1. Introduction

Conventional thermal light emitting sources, such as the sun or the filament of an incandescent light bulb, typically generate broadband and incoherent electromagnetic radiations [1]. Yet in various applications of light sources, including sensors, spectroscopy and energy conversion [2,3], it is essential to have a thermal light source with a narrowband thermal emission. Therefore, in recent years, there have been extensive works aiming to achieve narrowband thermal emission using a wide variety of nanophotonic structures [4–19]. In addition, it is desirable to control both the frequency and the linewidth of the narrowband emission while maintaining high peak emissivity. For example, in solar thermophotovoltaics [3,20–24], the emission wavelength needs to matches the band gap of the photovoltaic cell. And moreover there is an optimal emission bandwidth in order to achieve maximum power output [22].

In addition to the need for thermal emission control and energy conversion applications, there is a challenge to design thermal emitters that maintain structural stability at elevated temperatures. From basic thermodynamic efficiency considerations, the energy conversion efficiency improves as the temperature of the thermal source increases. For solar thermophotovoltaics applications, the optimal temperature for the emitter may exceed 2000K [22]. Since tungsten has a high melting point, there have been significant efforts in exploring tungsten nanostructures for the control of thermal emissivity [3,5,7,11,15,16,22,24–28]. Unlike bulk tungsten however, it has been experimentally observed that tungsten nanostructure is far less stable at elevated temperature [25]. There is therefore a need to develop a mechanism to control the thermal emissivity of a flat tungsten surface without nanostructuring the tungsten.

In this paper, using the rigorous coupled wave analysis [29], we numerically demonstrate the control of thermal emissivity from a flat tungsten surface, by critical coupling to a dielectric photonic crystal guided resonance in the near field. We show that narrowband thermal emission with peak emissivity approaching unity can be achieved in these structures. Both the bandwidth and the center frequency of the thermal emission can be controlled by designing the structure. Importantly, in this design, the photonic crystal, which provides the capability for spectral control, can in principle be completely lossless and need not be maintained at high temperature. The tungsten region, which is the source of thermal emission and therefore needs to be at high temperature, is flat and therefore should not suffer from the thermal instability issues of tungsten nanostructures. While it has been previously shown that narrowband emission from a flat tungsten surface can be achieved by placing a one-dimensional crystal on top of it [26, 30], our design here possesses a much wider tunability in its emission bandwidth.

The paper is organized as follows: In Section 2, we briefly summarize previous methods on achieving narrowband thermal emission and point out the advantages of our design. In Section 3, we discuss the simulation results for our design using a one-dimensional dielectric photonic crystal slab in the vicinity of the flat tungsten surface. We first briefly review the physics of guided resonances. We then show how to tune the resonant frequency and linewidth of thermal emission based on the properties of guided resonances. We present the results for the emissivity spectra of two-dimensional photonic crystal slab coupled to tungsten surface in Section 4 and conclude in Section 5.

2. The concept of critical coupling and its applications for achieving narrow-band thermal emission

To achieve narrowband thermal emission, it is natural to consider structures that support resonances. From coupled mode theory [31, 32], for a single-mode resonator coupled to a single input/output port, its spectral absorption coefficient A is

Based on the concept of critical coupling, we now briefly review several existing approaches as well as our proposed approach in this paper for achieving narrowband thermal emission from tungsten structures. In all the structures shown in Fig. 1, one assume a tungsten region that is sufficiently thick, such that a resonance created near the top surface can only emit to the half space on top of the tungsten structure. And moreover, the structures are either uniform (Fig. 1(d)), or have a structural period that is sub-wavelength at the peak emission wavelength (Fig. 1(a) and (c)), such that no diffraction occurs. In these structures, for a normally incident light the reflected wave can only occur in the normal direction, the resonance therefore couples only to a single input/output port. The emission property of all these structures to the normal direction therefore can all be understood in terms of a single resonance coupling to an input/output port. For these structures then, peak unity emissivity is achieved by satisfying the critical coupling condition.

 

Fig. 1 (a) Schematic of the emitter structure consisting of a photonic crystal slab with one-dimensional periodicity separated from a flat tungsten surface by a vacuum gap. (b) Thermal emission spectra from the structure shown in Fig. 1(a) with tunable linewidth. Here we vary the geometric parameters of the photonic crystal slab and then tune the vacuum gap size to achieve critical coupling. The structured parameters for each of the curves are as follows: yellow curve: d=0.06μm, g=0.12μm; red curve: d=0.04μm, g=0.16μm; blue curve: d=0.02μm, g=0.23μm. Structures with smaller gap size has higher intrinsic losses, leading to larger linewidths. (c) Thermal emission spectrum from a tungsten photonic crystal slab as shown in the inset. The structure has a period of 2μm and a grating depth of 3μm. (d) Thermal emission spectra from structures consisting of multiple layers of dielectric films placed on top of a flat tungsten surface separated by a vacuum gap. We use two sets of dielectrics to form the top mirror. The first set consists of high index layers with n_H = 3.34, low index layer with n_L = 1.45, with 3 layers in total (HLH), and the second set consists of high index layers with n_H = 3.57, low index layers with n_L = 2.90, with 5 layers in total (HLHLH). The emissivity spectra and hence the emission linewidth of the two structures are almost the same.

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Most existing approaches for designing tungsten thermal emitter can be broadly categorized into two categories. In the first category, one patterns either a one-dimensionally or two-dimensionally periodic structure on a tungsten surface [7,23,27], forming a tungsten photonic crystal slab. The structure shown in Fig. 1(c) represents an example with a one-dimensional periodic structure introduced into tungsten. The tungsten photonic crystal slab has a period of 2μm and grating depth of 3μm. This structure is designed to achieve unity emissivity at 0.6eV. In general, in these structures, the peak frequency of the resonance can be controlled by changing the period. However, the achievable linewidth is relatively broad. The resonant modes in these structures are localized in the tungsten region as shown in Fig. 2(b), therefore the intrinsic loss rate is almost entirely controlled by the properties of tungsten with relatively little tunablity. Also, since the loss of tungsten is substantial, the linewidth of the resonance is also relatively broad. In addition, such patterned structures are not as stable as their bulk counterparts at high temperatures. It was experimentally shown that tungsten-coated colloidal crystal nanostructures were completely destroyed at 1500K [25]. Thus protective coatings are often required to enhance the thermal stability.

 

Fig. 2 Distributions of the electric field intensity at critical coupling of (a) structure shown in Fig. 1(a) with a gap size of 0.16μm, (b) tungsten photonic crystal slab in Fig. 1(c), and (c) one-dimensional multilayer structure in Fig. 1(d). The dashed lines represent the outline of various elements of the structures.

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In the second category, a dielectric multilayer stack is put on top of a flat tungsten surface with a vacuum gap in between [22, 26, 33]. The dielectric multilayer consists of alternating layers of a high index material and a low index material. In our example a quarter-wave layer design is chosen for illustration purposes. This structure is capable of realizing narrowband thermal emission with tunable peak frequency. The linewidth is much narrower compared to the structure shown in Fig. 1(c). This can be understood by examining the corresponding field distributions. For the structure shown in Fig. 1(c), its electric field strongly penetrates into the tungsten photonic crystal slab as shown in Fig. 2(b), whereas the mode in the structure of Fig. 1(d) is a Fabry-Perot mode formed through reflection between the tungsten layer and the multi-layer dielectric mirror as shown in Fig. 2(c). The field is therefore located somewhat away from the tungsten region, resulting in a smaller intrinsic loss rate and a narrower linewidth. Moreover, for such a Fabry-Perot mode, the intrinsic linewidth is controlled by the absorption coefficient for light normally incident from vacuum onto the flat tungsten surface, which is a constant at a given resonant frequency. Therefore, at critical coupling, the linewidth of the lowest-order Fabry-Perot resonance at a given frequency is completely fixed, as explained by Eq. (1). As an illustration, in Fig. 1(d), we choose two sets of dielectrics to form the top dielectric multilayer. The first set consists of high index layers with n_H = 3.34, low index layer with n_L = 1.45, with 3 layers in total (HLH), and the second set consists of high index layers with n_H = 3.57, low index layers with n_L = 2.90, with 5 layers in total (HLHLH). Despite the differences in dielectric constants and number of layers, the emissivity spectra of the two structures are almost the same, and the linewidths are nearly identical, which is consistent with the argument above.

In this work we also use the concept of critical coupling to achieve narrowband thermal emission. The geometry is shown in Fig. 1(a), where we place a silicon carbide (SiC) photonic crystal slab on top of a tungsten slab separated by vacuum gap. A similar geometry of a photonic crystal slab separated from a metal surface by a vacuum gap has been recently used as a mid-infrared thermal emitter [19]. The SiC photonic crystal slab supports guided resonances. Unlike the two previous cases above, the energy density of the resonances is located in the SiC region away from the tungsten surface. Since in the wavelength range of interest SiC is largely transparent, the intrinsic loss rate of the guided resonances arises from the overlap of the evanescent tail of these resonances with the tungsten region and thus is very sensitive to the size of the vacuum gap. At a given vacuum gap size, which largely fixes the intrinsic loss rate of the resonance, one can then achieve critical coupling by adjusting the geometric parameters of the photonic crystal slab which control the external loss rate. Alternatively, one can also vary the vacuum gap size to achieve critical coupling for given geometric parameters of the photonic crystal slab. As a result, unlike previous methods, here at critical coupling we can achieve a much wider range of resonance linewidth, including resonances with a much narrower linewidth, as compared to previous methods. Also, similar to the structure shown in Fig. 1(d), in our case the tungsten itself needs not to be patterned, and the structure that is patterned, i.e. the SiC region, needs not to be heated. As a result the structure should be more thermally stable as compared to the tungsten photonic crystal slab shown in Fig. 1(c).

3. Critical coupling with one-dimensional photonic crystal slabs

In this Section we provide an in-depth discussion of the thermal emission properties of the structure shown in Fig. 1(a). In Section 3.1, we first review the properties of guided resonance of the SiC photonic crystal slab. In Section 3.2, we then utilize the properties of such photonic crystal slab in the vicinity of a tungsten slab to enable critical coupling. Finally, in Sections 3.3 and 3.4, we show that the critical coupling can be maintained in this structure as we vary either the resonance linewidth or the resonant frequencies.

3.1. A brief review of guided resonances

We first examine the modal properties of a uniform dielectric slab, to which the spectral features of guided resonances of photonic crystal slabs are closely related [34]. In Fig. 3(a), we plot the four lowest order modes, TE₀, TM₀, TE₁, and TM₁ of a SiC slab with a thickness of 0.4μm. In the modal calculation, the dielectric constant of SiC is fixed to be 6.6, while in all the absorption calculations, we have considered both the dispersion and loss of SiC, with dielectric constant data taken from Ref. [35].

 

Fig. 3 (a) Optical modes of a uniform SiC slab with a thickness of 0.4μm. (b) Transmission spectrum of a SiC photonic crystal slab with a period of 1μm and a grating depth of 0.04μm. The spectrum has its lowest order resonance located around 0.6eV.

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Periodic patterning of the slab breaks the continuous translational symmetry, and as a result some of the guided modes can couple to free space, resulting in the creation of guided resonances. For applications such as thermophotovoltaics, one typically would like to achieve narrowband emission at the band gap of the photovoltaic cell, while suppress emission at all sub-bandgap frequencies. Therefore, in our design, we focus on the lowest order guided resonance. To design a photonic crystal slab that has the frequency of its lowest order guided resonance located at ω₀, we examine the lowest order band of the underlying uniform structure and identify the wavevector k associated with the mode of frequency ω₀ in this band. We then choose the period of photonic crystal slab to be approximately 2π/k. This enables the phase matching of the guided mode with normally incident plane wave. Moreover this period, by construction, is smaller than the free space wavelength, ensuring that the guided mode is phase matched only to the zeroth diffraction order, which is important in order to achieve critical coupling with such guided resonances.

For the uniform slab considered here, the lowest order mode is the TE₀ mode (Fig. 3). To place the lowest order guided resonance at the target frequency of 0.6eV, we choose a period of 1μm following the procedure outlined above. Such a guided resonance couples to s-polarized wave, which has electric field along the y-direction in Fig. 1(a), and as a result the incident wave is chosen to be s-polarized for one-dimensional photonic crystal slabs. The presence of a guided resonance in a photonic crystal slab manifests in the transmission spectrum as a Fano line shape superimposed on an otherwise smooth background. In Fig. 3(b), we show the transmission spectrum of a photonic crystal slab with a period of 1μm and a grating depth of 0.04μm. We do observe a Fano resonance at the target frequency, and no resonances below it.

3.2. Critical coupling

We next place the SiC photonic crystal slab as discussed above, with a period of 1μm and a grating depth of 0.04μm, on top of a flat tungsten surface separated by a vacuum gap (Fig. 1(a)). We find that the critical coupling into the lowest order guided resonance is achieved with a gap size of 0.16μm. In Fig. 4(a), we show the absorption spectrum of the tungsten region in this structure at critical coupling. We find a distinct peak with near unity peak absorption at the target energy of 0.6eV, and no resonances existing at lower frequencies. In Fig. 4(a), we also compare the spectrum with the SiC photonic crystal slab to that of a flat tungsten surface. Away from the two main resonances near 0.6eV, the structure with photonic crystal slab has lower absorptivity as compared to flat tungsten surface, in the frequency range below 0.55eV, and almost in the entire frequency range from 0.8eV to 1.4eV which is above the main resonance. Such a suppression of emissivity away from the resonance is advantageous from efficiency considerations in thermophotovoltaic systems. In Fig. 2(a), we plot the electric field intensity when the structure is excited by an incident plane wave with an energy of 0.6eV. We observe the excitation of the guided resonance as evident from strong field concentration inside the SiC regions.

 

Fig. 4 (a) Narrowband thermal emission peaking at 0.6eV. Parameters: period a = 1.00μm, grating depth d = 0.04μm, vacuum gap size g = 0.16μm. (b) The system goes from under coupling (yellow curve, g = 0.10μm), through critical coupling (blue curve, g = 0.16μm), to over coupling (red curve, g = 0.22μm) as the gap size increases. (c) Absorption spectrum as a function of photon energy and angle of incidence for s-polarized wave, where the structure has same parameters as subplot (a). (d) Angle averaged absorptivity as a function of photon energy for structure in subplot (c).

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To achieve critical coupling, the key control parameter is the gap size. In Fig. 4(b), we show the absorption spectra for the structures that are the same as the one shown in Fig. 4(a), except that the gap sizes take the values of 0.10μm, 0.16μm and 0.22μm. As the gap sizes increases, the spectra show the transition from under coupling, through critical coupling, to the over coupling regime.

In Fig. 4(c), we plot the absorptivity for the s-polarization as a function of photon energy and angle of incidence. We see that the peak of absorptivity can shift significantly as the incident angle varies. In particular, the resonance at 0.6eV splits into two modes as the angle of incidence increases. Also, as angle of incidence increases, some of these modes will deviate from the critical coupling condition. In Fig. 4(d), we plot the angle averaged absorptivity spectrum as a function of photon energy. The average is defined as $\overline{\alpha }=\frac{2}{\pi }{\int }_0^{\pi /2}\alpha (\theta )\text{cos}\theta \mathbf{d}\theta$, where θ is the polar angle. We see that, in spite of the strong angular dependence of the resonances, the angle averaged absorptivity still exhibits a spectral peak. Further improvement of angle selectivity can be accomplished, for example, by following the method as discussed in Ref. [16].

3.3. Control of linewidth

A key advantage of our design is the capability to achieve a wide variety of emission linewidth while maintaining critical coupling. In this system, the intrinsic loss rate of a guided resonance is strongly influenced by the gap size, whereas the external loss rate is largely controlled by the grating depth. Therefore, to vary the emission linewidth, we can vary the grating depth and then adjust the gap size to maintain the critical coupling condition. In this Section, when varying the geometric parameters, we fix the grating width (w) to be half of the period, and the effective thickness of the slab (b + d/2) to be 0.4μm.

As an illustration, in Fig. 1(b), we vary the grating depth d at 0.02μm, 0.04μm and 0.06μm, the gap sizes at critical coupling are 0.23μm, 0.16μm and 0.12μm, respectively. All three structures achieve critical couping at the photon energy of 0.6eV. The linewidth of the resonance gets larger as the grating depth increases. Alternatively, we can also choose a particular gap size, and then adjust the grating depth to achieve critical coupling. In Fig. 5(a), we show the linewidth at gap sizes of 0.05μm (blue curve), 0.10μm (red curve), 0.20μm (yellow curve), where the grating depths at critical coupling are 0.147μm, 0.073μm and 0.027μm, the periods are 1.048μm, 1.013μm and 0.991μm, respectively. The linewidth of the resonance gets smaller as the vacuum gap size increases.

 

Fig. 5 (a) Tuning of linewidth by varying vacuum gap size. Here we vary the vacuum gap size and then adjust the geometric parameters of the photonic crystal slab to achieve critical coupling. The structured parameters for each of the curves are as follows: yellow curve: g = 0.20μm, d = 0.027μm, a = 0.991μm; red curve: g = 0.10μm, d = 0.073μm, a = 1.013μm; blue curve: g = 0.05μm, d = 0.147μm, a = 1.048μm. (b) Tuning of peak position by varying the period. The structured parameters for each of the curves are as follows: red curve: a = 0.8μm, g = 0.18μm, d = 0.02μm; blue curve: a = 1.0μm, g = 0.16μm, d = 0.04μm; yellow curve: a = 1.5μm, g = 0.23μm, d = 0.24μm.

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The examples above demonstrate the ability to design a wide range of resonance linewidth while maintaining critical coupling in our system. In the two examples as illustrated in Fig. 1(b) and Fig. 5(a), we are able to straightforwardly vary the linewidth by more than a factor of five. Here we comment on the range of emission linewidth that can be achieved in our structure. In principle there is no lower limit of the external loss rate because an unpatterned slab has an external loss rate of exactly zero. Also, assuming the photonic crystal slab itself is lossless, and the loss arises only from the tungsten, there is then no lower limit of the intrinsic loss rate, either, since we can always increase the vacuum gap between the photonic crystal slab and tungsten. Thus theoretically we can achieve resonances with arbitrarily small linewidth while maintaining critical coupling. On the other hand, there is an upper limit for the linewidth that can be achieved. Fundamentally, the upper limit arises from the absorption coefficient of tungsten at the target photon energy. And moreover, in our planar geometry, there is a large impedance mismatch between vacuum and tungsten which limits the field penetration into the tungsten region. In our numerical simulation, the largest full width at half maximum for the resonance is approximately 0.02eV. This is comparable to the linewidth in the multilayer structure shown in Fig. 1 since both have a flat tungsten surface and is significantly smaller than the tungsten photonic crystal slab shown in Fig. 1(c), where the fields penetrate significantly into the tungten region as shown in Fig. 2(b).

3.4. Control of peak frequency

The resonant frequency can also be varied over a broad range by varying the period of the photonic crystal slab. As the period a increases, the wavevector of the lowest order guided mode that the normal incident wave can couple to, which is 2π/a, decreases. As a result, the resonant frequency decreases as well, as seen from the modal dispersion curves shown in Fig. 3(a). This redshift of resonant frequency affects the intrinsic loss rate, since both the material loss of tungsten, and the penetration of the evanescent waves into the tungsten region, vary as a function of frequency. Therefore, in order to maintain critical coupling, the external loss rate should be designed correspondingly to match the change of intrinsic loss rate.

We show an example for such design in Fig. 5(b), where we vary the period of the photonic crystal slab from 0.8μm (red curve), through 1.0μm (blue curve), to 1.5μm (yellow curve). For each period, we vary the grating depth as well as the vacuum gap size until the critical coupling is reached. As a result of such a tuning process, we are able to achieve critical coupling at the wavelengths of 0.71eV, 0.60eV, and 0.46eV, respectively. This examples indicates that our structure can be designed to exhibit narrowband emission with unity-emissivity peak located at a wide range of energies. Such a capability of wavelength tunability is potentially important for the design of narrowband thermal sources.

4. Critical coupling with two-dimensional photonic crystal slabs

In previous sections, we have illustrated the concepts by focusing on a class of thermal emitters where the photonic crystal structure has one-dimensional periodicity. Such a system has strong polarization dependency. On the other hand, one can apply the same concept to create thermal emitters that achieve narrowband emission with unity emissivity for both polarizations at normal emission direction. For this purpose, we instead consider a photonic crystal slab consisting of a square array of air holes introduced into the SiC slab, as illustrated in Fig. 6(a) where we show four unit cells. At normal incidence the two polarizations are degenerate by symmetry. The slab is again placed on top of a flat tungsten surface separated by a vacuum gap.

 

Fig. 6 (a) Schematic of the emitter structure consisting of a photonic crystal slab with two-dimensional periodicity separated from a flat tungsten surface by a vacuum gap. (b) Narrowband thermal emission with unity peak emissivity by critical coupling to the lowest order TE-like guided resonance located at 0.6eV. The thickness of the SiC slab is 0.45μm, the period is 1.08μm, the radius of vacuum hole is 0.18μm, the vacuum gap size is 0.07μm. (c) Narrowband thermal emission with unity peak emissivity by critical coupling to the lowest order TM-like resonance located at 0.6eV. The thickness of the SiC slab is 0.58μm, the period is 1.03μm, the radius of vacuum hole is 0.10μm, the vacuum gap size is 0.13μm.

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In general, such a photonic crystal slab supports both TE-like and TM-like guided resonances. These guided resonances correspond to TE and TM modes in the dielectric slab waveguides. The TE and TM-like resonances have mostly in-plane electric or magnetic field within the slabs, respectively [34]. Both of these resonances can couple to normally incident plane wave. In addition, the frequencies of these resonances are typically close to each other, since the effective index for the TE and TM modes of the same order in the dielectric slab waveguides are relatively close to each other for modes that are strongly confined within the slab. Therefore, for the lowest energy emission peak one may expect to see two resonances adjacent to each other. On the other hand, the quality factors of these resonances are in general quite different, and therefore by controlling the intrinsic loss rates of the resonance one can design the system such one of the resonances reaches critical coupling and hence has unity emissivity at its peak, whereas the other resonance does not satisfy the critical coupling condition and hence has a suppressed emissivity.

As an illustration, we consider two examples shown in Fig. 6. In Fig. 6(b), the thickness of the SiC slab is 0.45μm, the period is 1.08μm, the radius of vacuum hole is 0.18μm, the vacuum gap size is 0.07μm. In this structure, the normal incident wave is critically coupled to the lowest-order TE-like resonance, which results in a unity emissivity peak at 0.6eV, whereas the lowest-order TM-like resonance located at 0.637eV is under coupled and hence has a strongly suppressed thermal emissivity. In Fig. 6(c), the thickness of the SiC slab is 0.58μm, the period is 1.03μm, the radius of vacuum hole is 0.10μm, the vacuum gap size is 0.13μm. Here the lowest order TM-like resonance satisfies the critical coupling condition, resulting in an unity emissivity peak centered at 0.6eV, whereas the lowest-order TE-like resonance, located at 0.57eV, is over coupled with suppressed emissivity. These examples show that one can achieve critical coupling with two-dimensional photonic crystal slab system, and moreover generate a single emissivity peak that has its energy well separated from all other emissivity peaks in the system.

In general, a two-dimensional lattice has a more extensive set of reciprocal lattice vectors. And consequently there are more resonant peaks in the emissivity spectrum, as we indeed observe when we compare the plots shown in Fig. 6 with any of the spectra that we show for the one-dimensional case.

5. Conclusion

In summary, we show that narrowband thermal emission can be achieved, by placing a photonic crystal slab on top of a flat tungsten surface separated by a vacuum gap, and by designing the parameters of the structure such that the lowest order guided resonance reaches critical coupling. This method achieves strong modifications of the spectral emissivity without the need to create tungsten nanostructures, which is expected to significantly improve the structure stability of tungsten based thermal emitters. This class of structures also offer wide tunablity range for both the linewidth and the position of the emissivity peak while maintaining unity emissivity. Such a capability is of importance for thermal source design in general. While for illustration purposes we have focused on creating a narrowband tungsten emitter that emits strongly near an energy of 0.6eV, the same principle can be applied for other wavelength ranges and other material systems as well. Finally, the aspect of near-field coupling between a thermal emitter and a dielectric structure has been previously used to demonstrate a thermal exaction scheme [36]. Therefore, an interesting future opportunity is to combine the concept here with the thermal exaction scheme to achieve narrowband thermal emission with power beyond the apparent blackbody limit.