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Abstract
We report a resonance-enhanced nonreciprocal Goos-Hänchen (GH) shift for the wave reflected from the coupled gyromagnetic chains. We demonstrate that the Fano resonance enhances the GH shift with high reflectivity at normal incidence, and the resonance results from the interference between the leaky guided modes of the coupled chains. Furthermore, we show that the GH shift can be controlled by the number of stacked chains. The Fano resonance-enhanced GH shift offers a new efficiently way to enhance and control the GH shift for reflected wave beam. Such coupled gyromagnetic chains provide an extremely compact way for the devices such as unidirectional couplers and other integration photonic components, paving the way for the applications of nonreciprocal GH shift.
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1. Introduction
When optical beam incidents on the interface of two media with different refractive indices, the reflected light beam experiences a lateral shift along the interface at the position expected from the geometric optics. This phenomenon is known as the GH effect, which was first experimentally observed by Goos and Hänchen in total internal reflection [1]. This effect has been known for many decades, but it still attracts considerable attention due to the potential applications such as sensors [2,3], switchers [4], splitter [5,6] and near-field detections [7,8]. In the case of total internal reflection, the GH shift is usually comparable to the wavelength, which poses a challenge in measurements. Thus, enhancing the GH shift is highly desirable, and has been studied in various systems, such as metal-dielectric nanocomposites [9], epsilon-near-zero metamaterials [10,11], the graphene family materials [12–14], atomic coherence cavities [15], photonic crystals [16,17]. Meanwhile, the GH shift has also been studied under different kinds of incident wave beams, including Gaussian beams [18], Laguerre-Gaussian beams [19], Airy beams [20], and vortex beams [21], and so on.
The GH shift is considered as the result of phase variation of the reflection coefficient of the linearly polarized wave beam according to the stationary phase theory [22]. Generally, the GH effect is referred to the shift that occurred in the case of oblique incidence. In recent years, increasing attention has been paid to the study of the GH effect at normal incidence [23–25]. Reflected from the surface of magnetic materials near ferromagnetic resonance, the GH shift is firstly reported and interpreted in terms of the nonreciprocal reflective phase produced by time-reversal (T-) symmetry broken [24,26]. Both the sign and magnitude of the shift can be magnetically controlled, and giant GH effects are predicted around a resonant frequency f0. However, at the frequency of maximum GH shift, the reflectance presents a dip (i.e., transmittance peak). Away from f0, the reflectivity rises but the shift drops dramatically. The maximum GH shift is bounded by λ/π in the case of total reflection [25]. Later, to realize large GH shifts while keeping considerable reflectivity, the GH effects caused by exciting the leaky surface wave are reported for magnetic arrays [27–29]. Because the shift appears on the surface of the magnetic arrays, only limited (one to two times wavelength) enhancement is observed. Recently, we report that the MPC slab can support leaky guided wave modes with lateral energy flow along the slab at normal incidence when the MPC slab breaks spatial inversion (I-) symmetry, T-symmetry and mirror (M-) symmetry at the same time [30]. The lateral energy flow causes nonreciprocal GH shift of which the shift is not reversible when wave propagation direction reverses. The lateral energy flow of the leaky waveguide modes offers a promising and feasible venue to enhance the GH shift. The energy of the normally incident beams is transferred into the waveguide structures, which leaks back to the reflected beam after propagating a certain distance along the waveguide structure. Thus, the reflection beam may exhibit a large lateral shift with high reflectivity (close to unity).
Following this line, in this paper, we achieve controllable high reflective nonreciprocal GH shifts at normal incidence by coupled ferrite chains. Under bias DC magnetic field, the individual chain simultaneously breaks I-, M- and T-symmetries, which support the nonreciprocal leaky guided modes for normal incident wave. We demonstrate that originated from the interference between the leaky guided modes excited in the coupled chains, the Fano resonance results in enhancing GH effects. Further, we prove that GH shifts can be controlled by the number of periodic stacks a MPC slab. Our results provide a new avenue to control the reflection-type GH shift at normal incident and an extremely compact way to construct optical isolators, optical switches, unidirectional couplers, and so on.
2. Nonreciprocal GH shift by Fano resonance of coupled ferrite chains at normal incidence
We consider the two-dimensional magnetic chains as shown in Fig. 1(a). The primitives of the chains are composed of three YIG ferrite rods in the background air. The radius of the rods is r = 2 mm. The primitive is in periodic configuration and the period is d = 13mm. The distance between the two chains is also 13mm, from center-to-center of the chains. The material parameters of the YIG are the permittivity εr = 15.26, and the saturation magnetization 4πMs = 1884 Ga. The chains are in the Voigt configuration in which the bias DC magnetic field is along rods’ axes, defined as the z-axis. When fully magnetized by bias magnetic field, the permeability of the ferrite rods becomes a tensor written as [30]:
Fig. 1. (a) Schematic diagram of the coupled magnetic chains. The lattice constant is d = 13mm, and the radius of ferrite rods is r = 2mm. The arrows indicate the direction of wave propagation. (b) The dispersion diagram of the ferrite chain displayed in the inset. The light cone is shaded in yellow. (c) and (d) are the electric field and power flow distribution of the mode at kx = 0 and the frequency 0.449(c/d), corresponding to the upper chain and the lower chain in Fig. 1(a), respectively. The white arrows represent power flow.
The two-chain configuration in Fig. 1(a) can be consider as the coupled single chains. As reported in our previous papers [30,31], the primitives in Fig. 1(a) simultaneously break I-symmetry, T-symmetry and M-symmetry, and then the dispersion of the single chain is nonreciprocal. Under the bias magnetic fields H0 = 800Oe, Fig. 1(b) plots the dispersion diagram of the single chain whose configuration is displayed in the inset of Fig. 1(b). The finite element solver COMSOL was used in the calculation. In the figure, the yellow-shaded region represents the light cone, and the blue dot lines are the band curves. At the vicinity of the frequency 0.449 (c/d) and kx = 0, we observe the dispersion curve is asymmetrical, and group velocity ${v_g}\textrm{ = }(\partial \omega /\partial {k_x})$ is negative, indicating the chain guides the power flow along –x direction. Because this guide mode is within the light cone, it will be leaky. We plot in Fig. 1(c) the electric field and power flow vector of the mode at kx = 0 and the frequency 0.449(c/d). We see the field localization in the primitives and the power flow goes in the –x direction. Due to the mode is within the light line, the field also appears out of the chain. When the chain in the inset is upside down, the band structure will be a mirror reflection of Fig. 1(b) for x to –x, considering the symmetry of the primitives. In this case, the group velocity become positive at kx = 0 and the frequency 0.449(c/d), and the power flow vector of the guided mode is along the x-direction, as displayed Fig. 1(d).
When two chains are put together as the configuration in Fig. 1(a), Fig. 2(a) plots the band diagram of the two chains in black dot lines, where the red and blue lines are the bands of two individual chains and the yellow-shaded region is the light cone. We see two individual chains prevents the crossing of bands at kx = 0, causing the modes of individual chains split into two separated hybrid modes [32], and a narrow gap is formed in the frequencies from 0.447(c/d) to 0.455(c/d). The result shows two chains are strongly coupled. In the vicinity of kx = 0, the upper band of the gap is flat and the lower band is broadband. Although the coupled chains break I-symmetry and T-symmetry, meanwhile keep M-symmetry about the x-axis, which ensures the dispersion curves satisfy ω(kx)=ω(-kx) [30,33,34]. Thus, at kx = 0, the resultant group velocity vg for individual modes is zero, and the lateral power flow should be zero. We plot the mode field profile and energy flow at kx = 0 marked by purple and green arrows in Fig. 2(a). The results are shown in Figs. 2(b) and 2(c). On the two sides of the coupled chains, the field profile is symmetric in Fig. 2(b) and is antisymmetric in Fig. 2(c). The fields and power flows on the each chains are quite similar to that of the individual chains displayed in Figs. 1(c) and 1(d). Thus, the net power flow of the couple chain is zero because of the opposite directions of power flow in each chain.
Fig. 2. (a) The dispersion diagram of the coupled chains in Fig. 1(a). The blue and red dot lines are the bands of individual chains, which correspond to the upper chain and the lower chain in Fig. 1(a), respectively. The light cone is still shaded in yellow. (b) and (c) are the electric field and power flow distribution of the mode marked by the purple and green arrows in Fig. 2(a), respectively. (d) The normalized frequency dependence of the reflectivity for coupled chains at normal incidence, the resonance peak marked by the cyan dot line. When the normal-incidence plane wave is incident to the upper surface of the coupled chains, (e) is the electric field and power flow distribution at 10.48 GHz, and the red arrows represent wave transmission.
Although the net lateral power flow disappears at the states marked by arrows, the lateral power flows in each single chain still exist, going along the x direction or -x direction, which may contributes to the enhancement of non-reciprocal GH shift. Under the normal incidence, we plot the reflection spectrum of the coupled chains as the function of the frequency. As displayed in Fig. 2(d), a sharp peak at the frequency 0.454(c/d) appears within a broad resonance dip, exhibiting the Fano type of resonance [35,36]. It is known that Fano resonance results from the interference of a broadband ‘bright’ mode and a narrowband ‘dark’ mode, featuring an asymmetric resonance curve in transmission or reflection spectra [36–40]. The dark mode serves as a localized state, and coexists in a certain spectral range of the bright mode. Figures 2(a) and 2(d) show the Fano resonance occurs at the vicinity the flat band. This indicates the result comes from of the modes interference of the flat band, which serves as dark mode due to the big density of state [41], and the bright mode below the flat band. Figure 2(e) illustrates the electric field and power flow distribution at normal incidence at the Fano resonance. The figure shows the field profile is different from the dark eigenmode, meaning it is the combination of the dark and bright modes. We note that in Fig. 2(e) the incident wave basically penetrates into the upper chain, and dies out rapidly in the lower chain, and a strong reflection appears at the interface of the coupled chain and the air. The power flow in the upper chain is observed, and concentrated within the chain pointing to the –x direction. Thus, the upper chain serves as a one-way waveguide while the lower chain function as a reflective layer. As a result, the normal-incidence energy is coupled into the coupled chains, flowing in the waveguide and then reflected into the air, resulting in an enhanced GH shift with high reflection.
3. Nonreciprocal normal-incident GH shift for reflected wave beams
When a wave beam is polarized in the z-axis (see Fig. 1), the lateral beam shift Dr at the interface of chains is the displacement between the reflected beam and the incident beam. According to the stationary phase theory, the shift can be expressed as [22]
Fig. 3. (a) The reflectivity and the normalized GH shifts for reflected waves as a function of the frequency when the incident waves perpendicularly projects onto the chains from the air. (b)-(d) are simulations for a wave beam scattering from the coupled chains. (b) The incident field, (c) the scattered field at H0 = 800Oe, and (d) the scattered field with H0=-800Oe. The incident wave is a Gaussian beam with beam waist radius w0 = 4λ, working at the frequency 10.48GHz. The white line in the panels (c) and (d) plots the reference position, the centerline of the incident wave beam to the reflected one. The simulation is done by software COMSOL.
We note the enhanced nonreciprocal GH shift in our coupled chain only happens when the incident radiation is in normal incidence or nearly normal incidence. The reason is the coupling between the chains happens only when kx is very small because the one-way leaky guided modes in individual chains are in opposite directions [43]. We calculated the GH shift at different incident angles, and demonstrated the reduced GH shift with increase of incident angle, which is provided in Supplement 1.
4. Controlling GH shift by stacking chains
Normally, the field penetration depth will affect the GH shift when the incident wave beam excites a leaky guided wave or surface wave along the interface of a slab. Hence, GH shift is expected to be enhanced when penetration depth becomes larger. Therefore, the beam shift of reflected waves is expected to be increased if the number of chains increases.
We increase the number of chains on the top of the bottom chain. For the chains’ configuration in Fig. 4(a), we plot the electric field profile and the backscattering field in Figs. 4(b) and 4(c) when incident Gaussian beam normally projects on the chains. We see the normally incident wave is coupled into two chains, which serve as a waveguide, and the coupled wave propagates a certain distance. Due to the strong reflection at the bottom chain, the guided wave is leaky from the bottom chain is prohibited and strongly reflected to the upside air. Because effective field penetration depth increases, the beam shift of the reflected wave is apparently increased.
Fig. 4. (a) The configuration of two-chain slab on top of Chain 1, N is the total layer number of slab on top of Chain 1. (b) the incident field and (c) the scattered field are simulated for a wave beam scattering from the configuration in Fig. 4(a). The white line in the panels (b) and (c) plots the reference position, the centerline of the incident wave beam to the reflected one. The simulation is done by software COMSOL. (d) Intensity profiles of the beams reflected from the multilayer coupled chains under N = 1, 2, 3, 4, 5, 6. The dotted line at x = 0 displays the position where the peak of the incident beam appears. The peaks of the reflected beam under the different layer number are marked by other doted lines. The incident Gaussian beam works at the frequency f = 10.48GHz. and the incident beam width w0 = 4λ.
The GH shift will become even bigger when we stack more chains. Figure 4(d) plots the normalized intensity profiles of the reflected wave beams for a different number of upper chains. We see the beam center deviates from the position of the incident beam and deviation increases as the layer number of slab increases. Figure 4(d) shows the beam shift to the chain number. One may find the beam shift almost linearly increases with the chain number. Therefore, one can get desired beam shifts in a controllable way.
One may expect that the GH shift will be any value when the chain number is bigger and bigger. However, as the number of chains increases the reflected wave beam will distort severely so that it cannot be regarded as a wave beam. Thus, the wave beam shift will be physically meaningless. This indicates the enhanced beam shift still has a limitation. As an example, when the layer number is up to N = 6, we observe a clear distortion in the reflected wave beam profile shown in Fig. 4(d). The distortion is related to the width incident wave and the leaky-wave structure. The distortion is negligible when the GH shift is smaller than the beam width. However, the reflected wave beam will accompany the distortion when GH shift is larger than the beam width [44].
5. Conclusion
We have investigated the nonreciprocal GH shift of the coupled ferrite chains at normal incidence. We show the enhanced GH shift of the reflected wave beam is due to the Fano resonance that is related to the interference between the leaky guided modes excited in the coupled chains. The coupling between the chains results in one single chain as a leaky waveguide and the other chain as a reflector, leading to a high reflection and a big GH shift. Furthermore, we show that the GH shifts can be controlled by the number of stacked ferrite chains. Our study provides a new way to manipulate the reflection-type GH shift at normal incidence, which has potential applications in constructing new optical components, such as unidirectional couplers, optical isolators, and switches.
Funding
National Natural Science Foundation of China (61771237).
Acknowledgments
This work is supported by the Key Program for Excellent Young Talents of Higher Education Institutions in Anhui Province (gxyqZD2020041), and partially by the Fundamental Research Funds for the Central Universities, Jiangsu Key Laboratory of Advanced Technique for Manipulating Electromagnetic Waves, and the Key Program of Natural Science Research of Higher Education Universities of Anhui Province (KJ2019A0684).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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