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Abstract
This paper studies the dynamic response characteristics of the scanning angle in a liquid crystal cladding waveguide beam scanner. Based on liquid crystal dynamic theory, finite element analysis and vectorial refraction law, a dynamic response calculation model of scanning angle is constructed. The simulation results show that the dynamic responses of the scanning angle during the electric field-on and field-off processes are asymmetric, and exhibit “S”-shape and “L”-shape changing trends, respectively. In addition, by comparing with the bulk phase modulation response process of traditional liquid crystal devices, the intrinsic physical reason for the rapid light regulation of the liquid crystal cladding waveguide beam scanner is clarified to be that the liquid crystal close to the core layer has a faster rotation speed during the electric field-off process. Moreover, the liquid crystal cladding waveguide beam scanner is experimentally tested, and the experiment results are in good agreement with theoretical simulations.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Beam scanners are key components in the fields of vehicle-mounted lidar, space optical communications, laser imaging and remote sensing [1]. Traditional beam scanners generally rely on mechanical devices to achieve beam control by changing the direction of the optical axis, which have limitations in terms of speed, reliability, energy consumption and cost [2]. Therefore, a new beam scanning method is urgently needed to meet the growing application requirements. Waveguide-based beam scanners are becoming attractive for the advantages of integration and miniaturization [3–5]. One of the waveguide-based beam scanners is the steerable electro-evanescent optical refractor (SEEOR) [6], which is built on slab waveguide and liquid crystal (LC). The LC is located in the cladding layer and interacts with waveguide mode through evanescent waves. The effective refractive index of waveguide mode can be tuned based on field-induced rotation of the LC molecules. Combined with patterned electrodes and prism, solid-state continuous beam scanning can be achieved.
SEEOR was first proposed by Vescent Photonics in 2008 [7]. A 50°×15° two-dimensional continuous scanning was achieved in the near-infrared band [8] and a lidar system was built, which verified the application potential of SEEOR in the field of solid-state lidar [9]. In 2017, Frantz et al. expanded the working band of SEEOR from near-infrared to mid-infrared [10], and proposed a general design method of SEEOR in the mid-infrared band [11,12]. In 2020, Kołacz et al. theoretically analyzed the transmission characteristics differences between transverse electric (TE) and transverse magnetic (TM) modes in SEEOR and discussed the factors affecting the scanning angle [13]. In 2023 Bi et al. studied the prism coupling structure of SEEOR, proposed an extended prism coupling theory suitable for anisotropic cladding, and provided the configuration design of core and substrate layers with the best coupling efficiency [14]. In 2018, Spillmann and Gotjen et al. experimentally tested the scanning response time of SEEOR in mid-infrared, demonstrating its potential for high-speed beam scanning [15,16].
In order to deeply reveal the physical mechanism of the high-speed response of SEEOR, this paper establishes a scanning angle dynamic response model which satisfies the physical characteristics of SEEOR based on LC dynamics theory, finite element analysis and vectorial refraction law. Subsequently, theoretical simulations are employed to investigate the dynamic response differences in scanning angle during electric field-on and field-off processes. By comparing with the bulk phase modulation response process of traditional LC devices, the distinctive dynamic response characteristics of SEEOR is clarified. Finally, experimental tests are conducted on the SEEOR, and the test results are in good agreement with theoretical simulations. The research results indicate that the scanning angle responses of SEEOR during the electric field-on and field-off processes are asymmetric, which present “S” and “L” shaped trends, respectively. Benefiting from the faster rotation speed of LC near the core layer during the field-off process, the SEEOR can achieve rapid beam scanning.
2. Calculation model of scanning angle dynamic response in SEEOR
The structure of SEEOR is shown in Fig. 1, which can be divided into input coupling region, scanning control region and output coupling region. The beam scanning on the x-z plane is controlled by the scanning control region, which contains a four-layers waveguide, consisting of a substrate layer, a core layer, a LC alignment layer and a LC cladding layer with initial LC alignment along z-axis. When a voltage is applied to the scanning electrode, the angle between the long axis of the LC molecules and the z-axis, θLC, will change, causing a change in the effective refractive index of the transmission mode in the waveguide. This will result in different refractive indices at each side of the patterned electrode, achieving angular deflection of the transmission beam. By changing the magnitude of the applied voltage, dynamic control of the deflection angle can be achieved.
Fig. 1. Working principle of SEEOR: (a) physical structure (y-z plane) and (b) patterned electrode (x-z plane).
Based on the scanning principle of SEEOR, the simulation process could be divided into three steps, as shown in Fig. 2. At first, the LC dynamic modulation model is established to evaluate the LC orientation distribution along y-axis during its response process, θLC(y,t). Then, the finite element analysis method is used to translate this distribution to the effective refractive index change, neff(t). According to the vectorial refraction law, the scanning characteristics of the SEEOR over time, θSC(t), can be ultimately determined.
The orientation of LC, θLC(y,t), under the electric field can be determined by the following equations [17,18]:
The above set of equations can be solved through differential iteration method by selecting appropriate time steps and thickness intervals. At each time step, the orientation evolution speed can be calculated based on the initial orientation distribution and the corresponding electric field distribution. Then, the orientation distribution at the end of that time step can be obtained. By repeating this iterative process, the evolution of LC orientation distribution over time can be obtained.
Under different orientation distribution θLC(y,t), the corresponding refractive index profile in LC layer can be expressed in tensors form as follows:
Due to the complex refractive index distribution in the LC layer, it is hard to obtain analytical solutions directly from Maxwell’s equation. Therefore, finite element method [19] is employed to solve the waveguide mode. The computational model is a basic four-layer waveguide structure, shown in Fig. 3(a). The excitation and reception sources are set on both sides to generate and propagate fundamental TM mode. The entire structure is divided into units with size of 0.1 µm to ensure accurate optical field calculations and obtain the effective refractive index neff(t) of the waveguide mode.
Fig. 3. (a) Finite element model of four-layers waveguide; (b) beam scanning model of patterned electrode.
The scanning electrode pattern unit was shown in Fig. 3(b), which includes two refractive interfaces. When a voltage is applied to region I, the refractive index of that region will be changed from neff-0 to neff(t), causing a small effective refractive index difference between region I and region II. In Fig. 3(b), the propagation vector of the incident light is $\overrightarrow {{\textrm{A}_\textrm{0}}} \textrm{ = }{\textrm{n}_{\textrm{eff - 0}}}\textrm{(0,1)}$, and the normal vector of the first interface is $\overrightarrow {{{N}_\textrm{1}}} \textrm{ = (sin(}{\mathrm{\varphi }_1}\textrm{),cos(}{\mathrm{\varphi }_1}\textrm{))}$. The incident angle, θi0, is given by:
According to the vectorial refraction law, after passing through the first refractive interface, the vector of the refracted light is $\overrightarrow {{{A}_\textrm{1}}\textrm{(t)}} $:
For the second interface, the normal vector is $\overrightarrow {{\textrm{N}_\textrm{2}}} \textrm{ = ( - sin(}{\mathrm{\varphi }_2}\textrm{),cos(}{\mathrm{\varphi }_2}\textrm{))}$, based on the process described by Eqs. (3)–(5), the incident angle θi1 and the outgoing light vector $\overrightarrow {{{A}_\textrm{2}}\textrm{(t)}} $ can be obtained. The angle between the outgoing light $\overrightarrow {{{A}_\textrm{2}}\textrm{(t)}} $ and incident light $\overrightarrow {{{A}_\textrm{0}}} $ is defined as the scanning angle θSC-in(t) of the light beam within the waveguide, which is given by:
Considering that the scanning light will undergo another refraction after leaving the coupling prism, the scanning angle of the SEEOR ${\mathrm{\theta }_{\textrm{SC}}}\textrm{(t) = arcsin[}{\textrm{n}_{\textrm{pri}}}\textrm{sin(}{\mathrm{\theta }_{\textrm{SC - in}}}\textrm{(t))]}$, where npri is the refractive index of the prism. The scanning conditions of more interfaces can be obtained by referring to the accumulation operation of Eqs. (3)–(6).
3. Theoretical simulation of scanning angle dynamic response in SEEOR
On the basis of the calculation model established in the previous section, theoretical simulations are employed to analyze the scanning angle dynamic response in SEEOR. The specific parameters are as follows. The refractive indices and thicknesses of the waveguide layers are shown in Table 1. The parameters of LC material (SLC13p20, from Slichem Display Material) are K11 = 11.2 pN, K33 = 11.6 pN, γ1= 68 mpa.s, ε∥=15.06, ε⊥=3.70, Δε=11.36 and the pre-tilt angle is 2°. The operating wavelength λ is 1064 nm. The electrode structure with three refractive interfaces is used, shown in Fig. 4(a). The schematic diagram of the applied voltage waveform is shown in Fig. 4(b). During the electric field-on process, a square wave with an amplitude of 30 V and a frequency of 1 kHz is applied. During the electric field-off process, the voltage amplitude is 0 V. The durations of the electric field-on and field-off are 16 ms and 120 ms, respectively, ensuring sufficient rotation time for the LC molecules. The time step and thickness interval to solve the LC orientation distribution are 0.1 µs and 0.045 µm, respectively.
Fig. 4. (a) The diagram of the electrode structure; (b) the schematic diagram of the applied voltage waveform.
Table 1. Refractive index and thickness of the waveguide layer
Figures 5(a)-(b) depict the evolution of θLC(y,t) during the electric field-on and field-off processes. When the electric field is on, the LC molecules undergo field-induced rotation and the orientation angle gradually increases from the pre-tilt angle of 2° with increasing time. During the whole electric field-on process, the LC molecules at the two sides (y < 528 nm and y > 3972 nm) have smaller orientation angles than the one in the middle portion (528 nm ≤ y ≤ 3972 nm), which is attributed to the anchoring effect of the alignment layer. When the electric field is off, the LC molecules start to fall back and the orientation angle gradually decreases until it returns to the pre-tilt angle of 2°. In addition, the variation rates of θLC(y,t) during the electric field-on and field-off processes are shown in Figs. 5(c)-(d). During the whole electric field-on process, the rotation speed of the LC molecules in the middle portion is always slightly faster than that at the two sides. While during the electric field-off process, the rotation speed of the LC molecules at the two sides is significantly greater in the early stage, and gradually decrease over time and eventually becomes slower than that in the middle portion. At t = 0.8 ms, the driving force for the rotation of LC molecules mainly comes from the alignment layers. The effect of the anchoring energy generated by the alignment layers can be gradually transferred from two sides to the middle portion through the continuous elastic interaction between LC molecules. For the LC molecules at two sides, despite the molecules closer to alignment layer have stronger anchoring effect, but the smaller initial angles lead to smaller rotation range and slower variation rates. For the LC molecules in the middle portion, the LC molecules closer to the center of LC layer receive smaller impact of the anchoring energy, and thus have slower variation rates. At t = 2.4 ms, the effect of the anchoring energy for LC molecules in the middle portion gradually increases, so the variation rates in the middle portion increase and approach to the one at two sides. The difference in rotation speed of the LC molecules between middle portion and the two sides is mainly due to the stronger anchoring effects of the alignment layer to the LC molecules at the sides.
Fig. 5. The variations of θLC(y,t) during electric (a) field-on and (b) field-off processes. The variation rates of θLC(y,t) during electric (c) field-on and (d) field-off processes.
Figure 6 depicts the evolution of θSC during the electric field-on and field-off processes and show a clear asymmetry. During the electric field-on process, θSC exhibits an “S”-shaped distribution and reaches a maximum scanning angle of 2.2° when t = 1.2 ms. While during the electric field-off process, θSC shows an “L”-shaped distribution, in which the scanning angle rapidly decreases from 2.2° to 0.1° as t increases from 0 to 5.4 ms, and gradually approaches to 0° after t > 50 ms and reaches 0° when t = 110.4 ms.
Fig. 6. The variations of θSC(t) during electric (a) field-on and (b) field-off processes, the inset is a locally enlarged image from 0 ms to 50 ms.
Fig. 7. The variations of θSC(t) during electric (a) field-on and (b) field-off processes under different voltage, the inset is a locally enlarged image from 0 ms to 2 ms.
Figure 7 represents the evolution of θSC during the electric field-on and field-off processes under different voltages. A higher voltage leads to a larger θSC and a faster response, since the rotation amplitude and speed of LC molecules increase with the increasing voltage. To avoid sample breakdown caused by high voltage in the experimental section, 30 V is chosen for detailed discussion.
To gain a deeper understanding of the dynamic response process of the scanning angle in SEEOR, a comparison is made with the bulk phase modulation process in the LC layer, which is calculated based on the analysis method of traditional LC device [20], assuming that the light propagates along the direction of the LC layer thickness. For ease of comparison, the response processes of scanning angle and bulk phase modulation are normalized. The scanning angle is normalized by converting the process of 0°→2.2° to 0→1, while the bulk phase modulation is normalized by converting the processes of 0.93λ→0.04λ to 0→1. Figure 8 presents the comparison of the scanning angle and bulk phase modulation during the electric field-on and field-off processes. During the electric field-on process, the response of bulk phase modulation is slightly faster than that of the scanning angle. Conversely, the response of the scanning angle in SEEOR is significantly faster during the electric field-off process. The difference originates from the different rotation speeds of LC molecules between the two sides and middle portion (as shown in Figs. 5(c)-5(d)). In traditional LC devices, bulk phase modulation is the result of the combined effects of the LC from two sides and middle portion. While in the SEEOR, the waveguide mode can only interact with the LC within the penetration depth of the evanescent wave (about 200 nm), so the response characteristics of the scanning angle in SEEOR is mainly determined by the rotation speed of LC near the alignment layer. Benefiting from the faster rotation speed of the LC near the alignment layer during the field-off process, the SEEOR can achieve faster optical modulation than traditional LC devices.
Fig. 8. The normalized response comparisons of the scanning angle and bulk phase modulation during the (a) electric field-on and (b) field-off processes.
4. Experiments of scanning angle dynamic response in SEEOR
In this section, the dynamic characteristics of the scanning angle of SEEOR was experimentally tested. The preparation of SEEOR followed the procedure outlined in Ref. [14]. The parameters of waveguide materials and structures were consistent with the theoretical simulations. The experimental setup used is depicted in Fig. 9. A 1064 nm laser source was collimated and directed onto the prism of SEEOR. The laser was coupled into the waveguide through the prism. After modulating by patterned electrode, the scanning beam was coupled out from prism to free space. A high-speed camera with a sampling frequency of 40 kHz was employed to capture the laser beam and record its scanning process. The centroid of the beam in the captured images was calculated to determine the displacement distance of the beam during the scanning process. Then the scanning angle at different time was calculated based on the distance between SEEOR and the high-speed camera. Moreover, a 532 nm laser was used to measure the bulk phase modulation of the LC layer by passing through SEEOR perpendicularly along the direction of LC layer thickness with a pair of orthogonal polarizers. A signal generator was used to apply voltage on the SEEOR. Figure 9(b) depicts the SEEOR sample image under a polarized camera, specifically, the sample is placed in a pair of orthogonal polarizers and taken by a digital camera. When the electric field is on, the electrode pattern can be clearly observed, as shown in Fig. 9(c).
Fig. 9. (a) The experimental setup of SEEOR; images of (b) sample under a polarized camera and (c) electrode pattern under an electric field.
Figures 10(a)-(b) illustrate the evolutions of the scanning angle of SEEOR during the electric field-on and field-off processes. When the electric field is on, a square-wave signal with a frequency of 1 kHz and an amplitude of 30 V is applied to SEEOR. It can be observed that the maximum scanning angle is detected to be about 2.18°, which matches the theoretical expectation of 2.2°. There is a slight discrepancy between the theoretical simulations and the experimental measurement results in terms of the variation trend of the scanning angle. This is mainly attributed to the backflow effect of the LC molecules under high voltage condition [21–23]. The impact caused by the backflow effect can be equivalent to a change in the rotational viscosity of LC molecules, and thus affects the rotation process of the LC molecules and the dynamic scanning angle of SEEOR. Moreover, the evolutions of the scanning angle of SEEOR during the electric field-on and field-off processes are tested under a driving voltage with an amplitude of 6 V, which are shown in Figs. 10(c)-(d). It can be obtained that the theoretical simulation and experimental result are in good agreement when the backflow effect is small under lower voltage.
Fig. 10. The variations of the scanning angle of SEEOR during: (a) field-on process under 30 V; (b) field-off process under 30 V; (c) field-on process under 6 V; (d) field-off process under 6 V.
Figure 11 presents the experimental results for the normalized responses of the scanning angle and bulk phase modulation during the electric field-on and field-off processes. As predicted in theoretical simulations, compared to the response of the bulk phase modulation, the scanning angle is slightly slower during the electric field-on process but significantly faster during the electric field-off process.
Fig. 11. The experimental results for the normalized responses of the scanning angle and bulk phase modulation during the electric (a) field-on and (b) field-off processes.
It is worth noting that although the theoretical simulations and experimental results focus on the three refractive interfaces, which lead to a maximum scanning angle about 2.2°, the dynamic scanning characteristics obtained in this paper is also suitable for the larger scanning angle by increasing the number of refractive interfaces, since the maximum scanning angle is decoupled from the time characteristics which can be obtained in Section 2.
5. Conclusions
This paper investigates the dynamic response characteristics of the scanning angle in SEEOR from theoretical, simulation and experimental perspectives. Firstly, a calculation model was established based on LC dynamic theory, finite element analysis and vectorial refraction law. Secondly, theoretical simulations were applied to study the dynamic evolution differences of the LC direction during the electric field-on and field-off processes and its influence on the scanning angle. The simulation results demonstrate that the dynamic variation of scanning angle in SEEOR exhibits asymmetry during the field-on and field-off processes, presenting “S” and “L” shaped trends, respectively. Thirdly, it was clarified that the intrinsic physical mechanism of SEEOR to achieve faster light modulation was the rapid rotation of LC close to the alignment layer during the electric field-off process. Finally, the dynamic scanning angle in SEEOR was measured experimentally, and the results showed a good agreement with theoretical expectations. The above research results have important guiding significance for the high-precision scanning control in SEEOR.
Funding
Changchun State Key Laboratory Major Project (23GZZ07); National Key Research and Development Program of China (2021YFB3600300); Jilin State Key Laboratory Major Project (SKL202402022).
Acknowledgment
This research is supported by the Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, and the CAS Interdisciplinary Innovation Team.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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