## Abstract

The paper presents paraxial analysis of special types of zoom lenses, which are composed of four members with variable focal length. The position of optical center of these systems is required to be fixed for a given value of focal length (i.e., the position of the optical center does not depend on object distance for given value of focal length of the zoom). The formulas that enable the calculation of the optical powers of individual members of such a zoom lens are derived from--and the practical application of the derived formulas is demonstrated with--an example. Such optical systems represent a completely new family of zoom optical systems with applications in measuring systems in photogrammetry, computer vision, triangulation sensors, fringe projection techniques, surveying, machine vision, and so forth.

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

## 1. Introduction

The key problem in various optical imaging and measurement methods is to estimate 3D information from detected 2D camera image coordinates [1–4]. One has to know accurately the relationship between the 2D image coordinates and the 3D object coordinates. This relationship can be determined using various geometric camera calibration methods [1–7], where the unknown parameters of the camera model are estimated. The camera model, which is used frequently in various machine and computer vision methods and 3D metrology systems, is the so called pinhole camera model [1–4]. The concept of an optical center [1–4] is used in various methods dedicated to the calibration of optical imaging and measuring systems in photogrammetry, computer vision, triangulation sensors, fringe projection techniques, surveying, machine vision, etc. The optical center (camera center) is considered to be an intersection of a central ray of a homocentric ray bundle with the optical axis of the optical system of the camera lens. It is assumed that the line from an arbitrary object point, passing through the optical center of the optical system of the camera lens, intersects the image plane in the point, which is identical with an image of a point formed by an optical system using the lens equation. The optical center is one of the basic parameters, which is used in camera calibration techniques. Several methods were used for estimating the optical center of real camera systems [8–11]. The problem of paraxial analysis of position of optical center of optical system with fixed optical properties was analyzed in detail in our previous paper [12] and the information on the influence of aberrations on imaging quality of optical systems can be found in [13–18].

It was shown in [12] that the optical center of given optical system does not depend on the axial distance of the object from given optical system in case that the distance between the principal planes of the optical system is equal to zero (i.e. both principal planes coincide). The aim of our current paper is to extend the results of the paraxial analysis of optical systems with zero separation of principal planes, which were given in our previous paper [12], to zoom optical systems [19-25], which are composed of active optical elements with variable focal length (tunable focus lenses). Due to the fact that such active optical elements are already commercially available it would be possible to realize such optical systems in practice [26-31]. The main advantage of application of active optical elements with variable focal length in zoom systems is a substantial simplification of their optomechanical construction since there is no need for mutual movement of individual members of the zoom system during the change of its focal length. The mutual positions of individual optical elements of the zoom system are fixed and the change of its focal length is achieved by change of optical powers of individual optical elements. To our knowledge the problem of paraxial analysis of zoom systems with fixed position of optical center for given focal length was not investigated yet and there exist no references dealing with this problem in optical literature. Our work presented in this paper is therefore a theoretical contribution to the new area of these special types of zoom lens systems. Such optical systems create a completely new family of zoom optical systems with applications in measuring systems in photogrammetry, computer vision, triangulation sensors, fringe projection techniques, surveying, machine vision etc.

## 2. Paraxial properties of optical systems

As it is well-known from the geometrical optics theory every optical system can be characterized by its focal length ${f}^{\prime}$, the position of object focal point ${s}_{F}$ and the position of image focal point ${{s}^{\prime}}_{{F}^{\prime}}$. The optical power $\varphi $ of given optical system is given by relation $\varphi ={n}^{\prime}/{f}^{\prime}$where ${n}^{\prime}$ is the refractive index of image space. Assuming that both object and image space media is air ($n={n}^{\prime}=1$) and the optical system is composed of *N* thin lenses one can define Gaussian brackets [15,16]

*i*-th lens and ${d}_{i}$ is the axial distance between

*i*+ 1th and

*i*-th lens. The following rules hold for the calculation of the Gaussian brackets [15,16]in Eq. (1):

*s*is the axial distance of the object from the first element of the thin lens system, ${s}^{\prime}$ is the axial distance of the image from the last element of the thin lens system, ${s}_{P}$ is the axial distance of the object (front) principal point from the first element of the thin lens system, ${{s}^{\prime}}_{{P}^{\prime}}$ is the axial distance of the image (back) principal point from the last element of the thin lens system and

*m*is the transverse magnification of the system.

## 3. Four-component zoom lens with fixed position of optical center composed of members with variable focal length

Let us now deal with the analysis of paraxial properties of four-component zoom lens with fixed position of optical center (Fig. 1) composed of members with variable focal lengths. By the term “fixed position of optical center” we will understand that the position of the optical center of the optical system at a given focal length of the optical system will not depend on the distance of the object from the optical system.

Suppose now that we require the optical system to meet the following conditions:

- 1) continuously variable optical power $\varphi $ (in a given range),
- 2) zero separation of principal planes (coincident principal planes),
- 3) fixed value of position of image focal point ${{s}^{\prime}}_{{F}^{\prime}}$,
- 4) corrected Petzval sum ${S}_{IV}\approx 0$.

As it is obvious such optical system must have four free design parameters in order to satisfy those four requirements. The optical system based on individual members with a continuously variable focal length that have fixed positions during the operation of the zoom must therefore be composed of four members with optical powers ${\varphi}_{1},\text{\hspace{0.17em}}{\varphi}_{2},\text{\hspace{0.17em}}{\varphi}_{3},{\varphi}_{4}$(free parameters) with fixed value of mutual axial separations *d*_{1}, *d*_{2}, *d*_{3} (fixed parameters).

Using Eqs. (1)-(3), one obtains the following relations for the four-component optical system

The meaning of individual symbols in Fig. 1 is the following: *ξ* is the object plane (i.e. the plane where the imaged object is located), *ξ'* is the image plane (i.e. the plane where the image of the object created by the lens is located), F is the object focal point of the system, F' is the image focal point of the system, P is the object (front) principal point of the system, P' is the image (rear) principal point of the system, *D* is the axial separation of principal points, ${\varphi}_{1},\text{\hspace{0.17em}}{\varphi}_{2},\text{\hspace{0.17em}}{\varphi}_{3},{\varphi}_{4}$ are optical powers of individual members of the optical system, *s* is the axial distance of the object from the first lens of the system, *s*′ axial distance of the image from the last lens of the system, ${s}_{F}$ is the axial distance of the object focal point F from the first lens of the system, ${{s}^{\prime}}_{{F}^{\prime}}$ is the axial distance of the image focal point F' from the last lens of the system and *d*_{1}, *d*_{2} and *d*_{3} are the distances between individual members of the system. By changing the optical powers ${\varphi}_{1},\text{\hspace{0.17em}}{\varphi}_{2},\text{\hspace{0.17em}}{\varphi}_{3},{\varphi}_{4}$ of individual members of a given optical system the change of focal length of the optical system is achieved. Point A′ is the image of point A and the straight line connecting those points forms an optical axis of the system.

The distance *D* between the principal points of the system can be expressed as:

*D*between the principal point to be zero $(D=0)$ and the Petzval sum to be corrected ${S}_{IV}\approx {\varphi}_{1}+{\varphi}_{2}+{\varphi}_{3}+{\varphi}_{4}=0$. By utilizing these requirements and using Eqs. (5)-(10) one can obtain the following formulas for calculation of optical powers ${\varphi}_{1},\text{\hspace{0.17em}}{\varphi}_{2},\text{\hspace{0.17em}}{\varphi}_{3},{\varphi}_{4}$ of individual members of the system for chosen values of distances

*d*

_{1},

*d*

_{2},

*d*

_{3}, optical power $\varphi $ and position of image focal plane ${{s}^{\prime}}_{{F}^{\prime}}$, it holds:

As it can be seen from Eqs. (11) and (12) in order to obtain real solution the term *D* has to be nonnegative which limits the choice of the possible combinations of values of distances *d*_{1}, *d*_{2}, *d*_{3} and the position of the image plane ${{s}^{\prime}}_{{F}^{\prime}}$ for the required range of optical powers $\varphi $ of the zoom lens. Furthermore one can see that for given parameters *d*_{1}, *d*_{2}, *d*_{3},${{s}^{\prime}}_{{F}^{\prime}}$ and $\varphi $ always exist two different solutions which comes from the choice of the sign in front of the radical sign in coefficient ${c}_{8}$ of Eq. (12).

## 4. Example

Let us now show the application of the formulas derived in the previous section on an example of the calculation of the paraxial properties of the four-component zoom lens with fixed position of optical center and composed of members having a variable focal length according to optical scheme given in Fig. 1. We require the separation of the principal planes to be zero $(D=0)$ and ${S}_{IV}=0$ for given value of ${{s}^{\prime}}_{{F}^{\prime}}$. Table 1 shows the results of calculation using Eqs. (6)-(12).

All values in the Table 1 are given in the same normalized length units. In order to recalculate the values for the required range of focal length of the system one has to multiply all the values in Table 1 by a proper factor (for example if one wants to obtain system with focal length range from 20 mm to 160 mm one has to multiply all the values in the table by the factor of 100).

## 5. Conclusion

In our paper we focused on the problem of paraxial analysis of four component zoom optical systems with zero separation of principal planes which are composed of active optical elements with variable focal length (tunable focus lenses). The change of focal length is achieved by the change of optical powers of individual zoom lens members (elements with variable focal length) without mutual movement of individual lens members. Such an optical system maintain the position of its optical center fixed at given value of focal length for arbitrary position of the object. We performed the paraxial analysis of such systems and we derived the formulas Eqs. (11)-(12) that enable to calculate the optical powers of individual members of the system. Finally, we presented the application of the derived equations of the example of calculation of such a system. Our work presented in this paper is therefore a theoretical contribution to the new area of such special zoom lens systems. These optical systems form a completely new family of zoom lenses which can find applications especially in in measuring systems in photogrammetry, computer vision, triangulation sensors, fringe projection techniques, surveying, machine vision etc.

## Funding

Czech Technical University in Prague (SGS18/105/OHK1/2T/11).

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