## Abstract

Magnetic polariton is a significant mode in tailoring thermal radiative properties with micro/nanostructures metamaterials and can be explained by equivalent inductor-capacitor circuit model. However, the equivalent inductor-capacitor circuit model is out of operation when the magnetic polariton resonance frequency is close to the surface plasmon polariton excitation frequency and cases of oblique incidence. In this work, we present a mutual inductor-inductor-capacitor circuit model to describe magnetic polariton resonance conditions. The mechanism of coupling between the surface plasmon polariton and magnetic polariton is explained from the perspective of equivalent circuits. The interaction between the surface plasmon polariton and magnetic polariton is studied and considered as a mutual inductance in the MLC circuit model. This model is still applicable in the case of oblique incidence. Slit arrays with different geometric parameters and incident angles are calculated to verify the rationality of the mutual inductor-inductor-capacitor circuit model. This study may allow us to predict features and parameters and achieve tailoring of the thermal radiative properties of the micro/nano-structures metamaterials.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Tailoring thermal radiative properties with micro/nanostructures metamaterials has drawn lots of attention in recent years for its wide applications such as signal detection [1,2], energy harvesting [3–5] and radiation cooling [6,7]. Some mechanisms such as Magnetic polariton [8,9] (MP) and surface plasmon polariton [10–12] (SPP), are used to explain various transmission and absorption phenomena. Surface plasmon polaritons refer to the couplings between the incident electromagnetic waves with the collective oscillation of the surface charges between dissimilar materials whose real parts of permittivity have opposite signs, while magnetic polaritons are the couplings between incident electromagnetic waves with magnetic excitation inside the structures [3,13–15]. Micro/nanostructures with different patterns such us one-dimensional (1D) gratings [9], two-dimensional (2D) gratings [16–18], complex gratings [19,20], nanowires [21] and nanoparticles [22,23], have been investigated and SPPs and MPs are confirmed to exist in these structures.

The equivalent inductor-capacitor (LC) circuit method is an effective way to find the MP resonance conditions in micro/nanostructures [24–27]. Many researchers employed LC circuit model in different structures and confirmed the excitation of the MP resonance. Wang et al. [28] investigated the fundamental MP in slit arrays and LC circuit model can show good agreement with numerical calculation. Zhao et al. [9] employed LC circuit model in deep gratings and the relationship between the capacitance coefficient and the charge-distribution of the gratings’ grooves is studied. Wang et al. [13] studied 2D periodic film-coupled concave grating metamaterial and MP can still be predicted by LC circuit model in multi-layer structures. Chang et al. [21] investigated nanowires and found that LC circuit model is reliable in this case. However, LC model is out of operation when the MP resonance frequency is near to SPP resonance frequency due to fact that the strong interaction between MP and SPP is neglected. In addition, the LC circuit model cannot describe oblique incidence cases in former studies.

In this work, Finite-difference time-domain (FDTD) method is employed to calculate the radiative properties by solving Maxwell's equations numerically. The electromagnetic field distributions confirm the excitation of MP resonance and effects of MP in tailoring the radiative properties are investigated. Taking the interaction between MP and SPP into account, equivalent mutual inductor-inductor-capacitor (MLC) circuit model is proposed to describe magnetic polariton resonance conditions in the case of oblique incidence in slit arrays. The electric currents at the top/bottom surface of the strip caused by SPP will induce a magnetic field, and this magnetic field will generate an additional induced current at the sidewall of the strip. The additional induced current has a significant effect on the equivalent circuit at the slit and will significantly affect the MP resonant conditions. Therefore, the effect of SPP on MP is considered as the mutual inductance in MLC circuit model according to the electromagnetism concept. The mechanism of coupling between SPP and MP is explained from the perspective of equivalent circuits. Slit arrays with different geometric parameters and incident angles are calculated to verify the rationality of the improved model. The findings of this work may contribute to the prediction of thermal radiative properties in nanophotonics, plasmonics, nanolithography, and nanoscale energy devices.

## 2. Modeling and numerical method

As illustrated in Fig. 1, the 1D periodic sliver slit arrays surrounded by vacuum are considered in this paper. The slit array is described by a period *Λ*, height *h* and slit width *b* (strip width *w* = *Λ*-*b*). A set of base geometric values, such as *Λ* = 0.5μm, *b* = 0.05μm and *h* = 1μm, is considered for the slit array. A transverse magnetic (TM) wave of wavelength *λ* is incident onto the slit arrays at an incidence angle *θ*.

The FDTD simulation is implemented using a commercial package (Lumerical Solutions, Inc.). The spectral reflectance *ρ* and transmittance *τ* of Ag slit arrays are calculated with the frequency range from 2500cm^{−1} to 25000cm^{−1}. The spectral absorptance *α* can be calculated from *α* = 1-*ρ*-*τ* according to energy balance. The frequency-dependent complex dielectric functions of Ag is obtained by using Drude model [14]:

*ε*

_{∞}= 3.4 is the high-frequency constant,

*ω*

_{p}= 2.22 × 10

^{15}Hz is the plasma frequency and

*γ*= 4.30 × 10

^{12}Hz is the scattering rate [29]. Refined meshes with sizes of 1nm in both x and z directions are used to ensure the numerical convergence.

As mentioned before, MLC circuit model consider SPP’s effect on MPs. The dispersion relation of SPP at the Ag-vacuum interface is expressed as [14,30]:

where*c*is the speed of light in vacuum. According to the Bloch-Floquet condition, the wavevector can be expressed as

*k*= (

_{x}*ω/c*)sin

*θ +*(2π

*j*)/

*Λ*, where

*j*is an integer accounting for the diffraction order [30]. By solving

*k*=

_{x}*k*

_{SPP}, the SPP excitation frequency

*ω*

_{SPP}are obtained. Note that, due to fact that the wavevector is related to the incident angle and the period of the slit array, the SPP excitation frequency varies with the incident angle and period.

## 3. The spectral radiative properties and electromagnetic field distributions

Figure 2 shows spectral normal radiative properties of the slit array with the set of geometric values: *Λ* = 0.5μm, *b* = 0.05μm and *h* = 1μm at TM-wave incidence. The first six resonances, MP1, MP2, MP3, MP4, MP5, MP6, are caused by the excitations of the magnetic polaritons. The next mode is an SPP excited at approximately 20000 cm^{−1}, followed by the MP6 and MP7. Note that, MP6 appears twice and both of them can be confirmed from electromagnetic field distributions which means SPP truncates the MP6 to make it two. This phenomenon confirms the interaction between the SPP and MP. All the MP result in transmittance and absorptance enhanced and as the order of MPn gets higher, the transmittance enhancement is weakened. Fixing *Λ* = 0.5μm and *b* = 0.05μm, and setting *h* to be 400 nm, 200 nm and 100 nm, respectively, the spectral normal radiative properties are also calculated similarly and not repeat here. When *h* is 1000 nm, 400 nm, 200 nm and 100 nm, the resonant frequencies of MP1 are 3475 cm^{−1}, 7914 cm^{−1}, 13377 cm^{−1} and 17961 cm^{−1}, respectively.

The electromagnetic field distributions at the MP1 resonant frequencies with varied slit array heights at normal incidence are plotted in Fig. 3: (a) *h* = 1000nm at 3475 cm^{−1}; (b) *h* = 400nm at 7914 cm^{−1}; (c) *h* = 200nm at 13377 cm^{−1}; (d) *h* = 100nm at 17961 cm^{−1}. The contour plots indicate the magnitude of magnetic field and the arrows represent the direction of the electric field vectors. The electromagnetic field distributions confirm the excitation of MP resonances and reveal that the magnetic field is strongly localized inside the slit between two neighboring metal strips, and the induced electric current flows along the sidewall surface. Figure 3 shows that as the height decreases, the energy is no longer concentrated only in the slit, while the magnetic field at the top and bottom surfaces of the strip is enhanced. Due to fact that as the slit height decreases, MP1 resonant frequency increases and gradually approaches to SPP excitation frequency 20000 cm^{−1}, the magnetic field enhanced at top/bottom surface can be attributed to the SPP’s impact.

Table 1 demonstrates numerical values of the electric field and current density in Fig. 3. For location 1, the electric field vectors and current density only have z components; while for location 2, they only have x components. For high frequency electromagnetic waves, electromagnetic fields and induced electric currents can mainly exist in a thin layer (within the power penetration depth) on the surface which is called as skin effect. The power penetration depth *δ* = *λ*/4π*κ* is calculated to be 12nm approximately in the frequency range studied in this article so that 10nm is chosen as the distance from top surface and sidewall and locations a1-d1 and a2-d2 are given in Table. 1. According to the microscopic form of Ohm’s law, the current density vector is related to the electric field vector by ** J** =

*σ*

**, where $\sigma ={\sigma}^{\prime}+i{\sigma}^{\u2033}$ is the complex conductivity and has a relationship with complex dielectric function, i.e.,**

*E**σ =*-

*iωε*

_{0}

*ε*. As Table 1 shows that as the slit height decreases (from location a2 to d2),

**|**

*E*

_{x}

**|**dramatically increases, that is, the current density at the top/bottom surfaces

**|**

*J*

_{x}

**|**is dramatically enhanced. This can be attributed to the coupling of SPP because as slit height decreases, the MP resonant frequency is closer to the SPP excitation frequency. The current flowing at the top/bottom surfaces will generate an induced magnetic field and this induced magnetic field will generate an additional induced current flowing at the sidewall. From slit height 1000nm to 400nm, the current density at the sidewall

**|**

*J*

_{z}

**|**increases by 8.0%, while

**|**

*J*

_{z}

**|**increases by 98.6% from slit height 400nm to 100nm, which indicates the enhancement of

**|**

*J*

_{z}

**|**is mainly due to the coupling, not slit height.

Hence, the magnetic field induced by the current flowing at top/bottom surface of the strip will generate an additional induced current at the sidewall of the strip. The additional induced current at the sidewall which is caused by SPP has a significant impact on the MP resonant frequency. As the height decreases, the MP1 resonant frequency shifts to high frequency and gradually approaches SPP excitation frequency which causes the induced magnetic field and induced current at the sidewall become stronger. The closer the excitation frequency of SPP and MP are, the stronger additional induced current is, that is, the stronger the interaction between SPP and MP is.

To develop the mechanism of higher-order MP, the electromagnetic field distributions of MP4 at normal incidence are chosen as an example. Figure 4 shows the electromagnetic field distributions at the MP4 resonant frequencies: (a) *h* = 600nm at 19060 cm^{−1}; (b) *h* = 1000nm at 13474 cm^{−1}. Four current loops with alternating directions are induced inside the slits between two neighboring metal strips. Similarly, as slit height decreases from 1000nm to 600nm, the MP4 resonant frequency shifts from 13474cm^{−1} to 19060cm^{−1}. As the excitation frequency of SPP and MP4 approach, the energy density at the top and bottom surfaces of the strip are enhanced and that is similar to the phenomenon of the MP1. Hence, the interaction between SPP and higher-order MPn can be also considered as the impact of the additional induced current on the higher-order MPn resonant frequency.

## 4. Development of the MLC circuit model

#### 4.1. MLC circuit model for MP1

The LC circuit model has been successfully used to understand the physical mechanisms and the geometric effect on MPs [28,31,32]. However, the model is limited for prediction of resonant frequency of MPs without considering the interaction between SPP and MP. As the resonant frequency of MP and SPP approach, the agreement gets worse. In order to extend the applicable frequency range of LC circuit model, we develop an MLC circuit model to describe the resonance conditions for multi-order MPn. In addition, MLC circuit model can be used in the case of oblique incidence. As shown in Fig. 5, the slit serves as a dielectric capacitor *C* and the sidewall of the slit is treated as a conductor with an inductance *L*. The difference from the previous model is that, the effect of SPP on MP is treated as a mutual inductance *M* in the MLC circuit model in the present study. The flowing current at the top/bottom surface of the strip will generate an induced magnetic field which will induce an additional current at the side wall. The above process can be treated as a mutual inductance according to the electromagnetism concept [33]. The electromagnetic field and current configurations confirm the existence of inductance at the top and bottom of the strip *L _{w}* which will interact with the inductance of the sidewall

*L*and cause a mutual inductance

*M*.

The inductance of the strip’s sidewall *L* consists of the inductance of two parallel strips and kinetic inductance and is model as [9,25]:

*μ*

_{0}is the permeability of vacuum, ${\epsilon}^{\prime}$and ${\epsilon}^{\u2033}$ are the real and imaginary parts of the complex dielectric function of Ag, respectively.

*l*is the strip length in the y direction and can be treated as 1.

*δ*=

*λ*/4π

*κ*is the power penetration depth where is

*κ*the extinction coefficient and can be calculated from the complex dielectric function $\epsilon ={\epsilon}^{\prime}+i{\epsilon}^{\u2033}={(n+i\kappa )}^{2}$.

The capacitance between two strips is expressed as [9,25]:

where*ε*

_{d}is the relative permittivity of the vacuum 1, and

*ε*

_{0}is the free-space permittivity.

*c*

_{1}is a numerical factor accounting for the non-uniform charge distribution at the strip sidewall surfaces.

As a result of the excitation of SPP with an excitation frequency *ω*_{SPP}, the impedance at the top and bottom surfaces of the strip *Z _{w}* is expressed as ${Z}_{w}={R}_{w}-i\left({\omega}_{SPP}-\omega \right){L}_{w}$.

*Z*can be also expressed as: ${Z}_{w}=s/(\sigma A)$ [9,14,31], where

_{w}*σ =*-

*iωε*

_{0}

*ε*is the complex conductivity. Eliminate

*Z*and complex equation ${R}_{w}-i\left({\omega}_{SPP}-\omega \right){L}_{w}=s/(\sigma A)$ is obtained. The real and imaginary parts of the complex equation can be expressed as: ${\epsilon}^{\u2033}{R}_{w}-{\epsilon}^{\prime}({\omega}_{\text{SPP}}-\omega ){L}_{w}=s/(\omega {\epsilon}_{0}A)$ and $i[{\epsilon}^{\prime}{R}_{w}+{\epsilon}^{\u2033}({\omega}_{SPP}-\omega ){L}_{w}]=0$, respectively.

_{w}*s*is the total length of current path in the metal and for the situation of the top and bottom surface of the strip,

*s*=

*w*.

*A*is the effective cross-section area and equal to

*A*=

*lδ*. Eliminate

*R*, and

_{w}*L*can be obtained as:

_{w}The effect of SPP on MP is treated as a mutual inductance *M* in the MLC circuit model. *M* can be expressed as:

*k*

_{1}is coupling coefficient and numerical factor between 0 and 1. When mutual inductance occurs between two circuits, if there is no loss of magnetic energy,

*k*

_{1}should be 1 and Eq. (6) would degenerate to the tightly coupled situation. The value of

*k*

_{1}depends on the relative position of the two circuits and the winding form of two the inductors and is difficult to obtain under certain circumstances [33].

Hence, according to the Fig. 5, the total impedance *Z* of MLC circuit model at the slit can be obtained as follows:

By setting *Z* = 0, the MP1 resonance angular frequency can be obtained as:

Note that, ${\epsilon}^{\prime}$, ${\epsilon}^{\u2033}$, *δ*, *κ* are all frequency dependent, so it is an implicit equation that needs to be solved. It is also worth noting that, in Eq. (5), as the frequency approaches to *ω*_{SPP}, *L*_{w} tends to infinity as well as mutual inductance *M* which means SPP has an infinite effect on MP. As a result, the interaction between SPP and MP causes the *ω*_{SPP} and *ω*_{MP} shifting to each other, i.e., *ω*_{MP} approaches *ω*_{SPP} as *ω* approaches *ω*_{SPP}. This phenomenon is confirmed by Figs. 7 and 8, i.e., the curves of MP cannot pass through the curve of SPP and shift to the curve of SPP. The difference between the present MLC model and the previous LC models is that the previous LC models do not take the coupling into consider which means *M* = 0 in the previous LC models.

#### 4.2. MLC circuit model for MPn

As shown in Fig. 6, the MLC circuit model for multi-order MPn can be equivalent to the series of multiple LC circuits of MP1 which can be confirmed by electromagnetic field distributions as shown in Fig. 4. Each circuit should have a same height *h*/*n* and *c*_{1} and *k*_{1} are assumed to be equal of each circuit. For the situation of MPn, Eqs. (3), (4), and (6)-(8) change in the form:

*L*is the inductance of the strip’s sidewall and

_{n}*C*is the capacitance between two strips of each circuit. Note that, due to fact that the relative positions between the top/bottom inductance and each sidewall inductance are different, each circuit should have a different

_{n}*k*

_{1}. In addition, each circuit also should have a different

*c*

_{1}for reason that charge distribution at the sidewall of each circuit is slightly different. In the present study,

*c*

_{1}is treated to be equal of each circuit for simplicity and represents the average value of all the circuits as well as

*k*

_{1}.

This MLC circuit model can be used to explain the geometric effect and incident angle on the resonances which will be shown in the following section.

## 5. Results and discussion

#### 5.1. Geometric effect

Figure 7 demonstrates the comparison between the previous LC model and MLC model with the geometric values: (a) *Λ* = 0.5μm, *b* = 0.05μm and *h* varies from 2nm to 1μm; (b) *h* = 0.4μm, *b* = 0.05μm and *Λ* varies from 52nm to 1450nm. The contour plots show 1−*ρ* at normal incidence. The green lines are the results calculated from the previous LC model [28] and the cyan-blue lines are calculated from the MLC circuit model. Yellow line represents the excitation frequencies of SPP which can be calculated as Section. 2 states. The previous models do not take the coupling into account and cause the MP curves pass through the SPP curve. Due to considering the effect of interaction between SPP and MP, the MLC circuit model shows good agreement with numerical calculation especially when the excitation frequency of SPP and MP are close.

As shown in Fig. 7(a), the curves of MPs are divided into two branches by the curve of SPP which is consistent with two MP6 in the Fig. 2. Besides, as the slit height increases, the MPs resonance frequencies decrease. SPP is excited at 20000cm^{−1} approximately and is nearly independent of *h*. Due to the interaction between MP and SPP, the resonance frequencies for both SPP and MP shift near the intersections. Figure 7(b) shows that SPP excitation is related to the period significantly and *ω*_{SPP} decreases with the period. As period increases, more SPP mode appear and divided MP into branches. Though there is no factor of period *Λ* in Eqs. (3)-(13), *ω*_{SPP} varies with *Λ* so that the resonant frequencies of MPs are related to *Λ*. The resonant frequencies of MPs remain unchanged until they approach SPP curves. The enhancement of MPs become weaker as the times of being divided increases.

Note that, *c*_{1} is an adjustable value in various structures [24,27,28,31,32]. Each curve of MP in Figs. 7 and 8 corresponds to a set of *c*_{1} and *k*_{1} which means the change rule of radiative properties is same under the similar geometric parameters (fixing *h*, *b* and *Λ*) or incident angles. The mutual inductance generated by the current flowing at the top/bottom of the strip will change the charge distribution of the sidewall, that is, *c*_{1} and *k*_{1} are interrelated. *c*_{1} and *k*_{1} will affect the resonant frequency together and that is why for the geometric parameters *Λ* = 0.5μm, *b* = 0.05μm and *h* = 0.4μm, *c*_{1} and *k*_{1} have slight difference in Fig. 7(a) and 7(b). As shown in Table. 2 and Fig. 7 (a), fixing *b* and *Λ*, as the order of MPn increase, the charge distributions of the sidewall are similar at various slit heights and cause *c*_{1} unchanged. As the number of the circuit (order of MP) increases, the induced magnetic field impact on several circuits together and it causes the average *k*_{1} reduced. Table. 2 and Fig. 7 (b) indicate that as the order of MP increases, *k*_{1} decreases slightly and *c*_{1} keep unchanged with the same reason above.

#### 5.2. Effect of incident angle

Figure 8 demonstrates of incident angle effect on the magnetic resonance conditions with slit height *h* = 400nm, slit width *b =* 50nm and period *Λ* = 0.5μm. *c*_{1} = 0.17 and *k*_{1} = 0.05 for both MP1 curve and MP2 curve. The cyan-blue lines are calculated from MLC circuit model and yellow line is SPP curve calculated by the method in Section 2. SPP curve consists of two fold lines and divide MP curves into branches. Though there is no factor of incident angle *θ* in Eqs. (3)-(13), *ω*_{SPP} varies with *θ* so that the resonant frequencies of MPs are related to *θ*. MPs curves remain unchanged until they approach SPP curve. MPs do not change with incident angle according to the LC model, but it is applicable only when the interaction between SPP and MPs can be neglected. The essence of MPs change with the incident angle and period is that SPP is significantly correlated with the incident angle and period. Figures 7 and 8 show that MLC circuit model can describe the MP resonant conditions of geometric effect and effect of incident angle with reasonable explanation and the rationality of MLC model is verified.

## 6. Conclusion

In summary, MLC circuit model is proposed to describe MP resonant conditions in the case of oblique incidence. The interaction between SPP and MP can be treated as that, the magnetic field induced by the electric current flowing at the top/bottom surfaces generates an additional induced current at the sidewall of the strip which will significantly affect the MP resonant conditions. The relative value of electric field at the distances of power penetration depth from the sidewall surface increases as slit height reduces which confirms the additional current flowing at the sidewall. The closer the excitation frequency between MP and SPP is, the stronger the additional current is, that is, the stronger coupling effect is. The mechanism of coupling between SPP and MP is explained from the perspective of equivalent circuits and the effect of SPP on MP is considered as the mutual inductance in MLC circuit model. MLC circuit model shows good agreement with numerical calculation results by FDTD. Slit arrays with different geometric parameters and incident angles are calculated to verify the rationality of MLC circuit model. It is expected that our study may contribute to the prediction of thermal radiative properties in nanoscale energy devices especially the frequency-dependent emitters and filters.

## Funding

National Natural Science Foundation of China (51876049).

## Acknowledgments

Special thanks are given to the reviewers and people who suggest improvements of the manuscript.

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