The non-reciprocal transmission phenomenon, which breaks the symmetry of time inversion, has attracted more and more attention recently. The magneto-optical (MO) effect provides a stable, ultrafast, and high-resolution optical control method, and the non-reciprocity revealed by the Onsager–Casimir principle has a wide range of applications (Kharratian et al., 2020), such as optical isolators (Takeda and John, 2008; Bi et al., 2011; Zhang et al., 2019), all-optical signal processing, circulators (Smigaj et al., 2010; Dmitriev et al., 2012; Pintus et al., 2013), and fast imaging (Zhang et al., 2018). Thus far, however, structures have usually exhibited a weak one-way transmission effect even under some judicious designs as the strength of MO activity of the natural materials is extremely small. Metamaterials, artificial materials composed of subwavelength unit cells, provide a powerful platform for manipulating the propagation of light (Liberal and Engheta, 2017; Niu et al., 2018; Guo et al., 2022). Leviyev et al. (2017) proposed that the one-way surface plasmon polaritons (SPPs) can be realized based on the MO hyperbolic metamaterials. In particular, the photonic crystals (PCs) containing the metamaterials have become the focus of scientific research because of their unique characteristics of enhancing light–matter interactions (Guan et al., 2006). Guo et al. (2018) uncovered the non-reciprocal transmission in the one-dimensional (1D) PCs with the magnetized epsilon-near-zero (ENZ) defect. Moreover, they showed that the wavelength difference of transmission peaks along two opposite incident directions can be significantly enhanced over two orders of magnitude compared with the bismuth iron garnet MO defect. This large non-reciprocal effect does not resort to the enhancement of MO activity by the strong external magnetic field and opens novel routes to exploit advanced materials for steering the non-reciprocal transmission of electromagnetic waves in nano-scale structures (Sounas et al., 2013; Guo et al., 2018). We know that some special interference effects will significantly enhance the interaction between light and matter. Therefore, the interference effect combined with MO metamaterials and PCs is promising to further enhance the MO effect.

Fano interference is a ubiquitous scattering wave phenomenon, and it can describe the interaction of waves from quantum interference to classical physics (Fano, 1961; Miroshnichenko et al., 2010; Limonov et al., 2017). At present, Fano interference in classical systems has produced a series of interesting physical phenomena and applications, such as slow light effect (Wu et al., 2011), molecular recognition (Yanik et al., 2011; Wu et al., 2012), optical switching (Argyropoulos et al., 2012; Dabidian et al., 2015), sensing (Gupta et al., 2017), and coding (Manjappa et al., 2018). It is worth mentioning that Zhang et al. (2013) found that Fano interference can be used to significantly reduce the threshold value of optical switching and bistability. In addition, Singh et al. (2014) found new high-sensitivity sensors based on Fano interference, and the sensitivity even exceeds quadrupole resonance. Although Fano interference can be used to enhance the light–matter interaction, it is mainly realized by adjusting structural parameters (Vlasov et al., 2001; Rybin et al., 2009; Poddubny et al., 2012; Rybin et al., 2013; Pariente et al., 2016; Rybin et al., 2016; Tribelsky and Miroshnichenko, 2016). It is very meaningful to realize the Fano interference of active control (Ott et al., 2013).

In this work, aiming at the aforementioned problems, we study the Fano interference realized by heterostructures composed of ENZ material and truncated photonic crystals. In particular, we verify that the asymmetric transmittance profile of the heterostructure spectrum due to Fano-type interference can be flexibly tuned by controlling the direction of the external magnetic field. The MO enhancement properties of isotropic and anisotropic ENZ materials are similar. Therefore, our results not only reveal an actively controlled Fano interference using the external magnetic field but also demonstrate the field enhancement effect of the ENZ material, together with the asymmetric characteristic of the Fano-type transmission, can enhance the MO effect and realize non-reciprocal transmission. In addition to non-reciprocal transmission, the results of this study also provide new ways for bound states in the continuum (BIC) (Bogdanov et al., 2019; Tan et al., 2020) and topological state (Zangeneh-Nejad and Fleury, 2019) manipulation, which is expected to be used to construct new optical sensors (Guo et al., 2021a), lasers (Yu et al., 2021), and energy transfer devices (Pham et al., 2017; Guo et al., 2021b).

We consider the structure, shown in Figure 1, which is composed of a magnetized isotropic epsilon-near-zero (ENZ) material layer M and a truncated PC: (AB)_n. In particular, M, A, and B correspond to the Ag, SiO₂, and TiO₂, respectively. The electromagnetic waves with transverse magnetic (TM) polarization (E _x , H _y , E _z ) launch into the structure with an incident angle θ . We suppose that the external magnetic field is in the y-direction, and the permittivity of Ag can be written as follows (Yu et al., 2008; Tuz, 2016): ε M ¯ = ( ε x x 0 − i α 0 ε y y 0 i α 0 ε z z ) , (1) where ε x x = ε y y = ε z z = ε M = ε ∞ − ω p 2 ω ( ω + i γ ) . α denotes the strength of MO coefficient of the medium. Here, ε ∞ is the high-frequency permittivity with a value of 4.09 for Ag. ω P = ( N 0 e 2 ∕ ε 0 m ∗ ) 1 2 = 1.33 × 10 16   r a d / s is the plasma frequency, where N 0 , e , and m ∗ are the free electron density, charge, and effective electron mass, respectively. In particular, the MO coefficient can be expressed by α = i ω B ω ω p 2 ( ω + i γ ) 2 − ω B 2 (Yu et al., 2008; Leviyev et al., 2017). ω B = e B / m ∗ denotes the cyclotron frequency, which is dependent on the external magnetic field B . The diagonal permittivity is not affected by the magnetic field. γ = 1.13 × 10 14 is the damping frequency. For the wavelength, λ 0 = 286.5 nm, the permittivity of Ag is ε M ≈ 0 + 0.06   i , which corresponds to the epsilon-near-zero (ENZ) media. The refractive index of the SiO₂ and TiO₂ are 1.46 and 2.56, respectively. Considering the TM wave impacting the structure with θ = 50^°, the first bandgap of PC is determined by the Bragg condition ϕ = k A z d A + k B z d B = π , where k A z and k B z are the wavevectors in the layer A and layer B, respectively. In order to obtain the definite Fano-type interference between the ENZ layer and the PC structure, the center of bandgap should be near the ENZ wavelength λ 0 of layer M (Manjappa et al., 2018). Here, the thickness of non-magnetic layers of A and B is set as d A = 70 nm and d B = 25 nm, respectively.

For the magnetized Ag layer in Eq. 1, the TM and transverse electric (TE) polarization are completely decoupled, and the EMs will maintain their initial polarization during propagation. As a result, the concise 2 × 2 transfer matrix method can be used to study the non-reciprocal transmission of the heterostructure in Figure 1 (Guo et al., 2018). The incident angle of TM-polarized wave is selected to be θ = 50^°. Figure 2A shows the transmittance spectrum of the single magnetized isotropic epsilon-near-zero material layer M, the 1DPC: (AB)₃, and their composite structure M(AB)₃, which are marked by the dotted red line, dashed blue line, and solid green line, respectively. It can be seen that the single M layer [1DPC: (AB)₃] exhibits a sharp transmission dip (a broad forbidden band) at 286.5 nm (from 218 to 416 nm). It should be noted that the imperfect bandgap here is caused by the small number of photonic crystal layers. In particular, the interference between the discrete and broadband reflection pathway leads to the asymmetric Fano-type spectrum for the heterostructure M(AB)₃. The transmission peak and dip at 285.4 and 291.7 nm can be clearly identified, respectively.

The sign of the MO coefficient α will flip when the magnetic field in the backward directions is applied. In particular, the transmittance spectrum of the composite structure M(AB)₃ for different MO coefficient: α = 0 (solid green line), α = 0.1 (dashed blue line), and α = −0.1 (dotted red line) are respectively shown in Figure 2B. We can find that the non-reciprocal transmission is realized in this heterostructure M(AB)₃. Moreover, the transmittance peak at the high frequency is more sensitive to the direction of the applied magnetic field than the transmission dip at the low frequency. For comparison, the transmittance spectrum of the heterostructures MA and MB for different MO coefficient is shown in Figures 2C,D, respectively. The thickness of layer A in Figure 2C and layer B in Figure 2D is the same as the total thickness of (AB)₃ in Figure 2B. We see that the transmitted spectrum with different MO coefficients nearly overlaps. Therefore, although the magnetized ENZ can realize the enhancement of the field (Guo et al., 2018), the associated MO effect is not evident in the heterostructures MA and MB with simple resonance. Fano interference provides an effective way to enhance the MO effect in heterostructures composed of epsilon-near-zero materials and truncated photonic crystals, as shown in Figure 2B.

In order to realize the MO effect based on the Fano-type interference effect in the heterostructure composed of epsilon-near-zero materials and truncated photonic crystals, we further study the dependence of the transmittance spectrum of the heterostructure M(AB)₃ on the incident angle, the thickness of layer M, and the strength of MO coefficient α . When d _M = 10 nm, we can find that non-reciprocal transmission will become more evident as the angle increases, as shown in Figures 3A–C. In particular, the different MO coefficients: α = 0, α = 0.1, and α = −0.1 are marked by the solid green line, dashed blue line, and dotted red line, respectively. Similarly, when we set the incident angle θ = 50^°, the MO effect will be enhanced by changing the thickness of layer M from 10 to 30 nm, which is shown in Figures 3D–F. In addition, when we set the incident angle θ = 50^° and d _M = 10 nm at the same time, Figures 3G–I give the transmittance spectrum of the composite structure M(AB)₃ for the MO strength. It can be clearly seen that the MO effect will be significantly enhanced with the increase of MO strength. When the sign of optical coefficient α changes, due to the homogeneous property of Ag-based ENZ material, the transmittance dip of the Fano-type transmittance spectrum always appears at the zero point of permittivity of M layer. However, the transmittance peak will blueshift when α changes from negative to positive. In order to quantitatively analyze the dependence of MO effect on the aforementioned introduced different parameters, the difference between the α > 0 and α < 0 at the transmission peak and transmission dip of the Fano-type spectrum are denoted as Δ T p and Δ T d , respectively. Δ T p and Δ T d for different cases in Figure 3 are summarized in Table 1.

Next, we study the dependence of MO effect on the incident angles, while the thickness of layer M and the strength of MO coefficient are fixed at d _M = 10 nm and | α | = 0.10, respectively. In particular, in order to determine the difference of transmission on the MO coefficient, Δ T 1 = T | α = 0.1 − T | α = 0 and Δ T 2 = T | α = 0.1 − T | α = − 0.1 as a function of the wavelength and the incident angle are shown in Figures 4A,B, respectively. Here, the forward (backward) incident TM-polarized waves are denoted by θ > 0 (θ < 0). From Figure 4A, it can be seen that the transmittance will be strongly modified after applying the external magnetic field in this heterostructure with magnetized ENZ material. Moreover, the transmittance difference between two magnetic field directions Δ T 2 is studied, and the corresponding enhancement of the MO effect on the incident angle can be determined in Figure 4B. It can be seen that the forward incident Δ T 2 is opposite to the backward incident Δ T 2 (i.e., Δ T 2 ( θ ) is equal to Δ T 2 ( − θ ) ). The dashed line denotes the position of θ = 0^° . The maximum of | Δ T 2 | is marked by the white crosses, which are marked by the white crosses at θ = ± 58^° and θ = ± 67^°, respectively.

The Fano profile of the heterostructure M(AB)₃ with magnetized ENZ material can also be flexibly controlled by changing the external magnetic field. Figures 5A,B show the maximum of Δ T 2 and - Δ T 2 as the function of the thickness of layer A and layer B for different incident angles and wavelengths. The dashed lines denote the sensitivity of the heterostructure on the structure parameters that the maximum of Δ T 2 and - Δ T 2 are very close. We take two cases P and Q for example, which are marked by pink and blue stars, respectively. The variation of the direction of magnetic field can lead to different Fano asymmetric parameters and different asymmetry Fano resonance spectra, as shown in Figures 5C,D For the heterostructure P with d _A = 50 nm and d _B = 32 nm in Figure 5C, the Fano asymmetry parameter of the weak Fano-type spectrum is positive without the external magnetic field, as shown by the solid green line. After considering the positive magnetic field α >0, the Fano-type spectrum becomes clear (marked by the red dashed line) and the Fano asymmetry parameter is still positive because of the transmittance dip and peak appearing at low and high frequencies, respectively. However, when the magnetic field changes to negative α <0, the Fano asymmetry parameter is still negative because the transmittance dip and peak appear respectively at high and low frequencies. A similar property is demonstrated for the heterostructure Q with d _A = 63 nm and d _B = 20 nm, as shown in Figure 5D. Therefore, we can clearly see that the asymmetric transmittance profile of the heterostructure spectrum due to Fano-type interference can be flexibly tuned by controlling the direction of the external magnetic field. In addition, the behavior of enhanced magneto-optical activity for a TM wave in ENZ media is also suitable for an incident TE wave in mu-near-zero (MNZ) configuration.

The Fano-type interference effect in heterostructures actually exists in both isotropic and anisotropic ENZ components, regardless of the effective electromagnetic parameters. The heterostructures C(AB)_n composed of an effective anisotropic magnetized ENZ layer and a truncated PC are studied. Similar to Figure 1, the schematic of the heterostructure with the effective anisotropic ENZ layer is shown in Figure 6A. Here the anisotropic ENZ layer is mimicked by subwavelength SiO₂/Ag/SiO₂ stacks as (DEF)_M. The unit cell of SiO₂/Ag/SiO₂ ensures that Ag is always clamped by SiO2, and the SiO2 layer plays a role in protecting the Ag layer. The thickness of the layers D, E, and F are 5 nm, 3, and 2 nm, respectively. Because the thickness of unit cell d = d D + d E + d F is far less than the wavelength of electromagnetic wave in the structure, the structure (DEF)_M can be homogenized based on the effective medium theory (EMT) (Guo et al., 2022): ε x x = ε y y = ε / / = ε D f D + ε E f E + ε F f F , (2) ε z z = ε ⊥ = ( ε D / f D + ε E / f E + ε F / f F ) − 1 , (3) where f i   = d i / d , ( i = D ,   E ,   F ) denotes the filling ratio of component layer of anisotropic ENZ material. Based on Eqs 2, 3, the effective permittivity of the C layer is shown in Figure 6B. The complex ε / / and ε ⊥ are marked by blue and red, respectively. The real part and imaginary part of the permittivity are shown by the solid line and dotted line, respectively. The magnetized anisotropic ENZ with negligible loss is marked by the black dashed line. In addition, to ensure consistency, the truncated PC (AB)_n is the same as introduced before.

Here the enhanced MO effect by Fano interference is studied based on the narrow discrete resonance of the effective anisotropic ENZ layer and the broadband reflection of the truncated PC, as shown in Figure 7A. The incident angle is set as θ = 50^°. Similar to Figure 2A, the constructive and destructive interference of the Fano-type spectrum is given based on the single C layer and the truncated PC (The transfer matrix of the effective anisotropic MO ENZ material can be found in the Supplementary Material). It can be seen that for the effective anisotropic ENZ layer, the transmittance spectrum exhibits a sharp transmission dip at 286 nm, which corresponds to a narrow discrete resonance, as shown by the dashed line in Figure 7A. Although EMT will become more effective with the increase of the number of unit cells, in the actual structure, we use limited unit cells m = 3 to illustrate the applicability of EMT. The transmittance spectrum of the (DEF)₃ is marked by the red dotted line, which met well with the results of effective single C layer. In addition, the transmittance spectrum of the truncated PC (AB)₃ is given by the blue dashed line, which exhibits a broadband reflection from 218 to 416 nm. As a result, the interference between the discrete and broadband reflection pathway leads to the asymmetric Fano-type spectrum for the heterostructure. The transmission peak and dip at 285.4 and 295.1 nm can be clearly identified, respectively. The transmission spectrum of C(AB)₃ and (DEF)₃(AB)₃ nearly overlaps which further confirms the applicability of EMT. In particular, the transmittance spectrum of the composite structure C(AB)₃ for different MO coefficient α = 0.1 and α = −0.1 are shown in Figure 7B, which are marked by the dashed blue line and dotted red line respectively. Similar to the isotropic case, the transmittance peak at the high frequency of the heterostructure C(AB)₃ with anisotropic magnetized ENZ material is more sensitive to the transmission dip at the low frequency after changing the direction of the applied magnetic field. The dependence of Δ T 1 = T | α = 0.1 − T | α = 0 and Δ T 2 = T | α = 0.1 − T | α = − 0.1 on the wavelength and the incident angle are also shown in Figures 7C,D, respectively. The maximum of | Δ T 2 | is marked by the white crosses in Figure 7D, which are marked by the white crosses at θ = ± 57^° and θ = ± 69^°, respectively. The dependence of the transmittance spectrum of the heterostructure C(AB)₃ on different parameters (i.e., the incident angle, the thickness of MO layer, and the strength of MO coefficient) is also systematically studied (The dependence of transmittance spectrum of the anisotropic MO ENZ material on different parameters can be found in the Supplementary Material).

From the maximum of Δ T 2 and - Δ T 2 as the function of the thickness of layer A and layer B in Figures 8A,B, we take two cases P and Q to demonstrate the external magnetic field-controlled Fano profile based on the effective anisotropic ENZ material. The heterostructure at P with d _A = 35 nm and d _B = 39 nm and at Q with d _A = 50 nm and d _B = 26 nm are marked by pink and blue stars, respectively. The variation of the direction of magnetic field can lead to different Fano asymmetric parameters and different asymmetry Fano resonance spectra, as shown in Figures 8C,D. The transmittance spectrum for α = 0, α = 0.1, and α = −0.1 is marked by the solid green line, dashed blue line, and dotted red line, respectively. Similar to the isotropic case in Figure 5, the external magnetic field-controlled asymmetric transmittance profile of the heterostructure spectrum is demonstrated in the heterostructure with effective magnetized anisotropic ENZ material in Figures 8C,D. At last, it should be noted that although the Fano interference can enhance the MO effect, the loss around the Fano interference dip is large, which may limit the application of the research results in some specific scenarios.

In summary, we investigated the Fano-type interference effect in the heterostructure composed of a homogenous isotropic (or an effective anisotropic) magnetized ENZ material layer and a truncated dielectric photonic crystal. The ENZ material provides a narrow reflection pathway, and the photonic crystal provides a broadband reflection pathway, which gives rise to the asymmetric Fano-type spectrum. By means of the field intensity enhancement of ENZ material and the Fano interference of the heterostructure, the enhanced MO activity is demonstrated from the non-reciprocal transmission spectrum. In particular, the MO effect which depends on the incident angle, the thickness of MO layer, and the MO coefficient is studied. Moreover, the asymmetric transmittance profile of the heterostructure spectrum can be flexibly tuned by controlling the direction of the external magnetic field. These results provide a new perspective to design novel magneto-optical devices with coherence mechanisms, such as optical isolators, high-sensitivity sensors, and switches.