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Photonic crystal-based circulators with three and four ports for sub-terahertz region

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Abstract

Three- and four-port circulators based on resonators in 2D photonic crystals with square unit cell possessing a low symmetry are investigated. The three-ports are described by only one specific element named antiplane of symmetry. On the other hand, the four-port circulators formed by cascading these two three-ports can have one of the two symmetries. One of them is described by the antiplane of symmetry, and the other symmetry corresponds to a twofold rotational axis. The theoretical part of our paper concerns scattering matrix analysis of the devices with different symmetries and also the operation of the four-port circulator as a single-pole triple-throw switch. Finally, the calculated frequency responses of two circulators are presented.

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Acknowledgments

This work was supported by the Brazilian agency CNPq.

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Correspondence to Leno Martins.

Appendices

Appendix 1: Properties of the nickel–zinc ferrite rod

The resonator of the proposed circulator is based on a NiZn ferrite rod (see Trans-Tech and its product TT2-111 [14]). Biasing this rod with a DC magnetic field \({\mathbf{H}}_0\) normal to the plane of the 2D PhC, one can change its physical parameters, namely the magnetic permeability tensor [\(\mu \)]. The magnetic permeability and the electric permittivity of this material are defined as follows:

$$\begin{aligned}{}[\mu ]= \mu _0\left( \begin{array}{lll} \mu &{}\quad \,-i\kappa &{}\quad 0\\ i\kappa &{}\quad \mu &{}\quad 0\\ 0 &{}\quad \,0 &{}\quad 1 \end{array}\right) ;\,\, \epsilon =12.5\epsilon _0 \end{aligned}$$
(5)

where the diagonal term \(\mu \) and the off-diagonal term \(\kappa \) are defined by the following expressions:

$$\begin{aligned} \mu= & {} 1 + \frac{{{\omega _m}\left( {{\omega _i} + j\omega \alpha } \right) }}{{{{\left( {{\omega _i} + j\omega \alpha } \right) }^2} - {\omega ^2}}}\ \end{aligned}$$
(6)
$$\begin{aligned} \kappa= & {} \frac{{{\omega _m}\omega }}{{{{\left( {{\omega _i} + j\omega \alpha } \right) }^2} - {\omega ^2}}} \end{aligned}$$
(7)

The terms \(\omega _m\) and \(\omega _i\) are defined as:

$$\begin{aligned} \omega _m= & {} \gamma {M_0} \end{aligned}$$
(8)
$$\begin{aligned} \omega _i= & {} \gamma {H_0} \end{aligned}$$
(9)

In Eqs. 59, \(M_0\) is the saturation magnetization (398 kA/m), \(\gamma \) is the gyromagnetic ratio (\(2.33 \times 10^5\,\hbox {rad/s per A/m}\)), \(\alpha \) is the damping factor (0.03175), \(\omega \) is the angular frequency (rad/s), \(\mu _0\) is the free space magnetic permeability (\(4\pi \times 10^{-7}\,\hbox {H/m}\)), \(\epsilon _0\) is the free space electric permittivity (\(\approx \)8.85 \(\times 10^{-12}\,\hbox {F/m}\)), and \({\mathbf{H}}_0\) is the applied DC magnetic field (862 kA/m). Whenever the direction of the DC magnetic field \({\mathbf{H}}_0\) is changed, the magnetic permeability tensor presented in Eq. 5 is transposed.

Appendix 2: Details of the resonator geometry

In order to adjust the performance of the proposed circulators, the radius or the position of the numbered rods in Fig. 15 was changed (in relation to the nonperturbed PhC). The optimum values for these parameters were found by performing a parametric optimization.

Fig. 15
figure 15

Details of resonator design

Table 2 summarizes the geometrical parameters of the numbered rods. Their centers are referred to the origin of the Cartesian coordinates system shown in Fig. 15, which in turn coincides with the center of one of the dielectric rods of the nonperturbed PhC.

Table 2 Details of the optimized geometry

Appendix 3: Details of cascading

The proposed four-port circulator is based on cascading of the two three-port circulators. The radius of the black dielectric rods located in the connection zone is different when compared to the dielectric rods of the nonperturbed PhC (see Fig. 16). The optimum value for this parameter obtained by a parametric optimization is 0.234a.

Fig. 16
figure 16

Details of cascading scheme

It is noteworthy that the two resonators of the four-port circulator have the same geometrical parameters given in Table 2. In this case, solely the geometrical parameters of the connection zone were subjected to the optimization process.

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Dmitriev, V., Portela, G. & Martins, L. Photonic crystal-based circulators with three and four ports for sub-terahertz region. Photon Netw Commun 33, 303–312 (2017). https://doi.org/10.1007/s11107-016-0641-4

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