Since color filters allow high selectivity of individual colors and simple image processing, using color filters to separate red (R), green (G), and blue (B) bands has always been a research hotspot in color imaging^[1–3]. As the pixel sizes of color image sensors are becoming smaller and smaller to meet higher-resolution color imaging, the incoming average photon count per pixel is also becoming less and less^[4,5]. This trend brings challenges to color filters for its inherent $\sim 2/3$ light losses in each color pixel filter, whether in one-dimensional (1D) color filters such as RGB arrays with RGB unit cells or two-dimensional (2D) color filters such as Bayer filter arrays with RGB unit cells^[6].

In the past decade, the color splitter (CS), having sub-wavelength structure and full-band transmission characteristics in visible light, was proposed to provide an alternative to the color filter and received a lot of attention from researchers^[7,8]. In theory, the splitter can increase the photon collection per pixel up to three times, making it possible to achieve a higher optical efficiency than a traditional filter. However, due to the complexity in color splitting and diffracting, the design of the splitter with high efficiency, high resolution, and easy processing is always a thorny problem, even in the simplest 1D-CS with RGB unit cells. For example, the device’s designs by introducing traditional nanostructures have difficulties in multiwavelength color splitting according to specific bands like R, G, and B wavelength bands and have a low average spectral efficiency in each pixel^[9,10].

To date, metasurfaces^[11–18] have been widely used to study various optical effects at the sub-wavelength scale and performed exceptional functions. NTT Device Technology Laboratories in Japan proposed a pixel-scale 1D-CS based on a dielectric metasurface that only has an average spectral efficiency of $\sim 40\%$ in each pixel, limited by the design methods^[19]. Then, Philip’s team has successfully demonstrated an ultra-efficient ($>80\%$) 1D-CS by increasing the design freedom of the device, in which they combine 1D-CS with an inverse design based on adjoint optimization^[20–23]. However, this 1D-CS is hard to fabricate and is compatible to current complementary metal-oxide-semiconductor (CMOS) processing technology due to its complicated 3D spatial structure. Recently, designed by an inverse design with a genetic algorithm (GA), a 1D-CS with an efficiency of $\sim 45\%$ and with a 2D structure of a quick response (QR) code has been demonstrated^[24], striking a balance between efficiency and processing. However, there are still many problems in this 1D-CS that need to be improved, such as increasing the splitter’s efficiency, reducing the thickness and geometric complexity of the device to make it easier to process, reducing the crosstalk from different frequency bands, essentially instead of combining the splitter and filter to increase the cost, and speeding up the design process.

Here, we propose two 1D structure 1D-CSs based on ${\mathrm{TiO}}_2$ metasurfaces and show a single-to-double-layer design to further improve the 1D color splitting efficiency of the 1D-CS in color imaging. The size of the 1D-CS unit cell is only 3 µm in width and 0.3 µm in thickness, and, because of its simple 1D structure, its length can be arbitrary, it requires less computing resources than other 1D-CSs with 2D or 3D structures, and it is easier to process theoretically. We show in simulations that the single-layer 1D-CS has an average efficiency of up to 46.2% across the entire visible spectrum (380–780 nm), and the average efficiency of the double-layer 1D-CS is up to 48.2%; both of them are significantly higher than that of traditional color filters ($\sim 33.3\%$, theoretical maximum value without considering other optical loss). The total design time including the single-layer and the double-layer 1D-CS is within a week. Moreover, we evaluate the double-layer 1D-CS’s performances in terms of incident angle sensitivity, polarization angle sensitivity, and assembly tolerance, demonstrating a 1D-CS with application value.

Device geometry in 2D and 3D is widely adopted to design 1D-CSs, especially in the methods of inverse design, which certainly increases the design freedom of the device by exploring more unknown structures^[20,24]. But, they also slow down the design process, caused by a large number of optimization data. What is more, there is no need to use such complex structures to design 1D-CSs because their light wave is only deflected or diffracted in the 1D direction. In contrast, device geometry with 1D refractive index distribution is excellent enough for 1D-CSs according to the principles of optical diffraction, which to our knowledge has not been investigated previously. Therefore, inspired by the design of 1D grating couplers^[25,26], we create a 1D structure consisting of a high refractive index material and a low refractive index material to design our 1D-CSs, as depicted in S1 in Fig. 1(b). We take titanium dioxide (${\mathrm{TiO}}_2$) as this high refractive index material ($n$ decreases from 2.399 to 2.098 in the wavelength range of 380–780 nm, according to Ref. [27]) for its high transmittance in the full-color band and mature processing technology^[28–30], and it would be better to take air as the low refractive index material ($n\approx 1$) for the same reasons. The whole RGB unit cell shown in Fig. 1(a) reflects the fundamental color splitting schematic and compactness of our 1D-CS unit cell. The full-color band (380–780 nm) transmitting through the unit cell with a width of 3 µm will be focused on three different pixels with a width of 1 µm according to its wavelength. For increasing the design freedom of the 1D-CS unit cell and considering the current fabrication capability at the same time, this 1D-CS unit cell consists of 60 bars, and each bar that can be ${\mathrm{TiO}}_2$ or air randomly has a cross-section dimension of $50\mathrm{nm}\times 300\mathrm{nm}$, where 300 nm is the thickness of the 1D-CS. This thickness can provide about half-wavelength (or phase $\pi$) modulation under the visible band when we change the refractive index of the bar. To better reflect the color characteristics of the band, the R, G, and B bands here are defined as 580–780 nm, 480–580 nm, and 380–480 nm, respectively.

Undoubtedly, the 1D structure above is simple enough that any current inverse design methods for multidimensional space optimization are adequately suited for the optimization of this 1D-CS to obtain good performance. Here, the simulated annealing (SA) algorithm is skillfully applied to this 1D-CS, which is our previous experience in designing an orbital angular momentum (OAM) emitter^[31,32]. The SA inspired by the principle of solid annealing is one of the global optimization algorithms, and it utilizes a cooling schedule (initial temperature, cooling rate, chain length, and termination condition) to control the optimization process. In implementation, the 60 bars in the 1D-CS unit cell will be replaced by a set of binary strings consisting of 0 and 1. During the iteration, better generation for the device structure is selected according to random probability based on Metropolis criterion that is affected by real-time temperature. We set the number of iterations at each temperature as 500 and the cooling equation as $T_n=0.{95nT}_0$, where the cooling rate is 0.95, $T_0=0.02$ is the initial temperature, and $T_n$ is the temperature for the $n$th iteration. By inverse design with SA, a final structure of the single-layer splitter with 1D color splitting performance will be obtained from a 1D random structure, as shown in the process of S1 to S2 in Fig. 1(b). To achieve more precise phase modulation, we need to implement the optimization search in a wider solution space. Inspired by the segmented hierarchical evolutionary algorithm^[32], we segment the single-layer 1D-CS into a double-layer 1D-CS without changing its total thickness, as shown in the process of S2 to S3 in Fig. 1(b). Then, taking S3 as a new initial structure, a final structure of the double-layer splitter with better 1D color splitting performance will be obtained by adopting the similar method above, as shown in the process of S3 to S4 in Fig. 1(b). The entire process in Fig. 1(b) can be finished within a week by finite-difference time-domain (FDTD) and a single server with the Dell T7920 model and NVIDIA Quadro P2000 graphics cards.

The color responses of the designed 1D-CSs are shown in Fig. 2. An outstanding color splitting performance is achieved by the designed single-layer 1D-CS with structure S2, demonstrated by its simulated spectra [Fig. 2(a)] and by its simulated field distributions in the cross section under three bands [Figs. 2(c)–2(e)]. By calculation, the average efficiency of this device is 46.2% across the entire visible spectrum (380–780 nm), and that of each color band is 45.2%, 45.4%, and 48.0%, respectively, where the efficiency is defined as the ratio between the flux of energy for corresponding color pixel and the flux of energy of incident light for the 1D-CS unit cell, showing the spectrum focusing ability of the splitter. In contrast, the designed double-layer 1D-CS with structure S4 has a slight improvement in focusing performance by its simulated spectra and field distributions, as shown in Figs. 2(b) and 2(f)–2(h), and it has an average efficiency of 48.2% across three bands (48.4%, 47.1%, and 49.2% for each band, respectively). Compared to schemes that combine splitters and filters to reduce crosstalk, both of our 1D-CSs exhibit a low crosstalk, making it possible to be filter-free. In terms of commercialization costs, the single-layer 1D-CS is excellent enough because it is also the simplest structure at present, which means that it has the simplest preparation process under such compactness.

The angle sensitivity is another important factor to evaluate the 1D-CS’s performances, including incident angle sensitivity and polarization angle sensitivity. Except for some special application scenarios like polarization cameras, in most cases, the improvement of average efficiency across the entire visible band at a cost of large incident angle sensitivity and polarization angle sensitivity is not expected. Taking the double-layer 1D-CS as an example, we analyze its sensitivity characteristics in the incident angle and polarization angle shown in Fig. 3, calculated from the software named FDTD Solutions. Here, we define the average efficiency above 40% across three bands (RGB) as acceptable efficiency. This 1D-CS has an acceptable incident angle range of $-4.7°$ to 3.3° [Fig. 3(a)] along the splitting direction ($x$) that is the most sensitive direction, and we believe it could have a better range than that if we consider all directions. The simulated field distributions in the cross section under the incident angles of $-4.7°$ and 3.3° along the splitting direction are shown in Figs. 3(c)–3(e) and 3(f)–3(h), respectively. Compared with the field plots in Figs. 2(f)–2(h), these field distributions maintain similar profiles but slightly shift in position, resulting in a decline in splitting functionality. Moreover, this 1D-CS exhibits a considerable polarization independence, as shown in Fig. 3(b), where the average efficiency across three bands is higher than 40% for most of the polarization angle. We select two representative polarization angles of 45° and 90° to demonstrate their splitting characteristics through their field distributions, as shown in Figs. 3(i)–3(k) and 3(l)–3(n), respectively. Compared with the field plots in Figs. 2(f)–2(h), these field distributions still work well in the R band, slightly bad in the G band, and obviously bad in the B band, and that of the polarization angle of 90° is worse than that of the polarization angle of 45°, corresponding to the effect shown in Fig. 3(b). For different color bands, it will have distinctly different responses, both in incident and polarization angle sensitivity, resulting in uneven color splitting, which is a common problem that exists in all 1D and 2D-CSs, due to the different optical paths resulting from its wavelength.

To further evaluate the application value of our 1D-CS, we explore the assembly tolerance of our double-layer 1D-CS in simulation, as shown in Figs. 4(a) and 4(b). Still taking 40% as an acceptable efficiency, we show a 1D-CS with an excellent lateral tolerance of [$-0.28\mathrm{\mu m}$, 0.55 µm], which means that our device can be misaligned with pixels along the color splitting direction and still maintains a good performance within large error. In longitudinal (diffraction direction) tolerance, our 1D-CS exhibits an excellent tolerance of [$-2.9\mathrm{\mu m}$, $-1.6\mathrm{\mu m}$] as well, and, different from the angle sensitivity above, the three color bands show a similar behavior in longitudinal tolerance. Besides, the Talbot effect can be found in the position of 9.05 µm below our 1D-CS, and the average efficiency is up to 41.1% across the three bands in this position, which is likely to be applied in future research.

The 1D structure 1D-CSs including the single-layer and double-layer are proposed for high-efficiency color imaging, and a single-to-double-layer design is presented by introducing the inverse design two times. From the perspective of device cost, the single-layer 1D-CS has great potential commercial value in high-efficiency, high-resolution color imaging for its simpler structure than the current 1D-CS, more compact size, and high efficiency of 46.2%. Although the performance of the double-layer 1D-CS is only slightly improved over that of the single-layer one, we still believe that the single-to-double-layer design method is promising not only in CSs but also in other photonic devices.