In recent years, liquid crystal (LC) devices have increasingly been used in many non-display applications. LC characteristics such as the Frederick’s effect and high birefringence are crucial for constructing small and lightweight devices that can be controlled by low voltages without requiring any mechanical components. By exploiting such characteristics, LC devices have been notably applied in the fields of optical communications as modulators, switches, filters, photonic optical fibers, and so on; adaptive optics for wavefront correction, beam shaping, and optical tweezers; and liquid crystal lenses.
Liquid crystal lenses were first reported more than 30 years ago, and they remain an active field of research even today, as evidenced by recent applications such as tunable-focusing optical zoom systems (cell phones, cameras, picoprojectors, night vision of hand-carried weapons) [1
1. Y. H. Lin and M. S. Chen, “A pico projection system with electrically tunable optical zoom ratio adopting two liquid crystal lenses,” J. Disp. Technol. 8(7), 401–404 (2012). [CrossRef]
] and spectacles and autostereoscopic devices [2
2. V. Urruchi, J. F. Algorri, J. M. Sánchez-Pena, M. A. Geday, X. Q. Arregui, and N. Bennis, “Lenticular arrays based on liquid crystals,” Opto-Electron. Rev. 20(3), 260–266 (2012). [CrossRef]
]. Over this period, many new topologies have been proposed, such as polymer gel stabilization, patterned electrode, surface relief profile, Fresnel lens, immersed lens, and modal control.
In this study, we develop a novel device based on modal control that has a lens array formed by a six-electrode structure with two high-resistivity layers. This advanced approach combines, in the same device, some of the improvements of modal devices previously reported. Specifically, some smart driving complex signals, with opposite electrical phase shifts, are applied to a minimum modal device based on a few electrodes. This device acts as a reference for suggesting that its electrooptic response can be extrapolated to 1D and 2D arrays of higher size. The remainder of this paper is organized as follows. In section 2, we present the theoretical basis for this device. In section 3, we describe its structure and device operation. In section 4, the experimental setup of the developed device is presented. In section 5, we compare the obtained experimental results with those obtained using a simulation program. Finally, in section 6, we summarize the conclusions of our study.
2. Modal device theory
Modal lens theory has been widely studied since it was first reported. Vdovin et al
. conducted the first studies of the physical processes occurring in the resultant distributed resistive-capacitive system [5
5. G. V. Vdovin, I. R. Guralnik, S. P. Kotova, M. Y. Loktev, and A. F. Naumov, “Liquid-crystal lenses with a controlled focal length. I. Theory,” Quantum Electron. 29(3), 256–260 (1999). [CrossRef]
]. For describing this type of system, the voltage distribution equation across the lens diameter is derived by considering the equivalent electric circuit of a LC at mid-range frequencies (a distributed parallel-plate capacitor). One of the layers is considered an equipotential surface, while the other contains non-uniformly distributed free charges. Following the law of charge conservation and Ohm’s law and ignoring the electrical field outside the lens diameter, the resulting voltage distribution equation is a second-order two-dimensional partial differential equation (PDE),
is the voltage distribution in the high-resistivity layer, G and C are distributed conductances and capacitors, respectively, that model the LC layer, per unit area, and Rsq
is the sheet resistance to model the control layer. If only one harmonic is considered (U(x, t) = U(x) eiωt
), this equation can be considerably simplified as follows:
This equation is strongly dependent on the boundary conditions (voltage at diameter ends). Considerably different analytical expressions are obtained when complex boundary conditions (described in terms of voltage amplitude and phase) are employed. Several studies have investigated the use of harmonics [6
6. A. F. Naumov, G. D. Love, M. Y. Loktev, and F. L. Vladimirov, “Control optimization of spherical modal liquid crystal lenses,” Opt. Express 4(9), 344–352 (1999). [CrossRef] [PubMed]
] or different electrical phase shifts [7
7. A. K. Kirby, P. J. Hands, and G. D. Love, “Liquid crystal multi-mode lenses and axicons based on electronic phase shift control,” Opt. Express 15(21), 13496–13501 (2007). [CrossRef] [PubMed]
] to improve the lens quality or to simply achieve different wavefront modulations than typical spherical or cylindrical shapes. In recent years, other types of modal controls have been proposed. Some of these are based on the use of two high-resistivity layers, one of them instead of the equipotential layer on the ground plane. The latter has been demonstrated to reduce lens aberrations [8
8. N. Fraval and J. L. de la Tocnaye, “Low aberrations symmetrical adaptive modal liquid crystal lens with short focal lengths,” Appl. Opt. 49(15), 2778–2783 (2010). [CrossRef] [PubMed]
] if electrodes of one substrate are placed parallel to each other. On the other hand, if substrates are arranged so that their electrodes are oriented orthogonal to each other, a tunable focusing device can be achieved [9
9. S. P. Kotova, V. V. Patlan, and S. A. Samagin, “Tunable liquid-crystal focusing device. 2. Experiment,” Quantum Electron. 41(1), 65–70 (2011). [CrossRef]
]. The voltage distribution in these two structures can be easily explained with a system using two versions of Eq. (1)
, one for each layer [10
10. S. P. Kotova, V. V. Patlan, and S. A. Samagin, “Tunable liquid-crystal focusing device. 1. Theory,” Quantum Electron. 41(1), 58–64 (2011). [CrossRef]
3. Structure and modal device operation
The modal device proposed in this study is based on nematic liquid crystal technology. Figure 1(a)
Fig. 1 Modal liquid crystal device proposed. Note drawings are not to scale. (a) Electrode layout and (b) device arrangement.
shows the structure comprising two high-resistivity layers. Figure 1(b)
shows that every high-resistivity layer consists of a very thin layer of titanium oxide with a sheet resistance of 11–14 MΩ/sq. The design of the electrode pattern consists of six striped ITO electrodes (three on the top and three on the bottom substrate); the spacers are 20 µm ± 10%. The substrates are arranged so that their electrodes are oriented orthogonal to each other. The active area is 1 cm2
and the electrodes are 1mm width.
As noted above, the two high-resistivity layers result in a two second-order two-dimensional coupled PDE with six initial conditions corresponding to each electrode,
correspond to the voltage distribution in each high-resistivity layer and Rsq1
, the sheet resistances of the two high resistivity layers. To solve this problem, a finite difference method could be implemented; however, issues such as Neumann boundary conditions or the need for initial values very close to the final result in order to converge correctly, may make this approach inaccurate. For better results, the finite element method is employed. Despite being a more complex method, MATLAB software has some useful functions such as mesh generation and PDE solvers for treating certain cases. We used MATLAB to develop a simulation program with six initial conditions, mixed boundary conditions (Dirichlet and Neumann) and a refined mesh. This one also includes a molecular liquid crystal distribution simulation program that minimizes the Gibbs free energy, FG
. This energy comprises the contribution of two energies: electric energy, Fe
, given by the dot product of the displacement vector (D) and the electric field (E), and the deformation free energy, Fd
, determined by the Frank-Oseen equation. By combining the voltage distribution and molecular director, the simulation program estimates the interference pattern and the unwrapped phase generated by the modal device. As the LC reorients the permittivity changes, affecting the electric field. The coupling between director and electric field makes it almost impossible to get a direct solution of the problem. In this case, this is solved by an iterative process. The software also provides aberration tests, obtaining the 36 Zernike coefficients of each individual optical element for a specific arrangement, to quantify the optical quality in terms of the phase deviation from a reference sphere.
4. Experimental setup
As noted before, one of the most interesting operation regimes in modal devices is obtained using complex voltages, defined by amplitude and phase. The simulations confirm the need to use this type of voltage to completely exploit the device characteristics. For this purpose, a driving module has been designed and built with four external outputs. It consists of a custom phase shift waveform generator that uses an NI USB 6259 module to permit four analog outputs ( ± 10 V) and a maximum speed of 1.25 MS/s in combination with a 18F45k20 PIC microcontroller as a master reference clock source to solve the problem of temporization and synchronization; the latter works as a tunable clock source to maintain a constant number of samples per waveform. The modal device is driven by the NI USB outputs and controlled using a computer.
5. Simulation and experimental results
The results of this work are presented and described in two steps. In a first stage, the validation of simulation program is established by the comparison of some simulations of a basic approach with their respective experimental measure. The simulation program uses the following input parameters about the device structure and the nematic LC features: thickness = 23 µm, K11 = 1.8, K33 = 6, birefringence = 0.18, dielectric anisotropy = 20.1 and resistivity of high-resistivity layer = 12 MΩ/sq. Differences in the two high resistivity layer thickness, achieved in the fabrication process, are negligible. So that, they are considered equal for simplicity in simulation. In a second phase, a set of advanced arrangements are considered. The device layouts are focused fundamentally to create two kinds of arrays: a one-dimensional array of optics elements working as tunable LC prisms and a two-dimensional matrix with a tunable LC axicon array functionality. Here, we describe the experimental results for validation via a comparison with some simulations carried out previously. The general driving scheme consists of the application of four electrical square signals, with until four different electrical phase shifts, to each electrode. These set ups use opposite electrical phase shifts applied to couples of electrodes. It is required to select one electrode of reference for phase; being 0° the phase of that electrode.
5.1 Validation of the simulation program
In this experiment, identical voltage amplitude without electrical phase shifts between electrodes is applied to electrodes of the top substrate (V1
= 4 Vpkpk
). The bottom electrodes are connected to ground (V3
= 0 Vpkpk
). This causes a hyperbolic voltage distribution and small voltage gradients, suffering from several aberrations. Figure 4
Fig. 4 (a) Experimental and (b) simulated interference patterns when 4Vpkpk are applied to the top substrate.
shows a comparison of the interference pattern of the obtained simulations with experimental measurements. The active area shows a surrounding effect caused by the optical glue in the manufacturing process that generates a constant optical phase shift in those zones. This inconvenience causes a light difference in size between experimental and simulated data. Despite this, simulations show acceptable agreement with the experimental results.
Phase unwrapping is realized by a phase-shifting technique from interference patterns. Figure 5
Fig. 5 (a, c) Experimental and (b, d) simulated unwrapped phase when 4Vpkpk are applied to the top substrate.
shows the unwrapped phase in XY and YZ planes. The glass substrate plane is used as the reference XY plane. The phase unwrapping and simulations reveal a peak-to-value optical phase shift over the active area of 1.6λ for λ = 632.8 nm (around 7.5π). The differences between the simulations and the experimental measurements could be caused by deviations in the input parameters considered just like the hypothesis of thickness uniformity of the high resistivity layers, in simulation, or a high sensitivity to temperature of the experimental device.
The comparison of the unwrapped phases, experimental and simulated, validates the capacity of the simulation program to reproduce these profiles. However, the experimental reconstruction of the phase by the Zygo interferometer is not a simple task for this active area due to the restricted resolution of the interference pattern images. That restriction motivated the interference patterns were the only parameter obtained experimentally. On the contrary, simulation program modeled all the parameters such as the voltage distribution or the unwrapped phase and could be considered to test the device and to adjust better-quality wavefronts.
The quality of the optical elements of this arrangement was quantified by the aberration tests. The aberration coefficients were calculated by comparing the phase profile of one optical element with that of a reference sphere whose radius was the maximum phase shift for this approach. The result reveals that this first configuration is restricted mainly by tetrafoil (Z14
) and spherical (Z12
) aberrations. Figure 6
The Zernike coefficients of an individual optical element for the arrangement whose unwrapped phase is shown in Fig. 5
shows a graph with the magnitudes of the 36 Zernike coefficients where the spherical aberration is especially noticeable. The tetrafoil aberration is mainly caused by the square aperture of the active area in this scheme with the substrates arranged so that their electrodes are oriented orthogonal to each other. Finally, some coma aberrations (Z7
) seem less relevant to the phase deviation.
5.2 A one-dimensional array of tunable LC prisms
An attractive optical unwrapped phase is produced via the use of opposite electrical phase shifts, that is, opposite signal polarity, in contiguous electrodes of the same substrate. This setting forces the voltage between the two electrodes to cross zero volts and to create a pyramidal voltage profile [see Fig. 7(a)
Fig. 7 (a) Voltage distribution, |U1-U2|, for a one-dimensional array of LC prisms. Experimental (first row) and simulated (second row) for (b) the horizontal and (c) vertical distributed interference patterns.
]. The result of applying this control is indeed a controllable one-dimensional LC prism array. Figures 7(b)
illustrate the comparison of the experimental and simulated interferograms, for the horizontal and vertical distributed interference patterns. Driving signals set up are (V3
= 6 Vpkpk
= 0° and Φ4
= 180°) and (V1
= 6 Vpkpk
= 0° and Φ2
= 180°), respectively. The frequency remains constant, at 1 kHz, in all the experiments. Both examples confirm the simulation model; the comparison reveals that they are fairly in agreement. In this case, low frequency signals are preferable (lower than 1 kHz) in order to operate with a small modal parameter. Both examples confirm the validation of the simulation model; the comparison reveals that they are fairly in agreement.
The recorded sequence of the prism operation starts with all electrodes connected to ground (0 volts, 0°). In a first step, with a constant electrical phase shift (Φ1 = 180°, Φ2 = 180° and Φ1bis = 0°), the amplitudes of electrodes V1 = V2 = V1bis are increased from 0 to 6 Vpkpk, while V3 = V4 = V3bis = 0 Vpkpk. During this interval, one of the prisms does not appear while the other shows a tunable profile. The second step was a customized design with a fixed active prism (the one placed on the right) and the second one varying its slope. All the parameters are constant (V1 = V2 = V1bis = 6 Vpkpk, Φ2 = 180°, Φ1bis = 0°) instead Φ1 that decreased from 180° to 0°. The third and last sequences are the complementary, with symmetric parameters, to the previous ones. That is, in the third sequence only Φ1bis controls; it is increased from 0° to 180°. And the last sequence has a constant electrical phase shift (Φ1 = 0°, Φ2 = 180° and Φ1bis = 180°) and the amplitudes of electrodes V1 = V2 = V1bis are decreased from 6 to 0 Vpkpk. This whole sequence demonstrates complete control over the prism array. This prism has a maximum optical phase shift of 10.6π.
The quality of one prism of the one-dimensional array was checked by evaluating the RMS wavefront deviation. An ideal prism has been taken as a reference prism. Figure 9
Fig. 9 The phase profiles for both an ideal prism (dashed line) and the proposed prism (solid line) for 6 Vpkpk.
shows the comparison of both prism profiles for the higher voltage, 6 Vpkpk
This voltage amplitude has been chosen because it leads to the higher RMS wavefront deviation from the ideal prism; a deviation of λ/10 has been measured. As with the previous device arrangement, a similar explanation can be argued; the effect of the non-linear LC birefringence is a key for voltages out of the linear dynamic range. Pursuing the same goal, a threshold voltage could be considered as the lower limit of the dynamic range instead 0 volts. This initial condition would lead to a perfect valley at the center of the lens with imperceptible plateau zones. Again, the small defects that are noticed in the phase prolife also could be solved by new materials and manufacturing technologies of modal control taking into account for optimizing the response.
5.3 A two-dimensional matrix of tunable LC axicons
The previous setting exploits only the tuning features of the device in one dimension. By adding a new control dimension, the one-dimensional LC array described before, evolves into a new two-dimensional LC array. Table 1
Table 1. Batch of measures carried out when an electrical signal is applied to the device with some constant electrical phases and voltage amplitudes in Vpkpk.
summarizes the bath of measures carried out whose results are shown in Fig. 10
Fig. 10 Experimental (first row) and simulated (second row) interference patterns for a two-dimensional matrix of tunable LC axicons.
. It also lists the applied voltage amplitudes in Vpkpk
for a set of constant electrical phases. The experiment is based on the use of opposite electrical phase shifts between electrodes in both substrates; being Φ1
= 0° and Φ2
= 180°, in one substrate, and Φ3
= 90° and Φ4
= 270°, in the other. Optical tunability is reached, for a constant phase shift, by varying the voltage amplitude of the signal. Figures 10(a)
show some interference patterns in XY plane for a fixed amplitude V1
. The evolution of interference patterns, as amplitudes of some chosen electrodes increase, is shown in Figs. 10(d)
. In Table 1
are described the conditions which are then shown in Fig. 10
The frequency remains constant, at 1 kHz, in all the experiments. This driving scheme generates the most interesting wavefronts identified in this work with respect to the tuning versatility and the potential application of the device. The device results in a 2 × 2 two-dimensional matrix of tunable rotationally symmetric prisms or axicons, that is, lenses featured by a conical surface. The simulations reveal a cone-type voltage that generates concentric fringes that can be controlled by Vpkpk
at each electrode. Further, the focus point of each axicon can be tuned over the active area. The videos of Fig. 11
A two-dimensional matrix of tunable LC axicons: (a) Experimental interference patterns (Media 2
), (b) 2D simulated in XY plane (Media 3
) and (c) 3D simulated unwrapped phase (Media 4
illustrate a specific example of the device control: the experimental interference patterns in Fig. 11(a)
, the 2D and 3D simulated optical phase shifts in Figs. 11(b)
. In this experiment, the electrical phase shifts (Φ1
) and also the amplitudes in the external electrodes, V1
, remain constant again, while the amplitudes in the central electrodes, V2
, change. The sequence that V2
follows is: First, beginning with a voltage of 5 Vpkpk
in each electrode and a maximum focus point (optical phase shift 12.9π) at the center of each sub-lens. Then, increasing V2
, until 7.4 Vpkpk
, that results in the focus moving toward the outside of the active area. And finally, decreasing V2
, until 1 Vpkpk
, that displaces the focus toward the center of the active area.
The prospective applications of the LC device proposed for adaptive optics can be performed for converting a parallel laser beam into four rings so as to create a set of four non-diffractive Bessel beams or for focusing a parallel beam into four controllable long focus depths that can be electrically controlled.
The quantification of the optical quality of an axicon of the two-dimensional matrix could be analogously obtained as with the prism arrangement. Two planes, length and width, must be considered, instead of only the width plane. Once more, an ideal axicon could be achieved by optimizing the manufacturing techniques.
In summary, we have described a novel configurable and tunable modal LC array of optical elements and presented experimental and simulation results of its electrooptic behavior. The wavefront generation using this device can be improved by using multiphase shift control voltages. The application of smart driving complex signals with opposite electrical phase shifts between electrodes in one substrate has allowed us to simulate a configuration that works as a one-dimensional array of tunable LC prisms. In an advanced approach with a similar strategy applied in both substrates, a two-dimensional matrix of tunable LC axicons, configured from the proposed device, has also been demonstrated experimentally. The simulation program, specially developed for achieving distinctive configurations of the device, predicts the voltage distribution inside the LC layer. Simulation results can be extrapolated to 1D and 2D matrices of higher size. Additionally, driving controls based on non-regular patterns of tunable voltages, in each electrode, has been attempted and have to be further investigated. By limiting the device operation to the linear range of the LC birefringence characteristic has been identified as key point to reduce aberrations.