## Abstract

In a recent publication [Appl. Opt. **57**, 8087 (2018).] a zoom system based on rotating toroidal lenses had been theoretically suggested. Here we demonstrate two different experimental realizations of such a system. The first consists of a set of four individually rotatable cylindrical lenses, and the second of four rotatable diffractive optical elements with phase structures corresponding to "saddle-lenses". It turns out that image aberrations produced by the refractive zoom system are considerably reduced by the diffractive system.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Optical zoom systems typically consist of a set of lenses which are shifted along the optical axis (z-axis) in order to magnify or demagnify an image of an object. However, using at least two lenses with an adjustable optical power it is also possible to obtain a zoom effect without the need of shifting any lens [1–4]. Here it is demonstrated that zooming can also be achieved by changing the mutual rotation angle of a set of four rotationally asymmetric lenses, namely of four cylindrical lenses, or of four diffractive saddle lenses. Both, cylindrical lenses and "saddle-lenses" are a sub-group of toroidal lenses. A saddle lens [5] has the transmission function of two crossed cylindrical lenses with opposite optical powers. The transmission function of a cylindrical lens may be obtained from that of a saddle lens by combining it with an adjacent spherical lens [6]. A general toroidal lens may be assumed to be composed of a rotationally symmetric lens, and a saddle lens.

Interestingly, a set of two adjacent saddle lenses, which is called in the following a combined saddle lens, still has the transmission function of a single saddle lens, however with a new optical power, which is tunable by a mutual rotation of its two components [7]. Since the transmission function of a saddle lens is a continuous and smooth two-dimensional function, it can be produced both as a refractive or a diffractive optical element. Furthermore the transmission function of a rotationally adjustable combined saddle lens does not have any undesired discontinuities. This is in contrast to so called Moiré lenses [8–14], which are rotationally tunable spherical lenses. However, the two sub-elements of a Moiré lens consist of discontinuous structures, which results in an undesired sector of a different optical power, or in undesired residual diffraction orders as a function of the rotation angle.

A single toroidal lens (e.g. a saddle lens or a cylindrical lens) typically cannot be used for imaging, since it affects the focal length in two orthogonal planes in a different way. However, this issue can be resolved by using *two* toroidal lenses (or two sets of combined, tunable toroidal lenses) placed at different positions within an optical system. In an earlier, theoretical investigation [6] it was shown that such a system may simultaneously act as a cylindrical Kepler- and a cylindrical Galilei telescope in two orthogonal planes, respectively. Using tunable combined toroidal lenses, such a setup can be used as a zoom system, magnifying or demagnifying an image in a broad zooming range by just rotating the included toroidal lenses. The setup has the additional advantageous feature that the object- and the image planes are stationary during zooming, - a feature which is hard to achieve with standard zoom systems based on axially shifting lenses. Furthermore the image can be continuously rotated by a rotation of all included toroidal lenses. The transmission function of saddle lenses can also be generated by a combination of standard cylindrical lenses (with both positive and/or negative optical power) with spherical lenses. Thus a zoom system can also be realized with arrangements of cylindrical lenses used as the rotationally tunable elements.

The underlying principles of a zoom system based on toroidal lenses have been described both in a ray-optic and a wave-optic regime in [6]. Here we will experimentally investigate one of the suggested implementations. For the theoretical considerations it will be assumed that all included optical elements are thin, such that the combined transmission function of two adjacent elements is just the product of the two individual transmission functions. Clearly this assumption can be easier satisfied if the respective optical elements consist of diffractive optical elements, rather than of more bulky refractive ones.

## 2. Combined toroidal lenses

Assuming a thin lens approximation, the surface profile (height) $H_t(x,y)$ of a semi-planar toroidal lens is represented in cartesian $(x,y)$-coordinates by:

where $n$ is the refractive index of the lens material. There, only quadratic terms are considered (though correction terms of different order may be present in practical implementations), and it is assumed that the optical axes of the lens are aligned parallel to the $x-$ and $y-$ axes. $F_x$ and $F_y$ are the cylindrical optical powers of the lens in the $xz-$ and $yz-$ planes, respectively, and may be positive (convex lens), or negative (concave lens).In a thin lens approximation the corresponding transmission function for light with a wavelength of $\lambda$ is given by $T_t=\exp [-i (n-1) H_t 2 \pi /\lambda$], which yields

This transmission function may be factorized:The transmission functions of the spherical lens and of the saddle lenses may also be expressed in cylindrical coordinates $(r,\varphi )$, which yields:

In the following we consider the combined transmission function $T_T$ of a set of two adjacent toroidal lenses $T_{t1}$ and $T_{t2}$, which may be rotated by an angle $\theta$ with respect to each other. In order to keep the optical axes of the combined element stationary, the rotation is done symmetrically, rotating one of the elements by an angle $\theta /2$ clockwise, and the other by the same angle counter-clockwise. Furthermore we assume that the quadrupole optical powers of the two elements are identical, i.e. $F_{q1}= F_{q2}:=F_q$. Thus the combined transmission function yields:

Since the operational principle of combined toroidal lenses can be separated into a rotationally independent spherical component, and a rotationally adjustable quadrupole component, it is convenient to separate each general toroidal lens into a pure spherical lens, and a pure saddle lens. Thus we can confine our following investigations without loss of generality to optical systems which just consist of pure spherical and pure saddle lenses. Nevertheless, at the end the transmission functions of adjacent lenses may be recombined into that of general toroidal lenses. Thus a final zoom system can be based on the rotation of pure saddle lenses, but also on the rotation of (positive and/or negative) cylindrical lenses, or on general toroidal lenses, which might be implemented as more sophisticated freeform- or diffractive optical elements.

## 3. Basic setup of a zoom system with adjustable toroidal lenses

Different implementations of zoom systems with toroidal lenses have been theoretically described in [6]. They basically consist of setups, where the optical Fourier transform of a first adjustable combined saddle lens is projected onto the plane of a second adjustable combined saddle lens. The optical Fourier transform may be performed in different ways using arrangements of spherical lenses [15]. In the following we discuss one of the possible setups, which is a compact realization of this principle. The basic system is shown in Fig. 1.

It consists of a first pair $S_1$ and $S_2$ of individually rotatable saddle lenses with the same quadrupole optical powers of $F_{q1}$ in combination with a spherical lens $L_1$ with optical power $F_{l1}$. $S_1$ and $S_2$ are combined into a rotationally adjustable combined toroidal lens $Q_1$, which has the transmission function of a single saddle lens with a new quadrupole optical power $F_{Q1}$, which is adjustable between $-2 F_{q1}$ and $+2 F_{q1}$ according to Eq. (7). After a distance $d$, another system of three adjacent lenses is located, namely a spherical lens $L_2$ with optical power $F_{L2}$, and another set of two rotatable saddle lenses $S_3$ and $S_4$, which both have the same quadrupole optical powers of $F_{q2}$. The saddle lenses $S_3$ and $S_4$ are combined into another combined saddle lens $Q_2$, with an optical power adjustable between $-2 F_{q2}$ and $+2 F_{q2}$.

In [6] it has been shown that this system corresponds to an afocal zoom system (object- and image planes at infinity) with an angular magnification of $m$ if the conditions

andA straightforward way to convert the afocal system sketched in Fig. 1 into an imaging setup with object and image planes at finite distances of $o$ and $i$, respectively, is shown in Fig. 2.

There, another spherical lens $L_3$ is added at the input side of the zoom system, which has an optical power of $F_{l3} = o^{-1}$, i.e. its focal length corresponds to the distance between the lens and the object plane. Furthermore, another spherical lens $L_4$ with an optical power of $F_{l4}=i^{-1}$ is added at the exit of the zoom system. Such a focal zoom system still has the same total zoom factor $Z$ as the previously described afocal system, but the absolute image magnification additionally scales with the ratio $i/o$ between image distance $i$ and object distance $o$.

The sketch in Fig. 2 is just an equivalent optical diagram which helps to clarify the basic operation principle of the zoom system. However a practical setup based on this principle may look significantly different. This is due to the fact that in a thin lens approximation the positions of the individual lenses within the two main lens groups ($L_3$, $S_1$, $S_2$, $L_1$, and $L_2$, $S_3$, $S_4$, $L_4$, respectively) may be arbitrarily exchanged. Furthermore, the optical powers of the individual lens elements may be redistributed between them. For example, the optical powers of lenses $L_3$ and $L_1$ may be combined into a single lens. Alternatively, the spherical optical powers of lenses $L_1$ and $L_3$ may be combined with the quadrupole optical powers of the saddle lenses $S_1$ and $S_2$, respectively, which results in just two mutually rotatable general toroidal lenses at the front end of the zoom system. Analogously the lenses at the exit part may be combined. In this case the whole zoom system would consist of only four rotatable general toroidal lenses. For special cases such a zoom system can be realized by a set of four rotatable cylindrical lenses, without any additional spherical lenses, as will be shown in the following.

## 4. Refractive zoom system with rotating cylindrical lenses

The transmission function of a cylindrical lens oriented along the $x-$axis with an optical power of $F_c$ is given by:

This can be expressed as the transmission function of a general toroidal lens, writingFor a practical implementation we choose four identical cylindrical lenses $C_1 - C_4$, each with a cylindrical optical power of $F_c=20$ Dpt. We choose a separation between the two cylindrical lens groups of $d=$120 mm. According to the previous considerations the correct distances for the object- and the image plane are $o=i=(F_c-d^{-1})^{-1}=$86 mm. The magnification of such a setup is expected to be tunable in a range between $m_{\mathrm {min}}=$0.41 and $m_{\mathrm {max}}=$2.4, which yields a total zoom factor $Z$ of 5.8.

A picture of the corresponding experimental setup is shown in the upper part of Fig. 3. A schematic of the setup is sketched below, indicating the positions of the four lenses, and of object and image planes.

The four plano-convex round cylindrical lenses (Thorlabs LJ1695RM-A) with respective focal lengths of 50 mm and with diameters of 25.4 mm are mounted in four computer controlled rotation stages (PI U-651.03), respectively. Each lens can be individually rotated under computer control. The cylindrical lens pairs $C_1-C_2$ and $C_3-C_4$ are mounted as close as possible with a relative distance of about 1 mm within their corresponding pairs of rotation stages. The object to be imaged was a transmissive USAF resolution target, illuminated from the rear side at a distance of 30 cm by a white light halogen bulb. Images were recorded by a Canon EOS 5D Mark II camera with removed objective. The respective distances (indicated in the figure) were $o=86$ mm, $d=120$ mm, and $i=86$ mm. Thus the object- and image sided numerical apertures both correspond to $NA=0.15$. The optical magnification factor $m$ was adjusted by rotating the two lenses of each lens pair symmetrically by the angles $\theta _1/2$, and $\theta _2/2$ into opposite directions. The quadrupole optical powers $F_{Q1}$ and $F_{Q2}$ of the two combined lens pairs, and the corresponding rotation angles $\theta _1/2$ and $\theta _2/2$ for obtaining a certain magnification factor $m$ are calculated according to Eq. (9). There it has to be considered that the factor $2 F_q$ in Eq. (9) now corresponds to $2 F_q=F_c=20$ Dpt.

The Figs. 3(a)–3(f) show 6 examples of images recorded with magnification factors of 0.4, 0.8, 1.2, 1.6, 2.0, and 2.4, respectively. A qualitative evaluation of the pictures shows that the low magnification factor 0.4 in Fig. 3(a) results in significant image aberrations (mainly pincushion distortion and chromatic aberration). For higher magnification factors ($m\;>\;1$) the image quality within the same area of the camera chip improves continuously. This is mainly due to the corresponding reduction of the viewing angle, which is $\pm 6^{\circ }$ in Fig. 3(a), and reduces continuously to $\pm 0.8^{\circ }$ in Fig. 3(f). Anyhow the pictures still show significant chromatic aberrations at the edges of the images. A quantitative evaluation of the image magnification shows that it reproduces the theoretically expected zoom factors within a small experimental error of 2.5% (on the average). The system provides a total optical zoom range of $Z=6$, as expected from the experimental parameters. It was also observed that a rotation of all cylindrical lenses by an angle $\alpha$ into the same direction leads to a rotation of the image by the doubled angle $2\alpha$, as expected.

It is assumed that the image distortions are mainly due to the fact that the employed cylindrical lenses, which have a center thickness of 6.7 mm, do not satisfy the thin lens approximation. Thus a pair of adjacent cylindrical lenses does not satisfactorily produce the desired transmission function of a thin saddle lens, which should be located in one plane. The chromatic aberrations are mainly due to the fact that the employed standard cylindrical lenses (made from BK-7 glass) are not achromatic. For future implementations it may be assumed that the chromatic aberrations may be strongly reduced by using achromatic cylinder lenses. Furthermore it might be possible to reduce image distortions using optimized cylindrical lenses, as e.g. sagged meniscus lens profiles (instead of semi-convex lenses). This allows one to superpose the principal planes of a pair of cylindrical meniscus lenses in a single plane, leading to a common transmission function which reproduces that of an ideal combined saddle lens with considerably reduced errors, which should then significantly reduce the image aberrations. For practical applications these optimizations of the optical layout may be conveniently performed with professional ray-tracing software.

## 5. Zoom system with rotating diffractive saddle lenses

As mentioned in the previous discussion, zoom systems with refractive toroidal lenses do not satisfactorily fulfill the thin lens approximation, which is supposed to produce the observed image distortions. Thus it may be expected, that a zoom system based on planar diffractive saddle lenses should have a significantly improved imaging performance. This will be investigated in the following.

For that purpose 4 diffractive saddle lenses have been produced by electron beam lithography and subsequent etching by a specialized company in contract work. An image of one of these lenses is shown in Fig. 4.

The diffractive surface profile $H$ was structured according to the formula:

Lower part: Examples of 6 images of a USAF resolution chart recorded with the setup sketched above. The object was diffusely illuminated from the rear side by a green LED and a subsequent diffuser. The corresponding magnification factors $m$ are indicated in the figure. In all recordings the zoom factor has been solely adjusted by rotating the four diffractive saddle lenses, whereas the positions of the object and the camera have been stationary.

A sketch of the experimental setup is shown in the upper part of Fig. 5 (not to scale). The parameters of the setup (distances and optical powers) are indicated in the figure. Two exemplary beam paths in the $xz-$ and $yz-$ plane are sketched. For the object we again choose a transmissive USAF resolution target, which was illuminated from the rear side with diffuse light form a green LED, followed by a diffuser. The images were recorded by a monochrome pco.4000s camera without objective. Six exemplary pictures with magnification factors in a range between $m=0.40$ and $m=2.32$ are displayed below (respective magnification factors indicated in the figure). As before, the zoom factor was solely adjusted by a rotation of the four diffractive saddle lenses, with respective rotation angles calculated according to Eqs. 9.

Qualitatively it turns out that the images show a slight shear, which indicates that the optical axes of the combined saddle lenses in the two lens groups have not been perfectly parallel. Furthermore there is some diffuse light in the center of the images, which is an artefact due to stray light passing through the non-structured edges of the diffractive saddle lenses. The corresponding image resolutions were estimated by evaluating the images of the USAF resolution target, selecting the smallest object group which is still resolved in each image. For the examples shown in the figure, the magnification factors were $m=0.40$, 0.47, 0.65, 1.03, 1.53 and 2.32, with respective image resolutions of 100 $\mu$m, 44 $\mu$m, 25 $\mu$m, 18 $\mu$m, 16 $\mu$m, and 15 $\mu$m. Thus, it turns out that the image is diffraction limited for the highest magnification factor, considering that the numerical aperture at the object side is 0.021, which yields a theoretical image resolution of 15 $\mu$m for a wavelength of 532 nm. The resolution becomes worse for lower magnification factors, which is due to vignetting of the light field at the aperture of the second lens group. This effect becomes more pronounced with decreasing magnification factors.

Comparing these results with those obtained with the cylindrical lens zoom system of Fig. 3, which cover a similar zoom range, it turns out that the diffractive system does not produce the pincushion distortion, which was observed before. This observation agrees with the consideration that the image distortions observed before are due to the fact that the thin lens approximation was only insufficiently satisfied for a system of (relatively thick) refractive lenses, whereas it is much better satisfied for a system of planar diffractive saddle lenses. Anyhow, a disadvantage of the diffractive zoom system is that its current implementation only works for monochromatic light. However, this issue might be resolved in future by employing multi-order diffractive saddle lenses, which may be achromatic and provide high diffraction efficiency over the whole visible wavelength range [16–20].

## 6. Conclusion

Two implementations of compact zoom systems based on rotating refractive cylindrical lenses, and on rotating diffractive saddle lenses, have been experimentally investigated. It turns out that the refractive system shows both chromatic aberrations and image distortion. It is assumed that in future applications the chromatic aberrations may be reduced by using achromatic toroidal lenses. Furthermore, optimized cylindrical lenses with meniscus profiles may help to reduce the image distortions, which are assumed to be due the thickness of the refractive optical elements. Finally, the performance might be improved by using optimized general toroidal lens profiles which correspond to freeform optical components.

A zoom system using rotating diffractive saddle lenses has been shown to avoid the image distortions of the refractive system, and it yields a diffraction limited imaging performance (for high magnification factors). Its current implementation still has the disadvantage that it is only suited for monochromatic light. However, in earlier publications it has been demonstrated that multi-order diffractive elements may resolve this issue [16–20]. In future this will allow one to construct a zoom system based on multi-order diffractive toroidal lenses, which can be used for color imaging. Zoom systems based on rotating toroidal lenses have the advantageous features that the object- and image planes are exactly stationary during zooming, and that the image can be arbitrarily rotated. Furthermore using lightweight diffractive toroidal lenses, zooming by just rotating these elements can be fast and accurate.

The zoom systems demonstrated so far are examples of a wider class of systems, which can be constructed by general toroidal lens elements. Since the operational principle of the zoom system is based on the fact that an optical Fourier transform of one tunable combined saddle lens is projected onto the plane of a second tunable saddle lens, there are also other possible implementations of such a system, as. e.g. positioning the two sets of saddle lenses in the front- and rear focal planes of a single central lens, respectively, which also performs an optical Fourier transform [6,15]. By rearranging the transmission functions of the individual lens elements in the equivalent optical diagram of Fig. 5, it is possible to get various advantageous optical implementations of the basic operation principle. For example, it is in principle possible to construct an extremely compact zoom system, acting between a fixed object- and image plane, which consists of just two sets of rotationally adjustable combined toroidal lenses, without using any other optical components.

A further interesting application of a refractive or diffractive zoom system based on rotating toroidal lenses is to use it as a compact variable beam expander for high power lasers. There, the rotationally asymmetric optical layout has the advantage that it produces no focal spot within the beam path. Instead it just produces an internal focal line (like a cylindrical lens), which has a peak intensity which is by orders of magnitude lower than that of a focussed spot. In high power laser applications this significant reduction of the peak intensity is advantageous to prevent undesired plasma generation ("optical breakthrough") within the beam path.

## Funding

Österreichische Forschungsförderungsgesellschaft (864729).

## Disclosures

The authors declare no conflicts of interest.

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