# Introduction

The previous discussions on Radar Cross-Section (RCS) and electromagnetic (EM) coupling suggested that radiation from any number of random sources can be coupled onto, and radiate from, small features with specific geometries. But this turns out to be true at a matter of scales. For instance, while RF radiation, may couple onto various components in an aircraft, much shorter wavelength radiation can couple onto much smaller objects.

The figures and plot below illustrate the electromagnetic response of a split-ring micro-resonator (SRR). The illustrations on the left depict a single unit cell containing a single vertical split-ring resonator. The unit cell is characterized by silicon walls on a pink silicon nitride base, resting on top of a gold platform. The height of these layers is described by [math] h_w [/math], [math] t_m [/math], and [math] t_{au} [/math], respectively. The X and Y period of the unit cell is given by [math] w_l [/math] and [math] w_{ftb} [/math], while the wall thickness is defined by [math] t_w [/math].

Similarly, the split-ring resonator can be characterized by a number of parameters including vertical height [math] h_s [/math], horizontal width [math] w_s [/math], resonator thickness [math] t_s [/math], arm width [math] w_t [/math], gap width [math] w_g [/math], and central rotation angle [math] r_b [/math].

The right side of the illustration shows the electromagnetic response of the SRR. Specifically, the figure plots absorption as a function of wavelength for each of the layers in the unit cell. The black line plots total absorption, while the green line plots the absorption of the bottom gold layer; the low values of the gold layer make sense, because metal is not expected to absorb light. The red line plots absorption of the silicon nitride ([math]Si_3N_4[/math]) layer, while the blue line plots absorption of the vertical split-ring resonator (VSRR).

# Modeling implications

The simulation results tell a great deal about the geometry of the VSRR. For instance, the frequencies correlated with each of the three absorption peaks ([math] r_1 [/math], [math] r_2 [/math], and [math] r_3 [/math]) likely associated with phenomenon that cause the absorption to occur such as standing modes along the surface of the SRR. Similar to examining surface currents on metal surfaces after calculating RCS, fields surrounding the SRR and their surface currents can be calculated for each absorption peak.

The figures above illustrate surface current plots associated with peaks [math] r_1 [/math], [math] r_2 [/math], and [math] r_3 [/math], respectively. The arrows of the surface current show the circulation direction. Furthermore, the background colors illustrate the field modes traveling on the surface of the VSRR.

As hypothesized, each resonant frequency seems to be associated with a standing wave. For peak [math] r_1 [/math], the mode is a fundamental mode, and the field mainly resides along both of the vertical arms. For peak [math] r_2 [/math], the field modes include an extra field maximum and minimum — still a standing wave, but with a different geometry than the mode of [math] r_1 [/math]. Lastly, frequency [math] r_3 [/math] also seems to be associated with a standing wave similar to that of [math] r_2 [/math], but with a much less pronounced amplitude. This observation makes sense, because the magnitude of the absorption of [math] r_3 [/math] is less than the absorption magnitude of [math] r_2 [/math].

One final observation of the three field plots is the location of where the field is the strongest. Although not immediately apparent in [math] r_1 [/math], the field strength significantly increases in the gap between the VSRR arms in [math] r_2 [/math], and much less so in [math] r_3 [/math].

# Although Parameter trade studies

Like any complex electromagnetic system, any of the twelve parameters used to define the individual unit cell can be varied and plotted as a function of wavelength. The “heat maps” produced by the results further provide intuition as to how the resonators act — a slice through the N-dimensional parameter space each geometry resides in.

For instance, the upper left plot shows the spectral absorption of a VSRR oriented with its gap on top. An incident wave with wave vector **k** angled at [math] /theta [/math] with respect to the normal does not appear to have a large impact on the first resonant peak, but appears to decimate the second, strong resonant peak. This makes sense intuitively, because the first absorption peak is largely due to the VSRR and the electric field’s excitation of the VSRR changes little based on the polarization of the incident field (which is parallel to the gap). As the unit cell rotates, the incident field continues to excite this resonance hence its robustness with angle of incidence.

In the upper right plot, the same VSRR geometry (with the gap on the top side) is extended vertically in length. Not only does this drastically change the conditions for a resonant mode to establish on the VSRR, but it also changes the distance between the bottom of the VSRR and the silicon nitride and gold layers. Therefore, as the height of the VSRR changes, a nonlinear shift of the first peak resonance is seen — a trait that may not be intuitive at first glance of a one-dimensional absorption plot.

Although not explicitly discussed, the analysis for the other plots is similar, and each provides unique intuition that can help identify locations within the complex parameter space to inform and define the final model of the VSRR.

# Conclusion

Whenever electromagnetism is involved, the parameter space tends to grow quickly and become very complicated. Oftentimes, the best analysis that can be done outside of the fabrication lab is computational. Within this space — given enough time and resources — a single design can often be found within the myriad of geometric combinations comprising the parameter space. The difference in cost of performing high-fidelity Mod/Sim before fabricating the ideal product is priceless, as the latter can result in an enormous amount of frustration and resources.