1. Introduction

Tailored thermal electromagnetic emission and absorption properties (i. e. spectral and directional characteristics) in the spectrum ranges from THz to IR are considered for the applications in sensing, spectroscopy, energy harvesting, radiative temperature control, and others [1] [2]. Multitude of methods for modifying the electromagnetic radiation profiles rely on surface polaritons (SPs) via employment of either surface plasmon polaritons (SPPs) [3,4] or surface phonon polaritons (SPhPs) [5,6]. Hybridization of these quasiparticles and dispersion characteristics of the surface plasmon-phonon polaritons (SPPhPs) has been found recently in the n-GaN surface relief grating (SRG) of linear shape [7,8]. The excitation of SPPhP resonances were also demonstrated in doped polar semiconductor SRGs [9–11], also by utilizing metallic nanostructures on semiconductor surface [12,13] or by using multilayer 2D materials which support plasmon/phonon oscillations, for example, semiconductor/graphene [14,15], h-BN/metal [16], and h-BN/graphene [17]. The promise of practical SPPhP applications lies in the strong enhancement of the electric field at the interface causing strong light - matter interaction. In the view of practical implementation, the grating couplers are easily processed and used for far field coupling instead of using the prism or another dispersive component. Hybrid nature of SPPhP modes benefit from high spatial and temporal coherences, providing a narrow spectral linewidth, long polariton traveling distance, and directional thermal radiation profile at the resonant frequency selected by the design [8,18].

In general, the angular frequency dispersion of a propagating SP mode allows the observation of directive emission features [19] polarization of which is defined by the linear grating coupler (i.e. electric field of radiated beam is parallel to the grating wavevector). On the other hand, so called angular “rainbow” emission imposes limitation on the experimental setup, requiring the angular field of view (AFV) to be small in order to distinguish a high spectral quality of SPPhP resonant features [8,18]. Directional thermal emission of SPhP and SPP modes has been demonstrated by various groups including polarized emission from 1D gratings on polar semiconductors [18] and on metals [20], with additional polarization control implementing a 2D cross polarized grating on surface of silicon carbide wafers [21]. The unpolarized beaming approach with radial shape, so-called “bulls-eye”, grating structure was proposed theoretically [22] and was realized experimentally [23]. Here SPPs were excited in metallic gratings with a radial symmetry for the beaming of thermal radiation in the direction normal to the sample surface. Results showed radiated beam directivity of 1.4 deg with the spectral linewidth of 55 nm (around 40 cm⁻¹), resulting to quality factor values of Q = ν / FWHM ≈ 55 at the wavelength of 3.6 µm (2778 cm⁻¹) [23]. Differently in this work, we proposed the usage of hybrid SPPhPs in two types of linear and circular shape SRGs of optimal design made of a wide-bandgap heavily doped n-GaN semiconductor providing a medium of high crystalline quality coupled with high mechanical and chemical rigidities and high thermal and electrical conductivities useful for many practical applications.

The intrinsic losses of SPPhP are lower in comparison to that of SPP and SPhP therefore hybrid polaritons are expected to offer an improved thermal radiation coherency and beaming. The linear shape n-GaN SRG with 50% filling factor demonstrated the SPPhP resonances with the linewidth values down to 5 cm⁻¹, resulting in quality factor values up to Q ≈ 114 in the frequency range close to transverse optical (TO) phonon of GaN (∼560 cm⁻¹) in both modelling and experiment employing AFV = 0.57 deg [8]. However the numerical investigations of the dispersion characteristics revealed that the damping of hybrid SPPhP modes in n-GaN should be down to values below 1 cm⁻¹ [8,19]. Therefore, predicted limits of high spectral quality and beaming of hybrid SPPhP radiation from SRG have not been realized yet to the best of our knowledge.

In this work the geometry of n-GaN SRGs were thoroughly optimized in order to minimize the propagation losses and to maximize spatial and temporal radiation quality of the selected SPPhP modes in the direction normal to the sample surface. For this reason, the grating period was fixed at P = 17.5 µm, preserving a condition where the grating wavevector is close to the TO phonon oscillation frequency where the damping of SPPhP modes is expected to be small, i.e., operation frequencies are considered in region of the intersection point of light line with the dispersive characteristic of hybrid SPPhPs. The emission characteristics of SPPhPs were maximized for the linear shape SRGs with various filling factors, with a help of rigorous coupled wave analysis (RCWA) optimizing the linewidth, quality factor, and spatial coherence length of the resonant features. Later, one linear and three radial shape SRG samples of optimal design were fabricated in order to validate the numerical findings and to explore coherent radiation and beaming of hybrid SPPhPs emission in the experiments without the use of focusing optics (i.e. parabolic mirrors). Our results confirm that the thermal beaming and emission in narrow spectrum range are possible to achieve under excitation and constructive interference of the same number but opposite sign of hybrid SPPhP modes.

2. Samples and methods

The samples were fabricated of a 500 µm thick c-plane crystal orientation n-type GaN wafer with the doping of around n = 1.55 × 10¹⁹ cm⁻³. A set of four different SRG samples was fabricated next to each other on the same wafer side using UV lithography and Cl-based reactive ion etching techniques as previously described [7,8]. A photo of the fabricated samples is shown in Fig. 1(a) with the scanning electron microscopy (SEM) photographs of the grating centers shown in Fig. 1(b). Design parameters are summarized in Table 1. The first sample in area of 5 × 5 mm² was designed as an experimental linear grating (ELG) with the filling factor FF = W/P (where W is the ridge width) of 25%, it was labeled as ELG 25. The other three samples were experimental circular gratings (ECG) of the same 5 mm diameter but with different FF values being of 25% 50% and 75%, labeled as ECG 25, ECG 50, and ECG 75, respectively. The relief depth h = 1 µm was optimized for efficient excitation of hybrid SPPhP modes previously in Ref. [7]. The step profile of fabricated samples was measured by atomic force microscopy (WITec Alpha 300). The scans across grating of each sample are shown in Fig. 1(c). Smooth grating profile without a notable fabrication defect was found for all samples. An average value of h = 1020 ± 10 nm revealed good accuracy of fabrication processes. The quality and side wall inclination in fabricated SRG samples were also inspected from (SEM) pictures. Thorough investigation revealed that the ridge width at the top (W_TOP) and the ridge width at the bottom (W_BOT) was wider by a 400 nm and 1000 nm, respectively (see Fig. 1(c)) than the initial size of the ridges specified in the design. Found inclination of walls was included in the SRG models for numerical experiments as explained elsewhere [7,8].

 

Fig. 1. a) a photograph of fabricated samples located in four zones. b) SEM images at the center area of the respective samples. Note: that all radial shape SRG samples are designed to have a valley in the center. c) Step profiles across the gratings of the fabricated samples.

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Table 1. Notations and geometries of different experimental and theoretical structures.

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Numerical computation was used to optimize optical characteristics of SRG samples composed of infinitely periodic straight lines (1D grating) employing RCWA method. This technique allows for the evaluation of transmissivity and reflectivity coefficients with the absorptivity (equal to emissivity) according to Kirchhoff’s law of thermal radiation as A = E = 1-R-T. We assumed T = 0 in the whole spectral range of interest due to the large doping of the n-GaN material. The modelled structure was described as two-dimensional problem composed of three regions of 100 µm thick vacuum, 1.0 µm thick n-GaN grating, and a 500 µm thick n-GaN substrate layer. The dielectric function of the semiconductor was described by the equation encompassing the contribution of lattice atom and free-electron vibrations:

Here the $\nu_{LO}$, $\nu_{TO}$, $_p$ and $\nu_{LO}$, $\nu_{TO}$, $_p$ correspond to the oscillation frequencies and damping factors of the longitudinal and transverse optical phonons and plasmons, respectively. The frequency of plasmon oscillation is given by $\nu_p\; = 1/2\pi \sqrt {{e^2}{n_0}/{m^\ast }{\epsilon _\infty }} $, where ${n_0}$ is the free electron density, value of which is 1.55E19 cm⁻³, ${m^\ast }$ = 0.22 is the electron effective mass, and ${\epsilon _\infty }$ = 5.3 is the high-frequency dielectric permittivity of GaN. The values of all oscillator parameters were found by fitting experimental optical spectra of unprocessed semiconductor wafer as described elsewhere [8].

The TM and TE polarization spectra were calculated, and the averaged spectrum of both was used to describe unpolarized beam characteristics of SRG samples. It is worth noting at this point that a linear shape grating was used as an approximation of the radially symmetric one as they both were assumed to produce a similar shape spectrum of unpolarized thermal radiation. Further investigation is needed to describe different effects caused by radial geometry, for example: focusing of hybrid polaritons in the center and polariton reflection at the outer perimeter of the circular grating. The rigorous theoretical treatment of the difference between the properties of circular and linear geometries requires separate investigation which was out of the scope of this work.

The emission spectra were measured with a Fourier transform infrared (FTIR) spectrometer (Thermo Scientific Nicolet 8700) in a MID-IR configuration equipped with a KBr beam-splitter and a room temperature DLaTGS pyroelectric detector and a KRS-5 holographic wire-grid polarizer.

The samples were mounted on a rotation-translation stage, providing a controlled heating up to 500 C. Two setups for characterization of SRG samples were developed. Setup I (see Fig. 2(a)) was used for the investigation of emission properties of linear shape SRG samples (see Table 1). Here emitted thermal radiation was collected into the optical path of the FTIR spectrometer by gold-coated off-axis parabolic (OAP) mirror with diameter and focal distance of 50 mm and 100 mm, respectively. If required, the polarizer was installed inside FTIR spectrometer. The collected beam intensity and angular resolution were controlled by placing a vertical slit at the input port of FTIR spectrometer in the direction parallel to the grating ridges. In such experiments, the AFV value was estimated to be of approximately 0.6 deg.

 

Fig. 2. Experimental Setup I a) and Setup II b) for measurement of emission characteristics of SPPhPs by FTIR spectrometer. The off axis parabolic mirror (OAP) in Setup I was used to collimate thermal radiation of the sample placed on the heater, shown as reddish color box. A vertical slit in a case of linear shape SRG samples was oriented in the direction parallel to the grating ridges Two iris apertures (Iris 1, 2) in Setup II were used to control the angular field of view (black dashed lines) and limit collection of background radiation from the edges of heater and sample (grey dashed lines). Note that we used the same diameter values of iris apertures as that of SRG samples of radial shape.

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In the Setup II shown in Fig. 2(b), all radiation from the whole emitter area was directed to the FTIR spectrometer without the use of collimating optics in front of SRG samples. The off-axis emission from edges of the heater and the wafer was limited with two metal-iris apertures with the diameters of 5 mm positioned at distances of L₁ = 3 cm and L₂ = 13 cm. The smallest diameter of iris was limited to value of 5 mm because of the available sensitivity of FTIR detector employed, estimating AFV = 2 deg of the Setup II.

3. Results

First, we numerically investigated the SPPhP emission characteristics by varying the filling factor value of fixed period grating. We performed calculations in whole range of 0 < FF <1 and found gradual changing in the SPPhP emission profile. We further report three cases: FF = 50%, as the most common case seen in literature; and two extreme cases of FF = 75% and FF = 25%, where the last one revealed the smallest damping of the hybrid SPPhP oscillations with high emissivity values. The angular emissivity contour plots of linear gratings with three different filling factors of 50, 25 and 75% are shown in Fig. 3(a), (b), and (c), respectively. Corresponding geometries are further abbreviated as TG 50, TG 25, and TG 75 (Table 1). At the intersection point of polariton branch with the light line, k_SPPhP = 2πν_TO, the hybrid SPPhP modes have smallest values of the damping factor (0.68 cm⁻¹) and the propagation losses (2.8 cm⁻¹) as discussed in Ref. [8]. For this reason, the period of gratings was selected to be of 17.5 µm corresponding to the spectral region with lowest polariton damping. The dispersion curves, obtained by solving the grating equation (not considering the specific SRG geometry) are also shown with particular mode number M next to them. The origin of slight shift between the modelled and numerically calculated dispersions was reported elsewhere [8].

 

Fig. 3. Calculated angular dependencies of the TM polarized emissivity for linear shape gratings: a) TG 50, b) TG 25, and c) TG 75. Corresponding TM polarized emissivity spectra at selected angles of d) φ = 50 deg, e) φ = 10 deg, and f) φ = 0 deg. Note difference in frequency scale for plots shown in d-e) and f). Highlighted squares in b) mark the regions represented in the experimental emission measurement results of which are shown in Fig. 4(a).

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The excitation of SPPhP modes is observed in each contour plots shown in Figs. 3(a-c). No resonant dispersive features were observed in TE polarization (not shown). In case of TG 50 structure (Fig. 3(a)), intense excitation of fundamental (M = ±1) SPPhP mode was observed. The second (M = -2) SPPhP mode remains silent in all spectrum region, causing a “gap” in the dispersion of M = + 1 mode at angle around φ = 20 deg, and weak emission at angles larger than φ = 50 deg, where symmetry is broken by the incidence geometry. Notable emission of the third (M = -3) SPPhP mode is also observed in this structure at angles larger than φ = 40 deg. In the results of TG 25 structure (Fig. 3(b)), the excitation of both fundamental and second SPPhP modes is observed due to large asymmetry in the grating profile. The amplitude of the fundamental mode is about two times lower in comparison to the case of TG 50 sample for all angles except those which are close to surface normal. The third SPPhP mode is silent in TG 25 structure without notable features in the emissivity spectra. The results of the TG 75 sample are shown in Fig. 3(c). Here the excitation of fundamental, second and third SPPhP modes is observed demonstrating amplitude values to be comparable to that of TG 25 structure. Amplitude spectra at selected angles of polarized SPPhP emission from different SRG structures are shown in Figs. 3(d-f). It is worth noting that the broadband non resonant feature from 400 cm⁻¹ to 550 cm⁻¹ is associated with the dielectric function peculiarities near the TO edge of the GaN Reststrahlen band. Results demonstrate a hybridized behavior of both SPPhP modes M = ±1, seen as the reduction in the peak linewidth and the redshift of the peak position in the spectrum with the decrease of angle. At normal angle, i.e. φ = 0 deg, an optimal emission was achieved demonstrating the smallest linewidth value for TG 25 structure to be of about 1 cm⁻¹ (Fig. 3(f)).

We look more closely at the results shown in Fig. 3(f). While at larger angles the fundamental SPPhP mode excited in TG 25 and TG 75 has half of the amplitude than that in TG 50 structures, at normal direction. Both TG 25 and TG 50 display emissivity close to 1, while the TG 75 – only of about 0.8. The results also show the fundamental mode frequency shift and widening of peak linewidth with the change of grating filling factor value from 25% (TG 25) to 75% (TG 75). In particular, the full width at half maximum (FWHM) values were of 6.5 cm⁻¹, 5.8 cm⁻¹, and 1 cm⁻¹ resulting in the SPPhP quality factors of 87, 97, and 569 were obtained for the TG 50, TG 75, and TG 25, respectively. The minimum linewidth value of 1 cm⁻¹ for TG 25 structure, is about 7 times smaller as compared to the case of TG 50 meanwhile high amplitude of the emissivity is maintained. The TG 75 sample demonstrates weaker emission despite having slightly smaller linewidth than TG 50. The observed drastic quality increase is achieved by the minimization of propagation losses for selected SPPhP modes M = ±1, made to interfere constructively in SRG. This is the main result of this work. We also note that the spectral characteristics of SPPhP modes at larger angles (for example at φ = 10 deg) show very similar data for TG 25 and TG 75. Meanwhile, proper selection of SPPhP modes for constructive interference results in a significantly smaller propagation losses of hybrid polaritons in case of TG 25. The obtained peak linewidth value of SPPhP resonance, of about 1 cm⁻¹, is close to the theoretical limit discussed previously in Ref. [8].

The amplitude and quality factor values of fundamental SPPhP mode resonance in different grating structures were found. For this reason, the TE polarized emissivity spectra, containing the incoherent information, were subtracted from the TM polarized emissivity spectra at corresponding observation angle. Resulting spectrum was used to obtain the frequency (${\nu _{\textrm{SPPhP}}}$), amplitude (A) and FWHM value of the peak resonance of hybrid SPPhP mode. The quality factor was defined as Q = ${\nu _{\textrm{SPPhP}}}$/FWHM. Found characteristics of A and Q for all linear gratings are shown in Fig. 4(a) and (b) respectively. Resonant increase of both amplitude and quality factor in hybrid-polariton emission was found at the frequency around 570 cm⁻¹. At the normal angle, both +1 and -1 hybrid polariton modes are of the same frequency and preserve the phase therefore they can interact constructively and produce temporally and spatially coherent radiation profile. Therefore, TG 25 structure demonstrated maximum emissivity at normal angle (see also Fig. 3(f)). On the other hand, the quite similar behavior of the quality factors for different structures were obtained which was governed by the dispersion of hybrid SPPhP modes. Moreover, maximum value of Q for all SRG structures was found at the frequency of around 570 cm⁻¹, despite fact that polaritons emissivity (amplitude value) may decline. Here we propose to use the figure of merit, defined as FOM = A · Q, in order to evaluate both effects and illustrate the applicability of SPPhP emission in high quality far field applications. The obtained FOM characteristics are shown in Fig. 4(c). A strongest emission with a highest quality was obtained for TG 25 structure demonstrating values of FOM ≈ 530. Another observation is the gap in all characteristics which demonstrates the bandgap in the excitation of hybrid SPPhPs in SRG of n-GaN [20,22].

 

Fig. 4. Calculated amplitude a), quality factor b), and figure of merit (FOM) c) of fundamental SPPhP mode dependencies on its frequency for all TG structures.

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The beam of hybrid SPPhP emission possesses high directivity, value of which can be evaluated at each frequency. Directivity is an important parameter showing the possibility to transfer the energy in beam to the far field as well as revealing the spatial coherence length of the polariton propagations in SRG. Spatial coherence is important in making emitters applicable in practice, for example, describing their applicability in transfer of quasi-optical signal.

Calculated angular dependencies of the directivities and their FWHM values of different samples are shown in Fig. 5(a). The change of grating filling factor from 50% to 25% modifies an angular profile and decreases the directivity value at FWHM level (_Δφ) from 1.1 deg down to 0.11 deg. The improvement is around 10 times. The coherence length, L_SC = 180·λ/π_Δφ, of hybrid SPPhP modes in TG25 sample corresponds to value of 9.1 mm or 517 λ. The obtained value is close to the one reported by Biener et al. for 1D plasmonic bandgap structures on the surface of Gold [20]. Large coherence length demonstrates a real possibility of fabricating globally coherent SPPhP emitter with a size up to tens of millimeters. Another implication of this result is the possibility of using periodic structures of optimal design to transfer information coherently over record long up to 9 mm distances employing hybrid SPPhP modes. The coherence length dependence on peak frequency of SPPhP resonance is shown in Fig. 5(b). The result demonstrates a sharp increase in the hybrid polariton coherence at the frequency region where propagation losses of the SPPhPs are small. These results suggest an extraordinary characteristics of hybrid SPPhP emission in normal direction, which encouraged fabrication of the linear and radial shape SRG samples with different filling factors in order to validate modeling findings in experiment.

 

Fig. 5. a) Calculated emission directivities of investigated TG structures at their peak emission frequency towards normal direction. The FWHM values are marked near the corresponding curves. b) Calculated coherence length dependency on the SPPhP frequency for the TG 25 grating.

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The emission contour plots of the ELG 25 sample were measured in Setup I. The results are shown in Fig. 6(a). The experimental data of SPPhP dispersion demonstrated good qualitative agreement with theory (see Fig. 3(b)). At the emission angles near normal direction the increase of the peak amplitude was also observed in the experiment. Figure 6(b) shows calculated emissivity of TM polarization at various angles close to normal direction for ELG 25 sample. Here the emissivity peak amplitude was found to grow from 0.25 at φ = 0.4 deg up to 0.93 at φ = 0 deg. The increase of resonance amplitude was not so pronounced in experiments due to the usage of limited size angular aperture. Due to this effect, in modeling we have to take into account AFV with values of approximately 2 deg; calculated emissivity spectra for angles ranging from -1 to 1 deg were averaged in order to obtain an effective spectrum of SPPhPs emission in the normal direction. The result labeled “Average -1:1 deg” is shown in Fig. 6(c). It compares well with normalized experimental emission spectrum of ELG 25 sample shown in the same graph by black color symbols.

 

Fig. 6. a) The angular emission spectra contour plot found measuring the difference between results in TM and TE polarizations for the ELG 25 sample in Setup I. b) Result of averaging (labeled as “Average -1:1 deg”) of modeled spectra which were numerically found for indicated angle values (colored lines) in order to take into account AFV = 2 deg. Note that there are no difference between negative and positive angle spectra due to incidence symmetry. c) The emission spectrum of fundamental SPPhP mode in ELG 25 sample at normal 0 deg angle. The “Average -1:1 deg” spectrum is also shown here for comparison purposes. d) Measured emission directivity of the different samples at selected frequency of 570 cm⁻¹ (dots). The Lorentz fits of the directivity data (red lines) were obtained using linewidth values of 1.4 ± 0.1 deg, 1.4 ± 0.3 and 3.2 ± 0.6 deg for samples ELG 25 in Setup I, ELG 25 in Setup II and ECG 25 in Setup II, respectively.

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Emission spectra and experimental directivity values at the frequency of around 570 cm⁻¹ from ELG 25 and ECG 25 samples were investigated in experimental Setup II. The results are shown in Fig. 6(d). While directivity profiles were similar for the ELG 25 sample in both setups, the results in Setup II displayed larger noise and directivity values due to significantly larger AFV employed. Nevertheless, the hybrid SPPhP emission spectra of ELG 25 and ECG 25 samples were also measured in Setup II without focusing optics, demonstrating the experimental directivity values of about 1.4 ± 0.3 deg and 3.2 ± 0.6 deg, respectively.

At this point, it is worth addressing the limitations imposed by used model and experimental setups. In an ideal case with a point emitter, the source would be placed on the optical axis of FTIR spectrometer with detector measuring the beam emitted strictly at the specified angle (φ = 0 deg for a current case). The detector requires a finite AFV in order to measure signals with an acceptable signal to noise ratio. On the other hand, a finite surface area of the emitter and an angular dispersion of hybrid SPPhPs impose limits on the spectral measurements of resonant emission peak. We introduced those limitations by averaging spectra modeled in the angles range of 2 deg. Found averaged spectrum corresponds to the experimental data as shown in Fig. 6(c). Those limitations do not originate from the physical aspect of hybrid SPPhP modes. Therefore, the hybrid polariton dispersion and emission characteristics were properly modelled and can be used as ideal case (AFV = 0 deg) and in further experiments. Potentially other techniques, such as knife edge method with MIR microscopy, could be used to measure modeled coherence characteristics of hybrid polaritons more accurately.

To take a step towards practical implementation of SPPhP emission, a linear and circular shape SRG samples were investigated directly in front of FTIR spectrometer without focusing optics, i.e. in Setup II. For this case the polariton emission and beaming were investigated also without the polarization components. Since at the angles different from 0 deg grating geometry projected at the optical axis is not held, the amplitude associated with hybrid SPPhP emission at the resonant frequency is expected to drop with the increase of observation angle [22]. In this way an improvement in beam directivity can be achieved while simultaneously obtaining the emission with reduced linewidth at a desired frequency. All samples emission spectra measured at the normal direction (φ ≈ 0 deg) are shown in Fig. 7. Next to the experimental results we show calculated emissivity spectra, for the respective grating parameters considering AFV = 2 deg. Here calculated TM-polarized and unpolarized (T0) emissivity spectra are shown by orange and green lines, respectively. The modeled spectra, in theory, fundamentally differ from that obtained in the experiment as Emission = Emissivity · BB · T, where BB is the black body radiation of the sample which needs additional consideration and T is the combined transmission coefficient of the atmosphere and spectrometer components. In the spectral range of interest, the BB and T modify only the slopes, without spectral signatures. To exclude unknown BB and T characteristic from analysis of measured data, the amplitude of theoretical emissivity spectra was normalized to the TO phonon edge (at 500 cm⁻¹), and a linear spectrum slope was subtracted to fit the region above the wavenumbers of 600 cm⁻¹.

 

Fig. 7. Measured unpolarized (dots) and modelled different polarization (curves) thermal beaming spectra different grating structures with circular designs and FF values of 25%, 50%, and 75% in a), b), and c), respectively. d) comparison between the measured linear (ECG 25) and circular (ELG 25) design grating thermal beaming.

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In case of ECG 25 sample (Fig. 7(a)) good agreement between theoretical TM polarized emissivity and measured emission spectra was found. The spectral position, the linewidth, and the peak profile are accurately modeled. In other two cases of ECG 50 and ECG 75 samples (see Figs. 7(b-c)), mismatch in polariton resonant position is observed with the feature in modeled spectrum being blue-shifted by several inverse centimeters in respect to measurements in experiment. Nevertheless, the overall experimental shape of SPPhP resonances matches those of numerical calculations. Another noteworthy feature of the obtained results lies in the amplitudes of resonance peaks. While the measured unpolarized spectra of the ECG 25 and ELG 25 correspond to the TM polarized emissivity in their amplitude, ECG 50 and ECG 75 SPPhP features show notably smaller experimental amplitudes as compared to the analogous modelled spectra. This feature might hint at additional resonant increase in the thermal beaming amplitude at the condition of optimized coupling or at some nonlinear phenomena present in the experimental excitation of hybrid SPPhPs.

Finally, the emission spectra of ECG 25 and ELG 25 samples are compared and shown in Fig. 7(d). Both samples demonstrated a similar amplitude and shape of the resonant peak. Nevertheless, smaller linewidth (6.2 vs 7.2 cm⁻¹) was obtained for the sample with a radial shape SRG.

4. Conclusions

We demonstrated a coherent and directive thermal emission of hybrid plasmon-phonon polaritons excited in n-GaN surface relief gratings with linear and circular geometries of optimized design. Numerical modeling of hybrid plasmon-phonon polariton dispersion characteristics allowed us to find the optimal grating period and filling factor values resulting in narrow SPPhP emission peak linewidth of down to 1 cm⁻¹ and emission directivity down to 0.11 deg leading to spatial coherence length values up to 9.1 mm. The samples of optimal design were fabricated validating the numerical findings. Coherent radiation and beaming of hybrid SPPhPs were demonstrated in emission experiments without the use of focusing optics, where the experimental results closely followed modeling data when accounting for the limitations of the angular field of view needed for experiments.