1. Introduction

Optical systems with variable optical parameters (zoom lens systems) find a large number of applications in many areas and a high importance is paid to their development [1–27]. Individual optical elements of these classical optical systems mainly consist of glass lenses and the change of their optical parameters is done by translation of several elements of such optical systems [6–8]. Recently, optical elements with a continuously variable focal length are available, which makes possible to design special optical systems with variable focus and no mechanical movement of optical elements inside the optical system [11–30]. We performed a detailed theoretical analysis [29] of a two-element zoom lens with the variable focal length. The aim of this work is to perform a detailed theoretical analysis of a three element optical system (triplet) with the variable focal length and apply the derived relations to the optical design of such system composed of commercially available refractive variable-focus lenses [14,15] and classical lenses. The derived formulas, described in the following sections of our paper, enable to calculate parameters of the triplet with a variable focal length, which may serve as input parameters (starting point) for further optimization of the triplet using appropriate optimization procedures.

2. Imaging properties of triplet lens

Figure 1 presents schematically three-element optical system composed of thin lenses. As it is well known, the following relations hold for imaging using such an optical system [5]

 

Fig. 1 Scheme of three-element optical system (ξ – object plane, ξ' – image plane, A – axial object point, A' – image of the point A, B – off-axis object point, B' – image of the point B, P – entrance pupil centre, ${y^{\prime }}_0$- paraxial image height, s – object distance, $s^{\prime }$- image distance, ${\phi }_1,{\phi }_2,{\phi }_3$ - individual powers of the lenses, $d_1,d_2$- separations of the lenses, $\overline{s}$- position of the entrance pupil, $h_1$- incidence height of the aperture ray, ${\overline{h}}_1$ - incidence height of the principal ray.

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3. Chromatic aberration of triplet lens

Let us focus on a problem of achromaticity of a triplet lens. As it is well-known, longitudinal chromatic aberration of a thin lens system is given by relation [2–8,31–34]

In case of the triplet where the aperture stop is close behind the middle element we choose ${\overline{h}}_2=0$. As it can be seen from relations (3) and (4), the conditions for achromaticity of a triplet lens are given by

Equation (2) for the power $\phi =-\gamma$of a triplet can be expressed as

The Petzval sum P of a triplet is then given by [2–8, 31–34]

If we assume that it holds approximately $n_1=n_2=n_3=n$, we can simplify previous relation to

The above mentioned equations can be completed with the formula for an approximate correction of distortion, which has the following form [31]

The values p and D are parameters, which are can be chosen appropriately. Using these parameters one can affect distortion and Petzval sum of the triplet. These parameters can be used for optimization of the triplet. Using Eqs. (6), (7), (8) and (9) with five unknown parameters (${\phi }_1,{\phi }_2,{\phi }_3,d_1,d_2$) one can determine the values of these parameters in order to made the triplet achromatic, assuming that we know the Abbe numbers $({\nu }_1,{\nu }_2,{\nu }_3)$ and refractive indices $(n_1,n_2,n_3)$ of materials of individual elements of the optical system. For simplicity and without loss of generality we set $\phi =1$in the following text. It holds for distances $d_1,d_2$ between the lenses, position of the image plane $s^{\prime }$, and position of the exit pupil ${\overline{s}}^{\prime }$ of the optical system ($\phi =1$, $h_1=1$)

We obtain using Eqs. (7), (8), and (9)

Moreover, we can express using conjugate equations

By substitution of relations (11) and (12) into Eq. (6) we obtain a set of two equations for incidence heights h₁ and h₂. It holds

The coefficients in Eq. (13) are given by the following formulas

If we substitute Eq. (14) into Eq. (15), we obtain (after a longer calculation) the following nonlinear equation for the calculation of the incidence height $h_3$. It holds

By solving Eq. (16) we can determine the incidence height $h_3$. By inserting this value into formulas (13) we can calculate the incidence height$h_2$. Then, from Eqs. (11) and (10) we can calculate the powers ${\phi }_1,{\phi }_2,{\phi }_3$ and distances $d_1,d_2$ between the individual lenses of the triplet. Our analysis is more complex and exhaustive than the analysis presented in papers [31–34].

Assume now that commercially available variable-focus (tunable-focus) lenses from companies Optotune and Varioptic [14,15] will be used for individual elements of the triplet lens. We will assume the lenses to be infinitely thin in the first approximation. If we want to design the triplet lens similar to classical triplets used in photography, i.e. having 6 free radii of curvature, we have to use 4 lenses Optotune to create two outer positive elements and 2 lenses Varioptic for the inner negative lens. A detailed theoretical analysis of refractive variable-focus lenses from the company Varioptic is performed in the papers [28,30]. For example, the lens ARTIC 416 [14] consists of two immiscible liquid lenses formed by fluids (PC200B, H100), having refractive index n_d and Abbe number ν_d with following values: PC200B - n₁ = 1.3999, ν₁ = 58.7 and H100 - n₂ = 1.489, ν₂ = 38.4 for the wavelength λ_d = 589 nm. The following expression is valid for the equivalent Abbe number ν_E of this lens

Refractive variable-focus lenses OL1024 and OL0901 from the company Optotune [15] have plano-convex shape (positive focal length) and consist of material having refractive index n_d and Abbe number ν_d with following values: OL1024 - n_d = 1.30012, ν_d = 100.177 and OL0901 - n_d = 1.55872, ν_d = 30.276 for the wavelength λ_d = 589 nm. Chromatic aberration of the Optotune lenses is lower compared with the lens ARTIC 416 at the same focal length. The disadvantage of the Optotune lenses compared to the lens ARTIC 416 is that they only can form a positive power, while ARTIC 416 can be both positive and negative lenses depending on the applied voltage.

Table 1 presents some calculation results of the achromatic (C_I = 0, C_II = 0) triplets having the unit focal length ($\phi =1/f^{\prime }=1$) and for the object located at infinity (m = 0), where ${f^{\prime }}_1,{f^{\prime }}_2,{f^{\prime }}_3$ are focal length values of individual lenses of the triplet, d₁ and d₂ are separations between individual lenses. The triplets 1 and 2 have the lens OL1024 as the first and third lens, the second lens is made of Schott glass N-BK7 and N-ZK7. The triplet 3 has the lens Optotune OL0901 as the first and third element, the second lens is made of Schott glass SF67-P. The triplet 4 has lens OL1024 as the first elements, the second elements is the lens ARTIC 416, and the third elements is the lens OL0901.

Table 1. Basic Parameters of Triplets

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4. Example of three-element zoom lens based on refractive variable-focus lenses

In the previous section, we derived formulas for calculating powers and distances of individual lenses of the triplet. We require now that this triplet can change its power (focal length $f^{\prime }=1/\phi$), without moving its individual elements. Another requirement is that the triplet lens will not change the position of the image plane with respect to the last member of the triplet. The continuous change of focal length is achieved by changing the focal length of its two elements. Because the magnitude of powers of commercially available refractive variable-focus lenses (Optotune, Varioptics) is limited, we choose the first and third lenses as the lenses with a continuously variable power. We obtain for the powers ${\phi }_1$ a ${\phi }_3$ of the first and the third element using Eqs. (1) and (2) the following formulas

Table 2 shows the results of calculating the three-element zoom lens based on two refractive variable-focus lenses (first and third element) Optotune OL1024. The second lens is formed by a conventional glass lens made of glass N-ZK7 (triplet 2 - Table 1). The triplet 2 in Table 1 was chosen arbitrarily only as an example. One could choose any other triplet in the same way. The ratio of focal distances is ${f^{\prime }}_{\max }/{f^{\prime }}_{\min }=1.5$. Moreover, Table 2 presents values of incidence heights h, $\overline{h}$ of paraxial aperture and principal rays on individual lenses of the triplet.

Table 2. Values of Paraxial Incidence Height and Focal Length - Triplet OL1024/N-ZK7/OL1024

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Optotune lens OL1024 can change its focal length from ${f^{\prime }}_{\min }=30\text{mm}$ to ${f^{\prime }}_{\max }=100\text{mm}$ [15]. As is evident from Table 2, the zoom lens will change its focal length from ${f^{\prime }}_{\min }=67\text{mm}$ to ${f^{\prime }}_{\max }=100\text{mm}$. Thus, the focal length of the second (glass) lens will be ${f^{\prime }}_2=-29.5\text{mm}$.

Table 3 presents chromatic aberration coefficients (C_I, C_II) and Seidel aberration coefficients ($S_I,S_{II},S_{III},S_{VI},S_V$) for different values of the focal length, where $S_I$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 sum, and $S_V$is the coefficient of distortion. Moreover, as it is evident from Table 4 , the longitudinal (C_I) and transversal (C_II) chromatic aberrations coefficients remain well-adjusted when changing the focal length, and chromatic aberrations of the triplet do not almost change. Table 4 also describes calculated ray aberrations for different values of the focal length, where $\delta s^{\prime }$ is the longitudinal spherical aberration, $K_t$ is the tangential coma, $\delta {s^{\prime }}_{ts}$ is astigmatism, $\delta {s^{\prime }}_s$ is the sagittal field curvature, $\delta {s^{\prime }}_t$ is the tangential field curvature, and $100\delta y^{\prime }/{y^{\prime }}_0$ is the relative distortion in percents. Linear dimensions in Table 3 and 4 are given in millimeters and the calculation is performed for the triplet consisting of thin lenses. Aberrations in Table 4 are calculated for the object at infinity, the angle of field of view 10°, and the incidence height of the aperture ray at the first lens H = 0.05 mm. The first and the third lens have a plano-convex shape [15]. The shape of the second lens can be calculated using methods and relationships described in detail in papers [8,9,31] and therefore this issue is not addressed in this work. Table 3 also shows the resulting shape parameters (bending factors) [9] X₁, X₂, and X₃ of each lens. The lens Optotune OL1024 has diameter 10 mm and thus the F-number of the zoom lens will have value $F\approx 10$. Using the previous analysis initial design parameters of the triplet can be calculated, which may serve as the starting point for further optimization using appropriate software such as, for example, ZEMAX, OSLO, etc. The thickness of individual lenses can be calculated by the approach mentioned in Ref [8].

Table 3. Aberration Coefficients of Triplet - OL1024/N-ZK7/OL1024

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Table 4. Residual Aberrations of Triplet - OL1024/N-ZK7/OL1024

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6. Conclusion

General formulas for calculating basic parameters of three-element zoom lens based on refractive variable-focus lenses were derived in this work. These relations can be used for the calculation of powers of the individual elements of the zoom lens and their distances, so that the zoom lens has corrected longitudinal and transverse chromatic aberration. None of the three elements of the zoom lens changes its position when the focal length of the zoom lens is changing. Thus, there is no need to use a complicated mechanical system, such as in the conventional zoom lenses. The analysis of the three-element optical system is more complex and exhaustive than the analysis presented in papers [31–34] because it also makes possible to affect distortion of the triplet by the parameter D. It also is not necessary to use an iterative process as suggested, for example, in Ref [2]. Using derived formulas we obtain all solutions of a given problem, which is not possible with the iterative method. The disadvantage of the iterative method is the fact that it provides only one solution near the starting point if the method converges. Derived formulas in our work can be used very easily, e.g. with the MATLAB system, the calculation is much faster and more complex than iterative methods, because all solutions are find. Furthermore, the proposed method makes possible to find out if a real solution exists (Eq. (16) must have real roots) for given input parameters, which is not possible using the iterative method. The example presented the calculation method for three-element zoom lens, where the first and the last lenses are commercially available lenses with a variable focal length from the company Optotune. Given the parameters of the Optotune lenses the focal length of the zoom lens can vary from ${f^{\prime }}_{\min }=67\text{mm}$ to ${f^{\prime }}_{\max }=100\text{mm}\text{.}$ It is evident from the presented example that the change of the focal length of the optical system does not vary its chromatic aberrations significantly. The position of the image plane is constant as well. Using the previous analysis one can obtain initial design parameters of the triplet, which can be used as the starting point for further optimization using appropriate optical design software such as, for example, ZEMAX, OSLO, etc.