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Abstract
Replacing mechanical optical beam steering devices with non-mechanical electro-optic devices has been a long-standing desire for applications such as space-based communication, LiDAR and autonomous vehicles. While promising progress has been achieved to non-mechanically deflect light with high efficiency over a wide angular range, significant limitations remain towards achieving large aperture beam steering with a tunable steering direction. In this paper, we propose a unique liquid crystal based Pancharatnam Phase Device for beam steering which can provide both tunability and a fast response times in a format scalable to large apertures. This architecture employs a linear array of phase control elements to locally control the orientation of the liquid crystal director into a cycloidal pattern to deflect transmitted light. The PCEs are comprised of a fringe field switching electrode structure that can provide a variable in-plane electric field. Detailed modeling of the proposed design is presented which demonstrates that such a device can achieve a high degree of uniformity as it rotates the LC molecules over the 180 ° angular range required to create a Pancharatnam phase device.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
An increasing demand for fast and efficient beam steering techniques within emerging applications such as LiDAR, space based optical communications, and autonomous vehicles has led to a desire to replace mechanical steering devices with less costly non-mechanically scanned ones. Many of the non-mechanical beam steering devices explored to date use liquid crystals (LC) for their inherent birefringence and responsiveness to electric fields. LC based beam steering devices include optical phased arrays (OPA) [1–4], volume holographic gratings [5,6], Pancharatnam-Berry phase based [7–9],continuous phase, optical phased array (V-COPA) device [10,11] and potential emerging approaches [12–14]. While OPA beam steering is the most mature of the options, it is limited in transmission efficiency for steering angles above about ± 1 degree due to the fringing fields at the 2π phase resets.
LC based beam steering devices that are utilizing a type of geometric phase called the Pancharatnam-Berry Phase rather than the optical path delay to steer light, are named Pancharatnam Phase Devices (PPDs). PPDs can create a continuous and unbounded optical phase delay across an arbitrary size aperture without requiring any phase resets. They have the remarkable property of being able to diffract circularly polarized light into a single diffracted mode in the far field with very high efficiency approaching 100%.
PPDs are usually designed as half-wave retarders with a cycloidal anisotropy pattern written into an LC polymer. The optical axis of the birefringent material in these devices is perpendicular to the light propagation vector and rotates by an angle β that varies linearly with x position across the aperture (see Fig. 1). For a distance Λ over which the optical axis of the LC director rotates by 180° in this cycloidal pattern, the PPD will diffract circularly polarized light of wavelength λ to an angle of $\theta ={\pm} {\sin ^{ - 1}}({\lambda /{\Lambda }} )$ with the sign depending on if the light is right or left handed circularly polarized. Because the PPD acts as a half-wave retarder, the output beam obtains the opposite polarization handedness to the input light. More generally, the relative phase shift of the circularly polarized light exiting the aperture at two points separated by x, will be Γ=2 $\beta (x )$ [15].
Fig. 1. Front and top views of the director structure in a beam steering PPD device. Different colors in the picture depicts opposite sides of LC molecules.
There has been work on liquid crystal based PPDs since 2005 [16–18]. Honma and Nose [16] used the Pancharatnam concept to build a liquid crystal Fresnel zone plate with a fixed focal length. Crawford et al. proposed a surface patterning method to fabricate PPDs using polarization holography [17]. Escuti et al. used PPDs to build a modulator for unpolarized incident light with high contrast ratio (600:1) and 2ms switching time for a monochromatic light [19].
Although PPDs can diffract light with very high efficiency, they are not inherently tunable. Dynamic beam steering with these devices is currently only accomplished in a stepwise fashion, whereby several PPDs are stacked in series with polarization switches in between each layer. As the switches toggle the light between right and left circularly polarized light, consecutive diffraction angles either add or subtract to give the final steering angle of the stack. This approach is practically limited because each successive stage of steering reduces the efficiency from transmission losses and beam walk-off. Research on continuous steering with high diffraction efficiency in PPD was the topic of interest to Lei Shi et al. [10]. In order to overcome the tunability challenge, they created a beam steering device called the Vertically Aligned Continuous Optical Phased Array (V-COPA). While the VCOPA approach provides a defect-free topology where a continuous tuning is allowed, it suffered from a slow response time on the order of seconds which makes the device impractical for applications where a rapid response is critical. In contrast to this approach, conventional FFS LCD devices have much faster response time in the order of several milliseconds. Although the response time depends on several factors such as cell gap, electrodes spacing, rubbing angle, LC material, S.H. Lee. et al. [20] showed that a response time of around 10 ms is expected for a FFS mode device with negative dielectric anisotropy.
In this paper we describe a much faster approach to tunable PPD beam steering which can steer to multiple angles with a single layer by tuning the cycloidal pitch Λ with a series of electrically controlled Phase Control Elements (PCEs). We will outline the device concept which implements PCEs by the fringe field switching (FFS) method, present modelling and simulation results predicting the ability to electrically reconfigure the LC molecules, and describe the ability to steer narrow bandwidth light in the presence of topological defects in the cycloidal pattern which naturally arise with this approach.
2. Device concept
The design concept for our device is to have individual PCEs control the sign and magnitude of the in-plane rotation angle of the director, β(x), in a local area through the application of an electric field. High efficiency beam steering with a PPD requires a uniform director angle throughout the thickness of the cell and the ability to set this rotation angle, β, so that any value between -90 and + 90 degrees is achievable. The basic electrode structure we used for the PCEs consists of a fringe field switching (FFS) electrode structure with a series of linear electrodes separated from a ground plane by a thin insulator layer [21]. Each PCE consists of a pair of sub-elements located on interior side of both surfaces of the device that are capable of providing an in-plane electric field. As shown in Fig. 2, the layers going from the glass to the LC are: the common electrode; an insulator; the drive electrode (sub-element of PCEs); and the LC alignment layer. Typical thickness of the glass substrate is 100s of microns, and the other layers are from 10-100s of nano meters. The presence of the ground plane so close to the charged electrodes generates strong electric fields in the vicinity of the electrode corners which enacts a torque on liquid crystal molecules causing rotation against the elastic torque from neighboring molecules.
Fig. 2. The electrostatic potential in the cell with dashed lines indicating electric field lines. The electric field is stronger near edges of electrodes and weaker in the middle of electrodes and gaps.
In Fig. 2, d is the cell thickness that contains the LC and is a few microns and W is width of an electrode and L is width of a gap. The electrodes are typically transparent ITO; the insulator is SiO2 and the alignment layer is a commercial polyimide or a photo-alignment type of layer. All aspects of the basic structure are well known and used in the display industry.
The important aspect of a PCE is its ability to provide a strong electric field to provide a torque that causes the director to rotate, about the layer normal, away from the symmetry axis. As the liquid crystal considered here, has a negative dielectric anisotropy, the symmetry axis is parallel to the electric field.
The surface boundary conditions along the top and bottom electrode layers set the director orientation in the absence of an electric field. Here, we align the director angle to be along an axis defined as +$\; \epsilon $ on one surface and -$\; \epsilon $ on the other in relation to a defined axis of symmetry (Fig. 3(a)). This boundary condition sets an intrinsic bias for the sign of the angle and hence the direction, i.e. clockwise or counterclockwise, it will rotate when an electric field is applied from the strong fringing fields. As a result, the sign of the angle of rotation of the director away from the symmetry axis is determined by the electric field applied to the top or bottom electrodes.
Fig. 3. The “two-step method” for voltage application. (a) The alignment layer rubbing direction of -ɛ and +ɛ on the top and bottom substrates, respectively. (b) The first step sets the direction of rotation based on activating either the top or bottom electrodes. (c) In the second step, a gradient of voltages is applied to the PCEs on both surfaces to make a full cycloidal pattern across the aperture. Black arrows show the handedness of the rotation in each section. (d) signal scheme of RMS voltages applied to PCEs.
To reach from the initial configuration defined by the rubbing angles to the final structure, voltages can be applied in a so called “two-step method” (Fig. 3). If it is desired to have a positive angle of rotation, the electrode surface with positive bias angle +$\epsilon $ is activated first. The magnitude of the field from the selected sub-element is increased until the electric torque on the molecules overcomes the elastic coupling forces to the director on the opposite surface and rotates all the molecules throughout the thickness to a positive angle (1st step). Then the other sub-element on the opposite surface is activated causing the director field between the sub elements of the PCE to uniformly rotate to the positive direction (2nd step). Applying the same process but to the -$\epsilon $ surface causes the director field to rotate in the negative direction. The final rotation angle determined by the balance of torques between the electric torque and the surface anchoring torque. Therefore, by controlling the magnitude of the electric field generated by the PCE, the desired magnitude of the rotation angle can be achieved. As can be seen if Fig. 3(b), in the first step of the two-step process the director rotates in the opposite direction on either side of the dashed line. In the second step of the two-step process the voltage on the electrodes is lowered to provide the desired spiral structure. In the second step of the process as shown in Fig. 3(c) the director now is seen to have a single rotational sense that causes the light to deflect so a single order. This is accomplished by trapping a thin wall of a pi director rotation in the region between the two highest voltage electrodes shown by the dotted line and black arrow. The high efficiency of this device is obtained by squeezing this wall, due to the torque exerted by the high voltage electrodes, to an extent where its optical effect is made to be very small. Figure 3(d) show signal scheme of RMS voltage applied to PCEs activated in Figs. 3(b) and 3(c). The letters designate the top electrodes (A) or bottom electrodes (B). In the first step, shown in Fig. 3(d), electrodes 1A-7A and 8B-14B have high voltage while rest of electrodes has no voltage. This step lasts for about 1 msec. In the 2nd step, the top and bottom electrodes of each PCE have the same voltage applied. The magnitude of the voltage on each PCE corresponds to the desired rotational angle of the director.
In our example device, the electric field of PCE’s on each surface is applied by the electrode structure shown in Fig. 3. The structure is similar to what is known in the display industry as the Fringe Field Switching “FFS” structure [21], but the requirements and use in our embodiment are much different than in those applications.
3. Modeling
3.1 Configuration
In this section, we are going to model the director configuration using the structure of Fig. 3. Consider an active liquid crystal cell where the material is sandwiched between two thin film structures deposited on the glass surface.
The director profile is simulated by a 2-dimensional LC model using vector field method. Solving the Euler-Lagrange derivative of Frank-Oseen free energy density, the director orientation is relaxed at each sampling point taking into account the initial voltage on the system, elastic free energy, electric free energy density and viscous torque on every local point.
Upon applying a voltage on the driving electrodes (PCE’s), fringing fields form at the edges of electrodes due to the voltage difference between PCEs and common electrode. Consequently, the LC director at the locations near the edge of electrodes go under an in-plane rotation by the dielectric torque between LC and the electric field. This is one of the main mechanisms that switch the LC director. The other one is the elastic torque between LC molecules which tends to make the director field uniform. Figure 2 shows the calculated equipotential lines in the regions between the substrates. As seen, there is high in-plane potential difference at the edges of electrodes while there is not such a potential difference at the center of electrodes or the center of the gap. Also, the electric field lines are drawn perpendicular to the equipotential lines to show the in-plane component of electric field resulted from FFS structure [Fig. 2]. Note that the maximum torque (maximum potential gradient), from the fringe filed is not exactly at the edge of an electrode but slightly further from the edge toward the gap. This makes the effective width of the gap smaller. In our modeling, we will take this into account by choosing the value of L larger than W.
3.2 Simulation method
In order to form a PPD structure within the LC molecules, we must control both the twist direction and magnitude. Using the “two-step voltage method” described in Section 2, we define regions with a desired twist direction depending on if we activate the top or bottom electrodes first. We control the local magnitude of the twist by applying different voltages on PCEs. The combination of these tools enables us to create the PPD structure.
Figure 4(a) shows a top view of the device with a voltage gradient on a linear array of PCEs. The width of each electrode is 2 µm which equals the gap width between electrodes. Figure 4(b) shows a top view of the director field at a location midway between the two substrates for the case of the voltages applied in Fig. 4(a) and an infinite anchoring energy on both surfaces. This example shows the basic device principle in operation whereby the in-plane director profile can be controlled by the PCEs. Figure 4(c) shows the azimuthal rotation angle of the director field shown in Fig. 4(b). The phase retardation as a function of coordinate x follows Fig. 4(c) because of the linear dependency Γ=2 $\beta (x ).$
Fig. 4. (a)Top view of the cell showing position of the electrodes with the voltage spectrum shown (b) Top-view of director profile at the middle layer. (c) Angular rotation of the director field shown in (b). Dashed lines show the locations of discontinuities.
The voltages have been applied to provide 3 cycles of a 180 degree azimuthal angle change of the director. As seen, the director rotation angle follows a fairly linear slope in the azimuthal angle across the cell with an average variation of around 10 °. Since these variations are within 10% change of a full π rotation of the director, their total effect would be to produce a rather small phase ripple of about 2 π /10 radians. The effects of these small nonlinearities will be addressed in the Section 3.3. Another issue apparent in Fig. 4 is the sudden change in the director angle from + 90 to -90 ° between highest voltage PCEs (8 V in this example) meet each other (dashed lines in Fig. 4). So, the director at the center of this region is trapped between two neighbors with oppositely twisted director profiles. This is a discontinuity in the angular profile introduced by the geometry of the device. The director in the trapped region undergoes a tight 180 rotation of the director. Light deflected from this region will not be directed to the correct angle and will cause a lowering of the device efficiency. McMannamon [1] found the effect of a defect in the phase profile similar to what we have here is equal to the square of 1 minus the ratio of the defect area to the total area included between two defects. Therefore, it is desired to have the area of this trapped region to be as small as possible. A more complete discussion of the effect of this wall on the optical efficiency, will be presented in a subsequent publication. The next section investigates the effect of design parameters in obtaining a more ideal director configuration for a PPD device.
3.3 Uniformity of the desired director profile
One particular difficulty of FFS devices for beam steering is getting the LC director to rotate monotonically across the aperture (i.e. along the x-axis) due to the presence of the periodic fringing fields above the edges of electrodes. Above the center of an electrode or gap where there is no in-plane field the LC director will still rotate as a result of elastic torque between neighboring LC molecules, but the amount of rotation will be less than on top of an electrode edge. The uniformity of the director rotation angle will be related to the width of the electrodes and the gap between them with smaller widths yielding a more uniform angle [22]. However, optical patterning of features less than 1 micron is difficult. As explained, the effective width of a gap is smaller than the effective width of an electrode. So, by taking the value L, larger than W, we can take this effect into account. With these considerations, we have considered an example design where the electrode width is 1 micron and the gap between electrodes is 2 microns.
From the modeling similar to Fig. 4, it was found that at least 5 electrodes and gaps are needed to reach a fairly linear slope in the azimuthal angle of the director across the cell. Therefore, with the considered 3µm center to center electrode spacing in this example, the minimum periodicity is 15µm. Correspondingly, the steering angle for 1.5 µm wavelength light is $\theta ={\pm} {\sin ^{ - 1}}({\lambda /{\Lambda }} )={\pm} 5.7^\circ $ and for visible light with 0.6 µm wavelength is 2.3 °. Similarly, larger pitches controlled with more number of electrodes would result in lower deflection angle with no practical limitation.
We have also investigated the significant effects of the azimuthal anchoring energy on the director uniformity. We modeled the effect of both an infinite and weak (10−5 J/m2) azimuthal anchoring energy on the values of the azimuthal angle of the director over the center of the electrode (${\phi _{me}}$), the center of the gap (${\phi _{mg}}$), and over the edge of the electrode(${\phi _{ee}}$) while the thickness of the device was fixed at 3.1 microns. We found that for the large anchoring energy the values were: ${\phi _{me}}$=49°; ${\phi _{mg}}$ = 54°;${\phi _{ee}}$=88°; while for the lower anchoring energy, the values were ${\phi _{me}}$=72°; ${\phi _{mg}}$ = 74°;${\phi _{ee}}$=90°. This shows the degree to which a lower anchoring energy provides a more uniform in-plane azimuthal twist angle of the director.
For a PPD, it is desirable that the director field has the same orientation through the thickness of the cell. Furthermore, the director should be perpendicular to the z-axis everywhere so that the condition of the device being a half wave retarder is fulfilled at all points across the aperture.
To test the performance of our simulated device, we can consider the light transmission through the PPD that is between a crossed polarizer and analyzer. If the director is in a plane normal to z-axis yet uniformly rotated by an angle $\alpha $ along z-axis, we expecte that the light transmission will be zero for all wavelengths of light if the polarizer is also rotated by the same angle $\alpha $. Also, we want to check if a PPD is a half-wave retarder for any in-plane rotation angle. This condition will be met if the light transmission is maximized and the same for any voltage for the case when the polarizer is at 45 ° to the extinction angle.
Figure 5 shows graph of normalized intensity as function of the angle of the polarizer relative to rubbing direction of the cell at two locations of center of the electrode and close to edge of an electrode where the torque is maximum. This is shown for red light (620 nm), green light (550 nm) and blue light (460 nm) wavelengths for an applied voltage of 10 volts that gives a director rotation of 90°. When the polarizers are set to the minimum transmission level, we see that the light transmission is close to zero for all wavelengths. And it is seen that the maximum light intensity is achieved for the polarizer angle of 45° from the extinction angle. This demonstrates that the device is acting as a half wave plate whose optic axis is rotated to 90°.
Fig. 5. Plots of Normalized Intensity as function of the rotational angle of the crossed polarizers $\alpha $ relative to rubbing direction of the cell for RGB light at (a) the edge of an electrode (black arrow) and (b) middle of an electrode (black arrow). Plots (c),(e) correspond to the boxes shown in Fig. 6(a) and (d),(f) correspond to boxes shown in Fig. 6(b)
Figure 6(a) shows a graph of the extinction angle vs. applied voltage for 620 nm and 460nm light. That the extinction angle for these different wavelengths occurs at the same angle, for a given voltage, indicates that the director profile along Z direction is effectively uniform (not twisted which would cause the extinction angle to be wavelength dependent).
Fig. 6. (a)Extinction angle vs. applied voltage for red (620nm) and blue light (460nm) at both the edge and middle of the electrodes (b) The phase retardation vs voltage for red and blue light at the locations as indicated.
Figure 6(b) shows the retardation of a PCE as a function of the applied voltage which controls the director rotation angle. It is seen that the retardation is constant over the required voltage range, which indicates the device can maintain the required half-wave retardation for any rotation angle. These data in Fig. 6 verify that the modeled device, based on our tunable PPD concept, is capable of providing the desired Pancharatnam beam steering device structure shown in Fig. 1.
3.4 Optical power efficiency calculation
We used a Finite Difference Time Domain (FDTD) calculation to predict the resultant efficiency of the diffracted light for the director configuration used in the above example. FDTD is a precise numerical analysis method that uses grid-based differential approximations to solve the time-dependent Maxwell’s equations in 2 or 3 dimensions. The electric and magnetic field vector components are repeatedly solved over a volume of space until a steady-state is reached. Then, the near field data is collected and transformed to far-field by Kirchhoff integral theorem in order to do an efficiency analysis of the diffraction limited light. A Guassian beam whose beam waist is much larger than the aperture size was chosen to construct a nearly top hat beam as an incident light.
As reference, a calculation was done for an ideal case where the mathematically generated director profile of the pancharatnam-Berry geometry goes under a continuous π rotation in each cycle without any trapped wall or out of plane tilt. The calculation was repeated for three different pitch sized of ${{\Lambda }_1} = 45\; \mu m$ (controlled by 15 PCEs) over an aperture of x = 180 µm, ${{\Lambda }_2} = 21\; \mu m$ (controlled by 7 PCEs) over an aperture of x = 84 µm and ${{\Lambda }_3} = 15\; \mu m$ (controlled by 5 PCEs) with an aperture of 60 µm, knowing that PCE’s periodicity is 3 µm.
In each case, the aperture size was set to include four pitches (and four walls) and was compared to an ideal case (four pitches with no wall) with the same number of points over the same aperture size of that specific case. Similar to section 3.4, 1 µm electrodes and 2 µm gaps with d = 3.1 µm and weak anchoring energy of ${w_a} = {w_p} = \frac{{{{10}^{ - 5}}J}}{{{m^{2\; }}}}$ was considered. For visible light with $\lambda = 620nm$, the corresponding deflection angles for these pitches are ${\theta _1} = 0.8^\circ $, $\; {\theta _2} = 1.7^\circ $ and ${\theta _3} = 2.4^\circ $ respectively. In this case the birefringence of the material was assumed to be ${\Delta }n = 0.1$ which would make the 3.1 µm thick cell, a half retarder for visible light with $\lambda = 620nm$, according to the equation ${\Delta }n.d = \lambda /2$. If we consider $\lambda = 1\mu m$, the corresponding deflection angles are $\theta {^{\prime}_1} = 1.3^\circ $, $\theta {^{\prime}_2} = 2.7^\circ $ and $\theta {^{\prime}_3} = 3.8^\circ $.
Figure 7 shows far filed intensity of a diffracted light for PPD structure correspond to the $\; {{\Lambda }_2} = 21\; \mu m,\; $normalized to the ideal case. The efficiency was then estimated as the ratio of the peak intensity for each case relative to the ideal case.
Fig. 7. Normalized far filed intensity of a diffracted light for the example device with ${\Lambda } = 21\; \mu $. The solid black curve is for the case of an ideal continuous spiral structure with an aperture of 84 microns and the dashed red curve is for the numerically calculated director configuration for the example device described in the text with the same aperture and pitch.
Final optical diffraction efficiencies are summarized in Table 1.
Table 1. Efficiency results for different deflection angles
4. Summary
We have introduced a novel approach to build a fast switching, tunable beam steering device based on Pancharatnam phase. The basic idea is using an array of PCEs to control the in-plane twist of the director field at every location along the aperture (x direction). Initial modeling has shown a promising result where we have enough control over the twist angle and uniformity of the twist. Although design parameters and voltages shown are not optimized, the result clears the way for the subsequent device optimization effort to achieve a tunable Pancharatnam based tunable beam steering device relatively short term. Moreover, we can consider a 2-D array of square PCEs in the future to make to two-dimensional electrically controllable phased array.
Funding
U.S. Air Force (FA8650-19-C-1834).
Acknowledgment
The authors would like to thank Dr. Paul McMannamon and Douglas Bryant for many helpful discussions and suggestions.
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