All-dielectric Schrödinger colours across the visible spectrum

Domain: physics
Published: 2025-12-05
Editorial outcome: major_revision

Abstract
Structural colours, which arise from nanostructures, are pervasive in nature and play a pivotal role in numerous modern applications. Traditional approaches usually deviate from the concept of Schrödinger’s pixels and thus have restricted performance in colour gamut and chiaroscuro presentation. Here, we harness the interactions between optical modes to achieve an unprecedented structural colour using SiO2-Si3N4 stacked metasurfaces. The mode interactions produce a high-reflection band formed by two Fano resonances with near-unity reflectance and sharp edges. The destructive interference between the tails of resonances and multiple reflections at the interfaces effectively suppresses the background reflection, creating an almost ideal stopband. Consequently, nearly perfect Schrödinger colours have been experimentally realized across almost the entire visible spectrum, setting a new colour gamut record of 1.06 times Rec. 2020. By introducing local size deviation, higher-order diffraction emerges and the colour brightness has also been continuously adjusted for direct chiaroscuro presentation for the first time.

Introduction
In the Diamond Perfection of Wisdom Sutra, the Buddha told Subhuti that “Everything with form is illusory; if all forms are viewed from their essence, the Tathagata will be perceived.”1 Colour is one prominent example. In appearance, it carries the main information and makes our world colourful. Essentially, different brilliant colours are closely related to light-matter interactions such as the inherent absorption, radiation, dispersion of materials, and wavefront changes in micro- or nano-structures2–5. The latter, also known as structural colour, was first discovered by Robert Hooke in 1665 and has been explored for centuries to explain the amazing colours of nature, e.g., birds, butterflies, beetles and even marbled berries6–9. With the advancement of nanofabrication technology, the colours of artificial nanostructures have been developed rapidly10–18. Compared with its counterparts, nanostructures can produce bright and vivid colours without additional energy consumption, so it is very attractive and promising in many practical applications such as anti-counterfeiting, display and many other fields19–24.

From an application perspective, highly saturated red, green, and blue colours are crucial, depending on the reflectivity and colour gamut of the nanostructures. Over the past decade, great efforts have been made in this regard. By replacing the metallic nanostructures with loss-free dielectric ones, the brightness of the structural colour has been significantly improved25–29. Meanwhile, the structural colour gamut has been further expanded by using stacked dielectric multilayer nanostructures or refractive index matching layers30–32. Despite the progress, the state-of-the-art colour gamut is still well below the Rec. 2020 standard (0.68- and 0.97-times Rec. 2020)30,31. This is because the optical response of conventional nanostructure deviates from Erwin Schrödinger’s considerations33, i.e., residual reflection in the stopband and the slow transition between passband and stopband greatly degrade the colour presentation34,35. While polarization-crossed resonance can solve the background problem, their transitions are still not sharp enough and an additional polarizer is required36,37. Very recently, Dong et al. ingeniously tackled this problem and realized the first Schrödinger red pixel by utilizing quasi-bound states in the continuum and the absorption of Si at short wavelengths38. However, their approach exhibited limited performance in the green and blue spectral regions due to the inherent short-wavelength absorption of Si, which consequently restricted the achievable colour gamut of the architecture. Although bound states in the continuum (BIC) employing lossless dielectric materials can also achieve the requisite passband, the fundamental realization of Schrödinger colour across the entire visible spectrum remains elusive due to limitations imposed by the requirement for zero reflectance outside the passband29. At the same time, a mechanism to directly control colour brightness is still missing so far. Here, we report a universal mechanism for constructing Schrödinger colours across the entire visible spectrum using lossless dielectrics. Record-breaking colour gamut (106% of Rec. 2020) and chiaroscuro presentation with direct brightness manipulation have been experimentally demonstrated.

Results
The working principle of all-dielectric Schrödinger colours
According to Schrödinger’s consideration, an ideal colour pixel should have the following three elements: near-unity reflectance in the desired spectral range, zero reflectance outside, and abrupt transitions between them33. In general, high reflection band and sharp edges can be obtained by adjusting the resonances in the nanostructure39,40. Broadband zero-reflection spectroscopy, however, remains a severe challenge, especially for the lossless dielectrics without the assistance of external polarizer41,42. Although anti-reflection coatings have been successfully achieved with transparent materials, their performance usually degrades when the refractive index increases43,44. Crucially, conventional designs, such as thin-layer stacks, graded-index nanostructures, and random nanostructures, are incompatible with the three requirements for high-brightness structural colours43,45.

In this research, we combine thin layer stacking and patterned nanostructures into a metasurface to achieve the desired Schrödinger colour (Fig. 1a). In the vertical direction, four layers of ITO, SiO2, Si3N4 and SiO2 are stacked sequentially on a glass substrate. Their thicknesses are h1=13nm, h2=54nm, h3=256nm, h4=110nm, respectively. In the transverse plane, a hexagonal lattice array of dielectric nanopillars is patterned in top two SiO2 and Si3N4 layers. Each nanopillar has an elliptical-shaped cross-section with major and minor axis length of a and b. The lattice size is p=381nm. A gradient descent algorithm as integrated with FDTD simulation package to design and optimize the geometric parameters of the metasurfaces has been developed and used to optimize the geometry toward achieving the CIE plot rim at three corners. Two cost functions, the distance on the CIE diagram from the achieved structural colour coordinate to the target Schrödinger colour position of the CIE plot and the maximized “CIEx or y coordinate” are used in optimization. The target colours positions correspond to a monochromatic light source with a full width at half maximum of 20nm at wavelengths of 460nm, 520nm, and 620nm in CIE plot. These two cost functions toggle from one to the other to avoid the optimization to be stuck at a local minimum. The detailed optimization methods and procedures are described in Supplementary Fig. 1. One example with a=350nm and b=220nm is depicted in Fig. 1b. Moreover, the resonance wavelength can be tuned across the entire visible region by scaling the period and geometric dimensions with a scaling factor S, which is set to S=1 in the example provided. A high reflection band with near unity reflectance and a FWHM of 20nm can be seen at ~515nm. As the incident wavelength deviates from the high-reflection region, the reflectance drops sharply to below 1% and remains so over almost the entire visible spectrum (the response shown in Fig. 1b corresponds to the orange solid line in Fig. 1c, both representing simulation reflection of the same structure). All such phenomena match the requirements of Schrödinger pixel very well. The corresponding coordinates in the CIE 1931 chromaticity diagram are thus very close to the ideal Schrödinger green (inset in Fig. 1b, with details of the CIE coordinates for blue and red provided in Supplementary Fig. 2), and this is the first time that a Schrödinger pixel can be realized using transparent materials.

To understand the formation of the nearly ideal Schrödinger colour, we decomposed the resonances in Fig. 1b and explored their respective contributions. In principle, the reflection spectra contain two parts, i.e., the radiation of the resonances and the reflection at interfaces. The latter one is strongly dependent on the thickness of each layer and their effective refractive indices (different from a uniform film). To produce high-purity colour, it is essential to suppress the reflection at non-resonant wavelengths. The decomposed spectra are plot in Fig. 1c. Two eigenmodes at 508nm and 520nm of the nanostructure in the visible region are crucial for constructing high reflection bands (see field distributions in Supplementary Fig. 3). These resonant states are leaky eigenmodes (guided-mode resonances) supported by the high-index Si3N4 layer and periodically coupled to free space. These discrete modes possess complex eigenfrequencies and finite radiative lifetimes, producing Fano-type interference with the non-resonant background. The excitation of a given resonant state depends on polarization: it can only be excited when the polarization of the incident light is non-orthogonal to the time-reversed radiation polarization of that state. Such resonant and polarization-selective behaviours are well established in dielectric nanoresonators and guided-mode metasurfaces25–29,39–41,46,47. Returning to the spectral construction, the constructed reflection spectrum (dashed green line in Fig. 1c) fits the two reflection peaks well, but the non-resonant reflectance is still much higher than the numerical simulation (orange solid line in Fig. 1c) within the spectral range from 475nm to 625nm, strongly spoiling the colour performances. This is because the contribution of multiple reflections of light at the interface, that is not coupled to resonances, has not been included by two eigenmodes. The solid blue line in Fig. 1c shows the reflection spectrum of the uncoupled field obtained by mode decomposition. Similar to the eigenmodes, it also has much higher reflectance than the simulation at non-resonant wavelengths. Consequently, the phases of different components play a vital role here. As depicted in Fig. 1d, the phase difference between the two eigenmodes and the uncoupled field ranges from 90 to 270 degrees over almost the entire visible region. Then the destructive interference between them effectively suppresses the reflection outside the high reflection band. Such interference can be clearly seen from the vectorial superposition in Fig. 1e, h, where the amplitude of the total reflection is greatly reduced to ~6.6 and 3.5 by the destructive interference between mode-1 and uncoupled fields. In the high reflection band, the interaction mainly occurs between two eigenmodes and produces total field more than an order of magnitude higher (Fig. 1f, g). Such interaction forms two Fano resonances in opposite directions, resulting in high reflectance and sharp edges simultaneously (insets in Fig. 1f, g). To further elucidate the spatial origin of the radiated fields presented in Fig. 1e–h, we note that the electric field components are derived from the backward radiation of both the resonant and uncoupled modes, evaluated on a reference plane (S1) situated above the metasurface (see Supplementary Note 18 for details). The radiated field of the extended state is calculated by spatially integrating the electric field over S1, E(rad)(ω) = 1/S ∫ E~r;ω d2r, and so are the resonant states’ radiation field E(rad)(1ω)1 and the uncoupled field’s radiation E(rad)(ω)0. Under x-polarized illumination, only resonant states emitting x-polarized radiation are excited. Then only the x-component of the backward field, E(rad)(ω)x = E(rad)(n,x)(ω) + E(rad)(0,x)(ω), is considered.

With the above analysis, we know that the coupling between different modes and their interaction with multiple reflections at the interfaces build the Schrödinger colour. Since no material absorption is used to suppress the stopband reflectance in Fig. 1, the mechanism of Schrödinger colour can be generalized and extended to other colours as well. By increasing the scaling parameter S from 0.8 to 1.2, the reflection spectra are optimized. The numerically calculated reflection spectra with x-polarization are summarized in Fig. 2a. To meet the needs of multi-colour monolithic manufacturing, here the thicknesses of layers in all samples are the same as Fig. 1. Although the wavelength of the high reflection band increases with the scaling parameter, near-zero reflection bands and sharp edges have been produced in all metasurfaces. As a consequence, our mechanism can achieve nearly ideal Schrödinger pixels in almost the entire visible light region. The spectra of different samples are plotted as dots in CIE 1931 chromaticity diagram (Fig. 2b). The colour performances of our metasurfaces have been improved close to the boundary. For a direct comparison, Rec. 2020, a standard for ultra high-definition television (UHDTV) since 2012 has also been plotted as dashed triangle in Fig. 2b. The colour gamut of our Schrödinger pixels is almost 120% of Rec. 2020, which is a new record of structural colour.

Considering the elliptical-shaped cross-section of the nanopillar, the reflection spectra and the corresponding structural colours exhibit polarization dependence. For y-polarized incident light, the reflection spectrum shows a red shift of the primary resonance. At the same time, the emergence of an additional peak at shorter wavelengths (as depicted in Supplementary Figs. 3 and 4) will reduces the colour saturation compared to the x-polarized case, though relatively bright structural colours are still achievable. Such polarization dependence can be utilized to realize versatile colours in one metasurface, which is essential for many practical applications such as sensing36,48. One example is illustrated in Supplementary Fig. 5. For x-polarized incident light, the colour can be continuously and directly tuned from 514nm to 568nm by rotating the axis of each meta-atom without the aid of an external polarizer.

More intriguingly, the proposed structural colours also hold the potential for chiaroscuro presentation, which is another challenge for colour nanoimprinting. Such a function is typically realized by cross-polarized resonances, but with the assistance of an external polarizer. Our Schrödinger colour can be completely different and provide a way to directly manipulate the colour brightness. As depicted in Fig. 2c, a size deviation parameter Δr can be locally introduced to one nanopillar. In this case, the hexagonal lattice changes to a rectangular supercell and higher-order diffraction thus occurs (see Supplementary Note 7 for details). Such diffractions take away some of the energy and reduce the brightness of the structural colours. Figure 2d summarizes the reflection spectra of our metasurface as a function of size perturbation when a single, global Δr is applied across the entire lattice. With the increase of Δr from 0 to 100nm, the reflectance gradually decreases and nearly vanishes at Δr~100nm. Meanwhile, the corresponding resonant wavelengths experience a slight blue shift relative to their initial positions due to the reduction in effective refractive index. Nevertheless, this shift can be readily compensated for by adjusting parameters such as the S or the minor axis b, thereby maintaining the resonance wavelength at the desired position. This controllable adjustment of brightness with stable hue well satisfies the requirements for direct chiaroscuro presentation (As depicted in Supplementary Fig. 18).

Experimental demonstration of all-dielectric Schrödinger colours
The designed metasurfaces have been experimentally fabricated on a 13nm ITO-coated glass substrate with a combined process of plasma enhanced chemical vapour deposition (PECVD), electron-beam (E-beam) evaporation, E-beam lithography, lift-off and inductively coupled plasma reactive ion etching (ICP-RIE, see details in Methods and Supplementary Fig. 8). Left column of Fig. 3a shows the top-view scanning electron microscope (SEM) images of the metasurfaces with different scaling parameters. The hexagonal lattice, elliptical atom, and the detailed size parameters follow the numerical design very well. For metasurfaces made of stacked different materials, a key technical challenge is to achieve vertical and smooth sidewalls. In our research, a two-step dry etching recipe has been developed. The top SiO2 layer is etched with a mixture of CHF3 (25sccm) and Ar (30sccm). This is a physical etching-dominated process that can effectively minimize undercutting under the mask. Then the Si3N4 layer is chemically etched with O2 (4 sccm) and CHF3 (37 sccm). During this process, a fluorocarbon polymer film is produced to passivate and protect the sidewalls of SiO2 and Si3N4. Meanwhile, the etching depth can be precisely controlled by the etching speed of 3nm/s. The tilt-view SEM images of the metasurfaces are shown as right column of Fig. 3a. It is easy to see that the sidewalls are straight and smooth with an angle of 86o. No obvious undercutting and roughness can be observed. This is completely different from conventional stacked metasurfaces30 and thus high-performance structural colours can be expected.

The characteristics of the metasurfaces are optically characterized with a home-built bright field optical microscope (see Methods). With an x-polarized incident light, the reflection spectra have been directly recorded without using an additional polarizer (Fig. 3b). Some samples in Fig. 3b exhibit only a single resonance mode, which we attribute to minor fabrication deviations. Nevertheless, this does not compromise the colour saturation, as confirmed by the results shown in Supplementary Fig. 15. For the case of S=1, two reflection peaks can be clearly seen at 504.21nm and 512.79nm, respectively (Panel-IV in Fig. 3b). The maximal reflectance is as high as 85% and the FWHM is only 16.25nm. Most importantly, the background reflectance at undesired wavelengths remains below 1% from 418.25nm to even 800nm (Supplementary Fig. 9). All these experimental observations are consistent with the numerical design very well. Slight difference in peak wavelengths arises from size deviations in nanofabrication. As a result, a bright and vivid green colour can be directly obtained in the brightfield microscope image (inset). The corresponding location is marked as a star in CIE 1931 chromaticity diagram (Fig. 3c), which is beyond the UHDTV triangle and very close to the ideal Schrödinger green colour. As a consequence, we know that high performance structural colour has been experimentally achieved using transparent materials.

Then the performances of all metasurfaces with different scaling parameters have been characterized. With the increase of scaling parameter from 0.8 to 1.2, the resonant wavelength continuously shifts from 462.94nm to 630.18nm, covering the colours from blue to red very well (Fig. 3b). The maximal reflectance gradually increases from 79.36% at blue colour to 89.20% at red colour. And the FWHM of the high-reflection band in all metasurfaces varies between 8.89nm and 16.27nm. The averaged FWHM and peak reflectance are thus around 83% and 13.39nm, respectively. Meanwhile, we find that the reflectance at non-resonant wavelengths of all metasurfaces are well below 1% in almost entire visible spectrum (Supplementary Fig. 9). Thus, nearly ideal different Schrödinger colours have been experimentally realized at many different wavelengths (insets in Fig. 3b).

Figure 3c summarizes the colour performances of 18 samples in CIE 1931 chromaticity diagram. All colours are distributed near the boundaries of the CIE 1931 map, and the blue and green are well beyond the Rec. 2020 area. As consequence, the colour gamut of our Schrödinger colours is about 106% of Rec. 2020. This is the first time that artificial structural colour has surpassed the performance of UHDTV in experiment. The FWHM, average reflectance, and colour gamut of all bright-field structural colours in literatures are summarized in Supplementary Fig. 10 for a direct comparison with our experimental results. It is apparent that our Schrödinger colour has obvious advantages in these indicators. Based on a similar process, we have fabricated 126 SiO2-Si3N4 stacked metasurfaces with smaller intervals of scaling parameters and experimentally recorded their colour performances. The corresponding results are summarized in Supplementary Fig. 11. Vivid rainbow colours are directly presented, indicating the potentials of stacked metasurfaces in full-colour printing. With all the above results, we can now confirm that our mode interaction mechanism can enhance the reflection of the designed wavelength and suppress the reflection of the non-resonant wavelength, thus producing nearly ideal Schrödinger colours in almost the entire visible spectrum. The corresponding reflection spectra and colour performances in CIE 1931 chromaticity diagram of all above samples with y-polarized incident light were also recorded and plotted in Supplementary Figs. 11 and 12. All the experimental results are perfectly consistent with the simulation too.

Similar to the numerical simulation, the potential of our structural colours in direct chiaroscuro presentation has also been investigated. Figure 3d shows red, green, and blue pixels with different size deviation parameters integrated across entire lattices. As Δr increases, the reflectance decreases by more than an order of magnitude, while the resonant wavelengths show a slight blue shift. This shift can be readily compensated by adjusting parameters such as the S or the minor axis b, thereby preserving a consistent hue. This possibility has been experimentally verified by fabricating 117 samples with different scaling parameters S and size deviation parameters Δr. The recorded colours are summarized in Fig. 3e. We can see that S and Δr determine the colour and its brightness respectively. All colours can be adjusted gradually from bright to near-dark by tuning Δr. We emphasize that no additional polarizer is employed in our scheme. Although the reflectance cannot be reduced to absolute zero—unlike in polarization-based extinction methods—the measured reflectance of our optimized low-reflection state is suppressed below 3% at the resonance and below 1% in off-resonant bands (see Supplementary Fig. 17 for details), which is sufficiently dark for high-contrast display applications. The key advantage of our approach lies in its direct control of brightness via structural tuning, which significantly simplifies the device architecture and improves optical efficiency compared to polarizer-dependent systems.

High-performance colour printing
Compared with Ref. 38, the nearly ideal structural colours arise from the mode interactions instead of the BIC. As a consequence, it doesn’t require an infinite size in at least one dimension and thus is applicable to small pixels. To verify this possibility, we have fabricated a series of metasurfaces with different pixel sizes and characterized their colour properties. All the results are summarized in Supplementary Fig. 13. We can see that the designed structural colours and the corresponding reflection spectra can still be well preserved with the reduction of pixel size from 8 μm of 3 μm. This information, associated with the colour and brightness control, enables the generation of high-quality colourful images with the near ideal Schrödinger pixels.

To demonstrate high-resolution colour printing, we designed a monochrome sketch (the original design is shown in Supplementary Fig. 21a) as a grayscale pattern. In experiment, structural colours with 70 precisely controlled brightness steps (with size deviations) have been realized to mimic the intensity variations (see bottom insets in Fig. 4(a)) and construct the designed Sketch. The total size of the sample is 1mm × 1mm, containing ~7,864,320 nanopillars. Figure 4a shows the bright-field microscope image of the sample under the illumination of a white-light source. While no additional polarizer is used, we can see that the designed image has been successfully reproduced by our metasurface pixels. The enlarged figure of the sample is measured by a 100× objective lens and shown as top-inset in Fig. 4a. Both the brightness variation and the detailed features match the design very well, clearly demonstrating the capability of our metasurface colour pixel in direct chiaroscuro presentation.

By simultaneously tuning the scaling parameter and size deviation, the Si3N4-SiO2 stacked metasurface can also be employed to construct a colourful image with fine brightness differences. For the case of Schrödinger colours, we designed an image of CAT (the original design is provided in Supplementary Fig. 21b), and constructed it with ~80,771,452 Si3N4-SiO2 stacked nanopillars. Under the illumination of white-light source, the bright-field image of the sample is almost identical to the design (Fig. 4b). With a 100× objective lens, the detailed variations in both colours and brightness of the enlarged starry night have been reproduced too (insets in Fig. 4b). Since no extra polarizer has been utilized in our experiments, we can confirm now that this new approach can be an all-in-one solution for high-performance colour nanoprinting.

Methods
Numerical simulation
The simulations of the reflectance spectrum for the metasurface is based on periodic boundary conditions under plane wave illumination. The simulation the commercial finite element method (FEM) software COMSOL Multiphysics and Lumerical FDTD Solutions for simulation. Perfectly matched layers are applied in the transmission and reflection directions to absorb the outgoing waves. The refractive index of all materials is taken from the experimental result by ellipsometer. A Python code is used to calculate the xy coordinates, which are plotted in the International Commission on Illumination (CIE) 1931 chromaticity diagram (Supplementary Fig. 14). The resonance contributions are analyzed with the mode decomposition (see Supplementary Note 4).

Sample fabrication
The SiO2-Si3N4 stacked metasurfaces are fabricated with a combined process of plasma-enhanced chemical vapour deposition (PECVD), electron-beam (E-beam) evaporation, E-beam lithography, lift-off and inductively coupled plasma reactive ion etching (ICP-RIE). Briefly, amorphous multi-layer SiO2/Si3N4/SiO2 films were grown onto the 13nm ITO-coated glass using the PECVD method. After that, 120-nm-thick PMMA A2 was spin-coated on the sample surface, followed by EBL and lift-off processes. The thicknesses of ITO, SiO2, Si3N4, and SiO2 are measured by a spectroscopic ellipsometer and the fitting result is 13nm, 54nm, 256nm and 110nm. The Cr pattern after lift-off is used as hard mask for SiO2 and Si3N4 etching. A new etching method is developed to etch SiO2 and Si3N4 to ensure that the nanostructures have a very good verticality. The detailed fabrication process is described in Supplementary Note 8 with corresponding process parameters.

Optical measurement
The metasurfaces are placed onto an optical microscope (ZEISS, Axio Scope A1) stage associated with a home-made optical setup to control the polarization. The reflection spectrum is recorded with a spectrometer under the ×5 objective lens (NA = 0.12).

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript proposed and experimentally demonstrated a metasurface composed by transparent materials to construct\nhigh quality Schrödinger colours. The authors reveal that mode interactions in stacked nanostructures can effectively\nenhance the reflectance of resonant modes and suppress reflections in non-resonant spectral regions. The fabrications are\ngood and the experimental results are nicely produced. However, given the recent advances in the field of BIC color\nmatasurface, the manuscript could be considerable for Nature Communications after the authors addressing the following\ncomments:\n1. According to the definition of the CIE 1931 coordinate, the authors need to re-calculate the CIE coordinates in Figure 2b\nand corresponding gamut based on the full spectrum data from 380 nm to 780 nm.\n2. The author stated that “The reflectance keeps decreasing by more than an order of magnitude with the increase of Δr,\nwhereas the resonant wavelengths are well remained.” However, from the results in Fig.3e and Fig.S6, it is obvious that both\nthe brightness and color hue are determined by Δr. The resonant wavelengths are NOT remained when Δr is altered. In\naddition, the brightness could NOT be decreased to Zero with this method. As a result, the chiaroscuro control in this\nmanuscript can NOT be comparable to Refs. 32, 33.\n3. All the experimentally recorded reflection spectra of the metasurfaces with different Schrödinger pixels exhibit a single\nsharp peak. But all the simulated reflection spectra of different metasurfaces have two peaks, as explained by two\neigenmodes. What causes such discrepancy?\n4. What is the viewing angle tolerance of this metasurface?\n5. The author stated that no additional polarizer is an advantage of such transparent- materials metasurfacce. My concern is\nthat there will be broadband reflectance from the samples without nanostructures. Figure 3e showcase experimentally\nrecorded structural colours under a bright field microscope. 117 samples with different scaling parameters and size deviation\nparameters are fabricated to show different colour hues and brightness with a black background. This presentation approach\nhides the background reflectance problem and it is misleading to the readers to some degrees.", "reviewer_2": "In this paper the authors propose and experimentally realize transparent nanostructures that exhibit very high-quality colours\nacross the whole visible spectrum. Mode interactions between resonant modes produce high reflectance and, at the same\ntime, reduce reflections in non-resonant spectral regions. These structures can be implemented even in small size pixels of\n3 microns. Besides, they allow fine control of the brightness without the use of extra polarizers.\nI consider the topic and the reported findings are relevant. However, there are certain points that should be addressed before\npublication.\nIn my opinion, the physical mechanism underlying the singular performance of the proposed nanostructure is not completely\nclear. This is of vital importance to replicate this effect in different types of nanostructures. The authors explain that the\nreported effect is based on the interaction between resonant modes, and in the Suppl. Material they derive the formalism for\nmode decomposition. However, the analysis made on their respective contributions, is not clear. The authors should expand\nthe explanation of their approach, its assumptions and validity limits, in order to validate the analysis in terms of mode\ncontributions. Only with a clear explanation of the physical mechanisms underlying the generation of the Schrodinger\ncolours this article has a real value.\nIn what follows I list minor comments that should also be addressed before publication:\n- line 115: the definition of the scaling factor S is missing.\n- There is a big confusion about the curves shown in Fig. 1. In lines 128-130 the authors speak of “constructed reflection\nspectrum”, “numerical simulation” and “non resonant reflectance”. It is not clear how each of these curves is obtained, and\nhow do they correlate with the legend in Fig. 1(c).\n- Is the curve in Fig. 1(b) the same as the orange one in Fig. 1(c)? If this is correct, it should be said explicitly, as the change\nof scale in Fig. 1(c) makes it difficult to realize of that.\n- lines 133-135 describing Fig. 1(c) are also not clear. What’s the physical meaning of “reflection spectrum of the uncoupled\nfield”? Also, the correlation between the sentence “Similar to the eigenmodes, it also has much higher reflectance than the\nsimulation at non-resonant wavelengths.” and the curves in Fig. 1(c) is not clear. The authors should explain more clearly\nand explicitly the details of such curves.\n- line 173: the authors state that, according to Fig. S4, “the y-polarized reflection peaks slightly redshift without destroying the\nbright and vivid colors”. However, in Fig. S4 it is evident that additional peaks with significant intensity appear at shorter\nwavelengths, which are expected to influence the obtained colors. Moreover, in the Supp. Mat. the authors even say so and\nadmit that these peaks destroy the high quality of the colors. Therefore, the authors should be honest and explain clearly the\ndegree of influence of these additional peaks in the reflected color, within the main text.\n- lines 184-187: the authors state that diffraction effects reduce the brightness of the structural color. However, the underlying\nphysical mechanism responsible for this effect is not clear at all. The authors should explain more deeply how diffraction\ntakes place in this system, and its role regarding the color generation. Also, a more explicit sentence should be given in the\nConclusions section (see lines 329-330).\n- line 189: it says Δr = 100 nm, but in Fig. 2(d) the axis showing Δr is in percentage and not in units of nm. Check.\n- In Fig. 3(d) the legend is missing. Although the caption says that it shows spectra as a function of size deviation parameter\nΔr, the meaning of the different shades in each panel of Fig. 3(d) is not clear.\n- In Fig. 3(e) it can be observed that the brightness cannot be completely controlled by varying Δr, especially for several\nhues. Moreover, in some cases the hue changes significantly. For instance, in column 9 the hue changes from dark orange\nto green. The authors should comment on this behavior and on how this tendency could be overcome.\n- line 287: In Fig. S12 it is shown that the peak intensity decreases significantly as the pixel size decreases. This should\ninfluence the colour brightness, although the authors do not mention this.", "reviewer_3": "This manuscript reports on the high performance of color pixels achieved by employing dielectric photonic crystal slab\nstructures. The work is well organized and state-of-the-art, combining numerical analysis with experimental demonstration.\nHowever, It appears that suppression of sidebands is the main contribution for achieving Schrodinger colors, but this\nremains ambiguous without quantification of the effects. Additionally, some major questions must be addressed for\nacceptance in this journal. All answers to those questions should be clearly explained in revised manuscript.\nQ1. Figure 1 seems primarily designed to explain sideband suppression. In Fig. 1c, the estimation shows that the resonance\nmode reduces reflectance outside the resonance by as much as 2% and as little as about 0.1%. How much does this\ncontribute to the implementation of the Schrodinger pixel? Also, the explanation in the main text for Fig.1e–1h is insufficient.\nFrom the Supporting Information, it appears that the radiated E-field components are calculated for each decomposed\nresonance mode and background mode. The main text needs to include an explanation of where spatially the radiated E-\nfield components originate for clear understanding.\nQ2. This focuses primarily on the interaction between resonance modes and uncoupled modes, but it lacks a fundamental\nexplanation of the resonance itself. This is critical and should be appropriately discussed in the Introduction with proper\nreferences.\nQ3. In the Introduction, the statement \"Their approach, however, suffers from noticeable performance degradation at green\nand blue colours [34]\" may be misleading. Reference 34 focuses on the red color, not the full color range. It cannot be\nconcluded whether the method using quasi-BIC is efficient; it may have the potential to be effective for blue and green colors\nwith proper design or adjustment of the scaling factor.\nQ4. The authors emphasized the performance achieved by comparison with Rec.2020 UHDTV, which has been done in\nsome articles in this research field. This proposed technology is based on reflectance type, but Rec. UHDTV is mostly\ntransmittance type. The author should explain the performance considering this difference, as well as any related limitations\nor drawbacks.\nQ5. Regarding the hexagonal array, what is the main reason for its use? Due to the symmetry property, it should be\npolarization-dependent. Thus, the authors emphasize the performance mainly for x-polarized input in the main text, rather\nthan for all polarizations. Also, to control brightness, the polarization dependence is likely to increase with delta_r; data or\nexplanation supporting this should be provided.\nQ6. In Fig. 2a, it can be seen that the resonance wavelength is tuned by the scaling factor, causing two modes to shift\ntogether. In the experimental results of Fig. 3b, what is the reason that only one mode appears for some colors?\nQ7. In Fig. 4b, the picture has a yellow tint. This study is about colorful pixels, but the example does not reflect that. In the\nsection \"High-performance colour printings,\" the authors should present examples of high-performance colour printings.\nQ8. What is the red dot on the left side of the scale bar in Fig. 4b?"}, "author_response": "Reply to reviewers’ reports\nWe thank Reviewers much for many constructive comments and useful suggestions.\nWe have carefully revised the manuscript and improved the technical quality of our\nmanuscript making it more appealing for general readership. Please find below our\npoint-to-point responses to Reviewers’ comments. All the changes in the revised\nmanuscript and highlighted below are marked with blue color.\nResponse to Reviewer #1\nThis manuscript proposed and experimentally demonstrated a metasurface composed\nby transparent materials to construct high quality Schrödinger colours. The authors\nreveal that mode interactions in stacked nanostructures can effectively enhance the\nreflectance of resonant modes and suppress reflections in non-resonant spectral regions.\nThe fabrications are good and the experimental results are nicely produced. However,\ngiven the recent advances in the field of BIC color metasurface, the manuscript could\nbe considerable for Nature Communications after the authors addressing the following\ncomments:\nOur Response: We are deeply grateful to Reviewer #1 for his/her highly positive\nassessment of our manuscript and for the supportive recommendation regarding its\npotential suitability for Nature Communications following revisions. We sincerely\nappreciate the recognition of our experimental fabrication quality and results.\nWe have carefully considered all the comments provided and have undertaken extensive\nrevisions to address them thoroughly. Our revisions include additional experimental\ndata, modifications to the manuscript, and the addition of supplementary information.\nWe hope that these significant enhancements have substantially improved the\nmanuscript's clarity, rigor, and overall impact.\nOur detailed point-by-point responses to each comment are provided below. We have\nincorporated all corresponding changes into the revised manuscript and believe it can\nmeet the high standards expected for publication.\nComment-1：According to the definition of the CIE 1931 coordinate, the authors need\nto re-calculate the CIE coordinates in Figure 2b and corresponding gamut based on the\nfull spectrum data from 380 nm to 780 nm.\nOur Response: We thank the reviewer for raising this important point regarding the\ncalculation of the CIE 1931 color coordinates. We agree that a rigorous calculation\nbased on the full visible spectrum is essential. Following the reference (Fairman, H. S.;\nBrill, M. H.; Hemmendinger, H. (February 1997). \"How the CIE 1931 Color-Matching\nFunctions Were Derived from the Wright–Guild Data\". Color Research and Application.\n22 (1): 11–23), the calculation of CIE coordinates for reflectance spectra is similar to\nthat for emission spectra, using the reflectance spectrum 𝑅(𝜆) to replace the emission\nspectrum and multiplying it by the spectral power distribution of the illuminant 𝑆(𝜆).\n780\n𝑋 = 𝑘 ∫ 𝑆(𝜆)𝑅(𝜆)𝑥̅(𝜆)𝑑𝜆\n380\n780\n𝑌 = 𝑘 ∫ 𝑆(𝜆)𝑅(𝜆)𝑦̅(𝜆)𝑑𝜆 (𝑅1)\n380\n780\n𝑍 = 𝑘 ∫ 𝑆(𝜆)𝑅(𝜆)𝑧̅(𝜆)𝑑𝜆\n380\n𝑥̅(𝜆),𝑦̅(𝜆),𝑧̅(𝜆) in Eq. (R1) are called the color-matching functions, as shown in Fig.\nR1(a), where k constant is chosen for a normalized value Y = 100. Base on the XYZ\ntristimulus values, we can define the chromaticity coordinates x y, as:\n𝑋 𝑌\n𝑥 = ; 𝑦 = (𝑅2)\n𝑋+𝑌+𝑍 𝑋+𝑌+𝑍\nFig. R1 (Figure S14, Supplementary Information). Recalculation of CIE 1931\ncolor coordinates. (a) The CIE 1931 standard color-matching functions. (b) A set of 33\nrepresentative simulated reflection spectra from the designed metasurface. (c) The CIE\ncoordinates corresponding to the spectra in (b).\nOur original color coordinate calculations were performed using a normalized equal-\nenergy illuminant (S(λ) = 1), integrated over the full visible spectrum (380–780 nm).\nFigure 2b is calculated using the above definition of CIE 1931 coordinate, eleven\nmetasurfaces, including seven designs in Fig. 2a, with resonances at different\nwavelength is plot and have been close to the boundary. To verify the larger colour\ngamut of the Schrödinger pixels, Fig. R1(b) presents 33 representative reflection spectra\nfrom our structures, covering resonance wavelengths from 440 nm to 650 nm, all\nexhibiting strong resonance peaks and suppressed sidebands. The chromaticity\ncoordinates for these spectra, plotted in the CIE 1931 diagram in Fig. R1(c), are\nconsistent with those originally shown in Fig. 2(b) of the manuscript.\nThe above information has been added as a new section of “Supplementary Note 14:\nCalculate method of the CIE 1931 coordinate” in the revised Supplementary\nInformation.\nComment-2：The author stated that “The reflectance keeps decreasing by more than\nan order of magnitude with the increase of Δr, whereas the resonant wavelengths are\nwell remained.” However, from the results in Fig.3e and Fig.S6, it is obvious that both\nthe brightness and color hue are determined by Δr. The resonant wavelengths are NOT\nremained when Δr is altered. In addition, the brightness could NOT be decreased to\nZero with this method. As a result, the chiaroscuro control in this manuscript can NOT\nbe comparable to Refs. 32, 33.\nOur Response: We thank the reviewer for pointing out this critical comment. The\nreviewer is absolutely correct that with the increase of Δr, the resonant wavelength will\nshift slightly as shown in Fig. 3e. This is because the effective refractive index of the\nmode changes with the increase of Δr, resulting in a slight blue-shift of the resonance。\nHowever, the blue-shift of the resonance can be suppressed by adjusting other geometry\nparameters such as scaling factor (S) or short axis (b). The main idea is to maintain the\neffective refractive index of resonance mode. For example, when a larger Δr is used to\nlower the brightness, a slight increase in S or b will induce a compensating redshift,\npreserving the original color tone. This method of coordinated adjustment successfully\ndecouples the control of the two variables, as shown in Fig. R2.\nFig. R2 (Figure S18, Supplementary Information). Reflection spectrum with\ndifferent brightnesses under fixed hue.\nWe agree with the reviewer that our polarization-free method cannot reduce reflectivity\nto zero, unlike the principle of complete extinction in crossed polarizers in Refs. 32-33.\nHowever, the practical advantage of our method is its ability to achieve a reflectance\nlevel that is visually imperceptible—the true standard for a \"black state\" in displays.\nAs evidenced in Figure R3, our samples exhibit reflectance below 1% in non-resonant\nregions and in resonance regions stay under 3% when Δr = 100 nm. This performance\nis negligible compared to conventional samples and fully meets the requirements for a\nhigh-quality dark state. The principal contribution of our work is a brightness control\nmethod that enables independent, continuous tuning without relying on extra polarizers.\nThis capability offers a significant advantage: a simpler architectural design and\nsuperior optical efficiency compared to polarizer-based schemes.\nFig. R3. Comparison of reflection spectra. Experimentally measured reflection\nspectra of a standard sample and a low-reflection sample.\nIn the revised manuscript, we have corrected the inaccurate statement regarding the\n“resonant wavelengths are well remained”. This clarification has been provided in\nPara-2, Page-6, and Para-3, Page-9, with the revised text reading as follows:\n“Such diffractions take away some of the energy and reduce the brightness of the\nstructural colours. Figure 2(d) summarizes the reflection spectra of our metasurface as\na function of size perturbation when a single, global Δr is applied across the entire\nlattice. With the increase of Δr from 0 to 100 nm, the reflectance gradually decreases\nand nearly vanishes at Δr ~ 100 nm. Meanwhile, the corresponding resonant\nwavelengths experience a slight blueshift relative to their initial positions due to the\nreduction in effective refractive index. Nevertheless, this shift can be readily\ncompensated for by adjusting parameters such as the S or the minor axis b, thereby\nmaintaining the resonance wavelength at the desired position. This controllable\nadjustment of brightness with stable hue well satisfies the requirements for direct\nchiaroscuro presentation (As depicted in Fig. S18).”\n“Similar to the numerical simulation, the potential of our structural colours in direct\nchiaroscuro presentation has also been investigated. Figure 3(d) shows red, green, and\nblue pixels with different size deviation parameters integrated across entire lattices. As\nΔr increases, the reflectance decreases by more than an order of magnitude, while the\nresonant wavelengths show a slight blueshift. This shift can be readily compensated by\nadjusting parameters such as the S or the minor axis b, thereby preserving a consistent\nhue.”\nIn the revised manuscript, we also have added a clarification regarding the inability to\nachieve zero brightness in Para-3, Page-9. The specific revised text is as follows:\n“This possibility has been experimentally verified by fabricating 117 samples with\ndifferent scaling parameters S and size deviation parameters Δr. The recorded colours\nare summarized in Fig. 3(e). We can see that S and Δr determine the colour and its\nbrightness respectively. All colours can be adjusted gradually from bright to near-dark\nby tuning Δr. We emphasize that no additional polarizer is employed in our scheme.\nAlthough the reflectance cannot be reduced to absolute zero—unlike in polarization-\nbased extinction methods—the measured reflectance of our optimized low-reflection\nstate is suppressed below 3% at the resonance and below 1% in off-resonant bands (see\nFig. S17 for details), which is sufficiently dark for high-contrast display applications.\nThe key advantage of our approach lies in its direct control of brightness via structural\ntuning, which significantly simplifies the device architecture and improves optical\nefficiency compared to polarizer-dependent systems.”\nComment-3：All the experimentally recorded reflection spectra of the metasurfaces\nwith different Schrödinger pixels exhibit a single sharp peak. But all the simulated\nreflection spectra of different metasurfaces have two peaks, as explained by two\neigenmodes. What causes such discrepancy?\nOur Response: We thank the reviewer for raising this important point regarding the\ndiscrepancy between the simulated double-peak spectra and the experimentally single-\npeak spectra. The disappearance of the second peak in experiments is primarily\nattributable to fabrication deviations.\nAs illustrated in Fig. R4(a), two primary deviations will usually happen during the\nfabrication process, one is the radius variations (especially the short-axis radius of the\nellipses) and another one is the over-etching depth. To systematically demonstrate this,\nwe simulated the impact of each deviation for a representative structure with S=1.\nFigure R4(b) demonstrates that a slight radius deviation is sufficient to drastically\nsuppress Mode 2. By comparison, the effect of over-etching alone, shown in Fig. R4(c),\nis relatively minor. This indicates that Mode 2 is far more sensitive to dimensional\nvariations than Mode 1. Crucially, Fig. R4(d) shows that when both deviations are\npresent—a realistic scenario—their combined effect causes the complete disappearance\nof the more sensitive Mode 2, leaving only the robust Mode 1. This finding directly\nexplains the single-peak spectra observed in our experiments.\nFig. R4 (Figure S15, Supplementary Information). Fabrication Tolerance of\nResonant Modes. (a) Main fabrication deviations: radius deviation and over-etching\ndepth. (b, c) Individual impact of (b) radius deviation and (c) over-etching on the\nreflection spectrum for structure with S=1. (d) Simulated reflection spectrum from\ncombined deviations.\nThe above information has been added as a new section of “Supplementary Note 15:\nThe effects of fabrication deviations” in the revised Supplementary Information.\nComment-4：What is the viewing angle tolerance of this metasurface?\nOur Response: We thank the reviewer for this important question. We systematically\nevaluated the angular tolerance through both simulation and experiment, using\ntransmittance spectra for experimental convenience. As shown in Fig. R5(a, b), the\nresonant wavelength redshifts by approximately 20 nm as the incident angle increases\nfrom 0° to 15°. The corresponding reflective color variation observed under an optical\nmicroscope (Fig. R5(c)) confirms that the color is relatively sensitive to the incident\nangle.\nTo improve angular tolerance, we optimized the nanostructure geometry with a circular\ncross-section. For the S=1 design with a 120 nm radius, the reflection spectrum remains\nstable under oblique incidence (Fig. R5(d)), exhibiting a resonant shift of less than 10\nnm at 15° while maintaining over 80% reflectance. Consequently, the change in CIE\ncolor coordinates with angle is minimal, as summarized in Fig. R5(e).\nFig. R5 (Figure S16, Supplementary Information). Analysis and optimization of\nthe metasurface's angular tolerance. (a, b) The angular dependence of the original\ndesign characterized by (a) simulated and (b) experimental transmission spectra. (c)\nOptical microscopy images corresponding to the angular. (d, e) Performance of the\noptimized design with circular nanoposts (radius = 120 nm for S=1): (d) Reflection\nspectra at incidence angles from -25° to 25°. (e) The CIE coordinates corresponding to\nthe spectra in (d).\nThe above information has been added as a new section of “Supplementary Note 16:\nThe angular dependence” in the revised Supplementary Information.\nComment-5：The author stated that no additional polarizer is an advantage of such\ntransparent- materials metasurface. My concern is that there will be broadband\nreflectance from the samples without nanostructures. Figure 3e showcase\nexperimentally recorded structural colours under a bright field microscope. 117 samples\nwith different scaling parameters and size deviation parameters are fabricated to show\ndifferent colour hues and brightness with a black background. This presentation\napproach hides the background reflectance problem and it is misleading to the readers\nto some degrees.\nOur Response: We thank the reviewer for raising this important point regarding the\npotential broadband reflectance from un-patterned areas. The reviewer is correct that a\nbare glass substrate will exhibit a broad-band reflectance of approximately 4-6% across\nthe visible wavelength range. Then 117 structural colour samples under a white\nbackground is shown in Fig. R6. Although the reflection of the background may cause\na certain degree of contrast reduction, the reflection of background is broadband and\nuniform across the entire visible light spectrum a bright and vivid colours without\ndistortion can be still directly obtained in the image.\nFig. R6. The experimentally recorded structural colours under a bright field\nmicroscope. Here 117 samples with different scaling parameters and size deviation\nparameters are fabricated to show different colour (horizontal axis) and brightness\n(vertical axis). The colour images have been cropped and arranged on a white\nbackground.\nThe presentation of samples against a black background in Fig. 3e will increase the\ncontrast. In order to avoid the misleading to the reader, we have provided clear\nexplanations on the annotations of the figure as below.\n“(e) The experimentally recorded structural colours under a bright field microscope.\nHere 117 samples with different scaling parameters and size deviation parameters are\nfabricated to show different colour (horizontal axis) and brightness (vertical axis). The\ncolour images have been cropped and arranged on a black background.”\nTo meet the requirement for a low-reflection \"black\" state in experiment, we designed\nstructures to suppress background reflection. Fig. R7(a) shows a fabricated pattern with\nRGB color elements and dark regions, which are composed by low-reflectance\nstructures created by tuning the parameter Δr. The SEM image in Fig. R7(b) clearly\nreveals distinct nanostructure differences between the bright and dark regions.\nFurthermore, even under 50× magnification with consistent illumination source, the\nlow-reflection area maintains a dark gray appearance, as shown in Fig. R7(c).\nReflection spectra from these dark regions (Fig. R7(d)) verify successful suppression,\nwith reflectance below 3% across most of the visible spectrum. Specifically, the dark\narea next to the red pixel exhibits only a minor peak (~2.5%) below 430 nm, while\nreflectance intensities elsewhere in the visible range remain under 0.5%. This effective\nminimization of broadband background reflection confirms that the black backgrounds\nin the microscope images accurately represent the structures' performance.\nFig. R7 (Figure S17, Supplementary Information). Engineering a low-\nbackground-reflection state. (a) Optical micrograph (5×, bright-field, NA = 0.15) of\na patterned sample featuring RGB color elements and dark regions. (b) SEM image of\nthe area marked by the white box in (a); the white dashed line demarcates the bright\nand dark nanostructures. (c) Magnified view (50×, bright-field, NA = 0.8) of the region\nindicated by the yellow box in (a). (d) Measured spectra from the dark areas confirm\nminimal reflectance (<3%).\nThe above information has been added as a new section of “Supplementary Note 17:\nChiaroscuro” in the revised Supplementary Information.\nResponse to Reviewer #2\nIn this paper the authors propose and experimentally realize transparent nanostructures\nthat exhibit very high-quality colours across the whole visible spectrum. Mode\ninteractions between resonant modes produce high reflectance and, at the same time,\nreduce reflections in non-resonant spectral regions. These structures can be\nimplemented even in small size pixels of 3 microns. Besides, they allow fine control of\nthe brightness without the use of extra polarizers.\nI consider the topic and the reported findings are relevant. However, there are certain\npoints that should be addressed before publication.\nIn my opinion, the physical mechanism underlying the singular performance of the\nproposed nanostructure is not completely clear. This is of vital importance to replicate\nthis effect in different types of nanostructures. The authors explain that the reported\neffect is based on the interaction between resonant modes, and in the Suppl. Material\nthey derive the formalism for mode decomposition. However, the analysis made on\ntheir respective contributions, is not clear. The authors should expand the explanation\nof their approach, its assumptions and validity limits, in order to validate the analysis\nin terms of mode contributions. Only with a clear explanation of the physical\nmechanisms underlying the generation of the Schrodinger colours this article has a real\nvalue.\nOur Response: We sincerely thank Reviewer #2 for the positive assessment of our\nwork and for acknowledging the relevance of our findings. We are particularly grateful\nfor the insightful comments and constructive suggestions, which have helped us\nsignificantly improve the clarity and impact of our manuscript.\nWe fully agree with the reviewer that a clear explanation of the underlying physical\nmechanisms is critical for the value and reproducibility of the work. In direct response\nto this central point, we have provided a more comprehensive and explicit analysis of\nthe physical mechanisms, with multiple additions made throughout the main text to\npresent a clearer physical picture. We believe these revisions have substantially\nstrengthened the mechanistic insight and the overall value of the article.\nOur point-by-point responses to the specific comments are provided below. We have\naddressed all the raised points and incorporated the corresponding changes into the\nmanuscript.\nComment-1：line 115: the definition of the scaling factor S is missing.\nOur Response: We thank the reviewer for pointing out the omission in the definition\nof the scaling factor S. In the revised manuscript, we have added a clarification in Para-\n2, Page-4 with the following specific amendment.\n“One example with a = 350 nm and b = 220 nm is depicted in Fig. 1(b). Moreover, the\nresonance wavelength can be tuned across the entire visible region by scaling the period\nand geometric dimensions with a scaling factor S, which is set to S = 1in the example\nprovided.”\nComment-2：There is a big confusion about the curves shown in Fig. 1. In lines 128-\n130 the authors speak of “constructed reflection spectrum”, “numerical simulation” and\n“non resonant reflectance”. It is not clear how each of these curves is obtained, and\nhow do they correlate with the legend in Fig. 1(c).\nOur Response: We thank the reviewer for this valuable comment. We apologize for the\nconfusion in our previous statement. To design the structure for ideal Schrödinger\ncolors, a gradient descent algorithm integrated with FDTD simulation package is firstly\nused to optimize the geometry toward achieving the CIE plot rim at three corners. For\nexample, a geometry with a = 350 nm and b = 220 nm is obtained after the optimization\nprocess and spectrum depicted in Fig. 1(b) is realized to achieve saturated green color.\nThen, in order to understand the mechanism of the optimized design, we performed\nmode decomposition on this spectrum and explored respective contributions from each\nmode. From the mode decomposition, the reflection of the nanostructure comes from\nthree parts, including two eigenmodes at 508 nm and 520 nm and the residual field\nexcluded from the eigenmodes. To plot the spectra for eigenmodes and the residual field,\nthe relevant concepts and calculations are elaborated here.\nThe extended state of the structure occurs when light is externally incident upon it, with\nan assumed input electric field denoted as 𝐸⃗ (in) . The electromagnetic fields of this\nextended state at angular frequency ω, represented as 𝐸⃗ (𝑟 ;𝜔) and 𝐻⃗⃗ (𝑟 ;𝜔) , are\ndirectly obtained from COMSOL Multiphysics. We also observe resonant states,\ncharacterized by electromagnetic fields 𝑒 (𝑟 ) and ℎ⃗ (𝑟 ) for n = 1, 2, 3, …, excited\n𝑛 𝑛\nwithin the extended state. The extended state can be expressed as\n𝐸⃗ (𝑟 ;𝜔) = ∑𝑎 (𝜔)𝑒 (𝑟 )+𝐸⃗ (𝑟 ;𝜔)\n𝑛 𝑛 0\n𝑛 , (𝑅1)\n𝐻⃗⃗ (𝑟 ;𝜔) = ∑𝑎 (𝜔)ℎ⃗ (𝑟 )+𝐻⃗⃗ (𝑟 ;𝜔)\n𝑛 𝑛 0\n{\n𝑛\nwhich corresponds to Eq. (1) in the SI. In Eq. (R1), 𝐸⃗ (𝑟 ;𝜔) represents the uncoupled\n0\nfield, defined as the residual field when resonant states are excluded from the extended\nstate. The coefficients 𝑎 (𝜔) of the resonant states and the uncoupled field 𝐸⃗ (𝑟 ;𝜔)\n𝑛 0\ncan be determined using the method outlined in Section 4 of the SI.\nThe backward radiation's electric field for the extended state is 𝐸⃗ (rad)(𝜔) , and the\n2 2\nreflection spectrum is calculated as 𝑅(𝜔) = |𝐸⃗ (rad)(𝜔)| ⁄|𝐸⃗ (in)| , referred to as the\nnumerically simulated reflection spectrum and depicted as the orange curve in Fig.\n1c of the main text.\n(rad)\nSince the electric fields 𝑒 of the n-th resonant state’s radiation opposite to the input\n𝑛\nlight is known from the simulation, the electric field of the reflected light attributable\nsolely to the resonant states in the extended states is calculated as:\n𝐸⃗ (rad) (𝜔) = ∑𝑎 (𝜔)𝑒 (rad) . (𝑅2)\nmode 𝑛 𝑛\n𝑛\nThe reflectance due exclusively to the resonant states, referred to as the constructed\nreflection spectrum, is given by 𝑅 (𝜔) = |𝐸⃗ (rad) (𝜔)| 2 ∕|𝐸⃗ (in)| 2 , and is depicted\nmode mode\nby the green dashed curve in Fig. 1C of the main text.\nSince the uncoupled field 𝐸⃗ (𝑟 ;𝜔) is known, the electric field of its backward\n0\nradiation 𝐸⃗ (rad) (𝜔) can be derived and the reflectance in the extended state only from\n0\nthe uncoupled field, i.e., non-resonant reflectance, is calculated as 𝑅 (𝜔) =\n0\n|𝐸⃗ (rad) (𝜔)| 2 ∕|𝐸⃗ (in)| 2 , shown as the blue curve in Fig. 1C in the main text.\n𝐸⃗ (rad)(𝜔) is given by\n𝐸⃗ (rad)(𝜔) = 𝐸⃗ (rad) (𝜔)+𝐸⃗ (rad) (𝜔). (𝑅3)\nmode 0\nThen, the relation of the reflectance spectrums of the extended states, resonant states\nand the uncouple field can be expressed as\n2\n|𝐸⃗ (rad)(𝜔)|\n𝑅(𝜔) =\n2\n|𝐸⃗ (in)|\n2 2\n|𝐸⃗ (rad) (𝜔)| |𝐸⃗ (rad) (𝜔)| 𝐸⃗ (rad) ∙𝐸⃗ (rad) ∗ +c.c.\n= mode + 0 + mode 0\n2 2 2\n|𝐸⃗ (in)| |𝐸⃗ (in)| |𝐸⃗ (in)|\n= 𝑅 (𝜔)+𝑅 (𝜔)+𝑅 (𝜔), (𝑅4)\nmode 0 interf\nwhere 𝑅 (𝜔) = (𝐸⃗ (rad) ∙𝐸⃗ (rad) ∗ +c.c.)⁄|𝐸⃗ (in)| 2 is the interference reflectance\ninterf mode 0\nspectrum between resonant states and the uncoupled field. Eq. (R4) depicts the relation\nbetween different curves in Fig. 1c in the main text.\nIn the revised manuscript, we provided the detailed derivation and calculations in the\nSupplementary Information “Supplementary Note 4: Mode Decomposition”, the\nspecific changes are as follows.\n“The extended state of the structure occurs when light is externally incident upon it,\nwith an assumed input electric field denoted as 𝐸⃗ (in). The electromagnetic fields of this\nextended state at angular frequency ω, represented as 𝐸⃗ (𝑟 ;𝜔) and 𝐻⃗⃗ (𝑟 ;𝜔) , are\ndirectly obtained from COMSOL Multiphysics. We also observe resonant states,\ncharacterized by electromagnetic fields 𝑒 (𝑟 ) and ℎ⃗ (𝑟 ) for n = 1, 2, 3, …, excited\n𝑛 𝑛\nwithin the extended state. The extended state can be expressed as\n𝐸⃗ (𝑟 ;𝜔) = ∑𝑎 (𝜔)𝑒 (𝑟 )+𝐸⃗ (𝑟 ;𝜔)\n𝑛 𝑛 0\n𝑛 , 11)\n𝐻⃗⃗ (𝑟 ;𝜔) = ∑𝑎 (𝜔)ℎ⃗ (𝑟 )+𝐻⃗⃗ (𝑟 ;𝜔)\n𝑛 𝑛 0\n{\n𝑛\nwhich corresponds to Eq. 1). In Eq. 11), 𝐸⃗ (𝑟 ;𝜔) represents the uncoupled field,\n0\ndefined as the residual field when resonant states are excluded from the extended state.\nThe coefficients 𝑎 (𝜔) of the resonant states and the uncoupled field 𝐸⃗ (𝑟 ;𝜔) can\n𝑛 0\nbe determined using the Eqs.8) and Eq.10).\nThe backward radiation's electric field for the extended state is 𝐸⃗ (rad)(𝜔) , and the\n2 2\nreflection spectrum is calculated as 𝑅(𝜔) = |𝐸⃗ (rad)(𝜔)| ⁄|𝐸⃗ (in)| , referred to as the\nnumerically simulated reflection spectrum and depicted as the orange curve in Fig.\n1c of the main text.\n(rad)\nSince the electric fields 𝑒 of the n-th resonant state’s radiation opposite to the input\n𝑛\nlight is known from the simulation, the electric field of the reflected light attributable\nsolely to the resonant states in the extended states is calculated as:\n𝐸⃗ (rad) (𝜔) = ∑𝑎 (𝜔)𝑒 (rad) . 12)\nmode 𝑛 𝑛\n𝑛\nThe reflectance due exclusively to the resonant states, referred to as the constructed\nreflection spectrum, is given by 𝑅 (𝜔) = |𝐸⃗ (rad) (𝜔)| 2 ∕|𝐸⃗ (in)| 2 , and is depicted\nmode mode\nby the green dashed curve in Fig. 1C of the main text.\nSince the uncoupled field 𝐸⃗ (𝑟 ;𝜔) is known, the electric field of its backward\n0\nradiation 𝐸⃗ (rad) (𝜔) can be derived and the reflectance in the extended state only from\n0\nthe uncoupled field, i.e., non-resonant reflectance, is calculated as 𝑅 (𝜔) =\n0\n|𝐸⃗ (rad) (𝜔)| 2 ∕|𝐸⃗ (in)| 2 , shown as the blue curve in Fig. 1C in the main text.\n𝐸⃗ (rad)(𝜔) is given by\n𝐸⃗ (rad)(𝜔) = 𝐸⃗ (rad) (𝜔)+𝐸⃗ (rad) (𝜔). 13)\nmode 0\nThen, the relation of the reflectance spectrums of the extended states, resonant states\nand the uncouple field can be expressed as\n2\n|𝐸⃗ (rad)(𝜔)|\n𝑅(𝜔) =\n2\n|𝐸⃗ (in)|\n2 2\n|𝐸⃗ (rad) (𝜔)| |𝐸⃗ (rad) (𝜔)| 𝐸⃗ (rad) ∙𝐸⃗ (rad) ∗ +c.c.\n= mode + 0 + mode 0\n2 2 2\n|𝐸⃗ (in)| |𝐸⃗ (in)| |𝐸⃗ (in)|\n= 𝑅 (𝜔)+𝑅 (𝜔)+𝑅 (𝜔), 14)\nmode 0 interf\nThen, the reflectance spectrum 𝑅 (𝜔) solely from the uncoupled field is the so-called\n0\nreflection spectrum of the uncoupled field in the main text, depicted as the blue curve\nin Fig. 1c in the main text. The simulated reflectance spectrum 𝑅(𝜔) and the\nreflectance spectrum 𝑅 (𝜔) solely from resonant states are represented as the\nmode\norange and the green dashed curves, respectively. where 𝑅 (𝜔) =\ninterf\n(𝐸⃗ (rad) ∙𝐸⃗ (rad) ∗ +c.c.)⁄|𝐸⃗ (in)| 2 is the interference reflectance spectrum between\nmode 0\nresonant states and the uncoupled field. Eq. 14) depicts the relation between different\ncurves in Fig. 1c in the main text.”\nComment-3：Is the curve in Fig. 1(b) the same as the orange one in Fig. 1(c)? If this\nis correct, it should be said explicitly, as the change of scale in Fig. 1(c) makes it\ndifficult to realize of that.\nOur Response: We thank the reviewer for this careful observation regarding the\nrelationship between Fig. 1(b) and the orange curve in Fig. 1(c). The reviewer is correct\nto note that these datasets are, in fact, identical. Both curves represent the optical\nresponse of same optimized nanostructure, simulated using two distinct numerical\nmethods for verification. Fig. 1(b) was generated using Finite-Difference Time-Domain\n(FDTD) simulations optimized by the gradient descent algorithm, while the orange\ncurve in Fig. 1(c) was obtained from COMSOL Multiphysics using the identical\ngeometry. To directly demonstrate their consistency, we have included a new\nSupplementary Figure (Fig. R1) that overlays the two curves, showing their excellent\nagreement. Any minor discrepancies can be attributed to the inherent methodological\ndifferences between the two computational techniques.\nFig. R1. Validation of simulation consistency. Direct comparison of the FDTD result\n(from Fig. 1b) and the COMSOL result (orange curve in Fig. 1c) for the same structure.\nIn response to this comment, we have revised the manuscript (Para-2, Page-4) to\nexplicitly state this correspondence as below.\n“As the incident wavelength deviates from the high-reflection region, the reflectance\ndrops sharply to below 1% across most of the visible spectrum (the response shown in\nFig. 1(b) corresponds to the orange solid line in Fig. 1(c), both representing simulation\nreflection of the same structure).”\nComment-4：lines 133-135 describing Fig. 1(c) are also not clear. What’s the physical\nmeaning of “reflection spectrum of the uncoupled field”? Also, the correlation between\nthe sentence “Similar to the eigenmodes, it also has much higher reflectance than the\nsimulation at non-resonant wavelengths.” and the curves in Fig. 1(c) is not clear. The\nauthors should explain more clearly and explicitly the details of such curves.\nOur Response: We thank the reviewer for this valuable comment. To design the\nstructure for ideal Schrödinger colors, a gradient descent algorithm integrated with\nFDTD simulation package is firstly used to optimize the geometry toward achieving\nthe CIE plot rim at three corners. For example, a geometry with a = 350 nm and b =\n220 nm is obtained after the optimization process and spectrum depicted in Fig. 1(b) is\nrealized to achieve saturated green color. Then, in order to understand the mechanism\nof the optimized design, we performed mode decomposition on this spectrum and\nexplored respective contributions from each mode. From the mode decomposition, the\nreflection of the nanostructure comes from three parts, including two eigenmodes at\n508 nm and 520 nm and the residual field excluded from the eigenmodes.\nThe uncoupled field, 𝐸⃗ (𝑟 ;𝜔), represents the residual component of an extended state\n0\nwhen all resonant states are excluded. Physically, the reflection spectrum of the\nuncoupled field is the reflection at interfaces, which is strongly dependent on the\nthickness of each layer and their effective refractive indices (different from a uniform\nfilm). In an extended state with no excited resonances, this field is directly termed the\nuncoupled field. Notably, the uncoupled field persists even when resonant states are\nstrongly excited. In such scenarios, it is also referred to as the \"background field,\" and\nit combines with the excited resonant states to form the complete extended state. The\ninterference between the uncoupled field and the resonant state radiations produces the\ncharacteristic Fano line shape in the scattering spectrum [1].\nIn our manuscript, the uncoupled field is defined as the field orthogonal with all\nresonant states, given by\n∭𝑑𝑉(𝜀 𝜀(𝑟 )𝑒 ∗ (𝑟 )∙𝐸⃗ (𝑟 ;𝜔)+𝜇 ℎ⃗ ∗ (𝑟 )∙𝐻⃗⃗ (𝑟 ;𝜔) = 0, (𝑅5)\n0 𝑚 0 0 𝑚 0\nwhere 𝑒 (𝑟 ) and ℎ⃗ (𝑟 ) are the electromagnetic field of the m-th resonant state.\n𝑚 𝑚\nThe extended state is composed of resonant states and the uncoupled field, depicted as Eq.\n(R6).\n𝐸⃗ (𝑟 ;𝜔) = ∑𝑎 (𝜔)𝑒 (𝑟 )+𝐸⃗ (𝑟 ;𝜔)\n𝑛 𝑛 0\n𝑛 , (𝑅6)\n𝐻⃗⃗ (𝑟 ;𝜔) = ∑𝑎 (𝜔)ℎ⃗ (𝑟 )+𝐻⃗⃗ (𝑟 ;𝜔)\n𝑛 𝑛 0\n{\n𝑛\nTherefore, the backward radiation of the extended state can be given by Eq. (R7).\n𝐸⃗ (rad)(𝜔) = 𝐸⃗ (rad) (𝜔)+𝐸⃗ (rad) (𝜔). (𝑅7)\nmode 0\nThen, the simulated reflectance spectrum 𝑅(𝜔) of the extended state is given by Eq.\n(R8).\n2\n|𝐸⃗ (rad)(𝜔)|\n𝑅(𝜔) =\n2\n|𝐸⃗ (in)|\n2 2\n|𝐸⃗ (rad) (𝜔)| |𝐸⃗ (rad) (𝜔)| 𝐸⃗ (rad) ∙𝐸⃗ (rad) ∗ +c.c.\n= mode + 0 + mode 0\n2 2 2\n|𝐸⃗ (in)| |𝐸⃗ (in)| |𝐸⃗ (in)|\n= 𝑅 (𝜔)+𝑅 (𝜔)+𝑅 (𝜔), (𝑅8)\nmode 0 interf\nThen, the reflectance spectrum 𝑅 (𝜔) solely from the uncoupled field is the so-called\n0\nreflection spectrum of the uncoupled field in the main text, depicted as the blue curve\nin Fig. 1c in the main text. The simulated reflectance spectrum 𝑅(𝜔) and the\nreflectance spectrum 𝑅 (𝜔) solely from resonant states are represented as the\nmode\norange and the green dashed curves, respectively.\nIn our project, we need to suppress the reflectance spectrum at non-resonant\nwavelengths, which is achieved through destructive interference between the radiation\nfrom resonant states and the uncoupled field. The phases of the two resonance modes\nand the uncoupled field are illustrated in Fig. 1d of the main text. Since the uncoupled\nfield comes from the surface reflection and is uniform across the entire wavelength\nrange, we define its phase as zero. Then relative phases for mode 1 at 520 nm and mode\n2 at 508 nm can be also plot. the phase difference between the two eigenmodes and the\nuncoupled field ranges from 90 to 270 degrees over almost the entire visible region.\nConsequently, the simulated reflectance spectrum, which arises from this interference,\nexhibits lower values than the reflectance spectra attributable solely to the resonant\nstates or the uncoupled field at these wavelengths. This effect not only suppresses\nreflectance at non-resonant wavelengths but also sharpens the reflection peaks, thereby\nenhancing performance. Fig. 1c in the main text demonstrates that within the non-\nresonant wavelength in the range of 490 nm to 600 nm, the orange curve, representing\nthe simulated reflectance spectrum, falls below the green dashed curve, which\nrepresents the reflectance spectrum of the resonant states alone. Additionally, the\norange curve lies below the blue curve, which represents the reflectance spectrum of\nthe uncoupled field, for wavelengths greater than 560 nm and less than 490 nm. This\ncorresponds to “Similar to the eigenmodes, it also has much higher reflectance than the\nsimulation at non-resonant wavelengths” in the main text. These observations confirm\nthat interference between the radiations of resonant states and the uncoupled field\neffectively suppresses the reflectance spectrum.\nIn order to explain all these curves in Fig. 1 in details, more description is added in the\nSupplementary Information “Supplementary Note 4: Mode Decomposition”, the\nspecific changes are as follows.\n“The extended state 𝐸⃗ (𝑟 ) can be deemed as the superposition of the resonant states\n𝑒 (𝑟 ) and the uncoupled field 𝐸⃗ (𝑟 ) . The uncoupled field, 𝐸⃗ (𝑟 ) , represents the\n𝑛 0 0\nresidual component of an extended state when all resonant states are excluded.\nPhysically, the reflection spectrum of the uncoupled field is the reflection at interfaces,\nwhich is strongly dependent on the thickness of each layer and their effective refractive\nindices (different from a uniform film). In an extended state with no excited resonances,\nthis field is directly termed the uncoupled field. Notably, the uncoupled field persists\neven when resonant states are strongly excited. In such scenarios, it is also referred to\nas the \"background field,\" and it combines with the excited resonant states to form the\ncomplete extended state. The interference between the uncoupled field and the resonant\nstate radiations produces the characteristic Fano line shape in the scattering spectrum.”\n“In our project, we need to suppress the reflectance spectrum at non-resonant\nwavelengths, which is achieved through destructive interference between the radiation\nfrom resonant states and the uncoupled field. The phases of the two resonance modes\nand the uncoupled field are illustrated in Fig. 1d of the main text. Since the uncoupled\nfield comes from the surface reflection and is uniform across the entire wavelength\nrange, we define its phase as zero. Then relative phases for mode 1 at 520 nm and mode\n2 at 508 nm can be also plot. the phase difference between the two eigenmodes and the\nuncoupled field ranges from 90 to 270 degrees over almost the entire visible region.\nConsequently, the simulated reflectance spectrum, which arises from this interference,\nexhibits lower values than the reflectance spectra attributable solely to the resonant\nstates or the uncoupled field at these wavelengths. This effect not only suppresses\nreflectance at non-resonant wavelengths but also sharpens the reflection peaks, thereby\nenhancing performance. Fig. 1c in the main text demonstrates that within the non-\nresonant wavelength in the range of 490 nm to 600 nm, the orange curve, representing\nthe simulated reflectance spectrum, falls below the green dashed curve, which\nrepresents the reflectance spectrum of the resonant states alone. Additionally, the\norange curve lies below the blue curve, which represents the reflectance spectrum of\nthe uncoupled field, for wavelengths greater than 560 nm and less than 490 nm. This\ncorresponds to “Similar to the eigenmodes, it also has much higher reflectance than the\nsimulation at non-resonant wavelengths” in the main text. These observations confirm\nthat interference between the radiations of resonant states and the uncoupled field\neffectively suppresses the reflectance spectrum.”\nComment-5：line 173: the authors state that, according to Fig. S4, “the y-polarized\nreflection peaks slightly redshift without destroying the bright and vivid colors”.\nHowever, in Fig. S4 it is evident that additional peaks with significant intensity appear\nat shorter wavelengths, which are expected to influence the obtained colors. Moreover,\nin the Supp. Mat. the authors even say so and admit that these peaks destroy the high\nquality of the colors. Therefore, the authors should be honest and explain clearly the\ndegree of influence of these additional peaks in the reflected color, within the main text.\nOur Response: We thank the reviewer for this careful observation and highlighting the\ninconsistency in our description. The reviewer is absolutely correct. Although the\nbackground reflection at non-resonant wavelength is well suppressed too for y-\npolarized light, additional peaks in the y-polarized spectrum (Fig. S4) do indeed have\nrelative impact on color purity. A strong, additional resonant mode at shorter\nwavelengths will appear for the y-polarization. Since the wavelength difference\nbetween peaks is around 70 nm, these two peaks correspond to two different colours\nand thus spoil the overall colour performance. As a result, the color saturation will be\nunavoidably reduced compared to the high-purity x-polarized case. This can be seen\nfrom the colour coordinates in CIE 1931 chromaticity diagram in Fig. S4(b). The\ncorresponding colour gamut covers around 74.5% of Rec. 2020 for UHDTV.\nIn response, we have revised the statement on Para-1, Page-6 of the main text to provide\nan honest and clear explanation as below.\n“Considering the elliptical-shaped cross-section of the nanopillar, the reflection spectra\nand the corresponding structural colours exhibit polarization dependence. For y-\npolarized incident light, the reflection spectrum shows a redshift of the primary\nresonance. At the same time, the emergence of an additional peak at shorter\nwavelengths (as depicted in Figs. S3 and S4) will reduces the color saturation compared\nto the x-polarized case, though relatively bright structural colors are still achievable.”\nComment-6：lines 184-187: the authors state that diffraction effects reduce the\nbrightness of the structural color. However, the underlying physical mechanism\nresponsible for this effect is not clear at all. The authors should explain more deeply\nhow diffraction takes place in this system, and its role regarding the color generation.\nAlso, a more explicit sentence should be given in the Conclusions section (see lines\n329-330).\nOur Response: We thank the reviewer for raising this important point regarding the\nphysical mechanism behind brightness reduction. In our architecture, the introduction\nof Δr acts as a perturbation that breaks the in-plane symmetry of the perfect hexagonal\nlattice, effectively transforming it into a rectangular supercell. This symmetry breaking\ncan be understood as a perturbation to symmetry-protected BIC, enabling the excitation\nof higher-order diffraction channels.\nTo elucidate how Δr modulates brightness, we analyzed the energy distribution among\ndiffraction orders for two representative cases: Δr = 0 nm and Δr = 40 nm. As shown in\nFig. R2(a-b), the pristine lattice (Δr = 0 nm) maintains its periodic boundary conditions.\nAt resonance, energy is predominantly confined to the specular (0th) reflection order,\nwith negligible power in higher orders. In contrast, when Δr is increased to 40 nm (Fig.\nR2(c-d)), the broken lattice symmetry enables higher-order diffraction modes.\nConsequently, a significant portion of energy is diverted into these additional channels\n(e.g., 1st-order reflection and transmission), directly reducing the energy in the 0th\nreflection order and thus dimming the brightness. Quantitatively, by integrating the\nreflection spectra over the resonant wavelength range of 480–540 nm (light green area\nin Fig. R2(c, d)), we observe that for Δr = 0, the integrated energy in the 0th reflection\norder is 16.33, while the energy in the 1st diffraction orders remains on the order of\n110-6, which is negligible. When Δr is increased to 40 nm, the integrated energy in the\n0th order decreases to 13.40, only 82% of the value at Δr = 0. Meanwhile, the integrated\nenergies in the 1st-order reflection and transmission increase to 0.92 and 5.76,\nrespectively.\nFig. R2 (Fig. S7, Supplementary Information). Mechanism of brightness control\nvia diffraction engineering. Energy distribution among diffraction orders for (a, b) Δr\n= 0 nm and (c, d) Δr = 40 nm. (a, c) show the 1st-order reflection, and (b, d) show the\n1st-order transmission. For the symmetric lattice (Δr = 0), energy at resonance is\npredominantly confined to the 0th order, with negligible power in higher orders. In\ncontrast, symmetry breaking induced by Δr = 40 nm enables higher-order diffraction,\ndiverting a significant portion of energy into the 1st-order reflection and transmission\nchannels, thereby reducing the specular (0th order) reflection and dimming the\nperceived brightness.\nThis analysis confirms that brightness tuning is achieved by controllably redistributing\noptical energy into higher diffraction orders through deliberate lattice symmetry\nbreaking. We have revised the sentence in conclusion on Para-3, Page-11 of the revised\nmain text to provide a more explicit explanation as below.\n“This mechanism does not require the use of material absorption and the BIC resonance,\nand can be generalized to the entire visible light spectrum as well as small pixel sizes.\nBy introducing a size deviation to break the in-plane symmetry of the lattice, we\ncontrollably excite higher-order diffraction, which redistributes optical energy from the\n0th order to other diffraction channels, thereby enabling direct tuning of the color\nbrightness.”\nThe above description about the contribution of diffraction to color generation has been\nadded in Supplementary Information “Supplementary Note 7: Mechanism for\nbrightness control” as below.\n“To elucidate how Δr modulates brightness, the energy distribution among diffraction\norders is analyzed for two representative cases: Δr = 0 nm and Δr = 40 nm. As shown\nin Fig. S7 (a-b), the pristine lattice (Δr = 0 nm) maintains its periodic boundary\nconditions. At resonance, energy is predominantly confined to the specular (0th)\nreflection order, with negligible power in higher orders. In contrast, when Δr is\nincreased to 40 nm (Fig. S7 (c-d)), the broken lattice symmetry enables higher-order\ndiffraction modes. Consequently, a significant portion of energy is diverted into these\nadditional channels (e.g., 1st-order reflection and transmission), directly reducing the\nenergy in the 0th reflection order and thus dimming the brightness. Quantitatively, by\nintegrating the reflection spectra over the resonant wavelength range of 480–540 nm\n(light green area in Fig. S7 (c, d)), we observe that for Δr = 0, the integrated energy in\nthe 0th reflection order is 16.33, while the energy in the 1st diffraction orders remains\non the order of 1 x 10-6, which is negligible. When Δr is increased to 40 nm, the\nintegrated energy in the 0th order decreases to 13.40, only 82% of the value at Δr = 0.\nMeanwhile, the integrated energies in the 1st-order reflection and transmission increase\nto 0.92 and 5.76, respectively.”\nComment-7：line 189: it says Δr = 100 nm, but in Fig. 2(d) the axis showing Δr is in\npercentage and not in units of nm. Check.\nOur Response: We thank the reviewer for this careful review. The Δr is in units of nm.\nIn the revised manuscript, we have corrected this error in Fig. 2(d), the specific changes\nare as follows.\nFig. R3 (Fig. 2). Numerical simulation of Schrödinger colour across the visible\nspectrum. (d) The x-axis label error has been corrected.\nComment-8：In Fig. 3(d) the legend is missing. Although the caption says that it shows\nspectra as a function of size deviation parameter Δr, the meaning of the different shades\nin each panel of Fig. 3(d) is not clear.\nOur Response: We thank the reviewer for this careful review. In the revised manuscript,\nwe have corrected this description in Fig. 3(d), the specific changes are as follows.\nFig. R4 (Fig. 3). Experimental demonstrations of different Schrödinger colour. (a)\nTop-view and tile-view SEM images of SiO -Si N stacked metasurfaces with scaling\n2 3 4\nparameters from top to bottom of S = 0.80, 0.85, 0.95, 1.00, 1.05, 1.10, and 1.20. All\nthe other parameters are the same Fig. 1. (b) Experimentally recorded reflection spectra\nof the metasurfaces with different Schrödinger pixels. Insets are their corresponding\nbright-field microscope images. (c) Locations of all experimentally recorded\nSchrödinger pixels in CIE 1931 chromaticity diagram. Rec. 2020 for UHDTV is also\nplotted for a direct comparison. (d) Experimentally recorded spectra of red (top),\ngreen (middle), and blue (bottom) Schrödinger pixels as a function of size deviation parameter\nΔr. (e) The experimentally recorded structural colours under a bright field microscope.\nHere 117 samples with different scaling parameters and size deviation parameters are\nfabricated to show different colour (horizontal axis) and brightness (vertical axis). The\ncolour images have been cropped and arranged on a black background.\nComment-9：In Fig. 3(e) it can be observed that the brightness cannot be completely\ncontrolled by varying Δr, especially for several hues. Moreover, in some cases the hue\nchanges significantly. For instance, in column 9 the hue changes from dark orange\nto green. The authors should comment on this behavior and on how this tendency could\nbe overcome.\nOur Response: We thank the reviewer for these insightful observations regarding both\nthe limits of brightness control and the hue shift observed in Fig. 3(e).\n1. Regarding the control of brightness:\nWe agree with the reviewer that that our diffraction-based method, unlike polarization-\nbased extinction, cannot achieve zero reflectance. To meet the requirement for a low-\nreflection \"black\" state in experiment, we designed structures to suppress background\nreflection. Fig. R5(a) shows a fabricated pattern with RGB color elements and dark\nregions, which are composed by low-reflectance structures created by tuning the\nparameter Δr. The SEM image in Fig. R5(b) clearly reveals distinct nanostructure\ndifferences between the bright and dark regions. Furthermore, even under 50×\nmagnification with consistent illumination source, the low-reflection area maintains a\ndark gray appearance, as shown in Fig. R5(c).\nFig. R5 (Figure S17, Supplementary Information). Engineering a low-\nbackground-reflection state. (a) Optical micrograph (5×, bright-field, NA = 0.15) of\na patterned sample featuring RGB color elements and dark regions. (b) SEM image of\nthe area marked by the white box in (a); the white dashed line demarcates the bright\nand dark nanostructures. (c) Magnified view (50×, bright-field, NA = 0.8) of the region\nindicated by the yellow box in (a). (d) Measured spectra from the dark areas confirm\nminimal reflectance (<3%).\nReflection spectra from these dark regions (Fig. R5(d)) verify successful suppression,\nwith reflectance below 3% across most of the visible spectrum. Specifically, the dark\narea next to the red pixel exhibits only a minor peak (~2.5%) below 430 nm, while\nreflectance intensities elsewhere in the visible range remain under 0.5%. This effective\nminimization of broadband background reflection confirms that the black backgrounds\nin the microscope images accurately represent the structures' performance. This\ndemonstrates a continuous brightness tuning range from approximately 83% down to a\nvery dark state, which is sufficient for high-contrast display applications. The key\nadvantage of our approach is its ability to directly control brightness via structural\ntuning, which significantly simplifies the device architecture and improves optical\nefficiency compared to polarizer-dependent systems.\n2. Regarding the change of hue with Δr:\nThe color shift occurs because a global change in Δr alters the resonant mode's effective\nrefractive index, resulting in a blueshift. To achieve independent control of color and\nbrightness, we must compensate for this Δr-induced shift. This is effectively done by\nco-optimizing another geometric parameter, such as the scaling factor (S) or short axis\n(b). For example, when a larger Δr is used to lower brightness, a slight increase in S or\nb will induce a compensating redshift, preserving the original color tone. This method\nof coordinated adjustment successfully decouples the control of the two variables. As\nshown in Fig. R6.\nFig. R6 (Figure S18, Supplementary Information). Reflection spectrum with\ndifferent brightnesses under fixed hue.\nThe above information has been added as a new section of “Supplementary Note 17:\nChiaroscuro” in the revised Supplementary Information.\nIn response to these comments, we have revised the manuscript to include these\nclarifications. On Para-3, Page-9, we discuss the brightness control limits and\nadvantages. Furthermore, on Para-2, Page-6, we have added an explanation of the hue\nshift and the compensation strategy in the context of Fig. 2(d). The specific revised text\nis as follows.\n“This possibility has been experimentally verified by fabricating 117 samples with\ndifferent scaling parameters S and size deviation parameters Δr. The recorded colours\nare summarized in Fig. 3(e). We can see that S and Δr determine the colour and its\nbrightness respectively. All colours can be adjusted gradually from bright to near-dark\nby tuning Δr. We emphasize that no additional polarizer is employed in our scheme.\nAlthough the reflectance cannot be reduced to absolute zero—unlike in polarization-\nbased extinction methods—the measured reflectance of our optimized low-reflection\nstate is suppressed below 3% at the resonance and below 1% in off-resonant bands (see\nSupplementary Fig. S17 for details), which is sufficiently dark for high-contrast display\napplications. The key advantage of our approach lies in its direct control of brightness\nvia structural tuning, which significantly simplifies the device architecture and\nimproves optical efficiency compared to polarizer-dependent systems.”\n“Such diffractions take away some of the energy and reduce the brightness of the\nstructural colours. Figure 2(d) summarizes the reflection spectra of our metasurface as\na function of size perturbation when a single, global Δr is applied across the entire\nlattice. With the increase of Δr from 0 to 100 nm, the reflectance gradually decreases\nand nearly vanishes at Δr ~ 100 nm. Meanwhile, the corresponding resonant\nwavelengths experience a slight blueshift relative to their initial positions due to the\nreduction in effective refractive index. Nevertheless, this shift can be readily\ncompensated for by adjusting parameters such as the S or the minor axis b, thereby\nmaintaining the resonance wavelength at the desired position. This controllable\nadjustment of brightness with stable hue well satisfies the requirements for direct\nchiaroscuro presentation (As depicted in Fig. S18).”\nComment-10：line 287: In Fig. S12 it is shown that the peak intensity decreases\nsignificantly as the pixel size decreases. This should influence the colour brightness,\nalthough the authors do not mention this.\nOur Response: We thank the reviewer for this important observation. The reviewer is\ncorrect that the decrease in peak intensity with reduced pixel size would influence the\ncolour brightness. This phenomenon occurs because the establishment of resonance\nmodes relies on sufficient periodic support. As the pixel size decreases, the periodic\nboundary conditions become increasingly compromised, leading to a gradual reduction\nin resonance intensity. This is a fundamental characteristic of such photonic crystal slab.\nIn the revised manuscript, we have corrected this description in the Supplementary\nInformation “Supplementary Note 13: Optical characterization of pixel arrays”,\nthe specific changes are as follows.\n“Pixel size is also an important indicator for structural colour. In Fig. S13(a), we show\nthe bright-field microscopic images of blue, green, and red pixel array samples as the\npixel size increases continuously from 3 μm to 8 μm. It can be seen from the figure that\nat 3 μm, this sample has already shown good colour effects, and as the pixel size\nincreases, the colour gradually approaches the ideal effect. The corresponding reflection\nspectra of the samples are shown in Fig. S13(b). While the samples maintain sharp\nresonance peaks and good suppression of reflection at non-resonant wavelengths across\ndifferent pixel sizes, it is noteworthy that the peak intensity shows a gradual decrease\nas the pixel size reduces. This is attributed to the breakdown of periodic boundary\nconditions in smaller pixels, which affects the completeness of resonance mode\nformation.”\nResponse to Reviewer #3\nThis manuscript reports on the high performance of color pixels achieved by employing\ndielectric photonic crystal slab structures. The work is well organized and state-of-the-\nart, combining numerical analysis with experimental demonstration. However, it\nappears that suppression of sidebands is the main contribution for achieving\nSchrodinger colors, but this remains ambiguous without quantification of the effects.\nAdditionally, some major questions must be addressed for acceptance in this journal.\nAll answers to those questions should be clearly explained in revised manuscript.\nOur Response: We sincerely thank Reviewer #3 for the positive evaluation of our work\nas \"well organized and state-of-the-art\" and for acknowledging the effective\ncombination of numerical and experimental analyses. We are also grateful for the\nconstructive comments, which have been instrumental in strengthening the manuscript.\nWe agree with the reviewer that a quantitative analysis of the sideband suppression is\ncrucial for unequivocally demonstrating its contribution. In direct response to this key\npoint, a dedicated quantitative assessment, comparing the average reflectance in the off-\nresonant bands of our structures against conventional designs, has been included in the\nrevised manuscript.\nFurthermore, we have thoroughly addressed all the major questions raised by the\nreviewer in the subsequent list. We have provided detailed explanations, supported by\nadditional data and analysis where necessary, and have incorporated these clarifications\nthroughout the revised manuscript.\nWe believe that the revisions made in response to the comments have significantly\nenhanced the manuscript, and we are confident that it now fully meets the journal's\nstandards.\nComment-1：Figure 1 seems primarily designed to explain sideband suppression. In\nFig. 1c, the estimation shows that the resonance mode reduces reflectance outside the\nresonance by as much as 2% and as little as about 0.1%. How much does this contribute\nto the implementation of the Schrodinger pixel? Also, the explanation in the main text\nfor Fig.1e–1h is insufficient. From the Supporting Information, it appears that the\nradiated E-field components are calculated for each decomposed resonance mode and\nbackground mode. The main text needs to include an explanation of where spatially the\nradiated E-field components originate for clear understanding.\nOur Response: We thank the reviewer for this critical question regarding the practical\nimpact of sideband suppression on achieving a Schrödinger pixel. As per its definition,\nan ideal Schrödinger pixel requires near-zero reflectance outside the resonant band. Any\nnonzero reflection in the sidebands leads to a rapid degradation of color saturation.\nTo quantitatively evaluate the contribution of sideband suppression in our design, we\nconstructed the reflection spectrum without such suppression by directly summing the\nnon-resonant background and the resonant mode contributions from Fig. 1c. The\nresulting CIE color coordinates were then calculated and are plotted in Fig. R1. As\nshown, without effective sideband suppression, the color coordinate for this wavelength\nrapidly degrades to (0.14, 0.69). In stark contrast, with the sideband suppression\nenabled by mode interference, the coordinate achieves a much higher saturation at (0.10,\n0.76).\nFig. R1. Role of sideband suppression. The CIE coordinates show that effective\nsideband suppression is essential for high color saturation, enabling the realization of a\nhigh-performance Schrödinger pixel.\nCalculate method of the radiated E-field components\nIn principle, the reflection spectra contain two parts, i.e., the radiation of the resonances\nand the reflection at interfaces. The latter one is strongly dependent on the thickness of\neach layer and their effective refractive indices (different from a uniform film). To\nproduce high-purity color, it is essential to suppress the reflection at non-resonant\nwavelengths. This is the main goal of Fig. 1e-1h to illustrate. Since the effective\nrefractive index is determined by the sizes of nanopillars and their spatial distribution,\nit is very difficult to precisely (in both amplitude and phase) describe the reflection at\ninterfaces with a transformer matrix. We tried to solve everything analytically but\ncannot achieve such a suppression of background reflection (at non-resonant\nwavelengths). The spectra can only be similar, but the position in color gamut is much\nworse than the numerical one and the experiment. So, in Fig. 1e-1h, we used the\nfollowing method to calculate the radiated E-field components using the numerical\nsimulation result.\nAssuming the SiO₂ substrate lies in the plane z = 0, plane S₁ is defined as the hexagonal\nplane within one unit cell of the structure, oriented parallel to the XY plane at a z-\ncoordinate of z₁ = 1660 nm. The electric field of the resonant state 𝑒 (𝑟 ),𝑛 = 1,2, at\n𝑛\nthe plane S is 𝑒 (𝑥,𝑦,𝑧 ). The electric field of the resonant state’s radiation in this\n1 𝑛 1\ndirection is calculated as\n∬ 𝑑2𝑟 𝑒 (𝑥,𝑦,𝑧 )\n𝑒\n(rad)\n=\n𝑆1 𝑛 1\n, (𝑅1)\n𝑛 𝑆\n1\nwhere 𝑆 is the area of plane S . Plane S is defined as the hexagonal plane within one\n1 1 2\nunit cell of the structure, oriented parallel to the XY plane at a z-coordinate of z = 1760\n2\nnm. The distance between plane S and S is d = 100 nm. In the COMSOL simulation\n1 2\nof the extended state 𝐸⃗ (𝑟 ;𝜔), the input port is set at plane S with the power 1 W. The\n2\nshapes and positions of plane S and S are depicted in Fig. R2. The electric field of the\n1 2\nextended state’s backward radiation is calculated as\n𝜇 𝑣\n𝐸⃗ (rad)(𝜔) = ∬ 𝑑2𝑟 (𝐸⃗ (𝑥,𝑦,𝑧 ;𝜔)−√2[W]× 0 exp(−𝑖𝑘𝑑))⁄𝑆 ,\n1 𝑆 1\n𝑆1 1\nwhere √2[W]×\n𝜇0𝑣\nexp(−𝑖𝑘𝑑) is the electric field of the input light on plane S\n1\nand\n𝑆1\nthe phase convention is exp(𝑖𝜔𝑡). k is the wavevector of the input light in the cladding\nmaterial, 𝜇 is the permeability in the vacuum, and v is the light speed in the cladding\n0\nmaterial. From section 4 in SI, the extended state is the combination of resonant states\nand the uncoupled field and is given by\n𝐸⃗ (𝑟 ;𝜔) = ∑𝑎 (𝜔)𝑒 (𝑟 )+𝐸⃗ (𝑟 ;𝜔), (𝑅3)\n𝑛 𝑛 0\n𝑛\nwhere 𝑎 (𝜔) can be calculated with the method introduced in section 4 of SI.\n𝑛\nTherefore, relation between the backward radiations of the extended state, resonant\nstates and the uncoupled field is\n𝐸⃗ (rad)(𝜔) = ∑𝑎 (𝜔)𝑒 (rad) +𝐸⃗ (rad) (𝜔)\n𝑛 𝑛 0\n𝑛\n= ∑𝐸⃗ (rad) (𝜔)+𝐸⃗ (rad) (𝜔), (𝑅4)\n𝑛 0\n𝑛\nwhere 𝐸⃗ (rad) (𝜔) can be calculated and 𝐸⃗ (rad) (𝜔) = 𝑎 (𝜔)𝑒 (rad) is the electric field\n0 𝑛 𝑛 𝑛\nof the n-th resonant state’s backward radiation in the extended state.\nFig. R2 (Figure S19, Supplementary Information). Schematics of Plane S1 and\nS2.\nIn our design, the input light is polarized along the x direction and resonant states only\nwith the x polarized radiation are excited. As a result, we only need to care about the x\ncomponent of the backward radiations’ electric field of the extended state, resonant\nstates and the uncoupled field, i.e., 𝐸\n(rad)\n(𝜔), 𝐸\n(rad)\n(𝜔) and 𝐸\n(rad)\n(𝜔). Relation of\n𝑥 (𝑛,𝑥) (0,𝑥)\nthese x-component of electric fields is\n𝐸\n(rad)\n(𝜔) = ∑𝐸\n(rad)\n(𝜔)+𝐸\n(rad)\n(𝜔), (𝑅5)\n𝑥 (𝑛,𝑥) (0,𝑥)\n𝑛\nwhich is plotted in Fig. 1e-h at different wavelengths.\nThe calculation result is plot in Fig.1e–1h. In the wavelength out of resonance at 400\nnm and 600 nm in Figs. 1(e) and 1(h), the amplitude of the total reflection is greatly\nreduced to ~ 6.6 and 3.5 by the destructive interference between resonant modes and\nuncoupled fields. While in the high reflection band, the interaction mainly occurs\nbetween two eigenmodes and produces total field more than an order of magnitude\nhigher (64.9 in Figs. 1(f) and 74.8 in 1(g)).\nThe above information has been added as a new section of “Supplementary Note 18:\nCalculate method of the radiated E-field components” in the revised\nSupplementary Information.\nIn the main text of the revised manuscript, we have added the above information in\nPara-1, Page-5:\n“In the high reflection band, the interaction mainly occurs between two eigenmodes\nand produces total field more than an order of magnitude higher (Figs. 1(f) and 1(g)).\nSuch interaction forms two Fano resonances in opposite directions, resulting in high\nreflectance and sharp edges simultaneously (insets in Figs. 1(f) and 1(g)). To further\nelucidate the spatial origin of the radiated fields presented in Figs. 1(e)–1(h), we note\nthat the electric field components are derived from the backward radiation of both the\nresonant and uncoupled modes, evaluated on a reference plane (S₁) situated above the\nmetasurface (see Section 18 in the SI for details). The radiated field of the extended\nstate is calculated by spatially integrating the electric field over S₁, 𝐸⃗ (rad)(𝜔) =\n1 ∫ 𝐸⃗ (𝑟 ;𝜔)𝑑2𝑟 , and so are the resonant states’ radiation field 𝐸⃗ (rad)(𝜔) and the\n𝑆1 𝑆1\nuncoupled field’s radiation 𝐸⃗ (rad) (𝜔). Under x‑polarized illumination, only resonant\n0\nstates emitting x‑polarized radiation are excited. Then only the x‑component of the\nbackward field, 𝐸\n(rad)\n(𝜔) = ∑ 𝐸\n(rad)\n(𝜔)+𝐸\n(rad)\n(𝜔), is considered.”\n𝑥 𝑛 (𝑛,𝑥) (0,𝑥)\nComment-2：This focuses primarily on the interaction between resonance modes and\nuncoupled modes, but it lacks a fundamental explanation of the resonance itself. This\nis critical and should be appropriately discussed in the Introduction with proper\nreferences.\nOur Response: We thank the reviewer for this valuable comment. In this structure,\nresonant states correspond to the leaky eigenmodes of the Si₃N₄-SiO₂ nanopillar array,\nwhich represent guided-mode resonances supported by the high-index Si₃N₄ layer and\nperiodically coupled to free space through the lattice. These so-called guided-mode or\nleaky resonances correspond to discrete eigenstates with complex eigenfrequencies and\nfinite radiative lifetimes. The photonic slab exhibits mirror symmetry with respect to\nboth the XZ and YZ planes. Consequently, non-degenerate resonant states are either\nmirror-symmetric or anti-symmetric relative to both planes. This symmetry dictates that\nradiation from non-degenerate resonant states is polarized along the x-axis, y-axis, or\nabsent entirely. Given the x-polarized input light, only resonant states with x-polarized\nradiation can be excited. Within the 450–700 nm spectral range, only the resonant states\nat 508 nm and 520 nm exhibit x-axis polarization and are thus excitable. These states\nare anti-symmetric with respect to the YZ plane and symmetric with respect to the XZ\nplane. The spectral positions and linewidths of the high-reflection bands in Fig. 1 in the\nmain text are determined by the near-field confinement and far-field radiation of these\nmodes. Such resonant behaviors have been extensively studied in Mie-type dielectric\nnanoresonators and guided-mode metasurfaces.\nIn the revised manuscript, we have added the above information to Para-2, Page-4 and\nreference list in the main text respectively:\n“Two eigenmodes at 508 nm and 520 nm of the nanostructure in the visible region are\ncrucial for constructing high reflection bands (see field distributions in Fig. S3). These\nresonant states are leaky eigenmodes (guided-mode resonances) supported by the high-\nindex Si₃N₄ layer and periodically coupled to free space. These discrete modes possess\ncomplex eigenfrequencies and finite radiative lifetimes, producing Fano-type\ninterference with the non-resonant background. The excitation of a given resonant state\ndepends on polarization: it can only be excited when the polarization of the incident\nlight is non-orthogonal to the time-reversed radiation polarization of that state. Such\nresonant and polarization-selective behaviors are well established in dielectric\nnanoresonators and guided-mode metasurfaces23-27, 37-39, 44. Returning to the spectral\nconstruction, the constructed reflection spectrum (dashed green line in Fig. 1(c)) fits\nthe two reflection peaks well, but the non-resonant reflectance is still much higher than\nthe numerical simulation (orange solid line) within the spectral range from 475 nm to\n625 nm, strongly spoiling the colour performances.”\nReference\n“44. Fan, S. & Joannopoulos, J.D. Analysis of guided resonances in photonic crystal\nslabs. Physical Review B 65, 235112 (2002).”\nComment-3：In the Introduction, the statement \"Their approach, however, suffers from\nnoticeable performance degradation at green and blue colours [34]\" may be misleading.\nReference 34 focuses on the red color, not the full color range. It cannot be concluded\nwhether the method using quasi-BIC is efficient; it may have the potential to be\neffective for blue and green colors with proper design or adjustment of the scaling factor.\nOur Response: We thank the reviewer for this valuable comment. The reviewer is\ncorrect that Reference 34 (Reference 36 in the revised manuscript, Sci. Adv. 8,\neabm4512 (2022)) focuses on the red color. And in Fig. S19 of the Supplementary\nInformation in Reference 36, optimized Si nanostructures by gradient descent\noptimization algorithm under y-polarized incidence condition has also been\ndemonstrated to achieve green color pixel. But due to the relatively large ohmic loss of\nthe amorphous silicon at the blue wavelength of below 480 nm, the suppressing\nreflectance at blue wavelength range becomes harder and the corresponding coordinates\nin the CIE 1931 chromaticity diagram is (0.218, 0.629), which is far away from the\nSchrödinger green. The authors also mentioned in the discussion part that it was\nchallenging to use q-BIC resonance to achieve the highly saturated blue pixel based on\namorphous silicon also due to the large ohmic loss. The authors also mentioned that\nengineering q-BIC resonance may realize achieve highly saturated blue pixel using\nother dielectric materials with lower ohmic loss at blue. In order to make our expression\nmore accurate, we have added the above information in the revised manuscript in Para-2,\nPage-2 as below.\n“Very recently, Dong et al. ingeniously tackled this problem and realized the first\nSchrödinger red pixel by utilizing quasi-bound states in the continuum and the\nabsorption of Si at short wavelengths34. However, their approach exhibited limited\nperformance in the green and blue spectral regions due to the inherent short-wavelength\nabsorption of Si, which consequently restricted the achievable color gamut of the\narchitecture. Although bound states in the continuum (BIC) employing lossless\ndielectric materials can also achieve the requisite passband, the fundamental realization\nof Schrödinger colour across the entire visible spectrum remains elusive due to\nlimitations imposed by the requirement for zero reflectance outside the passband.”\nComment-4：The authors emphasized the performance achieved by comparison with\nRec.2020 UHDTV, which has been done in some articles in this research field. This\nproposed technology is based on reflectance type, but Rec. UHDTV is mostly\ntransmittance type. The author should explain the performance considering this\ndifference, as well as any related limitations or drawbacks.\nOur Response: We thank the reviewer for providing this valuable and insightful\nopinion. The reviewer has correctly pointed out the fundamental mode difference\nbetween our reflective display technology and the Rec.2020 standard, which mainly\ntargets transmissive systems. We fully agree that this must be fully considered when\nconducting performance comparisons.\nWe chose Rec.2020 as the benchmark mainly because it represents the widest color\ngamut standard currently available and has become a universal indicator for measuring\nthe color performance of new display technologies, regardless of whether the\ntechnology is reflective or transmissive. Our core goal is to demonstrate that the\ndesigned nanostructures can generate sufficiently saturated colors to cover this vast\ncolor gamut, thereby demonstrating their potential in color generation ability. We\nacknowledge that direct comparison does have the limitations the reviewer mentioned.\nThe brightness of reflective displays depends on ambient lighting, and their absolute\nbrightness cannot be compared to self-luminous transmissive or OLED displays.\nHowever, the core advantages of reflective technology lie in its ultra-low static power\nconsumption and excellent outdoor visibility, making it suitable for application\nscenarios that are completely different from transmissive displays (such as smart\nelectronic paper, energy-saving display boards, etc.).\nIn the revised Supplementary Information, we added a paragraph in the discussion\n(“Supplementary Note 14: Calculate method of the CIE 1931 coordinate”) to\nspecifically elaborate on the limitations of this comparison as below.\n“Although this study compared the color performance with the Rec.2020 standard, it\nmust be noted that there are differences in the technical principles between the two,\nRec.2020 mainly targets transmissive displays, while this work is based on reflective\nmodes. Therefore, this comparison primarily aims to evaluate the inherent ability of this\ntechnology in generating high saturation colors. The actual visual experience of\nreflective displays is highly dependent on ambient lighting, and its absolute brightness\ncannot be compared to backlit displays. However, its core value lies in the energy-\nsaving advantage of not requiring backlighting and its visibility under strong light,\nwhich has found a unique positioning for it in specific application scenarios such as\nportable energy-saving devices.”\nComment-5：Regarding the hexagonal array, what is the main reason for its use? Due\nto the symmetry property, it should be polarization-dependent. Thus, the authors\nemphasize the performance mainly for x-polarized input in the main text, rather\nthan for all polarizations. Also, to control brightness, the polarization dependence is\nlikely to increase with delta_r; data or explanation supporting this should be provided.\nOur Response: We thank the reviewer for these insightful questions regarding the\nchoice of the hexagonal lattice and its polarization dependence.\nThe hexagonal lattice was selected for two primary reasons. First, as detailed in\nSupplementary Information “Supplementary Note 7: Mechanism for brightness\ncontrol”, it provides a platform where the parameter Δr can be effectively introduced\nto break the periodicity and enable brightness modulation. Second, compared to a\nsquare lattice, the hexagonal configuration permits for a larger structure spacing while\nmaintaining a high effective refractive index. This property is advantageous as it relaxes\nfabrication constraints, thereby enhancing the structure's experimental feasibility\nHowever, the intrinsic symmetry of the perfect hexagonal lattice will lead to\npolarization-dependent response, as the reviewer correctly noted. So, we emphasized\nthe performance mainly for x-polarized input in the main text.\nIn order to control the brightness, Δr is introduced and the structural symmetry is broken.\nThe relationship between Δr and polarization dependence is important and the spectral\nresponse under different polarization angles θ is further investigated as shown in Fig.\nR3. When the incident light is y-polarized (θ = 90°), the evolution of the reflection\nspectrum with increasing Δr is shown in Fig. R3(b). As Δr gradually increases, the\nprimary mode under y-polarization exhibits a clear blueshift. The resonant wavelength\nof the additional mode at shorter wavelengths remains unchanged with Δr, but both\nresonant modes gradually weaken with increasing Δr until they eventually vanish. To\nfurther clarify the relationship between polarization dependence and Δr, Figs. R3(c) and\n(d) detail the variation of the reflection spectrum as θ increases from 0° to 90° for fixed\nΔr values of 25 nm and 65 nm, respectively. As the polarization angle θ varies, the y-\npolarized mode begins to appear at around θ = 24°, and by θ = 66°, the x-polarized\nmode completely disappears, resulting in a full transition to the y-polarized mode. This\nbehavior is consistent with the observations in Fig. S5(a) in the Supplementary\nInformation (Δr = 0 condition), where different Δr values only result in slight shifts in\nthe resonant wavelength and changes in resonance strength, while the polarization\ndependence remains unchanged.\nThis analysis confirms that polarization dependence is an inherent feature of the design.\nThe parameter Δr primarily controls brightness, and the polarization dependence\nremains consistent across all values of Δr until the resonant modes are completely\nsuppressed.\nFig. R3 (Figure S20, Supplementary Information). Polarization-dependent\nresponse under varying Δr. (a) Schematic of the rectangular supercell, where a and b\ndenote the major and minor axes of the ellipse, respectively. The red arrow indicates\nthe electric field direction of the incident linearly polarized light. The polarization angle\nθ is defined as the angle between the electric field and the x-axis. (b) Reflection spectra\nas a function of Δr at a fixed polarization angle θ = 90°. (c, d) Reflection spectra as a\nfunction of θ (0° to 90°) for fixed Δr values of (c) 25 nm and (d) 65 nm, corresponding\nto the orange and gray dashed curves in (b), respectively.\nThe above information has been added as a new section of “Supplementary Note 19:\nPolarization Dependence and the effect of Δr” in the revised Supplementary\nInformation.\nComment-6：In Fig. 2a, it can be seen that the resonance wavelength is tuned by the\nscaling factor, causing two modes to shift together. In the experimental results of Fig.\n3b, what is the reason that only one mode appears for some colors?\nOur Response: We thank the reviewer for this important point regarding the\ndiscrepancy between the simulated double-peak spectra and the experimentally\nobserved single-peak spectra. The disappearance of the second peak in experiments is\nprimarily due to the fabrication imperfections.\nAs illustrated in Fig. R4(a), two primary deviations will usually happen during the\nfabrication process, one is the radius variations (especially the short-axis radius of the\nellipses) and another one is the over-etching depth. To systematically demonstrate this,\nwe simulated the impact of each deviation for a representative structure with S=1.\nFigure R4(b) demonstrates that a slight radius deviation is sufficient to drastically\nsuppress Mode 2. By comparison, the effect of over-etching alone, shown in Fig. R4(c),\nis relatively minor. This indicates that Mode 2 is far more sensitive to dimensional\nvariations than Mode 1. Crucially, Fig. R4(d) shows that when both deviations are\npresent—a realistic scenario—their combined effect causes the complete disappearance\nof the more sensitive Mode 2, leaving only the robust Mode 1. This finding directly\nexplains the single-peak spectra observed in our experiments.\nFig. R4 (Figure S15, Supplementary Information). Fabrication Tolerance of\nResonant Modes. (a) Main fabrication deviations: radius deviation and over-etching\ndepth. (b, c) Individual impact of (b) radius deviation and (c) over-etching on the\nreflection spectrum for structure with S=1. (d) Simulated reflection spectrum from\ncombined deviations.\nThe above information has been added as a new section of “Supplementary Note 15:\nThe effects of fabrication deviations” in the revised Supplementary Information.\nComment-7：In Fig. 4b, the picture has a yellow tint. This study is about colorful pixels,\nbut the example does not reflect that. In the section \"High-performance colour\nprintings,\" the authors should present examples of high-performance colour printings.\nOur Response: We appreciate the valuable feedback from the reviewers regarding the\ncolor performance demonstration. It should be noted that Figure 4b adopts a\nmonochrome grayscale image, and its core purpose is to clearly demonstrate the precise\nbrightness control ability through a monochrome pattern - this is one of the key\ninnovation points of this work. This design allows us to focus on evaluating the\ncharacteristics of reflectance changes without interference from hue variations.\nFigure 4 (d) is used to directly demonstrate the vibrant full-color display performance\nof the sample. It presents an optical micrograph of a high-resolution full-color pattern.\nThis figure clearly demonstrates that color tone and brightness can be independently\ncontrolled. Different bright colors can be obtained in different color blocks. The\nenlarged view on the right further reveals that even within the same color block, pixels\ncan exhibit different brightness levels, which strongly confirms that we have achieved\ndecoupling control of color and brightness throughout the entire visible spectrum range.\nComment-8：What is the red dot on the left side of the scale bar in Fig. 4b?\nOur Response: We thank the reviewer for the careful observation. The red dot on the\nleft side of the scale bar in Fig. 4(b) is a localized fabrication defect, caused by a minor\nimperfection in the resist film or the etching process. This caused a slight deviation\nfrom the intended nanostructure at that specific site. We confirm that this is an isolated\nanomaly that does not affect the structural integrity or optical performance of the\nsurrounding pixels, nor any of the results or conclusions presented in the study.\nReference\n1. Fan, S., Suh, W. & Joannopoulos, J. D. Temporal coupled-mode theory for the Fano\nresonance in optical resonators. J. Opt. Soc. Am. A 20, 569 (2003).\nReply to reviewers’ reports\nWe sincerely thank all the reviewers for their positive feedback and valuable comments\non our revised manuscript. We are pleased to know that the reviewers support the\npublication of our work in Nature Communications. Below, we provide point-to-point\nresponses to the final minor suggestions raised by reviewer #3. All corresponding\nchanges in the manuscript have been highlighted in blue for easy identification.\nResponse to Reviewer #3\nWe deeply appreciate the reviewer #3 thorough assessment and constructive\nsuggestions. We have fully addressed the two minor points as follows:\nComment-1: The authors have clearly explained the physical mechanism in terms of\nguided-mode resonance (GMR). However, there is no citation of the original paper on\nGMR: [Ref] S. S. Wang and R. Magnusson, “Theory and applications of guided-mode\nresonance filters,” Appl. Opt. 32, 2606–2613 (1993).\nOur Response: We agree and thank the reviewer for highlighting this important\nreference. We have now cited the seminal paper by S. S. Wang and R. Magnusson (Appl.\nOpt.1993, 32, 2606–2613) in Para-1, Page-4:\n“The excitation of a given resonant state depends on polarization: it can only be excited\nwhen the polarization of the incident light is non-orthogonal to the time-reversed\nradiation polarization of that state. Such resonant and polarization-selective behaviors\nare well established in dielectric nanoresonators and guided-mode metasurfaces25-29, 39-\n41, 46, 47.”\nReference\n“47. Wang, S.S. & Magnusson, R. Theory and applications of guided-mode resonance\nfilters. Appl. Opt. 32, 2606-2613 (1993).”\nComment-2: Regarding Comment 6, it would be helpful to add a brief explanation in\nthe revised manuscript.\nOur Response: We appreciate this suggestion. A clarifying sentence has been added in\nPara-2, Page-6, noting that occasional single-mode responses in some samples are due\nto fabrication variations but do not affect color saturation. The specific revised text is\nas follows:\n“The characteristics of the metasurfaces are optically characterized with a home-built\nbrightfield optical microscope (see Methods). With an x-polarized incident light, the\nreflection spectra have been directly recorded without using an additional polarizer (Fig.\n3(b)). Some samples in Fig. 3b exhibit only a single resonance mode, which we attribute\nto minor fabrication deviations. Nevertheless, this does not compromise the color\nsaturation, as confirmed by the results shown in Supplementary Fig. 15.”"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have addressed all my concerns and the manuscript can be accepted.", "reviewer_2": "The authors have addressed all my first review comments, and therefore I consider the manuscript is acceptable for\npublication in Nature Communications.", "reviewer_3": "All my comments and concerns have been adequately addressed in the revised version. I recommend acceptance of this\npaper in the journal, with two minor suggestions.\n1. The authors have clearly explained the physical mechanism in terms of guided-mode resonance (GMR). However, there\nis no citation of the original paper on GMR:\n[Ref] S. S. Wang and R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32, 2606–2613\n(1993).\n2. Regarding Comment 6, it would be helpful to add a brief explanation in the revised manuscript."}, "author_response": "Reply to reviewers’ reports\nWe sincerely thank all the reviewers for their positive feedback and valuable comments\non our revised manuscript. We are pleased to know that the reviewers support the\npublication of our work in Nature Communications. Below, we provide point-to-point\nresponses to the final minor suggestions raised by reviewer #3. All corresponding\nchanges in the manuscript have been highlighted in blue for easy identification.\nResponse to Reviewer #3\nWe deeply appreciate the reviewer #3 thorough assessment and constructive\nsuggestions. We have fully addressed the two minor points as follows:\nComment-1: The authors have clearly explained the physical mechanism in terms of\nguided-mode resonance (GMR). However, there is no citation of the original paper on\nGMR: [Ref] S. S. Wang and R. Magnusson, “Theory and applications of guided-mode\nresonance filters,” Appl. Opt. 32, 2606–2613 (1993).\nOur Response: We agree and thank the reviewer for highlighting this important\nreference. We have now cited the seminal paper by S. S. Wang and R. Magnusson (Appl.\nOpt.1993, 32, 2606–2613) in Para-1, Page-4:\n“The excitation of a given resonant state depends on polarization: it can only be excited\nwhen the polarization of the incident light is non-orthogonal to the time-reversed\nradiation polarization of that state. Such resonant and polarization-selective behaviors\nare well established in dielectric nanoresonators and guided-mode metasurfaces25-29, 39-\n41, 46, 47.”\nReference\n“47. Wang, S.S. & Magnusson, R. Theory and applications of guided-mode resonance\nfilters. Appl. Opt. 32, 2606-2613 (1993).”\nComment-2: Regarding Comment 6, it would be helpful to add a brief explanation in\nthe revised manuscript.\nOur Response: We appreciate this suggestion. A clarifying sentence has been added in\nPara-2, Page-6, noting that occasional single-mode responses in some samples are due\nto fabrication variations but do not affect color saturation. The specific revised text is\nas follows:\n“The characteristics of the metasurfaces are optically characterized with a home-built\nbrightfield optical microscope (see Methods). With an x-polarized incident light, the\nreflection spectra have been directly recorded without using an additional polarizer (Fig.\n3(b)). Some samples in Fig. 3b exhibit only a single resonance mode, which we attribute\nto minor fabrication deviations. Nevertheless, this does not compromise the color\nsaturation, as confirmed by the results shown in Supplementary Fig. 15.”"}]

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