1. Introduction

Optical systems with variable optical parameters (zoom lenses) play an important role in various types of optical imaging systems such as cameras [1,2]. Optical power, shape and material are fundamental optical parameters of the lens which affects its imaging properties [3–6]. Conventional optical systems with variable optical characteristics are composed of several optical elements and some of these elements are movable [7–19]. A motorized change of position of individual elements (or group of elements) then makes possible to achieve adjustable optical properties of zoom lens systems in the desired range, e.g. focal length or magnification. A disadvantage of such systems is that individual elements of these optical systems have to move very precisely along calculated trajectories, special driving motors must be used to provide the desired precise control over the mechanical positions of individual elements, and finally movements of individual elements must be synchronized, which results in high quality requirements on mechanical construction of such systems [20]. Such traditional zoom lens systems are complicated, and thus expensive. In addition, it is inconvenient to miniaturize such classical zoom lens systems, which is needed in various modern applications, such as in mobile phone cameras. Today’s technology puts more stringent requirements on image quality and functionality of modern systems which use zoom lenses. There is the need for simpler, smaller, lighter, and more compact optical devices with a fast zoom. Classical zoom lens systems need several lens groups to move back and forth to adjust the magnification. It is sometimes very difficult to move lenses in a traditional zoom lens system to obtain a desired magnification of an image, especially in very small imaging devices. Even for larger devices such as cameras, camcorders, etc., an alternative to mechanical motion would be beneficial by reducing the complexity of zoom systems.

A novel design of zoom lens systems without moving parts should be promising for future zoom lens systems with respect to a lower complexity and costs, a possibility for miniaturization, better robustness and a faster adjustment of optical parameters of zoom systems. These novel zoom lens systems can be based on active optical elements with tunable optical parameters such as the focal length of a lens. A fundamental change in the optical design of zoom lens systems can occur if one considers adjustable lenses which alter their shape or refractive index distribution to produce a continuous change of the focal length, without translational movement of lenses. Such lenses with a tunable focal length in a wide range make possible to design optical systems with functions that are difficult to combine using conventional approaches. The development of tunable-focus lenses is of great importance for a number of practical applications, ranging from adaptive eyeglasses for vision correction [21] to fast non-mechanical zooming devices in various cameras, camcorders, and mobile phones [22–25]. Different types of tunable lenses with variable focal lengths were developed in recent years and some of them are offered commercially today [25,26]. The technology of variable power lenses is inspired with the principle of the human eye. Several different approaches were developed for controlling the focal length of lenses. Variable power lenses can use the principle of voltage-controlled liquid crystals as active optical elements [27,28], the controlled injection of fluid into chambers with deformable membranes [29,30], thermooptical or electroactive polymers [31], and electrowetting [32–34]. The tunability of lens parameters provides an additional degree of freedom in the optical design process allowing a significant simplification of many existing zoom systems.

In this paper, we demonstrate that the characteristics of lenses with a variable focal length enable new designs for zoom lens systems by eliminating the mechanical movements required in conventional systems. Such systems are attractive for various applications due to their focal length tunability, small size, and low cost. Some authors published results about zoom lens systems with variable power lenses and their analysis [22,24,28,29,34–38]. Our work deals with the paraxial and third order aberration analysis of possible designs of zoom systems based on variable power lenses and it is more complex than analyses implemented in Refs [35–38]. First order chromatic aberrations of lenses with a variable focal length are also analyzed. It is shown a possible optical design of zoom systems without changing relative lens location which use variable power lenses on an example of the two-element optical zoom system. Detailed calculations and simulation examples were carried out showing that such zoom lens systems appear promising as the next-generation zoom lenses which may reduce costs, size and weight and which may enhance the overall system performance.

2. Two-element zoom system

Two-element optical systems with variable optical characteristics are frequently used in practice. These are usually subsystems of more complex optical systems where they serve as optical elements for providing continuous changes of optical characteristics of the whole systems, e.g. focal length or magnification. Consider a two-element optical system (Fig. 1 ), where the power of the first and second element are denoted as φ₁ and φ₂. Moreover, d is the distance of the object principal plane of the second element from the image principal plane of the first element, and $n_i$ (i = 1,2,3) denotes values of indices of refraction of i-th optical media. The meaning of other symbols is evident form Fig. 1. The following formulas hold for raytracing a paraxial ray through the optical system which consists of N elements (e.g. lenses) [4,5]

 

Fig. 1 Two-element optical system

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It is most frequent in practice that individual optical elements of the optical system are situated in air, where $n_1=n_2=n_3=1$. It holds for the two-element optical system in air [1–6]

Furthermore, we can write for the distance L between the object and its image

By solving the previous quadratic equation we obtain

As one can see from Eq. (5) a real solution of Eq. (4) does not exist in general. We can also express the distances a ₁ and a ₂´ using L* as [5]

In the case of two-element optical system using a thin lens approximation we can set ${\Delta }_{H_1}=0$, ${\Delta }_{H_2}=0$ and L* = L. Previous relations are used for calculations of classical two-element optical systems with a variable magnification. Given focal length values${f^{\prime }}_1$,${f^{\prime }}_2$ and the distance L we can calculate the distance d of both elements for different values of transverse magnification m from Eq. (5). Distances $a_1$ and ${a^{\prime }}_2$ are then calculated from Eq. (2) or Eq. (6).

In case that both elements of the optical system are characterized by a variable focal length, then the calculations are different from the conventional systems with variable magnification. From given values $a_1$, ${a^{\prime }}_2$ and d one have to determine the focal length values ${f^{\prime }}_1$ and ${f^{\prime }}_2$ in dependence on the transverse magnification m of the optical system. Using Eq. (1) and Eq. (2) we can derive the following equation for the power ${\varphi }_2$ of the second lens

We can express the discriminant Δ of the quadratic Eq. (7) as

One can see that the discriminant Δ is positive and thus it always exists a real solution of the problem. The roots of Eq. (7) are given by the following formulas. The first root can be expressed as

We can see from the previous formula that the power ${\varphi }_2$ of the second element is a linear function of the transverse magnification m of the optical system. The second root is given by ${\varphi }_2=(1-1/m)/d$. We can simply verify by substitution of the root into Eq. (2) that $a_1=0$ and the second root does not satisfy initial requirements of the solved problem. The power ${\varphi }_1$ of the first element of the optical system can be calculated from

Equations (8) and (9) represent the solution of our problem for the case of imaging the point situated at a finite distance from the first element of the optical system.

Consider now the case of imaging the point at infinity by the two-element system with variable focal length $f^{\prime }$, which consists of elements with continuously tunable focal length. Values ${a^{\prime }}_2$ and d are constant and the focal length changes within a chosen range $f^{\prime }\in 〈{f^{\prime }}_{\min },{f^{\prime }}_{\max }〉$. We have to calculate values of focal length ${f^{\prime }}_1$ and ${f^{\prime }}_2$ in dependence on the focal length $f^{\prime }$of the optical system. Using Eq. (1) we obtain$\varphi ={\varphi }_1+{\varphi }_2-d{\varphi }_1{\varphi }_2$ and ${a^{\prime }}_2\varphi =\left(1-{\varphi }_{1}d\right)$. Powers can be calculated by solving either previous relations or Eqs. (8) and (9) ($\underset{a_1\to \infty }{\lim }(a_{1}m)=f^{\prime }$)

Equation (10) enables to determine the optical power of both elements of the two-element optical system with a variable focal length in the case of imaging a point at infinity.

3. Third order aberrations of variable power lenses

Consider a rotationally symmetric optical system (Fig. 2 ) that consists of spherical lenses. In case we know radii of curvature of lenses, their thicknesses, indices of refraction and distances between individual lenses we can calculate aberration coefficients of the third order. Firstly, we calculate two paraxial (auxiliary) rays through the optical system, namely the paraxial marginal ray and the paraxial chief ray. The following relations hold for raytracing the paraxial marginal ray through the optical system having K surfaces [4–6]

 

Fig. 2 Principal and aperture paraxial rays

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The following relations are valid for raytracing the paraxial chief ray through the optical system having K surfaces

Considering $(x_P,y_P)$ as coordinates of the intersection of the ray with the entrance pupil plane, s ₁ as the distance of the object plane from the first surface of the optical system, ${\overline{s}}_1$ as the distance of the entrance pupil from the first surface of the optical system, and $y_0$ as the image height, then transversal ray aberrations $\delta x^{\prime }$ , $\delta y^{\prime }$of the third order of the rotationally symmetric optical system can be calculated from [2–5,9,10]

In previous relations we denoted $\Delta {\sigma }_i={{\sigma }^{\prime }}_i-{\sigma }_i={\sigma }_{i+1}-{\sigma }_i$ and similarly for other differences. Furthermore, it holds that ${\overline{\sigma }}_1={\overline{h}}_1/{\overline{s}}_1=y_0/({\overline{s}}_1-s_1)$. Individual aberration coefficients of the third order have the following meaning: S_I is the coefficient of spherical aberration, S_II is the coefficient of coma, S_III is the coefficient of astigmatism, S_IV is the Petzval coefficient, and S_V is the coefficient of distortion. The quantity H is the Lagrange-Helmholtz invariant. One may use an arbitrary choice of input parameters $(h_1,{\sigma }_1=h_1/s_1,{\overline{h}}_1,{\overline{\sigma }}_1={\overline{h}}_1/{\overline{s}}_1)$ for the calculation of aberration coefficients of the third order for a given object distance $s_1$ and a position ${\overline{s}}_1$ of the entrance pupil. We can choose for simplification, e.g. $h_1=1$, ${\overline{h}}_1=1$, ${\sigma }_1=1/s_1$, ${\overline{\sigma }}_1=1/{\overline{s}}_1$, and we trace both paraxial rays through the optical system by Eqs. (11) and (12). Then we multiply obtained aperture angles ${\sigma }_i$ by $s_1$ and all obtained angles ${\overline{\sigma }}_i$ of the principal ray by ${\overline{s}}_1$. By this normalization we get the input parameters $h_1=1$, ${\sigma }_1=1,$ ${\overline{h}}_1=1,$ ${\overline{\sigma }}_1=1$, and H = 0. We can determine aberration coefficients using Eq. (14) and Eqs. (13) are simplified for practical calculation.

We can apply above-mentioned formulas on an optical element (lens) with a variable focal length which consists of three surfaces. The outer surfaces are planar ($r_1=r_3=\infty$) and the inner surface is a spherical surface with the radius $r_2$ which can be changed in a continuous way. It is the case of variable power liquid lenses based on electrowetting phenomena, in which an electrically induced change in surface-tension changes the surface curvature of liquid [25]. Figure 3 presents an optical scheme of such lens. Using Eq. (14) we obtain for aberration coefficients of the thin lens in air (d ₁ = 0, d ₂ = 0, n ₁ = 1, n ₄ = 1) the following formulas

 

Fig. 3 Simple variable power lens

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As one can see from relations (15), (16), and (17) we expressed the third order aberration coefficients using parameters A and B, which depend only on refractive indices of fluids forming the variable power lens and do not depend on the optical power ϕ of the lens.

We can write for the entering paraxial aperture angle $\sigma ={\sigma }_1$ and the exiting paraxial aperture angle ${\sigma }^{\prime }={\sigma }_4$ (Fig. 3) using Eq. (11) ${\sigma }^{\prime }-\sigma =h\varphi$, where $\varphi =(n_3-n_2)/r_2$ is the power of the lens. Moreover, we can determine aberrations of the third order of an arbitrary system of lenses using (13),(15),(16), and (17). We obtain for a system of K elements

The provided analysis may serve for the initial design of zoom optical systems, whose parameters can be used for its further optimization using optical design software.

4. First order chromatic aberrations of variable power lenses

Consider now chromatic aberrations of variable power lenses. We obtain for the longitudinal chromatic aberration $\text{d}{s}^{\prime }$

We can express the distance s' for thin lens (d ₁ = 0, d ₂ = 0) using Eq. (11) by the formula

The longitudinal chromatic aberration can be calculated by differentiation of Eq. (20) and substitution into Eq. (19)

5. Examples of two-element zoom systems with variable power lenses

A detailed theoretical analysis on two-element zoom lens systems, which use variable-focus lenses, was performed and computer simulation examples and detailed calculations were carried out on the basis of the described analysis. Such zoom systems are a promising alternative to the conventional zoom lens system.

We will show an example of a possible design of the two-element zoom lens system with a variable focal length of individual adjustable lenses and a continuous change of transverse magnification. We consider individual elements as thin lenses and we choose the following indices of refraction: $n_2=1.3$ and $n_3=1.6$.

Example 1

The two-element zoom lens, which uses two lenses with a variable focal length, is described by the following parameters: $a_1=-150\text{\hspace{0.17em} mm}$, ${a^{\prime }}_2=300\text{\hspace{0.17em} mm}$, $d=100\text{\hspace{0.17em} mm}$, $m\in 〈-4,-1〉$. We calculated the dependence of the optical power of individual tunable lenses on the transverse magnification of the zoom lens system. This dependence is shown in Fig. 4 . Table 1 presents optical parameters of the zoom lens system (focal length of individual lenses and the whole zoom system in millimeters).One can use variable power lenses such as lenses ARCTIC 416 SL V3 from Varioptics [25], which make possible a change of power in the range from – 10 dpt to + 20 dpt.

 

Fig. 4 Dependence of power on transverse magnification (example 1)

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Table 1. Parameters of zoom lens system

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Example 2

The second example is the two-element zoom lens with the following parameters: $a_1=s_1=-150\text{\hspace{0.17em} mm}$, ${a^{\prime }}_2={s^{\prime }}_6=15\text{\hspace{0.17em} mm}$, $d=d_3=60\text{\hspace{0.17em} mm}$, $m\in 〈-0.6,-0.15〉$, $p_c=60\text{\hspace{0.17em} mm}$. The numerical aperture is $\text{N .A . }=-0.02$, and $p_c$ is the distance of the aperture stop from the first element of the optical system. The dependence of the power of individual tunable lenses on the transverse magnification of the zoom lens system is shown in Fig. 5 $\left(h_1=s_{1}m,{\sigma }_1=m,{\overline{h}}_1={\overline{s}}_1\overline{m},{\overline{\sigma }}_1=\overline{m}\right).$ Table 2 presents numerical values of the transverse magnification in pupils $\overline{m}$, focal lengths ${f^{\prime }}_1$, ${f^{\prime }}_2$, ${f^{\prime }}_{}$, and the position of the entrance pupil ${\overline{s}}_1$.

 

Fig. 5 Dependence of power values on transverse magnification (example 2)

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Seidel aberration coefficients of the third order$S_I$,…,$S_V$, and gyration radius $r_g={\displaystyle \sum (\text{d}{x}^2+\text{d}{y}^2)/M}$ (dx,dy are transverse ray aberrations, M is the number of rays) corresponding to the transverse magnification m of the optical system are presented in Table 3 .

Table 3. Aberration coefficients of optical system

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Table 2. Parameters of optical system

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Seidel aberration coefficients are calculated for input parameters: $h_1=s_{1}m$, ${\sigma }_1=m$, ${\overline{h}}_1={\overline{s}}_1\overline{m}$,

Figure 6 shows spot diagrams of the optical system for different values of the transverse magnification m and imaging of the axial object point.

 

Fig. 6 Spot diagrams for imaging of axial point in dependence on transverse magnification m (thin lens system). Transverse ray aberrations dx, dy are given in mm, numerical aperture is N.A. = - 0.02.

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Previous results were calculated for thin lenses (d ₁ = d ₂ = d ₄ = d ₅ = 0, d ₃ = d). We also considered elements with a finite thickness in our analysis (d ₁ = d ₂ = d ₄ = d ₅ = 0.5 mm, d ₃ = d) and we can conclude that a consideration of elements with a finite thickness does not mean almost any changes in spot diagrams. Thus, an analysis using thin lenses is satisfactory for the initial optical design of the optical system.

6. Summary

The work describes a method of calculation of paraxial parameters of the optical system with variable optical characteristics. Equations are given both for the calculation of parameters of a classical two-element optical system with variable optical characteristics, where we determine the mutual position of individual lenses on the transverse magnification, and for the calculation of parameters of the optical system with variable power lenses, where we determine values of the focal length of individual elements with respect to the transverse magnification. Mentioned equations are valid both for finite and infinite distances of the object from the optical system. Moreover, formulas for a complex analysis of monochromatic aberrations of optical systems with variable power elements were described. Chromatic aberrations can be calculated using formulas (21) and (22). The described method was demonstrated on examples of zoom lens systems composed of two lenses of a variable focal length. Computer simulation examples show that variable-focus lenses are suitable to be used in modern zoom lens systems and one can achieve optical designs without any mechanical movement of lenses. The analysis provided in this work may serve for the initial design of optical systems with variable optical characteristics. The calculated parameters can be used as initial values in the optimization process with the optical design software.