4.4 Transmission and Phase Spectra for Multiple Unit Cells
4.4.2 Unit cell geometry variation for both full transmission and 2π phase
For the specific design of our metasurface, we perform simulation with the actual
cell by varying the geometrical variables. We first evaluate the characteristic impedance and phase velocity of a single unit cell by substituting the retrieved effective properties into theoretical Eqs. (4.6) and (4.11), respectively. The corresponding results are shown in Fig. 4.14(a). Here, the impedance matching line can be drawn as similar to the theoretical analysis in Fig. 4.12. Additionally, the phase velocity plot shows that the impedance matching curve passes through wide range of phase velocities.
In Fig. 4.14(a), we plot the transmission and phase contours on the l-w plane and the 12 points that can shift the phase by 30°×n (where n varies from 1 to 12). The contour plots in Fig. 4.14(a) were drawn with increments of 50 μm and 2 μm for l and w, respectively. The impedance matching and tunneling curves based on the obtained effective parameters are also illustrated in this figure. For choosing specific subunits, we use 12 unit cells of varying configuration to achieve the 2π phase span along the y-direction. The geometric values for the 12 unit cells can be selected near the impedance matching curve as shown in Fig. 4.14(b). It is worth noting that the dimensions and step sizes of the geometrical parameters have been selected in such a way that they are feasible with the current manufacturing technology. Thus, some feasible candidates are chosen outside the impedance matching curve. It is also worth pointing out that slight discrepancies exist between the analytically derived curves (which are calculated by substituting the numerically retrieved parameters into Eqs. (4.9) and (4.10)) and the transmission spectra, where the values possess full transmission on the curves. The main reason
can be attributed to the local interactions among the multiply stacked continuum unit cells, which cannot be fully expressed by the analytical discrete mass-spring model. Nevertheless, the proposed metasurface achieves almost full transmission as well as complete 2π phase shift if the design parameters (l,w) of the substructured unit cell are properly chosen. The specific values of (l,w) for the 12 unit cells are summarized in Table 4.2.
The transmission ratio and the amount of phase shift for each of the selected unit cells are plotted in Fig. 4.14(c). Representatively, unit cells #1 and #11, which have quite different geometric parameters, are shown within the graph. For all the chosen unit cells, the transmissions are guaranteed to be higher than 0.9, in addition to their phase intervals being well matched to the desired phase profiles. It should be noted that the discrepancy in the theoretical calculation (where transmission should be unity) can be attributed to the interactions of the elastic unit cells, whose original characteristics as a single unit cell are affected when stacked together. On the other hand, it should also be noted that various other candidates with different phase shifts could be chosen as well to obtain an elastic metasurface. Such flexibility provides a concrete design chart for various properties in numerous cases.
Fig. 4.15(a) shows the phase shift plot in an elastic medium where the designed substructured unit cells of various geometries are inserted. The results are for unit cells whose phases are shifted by multiples of 60. The incident wave is a plane elastic longitudinal wave. The cases with phase shifts of multiples of 45 are
presented in Fig. 4.15(b). Figs. 4.15(a-b) clearly show how effectively and accurately the designed unit cells achieve the required phase shift by slowing the wave to accumulate the phase within them. Since each of the designed unit cells has a transmission ratio of over 90%, they can serve as excellent elements for exclusively tuning the elastic longitudinal mode.
In summary, it would be worthwhile noting that the main purpose of the analytical lumped model is to “facilitate the design of the unit cell and to provide a solid physical explanation.” With the help of our analytic model, we were able to find the essential effective parameters of the unit cell, and then realize them by the substructuring mechanism.
Furthermore, we were able to estimate the impedance matching condition (which is explicitly derived by our model for a discrete system) and the range of phase shift within a given range of mass and stiffness properties. Even when they were multiply aligned, we could calculate the transmitted wave characteristics with the transfer matrix method. Inversely, we were able to calculate the necessary values of mass and stiffness that a unit cell should possess to induce the impedance matching condition as well as ensuring 2π phase change.
We can summarize how our analytical model is utilized to estimate any desired conditions as below.
1. By Fig. 4.12, we theoretically showed the relation between impedance (transmission) and phase velocity (phase shift within the structure) with respect to the effective mass and stiffness, to conveniently visualize the desired values.
2. The transmission and phase shift of metasurfaces consisting of only one kind of substructure (shown in Fig. 4.5) are analyzed both numerically and theoretically.
This figure also justifies the necessity of both substructures to construct a metasurface structure for full transmission.
3. The minimum number of unit cells to cover the 2π phase shift range was estimated (shown in Fig. 4.13).