## 2 Design and Modeling
### Structural Color Theory  
According to color science [20; 21], any color in nature can be displayed as a combination of the three primary colors of red, green, and blue in a specific proportion. Measuring the color corresponding to the simulated spectrum can predict the range of color gamut that can be covered by the designed structure, so as to provide guidance for optimizing the structural parameters to obtain the widest color gamut. At the same time, it can also identify the types and corresponding performances of each structure and device according to the color index. In the second-field-of-view XYZ system, if the object is in the relative spectral power distribution of \(D(\lambda)\), the transmittance at different wavelengths is \(T(\lambda)\); the \(X\), \(Y\), and \(Z\) corresponding to the transmission color of the object can be calculated according to the following formula [22; 23]:  
\[X=k\sum_{\lambda}T(\lambda)I(\lambda)x^{\prime}(\lambda), \tag{1}\]  
\[Y=k\sum_{\lambda}T(\lambda)I(\lambda)y^{\prime}(\lambda), \tag{2}\]  
\[Z=k\sum_{\lambda}T(\lambda)I(\lambda)z^{\prime}(\lambda), \tag{3}\]  
where the wavelength integration range shall be within the visible light band (400-700 nm); \(I(\lambda)\) is the spectrum of the standard illuminant incident light source (D65) closest to sunlight; \(T(\lambda)\) is the transmission spectrum; \(x^{\prime}(\lambda)\), \(y^{\prime}(\lambda)\), and \(z^{\prime}(\lambda)\) represent the stimulus value of the three primary colors; \(Y\) represents not only the stimulus value of the green primary color to the human eye, but also the brightness of the object color; and \(k\) is the normalization coefficient, and its definition value is: adjust the \(Y\) stimulation value of the perfect reflector (reflectivity: 1) or ideal transmissive material (transmittance: 1) to 100, so the calculation formula of \(k\) is:  
\[k=100/\sum_{\lambda}I(\lambda)z^{\prime}(\lambda), \tag{4}\]  
Figure 1: Unit structure diagram.  
In order to express the relative proportion of the stimulus amount of the three primary colors \(X\), \(Y\), and \(Z\) in the total amount of \(X+Y+Z\), the international lighting committee introduced the chromaticity co-ordinates x, y, and z to represent the proportion of the primary colors and formulated the chromaticity diagram. For nonluminous objects, the calculation formula of the chromaticity co-ordinates x, y, and z is:  
\[\texttt{x}=\
icefrac{{X}}{{(X+Y+Z)}}, \tag{5}\]  
\[\texttt{y}=\
icefrac{{Y}}{{(X+Y+Z)}}, \tag{6}\]  
(\(x\), \(y\)) is the chromaticity co-ordinate of the transmission spectrum; thus, we only need to substitute the simulated or experimental transmission spectrum into the formula for calculation, and the color of the obtained structural color pixel can be accurately and quantitatively described by the tristimulus value, that is, the chromaticity value \(X\), \(Y\), and \(Z\) of the color presented by the object. Meanwhile, in the chromaticity diagram, the physical meaning of the \(X\)-axis co-ordinate is to reflect the relative proportion of the red primary color; the physical meaning of the \(Y\)-axis co-ordinate is to represent the relative proportion of the green primary color; and the point outside the horseshoe shape represents the chromaticity of the color that does not exist physically. We know that different colors have different chromaticity co-ordinates, and they occupy different positions in the chromaticity diagram. A group of chromaticity co-ordinates (\(x\), \(y\)) in the figure represent common characteristics of colors with the same tone and saturation but different brightness. Devices or structures that have a transmission or reflection spectrum can have their colors accurately displayed through spectral inversion in the visible light band. Essentially, each device or structure has its own unique spectral distribution, resulting in distinct color forms.