Four-group stabilized zoom lens design of two focal-length-variable elements

Qun Hao; Xuemin Cheng; Ke Du

doi:10.1364/OE.21.007758

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:

\begin{matrix} {}^{i}A_{j} = [ϕ_{i}, - e_{i}, ϕ_{i + 1}, - e_{i + 1}, \cdot \cdot \cdot, ϕ_{j - 1}, - e_{j - 1}], \\ {}^{i}B_{j} = [- e_{i}, ϕ_{i + 1}, - e_{i + 1}, \cdot \cdot \cdot, ϕ_{j - 1}, - e_{j - 1}], \\ {}^{i}C_{j} = [ϕ_{i}, - e_{i}, ϕ_{i + 1}, - e_{i + 1}, \cdot \cdot \cdot, ϕ_{j - 1}, - e_{j - 1}, ϕ_{j}], \\ {}^{i}D_{j} = [- e_{i}, ϕ_{i + 1}, - e_{i + 1}, \cdot \cdot \cdot, ϕ_{j - 1}, - e_{j - 1}, ϕ_{j}], \end{matrix}

where

{}^{i}A_{j}

,

{}^{i}B_{j}

,

{}^{i}C_{j}

,

{}^{i}D_{j}

are the Gaussian constants between groups

i

and

j

(as described in the Gauss bracket method). The terms

ϕ_{i}

and

ϕ_{j}

(i < j)

are the optical power of groups

i

and

j

, while

e_{i}

and

e_{j}

are the separations between the consecutive principle planes in each group.

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:

Fig. 1 Optical layout of four-group stabilized zoom system.

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Z = {}^{1}A_{5} = [ϕ_{1}, - e_{1}, ϕ_{2}, - e_{2}, ϕ_{3}, - e_{3}, ϕ_{4}, - e_{4}] = 0.

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

{}^{1}C_{4} = [ϕ_{1}, - e_{1}, ϕ_{2}, - e_{2}, ϕ_{3}, - e_{3}, ϕ_{4}] = Φ,

in which

Φ

is the optical power of the zoom system measured at any zoom position,

ϕ_{1}

and

ϕ_{3}

are the optical power of the fixed groups,

ϕ_{2}

and

ϕ_{4}

are the optical power of the stabilized focal-length-variable groups, and

e_{4}

is the distance between the last group and the image plane. The object distance is set to infinity.

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

Z = {}^{1'}B_{5} = [- e_{1'}, ϕ_{2}, - e_{2}, ϕ_{3}, - e_{3}, ϕ_{4}, - e_{4}] = 0.

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:

{\begin{cases} m_{2} = 1 / (1 - e_{1'} ϕ_{2}), \\ m_{3} = 1 / (1 - (m_{2} e_{1'} + e_{2}) ϕ_{3}), \\ m_{4} = - e_{4} / (m_{3} (m_{2} e_{1'} + e_{2}) + e_{3}) . \end{cases}

From Eq. (5) we obtain

d m_{3} / d m_{2} = e_{1'} ϕ_{3} / {[1 - (m_{2} e_{1'} + e_{2}) ϕ_{3}]}^{2},

d m_{4} / d m_{2} = e_{1'} e_{4} / {[{}^{2}B_{4} - m_{2} e_{1'} (1 - e_{3} ϕ_{3})]}^{2},

where

e_{1'}

,

ϕ_{3}

and

e_{4}

are non-zero system parameters in the zooming process, such that the gradients

d m_{3} / d m_{2}

and

d m_{4} / d m_{2}

appear as non-zero with their signs unchanged, therefore no extreme point exists for the functions

m_{3} = f (m_{2})

and

m_{4} = f (m_{2})

. The functions

m_{3} = f (m_{2})

and

m_{4} = f (m_{2})

do however have breakpoints as expressed in Eq. (5) and the functions are monotonic within each interval. . For example, we have the break point

e_{1'} ϕ_{2} = 1

where

m_{2} \to \propto

, and

m_{3} \to 0

. And the gradient

d m_{2} / d m_{3} \to \propto

as

m_{3} \to 0

, as derived from Eq. (6), which shows that there is a significant increase in the values of derivatives near the breakpoint.

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

{\begin{cases} m_{4} = 1 - e_{4} ϕ_{4}, \\ m_{3} = 1 - e_{3} m_{3} - e_{4} ϕ_{3} / m_{4}, \\ m_{2} = - (e_{3} m_{4} + e_{4} + e_{2} m_{3} m_{4}) / (m_{3} m_{4} e_{1'}) . \end{cases}

From which we obtain

d m_{2} / d m_{4} = e_{3} / {e_{1'} {[m_{4} (1 - e_{3} ϕ_{3}) - e_{4} ϕ_{3}]}^{2}},

d m_{3} / d m_{4} = e_{4} ϕ_{3} / m_{4}^{2},

where

e_{3}

,

e_{4}

, and

ϕ_{3}

are non-zero system parameters in the zooming process, such that the gradients

d m_{2} / d m_{4}

and

d m_{3} / d m_{4}

also appear as non-zero with their signs unchanged, which shows that

m_{2} = f (m_{4})

and

m_{3} = f (m_{4})

have no extreme point since is used as the variable.

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 $(ϕ_{2}, ϕ_{4})$ . The partial derivatives can be deduced from Eq. (2) as follows,

{\begin{matrix} \frac{\partial Z}{\partial ϕ_{2}} = {}^{1}A_{2} \cdot {}^{2}B_{5}, \\ \frac{\partial Z}{\partial ϕ_{4}} = {}^{1}A_{4} \cdot {}^{4}B_{5}, \\ \frac{\partial^{2} Z}{\partial ϕ_{2}^{2}} = 0, \\ \begin{array}{l} \frac{\partial^{2} Z}{\partial ϕ_{4}^{2}} = 0, \\ \frac{\partial^{2} Z}{\partial ϕ_{2} \partial ϕ_{4}} = {}^{1}A_{2} \cdot {}^{2}B_{4} \cdot {}^{4}B_{5} . \end{array} \end{matrix}

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

{\begin{matrix} \frac{d ϕ_{2}}{d ϕ_{4}} = - \frac{{}^{1}A_{4} \cdot {}^{4}B_{5}}{{}^{1}A_{2} \cdot {}^{2}B_{5}}, \\ \frac{d^{2} ϕ_{2}}{d ϕ_{4}^{2}} = 0. \end{matrix}

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

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

{\begin{matrix} \frac{d ϕ_{4}}{d ϕ_{2}} = - \frac{{}^{1}A_{2} \cdot {}^{2}B_{5}}{{}^{1}A_{4} \cdot {}^{4}B_{5}}, \\ \frac{d^{2} ϕ_{4}}{d ϕ_{2}^{2}} = 0. \end{matrix}

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 $Φ$ changes:

ϕ_{2} = \frac{e_{4} Φ - [ϕ_{1}, - (e_{1} + e_{2}), ϕ_{3}, - e_{3}]}{(1 - e_{1} ϕ_{1}) \cdot {}^{2}B_{4}},

ϕ_{4} = \frac{1 - e_{1} ϕ_{1}}{e_{4} Φ \cdot {}^{2}B_{4}} + \frac{{}^{2}B_{4} - e_{4} [- e_{2}, ϕ_{3}]}{e_{4} \cdot {}^{2}B_{4}} .

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

Δ ϕ_{2} = \frac{e_{4}}{(1 - e_{1} ϕ_{1}) \cdot {}^{2}B_{4}} Δ Φ,

Δ ϕ_{4} = \frac{1 - e_{1} ϕ_{1}}{e_{4} \cdot {}^{2}B_{4}} Δ (\frac{1}{Φ}),

where

Δ Φ = Φ_{s} - Φ_{l}

is the variation of the system optical power,

Δ (\frac{1}{Φ}) = \frac{1}{Φ_{s}} - \frac{1}{Φ_{l}}

,

Φ_{s}

and

Φ_{l}

represent the system optical power at the wide-angle end and the telephoto end respectively,

Δ ϕ_{2}

and

Δ ϕ_{4}

are the variation needed to achieve a given zoom range

\frac{1}{Φ_{s}} ~ \frac{1}{Φ_{l}}

. The first derivatives of the system focal length with respect to the optical power will be larger as the expressions

\frac{e_{4}}{(1 - e_{1} ϕ_{1}) \cdot {}^{2}B_{4}}

and

\frac{1 - e_{1} ϕ_{1}}{e_{4} \cdot {}^{2}B_{4}}

take smaller values. Therefore we can calculate the optimal parameters of the fixed groups to satisfy the specific value of the sensitivity of the system focal length change to variations in optical power of the focal-length-variable lens groups using the expressions

e_{4} / (1 - e_{1} ϕ_{1})

and

{}^{2}B_{4} = [- e_{2}, ϕ_{3}, - e_{3}]

.

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 $ϕ_{2}$ and $ϕ_{4}$ fixed, to calculate $Δ ϕ_{2} / Δ Φ$ and $Δ ϕ_{4} / Δ (\frac{1}{Φ})$ . 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 $\frac{e_{4}}{(1 - e_{1} ϕ_{1}) \cdot {}^{2}B_{4}}$ and $\frac{1 - e_{1} ϕ_{1}}{e_{4} \cdot {}^{2}B_{4}}$ converge to the target values, through the usage of the expressions $e_{4} / (1 - e_{1} ϕ_{1})$ and ${}^{2}B_{4}$ . 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 $(ϕ_{2}, ϕ_{4})$ , 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 $(ϕ_{2}, ϕ_{4})$ . 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.

Table 1. Gaussian parameters of 2:1 zoom system for example 1 ^a

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Table 2. Gaussian parameters of 2:1 zoom system for example 2 ^b

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Fig. 2 The relationships of $m_{2}$ , $m_{3}$ and $m_{4}$ in 2:1 zoom systems: (a) for example 1 and (b) for example 2.

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Fig. 3 The layout of 2:1 zoom system: the tele-angle (a) and wide-angle (b) positions for example 1.

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Fig. 4 The layout of 2:1 zoom system: the tele-angle (a) and wide-angle (b) positions for example 2.

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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.

Table 3. Gaussian parameters of 5:1 zoom system for example 3^c

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Table 4. Gaussian parameters of 5:1 zoom system for example 4^d

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Fig. 5 The relationships of $m_{2}$ , $m_{3}$ and $m_{4}$ in 5:1 zoom systems: (a) for example 3 and (b) for example 4.

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Fig. 6 The layout of 5:1 zoom system: the tele-angle (a) and wide-angle (b) positions for example 3.

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Fig. 7 The layout of 5:1 zoom system: the tele-angle (a) and wide-angle (b) positions for example 4.

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The results also prove that the zoom ratio of the stabilized zoom system is increased from two to five when the expressions $\frac{e_{4}}{(1 - e_{1} ϕ_{1}) \cdot {}^{2}B_{4}}$ and $\frac{1 - e_{1} ϕ_{1}}{e_{4} \cdot {}^{2}B_{4}}$ 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: $| \frac{e_{4}}{(1 - e_{1} ϕ_{1}) \cdot {}^{2}B_{4}} | = 0.40$ , and $| \frac{1 - e_{1} ϕ_{1}}{e_{4} \cdot {}^{2}B_{4}} | = 0.008$ . As for 5:1 zoom systems, Tables 3 and 4 can be used to calculate: $| \frac{e_{4}}{(1 - e_{1} ϕ_{1}) \cdot {}^{2}B_{4}} | = 0.25$ , and $| \frac{1 - e_{1} ϕ_{1}}{e_{4} \cdot {}^{2}B_{4}} | = 0.002$ .

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 $m_{3}$ , $e_{4} / (1 - e_{1} ϕ_{1})$ and ${}^{2}B_{4}$ . 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 $\frac{e_{4}}{(1 - e_{1} ϕ_{1}) \cdot {}^{2}B_{4}}$ and $\frac{1 - e_{1} ϕ_{1}}{e_{4} \cdot {}^{2}B_{4}}$ 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 $e_{4} / (1 - e_{1} ϕ_{1})$ and ${}^{2}B_{4}$ . 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

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	$1 / Φ$ ( $m m$ )	$ϕ_{2} (m m^{- 1})$	$ϕ_{4} (m m^{- 1})$	$m_{2}$	$m_{3}$	$m_{4}$
Wide	5	−0.04	0.04	0.421	−0.574	0.680
Middle	7.14	−0.016	0.023	0.645	−0.445	0.817
Tele	10	0.0	0.0	1.00	−0.328	1.0

	$1 / Φ$ （ $m m$ ）	$ϕ_{2} (m m^{- 1})$	$ϕ_{4} (m m^{- 1})$	$m_{2}$	$m_{3}$	$m_{4}$
Wide	5	0.00	0.00	1.00	-0.164	1.000
Middle	7.14	0.023	-0.017	5.142	-0.040	1.132
Tele	10	0.04	-0.04	-2.673	0.094	1.320

	$1 / Φ$ ( $m m$ )	$ϕ_{2} (m m^{- 1})$	$ϕ_{4} (m m^{- 1})$	$m_{2}$	$m_{3}$	$m_{4}$
Wide	5	−0.04	0.04	0.064	−0.395	0.547
Middle	11.18	−0.012	0.028	0.181	−0.249	0.687
Tele	25	0.0	0.0	1.0	−0.069	1.0

	$1 / Φ$ ( $m m$ )	$ϕ_{2} (m m^{- 1})$	$ϕ_{4} (m m^{- 1})$	$m_{2}$	$m_{3}$	$m_{4}$
Wide	5	−0.04	0.04	−0.046	−0.341	0.556
Middle	11.18	−0.012	0.028	−0.166	−0.170	0.693
Tele	25	0.0	0.0	1.000	0.044	1.000

	$1 / Φ$ ( $m m$ )	$ϕ_{2} (m m^{- 1})$	$ϕ_{4} (m m^{- 1})$	$m_{2}$	$m_{3}$	$m_{4}$
Wide	5	−0.04	0.04	0.421	−0.574	0.680
Middle	7.14	−0.016	0.023	0.645	−0.445	0.817
Tele	10	0.0	0.0	1.00	−0.328	1.0

Four-group stabilized zoom lens design of two focal-length-variable elements

Abstract

1. Introduction

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

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

4. Design examples

4.1 Examples for a 2:1 zoom system

4.2 Examples for a 5:1 zoom system

5. Conclusion and future work

Acknowledgments

References and links

Cited By

Figures (7)

Tables (4)

Equations (17)

Optics Express