1. Introduction

Thermal radiation is a spectrum of electromagnetic waves radiated by objects due to the change of the thermal state of the particles when the objects’ temperature is higher than absolute zero. This has important applications for thermal switching [1], imaging [2], and thermophotovoltaic (TPV) power generation [3–5]. The characteristic of thermal radiation depends on both physical properties and structural features of the radiating object. Because of these two dependencies, it is possible to manipulate the thermal radiation of objects [6–11]. However, because changing the properties of a given material is difficult, structural modification to change thermal radiation is an important alternative method. Recent studies have largely focused on periodic subwavelength-scale patterning of metallo-dielectric structures in many spectral ranges, which can also modify radiation spectra [12–16]. The thermal radiation can be as high as 0.8 at some frequencies, but is very low at other frequencies within their respective research ranges to achieve several radiation peaks, which can even be tuned to achieve the theoretical limit at particular frequencies by adjust the structure permittivity accurately [14,15,17,18]. But the precise periodicity and incident angle play key roles in determining the types of physical phenomena. In addition, near-field radiation is different from far-field radiation because Surface Plasmon Polaritons (SPPs) and other evanescent waves mainly exist in the near filed region when dispersive metallic materials are involved. In other words, structural features with different physical effects have a greater effect on the emitted near-field thermal radiation.

In this paper, we survey the level of thermal radiation coming from a single subwavelength slit. We then focus on the high-level near-field radiation efficiency within a wide spectrum that can be achieved using an ultra-thin multi-slit structure. Via finite-difference time-domain (FDTD) simulation, we study this system numerically and demonstrate the exact dependence of the radiation on the structural parameters. We also investigate how a combination of substructures can change the thermal radiation spectra. This enables us to analyze the radiation spectra of single- and multi-slit structures to identify the underlying physical processes. This makes it possible to tailor thermal radiation properties to specific design needs.

2. Simulation setup

The schematic profile of the sample structure is a self-standing single subwavelength slit, where the metallic film is silver (see inset of Fig. 1a). The depth and width of the slit are denoted by d and w, respectively. A cosine modulated Gaussian pulse with a center frequency of ${\omega }_c=300THz$ illuminates the structure at the normal angle, with a spectral width of $\Delta \omega =300THz$. The transverse distribution of the pulse is a Gauss type, whose spatial full-width at half-maximum (FWHM) is $\delta =530nm$. The incident beam can be described using this following expression:

 

Fig. 1 Thermal radiation of a single subwavelength slit with a variant depth d at w = 120 nm (a), and a variant width w for d = 600 nm (b). Inset in (a) shows schematic plot of a single subwavelength slit structure. T_line and R_line are the cross sections to calculate transmittivity and reflectivity with 150 nm distances away from the structure.

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Here the FDTD method is used to simulate the near-field electromagnetic field distributions around the structure. The grid size in the calculation is 5 nm. The simulation region is bordered with perfectly matched layers to absorb scattering lights. The permittivity of silver is described by the Drude model, whose parameters are taken from Ref. 19.

According to Kirchhoff’s law, the thermal radiation efficiency E equals to the absorptivity A for thermal equilibrium. Using the FDTD simulation, both transmittivity T and reflectivity R can be calculated, and the absorptivity can be obtained according to the energy conservation law, i.e. $A=1-R-T$. Here, the calculated transmitted and reflective cross sections are located 150 nm away from the structure, which are denoted by dashed lines of T_line and R_line in inset of Fig. 1(a).

3. Simulation results

First, the thermal radiation of a single subwavelength slit with w = 120 nm is simulated, and its slit depth d is increased from 100 nm to 700 nm. The results are summarized in Fig. 1(a). When slit depth d is small, the thermal radiation is gradually increased with frequency. When the slit depth d increases, a radiation peak appears that shifts to the left with increasing d. When d becomes thicker than 400 nm, another radiation peak appears in the high frequency region, and the whole radiation spectrum shifts left continually. This tendency to change indicates that the radiation wavelengths may follow a Fabry-Perot (F-P) resonance in the structure. We use FWHM to characterize the radiation peak quantitatively. If the spectral line is not complete, we use the half-width at half maximum to calculate FWHM. Hence, the obtained FWHM equal to 75, 65.7, 45.3 and 44.4 for d = 400 nm, 500nm, 600 nm, 700 nm in low frequency region, respectively; and to 229.2, 115.2, 97.2 and 93.6 for d = 300 nm, 500 nm, 600 nm and 700 nm in high frequency region, respectively, where the unit is terahertz. This fact indicates that a single resonator with a litter FWHM helps to achieve high radiation-efficiency.

To investigate the effect of the slit width on the radiation, we use different single slits but with a fixed depth of d = 600nm. We also consider that the highest radiation peaks and minimum FWHM values can be achieved for this condition, together with a flat and wide spectrum between them. The slit width w changes from 50 nm to 150 nm, and the corresponding radiation is summarized in Fig. 1(b). We find that the radiation spectra for a width exceeding 100 nm are almost identical. Specifically, the left narrow peaks almost coincide, and the right and broader peaks shift slightly right, while the values improve slightly for an increasing w. We find the same variant tendency for the radiations between these two peaks. This is different, however, for w = 50 nm, when the radiation peaks become small and narrow and the radiation located in the middle shows ups and downs. In addition, the right peak has a tendency to split. For a more systematic comparison, the radiation of another broad slit of w = 300 nm is given. The features are consistent with the aforementioned variant tendency. Hence, it can be confirmed that the slit width has a weak effect on radiation or absorption when w is smaller than half of the minimum wavelength.

For a slit with a fixed width, the intensity of the excited SPPs resonance in the slit remains unchanged, and the total field intensity is modulated by the slit depth to form an F-P resonance. However, for a slit with a fixed depth, the wavelength and intensity of the SPPs varies with the slit width, and so does the strength of the resonance. Considering the radiation efficiencies shown in Fig. 1(a) and 1(b), it is found that only if w belongs to the deep subwavelength range, the slit depth has a greater effect on radiation than the slit width. This is because it determines whether the intensity of the SPPs resonance is enhanced or suppressed by F-P interference.

For a single subwavelength slit, the thermal radiation is neither strong enough nor narrow across the whole spectrum. There are, however, radiant peaks associated with monochromatic radiation. To emit strong radiation across the full spectrum, we need a complex multi-slit structure formed by several slit units with identical or different parameters. We first construct a three-slit structure with the same unit cell (w = 120 nm and d = 600 nm) accompanied with an intermediate layer (t = w) between them. Great changes can be seen in the radiation spectrum shown in Fig. 2(a). Although the FWHM values of the radiation peaks increase, the overall radiation efficiency increases significantly: the left peak decreases by 7.8%; the right peak increases by 10% and becomes wider; and the radiation between these peaks increases up to 140%. These improvements are very beneficial for applications requiring high radiation across the full spectral range. The reason for not choosing w = 300 nm as the unit cell is that the radiation efficiency of the combined three-slit structure is very low regardless of the thickness of the intermediate layer (the spectra are not shown here).

 

Fig. 2 Thermal radiation of the three-slit structure for identical unit cell (w = 120 nm and d = 600 nm) but different intermediate thicknesses of the metal layer t, ranging from 2w to 0.5w (a). Inset shows schematic plot of multi-slit structure. Radiation spectra of structures for different numbers N of unit cells at t = 0.5w (b).

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The effect of the thickness of the intermediate layer t on thermal radiation with three identical unit cells is investigated when it is reduced from 2w to 0.5w. The results are summarized in Fig. 2(a). It is seen that the radiation efficiency shifts down due to the increased distance between the slits, while it moves upon reducing the distance. The shape of the radiation curve remains almost unchanged, and only the distance between the peaks increases, i.e., the left peak shifts left and the right peak shifts to the right while narrowing t. Neither single left slit nor single middle slit separately shows these radiation spectral properties. As a result, we can conclude that this kind of radiation is due to both interactions of the thermal radiation fields of each slit and fields on the intermediate layer’s output interface. Moreover, using t = 0.5w seems special because the intermediate layers are so thin that cavity modes in two adjacent slits can directly interact with each other. This may be another factor leading to the complex radiation spectrum.

We further investigate the effect of the unit cells number on the radiation efficiency for w = 120 nm, d = 600 nm and t = 0.5w. In order to facilitate such comparison, the radiation of a single slit, i.e. N = 1 is also shown. Compared with these curves shown in Fig. 2(b), we find that the thermal radiation increases with the increase of N at first, and the highest efficiency can be achieved for N = 3. However, if N continues to grow, the efficiency begins to decline and the radiation peaks disappear. In other words, uniform radiation can be achieved for the overall spectrum at the expense of efficiency at some frequencies for increasing N. Changing the depth of the unit cells yields the same calculated results. Although the incident wave not illuminating both sides of slits efficiently may be a factor, we can still conclude that the multi-slit structure must have a finite number of unit cells to generate optimal radiation.

Additionally, the influence of geometric parameters with slight difference between unit cells in a multi-slit structure on the thermal radiation is considered. First, we chose three widths (50 nm, 120 nm and 150 nm) to form complex multi-slit structures with a fixed depth (d = 600 nm) and intermediate layer (t = 60 nm). After illumination with a Gauss pulse, the obtained thermal radiation for different slit combinations are calculated - see Fig. 3(a). Different combinations produce different spectra. Generally speaking, only if the complex structure has a unit cell of w = 50 nm with larger gaps than the others, its radiation is less efficient in specific spectral areas. This phenomenon can be explained easily: a single slit can be regard as a point source of thermal radiation, whose radiation spectrum can be seen in Fig. 1(a). The radiation spectra of several point sources interfere to produce that of multi-slit structure. Therefore less efficient radiation of this structure mainly attributes to the spectrum of w = 50 nm having relatively large difference. For w>100nm, when the differences between unit cells are small, the radiation differences between this structure and a structure with uniform unit cells are small. This is the case for 150-120-150 nm, 120-120-120 nm, and 150-150-150 nm. Next, we consider this complex structure for 150-120-150 nm with one or two asymmetric slit edges. The radiation spectra of these specific combinations are shown in Fig. 3(b). Compared with the radiation of symmetric slits, it is found that asymmetric slit edges have a small effect on the radiation. It is because that there is little difference of thermal spectra between asymmetric and symmetric slit edges for a single slit only if the locations of cross sections keep unchanged – see inset of Fig. 3(b). This result indicates that, unlike refs. 17 and 18, the high-level thermal does not strictly depend on the geometrical parameters of the structure. However, if the difference in asymmetry is relatively large, the impact becomes large. In conclusion, regardless whether a multi-slit structure has the same unit cells or not, the thermal radiation improves across the whole spectrum only if the difference between the unit cells is small.

 

Fig. 3 (a) Thermal radiation of a three-slit structure with different combinations but identical depth (d = 600 nm), wherein the thickness of the intermediate metal layer t is fixed at 60 nm. Inset shows schematic plot of this spectral multi-slit structure. Legend shows w distribution from left to right of the structure. (b) Thermal radiation of a three-slit structure with w = 150 nm, 120 nm, 150 nm and t = 60 nm, but different slit depth d, whose distribution from left to right is shown by legend. Inset shows radiation spectra of a single slit with symmetric and asymmetric edges.

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4. Discussion

We use the electromagnetic (EM) field distribution in the slit to understand the physical mechanism of the emitted thermal radiation. First, we extract the steady-state modulus |Hz| of magnetic field, which is normalized by that of incident, propagated along the centerline of a single slit with w = 120 nm, d = 600 nm at two radiation peaks and one valley. The results are shown in Fig. 4(a). It seems that integer multiples of F-P interference periods correspond to the radiation peaks, and half integer multiples to radiation valleys. Calculated using the F-P interference lines, the wavelengths of the cavity modes are 1200 nm, 1060 nm and 680 nm, and the corresponding frequencies are larger than f₁, f₂ and f₃. This indicates that the resonant wavelength in the slit and the radiation wavelength do not strictly correlate one-to-one, which is different from what we expected. F-P resonant in the slit is regarded as the source of thermal radiation. While the frequency differences, i.e. energy differences, indicate that not all of energy convert from F-P resonant to thermal radiation, but rather a tiny part of energy converts into the Ohmic loss of metal.

 

Fig. 4 Normalized modulus |Hz| of magnetic field propagated along the centerline of a single slit with w = 120 nm and d = 600 nm for three frequencies: $f_1=178.8THz$ (peak), $f_2=252.6THz$ (valley) and $f_3=396.6THz$ (peak) (a). Normalized modulus |Hz| of magnetic field propagated along the centerline of the middle slit of ultra-thin multi-slit structures with N = 3, w = 120 nm d = 600 nm and t = 0.5w for three frequencies: $f_4=183.1THz$ (peak), $f_5=300THz$ (valley) and $f_6=436.2THz$ (peak) (b). The corresponding magnetic field distributions are shown in (c)-(e).

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For a complex structure with the same unit cells, the magnetic field in the middle slit shows the same characteristics - see Fig. 4(b): the corresponding wavelengths of the cavity modes are 880 nm and 630 nm, with frequencies that are little larger than f₅ and f₆, but with smaller gaps. This indicates that the radiation wavelengths are closer to the resonant wavelengths in the slit. Figures 4(c) - 4(e) show the corresponding magnetic field distributions illuminated by a sine continuous Gaussian wave with special frequencies of f₄, f₅ and f₆. The cavity mode for each slit is uniform for the transverse and periodic variations in longitudinal direction, but there is little magnitude difference between slits. Especially for f₄, the field in the bilateral slits is stronger. When the frequency increases, the differences gradually disappear. In addition, since SPPs at the slit edges can penetrate into metal, strong interaction of SPPs between adjacent slits can be seen in the intermediate layers, which leads to the changes of internal charges and field distributions. These, in turn, affect the distributions in the slits. For example, for SPPs’ wavelength reduction, the period of F-P interference becomes small. This follows a longitudinal shift of the periodic distributions in each slit, which makes this multi-slit structure with ultra-thin intermediate layers act like an ultra-wide slit that radiates a spectrum with broad peaks. In fact, the radiation of the ultra-thin multi-slit structure is neither similar to that of the single slit nor resembles a traditional multi-slit structure with very thick layers. Hence, we can conclude that the interaction of SPPs between adjacent slits and F-P interference in each slit play important roles with regard to its spectral characteristic.

We investigate the magnetic phase distributions in slits for different conditions – see Fig. 5. For a single slit, the phase continues to change without sudden variations for the slit under f₁ illumination (see Fig. 5a), which produces peak radiation. However, a phase jump appears for f₂ illumination that corresponds to a radiation valley (see Fig. 5b). When a multi-slit structure, comprised of several slits with thin interval layers, is illuminated, the associated phase distribution changes. For a three-slit structure with the same unit cells and f₁ illumination, the phase distribution is different – see inset in Fig. 5(a): bilateral slits show the same phase, which remains constant among the slits but changes suddenly at the exits. The phase in the middle slit jumps before the slit exit, however. Regardless of the changes in the slits, the phase at the exit interface of the structure remains constant. This is the reason that the radiation of multi-slit structure does not decrease unlike that of a single slit. When the frequency increases a little to f₄ (see Fig. 5c), the variation of the phase distribution becomes clear, especially in the middle slit. However, since phase matching is satisfied at the exit interface of this structure, the corresponding radiation increases slightly to produce a new radiation peak. For f₅ illumination, even though the phase jumps in the longitudinal direction of each slit, the phases of the magnetic field in all unit cells and intermediate layers remains uniform on the transverse (see Fig. 5d). In other words, phase matching is satisfied, which produces improved radiation. Therefore, we can conclude that no matter how the phase of a single slit change, as long as phase matching at the slits exit interface can be obtained, strong and uniform radiation can be generated with a multi-slit structure.

 

Fig. 5 Phase distributions of magnetic fields in the single slit (a) and (b) and ultra-thin multi-slit structure (c) and (d) with N = 1, N = 3, w = 120 nm, d = 600 nm, and t = 0.5w at frequencies of radiation peaks and valleys. The inset of (a) shows the phase distribution of three-slit structure under f₁ illumination. The white dashed lines represent the metal blocks.

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5. Conclusion

We numerically studied the thermal radiation of a self-standing single subwavelength slit in air to identify the underlying physical mechanism of radiation generation. Simulations using the FDTD method revealed that there are one or two strong radiation peaks in the near infrared frequency regime. These are the result of SPPs resonant and F-P interference in the slit. A structure with a radiation peak and a smaller FWHM value was chosen as unit cell to construct a multi-slit structure. Regardless whether the unit cells are identical or not, the thermal radiation efficiency of the multi-slit structure can improve considerably only if the intermediate layers are ultra-thin, where the interaction with the SPPs occurs. By studying the magnetic field patterns and phase distributions, we found that phase matching is the main physical mechanism determining the radiation output of a multi-slit structure. SPPs based thermal radiation helps understand and improve near infrared thermal radiation sources.