Mechanism of color change of flexible metafilm with structural parameters and stretching methods

Color change metafilm is promising for the color printing of displays and imaging. A metallic array on an elastic film can realize dynamic color change by mechanical extensions, such as uniaxial or biaxial stretching. In this study, an electromagnetic model of a flexible metafilm composed of a microscale Al cylindrical array on a polydimethylsiloxane elastic film was constructed to study the underlying optical mechanism of color change, especially brightness and saturation changes, of the flexible metafilm with structural parameters (diameter and height of the Al cylinders) and stretching methods (uniaxial or biaxial stretching). The 3D finite time domain difference method was used to simulate the propagation behavior of electromagnetic waves through metafilm. With increasing diameter, the lightness increases while the saturation decreases, which is due to the change of surface plasmon resonance from local surface plasmon resonance (LSPR) to propagation surface plasmon (PSP) and Wood Anomaly (WA) and finally to magnetic polariton excitation. With increasing height, lightness first decreases and then increases, while the change in saturation is opposite, which is due to the first increasing and then decreasing of the intensities of PSP and WA. By comparing the dynamic color changes of metafilm under uniaxial stretching and biaxial stretching, it is found that uniaxial stretching achieves lower lightness and saturation under small strain and higher lightness and saturation under large strain, which is caused by the enhanced WA and PSP on the Al-air interface at short wavelengths and the weakened LSPR on the Al-PDMS interface at long wavelengths. This discovery paves the way for practical applications of structural color display with high saturation and brightness.


Introduction
Colors of objects can be classified as chemical or structural color. The former is generated by reflected light, which is modulated by the absorption or emission of pigments or dyes. The chemical color is usually angleinsensitive, but it undergoes photochemical degradation over time, accompanied by color fading. The structural color is produced by the reflected light modulated by the interaction between incident light and the surface with the micro-and/or nanoscale structure. Compared with chemical color, structural color is of high durability, environmental friendliness, and superior pattern resolution [1,2]. There are two categories of optical mechanisms to produce structural color: Bragging scattering and surface plasmon resonance (SPR). Because the band gap of Bragg scattering in photonic crystals prevents the transmission of light by complete reflectance, light interference occurs at the top surface, producing structural color [3,4]. SPR refers to the coupling of the incident wave with the collective charge density oscillations on the conductor, which leads to the selective absorption of light so that the reflected and/or transmitted light generates structural color [5][6][7]. Recent advances in micro-and nanoscale fabrication technology offer a new platform for developing structural color on films with artificial structures, i.e., so-called metafilms. Bragging scattering is usually exploited in dielectric metafilms (DMFs), and SPR is usually exploited in metallic metafilms (MMFs) [8]. Compared with DMFs, MMFs are usually thinner [9] and less angular dependence [10].
Color change materials play important roles in the color printing of display and imaging [11]. Structural color is related to the material properties and structural parameters, so color change can be achieved by altering them [12]. Material parameters such as refractive index can be altered by using electrochemical reaction materials [13,14] or phase change materials [15,16]. For example, repeatable dynamic color change has been generated by cyclically controlling the reversible hydrogenation/dehydrogenation of the constituent magnesium nanoparticles of a metasurface [17] and the amorphous/crystalline phase of the Ge 2 Sb 2 Te 5 layer of a multilayer metamaterial [16]. Structural parameters, for example, the period of a micro/nano metal array, have been modulated by using ductile hydrogels [18] or flexible polymer materials [19]. Satoru Hamajima et al [18] prepared gold arrays with sub-100-nm dots on silicon substrates by electron beam lithography and transferred them onto the surface of the ductile hydrogel. The gel size changed with the concentration of the ion solution, and determined the pattern area, resulting in dynamic color change. However, due to the heterogeneity of the polymer network in the gel, the transformation is not uniform. Using a flexible polymer as a tensile substrate for metafilm can actively tune the structural color by mechanical extension. The polymers show a quick response to the extension, which is repeatable [20][21][22]. Ming-Lun Tseng et al [23] reported a device integrating a subwavelength-scale Al rectangular array on an elastomeric polydimethylsiloxane (PDMS) substrate. By uniaxial stretching of the substrate along either of two perpendicular periodic directions, the resonance wavelength of the scattering spectrum has been shifted. The authors reported the experimental result of dynamic color change of the device but did not explore the optical mechanism. Shichao Song et al [24] developed a composite structure made up of a nano Al cylindrical array and a PDMS substrate. The dynamic color change of the structure was experimentally achieved by uniaxial stretching along the period of Al cylindrical nanoparticles. Moreover, they experimentally demonstrated that the structure could also implement a vivid dynamic color change by biaxial stretching along the periods in two vertical orientations. However, the optical mechanism underlying the phenomenon of the dynamic color change under different stretching methods has not been elucidated. On the other hand, the three inherent merits of color are lightness, saturation, and hue. The device reported in Ming-Lun Tseng et al [23] achieved a full-color display, which indicates that different hues of metamaterial have been achieved [23]. However, the saturation and lightness have not been evaluated in previous papers [21,23,24]. The saturation and lightness are vital in display, imaging, and colorimetric sensors [25].
In this study, a metafilm with a 3D structure composed of a microscale Al cylindrical array on a PDMS substrate is modeled. The electromagnetic fields and reflectance/absorption spectra of the metafilm are solved using the 3D finite time domain difference method (FDTD). Structure colors, especially the lightness and saturation, and the optical mechanism of color change of the metafilm with diameters and heights are discussed. Then, the optical mechanism of the dynamic color change of metafilm under uniaxial and biaxial stretching modes is discussed.

Model
The metafilm, of which the schematic is shown is in figure 1(a), is composed of a microscale Al cylinder array on a PDMS substrate. The top surfaces and the peripheral surfaces of the Al cylinders are embedded in a homogeneous dielectric, i.e., air. Coordinates are chosen to make the array lie on the X-Y plane. The structural parameters of the metafilm are the diameter (D), height (H) of the Al cylinders, and periods (P) between the adjacent cylinders, which are further demonstrated as P x or P y along the X or Y axis, respectively. A plane electromagnetic wave is used as a source with a wavelength band from 380 nm to 780 nm (visible band) to simulate the visible spectrum of Sunlight. The magnetic field (  H in ), electric field (  E in ), and wave vector (  K in ) of the source are along the Y, X, and Z axes, respectively.
When the plane electromagnetic wave is incident on the top surface of the metafilm, a part of the incident wave is absorbed because of coherent oscillations between the wave and the electrons of the metallic conduction band of Al cylinders, while a part is reflected by the top surface, and the rest enters the transparent PDMS substrate and reaches the PDMS/air interface. When the wave reaches the interface, because the reflectivities in the visible band calculated by Fresnel's law is below 0.03 [26], it is assumed that the reflected wave at the interface is small enough to be ignored, and all the wave transmitted is absorbed by the surrounding environment of the space. In the above processes, the propagation of the electromagnetic wave in materials is governed by Maxwell's curl equations: where  E and  H are the electric field intensity and magnetic field intensity, respectively.  D and  B are the electric displacement field and magnetic flux density, respectively.   J is the conduction current density. The constitutive relations of the electromagnetic property of a material include     s = J E and the polarization and magnetization responses. In this study, it is assumed that there is no conductive current in the materials, that is,   J = 0. Al and PDMS are linear, isotropic, and nonmagnetic materials, in which the polarization and magnetization responses can be expressed as where ε is the dielectric constant of the material. The dielectric constant of Al was taken from the data of Palik [27], and that of PDMS was set as a constant of 2 [28]. ε 0 = 8.854 × 10 −12 F m −1 is the vacuum dielectric constant. μ 0 = 4π × 10 −7 N A −2 is the vacuum permeability. Thus, equations (1) and (2) can be respectively expressed as The evolution of the initial electromagnetic field with time is governed by equations (5) and (6) in conjunction with the boundary conditions. Figure S1 shows the diagram of the initial electromagnetic field and boundary conditions on the x-z plane. The plane electromagnetic wave is taken as the initial electromagnetic field, which is above the metafilm. For the boundary conditions, perfectly matched layer (PML)-absorbing boundary conditions were applied to the air and the PDMS substrate along the Z axis, whereas periodic boundary conditions were used at boundaries along the X and Y axes. Equations (5) and (6) are solved by the finite-difference time-domain (FDTD) method, in which both time-and space-derivatives are approximated with central finite differences. Spatial discretization is carried out in the form of Yee cells, as presented in figure  S2. The electric field is defined at the edge of a cell and the magnetic field is defined at the face of the cell.  E and  H are calculated by the alternating sampling method, in which each sampling time interval is half a unit of time (Dt ).Thus, equations (5) and (6) are discretized into , , , , where i, j, and k represent a spatial nodal of the material in the x, y, and z directions, as shown in figure S3. The nodal electric field is located exactly halfway between magnetic field nodes. Every electric field (magnetic field) component changes within a unit of time will change the four surrounding magnetic field (electric field) components at half a unit of time. For example, when the E x represents the dielectric constant of the spatial nodal. The electric field and magnetic field change with the change of ε. The ε of the material in the corresponding space is determined by the geometric parameters of Al nanocylinders, so the electric field and magnetic field are related to the geometric parameters. As mentioned above, when the initial electromagnetic field and boundary conditions are given, the variation in the electromagnetic field with time can be obtained by iteratively solving equations (7) and (8). In this study, Eastwave V6.0 software with the FDTD method was employed to solve the electromagnetic field. In the software, the power monitor of the reflectivity (R) was above the wave source, and that of the transmissivity (T) was between the top and the bottom surfaces of the PDMS substrate: where source power is the power of the plane electromagnetic wave.
real monitor is the time-averaged power flowing across the surface of the power monitors of R or T. dS is the surface of the power monitor of R or T. The factor of 1/2 is related to the time averaging of the continuous wave fields. P is the complex Poynting vector, which is calculated by where the symbol * denotes the complex conjugate. Absorptivity (A) was calculated by the formula A = 1 − R − T. The validations of grid independence and the numerical model are presented in figure S4. The reflected structural color can be calculated from the reflectance spectrum (R(λ)), which is described in detail in the supporting information. In this study, the CIE 1931 color space chromaticity diagram is adopted to qualitatively represent the color, and the RGB color is adopted to represent the real color as humans see in the natural world. Moreover, the lightness, saturation, and hue of the color are presented to quantitatively describe the color.

Results and discussion
The color of the metafilm can be tuned by changing the structural parameters (diameter, height, and period) of the microscale Al cylinder array. The period can be changed by stretching the metafilm, while the diameter and height are usually fixed and need to be determined in advance. Thus, the effects of diameter and height on the color characteristics of metafilm are studied first. Then, the dynamic color change of metafilm with uniaxial or biaxial stretching is discussed.

Effects of diameter and height on the color characteristics of metafilm
When the initial periods are 320 nm (≈0% strain) and the maximum periods are 500 nm (≈55% strain) in the stretching of metafilm, the deformation belongs to linear elastic deformation [29], and the wavelengths corresponding to the reflectivity peak are at 478 nm and 705 nm [24], respectively, which are all in the visible spectrum. Thus, the effects of diameter and height on the color characteristics of metafilm are studied first with an initial period of P x = P y = 320 nm. The colors of the metafilm are calculated by the reflectance spectra. Figure 1(b) shows the calculated RGB colors of the metafilm corresponding to different combinations of diameters (100 nm to 300 nm, in steps of 20 nm) and heights (25 nm to 275 nm, in steps of 25 nm). When the diameters are less than 100 nm, the lightness of the metafilm are very dark and thus are not shown. When the diameters are larger than 300 nm, the line width between the two Al cylinders are too small to fabricate metafilm. When the heights are large, the Al cylinders easily topple when metafilm is stretched, so suitable heights from 25 nm to 275 nm are discussed. As shown in figure 1(b), when the diameter is 100 nm, the colors are dark purple at various heights. When diameters and heights are different, the hues of RGB colors are different, which can also be seen from figure S5. As shown in figure 1(c), the lightness of metafilm increases with diameter. With increasing height, lightness first decreases and then increases. Lightness greater than 80 can be obtained when the diameters are greater than 240 nm and the heights are lower than 100 nm or greater than 225 nm. As shown in figure 1(d), the saturation of metafilm decreases with diameter, and saturation first increases and then decreases with height. Moreover, when the diameters are lower than 120 nm, the saturation is greater than 80. Figure 1(e) shows examples of reflectance spectra of different diameters with H = 125 nm, and their RGB colors correspond to the colors in the red dashed lines in figure 1(b). Figure 1(f) shows examples of reflectance spectra of different heights with D = 200 nm, and their RGB colors correspond to the colors in the black dashed lines in figure 1(b). As shown in equations (S5) and (S8), lightness (L) increases as the tristimulus value (Y) increases. Y of an object is shown in equation (S1), which depends on the integration of ( ) ( )¯( ) l l l D R y from 380 nm to 780 nm. The distribution of ( )¯( ) l l D y is shown in figure S6. The wavelength of the peak of ( )¯( ) l l D y is 550 nm. Thus, high reflectance at the wavelength of approximately 550 nm, means large Y and L. The spectral reflectance at approximately 550 nm increases with diameter, while it first decreases and then increases with height, which leads to the change rule of lightness discussed above. Saturation is related to the spectral bandwidth of the reflectance spectrum. When the spectral bandwidth is narrow, saturation is high [30,31]. However, the bandwidth increases with diameter, while it decreases first and then increases with height, so the change trend of saturation is opposite to that of lightness.
The reflectance spectrum is caused by the selective absorption of light, which is generated by the coherent oscillation between conduction band electrons of the microscale Al cylinder array and the incident light, so it is more convenient to understand of the reflectance spectrum from the view of absorption. The absorptance spectra of different diameters with H = 125 nm are plotted in figure 2(a), and the structural parameters are consistent with those in figure 1(e). The absorptivity peak increases as the diameter increases. The wavelength corresponding to the absorptivity peak (λ max ) shifts toward the red band with diameter. This phenomenon can be explained by using the Maxwell-Garnett theory (MGT) [32], in which it is assumed that there is an effective complex dielectric constant ε e for a composite film containing a metallic array and dielectrics [33]: where q = D 2 /P 2 is the particle filling factor (or volume fraction) of the Al cylinders [34]. ε av is the average dielectric constant of the PDMS substrate and the surrounding air [35]. ε m is the dielectric constant of Al. κ is the shape-dependent screening parameter ranging from unity to infinity. κ approaches unity when long needlelike particles are oriented with their axes of revolution parallel to the direction of incident light and approaches infinity for thin flat disks oriented with their axes of revolution perpendicular to the incident light [36]. The surface plasmon resonance (SPR) wavelength (λ SPR ) is calculated by the expression, λ SPR = 2πc/ω SPR , where ω SPR is SPR frequency, which is defined as the frequency for which real part of dielectric constant is zero (Re(ε e ) = 0) [37], then [38] where λ p = 79 nm, which is calculated by λ p = 2πc/ω p . Here, ω p = 2.4 × 10 16 rad s −1 is the plasma frequency of Al. λ SPR shifts to the red band with an rising q. Because q is proportional to the diameter, λ SPR shifts toward the red band with increasing diameter. The κ of a nanocylinder is 4 when the aspect ratio (height/diameter) is 1.25, and κ decreases when the diameter increases [39]. In this study, according to the calculation results in figure 2(a), as the diameter increases from 100 nm to 300 nm, the aspect ratio decreases from 1.25 to 0.42, and κ calculated by equation (12) decreases from 19.4 to 8.7 [39]. Therefore, the redshift of λ max may be caused by the SPR on metafilm. More specifically, the SPR can be caused by different resonance phenomena, such as propagation surface plasmon (PSP) [40], local surface plasmon resonance (LSPR) [32], and Wood's anomaly (WA) [41]. To identify the resonance type, the electromagnetic fields at diameters of 100, 200, and 300 nm are presented from figures 2(b)-(e), in which the squared magnetic fields are shown by contours and the electric field vectors are indicated by arrows. When the diameter is 100 nm, the wavelength corresponding to the absorptivity peak is λ = 450 nm, as illustrated in figure 2(a). Accordingly, the electric field is resonantly excited along the direction of the incident polarization (X axis), and the magnetic field is distributed on the Al-PDMS interface below the Al cylinder, demonstrating that the absorptivity peak originates from the LSPR [42], as seen from figure 2(b). When the diameter increases to 200 nm, the magnetic field is inspired at the air gap between the Al cylinders and on the Al-PDMS interface below the Al cylinder, which is characteristic of the PSP and WA, as illustrated in figure 2(c). However, when the diameter increases to 300 nm, which is close to the period of 320 nm, a strong magnetic field is confined in the air gap between the neighboring Al cylinders, accompanied by an electric current loop, which is the typical characteristic of magnetic polariton (MP) excitation, as clearly shown in figures 2(d) and (e) [43][44][45]. MP is the resonance of magnetic oscillation on the material surface coupled with incident light, in which the material response resonantly enhances the total magnetic field due to the occurrence of strong diamagnetism, altering the real part of the magnetic permeability to be negative [46]. As shown in figures 2(d) and (e), the occurrence can be explained as follows: the incident light produces an oscillating magnetic field, which in turn induces a current in the periodic array of Al cylinders along one direction and another current near the surface of the PDMS substrate in the opposite direction [47]. These anti-parallel currents induce the diamagnetic response, which excites the magnetic polariton. Two absorptivity peaks are generated when the diameters are 280 nm and 300 nm, as depicted in figure 2(a). To clearly explain the mechanism of these two peaks, the electric fields of the peaks at 452 nm and 537 nm with diameters of 300 nm are shown by contours in figures S7(a) and (b), and it is obvious that the electric fields mainly focus on the edges of the Al cylinder on the Al-air interface and the Al-PDMS interface, respectively. Thus, the peak at 452 nm is caused by the MP located mainly on the Al-air interface, while the peak at 537 nm is caused by the MP located mainly on the Al-PDMS interface. According to the above analysis, it is concluded that the redshift of λ max is caused by the SPR or MP on the metafilm. Figure 2(f) shows the shift of λ max with height. λ max first blueshifts when the height increases from 25 nm to 75 nm, then redshifts when it increases from 75 nm to 200 nm, and finally blueshifts when the height is greater than 200 nm. The electromagnetic fields at 25, 75, 200, and 225 nm are presented from figures 2(g)-(j), respectively, which proves that the relative proportion changes of LSPR, PSP, and WA lead to complicated λ max shift behavior. For instance, LSPR on the Al-air interface dominates the absorption of metafilm when the height is 25 nm, while LSPR on the Al-PDMS interface and PSP and WA are strong when the height is increased to 75 nm, which corresponds to the higher absorption peak in figure 2(f). However, when the height is 225 nm or 275 nm, LSPR, PSP, and WA are motivated simultaneously, but the magnetic field intensity caused by PSP and WA is lower than that of metafilm with 75 nm. According to the analysis, the results above can be used to predict the absorption characteristics with different diameters and heights. Moreover, it can be concluded that the resonance phenomenon changes from LSPR to PSP and then to MP with increasing diameter, which leads to an increase in lightness and a decrease in saturation. Because the intensities of PSP and WA are first increase and then decrease with increasing height, lightness first decreases and then increases, while the change in saturation is the opposite.

Comparison of the dynamic color change of metafilm with uniaxial and biaxial stretchings
In the previous section, the change in static color with height and diameter was discussed. In this section, the focus is on the dynamic color change comparison of metafilm with uniaxial and biaxial stretchings, as schematically shown in figures 3(a1) and (b1), respectively. Uniaxial stretching is assumed along the X axis, and the deformation characteristics of the metafilm along the Y axis are determined by the mechanical properties of the PDMS substrate. As mentioned above, when the strain of the PDMS substrate is between 0 and 55% with uniaxial stretching, experiments have proven the behavior of linear elastic deformation of metafilm [29]. Thus, the relationship between P x,u and P y,u is: where v is Poisson's ratio, which is 0.5 [29]. P x0 and P y0 are the initial periods in the X axis and Y axis, respectively, and P x0 = P y0 = 320 nm. The maximum period of P x,u is 500 nm, and the corresponding P y,u is 230 nm. The maximum period P x,b is 500 nm (≈55% strain), and the corresponding P y,b is 500 nm. Under uniaxial stretching, the RGB colors of the metafilm with different diameters at H = 125 nm and different heights at D = 200 nm are shown in figure 3(a2). Under biaxial stretching, the RGB colors of metafilm with different diameters at H = 125 nm and different heights at D = 200 nm are shown in figure 3(b2). The change in RGB colors is obvious under both stretching modes. Under the two stretching methods, the hues of RGB colors both change from blue to green, then to red, and finally to blue, which is also shown in figure S8. The hue angle decreases first, then increases, and finally decreases as P x,u or P x,b increases. For example, when H = 125 nm and D = 200 nm, the changes in colors show similar trends of lilac, green, red, and purple, although the color is blue when P x,u reaches 460 nm. However, the color difference generated between the different stretching methods increases with period. It is shown that the relatively higher lightness and saturation of the color of the metafilm can be obtained when the diameter is 200 nm and the height is 125 nm, as illustrated in figures 3(b), 1(c), and (d). Thus, these geometrical parameters are selected for further discussion. To further investigate the cause of the hue change trend and color difference, reflectance spectra under uniaxial stretching and biaxial stretchings are displayed in figures 3(a3) and (b3), respectively. Under the two stretching methods, the wavelength corresponding to the reflectivity peak shifts toward the red band, and the second peak wavelength appears in the blue band, so the proportion of spectrum integration in the long wavelength band first increases and then decreases. Thus, the change trend of hues is similar, as presented in figures 3(a2), (b2), and (c). The RGB color difference is due to the diverse reflectivity distribution with wavelengths from 380 nm to 780 nm. For example, with P x,u = P x,b = 400 nm, compared with biaxial stretching, the reflectivities of metafilm under uniaxial stretching are larger at wavelengths from 380 nm to 520 nm and are lower at long wavelengths from 520 nm to 780 nm.
The lightness and saturation of metafilm under different stretching methods are shown in figures 3(d) and (e), respectively. P x is applied to represent P x,u and P x,b , which are equal under the same strain. As seen from figure 3(d), there is a critical period of 460 nm. Below the critical value, the lightness of the metafilm by biaxial stretching is higher than that by uniaxial stretching as P x increases. Above the critical value, the result is the opposite. The spectral reflectance of metafilm under biaxial stretching approximately 550 nm is larger than that under uniaxial stretching when the period is less than 460 nm, so the lightness is higher. However, under uniaxial stretching, as the period increases, the reflectivity peak at the short band is higher, and the corresponding wavelength gradually approaches 550 nm, so the lightness of uniaxial stretching is larger than that of biaxial stretching. As illustrated in figures S9(a) and (b), the critical period is smaller with increasing diameter and larger with increasing height, so high lightness can be obtained by biaxial stretching of metafilm with small strain (P x from 320 nm to the critical period) and can also be obtained by uniaxial stretching of metafilm with large strain (P x > the critical period). It is shown in figures 3(a3) and (b3) that the overall spectral bandwidth mostly increases as P x increases, so the overall saturation mostly increases, which is shown in figure 3(e). Compared with uniaxial stretching, biaxial stretching achieves larger saturation at P x of 320, 340, 360, 380 and 500 nm and lower ones at P x of 400, 420, 440 and 460 nm. As illustrated in figures S9(c) and (d), with different diameters and heights, biaxial stretching achieves larger saturation of the metafilm at small strain, and uniaxial stretching achieves larger saturation of the metafilm at large strain. Thus, it is concluded that structural colors of high saturation and lightness can be obtained by biaxial stretching under small strain and uniaxial stretching under high strain.
The resonance mechanism is presented to explain the color difference of metafilm under different stretching methods with normal incidence. The absorptance spectrum is displayed in figure 4(a). With the increase in P x , it is found that the wavelength corresponding to the absorptivity peak (Peak 1) shifts toward the red band, and Peak 1 decreases, while another absorptivity peak (Peak 2) appears in the blue band and then shifts toward the red band for both stretching methods. Moreover, under different stretching methods, Peak 1 is generated at The resonance wavelengths of first-order PSP and WA (line) and the peak wavelengths (point) corresponding to (a). (c) Electromagnetic field distribution in the X-Z plane (y = 0 nm) at λ = 401 nm under uniaxial stretching when P x-u is 400 nm. (d) Electromagnetic field distribution in the X-Z plane (y = 0 nm) at λ = 431 nm under biaxial stretching when P x-b is 400 nm. (e) Electromagnetic field distribution in the X-Z plane (y = 0 nm) at λ = 561 nm under uniaxial stretching when P x-u is 400 nm. (f) Electromagnetic field distribution in the X-Z plane (y = 0 nm) at λ = 572 nm under biaxial stretching when P x-b is 400 nm. almost the same wavelengths. As illustrated in figure 2(c), it is known that PSP and WA phenomena are generated on the Al-PDMS interface at the initial period. To scrutinize the contributions made by the PSP and WA phenomena at different periods, resonance absorption wavelengths can be calculated by [40,[48][49][50] ( ) where n x and n y are integer numbers signifying the scattering orders of resonance in the X axis and Y axis, respectively. The incident light is along the X axis, so n x = 1 and n y = 0. ε m and ε d are the dielectric constants of the metal and medium, respectively. Peak 1 (triangle spots) at short wavelengths, Peak 2 (circle spots) at long wavelengths, and resonance absorption wavelengths (lines) are shown in figure 4(b). Peak 1 is caused by the resonance absorption of the first-order WA and PSP on the Al-air interface, and Peak 2 is related to the firstorder WA and PSP on the Al-PDMS interface. However, as shown in figure 4(a), the difference in absorptivity between different stretching methods increases with increasing P x . The distribution of electromagnetic fields of the peaks with P x = P x,u = P  Figure 4(c) illustrates the unique feature of the magnetic field at the cylinder's top surface and the air above the adjacent cylinders, which is caused by PSP and WA on the Al-air interface. In figure 4(d), the magnetic field mainly focuses on the top surface of the Al cylinder on the interface of Al and air, which indicates that the peak is mainly caused by LSPR. Figures 4(e) and (f) show that magnetic fields are mainly located on the bottom surface of the Al cylinder and space between the adjacent cylinders on the Al-PDMS interface. However, compared with uniaxial stretching, the magnetic fields are stronger on both sides of the Al cylinder under biaxial stretching, which is caused by the stronger PSP and WA on the Al-PDMS interface. As shown in figures 3(a3) and (b3), the wavelength corresponding to the first reflectivity peak shifts toward the red band, and the second reflectivity peak appears in the blue band, which is similar to the change in absorptance spectra in figure 4(a). Thus, it is concluded that the transverse narrowing behavior of metafilm on the Y axis under uniaxial stretching enhances PSP and WA on the Al-air interface, weakens LSPR on the Al-PDMS interface at short wavelengths, and weakens PSP and WA on the Al-PDMS interface at long wavelengths, which results in stronger absorptivity and reflectivity at short wavelengths and weaker absorptivity and reflectivity at long wavelengths when compared with biaxial stretching. The RGB colors of metafilm with different incident angles when P x increases are shown in figures S10(a) and (b). Under different incident angles, the dynamic color change of metafilm with an increased period is similar to that under normal incident light. This demonstrates that the colors of metafilm are angle anisotropic, which makes it promising in many circumstances, such as color displays and anticounterfeiting applications.

Conclusion
In summary, the structural color change of a metafilm formed by an Al micro/nanocylindrical array on a PDMS substrate was theoretically studied. It was found that the increase in diameter will diminish the space between Al cylinders, resulting in strong PSP and WA and weak LSPR, while MP is generated as the value of diameter approaches the value of period. Thus, the lightness of metafilm increases with diameter, while saturation decreases with diameter. The intensities of PSP and WA on the Al-PDMS interface first increase and then decrease with height, so the lightness of the metafilm first decreases and then increases, while the change in saturation is the opposite. Under the biaxial and uniaxial stretching methods, the wavelength corresponding to the reflectivity peak shifts toward the red band, and another reflectivity peak appears in the blue band, so the hues of metafilm change similarly. Compared with biaxial stretching, uniaxial stretching achieves larger reflectivity at short wavelengths and lower reflectivity at long wavelengths. This is caused by the enhanced firstorder PSP and WA on the Al-air interface at short wavelengths and the weakened first-order PSP and WA on the Al-PDMS interface at long wavelengths. Therefore, the RGB colors of metafilm under the two different stretching methods are different. More importantly, the structural color of higher saturation and lightness can be obtained by biaxial stretching under small strain and uniaxial stretching under high strain, which is useful in color printing of displays and imaging.