1. Introduction

Unlike a refractive optical device that generates phase pattern using the optical pathlength difference, liquid-crystal (LC)-based Pancharatnam-Berry optical element (PBOE), which can be regarded as a patterned half-wave plate, produces desired phase profile by spatially varying the LC directors [1–7]. With the advantages of high diffraction efficiency (nearly 100%), polarization selectivity, ultrathin formfactor, and dynamic switching, PBOE has found promising applications in near-eye display systems [8–11].

The underlying operation principle of PBOEs can be described by following equation using Jones matrix representation:

To obtain the desired LC distribution, several methods have been developed while surface alignment is the most established one for PBOE fabrication [16,17]. With the help of photo-alignment (PA) materials that are sensitive to the incident polarization, the desired phase pattern can be recorded using Mach-Zehnder Interferometer (MZI). However, the effective exposing wavelength is restricted by the absorption spectrum of the PA materials. For example, brilliant yellow (BY), a widely used PA material, shows an absorption peak at λ=432 nm [18], which means the exposure wavelength should be in the blue spectral region. The phase profile of a PB lens (PBL) with a focal length of ${f_0}$ is $2\pi \sqrt {{r^2} + f_o^2} /{\lambda _o}$, where ${\lambda _o}$ is the operating wavelength and r is the radius of the PBL. When designing and fabricating a PBL for a red wavelength, the mismatch between exposing and operating wavelengths will result in a phase deviation, which equals to $2\pi \left( {\sqrt {{r^2} + f_o^2} /{\lambda_o} - \sqrt {{r^2} + f_e^2} /{\lambda_e}} \right)$, where ${f_e}$ is the focal length of the template lens and ${\lambda _e}$ is the exposing wavelength. The incurred phase difference would degrade the imaging quality and limit its applications for AR/VR displays. Freeform lenses offer potential solutions to this problem and provide desired phase profile for the exposure of PBLs [19]. However, the design and fabrication process of freeform lenses are challenging. Meanwhile, each designed freeform lens is only applicable for the exposure of PBL with a fixed focal length and operating wavelength. These drawbacks limit the application of freeform lenses in the fabrication process of PBLs and more convenient approaches are needed to solve the wavelength-induced phase deviation.

In this paper, a new method for fabricating PBLs is demonstrated to correct the wavelength-induced phase deviation. In traditional PBL fabrication methods, the phase profile of only one template lens is recorded by the PA material. Here, we employ two different template lenses in different arms of MZI and record their phase profile difference by the PA material, which enables a more accurate phase profile control of the PBL. The intensity distribution of the PBL is measured and compared with simulation results. Good agreement is obtained. Moreover, the imaging characteristics of PBLs are investigated using a Maxwellian display system and the performance improvement is quite noticeable.

2. Design and fabrication

In a VR system, due to the etendue conservation [10], a small f-number imaging lens helps to reduce the depth of the headset. The f-number of a lens is defined as f/D, where f is the focal length and D is the diameter of the entrance pupil. To achieve a compact and lightweight VR display, PBLs with a small f-number are favorable, if the above-mentioned phase deviation problem can be alleviated. To obtain the desired phase profile, let us consider the interference of RCP and LCP beams with different phase patterns. The interference can be expressed using Jones matrix representation as:

Based on the above analysis, we design two lenses PBL-1 and PBL-2 for comparison. For PBL-1, the phase profile of a 5.56-cm lens (at λ = 457 nm) is used in the MZI to obtain a focal length of 4 cm at λ = 635 nm. In this case, either α or β is equal to 0, depending on which arm the lens is placed. To achieve the same focal length for PBL-2, two template lenses with focal lengths (4.32 cm and 19.41 cm at λ = 457 nm) are employed. The procedures to find the focal lengths of these two lenses are briefly outlined as follows. We first enumerate the possible combinations of focal lengths in the 1.5-cm to 20-cm range for the PBL with 4-cm focal length. For each case with two different focal lengths, the phase ϕ₁ to be recorded is the difference between the phase profiles corresponding to the two focal lengths according to Eq. (2). Then, each ϕ₁ is compared to the ideal phase profile ϕ₂ of the desired lens. Then the phase difference Δϕ = ϕ₁ – ϕ₂ can be calculated. By analyzing the distribution of Δϕ along radius, the combination with minimum variance can be obtained. Finally, the two focal lengths in this combination are the desired results. The phase profile of these two PBLs are compared with an ideal lens and their differences are plotted in Fig. 1(a). The phase deviation caused by the wavelength mismatch of PBL-1 increases dramatically as the lens radius (r) exceeds 5 mm. Meanwhile, for PBL-2, the difference of phase pattern is much smaller and remains in an acceptable range.

 

Fig. 1. (a) Simulated phase difference between PBL-1 and PBL-2 in comparison with an ideal lens (r = lens radius). Ray tracing around the focal point of (b) PBL-1 and (c) PBL-2.

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In addition to calculate the phase profiles, raytracing is also employed to simulate the performances of the designed lenses. Considering the case when a collimated incident beam traverses through the designed lenses, the ray bundle around the focal point is illustrated in Fig. 1(b) and 1(c). The rays that passing through an ideal lens are expected to converge to a single spot. However, for PBL-1 exposed using one lens, the rays diffracted by different positions of the lens spreads out in a large range. Compared to PBL-1, the diffracted ray bundle of PBL-2 is focused to a much tighter spot because the phase difference between PBL-2 and the ideal lens is relatively small, as Fig. 1(a) depicts.

In experiment, the lens pattens were recorded using surface alignment method. Firstly, the PA material BY was dissolved in dimethyl-formamide and then spin-coated on a clean glass [20]. The material ratio and spin-coating speed are listed in Table 1. After spin coating, the PA layer was exposed using the setup depicted in Fig. 2. The 457-nm laser beam first passed through an objective lens, a spatial filter, and then a collimating lens. The beam was then split into two components by the first beam splitter (BS₁). The two linearly polarized beams were converted into RCP and LCP, respectively, using the quarter-wave plates (QWP₁ and QWP₂). Let us assume RCP passing through L₁ whose focal length is 19.41-cm and LCP passing through L₂ whose focal length is 4.32-cm. A second beam splitter BS₂ was used to combine the two phase-modified beams and merge the two optical paths together. To record the phase pattern precisely, the optical paths should be finetuned, and the centers of the two lens patterns overlapped with each other. After optimizing the setup, the PA layer was exposed to a laser output at 150 mW/cm² for about 2 min. With the designed phase patterns recorded, two LC layers (Table 1) were spin-coated on the PA layer to satisfy the half-wave condition at λ = 635 nm. Afterwards, the spin-coated substrate was exposed to UV light to stabilize the polymerization process. In contrast to PBL-2, PBL-1 was also fabricated using the MZI setup shown in Fig. 2 but only with a 5.56-cm lens at λ = 457 nm.

 

Fig. 2. Mach-Zehnder interferometer for fabricating PBL-2.

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Table 1. Materials and coating speed for fabricating PBLs.

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3. Results and discussion

After fabrication, we evaluated the imaging behaviors of these two PBLs. A collimated red light (635 nm) from a laser diode was applied to measure the focal length of PBL-1 and PBL-2. The measured results are both around 4 cm, as expected. To further investigate the performances in a smaller scale, a CMOS camera was used to measure the light intensity distribution along the optical axis near the focal points. Meanwhile, simulation results were also obtained using Zemax to compare with the experimental data.

Figure 3 shows the normalized intensity distribution at different planes along the optical axis, and the experimental data agree well with the simulation results. For PBL-1, when coming from the lens side to the focal point, the intensity at the outer circle is stronger than that of the central area as shown in Fig. 3(a). However, the intensity at the plane behind the focal point exhibits an opposite distribution, as Fig. 3(b) shows. The high-intensity area is in the center while the intensity at peripheral area gradually decreases. Theoretically, the two symmetric planes along the optical axis should have the same intensity distribution if the collimating lens is an ideal lens. However, due to the wavelength mismatch, the phase profile of PBL-1 deviates from that of an ideal lens, especially when the lens diameter gets larger. For PBL-2 fabricated using the new method, the intensity of the two planes shows more similar distribution due to a much smaller deviation from the ideal lens phase profile. From Fig. 3, the maximum light intensity of our experimental results is lower than that of the simulated one. The main reason is due to the imperfect LC alignment caused by our two-layer spin-coating processes. In addition, some defects on each LC layer also degrade the PBL performance.

 

Fig. 3. Normalized experimental (top) and simulated (bottom) intensity distribution of PBLs at different positions: (a) 0.5 mm in front of PBL-1 focal point, (b) 0.5 mm behind PBL-1 focal point, (c) 0.5 mm in front of PBL-2 focal point, (d) 0.5 mm behind PBL-2 focal point.

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To investigate the imaging performance of our new lens, we built a Maxwellian display system as shown in Fig. 4(a) [21]. A laser projector was used as the light engine. The modulated output beam was then collimated by a lens, and a color filter was attached to the lens to avoid light leakage at other wavelengths. After passing through the PBLs, the light was focused to the pupil of a smartphone camera by PBLs, and the images were captured. The input image of vertical lines is depicted in Fig. 4(b), and the recorded images are shown in Fig. 4(c) for PBL-1 and Fig. 4(d) for PBL-2. The difference at the peripheral area is clearly observed. For Fig. 4(c), the vertical lines at the edges overlap with each other and only the lines at central area can be clearly observed. Some vertical lines in Fig. 4(c) and Fig. 4(d) are darker than other lines, which may result from the mismatched frame time between the laser projector and exposure time of the camera. In practical applications, the imaging performance may limit the field of view of the display system due to the degraded resolution caused by the phase deviation of PBL. The performance of PBL-2 is much better since the vertical lines at the border can still be distinguished by the camera. The lines in some peripheral area of Fig. 4(d) cannot be separated, which may originate from the alignment problem during exposure process. Compared to previous exposure method, our new approach provides a much better image quality.

 

Fig. 4. (a) Maxwellian display system, (b) original input image, (c) recorded images with PBL-1, and (d) recorded images with PBL-2.

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4. Conclusion

We demonstrated a new approach to correct the phase deviation of PBLs caused by the mismatched exposing and operating wavelengths. Using two lenses with different focal lengths in the exposure system, the newly designed PBL-2 exhibits a smaller phase difference from the ideal lens, which in turn leads to better imaging quality than the traditional PBL-1. The intensity of PBL-2 along optical axis shows more symmetric distribution about the focal point, which is like an ideal lens. The measured results agree well with the simulated intensity distribution. This new fabrication method is expected to make significant impact to AR/VR displays.