Effects of roughness and resonant-mode engineering in all-dielectric metasurfaces

2.1 Roughness and imperfection of fabricated metasurfaces

There are many applications of all-dielectric metasurfaces. One example is a full-color reflective display system based on such metasurfaces [13], [14], [15], which provides a promising solution to problems with current mobile device screens, such as high energy consumption and lack of readability under bright sunlight. The full-color reflective display system contains three tandem band reflectors (blue, green, and red), all of which are constructed by all-dielectric metasurfaces. However, due to imperfections in fabrication, there are differences between the designed metasurfaces (ideal metasurfaces) and the fabricated ones.

Figure 1A–C shows the schematic of an ideal all-dielectric metasurface, the schematic of the fabricated all-dielectric metasurface, and the SEM images of the fabricated three-color (blue, green and red) band reflector based on the all-dielectric metasurface, respectively. These images clearly indicate that the fabricated all-dielectric metasurfaces are different from the ideal one. As shown in Figure 1A, each nanopillar of the ideal all-dielectric metasurface is a frustum with smooth surfaces. However, the nanopillars in the fabricated all-dielectric metasurfaces (Figure 1B and C) are imperfect. There are some “bumps” on the nanopillar surface, which makes the surface rough.

Figure 1:

All-dielectric metasurfaces for visible light range applications.

(A) Schematic of the ideal all-dielectric metasurface. The nanopillar is a frustum with smooth surfaces. (B) Schematic of the fabricated all-dielectric metasurface. The surfaces of nanopillar are rough. (C) SEM images of fabricated band reflector based on all-dielectric metasurfaces for blue, green, and red light, respectively. (D) Calculated reflection spectra of the ideal all-dielectric metasurfaces (A). (E) Measured reflection spectra of the fabricated all-dielectric metasurfaces (C), which are obviously different from the calculated spectra.

For band reflectors used in the visible light range, the wavelength of the manipulated light is in the range 380–740 nm [19], which is much shorter than that of IR light. As a result, the roughness or imperfection of the all-dielectric metasurface has a large effect on its performance. For each band reflector, there are two reflection peaks in the calculated reflection spectrum (Figure 1D) when the ideal metasurface structure (Figure 1A) is used for numerical calculations. However, only one peak is observed in the measured reflection spectrum (Figure 1E). These data were obtained from the fabricated band reflector (Figure 1C). It is especially noticeable that all three band reflectors lose the reflection peak at longer wavelengths. This indicates that the effects of metasurface roughness are not even. Since the two reflection peaks represent different optical resonant modes, the metasurface roughness should have different effects on different optical resonant modes.

2.2 Effects and study of the metasurface roughness

To study how the roughness of the metasurface affects its performance, the red light band reflector based on an all-dielectric metasurface is taken as an example. Figure 2A and B shows the schematic of the all-dielectric metasurface for the red light band reflector and the corresponding calculated reflection spectrum at normal incidence, respectively. The incident light is unpolarized in Figure 2B. The all-dielectric metasurface contains a nanopillar array on a SiO₂ substrate, and each nanopillar has two layers (TiO₂ and SiO₂). There are two reflection peaks in the calculated spectrum at 652 and 710 nm. As mentioned above, the peak at the longer wavelength (710 nm) is absent in the measured spectrum (Figure 1E).

Figure 2:

Numerical calculations of the ideal all-dielectric metasurface.

(A) Schematic of the ideal all-dielectric metasurface for the red light band reflector, which has two layers (TiO₂ and SiO₂). (B) Calculated reflection spectrum of the ideal all-dielectric metasurface at normal incidence. There are two reflection peaks at 652 and 710 nm. (C) |E|² distributions of the ideal all-dielectric metasurface at different wavelengths (652 or 710 nm) in different planes (YZ-plane or XZ-plane). (D) Electrical field component (E_‖ or E_⊥) distributions of the ideal all-dielectric metasurface at different wavelengths (652 or 710 nm) in the YZ-plane, where E∥=Ex2+Ez2 and E_⊥=E_y. E_‖ represents the electrical field component that is parallel to the nanopillar surface, while E_⊥ represents the electrical field component that is perpendicular to the nanopillar surface.

Figure 2C shows the |E|² distributions of the ideal all-dielectric metasurface at different wavelengths (652 or 710 nm) in different planes (YZ- or XZ-plane). Both the YZ- and XZ-planes go through the nanopillar center. Because of the symmetry of our all-dielectric metasurface, the reflection spectrum is independent of the polarization at normal incidence [15]. To show the resonant modes clearly, the incident light is set to propagate along the z-axis and is y-polarized in Figure 2C and D. It is obvious that the electrical field in the YZ-plane is much larger than that in the XZ-plane; therefore, the field distribution in the YZ-plane is a better representation of the optical resonant modes. Therefore, the following discussion will be focused on the YZ-plane. Although the TiO₂ layer of the nanopillar is a frustum, the nanopillar sidewalls are almost perpendicular to the y-axis in the YZ-plane. Hence, E∥=Ex2+Ez2 and E_⊥=E_y are used here to represent the electrical field components that are parallel or perpendicular to the nanopillar surface, approximately. Figure 2D shows the distribution of the electrical field components (E_‖ or E_⊥) at 652 or 710 nm in the YZ-plane. Comparing the four field distributions in Figure 2D, it can be observed that E_‖ and E_⊥ contribute almost equally to the optical resonance at 652 nm, while E_‖ contributes much more to the optical resonance at 710 nm than E_⊥. In addition, at both wavelengths (652 and 710 nm), E_‖ overlaps with the nanopillar sidewalls significantly, while E_⊥ is mainly distributed inside the nanopillar.

The metasurface roughness can be regarded as “bumps” on the metasurface sidewalls. Intuitively, these “bumps” should have different effects on E_‖ and E_⊥ because the distributions of E_‖ and E_⊥ are different. In addition, E_‖ is oscillating along these “bumps”, so E_‖ receives larger scattering or absorption from these “bumps” compared to E_⊥. Also, Figure 2C and D shows that the electrical fields are mainly distributed in the TiO₂ layer. The refractive index of TiO₂ (around 2.6) is much higher than that of SiO₂ (around 1.5) [12], [14], [20]. As a result, the roughness of the TiO₂ layer has a larger effect on the metasurface’s performance. In the following discussion, only the roughness of the TiO₂ layer will be considered. Moreover, the nanopillar shape is adjusted to frustum in order to fit the real situations better.

As mentioned above, the FDTD method is adopted to carry out numerical calculations in this work. In these calculations, a periodic boundary condition is applied. If structural imperfection is added directly to the unit cell to simulate the roughness, this roughness becomes a part of the unit cell and hence periodic. The periodic features and random features scatter light very differently, so we cannot directly put the roughness feature into the unit cell to study the effects of surface roughness. Hence, a thin lossy layer is added to the nanopillar surface to mimic the surface roughness equivalently. Here, the volume-current method is adopted [18]. TiO₂ and the background (n=1) have high refractive index contrast, so the TiO₂ bump is in a high-index-contrast boundary with a small volume ΔV and a shift in the dielectric constant Δee. The electric field will induce a dipole moment in the bump by inducing charge density Δρ, as shown in Figure 3A. This induced dipole moment can be expressed as

Figure 3:

Effects of metasurface roughness.

(A) A “bump” on the interface between two materials ee₁ (TiO₂) and e₂ (background). An applied electric field E will induce a dipole moment on the surface (blue/red color denotes positive/negative charge). (B) Schematic of all-dielectric metasurface with a lossy shell. The 5-nm-thick lossy shell is outside the TiO₂ layer. (C) Calculated reflection spectrum of the all-dielectric metasurface with the lossy shell (B) at normal incidence. (D) Electrical field component (E_‖ or E_⊥) distributions of the all-dielectric metasurface with the lossy shell (B) at different wavelengths (652 or 710 nm) in the YZ-plane. (E) Calculated reflection spectra of the ideal all-dielectric metasurfaces (Figure 1A), measured reflection spectra of the fabricated all-dielectric metasurfaces (Figure 1B and C), and calculated reflection spectra of the all-dielectric metasurface with the lossy shell (B). All spectra are at normal incidence. It is obvious that the calculated reflection spectra of the all-dielectric metasurface with the lossy shell are much closer to the measured spectra.

where α_‖ and γ_⊥ are polarizability tensors, and E_‖ and D_⊥ are the surface parallel and perpendicular components. Polarizability tensors α_‖ and γ_⊥ are determined by the shape of the bump.

The oscillation of the induced dipole will cause energy loss [21]. The total time-averaged power radiated by a harmonically oscillating electric dipole is

where ω is the angular frequency, p is the dipole moment, μ₀ is the permeability in the vacuum, and c is the light speed in vacuum.

Since the material is linear and isotropic, D_⊥=E_⊥, where ε is the material permittivity. α_‖ and γ_⊥ are diageonal. Substituting Eq. (1) into Eq. (2), we get

Equation (3) shows that the parallel and perpendicular components of the electrical field (E_‖ and E_⊥) are scattered differently by the bumps (roughness); therefore they have different contributions to the energy loss. In other words, the bumps have different effects on the parallel and perpendicular components (E_‖ and E_⊥). Considering that the radiation loss due to the roughness and the material loss on the surface have the same effect on the optical resonant modes [22], the metasurface roughness effect can be considered equivalent to a lossy shell added to the nanopillar (Figure 3B). The index of the lossy shell is n_{lossy shell}=n+ik, where n is the refractive index of TiO₂, and k is the absorption constant. Since roughness has different effects on the parallel and perpendicular components, k=(kxx000kyy000kzz). By adjusting the values of k_xx, k_yy, and k_zz, the effect of roughness on the nanopillar sidewalls can be simulated equivalently.

Figure 3C shows the calculated reflection spectrum of the all-dielectric metasurface with the lossy shell (Figure 3B) at normal incidence. It is noticeable that the peak at longer wavelength (710 nm) receives much larger effects from the lossy shell, which fits the experimental results quite well. Figure 3D shows the distributions of the electrical field component (E_‖ or E_⊥) of the all-dielectric metasurface with the lossy shell (Figure 3B) at different wavelengths (652 or 710 nm) in the YZ-plane. Notice that the color scale bar in Figure 3B is same as in Figure 2D. Compared to those in Figure 2D, the distributions of the resonant modes have barely changed. However, the resonance intensity at 710 nm has decreased considerably, which is in accord with Figure 3C.

Figure 3E shows the calculated reflection spectra of the ideal all-dielectric metasurface (Figure 1A), the measured reflection spectra of the fabricated all-dielectric metasurface, (Figure 1B and C), and the calculated reflection spectra of the all-dielectric metasurface with the lossy shell (Figure 3B). These spectra indicate that the numerical calculations with the lossy shell fit the experimental results much better, which demonstrates the validity of our assumption and derivation.

From the above discussion, it is clear that our proposed volume-current method has established a way to simulate the effects of random roughness equivalently. By comparing the experimental spectra and numerically calculated spectra, the equivalent loss (i.e., k_xx, k_yy, and k_zz) of the lossy shell corresponding to the as-fabricated roughness level can be extracted. Based on those, the effects of different levels of roughness on the metasurface performance can be predicted. Assuming that the average roughness scale is ΔL, the bump volume is proportional to ΔL³. Figure 1C shows that the nanopillar sidewall is fully occupied by “bumps” (roughness). In this case, in unit area of the nanopillar surface, the total “bump” volume should be proportional to ΔL. Using Eq. (3), the radiated power should be proportional to ΔL²: in other words

The metasurface roughness is mainly determined by fabrication techniques. Our fabricated all-dielectric metasurfaces (Figure 1C) all have the same level of roughness. Using the fabricated red band reflector as an example, Figure 4A and B shows the top-view SEM image of the fabricated all-dielectric metasurface and the corresponding edge detection result (more images can be found in Supporting Information Figure S1). From the edge detection result, the average roughness (3σ) scale can be calculated, which is 8 nm. Using our proposed lossy shell method and Eq. (4), the effects of different level roughness on metasurface’s performance are calculated numerically (Figure 4C).

Figure 4:

Calculated results of effects of different level roughness.

(A) Top-view SEM image of all-dielectric metasurface. The roughness of nanopillars can be observed clearly. (B) Edge detection result of SEM image in (A). The average roughness scale is 8 nm (3σ). (C) Calculated reflection spectra with different levels of roughness and the measured reflection spectrum of the all-dielectric metasurface at normal incidence. (D) Resonant-mode extraction from fitting the measured reflection spectrum of fabricated all-dielectric metasurface. The three fitted peaks are at 594, 653, and 697 nm, respectively. Note that the second (653 nm) and third (697 nm) peaks correspond to the main optical resonances. The first peak (594 nm) represents the side lobe in the shorter wavelength region. (E) Resonant-mode extraction for the calculated reflection spectra in (C). Higher level of roughness has much more effect on the second peak, which is consistent with previous conclusion. (F) The change of the two fitting peaks in the reflection spectra with respect to the roughness level. The two red points are the experimental results, which demonstrate the effectiveness of the calculations.

Through fitting the reflection spectrum, the effects of different levels of roughness on different resonant modes can be revealed. Figure 4D shows the resonant-mode extraction using the measured spectrum of a fabricated all-dielectric metasurface as an example. Figure 4E and F shows the resonant-mode extraction results for Figure 4C and the change in value of two peaks in the reflection spectra with respect to the change in the roughness level, respectively. More separated resonant-mode extraction results can be found in Supporting Information (Figures S2–S9). It is obvious that a higher level roughness has much more effect on the resonant mode at the second main optical resonance, which is consistent with the above results (Figures 2C, D, and 3D). These analyses explain the effect of different roughness levels on the performance of all-dielectric metasurfaces, which could provide a guidance in designing more practical metasurfaces by taking into consideration the nanofabrication capability.

2.3 Engineering of resonant modes

Our experimental and theoretical data indicate that surface roughness has different effects on different resonant modes according to the field distribution of each resonant mode. This causes the absence of the peak at longer wavelengths in all three spectra of Figure 1E. Inspired by this, we propose a method to engineer the resonant modes by adding high-loss material in certain locations. Through precise design, we can keep the desired resonant modes while weakening the unwanted modes simultaneously.

Optical metasurfaces are usually sensitive to the incident angle [23], [24], [25], which limits their applications. For example, the incident angle sensitivity limits the viewing angle of a reflective display system based on all-dielectric metasurfaces [15]. Using the red light band reflector as the example, Figure 5A and B shows the schematics of the all-dielectric metasurface at oblique incidence and the calculated reflection spectra of the all-dielectric metasurface versus the incident angle, respectively. As shown in Figure 5B, the reflection spectra of the red band reflector change significantly with the change in the incident angle. Since the incident light for the red light band reflector in a real situation is unpolarized, the incident light was set as unpolarized in numerical calculations (Figure 5B). The wavelength for red light ranges from 625 to 740 nm [19], so the ideal red band reflector should reflect all the incident light in this range (625–740 nm) without reflecting any other wavelength. The red light band reflector based on all-dielectric metasurface works well at normal incidence (Figure 3E). However, its performance falls at oblique incidence (Figure 5B). This can be demonstrated by Figure 5C. The reflected light color changes from red to pink gradually with increasing incident angle from 0° to 39°. This phenomenon has two causes. First, side lobes less than 625 nm (unwanted resonant modes) appear in the reflection spectra with the increase in incident angle. Second, the reflection intensity of the light between 625 and 740 nm (desired resonant modes) decreases with the increase in the incident angle.

Figure 5:

Resonant mode engineering.

(A) Schematic of the all-dielectric metasurface at oblique incidence. (B) Calculated reflection spectra of the all-dielectric metasurface (A) versus the incident angle. (C) The numerically calculated color of the all-dielectric metasurface (A) versus the incident angle shown in a CIE 1931 color map. (D) Schematic of the all-dielectric metasurface with an a-Si cap at oblique incidence. The thickness of the a-Si cap is 40 nm. (E) Calculated reflection spectra of the all-dielectric metasurface with an a-Si cap (D) versus the incident angle. (F) The numerically calculated color of the all-dielectric metasurface with an a-Si cap (D) versus the incident angle shown in a CIE 1931 color map. (G) |E|² distributions of the all-dielectric metasurface with (A) and without (D) an a-Si cap upon 30° incidence at 580 nm. The left and right field distributions correspond to the point “1” in (B and E), respectively. (H) |E|² distributions of the all-dielectric metasurface with (A)/without (D) an a-Si cap upon 30° incidence at 667 nm. The left and right field distributions correspond to the point “2” in (B and E), respectively.

Previously, several methods (e.g. plasmonic systems) have been demonstrated to improve the incident angle insensitivity [24], [26], [27], [28]. However, these methods cannot be used to improve the performance of all-dielectric metasurfaces. Fortunately, the intensities and distributions of the desired resonant modes and unwanted modes are different, as shown in the left field distributions in Figure 5G and H, respectively. It is noteworthy that the incident light is set as y-polarized here in order to provide the distributions of resonant modes clearly. If the incident light is set as x-polarized, the results are very similar. The unwanted resonant modes are mainly distributed at the top part of the TiO₂ layer, while the desired resonant modes are mainly distributed at lower positions. Amorphous Si (a-Si) is a high-index, high-loss material, and its absorption of short wavelength light is stronger than that of longer wavelength light. Hence, a method based on engineering the resonant modes is proposed (Figure 5D). A 40-nm a-Si cap is added to each nanopillar to construct a hybrid nanopillar. The unwanted resonant modes overlap with the a-Si cap, while desired resonant modes are located far from the a-Si cap. Also, the unwanted resonant modes have shorter wavelengths compared to the desired resonant modes. As a result, this method can weaken the unwanted resonant modes without significantly affecting the desired resonant modes. Figure 5E shows the calculated reflection spectrum of the all-dielectric metasurface with an a-Si cap versus the incident angle, and Figure 5F shows the reflected color of the all-dielectric metasurface with an a-Si cap versus the incident angle. It can be seen that the side lobes have decreased in the reflection spectra. In addition, the reflection in the red light range (625–740 nm) also has increased. This is because a-Si is also a high-index material. Although a-Si is highly lossy, its high index (around 4.0) [12] can confine the incident light effectively and enhance the optical resonance. The right field distributions in Figure 5G and H show that the intensities and distributions of the desired resonant modes and unwanted modes have huge differences after adding the a-Si cap. It is noticeable that the unwanted resonant modes almost disintegrate, while the desired resonant modes still have high intensities. Hence, the change in the reflected light color of the metasurface in Figure 5E is much smaller compared to Figure 5B. All these performance enhancements demonstrate the feasibility and validity of improving all-dielectric metasurface performance through engineering the resonant modes.

In conclusion, we proposed, demonstrated, and experimentally verified a method to analyze the effect of roughness on the performance of a metasurface. Using the volume-current method, the metasurface roughness can be considered as equivalent to a lossy shell outside the nanopillar. Based on the FDTD method, different resonant modes were also studied. We demonstrated that the different effects of the metasurface roughness on different resonant modes are due to the different resonance distributions. In other words, the all-dielectric metasurface’s performance can be affected specifically by disturbing specific resonant modes. Further, the effects of different levels of roughness on metasurface’s performance were predicted. This can help in designing more manufacturable all-dielectric metasurfaces based on the fabrication capability. Moreover, a method to enhance metasurface’s performance based on resonant-mode engineering was proposed. A high-index, high-loss dielectric material (a-Si) can be placed in an appropriate position to weaken unwanted resonant modes without affecting the desired resonant modes. An example of improving the incident angle insensitivity of the red light band reflector demonstrated the feasibility of this method. We anticipate that, through engineering the resonant modes precisely, all-dielectric metasurfaces can be used for a broad spectrum of applications in the near future.

3.1 Numerical calculations

In this work, all the numerical analyses were performed by the FDTD method implemented with a commercial software (FDTD Solutions, Lumerical Inc.). All numerical calculations were performed under periodic or Bloch boundary conditions. In Figures 1D, 2B, 3C, 3E, 4C, 5B, 5C, 5E, and 5F, the input light source was set as collimated and unpolarized, with wavelengths ranging from 400 to 800 nm. In Figures 2C, 2D, 3D, 5G, and 5H, the input light source was set as collimated and y-polarized, with the wavelength ranging from 400 to 800 nm. The values of k_xx, k_yy, and k_zz for each band reflector in Figure 3E are determined by fitting the measured spectra with the numerically calculated spectra.

3.2 Optical characterization

The reflection spectra were measured with an Ocean Optics USB4000 spectrometer.

3.3 Fabrication

The all-dielectric metasurfaces were fabricated via nanoimprint lithography (NIL) [29], [30], [31], [32], [33] employing the following series of processes. TiO₂ was first deposited on a clean SiO₂ substrate by sputtering. Then, the patterns on the nanoimprint mold were transferred to the TiO₂ surface using NIL followed by reactive ion etching, metal deposition, and lift-off. Finally, TiO₂ and SiO₂ were etched by reactive ion etching to get the metasurface. The metasurface nanoimprint molds were also obtained from a series of processes. First, one-dimensional Si gratings were fabricated via interference lithography. Then, the line width of the one-dimensional gratings were tuned to desired values by angled evaporation [34]. After that, two steps of NIL were carried out to obtain the metasurface molds with the line width-tuned one-dimensional grating mold.