1. Introduction

The well-known GH effect is one of the most fundamental phenomena in beam optics. Early in 1947, it was discovered that a light beam of finite width will undergo a lateral shift in the incident plane when totally internally reflected from a dielectric surface. Today this effect has been extended to reflection from or transmission through different materials and structures, such as metal-dielectric nanocomposites [1,2], epsilon-near-zero metamaterials [3], graphene [4], PT-symmetric medium [5,6], topological insulator [7] and various resonant materials [8–14].

In most cases the GH shift only occurs at oblique incidence. The unique exception so far reported is the effect observed in reflection from the surface of magnetic materials near ferromagnetic resonance [8–14]. The permeability of such a magnetic material takes the MO tensor form

However, such an interesting NIGH shift, quite large and well tunable, is accompanied by a vanishing reflectivity at the same frequency $f_c$. Away from $f_c$, the reflectivity rises but the shift drops dramatically. For example, the NIGH shift on reflection from the surface of yttrium-iron-garnet (YIG), one of the most conventional MO metamaterials, follows [8]

Is it possible to realize large NIGH shifts while keeping considerable reflectivity? The strong-field limit of $d_r^{{}^{\text{TR}}}$ suggests that homogenous MO materials with intrinsic ferromagnetic resonance (IFR) cannot serve as good candidates to achieve this goal. In this paper, we propose a new design of MO metamaterials where a MSP resonance with TRS-breaking is utilized as the underlying mechanism to produce large NIGH shifts even at total reflection.

The MSP phenomenon is the magnetic analog of the well-known “surface plasmon” effect in noble metals [15]. Gollub et al. [16] first introduced this concept into the design of metamaterials and develop a split-ring-resonator structure to successfully excite MSPs in a frequency band with negative effective permeability. Similar effects were also reported in subwavelength photonic crystals (PCs) made of ferrite rods [17], where the collective MSP resonance of the whole medium is a phenomenon emerging from the coupled localized-surface-plasmon-resonance (LSPR) modes of individual rods. The MSP effect in these ferrite arrays has attracted considerable interest since it is combined with the TRS breaking effect of magnetized ferrites so that a special bandgap is formed around the MSP resonance which contains optical chiral edge states along the surface of the metamaterial [18,19]. Experiments have successfully demonstrated that such an MO metamaterial working in the MSP bandgap can be utilized to fabricate omnidirectional waveguides with better performance than those based on conventional Bragg gaps of magnetic PCs [20].

By comparing MSP in ferrite arrays and IFR in homogenous ferrites, one can see that they are both magnetically-tunable resonance with broken TRS, but the former, as a metamaterial property, can be controlled by tailoring the composite structure. So it is reasonable to expect that the MSP resonance can induce the NIGH effect for a beam reflected from the surface of a metamaterial consisting of ferrite rods, just like what the IFR do for a homogeneous ferrite, and “good” structures may exist which can exhibit large NIGH shifts well beyond natural MO materials.

In previous studies, an effective-medium approximation (EMA) has been applied to retrieve the effective parameters of these MO metamaterials consisting of ferrite rods [21,22]. We note that MO-EMA method is focused on the characteristics of band structures, which is not sufficient for the discussion of the behavior of beam reflection from the effective MO medium. In nature, it is reduced to the cases for magneto-dielectric composites beyond the long-wavelength approximation [23]. So here we are motivated to present a new effective-medium theory which can be applied to obtain band structures and effective MO parameters simultaneously. The key idea is to establish a correspondence between the scattering coefficients of different orders of a MO material and those of a series of normal dielectrics. Then the problem of the concerned MO metamaterial is transformed into a series of conventional EMA models of normal dielectric. This transformation is not only helpful to obtain the effective parameters of a MO metamaterial in a much easier way, but also offers a deeper insight into the relation between the effective MO effects and the underlying MSP mechanism.

The new method based on the transformation of EMA (TEMA) is applied to calculate the NIGH shift in the MSP gap of a square array of ferrite rods. Both the results for structures in the quasi-static limit and structures with significant size effects are discussed. It is shown that, unlike the NIGH effect in homogeneous MO materials, the reflected shift near the MSP resonance from the surface of the array is dominated by the value of effective permittivity instead of the magnetic field. Then a large NIGH shift is predicted even on total reflection when the structure parameters (such as rod radius or lattice constant) are adjusted to produce a small effective permittivity near the MSP resonance. Numerical simulations are performed to calculate the intensity and shift of reflected beams and the results coincide well with those from the TEMA. Giant NIGH shifts (i.e. shift larger than the incident wavelength) [24] well beyond the strong-field limit in homogeneous ferrites and enhanced magnetic-switch effects of the shift sign are demonstrated by tailoring the metamaterial structure as theoretically predicted.

2. The TEMA method

Consider a periodic square lattice as illustrated in Fig. 1(a) which consists of cylindrical ferrite rods of radius $r_s$ separated by distance $a$. The whole lattice is embedded in a uniform dielectric background of relative permittivity ${\epsilon }_b$ and permeability ${\mu }_b$. The effective electromagnetic parameters of a metamaterial in this structure can be retrieved by solving the scattering problem of a coated particle embedded in the effective medium [22,23]. As shown in Fig. 1(b), an individual component particle of the metamaterial is surrounded by a coating layer of the background medium. The radius $r_0$ of the coated particle is determined by the lattice constant $a$ according to the relation $\pi r_0^2=a^2$. The total scattering cross section of such a coated particle embedded in the effective medium is given by [23]

 

Fig. 1 (a) Structure of a MO metamaterial consisting of a square array of ferrite rods. (b) A conventional EMA model where an individual ferrite rod surrounded by a shell of the background medium is embedded in the MO effective medium. (c) Non-MO transformed effective media of order m = 0, 1, −1.

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When the metamaterial is composed of MO rods magnetized along the cylinder axis (the z axis), both the particle permeability and the effective permeability are in the tensor form in Eq. (1), namely

The core region ($r<r_s$) and the effective-medium region ($r>r_0$) in Fig. 1(b) are characterized by $({\epsilon }_s,{\mu }_{s,vg},Q_s)$ and $({\epsilon }_e,{\mu }_{e,vg},Q_e)$, respectively. For an s-polarized incident wave of frequency $\omega$, the electric field in each region can be expanded in terms of Bessel functions

The expressions of magnetic field can be obtained by the Maxwell’s equations:

The relations between the coefficients in Eq. (6) are determined by the boundary conditions of $E$ and $H$. At the boundary $r=r_s$, we have

Comparing Eqs. (9) and (10), it is easy to find the scattering coefficient of the m-th order of the MO particle is equivalent to that of a non-MO particle $({\epsilon }_s^{(m)},{\mu }_s^{(m)})$, where ${\epsilon }_s^{(m)}$ and ${\mu }_s^{(m)}$ satisfy

At the boundary $r=r_0$, the coefficients for $m=0,\pm 1$ satisfy

If the metamaterial is composed of non-MO particles ($Q_s=0$, $Q_e=0$), the effective permittivity ${\overline{\epsilon }}_e$ and the effective permeability ${\overline{\mu }}_e$ can be obtained from the $m=0$ equation and the $m=1$ equation (or $m=-1$), respectively, as

For the metamaterial composed of MO particles $({\epsilon }_s,{\mu }_{s,vg},Q_s)$, we have

Equation (17) establishes a correspondence between the concerned MO metamaterial and the three non-MO metamaterials defined according to the transformation relations given by Eq. (11). Accordingly we have a three-step strategy to solve the problem of MO-metamaterials described as follows.

Step 1. Apply Eq. (11) to determine ${\epsilon }_s^{(m)}$ and ${\mu }_s^{(m)}$ for $m=0,\pm 1$ from the properties of the original MO particles.

Step 2. Replace the MO-particles by the non-MO ones of (${\epsilon }_s^{(m)}$,${\mu }_s^{(m)}$), without changing the underlying structure (characterized by $r_s$ and $a$), and find the effective parameters ${\epsilon }_e^{(0)}$, ${\mu }_e^{(+)}$, ${\mu }_e^{(-)}$ by using the conventional EMA results Eq. (14).

Step 3. Substituting (${\epsilon }_e^{(0)}$,${\mu }_e^{(+)}$,${\mu }_e^{(-)}$) into Eq. (17) to obtain the effective parameters $({\epsilon }_e,{\mu }_{e,vg},Q_e)$ for the original MO metamaterials.

If necessary, one can also deduce the effective permeability tensor.

Here we would like to mention that an obvious benefit of this new strategy is to avoid the complicated calculation of scattering when the MO tensors are involved. The results are easily extended to the cases beyond the long-wavelength limit $k_b{r}_0<<1$ by treating the transformed media with Eq. (14), the general EMA formula for non-MO metamaterials which is valid even when $k_b{r}_0>1$ [23]. And, since the EMA approach for non-MO cases has been thoroughly discussed and extensively applied, to transform an MO problem to non-MO ones is helpful to gain insight into the phenomena such as NIGH effects occurring in MO metamaterials.

3. NIGH shifts

Now we begin to consider a typical subwavelength photonic crystal consisting of YIG rods in a vacuum background (${\epsilon }_b=1,{\mu }_b=1$). As illustrated in Fig. 2, an electrically z-polarized paraxial beam is perpendicularly incident along the x-axis upon the surface of such a square array of YIG rods. When the effective medium approximation is applied, the NIGH shift of the reflected beam is determined by the effective parameters as [8,11]

 

Fig. 2 Schematic figure of the geometric arrangement to calculate the NIGH shifts. The incident beam is electrically z-polarized, normally incident upon the surface of the square lattice of YIG rods.

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3.1. Quasi-static structures

We first investigate structures composed of extremely fine particles so that $k_b{r}_0<<1$ and $k_s{r}_s<<1$ are satisfied simultaneously, i.e. structures in the quasi-static limit. Then Eq. (14), the general EMA result for a non-MO metamaterial, can be approximately replaced by the well-known Maxwell-Garnett equations

Figure 3 shows the variance of ${\mu }_e^{(\pm )}$ and ${\mu }_{e,vg}$ in the QSA with the Voigt permeability ${\mu }_{s,vg}$ of MO particles for $Q_s=0,0.5,2$. The particle concentration has been set to be $p=0.1$. For $Q_s=0$, i.e. the case of non-MO particles, the three permeability curves coincide since ${\mu }_{e,vg}={\mu }_e^{(+)}={\mu }_e^{(-)}$ when no MO effects are included. Typical permeability resonance is produced around ${\mu }_{s,vg}=-1$, a signature of the occurrence of the MSP resonanceas reported in the previous studies [16,18,19]. These MSP resonances of the total composite correspond to the poles in ${\mu }_{e,vg}$. For a single rod or a structure in the limit $p\to 0$ when $Q_s=0$, the MSP condition reduces to the conventional form ${\mu }_{e,vg}=-1$. While for non-vanishing $Q_s$, the resonance of ${\mu }_{e,vg}$ is split into two sub-bands, either of which corresponds to the resonance regime of ${\mu }_e^{(+)}$ or ${\mu }_e^{(-)}$. The distance between the sub-bands is enlarged with strengthened MO effects. Such a resonance splitting due to MO effects is a direct result of the imbalance between the scattering coefficients $b_{s,1}$ and $b_{s,-1}$ caused by the broken time-reversal symmetry of MO particles [25].

 

Fig. 3 Permeability ${\mu }_e^{(+)}$ (thin red line), ${\mu }_e^{(-)}$ (thin blue line) and ${\mu }_{e,vg}$ (thick black line) for various $Q_s$ when $p=0.1$ in a quasi-static structure.

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Band gaps, i.e., the negative-${\mu }_{e,vg}$ regions as shown in Fig. 3, are formed around each MSP resonance. These gap regions are also called “reststrahl regions” since the reflectivity is extremely high in them. EMA works quite well near the gap end with ${\mu }_{e,vg}=0$ (since $k_e{r}_0<<1$ is ensured in this region) and the NIGH shift can be appropriately estimated by

By comparing Eq. (24) and Eq. (3), it is found the MSP-induced shift $d_e^{(\pm )}$ is dominated by the effective permittivity rather than driven by strong magnetic fields as is the shift $d_r^{TR}$ associated with IFR. But the two shifts share the same upper limit ${\lambda }_0/\pi$ whenever the QSA can be applied to the MO metamaterial, since the effective permittivity in this approximation cannot be lowered to less than 1 according to Eq. (22). In other words, the key factor to break the restriction on the NIGH shift is to introduce proper size-effects beyond QSA into the MO structures. In the following sub-section, we will illustrate how to realize large NIGH shifts in “coarser” MO structures when particle-size and spacing are carefully adjusted to achieve small effective permittivity around MSP resonance.

3.2. Size effects

Nontrivial size effects on MSP bands and gaps have been demonstrated, though not emphasized, in previous studies of band structures for subwavelength YIG-PCs. A typical example is the experimental work of omnidirectional waveguides based on a square array of YIG rods with $r_s=0.0016\text{m}$ and $a=0.0145\text{m}$ [20]. It was shown that, a MSP-induced gap is opened around $f_g=9.6\text{GHz}$ when the YIG rods are magnetized at $h_0=2980\text{Oe}$. Though this gap exhibits distinctive MSP features, the observed frequency range deviates obviously from the predicted electrostatic resonance in the QSA. As shown in Fig. 3, the QSA leads to the prediction of the formation of two MSP gaps near the resonance of ${\mu }_e^{(+)}$ and ${\mu }_e^{(-)}$, respectively, which can be estimated by the condition

The red-shift of the MSP gap can be well-explained when we apply the TEMA method to this structure. The dispersion relations of the parameters (${\epsilon }_s^{(m)}$,${\mu }_s^{(m)}$) of the component particles for the m-th order transformed medium ($m=0,\pm 1$) are obtained by substituting the YIG permittivity and permeability into Eq. (11). The frequency dependence of ${\epsilon }_e^{(0)}$, ${\mu }_e^{(+)}$, ${\mu }_e^{(-)}$ are then calculated and illustrated in Fig. 4, together with the relevant scattering features of individual component particles for each transformed medium. It is shown that dense and sharp resonances between $\text{10GHz}$ and $\text{11GHz}$ will occur in all the three spectra, which are bulk modes originated from the singularity of the Voigt permeability ${\mu }_{s,vg}$ of YIG, and irrelevant to the concerned MSP band. The LSPR peak of the first-order particle, which is at $f_r^{(+)}=10.9\text{GHz}$ in the QSA, has now evolved into a broad Mie resonance around $\text{9}\text{.6GHz}$ due to the large particle radius, leading to a pole in ${\mu }_e^{(+)}$ near this frequency [Fig. 4(b)]. The large particle size also results in the resonance of ${\epsilon }_e^{(0)}$ associated with a monopole peak around $\text{8}\text{.0GHz}$ for the zeroth-order particles [Fig. 4(a)]. The shift of LSPR modes and the occurrence of monopole resonance are not surprising but common size effects extensively observed in non-MO particles [25–28]. But in our case of MO metamaterials, the two effects can be combined and controlled to produce an MSP gap with large NIGH shift.

 

Fig. 4 Frequency dependence of the effective parameters and the scattering coefficients for the transformed media of the m-th order. (a) ${\epsilon }_e^{(0)}$ and $b_s^{(0)}$ for the $m=0$ medium; (b) ${\mu }_e^{(+)}$ and $b_s^{(+)}$ for the m = 1 medium; (c) ${\mu }_e^{(-)}$ and $b_s^{(-)}$ for the $m=-1$ medium; Here $b_s^{(0)}$ is the zeroth-order scattering coefficient of an individual particle for m = 0 while $b_s^{(\pm )}$ the first-order coefficients for $m=\pm 1$. The right panels are the partial enlarged drawing of the high frequency part.

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Figure 5(a) shows the frequency dependence of ${\epsilon }_e$ and ${\mu }_{e,vg}$ for the array of YIG rods calculated by substituting the results of ${\epsilon }_e^{(0)}$, ${\mu }_e^{(+)}$ and ${\mu }_e^{(-)}$ into Eq. (17). The resonance of ${\mu }_e^{(+)}$ from the shifted LSPR mode contributes a zero-point of ${\mu }_{e,vg}$ at $f_m=9.685\text{GHz}$ while the monopole resonance of ${\epsilon }_e^{(0)}$ leads to a zero-point of ${\epsilon }_e$ at $f_e=9.255\text{GHz}$. In the frequency range between the two zero-points, $k_e{}^2={\epsilon }_e{\mu }_{e,vg}{\omega }^2/c^2$ is negative and an MSP gap takes place. Note that $\left|k_e{r}_0\right|$ is so close to zero [Fig. 5(b)] that the TEMA is an appropriate approximation in the whole gap region, and a large NIGH shift is expected since the value of ${\epsilon }_e$ is now considerably suppressed due to the monopole resonance at lower frequencies. Figure 5(c) gives the calculated NIGH shift (solid line) in the MSP-gap region, where the two edges of the gap, $f_e$ and $f_m$, are labeled by vertical dashed lines. It is shown that the normalized NIGH shift $d_r/{\lambda }_0$ reaches up to nearly 0.7 at the gap end $f_m$ with ${\mu }_{e,vg}=0$, more than twice the QSA limit of $1/\pi$.

 

Fig. 5 Comparison between the TEMA results and numerical simulations for different lattice constants. (a) & (d) & (g): Frequency dependences of ${\epsilon }_e$ and ${\mu }_{e,vg}$ for the array of YIG rods calculated from the TEMA method, $a_0=0.0145\text{m}$. (b) & (e) & (h): Comparison of the MSP-gap positions. Dashed lines: ${(100k_e{r}_0)}^2$ vs $f$ from the TEMA method; Solid lines with squares: the field magnitude at the center of the reflected beam from simulations. (c) & (f) & (i): Comparison of the NIGH shifts. Solid lines: results from the TEMA method; Squares: results from the numerical simulations using Comsol Multiphysics (a normally-incident Gaussian beam with finite width).

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The NIGH shift can be further increased by adjusting the lattice constant $a$. Figure 5 compares the calculated MSP gap and the associated NIGH shift for $a=a_0$ [Figs. 5(a)-5(c)], $0.9a_0$ [Figs. 5(d)-5(f)] and $0.8a_0$ [Figs. 5(g)-5(i)] with $a_0=0.0145\text{m}$. Slight broadening of the ${\epsilon }_e^{(0)}$ and ${\mu }_e^{(+)}$ resonances can be observed since the coupling between particles is strengthened with the decrease of lattice constant. As a consequence, a narrower MSP gap with lower ${\epsilon }_e$, and hence, larger NIGH shift, is formed in structures with smaller lattice constant. When $a$ is decreased to $0.8a_0$, the normalized NIGH shift $d_r/{\lambda }_0$ at the gap end with ${\mu }_{e,vg}=0$ rises to be nearly 1.5, which is the so-called “giant GH effect”, i.e. a GH-shift larger than the incident wavelength.

To verify the above TEMA results, we performed a finite-element analysis of these YIG arrays using the software Comsol Multiphysics 4.3a. In the simulations the normally-incident beam is a z-polarized Gaussian beam with a finite width. The electric-field profile of the reflected beam is calculated and the reflected GH-shift is obtained by comparing the central positions between the reflected beam and the incident beam [Fig. 6(a)], where the incident beam is set to be centrally located around the x-axis with half-width $w=6{\lambda }_0$. When the frequency approaches the gap end with ${\mu }_{e,vg}=0$, large NIGH shifts are observed as predicted in the TEMA. The distributions of electric field at different frequencies in the gap region are illustrated in Figs. 6(b)-6(d). With the frequency increasing, the field pattern in MO rods evolves into a typical surface dipole mode, corresponding to the dipole resonance shown in Fig. 4. The frequency dependences of the centroid field amplitude $\text{|}{E}_r|$ for the reflected beam in the simulations are presented in Figs. 5(b), 5(e) and 4(h), together with the values of ${(k_e{r}_0)}^2$ calculated from the TEMA method. It is shown that the incident beam in simulations is totally reflected for frequencies within the TEMA-predicted gap region (the region where ${(k_e{r}_0)}^2$ is negative), with a reflected shift in good agreement with the TEMA result as illustrated in Figs. 5(c), 5(f) and 5(i).

 

Fig. 6 (a) The electric-field amplitude of the reflected beam from numerical simulations for different incident frequencies with (dashed curves) and without (solid curves) absorption. Blue: f = 9.3GHz; Yellow: f = 9.5GHz; Red: f = 9.65GHz. (b) & (c) & (d): The patterns of electric fields for each frequency. Structure parameters are chosen to be $r_s=0.0016\text{m}$ and $a=0.0145\text{m}$.

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So far we have neglected the absorption of YIG in both the effective-medium analysis and the numerical simulations. Results for more realistic materials are presented as well in Fig. 6(a) (dashed lines) where the YIG damping factor $\alpha \text{=}\gamma dH/2\omega$with linewidth $dH=30\text{Oe}$ [20] is included in the calculation of the electric-field profiles of the reflected beam. The non-vanishing damping factor is shown to result in a moderate or even slight decrease of reflectivity with the NIGH shift almost unchanged. This holds even at frequencies very close to the MSP resonance. For example, at $f=9.65\text{GHz}$ for a structure of $r_s=0.0016\text{m}$ and $a=0.0145\text{m}$, the central field of the reflected beam is only reduced from 0.9 without absorption to 0.8 with absorption, while the reflected shift remains to be $0.617{\lambda }_0$. The reflected beam is still “bright” enough for the observation of large NIGH shifts when absorption occurs. Two reasons account for this result. First, the NIGH shift is generally insensitive to the absorption of MO materials, as confirmed in previous studies of various samples [8,10,11]. Secondly, the MSP resonance originates from the surface mode of MO particles, so that much less absorption will be observed than those due to conventional bulk modes [17,20].

Though all the shifts obtained above are positive (along the + y-axis), negative NIGH shifts can be easily achieved by reversing the direction of the applied magnetic field from ( + z)-axis to (-z)-axis. This magnetic-switch of the sign of reflected shift is an important feature of the NIGH effect in MO materials [10,11] and can be remarkably enhanced in MO metamaterials with the MSP resonance. Figure 7 shows the electric-field patterns at different magnetic fields for a structure of $a=0.8a_0$ and $r_s=0.0016\text{m}$ illuminated by an incident beam of half-width $w={\lambda }_0$ at frequency $f=9.75\text{GHz}$. Here we have adopted “narrow” beams to provide a clearer illustration of the beam shifts. It is shown that the reflected shift is switched from $1.02{\lambda }_0$ to $-1.02{\lambda }_0$ when the applied field is changed to opposite direction. Again, both the results for the cases of zero-absorption [Figs. 7(a)-7(c)] and nan-vanishing absorption [Figs. 7(d)-7(f)] are presented and the inclusion of absorption leads to a slightly-weaker reflected beam with almost the same shift.

 

Fig. 7 Enhanced magnetic switch of NIGH shifts near the MSP resonance. (a)-(c): Patterns of electric field amplitude for the incident beam and the reflected beams without absorption; (d)-(f): Patterns of electric field amplitude for the incident beam and the reflected beams with absorption. Structure parameters are chosen to be a = 0.8a₀ and r_s = 0.0016m. Incident frequency is set to be f = 9.75GHz.

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4. Conclusion

In conclusion, our analysis based on the TEMA method reveals that enhanced NIGH shifts on total reflection can be realized in the MO metamaterials when particle size and spacing are carefully adjusted to produce a vanishing effective permittivity around the MSP resonance. The theoretical prediction is verified by numerical simulations. Giant NIGH shifts and enhanced magnetic switching effects near the MSP resonance [29] are demonstrated and the effects of absorption are discussed. We hope this study could invoke more interests in such a special GH shift which is rooted in the combination of MSP resonances and TRS breaking, and provide new clues to practical applications of NIGH effects with designs of metamaterials.

Here some comments are in order. Through homogenization of an array of gyromagnetic cylinders, photonic Chern insulator may be created [30]. Such investigation achieves the electromagnetic analogs of quantum spin Hall systems, which provides a platform for us to explore the interplay between disorder and topology on GH shift of such metamaterials like photonic Chern insulator.