Wavelength-decoupled geometric metasurfaces by arbitrary dispersion control

Conventional multicolor metaholograms suffer from the fundamental limitations of low resolution and irreducible noise because the unit structure functionality is still confined to a single wavelength. Here, we propose wavelength-decoupled metasurfaces that enables to control chromatic phase responses independently in a full range from 0 to 2π for each wavelength. The propagation phase associated with the geometric phase of rectangular dielectric nanostructures plays a critical role to embed a dual phase response into a single nanostructure. A multicolor metahologram is also demonstrated to verify the feasibility of our method that breaks through the fundamental constraints of conventional multicolor metaholograms. Our approach can be extended to achieve complete control of chromatic phase responses in the visible for general dual-wavelength diffractive optical elements. Conventional multicolour metaholograms suffer from inherently low resolution and irreducible noise due to the single phase response of each unit structure. Here, a wavelength-decoupled multicolour metahologram is demonstrated overcoming the existing problems by a dual phase response in a single nanostructured unit.

Wavelength decoupling is a key element for versatile applications, especially of metalenses and metaholograms which require different functions by wavelengths. For the purpose, a unit structure of a metasurface should control chromatic phase responses independently. Achromatic metalenses have already demonstrated by compensation of group delay and its dispersion 35 . However, design principles of achromatic metalenses are invalid for general applications such as multicolor metaholograms because they require independent full-phase control by wavelengths. Although many multicolor metaholograms have been demonstrated based on interleaved subarrays [36][37][38][39][40] and multiple positions [41][42][43] , their unit structure functionality is still confined to a single wavelength resulting fundamental limitations of low resolution and irreducible noise.
Here, we present a design principle for wavelength-decoupled metasurfaces that enable independent full-phase control of different wavelengths. Then we experimentally demonstrate a multicolor metahologram to verify our design principle. The wavelength-decoupled metasurface features a dual phase response embedded in a single nanostructure. The principle is based on the propagation phase of rectangular dielectric nanostructures associated with their geometric phase. Rotation of the structure induces equal geometric phases for different wavelengths whereas disparate structures cause different propagation phase gradients by wavelengths. One structure generates continuous phase variation by rotation, and thereby a finite structure set enables full-phase manipulation for different wavelengths. Similar works on combined propagation phase and geometric phase have been demonstrated, but they aimed independent phase control by polarization directions at a single wavelength 20,21,[44][45][46][47][48] . The experimentally demonstrated multicolor metahologram verifies our design principle overcoming fundamental limitations of conventional multicolor metaholograms, so our approach to designing wavelength-decoupled metasurfaces is capable of realizing general dual-wavelength optical devices.

Results
Principle of independent phase manipulation. Using the propagation phase alone, we need m 2 structure configurations with appropriate dispersion to realize full-phase control for two wavelengths divided by m phase steps for each wavelength 49,50 . For example, if m = 6, then coverage of the whole phase range with a corresponding phase step of π/3 requires 36 structures. The number of necessary structures equals to the total number of permutations of 6 phase values for each wavelength. This is straightforward, but impractical because the existence of those 36 structures with the desired dispersion is not guaranteed under limited pitch sizes and structure configurations such as rectangles or ellipses. Simulation of all the configurations may identify such structures, but the computation cost is high. As m increases, the situation becomes increasingly difficult and complex. To avoid these complications, an alternative design approach is necessary.
The geometric phase provides the intuitive way to realize multicolor metaholograms by reducing the design complexity of the m 2 structures (Fig. 1a), because the geometric phase is applied to both wavelengths in the same manner and amount [51][52][53] . When left-handed circularly-polarized light (LCP) is incident to the rectangular dielectric nanorod, the Jones vector of the transmitted wave can be expressed as 54 where T L and T S represent complex transmission coefficients of the nanorod along the long and short axis, respectively, α is the rotation angle of the nanorod, and the argument of (T L −T S )/2 represents the propagation phase of the transmitted right-handed circularly-polarized light (RCP). For example, if we express a phase pair of a certain nanorod as (φ 1 , φ 2 ) which means that this nanorod generates a propagation phase of φ 1 for one wavelength and φ 2 for the other wavelength, then other arbitrary phase pairs of (φ 1 + 2α, φ 2 + 2α) can also be represented by rotating the nanorod by angle α (Fig. 1b) Fig. 1 Operation schematic of the wavelength-decoupled metasurface. a By illumination of green (λ = 532 nm) and red (λ = 635 nm) light, multicolor holographic images can be generated from the metasurface. It features a dual phase response embedded in a single nanostructure that enables independent phase control for two wavelengths. b Illustration of the design principle. The lower white line represents an arbitrary cross-polarization propagation phase curve of a certain dielectric nanorod while φ 1 and φ 2 indicate specific values at wavelengths of λ 1 and λ 2 , respectively. A phase pair of this state can be represented as (φ 1 , φ 2 ). If this nanorod rotates by the angle α, the phase curve moves parallel by 2α. Therefore, the phase pair becomes (φ 1 + 2α, φ 2 + 2α); hence, any phase pairs in which phase difference equals to (φ 1 − φ 2 ) can be represented by one nanorod with appropriate rotation angles. Dashed lines are to guide the eye the rotation direction and on the incident polarization. By carefully choosing three values of φ 1 , φ 2 and α, one structure with appropriate rotation angles can cover m phase pairs. Hence, the total number of structural configurations is reduced from m 2 to m. Here, we use the phase step of π/3 which is the case of m = 6, and two wavelengths of 532 nm and 635 nm.
To cover the whole phase range from 0 to 2π for both wavelengths, we need six-kind structures in which propagation phase difference (PPD) of two different wavelengths varies linearly at intervals of π/3. Hydrogenated amorphous silicon (a-Si:H) based rectangular nanostructures are used as unit structures (Fig. 2a), and the a-Si:H benefits from lower optical loss than typical amorphous silicon in the visible (Fig. 2b). Rigorouslycoupled wave analysis is used to calculate the PPD for the two wavelengths (Fig. 2c), and we have six-kind rectangular structures with PPDs that are linearly spaced from −π to 2π/3 (Fig. 2d). Cross-polarization transmission amplitudes are also calculated for both wavelengths (Fig. 2e, f). Although the conversion efficiency is not high enough for practical applications due to the optical loss of a-Si:H, it is sufficient to demonstrate our principle. Other structure sets can also be used if they satisfy the linear relationship of PPDs because absolute values of PPDs do not affect to the operation.
Propagation phase plots of the six structures demonstrate the phase pairs generated by each structure with its rotation. When two wavelengths have the same incident polarization state, structure rotation moves two phase values in the same direction. Therefore, phase pairs that differ by PPDs of a certain nanorod can be represented using this nanorod with an appropriate angle α. For example, S 1 nanorod produces propagation phases of −π/3 at λ = 532 nm, and of 2π/3 at λ = 635 nm without rotation ( Fig. 3a). Its phase pair can be represented as (−π/3, 2π/3). When this nanorod rotates by π/6, the phase pair changes to (0, π) because each phase value increases by π/3 (Eq. 1). Other phase pairs of (π/3, 4π/3), (2π/3, 5π/3), (π, 2π) and (4π/3, 7π/3) can be also represented as the S 1 nanorod rotates, so it produces 6 of the total 36 phase pairs. In the same manner, the other five structures from S 2 to S 6 generate the remaining 30 phase pairs ( Fig. 3b-f); hence, total 36 phase pairs can be achieved using six structures with their rotation ( Table 1).
Demonstration of multicolor metahologram. A multicolor metahologram is demonstrated to verify the calculation result. The typical Gerchberg-Saxton algorithm with a phase step of π/3 is used to derive two individual phase profiles for Fourier holograms corresponding two different wavelengths 55 . Original images and their phase profiles all have pixel numbers of 1200 by 1200. By comparing two phase profiles, necessary phase pairs for each pixel can be derived, and all the pairs belong to the 36 phase pairs that we have already calculated. The final unit cell design for the multicolor metahologram can be obtained by locating the structures with appropriate rotation on each pixel.
A fabricated metahologram produces the multicolor holographic images. Conventional electron beam lithography is used to fabricate our metahologram (Fig. 4a). Two laser beams, in which wavelengths are 532 nm and 635 nm, respectively, are incident on our metahologram (Fig. 4b). Both beams have the same circular polarization state before incidence. Two different images are encoded in our metahologram (Fig. 4c, d), and a multicolor image can be represented by superposition of two monochromatic images (Fig. 4e). We choose two images that occupy both sides of the zeroth order beam in the image plane in order to show that our multicolor metahologram can generate holographic images throughout the whole image plane without the space loss. Experimentally demonstrated holographic images agree well with both monochromatic simulation images (Fig. 4f,  g), and the multicolor image can be constructed by simultaneous illumination of two laser beams (Fig. 4h). The intermediate colors such as yellow and orange are represented by the intensity ratio between two monochromatic images. Our approach can be a promising metasurface design platform overcoming fundamental constraints of conventional multicolor metaholograms that are based on interleaved subarrays [36][37][38][39][40] and multiple positions [41][42][43] . The interleaved metasurfaces are comprised of spatial integration of a few metasurfaces. Since each structure of the interleaved metahologram has only one monochromatic phase response, the number of structures that produce one holographic image is always less than the total number of structures. This causes pixel loss reducing image fidelity. The fundamental efficiency is also limited by structures that have zero conversion efficiency. Since incident light on these structures cannot contribute to form holographic images, these structures are no more than an optical screen except for each target wavelength. In metaholograms based on multiple positions, undesired images must appear at the same image plane in Fourier holography, or different image planes in Fresnel holography. These images cannot be removed or darkened in principle because this type of metaholograms can have only a single phase profile; hence, the available space and fundamental efficiency are limited. In contrast, our metahologram possesses a dual phase profile corresponding to two different wavelengths, so every structure of our metahologram contributes to form all holographic images without pixel loss and image degradation. Moreover, our metahologram has no fundamental limitations on space and efficiency because holographic images appear at desired positions only.
Using sophisticated structures can also be beneficial to improve the efficiency because the single rectangular structures significantly limit our choices of the unit cell configuration to obtain desired propagation phase gradients with high conversion efficiency. Various kinds of sophisticated structures have been investigated to satisfy one's requirements not only in metaholograms, but also other metasurfaces such as metalenses. For example, Chen et al. used multiple nanorods to realize achromatic metalenses 35 , and Wang et al. demonstrated the achromatic metalens using inverse nanostructures for the desired dispersion 3 . We can also use optimization algorithms to figure out structure designs to meet our requirements such as phase dispersion or efficiency 57,58 . Any of these approaches are not exploited in this work because single rectangular nanorods are enough to satisfy our phase requirements. Our metahologram suffers from the low efficiency because we did not fully explore the design space yet, not because of any fundamental limitations; hence, the efficiency of our metasurface can be further improved using these approaches.
In conclusion, we present the wavelength-decoupled metasurface design principle based on the propagation phase associated with the geometric phase to embed independent dual-wavelength functions into a single nanostructure. The experimental demonstration of the multicolor metahologram based on this principle verifies the feasibility of our method that can be a significant breakthrough overcoming fundamental limitations in multicolor metaholograms.

Methods
Metasurfaces fabrication. The metasurface was fabricated using electron beam lithography 59 . A 300-nm-thick layer of a-Si:H was deposited on a glass substrate at 300°C by plasma-enhanced chemical vapor deposition. The flow rates were 10 sccm for silane gas and 75 sccm for hydrogen gas; the deposition rate was 1.3 nm s −1 . A positive resist (495 PMMA A2, MICROCHEM) was spin-coated by 2000 rpm on the a-Si:H resulting the resist thickness of 50 nm. Then the sample was baked on a hot plate at 180°C for 5 min. Conductive polymer was also spincoated by 2000 rpm on the resist to prevent charge accumulation during electron beam exposure. Then electron beam lithography was used to draw patterns on the resist. After the exposure, the sample was immersed into deionized water for 1 min to remove the conductive polymer, and development solution (MIBK:IPA=1 reaction speed enabling precise control of final pattern shapes. A 30-nm-thick chromium (Cr) was deposited on the sample by electron beam evaporation. Then, the sample was immersed into acetone for lift-off to define a Cr mask on the a-Si:H thin film, which was then etched along the Cr mask using inductively coupled plasma reactive ion etching. The mask residue was removed using Cr etchant. Full scanning electron micrographs of the final metasurface can be found in Supplementary Fig. 7.
Materials characterization. The optical property of the a-Si:H thin film was measured using ellipsometry. A silicon substrate with 100-nm-thick silicon dioxide layer on the top was used to increase measurement accuracy. The incident beam angle was 65°. We used the Tauc-Lorentz model which is suitable for absorptive amorphous material. The imaginary part of the dielectric function of the Tauc-Lorentz model is given by where A is the strength of the absorption peak, B is the broadening term, E 0 is the energy position of the absorption peak, and E G is the optical band gap energy. We used fitting parameters of A = 101.2592 eV, B = 1.247 eV, E 0 = 4.137 eV, and E G = 1.628 eV.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.