as a concave lens under applied voltages. The focal length of the zoom lens system can be readily adjusted by changing the external voltages without motorized movements. In the meanwhile, regulating the voltages properly can also ensure that the summation of conjugate distances of all the lenses keep unchanged. In other words the image plane of the zoom lens remains fixed.

## 3. Theoretical analysis

In this section, we take infinite object distances as example and examine the conditions under which the new type of zoom lens meets the requirements mentioned earlier. Let us first consider a special case when the liquid interfaces in both double-liquid lenses are just a plane.

When the liquid interface becomes a plane under some external voltage, the liquid lens can be treated as a plane-parallel plate, which has no contribution to the total focal power. In this case, two double-liquid lenses only change the focus position of the zoom lens. For the zoom lens shown in Fig. 3, the focus position, which is defined by *ĺ*, the distance from the focus *P* to the back surface of the second liquid lens, can be given by

where *n*
_{0} is the refractive index of air. For a given zoom lens system, *ĺ* is a constant. We mark the thickness of the first and second liquid lens as *d*
_{01} and *d*
_{02} respectively, and the volume percentage of liquid 2 for the first and second liquid lens is denoted by *k*
_{1} and *k*
_{2} times. That is *d*
_{2} = *k*
_{1}
*d*
_{01} and *d*
_{5} = *k*
_{2}
*d*
_{02}, and Eq. (2) can be rewritten as

In general cases, the interfaces in both double-liquid lenses are either convex or concave. To keep the focus position of the whole system invariable, one of the double-liquid lenses is a positive lens and the other is a negative one. In other words, the radii *r*
_{2} and *r*
_{5} should have opposite signs. Figure 4 shows such a zoom lens with a concave interface (i.e. negative *r*
_{2} ) for the first double-liquid lens under some external voltage. The focus position of the system in this case can be derived and given by

where

Since *r*
_{2} (or *r*
_{5}) and *d*́_{2} (or *d*́_{5}) are geometrically related, the latter can be expressed as a function of the former, i.e.

where *a*
_{1} and *a*
_{2} are the inner radius of the cylindrical chamber for the first and second liquid lens respectively. Equations (6) and (7) also shows that the absolute value for both *r*
_{2} and *r*
_{5} have a lower limit, that is |*r*
_{2}| ≥ *a*
_{1} and |*r*
_{5}| ≥ *a*
_{2}.

To keep the focus position fixed, one has

Substituting Eq. (4) into Eq. (8) and some rearrangement give rise to

where *α* is a function of *r*
_{2} and *r*
_{5} is implicated in *d́*
_{5}. When *r*
_{2}, which is determined by the external voltage, is given, Eq. (9) can be solved analytically. However there will appear a high-order equation in terms of *r*
_{5} and extraneous roots will occur. The relation between *r*
_{5} and *r*
_{2} is then achieved by a numerically iterative method.

Assume the voltages applied to the first and second liquid lens are *U*
_{1} and *U*
_{2} respectively. According to Eq. (1), we have

Since there exists one-to-one correspondence between the applied voltage and the interface radius, the relationship curve of *r*
_{5} and *r*
_{2} can be readily converted into that of *U*
_{2} and *U*
_{1}.

Under such circumstances, the combined focal length of the zoom lens system, denoted by *f*, can be expressed in *r*
_{2} and *r*
_{5} by

In the case that the interface of the first liquid lens is convex and that of the second is concave, similar deduction can also be performed.

## 4. Results and discussion

Taking infinite objects as example, we have shown the feasibility for the proposed zoom lens system consisting of two double-liquid lenses. By applying appropriate electrical voltages to the two double-liquid lenses, one can not only change the focal length of the system but also keep the position of the image plane unchanged. Therefore, this lens system can realize the function of zooming and focusing to observe a large area with small magnification or a small area with large magnification.

In order to illustrate the working conditions and focal property of the zoom lens, some simulation results are given below. A set of parameters are specified as follows: *n*
_{0} = 1, *n*
_{1} = 1.38, *n*
_{2} = 1.55, *d*
_{01} = *d*
_{02} = 2*mm*, *d*
_{1} = 0.05*mm*, *d*
_{4} = 2.95*mm*, *k*
_{1} = 1/3, *k*
_{2} = 2/3 and *f*
_{0} = 10*mm*. Using Eq. (9), we can plot *r*
_{5} as a function of *r*
_{2} in Fig. 5. In agreement with earlier analysis, the signs of *r*
_{5} and *r*
_{2} remain opposite. Due to the limit we mentioned earlier for *r*
_{2} and *r*
_{5}, there exists a gap on the curve in Fig. 5 corresponding to the limited range.

Furthermore, we can specify the inner radii (for example *a*
_{1}=*a*
_{2}= 1*mm*) and choose two liquids with an interface tension coefficient *γ*
_{12} =38.1×10^{-3}
*N/m* for two double-liquid variable-focus lenses. According to Eqs. (9), (10) and (11), the relation between voltage *U*
_{2} and voltage *U*
_{1} can be simulated for different dielectric thicknesses, and the results are shown in Fig. 6(a). Figure 6(b) shows the focal length of the system as a function of the applied voltage *U*
_{1} for two different thicknesses of the dielectric layer.

For a dielectric layer with larger thickness, we can see that larger voltages are required in order to obtain the same range of the focal length for the zoom lens. In our design, the continuous optical zoom of the system can reach up to 1:2.

It should be mentioned that all results presented here are achieved without considering any aberrations. Also, although the discussion in this paper is based on an infinite object, similar derivations can be made for finite objects. These issues will be left for future work.

## 5. Acknowledgments

This work was supported by the National Basic Research Program of China (Grant No. 2005CB724304), by the Shanghai Leading Academic Discipline Project (Project Number: T0501), and by the Shanghai Committee of Education, China (Grant No. 05EZ43).

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