Multi-functional liquid crystal devices based on random binary matrix algorithm

ABSTRACT Multi-functional diffraction gratings have found broad applications in laser processing, medicine, beam manipulation, etc. However, based on the existing schemes there are still some challenges in fabricating the diffraction gratings with the controllable and arbitrary diffraction intensity of each order, multiple phase distribution, and polarisation of the output. Here, we propose a new phase mixing algorithm based on the random binary matrix and describe the theoretical foundation and experimental realisation of the proposed scheme. Using the new algorithm, several phase-mixing devices were fabricated, and the experiment verified the relationship of the light intensity, polarisation, phase and the algorithm. Our approach is practical to mix the phases of several diffraction gratings to realise the controllable diffraction orders, offering an innovative idea for the research of the liquid crystal phase mixing devices. GRAPHICAL ABSTRACT


Introduction
Phase is one of the vital parameters of the light field [1][2][3][4][5][6].By spatially modulating the optical phase, a series of light fields with special properties can be obtained [1][2][3][4], which expands the applications of these light fields.Diffraction gratings, as the device of spatially modulating phase, play important roles in many applications including spectroscopy [7], optical communication [8], optical computing [9], augmented reality [10][11][12], etc. Conventionally the diffraction gratings are made via building three-dimensional surface structures using methods like lithography, machining and imprinting [13].So far, spatial light modulators [14], metasurfaces [15], and liquid crystals (LCs) [16] have been utilised to produce diffraction gratings.Recently, breaking through the limitation of diffraction order distribution in traditional diffraction gratings and realising the separate control of the phase and amplitude of each diffraction order have become critical topics of diffraction gratings.
In the research of the methods and materials applied in light field modulation, liquid crystal (LC) has been widely used in phase manipulation due to its unique birefringence characteristics and high controllability, such as LC vortex Q-plate, LC polarisation grating (PG) and LC geometric phase lens [17][18][19][20].Unlike the surface structured gratings, LC gratings draw more and more attention for their special properties namely electrical tunability [18], total flat grating surface with less scattering [17], ideally continuous phase profile [19], self-assembled three-dimensional LC structures with high diffraction efficiency in broadband and wideangle [21] and high resolution.The geometric phase device based on LC has the obvious advantages of low cost, electrically switchable ability and broadband spectral range in operation [17][18][19][20][21].In addition, unlike dynamic phase devices that rely on changes in optical path difference, the principle of geometric phase factors stems from the manipulation of spatial polarisation changes [22].The geometric phase factor determines the phase of the output light.The continuous phase change and higher fabrication accuracy make LC devices based on the geometric phase principle more efficient in conversion than other spatial light modulators [18,23].
With the above special properties, LC gratings are widely used as coupling-in-and-out gratings in augmented reality [11], spatially linear phase shifter in beam steering, multiplexer and demultiplexer in communication using optical vortex [24], and optical differentiator in optical edge detection [25].For the LC gratings used currently, the positions or diffraction angles of the diffraction orders follow the grating equation with a certain grating pitch and the intensities of the target diffraction orders are normally equal.Arbitrarydesigned nonuniform diffraction angles and intensities for different diffraction orders have not been proposed and realised yet [19,20].Phase profiles of different orders are normally the same as the original incident beam, with exceptions like the LC fork gratings which can generate multiple orders with different orbital angular momentum (OAM) values and be used in optical communications [26,27].But for these LC fork grating series, the phase profile is still limited to generating only the vortex OAM beam [18].The schemes mentioned above are all based on the phase superposition algorithm, but the results getting by this algorithm are fixed and the phase factors mixed can not be manipulated respectively.Therefore, to achieve the multifunctional gratings with the controllable phase and amplitude of each diffraction order, it is necessary to develop a new phase mixing algorithm.Here, we propose a new phase mixing algorithm based on the random binary matrix.Different from the traditional phase superposition algorithm, the new phase mixing algorithm based on the random binary matrix can manipulate the contributions of phase factors by changing the distribution probability of the random binary matrix to manipulate the intensity of the incident light.It's what the traditional phase superposition algorithm can not achieve.Meanwhile, different from the random phase algorithm applied in SLM [28], the new algorithm applied in PB-phase devices, is based on the random fitting of the binary matrix image of digital micromirror device (DMD), which is more practical and can be used in fabricating phase-mixing gratings and holographic device.As the order of the matrix is large enough and the distributions of the two phases mixed are nearly uniform and complementary, it can make sure that the diffraction beam has an excellent quality.It's quite simple and available for the phase mixing algorithm to manipulate the contributions of the phase factors by changing the distribution probability of the random binary matrix.Depending on this algorithm, the fabrication of the multiplexing holographic devices and the multi-functional diffraction gratings has been achieved.The new LC gratings are supposed to greatly extend the potential application areas of the current LC gratings.The device creatively combines two grating structures, which makes the controllable diffraction orders of the output up to four, and with the increase of controllable diffraction order, the diffraction grating has a broader application prospect.Our proposed LC gratings exhibit the high efficiency, good flexibility, and multiplexing output, and will create new opportunities for diffractive optics.Meanwhile, the phase mixing algorithm based on the random binary matrix will also offer an innovative idea for the optical phase mixing.

Theory
The working principle of the proposed LC PG can be deduced by the Jones matrix calculation.Considering the incident light as a horizontal polarised light, it can be described as the superposition of left and right circular polarisation (LCP and RCP), and its Jones vector can be abbreviated as: When the fork polarisation grating (FPG) satisfies the half-wave condition, its outgoing light is expressed as: where p is the preset period and q is the topological charge of the FPG [2].From the calculation above, it can be seen that expð�2iαÞ occurs, which is usually called the geometric phase factor.The distribution of LC optical axes obeys Equation (2) in the x-y plane:α ¼ qθ þ πx=P.The FPG satisfies the grating equation as: sin γ L ¼ λ 0 P for LCP and sin γ R ¼ À λ 0 P for RCP [2].Due to the different signs of the phase factors, the left circular polarisation and the right circular polarisation are diffracted to different angles, and the output intensity of left and right circular polarisation accounts for 50% respectively.The process called phase superposition will be described in detail in Appendix A.
It is a common acknowledgement that spatially modulating polarisation, intensity, and the phase of the electric field mainly depend on the local phase factor on the LC geometric phase device.Therefore, the key point of mixing different phases is how to make each sub-region on the LC geometric phase device randomly obtain one of the two different phases.From such a point of view, the arbitrary random binary matrix consisting of 0 and 1 is unveiled in this paper: the proportion of getting 1 in this matrix can be set to be any percentage ranging from 0 to 100%, and meanwhile, the rest elements in the matrix are accordingly set to be zero.By convolution integral, the phase factor is perfectly coupled with the random binary matrix, which means that the probability acquiring one of the two phases in each point can be any value ranging from 0 to 1.As shown in Figure 1, the new hybrid phase distribution can be obtained by adding the two randomly processed phase matrices.A random binary matrix can be used to acquire the phase distribution of LC device: whereT n is an n-order random matrix in which every element is 1 or 0 and the * is the Hadamard product.In this equation, the effects of ϕ 1 and ϕ 2 to the output are relatively independent.For instance, when the random matrix is a third-order matrix and the probability of 1 in the distribution is 50%, the results are as follows: The first FPG satisfies the following equation: � where q 1 ¼ 0:5.The phase angle distribution of the second FPG fits with the following equation: � where q 2 ¼ 0:5.The phase distributions of the two FPGs are coupled with two complementary random binary matrices respectively, then the newly generated twophase distribution matrices are added to obtain a new mixed phase, and the process is shown in Figure 1.The results of phase mixing between diffraction gratings with different phase distributions can also be simulated by this method.As shown in Figure 1, the phase angle distribution of double Airy beams satisfies ϕ Airy ðx; yÞ �.The diffraction results of the grating after mixing are directly reflected in Figure 1, which includes vortex beam and vortex beam, vortex beam and Airy beam.It is proved theoretically that the diffraction order of any phase distribution can be obtained by this method.
The principle of the phase plane in the diffracted light field is as follows: and the process of phase mixing will be described in detail in Appendix A.
Due to the random characteristics of the matrix, the distribution of the phase factor of each FPG in the twodimensional plane is very uniform.This method can be used to bring the designed phase distribution matrix into the simulation software for calculation.When linearly polarised Gaussian light is incident and the distribution probability of both 0 and 1 contained in the random binary matrix is 50%, the simulation results are shown in Figure 1, and the results are in good agreement with the theoretical expectation.More importantly, we can also control the intensity distribution of the outgoing light by controlling the probability distribution of the matrix.When the distribution probability of 1 in the matrix is 50%, 60%, 70%, 80%, or 90%, the simulation result is shown in Figure 2. The simulation results show that the change in the probability distribution of the matrix will result in the change of the light intensity.

Experiments and results
In order to realise the accurate multi-domain alignment of LC, the photoalignment technique [28] based on the DMD was employed in the experiment.One clean glass that was spin-coated with the photoalignment agent SD1 [29] was exposed to polarised UV light by the regional exposure method, after which the sample was spin-painted with RM 257 to fabricate the designed LC PG.Under crossed polarisers, the micrograph gives a vivid exhibition of the LC PG which is exhibited in Figure 3(b1-b4).The continuous variation of the director of the LC molecules causes a continuous change in the brightness.This reveals that the designed polarisation holograms are accurately transferred into the LC device.A CCD is mounted on a linear translation stage to capture the diffraction patterns.Since the new diffraction grating is a mixture of two diffraction gratings with a special phase, it can be seen that a variety of beams of different properties are generated in the experimental results.Figure 3 illustrates the optical setup to generate and record the experimental results.The diffraction result of ϕ 1 ðx; yÞ � (where q 2 ¼ 1) is shown in Figure 3(c1).The experimental results of other mixing phases are shown in Figure 3 (c2-c4).
By changing the proportional distribution of the random binary matrix, the intensity ratio of different diffraction gratings can be controlled.As verified by the simulation algorithm, when the probability distribution of the random binary matrix in the simulation program is modulated to 6:4, 7: 3, 8:2, and 9:1, bringing the phase distribution obtained from the simulation results into the DMD system to prepare the LC sample, the fabrication is completed under the half-wave condition of 633 nm.The far-field diffraction results obtained in the optical system of Figure 3(a) are shown in Figure 4(a).The pitch of diffraction gratings is set to 152 nm.It can be seen that the two different gratings have been perfectly fused through the random binary matrix algorithm, and the intensity between the two gratings can be arbitrarily manipulated by changing the probability of the random binary matrix.
The intensity proportions of the diffraction orders of different gratings can be manipulated by adjusting the probability of the random binary matrix, but the intensity proportions of ± 1 orders of the same grating can not be manipulated by this way.Fortunately, the intensities of the emergent light of the ± 1 diffraction orders are influenced by the polarisation of the incidence light.Therefore, we can manipulate the intensity distributions of ± 1 orders by switching the polarisation state of the input.The liquid crystal cell that acts as a phase retardation element can just meet this requirement excellently [30].The relation between LC cell and external voltage can be deduced by elastic continuum theory of LCs.The preparation process and principle of LC cell are detailed in Appendix B. The LC material we used is E7 liquid crystals and the thickness of LC cell is 4um.By changing the voltage across the liquid crystal cell, the polarisation state of the incident light can be stably controlled, and the intensity distribution of diffraction orders of different polarisations under different voltages is shown in Figure 4(a).In short, the intensity distribution of different diffraction gratings can be controlled by probability distribution, and the intensity distribution between±1st of the same diffraction grating can be controlled by changing the voltage of the liquid crystal cell.
Figure 4(b) shows the applications of this algorithm on optical holography.Based on the holographic principle and GS algorithm [31,32], we have designed and fabricated the multiplexing LC holographic devices.The phase corresponding to the airplane image in the diffraction results is mixed with the phase corresponding to the flag image, where the two phases are 300 µm apart horizontally.Figure 4(b) shows the far field diffraction results when the holographic phases are mixed under different ratios, which is consistent with the results of   diffraction gratings.It can be achieved to manipulate the diffraction intensity of the multiplexing optical holography, which brings an innovative scheme for this field.
Through the detailed test of the LC cell, the data curve of the relationship between the voltage and the delay of the LC cell is shown in Figure 5(a).The data on the intensity of each diffraction order and the voltage of the liquid crystal cell are recorded in Figure 5(b).LCP corresponds to the+1 order of the diffraction grating and RCP corresponds to the −1 order of the diffraction grating.It can be seen that the diffraction intensities of the ± 1 orders can be accurately controlled by voltage, which is consistent with the expected results.At the same time, the intensity of the corresponding diffraction order is controlled by changing the distribution of the random binary matrix.It can be obtained from the experimental results shown in Figure 5(c) that the proportional relation of the light intensity and the distribution probability of the random binary matrix satisfies I 1 : , in which I represents the light intensity and R represents the ratio between the two phases.Meanwhile, the interaction of this algorithm and the traditional holographic phase superposition method hold the advantage of phase stability in the holographic algorithm so that beams with better quality are produced.The parameters and positions of a single diffraction grating can be manipulated freely in the phase mixing process which confirms our previous hypothesis.Meanwhile, because the algorithm retains the advantage of the random binary matrix, the intensity relationship between multiple light spots can also be arbitrarily manipulated.The efficiency of diffraction grating is one of its important parameters.After mixing two diffraction gratings by the algorithm based on the random binary matrix, the measured efficiency of the new diffraction grating which has four diffraction levels is shown in Figure 5(d), where the efficiency is calculated by Obviously, the introduction of the uncertainty algorithm fills the gap of independent phase control in the field of LC optics, which is highly essential to the study of hybrid geometric phase elements.

Conclusion
In conclusion, a new phase mixing algorithm based on the random binary matrix has been proposed.A new type of LC PG device is fabricated by this algorithm, which can generate a new type of beams, carrying four sets of vortex phase information with opposite torque.We have described the theoretical foundation and experimental realisation of the proposed scheme.The experimental results are in good accordance with the simulation results.In addition to its simple configuration, compact size, and low cost, the new LC PG offers more promising features and merits: (1) The introduction of the random binary matrix makes the phase mixing of two FPGs easier to control [22,33].(2) The application of the four optical vortices to optical information carriers will not only greatly improve the channel capacity, but also provide a new idea for OAM vortex optical multiplexing communication technology.Moreover, it also has potential development in the fields like quantum communication, optical tweezers and so on [26,27].In summary, the phase mixing algorithm based on LC photoalignment technology renders an entirely novel perspective for laser beam shaping and steering, which could be massively applied to other diffractive optics [16,31] and real-time holography.

Figure 1 .
Figure 1.(Colour online) The process diagram of the proposed algorithm, the phase distribution of the mixed grating, and the simulation diagram of the diffraction results.The color varying from blue to red corresponds to the LC orientation varying from -π to π.The black and white squares represent 0 and 1 in the random binary matrix, and the pixel size of the matrix and the phase plane is 3000 × 3000 in the actual simulation calculation.

Figure 2 .
Figure 2. (Colour online) The simulation diagram of diffraction results when the random binary matrix takes different ratios.The color varying from blue to red corresponds to the intensity from 0 to 1.The distribution ratios of the random binary matrix corresponding to the gratings with different phase distributions are abbreviated as ϕ1 : ϕ2.

Figure 3 .
Figure 3. (Colour online) Schematic of experimental process and results.(a) The diagram of the optical setup.(b1-b4) The crosspolarized optical microscope images of fabricated samples with different phase distributions.All scale bars are 200 µm.(c1-c4) the photos of the exit light were captured by the CCD when the incident light is horizontally polarized.

Figure 4 .
Figure 4. (Colour online) Schematic of experimental results.When the polarizer is set to 45 °and the LC cell is set to 0 °, the diffraction result captured by the CCD is shown in the figure.White arrows represent incident polarization distributions.The voltage data in the figure indicates the voltage value of the alternating current sandwiched between the two ends of the LC cell.The distribution ratios of the random binary matrix corresponding to the gratings with different phase distributions are abbreviated as ϕ1 : ϕ2.

Figure 5 .
Figure 5. (Colour online) (a) The relationship between the voltage at both ends of the LC cell and the delay of the LC cell.(b) The relationship between the voltage at both ends of the LC cell and the intensity of ±1st diffraction order, the intensity value comes from the optical power meter.(c) The relationship between the proportion of preset random binary matrix and the intensity of output diffraction.(d) The relationship between diffraction grating efficiency and different phase ratios, the data are taken from the optical power meter values corresponding to the results in Figure 4(a).