Multi-frequency coherent emission from superstructure thermal emitters

Thermal emission from periodically patterned surfaces^1–3 has been exploited for numerous applications from non-dispersive infrared (NDIR) spectroscopy^4,5 to thermophotovoltaics.⁶ Such light sources are relatively simple, relying only on local heating for light generation, yet can produce significant amounts of emitted power in the mid-infrared (mid-IR) region. This overcomes some of the limitations of semiconductor optoelectronics for narrowband mid-IR light emitters, which can be expensive and become increasingly less efficient when operating at longer wavelengths. Spatially coherent thermal emitters have a further advantage in that they can have non-Lambertian emission profiles. Specifically, they can have highly directed emission, more analogous to a laser than a conventional thermal emitter. Coherent thermal emitters rely on a surface-confined wave that propagates along a periodically structured material in the near field, thereby inducing far-field spatial coherence via coupling to grating modes. One route to realizing such coherent thermal sources has been via exploiting surface polaritons, which are evanescent waves that propagate along the interface between an optical metal and a dielectric (e.g., a metal surface in air). Surface polaritons are induced through light coupling with an oscillating charge, with two of the predominant forms within the infrared region resulting from coupling to free electrons or polar optic phonons, forming surface plasmon polaritons (SPPs) and surface phonon polaritons (SPhPs), respectively.^7,8 While both are characterized by an evanescent field that propagates along a surface, in the IR, SPPs generally possess frequency tunability and short lifetimes (broad spectral linewidths), whereas SPhPs offer limited frequency tunability and long lifetimes (narrow linewidths).⁹ One of the first demonstrations of spatially coherent thermal emission employed a patterned silicon carbide surface,^10,11 but such effort has since been extended to include plasmonic structures^12–14 and highly dispersive dielectric resonators.¹⁵ Some of the best performance metrics achievable by periodic structures have been realized using simple 1D gratings¹³ or bullseye designs using metals,^12,16 demonstrating the potential for spectrally and spatially narrow emission.

Improving the degree of control of spatial coherence could significantly broaden the application space for thermal emitters, but for simple gratings, it is limited by the excitation mechanism. For a given grating, we can describe the coupling from a free-space wave (with wave vector k₀ = 2π/λ₀ and incident angle θ) to a propagating polariton mode (k_p) as

where $Gi=2πnΛ$ is the Bragg vector, n is an integer, and Λ is the grating period. All frequencies of light that meet these criteria are emitted, which results in emission that changes color continuously as a function of angle. Given that for most applications, only well-defined frequencies or angles are required, coherent thermal emitters will waste much of their emitted light at undesired frequencies or angles. Achieving a greater level of control of the emitted frequencies and/or angles, therefore, becomes a key problem for implementing such designs in solving real challenges. This will require breaking the symmetry of the system in some way, shape or form, so that the grating condition offers a greater degree of control. One possibility is the use of bullseye type structures, which exhibit emission parallel to the surface normal; however, this does not allow an arbitrary definition of both frequency and angle.^12,16 Recently, inverse design techniques have been used to design thermal emitters for the purpose of improved control of their spectral and spatial emission properties.^17–21 These use arbitrary arrangements of patterned areas on the surface more analogous to holograms^22–24 than conventional periodic structures. However, while numerous designs and techniques have been proposed, they have not yet been reduced to practice. This leaves numerous open questions about the efficacy of spatially coherent emitters, such as limitations on the device size, the angular linewidth of emission, and constraints on periodicity, and if more than one spatially coherent mode can be supported.

Here, we study the spectral and spatial thermal emission profiles of superstructure gratings (SSGs) fabricated into a semi-insulating 4H–SiC substrate. In an SSG, the structure of the unit cell is altered to form an extended “superperiod” (L), much longer than a conventional grating period (Λ). In principle, within the superperiod, an arbitrary grating profile can be chosen depending on the design methodology.²⁵ We are able to experimentally show that SSGs can support multiple coherent modes each of which disperses as a function of angle, extending the simple grating model to more complex architectures. The frequency and dispersion of these additional modes are dependent on the superstructure design and can be directly related to the Bragg vectors calculated by Fourier transform. By comparing multiple SSG gratings with different sampling periods, we are also able to show that as the superstructure period approaches the coherence length of the surface polariton, modes become poorly defined. This suggests a breakdown of spatial coherence in these devices. We assert that this provides the fundamental limit to which spatial coherence can be controlled from partially coherent thermal emitters. Thus, we propose that inverse design techniques should account for this coherence length as a limit when it comes to creating a well-defined emission profile. We highlight that this study provides the fundamental limits to the design space in any partially coherent system, and as such, it implies broad implications for the design of other structures to induce spatial coherence into other incoherent sources, such as light emitting diodes.

We begin with a study of one of the simplest types of SSG, the sampled grating,^26,27 which is formed by taking a square periodic grating profile and multiplying it by a second sampling function [Fig. 1(a)]. Here, the sampling function is chosen to be a second periodic square wave. The result of this transformation on the behavior of the grating emission can be understood by examining the discrete Fourier transform of the grating permittivity,²⁸ allowing us to calculate the Bragg vectors G_i in Eq. (1). To easily calculate G_i, we consider only a binary representation of the grating, digitizing it with a “1” representing a raised section and a “0” representing a lowered section, following previous work.²² With Fig. 1(a) as an example, a periodic grating would be represented as repetitions of “10,” while the sampling function would be represented as repetitions of “11 110 000” and superstructure 2 as repetitions of “10 100 000.” We consider repetitions a total of 240 digits in length for our Fourier analysis. The resultant Fourier spectra are normalized by appropriate scaling, here considering each digit to have a length of 3.5 μm. This can be generalized to more complex profiles, for instance, “1000” could be used to represent a Λ = 7 μm periodic grating with a ridge of 1.75 μm wide, with a corresponding change in the normalization length. While simplistic as a representation of a 3D permittivity profile, it has been shown that in the limit of weak scattering, this can fully represent the propagation of light in 1D,²⁹ such as along the surface of a grating. Further, discretization and Fourier transform of the permittivity profile forms the basis of the Fourier modal method for analyzing complex grating profiles.³⁰ Using the results of the Fourier transform, we see that a periodic grating possesses a single fundamental Bragg vector, (G₁), which will lead to a conventional grating-based thermal emission profile. This will be a single, spatially coherent mode that changes frequency as a function of angle. The sampling function has a longer period, and hence, a smaller Bragg vector, labeled G₀, has several harmonics supported in the FT profile. In the SSG, the sampling and periodic Fourier spectra become convolved, resulting in the creation of a spectrum with two G₂ and G₃ close to that of the periodic grating. We can tune the frequency of these additional peaks with the sampling period, becoming more closely spaced as the sampling period is increased [Fig. 1(b)]. The consequence of a grating with multiple G_i is that it should support multiple spatially coherent modes at a single frequency [Fig. 1(c)]. While this approach will not necessarily allow for the design of an arbitrary emission profile, possible using inverse design approaches, the structure is conceptually easy to understand. This makes it easy to verify the fundamental limitations associated with engineering spatial coherence, insight that is absent within such inverse-design approaches. Specifically, there is still a well-defined period L, which can be compared against the spatial coherence length of the propagating polariton l_c, allowing us to assess the limitations imposed by partial spatial coherence.

To test our hypothesis, a series of 5-mm x 5-mm area SSGs featuring 1-μm-tall grating ridges were fabricated using photolithography and reactive ion etching. Each element of the grating consisted of 2.7-μm-wide grating ridges [Fig. 2(a)]. The design of the grating structures [Fig. 1(b) SSG 1–5, all structures directly compared in Fig. 1] was performed using the traditional “sampled grating” forward-design methodology widely used in semiconductor laser design.²⁶ The thermal emission from these gratings was measured with the SSG heated to 266 °C [Fig. 2(a)] using a custom-built, angle-resolved thermal emission setup (more completely described in Ref. 4) In brief, a 1″ parabolic mirror with a focal length of 6″ (angular FW ∼4°, calibrated using a HeNe laser) collects light from a sample mounted on a rotatable (emitter angles between 0 and 70°) hotplate. This emitted light is passed through a polarizer (set to collect p-polarized light), which is then directed into an FTIR spectrometer (Bruker Vertex 70V, 2 cm⁻¹ resolution) equipped with an Mercury Cadmium Telluride (MCT) detector. Three pinholes are included in the optical path to reduce the amount of background IR radiation reaching the detector. In our technique, we measure calibrated directional emissivity ε_Ω as a function of angle, relative to a vertically aligned carbon nanotube reference sample that was previously calibrated to have an emissivity of 0.97. Additionally, for comparison, we provide the thermal emission spectra of a reference single period 1 μm-tall grating with 2 μm-wide ridges and Λ = 7 μm in Fig. S1.

Angle-dependent thermal emission spectra of SSG1 are represented in the contour plot provided in Fig. 2(b). We observe a spatially coherent mode that disperses as a function of angle in the spectral range of 797–870 cm⁻¹, in line with numerous earlier experiments.^10,11 The non-dispersive modes above 880 cm⁻¹ are localized modes confined to the grating ridges, which are also observed in previous reports.³¹ To verify that this mode can be attributed to the existence of an SPhP mode, we can overlay the analytical solution to the grating equation [Eq. (1)] associated with the multiple Bragg vectors present (G_i). Instead of presenting these vectors as reciprocal space values, we instead choose to present them in terms of the equivalent grating period Λ for clarity. We consider both the forward-propagating grating mode, which will have a positive dispersion with angle, and the reverse-propagating mode that will exhibit a negative dispersion. The spatially coherent mode for SSG1 closely matches the reverse propagating G₁ (Λ = 7 μm) mode from the grating equation with a slight detuning attributed to the finite height of the grating.³¹ We note that for SSG1, we also anticipate contributions from the Λ = 14 μm forward-propagating wave, which only manifests above 20° at higher frequencies >900 cm⁻¹. The forward-emitting wave interferes with the reverse-propagating wave, creating a frequency band with no coherent emission. This forward propagating wave also interferes with the non-dispersive modes, resulting in an angle dependence on the non-dispersing modes; however, quantification of this interaction is beyond the scope of this work. We compare the behavior of SSG1 with that of the response of SSG3 and 5, which both have two secondary Bragg vectors close in wavevector to the Λ = 7 μm mode. The emission from these two structures should, therefore, exhibit multiple dispersive modes, as shown in Figs. 2(c) and 2(d). These new frequency-dispersive modes exhibit a reduced emission intensity in comparison to the primary mode, as the corresponding G₂ and G₃ have a reduced amplitude [Fig. 1(b)], leading to a decrease in coupling to free space. Further, each mode matches reasonably well with the associated “effective periods” extracted from the Fourier spectrum for these Bragg vectors, that is, Λ = 6 and 8.6 μm for SSG3 and Λ = 6.33 and 7.7 μm for SSG5, respectively. These additional modes are also observed in SSG2 and 4 (shown in Fig. S2). This demonstrates that SSGs can indeed support more than one coherent, emissive mode, enabling the potential for design of the thermal emission spectra and angular distribution. To visualize the field distributions of the multiple spatially coherent modes observed in our experiments, we performed finite element method (FEM) simulations using CST Studio (Fig. S3). The simulated dispersion plot [Fig. S3(a)] agrees well with our measured emission plot for SSG 3 [Fig. 2(c)]. We should note that the forward-propagating grating modes (positive dispersion with the angle) are more apparent in the simulations. All the coherent modes [Fig. S3(b)–S3(d)] exhibit propagating characteristics, confirming the nature of propagating SPhPs launched by the grating elements.

The differences in the behavior of the devices can be more clearly compared by taking a single frequency and plotting the radiation pattern from the device, as shown in Fig. 3(a). We observe that the side “lobes” on the emission change significantly with design, but the primary emission frequency shows minimal tuning compared with the measured angular spread of approximately 9°. For further analysis and comparison to the k-space behavior of the grating, we can convert the angular data of Fig. 2(d) to wavevector units by using Eq. (1). We then normalize the data to the primary Λ = 7 μm emission frequency through an x-axis shift applied to each frequency independently creating a shifted wave vector k_s = k₀sin(θ)-2π/Λ. The purpose of this is to overlay the Fourier spectrum of Fig. 1 more effectively on top of our measured spectra accounting for changes in the angle and frequency, as shown for SSG5 in Fig. 3(b). We see that the secondary emission peak with the positive wavevector (corresponding to G₂) lines up extremely well with the secondary Bragg vector. The mode featuring the negative wavevector (corresponding to G₃) does not line up as closely; however, this is largely limited to the maximum collection angle of our variable-angle system (weak response at angles larger than 60°) and G₃ disperses mostly at high angles (Fig. 2). To further verify that these additional modes are due to the SSG pattern, we can also compare the angle-resolved data collected between SSGs 1 through 5, using the same normalization process, as shown in Fig. 3(c). We see that SSG2 through 5 exhibit a secondary emission peak with the positive wavevector (labeled G₂) that lines up well with the spacing predicted from the Fourier transform of the grating response. This provides a complete demonstration that SSGs can indeed be used to induce additional emission frequencies into the response of a grating-based thermal emitter with high degrees of spatial coherence.

One of the key questions that we can answer by comparing results for different gratings is the maximum possible SSG period that we can meaningfully use and still get coherent emission. This “longest period” could also be used as a template for the largest optical structure that could practically be used for any type of spatial coherence manipulation. We see from SSG5 [Fig. 3(a)] that the emissivity of G₂ and G₃ merge into the central mode at higher frequencies. We can compare the relationship between the grating period and the mode coherence length l_c using the following expression:^10,11

where $Δθ$ is the angular spread. By fitting a Lorentz curve to the data in Fig. 3, we extract $Δθ=9.2°$⁠, at 860 cm⁻¹. If we account for the angular spread of the collection optics (assuming a FW 4° box-car spread), we estimate the true angular spread for the primary peak as 8.3° through numerical deconvolution. Using this fitted value, we get a coherence length of l_c = 81 μm. This makes the super period of SSG5 close to this coherence length (as L = 70 μm), suggesting that this merging of the peaks can be attributed to operation close to the limit of spatial coherence. At other wavelengths, the coherence length is extended or reduced due to the corresponding polaritonic losses at that wavelength, and the three modes are well defined. This result suggests that the coherence length provides a fundamental limit to the superstructure period over which coherence can be manipulated. We envision that at longer periods, spatial coherence will completely break down and coherent emission is not observed. This result will be more broadly applicable to any manipulation of the partially coherent light source. All optical design methodologies must define an appropriate periodicity or total size for the optical element. Often, this choice is phenomenologically determined by multiple computationally intensive simulations examining different sizes of the structure. This is especially relevant for inverse-design methodologies, which offer little physical insight into an appropriate number of optical elements to include in the design process. The breakdown of coherence in our experimental results suggests that the coherence length can be used as an appropriate limit to the periodicity or total size of a coherent structure in such designs. This also acts as a useful tool for determining the appropriate choice of material system for a given directional application. For instance, this suggests that metal gratings with extremely long coherence lengths might act as a better platform for highly directional emission;¹³ however, greater power might be accessed using materials with shorter coherence lengths, broader emission, and more total emitted power. We also note that these designs could, in principle, be extended to two dimensions, as has been shown previously.³² 

In this work, we have studied how SSGs can be used to tailor the thermal emission spectrum of polaritonic thermal emitters. Our devices exhibit multiple coherent modes existing at different combinations of frequency/angle, controlled by the superstructure design. These correspond to the different diffractive modes revealed by the Bragg vectors of the 1D grating profile. While this approach offers a fairly limited scope in terms of creating a highly directed thermal emitter, it does suggest the limits over which thermal emission can be controlled. Our results suggest that the sample size or period must be shorter than the spatial coherence length of the propagating mode to be effective. Our work shows that the limits of directionality and manipulation are inexorably tied to the coherence length, identical to simple gratings. Ultimately, we believe that this promises to lead to the creation of sophisticated thermal sources, such as thermal lenses and holograms that offer applications in non-dispersive spectroscopy and/or thermal signature management.

See the supplementary material for the thermal emission from a reference periodic grating, thermal emission from additional superstructure gratings, and electromagnetic simulations for superstructure grating.

T.G.F. acknowledges funding provided through the School of Engineering at Vanderbilt University through the startup package of J.D.C. J.D.C. acknowledges support from the Office of Naval Research under Grant No. N00014-18-12107. The research at the Naval Research Laboratory was supported by the Office of Naval Research. A portion of this research was conducted at the Vanderbilt Institute of Nanoscale Science and Engineering. Funding for G.L. was provided through an STTR program provided by the National Science Foundation, Division of Industrial Innovation and Partnerships (IIP) (Award No. 2014798).