# High-Precision Beam Angle Expander Based on Polymeric Liquid Crystal Polarization Lenses for LiDAR Applications

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Working Principle

#### 2.1. BAE Module

_{1}) PB lens and a larger negative power (k

_{2}) PB lens. The angle magnification is dictated by the two lens powers, which can be simply calculated by the ray-tracing method. Figure 1 shows the system configuration and the parameters involved in the ray tracing, where f

_{1}and f

_{2}denote the focal length of each lens, respectively. The magnification factor can be derived as follows:

_{out}) can be expressed as follows:

_{in}), the distance d should be adjusted to satisfy the following relationship:

#### 2.2. PB Lens

_{±}indicates the Jones vectors of left- and right-handed circularly polarized light. The Jones matrix of a half-wave plate (HWP) with a spatially orientated angle φ(x) is expressed as follows:

## 3. Experimental Results

#### 3.1. Infrared PB Lens Fabrication and Characterization

_{1}λ

_{1}= f

_{2}λ

_{2}. In our experiment, the exposure wavelength (λ

_{1}= 457 nm) is much shorter than the laser beam steering wavelength (λ

_{2}= 1550 nm). As a result, the selection of TL power should be based on the designed optical power of the PB lenses at 1550 nm. The resulting interference pattern is recorded on the sample substrate coated with a photo-alignment layer. The photo-alignment material we used is Brilliant Yellow (BY), which was dissolved in a dimethylformamide (DMF) solution. Then, the substrate was exposed to interfering beams. BY is a kind of azo dye, whose azo molecules would rotate to align perpendicular to the polarization direction of the exposure light [21,22]. After exposure, we started the spin coating process. To achieve π phase retardation, the LC film thickness (d

_{LC}) should satisfy the following:

^{2}), covering the whole LC layer, was used for adequate polymerization, which is essential for the next coating. An elegant way to determine how many layers should be coated is to observe which incident wavelength meets the half-wave condition after coating each layer. We built a simple testing setup (Figure 2a) to find the optimal working wavelength by measuring the system transmittance. The incident light passing through the first circular polarizer (CP1) becomes RCP light. At the optimal wavelength, the PB lens will maximumly convert RCP to LCP light; otherwise, the leaked RCP light will pass through the second circular polarizer (CP2). Consequently, the optimal wavelength should have the lowest transmittance. In other words, we stopped coating when the lowest transmittance was found at 1550 nm. Although the spectral range of our spectrometer is only from 400 nm to 1100 nm, we could use a reliable method to indirectly determine if the lens efficiency reaches its highest at 1550 nm. The film thickness that induces a π phase change at 1550 nm also brings about a 3π phase change at ~589 nm, based on the phase retardation equation. Figure 2b indirectly proves that the efficiency has achieved its highest at 1550 nm, after 8 LC layers have been deposited.

#### 3.2. Angle Magnifier Performance

## 4. Discussion

_{1}and f

_{2}. However, Equation (7) reveals that there is a trade-off between the angle magnification and system volume. In other words, when it comes to a practical situation, the magnification should be dictated by the allowable system volume. The system volume of the BAE is determined by the diameter of the two employed lenses and the distance between them. It is, therefore, important to evaluate the trade-off between the magnification factor and system volume by taking some practical restrictions and assumptions into account.

_{1}) between the OPA and BAE, as listed in Table 2. The BAE can be more compact by selecting a shorter focal length for L

_{1}and L

_{2}, according to Equation (6). However, the whole design needs to follow the restrictions and assumptions discussed above. In Table 2, d

_{1}and f

_{1}are manually setting variables, and f

_{2}/# is set to 0.64 and 1.1 at different wavelengths. The other parameters can then be calculated according to Equations (1)–(7). The BAE is designed to accommodate a maximum incident angle of 5°. A larger incident angle requires redesign, since larger-sized lenses are needed. Assuming that the angular range of an OPA is ±5° at λ = 1550 nm, after integration with our BAE, the steering range can reach about ±45°. If the BAE is optimized at 905 nm, then the steering range is about ±25°.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic illustration of the BAE consisting of two PB lenses. The magnified part is a detailed ray-tracing process for one beam, as an example.

**Figure 2.**(

**a**) Transmittance measurement setup. CP: circular polarizer. (

**b**) Measured normalized transmission spectrum of the PB lens sandwiched between two CPs.

**Figure 3.**(

**a**) Photograph of the fabricated PB lens. The diameter is ~1.7 cm. Polarizing optical microscope images of the large optical power PB lens (

**b**) at the central area, and (

**c**) at the margin.

**Figure 4.**(

**a**) Schematic of the measurement setup. LP: linear polarizer; QWP: quarter-wave plate. The incident angle is tuned from 0° to 4°. (

**b**) Performance of the beam angle expander. Dots represent the measured data and solid lines are visual guides. (

**c**) Simulated angle-dependent efficiency of non-twist LC PB elements. Local grating period: 5 µm, 10 µm, and 20 µm.

Solute: Solvent | Coating Speed | |
---|---|---|

1st LC layer | 1:4 | 1000 (30 s) |

2nd–8th LC layer | 1:4 | 2000 (30 s) |

**Table 2.**The parameters involved in the calculation for the whole magnification system. The incident angle is set at 5°. d

_{1}: the distance between OPA and BAE; f

_{1}: focal length of the first PB lens (L

_{1}); f

_{2}: focal length of the second PB lens (L

_{2}); d: the distance between L

_{1}and L

_{2}; D

_{1}: diameter of L

_{1}; f

_{1}/#: f-number of L

_{1}; D2: diameter of L

_{2}; f

_{2}/#: f-number of L

_{2}; M: magnification factor. The minus sign of f

_{2}indicates L

_{2}is a negative lens.

d_{1} (cm) | f_{1} (cm) | f_{2} (cm) | d (cm) | D_{1} (cm) | f_{1}/# | D_{2} (cm) | f_{2}/# | M | |
---|---|---|---|---|---|---|---|---|---|

1550 nm | 1 | 1 | −0.11 | 0.88 | 0.17 | 5.71 | 0.17 | 0.64 | 8.87 |

2 | −0.21 | 1.78 | 0.17 | 11.43 | 0.33 | 0.64 | 9.37 | ||

2 | 1 | −0.12 | 0.87 | 0.35 | 2.85 | 0.19 | 0.64 | 7.87 | |

2 | −0.22 | 1.77 | 0.35 | 2 | 0.35 | 0.64 | 8.87 | ||

905 nm | 1 | 1 | −0.19 | 0.80 | 0.17 | 5.71 | 0.17 | 1.1 | 5.18 |

2 | −0.35 | 1.64 | 0.17 | 11.43 | 0.31 | 1.1 | 5.68 | ||

2 | 1 | −0.24 | 0.76 | 0.35 | 2.85 | 0.21 | 1.1 | 4.18 | |

2 | −0.38 | 1.61 | 0.35 | 2 | 0.35 | 1.1 | 5.18 |

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**MDPI and ACS Style**

Li, Y.; Luo, Z.; Wu, S.-T.
High-Precision Beam Angle Expander Based on Polymeric Liquid Crystal Polarization Lenses for LiDAR Applications. *Crystals* **2022**, *12*, 349.
https://doi.org/10.3390/cryst12030349

**AMA Style**

Li Y, Luo Z, Wu S-T.
High-Precision Beam Angle Expander Based on Polymeric Liquid Crystal Polarization Lenses for LiDAR Applications. *Crystals*. 2022; 12(3):349.
https://doi.org/10.3390/cryst12030349

**Chicago/Turabian Style**

Li, Yannanqi, Zhenyi Luo, and Shin-Tson Wu.
2022. "High-Precision Beam Angle Expander Based on Polymeric Liquid Crystal Polarization Lenses for LiDAR Applications" *Crystals* 12, no. 3: 349.
https://doi.org/10.3390/cryst12030349