1. Introduction

Tailoring of the spectral and directional radiative properties has numerous applications in photonic and energy conversion systems, such as photodetectors, solar cells and solar absorbers, thermophotovoltaic devices, and radiation filters and emitters [1]. Coherent thermal emission is characterized by the spectral and/or directional selectivity in the emissivity, and has been demonstrated by exciting surface polaritons using modulated structures. The surface polariton is a resonance phenomenon in which electromagnetic waves are coupled to the collective oscillation of electrons or optical phonons [2]. Recently, one-dimensional gratings [3], truncated photonic crystals [4–6], and bilayer structures composed of single-negative materials [7] have been proposed as coherent thermal-emission sources. The difficulty in creating negative permeability materials has hindered the realization of the concept of paired single-negative materials.

Rapid developments in the electromagnetic theory and nanotechnology have facilitated the manufacturing of metamaterials with unique electric and magnetic properties [8]. It is known that when a time-varying magnetic field is parallel to the axis of a spiral coil of metal wire, an induced magnetic field will occur due to the resultant current in the coil according to Lenz’s law. Such a diamagnetic response has been demonstrated with the split-ring resonator in the microwave region, resulting in a negative index medium with simultaneously negative permeability and permittivity [9]. With such artificial structures, sometimes called magnetic elements, electromagnetic waves can interact with materials via both the electric and magnetic fields; thus, the effective permittivity and permeability can be controlled. Recently, several groups have demonstrated a negative index behavior in the optical frequency using various structures [10–14]. However, most studies has focused on subwavelength imaging beyond the diffraction limit and invisible clocking, without addressing the thermal emission and absorption processes that have numerous applications in energy conversion and optical detection.

The present work describes an innovative idea for tailoring the thermal emission and absorption characteristics using a metallic grating coated on a dielectric spacer atop an opaque metal film. The magnetic response existing between the metallic strips and the film allows the structure to behave as a single-negative material, with a negative permeability (real part) and positive permittivity. At the same time, the bulk metallic film also serves as a nonmagnetic material with a negative permittivity (real part) and positive permeability in the near infrared. Consequently, surface magnetic polaritons [15] can be excited in the proposed structure, resulting in coherent thermal emission. Magnetic polaritons can be viewed as a resonance with a magnetic response (negative permeability) that is coupled to a nonmagnetic medium. The dispersion relation of the surface magnetic polariton may be described by μ ₁/k _1z + μ ₂/k _2z = 0, where μ’s are the relative permeabilities and k_z’s are the normal component of the wavevectors of the media [7]. The resonance condition of the magnetic response can be predicted by an analogy to an inductor-capacitor (LC) circuit. The predicted resonance conditions are compared with those obtained from the rigorous coupled-wave analysis (RCWA). Furthermore, the effect of geometric parameters on the resonance condition is also investigated.

2. Surface magnetic polaritons

Consider the structure that is made of periodic metal strips with a dielectric spacer deposited on a metallic film, as depicted in Fig. 1(a). Without the spacer, it is simply a binary grating. For simplicity, silver is selected as the material for the strips and (opaque) film, and silicon dioxide is used for the spacer. In practice, the silver film, whose thickness is much greater than the radiation penetration depth, can be deposited on a substrate, which is not shown in the figure. The geometry of the one-dimensional grating geometry is represented by period Λ, strip width w, and thickness h. The thickness of dielectric spacer is denoted by d. A linearly polarized electromagnetic wave is incident from air at an incidence angle θ. Similar to metallic strip pairs, the oscillating magnetic field parallel to the grating grooves can cause anti-parallel currents in the metal strips and the metal film surface. Since no magnetic response is associated with the transverse electric waves, only the transverse magnetic (TM) wave is considered in the present study.

The RCWA allows the calculation of the directional-hemispherical reflectance at any given wavelength. The electric and magnetic fields near the grating region can also be calculated using the RCWA [16]. The frequency-dependent dielectric functions of Ag and SiO₂ are obtained from the tabulated data in Ref. [17]. In the calculation, a total of 101 Fourier components are used to represent the dielectric function in the grating region. The calculated reflectance is shown in Fig. 1(b) in the wave number range from 3,000 to 20,000 cm^-1 at θ = 25°. The geometric parameters are selected as Λ = 500 nm, w = 250 nm, and h = d = 20 nm, to generate a close-to-unity emissivity in the considered wave number region. For the sake of comparison, the reflectance of a simple grating is shown as the dotted curve, by setting d = 0 with the remaining parameters unchanged. It is well known that gratings can support surface plasmon polariton. At the resonance condition described by the dispersion relation: ε ₁/k _1z + ε ₂/k _2z = 0 [2], where ε’s are the permittivities of the media (metal and air), the incident wave is coupled to the collective oscillation of electrons and is eventually absorbed by the Ag substrate. The result is a sharp reflectance dip near the wave number ν = 13,780 cm^-1. On the other hand, if a 20-nm SiO₂ spacer is added, several additional reflectance dips appear besides the one caused by the surface plasmon. These reflectance dips are attributed to the excitation of magnetic polaritons. Since the semi-infinite metallic film is opaque, one can regard the reflectance dip as the emissivity peak according to Kirchhoff’s law [1].

 

Fig. 1. (a). Schematic of the proposed metamaterial structure with coherent thermal emission characteristics. Here, Λ is the grating period, w is the width of the metallic strip, and h is the thickness of the metal strip or grating. The thickness of the dielectric spacer is d. The coordinates system is also shown for a TM-polarized plane wave with a wavevector k incident at the angle θ. (b) Reflectance of the proposed structure for the TM wave at θ = 25°. The geometric parameters are Λ = 500 nm, w = 250 nm, and h = d = 20 nm. The dotted line shows the structure without the dielectric spacer (d = 0).

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The underlying mechanism of magnetic metamaterials can be explained as follows. The oscillating magnetic field produces a current in the metal strip in the x direction and another near the surface of the metal film in the opposite direction. The anti-parallel currents result in a diamagnetic response [14]. The diamagnetic response is then coupled to the metallic film to cause a surface magnetic polariton with a fundamental mode at the wave number around 5,670 cm^-1. Magnetic polaritons of the second and higher order harmonics can also be excited. Therefore, the reflectance spectrum with spacer exhibits dips at ν = 5,670, 11,490, and 16,095 cm^-1, corresponding respectively to the fundamental, second, and third harmonic modes. Magnetic polaritons, however, are distinct from the surface plasmon such that the resonance frequencies depend strongly on the strip width but remain almost unchanged with the grating period Λ. This is because the magnetic polariton is not induced by the diffracted evanescent waves but induced by the magnetic element formed in the modulated structure.

Figure 2 shows the contour plot of the spectral-directional emissivity ε_ν,θ for the simple grating (a) and the proposed structure with the spacer (b) in terms of ν and the parallel wavevector component k_x (divided by 2π). Note that k_x = 2πνsinθ. In Fig. 2, darker colors represent lower emissivities, whereas brighter colors correspond to higher emissivities. The region outside the light line on the lower-right corner is left blank. As can be seen clearly from Fig. 2(a), the grating results in folding of the dispersion curves at k_x = π/Λ, 2π/Λ, etc. The emissivity is greatly enhanced when surface plasmons are excited. The branch at ν < 18,200 cm^-1 corresponds polaritons coupled with the -1 diffraction order and the high-frequency branch is associated with the +1 diffraction order [2]. The intersection of the surface plasmon dispersion line and the inclined white line, representing θ = 25°, is marked as SP and corresponding to the reflectance dip shown in Fig. 1(b) due to surface plasmon resonance. In general, the resonance condition of surface plasmon depends strongly on both ν and k_x; thus, the emissivity peak exhibits the spectral and directional selectivity [3].

The contour plot of the emissivity for the grating with spacer exhibits several additional bands with enhanced emissivity as shown in Fig. 2(b). The surface plasmon dispersion is very similar to that shown in Fig. 2(a). The multiple magnetic polariton branches correspond to the fundamental, second, and third harmonic resonances and their intersection with the line θ = 25° are denoted by MP1, MP2, and MP3, which are associated with the reflectance dips at ν = 5,670, 11,490, and 16,095 cm^-1, respectively shown in Fig. 1(b). In contrast to the surface plasmon, k_x has little effect on resonance conditions for the magnetic polaritons because the magnetic resonance conditions are largely determined by w rather than Λ. Furthermore, the magnetic polaritons are localized in the vicinity of metal strips and are not coupled with each other due to the 250-nm air gap [15]. Hence, the emissivity peak resulted from the magnetic polariton becomes nearly independent of the emission angle and exhibits diffuse characteristic that is desirable for thermophotovoltaic emitters. It should be noted that the resonance frequency can be easily tuned by varying the strip width w.

Figure 2(b) reveals additional interesting aspects of different modes of magnetic responses. The even-order magnetic polaritons (such as the second harmonic mode) can only be excited at oblique incidence, whereas the odd-order magnetic polaritons can be excited at normal incidence. Furthermore, surface plasmons can strongly interact with magnetic polaritons at certain ν and k_x values. The interaction of surface plasmons with magnetic polaritons can result in either enhancement or suppression of the emissivity. It can be inferred from Fig. 2(b) that if the surface plasmon dispersion curve intersect an even-order magnetic polariton, the corresponding emissivity is suppressed and the magnetic polariton dispersion line splits in to two curves, as illustrated in Fig. 2(b) for the second-harmonic magnetic polariton mode. On the other hand, the odd-order magnetic polaritons constructively interact with the surface plasmon, resulting in high emissivity values and a spectral broadening of the emissivity peak, as illustrated in Fig. 2(b) for the third-harmonic magnetic polariton mode. The above conclusions are drawn from numerous calculations with various geometric parameters not shown here.

 

Fig. 2. Contour plot of the spectral-directional emissivity of (a) the simple grating and (b) the Ag grating and Ag film separated by a SiO₂ spacer. The geometric parameters are the same as those in Fig. 1. At θ = 25°, surface plasmon resonance is labeled as SP, while the magnetic polaritons are labeled as MP1, MP2, and MP3 for the fundamental, second, and third harmonic modes, respectively.

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In order to further investigate the physical mechanism of the magnetic polariton, the magnetic field distribution inside the considered structure is calculated by the RCWA and plotted in Fig. 3. The three figures correspond to the resonance conditions MP1, MP2, and MP3 shown in Fig. 2(b) at the incidence angle of θ = 25°. Here, the z axis is pointed upwards so that the Ag strips appear to be below the Ag film. The background contour represents the logarithmic values of the square of the magnetic field magnitude, and the arrows indicate the electric field vectors. As shown in Fig. 3(a) for the fundamental mode of the magnetic response, anti-parallel currents in the metallic strips and the substrate confine strong magnetic field inside the dielectric spacer. The considered structure acts similarly to the metal strip pairs regarding the magnetic field distribution. However, the semi-infinite metal substrate employed here results in the enhanced absorption at resonance conditions. Although the magnetic field is not symmetric with respect to the center of the metallic strip, the second and third order magnetic resonances are clearly demonstrated such that two and three anti-nodes of the magnetic field are formed in the dielectric spacer underneath the metal strip, respectively. The corresponding electric field distribution further confirms the magnetic induction around the anti-nodes of the magnetic field distribution. Hence, the effective permeability of the considered structure exhibits a resonance like dispersion according to the electric and magnetic fields distribution in the dielectric spacer. It should be noted that the effective permeability can be calculated by averaging the magnetic moment of current loops [10]. As illustrated in Fig. 3(b), there are two induced current loops with opposite direction for the second harmonic mode. Therefore, the averaged magnetic moment is zero at normal incidence due to the symmetry, suggesting that even-order harmonic modes can only be excited at oblique incidence. In contrast to the magnetic polariton, the surface plasmon polariton (not shown in Fig. 3) generates enhanced magnetic field along the interface between the dielectric and metal film, as well as along the interface between metal strips and air. Furthermore, the field between the metal strips and the film is not enhanced when surface plasmon is excited.

 

Fig. 3. Contour shows the square of the magnitude of complex magnetic field in logarithmic scale and the arrows indicate the electric fields when the magnetic polariton is excited for conditions corresponding to MP1, MP2, and MP3 shown in Fig. 2(b). (a) MP1, ν = 5,670 cm^-1; (b) MP2, ν = 11,490 cm^-1; (c) MP3, ν = 16,095 cm^-1.

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3. Geometrical effects on resonance frequencies

As illustrated in the previous section, the magnetic field is strongly localized in the vicinity of the dielectric spacer underneath the metal strips at the resonance condition. In this regard, the considered structure can be approximated as metal strip pairs although a metallic film is used as the substrate. Hence, the resonance condition for the fundamental mode of the magnetic response can be predicted by the equivalent LC circuit model, as shown in Fig. 4. Here, the inductance L _m of two parallel plates separated by a distance d can be expressed as L _m = 0.5μ ₀ dw/l, where μ ₀ is the permeability of vacuum. Notice that l is the metallic strip length in the y direction, which is assumed to be finite in the LC circuit model. Due to the nanoscale dimension of metal strips, the contribution of the drifting electrons to the total inductance cannot be neglected [18] and is given by L _e = w/(γhlω ² _p ε ₀), where γ = 2/3 is a factor considering the effective cross-sectional area of the metal strip and ω _p = 1.364×10¹⁶ rad/s is the plasma frequency of silver. On the other hand, the capacitance C _m by two parallel plates sandwiching the dielectric spacer with thickness d is given as C _m = c ₁ ε _d ε ₀ wl/d, where c ₁ = 0.222 is a numerical factor that accounts for the effective area of the capacitor on the metallic strip [19], ε ₀ is the permittivity of vacuum, and ε _d is the dielectric function of SiO₂. In addition, the capacitance C _e, which accounts for the contribution of the air gap between Ag strips, can be approximated as two parallel wires of diameter h with a length l; hence, C _e = πε ₀ l/ln[(Λ - w)/h] . According to the circuit model illustrated in Fig. 4, the total impedance can be expressed as

where ω is the angular frequency. The magnetic resonance occurs when Z _tot = 0 ; thus, the resonance condition can be obtained from

 

Fig. 4. Schematic of the equivalent inductor and capacitor circuit for the one period of the structure shown in Fig. 1(a).

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Since L _m ∼ l ^-1, L _e ∼ l ^-1, C _m ∼ l, and C _e ∼ l, the resulting ω _R is independent of the assumed metal strip length l in the y direction. Therefore, the resonance condition obtained from Eq. (2) is also applicable to the proposed structure in which l → ∞.

The predicted resonance condition based on the equivalent LC circuit is compared with that obtained from the RCWA for different values of Λ and w when h = d =20 nm at normal incidence. Figure 5(a) shows the resonance condition for different w values from 50 nm to 480 nm when Λ = 500 nm. It can be clearly seen that the magnetic response largely depends on the metal strip width. As expected, the resonance wave number decreases as the metal strip width increases. However, the resonance condition is not inversely proportional to w, as predicted in Refs. [14] and [20] for the millimeter-size structure, due to the contribution of C _e. The resonance condition calculated based on the LC model exhibits good agreement with that obtained by the RCWA when w > 150 nm. On the other hand, when the metal strip width is less than 100 nm, the relative difference in the resonance conditions exceeds 10%. This error is probably caused by the factor c ₁. Recall that c ₁ accounts for the effective area of the capacitor and may need to be modified as the metal width becomes narrower. The effect of the grating period on the resonance condition is considered in Fig. 5(b) for different Λ values from 280 nm to 800 nm when w = 250 nm. In contrast to the strip width, the period does not much affect the resonance condition. The predicted resonance conditions agree well with that of the RCWA, but there exists relatively larger difference if Λ < 300 nm. This is because the magnetic resonances in neighboring metal strips can be coupled with each other for narrow air gap, as verified by the magnetic field distribution (not shown here).

 

Fig. 5. Comparison of the predicted resonance condition with that obtained from the RCWA: (a) for different w values at Λ = 500 nm; (b) for different Λ values at w = 250 nm.

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The LC circuit model can be easily extended to predict the resonance condition for higher harmonic modes. The additional calculations reveal that the ratio of resonance conditions between the second and the fundamental harmonic modes is approximately 2.1 from the LC circuit model and 2.0 from the RCWA, when the metal filling ratio w/Λ is between 0.2 and 0.8. Therefore, it can be inferred that the geometric parameters, such as Λ and w, affect the resonance condition of higher harmonic modes in a similar way to the fundamental mode. The good agreement between the LC circuit model and the RCWA calculation further confirms that the considered structure exhibits the magnetic responses, resulting in the excitation of magnetic polaritons that generate spectral emissivity peaks.

4. Concluding remarks

The present work has demonstrated that coherent thermal emission can be achieved by exciting magnetic polaritons using a metallic grating coupled with a metal film. Multiple modes of the magnetic polariton have been identified according to the electric and magnetic fields distribution in the dielectric spacer. It is evident that the magnetic polariton can strongly interact with the surface plasmon, resulting in either enhancement or suppression of the emissivity for different magnetic polariton orders. The resonance condition of the fundamental mode of the magnetic response predicted by the LC model agrees well with that obtained from the RCWA when the metal filling ratio is between approximately 0.2 and 0.8. The proposed structure can be readily constructed using available micro/nanofabrication techniques. This study will facilitate the development of wavelength-selective emitters/absorbers for thermophotovoltaic devices and infrared radiation detectors.