## Abstract

By integrating wave-type analysis and fluctuation-dissipation theorem, the enhancement of photon tunneling distance in near field thermal radiation through metallic nanopatterns with/without dielectric structures is theoretically studied. When metallic patterns are in the immediate proximity of the conductive emitter, substantial thermal electric enhancement at surface plasmon frequency is observed between the metallic patterns and the emitter when the periodicity of the thermal electric field along the emitter surface is around integer times of the period of the metallic patterns. The mechanism of field amplification is similar to Fabry–Perot type resonance between two reflecting surfaces. The strong thermal electric field from resonance allows long-distance photon tunneling observed in near field radiation at a ∼5 *μ*m separation distance when the same metallic patterns are placed on the collector surfaces. This value is nearly 50 times longer than that with bared emitter surfaces. This long-distance photon tunneling can also happen at a broader range of parallel wavenumbers (i.e., not determined by the period of the metallic patterns) at the surface plasmon frequency when the periodic metallic patterns' sizes are different each period. However, increasing the range of parallel wavenumbers in long-distance photon tunneling with this approach can reduce the strength of photon tunneling. The reduced tunneling strength can be brought up by attaching high refractive index dielectric resonators on top of the metallic patterns. The dielectric resonators on top of the metallic patterns show additional Mie-type resonance when displacement current is induced at the interface between the metallic patterns and the high refractive index dielectric. The higher intensity long-distance photon tunneling with a broad range of parallel wavenumbers can be valuable in harvesting the high intensity and high quality near field radiative energy with engineering feasible micron level vacuum gaps.

## Introduction

Thermal engines such as steam engines and gas turbine engines and their combinations are considered the most efficient way to harvest thermal energies. A combination of a steam and gas turbine engine with multistages intercooling and reheating can achieve up to 65% thermal energy conversion efficiency [1]. However, existing high-efficiency thermal engines are bulky and are mainly used in very large power plants. Efficiencies of small-scale high power density internal combustion engines, on the other hand, are limited to ∼20–30% [2]. Thermophotovoltaic (TPV) devices were considered as one of the high potential candidates to achieve high thermal energy conversion efficiency with full solid-state small-scale devices [3]. TPV uses a photovoltaic (PV) cell to convert thermal radiation from a hot object into electricity. TPVs have been predicted to provide a high energy density that is comparable to fuel cells and can be used for a wide range of fuels. Standard TPV devices provide ∼10–20% thermal energy conversion efficiency. The low energy conversion efficiency is because the broad emission spectrum from a hot object cannot be effectively harvested with the PV cells, leading to inefficient operations. By using a spectral selected emitter providing narrowband emission around the band edge of the PV cells, TPV can achieve ∼20–30% thermal efficiency at 1000 K level temperature. Theoretically speaking, close to Carnot efficiency (i.e., ∼70% at 1000 K) can be achieved with TPV by using a spectral selected emitter providing a very narrow emission band that exactly fits with the band edge of the PV cells [4]. In reality, TPV with a very narrow spectral selected emitter suffers from very low power density and a very low energy conversion efficiency of PV cells because of the relatively large dark current under low incoming radiation intensity [5].

Near field thermal radiation is proposed as a solution to current difficulties in TPV. Near field thermal radiation can provide hundreds of times of radiation intensity compared with blackbodies [6–8]. Moreover, the radiation spectrum can be quasi-monochromatic when the hot radiation emitter supports surface waves. Near field TPV was estimated to achieve >60% thermal energy conversion efficiency [6–9]. However, to effectively use near field radiation, PV cells should be placed within ∼100 nm away from the hot object in a vacuum environment, especially when we are interested in the near-infrared wavelengths that can be harvested with PV cells [10]. In demonstrated experimental study of near field thermal radiation, the 100 nm level vacuum gap between the hot object and the cold collectors (e.g., PV cells) was achieved with low conductivity spacers or piezo-electric stages [11–15]. Near field radiation heat flux is then determined as the measured heat flux from the hot to the cold object minus the conductive heat transfer rate from the hot to the cold domain, which can be estimated with Fourier's law [16,17]. Though in such configurations, the conductive heat transfer rate can be is nearly one order (or more) lower than the near field radiative heat transfer rate from the hot emitter surface to the cold collector surface, it is experimentally difficult, including the measurement of the gap size. One of the solutions of this difficulty will be increasing the separation distance between the hot emitter and the cold collector to a range larger than 1 *μ*m.

Thermal radiation can be considered as electromagnetic emission caused by randomly fluctuating thermal current inside a heated object [18]. Near field thermal radiation is mainly composed of the evanescent portion of the electromagnetic wave emission. We expect that optical resonators that can amplify the incident electric field through resonance can also be applied in amplifying the evanescent field of the thermal electric field. In return, the detection range of near field thermal radiation can be more than ten times improved. This study examined Fabry–Perot type resonance with metallic surfaces and Mie-type resonance with high refractive index dielectrics in amplifying the thermal electric field. To effectively use the available computational resources, we focus on 2D geometries and the associated spectral radiation intensity along the plane with the 2D geometries supporting field resonance. We expect the obtained results can be generalized for 3D cases. Theoretical modeling to calculate the spectral radiation field from a hot emitter to a cold collector with different field resonators is described in the Theory section.

## Theory

with $|ETE(kx,ky,z\u2032)|$ and $|HTM(kx,ky,z\u2032)|$ the amplitude of the thermal electric field in transverse electric (TE) modes and the amplitude of the thermal magnetic field in transverse magnetic (TM) modes, respectively; $\omega $ the angular frequency of the radiation; $\epsilon $ and $\mu $ the permittivity and permeability of the emitter; $kz=\omega 2\epsilon \mu \u2212kx2\u2212ky2$ with $kx$ and $ky$ the wavenumbers in the *x* and *y* directions, respectively; “*” the complex conjugate.

*z*= 0

^{+}plane) for a given emitter temperature

*T*can be expressed as follows with $kz\u2033=\epsilon 1\u2033\epsilon 0\mu \omega 2/2kz\u2032$ and $(kzkz*+kxkx+kyky)kz\u2032/\mu \omega 2=RE[\epsilon kz*]$

*z*= 0

^{+}to a parallel collector across a gap with thickness

*d*as illustrated in Fig. 1, the transported radiative heat flux $q\u20331\u22122$ can be expressed as

$FTE(kx,ky)$ and $FTM(kx,ky)$ are the radiation power ratios at $z=d\u2212$ (just arriving at the collector) and $z=0+$ (before leaving the emitter) surfaces of Fig. 1 in TE or TM modes, respectively. $18\pi 3\Theta (\omega ,T)$$RE[k1z\u2032kz]FTE(kx,ky)$ is equal to directional spectral radiation intensity in TE mode received by the collector when $(kx2+ky2)\u2264\omega 2\epsilon \mu $. Therefore, we define $18\pi 3\Theta (\omega ,T)$$RE[k1z\u2032kz]FTE(kx,ky)$ the generalized directional spectral radiation intensity in TE mode in the near field radiation heat transfer for all $kx$ and $ky$ values. Considering $18\pi 3\Theta (\omega ,T)$ is the directional spectral radiation intensity of a blackbody in terms of parallel wavenumbers $kx$ and $ky$, $RE[k1z\u2032kz]FTE(kx,ky)$ which is the ratio between the generalized directional spectral radiation intensity of a real surface and the directional spectral radiation intensity of a blackbody can be realized as the normalized generalized directional spectral radiation intensity of TE mode with respect to the blackbody radiation intensity. Following the same terminology, we define $18\pi 3\Theta (\omega ,T)$$RE[RE[\epsilon kz*]\epsilon kz*]FTM(kx,ky)$ and $RE[RE[\epsilon kz*]\epsilon kz*]FTM(kx,ky)$ as the generalized and normalized directional spectral radiation intensity in TM mode for the near field radiative heat transfer in this study.

$FTE(kx,ky)$ and $FTM(kx,ky)$ values, affected by the resonance structures between the emitter and the collector, can be determined with semi-analytical Fourier modal method [22,23]. To simplify the calculation and present the enhancement of near field radiation with 2D plots, we focus on radiative heat transfer in the *x*–*z* plane of Fig. 1. In this situation, periodic resonators are placed along the *x*-direction with infinite width in the y-direction to regulate the thermal electric field in the *x*–*z* plane. In other words, we mainly study the values of $18\pi 3\Theta (\omega ,T)RE[k1z\u2032kz]FTE(kx,0)$ and $18\pi 3\Theta (\omega ,T)RE[RE[\epsilon kz*]\epsilon kz*]FTM(kx,0)$ with the semi-analytical Fourier modal method. The obtained results are discussed in the Results and Discussions section. Note that it has been proven that our computational model based on Poyting vector theorem generates the same results as Rytov's theory for near field thermal radiation between parallel surfaces. Rytov's solution can also be derived analytically with the Poynting vector approach in previous reports [20,21,24,25]. It is also noted that there are many different numerical methods for near-field thermal radiation [26]. We picked Wiener-Chaotic expansion and rigorous coupled-wave analysis based on our convenience.

## Results and Discussions

To verify the theoretical modeling and provide reference results for this study's simulations, we first simulated the normalized directional radiation intensity from a conductive emitter to a conductive collector supporting short wave infrared surface plasmon. Both the emitter and the collector are considered as semi-infinitely thick with plasma frequency of $2.068\xd71014$ Hz and damping coefficient of $4.449\xd71012$ Hz, which are similar to a diluted gold compound supporting infrared surface plasmons [27]. Figure 2 shows the simulation results when the separation distances between the emitter and the collector are 0.1 *μ*m, 1 *μ*m, and 5 *μ*m. Note that for all the results presented in this study, we consider only the radiation from the emitter to the collector, equal to 0 K temperature for the collector. Also, because we focus on normalized directional spectral radiation intensities, temperature of the emitter does not show up in all the plots of flux.

As expected, photon tunneling (when $kx>\omega /c$) happens only under TM radiation from visible to short-wavelength infrared, stimulating surface plasmon waves around the frequency of surface plasmon resonance (SPR) of the emitter/vacuum and the collector/vacuum interfaces (Figs. 2(a) and 2(b)). We mainly focus on this wavelength range that photons can be harvested with photovoltaic devices. The range of $kx$ with photon tunneling decreases significantly when the separation distance between the emitter and the collector increases from 0.1 *μ*m to 1 *μ*m (Figs. 2(b) and 2(c)). Almost no photon tunneling can be observed when the separation distance between the two domains equals 5 *μ*m (Fig. 2(d)). In addition to the photon tunneling at the frequencies around the surface plasmon resonance when $kx>\omega /c$, the enhancement of far-field radiation heat transfer occurs in a format of dispersion curves when $k\u2264\omega /c$ (Figs. 2(c) and 2(d)). The number of dispersion curves with enhanced far-field radiative heat transfer when $k\u2264\omega /c$ increases with respect to the separation distance between the emitter and the collector surfaces. These dispersion bands showing enhancement of radiative heat transfer can be understood as a consequence of Fabry–Perot resonance between the reflecting emitter and the collector surfaces, which supports more resonance modes with large separation distances between the emitter and the collector. Figure 2(e) shows the normalized directional spectral radiation intensity between the emitter and collector with a 0.1 *μ*m separation distance (using Mathematica) from the analytical solution [21,25]. The plot generated from the well-adopted analytical solution agrees well with our computational results in Fig. 2(b), which supports the validity of the computational model adopted in this study.

After establishing the normalized directional radiation intensity with bared emitter and collector surfaces as the reference case, we calculated the normalized direction radiation intensity between the same emitters and collectors with additional periodic metallic patterns on top of the emitter and the collector surface, as illustrated in Fig. 3(a). Plasmon frequency and damping coefficient of the metallic patterns are $2.068\xd71015$ Hz and damping coefficient of $4.449\xd71012$ Hz, respectively, which are the same as gold. The reason for selecting pure gold for the nanostructures rather than diluted gold is because the higher plasma frequency can provide stronger reflection at near IR and the associated Fabry–Perot resonance. The width and the height of the period metallic patterns are 200 nm and 50 nm, respectively. The separation distance between the metallic patterns and the emitter/collector surface is equal to 50 nm, as illustrated in Fig. 3(a). This 50 nm gap provides strong photon tunneling based on our tests. Figures 3(b)–3(e) are the obtained normalized directional spectral radiation intensity between the emitter and the collector when their separations distances are equation to 0.1 *μ*m, 1 *μ*m, and 5 *μ*m. Photon tunneling also appears only for TM mode radiation when periodic metallic patterns appear. Though the tunneling strength rapidly decreases with respect to the separation distance between the emitter and the collector as in Figs. 3(b)–3(d) for most $kx$ values, the tunneling strength remains high when $kx$ are around 5*π* (1/*μ*m) and 10*π* (1/*μ*m) even when the separation distance reaches 5 *μ*m. These specific $kx$ conditions are equal to $2\pi /(2w)$ and $4\pi /(2w)$ with $2w$ the period of the metallic patterns, that is 400 nm in this study. In other words, very long distance photon tunneling in near field thermal radiation can be observed when the periodicity of the half period wide metallic pattern is around integer times of the modulation period of the thermal electric field in the *x*-direction. To better understand the origin of this long photon distance when $kx$ equals $2\pi /(2w)$ or $4\pi /(2w)$, we simulated the thermal electrical field distribution from the emitter to the collector with finite element method (Figs. 4(a) and 4(b)). When an incident radiation field with parallel wavenumber $kx$=$2\pi /(2w)$ at a frequency around SPR is delivered from the emitter domain, a strongly amplified electric field can be observed between the periodic metallic patterns and the emitter surface (Fig. 4(a)). This amplified electric field can be understood as a consequence of field resonance between the metallic patterns and the conductive emitter across the 50 nm gap, which is similar to far-field Pabry-Perot resonances.

Interestingly, the resonance adjacent to the emitter surface can stimulate a corresponding electric field resonance between the metallic patterns adjacent to the collector. This coupling of resonances allows long-distance transport of near field energy, which can be considered as long-distance photon tunneling around the SPR frequency when $kx$=$2\pi /(2w)$. Similar but weaker electric field resonance between the metallic patterns and the emitter/collector surface can be observed at $kx$= $4\pi /(2w)$ around the SPR frequency (Fig. 4(b)), which corresponds to the weaker photon tunneling around $kx$= $4\pi /(2w)$. In addition to a main photon tunneling band around $kx$=$2\pi /(2w)$ or $4\pi /(2w)$ at the SPR frequency, many thin photon tunneling bands with similar dispersion relation as far-field radiation enhancement bands due to Fabry–Perot resonance (Figs. 2(c) and 2(d)) can be observed around $kx$=$2\pi /(2w)$ or $4\pi /(2w)$ at the SPR frequency (Figs. 3(c) and 3(d)). These additional thin photon tunneling bands can be because the metallic patterns act as diffraction gratings when they are very close to the emitter surface. The gratings modify the momentum of near field radiation from the emitter with integer numbers of $kx$. Therefore, the photon transmission band for $kx\u2264\omega /c$ can appear periodically at higher $kx$ values similar to the electron band structures at high Brillouin zones. For the specific emitter and collector selected in this study, the enhanced photon transmission bands dominate the far-field radiation for $kx\u2264\omega /c$ as illustrated in Figs. 2(c) and 2(d).

After determining the abilities of metallic patterns having a width equal to half of the pattern periods in significantly improving the photon tunneling distance around $kx$=$2\pi /(2w)$ or $4\pi /(2w)$ at the SPR frequency in near field thermal radiation, we further explore the possibilities to increase the range of $kx$ values allowing long-distance photon tunneling around the SPR frequency similar to that in Fig. 2(b). The approach tested in this study is by slightly tweaking the geometry in Fig. 3(a) by reducing the metallic resonators' width from 200 nm to 150 nm every other period, as illustrated in Fig. 5(a). The obtained normalized directional radiation intensity in TM mode at different separation distances between the emitter and the collector are shown in Figs. 5(b)–5(d). Compared with metallic patterns in Fig. 3(a), long-distance near field radiation occurs not only around $kx$=$2\pi /(2w)$ and $4\pi /(2w)$ but also around $kx$=$\pi /(2w)$ and $3\pi /(2w)$ with a much weaker intensity. In other words, through breaking the periodicity of the sizes of the metallic patterns, we can trigger photon tunneling at additional $kx$ wavenumbers for broader $kx$ range long distance photon tunneling at SPR frequencies between the emitter and the collector. However, compared Figs. 5(b)–5(d) to 3(b)–3(d), photon tunneling intensity around $kx$=$2\pi /(2w)$ and $4\pi /(2w)$ are reduced as a tradeoff. Note that broader range photon tunneling is the basis of quasi-monochromatic near field radiation at the SPR frequency for bare emitters and collectors in Fig. 2(b). To better understand the increased $kx$ range for long-distance photon tunneling by adjusting the metallic patterns' size, electric field distribution between the emitter and the collector with the perturbed metallic patterns is again simulated with finite element method. Figure 6(a) shows the electric field distribution between the emitter and the collector at $kx$=$\pi /(2w)$, one of the wavenumbers providing strong photon tunneling. Compared with Fig. 4(a), a similar but weaker electric field between the metallic patterns and the collector under the same photon tunneling condition. So does the resulting photon tunneling strength when we compare Fig. 5(c) with Fig. 3(c). When we change $kx$ to $3\pi /(2w)$, the new photon tunneling condition when we break the exact periodicity of the metallic patterns, minimum field resonance is induced between the metallic patterns and the collectors. So does the resulting very weak photon tunneling in Fig. 5(d). Therefore, to extend the $kx$ range of long-distance photon tunneling by adjusting the sizes of the metallic patterns in each period, we must further enhance the resonance strength of the thermal electric field, especially at the $kx$ values different from $2\pi /(2w)$ and $4\pi /(2w)$.

The last condition we examined in this study is the possibility to enhance the intensity of long-distance photon tunneling in near-field radiation through a combination of metallic patterns and dielectric resonators. Based on the previous cases, we learned that the long-distance photon tunneling appearing at SPR frequencies is due to strong thermal electric field resonance and the associated amplification of the thermal electric field in the near field. Therefore, adding an additional dielectric resonator supporting Mie-type resonance on top of the metallic patterns may further improve the strength of the thermal electric field through the extra resonance and increase the tunneling strength of photons in near-field thermal radiation. Figure 7(a) illustrates the combination of rectangular dielectric resonators with the periodic metallic patterns in the previous case. The rectangular dielectric resonators with a high refractive index (*n* = 6) are attached on the top of the metallic patterns. The height of the dielectric resonator is selected as 130 nm, which shows better resonance results in our tests. Figures 7(b)–7(d) illustrate the simulation results. Compared with Figs. 5(b)–5(d), the strength of photon tunneling increases modestly for all $kx$ values. Figures 8(a) and 8(c) show the electric field distribution between the emitter and the collector with the metallic patterns and the dielectric resonators under the resonance condition. Additional field resonance similar to Mie-type resonance in dielectric resonators can be observed in the selected rectangular dielectric resonator, which enhances the thermal electric field arriving at the collector. The additional resonance in the dielectric resonator can be observed more obviously in the magnetic field plot (Figs. 8(b) and 8(d)). The higher-strength magnetic field can be observed inside the high refractive index dielectric resonators due to field resonance. We attribute the stronger photon tunneling with the integration of metallic patterns and rectangular dielectric resonators to the additional field resonance in the dielectric resonators. The combined resonance in the integration of metallic patterns and the rectangular dielectric start from near field Fabry–Perot-like resonance between the conductive emitter/collector and the metallic patterns as in the previous case. The amplified electric field from resonance causes a strong oscillating current in the metallic patterns, which induces displacement current at the interface of the metallic patterns and the high refractive index dielectric resonators. The displacement current and the resulting electromagnetic field resonance can happen in the dielectric resonators with appropriate geometries similar to the far-field Mie-type resonators [28]. Note that the observations are only along the *x*–*z* plane for TM mode near field thermal radiation.

By integrating the normalized directional spectral radiation intensity for all parallel wavenumbers, we can have the spectral radiation intensity between the emitter and the collector divided by $18\pi 3\Theta (\omega ,T)$. We name this quantity as modified spectral radiation intensity, which can be larger than one based on its definition (i.e., different from normalized spectral radiation intensity commonly adopted in other radiation studies). We calculated the modified spectral radiation intensity from the emitter to the collector without optical resonators, with metallic and dielectric optical resonators as in a configuration of Fig. 7(a), and with metallic only optical resonators as in the configuration of Fig. 5(a). The separation distance “*d*” is selected as 5 *μ*m to calculate these values along the x-z surface based on results in Figs. 2(d), 7(d), and 5(d), respectively, for a fair comparison. As illustrated in Fig. 9, a local peak appears around the SPR wavelength when strong thermal electric resonance occurs and the associated long-distance photon tunneling for both metallic only and metallic + dielectric resonator conditions. The spectral radiation is lower for other wavelengths when the emitter has optical resonators. The additional resonators block the radiative heat transfer if no resonance is induced. Moreover, metallic + dielectric optical resonators provide slightly stronger and wider tunneling bandwidth than metallic-only optical resonators.

## Conclusions

Based on wave type analysis of near field thermal radiation induced by randomly fluctuating thermal current described with fluctuation-dissipation theorem, we studied enhancement of near field thermal radiation and the associated photon tunneling between two metallic surfaces supporting near IR surface plasmon through periodic metallic patterns with/without dielectric resonators. For bared surfaces, photon tunneling when $kx>\omega /c$ becomes very weak when the separation distance between the two surfaces is larger than ∼1 *μ*m. When 50 nm thick gold-like metallic patterns are added on both the emitter and the collector surfaces with a 50 nm gap, photon tunneling at the surface plasmon frequency can still be observed around $kx$=$2\pi /(2w)$ and $4\pi /(2w)$ with $2w$ the period of the metallic patterns even at a 5 *μ*m separation distance. This long photon tunneling distance is nearly two orders longer than that in the previous study with bare metallic surfaces. Based on the corresponding finite element analysis of the electric field distribution, the enhanced photon tunneling distance at the specific $kx$ values is because of the electric field resonance between the highly reflecting gold-like metallic patterns and the base conductive material. This physical phenomenon is similar to Fabry–Perot resonance between two reflecting surfaces but with a much smaller gap (∼50 nm). When we adjust the size of the metal patterns by reducing their widths from $w$ to 0.75$w$ every other period, long-distance photon tunneling happens at additional $kx$ values around $0.5\pi /w$ and $1.5\pi /w$. However, the strength of photon tunneling is reduced compared with single-sized metallic patterns. The reduced tunneling strength can be brought up by attaching appropriate high refractive index dielectric resonators on top of the metallic patterns to have an additional Mie-type resonance. Based on the technique of using metallic patterns with high refractive dielectric resonators to combine Fabry–Perot type and Mie-type resonances, quasi-monochromatic near field thermal radiation reported between two metallic surfaces at a ∼100 nm separation distance at surface plasmon frequencies can be recovered at a ∼5 *μ*m separation distance. This much longer distance quasi-monochromatic near field thermal radiation between the emitter and the collector due to broad $kx$ range long-distance photon tunneling can allow easier usages of the higher power density and high quality near field radiative energy in thermophotovoltaic devices with engineering feasible μm level vacuum gaps.

Note that the nano-optical structures are assumed to be cold and do not participate in the emission process, considering we focus on their contributions as optical resonators to the thermal electric field from the semi-infinitely large emission/collection domain. The associated physics of this study is similar to energy transfer between an array of Fabry–Perot resonance emitting antennas and Fabry–Perot resonance collecting antennas (FPRA) to increase the photon tunneling distance in near field thermal radiation, which can be new compared with previous studies [29]. The nanopatterns with nonzero imaginary parts of the permittivity (e.g., the metallic patterns) can provide their own thermal emission. The roles of nanopatterns in near-field regime will be investigated in future studies. With consideration of both the radiation from the semi-infinite domain mediated with the nanopatterns and the radiation from the nanopatterns, the total emission power between the emission and the collection domain can be stronger than what we predicted in this study.

We expect the demonstrated approach to extend the photon tunneling distance in near field thermal radiation through adopting optical resonators to amplify the evanescent thermal electric field can be further improved in the future by using optical resonators with higher quality factors. These optical resonators can be fabricated with existing microfabrication techniques such as e-beam lithography and imprint lithography.

Also, these enhanced near field radiation and longer photon tunneling distance can be experimentally verified with near field scanning spectroscopies.

## Funding Data

National Science Foundation (CBET-2117953; Funder ID: 10.13039/100000001).