# Free-Space Nonreciprocal Transmission Based on Nonlinear Coupled Fano Metasurfaces

^{1}

^{2}

^{3}

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^{*}

## Abstract

**:**

## 1. Introduction

## 2. Principle of Operation of Nonlinearity-Based Nonreciprocal Devices Based on Coupled Fano Metasurfaces

^{2}. The second design is based on high quality-factor resonators, which are more sensitive to fabrication errors but require much lower input power levels. The results for the proposed designs are plotted with red asterisks in Figure 1b. Some of them lie in the red shaded region, thereby overcoming the fundamental limitation Equation (5).

## 3. Practical Implementations (for the Numerical Analysis, Please See Section ‘Full Wave Numerical Simulation’)

#### 3.1. Low Quality Factor Resonators

_{01}dielectric waveguide mode [42]. The quality factor of the resonance is in the order of 10, suggesting that the design is not very prone to fabrication errors and imperfections, and has a wide bandwidth.

#### 3.2. High Quality Factor Resonators

^{2}, due to the low-quality factor of the resonator, which may hinder its broad applicability. However, one possible solution is to employ materials with high nonlinearity, e.g., multiple quantum wells have shown extremely high nonlinear susceptibilities both in both the second and third orders [45,48,49]. Higher-Q Fano resonators can be used to reduce the required power, however, and for this reason we aim to maximize the field enhancement levels inside the resonators by designing a high-Q Fano resonator based on guided mode resonance coupling through narrow slits in a dielectric slab, as shown in inset of Figure 6a. The transmission coefficient for a normally incident plane wave, with an electric field in the z-direction (TE), is shown in Figure 6a, exhibiting a typical Fano line shape with zero transmission at ${\lambda}_{0}=1.5608\mathsf{\mu}\mathrm{m}$ and unitary transmission at the very nearby wavelength ${\lambda}_{0}=1.5598\mu \mathrm{m}$. In order to evaluate the quality factor of this resonator, we plot the average electric field enhancement inside the nonlinear material |E|

^{2}, as shown in Figure 6b, which shows a Lorentzian line shape with quality factor $Q=1700,$ compared to $Q=10$ of the design in the previous section. Additionally, the inset shows that the electric field distribution inside the slab is similar to the TE

_{01}waveguide mode [42]. The second resonator is formed by adding a glass slab (${n}_{glass}=1.5$) to another Si grating, as shown in the inset of Figure 6c. Similar to Figure 6a, the transmission exhibits a sharp Fano resonance; however, the average electric field enhancement is smaller than in Figure 5d. This is understood, as we chose the guided mode resonance to be formed inside the glass slab, and not in the Si layer, as confirmed by the electric field distribution in the inset of Figure 6d. We chose the second resonator to be attached to the glass substrate in order to control the field enhancement in the Si layer and hence obtain unitary transmission at the same power levels when the nonlinearity is included (similar to point $\aleph $ in Figure 4b). In addition, we plot the CMT standard model results in Figure 6a, and Figure 6c using the fitting parameters from Table 3 in Section 5.5 (Fano Resonator Parameters), which shows excellent agreement with the full wave simulation.

## 4. Conclusions

## 5. Materials and Methods

#### 5.1. Coupled Mode Theory

#### 5.2. Nonlinear Bistability

#### 5.3. Bistability Condition

#### 5.4. Effective Power of Coupled Nonlinear Fano Resonators

#### 5.5. Fano Resonator Parameters for Figure 3, Figure 4, Figure 6 and Figure 7

#### 5.6. Full Wave Numerical Simulation (We Used Full Wave Numerical Simulation to Obtain the Results in Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8)

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Schematic depicts the nonlinearity-based nonreciprocal metasurface bilayer presented in this work. The arrows represent high power plane waves pointing to the direction of propagation that are blocked and reflected when propagating from the top to the bottom (red arrow) and transmitted when propagating from the bottom to the top (blue arrows). The zoomed in box shows one unit cell of the metasurface top layer made of Si (the bottom layer is similar, but with different geometrical parameters). Upon the plane wave excitation, the transmission from each layer takes the Fano line shape in the dashed box, and can be explained by the interference between: (1) the dark narrow band mode similar to the waveguide mode inside the unperturbed slab (i.e., with filled Si in the dashed box) and (2) the bright wideband mode that resembles a Fabry–Pérot mode or background reflection, as shown in the top and bottom of the shadowed rectangle, respectively. $\left({\omega}_{1},{\omega}_{2}\right)$ are the resonance frequencies of the modes, while their complex amplitudes are $\left({a}_{1},{a}_{2}\right)$ and their coupling rates to the free space are (${\kappa}_{1},{\kappa}_{2}$ ) for the dark and bright mode, respectively. (

**b**) Transmission in the forward direction vs the nonreciprocal intensity range for the various nonlinear resonator designs presented in this work (red asterisks). The blue shaded region corresponds to the bound in Equation (5).

**Figure 2.**Nonlinear response of individual Fano resonators as a function of the input frequency and power. (

**a**) Bistability in the shift of the zero-transmission frequency, ${\delta}_{1}$, versus the frequency $\omega $ of the input plane wave, with fixed input power P = 0.5. For an incident wave with an increasing frequency, the shift follows the path through the point ${c}_{1}$ along the arrow direction to the top branch, and for a decreasing frequency the shift path follows the line passing through the point ${c}_{2}$ along the arrow direction to the lower branch. The dashed lines indicate the value ${\delta}_{1}$, such that the transmission coefficient goes to 0 dB. (

**b**) Similar to (

**a**), but with the transmission coefficient instead of shift ${\delta}_{1}$. The blue line plots the linear response of the resonator, i.e., at P = 0. (

**c**) Similar to (

**b**) but considering different power levels and only branches with an increasing frequency. (

**d**–

**f**) Similar to (

**a**–

**c**) but analyzing the variations with the input power at fixed frequency $\omega =0.995\times {\omega}_{0}$. The other parameters are constant for all of the subfigures, ${\omega}_{0}=1.1,{\kappa}_{2}=152$, ${\omega}_{2}=100$, and ${\kappa}_{1}=0.02$.

**Figure 3.**Operation of the device based on coupled shifted Fano resonators. (

**a**) Linear transmission coefficient for two individual resonators with different zero transmission frequencies. ${\omega}_{01}$ is the zero-transmission frequency of the first resonator, and ${\omega}_{02}$ is for the second resonator. (

**b**) Nonlinear transmission coefficient of the resonators in (

**a**) excited at the frequency dictated by dashed lines in (

**a**) with increased power P. (

**c**) Nonlinear transmission coefficient for coupled resonators, separated by a delay line $\theta $ with increasing P. The transmission from left to right is ${T}_{LR}$, while that from righ to left is denoted as ${T}_{LR}$. (

**d**,

**e**) Linear and nonlinear reflection phases of the individual resonators. (

**f**) The effective power seen by the second resonator when the device is excited from left for different electrical lengths $\theta $. The vertical dashed lines indicate the required input power for the transition to happen in the transmission curve for the corresponding thetas. We used the parameters for resonator 1 and resonator 2 as given in Table 3 (row 1 and row 2) in Section 5.5 (Fano Resonator Parameters).

**Figure 4.**Full wave simulation results for the design of an optical Fano resonator for free-space radiation at normal incidence, achieved by choosing p = λ/2, where λ is the free-space wavelength. (

**a**) Linear response of the transmission coefficient of individual resonators arranged in a periodic array, with period p excited with a normally incident plane wave, with polarization as shown in the inset. The red markers indicate the CMT results with the fitting parameters given in Table 3 in Section 5.5 (Fano Resonator Parameters). The color plot inset shows the z-component of the magnetic field distribution inside the slab, consistent with the guided mode inside the unperturbed dielectric slab of the same thickness, and the color bar is normalized to the maximum field. (

**b**) Transmission coefficient for individual resonators in dB versus the incident power for normal incidence assuming ${\chi}^{(3)}=2.8\times {10}^{-18}{m}^{2}/{V}^{2}$ at the resonance frequency.

**Figure 5.**Full wave simulation results for transmission from opposite ports (${T}_{12},{T}_{21}$) of the proposed bilayer metasurface shown in the inset of (

**a**), where the top resonator has dimensions (${d}_{1}=0.203\lambda ,{d}_{2}={d}_{1}/4$ ) and the bottom resonator has dimensions (${d}_{1}=0.18\lambda ,{d}_{2}={d}_{1}/3$ ) for the electrical distance $\theta =1.5\pi =2\pi /\lambda l,$ and l is the physical distance between the two resonators. The period of the two-groove array is $\lambda /2$ where $\lambda =1.55\mathsf{\mu}\mathrm{m}$. The transmission of a normally incident plane wave with increased input power ${P}_{inc}$ at (

**a**) wavelength ${\lambda}_{0}=1.55\mathsf{\mu}\mathrm{m}$, (

**b**) ${\lambda}_{0}=1.56\mathsf{\mu}\mathrm{m}$ and (

**c**) ${\lambda}_{0}=1.565\mathsf{\mu}\mathrm{m}$. The inset in (

**c**) shows the full wave simulation of the magnetic field when the device is excited from each side separately when ${P}_{inc}=18\mathrm{GW}/{\mathrm{cm}}^{2}$. (

**d**) Transmission from two sides versus the frequency at a fixed power level of 16.5 GW/cm

^{2}. The NRIR and transmission for the plots from (

**a**) to (

**c**) are NRIR = [0.27, 0.541, 0.984] dB, and the peak transmission is T = [−0.132, −0.04, −0.037] dB, respectively.

**Figure 6.**Full wave simulation results for the linear response of the Fano resonators. (

**a**) Linear transmission of a periodically patterned dielectric slab for a normally incident plane wave excitation with an electric field along the z axis. The red markers are the CMT standard model results using the parameters given in Table 3 (see Section 5.5 (Fano Resonator Parameters)) for resonator 1 (w/o glass). (

**b**) Average squared value of the electric field inside the Si material shown in (

**a**); the inset shows the electric field distribution at resonance, and the scale bar is normalized to the maximum field. (

**c**) The same as (

**a**), but with an added glass slab. The red markers are the CMT standard model results using the parameters given in Table 3 (see Section 5.5 (Fano Resonator Parameters)) for resonator 2 (w glass). (

**d**) The same as (

**b**) but for the structure in (

**c**). The geometry parameters used are d = 0.1179 μm, p = 1.4763 μm, dg = 0.45875 μm, and slit width = 0.0536 μm.

**Figure 7.**Full wave simulation results for excitation at frequency c/1.56104 [um], with a normally incident plane wave of power ${P}_{inc}$. (

**a**) Nonlinear transmission for isolated gratings as a function of the input power. The red markers indicate the results obtained using the CMT standard model with the parameters given in Table 3 (see Section 5.5 (Fano Resonator Parameters)). (

**b**) Transmission for the coupled metasurface design with θ = 6.4π shown in the inset. ${T}_{12}$ indicates the transmission from top to bottom, while ${T}_{12}$ indicates the transmission from bottom to top. The NRIR is 2.1 dB, and the peak forward transmission is −0.2 dB.

**Figure 8.**Full wave simulation results for the nonlinear device in the inset of Figure 7b, excited at frequency c/1.56104 [um] with a normally incident plane wave of power ${P}_{inc}$ for the different values of ϴ displayed in the bottom right corner of each figure. The horizontal axis is normalized by the period $p=1.475\mathsf{\mu}\mathrm{m}$ such that it has units of Watts.

**Figure 11.**(

**Left panel**) Setup of the frequency domain simulation showing that the structure is segmented in triangle areas with a maximum size of 38nm, and that it is excited by a periodic port either from the top or bottom, while keeping the periodic boundary condition on the left and right. (

**Right panel**) Distribution map of the area of each mesh element; the scale bar unit is m

^{2}.

Work | Breaking Reciprocity Due to: | Bandwidth/Center Frequency | Modulation Frequency (mf) or Total Gain of the Amplifiers (tga) or Kerr Nonlinearity Coefficient $\left({\mathit{\chi}}^{\left(3\right)}\right)$ | Thickness/Wavelength | Pump Power Per Unit Cell or Signal Power/Intensity | Isolation (or Transmission Contrast) at Best Insertion Loss | Frequency Conversion/Programmable | ||
---|---|---|---|---|---|---|---|---|---|

[11] | Time modulation | NA/ 5.28 [GHz] | MF | 50 [MHz] | 2.54 [mm]/56.8 [mm] | Modulation signal power or intensity | NA | NA | No/yes |

[13] | 0.3 [GHz]/ 8.97 [GHz] | 370 [MHz] or 600 [MHz] | 2 [mm]/33.33 [mm] | 10 dBm or 1 V | 5 dB loss, isolation of 30 dB | Yes/yes | |||

[14] | 5.77 [THz]/ 348.8 [THz] | 2.8 [THz] | 400 [nm]/860 [nm] | 15 GW/cm^{2} | NA | Yes/yes | |||

[17] | Unidirectional gain amplifiers | 6 [MHz]/ 944 [MHz] ^{¥} | TGA | 0 dB | 31.7 [mm]/317 [mm] | DC power of each amplifier in one layer, number of layers | NA, 2 | Isolation of −1.5 dB assuming 0 dB insertion loss ^{¥} | No/yes |

[18] | 0.17 [GHz]/ 5.9 [GHz] | 20 dB | 1.7 [mm]/50.8 [mm] | 0.18 [W], 2 | 17 dB transmission gain and 10 dB loss correspond to 27 dB. | ||||

[19] | 0.25 [GHz]/ 5.875 [GHz] | 10 dB- 30 dB | 1.82 [mm]/51 [mm] | 0.1–0.2 [W] ^{£}, 2 | 20 dB transmission gain and 20 dB loss correspond to 40 dB isolation | ||||

[20] | 0.05 [GHz]/ 5.5 [GHz] | 20 dB | 2.1 [mm]/54.54 [mm] | 0.1 [W], 2 | 13 dB of transmission gain and 32 dB isolation | ||||

[23] | Kerr nonlinearity | NA | ${\chi}^{\left(3\right)}$ (m ^{2}/V^{2}) | $2.8\times {10}^{-18}$ | 0.1 [um]/1.5 [um] | Signal intensity | 5 kW/cm^{2} | −17 dB at −1.2 dB over 4.77 dB * | No/no |

[24] | (2.7–6.15) [um]/1.53 [um] | (1.5–2) MW/cm^{2} | Isolation of −25.4 dB at insertion loss of −0.46 dB over NRIR of 2.79 dB ** Isolation of −35.7 dB at insertion loss of −0.41 dB over NRIR of 1.5 dB *** Isolation of −15.2 dB at −0.044 dB over NRIR of 1.52 dB **** ^{¿} | ||||||

This work | 0.6 [THz]/ 192 [THz] | (1.33–5.334) [um]/1.56 [um] | (16.8–0.001) GW/cm^{2} | −56 dB at −0.04 dB, −65 dB at −0.2 dB |

^{¥}From Figure 4 [16].

^{£}Based on the datasheet of the amplifier and the DC bias used in the paper and similar transistors used in previous publications [17], it is expected to be 0.1–0.2 W [18]. * From Figure 3c [23]. ** From Figure 3 [22]. *** From Figure 4 [22]. **** From Figure 5 [22].

^{¿}Notice that the NRIR of this metasurface does not break the limitation of the single Fano resonator nonlinear isolator, even though 1.52 dB NRIR goes beyond the limit, the isolation is not infinite, so a fair comparison cannot be guaranteed here.

**Table 2.**Parameters of the Fano resonance in Figure 4.

Structure 1 (${d}_{1}=0.18\lambda ,{d}_{2}={d}_{1}/3$) ($\omega ,\kappa $) are normalized by ($2\pi \times c/2p$) | ${\omega}_{1}=1.016$ | ${\omega}_{2}=1.3$ | ${\kappa}_{1}=0.04$ | ${\kappa}_{2}=0.138$ | $\theta =0$ |

Structure 2 (${d}_{1}=0.203\lambda ,{d}_{2}={d}_{1}/4$) ($\omega ,\kappa $) are normalized by ($2\pi \times c/2p$) | ${\omega}_{1}=0.977$ | ${\omega}_{2}=1.33$ | ${\kappa}_{1}=0.07$ | ${\kappa}_{2}=0.2$ | $\theta =0$ |

Structure 1 (w/o glass) ($\omega ,\kappa $) are normalized by ($2\pi \times {10}^{14}$) | ${\omega}_{1}=1.92204$ | ${\omega}_{2}=100$ | ${\kappa}_{1}=0.00191$ | ${\kappa}_{2}=152.2$ | $\theta =0$ | ${\left|{a}_{0}\right|}^{2}=4.5\times \frac{{\omega}_{1}}{{\kappa}_{1}^{2}}$ |

Structure 2 (w glass) ($\omega ,\kappa $) are normalized by ($2\pi \times {10}^{14}$) | ${\omega}_{1}=1.921628$ | ${\omega}_{2}=100$ | ${\kappa}_{1}=0.00062$ | ${\kappa}_{2}=144.4$ | $\theta =0$ | ${\left|{a}_{0}\right|}^{2}=10\times \frac{{\omega}_{1}}{{\kappa}_{1}^{2}}$ |

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Mekawy, A.; Sounas, D.L.; Alù, A. Free-Space Nonreciprocal Transmission Based on Nonlinear Coupled Fano Metasurfaces. *Photonics* **2021**, *8*, 139.
https://doi.org/10.3390/photonics8050139

**AMA Style**

Mekawy A, Sounas DL, Alù A. Free-Space Nonreciprocal Transmission Based on Nonlinear Coupled Fano Metasurfaces. *Photonics*. 2021; 8(5):139.
https://doi.org/10.3390/photonics8050139

**Chicago/Turabian Style**

Mekawy, Ahmed, Dimitrios L. Sounas, and Andrea Alù. 2021. "Free-Space Nonreciprocal Transmission Based on Nonlinear Coupled Fano Metasurfaces" *Photonics* 8, no. 5: 139.
https://doi.org/10.3390/photonics8050139