Design of an 8x Fourgroup Innerfocus Zoom System Using a Focus Tunable Lens
 Author: Lee Daye, Park SungChan
 Publish: Journal of the Optical Society of Korea Volume 20, Issue2, p283~290, 25 Apr 2016

ABSTRACT
This study presents an 8x fourgroup innerfocus zoom lens with onemoving group for a compact camera by use of a focus tunable lens (FTL). In the initial design stage, we obtained the powers of lens groups by paraxial design based on thin lens theory, and then set up the zoom system composed of four lens modules. Instead of numerically analytic analysis for the zoom locus, we suggest simple analysis for that using lens modules optimized. After replacing four groups with equivalent thick lens modules, the power of the fourth group, which includes a focus tunable lens, is designed to be changed to fix the image plane at all positions. From this design process, we can realize an 8x fourgroup zoom system having one moving group by employing a focus tunable lens. The final designed zoom lens has focal lengths of 4 mm to 32 mm and apertures of
F /3.5 toF /4.5 at wide and tele positions, respectively.

KEYWORD
Zoom lens , Zoom locus , Lens module , First orders , Optimization design

I. INTRODUCTION
A zoom system consists of several groups including a variator for zooming and a compensator for keeping the image plane stationary. Because the movement of a compensator is generally smaller than that of a variator, a compensator can be replaced by a liquid lens. Zoom systems having one moving group plus a liquid lens have been reported [1, 2], but the liquid lens actuated by electrowetting is composed of a cell filled with two immiscible liquids. One is an oillike electrical insulator, and the other is an electrolyte. Because of small refractive index difference between these materials, the magnitude of a changeable optical power obtained by this liquid lens is limited compared to that of a focus tunable lens.
Figure 1 shows the structure of the focus tunable lens (FTL) composed of a liquid and a thin elastic membrane enclosing it [3, 4]. An optical liquid is concealed in a container with at least one side being an elastic membrane. A change of pressure in the container causes the membrane to deflect, thus forming a lens. This tunable lens design has the advantage of being convertible from a convex to a concave shape. Thus, various focal lengths can be obtained by changing the curvature of a thin membrane. Since the focus tunable lens has also the big refractive index difference between the materials of air and liquid, not liquid and liquid, it can make greater changes in optical power than previous liquid lenses even for the same curvature. In addition, the focus tunable lens is known to be much more stable from the gravitational force at large aperture than the liquid lens is [4]. Because of these merits of a focus tunable lens, we can use it as a compensator in a compact zoom lens design.
Much research on optical systems using liquid lenses has been reported. Most of them are for autofocusing by a liquid lens [5, 6]. However, such lenses can be used to compensate for image position errors generated by changes in the power and the object distance.
Focus tunable lenses performing well as a compensator in zoom systems have also been reported [7, 8]. In 2013, Lee proposed a zoom lens using tunable lenses for a laparoscope, which has the apertures of
F /4.9 toF /6.04 at two extreme positions, a threegroup configuration, and a total track of 23.26 mm [8]. The tunable lens proposed by Lee is located at the first and third groups as variator and compensator, respectively. Because all groups are fixed during zooming process, the zoom ratio is limited to 4x at most.In this research, to achieve a higher zoom ratio of 8x, we propose a fourgroup innerfocus zoom system having one moving group by employing a focus tunable lens. We locate the FTL at the fourth group as a compensator. Because this group is always fixed, so the curvature of an FTL should be changed to compensate for image position errors induced by focallength changes at different zoom positions. This configuration enables the zoom lens to have a higher zoom ratio of 8x and a compact size.
II. ANALYSIS OF A FOURGROUP INNERFOCUS ZOOM SYSTEM WITH COMPENSATOR OF FOCUS TUNABLE LENS
The layout of the fourgroup innerfocus zoom system, consisting of a focus tunable lens, is shown in Fig. 2. The powers of the groups are denoted by
k _{1},k _{2},k _{3},k _{4w} at a wide position, andk _{4t} at a tele position. The zoom system is composed of a fixed first group, a moving second group, a fixed third group, and a fixed fourth group. Because the second group moves to the image side to have a longer focal length, the power of the fourth group should be varied to keep the image position stationary. This variable power of the fourth group can be realized by varying an FTL whose curvature is changed. Thus, the fourgroup innerfocus zoom system with an FTL has a structural advantage and the same total track at all zoom positions. This configuration is useful in obtaining a compact zoom system for a camera. The initial design of this zoom system with an infinite object can be formulated as follows [9, 10]:where
K is the optical power at a zoom position, andbfd is the back focal distance. Equations (1) and (2) can be rearranged as follows:Therefore, Eqs. (3) and (4) can be expressed in a matrix form for wide and tele positions as follows:
If the GaussJordan elimination method is applied, Eq. (5) is replaced by
where
b_{ij} is given as a function ofb_{ij} (K_{w} ,K_{t} ,z _{1w},z _{2w},z _{3w},bfd_{w} ,z _{1t},z _{2t},z _{3t},bfd_{t} ) withK_{w} andK_{t} being the optical powers at wide and tele positions, respectively. The parametersz_{ji} (j = 1, 2, 3) andbfd_{i} are the distances between adjacent principal planes at thei th zoom position. From Eq. (6), four equations are obtained:Plugging the condition of Eq. (9) into Eq. (10) results in an expression for the unknown parameter
k _{1}:where coefficients of
A ,B ,C ,D , andE are given as a function ofK_{w} ,K_{t} ,z _{1w},z _{2w},z _{3w},bfd_{w} ,z _{1t},z _{2t},z _{3t},bfd_{t} . Solving Eq. (11), we can getk _{1}. As a result,k _{2},k _{3} andk _{4} are sequentially determined from Eqs. (7)~(9). In common with other paraxial designs, solving the above equations by hand is difficult, so we developed a program tool to calculate the powers of each group at given zooming parameters ofz _{1},z _{2},z _{3},bfd , andK in Matlab, as shown in Fig. 3. As a result, obtaining physicallymeaningful initial zoom data through a paraxial design was possible [11].III. PARAXIAL DESIGN OF AN 8X FOURGROUP INNERFOCUS ZOOM SYSTEM
The paraxial design for a zoom system is carried out at two extreme positions. To reduce the amount of the curvature variation at the FTL, the powers of the fourth group including an FTL are set to be the same at wide and tele positions. This configuration can realize a simple and mechanically stable zoom system. For the initial design of an 8x fourgroup innerfocus zoom system, we input the proper distances between principal points and the targeted total powers at the wide and the tele positions. These starting values are empirically selected to satisfy the following basic requirements for an 8x compact zoom system:
By inputting these starting values for Eq. (12) into the program tool, we can immediately obtain the power of each group, as shown in Fig. 3. In that figure, the Result Data shows the four solution cases obtained from Eq. (7) to Eq. (11). These solutions for
k _{1},k _{2},k _{3},k _{4} are summarized in Table 1.The next step is to select the physicallymeaningful data among solutions given in Fig. 3. This process can be carried out by executing the ray trace and the ‘Check Data’ analysis of the program developed in this study. The case 4 of Table 1 is confirmed to fulfill the requirements for the available zoom system with 8x.
IV. INITIAL DESIGN FOR AN 8X FOURGROUP ZOOM LENS
We have set up a zoom lens system with four thicklens modules, for which the power of each lens group and the zoom loci inputs are taken from the case 4 of Table 1 and Eq. (12), respectively. The lens modules do not reflect higherorder aberrations, so reducing the aperture and the field size of the system is desirable. Thus, we took a zoom system with a half image size of 1 mm and fnumbers of
F /5 at the wide position toF /7 at the tele position. The air distances between modules were constrained to be longer than 0.5 mm for the mounting space.In order to get an initial zoom system, we optimized the lens module parameters so that the specific constraints were satisfied [12]. In Table 1 of the previous section, the thickness of each group and the air spaces are not presented. They can be derived by specifying the design variables of the lens modules, such as the front focal length (
MFF ) and the back focal length (MBF ) of each group, and the air distances between the modules. Figure 4 shows the initial zoom design composed of four lens modules obtained from this process. The focal lengths range from 4.0 mm to 32.0 mm. Table 2 shows the design data for each module and the air distances between modules.Numerically analytic approaches using extensive calculations to obtain zoom locus have been reported [1, 10, 11]. However, handling many equations given in Section 2 at the same time is very complicated and requires much more effort. First of all, handling analytically all the firstorders of an FTL at all zoom positions is hard work. In this research, therefore, we analyze the zoom loci at all positions by optimizing the lens modules.
First, we calculate the image position shifts induced by the variator’s moving at different zoom positions, as shown in Fig. 5. We then locate the focus tunable lens at the fourth lens group as a compensator. Because this group is always fixed, so the curvature of an FTL should be changed to compensate simultaneously for image position shifts induced by focallength changes at different zoom positions.
Figure 6 shows the lens module zoom system with a fixed compensator, of which focal lengths are changed from 14.23 mm to 16.34 mm to 14.23 mm. Figure 7 illustrates the focal lengths of the fourth group at various positions.
V. OPTIMIZED DESIGN FOR A ZOOM SYSTEM USING FOCUS TUNABLE LENS
5.1. Converting Lens Modules into Real Lens Groups Successively
In the lens module zoom system of Fig. 6, four lens modules were successively, not separately, converted into real lens groups using an automatic design method. The conversion of the lens module into a real lens group is based on the firstorder properties given in Table 2 and Fig. 7.
At first, the first lens module is converted into the real lens equivalent to the module within the firstorder properties, while the other three groups are lens modules and, unlike the first group, have not yet been carried through this converting process. By these iterative processes, the second group, third group, and fourth group are successively converted to real lens groups [13]. As a result, the lens module zoom system is group by group converted to the initial real lens zoom system without losing the equivalence to each other.
In the converting process of the fourth lens group, the curvature of a focus tunable lens should be changed to keep the image plane stationary at different zoom positions. From the structure of an FTL in Fig. 1, while the edge thickness (
e ) of an FTL is always fixed, but its central thickness is changed by shapechanging polymer membrane, as shown in Fig. 8 [4, 14]. Assuming that the back surface of the FTL is rigid and a plane in this study, the sag (S ) and thickness (T ) with radius (R ) of curvature of a membrane are given byThe focus tunable lens as a compensator is designed to satisfy the conditions for its sag and thickness given in Eqs. (13) and (14), respectively. Its edge thickness was constrained to be more than 0.4 mm so that the curvature of the membrane can be sufficiently changed. Figure 9 represents the initial real lens zoom system combined due to the zooming locus listed in Table 3. As compared Fig. 6 to Fig. 9, the agreement for the firstorder properties between both zoom systems is complete. This design procedure results in a real lens zoom system equivalent to the lens module zoom system. Table 3 shows the firstorder quantities and zooming loci of an initial real lens zoom system.
5.2. Optimized Design for Zoom System with an FTL
In the initial design of Fig. 9, to meet the current specifications for a zoom camera, the aperture and the field size should be increased. The fnumbers are extended to
F /3.5 at wide and toF /4.5 at tele positions. The half image size should be 2.2 mm for a 1/4inch CMOS image sensor. In order to improve the overall performance in an extended aperture and field system, we balance the aberrations of the starting data by using a lens design program.Figure 10 illustrates the sag of the finally designed FTL at various positions, and Table 4 lists its design data. To reduce the magnitude of the sag’s variation at the FTL in this study, the focal lengths of an FTL are set to be the same at wide and tele positions. The variation of the sag from a wide to narrow field is less than 0.189 mm. It is such a small value that this configuration can realize a simple and mechanically stable zoom system.
The layout of the finally designed zoom system is shown in Fig. 11. This system consists of 11 elements including an FTL and three aspherical lenses. The maximum diameter of the front group is 17.4 mm. Aberrations are significantly reduced. From Fig. 12, the modulation transfer function (MTF) at 180 lp/mm is more than 33% over all fields at three positions. The ratio of relative illuminations (RI) measured at margin fields is more than 57.1% over all positions, and distortions are sufficiently balanced, as shown in Fig. 13. Figure 14 illustrates the lateral color aberrations with zoom position, which are less than 3 μm within 0.7 fields.
The variation of the chief ray angle of incidence (AOI) from a wide to narrow field is less than 2 degrees. Because that is an extremely small value, a stable image quality for zooming can be realized. The total track of the zoom lens, even at a high zoom ratio of 8x, is less than 29 mm, which results in a compact zoom camera. Consequently, the performance of this zoom system satisfies current requirements for a zoom camera.
VI. CONCLUSION
For the design of an 8x fourgroup innerfocus zoom system having one moving group, a focus tunable lens has been suggested as a compensator, which is fixed for zooming.
In the initial design process, we obtained the powers of lens groups by paraxial design, and then setup the zoom system composed of four lens modules equivalent to each group. The numerical analysis of zoom locus has generally many difficulties in that lots of equations for various cases should be solved. To overcome these difficulties, we suggested the analysis of zoom locus using optimized lens modules. Here, four groups were replaced by equivalent lens modules. And to fix the image plane, we changed the powers of the fourth group at each position.
Next, four lens modules were converted to real lens groups step by step using an automatic design method. From this successive conversion process, the real lens of each group was quickly obtained by matching the firstorder quantities of lens modules.
After aberration balancing, we obtained an 8x fourgroup innerfocus zoom system having one moving group by employing a focus tunable lens. This zoom lens has the focal lengths of 4 mm to 32 mm and the fnumbers of 3.5 at wide to 4.5 at tele zoom positions. Through this study, we showed the design process of a fourgroup innerfocus zoom system with a fixed compensator and its zoom locus analysis using optimized lens modules.

[FIG. 1.] Structure of the focus tunable lens and variation of focal length: (a) positive and (b) negative focal lengths.

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[FIG. 2.] Layout of the fourgroup innerfocus zoom system with a focus tunable lens.

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[FIG. 3.] Windows for calculating the powers of groups and checking the data.

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[TABLE 1.] Focal lengths of groups at four solution cases (in mm)

[FIG. 4.] An 8x fourgroup zoom system consisting of lens modules.

[TABLE 2.] Design data for the lens modules and air distances between modules in the lens module zoom system (in mm)

[FIG. 5.] Zoom locus of an 8x fourgroup zoom system with no compensator.

[FIG. 6.] An 8x fourgroup lens module zoom system with a compensator having variable powers.

[FIG. 7.] Focal lengths of the fourth group with zoom position.

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[FIG. 8.] The shapechanging tunable lens and thickness variation by its curvature change.

[FIG. 9.] An 8x initial real lens zoom system with a fixed compensator realized by an FTL.

[TABLE 3.] Firstorder properties and zooming loci of an initial real zoom system with an FTL (in mm)

[FIG. 10.] Variations for sag of the FTL with zoom position.

[TABLE 4.] Design data for the FTL in an aberrationbalanced zoom system (in mm)

[FIG. 11.] Layout of an aberrationbalanced zoom system with an FTL at the fourth group. (ⓐ denotes the aspheric lenses).

[FIG. 12.] MTF characteristics of an aberrationbalanced zoom system with zoom position.

[FIG. 13.] Distortion and relative illumination of an aberrationbalanced zoom system with zoom position.

[FIG. 14.] Lateral color aberrations of an aberrationbalanced zoom system with zoom position (in mm).