Ultrathin Tunable Lens Based on Boundary Tension Effect

Solid and liquid lenses are commonly used in optical design. Such lenses have suitable thicknesses due to their working principle and processing mode. Thus, zoom optical systems comprising solid and liquid lenses are extremely large. This work presents a new ultrathin tunable lens (UTL) comprising two liquid film lenses (LFLs) obtained through aspheric deformation and produced from the surface of a micro-liquid under gravity and boundary tension. The UTL can flexibly change focal lengths between positive and negative lenses when the device thickness is merely 2.15 mm. The proposed lens has the advantages of small volume, light weight, simple fabrication, and independence from external force during zooming. This research makes up for the drawback that traditional solid and liquid lenses cannot further reduce their thicknesses. The proposed UTL provides a new lens form and fabrication method, and can be used to replace solid and liquid lenses for designing miniature zoom optical systems.


Introduction
Micro-optical systems have become a research hotspot in the field of optical design and manufacturing [1][2][3]. However, the shape of traditional solid optical elements is hardly changed after processing. Meanwhile, the central and edge thicknesses of negative and positive lenses must respectively have certain numerical values to ensure the necessary strength of solid-state optical elements and keep them from being easily deformed or damaged in processing. Therefore, this kind of lens cannot easily meet the small structure and large zoom range in modern optical system design [4]. Some scholars have provided new ideas and methods for manufacturing ultrathin lenses [5,6]. Compared with a traditional zoom system, the zoom system made of the metasurface and microelectromechanical system has a smaller volume and larger tunability [7,8]. These methods have the advantages of small volume and variable focal length when manufacturing an optical system. However, this lens type has the disadvantages of high manufacturing costs and complex technology, making it difficult to fulfill the required full-field-of-view clear imaging in large zoom ranges.

Mechanical Analysis
When a small amount of liquid is affected by boundary tension, the surface of the liquid becomes similar to an elastic film under uniform force, thereby tending to reduce the volume of the liquid and bending the LFS. The force and deformation of the LFL in the inverted state are shown in Figure 2a, and those for the positive state are in Figure 2b. In [28], a second-order, second-degree equation of force and deformation on a droplet surface is written as Equation (1). This formula can be used to analyze the surface profile of the droplet in Figure  2.

Mechanical Analysis
When a small amount of liquid is affected by boundary tension, the surface of the liquid becomes similar to an elastic film under uniform force, thereby tending to reduce the volume of the liquid and bending the LFS. The force and deformation of the LFL in the inverted state are shown in Figure 2a, and those for the positive state are in Figure 2b. form a thin curved surface that resembles a liquid film. The liquid film surface (LFS) will produce spherical deformation under the influence of gravity and boundary tension [24,27]. The curvature variation of the refracting interface changes the focus. Given this characteristic, liquid films can be used to replace solid and liquid materials in producing lenses, thereby reducing the volume and processing costs of lenses while increasing zoom ability.

Lens Design
The UTL is obtained by combining the LFL1 and LFL2, as shown in Figure 1b. LFL1 is the LFL in an inverted state, and LFL2 is the LFL in a positive state, as shown in Figure 1a. The structure of the LFL comprises a flat glass, the chamber, and the liquid, as shown in Figure1b. The groove on the chamber can be used to control the volume of liquid in the chamber. Controlling the volume of liquid can change the profile of the LFS, thus changing the diopter of the UTL.

Mechanical Analysis
When a small amount of liquid is affected by boundary tension, the surface of the liquid becomes similar to an elastic film under uniform force, thereby tending to reduce the volume of the liquid and bending the LFS. The force and deformation of the LFL in the inverted state are shown in Figure 2a, and those for the positive state are in Figure 2b. In [28], a second-order, second-degree equation of force and deformation on a droplet surface is written as Equation (1). This formula can be used to analyze the surface profile of the droplet in Figure  2.  In [28], a second-order, second-degree equation of force and deformation on a droplet surface is written as Equation (1). This formula can be used to analyze the surface profile of the droplet in Figure 2.
where x is the horizontal distance to the axis of the drop, z is the vertical coordinate measured away from an origin placed at the point where the axis of rotation cuts the surface of the drop, R is the curvature radius of the drop at the origin, γ is the boundary tension, ρ is the density of fluid, and g is the acceleration of gravity. The deformation model of concave LFS and convex LFS are shown in Figure 3. The boundary of the cavity is set as the x-axis, and the intersection point A of the LFS and the x-axis is selected as the origin. Any point A on the LFS and the origin A shows a difference in gravity. If the point A is higher than the origin A, then the gravity difference is positive, as shown in Figure 3a,d. If the point A is lower than the origin A, then the gravity difference is negative, as shown in Figure 3b,c.
where x is the horizontal distance to the axis of the drop, z is the vertical coordinate measured away from an origin placed at the point where the axis of rotation cuts the surface of the drop, R is the curvature radius of the drop at the origin, γ is the boundary tension, ρ is the density of fluid, and g is the acceleration of gravity. The deformation model of concave LFS and convex LFS are shown in Figure 3. The boundary of the cavity is set as the x-axis, and the intersection point A of the LFS and the x-axis is selected as the origin. Any point A′ on the LFS and the origin A shows a difference in gravity. If the point A′ is higher than the origin A, then the gravity difference is positive, as shown in Figure 3a,d. If the point A′ is lower than the origin A, then the gravity difference is negative, as shown in Figure 3b,c.

Numerical Solution of the Lfs Profile
The relationship between the ordinate z and abscissa x of the points on the LFS cannot be solved by Equation (1) because the radius of curvature R at the origin is unknown. Therefore, the solution formula of the volume of liquid is introduced.
where, Vi and hi are the volume of liquid and the sag of LFS, respectively, both of which are obtained by the ith solution. The positive and negative signs in the formula correspond to the volume of liquid larger than the cavity volume and smaller than the cavity volume, respectively. The numerical solution of the LFS profile can be obtained by combining Equation (1) and Equation (2). The solution process is as follows. (1), the corresponding relationship between ordinate z and abscissa x can be solved by giving a non-zero origin curvature Ri with ODE45, which is a numerical solution function of the ordinary differential equation in MATLAB. 2. With the aid of the trapezoidal numerical integration formula (TRAPZ) in MATLAB, the integral volume of the area surrounded by the LFS can be conducted for the numerical solution of the LFS profile, which is obtained in the first step. The volume of liquid Vi can then be calculated by Equation (2).

Numerical Solution of the Lfs Profile
The relationship between the ordinate z and abscissa x of the points on the LFS cannot be solved by Equation (1) because the radius of curvature R at the origin is unknown. Therefore, the solution formula of the volume of liquid is introduced.
where, V i and h i are the volume of liquid and the sag of LFS, respectively, both of which are obtained by the ith solution. The positive and negative signs in the formula correspond to the volume of liquid larger than the cavity volume and smaller than the cavity volume, respectively. The numerical solution of the LFS profile can be obtained by combining Equation (1) and Equation (2). The solution process is as follows.

1.
With Equation (1), the corresponding relationship between ordinate z and abscissa x can be solved by giving a non-zero origin curvature R i with ODE45, which is a numerical solution function of the ordinary differential equation in MATLAB.

2.
With the aid of the trapezoidal numerical integration formula (TRAPZ) in MATLAB, the integral volume of the area surrounded by the LFS can be conducted for the numerical solution of the LFS profile, which is obtained in the first step. The volume of liquid V i can then be calculated by Equation (2). 3.
The volume of liquid V i obtained by the second step is compared with the given volume of liquid V 0 , and their difference is calculated as Meanwhile, the volume ratio can be calculated as If m i < 10 −4 , then the calculated volume of liquid V i is consistent with the given volume of liquid V 0 and the output relationship between ordinate z and abscissa x.

5.
If m i > 10 −4 , then the calculated volume of liquid V i is inconsistent with the given volume of liquid V 0 . The origin curvature is therefore corrected as R i+1 = w i × R i . R i+1 is substituted into the first step, and Steps (1)-(4) are repeated until the relationship between ordinate z and abscissa x of the LFS profile satisfies the requirements.
The program flow diagram is shown in Figure 4. 3. The volume of liquid Vi obtained by the second step is compared with the given volume of liquid V0, and their difference is calculated as mi = |(Vi − V0) / V0|. Meanwhile, the volume ratio can be calculated as wi = Vi / V0. 4. If mi < 10 −4 , then the calculated volume of liquid Vi is consistent with the given volume of liquid V0 and the output relationship between ordinate z and abscissa x. 5. If mi > 10 −4 , then the calculated volume of liquid Vi is inconsistent with the given volume of liquid V0. The origin curvature is therefore corrected as Ri + 1 = wi × Ri. Ri + 1 is substituted into the first step, and Steps (1)-(4) are repeated until the relationship between ordinate z and abscissa x of the LFS profile satisfies the requirements.
The program flow diagram is shown in Figure 4. For calculation, we specify some parameters of LFL. The cavity diameter D is 5 mm. Water is used as the liquid material to solve the deformation of the LFS profile. The boundary tension of water γ is 72 mN/m, and the density of water ρ is 1 g/mL. The numerical solution of the LFS profile with different volumes of liquid can be obtained by using the above steps. The calculated profiles of LFS are illustrated in Figure 5.

Optical Simulation of the Ultrathin Tunable Lens (UTL)
The UTL can be regarded as a lens consisting of three elements, as shown in Figure 6. For calculation, we specify some parameters of LFL. The cavity diameter D is 5 mm. Water is used as the liquid material to solve the deformation of the LFS profile. The boundary tension of water γ is 72 mN/m, and the density of water ρ is 1 g/mL. The numerical solution of the LFS profile with different volumes of liquid can be obtained by using the above steps. The calculated profiles of LFS are illustrated in Figure 5.  For calculation, we specify some parameters of LFL. The cavity diameter D is 5 mm. Water is used as the liquid material to solve the deformation of the LFS profile. The boundary tension of water γ is 72 mN/m, and the density of water ρ is 1 g/mL. The numerical solution of the LFS profile with different volumes of liquid can be obtained by using the above steps. The calculated profiles of LFS are illustrated in Figure 5.

Optical Simulation of the Ultrathin Tunable Lens (UTL)
The UTL can be regarded as a lens consisting of three elements, as shown in Figure 6.

Optical Simulation of the Ultrathin Tunable Lens (UTL)
The UTL can be regarded as a lens consisting of three elements, as shown in Figure 6. Given that the profile of the deformed LFS is non-spherical, an even aspheric surface can be used to fit the profile of the deformed LFS. The expression of an even aspheric surface can be given as [29] where c is the vertex curvature of the LFS profile, k is the vertex conic constant (which is assumed as -1 (paraboloid)), and A1-A4 are the aspheric surface coefficient. The numerical solution of the LFS profile, which was obtained in Section 2, can be fitted with the even aspheric surface by using Equation (3). Meanwhile, the curvature radius r of the LFS profile can be calculated as r = 1/c.
The formula for calculating the focal length of the UTL can be derived from the formula of refraction between the spherical object-image relationship and the formula of the turning surface, which are expressed as where nk and lk are the index of refraction and the object distance in the object plane, respectively, and n'k and l'k are the corresponding values in the image space. rk is the radius of curvature of the kth surface, and dk is the distance between the kth surface and the (k + 1)th surface.
When the object is located at infinite distance, it is l1 = ∞. According to Equation (4), the expression of focal length of the UTL can be calculated as In this study, the optical parameters of the UTL are shown in Table 1. The relationship between the focal length of the UTL and the injection volume of two LFLs can be obtained by substituting the radius values (r1 and r2) of LFL1 and LFL2 from different volumes fit into Equation (3) and the parameters in Table 1 into Equation (5). The resulting calculation is shown in Figure 7. Given that the profile of the deformed LFS is non-spherical, an even aspheric surface can be used to fit the profile of the deformed LFS. The expression of an even aspheric surface can be given as [29] where c is the vertex curvature of the LFS profile, k is the vertex conic constant (which is assumed as −1 (paraboloid)), and A 1 -A 4 are the aspheric surface coefficient. The numerical solution of the LFS profile, which was obtained in Section 2, can be fitted with the even aspheric surface by using Equation (3). Meanwhile, the curvature radius r of the LFS profile can be calculated as r = 1/c.
The formula for calculating the focal length of the UTL can be derived from the formula of refraction between the spherical object-image relationship and the formula of the turning surface, which are expressed as where n k and l k are the index of refraction and the object distance in the object plane, respectively, and n' k and l' k are the corresponding values in the image space. r k is the radius of curvature of the kth surface, and d k is the distance between the kth surface and the (k + 1)th surface.
When the object is located at infinite distance, it is l 1 = ∞. According to Equation (4), the expression of focal length of the UTL can be calculated as In this study, the optical parameters of the UTL are shown in Table 1. The relationship between the focal length of the UTL and the injection volume of two LFLs can be obtained by substituting the radius  Table 1 into Equation (5). The resulting calculation is shown in Figure 7.

Experiment and Result Discussion
The fabricated UTL is shown in Figure 8. The material of the chamber is glass. The aperture and external diameter of the chamber are 5 and 12 mm, respectively. The thickness of the chamber is 1 mm. A groove for liquid flow is machined on the bottom. The groove is connected with an injector (50 μL) through the rubber conduit, and the gap at the joint is sealed with UV glue. The top chamber, bottom chamber, and flat glass are bonded by UV glue.

Experiment and Result Discussion
The fabricated UTL is shown in Figure 8. The material of the chamber is glass. The aperture and external diameter of the chamber are 5 and 12 mm, respectively. The thickness of the chamber is 1 mm. A groove for liquid flow is machined on the bottom. The groove is connected with an injector (50 µL) through the rubber conduit, and the gap at the joint is sealed with UV glue. The top chamber, bottom chamber, and flat glass are bonded by UV glue.

Experiment and Result Discussion
The fabricated UTL is shown in Figure 8. The material of the chamber is glass. The aperture and external diameter of the chamber are 5 and 12 mm, respectively. The thickness of the chamber is 1 mm. A groove for liquid flow is machined on the bottom. The groove is connected with an injector (50 μL) through the rubber conduit, and the gap at the joint is sealed with UV glue. The top chamber, bottom chamber, and flat glass are bonded by UV glue.  The experimental structure is shown in Figure 9. A resolution target is selected as the object and placed 20 mm from the UTL. The focal length of the UTL can be changed from a negative lens to a positive lens. Therefore, an imaging lens (MV-M1214-MP2-D, from Computer, Japan) with a focal length of 12 mm is combined with the UTL to form a combined lens that resembles the object plate. The image is received by a black and white camera (MV-U3B130GM, from MindVision, Shenszhen, China) with a pixel size of 3.75 µm and sensor size of 1/3 inch. The experimental process is as follows.

1.
The image height of the resolution target y 0 is obtained when no liquid is injected into the UTL, as shown in Figure 10a.

2.
To form a flat lens, 19.6 microbubbles are injected into LFL2.

3.
The zoom of the UTL is controlled by adjusting the volume of liquid injected into LFL1. Simultaneously, with the adjustment of the distance, a clear image and its height y n of the nth zoom can be obtained by the camera. The imaging results of the UTL with different focal lengths are shown in Figure 10b-i. The experimental structure is shown in Figure 9. A resolution target is selected as the object and placed 20 mm from the UTL. The focal length of the UTL can be changed from a negative lens to a positive lens. Therefore, an imaging lens (MV-M1214-MP2-D, from Computer, Japan) with a focal length of 12 mm is combined with the UTL to form a combined lens that resembles the object plate. The image is received by a black and white camera (MV-U3B130GM, from MindVision, Shenszhen, China) with a pixel size of 3.75 μm and sensor size of 1/3 inch. The experimental process is as follows.   The focal length of the UTL at the nth zoom can be calculated by where f n is the focal length of the UTL at the nth zoom, l is the distance between the object plane and the UTL, y 0 is the image height when f = ∞, and y n is the image height at different focal lengths of the UTL. The measured and calculated results of the focal length of the UTL are shown in Table 2, and the focal lengths of the UTL comparison among calculated and measured values are shown in Figure 11. In the experiment, the UTL had a very large zoom range when zoomed from the positive to negative lens. The measured results of focal length are consistent with the calculated values. Because the lens was not designed to eliminate aberration, the result has obvious distortion, as shown in Figure 10b,c.
The zoom results in Figure 10 show that the UTL has certain feasibility in practical applications. However, because the surface deformation of the UTL is based on gravity and surface tension, the UTL can only be applied in cases where gravity is collinear with the optical axis of the lens. In addition, the evaporation of liquids can be avoided by choosing different kinds of liquids (such as silicone oil, polydimethylsiloxane) or by using lens encapsulation. At the same time, the influence of temperature on zoom accuracy can be compensated for by establishing a zoom compensation system for the UTL. These methods can further improve the feasibility of a liquid lens in practical applications.
The liquid is injected into the chamber by an injector and the speed of tunability in Figure 10 is slow. Therefore, in the future research, in order to improve the zoom speed of the UTL, we plan to use a piezoelectric crystal instead of the injector as the liquid actuator, so as to improve the zoom speed and precision of the UTL.
where f′n is the focal length of the UTL at the nth zoom, l is the distance between the object plane and the UTL, y′0 is the image height when f′ = ∞, and y′n is the image height at different focal lengths of the UTL. The measured and calculated results of the focal length of the UTL are shown in Table 2, and the focal lengths of the UTL comparison among calculated and measured values are shown in Figure 11. In the experiment, the UTL had a very large zoom range when zoomed from the positive to negative lens. The measured results of focal length are consistent with the calculated values. Because the lens was not designed to eliminate aberration, the result has obvious distortion, as shown in Figure 10b,c.
The zoom results in Figure 10 show that the UTL has certain feasibility in practical applications. However, because the surface deformation of the UTL is based on gravity and surface tension, the UTL can only be applied in cases where gravity is collinear with the optical axis of the lens. In  The liquid is injected into the chamber by an injector and the speed of tunability in Figure 10 is slow. Therefore, in the future research, in order to improve the zoom speed of the UTL, we plan to use a piezoelectric crystal instead of the injector as the liquid actuator, so as to improve the zoom speed and precision of the UTL.

Conclusions
This paper studies the surface deformation of a micro-liquid under boundary tension. This research makes up for the drawback that traditional solid and liquid lenses cannot further reduce their thicknesses. The UTL proposed in this paper only relies on the effect of boundary tension to produce surface bending. Consequently, the focal length between positive and negative lenses can be flexibly changed while the device thickness remains only 2.15 mm. Compared with solid and liquid lenses, the UTL has the advantages of small volume, simple fabrication, and independence from complex institutions during zooming due to its new surface-forming method. Therefore, the UTL can be used instead of solid or liquid lenses to design optical zoom systems, reduce their volume and weight, and improve their zoom ability and accuracy. In future work, improvement of the structure and manufacturing technology of the UTL will be continued to enhance its imaging effect, and the combination zoom and aberration correction of several UTLs will be further discussed.

Conclusions
This paper studies the surface deformation of a micro-liquid under boundary tension. This research makes up for the drawback that traditional solid and liquid lenses cannot further reduce their thicknesses. The UTL proposed in this paper only relies on the effect of boundary tension to produce surface bending. Consequently, the focal length between positive and negative lenses can be flexibly changed while the device thickness remains only 2.15 mm. Compared with solid and liquid lenses, the UTL has the advantages of small volume, simple fabrication, and independence from complex institutions during zooming due to its new surface-forming method. Therefore, the UTL can be used instead of solid or liquid lenses to design optical zoom systems, reduce their volume and weight, and improve their zoom ability and accuracy. In future work, improvement of the structure and manufacturing technology of the UTL will be continued to enhance its imaging effect, and the combination zoom and aberration correction of several UTLs will be further discussed.