Optical varifocal is a fundamental requirement of many imaging application fields, such as consumer electronics^[1], medical instruments^[2], and machine vision apparatuses^[3]. Traditional varifocal lenses typically consist of several solid lens assemblies with fixed optical properties^[4]. Focal length tuning is usually obtained by displacing one or more constant-focus lenses over specific distances along the optical axis. Owing to the need for moving parts, the miniaturization of traditional varifocal lenses is typically complex and results in bulky and large dimensions, especially when a wide varifocal range is required. To address the demand for miniaturization of the varifocal lenses, various simple but effective solutions have been proposed, including liquid lenses^[5,6], liquid crystal lenses^[7], Moiré lenses^[8], and Alvarez lenses^[9–11]. However, liquid lenses always suffer from environmental effects, such as temperature, vibration, and gravity, because of the inherent features of liquid material. Liquid crystal lenses are usually polarization dependent, and their varifocal range is rather limited. Multiple liquid crystal layers need to be stacked to achieve a wide varifocal range, which in turn leads to long response time, low efficiency, and low resolution. The varifocal of the Moiré lenses requires rotational motion, and high alignment accuracy is required. However, their circular structure makes it difficult to align. The Alvarez lenses feature a unique design and achieve a wide varifocal length range through small relative lateral movement between two phase plates perpendicular to the optical axis. It has garnered significant attention due to their compact size and stable structure, making them particularly suitable for miniaturization imaging applications, such as endoscopes^[12], smart glasses^[13], head-up displays^[14], and others^[15–17].

The concept of the Alvarez lenses was proposed in 1967 by Luis Alvarez. With the development of machining technology in recent years, the fabrication of freeform phase plates is no longer a challenge^[18–20]. However, due to its unique varifocal principle, the actuation of the two-phase plates becomes a challenging issue that limits its application. Several drive structures of Alvarez lenses in recent years have emerged. The simple yet effective method is utilizing a displacement platform. Hou et al. used four displacement platforms to control the movement of the Alvarez lenses. By moving the Alvarez lenses along the positive and negative direction about 1 mm, the focal length is changed from 34.50 to 103.50 mm^[21]. However, the structure of the displacement platform may complicate the configuration and increase the dimension and cost, which is not applicable to miniaturization. Therefore, they also adopted a voice coil motor to drive the Alvarez lenses, which can provide a displacement of 3 mm, enabling the Alvarez lenses to achieve a 3× zoom capability^[22]. Zhou et al. introduced varifocal Alvarez lenses driven by a micro-electromechanical system (MEMS). The drive structure enables a displacement of 40 µm to change the focal length from 3 to 4.65 mm^[23]. To enlarge the displacements, the MEMS was combined with an amplifier, the maximum displacement reached 100 µm, and then a varifocal range from 28 to 65 mm was achieved^[24]. Additionally, they employed a piezo bender to drive the Alvarez lenses. The piezo bender provided a 110 µm displacement to achieve 3× zoom for endoscope imaging^[25]. Petsch et al. used the liquid crystal elastomer to drive Alvarez lenses to achieve a 178 µm displacement with a 3.30 mm change in focal length^[26]. Similar to a liquid crystal elastomer, another material such as a dielectric elastomer can be used to drive Alvarez lenses. Our team previously introduced a dielectric elastomer actuator to drive the Alvarez lenses. The dielectric elastomer offered a 1.15 mm displacement to realize a wide varifocal range from 6 to 180 mm^[27]. However, the dielectric elastomer needs an extra high voltage supply, which makes the structure bulky in size. From the above analysis, we can find that a driving mechanism for Alvarez lenses with a large varifocal range and a compact structure is pursued.

Here, we have developed a specialized cam-driven structure that is assembled with the Alvarez lens. Through the rotation of the driving ring of the cam-driver, the two phase plates of the Alvarez lenses move perpendicular to the optical axis to achieve varifocal capability. The proposed varifocal Alvarez lens has a compact overall size. It has an axial dimension of only 30.50 mm and a maximum aperture of 28.50 mm, and it achieves a wide varifocal range from 15 to 75 mm within a maximum displacement of 1.98 mm. The rest of this paper is organized as follows: Sec. 2 describes the varifocal principle and optical design of the Alvarez varifocal lenses. Section 3 presents the mechanical design of the cam-driver and the fabrication result of the proposed varifocal lens. Section 4 shows the experimental results. Section 5 is the discussion, and Sec. 6 is the conclusion.

The Alvarez lenses comprise two complementary phase plates positioned opposite each other. One side of the phase plate is a flat surface, while the other side is a freeform surface. When these two phase plates move relative to one another (the displacement is δ), as shown in Fig. 1, the Alvarez lens functions as a varifocal lens. For example, when δ=0, the light passes through the Alvarez lens in parallel. So the focal power of the Alvarez lens is 0, which is similar to a parallel plate, as shown in Fig. 1(b). When δ>0, the light is focused. The focal power of the Alvarez lens is greater than 0, which is similar to a convex lens, as shown in Fig. 1(a). When δ<0, the light diverges. The focal power of the Alvarez lens is smaller than 0, which is similar to a concave lens, as shown in Fig. 1(c).

The extended polynomial expression of the freeform surface of the Alvarez lenses is^[28]{Z1=A(X1Y2+13X3)+DX+EZ2=−A(X1Y2+13X3)−DX+E,(1)where A, D, and E are the constant parameters of the phase plates, and X, Y are the transverse coordinates perpendicular to the z-direction. Z represents the thickness of the phase plate at different points. The value of A determines the depth modulation of the freeform surface. The value of D determines the tilt of the freeform surface, and the value of E determines the center thickness of the lens.

When the two phase plates move relative to each other, the focal length of the Alvarez lenses (f′) can be expressed by^[29]f′=14Aδ(n−1),(2)where δ is the move displacement, and n represents the refractive index of the lens material.

In this paper, a varifocal Alvarez lens with a wide focal length range from 15 to 75 mm is designed. The specific parameters are shown in Table 1.

Before the optical design, the basic parameters of the Alvarez lenses need to be considered. From Eq. (1), we can find that A determines the focal length range of the Alvarez lenses. If the value of A is greater, the focal length range is wider. However, a great value of parameter A results in a large peak to valley (PV) value of the freeform surface, which is not conducive to the processing of the lens. Therefore, considering the surface shape and focal length range, the value of parameter A is determined to be 0.02 mm^-2. Because the PMMA material has good optical transmittance and can be easily turned by a five-axis diamond lathe, it is selected as the material for the Alvarez lenses. PMMA material has a refractive index of 1.49. To ensure the machinability of the Alvarez lenses, the thickness of the phase plate cannot be too small. In this paper, the central thickness (E) of the phase plate is 3 mm, and the parameter D is determined as 0.03. To ensure that the two phase plates do not collide with each other, the interval between the two phase plates is controlled at 0.70 mm. The proposed varifocal Alvarez lenses are first designed as an ideal model structure, and then the ideal lens is gradually replaced by a solid lens. In the process of gradual optimization, more and more high-order terms of the freeform surface of the Alvarez lenses are added. Finally, the high-order coefficient is selected up to the ninth order. To reduce the optical aberration, we also used the flat surface of the Alvarez lenses with a freeform surface. After repeated optimizations of the overall structure, the optical schematic of the final proposed varifocal Alvarez lenses is shown in Fig. 2. The proposed varifocal Alvarez lenses are composed of four groups of lenses. The first and fourth groups are single lenses. The second group is doublet lenses, which are used to correct the chromatic aberration. The third group is the Alvarez lenses, which are located after the diaphragm so that the aperture of the light entering the Alvarez lenses is small. This allows the light to pass through the central region of the Alvarez lenses during the varifocal process and also reduces the aperture of the Alvarez lenses. The coefficients of higher-order terms on the 8th and 13th surfaces are the same value, and the coefficients of higher-order terms on the 9th and 12th surfaces are the same value. The data of the optimized varifocal Alvarez lenses are shown in Tables 2 and 3.

Spot diagrams, modulation transfer function (MTF) curves, and image simulation are utilized to illustrate the overall imaging quality of the proposed varifocal Alvarez lenses. When the focal lengths of the proposed varifocal Alvarez lenses are 15, 45, and 75 mm, respectively, the spot diagrams at the three focal lengths are shown in Figs. 3(a)–3(c). From Fig. 3, we can find that when the focal length of the proposed varifocal Alvarez lenses is 15 mm, the root mean square (RMS) radius of each field of view is larger than the Airy spot radius, but it is still in single digits, and the maximum RMS radius of each field of view is 5.55 µm. At focal lengths of 45 and 75 mm, the RMS radii of each field of view are also smaller than the Airy spot radius. The maximum RMS radii are 4.85 and 6.01 µm, respectively. Figure 4 illustrates the MTF, and image simulations are at focal lengths of 15, 45, and 75 mm. According to the simulation results, it can be seen that the spatial resolutions of the designed system are 80, 60, and 40 lp/mm when the MTF is 0.20 at three focal lengths of 15, 45, and 75 mm, respectively. Image simulation results also show that the proposed varifocal Alvarez lenses can obtain clear images at different focal lengths. The analysis of field curvature and distortion in the proposed varifocal Alvarez lenses can be found in the Supplementary Material.

The proposed varifocal Alvarez lens driving structure is shown in Fig. 5. It has two phase plate frames, two fixing frames, two phase plates, and a drive ring. The structure uses the cam-driven principle to convert the rotational motion into a linear motion. There is a convex cam curve on the left and right sides of the circular driven ring. The width of the cam curve is consistent with the width of the depression at the edge of the phase plate frame, so that they can be assembled. The upper and lower parts of the phase plate frame have rectangular protrusions, which are matched with the grooves of the fixing frame. When the driving ring rotates, the cam curve drives the frame for placing the phase plates to move along the groove. The size of the inner rectangular part of the frame for placing the phase plates is consistent with the size of the phase plate.

The five-axis diamond lathe is used to fabricate the two-phase plates of the Alvarez lenses. The processed two-phase plates are shown in Fig. 6(a). The shape of the phase plate is rectangular, which is 5 mm in width and 7 mm in length. A contact measurement method was used for roughness analysis through test 307,200 points on the freeform surface. The measurement result is shown in Fig. 6(b). The results show that the roughness (Ra) of the fabricated phase plates is about 23 nm, which indicates the Alvarez lenses have a smooth and precise freeform surface. The interval between the phase plates is 0.70 mm after assembly. The assembled proposed varifocal Alvarez lenses are shown in Fig. 6(c). The dashed box part is the driving structure, and the Alvarez lenses are assembled inside. The axial length of the proposed varifocal Alvarez lenses is 30.50 mm, and the maximum aperture is 28.50 mm. The cam-driven process of the Alvarez lenses is shown in Visualization 1, and the assembly of the actuator structure can be found in the Supplementary Material. When the driving ring rotates, the left and right moving strokes of the phase plates are 1.20 and 0.80 mm, respectively.

The proximal end of the drive ring is positioned 6 mm from the center of rotation (COR). Upon rotating the ring by 100°, the distal end displaces to 8 mm from the COR, achieving a radial displacement of 2 mm. The corresponding cam profile equation is r(θ)=6+0.02θ(θ∈[0°,100°]).(3)

It can be concluded that for every 10° of cam rotation, the displacement of the follower is 0.02 mm. The theoretical value of its displacement curve is shown in the blue curve in Fig. 7. Displacement experiments were carried out on the processed driver, and the displacement of the driver was measured using a displacement sensor (BL-30NZ) with an accuracy of 0.01 mm for every 10° of rotation. Its experimental value is shown as the red curve in Fig. 7. It can be seen from the figure. The experimental results are consistent with the theoretical value, indicating that the drive can accurately control the displacement.

The focal length of the proposed varifocal Alvarez lenses is measured. According to the shear interferometer principle, a parallel beam of light passing through two lenses will have three states: focused, parallel, and diverging^[30,31]. We use this state of the outgoing parallel light to measure the focal length of the proposed varifocal Alvarez lenses. The schematic diagram is shown in Fig. 8. f1′ and f2′ represent the focal lengths of a single lens and the varifocal Alvarez lenses, respectively. The value of f1′ is 150 mm. A beam of parallel light hits a piece of the single lens. By adjusting the interval between the Alvarez varifocal lenses and the single lens and making the interval of the single and varifocal Alvarez lenses equal the sum of f1′ and f2′, the outgoing light is parallel. Therefore, the focal length of the proposed varifocal Alvarez lenses f2′ can be measured according to the interval and the value of f1′.

The overall experimental schematic is illustrated in Fig. 9. A green laser (DH-HN300) with a wavelength of 632.80 nm is used. Due to the thickness of the proposed varifocal Alvarez lenses, we are unable to accurately measure the position of their principal plane. Instead, the back focal length of the proposed varifocal Alvarez lenses is measured. We compare the measured back focal length with the simulation result to verify the accuracy of the focal length range. The driving ring is rotated to the leftmost and rightmost positions, which correspond to the shortest and longest focal lengths of the varifocal Alvarez lenses, respectively. The intervals in the two cases are 151.70 and 176 mm, respectively. Given that the focal length of the single lens is 150 mm, the shortest and longest back focal lengths of the varifocal Alvarez lenses are 1.70 and 26 mm, respectively. In the optical simulation results, the back focal lengths of the shortest focal length and the longest focal length are 1.80 and 25.80 mm, respectively. It shows that the fabricated varifocal Alvarez lenses conform to the design results.

The relationship among the object distance (u), the focal length of the imaging lenses (f), and the image distance (v) can be described by the Gauss formula, as shown in Eq. (4). 1f=1u+1v.(4)

From Eq. (4), we can find that, when the object distance is fixed, the image distance changes with the focal length. Also, when the image distance is fixed, the object distance changes as the focal length changes. In the following experiments. The distance between the object and the foremost part of the proposed varifocal Alvarez lenses is defined as l, and the distance between the backmost part of the proposed varifocal Alvarez lenses and the sensor is defined as d.

When the object distance is fixed, a resolution test chart (ISO 12233) is placed 500 mm in front of the proposed varifocal Alvarez lenses, i.e., l=500mm. A schematic of the experiment is shown in Fig. 10(a). The sensor model is OV2719. The focal length of the varifocal Alvarez lens is adjusted to achieve a clear image. Figures 10(b)–10(d) display the images captured at back focal lengths of 6, 14, and 19 mm, respectively. From the experimental results, it can be observed that as the back focal length increases, the imaging field of view decreases, which results in the central region of the test chart being magnified in the image. Among the three back focal lengths, the highest image resolution is achieved when the back focal length is 14 mm. When the back focal length is 6 mm, the field of view is large, but the image resolution is low. When the back focal length is 19 mm, the picture is a little blurred and has low contrast. The imaging quality is still in an acceptable range for optical lens requirements. Experiments performed for other resolution plates are shown in the Supplementary Material.

When the back focal length of the proposed varifocal Alvarez lenses is fixed, three marked objects are placed at different distances in front of the proposed varifocal Alvarez lenses. The blue object is marked with the letter “X” and is located at a distance of 130 mm from the lens. The red object is marked with the letter “W” and is positioned at a distance of 217 mm from the lens. The green object is marked with the letter “V” and is situated at a distance of 331 mm from the lens. Because the imaging field of view increases with the decrease in the back focal length, the back focal length of the lens is fixed at 6 mm to cover the three objects in the experiment. The experimental schematic is described in Fig. 11(a). Figures 11(b)–11(d) show the imaging results of the Alvarez varifocal lens for objects of different distances. The experimental results show that the Alvarez varifocal lens is capable of imaging objects at varying distances in front of it and can clearly distinguish the letters marked on different objects. The detailed varifocal process can be found in Visualization 2. Focusing experiments after replacing the sensor are shown in the Supplementary Material.

Based on the focal length measurement experiment, a wavefront sensor (WFS31-5C/M) is used to measure the wavefront of the emitted light. The experimental diagram is shown in Fig. 12(a). By analyzing the measurement results from the wavefront sensor, particularly the Zernike coefficients, various aberrations present in the system can be assessed^[32,33]. When the focal lengths of the proposed varifocal Alvarez lens are 75, 36.80, and 15 mm, respectively, the Zernike information is shown in Figs. 12(b)–12(d). From the measurement results, we can find that the most obvious aberration is astigmatism, and the other types of aberration coefficients are very small and almost negligible. However, they can also have an impact on image quality. All the aberrations combine to affect the image quality, and the core effects include resolution degradation, geometrical distortion, and focal plane shift. From the simulation results as well as the experimental results, the main problem of this lens is the degradation of the resolution, which may be caused by dispersion, spherical aberration, and coma aberration. Astigmatism is caused by the inconsistency of imaging characteristics of the varifocal lens in the tangential and sagittal directions. This inconsistency is caused by the non-rotational symmetry of the freeform surface. Therefore, light in more directions should be traced during the phase design process.

To meet the requirement of a compact structure and a wide varifocal range, a cam-driven varifocal Alvarez lens is proposed in this paper. A comparison of the Alvarez varifocal lenses in this paper with other references is shown in Table 4. The proposed varifocal Alvarez lens is compact with a large varifocal range. In previous studies, Hou et al. proposed a 3× zoom module with a focal length range from 14.10 to 4.65 mm and overall dimensions of 25mm×25mm×6mm^[22]. In comparison, our lens is slightly larger but has a 5× varifocal capability. Zou et al. has proposed a 3× zoom endoscope, and although the overall volume is not mentioned, the axial dimensions in the schematic are longer, and the varifocal range is smaller than that in this paper^[25]. In a comprehensive comparison, the lens proposed in this paper has a larger varifocal range while ensuring a compact structure. Compared to other references, the proposed varifocal Alvarez lenses in this paper have a large varifocal range and, at the same time, better image quality. However, some drawbacks were also found during the experiments. The first is the low resolution of the proposed varifocal Alvarez lenses, which is related to the initial optical design as well as processing errors. Second, the proposed varifocal Alvarez lenses are found to have large astigmatism in the wavefront detection experiments, which is closely related to the non-rotational symmetry of the Alvarez lens. Finally, it is found during use that the drive structure needs to be improved, including precise scale markings and addressing friction problems between the components. As the next step, we will first consider introducing other aspherical surfaces and using deep learning to trace and analyze more angles of light to further improve the performance of the lens. Second, we will be looking for more sophisticated methods of machining freeform surfaces and drive structures. Concerning friction, we will optimize the cam structure, use low-friction material combinations, coat the material surfaces, and use lubricants to reduce friction. Third, we will design a more suitable driving structure to reduce the alignment error of the Alvarez lens. After a comprehensive evaluation of the final image quality, we can consider introducing algorithms to further improve the image quality, including de-blurring and correction of chromatic aberration.

In this paper, a compact varifocal Alvarez lens with a large varifocal range is proposed. The lens effectively utilizes the advantages of the lateral movement of the Alvarez lens to meet the requirements of miniaturization and a large varifocal range. Through optical design, the varifocal range from 15 to 75 mm is obtained. When the contrast is 0.20, the resolutions at three focal lengths are 80, 60, and 40 lp/mm, respectively. The driving structure of the cam is designed, and the driving displacement is 2 mm. Through a series of experiments, it is further verified that the proposed varifocal Alvarez lenses achieve a focal length range from 15 to 75 mm and show good imaging quality in the experiment of imaging the object. The weight of the lens is 17.78 g, and the total volume of the lens is 30.5mm×28.5mm×17mm. The proposed varifocal Alvarez lenses successfully achieve a predetermined varifocal range and meet the needs of miniaturization in size. This result not only shows the effectiveness of the Alvarez lens in the design of varifocal lenses but also provides a new idea for the design of miniaturized varifocal lenses in the future. We will further focus on optimizing varifocal performance, reducing costs, and exploring its application potential in different fields. In a word, the lens represents a successful attempt to apply the Alvarez lens to the design of the varifocal lenses and meets the requirements of miniaturization and a large varifocal range. In the future, it will be effectively applied in unmanned aerial vehicle (UAV), virtual reality (VR), medical imaging, and other fields.