Diffraction characteristics of a non-mechanical beam steering system with liquid crystal polarization gratings

Zishuo Wang; Chunyang Wang; Chunyang Wang; Chunyang Wang; Shuning Liang; Xuelian Liu; Xuelian Liu

doi:10.1364/OE.452397

1. Introduction

The liquid crystal polarization grating (LCPG) is a non-mechanical beam steering device based on the geometric phase [1], which has the advantages of small size, low weight, flexibility, fast response times, and low power consumption [2,3]. It has a wide range of applications such as laser radar [4], optical communication [5], and polarization imaging [6]. The LCPG realizes beam steering by changing the applied voltage and the polarization state of the incident beam [7]. A sub-system consisting of a single LCPG and a wave plate can only produce three beam steering angles, Therefore, to achieve multi-angle and large-angle beam scanning, multiple sub-systems must be combined to construct a cascaded system [8]. Two cascade technologies are used in this beam steering system: the binary [9] cascade and the ternary cascade [10]. Both technologies use multiple sub-systems stacking, but differ in the geometric progressions of the cascaded LCPGs angles. The LCPGs angles of the binary system increases by 2 exponential times, and the ternary LCPGs angles increases by 3 exponential times.

Currently, the diffraction efficiency analysis of these two beam steering systems is based on the grating diffraction equation under normal beam incidence [11]. However, cascading multiple LCPGs cause the beam from one LCPG to obliquely enter the subsequent LCPGs. As a result of this oblique beam incidence, the diffraction efficiency of the LCPG no longer follow the grating diffraction equation [12]. Moreover, oblique incidence changes the polarization state of the beam incident on the subsequent LCPGs, and a change in the polarization state will further reduce the system’s diffraction efficiency. The LCPG diffraction efficiency is approximately 100% only when the beam is at normal incidence and is circularly polarized [13], In addition, the LCPG’s working voltage is the same at all steering angles in non-mechanical beam steering systems [9,10]. However, oblique beam incidence causes the working voltage of the system to differ depending on the steering angle. The cascaded LCPGs also exhibit strong coupling, which increases the complexity of voltage calibration. In addition, the high throughput is an important goal for the beam steering system, and both these cascade technologies use large numbers of devices, which greatly reduces the throughput of the system.

This paper uses the elastic continuum theory of liquid crystals [14,15] to simulate the tilt angle of liquid crystal (LC) molecules under different voltages. The transmission process of the beam in the system at oblique incidence is described with an extended Jones matrix [16–18], and the highest diffraction efficiency and working voltage of each LCPG at different steering angles are calculated by vector diffraction theory [19]. The effect of oblique beam incidence on the system diffraction efficiency is discussed. In addition, we use an improved binary cascade technology to design a LCPGs non-mechanical beam steering system with a steering angle of ±10° and an angular resolution of 0.67°. Compared with binary cascade, this technology can effectively reduce the number of cascaded devices and increases the system throughput under the same maximum beam steering angle and angular resolution. The proposed method can also be applied to the design of large-angle non-mechanical beam steering systems, which is convenient for LCPGs working voltage calibration and diffraction characteristics analysis of the system.

2. Basic principle and simulation method of LCPGs

The LC molecules in the LCPG are arranged between two glass substrates with a fixed period $\Lambda $[20,21], as shown in Fig. 1(a). When a circularly polarized beam is at normal incidence on the LCPG, the birefringence occurs, that is, the beam splits into an ordinary beam with a refractive index of ${n_\textrm{o}}$, and an extraordinary beam with a refractive index of ${n_e}$. Applying a driving voltage to the glass substrate tilts the LC molecules along the z-axis, thereby changing the birefringence $\Delta n$ of the LC [22,23]. When $\Delta nd$ (d is the thickness of the LC layer) equals half the wavelength of the incident beam $\lambda /2$, all the beams are diffracted to ±1st order, and the diffraction efficiency is approximately 100%. Because the LCPG has imparted a relative phase shift to the beam, rotation of the outgoing beam and the incident beam to have opposite states. That is, an incident right-handed circular (RHC) polarized beam becomes outgoing left-handed circular (LHC) polarized beam. When $\Delta nd$ is equal to the wavelength $\lambda$, the beams are all diffracted to the 0th order, and rotation of the outgoing beam does not change, as shown in Fig. 1(b). The LCPG can also change the polarization state of the incident beam to switch the outgoing beam between ±1st order. Therefore, the LCPG has both electro-optical and polarization controllability.

Fig. 1. Schematic diagram of beam steering of LCPG. (a) unload voltage and (b) load voltage.

Download Full Size | PDF

To simulate the tilt angle of LC molecules under different voltages, we introduce the elastic continuum theory of LCs. The LC molecular director is as follows [14]:

(1)$$\hat{n} = ({\cos \theta \cos \phi ,\cos \theta \sin \phi ,\sin \theta } ),$$

where $\theta$ is the tilt angle of the LC molecules, and $\phi$ is their azimuth angle. Introducing the LC molecular director into the Gibbs free energy density equation, the Gibbs free energy density f of LC molecules under the action of an external electric field can be expressed as [15]:

(2)$$\begin{aligned} f &= {\frac{1}{2}({{K_{11}}{{\cos }^2}\theta \textrm{ + }{K_{33}}si{n^2}\theta } ){{\left( {\frac{{\partial \theta }}{{\partial z}}} \right)}^2} + ({{K_{22}} - {K_{11}}} ){{\cos }^2}\theta \sin \phi \frac{{\partial \phi }}{{\partial x}}\frac{{\partial \theta }}{{\partial z}}}\\ &\quad + \frac{1}{2}{\cos ^2}\theta ({{K_{11}}{{\sin }^2}\phi + {K_{22}}{{\cos }^2}\phi {{\sin }^2}\theta + {K_{33}}{{\cos }^2}\phi {{\cos }^2}\theta } ){\left( {\frac{{\partial \phi }}{{\partial x}}} \right)^2}\\ &\quad - \frac{1}{2}({{\varepsilon_ \bot } + \Delta \varepsilon {{\sin }^2}\theta } ){\left( {\frac{{d\textrm{v}}}{{dz}}} \right)^2},\end{aligned}$$

where x and z are the coordinate axes; ${K_{11}} = 11.1 \times {10^{ - 12}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{N},{K_{22}} = 7.4 \times {10^{ - 12}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{N}$ and ${K_{33}} = 17.1 \times {10^{ - 12}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{N}$ are the corresponding of splay, twist and bend elastic coefficients; ${\varepsilon _ \bot } = 4.6 \times {10^{ - 11}}$ is the vertical permittivity; $\Delta \varepsilon = 1.2 \times {10^{ - 10}}$ is the anisotropy of the permittivity; and v is the voltage. The Gibbs free energy density of the LC molecule is minimum when it is in equilibrium. Assuming that the LC layer with a thickness of d is uniformly divided into M layers along the z-axis, and Eq. (2) is solved by the differential iteration method, we can obtain the director distribution of the LC molecules at different voltages, as shown in Fig. 2.

Figure 2 shows that the tilt angle of the LC molecules is related to the driving voltage and the position of the LC layer. Therefore, the function $\theta {}_z(v){\kern 1pt} {\kern 1pt} {\kern 1pt} ,(z = \frac{d}{M} \times j,j = 1,2,3, \cdots ,M){\kern 1pt} {\kern 1pt}$ is used to represent the tilt angle of the LC molecules at different voltages and positions.

Fig. 2. Director distribution of the LC molecules at different voltages.

Download Full Size | PDF

Because the beam steering system consists of multiple stacked stages of LCPGs sub-systems, the beam from one LCPG obliquely enters the subsequent LCPGs, as shown in Fig. 3. Therefore, to analyze the birefringence of LC molecules for oblique incidence, a three-dimensional coordinate diagram (Fig. 4) is established.

Fig. 3. Oblique incidence of beam entering into LCPGs.

Download Full Size | PDF

Fig. 4. Three-dimensional coordinate diagram of incident beam vector and LC director.

Download Full Size | PDF

where ${K_i}$ is the incident beam vector, and ${\theta _0}$ is the oblique angle of the incident beam. The LC molecules in the x-axis direction do not all have the same azimuth angle; therefore, the molecules are divided into n layers along the x-axis, and the azimuth angles of different positions are represented by ${\phi _n}$.The extraordinary refractive index ${\sigma _e}$ and ordinary refractive index ${\sigma _o}$ of the LC molecules are respectively

(3)$$\begin{aligned} {\sigma _e} &= \frac{{({n_e^2 - n_o^2} )\sin {\theta _z}(v)\cos {\theta _z}(v)\cos {\phi _n}\sin {\theta _0}}}{{{\gamma ^2}}}\\ &\quad + \frac{{{n_o}{n_e}}}{{{\gamma ^2}}}{\left\{ {\left. {{\gamma^2} - \left[ {1 - \frac{{n_e^2 - n_o^2}}{{n_e^2}}{{\cos }^2}{\theta_z}(v){{\sin }^2}{\phi_n}} \right]{{\sin }^2}{\theta_0}} \right\}} \right.^{\frac{1}{2}}}{\kern 1pt} ,\\ {\sigma _o} &= \sqrt {n_o^2 - {{\sin }^2}{\theta _0}} , \end{aligned}$$

where $\gamma = n_o^2 + (n_e^2 - n_o^2){\sin ^2}{\theta _z}(v)$, and ${n_0} = 1.5,{n_e} = 1.7$. Letting ${\sigma _e}$ take the average value after integration in the z-axis direction, we can then obtain the equivalent average value of the extraordinary refractive index ${\hat{\sigma }_e}$.

(4)$${\hat{\sigma }_e} = \frac{1}{z}\int_0^z {{\sigma _e}} dz.$$

Figure 3 shows that for oblique incidence, the actual optical length is greater than the thickness d of the LC layer, so the optical length of the beam propagating in the LC layer is as follows:

(5)$$L = \frac{d}{{\cos {\theta _0}}}.$$

Combining Eqs. (3), (4) and (5), we can obtain the phase retardation of the n-layer LCPG:

(6)$$\Gamma (n) = \frac{{2\pi }}{\lambda }({\hat{\sigma }_e} - {\sigma _o})L.$$

Based on the LCPG’s phase retardation, the extended Jones matrix is used to describe the transmission process of the beam in the LCPG. The extended Jones matrix of the LC is expressed as [18]:

(7)$${J_n} = {[{{R_2}G{R_1}} ]_n},$$

where $G = [1{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} ;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {e^{i\Gamma (n)}}]$ is the transmission matrix of the n-th layer of the LC, ${R_2}$ is the refraction matrix, and ${R_1} = R_2^\textrm{T}$ is the incident matrix.

(8)$${R_1} = \left[ {\begin{array}{{cc}} {\frac{{a \times b}}{{{A_e}}}}&{ - \frac{{a \times \sin {\phi_n}\cos {\theta_z}(v)}}{{{A_e}}}}\\ {\frac{{\sin {\phi_n}\cos {\theta_z}(v)}}{{{A_o}}}}&{\frac{b}{{{A_o}}}} \end{array}} \right],$$

where ${A_e} = \sqrt {[{n_0}^2{{\sin }^2}{\theta _0} + n_e^2({\theta _e}){{\cos }^2}{\theta _e} + v] \times {b^2}\; + {{\sin }^2}{\phi _n}{{\cos }^2}{\theta _z}(v) \times {a^2}}$ and ${A_0} = \sqrt {{{\sin }^2}{\phi _n}{{\cos }^2}{\theta _z}\textrm{(v) + }{b^2}}$ is the normalization constant. $a = {n_o}{\sin ^2}{\theta _0} + {n_e}({\theta _e})\cos {\theta _e}\cos {\theta _0}$, $b = \cos {\phi _n}\cos {\theta _z}(v)\cos {\theta _0} - \sin {\theta _z}(v)\sin {\theta _0}$, and ${\theta _e}$ is the refraction angle of extraordinary beam. ${n_e}({\theta _e})$ is the refractive index of the extraordinary beam in the ${\theta _e}$ direction. The extended Jones matrix of the LCPG is expressed as:

(9)$${J_{n\_k}}\textrm{ = }{\kern 1pt} {J_n} \times {J_{n - 1}} \cdot{\cdot} \cdot {J_2} \times {J_1},$$

where $k = 1,2,3,4, \cdots$ denote the LCPG serial number of different periods. When $k = 1$, the beam is normally incident on the $\textrm{LCP}{\textrm{G}_\textrm{1}}$ with a grating period of ${\Lambda _1}$, the extended Jones matrix ${E_{out\_k + 1}}$ of the outgoing beam is as follows:

(10)$${E_{out\_k + 1}} = {J_{n\_1}}{E_{in}},$$

where ${E_{in}}$ is the Jones matrix of the incident light. According to vector diffraction theory [19], the m-th order (m = 0, ±1) polarization characteristics and diffraction efficiency of the LCPG are determined by the vector Fourier transform coefficients of the outgoing beam Jones matrix. When the beam is normally incident on the $\textrm{LCP}{\textrm{G}_\textrm{1}}$, the Fourier transform coefficient ${D_{m\_1}}$ of the m-th order outgoing beam is

(11)$${D_{m\_1}} = \frac{1}{{{\Lambda _1}}}\int_0^{{\Lambda _1}} {{J_{n\_1}}} {E_{in}}\textrm{exp} (i2\pi mx/{\Lambda _1})dx.$$

Because multiple LCPGs are cascaded, the incident beam of a subsequent $\textrm{LCP}{\textrm{G}_k}$ ($k = 2,3,4, \cdots$) is the outgoing beam of the previous $\textrm{LCP}{\textrm{G}_{k - 1}}$, then

(12)$${E_{out\_k + 1}} = {J_{n\_k}}{E_{out\_k}}.$$

Therefore, the Fourier transform coefficient of the m-th order outgoing beam of the subsequent $\textrm{LCP}{\textrm{G}_k}$ is

(13)$${D_{m\_k}} = \frac{1}{{{\Lambda _k}}}\int_0^{{\Lambda _k}} {{J_{n\_k}}{E_{out\_k}}} \textrm{exp} (i2\pi mx/{\Lambda _k})dx{\kern 1pt} .$$

The ratio of the power of the m-th order outgoing beam of the LCPG to the sum of the power of the diffracted beams of each order is the diffraction efficiency of the m-th order, The power of the m-th order outgoing beam is proportional to ${|{{D_{m\_k}}} |^2}$, assuming that the proportionality factor is $\beta$, the m-th order diffraction efficiency ${\eta _{m\_k}}$ of the $\textrm{LCP}{\textrm{G}_k}$ can be expressed as:

(14)$${\eta _{m\_k}} = \frac{{{{|{{D_{m\_k}}} |}^2}}}{{{{|{{D_{ - 1\_k}}} |}^2} + {{|{{D_{0\_k}}} |}^2} + {{|{{D_{1\_k}}} |}^2}}}.$$

The diffraction efficiency of the LCPG beam steering system is then expressed as:

(15)$$\eta = {\eta _{m\_1}} \times {\eta _{m\_2}} \times \cdots \times {\eta _{m\_k}}.$$

This simulation method can calculate the optimal diffraction efficiency of the outgoing beam and the working voltage of each LCPG at different steering angles, thus simplifying LCPGs calibration.

3. Design and experimental details of the beam steering system

This paper used an improved binary cascade technology to design a LCPGs non-mechanical beam steering system with a steering angle of ±10° and an angular resolution of 0.67°. This technology uses the ability to add and subtract from the input angle (or leave it unchanged) at each stage, combined with the characteristics of the opposite (or leave it unchanged) rotation of the outgoing beam and the incident beam. By changing the angles of the cascaded LCPGs and rotating the last LCPG by 180°, the number of half-wave plates can be reduced while keeping the beam steering angle resolution and maximum steering angle unchanged, thereby increasing the throughput of the system. As shown in Fig. 5, the binary cascade beam steering system requires 2L devices, The improved binary system requires only L+1 devices, where L is the number of LCPGs.

Fig. 5. Improved binary beam steering system.

Download Full Size | PDF

The improved binary beam steering system can produce the total number of steering angles $\Omega $ and maximum steering angle ${\vartheta _{\max }}$ are respectively:

(16)$$\begin{aligned} \Omega &={2^{\textrm{L + 1}}} - 1,\\ {\vartheta _{\max }}&=\alpha ({{2^L} - 1} ), \end{aligned}$$

where $\alpha$ is the resolution of the system, that is, the steering angle of the $\textrm{LCP}{\textrm{G}_\textrm{1}}$. Other LCPG angles can be calculated using ${\vartheta _p}\textrm{ = }\alpha ({{2^k} - 1} )$, and the last LCPG angle is ${\vartheta _p}\textrm{ = }\alpha \times {2^{L - 1}}$. The system steering angle ${\vartheta _{out}}$ can be expressed as [10]:

(17)$$\sin {\vartheta _{out}} = \lambda ({m_1}/{\Lambda _1} + {m_2}/{\Lambda _2} + \cdots + {m_k}/{\Lambda _k}).$$

The grating period ${\Lambda _k}$ can be derived from the grating equation [24]

(18)$${\Lambda _k} = \frac{\lambda }{{\sin {\vartheta _p}}}.$$

To verify the effectiveness of the proposed simulation method and cascade technology, four LCPGs with grating periods of $91.0{\kern 1pt} {\kern 1pt} {\kern 1pt} \mathrm{\mu m}{\kern 1pt} {\kern 1pt} \textrm{(0}\textrm{.6}{\textrm{7}^ \circ })$, $30.4{\kern 1pt} {\kern 1pt} {\kern 1pt} \mathrm{\mu m\ (2}\textrm{.0}{\textrm{1}^ \circ })$, $13.0{\kern 1pt} {\kern 1pt} {\kern 1pt} \mathrm{\mu m\ (4}\textrm{.6}{\textrm{9}^ \circ })$, and $11.4{\kern 1pt} {\kern 1pt} \mathrm{\mu m\ (5}\textrm{.3}{\textrm{6}^ \circ })$ with the LC thickness of $3.7{\kern 1pt} {\kern 1pt} {\kern 1pt} \mathrm{\mu m}$ were prepared using the polarization holography technology. The system diffraction efficiency measurement platform, shown in Fig. 6, was composed of a 1064nm laser, a polarizer, a quarter wave plate, LCPGs, a controller, and a power meter. The laser emits a 1064nm continuous beam, and after passing through the polarizer and the quarter wave plate, the beam becomes LHC polarized light normally incident on the system, The voltage of the LCPG is adjusted using the controller to steer the beam to different angles. The power meter measures the total power and the diffraction order power of the outgoing beam, from which the actual beam diffraction efficiency can be calculated. To ensure accurate measurements, the diffraction efficiency for each angle was measured 10 times, and the average value was used.

Fig. 6. Beam steering system diffraction efficiency measurement platform. (a) measurement principle and (b) actual measuring platform.

Download Full Size | PDF

4. Results and discussion

Under normal beam incidence, the diffraction efficiencies of the prepared LCPGs were separately measured. The simulation method was used to calculate the diffraction efficiencies and working voltages of the LCPGs. The measured and calculated results are compared in Fig. 7.

Fig. 7. Comparison of simulated and measured results. (a) 0.67° LCPG’s diffraction efficiency. (b) 2.01° LCPG’s diffraction efficiency. (c) 4.69° LCPG’s diffraction efficiency and (d) 5.36° LCPG’s diffraction efficiency.

Download Full Size | PDF

The working voltage corresponding to the optimal diffraction efficiency calculated by the simulation method agrees with the actual result, 2.11 V, When the voltage is gradually increased, the +1st order diffraction efficiency gradually decreases. At 20 V, the 0th order diffraction efficiency is 98%, and the dynamic variation in the diffraction efficiency with the driving voltage is consistent with the measured results.

To verify whether the diffraction efficiency of the outgoing beam and the working voltage of each LCPG change with different steering angles, beam steering angles from 0° to +10.05° were simulated, and the simulation results (Table 1) were compared with the measured results, as shown in Fig. 8.

Fig. 8. The system diffraction efficiency at different steering angles.

Download Full Size | PDF

Table 1. LCPGs’ working voltage and diffraction efficiencies under different beam steering angles.

View Table

Figure 8 and Table 1 show that the diffraction efficiency of the outgoing beam varies with the steering angle, and that the LCPG working voltage gradually increases with increasing steering angle. At a steering angle of 10.05°, the working voltage of the 5.36° LCPG is 2.16 V. If the working voltage for 5.36° is set to 2.11 V, the measured diffraction efficiency of the outgoing beam is 75.54%, as shown in Fig. 9. This also verifies that the simulation method can be used to calibrate the LCPG working voltage and calculate the system’s optimal diffraction efficiency.

Fig. 9. Comparison of different working voltages on the 5.36° LCPG. (a) working voltage is 2.16 V and (b) working voltage is 2.11 V.

Download Full Size | PDF

The factors that reduce the diffraction efficiency of the system are now discussed. Table 1 shows that the greater number of LCPGs used to add and subtract ±1st order beam steering angles, the lower is the diffraction efficiency of the system, because the ±1st order diffraction efficiency is less than the 0th order diffraction efficiency. Although only two LCPGs are used at 10.05°, the increase in the oblique incident angle of the beam decreases the diffraction efficiency. Figure 10 shows the simulated diffraction efficiency for incidence on the 5.36° LCPG at different oblique angles. When the oblique incident angle is 20°, the outgoing beam diffraction efficiency decreases by 30.69%.

Fig. 10. Diffraction efficiency of the 5.36° LCPG at different oblique angles.

Download Full Size | PDF

The oblique incidence of the beam also changes the polarization state of light incident on the subsequent LCPG, further reducing the diffraction efficiency of the outgoing beam. The polarization state of the outgoing beam is characterized by its ellipticity, which can be calculated by bring the Jones matrix ${E_{out\_k + 1}}$ into the Stokes parameter equation [25]. Figure 11 shows the changes in the ellipticity and diffraction efficiency of the outgoing beam when a beam is incident on the 5.36°LCPG at different oblique angles. The larger the oblique angle, the smaller is the ellipticity of the outgoing beam and the lower the diffraction efficiency.

Fig. 11. The changes in the ellipticity and diffraction efficiency of the outgoing beam when a beam is incident on the 5.36°LCPG at different oblique angles.

Download Full Size | PDF

Figure 12 compares the effects of different ellipticities on the diffraction efficiency at the same oblique incident angle. When the beam oblique angle is 4.69°, the ellipticity is 41.9°, and the 5.36°LCPG’s diffraction efficiency is 86.62%. When the beam oblique angle is 4.69°, the ellipticity is 45°, and the 5.36°LCPG’s the diffraction efficiency is 90.01%, which has little effect on the overall performance of the system. However, when the oblique incident angle is 18°, the 5.36°LCPG’s diffraction efficiency is reduced by 7.28%. Therefore, when applying the improved cascade method to design a large-angle LCPG beam steering system, the influence of polarization state changes should be considered. If the system has a large oblique incident angle, an electronically controlled wave plate can be added before the last LCPG to adjust the polarization state of the beam to maximize the system's diffraction efficiency.

Fig. 12. Influence of the ellipticity on the 5.36°LCPG’s diffraction efficiency.

Download Full Size | PDF

To verify whether changing the ellipticity affects the working voltage of the LCPG, we simulated the effect of various ellipticities on the working voltage and diffraction efficiency of the LCPG, as shown in Fig. 13. The ellipticity and the diffraction efficiency of the outgoing beam decrease, but the working voltage of the LCPG remains unchanged. This result verifies that the working voltage of the LCPG is related only to the oblique incident angle of the beam, and is independent of the ellipticity.

Fig. 13. A normal incident beam to the 5.36°LCPG, and the influence of different ellipticities on the working voltage and diffraction efficiency.

Download Full Size | PDF

5. Conclusion

This paper proposed a simulation method to calculate the diffraction efficiency of the LCPG-based non-mechanical beam steering system. This method can calculate the optimal diffraction efficiency and working voltage of each LCPG for different steering angles. Moreover, an improved binary cascade technology was used to design a LCPGs non-mechanical beam steering system with a steering angle of ±10° and an angular resolution of 0.67°. Through a combination of numerical simulation and experimental verification, the influence of the oblique incidence of the beam on the system diffraction efficiency and the working voltage of the LCPGs was quantitatively analyzed. The results show that as the oblique incident angle of the beam increases, the working voltage of the LCPG increases, both the ellipticity of the outgoing beam and the system’s diffraction efficiency decrease. Moreover, the change in ellipticity does not affect the working voltage of the LCPG. When the improved binary cascade method is used to design a large-angle LCPG beam steering system, an electronically controlled wave plate can be added before the last LCPG to adjust the ellipticity of the beam to maximize the system's diffraction efficiency.

Funding

Department of Science and Technology of Jilin Province (20190302089GX).

Acknowledgement

We are indebted to Bo Xiao and Qingquan Zheng 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.

References

1. K. Zou, Y. Shi, and D. Luo, “A Review of Two-Dimensional Liquid Crystal Polarization Gratings,” Crystals 11(9), 1015 (2021). [CrossRef]

2. J. C. Chao, W. Y. Wu, and Y. G. Fuh, “Diffraction characteristics of a liquid crystal polarization grating analyzed using the finite-difference time-domain method,” Opt. Express 15(25), 16702–16711 (2007). [CrossRef]

3. J. Kim, J. H. Na, and S. D. Lee, “Fully continuous liquid crystal diffraction grating with alternating semi-circular alignment by imprinting,” Opt. Express 20(3), 3034 (2012). [CrossRef]

4. J. Kim, M. N. Miskiewicz, and S. Serati, “Demonstration of large angle non-mechanical laser beam steering based on LC polymer polarization gratings,” The International Society for Optical Engineering 8052, 80520T (2011). [CrossRef]

5. C. Hoy, J. Stockley, J. Shane, K. Kluttz, and S. Serati, “Non-mechanical beam steering with polarization gratings: a review,” Crystals 11(4), 361 (2021). [CrossRef]

6. H. Masterson, R. Serati, S. Serati, and J. Buck, “MWIR wide-area step and stare imager,” International Society for Optics and Photonics 8052, 80520N (2011). [CrossRef]

7. J. Buck, S. Serati, L. Hosting, R. Serati, and M. Miskiewicz, “Polarization gratings for non-mechanical beam steering applications,” The International Society for Optical Engineering 8395, 83950F (2012). [CrossRef]

8. S. Serati, C. L. Hoy, and L. Hosting, “Large-aperture, wide-angle non-mechanical beam steering using polarization gratings,” Opt. Eng. 56(3), 031211 (2016). [CrossRef]

9. J. Kim, M. N. Miskiewicz, S. Serati, and M. J. Escuti, “Non-mechanical laser beam steering based on polymer polarization gratings: design optimization and demonstration,” J. Lightwave Technol. 33(10), 2068–2077 (2015). [CrossRef]

10. J. Kim, C. Oh, S. Serati, and M. J. Escuti, “Wide-angle, nonmechanical beam steering with high throughput utilizing polarization gratings,” Appl. Opt. 50(17), 2636–2639 (2011). [CrossRef]

11. C. Oh and M. J. Escuti, “Numerical analysis of polarization gratings using the finite-difference time-domain method,” Phys. Rev. A 76(4), 043815 (2007). [CrossRef]

12. Z. Xiangjie, D. Jiazhu, Z. Dayong, L. Cangli, and L. Yongquan, “Oblique incidence effect on steering efficiency of liquid crystal polarization gratings used for optical phased array beam steering amplification,” Opt. Rev. 23(5), 713–722 (2016). [CrossRef]

13. C. Provenzano, P. Pagliusi, and G. Cipparrone, “Highly efficient liquid crystal based diffraction grating induced by polarization holograms at the aligning surfaces,” Appl. Phys. Lett. 89(12), 121105 (2006). [CrossRef]

14. S. Ping, Introduction to the elastic continuum theory of liquid crystals (Springer US, 1975).

15. R. K. Komanduri and M. J. Escuti, “Elastic continuum analysis of the liquid crystal polarization grating,” Phys. Rev. E 76(2), 021701 (2007). [CrossRef]

16. P. Yeh, “Extended jones matrix method,” J. Opt. Soc. Am. 72(4), 507–513 (1982). [CrossRef]

17. C. Gu and P. Yeh, “Extended Jones matrix method. II,” J. Opt. Soc. Am. A 10(5), 966–973 (1993). [CrossRef]

18. F. Yu and H. Kwok, “Comparison of extended jones matrices for twisted nematic liquid-crystal displays at oblique angles of incidence,” J. Opt. Soc. Am. A 16(11), 2772–2780 (1999). [CrossRef]

19. T. Li, J. Y. Ho, and H. Kwok, “Characterization of complex liquid crystal polarization gratings at oblique incidence using extended jones matrix method,” SID Symposium Digest of Technical Papers 44(1), 1335–1337 (2013). [CrossRef]

20. S. Z. Li, Y. G. Liu, C. J. Liu, and Z. W. Zhao, “Electrically tunable photo-aligned hybrid double-frequency liquid crystal polarisation grating,” Liq. Cryst. 45(12), 1753–1759 (2018). [CrossRef]

21. J. Xiong, R. Chen, and S. Wu, “Device simulation of liquid crystal polarization gratings,” Opt. Express 27(13), 18102–18112 (2019). [CrossRef]

22. S. Panezai, D. Wang, Z. Jie, Y. Wang, R. Lu, and S. Ma, “Study of oblique incidence characterization of parallel aligned liquid crystal on silicon,” Opt. Eng. 54(3), 037109 (2015). [CrossRef]

23. S. W. Ke, T. H. Lin, and Y. G. Fuh, “Tunable grating based on stressed liquid crystal,” Opt. Express 16(3), 2062–2067 (2008). [CrossRef]

24. R. K. Komanduri, W. M. Jones, C. Oh, and M. J. Escuti, “Polarization-independent modulation for projection displays using small-period LC polarization gratings,” J. Soc. Inf. Disp. 15(8), 589–594 (2007). [CrossRef]

25. K. L. Woon, M. O. Neill, G. J. Richards, M. P. Aldred, and S. M. Kelly, “Stokes parameter studies of spontaneous emission from chiral nematic liquid crystals as a one-dimensional photonic stopband crystal: experiment and theory,” Phys. Rev. E 71(4), 041706 (2005). [CrossRef]

Steering Angle	Incident beam rotation	LCPG (0.67°)	LCPG (2.01°)	LCPG (4.69°)	LCPG (5.36°)	Diffraction efficiency
0°	Left	0°/20v	0°/20v	0°/20v	0°/20v	92.14%
+0.67°	Left	+0.67°/2.11v	0°/20v	0°/20v	0°/20v	92.08%
+1.34°	Right	−0.67°/2.11v	+2.01°/2.11v	0°/20v	0°/20v	90.06%
+2.01°	Left	0°/20v	+2.01°/2.11v	0°/20v	0°/20v	89.71%
+2.68°	Right	0°/20v	−2.01°/2.11v	+4.69°/2.12v	0°/20v	86.57%
+3.35°	Left	+0.67°/2.11v	−2.01°/2.11v	+4.69°/2.11v	0°/20v	86.65%
+4.02°	Right	−0.67°/2.11v	0°/20v	+4.69°/2.11v	0°/20v	86.89%
+4.69°	Left	0°/20v	0°/20v	+4.69°/2.11v	0°/20v	87.12%
+5.36°	Left	0°/20v	0°/20v	0°/20v	+5.36°/2.11v	92.78%
+6.03°	Left	+0.67°/2.11v	0°/20v	0°/20v	+5.36°/2.11v	91.02%
+6.70°	Right	−0.67°/2.11v	+2.01°/2.11v	0°/20v	+5.36°/2.11v	89.09%
+7.37°	Left	0°/20v	+2.01°/2.11v	0°/20v	+5.36°/2.12v	87.54%
+8.04°	Right	0°/20v	−2.01°/2.11v	+4.69°/2.12v	+5.36°/2.13v	84.88%
+8.71°	Left	+0.67°/2.11v	−2.01°/2.11v	+4.69°/2.12v	+5.36°/2.14v	82.43%
+9.38°	Right	−0.67°/2.11v	0°/20v	+4.69°/2.11v	+5.36°/2.15v	81.81%
+10.05°	Left	0°/20v	0°/20v	+4.69°/2.11v	+5.36°/2.16v	80.90%

Diffraction characteristics of a non-mechanical beam steering system with liquid crystal polarization gratings

Abstract

1. Introduction

2. Basic principle and simulation method of LCPGs

3. Design and experimental details of the beam steering system

4. Results and discussion

5. Conclusion

Funding

Acknowledgement

Disclosures

Data availability

References

Data availability

Cited By

Figures (13)

Tables (1)

Equations (18)

Optics Express