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Abstract
The optical properties and performance of the two-dimensional photonic crystal (2D PhC) filters at normal incidence were simulated, and the best geometric parameters were obtained with the help of a global optimization program. The honeycomb structure has better performance, including high in-band transmittance, high out-band reflectance, and low parasitic absorption. The power density performance and conversion efficiency can reach 80.6% and 62.5%. Furthermore, the deeper cavity structure and multi-layer structure were designed to improve the performance of the filter. The deeper one can reduce the influence of transmission diffraction, increases the power density performance and conversion efficiency. The multi-layer structure reduces the parasitic absorption significantly and increases the conversion efficiency to 65.5%. These filters have both high efficiency and high power density, avoid the challenge of high-temperature stability faced by emitters, also easier and cheaper to fabricate compared to the 2D PhC emitters. These results suggest that the 2D PhC filters can be used in thermophotovoltaic systems for long-duration missions to improve the conversion efficiency.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Thermophotovoltaic (TPV) cells utilize locally emitted thermal radiation to generate electricity [1]. They can use various heat sources, such as solar absorbers [2,3], combustion sources [4], or decaying radioisotopes [5], leading to the potential for applicated in solar energy conversion, waste heat harvesting, or deep-space probes.
The challenge in high efficiency TPV system is the relatively low heat sources temperature compared to the bandgap of TPV cells. Only a small portion of radiated photons has the energy exceeding the bandgap, and reduce the bandgap of TPV or increase the fraction of in-band photons will influence the efficiency of the TPV cell itself. Considering the limited TPV cells available at present, it is important to use spectral control technologies to increase the fraction of in-band photons and reduce the low-energy photons absorbed by the TPV cells.
Spectral control technologies can be divided into selective emitters [6,7], selective filters [8–12], and back surface reflectors [1,13]. Selective emitters are working at elevated temperature approximately same as the heat source, facing the stringent challenge of thermal stability during the long operational time. The selective filters and back surface reflectors are attached to the TPV cells, working at a relatively low temperature. One-dimensional photonic crystal (1D PhC) filters have very low absorption and sharp cut-off edge [9,10]. But they will also transmit some far infrared photons, decrease the out-band reflectance, another problem is the bandgap of 1D PhCs will shift for different incident angles, causing the damage of in-band transmittance and out-band reflectance. Plasma filters use highly doped semiconductor layer [14], have good out-band reflectance, but the absorption is relatively high. Back surface reflectors use a highly reflective material applied to the backside of TPV cells [1,13], they show near perfect out-band reflectance. However, the spectral selectivity of this method is obtained by the TPV cells themselves, which means the additional limitation for the material of TPV cells, and the power density is relatively low.
Two-dimensional photonic crystals (2D PhCs) have shown great power density and spectral selectivity when applied to emitters [15–17], the in-band absorption can reach 90%, the out-band absorption is lower than 10%, and the bandgap is insensitive to the incident angle [18]. 2D PhCs also shown ability to act as light trapping and anti-reflective coatings when fabricated directly onto the photodiode [19]. However, there has been little work about using 2D PhCs as selective filters for TPV applications.
In this work, we try to design and improve 2D PhC filters with high in-band transmittance, high out-band reflectance, and low parasitic absorption, in order to obtain a filter with both high power density and high efficiency. The finite difference time domain (FDTD) method was used to calculate the transmittance spectra T(λ) and reflectance spectra R(λ) of the filters. We also calculate the absorption spectra A(λ) = 1- T(λ)- R(λ) to directly show the energy wasted by the parasitic absorption. In Section II, how to evaluate the performance of filters from T(λ) and R(λ), find the best geometric parameters with the help of a global optimization program are introduced. In Section III, the 2D PhCs made of different materials, and cavities in different shapes and arrangements are shown, the performances of these filters are compared and discussed. The honeycomb structure made of silver shows a good performance. Furthermore, two different designs based on the honeycomb structure are studied to further improve the performance. The conclusions are presented in Section IV.
2. Modeling and optimization
The traditional 2D PhCs consist of an array of air cavities in a flat metal. According to cavity resonances theory [15,20], the size and depth of cavities determine the cavity resonance frequency, creating the peak and cut-off edge of the transmittance spectrum. Some factors should be considered regarding the design of 2D PhCs filters. Because of the nonideal cut-off edge, a large cut-off wavelength will lead to a high in-band transmittance, but decreases the out-band reflectance. As a selective filter, the deeper cavities result in a better selectivity, but have a higher absorption. For grating structure, the grating equation shows that the diffraction is related to the period of structure, also influence the transmittance spectrum.
In order to balance the in-band transmittance, out-band reflectance and absorption, maximize the power density and efficiency of filters, we use FDTD method to calculate the optical properties of 2D PhC filters, and use a global optimization program to find the best geometric parameters of filters. The merit function of optimization program is defined as follow:
ηc is the conversion efficiency, ηpd is the power density performance, 0.9 and 0.1 are the weighting given to ηc and ηpd. The weigh factors are taken from literature [21], 0.9 for ηc because we are mainly concerned in obtaining the highest conversion efficiency, and 0.1 for ηpd is enough to balance the conversion efficiency and power density performance.The conversion efficiency ηc is the ratio of the energy contained in the electron-hole pairs in the TPV cell to the total thermal radiation received by the filter, similar to the efficiency of emitter [17], representing the potential of thermal radiation to generate electrical power in the PV cell.
The power density performance ηpd is the ratio of the energy contained in the electron-hole pairs in the TPV cell with spectral control technologies to the maximum energy that can be produced form blackbody spectrum, representing the influence of the spectral control technologies to the TPV system power density performance. For the blackbody emitter used in our calculation, this influence only attribute to the filter.
The permittivity of materials used in FDTD method come from Palik data [22]. Simulating optical properties at short wavelength is very time consuming because the mesh in FDTD method is related to the shortest wavelength in medium. Thus, we only calculate optical properties range from 1-6 µm. The transmittance and reflectance for wavelength shorter than 1 µm are neglected, for wavelength longer than 6 µm, they are assumed to be constant. According to our test, the errors of power density performance and conversion efficiency are less than 1% compared to the results of simulation range from 0.5-30 µm, which is acceptable in the optimization. In our calculation, we use an emitter of blackbody to evaluate the performance of filters. The temperature of emitter is 1200°C, same as the temperature of radioisotope heat source. The cut-off wavelength is 2.0 µm, corresponding to the bandgap of InGaAs cells. The global optimization program is based on the surrogate optimization algorithm from MATLAB. We use surrogate optimization algorithm to generate the geometric parameters of 2D PhC, then pass these parameters to the FDTD program, the merit function evaluates the performance of 2D PhC based on the simulation result, and pass the score back to the surrogate optimization program, the surrogate optimization program generates new parameters and start a new cycle. We choose the best parameters in 50 cycles.
3. Result and discussion
3.1 Choice of material
Gold and silver are wildly used metal film material with high reflectance in the infrared region. We simulated and compared the transmittance and absorption spectra of the gold and silver PhC filters on a fused silica substrate. The calculated results are shown in Fig. 1 and the structure of the PhC is shown in Fig. 2(a). Geometric parameters obtained by the optimization program are listed as follows, gold PhC: a = 1.23 µm, d = 1.14 µm, t = 1.6 µm, silver PhC: a = 1.25 µm, d = 1.15 µm, t = 1.7 µm, a is the period of cavities, d is the diameter of cavities, and t is the thickness of the metallic PhC. The average in-band transmittance (1-2 µm) of the gold PhC is 73.5%, the average in-band absorption is 9.2%, and the average out-band reflectance is 94.2%. For the silver PhC, these results are 75.6%, 7.8%, 94.6%, the silver PhC showing a better performance than the gold PhC. From these results, we can see the average out-band reflectance of 2D metallic PhC can reach more than 94%, which is beneficial to maintain the temperature of heat source and improve the conversion efficiency of TPV system.
Fig. 1. Spectral properties of the Au PhC and the Ag PhC. (a) Transmittance spectra. (b) Absorption spectra.
Fig. 2. Schematics and geometric parameters of four different cavity types. (a) Cylinders in a square lattice (Type A), (b) squares in a square lattice (Type B), (c) cylinders in a hexagonal lattice (Type C), (d) hexagons in a hexagonal lattice (Type D).
The absorption of 2D PhCs is caused by ohmic losses resulting from induced currents in the metallic film [8]. Because of the higher electrical conductivity than gold, the silver PhC has a lower absorption. It can be thicker to obtain better selectivity while maintaining lower absorption, especially in the wavelength of 1-2 µm. The silver PhC has a bit larger cavity, making the cut-off edge closer to the target wavelength, increases the in-band transmittance. Due to the better selectivity, the out-band transmittance only changes a little.
3.2 Cavity type
We tested four different cavity types as shown in Fig. 2, A. cylinders in a square lattice, B. squares in a square lattice, C. cylinders in a hexagonal lattice, D. hexagons in a hexagonal lattice. All these PhCs are made of silver. Geometric parameters obtained by the optimization program and the performance of the filters are listed in Table 1. The transmittance and absorption spectra of these four cavity types are shown in Fig. 3.
Fig. 3. Spectral properties of the filter with different cavity types. (a) Transmittance spectra. (b) Absorption spectra.
Table 1. Parameters and performances of the filter with different cavity typesa
For the square lattice, as shown in Fig. 3(a), diffraction appears when the wavelength shorter than the period of lattices for normally incident photons, resulting in a decreasing of transmission. In the same position, a strong absorption peak appears, as shown in Fig. 3(b). We called it reflection diffraction to make a distinction with transmission diffraction, which will be discussed in the next section. The transmittance of the square cavity in 1-2 µm is 80.8%, larger than that of 75.6% in the cylindrical cavity. The average in-band absorption and average out-band reflectance for these two types are close, as shown in Table 1. The average in-band absorption is 7.8%, and the average out-band reflectance is larger than 94%. Comparing the size of these two cavities in Table 1, we can see the d, t, and a of the square cavities are smaller than the cylindrical cavities when creating the same cut-off wavelength and a higher transmission. Square cavities are more compact than cylindrical cavities in square lattice, leading to the larger void ratio. The larger void ratio improves the transmittance, and the smaller period broadens the band of high transmittance. Thus, the conversion efficiency and power density performance in Type B are 61.3% and 80.4%, better than that of 60.5% and 75.6% in Type A.
For the hexagonal lattice, as shown in Fig. 3(a), reflection diffraction appears at the wavelength shorter than $\sqrt 3 /2$ times of the period, which we called equivalent period. The smaller equivalent period of the hexagonal lattice allows cylindrical cavities to have a sufficiently wide band of high transmittance. From Fig. 3 and Table 1, we can get the average in-band transmittance of Type C is 82.3%, larger than that in Type A and B. Because the hexagonal lattice also allows cylindrical cavities to be more compact, overcome the shortage of cylindrical cavities in Type A. Furthermore, the average in-band absorption is decreased from 7.8% to 6.7%. Thus, the conversion efficiency and power density performance are increased to 62.4% and 81.9%.
Type D is the hexagonal cavities in hexagonal lattice. From Fig. 3, we can see the optical properties are similar to that of Type C, with the size of the cavities slightly smaller. Since all the parameters are discrete, the optimal cut-off wavelength obtained by the optimization program has a slight difference, leading to the discrepancy of power density performance and efficiency between type C and type D. The smaller cavity of type D allows a smaller period while maintaining sufficient sidewall thickness. The advantage of type D doesn’t show here because the period is small enough for normally incident photons, but when optimizing for an omnidirectional filter, the structure with smaller period will have better performance, as is consistent with the 2D PhC emitter in literature [18]. The uniform and thicker sidewalls should also be easier to fabricate.
3.3 Influence of substrate
In applications, we need a substrate to support the thin film of 2D PhCs, but the substrate has some influence on the optical properties of filters. The transmittance spectra of filters without substrate and with different substrate refractive indexes were calculated. The structure of 2D PhC in this test is same as type D. The results are shown in Fig. 4.
Fig. 4. Influence of substrate material. Transmittance spectra of filters that consist of the type D PhC and a substrate with various refractive index. n is refractive index of the substrate material. For fused silica, n is about 1.44. Only PhC is the result from the filter without substrate.
The filter without substrate shows the best power density performance, and the transmittance slight decreases with the increase of refractive index. Some interesting phenomena were found. From the result with refractive index of 1.4, as shown in Fig. 4, we can see a turning point at the wavelength of 1.61 μm, equal to the product of equivalent period and refractive index of substrate. This character can be verified by substrates with refractive index of 1.44 and 1.5. According to the grating equation, diffraction appears when the wavelength in the medium shorter than the equivalent period of lattices for normally incident photons. So, we think the slightly decrease of transmittance before the turning point is caused by the transmission diffraction. It restricts the passage of photons through cavities, and this resistance increases when the substrate has higher refractive index. The transmission diffraction also causes the increase of transmittance after the turning point.
Based on the analysis above, we chosen fused silica as substrate for its low refractive index and excellent near-infrared transmittance. To further improve the performance of filters, we tried to make cavities a bit deeper than the metallic film. A SiO2 PhC was added between the metallic PhC and the substrate, as shown in Fig. 5. Geometric parameters obtained by the optimization program are listed as follows: a = 1.46 µm, d = 1.09 µm, t0 = 0.17 µm, t1 = 1.5 µm, t0 is the thickness of the SiO2 PhC, t1 is the thickness of the metallic PhC. The transmittance and absorption spectra are shown in Fig. 6. This new structure reduces the influence of transmission diffraction, and the transmittance from 1.4 µm to the cut-off edge can reach to 90%. Moreover, from Fig. 6(a), due to the reflection diffraction, the large period of 1.46 µm can reflect more high-energy photons without losing too much power density. And the lower void rate can also reduce out-band absorption, as shown in Fig. 6(b). These results are helpful to improve the conversion efficiency. The power density performance of this structure is 81.4% and conversion efficiency is 63.5%.
Fig. 6. Comparison of spectral properties between Type D and the deeper cavity structure. (a) Transmittance spectra. (b) Absorption spectra.
3.4 Multi-layer structure
The parasitic absorption of filters, especially the absorption peak near the cut-off wavelength, is the main issue of the further improvement of performance. We tried to combine 1D PhCs with 2D PhCs, using 1D PhCs to create a sharp and low absorption cut-off edge, and using 2D PhCs to obtain high out-band reflectance. The schematic of the multi-layer structure is shown in Fig. 7. The first layer above the substrate is Ag. The second SiO2 layer is a low refractive index layer to decrease the influence of reflection diffraction, and the third layer is Si with the refractive index of 3.4. The top SiO2 layer is a protective layer as the cut-off edge is very sensitive to the thickness of Si layer. These four layers are etched and filled with air. Geometric parameters obtained by the optimization program are listed as follows: a = 1.3 µm, d = 1.08 µm, t1 = 1 µm, t2 = 0.22 µm, t3 = 0.42 µm, t4 = 0.06 µm, t1 t2 t3 t4 is the thickness of each layer from bottom to top. The transmittance and absorption spectra are shown in Fig. 8.
Fig. 8. Comparison of spectral properties between Type D and multi-layer structure. (a) Transmittance spectra. (b) Absorption spectra.
From Fig. 8(a), we can see the transmittance of multi-layer from 1.37 µm to the cut-off wavelength is still larger than 80%, similar to the Type D, while the multi-layer structure has even better cut-off edge. For wavelengths shorter than 1.37 µm, the average transmittance is only 50%. The Si layer creates additional selectivity with little absorption. Thus, the absorption near the cut-off wavelength significantly decreases, but slightly increase at the wavelength longer than 4 µm, as shown in Fig. 8(b). The power density performance of this structure is 75.8% and the conversion efficiency is 65.5%. These results indicate the multi-layer structure can reduce the absorption near the cut-off wavelength significantly and improve the conversion efficiency of TPV system. More detailed study is in progress.
3.5 Discussion
The performances of filters we proposed in this work are shown in Fig. 9, also shown are prior results from the literature. The back surface reflector [1] shows the highest conversion efficiency, but its power density performance is the worst. The 2D PhC filters have a good balance between conversion efficiency and power density. The better power density performance means the TPV system can be more compact or lower the temperature of heat source for required power capacity, which will simplify the design of TPV system. The microstructure on silver is easier and cheaper to fabricate compared to the 2D PhC emitters made of refractory metals [16] or tungsten–carbon nanotube [17]. The microstructure on filter is also more stable for its lower operating temperature, and more suitable for long-duration missions. As an independent component, the 2D PhC filter can be easily combined with existing PV cells to improve conversion efficiency.
Fig. 9. The performances of filters we proposed in this work (orange), also shown are prior results from the literature: back surface reflectors (blue) [1,13], selective emitters (yellow) [16,17], and selective filters (green) [8,11].
4. Conclusions
In this article, we numerically analyzed and optimized selective filters for TPV applications, based on 2D PhCs. The shape and arrangement of cavities in 2D PhCs were studied. We obtained a honeycomb structure with high in-band transmittance, great spectral selectivity and low parasitic absorption compares to the traditional 2D PhCs. The larger void ratio improves the transmittance, and the smaller equivalent period reduces the influence of reflection diffraction, broadens the band of high transmittance. The power density performance of the honeycomb structure is 80.6% and the conversion efficiency is 62.5%. To reduce the influence of transmission diffraction, we made the cavities a bit deeper than the metallic film, further improve the in-band transmittance. The power density performance and conversion efficiency are increased to 81.4% and 63.5%. To lower the parasitic absorption of metallic PhCs, three layers dielectric PhCs were added in front of the metallic PhC. The absorption near the cut-off wavelength is significantly reduced. The power density performance of this structure is 75.8% and the conversion efficiency is 65.5%. These 2D PhC filters can be used to build high performance radioisotope TPV power system.
Funding
National Natural Science Foundation of China (Grants No. U20B2008); Project of Nuclear Power Technology Innovation Center of Science Technology and Industry for National Defense (No. HDLCXZX-2021-ZH-031).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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