Analysis of mutual couplings in a concentric circular ring plasmonic optical antenna array

In this paper, we report the analysis of a concentric circular ring plasmonic optical antenna (POA) array using a simple lumped coupled circuit (LCC) model. The currents in the circular rings of the POA array and their mutual couplings are analyzed using the LCC model. The results agree well with the numerical simulation using CST’s Microwave Studio®. The LCC model reveals the mutual couplings between the antenna rings. It is found that the mutual couplings are not only between the adjacent antenna rings, but also involve their second (2nd) nearest or farther neighbors. Since the near-fields of the optical antennas are related to the currents in the optical antennas, the LCC model provides a useful tool for the analysis of the near-field and their mutual interactions in the circular ring POA array.


Results and Discussion
Before modeling of the multiple element concentric circular ring POA array, we first analyze a single circular ring POA and its equivalent lumped RLC circuit. Figure 1(a) shows the schematic layout of the circular ring POA. It is a metallic (gold) ring on a GaAs substrate. The thickness of the antenna ring t m is 30 nanometers (nm), and the thickness of GaAs substrate is t d = 0.35 µm. The outer diameter and the width of the metallic ring are labeled as d, and w, respectively. The incident light is a surface-normal plane wave propagating in the −z direction (i.e. top illumination). Figure 1(b) illustrates the equivalent lumped circuit. It is a simple series RLC circuit. The current is I(ω) in the frequency domain.
The single POA was numerically simulated using CST's Microwave Studio ® . The incident light is a surface normal plane wave traveling in the -z direction (i.e. top illumination). The H-field is in the y-direction, and the E-field is in the x-direction with a magnitude of 1 V/m throughout the paper. Since the circular rings are symmetric, other linear polarizations give the same results. Open boundary conditions were used in all simulations. The incident light excites LSPR modes in the circular ring POA and generates surface current in the ring.
The resonant condition of the fundamental LSPR mode can be written as: sp where k sp is the wave vector of the surface plasmonic wave. At the resonance, the LSPR mode induced surface current shows a sinusoidal-type distribution along the ring with a maximum current I m (ω) at the top (i.e. x = 0, and y = d/2) and bottom (i.e. x = 0, and y = −d/2) of the ring. Figure 2 shows the simulated current distributions (dots, crosses, and diamonds) in the three individual rings, Ring 1, Ring 2, and Ring 3 with the outer diameters of 1.2 µm, 1.4 µm, and 1.6 µm, respectively. The currents are at their corresponding resonant frequencies of 30 THz, 26 THz and 23 THz, respectively. The widths of the rings and thickness of the substrate are kept at w = 0.05 µm and t d = 0.35 µm, respectively. Under the plane wave incidence, the current distributions in the upper and lower parts of each ring are symmetric. Therefore, only upper half ring (i.e. angle 0 ≤ φ ≤ 180°) is analyzed. The other half is the same due to the symmetry. The currents are also calculated using the sinusoid standing wave formula in a standard half-wave antenna, i.e. |I| = |Imax|sin(φ). The calculated sinusoidal current distributions (solid lines) are also plotted in Fig. 2. The current distributions follow the sinusoidal standing wave current distribution in a half-wave antenna. The surface current I(ω) in the equivalent lumped RLC circuit can be written as: where ω is the angular frequency of the incident light, R is the resistance, L is the inductance, C is the capacitance, and V 0 is the induced voltage in the lumped circuit. The resistances R are the Ohmic resistances of the individual rings: in out m sk , where ρ = 2.44 × 10 −8 Ω-m is the resistivity of gold (Au) 29 , and R in and R out are the inner and outer radius of the ring, respectively. t m,sk is the skin depth.
where f is the frequency, and µ is the permeability of the metallic (gold) rings. Since the skin depth t m,sk is frequency-dependent, the R also changes with the frequency.  The inductance of the circular rings with the rectangular cross section can be calculated using Eq. (6) given by Frederick W. Grover 30 :  The solid curves are calculation data using the lumped circuit models, and the points by the "o", "+" and "◊" are numerical simulation data points using CST's Microwave Studio ® .
where L u is the inductance in microhenries (µH), R and t m are in centimeters (cm), and K' is the is a dimensionless factor that depends upon the (2 R/t m ) ratio and the radius of the ring antennas. K' values are given by Frederick W. Grover 30 . The inductance L of the half ring is thus: The calculated inductances are 6.0 × 10 −13 H, 7.8 × 10 −13 H, and 9.7 × 10 −13 H, for Ring 1, Ring 2, and Ring 3, respectively.
The capacitance C can be obtained by: The current distributions are also fitted using Eq. (2) and plotted in Fig. 3 (solid curves) together with those from the CST simulation (dots, crosses, and diamonds). The comparison of the calculated R, V 0 , L and C values and those from the curve-fitting are listed in Table 1.
After obtaining the parameters R, L, C for individual single rings, we then analyze the mutual couplings in a concentric circular ring POA array. Figure 4 Table 1. Comparison of calculated and curve-fitting values. Note that in the calculation, the circular ring is treated as a perfect transmission line. No current distribution is considered across the width of the ring, whereas in the CST numerical simulation the current nonuniformity is counted. This causes the differences between the calcuted R, V 0 , L and C vlaues and those from the curve-fitting vlaues. As shown in Table 1, the values are quite close. In a matrix form the currents in coupled rings can be written as:     where is the total impedance of an individual uncoupled ring. Two different models are investigated to calculate the mutual induces. The first one is the mutual inductance of circular coils 31 using Nagaoka's formula 32 : where A and a are the radii of the two circular coils and q and ε are the geometric parameters of the circular coils.
where d is the distance between the two circular coils, and d = 0 for the concentric circular rings on the same plane.
The mutual inductances calculated using the Nagaoka's formula 32 do not agree well with the simulation results. This is possibly due to the plane-wave induced non-uniform current distributions.
The second method is the mutual inductances of two parallel wires given by Rosa 33 : where µ 0 is the permittivity of vacuum, l eff is the effective length of a wire, and s is the separation of the two wires.
The l eff can be written as:

eff out
where the factor 2 in the denominator is for the half-circle due to the plane-wave incidence induced symmetry, and the factor 2 accounts for the sinusoidal current distribution.  Table 2 lists the calculated mutual inductances M 12 of different coupled rings using Eq. (17) compared with the mutual inductances from the curve-fitting.
The results calculated from the parallel wire model using Eq. (17) agree well with the curve-fitting. This indicates that under a plane wave illumination, the current distribution in a ring is similar to that in a wire transmission line.
Figures 6(a) and 6(b) show the calculated real and imaginary parts of currents (solid curves) in a three-ring POA array from Eqs (10) and (11) compared with the numerical simulation (circles, crosses, and diamonds). Table 3 lists the calculated mutual inductances using Eq. (17) compared with the values from the curve-fitting.
The calculations agree well with the numerical simulation. The mutual couplings from the 2 nd nearest neighbors (i.e. M 13 , M 31 ) are on the same orders as the nearest couplings.

Conclusion
In conclusion, we develop an LCC model for the analysis of the mutual coupling in a concentric circular ring POA array. The current distributions in the circular rings and their mutual couplings are analyzed using the LCC model. The analytical calculations agree well the numerical simulation. The LCC model reveals the underlying mutual couplings between the rings in the POA array. It is found that the mutual couplings from the 2 nd nearest neighbors are not negligible. The LCC model provides a useful tool for the analysis of the near-field and their couplings in the circular ring POA array.