Nearly perfect resonant absorption and coherent thermal emission by hBN-based photonic crystals

In this paper, we numerically demonstrate mid-IR nearly perfect resonant absorption and coherent thermal emission for both polarizations and wide angular region using multilayer designs of unpatterned films of hexagonal boron nitride (hBN). In these optimized structures, the films of hBN are transferred onto a Ge spacer layer on top of a onedimensional photonic crystal (1D PC) composed of alternating layers of KBr and Ge. According to the perfect agreements between our analytical and numerical results, we discover that the mentioned optical characteristic of the hBN-based 1D PCs is due to a strong coupling between localized photonic modes supported by the PC and the phononic modes of hBN films. These coupled modes are referred as Tamm phonons. Moreover, our findings prove that the resonant absorptions can be redor blue-shifted by changing the thickness of hBN and the spacer layer. The obtained results in this paper are beneficial for designing coherent thermal sources, light absorbers, and sensors operating within 6.2 μm to 7.3 μm in a wide angular range and both polarizations. The planar and lithography free nature of this multilayer design is a prominent factor that makes it a large scale compatible design. © 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes: (010.1030) Absorption; (240.6690) Surface waves; (050.5298) Photonic crystals. References and links 1. R. Carminati and J. J. Greffet, “Near-field effects in spatial coherence of thermal sources,” Phys. Rev. Lett. 82, 1660 (1999). 2. A. Joulain, J. P. Mulet, F. Marquier, R. Carminati, and J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57, 59 (2005). 3. J. J. Greffet, R. Carminati, K. Joulain, J. P. Mulet, S. Mainguy, and Y. 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Introduction
Thermal emission originates from the totally uncorrelated process of spontaneous radiation inside matter in local thermal equilibrium.In spite of the randomness involved in the mechanism of emission, it has been shown that some thermal sources could present high spatial coherence in the near field [1,2].This spatial coherence can be used to design highly directional sources with angular divergence comparable to lasers at the same wavelength [3][4][5] or to strongly enhance or inhibit thermal radiation [6,7].Thermal emitters with modified spectral and directional emission properties have numerous potential applications such as thermophotovoltaic devices [7][8][9][10][11][12][13], infrared imaging and detection [14], sensing [15,16], radiative heat transfer [17], antennas [18], and radiation cooling by enhancing the emissivity of a device [19].Therefore, the investigation of mechanisms behind the spectral and directional control of thermal radiation will help us to design feasible devices suitable for the above-mentioned applications.One of the promising ways to achieve control over the direction and spectral properties of thermal emission is to employ a surface structure supporting electromagnetic eigenmodes [3,5,[20][21][22].To date, various coherent thermal sources have been studied.Researchers have made narrow-band thermal emitters from metallic photonic crystals [23,24], silicon photonic crystals coupled with quantum wells [25,26], arrays of deep metal grooves [27], and micropatches [12].Moreover, lithography free epsilon-near-zero metamaterials have been considered for controlling thermal emission at the near-IR region [28][29][30].Guided modes [31] and coupled resonant cavities [19] have also been proposed to generate spectral-directional coherence.Thermal excitation of delocalized modes is one notable way to obtain the coherence.In this case, electrons or phonons are no longer uncorrelated but semi-coherently oscillating along the decay length of the mode propagation.Therefore, their emitted fields are spatially coherent to a degree and, if properly coupled to the far field, can produce constructive interference along a certain direction in space.One practical way to achieve this is to use grating structures.There have been numerous studies about coupling surface waves to free space waves using one-dimensional (1D) [3][4][5][32][33][34][35][36][37] and two-dimensional (2D) gratings [22,38] and metasurfaces [39,40].To date, the articles mostly deal with the thermal emission of gratings made of polar materials and metals.In polar materials (e.g.SiC for mid-IR applications), the enabling mechanism is the coupling between the optical phonons and light.These surface phonon-polaritons exist only in the Reststrahlen band (RS band) and their emission wavelength can be tuned over that range [3,17,18,22,[35][36][37][38].It has been reported that combining 1D [37] and 2D [38] SiC-based gratings with 1D photonic crystals (1D PCs) can considerably improve the thermal radiative properties of the microstructures in mid-IR region.PCs are structures with dielectric constant periodically varying in space [41].In [37], it was proven that significantly enhanced thermal emission can be achieved in a broad spectral band due to the excitation of surface photon polaritons, PC modes, magnetic polaritons [35], and the coupling between them.Similar mechanisms play the role of enhancing the thermal emission of the 2D grating that is mediated with 1D PC [38].However, due to the 2D patterning of that structure, the enhanced emissivity was achieved for both transverse electric (TE) and transverse magnetic (TM) polarizations.On the other hand, from the fabrication point of view, the use of lithography free designs is a noticeable advantage over the elaborately patterned structures [42,43].Yeh et al. [44] proved that a 1D PC is capable of supporting surface waves (SWs) for both TE and TM polarizations.Surface waves are electromagnetic modes that can propagate along interfaces over long distances and are confined in the normal direction.Therefore, inspired by that point, Lee et al. [45,46] demonstrated that a coherent thermal source can be designed using an unpatterned film of SiC that is coated on a 1D PC.They showed that the excitation of surface waves at the interface of the coated layer and the photonic crystal results in highly spectral and directional thermal emission in the mid-IR for both TE and TM waves [45,46].Moreover, they explained that the excitation of the cavity resonance and Brewster modes can lead to the enhancement of the emission for the modes supported outside of the RS band of SiC [47].It is noteworthy that similar designs have been also used for the optimization of surface plasmon polaritons supported by graphene in terahertz frequencies [48] and directing fluorescence of Dye emitters in the visible region using Tamm plasmons supported by a thin film of metal on top of a 1D PC with a dielectric spacer [49,50].Moreover, it has been theoretically verified and experimentally proved that using metallic films on top of truncated 1D PCs [51,52] and also Fabrey-Perot cavities [53] sharp spectral peaks and narrow angular lobes can be onserved in the emissivity spectrum of the system.It should be noticed that in the metal-coated 1D PCs the electric fields for the Tamm plasmons are localized in the dielectric below the metal film.Unlike surface plasmons, Tamm plasmons can be created for both polarizations and the in plane wavevector can be zero.Moreover, the wavevector of Tamm plasmons is smaller than that of light in vacuum, so it can be directly excited from air without the aid of prisms or gratings [49][50][51][52].
Hexagonal boron nitride (hBN) is a natural hyperbolic material, for which the dielectric constants are the same in the basal plane ( t the normal one ( z ε ) in the mid-IR region [54][55][56][57][58][59].Owing to this property, finite-thickness slabs of hBN are capable of supporting sub-diffractional volume-confined polaritons [54,55] and can act as multimode waveguides for the propagation of hyperbolic phonon polariton (HPP) collective modes for TM polarization that originate from the coupling between photons and electric dipoles in phonons [55][56][57][58][59].It has been recently reported that, due to the support of HPPs, 1D grating of hBN is capable of supporting highly concentrated resonant absorption, and thereby thermal emission for TM polarization [60].Moreover, because of the coupling of magnetic polaritons in metal gratings with HPPs of hBN in TM polarization, perfect or nearperfect absorption can also be observed by an unpatterned film of hBN on top of a metallic grating [61].
In this paper, we design nearly perfect resonant absorbers that can act as coherent thermal emitter sources in the mid-IR region operating for both TE and TM polarizations within a wide angular region.The designs are made up of unpatterned films of hBN transferred onto a 1D PC composed of alternating layers of KBr and Ge with a Ge spacer later separating the 1D PC and the hBN films.First, we analytically demonstrate that, by the appropriate design of the photonic crystal structure, it is possible to couple phononic modes of the hBN films with photonic modes supported by the 1D PC; i.e. the support of Tamm phonons.Then, in perfect agreement with the analytical results, it is numerically verified by the FDTD method that, the mentioned coupling leads to widely angled nearly perfect resonant absorption and consequently the coherent emission of thermal radiation for both TE and TM polarizations.The proposed designs are made of planar lithography-free layers where both Ge and KBr layers can be deposited by thermal evaporation and hBN films can be transferred on top of the fabricated structure.To the best of our knowledge, this is the first analytical-numerical study of resonant absorption in hBN-based 1D PCs for both polarizations.

Math and equations
By applying the appropriate boundary conditions for TM polarization [62], the dispersion relation of HPPs supported by the film of hBN can be formulated as [54][55][56][57][58][59] ' h,TM ' tan h(q t) . 1 Notice that m ε becomes negative between / / However, for TE polarization, hBN behaves as an isotropic medium and the spatial dispersion of the supported modes becomes spherical that can be described as 2 It is well-known that the allowed and forbidden frequency regions (band gaps) of a 1D PC can be given by [58] Where , / where θ is the angle of incident light.As mentioned hereinabove, a 1D PC is capable of supporting surface waves for both TE and TM polarizations [44].These modes are supported in the bandgap of the system.Therefore, by appropriately designing the photonic crystal, i.e. matching the RS-II band of hBN and bandgaps of the 1D PC, it is possible to couple phonon polaritons of hBN and SWs of the 1D PC in the hBN-based 1D PC.This structure is schematically illustrated in Fig. 1(b); we call these coupled modes Tamm phonons.Considering y H as , , and applying the appropriate boundary conditions for TM polarization at different interfaces, the dispersion of the Tamm phonons supported by the 1D hBN-based PC can be obtained using the direct-matching procedure which is applicable for electronic and photonic layered devices [48,63] h,TM tan h(q t) . 1 (1 e X ) / ( 1 e X ) / Similar to plasmonic materials [64], phononic materials can support surface phononpholaritons only for TM polarization [65].However, symmetrically patterned structures, i.e. 2D grating phononic/plasmonic systems, are capable of supporting plasmonic/phononic resonances [22] for both TE and TM polarization.Moreover, using Tamm plasmons/phonons of plasmonic-/phononic-based 1D PCs, it is possible to excite plasmonic [49][50][51][52] or phononic [45][46][47] resonances in unpatterned films.Here, in order to investigate phononic-photonic resonance absorptions for TE polarization in the structure illustrated in Fig. 1(b), we also formulate TE Tamm polaritons supported by the 1D hBN-based PC.To this aim, considering y E as Eq. ( 7) and applying TE boundary conditions [59], we arrive at the following dispersion relation for the TE Tamm modes tan h(q t) .X X X X Here ( 1 e X ) / (1 e X ) and TE γ can be calculated using Eq. ( 7) by replacing TE F with TM F [66] where

Results and discussion
In this section, we illustrate and discuss the obtained results from the above mentioned analytical relations and compare them with the ones that are achieved by FDTD simulations [67].In our calculations, we take .Moreover, for the sake of simplicity, the material of the spacer layer is taken to be Ge in our calculations.It should be mentioned, by taking the appropriate value for the thickness of the spacer layer, similar results can be obtained for a different arrangement of the layers in the photonic crystal for which  ).On the other hand, as mentioned earlier, it is well-known that a 1D PC comprising dielectric layers is capable of supporting surface waves for both TE and TM polarizations [44]; these modes are supported in the bandgap of the PC.By properly choosing the thickness of Ge and KBr layers, it is possible to match the RS-II band of hBN and photonic bandgap of the 1D PC.Therefore, the phonons of hBN and SWs of the 1D PC can be efficiently coupled to create coupled photonic-phononic modes in the structure; i.e.Tamm phonons.As we mentioned earlier, our main aim in this paper is the investigation of light absorption/coherent thermal emission by the hBN-based 1D PC for different angles, which is schematically illustrated in Fig. 1(b).Consequently, in the next step, we examine the projected band structure and the supported Tamm phonons of the hBN-based 1D PC in Fig. 3 for incident angles smaller than 70 degrees that can be meaningfully realized by the ellipsometry measurements. .Due to the interactions between the surface waves supported by the 1D PC and the phonons supported by the covered hBN film, some coupled phononic-photonic modes, Tamm phonons, can be supported by the system for 0 90 o θ ≤ < .Depending on the thickness of the hBN films, these couplings can be efficiently achieved by the proper thickness of the Ge spacer layer in-between the 1D PC and the hBN film.For the dispersion curves illustrated in Fig. 3 [plotted using Eqs.( 6) and ( 8)], those thicknesses of the spacer layers are taken to be 1.08 μm, 1.03 μm, and 900 nm for 300 nm-, 500 nm-, and 1 μm-thick hBN cover layers, respectively.By these choices, the dispersions of Tamm phonons in Fig. 3 are more separated in the bandgap and, therefore, they are more illustratively recognizable.The procedures of the optimization of the spacer layer will be discussed later.Dashed-dotted black ( 300 ), dotted red ( 500  ] of a structure is the same as its spectral-directional absorptivity [ ( , )  A λ θ where 1 ( ) A R T = − + ]; as a couple of examples see [3,30,33,[36][37][38][45][46][47][48][49][50].Therefore, in the following, using FDTD simulations, we numerically illustrate the emissivity results of the mentioned structures that can also be interpreted as the absorption of the structures under our consideration.In addition, it should be noted that we consider 10 Ge-KBr pairs in our numerical analysis.Panels (a) and (b) of Fig. 4 show the calculated emissivity of the hBN-based 1D PC for 300 t nm = and 1.08 As is seen from panels (a) and (b) of Fig. 4, using a 300 nm-thick film of hBN on the 1D PC with Ge spacer of thickness 1.08 μm, it is possible to achieve nearly perfect coherent thermal emission (nearly perfect resonant absorption) for both TM and TE polarizations for angles smaller than 45 o .For 45 o θ > TM thermal emission of the structure decreases while, for TE polarization, the absorption and consequently thermal emission are perfectly kept.In order to get more insight into the mechanism of the nearly perfect absorption/coherent thermal emission of the hBN-based 1D PC, we compare the numerical results illustrated in Figs.4(a) and 4(b) with the analytical Tamm phonon dispersions of the structure that are illustrated by dotted-dashed black curves in Figs.3(a) and 3(b).By this comparison, we understand that the nearly perfect resonant absorption is exactly due to the support of Tamm phonons that are the coupled photonic-phononic modes of the system.While a film of hBN on a homogeneous substrate supports HPPs only for TM polarization, the hBN-based 1D PC can support the Tamm phonons for both TE and TM polarizations, and this coupling leads to the perfect thermal emission for a wide angular range.This characteristic resembles the directing fluorescence and coherent thermal emision by metal-based 1D PCs [49][50][51][52] and SiC-based 1D PCs [45][46][47].In order to get more insight into the coherent response of the system, the angular distribution of thermal emission of the structure at two different wavelengths ( ) thermal emission by the structure at 45 o .As mentioned earlier, the results illustrated in Fig. 4 were obtained for 1.08 This thickness is an optimal value for the thickness of Ge spacer in the hBN-based 1D PC for 300 t nm = .In Fig. 5, we explain how that value was taken as the optimal thickness of the Ge spacer layer.the system by changing the spacer thickness; i.e. by increasing s d the resonant frequency is red-shifted.Consequently, by appropriately choosing the thickness of hBN and the spacer layer, it is possible to achieve the desired resonant absorption within 6.2 μm to 7.3 μm.
In the case for which the thickness of hBN and the spacer layer are taken as 500 nm and 1.03 μm, respectively, the spectral-directional emissivity of the hBN-based 1D PC is represented in Fig. 6.The first point should be noticed about Figs. 6(a) and 6(b) is that by increasing the thickness of hBN from 300 nm to 500 nm, the perfect thermal emission region is shifted to shorter wavelengths.As it is obviously observed from these panels, the numerically calculated angular dispersion of the perfect thermal emission regions for TM and TE polarizations have been perfectly predicted by the analytically obtained dotted red dispersions of Tamm phonons in Fig. 3. Angular distributions of the spectral-directional emissivity of the structure for t = 300 nm and polarizations is more coherent (angular width~33 o ) than the previous case for which 300 t nm = . In order to obtain the optimal value for the thickness of the spacer layer for the case of 500 t nm = , we investigate the thermal emission of the system at 0 θ = ,  From the results shown in Fig. 7, it is seen that 1.025It is observed that, similar to the previous two cases, there is a perfect match between the analytically obtained dispersion of the Tamm phonons shown by dashed blue curves in Fig. 3 and the resonant perfect thermal emission illustrated in panels (a) and (b) of Fig. 9.This match once more recalls this point that the coupling between phononic modes of hBN and confined photonic modes of the 1D PC is responsible for the observation of nearly perfect resonant absorption/thermal emission by the structure.Moreover, Fig. 9(c) shows that it is possible to achieve a coherent thermal emission by this structure at Consequently, for this case, thermal emission by the structure is considerably more coherent than the previous cases for which 500 nm and 300 nm films of hBN were considered in the structure.Like the previous two structures, in order to be able to find the optimal thickness of the Ge spacer layer for  In contradiction with Tamm plasmons for which the electric field is localized in the dielectric below the metal film [49][50][51][52], mode profiles in Fig. 12 illustrate that for Tamm phonons the magnetic field is localized inside the spacer layer below the phononic film.It is seen from the mode profiles that the incident light can excite a Tamm mode that vanishes within five periods inside the structure.As a complementary discussion, it should be reminded that to the present, using isotropic phononic (e.g.SiC) [45][46][47] and plasmonic (e.g.Ag) materials [49][50][51][52], it has been theoretically and experimentally proved that the coherent thermal emission can be obtained by truncated 1D PCs operating in the far-IR and a window within the mid-IR region (10-12 μm).The physics behind all these works are the support of Tamm plasmons/phonons by the Ag-based and SiC-based 1D PCs.In the present study, first, we have explicitly derived dispersion of Tamm phonons supported by hBN as a uniaxially anisotropic material on a truncated 1D PC for both polarizations and verified our analytical results by FDTD simulations.Second, the operating wavelength of the considered structures in this manuscript (6.2 to 7.3 μm) makes the potential application of the hBN-based 1D PCs completely distinguishable from those similar Ag-based and SiC-based systems.Moreover, as mentioned earlier, due to the support of HPPs in the RS-I band of hBN, this material is also capable of light absorption at oblique incidences for TM polarization.By an appropriate design, this absorption can be enhanced up to 70 percent in the hBN-based 1D PCs.Thus, these hybrid systems can operate simultaneously in two separate absorption windows in the mid-IR range.

Conclusion
In conclusion, we have theoretically shown that hBN-based 1D PCs are capable of perfectly absorbing mid-IR light in a wide angular range and can behave as coherent thermal emitters for both polarizations.These optimized structures are made of unpatterned films of hBN on top of a 1D PC that is composed of alternating layers of KBr and Ge with a Ge spacer layer.
Based on analytical calculations, we have investigated coupled photonic-phononic modes supported by the mentioned hBN-based devices; i.e.Tamm phonon modes.In support of the analytical results, our numerical findings have proven that the perfect light absorption is achieved due to the support of Tamm phonons by the structure.We have found that, with optimal designs according to the thickness of hBN films, it is possible to resonantly absorb light within 6.2 μm to 7.3 μm for both polarizations in a wide angular range.However, the device based on 1 μm-thick film of hBN is the best candidate for a nearly perfect coherent thermal emitter.

Figure 1 (and 2 ε and a thickness of 1 d and 2 d
Figure 1(a) schematıcally represents a film of hBN of thickness t on top of a homogeneous Ge substrate with permittivity of s ε .In panel (b) of Fig. 1, the film of hBN is transferred onto a 1D PC that comprises alternating layers of materials with a permittivity of 1 ε and 2 ε and a thickness of 1 d and 2 d .The 1D PC and the film of hBN are separated with a Ge spacer layer of the width s d and a permittivity of 2 s ε ε = .

Fig. 1 .=
Fig. 1.Panels (a) and (b) illustrate a film of hBN with thickness t (placed in 0 z = ) that is transferred onto a Ge substrate and onto a 1D PC, respectively.As shown in panel (b), the hBN-based 1D PC comprises alternating layers of KBr and Ge layers with the thicknesses 1 d makes the inplane and out-of-plane dielectric functions of hBN possess opposite signs in the lower-(upper-) frequency band, that represents type-I (type-II) hyperbolicity[54][55][56][57][58][59].Therefore, in Figs.2(a) and 2(b), these bands are labeled as the RS-I and RS-II regions.Therefore, the spatial dispersion of the modes that can propagate inside a film of hBN with unbounded wavenumbers can be expressed as hyperbolic isofrequency surfaces ε = , which correspond to the permittivity values of KBr and Ge in the mid-IR region, respectively.These materials are lossless and exhibit almost constant refractive index within the range of our interest; i.e. 6.2 μm to 7.3 μm.In order to match the photonic bandgap of the 1D PC to the second RS band of hBN, we choose 1 using Eq.(2), we investigate the dispersion of HPPs supported by the films of hBN with different thicknesses, i.e. 300 t nm = , 500 nm, and 1 μm.The considered structure is schematically illustrated in Fig.1(a).

Fig. 2 .
Fig. 2. (a) and (b) illustrate real and imaginary parts of t ε and z ε of hBN within entire mid-IR region.Panels (c), (d), and (e), respectively, illustrate the dispersion of HPPs supported by 300 nm, 500 nm, and 1 μm films of hBN within the wavelength range of our interest.6.2 μm to 7.3 μm is the RS-II region in which hBN shows hyperbolic response of the second type.Type-I ( 0 z ε < and

Fig. 3 .
Fig. 3. Panels (a) and (b), respectively, show projected band structure of the 1D PC for TM and TE polarizations; the white and aqua regions depict photonic bandgaps and allowed the bands of the 1D PC.In these panels, dashed-dotted black ( 300 , 1.08 s t nmd m μ Figs. 3(a) and 3(b) correspond to the coupled modes supported in the bandgap of the hBN-based 1D PC.As it is clearly observed from these panels, these modes are supported in the RS-II region of hBN and are blue shifted by increasing the thickness of hBN, as it was expected from the results shown in Figs.2(c)-2(e).In panels (c) and (d) of Fig. 3, dispersions of those Tamm modes are magnified for illustrative purposes.

Fig. 4 .
Fig. 4. Spectral-directional emissivity of the hBN-based 1D PC with 300 t n m = and in Figs.4(c)-4(e).As expected from the results shown in panels (a) and (b) of Fig. 4, at 7.07 m λ μ = polar plot of the emissivity shows a wide thermal emission (angular width 70 o ∼ ) by the structure in the normal direction.Similarly, the angular distribution of the emissivity at 7 m 4(d)] and TE [Fig.4(e)] polarizations also show a wide (angular width 37 o ∼

Fig. 5 .
Fig. 5. Thermal emission of the 1D hBN-based PC with 300 t n m = for different thicknesses

Fig. 6 .
Fig. 6.Panels (a) and (b) respectively show TM and TE spectral-directional emissivity of the hBN-based 1D PC with 500 t n m = and panels (c)-(e) of Fig.6.Figure7(a) shows that using the mentioned structure, it is possible to emit thermal energy at 6.872 m λ μ = for normal incidence with angular width of almost 60 o .Moreover, at 6.8 m λ μ = thermal emission of this hBN-based 1D PC for TM [Fig.6(d)] and TE [Fig.6(e)]

Fig. 7 .
Fig. 7. Thermal emission of the 1D hBN-based PC with t 500 nm = for the different

8 .
order to keep thermal emission high enough for both polarizations and all anglesthickness of the Ge spacer layer.The spectraldirectional emissivity of the hBN-based 1D PC for 500 It is observed from this figure that using the mentioned system, nearly perfect resonant absorption and coherent thermal emission can be achieved for angles smaller than 50 o θ < .For 50 o θ > , while TE thermal emission is perfectly kept, TM emission is decreased but it is still more than 70 percent until 60 o θ = .

Fig. 8 .
Fig. 8. TM [panel (a)] and TE [panel (b)] spectral-directional thermal emission for the hBNbased 1D PC with 500 t n m = and the optimal value of thickness of the Ge spacer layer; i.e.

Fig. 9 .
Fig. 9. Spectral-directional emissivity of the hBN-based 1D PC with 1 t m μ = almost 36 o .Angular distribution of the thermal emission of the structure at 6.422 m λ μ = illustrates that coherent thermal emission by the angular width of 20 o [Fig.9(d)] and 15 o [Fig.9(e)] can be obtained by the hBN-based 1D PC for 1 t m μ = .

Fig. 10 .
Fig. 10.Thermal emission of the 1D hBN-based PC with t 1μm = for different thicknesses 1D PC, we investigate the thermal emission of the structure for the different values of s d at those three critical angles for both polarizations.According to panels (a), (b), (d), and (e) of Fig. 10, by taking s d as 0.92 μm, shows the best response for TM polarization at 60 o θ = [see Fig. 10(c)].Consequently, we choose 0.95 s d m μ = for the optimal thickness of the Ge spacer layer in order to get the best response from the hBN-based 1D PC with 1 t m μ = at all angles.TM and TE spectral-directional emissivity for the optimal design are represented in Fig. 11.