1. Introduction

Since Lippmann’s study in 1875 [1], electrowetting has been widely investigated as an effective means for small-scale liquid handling, due to the dominance of surface tension forces over body forces in micro/meso scales [2]. When an electric potential is applied between a liquid droplet and a solid electrode, charge re-distribution occurs, modifying the surface tension at the liquid–solid interface. This process results in the expansion of the surface area by reducing the work due to the like-charge repulsion. The resulting contact angle (θ) of a liquid droplet can be mathematically estimated using the Young-Lippmann equation as [3,4]:

In recent years, the electrowetting technology has become increasingly popular as an actuation mechanism of liquid prisms for beam steering [23–25]. The concept of this liquid-filled prism was first introduced by Heikenfeld’s group [26]. A prism was fabricated in a rectangular cuvette coated with a hydrophobic and dielectric layer on an electrode surface and then filled with water in ambient air as two immiscible fluids. By applying electric potentials, the contact angles of the water on the prism’s sidewalls were altered, as predicted by the Young-Lippmann Eq. (1), resulting in the air-water interface flat with a tilted angle (i.e., the prism’s apex angle) using the electrowetting effect. The refractive index difference (n_air ≠ n_water) between air and water caused an incident ray to refract at the fluidic interface tilted at the prism angle, enabling effective beam steering without the use of bulky mechanical moving parts. With the refractive index (n_water ≈ 1.359) of the water solution mixed with glycerin experimentally used, the beam steering performance can be estimated to be as large as ± 7°, but water evaporation was an issue as it was exposed to air [26]. To address this, a two-liquid prism was later developed using water and silicone oil as two immiscible liquids [27]. This binary combination provided greater stability and larger modulation of the prism angle compared to the air-water system. However, silicone oils typically used for electrowetting studies do not provide a high refractive index (n_silicone = 1.38 ∼ 1.45), compared to that of water (n_water ≈ 1.33). Consequently, the beam steering performance achieved by such a binary combination as water and silicone oil is very limited (less than 3.6°). In order to enable a wide beam steering capability for the liquid prism, our group proposed a new non-conductive oil, called 1-bromonaphthalene (1-BN), which has a refractive index as high as n = 1.657 [28]. By combining this high-refractive-index oil with water, the beam steering performance could be significantly enhanced as wide as -18.49° ≤ β ≤ 19.06° in a double-stacked prism configuration, demonstrating the highest beam steering performance ever achieved using electrowetting.

This liquid prism technology has been further employed in practical applications related to optics and solar energy [10–12,23,29,30]. For example, Enrico et. al. presented an optofluidic Fresnel lens configured in a linear array of liquid prisms [29]. A Fresnel lens is an important optical device that approximates the performance of a conventional bulk lens, while using far less lens material for both cost and space effectiveness [31]. However, the use of solid-type materials like glass or plastic restricts its optical performance such as a focal length once the lens geometry has been established. To provide a high degree of optical tunability for a Fresnel lens, liquid prisms were assembled into a linear array to replace the subsections of a solid Fresnel lens. By modulating individual prisms, incoming light rays could be steered to spatially control the focal point’s location. This optofluidic tunable Fresnel lens demonstrated a notable focal tunability along both longitudinal (263 mm ≤ f_long≤ ∞) and lateral (0 ≤ f_lat≤ 30 mm) directions. Another intriguing application of liquid prisms was in solar indoor lighting to reduce lighting electricity in buildings [10]. Rooftop sunlight is guided to individual rooms via a waveguide, on the bottom of which liquid prisms are linearly integrated. By controlling the prism’s apex angle, up to 98.6% of the input solar power can be utilized for interior illumination. However, all previous works [10,15,28,29,32] were limited to linear optical tuning, failing to fully demonstrate focal tunability in 3D free space, although there is a strong demand of rapid 3D focal control for advanced sensing and telecommunication applications.

In this study, we present a 3D focal control system consisting of an n × n optofluidic prism array. By dynamically controlling individual prisms via electrowetting, incoming rays can be rapidly manipulated to converge on a focal point at P_focal (f_x, f_y, f_z) located in 3D free space, without the need for bulky and complex mechanical moving parts. Detailed analytical studies have been carried out to precisely predict the prism operation required for achieving 3D focal control. For experimental concept demonstrations, we fabricated three liquid prisms aligned along x-, y-, and 45°-diagonal axes, representing segments of the arrayed optofluidic system. By modulating individual prisms symmetrically or asymmetrically, incoming light rays were manipulated to control the focal point’s location in 3D space. Experimentally, we demonstrated 3D focal tunability of the arrayed system by performing light focusing along lateral, longitudinal, and axial directions with ranges of 0 ≤ f_x ≤ 30 mm, 0 ≤ f_y ≤ 30 mm, and 500 mm ≤ f_z ≤ ∞. Compared to our preliminary work [33], this study presents significant advancements and improvements by providing mathematical derivations to precisely predict the optical performance for 3D focal tunability. Furthermore, the study incorporates in-depth experimental results, such as various prism operations in 3D space, axial focal control along the central lens axis, and 3D focal control featuring both lateral and longitudinal offsets while altering focal planes. This compact optofluidic device offers a novel lens capability for 3D focal control, potentially applicable in areas such as eye-movement tracking for smart displays, autofocusing of smartphone cameras, or smart solar tracking for compact concentrated photovoltaic systems.

2. Overview of an arrayed optofluidic prism system for 3D focal control

Figure 1 illustrates schematics of the optofluidic beam steering system that enables spatial focal control of incoming rays in 3D free space. It is designed as an n × n array of a tunable liquid prisms, each containing two immiscible liquids within a rectangular cuvette. Due to the Laplace pressure difference, the liquid-liquid interface naturally forms a curved meniscus with an initial contact angle on the prism sidewall surface [28]. When bias voltages are applied to the prism’s sidewalls, the electrowetting effect is induced, modifying the surface tension forces, and resulting in the contact angle changes. This alteration in surface energy allows the fluidic interface to be flat with the prism’s apex angle tilted in 3D. With the two liquids having different refractive indices, incoming light can be adaptively manipulated for 3D beam steering without the use of any mechanical moving components like gimbaled mirrors. To achieve focal tunability in 3D space, each prism in the arrayed optofluidic system is individually controlled to have its own prism angle using the electrowetting principle. Consequently, incoming rays can be spatially manipulated for light focusing on a focal point located at P_focal (f_x, f_y, f_z) in 3D space. This 3D optical tunability of the system enables flexible modulation of the lens power, which could not be achieved by solid-type optics without bulky and complex mechanical moving components.

 

Fig. 1. A schematic of an arrayed optofluidic system for 3D focal control. (a) An optofluidic beam steering system is configured in an n × n array of a liquid prism for 3D focal control. (b) A single liquid prism module is filled with two immiscible liquids in a rectangular cuvette. With the application of bias voltages to the prism’s sidewalls, the electrowetting effect modifies the surface tension forces, forming a straight profile of the fluidic interface with an apex angle of the prism. Due to the refractive index differences (n₁ ≠ n₂ ≠ n_air) between the media, an input ray is adaptively steered at the interfaces. By symmetrically or asymmetrically controlling individual prisms in the arrayed system, light focusing can be achieved in 3D free space, truly offering a new lens capability for 3D focal control without the need of bulky and complex mechanical moving components.

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3. Prism coordinates

We first introduce the prism coordinates for predicting the optical performances of the arrayed optofluidic system. Figure 2 shows the vertical plane at the azimuth angle (ω) where the beam path steered by the prism is placed on. A focal point is located at P_focal (f_x, f_y, f_z) in 3D space. The x-y plane at z = 0 is set to be placed at the center of the prism with a height of 2 h. Thus, its center position at P_C can be expressed in vector form as:

 

Fig. 2. Beam path steered by a liquid prism is viewed on the vertical plane at the azimuth angle (ω) from an x axis.

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Using the electrowetting principle, the interface between two immiscible liquids can be adjusted to create a flat surface with the prism’s apex angle (φ) tilted in 3D, as shown in Fig. 2. Assuming an incoming ray is projected perpendicularly from the prism bottom (i.e., α = 0°), the ray undergoes refraction at the interfaces due to the refractive index difference (n₁ ≠ n₂ ≠ n_air) between the media. The angles of incident and transmitted light are denoted by α and β at each interface, with the subscripts 1 and 2 representing liquid 1 and 2. Using the geometrical relations and Snell’s law, the angle β by which the ray is steered as it exits the prism can be calculated as [28]:

An intersection point of the ray on the top surface of the prism is denoted as P_A (see Fig. 2). The beam vector steered in the liquid 2 is denoted as $\vec{u}$, which can be expressed by multiplying its magnitude (i.e., the distance between two points of P_A and P_C) with a unit direction vector of the beam as:

Equation (6) provides a general expression for the azimuth angle, ω, estimated using the function, atan2. This function is a two-argument arctangent typically used for the angle measurement, considering the four-quadrant inverse tangent based on the two inputs of (f_y – y) and (f_x – x). Furthermore, the steered beam vector in air is denoted as $\vec{v}$, which can be expressed as:

By substituting Eqs. (2), (4), and (7) into Eq. (8), the focal point at P_focal (f_x, f_y, f_z) can be finally obtained in a vector form as:

With the given information on prism geometry, optical properties of liquids, prism location at P_C (x, y, 0), and desired focal point P_focal (f_x, f_y, f_z), Eq. (9) can be used to predict the 3D focal tunability of the arrayed optofluidic system.

4. Prism angle control in 3D

4.1 Analytical study

To accomplish 3D focal control in the arrayed system, it is crucial to modulate the prism’s apex angle (φ) in 3D. This section discusses both analytical and experimental studies on the control of the prism angle in 3D space. When bias voltages (V_F, V_B, V_L, and V_R) are separately applied at four (front, back, left, and right) sidewalls of the prism, respectively, the surface tension forces are correspondingly modified, leading to changes in the contact angles on each sidewall of the prism. Using the Young-Lippmann Eq. (1), the resultant contact angles (θ_F, θ_B, θ_L, and θ_R) altered from the initial angle θ₀ on four sidewalls are mathematically estimated as:

Since the fluid-fluid interface is spatially modulated, it is difficult to determine the orientation of the interface and the prism angle (φ) tilted in 3D space from experiments. To estimate these values, we have used two characteristic parameters simply measured from experiments. Figure 3 illustrates the left-side and front views of the prism when the fluidic interface is spatially modulated with the prism angle (φ). The fluidic interface appears as a dark stripe, due to reduced light transmission through the three-dimensionally manipulated interface. This dark stripe can be characterized by two tilted angles (δ_side and δ_front) observed on the left and front sidewalls of the prism, as presented in Fig. 3. To maintain the straight profile (i.e., θ_L+ θ_R = 180° and θ_F+ θ_B = 180°) of the interface with the prism angle (φ) in 3D, the contact angles are modulated in such a manner that fulfils the following relations:

 

Fig. 3. (a) Left-side and (b) front views of the prism. When bias voltages are separately applied to four sidewalls of the prism, the fluid-fluid interface is controlled via electrowetting to have its straight profile with the prism angle (φ) in 3D space. Due to less light transmission, the fluidic interface is visualized as a dark stripe on the left-side and front views of the prism. The tilted angles (δ_front and δ_side) of the dark stripe observed from experiments can be simply used to confirm the orientation of the interface and the prism angle (φ) achieved by controlling the prism in 3D.

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Using the geometrical relations, these two observation parameters, δ_side and δ_front, can be further expressed as a function of φ and ω as:

Thus, Eq. (12) informs us that two observation parameters, ${\delta _{\textrm{side}}}$ and ${\delta _{\textrm{front}}}$, can be simply used to confirm the prism angle (φ) and the azimuth angle (ω). To further mathematically express the orientation of the fluidic interface, a unit direction vector ($\hat{n}$) normal to the interface (see Fig. 2) can be similarly estimated as a function of φ and ω as:

Combining Eqs. (10), (11), and (12), one can know the bias voltages required to obtain the fluidic interface directed to a unit normal vector of $\hat{n}$ with the prism angle (φ) achieved by controlling the prism in 3D.

With such prism operations, an incoming ray can be spatially steered at an angle (β) away from a vertical line when the ray exits the prism in 3D space, as depicted in Fig. 2. Similar to the prism angle (φ), the value of β steered in 3D is not easily estimated from experiments. To determine the magnitude of the steered angle (β) and the direction ($\hat{v}$) the ray points out, we have similarly employed two observable parameters, the angles (β_side and β_front) of the ray steered away from a vertical line. These angles can be easily obtained from the left-side and front views of the prism, respectively (Fig. 3). Their relationships can be similarly expressed as:

Furthermore, a unit direction vector ($\hat{v}$) of the beam pointing out in an air medium can be obtained from Eq. (7) as:

Equations (14) and (15) tell us that the angle (β) the beam is steered in 3D and its orientation the beam points out can be simply confirmed using two parameters of β_side and β_front observed from experiments.

4.2 Experimental demonstrations for 3D prism operation

The above analytical study for 3D prism angle control has been experimentally demonstrated. The fabrication procedures of a liquid prism have been well described in our previous study [28]. In brief, four pieces of an indium tin oxide (ITO) substrate was assembled using a UV-curable epoxy (NOA 61, Norland) to create a hollow cuvette structure with an opening size of 10 mm × 10 mm. With this characteristic length in a prism size, the Bond number (Bo), a dimensionless quantity representing the ratio between gravitational forces and surface tension forces, was calculated to be much less than 1. This indicates that the surface tension force is dominant over gravitational effects for our prism study. To provide a hydrophobic and dielectric layer, this study utilized a dip-coating method (L2006A1, Ossila) with a 6% Teflon solution (AF 1601, Chemours). This one-step dip-coating process allows coating a Teflon layer over the entire surface of the hollow structure, which is much simpler than a spin-coating method requiring four repetitions for coating on each of the four sidewalls. At a withdrawal speed of 200 mm/min, a 670 nm thick hydrophobic layer was obtained, after which being cured on a hot plate at 110°C for 12 hours. The hollow cuvette was placed on a bottom ITO plate and then filled with water and silicone oil as two immiscible liquids for experimental demonstrations of 3D interfacial control of the prism using electrowetting.

Figure 4(a) shows the left-side and front views of the prism experimentally obtained, when direct current (DC) bias voltages (V_F = 143 V, V_B = 142 V, V_L = 160 V, and V_R = 110 V) were applied to four sidewalls of the prism. Using the electrowetting effect, the fluidic interface was modulated to have its straight profile, where the interface appeared as a dark stripe characterized with a tilted angle of δ_side = 0° on the left-side view, while δ_front = -5° on the front view, respectively. It is noted that a minus sign for δ_front indicates the angle clockwise. By substituting these two observation parameters of δ_side and δ_front into Eq. (12), it is simply obtained that φ = 5° and ω = 0°, indicating the magnitude of the prism angle as φ = 5° and the azimuth angle at ω = 0° (i.e., the beam is steered to point out along a positive x direction). Furthermore, using Eq. (13), a unit direction vector normal to the interface can be expressed as $\hat{n} = \left[ {\begin{array}{{ccc}} { - \sin ({5^\circ } ),}&{0,}&{\cos ({5^\circ } )} \end{array}} \right]$ to indicate its orientation. Thus, the prism operation presented in Fig. 4(a) demonstrates the successful modulation of the fluid-fluid interface via electrowetting to have the straight profile with ω = 0°, φ = 5°, and $\hat{n} = \left[ {\begin{array}{{ccc}} { - 0.087,}&{0,}&{0.996} \end{array}} \right]$. Figure 4(b) shows another prism operation with two tilted angles of the interface at δ_side = 8° and δ_front = -11° observed from the left-side and front views of the prism, when DC bias voltages (V_F = 162 V, V_B = 116 V, V_L = 158 V, and V_R = 116 V) were applied to four sidewalls of the prism. Similarly, using Eq. (12) with the two observation parameters of δ_side and δ_front in Fig. 4(b), we obtained ω = 35.9° and φ = 13.4°. These estimations imply that the magnitude of the prism angle is at φ = 13.4° and the azimuth angle of the vertical plane on which the beam vector is situated at ω = 35.9° from a x axis. Additionally, a unit direction vector normal to the interface can be correspondingly obtained as $\hat{n} = \left[ {\begin{array}{{ccc}} { - 0.188,}&{ - 0.136,}&{0.973} \end{array}} \right]$ using Eq. (13). Figure 4(c) shows the prism operation with its apex angle at φ = 0°. Both left-side and front views show two tilted angles as δ_side = 0° and δ_front = 0° when bias voltages (V_F = 145 V, V_B = 158 V, V_L = 150 V, and V_R = 163 V) were applied to four sidewalls of the prism. Consequently, the prism angle and the unit direction vector of the interface are given as φ = 0° and $\hat{n} = \left[ {\begin{array}{{ccc}}0,&{0,}&1 \end{array}} \right]$. This specific prism operation allows an incident ray to pass through the prism without any light refraction at the interfaces.

 

Fig. 4. Experimental validations of the fluid-fluid interface three dimensionally manipulated via electrowetting. (a) The fluidic interface is modulated to have its straight profile with the prism’s apex angle at φ = 5°, the azimuth angle at ω = 0°, and its unit direction vector of $\hat{n} = [{ - 0.088,\,\,0,\,\,0.996} ]$ which are ensured using the observation parameters of δ_side = 0° and δ_front = -5° observed from the left-side and the front views of the prism, respectively. (b) Control of the fluidic interface in 3D is visualized as a dark stripe that is characterized with two tilted angles, δ_side = 8° and δ_front = -11° observed from the left-side and front views. These two observation parameters ensure 3D prism control with ω = 35.9°, φ = 13.4°, and $\hat{n} = [{ - 0.188,\,\, - 0.136,\,\,0.973} ]$. (c) The prism operation showing δ_side = 0° and δ_front = 0° confirms that the fluidic interface was modulated to have φ = 0° and $\hat{n} = [{0,\,0,\,1} ]$ where an incoming ray passes through the prism with no light refraction.

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The experimental results presented in Fig. 4 successfully demonstrate various prism operations to implement 3D beam steering using the electrowetting principle. Moreover, by utilizing the two observation parameters, ${\delta _{\textrm{side}}}$ and ${\delta _{\textrm{front}}}$, obtained from experiments, we can simply confirm the azimuth angle (ω), the prism’s apex angle (φ), and the unit direction vector ($\hat{n}$) of the fluidic interface achieved for 3D focal control.

5. Axial focal control (f_z) along the lens axis

The ability to achieve axial focal control in the arrayed optofluidic system, specifically directing light to focus along the z-axis at a focal point denoted as P_focal (0, 0, f_z), is enabled by symmetric modulation of individual prisms within the system. This section delves into both analytical and experimental approaches to ascertain the necessary prism operations for accomplishing axial focal control.

5.1 Analytical study

A key consideration for axial focal control is to find f_z as a function of φ for the prism located at P_C (r, θ, 0) expressed in cylindrical coordinates, where r = $\sqrt {{x^2} + {y^2}} $ is a radial distance of P_C from an origin and θ = atan2 (y, x) is an azimuth angle of the prism location obtained using the four-quadrant inverse tangent (see Fig. 2). For axial focal control, several variables are constrained such that f_x = f_y = 0 and two vectors $\vec{r}$ and $\vec{v}$ are placed on the same vertical plane, but in opposite directions (i.e., $\vec{r}$ directs to outward, while $\vec{v}$ directs to inward to the origin). Therefore, the following constraint is additionally provided for the azimuth angles of two vectors $\vec{r}$ and $\vec{v}$ as:

With these constraints for axial focal control, f_z can be estimated from the geometric relations as:

By substituting Eqs. (3) and (5) into Eq. (17), the axial focal length f_z on the z axis can be graphically represented as a function of φ for various prism locations, r, in a radial direction, as shown in Fig. 5. For this analytical study, a liquid prism filled with binary liquids, 1-bromonaphthalene (1-BN) as a high-refractive-index oil (n₁ = 1.65) and water (n₂ = 1.33) [28], was assumed. The height of the prism is set at 2 h = 25 mm, which will be consistent with the subsequent experimental conditions. Figure 5 provides critical insight into the required prism angle (φ) for the prism positioned at r in the radial direction to achieve the ray focusing on a particular focal length, f_z, on a z axis. In general, a large apex angle is required for larger beam steering of incoming light, which leads to high lens power with a short focal length. Similarly, for the prisms with large r values (i.e., further from the origin), a high apex angle is required to compensate for their distance from the central axis. an incident ray to pass through the prism without any light refraction at the interfaces.

 

Fig. 5. Axial focal tunability of the arrayed optofluidic system. An axial focal length f_z on a z axis is presented as a function of the prism angle φ for various prism distances of r in a radial direction from an origin. At higher prism angles, more beam steering and thus shorter focal lengths can be achieved, resulting in higher lens power. Four data points added in the graph indicate the axial focal length f_z obtained from experimental tests shown in Fig. 7 (b) and (c).

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To illustrate this study, we may arbitrarily choose P_focal (0, 0, f_z = 600 mm) as the desired focal point on the z axis, which is indicated as a dotted line in Fig. 5. Each prism in the array system would be modulated to have the prism angle relying on the radial distance of the prism from an origin. Consider two prisms located at P_C1 (r₁ = 10 mm, θ₁ = 0°, 0) and P_C2 (r₂ = 30$\sqrt 2 $ mm, θ₂ = 45°, 0) expressed in cylindrical coordinates. To achieve the ray focusing at ƒ_z = 600 mm on the z axis, the prism angles required for these two prisms are obtained from the graph in Fig. 5 as φ₁ = 2.9° and φ₂ = 12.1°, respectively. To further present the directions normal to the fluidic interface of the prisms, Eq. (13) can be used to express their unit direction vectors as ${\hat{n}_1} = [{ - 0.051,\,0,\,0.999} ]$ for the prism at P_C1 and ${\hat{n}_2} = [{ - 0.148,\,\, - 0.148,\,\,\,0.978} ]$ at P_C2, respectively.

For any other focal length ƒ_z desired to be achieved, the required prism angles may be similarly determined for any prism location. For comparison with our analytical studies, four experimental data points are added to the graph in Fig. 5 when r = 30 mm and r = 30$\sqrt 2 $ mm to demonstrate the system’s capability for axial focal control. Detailed experimental results will be discussed in a following section.

5.2 Experimental demonstrations for axial focal control

For experimental demonstrations of axial focal tunability, an experimental setup was built up as presented in Fig. 6. To represent segments of the arrayed optofluidic system, three liquid prisms were fabricated and positioned at P_C1 (r₁ = 30 mm, θ₁ = 0°, z₁ = 0), P_C2 (r₂ = 30$\sqrt 2 $ mm, θ₂ = 45°, z₂ = 0), and P_C3 (r₃ = 30 mm, θ₃ = 90°, z₃ = 0) which are expressed in cylindrical coordinates. To enhance the beam steering performance, the prisms were then filled with water and 1-bromonaphthalene (1-BN) whose refractive indices are largely different each other (n_water= 1.33 and n_1-BN= 1.65) [28]. To independently control individual prisms, DC bias voltages were applied to the four sidewalls of each prism using a LabView-controlled multi-channel voltage source (PFR 100 M, GW INSTEK). Collimated laser beams at 543 nm wavelength were projected from the bottom of the three prisms to represent incoming rays. To visualize beam pathway through the three prisms, their views in the left and front sides were recorded using additional CCD cameras (CS135CU, Thorlabs), as presented in Fig. 6. Beam pathways through the prisms at P_C1 and P_C2 are presented in the left-side view, while the front view shows the beam pathways through the prisms at P_C2 and P_C3, respectively. A grid paper is placed perpendicular to the axial axis to show us a top view of the focal plane where three laser beams are converged after beam steering through the three prisms. respectively.

 

Fig. 6. An experimental setup for demonstration of 3D focal tunability of the arrayed optofluidic systems. Three liquid prisms are positioned at P_C1 (r = 30 mm, θ = 0°), P_C2 (r = 3$\sqrt 2 $ mm, θ = 45°), and P_C3 (r = 30 mm, θ = 90°,) at z = 0 to represent segments of the arrayed system. Laser beams are projected from the bottom of the three prisms to represent incoming light rays. To visualize 3D focal control of the laser beams, the side view shows beam pathways for the prisms located at P_C1 and P_C2, while the front view presents beam pathways through the prisms at P_C2 and P_C3, respectively. A grid paper was used to visualize a top view of the focal plane where three laser beams converges after refraction through the three prisms.

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Figure 7 shows experimental results to demonstrate the lens tunability for axial focal control along a z axis. With the bias voltages (V_F = 165 V, V_B = 165 V, V_L = 169 V, and V_R = 169 V) applied to the sidewalls of the three prisms, they were modulated to have the prism angles at φ₁ = φ₂= φ₃ = 0° and the direction vectors normal to the fluidic interfaces as ${\hat{n}_1} = {\hat{n}_2} = {\hat{n}_3} = [{0,\,\,0,\,\,1} ]$ for three prisms (Fig. 7(a)). As a result, incoming laser beams just pass through the prisms without any light refraction at the interfaces. Figure 7(a) shows the top view of a grid paper placed at z = 500 mm, where the three laser spots are positioned closely to the prism locations. As indicated in the graph of Fig. 5, a focal point is said to be at infinity, i.e., f_z ≈ ∞, for zero prism angles at φ₁ = φ₂= φ₃ = 0°. The beam pathways visualized in the left-side and front views of Fig. 7(a) support the fact that there was no refraction of the laser beams through the prisms by showing β₁ = β₂= β₃= 0° for all three prisms. Thus, unit direction vectors of the steered beams pointing out are given as ${\hat{v}_1} = {\hat{v}_2} = {\hat{v}_3} = [{0,\,\,0,\,\,1} ]$ from Eq. (15).

 

Fig. 7. Experimental demonstrations for axial focal control along the lens axis. (a) All prisms are modulated to have their apex angles at φ₁ = φ₂= φ₃ = 0°. Consequently, the incident rays just pass through the prisms without any light refraction to have the beam steering angles at β₁ = β₂= β₃= 0°, resulting in the axial focal length at f_z ≈ ∞. (b) A subsequent prism operation switched an axial focal length from f_z ≈ ∞ to ƒ_z ≈ 500 mm along a z axis by modulating the prisms to have φ₁ = 10.2°, φ₂= 14.6°, and φ₃ = 10.3°. With the projected angles of the transmitted beams, β_side and β_front, observed from the prism views in the left and front sides, the beam steering angles for each prism are estimated as β₁ = 3.54°, β₂= 4.8°, and β₃= 3.4°. (c) By further modifying the prism angles to φ₁ = 7.4°, φ₂= 9.1°, and φ₃ = 7.5°, the steered angles of the rays were controlled to be at β₁ = 2.3°, β₂= 3.3°, and β₃= 2.27°. As a result, an axial focal length increases from f_z ≈ 500 mm to f_z ≈ 750 mm to reduce the lens’ focusing power using the electrowetting principle without extra bulky mechanical moving parts.

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Figure 7(b) shows subsequent test results to demonstrate the axial focal control at f_z = 500 mm by symmetrically modulating the prisms. Using Eq. (16), the azimuth angles of the steered rays from the x axis for three prisms are given as ω₁ = 180°, ω₂ = 225°, and ω₃ = 270°, respectively. With the applications of the bias voltages to three prisms at P_C1 (V_F = 143 V, V_B = 147 V, V_L = 180 V, and V_R = 120 V), P_C2 (V_F = 220 V, V_B = 140 V, V_L = 230 V, and V_R = 123 V), and P_C3 (V_F = 180 V, V_B = 120 V, V_L = 143 V, and V_R = 147 V), the prism angles were modified to have φ₁ = 10.2°, φ₂= 14.6°, and φ₃ = 10.3° for the three prisms, which closely match with the values estimated from the graph of Fig. 5. The incoming laser beams were correspondingly steered for light focusing on P_focal = (0, 0, 500 mm). To further visualize it, a grid paper was placed at z = 500 mm where the top view of Fig. 7(b) shows three laser beams converged near to the origin to demonstrate the lens power tuned from f_z ≈ ∞ to f_z ≈ 500 mm along a z axis. The beam pathways manipulated by three prisms are also presented in the views of the left and front sides of Fig. 7(b), from which two observation parameters, β_side and β_front, are simply recognized for each prism as β_1-side = 0° and β_1-front = -3.54° for the prism at P_C1, β_2-side = 3.35° and β_2-front = -3.4° for P_C2, and β_3-side = 3.4° and β_3-front = 0° for P_C3, respectively. A small disparity between the values of β_2-side and β_2-front for the prism at P_C2 could be from an experimental error or uncertainty. Using Eq. (14) and the values of β_front and β_side observed from experiments, the steered angles of the rays for each prism can be estimated as β₁ = 3.54°, β₂= 4.8°, and β₃= 3.4°. Similarly, we can have unit direction vectors of the steered beam for three prisms as ${\hat{v}_1} = [{ - 0.062,\,\,\,0,\,\,\,0.998} ]$, ${\hat{v}_2} = [{ - 0.059,\,\, - \,0.059,\,0.997} ]$, and ${\hat{v}_3} = [{0,\,\,\, - 0.059,\,\,0.998} ]$ estimated from Eq. (15).

Figure 7(c) shows prism operations that further enable to reduce the lens’ focusing power by increasing an axial focal length from f_z ≈ 500 mm to f_z ≈ 750 mm. We have the same azimuth angles of the steered rays for three prisms as before. When bias voltages were applied to the prism at P_C1 (V_F = 139 V, V_B = 137 V, V_L = 165 V, and V_R = 130 V), P_C2 (V_F = 215 V, V_B =150 V, V_L = 230 V, and V_R = 162 V), and P_C3 (V_F = 165 V, V_B = 130 V, V_L = 139 V, and V_R = 137 V), the three prisms were symmetrically controlled to have the prism angels at φ₁ = 7.4°, φ₂= 9.1°, and φ₃ = 7.5°, which are very close to the ones estimated in the graph of Fig. 5. Such prism operations enabled to steer the incoming laser beams to focus on the focal point at P_focal = (0, 0, 750 mm). The top view in Fig. 7(c) supports the convergence of the beams on around the origin at z = 750 mm. The left-side and front views in Fig. 7(c) also shows the beam pathways with the steered angles of β_front and β_side observed for each prism. Using Eq. (14) and the observation values of β_front and β_side, the steered angles of the rays for each prism can be similarly estimated as β₁ = 2.3°, β₂= 3.3°, and β₃= 2.27°. Thus, unit direction vectors of the steered beam are given for three prisms as ${\hat{v}_1} = [{ - 0.04,\,\,0,\,\,\,0.999} ]$, ${\hat{v}_2} = [{ - 0.041,\,\,\, - 0.041,\,\,0.998} ]$, and $\hat{v} = [{0,\,\, - 0.058,\,\,\,0.998} ]$, respectively.

Experimental results in Fig. 7 show the concept demonstration for the tunable lens power by symmetrically modulating the prisms for axial focal control, ranging 500 mm ≤ f_z ≤ ∞ along the direction of the lens axis without the need of bulky and complex mechanical moving parts. In addition, the observation parameters, β_front and β_side, obtained from experiments can be simply used to ensure the angle (β) the beam is steered in 3D.

6. 3D focal control

This section will discuss both analytical and experimental studies on the prism operation required to achieve the lens power control in 3D through asymmetrical modulation of the prism array.

6.1 Analytical study

For the study of axial focal control discussed in the previous section, we could have several variables constrained as f_x = f_y = 0 and ω = θ + 180°. Thus, the axial focal length f_z on the z axis was expressed as a function of φ for the prism at a particular location (r) along a radial direction (see Fig. 5). However, for 3D focal control, all incoming rays would be converged on any location in 3D space. Thus, it is required to confine at least one variable for the study of 3D focal control. To address it, Eq. (8) was first used to find the location of a desired focal point at P_focal (f_x, f_y, f_z) relative to a prism location at P_C (x, y, 0), which can be expressed in a vector form as:

In Eq. (19), the vector, $\vec{\mu }$, is determined when prism geometry and optical properties of liquids are fixed. Thus, $\vec{\mu }$ can be expressed as a function of φ and ω. However, the beam vector, $\vec{v}$, steered in an air medium is not constrained. A light ray continues to travel infinitely in the direction of the unit vector, $\vec{v}$. To express $\vec{v}$ as a function of φ and ω, similar to $\vec{\mu }$, the magnitude A of the vector $\vec{v}$ (i.e., the distance between P_A and P_focal) must be constrained by specifying one of the values f_x, f_y, and f_z. For this study, a location of the axial focal plane (i.e., f_z) along the z axis was conveniently fixed to constrain the beam vector $\vec{v}$. Hence, we can have a circular plot created by (f_x – x) and (f_y – y) as a function of φ and ω on the focal plane specified at f_z, using Eq. (19).

Figure 8 shows a circular color gradient plotted on the given focal plane at f_z = 500 mm to present a possible location of the desired focal point relative to the prism. In the circular plot of Fig. 8, a radial length from the center indicates the magnitude of the prism angle φ as indicated in a color bar, while its polar angle from the x axis shows the azimuth angle ω of the beam manipulated by a liquid prism. For example, consider the desired focal point relative to the prism position to be (f_x – x) = 40 mm and (f_y – y) = 60 mm, which is presented as a white circular point in Fig. 8. Its radial distance from the center and its polar angle are calculated as φ = 23.6° and ω = 56.3°.By substituting these two values φ and ω into Eq. (12), the tilted angels of the fluidic interface observed on the left and front side views of the prism can be obtained as δ_side = 19.98° and δ_front = -13.63°, which indicates the prism operation required to achieve the desired beam steering performance. For comparison with the analytical studies, various experimental data points are added to the graph in Fig. 8 to determine the values of φ and ω required for prism operations. Detailed experimental results for 3D focal control at f_z = 500 mm will be discussed in a following section.

 

Fig. 8. The desired focal point relative to the prism position can be possibly located within the circular range plotted on the focal plane fixed at f_z = 500 mm as a function of φ and ω. A radial distance from the center indicates the magnitude of the prism angle φ, while its polar angle from the x axis shows the azimuth angle ω of the beam steered by a liquid prism. For comparison study, experimental test results shown in Fig. 9 are added as data points for each prism.

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6.2 Experimental demonstrations for 3D focal control at f_z = 500 mm

An asymmetrical modulation of the prism array enables focal control in 3D space. For this study, three liquid prisms were positioned as the same as the previous study in section 5.2, but their positions are expressed in Cartesian coordinates for P_C1 (x₁ = 30 mm, y₁ = 0, z₁ = 0), P_C2 (x₂ = 30 mm, y₂ = 30 mm, z₂ = 0), and P_C3 (x₃ = 0, y₃ = 30 mm, z₃ = 0). All other experimental conditions remain the same as before. To present 3D focal tunability of the arrayed system, light focusing tests were repeated on two different focal planes at f_z = 500 mm and f_z = 750 mm.

Figure 9(a) presents experimental demonstrations for light focusing on the focal point at P_focal (f_xa = 30 mm, f_ya = 0, f_z = 500 mm), when bias voltages were applied to the prisms at P_C1 (V_F = 167 V, V_B = 167 V, V_L = 170 V, and V_R = 170 V), P_C2 (V_F = 180 V, V_B = 120 V, V_L = 145 V, and V_R = 147 V), and P_C3 (V_F = 180 V, V_B = 125 V, V_L = 125 V, and V_R = 190 V). For the prism at P_C1, relative distances between the focal point and the prism location are estimated as all zeros, i.e., (f_xa – x₁) = (f_ya – y₁) = 0, which is identified as a black circular point at the center of the plot in Fig. 8. As mentioned earlier, its radial length from the center indicates the magnitude of the prism angle φ, while its polar angle from the x axis presents the azimuth angle ω of the steered beam. Thus, the prism operation required for P_C1 is to have φ₁ = 0° for no light refraction. For the prism at P_C2, (f_xa – x₂) = 0 and (f_ya – y₂) = -30 mm are calculated, which is presented as a black square point in Fig. 8. Similarly, the radial length of this point from the center indicates the prism angle at φ₂ = 10.3° and its azimuth angle at ω₂ = 270° being required for the prism at P_C2. A similar calculation was conducted for the prism at P_C3 and gave us (f_xa – x₃) = 30 and (f_ya – y₃) = -30 mm, which is identified as a black diamond point in Fig. 8. With this calculation point, we can have the required performance for the prism P_C3 as φ₃ = 14.4° and ω₃ = 315°. Furthermore, we observed the projected beam steering angles for each prism from the left-side and front views of the prisms as β_1-side = 0° and β_1-front = 0° for the prism at P_C1, β_2-side = 3.37° and β_2-front = 0° for P_C2, and β_3-side = 3.37° and β_3-front = 3.46° for P_C3, respectively. With these observed parameters, the beam steering angles for each prism can be derived as β₁ = 0°, β₂ = 3.37° and β₃ = 4.8° based on Eq. (14). Using these values and Eq. (15), we can calculate unit direction vectors of the steered beam for three prisms as ${\hat{v}_1} = [{0,\,\,\,0,\,\,\,1} ]$, ${\hat{v}_2} = [{0,\,\,\, - 0.588,\,\,\,0.998} ]$, and ${\hat{v}_3} = [{0.059,\,\,\, - 0.059,\,\,\,0.996} ]$.

 

Fig. 9. Experimental demonstrations for 3D focal control on the focal plane at f_z = 500 mm. (a) Three prisms were modulated to have at φ₁ = 0° with no light refraction for P_C1, φ₂= 10.3° and ω₂ = 270° for P_C2, and φ₃ = 14.4° and ω₃ = 315° for P_C3, respectively. As a result, the incoming laser beams are spatially steered to be focused on the focal point at P_focal (30 mm, 0, 500 mm). With the projected angles of the steered beams as β_side and β_front observed from the prisms viewed in the left and front sides, the beam steering angles for each prism are estimated as β₁ = 0°, β₂= 3.37°, and β₃= 4.8°. (b) Light focusing on P_focal (30 mm, 30 mm, 500 mm) was achieved by modulating the three prisms to have φ₁ = 10.2° and ω₁ = 90° for P_C1, φ₂= 0° with no light refraction for P_C2, and φ₃ = 10.1° and ω₃ = 0° for P_C3. From the observation views of the prism in the left and front sides, the beam steering angles for each prism are estimated as β₁ = 3.25°, β₂= 0°, and β₃= 3.37°. (c) The last focal point control was shifted to P_focal (20 mm, 15 mm, 500 mm) by operating three prisms to have φ₁ = 6.5° and ω₁ = 123.7° for P_C1, φ₂= 6.5° and ω₂ = 236.3° for P_C2, and φ₃ = 8.9° and ω₃ = 323.1° for P_C3. Similarly, the beam steering angles for each prism are estimated as β₁ = 1.7 °, β₂= 2.4°, and β₃= 3.0° from the observation angles, β_side and β_front.

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Figure 9(b) shows another prism operation for the focal point shifting from (f_xa = 30 mm, f_ya = 0) to (f_xb = 30 mm, f_yb = 30 mm), while keeping the same focal plane at f_z = 500 mm, with the bias voltages applied to three prisms at P_C1 (V_F = 180 V, V_B = 120 V, V_L = 145 V, and V_R = 147 V), P_C2 (V_F = 167 V, V_B = 167 V, V_L = 170 V, and V_R = 170 V), and P_C3 (V_F = 150 V, V_B = 145 V, V_L = 120 V, and V_R = 180 V). For this 3D focal control, the relative distance between the focal point and the prism was calculated as (f_xb – x₁) = 0 and (f_yb – y₁) = 30 mm for the prism at P_C1. This is identified as a pink circular point in the plot of Fig. 8. Thus, the prism operation required for P_C1 is to have φ₁ = 10.2° and ω₁ = 90°. For the prism at P_C2, (f_xb – x₂) = 0 and (f_yb – y₂) = 0 mm are calculated, which is presented as a pink square point at the center of the plot in Fig. 8. The required prism operation is given as φ₂ = 0° with no light refraction for P_C2. A similar calculation was conducted for the prism at P_C3 and gave us (f_xb – x₃) = 30 and (f_yb – y₃) = 0, which is presented as a pink diamond point in Fig. 8. This calculation point can be characterized to have a radial length as φ₃ = 10.1° and the angle as ω₃ = 0°, indicating the required performance for the prism at P_C3. From the side and front views of the prisms, we additionally observed the projected beam steering angle for each prism as β_1-side = -3.25° and β_1-front = 0° for the prism at P_C1, β_2-side = 0° and β_2-front = 0° for P_C2, and β_3-side = 0° and β_3-front = 3.37° for P_C3, respectively. Using these observation parameters, the beam steering angles for each prism can be derived as β₁ = 3.25°, β₂ = 0° and β₃ = 3.37° based on Eq. (14). Moreover, the unit beam vectors can be found as ${\hat{v}_1} = [{0,\,\,0.057,\,\,\,\,0.998} ]$, ${\hat{v}_2} = [{0,\,\,0,\,\,1} ]$, and ${\hat{v}_3} = [{0.059,\,\,0,\,\,0.998} ]$ based on Eq. (15).

Figure 9(c) shows an additional asymmetrical operation of three prisms for the focal point to be located at f_xc = 20 mm and f_yc = 15 mm on the same focal plane at f_z = 500 mm through the applications of bias voltages at P_C1 (V_F = 130 V, V_B = 145 V, V_L = 160 V, and V_R = 137 V), P_C2 (V_F = 184 V, V_B = 120 V, V_L = 160 V, and V_R = 137 V), and P_C3 (V_F = 184 V, V_B = 130 V, V_L = 135 V, and V_R = 175 V). For the prism at P_C1, the relative distance between the desired focal point and prism was calculated to be (f_xc – x₁) = -10 mm and (f_yc – y₁) = 15 mm, which is represented by a green circular point in Fig. 8. Thus, the prism operation required for P_C1 is to have φ₁ = 6.5° and ω₁ = 123.7°. The second prism located at P_C2 has the relative distance of (f_xc – x₂) = -10 mm and (f_yc – y₂) = -15 mm, denoted by a green square point in Fig. 8. The prism and azimuth angles, φ₂ = 6.5° and ω₂ = 236.3°, are required for the prism modulation at P_C2. Lastly, the relative distance between the focal point and prism at P_C3 is calculated to be (f_xc – x₃) = 20 mm and (f_yc – y₃) = -15 mm, represented by a green diamond point in Fig. 8. The required prism operation is found to be φ₃ = 8.9° and ω₃ = 323.1°. The projected beam steering angles are observed to be β_1-side = -1.23° and β_1-front = -1.1° for P_C1, β_2-side = 2.09° and β_2-front = -1.1° for P_C2, and β_3-side = 2.09° and β_3-front = 2.17° for P_C3 from the left-side and front views of the prisms. Using these values, we can obtain β₁ = 1.7°, β₂ = 2.4° and β₃ = 3.0°. In addition, based on Eq. (15), we can derive the unit beam vectors as ${\hat{v}_1} = [{ - 0.017,\,\,0.025,\,\,0.999} ]$, ${\hat{v}_2} = [{ - 0.023,\,\,0.035,\,\,0.999} ]$, and ${\hat{v}_3} = [{0.042,\,\, - 0.031,\,\,0.999} ]$.

6.3 Experimental demonstrations for 3D focal control at f_z = 750 mm

Subsequent experiments have been repeated, but on another focal plane at f_z = 750 mm. Figure 10(a) shows focal tuning of light rays for the desired focal point at P_focal (f_xa = 30 mm, f_ya = 0 mm, f_z = 750 mm), when the bias voltages were applied to the prisms at P_C1 (V_F = 167 V, V_B = 167 V, V_L = 170 V, and V_R = 170 V), P_C2 (V_F = 180 V, V_B = 132 V, V_L = 135 V, and V_R = 137 V), and P_C3 (V_F = 180 V, V_B = 143 V, V_L = 143 V, and V_R = 180 V). The relative distances from the focal point to each prism are calculated to be (f_xa – x₁) = (f_ya – y₁) = 0 mm for the P_C1 prism, (f_xa – x₂) = 0 and (f_ya – y₂) = -30 mm for P_C2, and (f_xa – x₃) = 30 and (f_ya – y₃) = -30 mm for P_C3. They are all presented as black circle, square, and diamond points in the plot of Fig. 10(c) to obtain the values of φ and ω analytically estimated on the focal plane of f_z = 750 mm using Eq. (19). The required operations for three prisms are to have φ₁ = 0° without light refraction for the P_C1 prism, φ₂= 7.6° and ω₂ = 270° for P_C2, and φ₃ = 9.5° and ω₃ = 315° for P_C3, respectively. With the observed angles of the steered rays, β_side and β_front, we can derive the beam steering angles as β₁ = 0°, β₂ = 2.35° and β₃ = 3.39° for each prism using Eq. (14). Moreover, the unit beam vectors pointing out can be also calculated to be ${\hat{v}_1} = [{0,\,\,0,\,\,1} ]$, ${\hat{v}_2} = [{0,\,\, - 0.041,\,\,0.999} ]$, and ${\hat{v}_3} = [{0.042,\,\, - 0042,\,\,0.998} ]$ using Eq. (15).

 

Fig. 10. Experimental demonstrations of 3D focal control on the focal plane at f_z = 750 mm. (a) Light focusing on P_focal (30 mm, 0, 750 mm) was achieved by modulating the prisms to have φ₁ = 0° without light refraction for P_C1, φ₂= 7.6° and ω₂ = 270° for P_C2, and φ₃ = 9.5° and ω₃ = 315° for P_C3, respectively. With the projected angles, β_side and β_front, of the steered beams observed from the prisms viewed in the left and front sides, the beam steering angles for each prism are estimated as β₁ = 0°, β₂= 2.35°, and β₃= 3.39°. (b) Another focal control has been demonstrated at P_focal (15 mm, 10 mm, 750 mm) by modulating the prisms to have φ₁ = 4.3° and ω₁ = 146.3° for P_C1, φ₂= 5.8° and ω₂ = 233.1° for P_C2, and φ₃ = 5.8° and ω₃ = 306.9° for P_C3. Similarly, the beam steering angles for each prism are estimated as β₁ = 1.28°, β₂= 1.76°, and β₃= 1.89°. (c) A plot of the color gradient shows the desired focal point relative to the prism position as a function of φ and ω on the focal plane fixed at f_z = 750 mm. Experimental data points obtained from Fig. 10(a) and (b) are added for comparison study.

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Figure 10(b) shows experimental demonstrations of the focal point shift to f_xb = 15 mm and f_yb = 10 mm, while keeping the same focal plane on f_z = 750 mm, with the bias voltages applied to three prims at P_C1 (V_F = 136 V, V_B = 152 V, V_L = 163 V, and V_R = 135 V), P_C2 (V_F = 176 V, V_B = 133 V, V_L = 160 V, and V_R = 138 V), and P_C3 (V_F = 179 V, V_B = 138 V, V_L = 140 V, and V_R = 184 V). Based on the location of the focal point and each prism, their relative distances are calculated to be (f_xb – x₁) = -15 mm and (f_yb – y₁) = 10 mm for the prism at P_C1, (f_xb – x₂) = -15 mm and (f_yb – y₂) = -20 mm for P_C2, and (f_xb – x₃) = 15 mm and (f_yb – y₃) = -20 mm for P_C3. Figure 10(c) shows them as pink circle, square, and diamond points in the color gradient plot to give us the information on φ and ω required for operation of each prism. Prism modulations were implemented to achieve φ₁ = 4.3° and ω₁ = 146.3° for P_C1, φ₂= 5.8° and ω₂ = 233.1° for P_C2, and φ₃ = 5.8° and ω₃ = 306.9° for P_C3. Similarly, from he observed angles of the steered rays, β_side and β_front, and Eq. (14), the 3D beam steering angles for each prism can be derived as β₁ = 1.28°, β₂ = 1.76° and β₃ = 1.89°. Correspondingly, the unit beam vectors are obtained, based on Eq. (15), as ${\hat{v}_1} = [{ - 0.019,\,\,0.012,\,\,0.999} ]$, ${\hat{v}_2} = [{ - 0.018,\,\, - 0.025,\,\,0.999} ]$, and ${\hat{v}_3} = [{0.020,\,\, - 0.026,\,\,0.999} ]$.

The experimental results presented in Figs. 9 and 10 demonstrate the adjustable lens power obtained through symmetric and asymmetric modulation of the prisms for 3D focal control, ranging 0 ≤ f_x ≤ 30 mm, 0 ≤ f_y ≤ 30 mm, and 500 mm ≤ f_x ≤ ∞ without the use of bulky and complex mechanical moving parts. This optofluidic beam steering technology exhibits a novel optical tunability for 3D focal control, enabling incident rays to be spatially steered without directional constraints within the focal range. Since 1-BN oil is a relatively new material for electrowetting studies, the electrowetting properties and compatibilities of 1-BN have not been fully studied yet. Operational uncertainties such as contact angle pinning and hysteresis when employing liquid prisms with 1-BN led to imperfect convergence of laser beams on a single point in the experiments. Consequently, future research involving liquid prisms would benefit from in-depth studies on the electrowetting properties of 1-BN and their potential impact on overall performance. To tackle a leaking issue of liquids as a potential concern, gaskets or O-rings are recommended as sealant materials, which are commonly used in mechanical systems.

7. Conclusion

Conventional solid-type optical devices often require bulky and complex moving parts for rapid and spatial beam steering. To address this, recent optofluidic studies have demonstrated wide-angle beam steering performance using liquid prisms driven by electrowetting. However, these devices were limited to linear tuning and failed to deliver comprehensive focal tunability in 3D space. This study presents a novel lens tunability for 3D focal control that can be achieved by an optofluidic system composed of n × n arrayed liquid prisms. When bias voltages are applied to the sidewalls of each prism, the fluid-fluid interface is modulated to have its straight profile with the prism’s apex angle controlled in 3D space. Such dynamic control of the fluidic interface via electrowetting enables an incident ray to be spatially manipulated for 3D beam steering. To achieve 3D focal control, individual prisms in the arrayed system are either symmetrically or asymmetrically modulated. As a result, incoming rays are rapidly manipulated for light focusing on an arbitrary location at P_focal (f_x, f_y, f_z) in 3D space. This 3D focal tunability enables to variously modulate the lens’ focusing power that couldn’t be implemented by solid-type optics without bulky and complex mechanical moving components.

Both analytical and experimental studies have been explored to demonstrate 3D focal tunability of the arrayed system. Analytical study predicted the prism operation required to achieve the desired beam steering performance. Experimental studies have further demonstrated 3D focal tunability of the system by simultaneously performing light focusing along lateral, longitudinal, and axial directions as much as ≤ f_x ≤ 30 mm, 0 ≤ f_y ≤ 30 mm, and 500 mm ≤ f_z ≤ ∞. This new lens capability for 3D focal control is of great potential to be used for advanced optical applications such as tracking eye movement for smart displays or solar tracking for smart compact concentrated photovoltaic systems.