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Abstract
The liquid crystal spatial light modulator (LCSLM) is an optical device that can realise non-mechanical beam scanning. However, the traditional integer-order model cannot adequately characterise the dynamic performance of LCSLM beam steering because of the viscoelasticity of liquid crystals. This paper uses the memory characteristics of fractional calculus to construct a fractional constitutive equation for liquid crystals. Combining this equation with the LCSLM beam steering principle, a fractional-order model of the beam steering system is established, and the Legendre wavelet integration operational matrix method is used to estimate the model parameters. In addition, we established a test platform for the dynamic characteristics of LCSLM beam steering system and verified the effectiveness of the established model through experiments. The fitting effects of the integer-order and fractional-order models are compared, and the influence of different model orders on the dynamic performance of beam steering is analysed. Experimental results show that the fractional-order model can accurately describe the dynamic process of beam steering, and this model can be applied to the study of LCSLM-based two-dimensional non-mechanical beam steering control strategies to achieve fast, accurate, and stable beam scanning.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The liquid crystal spatial light modulator (LCSLM) is an optical device that can realise non-mechanical beam scanning [1]. LCSLMs are characterised by their small size, light weight, low power consumption, high flexibility [2–4], and wide range of applications, such as beam shaping [5], space optical communications [6,7], lidar [8], and adaptive optics [9,10]. LCSLM change the effective refractive index of the liquid crystal molecules through electronic control, and then modulate the phase of the incident beam to achieve non-mechanical steering of the beam [11,12]. The LCSLM-based beam steering system can effectively avoid the problems of complexity of the control system, beam jitter, and the difficulty of miniaturisation of traditional mechanical steering parts [13,14].
In laser-phased array radar, fast, precise, and stable beam steering can achieve accurate target detection. However, the beam control reliant on the system model. Therefore, accurate modelling is the key to beam control. At present, research on LCSLM beam steering system modelling focuses on two main aspects. The first is mechanism modelling techniques, using numerical analysis to solve the Erickson–Leslie equation, modelling the liquid crystal’s response time and voltage [15], or using exponential function fitting to describe the liquid crystal’s relaxation phenomenon [16]. However, these methods only describe the dynamic response process of liquid crystals under the action of an electric field, and do not model the relationship between the LCSLM phase retardation and the beam steering angle. The LCSLM wave control model was established in [17], based on the principle of radar-phased array combined with Kirchhoff’s diffraction theory, but this method can only obtain the corresponding relationship between the steering angle of the beam and the phase retardation in the steady state, and fails to characterise the dynamic change of the beam’s steering angle. The second aspect is the system identification techniques. Lin et al. [18] used a high-speed CCD camera to collect the light intensity of the beam centroid, applied the first-order inertia link to establish an integer-order model, and estimated the model parameters using the two-point method. This method ignored the viscoelasticity of liquid crystals and the model did not fit well with the steering process of actual beam.
Previous research has yet to produce a dynamic model that can accurately describe the LCSLM phase retardation and beam steering angle. With the continuous improvement of fractional calculus theory, many researchers have used fractional calculus to establish more accurate models of viscoelastic materials [19–21]. In contrast to integer-order calculus, fractional-order calculus has memory characteristics and global correlation, and is an effective tool for viscoelastic materials modelling [22–24].
In this paper, we apply fractional calculus theory to construct a fractional constitutive equation of a liquid crystal. A fractional model of the phase retardation and the beam steering angle is then established, according to the LCSLM beam steering principle. Using the Legendre wavelet integration operational matrix technique to identify the model parameters. Finally, the validity of the fractional-order model is verified through experiments and the fitting effects of the integer-order model and the fractional-order model are compared, with the influence of different model orders on the dynamic performance of beam steering analysed.
2. Liquid crystal fractional characteristics and fractional constitutive equation
2.1 Liquid crystal fractional characteristics
Liquid crystals are a distinct class of material combining properties of solid and liquid states, possessing both the elasticity of solids and the viscosity of liquids. In a phase-only nematic LCSLM, the liquid crystal molecules are arranged parallel to the glass substrate when no driving voltage is applied. Under the action of a driving voltage, the liquid crystal molecules exhibit viscoelastic deformation, that is, splay and bending deformation, which is appears as tilt of the molecules along the Z-axis, as shown in Fig. 1. When the voltage is removed, the liquid crystal molecules return to their original orientation; the molecules are therefore said to have memory. That is, the deformation of liquid crystal molecules is not only related to their present but also to their historical state.
Fig. 1. The arrangement of liquid crystal molecules in LCSLM. (a) unload voltage and (b) load voltage.
Fractional calculus has memory characteristics and global correlation, making it a useful tool for describing the viscoelastic deformation of liquid crystals. According to fractional calculus theory, if the stress on a material does not exceed a certain limit, the viscoelastic properties of the material are approximately linear. In the LCSLM, the stress is the electric field force, with the magnitude of stress thus related to the driving voltage. The LCSLM upper voltage limit is 5 V and the tilt angle of the liquid crystal molecules is less than π/2 at this driving voltage. The viscoelastic properties of liquid crystal can therefore be approximated as linear. This linear response means that the tilt angle of the liquid crystal molecules changes proportionally with changes in the electric field force given sufficient time. Moreover, the process of change is only related to time, that is, it is only a function of time.
2.2 Liquid crystal fractional constitutive equation
In fractional calculus, the elasticity and viscosity of materials can be described by elastic and viscous elements, respectively, as shown in Fig. 2. The combination of these two elements can express the viscoelasticity of the material. The elastic element obeys Hooke’s law:
where $\sigma (t)$ is the stress, $\varepsilon (t)$ is the strain, and H is the elastic coefficient.The viscous element can be expressed as:
where K is the viscosity coefficient. ${D^\alpha }$ is the fractional differential operator, $\alpha > 0$ is the fractional order, and ${D^\alpha }f(t)$ is the Riemann–Liouville fractional differential, which is defined as [25]:For a single liquid crystal molecule, the deformation process under the action of the electric field force $F(t)$ can be described in parallel by elastic and viscous elements. That is, the viscous characteristic of splay is represented by the fractional viscous element ${\eta _1}{D^{{\alpha _1}}}\theta (t)$, and the elasticity characteristic is represented by ${k_1}\theta (t)$. Similarly, the viscosity of bending is expressed as ${\eta _2}{D^{{\alpha _2}}}\theta (t)$ and the elasticity of bending is expressed as ${k_2}\theta (t)$. Here, ${\eta _1}$ and ${\eta _2}$ are the viscosity coefficients of splay and bending, respectively. ${k_1} = 1.11 \times {10^{ - 11}}\, \textrm{N}$ is the splay elasticity coefficient and ${k_2} = 1.71 \times {10^{ - 11}}\, \textrm{N}$ is the bending elasticity coefficient. $\theta (t)$ is the tilt angle of the liquid crystal director from the Z-axis. We can therefore obtain the constitutive equation of a single liquid crystal molecule as:
The tilt angle of liquid crystal molecules can be calculated by the elastic continuum theory of liquid crystals, and the tilt angle is related to their position in the liquid crystal layer and the voltage, as shown in Fig. 3. The theory of linear fractional viscoelasticity allows the relationship between the voltage and the tilt angle of the liquid crystal director to be approximated as linear. Therefore, we take $\hat{\theta }(t)$ as the average value of the tilt angle of the liquid crystal molecules.
Because $\hat{\theta }(t)$ is the average of the tilt angle of all of the liquid crystal molecules, the electric field force is also averaged. Assuming a uniform electric field, then:
where $q = 1.6 \times {10^{ - 19}}\textrm{C}$ is the unit charge, j is the volume distribution charge coefficient of the liquid crystal molecules, and $x = 9.8\, {\mathrm{\mu} \mathrm{m}}$ is the thickness of the liquid crystal layer and $u(t)$ is the driving voltage. Substituting Eq. (5) and $\hat{\theta }(t)$ into Eq. (4), we can obtain the fractional constitutive equation of the liquid crystals:3. LCSLM-based beam steering system fractional-order model
3.1 LCSLM-based beam steering system fractional-order modelling
The LCSLM-based beam steering system, shown in Fig. 4, includes the following components: a laser, an expanding collimator, a polariser, an LCSLM, a beam splitter, a mirror, a computer, and a high-speed CCD camera.
The LCSLM adjusts the voltage distribution between adjacent electrodes to produce a constant phase retardation $\Delta \varphi $, forming a wavefront with a phase modulation of 2π. The liquid crystal phase modulation characteristic curve is used to calculate the driving voltage corresponding to the wavefront. The resulting greyscale image is then loaded onto the LCSLM electrode. Under the action of the driving voltage, the liquid crystal molecules are tilted along the Z-axis and the incident beam is phase-modulated to achieve beam steering [26]. The schematic diagram of the LCSLM beam steering is shown in Fig. 5.
According to the theory of phased array radar for a beam perpendicularly incident on the LCSLM, the steering angle ${\theta _p}$ of the outgoing beam is:
where $\lambda$ is the incident light wavelength, and d is the distance between adjacent electrodes.Figure 5 shows that although the phase modulation characteristic curve is nonlinear, we select the curve to acquire the best linearity in the range of 0 to 2π for calibration (The phase modulation characteristic curve of the LCSLM used in this paper corresponds to a driving voltage of 1-4 V). The phase and voltage can therefore be approximated as having a linear relationship. To form a constant phase retardation, different voltages need to be applied to adjacent electrodes. A higher voltage obtains a faster response speed from the liquid crystal [15]. The steering of the beam requires multiple electrodes to work together to achieve a phase modulation of 2π. Therefore, to ensure that the beam can be deflected, the minimum drive voltage is used as the system input.
To illustrate the relationship between the minimum voltage and the phase retardation, we provide an example. Assuming that the phase retardation is $\Delta \varphi = 0.4\pi$, five electrodes are needed to achieve a phase modulation of 2π. In this case, the voltages are ${u_1} = 1\, \textrm{V},\, {u_2} = 1.75\, \textrm{V},\, {u_3} = 2.5\, \textrm{V},\, {u_4} = 3.25\, \textrm{V}\, \textrm{and}\, {u_5} = 4\, \textrm{V}$; the minimum voltage is 1 V. The corresponding relationship between phase retardation and voltage is then:
It is worth noting that the linear scale factor L is not a fixed parameter. The difference is that the value of $\Delta \varphi$ is different, and the driving voltage of other electrodes is different between 1-4 V, and the scale factor L becomes smaller as $\Delta \varphi$ decreases. When $\Delta \varphi = \pi$, the steering angle of the system is maximum, and two electrodes are required to achieve a phase modulation of 2π, then ${u_1} = 2.5\textrm{v,}\, {u_2} = 4\textrm{v}$.
The driving voltage u is not only related to the phase retardation, but also to the tilt angle of the liquid crystal molecules. In Fig. 3, we can fit the voltage and the average of the molecules’ tilt angles $\hat{\theta }$ using a linear piecewise function, the fitting results are shown in Fig. 6.
Because the maximum amount of phase retardation is $\Delta \varphi = \pi$, the corresponding maximum voltage is 2.5 V. Adding the linear function corresponding to the 0 V to 3 V range in Eq. (9) into Eq. (8), we obtain:
Substituting Eq. (10) into Eq. (7), we obtain the relationship between the steering angle of the beam and the tilt angle of the liquid crystal molecules:
Substituting Eq. (8) and Eq. (11) into Eq. (6), we can obtain the fractional-order model of the beam steering angle of the LCSLM and the phase retardation:
The beam steering system can experience delays because data processing and transmission require time. Adding the total delay $\tau$ of the system into Eq. (12), we obtain the final LCSLM beam steering system fractional-order model as:
3.2 Identification of model parameters
In Eq. (13), the splay and bending viscosity coefficients of liquid crystal ${\eta _1},{\eta _2}$; the volume distribution charge coefficient of the liquid crystal molecule j ; fractional orders ${\alpha _1}$ and ${\alpha _2}$; and the time delay coefficient $\tau$ are unknown. We used the Legendre wavelet integration operational technology to identify the model parameters [27]. This technique is to construct the Legendre polynomial into the form of wavelet, and use the Legendre wavelet to transform the fractional order system into the integration operational matrix. The least squares method is used to solve the matrix, and then the model parameters are identified. The Legendre wavelet integration operational technology can simultaneously estimate model parameters and fractional order with high identification accuracy.
Assuming that ${\alpha _2}$ is the highest order in Eq. (13), by simultaneously dividing both ends of Eq. (13) by ${\alpha _2}$, we can obtain the fractional integral equation
Equation (17) can be simplified to $AX = B$, We use the least square method to solve the matrix X. First, assuming that the fractional orders ${\alpha _2},{\alpha _1}$ and delay operational matrix Z are known, then the matrix X can be solved by the following equation
Secondly, we bring the ${\alpha _2},{\alpha _1}$, time delay coefficient $\tau$ and the model parameters X into Eq. (13), and obtain the output of the identification model. The error between the system identification output ${\theta _a}(t)$ and the actual beam steering angle ${\theta _p}(t)$ is calculated by the following equation
where ${T_f}$ is the data collection time.Finally, we define the change interval of the fractional derivative orders ${\alpha _2},{\alpha _1}$ and delay matrix Z, circulate them within the defined interval, and then obtain their respective error values from Eq. (19). The differential orders, model parameters corresponding to the minimum error value are taken as the optimal solution.
4. Experiments and results
The LCSLM produced by Meadowlark Optics (USA) is used to establish the beam steering system. This device has 1920×1152 electrodes, its incident wavelength is $\lambda = 1064\,$ nm, the distance between its electrodes is $d = 9.8\, {\mathrm{\mu} \mathrm{m}}$, and the refresh frequency of its voltage is 1 kHz. The detector uses a UX50 high-speed CCD camera produced by Hamamatsu (Japan) with a sampling frequency of 10 kHz. The experimental diagram of the LCSLM-based beam steering system is shown in Fig. 7.
During operation, a greyscale image with a phase retardation of $\Delta \varphi = \pi /2$ is input to the LCSLM. The high-speed CCD camera then detects the position change of the beam’s centre of mass and identifies the model parameters, obtained by the Legendre wavelet integration operational matrix method. The identification results are ${\eta _2} = 7.3076 \times {10^{ - 11}},\, {\eta _1} = 6.9062 \times {10^{ - 11}},j = 695.0151,\, {\alpha _2} = 1.8,\, {\alpha _1} = 0.9\, \textrm{and}\, \tau = 3.027$. Figure 8 shows the fitting effects of the fractional-order and integer-order models.
Fig. 8. When $\Delta \varphi = \pi /2$, comparison results of the integer-order and fractional-order models. (a) load voltage and (b) remove voltage.
Figure 8(a) shows that when the loading phase retardation $\Delta \varphi = \pi /2$, the beam steering angle gradually increases from 0 rad to 0.0272 rad. The fractional-order model accurately characterises the dynamic performance of the beam steering. However, there is a large error in fitting the integer-order model to the actual data. Figure 8(b) shows that when the voltage is removed, the beam steering angle gradually decreases from 0.0272 rad to 0 rad, which can also be accurately simulated using the fractional-order model. This verifies the validity of the fractional-order model established in this paper.
To verify whether the model can still accurately fit the beam steering process at different steering angles, we input phase retardations of $\Delta \varphi = \pi /4,\, \Delta \varphi = \pi /8\, \textrm{and}\, \Delta \varphi = \pi /16$; the resulting beam steering angles are 0.0136 rad, 0.0068 rad, and 0.0034 rad, respectively. The comparison results of the integer-order and fractional-order models are shown in Figs. 9–11.
Fig. 9. When $\Delta \varphi = \pi /4$, comparison results of the integer-order and fractional-order models. (a) load voltage and (b) remove voltage.
Fig. 10. When $\Delta \varphi = \pi /8$, comparison results of the integer-order and fractional-order models. (a)load voltage and (b) remove voltage.
Fig. 11. When $\Delta \varphi = \pi /16$, comparison results of the integer-order and fractional-order models. (a) load voltage and (b) remove voltage.
Here, for all of the different beam steering angles, the fractional-order model more accurately describes the dynamic characteristics of the beam. Whether during loading or removing voltage, the fitting effect of the fractional-order model is better than that of the integer-order model.
To verify the influence of fractional order on the dynamic characteristics of the model, a quantitative analysis is performed. With ${\alpha _1} = 0.9$ and ${\alpha _2}$ increased from 1.4 to 2.2, the error between the fractional model at different orders and the actual beam steering process is calculated. We also calculate these errors for ${\alpha _2} = 1.8$ and ${\alpha _1}$ increased from 0.7 to 1.1. The errors for different orders during the voltage loading and voltage removal processes are record in Tables 1 and 2, respectively. The experimental results are shown in Figs. 12 and 13.
Fig. 12. When $\Delta \varphi = \pi /16$, comparison results of the voltage loading process for different orders of the fractional-order model.(a) fractional model of different order ${\alpha _2}$ and (b) fractional model of different order ${\alpha _1}$.
Fig. 13. When $\Delta \varphi = \pi /16$, comparison results of the voltage removing process for different orders of the fractional-order model.(a) fractional model of different order ${\alpha _2}$ and (b) fractional model of different order ${\alpha _1}$.
Table 1. Error between different fractional-order models and experimental data during voltage loading.
Table 2. Error between different fractional-order models and experimental data during voltage removing.
From the experimental results and Tables 1 and 2, the model error gradually decreases when ${\alpha _2}$ is increased from 1.4 to 1.8 (with ${\alpha _1}$= 0.9). However, when ${\alpha _2}$ is further increased from 1.8 to 2.2, the fractional-order model error gradually increases. Similarly, ${\alpha _1}$ have the same trend of change (with ${\alpha _2} = 1.8$). The model fits the actual curve best when the fractional order is ${\alpha _2}$= 1.8 and ${\alpha _1}$= 0.9.
Figure 14 compares the error between the fractional model of different orders and the actual beam steering process when the order varies widely. During both the voltage loading and removal processes, the error is smallest when ${\alpha _2}$= 1.8 and ${\alpha _1}$= 0.9. This model is closest to the process of actual beam steering, which further verifies the validity of the fractional-order model.
Fig. 14. When $\Delta \varphi = \pi /16$, comparison results of different order fractional-order models error.(a) load voltage and (b) remove voltage.
5. Conclusion
This paper applied the theory of fractional calculus to construct a fractional constitutive equation of liquid crystals. The beam steering principle of the LCSLM was analysed to determine the relationship between the beam steering angle and the tilt angle of the liquid crystal molecules, and then the beam steering system fractional-order model was established. The fitting effect of beam steering for different angles was compared, and the influence of different fractional orders on the dynamic characteristics of the model was analysed. The experimental results show that fractional-order model can more accurately characterise the dynamic response of the beam than the integer-model can, and fractional-order model fits the actual beam steering well, whether during the voltage loading or removal processes. This model could be applied to research on LCSLM-based non-mechanical beam steering control strategies to achieve fast, accurate, and stable beam scanning.
Funding
Department of Science and Technology of Jilin Province (20190302089GX).
Acknowledgement
We are indebted to Bo Xiao and Ding Kang for useful technical discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in the paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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