## Abstract

We present an 8x four-group zoom lens system for a compact camera without any moving groups by employing a focus tunable lens (FTL). We locate the FTLs at the second and fourth groups as a variator and a compensator. In the initial design stage, paraxial analysis for the zoom position was numerically determined by examining the solutions for various first group and third group powers, to achieve a physically meaningful and compact zoom system at a zoom ratio of 8x. The designed zoom lens has focal lengths of 4–31 mm and the apertures of F/3.5 to F/4.5 at wide and tele positions, respectively.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

A zoom system consists of several groups, including moving groups such as a variator for zooming and a compensator for keeping the image plane stationary. These moving groups are necessary to achieve different focal lengths, but they require additional space for moving [1, 2]. Because the movement of a compensator is generally smaller than that of a variator, the former can be replaced by a liquid lens. For a compact system, the development of a zoom system with one moving group in addition to a liquid lens as a compensator has been reported [3]. However, because of the small refractive index difference between liquid materials, the magnitude of change in the optical power for a liquid lens is limited compared to that of a focus tunable lens (FTL) [4].

The structure of an FTL is composed of a liquid enclosed in a thin elastic membrane [5–7]. A change of pressure in the container causes the membrane to deflect, thus forming a lens. Since this tunable lens has the advantage of being convertible from a concave to a convex shape (or vice versa) by changing the curvature of a thin membrane, negative and the positive focus-tunable performance can be obtained. Also, an FTL has a big refractive index difference between air and the liquid which makes up the lens, so that it can produce greater changes in optical power than a liquid lens for the same curvature.

Much research on zoom systems using FTLs has been reported. The excellent performance of FTLs as variators or compensators in zoom systems is well documented [8–10]. In 2013, Lee et al. proposed a zoom lens using tunable lenses for a laparoscope, which had apertures ranging from F/4.9 to F/6.04 at two extreme positions, and a three-group configuration [8]. The tunable lenses proposed by Lee were located at the first and third groups as a variator and a compensator, respectively. Because all the groups are fixed during the zooming process, the zoom ratio is limited to 4x at most. In 2016, Park et al. proposed an 8x four-group inner-focus zoom system with one moving group by employing an FTL at the fourth group as a compensator [9]. This configuration with one moving group facilitated a higher zoom ratio of 8x, which is better than the system proposed by Lee.

Many initial design approaches for zoom systems using FTLs or liquid lenses have been reported [8, 10–13]. These studies deal with the zoom system design based on thin lens theory but do not consider the shift of the principal points of the FTLs at different zoom positions. Since the principal points of FTLs moved during the curvature change which accompanies zooming and compensating, a rigorous analysis of the zoom position should be pursued to achieve physically-meaningful solutions for initial zoom systems.

In this study, we propose an 8x four-group zoom system with no moving groups by employing two FTLs as a variator and a compensator. To achieve a reasonable and a compact initial zoom system at a high zoom ratio of 8x, we numerically examined the paraxial solutions for zoom parameters by analyzing the shifts of the principal points with zoom position, for various and highly probable lens group powers of the first and third groups. This design approach enables us to design a compact camera with a high zoom ratio of 8x, even though all groups are fixed for zooming and compensating. This analytical design concept is a key point of this study.

## 2. Analysis of a four-group zoom system without moving groups

The general four-group inner-focus zoom system is composed of a fixed first group, a moving second group for zooming, a fixed third group, and a moving fourth group for compensating for image position errors induced by focal-length changes at different zoom positions [1, 2, 14]. Thus, all powers of each group are always fixed, and the variation of distances between principal planes causes a change in the focal length and magnification of the zoom system.

In this study, we propose a four-group zoom system with no moving group by employing FTLs, to achieve a compact zoom system for a zoom ratio of 8x. Thus, we locate the FTLs at the second group for zooming and the fourth group for compensating, as shown in Fig. 1. Because all groups are always fixed, the powers of each FTL should be changed to produce different focal lengths and compensate for image position errors induced by focal-length changes. The variable power of the second and fourth groups can be realized by varying the curvature of an FTL.

The paraxial design of this zoom system with an infinite object can be formulated as follows [1, 15]:

where ${k}_{1},{k}_{2},{k}_{3},{k}_{4}$ are the powers of the groups, and ${z}_{1},{z}_{2},{z}_{3},{z}_{4}$ are the distances between adjacent principal points at a given zoom position. ${h}_{i}$ is the axial ray height on the image plane, and $K$ is the total power of the zoom system. The subscripts*w*and

*t*denote the zoom positions at wide and narrow fields, usually designated as wide and tele positions, respectively. Finally [ ] represents the Gaussian brackets [16].

The curvature of an FTL should be changed to obtain the required power at various zoom positions, which consequently causes the position of the principal points to shift. Therefore, the distances between the adjacent principal points at the tele position ${z}_{it}(i=1,2,3,4),$ can be expressed using the amount of movement $\Delta {z}_{i}(i=1,2,3,4)$ of the principal points from the wide position, as follows:

Substituting Eqs. (5)–(8) in Eqs. (1)–(4) yields more detailed expressions for the zoom equations in a four-group zoom system with no moving groups:

## 3. Principal point shifts of FTLs for zooming and compensating

Figure 2 shows the cardinal points of a single lens. In this figure, the first$(H)$and the second$({H}^{\prime})$principal points of the lens are located as follows [17]:

where ${A}_{1}$ and ${A}_{2}$ are the apex points of the front and rear surfaces, ${k}_{1}$ and ${k}_{2}$ are the powers of these surfaces, $d$ is the central thickness of a lens, and $n$ denotes the refractive index .From Fig. 3, the sags$({S}_{1},\text{\hspace{0.17em}}{S}_{2})$ of the front and rear surfaces at the semi-diameter of $h$, depending on the sign of the radius$(R)$ of curvature, are expressed as [18]

To obtain a high zoom ratio of 8x, it is desirable to configure the zoom system to be a retro-focus type at the wide position and a tele-photo type at the tele position. Thus, the FTL at the second group should have a negative power at the wide position, and a positive power at the tele position.

On the contrary, the FTL at the fourth group should have a positive power at the wide position, and a negative power at the tele position, as shown in Fig. 1. This large change in power of FTLs can be obtained by shape-changing the polymer membrane from plano-concave to plano-convex (or vice versa), as shown in Fig. 4. This zoom configuration is one of the key points of this study. In this research, we can easily formulate simple expressions for the movement of the principal points due to curvature-changing of the FTL at two extreme positions, assuming that the back surface of the FTL is rigid and a plane.

Figure 4 illustrates the changes of the central thickness and the shifts of the principal points induced by curvature-changing at different zoom positions. For the FTL located at the second group, as shown in Fig. 4(a), the positions of the principal points at the wide and the tele positions respectively, are given by

By applying the same process to the fourth group composed of another FTL, as shown in Fig. 4(b), the principal points at the wide and the tele positions are as follows:

_{${A}_{1}$}and${A}_{4}$are the reference points which denote the maximum thicknesses of the FTLs at the second and fourth groups, respectively. Thus, regardless of the power change of an FTL, the distance from ${A}_{1}$ and ${A}_{3}$ is always fixed. It follows that the distance from ${A}_{4}$ and ${A}_{6}$ is also fixed. When the zoom position is switched from wide to tele, the amount of movement of the principal points of each group, $\Delta {z}_{i}(i=1,2,3,4)$, is calculated as follows:

To analyze the four zoom equations of Eqs. (9)–(12) in Section 2, the amount of movement of the principal points in the FTLs should be specified. In addition, the minimum central thickness and the aperture size of each FTL need to be determined initially. In order to obtain a compact and feasible FTL as shown in Fig. 4, the distances from ${A}_{1}$ to ${A}_{3}$ and from ${A}_{4}$ to ${A}_{6}$ are set to 3 mm in the proposed system.

The FTL of the second group is close to the first group so that its aperture is sufficiently large to realize a wide-angle system. Hence, its aperture diameter is specified as 8 mm. The thicknesses of the plane lenses that support the rear surfaces of both FTLs are each set to at 0.6 mm. In addition, the minimum central thicknesses of both liquid elements, are initially specified to be 0.4 mm, since they are concave FTLs. In this case, the FTL’s sag was set to −1 mm at the maximum negative power and 1 mm at the maximum positive power. The radius of curvature of the FTL at the second group as determined from Eqs. (15) and (16), ranges from $\text{-8}\text{.5}\text{\hspace{0.17em}}\text{mm}$at the wide to 8.5 mm at tele position.

Since the FTL at the fourth group is located in front of an image plane, its aperture should be large enough to cover the size of a 1/5-inch image sensor, for which a diameter of 6 mm is computed. Similar to the FTL at the second group, the radius of curvature of the FTL at the fourth group ranges from 5 mm at the wide to −5 mm at tele position, if we set the FTL’s sag to be 1 mm at the maximum positive power and −1 mm at the maximum negative power.

The liquid material for an FTL has an effect on the variations in color aberrations for zooming. In this zoom system, since the power of each FTL changes from positive to negative (or vice versa) based on zoom position, the liquid material with a typical Abbe number is preferred to balance the color aberrations over all zoom positions. Thus, the FTLs for the second and fourth groups are selected to be OL1129$({n}_{d}=1.38,\text{\hspace{0.17em}}{v}_{d}=65)$ manufactured by Optotune and LiCl solution$({n}_{d}=1.42,\text{\hspace{0.17em}}{v}_{d}=46)$ [6–8], respectively.

Next, by inputting the previous initial data of both FTLs into Eqs. (17)–(24), then Eqs. (25)–(28) yield the amount of movement of the principal points from the wide position as:

Also, the powers of the FTLs at the second and fourth groups are computed based on the aforementioned design approach, as follows:

## 4. Paraxial design of an 8x four-group zoom system without moving groups

To obtain a physically-meaningful four-group zoom system, the first and third groups should have reasonable powers. If the powers of the second and fourth groups are given by Eq. (30) along with $\Delta {z}_{i}(i=1,2,3,4)$ of Eq. (29), four parameters should be determined to satisfy the four zoom equations of Eqs. (9)–(12), among the six unknown parameters of ${k}_{1},{k}_{3},{z}_{1w},{z}_{2w},{z}_{3w},{z}_{4w}$.

The FTL’s power changes significantly, ranging from positive to negative values, which results in the changing distances ${z}_{i}(i=1,2,3,4)$ between adjacent principal points. Therefore, for various ${k}_{1}$ and ${k}_{3}$ values, the zoom data that enables an 8x compact system are selected from the solutions for ${z}_{1w},{z}_{2w},{z}_{3w},{z}_{4w}$ that simultaneously satisfy the four zoom equations of Eqs. (9)–(12).

A retro-focus system is desirable at the wide position to achieve a shorter focal length. When this zoom system consists of two groups, the front group composed of the first and second groups should have a negative power, and the rear group composed of the third and fourth groups should have a positive power. Thus, the first group’s power is slightly positive to collect the rays while the power of the second group is strongly negative to diverge the rays. The power of the third group should be positive to relay the rays onto the next group, and the final group should have a positive power to realize a retro-focus system at the wide position.

To obtain a longer focal length, it is desirable that the front group has a positive power, while the rear group has a negative power, which leads to a tele-photo system at the tele position. This configuration can be obtained by changing the signs of the curvatures of both FTLs, i.e., the positive FTL of the second group, and the negative FTL of the fourth group. As the third group relays the rays onto the final group, that group should have a positive power. In this study, we have investigated changes of the distance between adjacent principal points by changing the first group’s focal length${f}_{1}$, as shown in Fig. 5 for various${f}_{3}$. Figure 5 shows the distances between the adjacent principal points of an 8x four-group zoom system, which are obtained by solving the four zoom equations of Eqs. (9)–(12).

All solutions investigated in this study have a negative distance of${z}_{3w}$, which is useful to shorten the total track of zoom system by relaying the rays to the final group. Among them, the sum of ${z}_{3w}$and ${z}_{4w}$is constrained to be positive so that a physically-meaningful system can be identified. In Figs. 5(a), 5(b), and 5(d), as ${f}_{3}/{f}_{1}$gets smaller, ${z}_{1w}$ becomes dramatically longer. In order to have a compact zoom system, the solution that has the shortest and positive ${z}_{1w}$along with positive sum of ${z}_{3w}$ and ${z}_{4w}$, was selected. The solutions were investigated for four different focal lengths${f}_{3}$, being based on this selection rule for solutions. The best solutions at a given ${f}_{3}$ are designated in the box of Figs. 5(a)–5(d), and listed in Table 1. Among four cases of${f}_{3}$, the zoom system with${f}_{3}=16\text{\hspace{0.17em}}mm$and${f}_{3}/{f}_{1}=0.18$, is found to be the best solution that satisfy all requirements, which is designated in the box of Fig. 5(c). Thus, the initial zoom system is reasonably obtained using this numerically paraxial analysis for zoom parameters. This analytical design concept is a key aspect of this study, and the zoom data obtained in this process are listed, as follows:

## 5. Initial design for an 8x four-group zoom system without moving groups

We have set up an initial zoom lens system with four thick-lens modules, for which the focal length (EFL) of each group and the zoom position inputs are taken from the data given in Eq. (31). In Table 1 and Eq. (31), the specific first-order quantities such as the front focal length (FFL) and the back focal length (BFL) of each group, and the air distances between the groups, are not presented.

In this study, however, we proposed the paraxial design method that took into account the principal plane shifts when the radius of the tunable lens was changed. After a few numerical calculations, this proposed design method immediately yields the specific first-order quantities (EFL, FFL, BFL) of each group at given air distances, as shown in Table 2, which enables us to independently design the real lens group equivalent to each module within paraxial optics. Next, an actual initial zoom system can be realized by combining four real lens groups designed separately. Thus, this paraxial design concept, based on the analysis for principal plane shifts, reduces much effort required to obtain an actual zoom lens system, which is an advantage of this study.

The air distances between groups are constrained to be longer than 0.3 mm for the mounting space. In order to get a compact zoom system even at a high zoom ratio of 8x, we optimized the air distances so that the specific constraints were satisfied. Table 2 lists the specific first-order data for each group and the air distances between groups. Figure 6 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. This zoom system has fixed variator and compensator, of which the former focal lengths are changed from −22.37 mm to 22.37 mm, and the latter focal lengths are changed from 11.91 mm to −11.91 mm.

A real lens group is generally composed of several lens elements, which should be equivalent to the lens module group given in Table 2. In this paper, an optimization design method is used to design a real lens group equivalent to the module of each group. We choose an appropriate structure for each real lens group and scale it up or down so that the focal length of each real lens group is the same as that of the lens module. By using the optimization method of Code-V, we matched the first-order quantities (EFL, BFL, FFL) of the lens module to those of a real lens. For the conversion process, the design variables of the real lens are the radius of each surface, the thickness, and the refractive index. Therefore, the three constraints (EFL, FFL, BFL) given in Table 2 can be satisfied by specifying the lens design variables. After a few iterations, the real lenses of the groups are obtained. If a zoom system equivalent to the lens module zoom system is to be achieved, the air distances between groups should be set according to the zoom parameters of Table 2 at each position. This procedure results in a zoom system equivalent to the lens module zoom system within paraxial optics [14].

To reduce the number of lens elements, the first group is designed with two components, and the second and fourth groups consist of only one FTL for changing their powers, not for correcting aberrations. As seen in Eq. (31), to have a higher zoom ratio of 8x, the distances (${z}_{2w},\text{\hspace{0.17em}}{z}_{2t}$) between the principal planes of the second and third groups are set to be very long. Also, the distances (${z}_{3w},\text{\hspace{0.17em}}{z}_{3t}$) between those of the third and fourth groups are negative, which denotes the first principal plane of the fourth group is to the left of the second principal plane of the third group. This structure is required to realize a compact zoom system even at a high zoom ratio of 8x. Thus, the second principal points of the third group may be outside the third group. In addition to them, the third lens group is required to correct all residual aberrations, not corrected by other groups. This situation leads this group to be composed of many elements, as shown in Fig. 7.

In the initial design, reducing the aperture and the field size of the system is desirable. Thus, we take a zoom system with a half image size of 1 mm and f-numbers of F/5 at the wide position to F/7 at the tele position. Figure 7 illustrates an 8x initial real lens zoom system in the reduced aperture and field, obtained by combing the separately designed real lens groups according to the zoom positions of Table 2.

## 6. Optimized design for an 8x four-group zoom system without moving groups

In the initial design of Fig. 7, the aperture and the field size should be increased, to meet the current specification for a zoom camera. The f-numbers are extended to F/3.5 at the wide and to F/4.5 at the tele positions. The half image size should be 1.8 mm for a 1/5-inch 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 aspheric surfaces.

The layout of the final zoom system is shown in Fig. 8, which consists of 11 elements including two FTLs and four aspherical lenses. The maximum diameter of the front group is 11.28 mm. The ratio of the relative illumination (RI) measured at the margin fields is more than 79% over all positions, and the distortions are sufficiently balanced to less than 5%, as shown in Fig. 8.

Figure 9 illustrates the sags of the final FTLs at various zoom positions. To effectively improve the magnitude of the power variation at the FTL, the sags of the FTL are set to be symmetric with opposite signs at the wide and the tele positions. The variations of the sag from a wide to a narrow field are symmetric and less than 2 mm. It is such a small value that this configuration can realize a mechanically stable zoom system.

Figure 10 illustrates the lateral color aberrations with zoom position, which are less than 4 μm within 0.9 fields. Thus aberrations are significantly reduced. From Fig. 11, the modulation transfer function (MTF) at 180 cycles/mm is more than 30% over all fields at both extreme positions.

The variation of the chief ray angle of incidence (AOI) from a wide to narrow field is less than 5.25 degrees. Because this value is small, a stable image quality for zooming can be realized. The total track of this zoom lens, even at a high zoom ratio of 8x, is less than 35.5 mm. Consequently, the performance of this zoom system satisfies current requirements for a zoom camera.

## 7. Conclusion

In the design of an 8x four-group zoom system with no moving groups, FTLs have been suggested as a variator and a compensator, which are fixed during zooming and compensation.

In the initial design stage, numerous difficulties were encountered during the paraxial design of the group’s power and the zoom position, as it was necessary to solve many equations for the various cases. To overcome these difficulties, we developed a numerical approach for paraxial analysis of the zoom position by examining the distances between the principal points with zoom position, at highly probable lens group powers. Subsequently, the initial real zoom system was set up based on physically meaningful solutions.

After aberration balancing, an 8x four-group zoom system with no moving groups was realized by employing two FTLs. These zoom lenses had focal lengths of 4–31 mm and f-numbers of 3.5 at the wide to 4.5 at the tele positions. Through this study, we illustrated the design process of a four-group zoom system with a fixed variator and compensator, along with its zoom position, using a numerically paraxial design approach.

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