## Phased Array Antennas Floquet Analysis Essay

### Plasmonic nano-waveguide and nanoantenna

Existing candidate plasmonic waveguides to feed nanoantennas include surface plasmon channel^{16}, surface plasmon gap structure^{20}, surface plasmon nanowire^{18} and surface plasmon stripe^{19}. However, coupling light into these structures is always a problem due to the significant size and momentum mismatches^{32} between these surface plasmon waveguides (SPWs) and conventional photonic components. As a result, special techniques are required to excite propagating plasmons in these SPWs^{33,34,35,36}. In addition, these structures are unsuitable for the manipulation of a nanoantenna array, considering their highly confined fields and the lossy nature of plasmons. The above limitations prompt us to seek for an alternative for the POPAs.

The geometric sketch of the proposed SPW is shown in Fig. 1a. It is composed of a silver nanostrip, a SU-8 slab supporting the silver nanostrip and a silver film on the backside of the SU-8 slab. A plasmonic mode formed by hybridizing *ss*_{b}^{0} mode of a nanostrip^{37} and surface plasmon polaritons (SPPs) on a metal-dielectric interface is supported by this waveguide. The nomenclature *ss*_{b}^{0} is used to identify the mode symmetry with respect to the *z* and *y* axes. Detailed information about this nomenclature can be found in Section 1 of Supplementary Information. The electric field distributions of this mode at a cross section of the proposed SPW is depicted in Fig. 1b. The mode offers a significantly lower transmission loss than conventional SPWs (Supplementary Table S2) and thus it is attractive for the construction of a POPA. Detailed information of this SPW can be found in Section 1 of Supplementary Information.

The plasmonic nanopatch antenna was first proposed by R. Esteban^{38} and then investigated for applications like single photon emission^{39}, spontaneous emission controlling^{40}, optical polarization converter^{41} and holograms^{42}. Here, near-field interactions between the nanopatch and the nanostrip are exploited to excite the fundamental mode of the nanopatch. As depicted in Fig. 2, when the nanopatches are in proximity to the nanostrip, the localized surface plasmons (LSPs) on the nanopatches will be excited by the displacement currents around the nanostrip via capacitive coupling. The LSP intensities on the nanopatches are determined by the distance *D*_{p}, while the phase distribution among them can be controlled by the positions of the nanopatches along the *x*-axis.

Depending on the height *H*_{p} over the silver film, the operating mechanism of the nanopatch antenna can be either interpreted by the widely adopted cavity model^{43} or a nanoparticle backed by a plasmonic reflector when *H*_{p} is too large to sustain a cavity mode. The electric field distributions are presented in Fig. 2c to show the coupling behavior, where *E*_{z} is induced by the LSPs and confined near the nanopatches while *E*_{y} directly contributes to the emission. Due to the existence of the silver film, the potential energy leakage towards the substrate side is prohibited.

### Proposed POPA

The overall configuration of the proposed POPA is depicted in Fig. 3a. The nanostrip SPWs are arranged in parallel to the *x*-axis into an array to directly couple light from a fiber and also act as a power relay from the fiber to the nanopatches. The nanopatches then convert the energy from the nanostrip SPWs back into photons. The dimensions and relative positions of the nanopatches inside each unit cell of the nanostrip array are depicted in Fig. 3b and c.

Note that the *y*-polarized fields of the proposed nanostrip SPW have a natural symmetry with respect to the *xoz* plane and the electric field intensity is dominated by the *y* component (Supplementary Fig. S2). Benefiting from the two aforementioned unique modal properties, when the nanostrips are arranged together into an array, there is an in-phase superposed mode which is free of obvious distortions over the original mode characteristics of the nanostrip SPW (Supplementary Information Section 1.6). Moreover, by arranging the nanostrips into an array, the modal size of the nanostrip SPW is equivalently enlarged and the significant size mismatch between the SPW and the fiber is diminished.

Considering the aperture size of the fiber, a large area containing numerous nanostrips will be covered by the fiber. Usually it is impractical to obtain the solution of such an electrically large structure directly with full-wave simulations. However, due to the periodic nature of this case, the calculation domain can be reduced by applying periodic boundary conditions in a unit cell. Here, electric walls (*E*_{//} = 0, d*E*_{⊥}/d*n* = 0, *H*_{⊥} = 0, d*H*_{//}/d*n* = 0) are employed at the symmetrical planes of the adjacent nanostrips for simplicity, as depicted in Fig. 3b and c.

For demonstration purpose, a unit cell with 7 × 3 nanopatches is numerically investigated. The geometrical paramaters of the investigated POPA are as follows: *W*_{s} = 100, *T*_{s} = 100, *H*_{s} = 400, *h* = 700, *R*_{p} = 180, *T*_{p} = 100, *H*_{p} = 700, *D*_{p} = 600, *P*_{x} = 1200 and *P*_{y} = 1100 in nm. Note that depending on the distance between the adjacent nanostrips, there is a trade-off between the uniformity of the intensities of the excited LSPs along the *y*-axis and the amount of the nanopatches that can be covered within each unit cell (Supplementary Fig. S9). A 3.4 μm distance is selected here to reach a compromise between them. As an indicator of the LSPs intensities, the intensities of *E*_{z} on the bottom surfaces of the nanopatches are monitored along the reference line 1 in Fig. 3b. As seen from Fig. 4a, the intensities of the LSPs on three nanopatches are on the same level. With a uniform spacing along the *x-*axis, the nanopatches in the POPA are expected to exhibit a gradient phase variation along the positive *x-*axis given by *φ*_{step} = 2π (*P*_{x} − *λ*_{g})/*λ*_{g}, where *λ*_{g} is the guided wavelength of the nanostrips (λg = λ0 / neff, where λ0 is the free space wavelength and neff is the effective mode index of the nanostrip SPW). The numerically extracted phases of Ey component, i.e., the fields contributing to emission, on seven nanopatches along the *x-*axis are compared with the theoretically predicted phases in the inset of Fig. 4b. The reasonable agreement between them demonstrates the ability of the nanostrip to manipulate the phase of the nanopatches. With a gradient phase variation along the *x-*axis, the emissions of these nanopatches are expected to interfere constructively at a certain angle. The near-field distribution of the investigated POPA in the *xoz* plane is plotted in Fig. 4c. It clearly indicates that a coherent interference of the radiated power from the nanopatches can shape the far-field emission pattern into a directional one. The simulated far-field emission patterns in the *xoz* plane are shown in Fig. 4b. Five of the telecommunication windows (*E, S, C, L and U* bands) are covered by the investigated POPA with an access up to 1800nm. When the wavelength increases from 1428 nm to 1800 nm, the beam experiences a scanning from the backward to the forward region, covering an angular range of 21.7 degrees.

### Impacts of the delocalized surface plasmons

Another key issue that needs to be taken into account is the influence of the delocalized surface plasmons^{27,28,29,30,31}. Different from the non-propagating LSPs, the delocalized surface plasmons are the propagating electromagnetic waves bounded on the metal-dielectric interface, also known as propagating surface plasmons (PSPs). In certain circumstances, the PSPs can be excited by the additional momentum provided by the periodicity of the nanoparticle array. The interaction between LSPs and PSPs under plane wave illumination has been widely explored^{44,45,46} and it has been shown that their mutual coupling will result in a shift of the resonant wavelength of the LSPs^{27} or an anti-crossing behavior of the resonant wavelengths of the LSPs and PSPs^{45}.

For the investigated POPA, numerical results indicate that the coupling condition of the PSPs can be described by the interaction between the Floquet modes and the PSP modes^{47}:

where *m, n* denote the orders of the Floquet modes and *p, q* denote the orders of the PSPs in the *x-* and *y-* directions. In equation (1), *k*_{xmn} and *k*_{ymn} denote the locations the (*m*, *n*) Floquet circle in the *k* space and *βpsp* is the phase constant of the PSPs (, where *k0* is the free-space wavenumber, *εd* is the dielectric constant of SU-8 and *εm* is the dielectric constant of silver). As the high-order Floquet modes will give rise to the grating lobes or aliased lobes in the far-field^{47, 48}, only the fundamental Floquet mode (*m* = *n* = 0) is of our concern47. Hence equation (1) can be simplified to:

where *k*_{x0} is given by *k*_{x0} = 2π (*P*_{x} − *λ*_{g})/(*P*_{x}*λ*_{g}) and *k*_{y0} is equal to zero for the investigated POPA. The comparison between the theoretically predicted coupling wavelengths and the ones obtained by numerical simulations is shown in the inset of Fig. 5a. Taking the case with *Px* = 1 μm and Py = 0.97 μm as an example, in such a configuration, the PSP (0, ±1) (p = 0, q = ±1) modes (the index numbers are assigned in the order of x- and y- directions, respectively) are predicted at 1580 nm according to equation (2). Accordingly, in the full-wave simulations, the coupling is observed to occur at 1538 nm, showing an error of 2.66%. To provide a clear insight into this issue, the Floquet circle diagram associated with the investigated case (*P*_{x} = 1 μm, *P*_{y} = 0.97 μm) is plotted in Fig. 5b. The intersection of the fundamental Floquet mode circle and the PSPs (0, ±1) circle clearly indicates that their coupling condition is fulfilled in the investigated case.

In Fig. 5c, the field distributions of the investigated POPA at the coupling wavelength as well as two other wavelengths are depicted. An enhanced interaction between the nanopatch and the silver film via *E*_{z} is clearly observed at the predicted wavelength. Comparatively, at 1460 nm and 1715 nm, this strong dependence on *E*_{z} is not observed and the distribution is very close to that of the isolated nanopatch (see the first figure of the lower-half panel in Fig. 5c) where *E*_{z} is locally sustained by the LSPs around the nanopatch and the SPPs over the silver film, respectively.

For further verification, the results are also compared with the far-field illumination case in which the principles of light-PSPs coupling has been experimentally demonstrated^{45}. The field distribution at the theoretically predicted coupling wavelength under the plane wave illuminations is depicted in the second figure of the lower-half panel in Fig. 5c, and it is found to be highly consistent with what has been previously observed in the investigated POPA. This consistency provides a solid evidence of the emergence of the PSP mode.

A side effect resulting from the emergence of the PSPs is a significant drop of the radiation efficiency. In Fig. 5a, the simulated radiation efficiency versus wavelength is plotted. With the emergence of PSPs, the POPA radiation efficiency drops by about 50 percent compared to that at 1460 nm (see Supplementary Information Section 2 for the calculation method of the radiation efficiency). This is not counter-intuitive because as the PSPs occur, a part of the energy is transferred into the SPPs on the silver film and barely contributes to the emission. The discussion on the resonant wavelength anomaly for the investigated POPA can be found in Supplementary Information Section 5.

Considering the conditions of no aliasing in the far-field at the telecommunication wavelength (1550 nm) and the size of each nanopatch when the LSPs are in resonance, the above discussion just makes sense when both *P*_{x} and *P*_{y} fall into the range between 0.5 μm and 1.6 μm. Under this restriction, the calculated results according to equation (2) indicate that the coupling between the fundamental Floquet mode and only four PSP modes with indices (0, ±1) and (−1, ±1) are possible. The calculated results at various wavelengths close to 1550 nm for PSP modes (0, ±1) and (−1, ±1) are plotted in Supplementary Figs S10a and S10b, respectively.

### Fabrication and measurement

Based on the calculation results in Supplementary Fig. S10, with an exclusion of the assembles of *P*_{x} and *P*_{y} to avoid the PSP modes, a large-scale POPA is designed and fabricated (at least 5,826 nanopatches are excited according to the experimental results). The geometrical parameters are as follows: *W*_{s} = 100, *T*_{s} = 100, *H*_{s} = 700, *h* = 700, *T*_{p} = 100, *H*_{p} = 700, *R*_{p} = 180, *D*_{p} = 600, *P*_{x} = 1200 and *P*_{y} = 1100 in nm. The image of the fabricated sample under microscopy and the SEM image of the fabricated sample are shown in Fig. 6a and b, respectively. The details of the fabricated sample can be found in Section 6 of Supplementary Information.

The measurement setup is described in ref. 49. Based on the measured real-space image, the excitation area is over 56 λ in the *x-*axis and over 118 λ in the *y-*axis (Supplementary Fig. S14). It is thus impractical to obtain the solutions of such an electrically large structure directly with numerical simulations. As an alternative, a theoretical calculation based on the measured real-space image is developed to get the far-field results of the fabricated POPA (Supplementary Information Section 8).

Figure 6c shows the measured far-field emission pattern at 1540 nm with the theoretical one on its right side. The white lines in Fig. 6c indicate the far-field power distributions along two reference lines traversing the peak of the beam, as depicted in the insets of Fig. 6d and e. A reasonable agreement between the measured and theoretical beam shapes is observed. The measured and the theoretical power profiles along the two aforementioned reference lines are compared in Fig. 6d and e, respectively, where a beam scanning behavior is clearly observed as expected. Note that the actual phase distribution along the *y-*axis is very likely to deviate from an ideal quadratic phase distribution of a Gaussian beam. It is thus reasonable to believe that the deviation between the theoretical and measured profiles in Fig. 6e originates from the inaccurate *φ*_{n} (see the determination of *φ*_{n} in Supplementary Information Section 8).

The theoretical peak directivity of the fabricated POPA is plotted in the inset of Fig. 6e. It can be seen that a directivity of 38.7 dBi with a full width half maximum (FWHM) of 0.68° in the *x-*direction (Fig. 6d) is obtained near the telecommunication wavelength (1540 nm). With a wavelength-dependent projection angle and an ultra-small beam divergence angle, the proposed POPA has the potential to act as an integrated optical spectrometer^{50}, a frequency-division multiplexer^{51} and is promising for the construction of wireless data nanolinks for plasmonic ICs^{22}.

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