Engineering Aspheric Liquid Crystal Lenses by Using the Transmission Electrode Technique

The transmission electrode technique has been recently proposed as a versatile method to obtain various types of liquid-crystal (LC) lenses. In this work, an equivalent electric circuit and new analytical expressions based on this technique are developed. In addition, novel electrode shapes are proposed in order to generate different phase profiles. The analytical expressions depend on manufacturing parameters that have been optimized by using the least squares method. Thanks to the proposed design equations and the associated optimization, the feasibility of engineering any kind of aspheric LC lenses is demonstrated, which is key to obtain aberration-free lenses. The results are compared to numerical simulations validating the proposed equations. This novel technique, in combination with the proposed design equations, opens a new path for the design and fabrication of LC lenses and even other types of adaptive-focus lenses based on voltage control.


Introduction
Adaptive-focus lenses are capable of tuning their focal length by using some kind of external stimulus. In order to obtain optical power tunability, either the curvature of their interface or the spatial profile of the constituent material's refractive index need to be modified. Many techniques have been thus far used to obtain the aforementioned effects. They can be grouped in three main categories: sliding, shape-changing, and refractive index-controlled lenses. The first group is also known as Alvarez lenses. They are now gaining attention for miniature adaptive lenses [1] and high-speed focusing [2]. These lenses show fast switching in combination with broad optical power range. The shape-changing technique embraces several structures, for example, based on elastic membranes [3][4][5][6], ferrofluidics [7][8][9], soft electroactive actuators [10], and the electrowetting and dielectrophoretic effect [11,12]. For this reason, the characteristics vary widely depending upon the nature of the surface or the physical effect. In general, the main disadvantages associated with this approach are the usually high operating voltage, the increased thickness necessary for large aperture lenses (in order to avoid the capillary effect) and detrimental reflection and scattering due to multiple interfaces. Finally, adaptive-focus lenses based on the gradual variation of the refractive index do not need curved surfaces. Instead, such lenses are made by using a material that exhibits a spatial gradient of its refractive index. One of the most common materials for this purpose is liquid crystals (LC). Many of the topologies proposed for LC lenses are based on generating a gradual voltage drop across the lens surface. This voltage profile produces a refractive index gradient in the LC layer by means of the LC electro-optic effect, thus mimicking the optical behavior of a conventional lens with the great advantage of volume/weight reduction, and tunable focal distance.
Two of the most important approaches for the fabrication of adaptive-focus LC lenses are the patterned [48,49] and the modal [50] electrode techniques. The first category comprises multielectrode lenses, which are one of the most versatile solutions due to the fully configurable phase profiles. The main problem is the complex fabrication process (usually involving several layers of electrodes) and the complex voltage control (each electrode requires a distinct voltage). On the other hand, the modal control overpasses previous shortcomings by using a high resistivity layer that distributes homogeneously the voltage across the active area [51]. In this case, only one or two low voltage sources are required but fabrication is still an issue given the difficulty to obtain homogenous high resistivity layers (evaporation of thin layers over large apertures are necessary). In order to solve all of these problems a novel technique has been recently proposed, the transmission electrode technique. It combines all the advantages of previous structures but fabrication and voltage control are much simpler and straightforward. By exploiting this technique, we have recently demonstrated axicon and spherical lenses [52], cylindrical and Powell lenses [53] and mutioptical systems [54].
In this work, this technique is analytically studied in order to engineer any kind of aspheric LC lenses, which is crucial in order to obtain aberration-free lenses. We demonstrate that this versatile technique allows the design of the spatial phase profile by changing only two parameters of the electrode. Thanks to the proposed design equations, it is possible to adjust the profiles to different analytical mathematical curves, for example, parabolas, ellipses, and hyperbolas. The results are compared with numerical simulations showing excellent agreement. This novel technique, in combination with the proposed design equations, opens a new path to fabricate LC lenses and even other types of adaptive-focus lenses based on voltage control.

Analytical Study of the Transmission Electrode Technique
The transmission electrode technique is based on two main elements. The first one is a central transmission electrode that distributes the voltage between two voltage sources, as in Figure 1a,b. A low resistivity material can be used, a straightforward solution being commercial ITO-on-glass substrates. Despite the low resistivity, the total resistance between the contacts is high (kΩ-MΩ) due to the high aspect ratio of the transmission electrode (length/width). In the first example, shown in Figure 1a, a voltage divider is obtained. In this configuration, the central voltage, V C , can be controlled by V 2 [52]. In the second example, Figure 1b, when the applied voltages have fixed phase shifts V 1 = A 1 < 0 • ; V 2 = A 2 < 180 • , the voltage goes from A 1 to −A 2 , crossing in the middle by zero when the two amplitudes match (A 1 = A 2 ). Moreover, if necessary the central voltage can be offset from zero by using some special signals (see, for instance, Reference [54])). The profile of the voltage distribution between the lateral and central points x 1 and x 2 , respectively, can be designed by modifying the transmission electrode distributed resistance between these two points. Specifically, by modifying the shape of the electrode the voltage profile is also modified accordingly. As it will be demonstrated in the following sections, the optical phase profiles resulting from such voltage profiles can approach linear, parabolic, elliptical and hyperbolic profiles, enabling the generation of all kinds of aspheric lenses.
The second element consists of a series of parallel or concentric electrode stubs evenly arranged, as in Figure 1c,d, which show two standard configurations corresponding to spherical and cylindrical lenses. These electrodes distribute the voltage across the active area, behaving like the high resistivity layer of modal devices but with the advantage of being made by a simple ITO lithographic process and having low resistivity. The gaps between electrodes have to be as short as possible to avoid abrupt phase steps. This discontinuity effect is proportional to the gap/LC cell thickness aspect ratio [52]. As the gap can be brought down towards micrometric resolution using standard photolithography, the phase-discontinuity effect is negligible for LC cell thicknesses above 10 µm (i.e., roughly for aspect ratios above 1:10), which provides ample space for device engineering.
The combination of these two elements allows a simple study. By estimating the voltage in the central transmission electrode, the voltage across the active area can be inferred. In the next section, a simple Equivalent Electrical Circuit (EEC) is proposed in order to calculate the voltage at each point of the transmission electrode. Then, the equations for different electrode shapes are elaborated.

Voltage Distribution
As commented before, the parallel/concentric electrodes act like the high resistivity layer of a modal lens. Some years ago, some of the authors proposed a general EEC for modal/hole patterned lenses, described in Figure 2a [55,56]. The model resembles that of a transmission line with the inclusion of a coplanar capacitor (C 1 ) and a loss resistance (G). In the modal topology, the distributed resistors are R i = R sq , (sheet resistance of the high resistivity layer). Alternatively, in a hole-patterned structure R i = R LC , modelling the LC dielectric losses in the lens surface. In the case of the transmission electrode technique, this circuit is simplified considerably, as shown in Figure 2b. Unlike the case of modal control, R is very low so the rest of the components can be neglected. Thanks to this, reactive components, and consequently the temporal dependence, are dropped. There is no need to use differential equations and the system can be modeled by considering only the distributed resistance of the transmission electrode, as depicted in Figure 2b. This model is always valid when commercial ITO-on-glass substrates are used (to be in the modal control regime sheet resistances in the order of MΩ would be necessary). In order to have low current between electrodes, for example, in the order of µA, the resistance between contacts has to be in the order of several kΩ. For a fixed-width electrode, the resistance is equal to the sheet resistance (see also Section 2.2) multiplied by the aspect ratio L/w, where L and w is the electrode length and width, respectively. Considering a diameter of the active are of the proposed devices equal to 1 cm, and transmission electrode width of 10 µm, the aspect ratio is about 1000. The typical commercial ITO-on-glass sheet resistance ranges from 4.5 to 500 Ω/sq, corresponding to a thickness ranging from 310 to 15 nm respectively. This means that any typical commercial ITO-on-glass would be in principle valid for the proposed technique. Despite this, the thicker the electrode the higher the current, which could produce self-heating and losses. On the other hand, the lower the thickness the harder the fabrication due to, for instance, the difficulty to control the acid attack of thin ITO layers.
where R(x) the resistance in the segment of the electrode from x = 0 up to x. The current is calculated as where R t is the total resistance of the electrode from x = 0 to x = x c . The voltage V(x) is then equal to where R(x) is calculated by where ρ is the resistivity t the thickness, R sq = ρ/t is the sheet resistance, and w(x) is the (generally) variable width of the ITO transmission electrode. By controlling w(x) in the design of the photolithographic mask, different resistance and, hence, voltage profiles can be obtained, as studied in the next sections for three particular electrode shapes.

Rectangular Electrode
The first example is the simplest one. By using rectangular ITO electrodes, as in the structures shown in Figure 1a,b, a cuboid resistance is obtained, shown in Figure 3. The electrode has fixed width w(x) = w, thickness t, and length L = x 2 − x 1 . According to Equation (4), the resistance R(x) = R sq x/w and the total electrode resistance equals R t = R sq L/w. Therefore, the voltage V(x) is calculated according to Equation (3) This results in a linear equation where the voltage distribution is proportional to the position x.

Pyramid-Shaped Electrode
In order to obtain curved voltage profiles, a straightforward way from the design and fabrication point of view is by progressively increasing the electrode width towards the lens center, as depicted in Figure 4c. This can apply both in the voltage divider and the orthogonal configurations, shown in Figure 4a,b, respectively. As a consequence, a linearly decreasing resistance R(x) towards the lens center is obtained leading to a curved voltage drop profile. The curvature depends on the relation between w 1 and w 1 . In this case, the electrode width is described by the linear equation where L is the electrode length. The electrode resistance R(x) is calculated according to Equation (4) as follows where, without loss of generality, we have consider x 1 = 0, x 2 = L. The total electrode resistance R t is given for x = L and it equals Subsequently, the voltage profile V(x) for the pyramid-shaped transmission electrode is given by and it is a function of the Neperian logarithmic equation. By modifying the values w 1 and w 1 different voltage profile curvatures can be obtained.

Parabola-Shaped Electrode
In order to obtain other profiles, the width of the electrode can be modified according to, in principle, any analytical equation. As a next case we study the effect of a parabola-shaped electrode. As it will be demonstrated in the results presented in Section 3, both the parabola-shaped and the pyramid-shaped electrodes can closely approximate two selected target optical phase profiles, therefore the investigation of further electrode shapes is not included in this work. Nevertheless, topologies targeting other voltage profiles can be engineered by following the same steps employed before. The proposed structure is detailed in Figure 5. The variable electrode width is given by the parabolic equation which equals w 1 and w 1 at x = 0 and x = L, respectively. The resistance R(x) is calculated as and the electrode resistance equals The voltage profile is then calculated as For the case of higher degree polynomials, sums of Neperian logarithmic equations are expected. By modifying the values w 1 and w 1 , the voltage profile can be approximated to parabolic, hyperbolic/elliptical equations.

Rectangular Electrode
The rectangular electrode is the simplest case. The voltage is proportionally distributed along the transmission electrode, as in Equation (5). In order to validate the analytical equation, we simulated the structure by using the finite-element method (FEM) implemented in the commercial tool COMSOL Multiphysics TM . For both configurations the total diameter is 1 cm, the width of the transmission electrode is w 1 = w 1 = 10 µm and the ITO resistivity is R sq = 100 Ω/sq. In the case of the voltage divider, w 2 equals 60 µm and the value of V 2 is selected such that V c = 1 V.
The numerical results are depicted by dots whereas the analytical results by solid lines in Figure 6. Excellent agreement is observed. By using this equation, the phase profile of an optical beam propagating through a LC lens with the calculated voltage profiles can be estimated by means of the Frank-Oseen equations. Such results can be checked in the references commented before.

Pyramid-Shaped Electrode
In this case, the voltage can be analytically estimated by Equation (9). In order to validate the analytical equation, the structure is also simulated. For both configurations the total diameter is 1 cm, the width of the transmission electrode is w 1 = 1 µm and ITO resistivity is R sq = 100 Ω/sq. For the voltage divider configuration w 1 = 10 µm and w 2 = 60 µm, whereas for the orthogonal configuration w 1 = 20 µm. The results are shown in Figure 7.  (9). Structural parameters can be found in References [52,54].
As can be observed, as the value of w 1 increases the curvature also increases. The analytical equations agree with the numerical estimations validating the proposed EEC. Despite this, the results are far from optimal when the phase is compared to parabolic or hyperbolic profiles. For this reason, an optimization process is presented in Section 3.4.

Parabola-Shaped Electrode
In this case, the voltage can be analytical estimated by Equation (13). In order to validate the analytical equation, the structure is also simulated. In this case, only the orthogonal configuration is numerically solved to demonstrate the validity of the derived analytical equations. The used parameter are w 1 = 1 µm, w 1 = 100 µm and the ITO resistivity is R sq = 100 Ω/sq. The results are shown in Figure 8. As in the previous sections, the analytical equations agree with the numerical estimations validating the proposed EEC. We have observed that this electrode type produces a better approximation to parabolic and hyperbolic voltage profiles. Despite this, as the LC has a non-linear response this is not translated to better approximation to parabolic and hyperbolic optical phase profiles. The optimization is detailed in next section.

Parametric Optimization to Different Aspheric Phase Profiles
In order to obtain some classic aspheric profiles as parabola or hyperbola, an optimization process has to be carried out. The required voltage profile for aspheric phase profiles is firstly estimated by using the Frank-Oseen Equations (LC MDA-98-1602) [57]. The key properties of the LC are reported in Table 1, where the birefringence and the permittivity are measured at 633 nm and 1 kHz, respectively. For the aspheric phase profiles, two unsaturated points from the effective birefringence are taken into account (from 0.05 to 0.25) as limit values, shown in Figure 9a. Then, the reference phase profiles are obtained following the canonical equations of the parabola and hyperbola, that is, the solid lines in Figure 9b. From this data, the required voltages to obtain these phase profiles, considering the Frank-Oseen equation, are estimated, shown as dashed lines in Figure 9b.  As can be observed, the required voltage has a non-linear curvature with respect to the phase profile caused by the non-linear voltage response of the effective birefringence. In order to determine the best manufacturing parameters the analytical expressions proposed in previous sections are adjusted to these voltage profiles. Depending on the manufacturing constraints, different parameters can be left as unknown variables. The lsqcurvefit from MATLAB is used to solve the nonlinear least-squares (nonlinear data-fitting) problem. In the following example, the square resistance and electrode length are fixed at R sq = 100 Ω/sq and L = 1 cm, while w 1 and w 1 are free to vary. In fact, the important data is the relation between these two parameters, but they are free in order to allow the program to converge more easily. For this reason, the results are shown by fixing the parameter w 1 to 10 µm.
As can be observed in Figure 10, the pyramid-shaped electrode shows a perfect approximation to the desired aspheric profiles. Specifically, for the hyperbolic phase profile the electrode with w 1 = 10 µm and w 1 = 60 µm results in a R 2 = 0.9992. In the case of parabolic phase profile, w 1 = 10 µm and w 1 = 165.8 µm results in a R 2 = 0.9993. On the other hand, the parabola-shaped electrode results in R 2 = 0.9911 for the hyperbolic phase profile (with w 1 = 10 µm and w 1 = 36.9 µm) and in R 2 = 0.9908 for the parabolic phase profile (with w 1 = 10 µm and w 1 = 93 µm). The coefficient of determination R 2 reveals that pyramid-shaped electrodes produce a better approximation for these two aspherical profiles. Moreover, the design and fabrication are simpler than parabola-shaped electrode. Despite this, the latter could be useful for other custom phase profiles or even for other devices on which the electro-optic response is linear and more parabolic/hyperbolic voltage profiles are required.

Tunability of the Optimized Designs
The phase shift modulation across the LC lens can be estimated by the optical path length (d∆n eff ) multiplied by 2π/λ, where d and λ are the LC cell thickness and the working wavelength. We define ∆(∆n eff ) as the difference between the effective birefringence in the lens center and at the lens side. In the previous section, both pyramid and parabola-shaped electrodes were optimized for obtaining a maximum phase modulation, which corresponds to ∆(∆n eff = 0.2). However, it is a known fact that LC lenses usually have different aberrated responses as the maximum phase shift modulation is modified. For this reason, a study of the tunability response of the designed electrodes is carried out in this section. The optimization process is twofold, aiming to find the best combinations of both maximum effective birefringence/corresponding voltages and structural dimensions at once to obtain the best approximation to a parabolic profile (in terms of R 2 ). For the pyramid-shaped electrode, the results (Table 2) confirm that the structural parameters of previous section behaves better for high phase shifts. There is a reduction in the fit quality as the maximum phase shift is reduced but is not considerable until ∆(∆n eff = 0.05). Therefore, the same electrode, which was optimized for ∆(∆n eff = 0.2), can perform satisfactorily for ∆(∆n eff ) values down to 0.05, meaning that there is no need to modify the device in order to operate in different regimes of effective birefringence that produce parabolic phase profiles. On the contrary, it was found that the parabola-shaped electrode behave considerably better for larger ranges of modulation of the effective birefringence. The resulting optimized structural parameters are different from those reported in Section 3.4, as these were optimized for a range of ∆(∆n eff ) values (contrary to a sole value 0.02 in the previous Section). The optimal structural parameters are w 1 = 10 µm and w 1 = 41 µm, which yielded the best behaviour for ∆(∆n eff ) = 0.1. The corresponding results are summarized in Table 3. These results reveal that parabola-shaped electrodes have a more stable response to phase shift changes (tunability). The pyramid-shaped electrodes have the best response for high phase shift modulation but a considerable reduction of the fitting quality as this phase shift is reduced. For these reasons, when tunability prevails over optical power, parabola-shaped electrodes are recommended.

Conclusions
An equivalent electric circuit and new analytical expressions based on the recently proposed transmission electrode technique have been studied and numerically demonstrated. Thanks to the modification of only one structural parameter (the width of one part of the electrode), it is possible to engineer all kinds of aspheric LC lenses (key to obtain aberration-free lenses). The results reveal that pyramid-shaped electrodes produce a better approximation for two aspherical phase profiles. In addition, this solution is easy to design and fabricate. On the other hand, the parabola-shaped electrode have a better response when tunability is required, especially for lower optical powers. In particular, the parabola-shaped electrodes can produce a broader range of parabolic voltage profiles so they can be useful in other devices with a linear electro-optic response. The optimization process yielded the optimum structural parameters and voltages to obtain almost perfect profiles with a single electrode structure. Moreover, the parabola shaped electrodes can produce a more parabolic voltage profiles so can be useful in other devices with a linear electro-optic response. For these reasons, this novel technique in combination with the proposed design equations could open new avenues of research in adaptive-focus lenses based on voltage control.