## Abstract

We present a theoretical method for analyzing the first-order optics of stabilized zoom lenses with two focal-length-variable elements. The zoom equations are established through the use of the Gaussian brackets method. This is done because the optical power of the focal-length-variable elements varies during the zooming process. The first and second derivatives and the Hessian matrix of the zoom equations with respect to the Gaussian parameters are determined using the equations. These parameters could represent the sensitivity of the zoom ratio of the system to changes in the corresponding system variables. We select the initial values of these system variables, i.e. the magnification of the focal-length-variable element and the structure parameters of the fixed lens group, to be close to the steepest gradient direction. Here the sensitivity of the system focal length is high with respect to variations in the zoom variables. This process leads to an increase in the zoom ratio of the zoom system. The results show successful four-group stabilized zoom lens designs with 2:1 and 5:1 zoom ratios, using two deformable mirrors as focal-length-variable elements. This system, with the inherent characteristics of a steepest gradient, could miniaturize zoom systems.

© 2013 OSA

## 1. Introduction

Zoom systems are widely used in the fields of aviation and aerospace. Industrial and other high-performance imaging instruments found on air and spacecraft are capable of wide-range region searching at the telephoto end and high image performance at the wide angle end [1, 2]. In conventional zoom systems, the separation between adjacent groups is allowed to vary. Typically, the optical system focal length is varied continuously by the displacements of lens groups, and a complex mechanical cam system is required in the process of zooming [3–6]. This mechanism may be noisy, bulky, and often consumes excess power which can be problematic for certain applications, such as deep-sea submersible and aerial vehicle surveys.

Many researchers have considered the usage of focal-length-variable opto-electronic components, such as deformable mirrors, liquid lenses, etc., in zoom lens designs. These techniques take advantage of stabilized zooming to acquire much higher zoom speeds with less noise than the cam system in traditional zoom optics [2, 7–10]. This approach has the potential to significantly simplify the opto-mechanical system of the zoom lens. Various authors have also investigated the construction of the zoom equation, the balance of the third-order aberration, and the relationship between the component parameters and the system structure [11–16]. However, there remain difficulties in the realization of high zoom ratios for stabilized zoom systems using optical power as a zoom variable. For example, the focal-length-variable elements should compensate for the image shift while taking over the variations of the focal length of the zoom system.

In this paper, we propose to analyze the paraxial properties and investigate the quick zooming characteristics of four-group stabilized zoom systems with two focal-length-variable elements, using the optical power of the optical element as the independent variable in the zoom equation. Firstly, we calculate the suitable initial point of first-order optics and determine the Gaussian parameters. Secondly, the optimization for the sensitivity of the system focal length change to variations in zoom variables will be executed along the steepest descent gradient, which miniaturizes the stabilized zoom system. Miniature and compact imaging modules integrated in mobile devices, aerial vehicles, and deep-sea submersibles, could help reduce the load capacity of these instruments.

When designing and optimizing the stabilized zoom system, the effects of a given variable (e.g., the magnification, the optical power, etc.) on the zoom ratio could be evaluated through calculating the derivatives of the zoom equations. The derivatives and Hessian matrix of the image plane displacement equation describe the influence of these variables on the compensation of the image shift. The derivatives of the optical power equation describe the sensitivity of the zoom ratio to the zoom variables as demonstrated in sections 2 and 3. In performing such analyses, the steepest descent gradient with respect to the aforementioned variables is determined by computing the critical points. The most influential variables can be found and the critical points can be used to identify and calculate a set of initial optimal parameters for the zoom system according to the requirements of the zoom ratio. The zoom systems may then be optimized to reach a local optimal solution, near the initial point in the parameter space, quickly along the direction of the steepest descent gradient.

## 2. The zoom equation and its derivatives with respect to magnification

In this section we discuss the application of Gaussian brackets to thin lens optical systems. The paraxial characteristics associated with these systems [17] can be described as follows:

The optical layout of the four-group stabilized zoom system is shown in Fig. 1 . The image plane displacement equation for this system can be stated as follows:

The optical power equation of the zoom system is expressed as:

As part of our analysis, we first consider derivatives of these equations with respect to magnification. We examine the three-group system $({\varphi}_{2},{\varphi}_{3},{\varphi}_{4})$of finite object distance ${e}_{1\text{'}}$, for which the zoom equation is represented by:

Functions of the group magnification ${m}_{i}(i=2,3,4)$ may then be established in order to investigate critical points in the zoom range. The ${m}_{3}$ and ${m}_{4}$ terms are expressed as functions of ${m}_{2}$ as follows:

From Eq. (5) we obtain

Similarly, ${m}_{2}$ and ${m}_{3}$ can be defined in the form:

From which we obtain

Thus the exact solutions to the zoom equations could not be calculated as functions ${m}_{3}=f({m}_{2})$, ${m}_{4}=f({m}_{2})$, ${m}_{2}=f({m}_{4})$ and ${m}_{3}=f({m}_{4})$ have no extreme point in the zoom process. The breakpoints in ${m}_{3}=f({m}_{2})$ and ${m}_{4}=f({m}_{2})$ and the characteristics of the large derivatives near the breakpoint, however, can be used to select the initial points of variables in the system for the convenience of acquiring a higher zoom ratio as well as image shift compensation.

## 3. Derivatives of the zoom equation with respect to optical power

We examine the quick varifocal characteristics of the zoom equations using the optical power of the focal-length-variable groups $({\varphi}_{2},{\varphi}_{4})$. The partial derivatives can be deduced from Eq. (2) as follows,

The Hessian matrix $H({\varphi}_{2},{\varphi}_{4})$ of the zoom equation is negative as calculated from Eq. (11), $\partial Z/\partial {\varphi}_{2}\ne 0$, and the values of $d{\varphi}_{2}/d{\varphi}_{4}$ and ${d}^{2}{\varphi}_{2}/d{\varphi}_{4}{}^{2}$ can be obtained as follows:

The extremum characteristics of the optical power of the focal-length-variable group$v{f}_{1}$ during zooming can then be obtained.

Similarly for $\partial Z/\partial {\varphi}_{4}\ne 0$, the extremum characteristics of the optical power of the focal-length-variable group$v{f}_{2}$are found as follows:

As deduced from Eqs. (11)‒(13), the extremum point does not exist. We can however find the quick varifocal position along the direction of steepest descent since the function is monotonic.

Therefore, the extremum characteristics can be used to analyze the parameters of the fixed group in four-group stabilized zoom systems and determine the initial point of the Gaussian parameters. Equations (2) and (3) lead to obtaining expressions (14) and (15), which can be used to calculate the optical power of the focal-length-variable group as the system focal length $\Phi $changes:

As the function is monotonic, the variation of ${\varphi}_{2}$ and ${\varphi}_{4}$can be obtained as follows:

## 4. Design examples

If the zoom ratio of an optical system is fixed, we can calculate the Gaussian parameters of the zoom system. We can restrict the sign of ${m}_{3}$, i.e. the magnification of the fixed group, to a fixed or changeable value during zooming. The latter exhibits a larger derivative near the breakpoint, as stated in section 2, and is relevant to systems with the inherent characteristic of a quick varifocal position. At the same time, we can make the variation of ${\varphi}_{2}$ and ${\varphi}_{4}$ fixed, to calculate $\Delta {\varphi}_{2}/\Delta \Phi $ and $\Delta {\varphi}_{4}/\Delta (\frac{1}{\Phi})$. Then we optimize the Gaussian parameters of the lens groups, especially the optical power of the fixed group, to calculate and make the values of $\frac{{e}_{4}}{(1-{e}_{1}{\varphi}_{1})\cdot {}^{2}B{}_{4}}$ and $\frac{1-{e}_{1}{\varphi}_{1}}{{e}_{4}\cdot {}^{2}B{}_{4}}$converge to the target values, through the usage of the expressions ${e}_{4}/(1-{e}_{1}{\varphi}_{1})$ and ${}^{2}\text{B}{}_{\text{4}}$. The characteristics of ${m}_{3}$ and the corresponding expressions are illustrated in the design examples for 2:1 and 5:1 stabilized zoom systems.

#### 4.1 Examples for a 2:1 zoom system

Firstly, the optical design of a 2:1 stabilized zoom system is investigated. The zoom range is 5-10mm. The object distance is set to infinity. We use two deformable mirrors as the focal-length-variable groups$({\varphi}_{2},{\varphi}_{4})$, with the optical power range of the deformable mirror being 0.04 $m{m}^{-1}$ [12, 13]. The values of the Gaussian parameters are listed in Table 1 for example 1 and Table 2 for example 2. These show that the sign of ${m}_{3}$ in example 1 is fixed and that it varies in example 2 during zooming, which can also be seen in the relationship between ${m}_{2}$, ${m}_{3}$ and ${m}_{4}$ as shown in Fig. 2 . The optical layout of example 1 is shown in Fig. 3 and that of example 2 in Fig. 4 . The red lines are included to delineate the focal-length-variable groups $({\varphi}_{2},{\varphi}_{4})$. The plots are displayed at the same scale, which shows that the stabilized zoom system with the inherent characteristics of a quick varifocal position can have a miniaturized layout. The value of ${m}_{2}$ varies from a large positive number to a large negative number when the sign of ${m}_{3}$ changes.

#### 4.2 Examples for a 5:1 zoom system

We calculate and propose a design for two 5:1 stabilized zoom systems with the same optical power range of the deformable mirror. The zoom range is 5-25mm and the values of the Gaussian parameters for examples 3 and 4 are list in Tables 3 and 4 respectively. The sign of ${m}_{3}$ is fixed in example 3 and varies in example 4 during zooming. The relationships between ${m}_{2}$, ${m}_{3}$ and ${m}_{4}$ in examples 3 and 4 are shown in Fig. 5 . The optical layout of examples 3 and 4 are plotted in Figs. 6 and 7 respectively. The results demonstrate that the proposed method could be used to design a stabilized zoom system and that the optical layout can be miniaturized as ${m}_{3}$ has a larger derivative near the breakpoint.

The results also prove that the zoom ratio of the stabilized zoom system is increased from two to five when the expressions $\frac{{e}_{4}}{(1-{e}_{1}{\varphi}_{1})\cdot {}^{2}B{}_{4}}$ and $\frac{1-{e}_{1}{\varphi}_{1}}{{e}_{4}\cdot {}^{2}B{}_{4}}$ take smaller values. A larger focal length at the telephoto end is achieved to be able to distinguish the same target in a wider range area as the focal length at the wide angle end is set equal for 2:1 and 5:1 zoom systems. For 2:1 zoom systems, the following can be calculated from Tables 1 and 2: $\left|\frac{{e}_{4}}{(1-{e}_{1}{\varphi}_{1})\cdot {}^{2}B{}_{4}}\right|=0.40$, and $\left|\frac{1-{e}_{1}{\varphi}_{1}}{{e}_{4}\cdot {}^{2}B{}_{4}}\right|=0.008$. As for 5:1 zoom systems, Tables 3 and 4 can be used to calculate: $\left|\frac{{e}_{4}}{(1-{e}_{1}{\varphi}_{1})\cdot {}^{2}B{}_{4}}\right|=0.25$, and $\left|\frac{1-{e}_{1}{\varphi}_{1}}{{e}_{4}\cdot {}^{2}B{}_{4}}\right|=0.002$.

## 5. Conclusion and future work

In this paper, we have presented a novel method for analyzing the paraxial characteristics of four-group stabilized zoom systems with two focal-length-variable lens groups. This technique established the zoom equations through the use of Gaussian brackets. During the zooming process, the optical power of the focal-length-variable elements is inconstant and Gaussian brackets provide a means of understanding the effects of such variations on the system. The first and second derivatives and the Hessian matrix of the zoom equation with respect to the magnification and the optical power were then calculated, to describe the influence of variables on the compensation of the image shift and the sensitivity of the zoom ratio to the zoom variables. Then the critical points can be identified. It was deduced that the extremum point does not exist and the exact solutions to the zoom equations could not be calculated. However, a larger derivative near the breakpoint can be used to select the initial points in the variable space. The quick varifocal position, the direction of steepest descent, could be determined as the function was monotonic. The quick varifocal position was then found using the expressions${m}_{3}$, ${e}_{4}/(1-{e}_{1}{\varphi}_{1})$ and ${}^{2}\text{B}{}_{\text{4}}$. The system exhibits the inherent characteristic of a quick varifocal position when the sign of ${m}_{3}$ varies in the process of zooming. The zoom ratio of the stabilized zoom system can be increased when the expressions $\frac{{e}_{4}}{(1-{e}_{1}{\varphi}_{1})\cdot {}^{2}B{}_{4}}$ and $\frac{1-{e}_{1}{\varphi}_{1}}{{e}_{4}\cdot {}^{2}B{}_{4}}$ take smaller values. The Gaussian parameters of the lens groups, especially the fixed group, can then be calculated and optimized through the use of the expressions of ${e}_{4}/(1-{e}_{1}{\varphi}_{1})$ and ${}^{2}\text{B}{}_{\text{4}}$. The design results indicate that we have successfully achieved a four-group stabilized zoom lens design of 2:1 and 5:1 zoom ratios. This indicates that the deformable mirrors can act as the focal-length-variable group and the stabilized zoom lens of higher zoom ratio can be achieved with the same optical power range as the corresponding expressions take smaller values, as illustrated in section 3. We have also demonstrated that the stabilized zoom system can be miniaturized by showing that the magnification of the fixed group has a larger-value derivative near the breakpoint, with the inherent characteristic of a quick varifocal position. This method could prove to be a valuable tool in the design of future zoom systems. Stabilized zoom systems exhibiting a larger zoom ratio and the realization of the real optical structure of the stabilized zoom system using the focal-length-variable element has yet to be discussed, further improvements on this method could be investigated in future studies.

## Acknowledgments

This research was supported by the grant from the National Natural Science Foundation of China (No. 61275003), Research Fund for the Doctoral Program of Higher Education of China (20101101110016) and Shenzhen Science and Technology Projects (JC201005310719A).

## References and links

**1. **K. Yamaji, “Design of zoom lenses,” in *Progress in Optics*, Vol. **6**, E. Wolf, ed. (North-Holland, 1967), pp.105–170.

**2. **S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett. **85**(7), 1128–1130 (2004). [CrossRef]

**3. **C. Tao, “The varifocal equation for a zoom system,” Kexue Tongbao **22**(Z1), 207–213 (1977).

**4. **T. ChunKan, “Design of zoom system by the varifocal differential equation. I,” Appl. Opt. **31**(13), 2265–2273 (1992). [CrossRef] [PubMed]

**5. **K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. I: Four-component type,” Appl. Opt. **21**(12), 2174–2183 (1982). [CrossRef] [PubMed]

**6. **K. Tanaka, “Paraxial theory in optical design in terms of gaussian brackets,” in *Progress in Optics*, Vol. **23**, E. Wolf, Ed. (North-Holland, Amsterdam, 1986), pp.63–111.

**7. **E. Betensky, “Forty years of modern zoom lens design,” Proc. SPIE 586506 (2005). [CrossRef]

**8. **S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE **6466**, 64660F, 64660F-7 (2007). [CrossRef]

**9. **Q. Hao, X. Cheng, and Y. Song, “Zoom system of MOEMS elements,” PRC Patent 200810119431.4 (18 February 2008).

**10. **P. Valley, M. Reza Dodge, J. Schwiegerling, G. Peyman, and N. Peyghambarian, “Nonmechanical bifocal zoom telescope,” Opt. Lett. **35**(15), 2582–2584 (2010). [CrossRef] [PubMed]

**11. **R. Peng, J. Chen, C. Zhu, and S. Zhuang, “Design of a zoom lens without motorized optical elements,” Opt. Express **15**(11), 6664–6669 (2007). [CrossRef] [PubMed]

**12. **Y. H. Lin, Y. L. Liu, and G. D. Su, “Optical zoom module based on two deformable mirrors for mobile device applications,” Appl. Opt. **51**(11), 1804–1810 (2012). [CrossRef] [PubMed]

**13. **B. M. Kaylor, C. R. Wilson, N. J. Greenfield, P. A. Roos, E. M. Seger, M. J. Moghimi, and D. L. Dickensheets, “Miniature non-mechanical zoom camera using deformable MOEMS mirrors,” Proc. SPIE **8252**, 82520N, 82520N-7 (2012). [CrossRef]

**14. **A. Mikš and J. Novák, “Third-order aberrations of the thin refractive tunable-focus lens,” Opt. Lett. **35**(7), 1031–1033 (2010). [CrossRef] [PubMed]

**15. **A. Miks and J. Novak, “Analysis of two-element zoom systems based on variable power lenses,” Opt. Express **18**(7), 6797–6810 (2010). [CrossRef] [PubMed]

**16. **A. Miks, J. Novak, and P. Novak, “Chromatic aberrations of thin refractive variable-focus lens,” Opt. Commun. **285**(10-11), 2506–2509 (2012). [CrossRef]

**17. **M. Herzberger, “Gaussian optics and Gaussian brackets,” J. Opt. Soc. Am. **33**(12), 651–655 (1943). [CrossRef]