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Abstract
Dispersion characteristics of hybrid surface plasmon-phonon-polaritons (SPPhPs) on the air/polar semiconductor interface were investigated by means of shallow surface relief grating using emission spectroscopy methods. A set of grating structures with optimal 1 µm depth and periods from 8 to 22 µm was developed on a heavily-doped GaN crystal. The SPPhPs were excited by thermal heating or electrical biasing of the samples which radiated directive polarized features in an extremely narrowband spectrum range. Detailed analysis of damping factors and propagation losses revealed maximum values of quality factor and spatial coherence of hybrid SPPhPs modes. Highest quality factor was found to be practically independent on the period of the shallow grating, as it was always detected near the frequency of transverse optical phonon, demonstrating values as high as 88 and 200 in experiment and theory, respectively. Meanwhile, the largest values of coherence length strongly depended on the grating as the propagation losses of hybrid SPPhP modes showed a tendency to accumulate with the wavevector increase. The sample with 22 µm grating period demonstrated the highest coherence of hybrid polaritons with the experimental (theoretical) coherence length values as high as 1.6 mm (2.3 mm).
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Research and development of coherent thermal emitters for terahertz (THz) and infrared (IR) bands remains a hot topic for solid state physics [1,2]. Surface polaritons have long been shown to be excellent means of light localization [3], manipulation [4], non-linear effect investigation, and narrowband thermal emission [5]. Surface phonon polaritons (SPhPs) demonstrate nearly an order of magnitude higher quality factors than their surface plasmon polariton (SPP) counterparts. Low intrinsic damping of phononic oscillations allows observing narrowband excitations with tunable frequency within a Restsrahlen band (RB) of semiconductor [5,6]. Surface polariton waves always possess a wavevector mismatch requiring the utilization of coupling devices for radiation extraction such as prisms [7] or gratings [6,8]. In case of polarization insensitive coupling, more complex two-dimensional gratings [9], arrays of holes [10] can be employed.
The spatial coherence, frequency, and polarization selectivity of thermal polaritonic emitters based on SiC [6,11,12], tungsten [13], and GaN [8] have been reported. All of them reveal a noticeable wavevector dispersion that can be taken advantage of. Moreover, coherent nature of thermal SPhP emission was demonstrated, allowing to consider SPhP elements for the use in coherent signal transfer [6]. Applications of polaritonic emitters in thermophotovoltaics, radiative cooling [14], chemical sensing [15,16], biosensing [17], ultrafast optical computing systems [18], and other fields have been demonstrated. The coherent thermal sources may be utilized for the purposes of on-chip intercomponent communication, and directional properties of the emission could be beneficial for open-space communication [2]. Recently, a radiative-cooling polariton-photonic circuitry feeding system powered by computer processor waste heat has been suggested [2]. Most prominent drawback of SPhP-based elements is the operation bandwidth, which is limited to the RB of the selected semiconductor [5]. Multilayer structures [19,20] or heavily-doped semiconductors [21,22] were demonstrated to be promising options for overcoming these limitations.
Surface plasmon-phonon-polaritons (SPPhPs), i.e., quasi-particles comprising phonons of the crystal lattice interacting with free-carriers of the conductive medium, can form surface waves propagating along the inter-media boundary. Polaritons cannot be radiated as photons because they are not coupled with the light in free space [8,11]; nevertheless, their radiation has been experimentally obtained from various heavily-doped polar semiconductors at so-called longitudinal optical (LO) Phonon-Plasmon (LPP) mode frequencies [5,21,22], where the dispersion characteristics of the polaritons experience an anticrossing behaviour as it will be discussed in this work. Until recently, in most papers, the investigations were restricted to the dispersion analysis of the corresponding mode oscillation frequency neglecting the damping rate and the propagation losses [23,24] which determine the quality factor and the length of spatial coherence. The quality factor, Q, can be defined as the ratio between the oscillation frequency and the linewidth of the resonance. The directivity is known to be associated with spatial coherence of the investigated source. The spatial coherence length, LSC, can be related with the angular broadening of the resonant feature, ΔΦ, at the wavelength, λSPPhP, as LSC=λSPPhP/ΔΦ [6,8].
In this work, we provide a comprehensive analysis of the dispersion characteristics of hybrid SPPhP modes excited on the air/polar semiconductor interface by means of the shallow surface relief grating (SRG). The emission spectroscopy allowed to probe the dispersion curves of these quasi-particles without a limitation on the observation angle in contrast to the measurement restrictions common in the reflection spectroscopy [8]. The set of SRGs with 1 µm depth and various periods was developed on a heavily-doped GaN crystal for the investigation of dependence of emission characteristics on the observation angle and on grating period, i.e., wavevector. The SPPhPs were excited by thermal heating or electrical biasing of the samples with the polarized feature emission in the RB region of n-type GaN. The experiments were supported by the spectral calculations using rigorous coupled wave analysis (RCWA) method, which revealed a significantly decreased damping factor and propagation loss values of hybrid SPPhP modes at the frequency of transverse optical (TO) phonons. The directive emission of polaritons in an extremely narrowband spectrum range was found with experimental values of the angular broadening and the linewidth as small as 0.66 deg and 6.4 cm−1, respectively. The narrowest linewidth of the emission peak was always detected near the TO phonon frequency demonstrating very similar quality factor of the SPPhP modes in various SRG samples with the highest values to be as high as 88 and 200 in the experiment and theory, respectively. Meanwhile, the coherence length was strongly dependent on the period as the propagation losses of hybrid SPPhP modes showed a tendency to accumulate with the increase of the wavevector. The highest experimental (theoretical) coherence lengths were found to be as high as 87 λ (130 λ), exceeding the maximum values reported in the literature.
2. Samples and methods
A set of surface-relief gratings was fabricated on a 500 µm-thick c-plane n-type GaN crystal (n = 1.55 × 1019 cm−3) using UV lithography and reactive ion etching in chlorine-based plasma. The depth of the gratings with 50% filling factor was optimized to be approximately 1 µm for efficient excitation of SPPhPs in n-GaN [8]. A set of four gratings with the periods of P = 11, 16, 22, and 8 µm and areas of 4 × 4 mm2 each was fabricated on the surface of a single crystal, labelling the samples SRG1, SRG2, SRG3, and SRG4, respectively. Scanning electron microscope (SEM) image of the fabricated SRG4 sample is shown in Fig. 1(a).
Fig. 1. (a) SEM image of selected sample (SRG4) with marked period P and ridge width W. Dotted line represents the profile of the fabricated grating surface, with marked grating height h. (b) A schematic view of the experimental setup with the sample mounted on the translation rotation stage. An off-axis parabolic mirror (OAP) is used to collect sample radiation and direct it into the FTIR spectrometer through the vertical slit limiting the angular field of view (AFO). (c) SPPhP emission spectra of SRG3 sample measured with different widths of the vertical slit. Measured FWHM values of SPPhP emission peak are given for each aperture size. (d) Comparison of SPPhP emission spectra in case of sample excitation by heater (black curves) and direct current flow (red curves). Spectra are vertically shifted for clarity.
Modeling of the dispersion and spatial coherence characteristics of hybrid SPPhP modes was performed using RCWA method. The spectra of transmissivity, T, reflectivity, R, and absorptivity, A, were obtained, with the latter one governed by the energy conservation law as A = 1 – T – R. In the spectrum range of interest, all SRG samples were considered to be opaque (T = 0) due to high doping of the n-GaN crystal, so, in accordance with Kirchhoff’s law, the emissivity, E, which is equal to absorptivity, A, was evaluated as E = 1 – R. Calculations were carried out for TM and TE polarizations, and the difference between TM and TE spectra was used for analysis. Dependences of the optical characteristics on the periodicity of the grating and the observation angle were investigated at each frequency allowing for the investigation of SPPhP resonant features.
The model structure was approximated as 100 µm-thick free space, 1 µm-thick grating and 499 µm-thick substrate layers. Owing to the fact that the profile of the fabricated SRG displayed slight sidewall inclination, the grating layer was additionally divided into 5 sublayers. Sublayers had incremental ridge widths which were 0.2 µm narrower at the ridge top and 0.6 µm wider at the ridge bottom compared to the initial layer with 50% filling factor. The grating region was approximated by 50 Fourier modes, giving good convergence behaviour without notable calculation artefacts.
The emission spectra were measured using Fourier-transform infrared (FTIR) spectrometer (Thermo Scientific Nicolet 8700) comprising KBr beamsplitter and DLaTGS pyroelectric detector optimized to operate in the mid-IR spectrum range. A KRS-5 holographic wire-grid polarizer with a flat transmission curve and extinction ratio higher than 150:1 over the whole spectrum range of interest was used. The TM polarized spectra were analysed with TE-polarized emission subtracted from them. Such subtraction allows for the analysis of hybrid SPPhP modes only, by removing unpolarized broadband thermal radiation of the sample.
The sample was mounted either on a custom-built heater assembly with precise temperature control in the range from room temperature up to 500 °C or on a customized sample holder allowing application of high direct current, both of them coupled to an automated rotating stage to ensure the precise variation of the observation angle with respect to the sample surface plane. Sample radiation was guided into an external input of the spectrometer using off-axis gold-coated parabolic mirror (OAP) with diameter and focal distance of 50 mm and 100 mm, respectively. The schematic view of the experimental setup is shown in Fig. 1(b).
A vertical slit with adjustable width was installed at the input of spectrometer in order to limit the angular field of view (AFO). Its limitation allows probing the angular dispersion of SPPhPs with limited broadening of the resonance feature in the experiment. Figure 1(c) illustrates the polarized SPPhP emission under thermal excitation of SRG3 sample, measured at an arbitrary, fixed observation angle of 12 deg with different widths of the vertical slit. The emission peak exhibited full width at half maximum (FWHM) values of 11, 14 and 20 cm−1 for the slit widths of 1, 2, and 3 mm, respectively. The linewidth dependence on the size of AFO is natural because a larger section of the SPPhP dispersion characteristic is probed with larger slit aperture, resulting in a detection of integral response of neighbouring resonant features with different intensities and peak positions [8]. A balance between the size of the aperture and acceptable signal-to-noise ratio had to be found. In our case, 1 mm slit was selected for all experiments, providing the value of AFO = 0.57 deg.
Hybrid SPPhPs can be excited either by heating or electric current flow through the sample resulting in emission powers of approximately 200 nW [25]. Comparison of both excitation regimes is shown in Fig. 1(d). Each of SRG1 and SRG3 samples was heated to the temperature T = 500 °C, and emission was probed at arbitrarily selected angles of 30 and 16.5 deg, respectively. The respective spectra were also obtained using an electrical current excitation at the same setup arrangement. Direct current of up to 7 A was applied to the samples providing a similar temperature, magnitude of which was estimated from the comparison of thermal emission spectra. Comparison of the SPPhP features obtained under different excitation regimes revealed that both types of excitation are equivalent, showing the same spectral line shape and intensity. Thermal excitation of SRG samples was utilized further in this work. However, the excitation with electric pulses together with a thinner substrate for the SRG has to be considered in future studies.
3. Dispersion characteristics of hybrid SPPhP modes
Longitudinal component of the incident electromagnetic field is used to describe the excitation of evanescent waves along the air/semiconductor interface with the wavevector of [11,26]:
where ${\epsilon _0}$ and ${\epsilon _s}$ are the dielectric permittivities of air and semiconductor, respectively; ${k_0} = 2\pi {\nu _0} = \frac{{{\omega _0}}}{c}$ is the wavevector of light with an angular frequency ${\omega _0}$, which is directly proportional to the wavenumber ${\nu _0}$; and c ≈ 3 × 1010 [cm/s] is the speed of light. A scalar form of the dielectric function inherent for polar materials is used here, neglecting the anisotropy effects in GaN [22]:The first and second terms describe the polarization of the lattice and electron subsystems, respectively. The $\nu_{LO}$, $\nu_{TO}$, $\nu_p$ and $\nu_{LO}$, $\nu_{TO}$, $\nu_p$, are the oscillation frequencies and the damping factors of the longitudinal, transverse optical phonons and bulk plasmons, respectively. The angular oscillation frequency of bulk plasmons is described as ${\omega_p}\; = \sqrt {{e^2}{n_0}/{m^\ast }{\epsilon _\infty }} ,$, where ${n_0}$ is the free electron density, ${m^\ast }$ is the electron effective mass, and ${\epsilon _\infty }$ = 5.3 is the high-frequency dielectric permittivity of GaN. Used material parameters as well as high-frequency dielectric permittivity value were obtained by fitting the experimental reflection spectrum of the n-GaN without the grating [8]. Best fit parameters are listed in Table 1.
Table 1. Lattice and free-carrier parameters of the studied n-type GaN crystal.
The evanescent waves through the interaction with the periodic grating profile can radiate into free space only at the characteristic frequency which satisfies the phase matching condition:
here $\varphi $ is the incidence angle, M is the integer mode number ($M$ = ±1, ±2,…), and ${k_G} = 2\pi /P$ is the wavevector of the SRG.RCWA characterization of the spectral features of reflectivity, transmissivity, and absorptivity under uniform sample illumination was performed. The calculated emissivities of selected samples are shown in a left-hand side column of the Fig. 2. The white dashed lines indicate dispersion of the fundamental and higher order polariton modes calculated using Eqs. (1–3). The SRG4 sample was designed to probe the dispersion of the M=−1 mode in the spectral region overlapping with that for SRG1 sample, so we will analyze the results of characteristic features of SPPhPs only in the latter sample.
Fig. 2. The calculated emissivity (left column) and experimental emission (right column) spectra, obtained as the difference between TM and TE polarizations $({E_{TM}} - {E_{TE}}$), for the SRG samples with the grating period of 11 µm (top row), 16 µm (middle row), and 22 µm (bottom row). White dashed lines show the dispersion of SPPhP modes M = ±1, –2, –3 as the solutions of Eqs. (1–3). Experimental spectra were normalized to the maximum emission intensity of the corresponding sample, namely, 0.284, 0.187, and 0.162 for the SRG1, SRG2, and SRG3 samples, respectively.
Experimental results are shown in a right-hand side column of the Fig. 2. A good agreement between the experiment and the theory was found. Indeed, experimental emission spectra confirm the theoretical modelling. The fundamental mode of SPPhPs was excited in all cases. In case of SRG3, emission starts in the spectral region of TO phonons and exhibits a blue shift far above the frequency of LO-phonons. In addition, the linewidth of SPPhP feature decreases from 180 cm−1 down to 6.7 cm−1 when its resonance frequency red shifts to the TO-phonon frequency. The obtained minimum and maximum values of the linewidth and quality factors for all SRG samples are summarized in Table 2. Note that smallest experimental values of the FWHM were found to be smaller than the damping factor of the TO phonons (for ΓTO see Table 1).
Table 2. Experimentally measured maximum and minimum values of linewidth, FWHM, and quality factor, Q, of fundamental SPPhP mode for all samples under investigation.
Analysis of the angular dependencies of emission intensity is demonstrated in Fig. 3. All the experimental spectra of SPPhP emission were described well with the Lorentzian function. The results of fitting the experimental data for three different SRG samples are shown in Fig. 3(a). Meanwhile, emission spectra, calculated by RCWA method for different AFO values (see Fig. 3(b)), demonstrate an asymmetric line-shape of SPPhP features which at AFO = 0 can be described by a Fano resonance model [23]. However, it demonstrated more symmetrical shape when AFO = 0.6 deg was employed (see red dashed lines in Fig. 3(b)). Other possible reasons of the peak shape change were attributed to the mechanical vibrations within the setup and variations in the environment during the measurement. The experiment did not demonstrate pronounced asymmetry of the line-shape, therefore, the use of Lorentz function approximation was considered to be sufficient for further result analysis in this work.
Fig. 3. (a) Experimental angular dependencies of SPPhP emission peak together with Lorentzian fitting curves. (b) RCWA data demonstrating the effect of limited size of vertical slit aperture. A SPPhP emission peak from two SRG samples was modeled with the slit aperture width of 1 mm (AFO=0.6 deg) and “closed slit” (AFO=0 deg).
Figure 4 shows the results of comparison of experimental oscillation frequencies, $\nu$, (top panel), and the linewidth, FWHM, (bottom panel), to the respective theoretical data of RCWA model. Again, a good quantitative agreement between experimental and theoretical values of both parameters is observed for all SRG samples. Numerical solution of Eqs. (1–2) with the parameters of the investigated material were found in a form of the complex-valued angular frequencies, $\nu_m(k )= \nu_m^{\prime}(k )+ \; i \nu_m^{\prime\prime}(k )$, taking for calculation the real-valued wavevectors [11,24]. In total, three branches (m is the index number 1,2,3) with different dispersion characteristics and fundamental properties were observed with a particular interest in the branch $\nu_2$, which is monitored in the emission experiments.
Fig. 4. The measured (symbols), RCWA modeled (color line), and numerically calculated (black line) characteristic frequency, ν, (top panel), and damping factor, Γ, (bottom panel) of the SPPhP modes observed in polarized emission spectra of various SRG samples. A dash-dotted blue line stands for the light line, ν’=ν0.
The $\nu^{\prime}$ plot demonstrates (see Fig. 4, top panel) that $\nu_2$ branch remains unguided for all small values of the wavevectors until it intersects with the light line. Analysis of Eq. (2) suggests that the intersection point, νINT, is at the position of kINT = 2πνTO with the resulting value of νINT = 557-i3.5 cm−1, i.e. the oscillation frequency and half of the damping factor of TO phonons. The $\nu^{\prime\prime}$ plot demonstrates that the minimum losses of $\nu_2$ branch are in a vicinity of the intersection point, however with the damping value of $\nu_2^{\prime\prime}$(557) = 0.68 cm−1, which is considerably smaller than ΓTO/2. Even more, all values of the branch $\nu_2^{\prime\prime}$ differ from those measured in the experiment and modelled by RCWA method. The difference was attributed to the propagation losses, ΓPL, which can be estimated as ΓPL = Γ∑ - $\nu_2^{\prime\prime}$, where Γ∑ is the total damping rate measured as the linewidth (FWHM) of the SPPhP peak emitted from the shallow n-GaN grating. The experimental Γ∑ characteristic, found by averaging FWHM data of respective SRG samples, demonstrates that the propagation losses tend to accumulate with the increase of the wavevector value. Moreover, the accumulation rate increases after the Γ∑ characteristic crosses the branches $\nu_1^{\prime\prime}$ and $\nu_3^{\prime\prime}$, thus demonstrating a very different behaviour of the hybrid polaritons in the experiment. Interesting to note that the propagation losses of hybrid polaritons are not zero even for a plane surface of GaN crystal, which demonstrates the value of ΓPL = 2.8 cm−1 at the intersection point. For comparison, smallest Γ∑ value for investigated SRG samples was found in theory to be of 4.0 cm−1, demonstrating the losses as large as 3.3 cm−1. One can also notice a small deviation in $\nu^{\prime}$ plot between the calculated oscillation frequencies $\nu_2^{\prime}$ and RCWA or experimental data, which is more pronounced at larger wavevectors indicating that numerical calculation of dispersion characteristics is mostly suitable for analysis of crystals with plane surface.
It is also worth to note the anticrossing region in a $v^{\prime\prime}$ plot where branches $\nu_1$ and $\nu_2$ intersect with each other and the damping factors of both are equal at the wavevector of k0 = 3010 cm−1. However, these two branches do not cross in a $v^{\prime}$ plot. Interesting feature of the anticrossing region is the degeneration of the oscillation frequencies seen in a $v^{\prime}$ plot, meaning that the same oscillation frequency corresponds to two complex-valued wavevectors which belong to different polaritonic branches, namely, $\nu_1$ and $\nu_2$. The result is that hybrid polaritons intersect the light line making possible the radiation of surface waves from a plane surface of heavily doped polar semiconductor (so called LPP modes) with the peak position localized at the frequency of ν- = $\nu_2^{\prime}$(k=0) and the linewidth of FWHM = $\nu_3^{\prime\prime}$(k=0) as observed in different experiments [8,21,22]. The exact electrodynamic simulations of the resonant behaviour of three branches with analytical analysis of asymptotic values and the anticrossing region are out of scope of this work [24] and will be analysed elsewhere.
4. Discussion
The emission spectra of hybrid SPPhP modes exhibit either a narrow linewidth in the longwave spectrum part (phonon-like behaviour) or a wide linewidth in the shortwave region (plasmon-like behaviour). The variation of the observation angle allows for characterization of SPPhP dispersion measuring the central frequency and the linewidth (FWHM) of the emission peak. At normal observation angle, the wavevector of incident light has zero angular component (see Eq. (3)), and SRG samples radiate polarized SPPhPs with wavevector governed only by the diffraction grating wavevector kG. With the increase in incidence angle, the light wavevector component, k0*sin $(\varphi)$, starts to be significant and splits SPPhPs into +1 and −1 branches (see Fig. 2). The change in the grating wavevector kG of different SRG samples results in different emission frequencies observed at normal direction. For example, SRG1 and SRG2 samples demonstrated a fundamental mode resonance at respective frequencies of about 850 and 600 cm−1, which, with the increase of the observation angle, split into two branches with positive and negative slopes. There was no emission of SPPhP features from the SRG3 sample at normal observation angle, however, M = +1 mode appeared in the spectrum at angles larger than 10 deg in the experiment.
The linewidth of the resonant feature at short-wavelength region is large with the values as high as 140 cm−1 and 180 cm−1 (at wavenumber of 800 cm−1) for samples SRG1 and SRG3, respectively (see also Table 2). Due to hybrid polariton nature, the linewidth reduces down to the values of 4.0 cm−1 (6.3 cm−1) in theory (experiment) when the SPPhPs shift to the long-wavelength region near the TO phonon frequency. The minimum FWHM value of SPPhP emission peak for all SRG samples was found to be smaller than the damping factor of phonons, ΓPh, but higher than the damping factor of branch $\nu_2$ at the intersection point with the light line, i.e. the value of $\nu_2^{\prime\prime} = 0.68$ cm−1. Results in Table 2 reveal that the maximum quality factors with similar values of up to 88 were observed for all samples except SRG4 sample. The latest was probed far from optimal wavevector values (only above |k0| > 4250 cm−1) owing to the limitation of the observation angle in our setup to the range of 60 deg.
Excitation of higher order SPPhP modes such as M=−2 and M=−3 was also possible in experiment (see Fig. 2) which is the result of a good crystal quality and an accurate fabrication of SRG samples. The M=−3 mode in the emission spectra of the SRG3 sample was directly observed, intersecting with the fundamental mode at φ ≈32 deg. Excitation of even modes is prohibited by the selection rules for symmetric gratings, as all SRG samples were developed with the filling factor of 50%. Nevertheless, M=−2 appears as a “silent” mode measured as the perturbation in M=+1 mode dispersion owing to weak emission of the SRG2 and SRG3 samples at φ ≈ 20 deg (respective wavenumbers of 820 cm−1 and 670 cm−1).
The polarization characteristics of the SPPhP emission were also investigated. The results are shown in Fig. 5(a), where symbols stand for experimental data and solid line is the fitting function in form of ${\cos ^2}\mathrm{\alpha }$, where α is the angular position of polarizer. The emission was found to be linearly polarized with its angular dependence following the Malus’ law.
Fig. 5. (a) Characteristic polar plot of linearly polarized SPPhP emission. Solid curve represents the Malus’ law: $I = {I_0}{\cos ^2}\alpha $, where $\alpha $ is the polarizer rotation angle. (b) The directivity diagrams of three SRG samples obtained at the wavenumber of 560.2 cm−1. Note that observation angles are in the range of 0-40 deg. Symbols denote experimental results, and solid lines are the results of the RCWA calculations.
The measurements of angular characteristics of emission demonstrate a strong directivity of the SPPhP modes at fixed frequencies and especially in the spectral region near the frequency of TO phonon. Normalized experimental and RCWA data in polar coordinates are shown in Fig. 5(b). The change of the grating period modifies the angle value at which highest directivity can be observed. A good agreement between theory and experiment is found. The angular broadening ΔΦ of the SPPhP modes was estimated from the angular characteristics of emission plotted at certain frequencies. The minimum angular broadening detected experimentally for samples SRG1, SRG2, and SRG3 were of 1.79, 0.73, and 0.66 deg, respectively.
The coherence lengths were found for all SRG samples as a function of the oscillation frequency of SPPhP modes. The results are shown in Fig. 6(a). Taking into account Eq. (3), a corresponding dependence on the wavenumber was found. The results are shown in Figs. 6(b)–6(d). Symbols stand for the experimental values while lines show the RCWA data. Note that modeling data in Figs. 6(a) and 6(b)–6(d) are presented for ideal case and the case with limited size for angular aperture, respectively.
Fig. 6. (a) Coherence length dependence on the observation angle. Symbols denote the experimental results and solid lines represent the RCWA data taking into account that AFO=0 deg. (b) –(d) Coherence length dependence on frequency for samples SRG3, SRG2, and SRG1, respectively. Symbols denote the experimental results and solid lines represent the RCWA data taking into account the AFO=0.6 deg. Note that maximum of the coherence length is found in the vicinity of TO phonon frequency for all SRG samples.
In the experiment, all samples demonstrated the coherence length values approximately 30-40% smaller than the values modeled without taking the AFO value into account. An insignificant difference between ideal case and RCWA modeled values of ΓPL (2.8 cm−1 vs 3.3 cm−1) indicates either an existence of a systematic measurement error or some aspects of SPPhP behavior which were not accounted for in modeling procedure. The former effect might be associated with the presence of an angular aperture in the measurements. Indeed, RCWA data, with the value of AFO = 0.6 deg taken into account, fitted the experimental results better in terms of the LSC values. More detailed analysis is outside of a scope of this work and will be investigated in a future.
Numerical modeling of spatial coherence for the SRG1 sample suggested the values up to 0.9 mm (55 λ) with its maximum at the angle of 37 deg and frequency of 560 cm−1. In the experiment SRG1 sample demonstrated a highest value to be a bit smaller, i.e. LSC ≤ 0.58 mm, although the polaritonic feature was weakly observed in the spectrum and the result extraction was complicated near the TO phonon frequency (see also Fig. 3(a)). In comparision, SRG2 sample in experiment demonstrated up to three times higher values of the coherence length, reaching level up to 1.4 mm at the observation angle of 7 deg. For this sample the theoretically predicted value was in the range of 1.9 mm. Finally, the SRG3 sample demonstrated the best performance among others with its highest coherence length at an angle of 12 deg reaching values up to 2.4 mm and 1.5 mm in theory and experiment, respectively. The SRG4 sample was suitable for the observation of hybrid SPPhP modes emission with relatively weak spatial coherence, which value was below of 0.09 mm. It is worth to note, that despite the difference in highest values of spatial coherence length between various SRG samples, the best performance was found at the similar wavenumbers which correspond to the TO phonon frequency (see Fig. 6(b)-(d)). At the intersection point of polariton branch with the light line, kSPPhP = 2πνTO, the hybrid SPPhP modes have the smallest values of the damping factor $\nu_2^{\prime\prime}$ = 0.68 cm−1 and the propagation losses ΓPL = 2.8 cm−1 (see also Fig. 4). Indeed, the spatial coherence decreases very rapidly going away from TO phonon frequency towards a short-wavelength region where the propagation losses of SPPhP modes tend to accumulate more rapidly with the increase of wavevector value.
Maximum values of the LSC parameter are summarized in Table 3. The SRG3 sample demonstrated the highest directivity among others. The highest coherence length value of 130 λ (87 λ), was found in this paper for modelled (measured) performance of SPPhPs in GaN, which already exceeds the maximum values of 60 λ and 104 λ measured for SiC [6] and estimated for w-GaN [5], respectively.
Table 3. Highest values of theoretical and experimental coherence lengths expressed in wavelengths and in millimetres. Theoretical values are found for RCWA data without (with) taking into account the AFO = 0.6 deg.
5. Conclusion
The spatial coherence of hybrid surface plasmon-phonon-polaritons excited in the shallow surface relief n-GaN grating was investigated using emission spectroscopy methods supported by numerical rigorous coupled wave analysis. Found dispersion of the oscillation frequencies, damping factors, propagation losses, and angular profiles of polarized directive emission features revealed the independence of quality factor and dependence of spatial coherence length of hybrid polariton modes on the grating wavevector. The largest coherence length values up to 1.6 mm (2.3 mm) were found experimentally (theoretically) for the sample with grating period of 22 µm. Good agreement was achieved between experiment and theory paving a way for development of narrowband thermal sources of coherent radiation in IR and THz ranges.
Funding
Lietuvos Mokslo Taryba (DOTSUT-247); European Regional Development Fund (No. 01.2.2-LMT-K-718-0047).
Acknowledgments
The authors thank Dr. V.V. Korotyeyev and Dr. L. Minkevičius for fruitful discussions of the various aspects of this work. We acknowledge the support from the Research Council of Lithuania (Lietuvos mokslo taryba) through the “KOTERA-PLAZA” Project (Grant No. DOTSUT-247) funded by the European Regional Development Fund according to the supported activity “Research Projects Implemented by World-class Researcher Groups” under the Measure No. 01.2.2-LMT-K-718-0047.
Disclosures
The authors declare no conflicts of interest.
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