## Abstract

This work performs a paraxial analysis of three-component zoom lens with a fixed position of image-space focal point and a distance between object and image points, which is composed of three tunable-focus elements. Formulas for the calculation of paraxial parameters of such optical systems are derived and the calculation is presented on examples.

© 2014 Optical Society of America

## 1. Introduction

Zoom lenses, which enable to change focal length and magnification of the optical system, find many applications in different areas and various papers [1–23] are dedicated to their optical design. The change of a focal length or a transverse magnification can be achieved by the change of the position of individual elements of the optical system. The position of the image plane is required to be fixed during the change of the focal length in order the image was sharp in the whole range of the focal length change. If it is required that the image should be located at a specific constant distance from the object for a given range of magnification, then the position of the image-space focal point of the classical zoom lens is not fixed and changes its position during the magnification change. A very precise optomechanical design is needed for the movement of individual elements of the classical zoom lens. Individual elements of such a zoom lens do not enable to change its basic optical parameters, for example, its focal length. On the other hand, refractive fluidic lenses with a variable focal length [24–39] became in last years a promising technology for future optical and microoptical systems, especially due to a possibility for a size reduction, lower complexity and costs, better robustness, and faster adjustment of optical parameters of such systems. These lenses can be also used in the zoom lens optical design, where individual elements stay in fixed position and the complicated mechanical design of conventional zoom lenses can be eliminated. The change of the focal length of individual elements can be realized by different physical principles, which are described in detail in [24,25], and we will not deal with them in this work. As it is well known [40–53], paraxial parameters of the optical systems are fundamental for the primary design of any type of optical systems and therefore our work is devoted to this issue. This is important even more for the design of zoom lens systems, because using paraxial parameters of the optical system (i.e. focal lengths, distances between individual elements, f-numbers, etc.) one can deduce how complex the individual elements should be and how complicated will be to correct aberrations and design the mechanical construction of the zoom lens system in further stages of the optical design process. The paraxial analysis also provides an excellent insight into the tolerance sensitivity and manufacturability of the underlying design form.

In recent work [40] the problem of the calculation of paraxial parameters of zoom lenses with a fixed distance between object and image points and a fixed position of the image-space focal point is solved using conventional fix-focus elements, where the change of the magnification of the zoom lens is performed by the change of distances between individual elements of the optical system. The aim of this work is to analyze and calculate paraxial parameters of zoom lenses with a fixed distance between object and image points and a fixed position of the image-space focal point, which are composed of optical tunable-focus elements. Thus, it is an entirely different problem than the method described in [40]. Further, formulas for the calculation of paraxial parameters are derived for such type of zoom lenses. The advantage of these optical systems is the fact that complicated the mechanical design is not needed as in conventional zoom lenses with fix focus elements. Due to the fact that active optical elements with tunable optical parameters [24,25] are commercially available [53,54] and started to be used in the design of optical systems, it is important to analyze properties of different optical systems composed of such optical elements. Our work represents a theoretical contribution to the optical design using active elements with tunable focal length. As far as we know the analysis of such type of the zoom lenses that are composed of optical tunable-focus elements was not published yet.

## 2. Paraxial imaging properties of three-component zoom lens

Consider a three-component zoom lens in air composed of tunable-focus lenses (Fig. 1) with optical powers ${\phi}_{1},\text{\hspace{0.17em}}{\phi}_{2},\text{\hspace{0.17em}}{\phi}_{3}$ and fixed distances *d*_{1}, *d*_{2} between components of the zoom lens. We can write for such zoom lens with thin lenses the following formulas using Gaussian brackets [41,42]

*s*is the distance of the object from the first element of the optical system, ${s}^{\prime}$ is the distance of the image from the last element of the optical system, φ is the optical power, and

*m*is the transverse magnification of the optical system. Consider an optical system presented in Fig. 1, where ξ is the object plane, ξ' is the image plane and

*m*is the transverse magnification of the optical system. The distance $L=\overline{A{A}^{\prime}}$ between planes ξ and ξ' (Fig. 1) can be expressed asand mutual distance $D=\overline{A{F}^{\prime}}$ between points

*A*and ${F}^{\prime}$(image-space focal point) can be expressed aswhere ${s}_{F}$is the position of the object focal point and ${{s}^{\prime}}_{{F}^{\prime}}$ is the position of the image focal point. In further analysis we require that the distances $L=\overline{A{A}^{\prime}}$ and $D=\overline{A{F}^{\prime}}$ remain constant during the change of magnification

*m*.

By substitution of Eqs. (1)-(5) into Eqs. (6) and (7) one obtains after a tedious derivation the following formulas for the calculation of optical powers ${\phi}_{1},\text{\hspace{0.17em}}{\phi}_{2}$ and ${\phi}_{3}$

The next step of the lens calculation is the determination of the parameters (radii of curvature, index of refraction, etc.) of individual elements of the zoom lens in order to minimize aberrations [43–51]. This problem, which is described in detail in [3,44–51], is not the aim of our work and we will not deal with aberration compensation of the zoom lens.

## 3. Example of calculation zoom lens parameters

We present an example of the calculation of basic parameters of the three-component zoom lens with the fixed distance between object and image points and the fixed position of the image-space focal point, where the change of the focal length is performed using the elements with continuously variable focal length. The result of the calculation for different values *m* of the transverse magnification is presented for a variant of the zoom lens in Table 1. Figure 2 presents the graph of optical power ${\phi}_{1},\text{\hspace{0.17em}}{\phi}_{2},\text{\hspace{0.17em}}{\phi}_{3}$ (in diopters) of individual elements of the zoom lens with respect to the transverse magnification *m*, evaluated for the optical system with parameters given in Table 1. As one can see from Fig. 2, the change of optical power is continuous for all three elements. Table 1 also presents incidence heights *h* of the aperture rays and incidence heights $\overline{h}$ of the principal rays [43–47] on individual elements of the zoom lens in dependence on the transverse magnification *m*. The calculation is performed for the image size ${y}^{\prime}=5$ mm. The f-number [43,45–47] of the zoom lens in the object space is *F* = 4 for all values of the transverse magnification $m\in \u3008-5,\text{\hspace{0.17em}}-2\u3009$. For clarity, the calculation is performed for two positions of the entrance pupil. Namely, the first case corresponds to the position identical with the first element of the zoom lens (${\overline{h}}_{1}=0$) and the second case corresponds to the position of the entrance pupil at infinity (${\overline{h}}_{3}^{\infty}={y}^{\prime}{{s}^{\prime}}_{F}/m{f}^{\prime}$), i.e. for the telecentric path of the chief ray in the object space (denoted by the superscript $\infty $). The effective f-number $F$of the individual optical element, i.e. the f-number for a given value of the transverse magnification *m* of the zoom lens can be approximately ($\mathrm{sin}\alpha \approx \mathrm{tan}\alpha $) calculated from the formula: ${F}_{i}=0.5\text{\hspace{0.17em}}{d}_{i}/({h}_{i}-{h}_{i+1})$, where $i=0,1,2,3$ and ${d}_{0}=-s$, ${d}_{3}={s}^{\prime}$, ${h}_{0}={h}_{4}=0$. One obtains two f-numbers for each optical element. The first f-number is given for the space in front of the element and the second one is given for the space behind the element. As one can calculate simply from Table 1, the minimum effective f-number for the first element is ${F}_{1\mathrm{min}}=2.04$, for the second element ${F}_{2\mathrm{min}}=2.04$ and for the third element${F}_{2\mathrm{min}}=3.69$. Thus, these individual elements can be realized in the form of the doublet or triplet [37,49–51] in the further steps of the design process. All linear dimensions in Table 1 are given in millimeters.

## 4. Conclusion

We presented a theoretical analysis of paraxial optical properties of the three-component zoom lens with the fixed distance between object and image points and the fixed position of the image-space focal point, which is composed of three tunable-focus elements. Formulas (Eq. (8) for the calculation of paraxial parameters of such zoom lenses were derived. The procedure of the calculation of parameters of zoom lenses was presented on examples. The derived formulas can be used in the initial stage of the optical design of such zoom lenses with tunable-focus elements. Zoom lenses with the fixed distance between object and image points and the fixed position of image-space focal point may find, for example, their applications in optical systems for information processing [55–57], where it is possible to affect the amplitude, phase and polarization using the spatial filter, which is positioned in the fixed focal plane of the zoom lens. The advantage of the proposed solution, i.e. the zoom lens with the fixed distance between object and image points and the fixed position of the image-space focal point, is the fact that one can affect, for example, Fourier spectra of objects [55–57], which are characterized by the different spatial structure, with just one spatial filter. The position of the image-space focal point does not change with the change of focal length or the transverse magnification of the zoom lens and the position of the investigated object stays also fixed. The problem of influence of aberrations of optical system was not studied in our work. We focused on the calculation of the paraxial parameters of the zoom system because the paraxial parameters are fundamental parameters for design of the any optical systems. Those, which are interested in the influence of aberrations on the transformed beam and design parameters (radii, thickness and refractive indices of glasses) of individual elements from the point of view of the theory of aberrations, can find detailed information, e.g. in Refs [3,44–51].

## Acknowledgment

This work has been supported by Czech Science Foundation from the grant 13-31765S.

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