All-in-One Collimating Splitter Based on a Meta-Fiber Platform

Featured Application

Face identification, motion sensing and free-space optical communication.

Abstract

The use of array generators has become ubiquitous in various applications such as laser fabrication, face identification, and motion sensing. The Dammann grating, a diffractive optical element, is the mainstream approach for generating uniform spot arrays. However, its limited capability and the contradiction between the performance and the complexity of fabrication hinder its application. To address this issue, an all-in-one collimating splitter based on metasurfaces is theoretically proposed by synthesizing the phase of an inverse-optimized Dammann grating and a collimating lens. Leveraging both the diffraction effect of Dammann grating and the Fourier transformation of the collimating lens, the number of spot arrays can be largely increased with a single lenslet. The proposed design shows a large field of view of 62° × 62° and a high uniformity of 1.29% in generating a spot array of 3 × 3 on a single-fiber platform, confirmed by both the scalar and full-wave simulation. Further, a larger spot array up to 15 × 15 is also derived in the far field by integrating the proposed metasurface on a 5 × 5 fiber array platform, confirmed by the scalar simulation. Our design may be transplanted to the vertical-cavity surface-emitting laser platform, and shows great potential in various applications including face identification and motion sensing.

1. Introduction

Array generators were proposed in the last century for use in laser fabrication to improve the efficiency of this process [1]. With the development of smart devices, the array generator has found new applications in depth sensing for face identification and motion sensing [2,3]. In a typical spot array generation architecture, light from a laser source is initially collimated by a lens and then directed to a diffractive optical element (DOE), which duplicates the light into a large array across a wide field of view [4]. The most common and efficient approach for generating such a DOE spot array is through the use of Dammann grating. In 1971, Dammann introduced a binary-phase grating. By precisely selecting the abrupt phase change points, the phase and amplitude of the spot array in the far-field can be uniformly distributed across different diffraction orders [1]. However, using a binary phase for complex designs is inefficient. Therefore, multilevel-phase gratings were then introduced to improve the diffraction efficiency of the DOE [5]. Nonetheless, as the desired number of spot arrays increases, the design becomes more intricate, which directly lowers the final efficiency and significantly increases the fabrication complexity. Furthermore, the separated configuration of the collimating lens and the DOE is cumbersome and limits its application in smart devices, such as phones.

Compared to conventional DOEs, a metasurface consists of planar subwavelength structures, which usually induce local resonance [6,7]. This attribute confers great flexibility for manipulating the amplitude, phase and polarization of the light waves [8]. In addition, the fabrication of the planar nanostructures of the metasurfaces enjoys good CMOS compatibility, which is much simpler than the multilevel DOE fabrication process. Thanks to the flexibility in manipulating light and the good CMOS compatibility in terms of fabrication, various applications of metasurfaces have been demonstrated, including imaging [9,10,11,12], holography [13,14,15,16], beam steering [17,18,19,20,21,22], beam shaping [23,24] and so on. Recently, metasurfaces based on Dammann grating have been proposed for realizing versatile beam functions in different diffraction orders [25,26,27,28] or switchable functions empowered by tunable materials [29,30]. Combining the phase of the Dammann grating and the special beams to generate special beam arrays using metasurfaces has been demonstrated, such as the Airy beam array [31,32], Bessel beam array [33,34] and vortex beam array [25]. Moreover, a Dammann grating metasurface has been designed for generating spot arrays within a large field of view in the far field [35]. However, the laser source in these designs needs to be collimated by a traditional lens before being directed to the Dammann metasurface, which increases the bulkiness of the entire system for practical use. Additionally, the scale of the spot array generated by a traditional Dammann grating is limited [36], and scaling up is challenging without additional optical elements, such as cascading another Dammann grating [37].

In this study, we proposed a metasurface-based all-in-one collimating splitter with high uniformity and a large field of view by synthesizing the phase of the Dammann grating and the lens. By optimizing the phase profile using an inverse design method, we realize a 3 × 3 Dammann grating, achieving a large field of view (FOV) of 62° × 62° and a uniformity of 1.29%. Through the joint effect of the space-spectrum Fourier transformation of the lens and the diffraction effect of the Dammann grating, we demonstrate that a singlet metasurface chip can greatly increase the number of spot arrays in the far field by N × N times on the N × N fiber array platform. This design can be easily transplanted to the vertical-cavity surface-emitting laser (VCSEL) platform to derive a more compact spot array generator for depth-sensing applications on smart devices.

2. Design of the All-in-One Collimating Splitter

2.1. Concept of the All-in-One Collimating Splitter

Figure 1 shows the concept of the all-in-one collimating splitter based on a single-mode fiber. In the traditional design shown in Figure 1a, the light output from the fiber should first be collimated by a collimator and then directed to the Dammann splitter to generate a beam array in the far field. The traditional optical system consists of discrete optical components with bulky size and heavy weight, which hinders their applications, such as face identification, with smart devices. Here, utilizing the flexibility to manipulate beams, an all-in-one collimating splitter based on metasurfaces is proposed, with a compact configuration and excellent optical performance, as shown in Figure 1b. The designed metasurface combines the functions of the collimator and the splitter and can be conveniently integrated on the end face of the single-mode fiber using methods like bonding [17,38].

2.2. Design of the Phase Profile

As shown in Figure 1b, the metasurface consists of silica substrate and silicon nano-cuboids. The laser beam from the single-mode fiber travels for about 500 μm ($f=500 \mathsf{\mu }\mathrm{m}$) in the silica substrate and is diffracted by the designed metasurface, forming a collimated 3 × 3 spot array in the far field. The core component of the collimating splitter, i.e., the metasurface, is designed by synthesizing the phase ($\varphi \left(x,y\right)$) of a collimating lens (${\varphi }_L\left(x,y\right)$) and the phase of an optimized Dammann grating (${\varphi }_D\left(x,y\right)$):

$$\varphi \left(x,y\right)={\varphi }_L\left(x,y\right)+{\varphi }_D\left(x,y\right),$$

(1)

where the collimating lens follows the hyperbolic profile without spherical aberration to collimate the diverging Gaussian beam from the single-mode fiber:

$${\varphi }_L\left(x,y\right)=-k_0\cdot n_{sub}\cdot \left(\sqrt{x^2+y^2+f^2}-f\right),$$

(2)

where $k_0$ refers to the wavevector in the vacuum, $n_{sub}$ refers to the refractive index of the substrate, i.e., $n_{sub}=n_{Si{O}_2}=1.447$, and the designed focal length $f$ of the collimating lens equals the thickness of the substrate, i.e., 500 μm in this case. The designed wavelength is 1.55 μm. Figure 2a shows the phase profile of the collimating lens (${\varphi }_L$).

The Dammann grating consists of periodically arranged supercells. One supercell consists of $5\times 5$ unit cells with the same period and different-sized nanostructures located in the center of each cell. The period of the supercell ($P$) is chosen according to the grating equation, so that a diffraction order larger than 2 is eliminated:

$$P_x\sin {\theta }_x=m\lambda , P_y\sin {\theta }_y=n\lambda ,$$

(3)

where $P_x$ and $P_y$ are the period of the supercell in the $x$ and $y$ direction, respectively, and ${\theta }_x$ and ${\theta }_y$ are the diffraction angles of the $m$th and $n$th grating order along the $x$ and $y$ direction, respectively. In our design, we set $P_x=P_y=3 \mathsf{\mu }\mathrm{m}$; thus, the diffraction angle (${\theta }_x, {\theta }_y$) of the first order ($m=n=1$) is (${31.1}^{°}, {31.1}^{°}$). The period of the unit cell ($p_x, p_y$) is chosen such that the interaction between the adjacent unit cells is small. The small interaction ensures that we can discretize the supercell and optimize the discrete phase profile of the supercell. Here, we set $p_x=p_y=0.6 \mathsf{\mu }\mathrm{m}$, and $5\times 5$ unit cells are distributed in one supercell. The phase profile of the supercell is designed to have $C_4$ symmetry so that the grating can be polarization-independent. To save the time during the device simulation, mirror symmetry is also introduced in the design.

Under the predefined constraints, the phase distribution of the supercell is optimized following an inverse design method. A random phase profile is chosen as the initial value, and the diffraction efficiency of each order is calculated using the angular spectrum propagation theory. The cost function is defined according to Equation (4) to derive a uniform intensity distribution of different diffractive orders in the far field and high efficiency:

$$cost=\sqrt{\frac{1}{N}{\displaystyle \sum }_{i=1}^N{\left(I_i-\frac{1}{N}\right)}^2},$$

(4)

where $I_i$ is the diffraction efficiency of the $i$th order, and $N$ denotes the total number of the diffraction orders. The diffraction efficiency is defined as the fraction of the incident power diffracted to the specific order. (Actually, in angular propagation theory, the metasurface is simplified to a phase profile with unity transmittance during the optimization; thus, the total transmission is assumed to be 1.) A sequential quadratic programming algorithm is used to minimize the cost function above [39]. The optimized phase profile of the Dammann grating (${\varphi }_D$) is shown in Figure 2b. The inset shows an enlarged phase profile of one supercell of the Dammann grating, which has $C_4$ and mirror symmetry as previously defined. By synthesizing the phase profile of the collimating lens (${\varphi }_L$) and the Dammann grating (${\varphi }_D$), the profile of the final collimating splitter ($\varphi$) is derived, as shown in Figure 2c. The final results are characterized by the total diffraction efficiency ($\eta$) and uniformity, which is defined as the standard deviation (SD):

$$\eta ={\displaystyle \sum }_{i=1}^N{I}_i, SD=\sqrt{\frac{1}{N}{\displaystyle \sum }_{i=1}^N{\left(I_i-\frac{\eta }{N}\right)}^2},$$

(5)

The final efficiency and uniformity of the Dammann grating shown in Figure 2b reach 96.27% and 0.011642%, respectively. The diffraction efficiency is calculated by integrating the intensity at the corresponding diffraction order within the directional cosine radius of 0.05 in the angular spectrum. Thus, the total value ($\eta$) can be slight deviated from unity.

2.3. Design of the Unit Cell

The square unit cell consists of silicon nano-cuboids with different widths ($d$), shown in Figure 3a. The height ($h$) and period ($p=p_x=p_y$) of the nano-cuboid are fixed at 1 μm and 0.6 μm, respectively, and the width ($d$) varies from 150 to 450 nm. The refractive index of silicon is set as 3.6451 at the wavelength of 1.55 μm according to the experimental results. The transmittance and the normalized phase (${\varphi }_T\left(d\right)$) of the unit cell are calculated with Reticolo [40] using rigorous coupled wave analysis (RCWA). The results are plotted in Figure 3b. According to the optimized phase profile ($\varphi \left(x,y\right)$) above, the nano-cuboid with the proper width ($d$) is chosen such that the corresponding cost function ($f\left(x,y\right)$) is minimized at each location:

$$f\left(x,y\right)=\left|e^{i{\varphi }_T\left(d\right)}-e^{i\varphi \left(x,y\right)}\right|,$$

(6)

The width profile of the final results is plotted in Figure 3c for demonstration.

3. Results and Analysis on a Meta-Fiber Platform

3.1. Analysis Using Angular Spectrum Propagation Theory on a Single-Fiber Platform

After choosing nano-cuboids with the proper size (Figure 3c), the far-field performance of the device is evaluated with the corresponding transmittance and phase according to Figure 3b using the angular spectrum propagation theory. The parameters of a Corning SMF 28e single-mode fiber are used for the simulation. The angular spectrum in the far field is calculated after Fourier transformation. Figure 4a,b shows the far-field angular spectrum profile without collimation. The divergence angle is defined when the intensity drops to $e^{-2}$, which is about 11.06° according to Figure 4b. By adding a collimating metalens (${\varphi }_L$), the divergence angle can be reduced to 1.97°, as shown in Figure 4c,d. Figure 4e,f shows the metasurface-based 3 × 3 collimating splitter, which synthesizes the collimating phase and Dammann grating together ($\varphi$). The incident large-divergence Gaussian beam is collimated and splits into nine beams with uniform intensity within the FOV of 62° × 62°, matching well with the prediction using Equation (3). The calculated diffraction efficiency and uniformity of the collimating splitter reach 99.1% and 0.21%, respectively. Compared with the optimized results of the Dammann grating, the uniformity gets worse mainly for two reasons: first, the phase profile after synthesizing will deviate from the optimized phase for splitting; second, there will be a difference between the practical complex amplitude of the chosen silicon structures (according to Figure 3b) and the profile ($\varphi \left(x, y\right)$) with unity transmittance.

The diffraction efficiencies of the metasurface collimating splitter extracted from different diffraction orders are extracted and shown in Figure 4h. The diffraction efficiencies of the previously optimized Dammann grating are also extracted for comparison in Figure 4g. Both of the results show good uniformity and near-unity efficiency, which confirms the validity of the phase synthesis.

3.2. Analysis Using FDTD on a Single-Fiber Platform

To further verify the results, we conduct a full-wave simulation using Lumerical FDTD. The results are shown in Figure 5. The metasurface collimating splitter with a total size of 80 μm × 80 μm is simulated and a Gaussian beam with a waist diameter of 10.4 μm and single wavelength of 1.55 μm is incident from the substrate, about 500 μm below the metasurface. The conditions of all the boundaries are set to perfect matched layers (PMLs). Due to the symmetry of the phase profiles and the unit cell, the device is polarization-independent, as shown in Figure 5a–d. The incident Gaussian beam is well collimated and splits into nine different orders for both polarizations, matching well with the results simulated using angular spectrum propagation theory. The same diffraction efficiency and uniformity are derived for both polarizations, i.e., about 57.87% and 1.29%, respectively. The diffraction efficiency and uniformity results are both worse than the results derived using angular spectrum propagation theory. For further investigation, we extract the diffraction efficiency for each order, shown in Figure 5e,f. The deterioration of the uniformity comes from the decrease of the diffraction efficiency at the four corner orders, i.e., (−1, −1), (−1, 1), (1, −1) and (1, 1), which correspond to much larger diffraction angles (46.94°) than the other orders (31.1°). This will bring strong coupling between the adjacent nano-cuboids, making the practical phase deviate from the designed value. Moreover, the coupling also leads to a drop in the diffraction efficiency by introducing a reflection of about 31.86%, as shown in the FDTD simulation.

The deviation between the results of the scalar simulation and the vectorial simulation is also deeply discussed in Ref. [35]. To eliminate the deviation requires a decrease in strong coupling between the adjacent nano-cuboids. First, the diffraction angle can be designed to be much smaller by increasing the period of the supercell (even though more diffraction orders may be involved) to avoid the strong coupling. This is suitable to most of the design methods. Second, the Dammann grating should be optimized using a vectorial method, like RCWA or FDTD. The coupling effect between the unit cells is considered and optimized in these methods. The drawback is that the phase synthesis can be no longer suitable, since it works assuming that there are small interactions between the unit cells and that the phase profile can be discretized by the unit cells.

3.3. Analysis of the Influence of the Fabrication Errors Using FDTD

To further show the experimental feasibility of the proposed design, the influence of fabrication errors on optical performance influenced is studied. Based on the fabrication experience, the variation of the diameter and the sidewall angle exert the greatest influence. These deviations are usually introduced during the electron beam lithography, the lift-off, and especially the etching process. Typically, the largest sidewall angle deviation ($\alpha$) is less than 3° (Figure 6a) and the diameter variation ($\delta D$) is less than 20 nm (Figure 6e).

The influence of the fabrication error on the divergence, uniformity and the total diffraction efficiency of the device was investigated based on the typical error values above using the FDTD method. The existence of the sidewall angle or the variation of the diameter brings disturbance to the optimized phase profile; thus, the incident beam cannot be perfectly collimated by the beam splitter and the divergence angle increases, as shown in Figure 6b,f. The interference between the inefficient deflected beam and the diffracted beam at the (0, 0) order leads to an interference pattern at the (0, 0) order. This directly deteriorates the uniformity and the efficiency of the output, as shown in Figure 6c,d,g,h. Apart from the phase change and the interference, the lower efficiency can also be attributed to an about 5–10% increase in the reflection. Compared to the sidewall angle formed during the fabrication process, the diameter variation brings much less influence on the final optical performance. During the fabrication, the etching process should be carefully optimized to minimize the sidewall angle for better optical performance.

3.4. Large-Scale Spot Array Generation on a Fiber Array Platform

By cascading the designed metasurface collimating splitter to a fiber array, a larger matrix of spot arrays can be generated in the far field, as shown in Figure 7a. The lens can be used to transform the lateral shift in the focal plane to the angular deflection in the far field, functioning as a Fourier transformer. The incident light is collimated and deflected by the lens component at a smaller angle, while it is deflected by the Dammann grating component at a larger angle. The Fourier function of the lens can be expressed as:

$$l_x=\frac{f}{n_{sub}}\cdot u_x, l_y=\frac{f}{n_{sub}}\cdot u_y$$

(7)

where $l$, $f$, $n_{sub}$ and $u$ denote the lateral distance to the optical axis on the focal plane, the focal length, the refractive index of the substrate and the direction cosine. The refractive index of the substrate is the same as in the design above, while the focal length is increased to 2000 μm to make sure that the generated spots are within the light cone. A 3 × 3 collimating splitter cascading to a 5 × 5 fiber array, which is able to generate up to a 15 × 15 spot array, is demonstrated and simulated using the angular spectrum propagation theory, as shown in Figure 7b. In the simulation, a 5 × 5 single-mode fiber array is placed at an adjacent distance of 127 μm, which is the typical value in a commercial fiber array. The metasurface is designed with a size of 900 μm × 900 μm to cover the effective area of the incidence. The Gaussian beam from each of the 5 × 5 fiber arrays is collimated and diffracted into a 3 × 3 spot array by the metasurface, resulting in the final 15 × 15 spot array in the far field. The deflected angle can be estimated by adding the deflection angle of the Dammann grating according to Equation (3) and the deflection angle of the collimating lens according to Equation (7). The compact design has great potential for generating large-scale spot arrays and it can be transplanted to a VCSEL platform for higher integration, meeting the requirements of small size, weight and power (SWaP) of smart devices.

4. Conclusions

In conclusion, an all-in-one collimating splitter base on metafibers is proposed. Compared to the traditional configuration consisting of discrete optical components that are bulky and heavy, the metafiber design enjoys more compactness, which is beneficial for practical applications, like face identification. By synthesizing the phase of a collimating lens and a Dammann grating and integrating the final metasurface with a single-mode fiber, the Gaussian incidence can be collimated and split into multiple beams with high uniformity with only a single lenslet. Compared to the global optimization method, the phase synthesis design with excellent optical performance consumes less computing resources and is convenient for large-scale design on the fiber array platform. A 3 × 3 metasurface collimating splitter with a large FOV of 62° × 62° is designed and simulated using the angular spectrum propagation theory and FDTD. The uniformity reaches 1.29%, characterized by the standard deviation. Combining the deflection effect of the Dammann grating and the Fourier transformation of the collimating lens, the proposed metasurface can enlarge the scale of the spot array by N × N times by simply integrating it with the N × N fiber array. The compact design can also be transplanted to a VCSEL platform for higher integration. The proposed design, with high compactness and flexibility, shows great potential in the applications of face identification and motion sensing.