Graphene-based four-port circulator with an elliptical resonator for THz applications

Abstract

We suggest a new graphene-based four-port circulator for THz applications that consists of two parallel graphene waveguides coupled to an elliptical graphene resonator between them. The graphene elements are deposited on a dielectric substrate. An external constant magnetic field is applied normal to the resonator. The frequency response of the circulator obtained from numerical simulations performed with COMSOL Multiphysics software is in good agreement with those obtained from the ad hoc temporal coupled-mode theory. The analysis shows that the bandwidth of the circulator is about 5.7% around the central frequency 5.03 THz with the applied constant magnetic field of 0.8 T.

Appendices

Appendix 1: Calculation of the resonator radius

The power transfer through the circulator is possible because of the excitation of surface plasmon-polaritons (SPPs) in the graphene layer [45]. By considering some assumptions, such as the graphene layer with infinite size, one can determine the following formula for the calculation of the propagation constant $\beta _{spp}$:

$$\begin{aligned} \beta _{spp}=\dfrac{(1+\varepsilon _1)(\omega \hbar )^2}{4\alpha E_{\rm F}\hbar c}\biggl (1-\dfrac{\omega _{c}^2}{\omega ^2}\biggr ), \end{aligned}$$

(8)

where $\alpha \approx 0.01$ is the fine-structure constant and $\varepsilon _1$ is the dielectric constant of the substrate.

The wavelength $\lambda _{spp}$ of the SPP mode can be obtained by the expression $\lambda _{spp}=2\pi /\beta _{spp}$, while the resonator radius R for the dipole TM mode can be calculated by the expression $2\pi R=\lambda _{spp}$. Therefore, the following formula for the calculation of R can be derived:

$$\begin{aligned} R=\frac{1}{\beta _{spp}}. \end{aligned}$$

(9)

For the considered operating frequency range and material parameters, the estimated resonator radius obtained from Eqs. (8) and (9) is 600 nm. Since we have adopted an elliptical shape instead of a circular one for the resonator, we have used this value as an initial guess for the semi-major axis of the disk resonator (a). As one can see, the initial guess is close to the optimum value of a (618 nm).

Appendix B: Temporal coupled-mode theory based model

The circulator can be analytically modeled by using a TCMT-based approach. The general TCMT equations that describe the variation of the energy stored in the resonator and the power flow in the waveguides are [31]:

$$\begin{aligned}&\frac{d\mathbf{a}}{dt}=(j\Omega -\Gamma )\mathbf{{a}} + K^t|s_{\rm in}\rangle , \end{aligned}$$

(10)

$$\begin{aligned}&|s_{\rm out}\rangle =C|s_{\rm in}\rangle +D\mathbf{{a}}. \end{aligned}$$

(11)

In Eqs. (10) and (11), a is the matrix that represents the magnitude of the resonant modes, while the matrices $|s_{\rm in}\rangle$ and $|s_{\rm out}\rangle$ describe the incoming and outgoing waves, respectively. The square matrices $\Omega$ and $\Gamma$ represent the resonant frequencies and the decay rate of modes, respectively. These matrices are defined as:

$$\begin{aligned}&\mathbf{a}= \left( \! \begin{array} {l} a_+ \\ a_- \end{array} \!\!\right) , \end{aligned}$$

(12)

$$\begin{aligned}&|s_{\rm in}\rangle = \left( \! \begin{array} {l} s_{1+} \\ s_{2+} \\ s_{3+} \\ s_{4+} \end{array} \!\!\right) , \end{aligned}$$

(13)

$$\begin{aligned}&|s_{\rm out}\rangle = \left( \! \begin{array} {l} s_{1-} \\ s_{2-} \\ s_{3-} \\ s_{4-} \end{array} \!\!\right) , \end{aligned}$$

(14)

$$\begin{aligned}&\Omega = \left( \! \begin{array} {ll} \omega _+ &{} 0 \\ 0 &{} \omega _- \end{array} \!\!\right) , \end{aligned}$$

(15)

$$\begin{aligned}&\Gamma =\Gamma _{\rm port}+\Gamma _{i}= \left( \! \begin{array} {ll} 2\gamma _+&{} 0 \\ 0 &{} 2\gamma _- \end{array} \!\!\right) + \left( \! \begin{array} {ll} 2\gamma _{i+} &{} 0 \\ 0 &{} 2\gamma _{i-} \end{array} \!\!\right) , \end{aligned}$$

(16)

where the matrices $\Gamma _{\rm port}$ and $\Gamma _{i}$ define the decay rates of the counter-rotating modes related to waveguide coupling and intrinsic losses, respectively, and the parameter $s_{m+}$ ($s_{m-}$) represents the field amplitude of the incoming (outgoing) wave at the mth port.

Matrices K and D describe the coupling of the input power to the resonator and the decay rate of the resonant mode energy to the output ports, respectively, while matrix C represents the direct coupling that takes place between the input ports and the output ones. They are defined below [31]:

$$\begin{aligned}&K=-\sqrt{2} \left( \! \begin{array} {ll} \sqrt{\gamma _+}e^{-j\left( \frac{\varphi +\pi }{2}\right) } &{} \quad 0 \\ 0 &{} \quad -\sqrt{\gamma _-}\\ \sqrt{\gamma _+}e^{-j\left( \frac{\varphi -\pi }{2}\right) } &{} \quad 0 \\ 0 &{} \quad \sqrt{\gamma _-} \end{array} \!\!\right) , \end{aligned}$$

(17)

$$\begin{aligned}&D=\sqrt{2} \left( \! \begin{array} {ll} 0 &{} \quad \sqrt{\gamma _-} \\ \sqrt{\gamma _+}e^{j\left( \frac{\varphi -\pi }{2}\right) } &{} \quad 0 \\ 0 &{} \quad -\sqrt{\gamma _-} \\ \sqrt{\gamma _+}e^{j\left( \frac{\varphi +\pi }{2}\right) } &{} \quad 0 \end{array} \!\!\right) , \end{aligned}$$

(18)

$$\begin{aligned}&C= \left( \! \begin{array} {cccc} 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 \\ 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 \\ 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 \\ 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{array} \!\!\right) . \end{aligned}$$

(19)

From Eqs. (10)–(19), the following formulas for the transmission T and reflection R coefficients can be derived:

$$\begin{aligned}&T_{1\rightarrow 2}(\omega )=\left| \frac{s_{2-}}{s_{1+}}\right| ^2=\left| \frac{2\gamma _+}{j(\omega -\omega _+)+2\gamma _+ + 2\gamma _{i+}}\right| ^2, \end{aligned}$$

(20)

$$\begin{aligned}&T_{1\rightarrow 4}(\omega )=\left| \frac{s_{4-}}{s_{1+}}\right| ^2=\left| 1-\frac{2\gamma _+}{j(\omega -\omega _+)+2\gamma _+ + 2\gamma _{i+}}\right| ^2, \end{aligned}$$

(21)

$$\begin{aligned}&T_{2\rightarrow 1}(\omega )=\left| \frac{s_{1-}}{s_{2+}}\right| ^2=\left| \frac{2\gamma _-}{j(\omega -\omega _-)+2\gamma _- + 2\gamma _{i-}}\right| ^2, \end{aligned}$$

(22)

$$\begin{aligned}&T_{2\rightarrow 3}(\omega )=\left| \frac{s_{3-}}{s_{2+}}\right| ^2=\left| 1-\frac{2\gamma _-}{j(\omega -\omega _-)+2\gamma _- + 2\gamma _{i-}}\right| ^2, \end{aligned}$$

(23)

$$\begin{aligned}&T_{1\rightarrow 3}(\omega )=T_{2\rightarrow 4}(\omega )=R_{1}(\omega )=R_{2}(\omega )=0. \end{aligned}$$

(24)

In the ideal case ($T_{1\rightarrow 2}(\omega _+) = 1$ and $T_{2\rightarrow 1}(\omega _+) = 0$), the condition $\gamma _{i+}=0$ must be satisfied. However, this condition cannot be satisfied by real circulators, since it implies that the resonator should be lossless (no intrinsic losses). In practice, it is sufficient to ensure that $\gamma _+ \gg \gamma _{i+}$ for the proper operation of the circulator.

Assuming that this condition is also met by the clockwise rotating mode, that is, $\gamma _- \gg \gamma _{i-}$, one can directly derive the condition $|\omega _+ - \omega _-|^2 \gg 0$, that is, the frequency splitting between the two counter-rotating resonant modes must be high enough in order to ensure the adequate functioning of the device.

In addition, for the case wherein the excitation is connected to port 1, one can derive, from Eqs. (20) and (21), the following formulas for the operating bandwidth of the circulator:

$$\begin{aligned}&\Delta \omega _{o} = 4\sqrt{\left( 10^{-0.1n_o}\right) \gamma _+^2-\left( \gamma _+ + \gamma _{i+}\right) ^2}, \end{aligned}$$

(25)

$$\begin{aligned}&\Delta \omega _{i} = 4\sqrt{\frac{\left( 10^{0.1n_i}\right) \left( \gamma _+ + \gamma _{i+}\right) ^2-\gamma _{i+}^2}{1-10^{0.1n_i}}}, \end{aligned}$$

(26)

where $\Delta \omega _{o}$ is the bandwidth measured in the output port and $\Delta \omega _{i}$ is the bandwidth measured in the isolated port. The parameters $n_o$ and $n_i$ are the reference levels (in dB) considered for the calculation of $\Delta \omega _{o}$ and $\Delta \omega _{i}$, respectively.

The results given in Sect. 4.2 are derived from the TCMT Eqs. (20)–(26) and refer to the following set of parameters: $\omega _{ + } = 3.1604 \times 10^{{13}}\;{\text{rad}}\;{\text{s}}^{{ - 1}}$, $\omega _{ - } = 3.8327 \times 10^{{13}}\;{\text{rad}}\;{\text{s}}^{{ - 1}}$, $\gamma _{ + } = \gamma _{ - } = 10 \times 10^{{11}} \;{\text{rad}}\;{\text{s}}^{{ - 1}}$, and $\gamma _{{i + }} = \gamma _{{i - }} = 2.5 \times 10^{{11}} \;{\text{rad}}\;{\text{s}}^{{ - 1}}$.

Appendix 3: Calculation of the total decay rate of the counter-rotating modes

The quality factor of the $a_+$ mode ($Q_+$) can be calculated from the $S_{21}$ curve obtained from COMSOL Multiphysics by using the formula

$$\begin{aligned} Q_+=\frac{f_0}{\Delta f} \end{aligned}$$

(27)

where $f_0$ is the resonant frequency and $\Delta f$ is the -3 dB resonance width (or, equivalently, the full width at half maximum—FWHM).

The parameter $Q_+$ can also be described in terms of the total decay rate of the $a_+$ mode. Assuming the decay rate matrix $\Gamma$ given in Eq. (16), the quality factor is defined as follows:

$$\begin{aligned} Q_+=\frac{\omega _0}{2\times 2\gamma _{t+}}, \end{aligned}$$

(28)

where $2\gamma _{t+}$ is the total decay rate of the mode.

By combining Eqs. (27) and (28), it is possible to obtain the following formula for the calculation of $\gamma _{t+}$:

$$\begin{aligned} \gamma _{t+}=\frac{\pi \Delta f}{2}. \end{aligned}$$

(29)

From Fig. 11, one can obtain $\Delta f \approx 880$ GHz. Therefore, $\gamma _{{t + }} \approx 13.823 \times 10^{{11}} \;{\text{rad}}\;{\text{s}}^{{ - 1}}$. This value is very close to the sum of the considered values for $\gamma _{+}$ ($10 \times 10^{{11}} \;{\text{rad}}\;{\text{s}}^{{ - 1}}$) and $\gamma _{i+}$ ($2.5 \times 10^{{11}} \;{\text{rad}}\;{\text{s}}^{{ - 1}}$). Since the field profile of the counter-rotating modes $a_+$ and $a_-$ is the same except for the rotation direction and the resonance curves of the modes are similar, one can assume that the total decay rates of the modes are approximately equal ($\gamma _{t+}\approx \gamma _{t-}$).