1. Introduction

High performance incandescent light bulbs (ILBs) rely on the tailored emission of thermal radiation from an emitter towards the human eye. By tailoring the emission spectrum to match the human eye sensitivity, ILBs have the potential to overcome current LEDs in terms of both luminous efficiency and color rendering index (CRI) [1,2]. Although their potential to surpass LEDs is attractive, experimental non-idealities in the thermal-to-optical energy conversion have severely limited the efficiency of real devices below 15% [3] of the theoretical limit, which is not competitive with current LEDs. Similarly, the performance of systems such as thermophotovoltaic (TPV) devices that rely on selective emission from a thermal emitter to generate electrical power using a photovoltaic (PV) cell is also affected by system-level non-idealities and have yet to surpass [4] the Shockley-Queisser limit [5] of single junction PV cells.

Many studies [3,6–18] have focused on improving different aspects of emitters and cold-side filters (e.g., high temperature stability, high spectral selectivity and sharp cutoff) to achieve spectrally-tailored emission. While these improvements are necessary to achieve high conversion efficiency [19] of thermal radiation to useful light, the overall system efficiency is often significantly lower than expected based on the emitter and filter characteristics due to imperfections such as radiation leakage and parasitic heat losses. As an example, Fig. 1 illustrates the importance of view factor F between a tungsten emitter at 2800 K and different cold-side filters (reflectivity R_VIS = δ in the visible and R_IR = 1-δ in the infrared) on the conversion efficiency of broadband thermal radiation to emission in the visible spectrum. Here, the view factor is defined as the proportion of diffuse radiation leaving the emitter directly reaching the filter, with F = 1 representing the ideal case (i.e., all the emitted radiation reaches the filter) and F = 0 representing the worst case (i.e., none of the emitted radiation reaches the filter). The efficiency is defined as the ratio of visible light (400 nm < λ < 700 nm) transmitted through the cold-side filter and the total energy radiated by the emitter:

 

Fig. 1 Influence of view factor between a tungsten emitter and different cold-side filters on the system efficiency as defined by Eq. (1). (a) The optical properties of the tungsten emitter at 2800 K and filter (reflectivity R_VIS = δ in the visible and R_IR = 1-δ in the infrared; absorption in the filter is neglected). (b) Efficiency of different emitter-filter systems as a function of view factor and non-ideality factor δ in the filter. The efficiency is strongly dependent on the view factor, particularly for δ = 0.01 and F > 0.95, due to the high average number of reflections that occur between the emitter and filter before a photon is absorbed or escapes the system.

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Figure 1(b) shows that the view factor between the emitter and filter significantly affects the system efficiency with most gains in efficiency achieved with high-performance cold-side filters at high view factors (3 × increase in efficiency from F = 0.95 to F = 1 for δ = 0.01). However, due to practical challenges in achieving precise alignment and due to emitter deformation with time, view factors ≤ 0.95 are typically observed in practice [2,3]. While improving the cold-side filter certainly helps increase the system efficiency, merely focusing on the visible and infrared reflectivities of the filter alone while neglecting improvements in the view factor will only yield modest gains in the system performance. Thus it is important to strive to achieve view factors as close to unity as possible to maximize the efficacy of the emitter-filter system.

Several approaches already exist to increase the effective view factor between the emitter and the filter. Perhaps the most simplistic approach for a planar geometry is to decrease the spacing between the two surfaces. However, as was previously mentioned, it is practically difficult to achieve F > 0.95 and reaching F = 1 would require placing the emitter in contact with the cold-side filter which will lead to thermal degradation of the filter [20,21]. Another method to increase F is to use a directional emitter [22–25] that preferentially emits towards the cold-side filter, thus reducing the number of photons escaping through the gap between the filament and filter. Although this approach could provide some gains in efficiency, the challenges associated with the fabrication and thermal stability [26,27] of the emitter make it difficult to implement. Another approach [28], which was recently explored theoretically, consists of inserting a low thermal conductivity waveguide between the emitter and the cold-side filter to redirect otherwise “lost” photons back to the filament or filter. Although this method allows for great improvement in the view factor, it poses challenges in the temperature stability of the waveguide material, especially for ultra-high temperature emitters (i.e., incandescent lamp bulbs with T_emitter ≈2800 K), and the increased thermal load on the cold-side filter due to conduction through the waveguide can lead to the thermal degradation of the filter [29]. Replacing the waveguide with specular reflectors between the emitter and filter could provide similar improvements in the view factor, without the added thermal load on the filter and thermal stability concerns.

To quantify the benefits of using a waveguide, earlier work [28] proposed an approximate solution to obtain the view factors between the emitter and filter as a function of the waveguide dimensions and reflectivity. However, due to the specular nature of reflections, a more accurate method to calculate the radiative heat transfer between specular surfaces involves solving for specular view factors. While diffuse view factors between parallel plates can be determined using well-known analytical solutions, the calculation of specular view factors is challenging and not always available in the literature, especially for fully specular 3-D enclosures. Previous work has proposed different numerical [30,31] or analytical [32–34] solutions for specular view factors for 2-D and 3-D geometries, however partially specular enclosures were assumed. Though the approach to add specular reflectors or a waveguide between the emitter is promising, a detailed analytical model that captures the specular nature of the reflections as well as an experimental demonstration of the concept are still lacking.

In this work, we theoretically and experimentally investigated an approach that utilizes specular reflectors to increase the effective view factor between the emitter and cold-side filter resulting in significant improvement in the system efficiency. Specular reflectors, such as mirror-polished metallic surfaces, are relatively low-cost, can easily be added to an existing system and do not induce additional thermal load on the filters. We developed an analytical model to obtain the heat flux at each surface of 2-D and 3-D fully specular rectangular enclosures and then derive analytical solutions to the specular view factors inside the cavity. Finally, we demonstrated the potential of our approach and validate the model with experiments using an emitter-filter setup designed for lighting applications.

2. Theoretical model

In order to accurately model and understand the performance enhancement due to the addition of specular reflectors between a thermal emitter and a cold-side filter, we first develop an analytical model to calculate the radiative transfer between all the interacting surfaces. We assume that all surfaces form a rectangular enclosure and act as specular reflectors, as would be the case for a polished metal emitter and reflectors, and a multilayer dielectric filter.

When calculating the radiative heat transfer between surfaces of an enclosure, it is usually assumed that all surfaces reflect diffusely and that the radiosity is independent of the direction of the irradiance. However, in practice, the reflectance ρ of surfaces can be approximated as a combination of diffuse ρ^d and specular ρ^s components of reflectivity such that ρ = ρ^d + ρ^s. In an enclosure with one or multiple specular surfaces, the underlying assumptions for diffuse view factors between the surfaces no longer hold and differences in radiative transfer can therefore occur due to the specular nature of the reflections and the directional dependence of the reflected rays. To accurately account for specular reflections, which carry additional information about the directionality of the incoming ray of light, we use a specular view factor [35]:

The specular view factor $F_{i-j}^s$ may be viewed as the sum of the diffuse view factor and every other way that radiation from A_i can reach A_j by mirror-like reflections on specular surfaces. Using specular view factors, we can compute the radiative transfer between N surfaces in an enclosure with partially-specular surfaces by simultaneously solving [35]:

We now consider a 2-D or 3-D enclosure consisting of a hot side emitter (A_e) with a cold-side filter (or photovoltaic cell; A_f) and side reflectors (A_ref) with specular-only components of reflections (Fig. 2), similar to an incandescent lighting or TPV system. Equation (3) can then be simplified by assuming E_e >> E_f and E_e >> E_ref (i.e., the emitter is at a much higher temperature than the surrounding surfaces), no external radiation source, and only specular reflections (e.g., highly polished emitter and side reflectors, and a multilayer dielectric filter) such that the heat flux from the hot emitter (or the total power consumption per emitter area) becomes:

 

Fig. 2 Schematics showing fully-specular (a) 2-D and (b) 3-D enclosures for radiative heat transfer model with the emitter (A_e), the cold-side filter or photovoltaic (PV) cell (A_f) and side reflectors (A_ref). Only three of the four surfaces of the side reflectors are shown in the 3-D enclosure (b) for visualization.

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Using Eqs. (5)-(6), the power consumption of the emitter and the heat flux at the cold-side filter can be calculated if we assume gray optical properties for all the surfaces. To better account for spectrally-dependent optical properties of real surfaces, wavelength-dependent versions of Eqs. (5)-(6) may also be used to obtain the spectral heat flux at the emitter or filter. Finally, using the previously developed equations, we calculate the efficiency of a system having an emitter, cold-side filter and side reflectors to convert thermal radiation to visible light (or any spectral band) using Eq. (7):

Interestingly, Eq. (7) shows that the efficiency can be increased by maximizing $F_{e-f}^s$ in the visible spectrum (i.e., more visible radiation is transmitted through the cold-side filter or absorbed by the photovoltaic cell) and $F_{e-e}^s\left(1-{\rho }_e^s\right)$ at all wavelengths to increase reabsorption at the emitter.

To solve Eq. (7), specular view factors must still be calculated for the particular geometry and optical properties of the interacting surfaces. Although several analytical solutions and numerical methods to calculate specular view factors have been proposed in the literature, analytical solutions to 2-D and 3-D specular view factors in a fully specular rectangular enclosure have not yet been described. We have thus derived analytical solutions to these specular view factors (see Appendix A and B) to evaluate the heat flux at the emitter [Eq. (5)] and filter [Eq. (6)], and the system efficiency [Eq. (7)]. The analytical expressions of the specular view factor in 2-D and 3-D enclosures were validated using ray tracing simulations for a range of parameters (see Appendix C). A comparison between diffuse and specular only reflections as well as between 2-D and 3-D enclosures is presented in Appendix D.

3. Modeling results

Using Eq. (7) and specular view factor expressions (Appendix B), we model the thermal-to-optical efficiency of an incandescent thermal emitter (A_e) coupled with a selective cold-side filter (A_e) and specular side reflectors (A_ref) as illustrated in Fig. 2(b). For the results presented in this section, it is assumed that the surfaces have a specular-only component of reflectivity (i.e., ρ^d = 0), the filter has a specular reflectivity ${\rho }_{IR}^s$ = 0.85 in the infrared and perfect transmissivity τ_VIS = 1 in the visible, the emitter is gray and at 2800 K unless specified, and the side reflectors have a gray specular reflectivity of ρ^s = 0.95, as could reasonably be achieved using polished silver or copper side reflectors.

Figure 3(a) shows the influence of side reflectors on the efficiency of the system for the case of gray emitters. At the same time, the influence of geometry (ratio of emitter-filter gap spacing to length of the emitter defined by the dimensionless parameter S*) and optical properties of the emitter on the system efficiency are analyzed. Significant gains in efficiency are observed for most values of S* by adding specular reflectors, as these “mirrors” redirect the otherwise lost thermal radiation (radiation not directed towards the filter or emitter) back to the filter or emitter for increased infrared radiation recycling. Figure 3(a) also shows that the plateau of maximum efficiency at every gray emitter emissivity occurs at more than an order of magnitude larger S* with reflectors than without. This can have important implications in devices where S* < ~10⁻³ is challenging to achieve due to system constraints and alignment uncertainty. Adding highly reflecting side reflectors to an existing system can therefore significantly improve the efficiency of the system while allowing for larger S*. Figure 3(a) also shows that for gray emitters, a higher emissivity translates into higher efficiency. This is expected since the ratio of visible to infrared light emission is always constant for gray emitters and the higher emissivity (absorptivity) results in greater reabsorption of infrared radiation reflected by the filter. For example, with a blackbody emitter, all the radiation reflected by the filter reaching the emitter will be reabsorbed, while for a low emissivity emitter, most of that radiation will be reflected by the emitter and eventually lost due to non-idealities in the filter and side reflectors.

 

Fig. 3 Influence of specular side reflectors (ρ^s = 0.95) in a 3-D rectangular enclosure on the efficiency for different (a) gray emitters and (b) selective emitters as a function of dimensionless gap spacing S*. Selective emitters have blackbody emission in the visible and emissivity ε_IR in the infrared.

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Figure 3(b) demonstrates the impact of specular side reflectors on the efficiency of a system having a selective emitter (ε_vis = 1 in the visible and ε_IR in the infrared) as opposed to a gray emitter. Significant improvements in efficiency are observed with the addition of side reflectors to the system, similar to the case with a gray emitter in Fig. 3(a). However, comparatively higher efficiencies are observed due to the lower infrared emissivity of the emitters. Even though low ε_IR emitters have a lower potential to reabsorb the infrared radiation reflected by the filters, they emit significantly less infrared radiation to begin with (radiation not emitted is radiation that does not need to be recycled), thus increasing the efficiency.

It is also worth noting that for both gray and selective emitters, an increase in the reflectivity of the side reflectors shifts the transition from high efficiency to low efficiency towards larger values of S* as illustrated in Fig. 4. Ultimately, this transition would occur at an infinitely large value of S* for side reflectors with perfect reflectivity (ρ^s = 1). This highlights the strong dependence of efficiency on the side reflector reflectivity, particularly as it approaches unity.

 

Fig. 4 Influence of specular side reflectors reflectivity on system efficiency. Same optical properties of the gray emitter (ε = 0.2) and filter (ρ_IR = 0.85 in the infrared and τ_VIS = 1 in the visible) were assumed for all curves.

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4. Experimental demonstration

While we have theoretically demonstrated the advantages of using specular side reflectors, we also experimentally examined the power consumption of a planar tungsten emitter (T_e = 2000 K) coupled with nanophotonic filters at various spacings with and without specular side reflectors.

As shown in Fig. 5(a) and 5(b), the experimental setup consisted of a 6 mm × 10 mm tungsten emitter symmetrically sandwiched between two planar optical filters held and maintained at a controlled gap (0.5 mm, 1 mm, 3.3 mm and 25.4 mm) by copper supports [2,29]. Molybdenum electrical feedthroughs allowed for support and resistive heating of the emitter to T_e = 2000 K inside a bell jar vacuum chamber maintained at a pressure below 10⁻⁶ torr. A four-wire measurement of the emitter resistance and PID control were used to accurately measure and control the resistance (i.e., temperature) of the emitter. The specular side reflectors were made of silver-coated polished copper, and were inserted between the filters to surround the emitter. For reference, the optical properties of the tungsten emitter, the nanophotonic filters (optimized to transmit visible light while reflecting infrared light) and the silver-coated copper side reflectors are presented in Appendix E (Fig. 11).

 

Fig. 5 (a) Schematic of the experimental setup. A planar 6 mm × 10 mm tungsten emitter is sandwiched between two cold-side nanophotonic filters (see Appendix E for optical properties) maintained at a fixed spacing by copper supports. The experiment was set up inside a bell jar vacuum chamber to minimize the effects of contaminants. (b) Close-up view of the assembled system consisting of the emitter, filters, and side reflectors. (c) Comparison of power consumption of a tungsten emitter coupled with nanophotonic filters at different spacings with and without silver coated side reflectors. Dimensionless emitter-filter gap spacing (S*) assumes a characteristic emitter length of 7.5 mm corresponding to 4 × Area/Perimeter. Due to the limited vacuum chamber size, the experimental measurement reported at an emitter-filter gap spacing of ∞ mm was performed without filters, equivalent to filters being so far to not interact with the emitted radiation (i.e., $F_{e-e}^s$ = 0 indicating that no emitted radiation is reflected back to the emitter), as suggested by the theory for emitter-filter gap spacing > 10³ mm. The shaded areas represent the upper and lower bounds of the 3-D theoretical model when modeling for absolute changes of ± 0.01 in the reflectivity of the surfaces and ± 20 K in the emitter temperature. The error bars for the data points, including experimental error of the power consumption (current and voltage) and emitter-filter spacings, are smaller than the markers.

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Figure 5(c) shows the power consumption as a function of emitter-filter spacing, with and without side reflectors. Similar to the modeling results shown earlier, the power consumption decreases with smaller emitter-filter spacing as more and more infrared radiation is recycled back to the emitter (i.e., an increasing proportion of the emitted radiation only interacts with the filter and emitter leading to a higher and converging view factor $F_{e-e}^s$ for both No reflector and With reflector cases). We also observe a shift of the power consumption curve towards larger spacings when side reflectors are added, indicating more effective recycling of infrared radiation at the emitter. Moreover, while emitter-filter spacings below 500 µm are difficult to achieve practically due to the deformation and alignment of the emitter, we see that the addition of side mirrors reduces power consumption by >10% – from 18.6 W to 16.4 W at the smallest spacing of 500 µm. Furthermore, only a small increase in power consumption is observed when going from 500 µm (16.4 W) to 3.3 mm (17.4 W) spacing with side reflectors – still smaller than the power consumption at 500 µm (18.6 W) without reflectors, which suggests that the addition of side reflectors provides more flexibility in determining the emitter-filter spacing.

The results in Fig. 5(c) show good agreement between the theoretical model and the experimental data points. The model results, represented by the shaded area in Fig. 5(c), show a range of power consumption corresponding to ± 0.01 absolute change in the reflectivity of all surfaces and ± 20 K uncertainty on the emitter temperature, to demonstrate the sensitivity of results to input variables. Fluctuations between the model and the experimental data can be attributed to the small gap between the emitter and the side reflectors to prevent physical contact between the two as well as uncertainty in the optical properties of the surfaces. These experimental results validate our theoretical model and demonstrate the benefits of adding side reflectors to a typical emitter-filter system.

While we have shown that specular side reflectors can improve the efficiency of existing systems, this approach may also help alleviate the need for high-precision alignment and minimize emitter-filter spacing, allowing for a simpler system. Furthermore, systems in which noble gases are typically added (e.g., inert gas in incandescent light bulbs to reduce the evaporation of the emitter) could significantly benefit from a larger emitter-filter spacing that side reflectors allow as it will decrease the conduction and convection heat transfer between the emitter and filter, thus minimizing the filter temperature and improving its long-term stability. Adding side reflectors to current incandescent lighting and TPV systems therefore represents an opportunity to increase system efficiency and filter thermal stability at a relatively low cost and level of complexity.

5. Conclusion

In summary, we investigated the use of specular side reflectors in existing systems converting thermal radiation to a spectrally tailored spectrum as a simple approach to achieving significant gains in efficiency while relaxing the need for high-precision alignment between the emitter and the filter. We developed analytical solutions for specular view factors in 2-D and 3-D for fully specular enclosures, which allow for a better understanding of the parameters affecting the system efficiency and efficient computation. Finally, we validated our proposed approach and analytical model using an experimental setup comprising of a thermal emitter and nanophotonic filters. We demonstrated >10% reduction in power consumption at the smallest emitter-filter gap of 500 µm when using silver-coated polished side reflectors. The easy-to-implement approach presented in this work could lead to significant improvement in the efficiency of existing TPV and incandescent lighting systems, as well as in other systems involving thermal radiation management at a low cost.

Appendix A 2-D specular view factors in a fully specular rectangular enclosure

To accurately account for specular reflections, specular view factors were defined by Eq. (2). In order to calculate specular view factor $F_{i-j}^s$, one must therefore find and account for all the ways that radiation diffusely emitted from A_i can reach A_j through direct travel or any mirror-like reflections on specular surfaces. Specular surfaces create virtual images of their neighboring physical and virtual surfaces and therefore increase the number of ways a photon leaving surface A_i can reach surface A_j.

For simplicity, we start our calculation of specular view factors for a 2-D fully specular rectangular enclosure, as shown in Fig. 6(a).

 

Fig. 6 (a) 2-D rectangular enclosure for the derivation of the specular view factors. (b) 3-D rectangular enclosure for the derivation of the specular view factors. For illustration purposes, surface A₃ (parallel to A₄) is hidden.

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To account for all possible reflections, we create a map of physical and virtual images using the imaging technique [32]. We illustrate a small portion of this map in Fig. 7, where the notation A_x(y-z-w) is used to represent the virtual image of A_x as reflected by surfaces y-z-w (in that order). Only a sample of the virtual images is represented in Fig. 7 for clarity; the map of images extends to infinity in all direction due to the specular nature of all surfaces. More details and examples on how to create this map can be found in References [32,33,35].

 

Fig. 7 Map of physical (A₁ to A₄) and virtual images (A_x(y-z-w)) due to specular reflections. Only a sample of the map is shown as the map extends to infinity in all directions. Using this map, we can find all possible real and virtual images with their corresponding reflecting surfaces as seen by a real image. A color code is used to show which images surface A₁ sees either through direct travel or through reflections (green: A₁; red: A₂; purple: A₃; blue: A₄; dark: not visible from A_1, assuming the face of A₁ facing the inside of the rectangular cavity)

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Using the map of physical and virtual images, we calculate the specular view factor between the emitter A₁ and the filter A₂ ($F_{1-2}^s$) by summing the diffuse view factors between the real surface A₁, and the real and virtual surfaces of A₂ multiplied by the specular reflectivity (the superscript s of specular reflectivity ρ^s has been omitted in the following equations for clarity) of the surfaces on which the rays had to be reflected to reach the corresponding virtual image of surface A₂. A color code is used in Fig. 7 to show which images surface A₁ sees either through direct travel or through reflections (green: A₁; red: A₂; purple: A₃; blue: A₄; dark: not visible from A₁). Because all surfaces are specular, infinite sums accounting for an infinite number of reflections will appear in the solution of the specular view factor. A similar approach is used to calculate the specular view factor of the emitter A₁ and itself, $F_{1-1}^s$ and for $F_{1-3}^s$. Expressions for $F_{1-1}^s$, $F_{1-2}^s$ and $F_{1-3}^s$ are presented in Eqs. (9)-(11):

(9)$$F_{1-1}^s= {\displaystyle \sum }_{M=0}^{\infty }{\displaystyle \sum }_{N=0}^{\infty }{\rho }_1^M{\rho }_2^{M+1}\left[\begin{array}{l}{F}_{1\left(1^M-2^{M+1}\right)-1}+2{\left({\rho }_3{\rho }_4\right)}^{N+1}{F}_{1\left(1^M-2^{M+1}-3^{N+1}-4^{N+1}\right)-1}\\ + {\rho }_3^{N+1}{\rho }_4^N{F}_{1\left(1^M-2^{M+1}-3^{N+1}-4^N\right)-1}+{\rho }_3^N{F}_{1\left(1^M-2^{M+1}-3^N-4^{N+1}\right)-1}\end{array}\right]$$

(10)$$F_{1-2}^s={\displaystyle \sum }_{M=0}^{\infty }{\displaystyle \sum }_{N=0}^{\infty }{\left({\rho }_1{\rho }_2\right)}^M\left[\begin{array}{l}{F}_{1\left(1^M-2^M\right)-2}+2{\left({\rho }_3{\rho }_4\right)}^{N+1}{F}_{1\left(1^M-2^M-3^{N+1}-4^{N+1}\right)-2}\\ +{\rho }_3^{N+1}{\rho }_4^N{F}_{1\left(1^M-2^M-3^{N+1}-4^N\right)-2}+{\rho }_3^N{\rho }_4^{N+1}{F}_{1\left(1^M-2^M-3^N-4^{N+1}\right)-2}\end{array}\right]$$

(11)$$F_{1-3}^s= {\displaystyle \sum }_{M=0}^{\infty }{\displaystyle \sum }_{N=0}^{\infty }\left\{\begin{array}{l}{\left({\rho }_1{\rho }_2\right)}^M\left[{\left({\rho }_3{\rho }_4\right)}^N{F}_{1\left(1^M-2^M-3^N-4^N\right)-3}+{\rho }_3^N{\rho }_4^{N+1}{F}_{1\left(1^M-2^M-3^N-4^{N+1}\right)-3} \right]\\ +{\rho }_1^M{\rho }_2^{M+1}\left[{\left({\rho }_3{\rho }_4\right)}^N{F}_{1\left(1^M-2^{M+1}-3^N-4^N\right)-3}+{\rho }_3^N{\rho }_4^{N+1}{F}_{1\left(1^M-2^{M+1}-3^N-4^{N+1}\right)-3} \right]\end{array}\right\}$$

where the notation $F_{A\left(B^N-C^M\right)-D}$ denotes the diffuse view factor from surface A to surface D as mirrored by surfaces B and C, N and M number of times respectively. Due to the symmetry of the geometry, $F_{1-4}^s$ can be calculated using the same formula as for $F_{1-3}^s$ if indices 3 and 4 in Eq. (11) are switched. The diffuse view factors between two lines (or infinitely long planes) can easily be calculated using the cross-string method or using other analytical solution.

Appendix B 3-D specular view factors in fully specular rectangular enclosure

Until now, we have assumed that the surfaces extended to infinity in the out-of-plane direction (width W_i much larger than the length L_i) such that the 2-D assumption is valid. We now consider the case where the width and the length of the surfaces of the enclosure are comparable, thus requiring the consideration of the 3-D aspect of the geometry, as depicted in Fig. 6(b).

The calculation of the 3-D specular view factors $F_{1-1}^s,$ $F_{1-2}^s$ and $F_{1-3}^s$ is similar to the 2-D case except that we now add another dimension in which reflections are possible (specular reflections on surfaces A₅ and A₆). The previously developed 2-D map of real and virtual images (Fig. 7) can serve as a starting point for the 3-D version. The addition of specular surfaces A₅ and A₆ now imposes this 2-D map to be reflected an infinite number of times over these surfaces as shown in Fig. 8.

 

Fig. 8 Reflections over side reflectors A₅ and A₆ of the 2-D map of physical and virtual images from Fig. 7 (here shown as the blue middle plane).

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Similar to the 2-D case, $F_{i-j}^s$ includes the sum of the diffuse view factors between the real surface A_i and the real and all virtual images of A_j multiplied by the reflectivity of the surfaces on which light had to be reflected to reach the corresponding virtual image of surface A_j. Solutions for the 3-D specular view factors $F_{1-1}^s$, $F_{1-2}^s$ and $F_{1-3}^s$ are presented in Eqs. (12)-(14).

(12)$$F_{1-1}^s={\displaystyle \sum _{M=0}^{\infty }{\displaystyle \sum _{N=0}^{\infty }{\displaystyle \sum _{P=0}^{\infty }{\rho }_1^M{\rho }_2^{M+1}\left\{\begin{array}{l}{F}_{1\left(1^M-2^{M+1}\right)-1}+2{\left({\rho }_5{\rho }_6\right)}^{P+1}{F}_{1\left(1^M-2^{M+1}-5^{P+1}-6^{P+1}\right)-1}\\ +{\rho }_5^{P+1}{\rho }_6^P{F}_{1\left(1^M-2^{M+1}-5^{P+1}-6^P\right)-1}+{\rho }_5^P{\rho }_6^{P+1}{F}_{1\left(1^M-2^{M+1}-5^P-6^{P+1}\right)-1}\\ +{\rho }_3^{N+1}{\rho }_4^N\left[\begin{array}{l}{\rho }_5^P{\rho }_6^{P+1}{F}_{1\left(1^M-2^{M+1}-3^{N+1}-4^N-5^P-6^{P+1}\right)-1}\\ +{\rho }_5^{P+1}{\rho }_6^P{F}_{1\left(1^M-2^{M+1}-3^{N+1}-4^N-5^{P+1}-6^P\right)-1}\\ +2{\left({\rho }_5{\rho }_6\right)}^{P+1}{F}_{1\left(1^M-2^{M+1}-3^{N+1}-4^N-5^{P+1}-6^{P+1}\right)-1}\\ +F_{1\left(1^M-2^{M+1}-3^{N+1}-4^N\right)-1}\end{array}\right]\\ +{\rho }_3^N{\rho }_4^{N+1}\left[\begin{array}{l}{\rho }_5^P{\rho }_6^{P+1}{F}_{1\left(1^M-2^{M+1}-3^N-4^{N+1}-5^P-6^{P+1}\right)-1}\\ +{\rho }_5^{P+1}{\rho }_6^P{F}_{1\left(1^M-2^{M+1}-3^N-4^{N+1}-5^{P+1}-6^P\right)-1}\\ +2{\left({\rho }_5{\rho }_6\right)}^{P+1}{F}_{1\left(1^M-2^{M+1}-3^N-4^{N+1}-5^{P+1}-6^{P+1}\right)-1}\\ +F_{1\left(1^M-2^{M+1}-3^N-4^{N+1}\right)-1}\end{array}\right]\\ +2{\rho }_3^{N+1}{\rho }_4^{N+1}\left[\begin{array}{l}{\rho }_5^P{\rho }_6^{P+1}{F}_{1\left(1^M-2^{M+1}-3^{N+1}-4^{N+1}-5^P-6^{P+1}\right)-1}\\ +{\rho }_5^{P+1}{\rho }_6^P{F}_{1\left(1^M-2^{M+1}-3^{N+1}-4^{N+1}-5^{P+1}-6^P\right)-1}\\ +2{\left({\rho }_5{\rho }_6\right)}^{P+1}{F}_{1\left(1^M-2^{M+1}-3^{N+1}-4^{N+1}-5^{P+1}-6^{P+1}\right)-1}\\ +F_{1\left(1^M-2^{M+1}-3^{N+1}-4^{N+1}\right)-1}\end{array}\right]\text{}\end{array}\right\}}}}$$

Now, we have assumed that the surfaces extended to infinity in the out-of-plane direction (width W_i much larger than the length L_i) such that the 2-D assumption is valid. We now consider the case where the width and the length of the surfaces of the enclosure are comparable, thus requiring the consideration of the 3-D aspect of the geometry (as depicted in Fig. 3).

(13)$$F_{1-2}^s={\displaystyle \sum _{M=0}^{\infty }{\displaystyle \sum _{N=0}^{\infty }{\displaystyle \sum _{P=0}^{\infty }{\left({\rho }_1{\rho }_2\right)}^M\left\{\begin{array}{l}{F}_{1\left(1^M-2^M\right)-2}+2{\left({\rho }_5{\rho }_6\right)}^{P+1}{F}_{1\left(1^M-2^M-5^{P+1}-6^{P+1}\right)-2}\\ +{\rho }_5^{P+1}{\rho }_6^P{F}_{1\left(1^M-2^M-5^{P+1}-6^P\right)-2}+{\rho }_5^P{\rho }_6^{P+1}{F}_{1\left(1^M-2^M-5^P-6^{P+1}\right)-2}\\ +{\rho }_3^{N+1}{\rho }_4^N\left[\begin{array}{l}{\rho }_5^P{\rho }_6^{P+1}{F}_{1\left(1^M-2^M-3^{N+1}-4^N-5^P-6^{P+1}\right)-2}\\ +{\rho }_5^{P+1}{\rho }_6^P{F}_{1\left(1^M-2^M-3^{N+1}-4^N-5^{P+1}-6^P\right)-2}\\ +2{\left({\rho }_5{\rho }_6\right)}^{P+1}{F}_{1\left(1^M-2^M-3^{N+1}-4^N-5^{P+1}-6^{P+1}\right)-2}\\ +F_{1\left(1^M-2^M-3^{N+1}-4^N\right)-2}\end{array}\right]\\ +{\rho }_3^N{\rho }_4^{N+1}\left[\begin{array}{l}{\rho }_5^P{\rho }_6^{P+1}{F}_{1\left(1^M-2^M-3^N-4^{N+1}-5^P-6^{P+1}\right)-2}\\ +{\rho }_5^{P+1}{\rho }_6^P{F}_{1\left(1^M-2^M-3^N-4^{N+1}-5^{P+1}-6^P\right)-2}\\ +2{\left({\rho }_5{\rho }_6\right)}^{P+1}{F}_{1\left(1^M-2^M-3^N-4^{N+1}-5^{P+1}-6^{P+1}\right)-2}\\ +F_{1\left(1^M-2^M-3^N-4^{N+1}\right)-2}\end{array}\right]\\ +2{\rho }_3^{N+1}{\rho }_4^{N+1}\left[\begin{array}{l}{\rho }_5^P{\rho }_6^{P+1}{F}_{1\left(1^M-2^M-3^{N+1}-4^{N+1}-5^P-6^{P+1}\right)-2}\\ +{\rho }_5^{P+1}{\rho }_6^P{F}_{1\left(1^M-2^M-3^{N+1}-4^{N+1}-5^{P+1}-6^P\right)-2}\\ +2{\left({\rho }_5{\rho }_6\right)}^{P+1}{F}_{1\left(1^M-2^M-3^{N+1}-4^{N+1}-5^{P+1}-6^{P+1}\right)-2}\\ +F_{1\left(1^M-2^M-3^{N+1}-4^{N+1}\right)-2}\end{array}\right]\text{}\end{array}\right\}}}}$$

(14)$$\begin{array}{r}{F}_{1-3}^s={\displaystyle \sum _{M=0}^{\infty }{\displaystyle \sum _{N=0}^{\infty }{\displaystyle \sum _{P=0}^{\infty }{\left({\rho }_1{\rho }_2\right)}^M\left\{\begin{array}{l}{\left({\rho }_3{\rho }_4\right)}^N\left[\begin{array}{l}{\rho }_5^P{\rho }_6^{P+1}{F}_{1\left(1^M-2^M-3^N-4^N-5^P-6^{P+1}\right)-3}\\ +{\rho }_5^{P+1}{\rho }_6^P{F}_{1\left(1^M-2^M-3^N-4^N-5^{P+1}-6^P\right)-3}\\ +2{\left({\rho }_5{\rho }_6\right)}^P{F}_{1\left(1^M-2^M-3^N-4^N-5^P-6^P\right)-3}\\ -F_{1\left(1^M-2^M-3^N-4^N\right)-3}\end{array}\right]\\ +{\rho }_3^N{\rho }_4^{N+1}\left[\begin{array}{l}{\rho }_5^P{\rho }_6^{P+1}{F}_{1\left(1^M-2^M-3^N-4^{N+1}-5^P-6^{P+1}\right)-3}\\ +{\rho }_5^{P+1}{\rho }_6^P{F}_{1\left(1^M-2^M-3^N-4^{N+1}-5^{P+1}-6^P\right)-3}\\ +2{\left({\rho }_5{\rho }_6\right)}^P{F}_{1\left(1^M-2^M-3^N-4^{N+1}-5^P-6^P\right)-3}\\ -F_{1\left(1^M-2^M-3^N-4^{N+1}\right)-3}\end{array}\right]\text{}\end{array}\right\}}}}\\ +{\displaystyle \sum _{M=0}^{\infty }{\displaystyle \sum _{N=0}^{\infty }{\displaystyle \sum _{P=0}^{\infty }{\rho }_1^M{\rho }_2^{M+1}\left\{\begin{array}{l}{\left({\rho }_3{\rho }_4\right)}^N\left[\begin{array}{l}{\rho }_5^P{\rho }_6^{P+1}{F}_{1\left(1^M-2^{M+1}-3^N-4^N-5^P-6^{P+1}\right)-3}\\ +{\rho }_5^{P+1}{\rho }_6^P{F}_{1\left(1^M-2^{M+1}-3^N-4^N-5^{P+1}-6^P\right)-3}\\ +2{\left({\rho }_5{\rho }_6\right)}^P{F}_{1\left(1^M-2^{M+1}-3^N-4^N-5^P-6^P\right)-3}\\ -F_{1\left(1^M-2^{M+1}-3^N-4^N\right)-3}\end{array}\right]\\ +{\rho }_3^N{\rho }_4^{N+1}\left[\begin{array}{l}{\rho }_5^P{\rho }_6^{P+1}{F}_{1\left(1^M-2^{M+1}-3^N-4^{N+1}-5^P-6^{P+1}\right)-3}\\ +{\rho }_5^{P+1}{\rho }_6^P{F}_{1\left(1^M-2^{M+1}-3^N-4^{N+1}-5^{P+1}-6^P\right)-3}\\ +2{\left({\rho }_5{\rho }_6\right)}^P{F}_{1\left(1^M-2^{M+1}-3^N-4^{N+1}-5^P-6^P\right)-3}\\ -F_{1\left(1^M-2^{M+1}-3^N-4^{N+1}\right)-3}\end{array}\right]\text{}\end{array}\right\}}}}\end{array}$$

The diffuse view factors ($F_{A\left(B^N-C^{N+1}\right)-D}$) in Eqs. (12)-(14) can be calculated using available analytical correlations [36] or numerical methods [37]. The number of summations (M, N and P) in the specular view factor equations for both 2-D and 3-D view factors are chosen to reach convergence of the view factors, satisfying the summation rule [Eq. (4)].

Appendix C validation of specular view factors using ray tracing simulations

In addition to verifying the calculated specular view factors using the summation rule [Eq. (4)], we also validated our results using ray tracing simulations. Ray tracing simulations were performed for geometries and optical surfaces similar to those calculated using our analytical model. The ray tracing simulations assume a rectangular cavity (Fig. 6) with diffuse emission of 10⁶ rays at surface A₁ and specular reflections on all surfaces A₁ to A₆. To calculate the specular view factors $F_{i-j}^s$ from the ray tracing simulations, we count the average number of times a ray initially emitted at i hits surface j, including absorption and all intermediate reflections on the latter surface. Validation of the analytical view factors for different systems configuration is presented in Fig. 9, demonstrating the validity of the analytical model for different optical properties of the surfaces [Fig. 9(a) and 9(b)] and dimensions of A₁ [Fig. 9(a) and 9(c)].

 

Fig. 9 Validation of analytical specular view factors with ray tracing simulation for (a) A₁ square with ρ₁ = 0.99, ρ₂ = 0.85 and ρ_3-6 = 0.95; (b) A₁ square with ρ₁ = 0, ρ₂ = 0.85 and ρ_3-6 = 0.95; (c) A₁ rectangular (1 × 2) with ρ₁ = 0.99, ρ₂ = 0.85 and ρ_3-6 = 0.95. Because A₁ is rectangular, $F_{1-3}^s$ and $F_{1-5}^s$ are different.

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Appendix D comparing the assumption of specular and diffuse reflections, and 2-D and 3-D enclosures

Earlier, we motivated our efforts to model specular reflections within our 3-D fully specular rectangular enclosure by stipulating that modeling error could occur when assuming diffuse only surfaces and a 2-D enclosure. We now compare in Fig. 10 the differences between the assumptions of diffuse only or specular only reflections for a 3-D rectangular enclosure as well as the differences between the assumption of 2-D and 3-D enclosures. As in the main text, it is assumed that the emitter is square for the 3-D geometry, the filters have a reflectivity ρ_IR = 0.85 in the infrared and perfect transmissivity τ_VIS = 1 in the visible, the emitter is gray and at 2800 K, and the side reflectors have a gray reflectivity of ρ = 0.95, as could reasonably be achieved using polished metallic side reflectors.

 

Fig. 10 (a) Comparison between specular and diffuse reflections in the calculation of the efficiency of different gray emitters at 2800 K when combined with an optical filter and side reflectors. (b) Comparison between 2-D and 3-D assumptions in the calculation of the efficiency of different gray emitters at 2800 K when combined with an optical filter and side reflectors.

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In Fig. 10(a), we observe that for the diffuse reflections assumption, we underestimate the efficiency for a range of dimensionless spacings S* and emitter gray emissivities. However, both assumptions give similar results for S* < 10⁻² or S* > 10². These results suggest that for specular surfaces with the given optical properties and for 10⁻² < S* < 10², a model accounting for specular reflections should be used for accurate results. In Fig. 10(b), we observe that the 2-D model generally overestimates the efficiency of the system for a wide range of emitter emissivity, as could be expected since it assumes that the geometry is infinitely long in the out-of-plane direction. However, for S* < 10⁻² or S* > 10², both assumptions give similar results. These results suggest that for the given surfaces optical properties and for 10⁻² < S* < 10²_, it is important to model the geometry in 3-D if the emitter has comparable length and width.

Appendix E optical properties

The tungsten planar emitter (6 mm × 10 mm) was laser cut from a polished pure (99.95%) tungsten sheet (0.003 in thick) from H.C. Stark. The copper reflectors were machined on a CNC mill (280 µm larger than the emitter to allow for alignment and avoid short circuit with the emitter) at different lengths (0.5 mm, 1 mm, 3.3 mm and 25.4 mm) corresponding to the four different emitter-filter gap spacings and then polished using 400, 800, 1000 grit sandpaper followed by lapping compound and diamond paste (0.5 µm grain size). A 150-nm thick layer of silver was then deposited on the polished copper by electron-beam physical vapor deposition. The optical filter consisted of 90 alternating layers of Ta₂O₅ and SiO₂ deposited on a 20 mm × 20 mm and 1-mm thick fused silica substrate, optimized to transmit visible light and reflect infrared radiation for incandescent lighting applications [3]. The optical properties of the emitter, side reflectors and filters are shown in Fig. 11.

 

Fig. 11 (a) Tungsten emitter emissivity, (b) side reflector direct-hemispherical reflectance at normal incidence angle and (c) filter reflectivity averaged over the range of incident angles 0° to 70° by increments of 5°.

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Funding

Solid-State Solar Thermal Energy Conversion (S3TEC) Center, Energy Frontier Research Center, U.S. Department of Energy, Office of Science, Basic Energy Sciences (DE-SC0001299/DE-FG02-09ER46577); Fonds de Recherche du Québec – Nature et Technologies (FRQNT).

Acknowledgments

The authors would like to thank Prof. Marin Soljačić and Dr. Ognjen Ilic for sharing their equipment and nanophotonic filters. This work was supported as part of the Solid-State Solar Thermal Energy Conversion (S3TEC) Center, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award No. # DE-SC0001299/DE-FG02-09ER46577. A. Leroy acknowledges a graduate fellowship from the Fonds de Recherche du Québec – Nature et Technologies (FRQNT).

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