# Ultra-Thin Metamaterial Beam Splitters
## 3 Polarizing Beam Splitter  
By properly tuning the transmittance and reflectance, PBS is obtained. This is achieved when the geometric parameters are \(w_{x}=400\) nm, \(w_{y}=50\) nm, \(\Lambda_{x}=600\) nm, \(\Lambda_{y}=600\) nm, \(t_{Ag}=15\) nm, and \(t_{MgF_{2}}=30\) nm. The electromagnetic wave is incident to the fishnets at 45\({}^{\circ}\) oblique incidence. Figure 2 shows the electric field distribution of the incident field, reflected field and transmitted field, respectively. TM-polarized light (\(E_{x}\) component) is transmitted as shown in Figure 2a, and TE-polarized light (\(E_{y}\) component) is reflected as shown in Figure 2b. The designed fishnet structure achieves the goal of polarization splitting.  
To explore the performance of this PBS, extinction ratio and insertion loss are calculated by  
\[\begin{cases}ER_{r}=10 log_{10}(\frac{R_{TE}}{R_{TM}}),\\ ER_{t}=10 log_{10}(\frac{T_{TM}}{T_{TE}}),\end{cases} \tag{2}\]  
and  
\[\begin{cases}IL_{r}=10 log_{10}(\frac{1}{R_{TE}}),\\ IL_{t}=10 log_{10}(\frac{1}{T_{TM}}),\end{cases} \tag{3}\]  
respectively. The extinction ratio for the TE-polarization \(ER_{r}\) is the ratio of the TE-polarization reflectance (\(R_{TE}\)) to the TM-polarization reflectance (\(R_{TM}\)) in logarithmic scale. Similarly, the extinction ratio for the TM-polarization \(ER_{t}\) is the ratio of the TM-polarization transmittance (\(T_{TM}\)) to the TE-polarization transmittance (\(T_{TE}\)) in logarithmic scale. Then the results are \(ER_{r}=22.5\) dB, \(ER_{t}=14.5\) dB, \(IL_{r}=0.25\) dB, and \(IL_{t}=0.26\) dB. Compared with other PBS designs [9; 10; 11], these values indicate the good performance of our designed PBS. For example, experimental results for the PBS with gratings etched in a TiO\({}_{2}\) layer indicated more than 80% diffraction efficiency and above 20 dB extinction ratio [11]. The perforated GaAs photonic crystal based PBS design [10] was shown to have a theoretical extinction ratio of around 20 dB. The reflecting PBS design based on a Ag mirror embedded in silica grating [9] can theoretically give around 95% diffraction efficiency and high extinction ratios up to 60 dB for some diffraction orders. However, an extinction ratio \(>20\) dB is already considered sufficiently a good ratio [9].  
Figure 2: (a) x-component of Electric field \(E_{x}\) and (b) y-component of electric field \(E_{y}\) at 1.55 \(\upmu\)m. The fields are incident from the \(-z\) direction.  
Following the retrieval procedure described in ref [29], we can calculate the effective impedance of this beam splitter under different polarization incidence. Figure 3a,b show the effective impedance (\(Z=Z^{\prime}+iZ^{\prime\prime}\)) under the TE-polarized light and TM-polarized light incidence, respectively. Under the TE-polarized incidence, the real part of effective impedance is almost 0 around 1.55 \(\upmu\)m, while the impedance of the background material air is 1. So the large impedance mismatch leads to \(R_{TE}=0.94\). In contrast, when the TM-polarized light is incident to the structure, the real part of the effective impedance is 0.2 around 1.55 \(\upmu\)m. The impedance mismatch is reduced, so that 94% of the power is transmitted. Meanwhile, we should note that the inherently lossy metal components (i.e., 2.3% and 5.3% absorbance (A) in metallic regions for the TE-polarized and TM-polarized light, respectively) in the metamaterials result in nonzero insertion loss (see Figure 3c). Since the absorption increases around the resonance, it is desirable to be sufficiently away from the resonance for an efficient beam splitter.