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Abstract
This paper proposes a liquid crystal-based order adjustable q-plate system. The system, which is solid-state and electrically controlled without any mechanical components, consists of several bit cells and one symbol cell. The bit cells can be electrically selected whether to modulate the beam. The magnitude of the order of the q-plate system can be controlled by activating specific bit cells. And the sign of the order can be changed by controlling the voltages in the symbol cell. The whole system can realize the function of the order adjustable q-plate with the order ranging from −2n + 1 to 2n-1 with n bit cells. In our experiment, the system with 4 bits is verified. Based on the q-plate system, the vector beams and optical vortexes with the orders ranging from −15 to 15 can be generated.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
For several decades, optical vortex [1,2] and vector beam [3,4], as the landmarks of singular optics corresponding to the phase singularity and the polarization singularity, have attracted considerable attention. Both of them has unique focusing properties in the presence of tight focusing [5–8]. Meanwhile, there are also the promising candidates for oncoming generation of optical communication by improving the capacity of optical communication systems [9–12]. In order to generate optical vortex or vector beam, Pancharatnam–Berry phase optical element (PBOE) is the main management by modulating the space-variant geometric phase associated with the polarization of light [13], which can be also used for imaging [14,15], beam shaping [16,17] and optical modes sorting [18,19]. The PBOE can be realized based on self-assembled nanostructures [20], meta-surfaces [21,22] and photo-aligned liquid crystal (LC) [23–25]. Compared with the other methods, the LC-based PBOE can be applied with the external voltage to change the phase retardation, then the output can be switched electrically [26–28].
The q-plate, frequently-used PBOE, is an optical retarder with the azimuthal rotation of the optical axes, which can create vector beams and let the beams to carry orbital angular momentum (OAM) [28,29]. However, in most situations, the order of the q-plate is fixed, with the q-value defined in the fabrication design. Although a high-speed switching system is proposed, it only switches within two states [30]. And another dynamically switched system for arbitrary order is realized through encoding q-plates onto a spatial light modulator [31].
In this paper, we propose an LC-based order adjustable q-plate system. The system consists of one symbol cell and several bit cells, which are all LC-based. Using the character of the LC that the phase retardation can be controlled by externally applied voltages, the bit cells can be electrically selected whether to modulate the beam. The magnitude of the order of the q-plate system can be controlled by activating specific bit cells. While n bit cells are sandwiched with the symbol cell, the sign of the order can be controlled by manipulating the voltages in the symbol cell. Finally, the whole system acts as an order adjustable q-plate with the order ranging from −2n + 1 to 2n-1. In the experiment, the system with 4 bits is verified. Based on the q-plate system, the vector beams and the optical vortexes with the orders ranging from −15 to 15 can be generated. The system is solid-state and electrically controlled without any mechanical components.
2. Principles for the order adjustable q-plate
Figure 1 shows the architecture for the proposed system. The system consists of several bit cells and one symbol cell, both of which are LC-based. Each bit cell is composed of an LC-based q-plate and a conventional LC-based wave plate. In order to describe the cell for convenience, we use a Jones matrix under circular polarization bases [19] to describe the modulation of LC-based q-plates and LC wave plates. And we can convert Jones matrix Mxy under linear polarization bases to that under circular polarization bases by
with C = [1, j; 1, -j]/21/2. Because the phase retardance can be changed through applied voltage, the matrix under circular polarization bases of the LC-based q-plate at the half-wave voltage (v = 1) and the full-wave voltage (v = 0) can be described aswhere θ is the azimuth, and the order of the q-plate m = 2q. Here, q is the repetition number of optical axis change along the azimuthal axis, which is defined in the fabrication design. When m = 0, the matrix degrades into the modulation of the LC-based wave plate, which is denoted as H(v) = Q(0,v). If the q-plate and the wave plate in each bit cell are at the same phase retardance, we can get the matrix of single bit cell asAs we can see from the Eq. (3), when the two elements are both at the full-wave voltage (v = 0), the bit cell does not modulate the beam. And it will produce modulation when both of them are at the half-wave voltage (v = 1). Therefore, by controlling the voltages, the bit cell can be electrically selected whether to modulate the beam. The matrix for all bit cells iswhere i is the index of the bit cell counted from 1. If the mi of each q-plate are selected as 2i-1 that for binary coding, we can change the positive V from 0 to 2n-1 by controlling the vi of each q-plate corresponding to the voltages applied, where n is the number of total bit cells. It is worth mentioning that Eq. (4) also reveal that the bit cells are reciprocal. In order to realize the negative order, the symbol cell can be set as two LC wave plates, and apply different phase retardation through the voltages control. For the positive situation (sign = 0), the first wave plate is set at half-wave retardation (v = 1) and the second one is at full-wave retardation (v = 0). At this time, the system matrix isFor the negative situation, the phase retardations of the two wave plate exchanges (sign = 1), then the system matrix isFrom Eqs. (5) and (6), it can be seen that the matrixes are the same form as the matrix of a q-plate. The sign of the order can be controlled by the voltages, and the order of the adjustable q-plate system can be expressed asTherefore, the order of the electrically controlled q-plate ranges from -2n + 1 to 2n-1 by controlling vi of each bit cell and sign of symbol cell corresponding to the applied voltages.3. Experimental arrangement and results
In our experiment, the used linearly polarized laser is from a He-Ne laser of 632.8 nm with a polarizer. The adjustable q-plate system is with 4 bit cells and the orders of the LC-based q-plates in each cell are 1, 2, 4 and 8. The LC-based q-plates are fabricated by polarization-sensitive alignment [32, 33], and the LC-based wave plates are by rubbing alignment. The used LC material of each plate is the nematic LC E7. And half-wave voltages and full-wave voltages of each plate in the system are well calibrated, which are about 3.0 V and 2.0 V correspondingly. The response times of the wave plates and the q-plates are about 20 ms and 30 ms. Each q-plate is mounted in a XY translation optical mount for the center alignment and all the plates are mounted together with an optical cage system, as shown in Fig. 2.
As one application of the q-plate, we use our adjustable q-plate system for generating vector beam as shown in Fig. 3. After the linear polarized laser passing after the system, the vector beam with an order of M is generated. With an analyzer, the CCD will obtain a 2|M| petals beam. And the petals should be symmetrical when the q-plates are well positioned. Based on this feature, we can align the positions of the q-plates. Before center alignment, all the bit cells are set at the full-wave voltages initially. Then the first bit cell is activated and the position of Q1 is adjusted for obtaining symmetrical petals, as shown in Figs. 4(a) and 4(b). After that, Q2 can be adjusted under the first two bit cells activated for obtaining another set of symmetrical petals, as shown in Figs. 4(c) and 4(d). Q3 and Q4 are adjusted under activating the first three bit cells and all the bit cells.
Fig. 4 Beam patterns under center alignment, Q1 is mismatched (a); Q1 is matched (b); Q2 is mismatched (c); Q2 is matched (d).
By rotating the analyzer, the petals also rotate. If the rotational directions of the analyzer and the petals are consistent, the order of the vector beam is positive. Otherwise, it is negative. By adjusting the voltages of the system, we activate only one bit cell at a time and symbol cell is set at positive. Then we can get the vector beams with the orders of 1, 2, 4, and 8, respectively. The images of the beam after the analyzer are acquired, as shown in Fig. 5, while the red intensity distribution represents the analyzer at 0° and green represents the analyzer at 45° with anticlockwise. The numbers of petals are 2, 4, 8 and 16 corresponding to the order of 1, 2, 4, and 8. It can be seen that the patterns rotated along the same direction with the analyzer showing the orders of the generated vector beams are both positive.
Fig. 5 Intensity distribution of vector beam with the orders of 1, 2, 4, and 8 after an analyzer (The red intensity distribution represents the analyzer at 0° and green represents the analyzer at 45° with anticlockwise).
According to the binary coding, we can realize the q-plate of any orders ranging from −15 to 15 by setting the states of the bit cells and the symbol cell. For example, we can realize the order of 3, by activating the first- and second-bit cells, and disabling the others, or activating all the cells to get the order of −15. The generated vector beam with the orders of ± 3, ± 6, ± 11 and ± 15 by adjusting the voltages in the system according to the binary codes of each order. As shown in Fig. 6, the numbers of petals are 6, 12, 22 and 30 corresponding to the orders of ± 3, ± 6, ± 11 and ± 15. And the positive or negative of order can be distinguished by the direction of pattern rotation. These results are accordant to our setting states of the cells, which verify our system works correctly.
Fig. 6 Intensity distribution of vector beam with the orders of ± 3, ± 6, ± 11 and ± 15 after an analyzer (The red intensity distribution represents the analyzer at 0° and green represents the analyzer at 45° with anticlockwise).
As another application of the q-plate, we use the adjustable q-plate for generating optical vortex as shown in Fig. 7. Same as traditional q-plate, the optical vortex can be generated by the incidence of circularly polarized beam. In our experiment, the circularly polarized beam is generated by the linear polarized laser after passing a quarter-wave plate (QWP). The generated M order optical vortex passing through a cylindrical lens whose optical axis is horizontal, then the beam will transform to a slanted beam with |M| + 1 petals while the direction of slope corresponds to the sign of order. Figures 8(a)-8(d) shows the intensity distributions of the generated optical vortex with the orders of −3, 6, −11 and 15. The corresponding petals numbers of intensities after a cylindrical lens are 4,7,12 and 16, while the positive or negative of orders can be distinguished by the direction of slope, as shown in Figs. 8(e) and 8(h). These results are also accordant to our setting of the state of the bit cells and the symbol cell, which verify our system works properly again.
Fig. 8 The intensity distribution of the optical vortexes with the orders of −3, 6, −11 and 15 (a-d) and the corresponding intensity after a cylindrical lens (e-f).
Because the laser beam is modulated by a series of q-plates, the system will not be able to generate the corresponding beam well due to the position errors of the q-plates. In order to evaluate the tolerances for each cells, we make a simulation for making one q-plate to have a mismatched position, then the corresponding result is compared with the well-positioned output and the field similarity coefficient is calculated as
where the A is for the corresponding mismatched output and B is the well-positioned output. The field similarity coefficient represents the percentage of A containing B. Figure 9 shows how the field similarity coefficient varies with the position errors of q-plates (The distances between q-plates are set as 3.5 cm according to our experimental system). The full lines and dotted lines represent the input beams with radius of 1 mm and 2 mm respectively. And the different colors refer to the mismatched q-plates. From the Fig. 9, we can see the position error of high order q-plates have great influence than the low order ones. And it can be seen that the tolerances is also affected by input beam radius. For the larger radius, the system can tolerate the greater position error.4. Conclusions
Summarily, this paper proposes an electrically controlled order adjustable q-plate system. The system consists of several bit cells and one symbol cell, all of which are LC-based. Using the character of the LC that the phase retardation can be controlled by externally applied voltage, the bit cells can be selected whether to modulate the beam. The magnitude of the order of the q-plate system can be flexibly controlled through activating specific bit cells, while the sign of the order can be changed by adjusting the voltages in the symbol cell. The whole system can realize the function of the order adjustable q-plate with the order ranging from −2n + 1 to 2n-1. In our experiment, the system with 4 bits is realized to generate vector beams and optical vortexes with the orders ranging from −15 to 15. The system is solid-state and electrically controlled without any mechanical components.
Funding
National Natural Science Funds of China (61775142, 61490710), Shenzhen basic research project on subject layout (JCYJ20170412105812811), China Post-doctoral Science Foundation (2017M612726), and Natural Science Foundation of Guangdong Province (2018A030310662).
Acknowledgments
The authors thank Prof. Yanqing Lu and Prof. Wei Hu's group in Nanjing University for the kind offering of q-plates and wave plates. Correspondences about this paper should be addressed to Prof. Shixiang Xu or Dr. Shuiqin Zheng.
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