Polarization-independent split bull’s eye antennas for infrared nano-photodetectors

Split bull’s eye (SBE) antennas exhibit much larger extraordinary optical transmission and strong polarization dependence rather than bull’s eye (BE) antennas in the infrared range due to the introduced sub-wavelength slit. Here, we demonstrate a dual-split bull’s eye (DSBE) antenna, which consists of two sub-wavelength slits crossing through the center of the BE antenna with an intersection angle θ. The polarization dependence in transmission can be flexibly tailored by adjusting the intersection angle, following a cos2 (Φ + θ/2) angular dependence on polarization angle Φ. When θ = 90°, the DSBE antenna yields high and polarization-independent transmission enhancement over the entire infrared spectrum. It presents highly promising applications for polarization-insensitive photodetectors and other optoelectronic devices.

are same with our previous work from near to mid infrared 9 . Similar to ref. 12., we assume that the aperture size w is fixed at 0.3 μ m for the intention of high-speed response in nano-photodetectors. The incident light, which is linearly polarized plane waves, impinges normally on the surface of the antennas (i.e., incident along z-axis). The polarization angle Φ is defined as the angle between the electric field and the x-axis ( Fig. 1(c)).
Basically, the absorption of the active region beneath the antennas (see Fig. 1(b)) can be estimated by the transmission enhancement associated with the EOT effect 13 , which has been illustrated through an equation in Method part. As reported, the transmission enhancement, η, of the aforementioned antennas satisfied the relation 14 ce co It means that the transmission enhancement η is determined by two independent factors: the coupling efficiency of the grooves, f ce , and the cutoff function of the central aperture, f co . Either of them can produce polarization dependence in EOT through the antennas. The example schemes can be constructed as (i) circular hole surrounded by elliptical or fan-shaped grooves 15,16 , wherein the factor f ce is polarization dependent, (ii) elliptical or bowtie central apertures surrounded by circular grooves [17][18][19] , wherein f co is polarization dependent, or (iii) polygonal apertures surrounded by polygonal grooves, wherein both f ce and f co are polarization dependent [20][21][22] . To further understand the impacts of light polarization, we define the PER as the ratio of the maximum to the minimum transmission enhancement for all the polarization states. As studied in ref. 9, a SBE antenna with a subwavelength slit belongs to scheme (ii) and the EOT intensity obeys a cos 2 Φ angular dependence characteristic. In this case, PER can be calculated by η SBE (Φ = 0°)/η SBE (Φ = 90°), where η SBE (Φ = 0° or 90°) corresponds to the transmission enhancement of SBE antenna with Φ = 0° or 90° incidence. Figure 2(a) shows η SBE (Φ = 0°) and η SBE (Φ = 90°) of SBE antennas with respect to the metal thickness when λ = 1.31 μ m. The groove period a is fixed at 1.29 μ m to ensure strong coupling of surface plasmons in accordance with Bragg condition. The value of η SBE (Φ = 0°) is extremely high and varies periodically with the growing film thickness. The maxima correspond to F-P resonances caused by the reflection of the waveguide modes in the subwavelength slit 9 . The value of η SBE (Φ = 90°) is quite lower (< 1) and goes down exponentially with film thickness increasing, which can be understood from the approximate formula of electric field in a single sub-wavelength slit with Φ = 90° incidence 23

Polarization dependence of split bull's eye antennas.
here m is a non-zero integer. When λ > 2w, all the modes in the slit are evanescent and the EM intensity follows an exponential decay with film thickness, resulting in a rapid increase in PER. Table 1 lists PER values at h = 0.38 and 0.9 μ m, corresponding to the 1st or 2nd-order F-P resonances at λ = 1.31 μ m. It is clear that the SBE antenna with a narrower slit (w = 0.1 μ m) has a larger PER, which is mainly attributed to two reasons. First of all, the SBE antenna with a narrower central slit will get a higher η SBE (Φ = 0°) as previously reported in ref. 9. Secondly, a smaller w will lead to a lower η SBE (Φ = 90°) due to a larger imaginary part of the propagation constant according to equation (2). In addition, the value of PER will be larger when towards longer wavelengths. It has been studied in details that the value of η SBE (Φ = 0°) monotonously increases when the wavelength extends from near to mid-infrared ( Fig. 2(b)). The reduction of η SBE (Φ = 90°) is raised by the larger imaginary part of the propagation constant (equation (2)) as well as the larger film thickness h, as it should be increased to support F-P resonances for the case of Φ = 0° incidence at longer wavelengths 9 . As a result, a very high PER of 89 dB is achieved at λ = 4 μ m due to the large contrast between η SBE (Φ = 0°) and η SBE (Φ = 90°).

Polarization tunability of transmission enhancement in dual-split bull's eye antennas. In order
to overcome the issue of polarization dependence caused by the symmetry breaking, we introduce an additional sub-wavelength slit into the SBE antenna to form a DSBE antenna. The polarization dependence in EOT can be modulated by tuning the intersection angle θ as shown in Fig. 3(a) with h = 0.38 μ m. Specially, the case of θ = 0° corresponds to a SBE antenna. It is found that the transmission enhancement of the DSBE antenna, η DSBE , obeys a cos 2 (Φ + θ/2) angular dependence characteristic, which can be roughly estimated by treating the DSBE structure as a superposition of two independent SBE antennas. As mentioned above, DSBE , here f ce DSBE is polarization independent. Thus, the polarization dependence of a DSBE antenna is mainly determined by its cut-off function f co DSBE , which can be written as , ′ f co is the corresponding maximum cutoff function of each SBE antenna. Figure 3(a) shows the simulated angular distribution of transmission enhancement in polar coordinate for DSBE antennas with different intersection angles θ. It is found that η DSBE reaches its maximum at Φ = − θ/2 or π − θ/2, which is in good agreement with equation (3). In particular, when the two crossed slits are orthogonal to each other, i.e., θ = 90°, the value of η = ′ f f DSBE ce DSBE co is independent on the polarization angle Φ. It is concluded that, in such a way, the polarization properties of our DSBE structure can be effectively modulated, including the transmitted field intensity and polarization state.
As depicted in Fig. 3(b), if sweeping the intersection angles θ from 0 to 90°, the maximum value of η DSBE simulated by finite-difference time-domain (FDTD) method will increase to a saturation point at ~20° and then decrease. Such a feature cannot be precisely described by the superposition model equation (3), in which the maximum value of η DSBE (i.e., η DSBE (Φ = − θ/2 or π− θ/2)) equals , obeys a monotonous decrease function. In Fig. 3(c), we plot the normalized (with respect to the incident light) and time-averaged |E| intensity distributions around the center of the antennas on the z = 0 plane when Φ = − θ/2. It is found that the surface charge oscillations, i.e., the so-called localized surface plasmon (LSP) resonances, take place at the sharp corner of the crossed slits when the intersection angle θ > 0°. The induced strong near-filed enhancement makes contribution to the cut-off functions 13 , which has not been involved in superposition model. When θ increases   To further understand the EOT behaviors of the polarization-independent DSBE antenna (i.e., θ = 90°), we investigate how the transmission enhancement depends on the metal thickness h at λ = 1.31 μ m (Fig. 4(a)). With h increasing, the transmission enhancement of DSBE (θ = 90°) antenna shows similar periodicity with SBE antenna (under Φ = 0° incidence), which confirms the existence of propagating modes in the central aperture. In addition,  according to F-P resonance condition, the F-P resonance modes will not shift in the DSBE (θ = 90°) antenna, as the metal thicknesses are unchanged when introducing the second slit into the SBE antenna. The inserts shows the cross-sectional distribution of the normalized (with respect to the incident light) and time-averaged magnetic field |H| for SBE and DSBE (θ = 90°) antennas with h = 0.38 μ m, Φ = − θ/2, and λ = 1.31 μ m. One can see that the magnetic field is confined to the gaps with the maximum field occurring along slit walls, which is due to the existence of waveguide mode and its F-P resonance 9 . As explained in refs 24-26, when the EM energy transmits through the central slit as a charge-density wave, surface currents will be formed on the slit walls and thus the magnetic field is especially strong there. As depicted in Fig. 4(b) with h = 0.38 μ m, the transmission peaks associated with the 1st-order F-P resonance occurs at λ = 1.31 μ m in both structures.

Discussion
On the basis of the above analysis, the operating wavelength of DSBE (θ = 90°) antenna can be easily extended to mid-infrared if using the same geometrical parameters of SBE antennas. As shown in Fig. 5(a), in spite of the incident polarization, the transmission enhancement of the DSBE (θ = 90°) antenna increases monotonically with the operating wavelength and is about 6 orders of magnitude higher than the traditional polarization-independent BE antenna at λ = 4 μ m. On the contrary, the transmission enhancement of the BE antenna exponentially decreases with the operating wavelength increasing. The DSBE (θ = 90°) antenna is therefore the prior of choice for nano-photodetectors with unpolarized or arbitrarily polarized light incidence at infrared wavelengths.
As seen from Fig. 4, the transmission enhancement of DSBE (θ = 90°) antennas will be reduced in half when compared to SBE antenna, which also can be evaluated by equation (3). However, the ratio of η η Φ =°θ =°/  of the slits. When θ = 0°, the antenna (i.e., SBE antenna) is strong polarization dependent. The large PER can be enhanced by using a narrower sub-wavelength slit or extending the operating wavelength to mid-infrared. The maximum PER = 89 dB is achieved at λ = 4 μ m, exhibiting excellent ability for polarization-sensitive photodetectors that only sense the signal intensity. When θ = 90°, the transmission enhancement of the antenna is completely polarization independent. Moreover, DSBE antennas can maintain the EOT behaviors of SBE antennas. For instance, when λ = 4 μ m, the transmission enhancement of the DSBE (θ = 90°) antenna is 6 orders of magnitude higher than the traditional polarization-insensitive BE antenna. This may offer the possibility to unyoke the limitation of SBE antennas in polarization independence and facilitate the applications of optical antennas in optical communication devices.

Methods
Numerical simulations. Three-dimensional FDTD simulations were performed using a commercially software package (Rsoft FullWAVE) with perfectly matched layer (PML) boundary conditions. The transmission enhancement is defined as the ratio of the integrated z-component of the Poynting vector S (S = ½Re(E × H * )) over the output and input apertures. The integral domains of DSBE and SBE antennas are 0.3 μ m × 0.3 μ m square areas at the center of the antennas. The permittivity of Ag follows the Lorentz-Drude model at all operating wavelengths. A non-uniform orthogonal mesh grid is used to reduce the computational costs. The mesh size at the material interfaces is set to be 5 nm which is much smaller than the element sizes and the operating wavelength, and the calculations converged well.