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Broadband mirrors for thermophotovoltaics

Zunaid Omair, Sean Hooten, Varun Menon, Patrick Oduor, Kwong-Kit Choi, and Achyut K. Dutta

Open Access Open Access

Abstract

We present an innovative solution to improve the efficiency of thermophotovoltaic (TPV) devices by tackling the problem of sub-bandgap photon losses. We propose an optimized design for thin-film mirrors using inverse electromagnetic design principles, thereby enhancing the average reflectivity and photon re-use. Our method surpasses the traditional Bragg mirror by employing a gradient-descent based optimization over Bragg mirror geometrical parameters, leveraging the transfer matrix method for derivative calculations. The optimized structure, based on continuously chirped distributed Bragg reflectors proposed herein demonstrates a remarkable increase in reflectivity beyond 98%, over an almost three-octaves bandwidth (0.1eV-0.74eV). We show that the incident power loss in InGaAs TPV cells at an emitter temperature of 1200°C is significantly reduced. While our work shows considerable promise, further exploration is needed to ascertain the practicability and robustness of these designs under various operational conditions. This study thus provides a major step forward in TPV technology, highlighting a new route towards more effective energy conversion systems.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Thermophotovoltaic (TPV) devices are a promising technology that converts thermal radiation into electricity using photovoltaic cells [1–6]. TPV devices convert heat to electricity, and the goal is to maximize the electricity generation for a given amount of absorbed heat [1,5,6,7], known as thermophotovoltaic efficiency. However, the efficiency of these devices is often limited due to the substantial number of low-energy photons emitted that are not absorbed by the photovoltaic cells and are consequently wasted. This paper presents an innovative approach to address this issue and improve TPV efficiency. Previous studies [6–8].have suggested the use of a rear mirror to reflect these low-energy photons to the emitter, allowing them to be absorbed and regenerated. However, this approach requires mirrors with a reflectivity of at least 98% to achieve substantial efficiency gains.

Several attempts have been made to enhance the mirror reflectivity. For instance, the work by Fan et al. employed ‘air-bridge’ as mirrors, achieving the maximum reflectivity possible with a single dielectric-metal interface [7,8]. Achieving reflectivity higher than that would require the use of multiple layers of dielectric like widely-known distributed Bragg reflectors. However, for thermophotovoltaics, we need to achieve high reflectivity over a broad spectrum [5]. This requires complex optimization techniques, as used in optical filters [9–14]. In this paper, we present a simpler alternative, based on inverse electromagnetic design [15–17] to extend the bandwidth and increase the average reflectivity of thin-film mirrors. This method involves identifying a suitable initial structure for optimization, determining a figure of merit (FOM), and calculating the derivatives of the FOM with respect to the design parameters to be used in a gradient-descent based parameter update.

Our research demonstrates that the continuously chirped distributed Bragg reflector (DBR) serves as an excellent starting point for this optimization procedure. We then detail our method for calculating the derivatives required for the optimization using the transfer matrix method.

The findings presented in this study offer a promising pathway toward significantly improving the power conversion efficiency of thermophotovoltaic devices through the intelligent design of high-reflectivity mirrors.

2. Broadband mirrors for thermophotovoltaics

The hot emitter in thermophotovoltaics emits a significant number of low-energy photons, unusable in a single-junction photovoltaic cell. As an example, we show the Planck spectrum for a 1200°C blackbody Fig. 1. However, 86% of the incident power is below the bandgap of a typical indium gallium arsenide (InGaAs) thermophotovoltaic cell. These low-energy photons are not absorbed in the InGaAs cell and are usually wasted in the substrate. TPV efficiency is defined as the ratio of the total useful power generated (depicted as the blue region in Fig. 1) to the total power absorbed, which includes the PV power, the mirror loss, and the PV loss in Fig. 1, in line with the definitions given in [1,5,6,8,9]. It is essential to note that the regenerated power, which reflects to the emitter, is not considered as absorbed power. However, absorption of the low-energy photons in the mirror penalizes efficiency. This photon loss limits the maximum power conversion efficiency to only 8% for InGaAs cells with no mirror and 1200°C emitters.

 figure: Fig. 1.

Fig. 1. Mirrors for ultra-efficient thermophotovoltaic energy conversion. An InGaAs thermophotovoltaic cell receives radiation from a 1200°C hot emitter. Only a portion of the high-energy photons (blue area) is converted to electricity. A highly reflective mirror can reflect low-energy photons that would otherwise be wasted (red area), thereby increasing the power conversion efficiency. Any imperfection in the mirror creates mirror loss. Thermalization of high-energy carriers is the main contributor to PV loss for high-energy photons.

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Wernsman et al. [18] proposed using a rear mirror to reflect these low-energy photons to the emitter, where they can be absorbed and regenerated. In [5], it was demonstrated that an optimal choice of cell bandgap, and emitter temperature can lead to >50% thermophotovoltaic power conversion efficiency. However, we need mirrors with reflectivities ≥98%, to achieve this efficiency.

We show the benefit of mirror reflectivity in Fig. 2(a). The optimal sub-bandgap spectrally averaged reflectivity of an InGaAs cell on top of a metal (gold/silver) mirror is 94.5%. Accounting for resistive Joule heating and the material recombination in the cell, the resulting efficiency as a function of emitter temperature is the blue line. As we increase mirror reflectivity, the thermophotovoltaic efficiency also increases, and there is an increasing return of thermophotovoltaic efficiency with mirror reflectivity. This is due to enhanced luminescence extraction.

We also assume an anti-reflection coating is present on top of the thermophotovoltaic cell. This ensures efficient absorption of above-bandgap photons to generate electricity. We note that for TPV, obtaining an above-bandgap anti-reflection coating is less difficult compared to sub-bandgap mirror. For example, in lattice-matched InGaAs with 1200°C emitter, only a small part of the incident spectrum (1.09µm−1.65 µm) needs to be efficiently absorbed. On the other hand, a much wider part of the incident spectrum (> 1.65 µm) need to be reflected for sub-bandgap photons [5].

 figure: Fig. 2.

Fig. 2. Increasing returns of a better mirror. As mirror reflectivity gets closer to 100%, there is an increasing return of thermophotovoltaic efficiency, in (a). The voltage increases due to enhanced luminescence extraction, and parasitic low-energy photon absorption is reduced. However, the minimum mirror bandwidth needed for thermophotovoltaics is much wider than can be achieved with a traditional distributed Bragg reflector, as we show in (b). For (a), we assume the TPV cell has an anti-reflection coating for above-bandgap photons, to minimize photon loss.

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We face a unique challenge in designing mirrors with >98% reflectivity for regenerative thermophotovoltaics. Ideally, the mirror should reflect all sub-bandgap photons. However, the sub-bandgap photons span greater than three-octaves bandwidth 0-0.74 eV for practical emitter temperatures in thermophotovoltaics (1000°C-1200°C). We can achieve a maximum reflectivity of 94-95% reflectivity with a simple metal (gold/silver) mirror under an III-V thermophotovoltaic cell, limited by the losses in metals. We can increase the reflectivity by inserting a low-index dielectric between the cell and the metal layer. Forrest et al. [7] demonstrated a 98% average reflectivity over 0.3-0.7 eV for InGaAs thermophotovoltaic cells using an air-gold layer as a mirror.

Distributed Bragg reflectors (DBR) can provide excellent reflectivity but over a small, insufficient bandwidth for TPV. Figure 2(b) shows the bandwidth needed in thermophotovoltaics vs the bandwidth available with a 20-pair Si-SiO2 DBR, centered at 0.4eV. Szipocs et al. [5] proposed a modified version of a DBR to achieve greater bandwidth, as we show in Fig. 3(a). The layer thicknesses, instead of being fixed, are continuously varied from across the depth of the mirror, providing quarter wave thickness for reflections over a broader range of wavelengths than possible with a Bragg mirror. The thickness should be ordered to reflect any lossy part of the spectrum near the top, rather than the depth of the mirror. Figure 2(b) shows the bandwidth needed in thermophotovoltaics vs the bandwidth available with a 20-pair Si-SiO2 DBR, centered at 0.4 eV. In this continuously chirped DBR, the thickness of the layers gradually decreases along the depth of the DBR. The different wavelengths in the below-bandgap spectrum of incident light are reflected more effectively at various depths of the mirror. This can potentially boost the bandwidth, and therefore the average reflectivity thermophotovoltaic mirror.

 figure: Fig. 3.

Fig. 3. Continuously chirped broadband mirror. Sjopek et. al. proposed a continuously chirped Bragg reflector, as we show in (a). The broad spectrum of wavelengths gets reflected at different depths. However, the resulting interference pattern is impossible to track mathematically. This gives rise to nulls in the reflection spectrum within the target spectral band, in (b). The nulls penalize the average reflectivity severely. The nulls can reach even 0% reflectivity for the sub-bandgap photons.

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We show the reflectivity of a 20-paired continuously chirped DBR on top of a gold substrate in Fig. 3(b). Here we focus on the spectral range 0.1-0.75 eV corresponding to 90% incident sub-bandgap energy from 1200°C emitter. The 20 pairs of DBR consist of layers with alternating refractive index, representative of Si (n = 3.5) and SiO2 (n = 1.45). The thickness of the top layer (3 µm, facing the emitter) and bottom layer (400 nm, on top of gold) satisfy the quarter wavelength thickness criterion to reflect incident light with photon energy 0.75 eV and 0.1 eV, respectively. We only show reflectivity for incident p-polarized photons, accounting for Brewster's angle. Nulls from destructive interference are present throughout the reflectivity spectrum. In a DBR with a constant optical thickness of all the layers, we can identify these nulls’ spectral location analytically. We can thus choose the DBR thickness to avoid the nulls within the desired part of the spectrum. But in a continuously chirped DBR, the spectral location of the nulls is challenging to predict because each photon wavelength sees a different phase shift after reflection from the mirror. Consequently, there is no known analytical solution, and hence it is difficult to avoid nulls in a manually tuned design. The nulls can appear inside the 0.1-0.75 eV range, severely penalizing the average reflectivity.

In the 20-paired continuously chirped DBR of Fig. 3(b), only 98% average reflectivity is achieved within the 0.1-0.75 eV range. Due to the origin of destructive interference, simply increasing the number of layers in the mirror cannot solve the null problem. Moreover, in the presence of material losses, increasing the number of layers would increase the parasitic absorption and penalize the reflectivity. Instead, we need a more thoughtful design approach to avoid the nulls in the reflectivity spectrum.

3. Inverse design of broadband mirrors

Inverse electromagnetic design is a powerful tool for achieving efficient photonic devices. Previous authors have used inverse design for waveguide splitters [19], grating couplers [17], textured solar cells [20], etc. In this section, we apply inverse design inspired techniques to extend thin-film mirrors’ bandwidth and average reflectivity.

There are three main steps in an inverse design problem: (i) identifying a physically meaningful initial structure for the optimization, (ii) identifying a figure-of-merit (FOM), and (iii) a method to calculate derivates of the FOM with respect to the 'design parameters for gradient descent based update.

We chose the continuously chirped DBR as a starting point. The continuously chirped DBR has a physically meaningful structure to reflect a broader spectrum of incident radiation. Then the objective of the inverse design is to remove/reduce the nulls from the reflection spectrum and increase the spectrally averaged reflectivity.

For thermophotovoltaic energy conversion, the mirror should reflect sub-bandgap photons over the entire hemisphere and both polarizations. The sub-bandgap spectrum also depends on the temperature of the thermal emitter. As such, our FOM would be to maximize the spectral, hemispherical, and polarization-averaged sub-bandgap reflectivity. We express this FOM in the following equation:

$$FOM\; = \; \frac{{2\pi \mathop \smallint \nolimits_0^{{\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 2}}}\!\lower0.7ex\hbox{$2$}}} \mathop \smallint \nolimits_0^{{E_g}} \frac{1}{2}({{{|{{r_s}({E,\theta } )} |}^2} + {{|{{r_p}({E,\theta } )} |}^2}} ){b_s}({E,{T_s}} )sin(\theta )\; cos(\theta )dEd\theta }}{{2\pi \mathop \smallint \nolimits_0^{{\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 2}}}\!\lower0.7ex\hbox{$2$}}} \mathop \smallint \nolimits_0^{{E_g}} {b_s}({E,{T_s}} )sin(\theta )\; cos(\theta )dEd\theta }}$$

In Eq. (1), rs and rp are the complex reflectivities of the mirror for s and p polarization of incident light, bs (E, Ts) is the power flux emitted by the thermal emitter at Ts (in units of W/m2/Str) as a function of photon energy E, and Eg is the bandgap energy of the thermophotovoltaic cell. The factor of sin(θ) comes from the spherical differential, and cos(θ) accounts for a Lambertian photon flux from the thermal emitter. The azimuthal angle θ variation is taken from 0 to π/2 since we are interested in measuring the reflectivity, a surface property. Thus, we address the FOM issue in point (ii).

For continuously chirped broadband mirrors, we can identify four design parameters: high and low-index material for the mirror, number of layers (or equivalently number of pairs of alternating high and low-index materials), and the thickness of the individual layers. The choice of physically compatible high and low-index materials is rather limited, e.g. AlGaAs (n = 3.5)-AlOx (n = 1.42), Silicon (n = 3.5)- SiO2 (n = 1.45). For our design purpose, we can fix the material pairs and the number of layers N. We can then optimize the thickness of each layer to maximize the FOM of Eq. (1). We can now write the optimization procedure of our mirror as follows:

$$\textrm{In}\;\textrm{the}\;k - \textrm{th}\;\textrm{step},\;d_i^k = d_i^{k - 1} + \alpha \; \frac{{\partial FOM}}{{\partial d_i^{k - 1}}}$$
where i = 1,…N, for N layers in the mirror. We can find the step size α through a line search. We have previously mentioned how to obtain the di's for the initial step i = 1, namely by assuming a continuously chirped DBR structure. Thus for our optimization, we need the derivatives $\frac{{\partial FOM}}{{\partial {d_i}}}$ in Eq. (2).

In photonics, the commonly used techniques for calculating derivatives of FOM are the adjoint method [8], automatic differentiation [9], parameter sweep, etc. For the one-dimensional broadband mirror of thermophotovoltaics, we use a new method of obtaining the derivative $\frac{{\partial FOM}}{{\partial {d_i}}}$ using the well-known transfer matrix method. The underlying mathematics is simple to track. We can significantly parallelize the derivative calculation for a large spectral and angular bandwidth. We provide a complete mathematical derivation of $\frac{{\partial FOM}}{{\partial {d_i}}}$ in the Appendix.

4. Methods and results

We can now implement the following strategy for mirror optimization:

  • i. Start with an initial structure. This will include a given thickness of the active layer of the thermophotovoltaic cell (superstrate of the mirror) and the mirror's high and low index material.
  • ii. Define an initial thickness profile of the mirror, with a given number of layers N. This will be a continuously chirped structure on top of a gold layer.
  • iii. Calculate the derivatives $\frac{{\partial FOM}}{{\partial {d_i}}}$ for all i = 1,..N. We can independently obtain the derivatives for the entire spectral and angular range with our transfer-matrix-based approach. We implement this parallel scheme on a 64-core server.
  • iv. Use Eq. (2) to update the thickness of each layer of the mirror.
  • v. Repeat steps iii and iv until the FOM reaches saturation.

We show the results of our optimization in Fig. 4. We performed the optimization for varying numbers of Si-SiO2 pairs. We obtain reflectivities > 99.0%, with a with a 4-pair optimized mirror, compared to 98% reflectivity with a 20-pair continuously-chirped mirror. Reflectivities improve with higher number of pairs with more constructive interferences inside the mirror. However, with thicker mirror, any parasitic absorption in the material will significantly penalize the reflectivity. This is a tradeoff to be considered while selecting the optimal number of pairs to use in the mirror. We show the improvement of reflectivity with a five-paired mirror in Fig. 4(b). There are far fewer nulls in the optimized mirror, resulting in a higher averaged reflectivity.

 figure: Fig. 4.

Fig. 4. Excellent broadband reflectivity with optimized mirrors. In (a) we show the average (angle, energy, and polarization) sub-bandgap reflectivity for a given number of Si-SiO2 pairs under the InGaAs thermophotovoltaic cell. We optimize the thickness using our proposed Fresnel-propagation based scheme. We can achieve >99% reflectivity using 4 pairs, as compared to 98% with a 10-paired continuously chirped mirror. We also show the total thickness, for the optimized mirror in (a). We show the comparison between original non-optimized mirror, and the optimized mirror for the 5 pair case in (b), at 89° incident angle, and p-polarized light. We see fewer nulls below the bandgap, for the optimized mirror.

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We show the optimization results for the five-paired mirror in Fig. 5. The thermophotovoltaic superstrate is InGaAs, and the mirror material is Si (n = 3.5)-SiO2 (n = 1.45). The FOM converges within 200 iterations, as we show in Fig. 5(a). The initial reflectivity is 97.5%, and the optimized reflectivity is 99.2%. As opposed to the continuously chirped structure of the initial mirror, the final design in Fig. 5(b) is approximately two DBRs in tandem. We call this a discreetly chirped DBR instead of a continuously chirped DBR. The discreetly chirped DBR significantly removes the nulls in the reflectivity spectrum, as we show in Fig. 5(c). If the nulls cannot be removed, the discreetly chirped DBR narrows the nulls. We repeated the optimizations for a different number of layers and other mirror materials. In all cases, the optimized structure is discretely chirped (two DBRs in tandem). For thermophotovoltaic broadband mirrors, discretely chirped DBR is the optimal mirror that achieves the best reflectivity. In the absence of losses in material, we can achieve a 99% reflectivity with just four pairs of alternating Si and SiO2. We can achieve this reflectivity across near three-octave bandwidths 0.1-0.75 eV, and the hemispherical angle 0-90°.

 figure: Fig. 5.

Fig. 5. Discretely chirped broadband mirror. The optimization converges within 200 iterations, in (a) for the case of a 4-paired mirror. Final structure is two DBRs on top of each other, a discretely chirped structure as opposed to the continuously chirped mirror, in (b). The optimization removes most nulls from the reflectivity spectrum, or narrows the nulls to improve the average reflectivity, in (c).

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Intuitively, discrete DBRs reflect across a wider range of wavelengths. By using multiple discrete DBRs with overlapping reflection spectra, we can fine-tune and enhance reflectivity across the entire desired spectrum without needing an infinite layer variation. On the other hand, continuous DBRs reflect most effectively at specific wavelengths, but not as effectively across a broad spectrum. Achieving consistent reflection over the entire incident spectrum with a continuous DBR would require an infinite number of layers with varying thicknesses. Otherwise, gaps in reflection (nulls) occur, lowering overall reflectivity.

When assessing fabrication feasibility, it is crucial to consider the collective variability in optical thickness, rather than just individual layer thickness. This will help in setting realistic bounds on integrated reflectivity during practical applications. Additionally, for a more realistic mirror, material dispersion should be taken into account, and the proposed method enables that with a simple optimization method. Future studies will take material losses and fabrication tolerances into account.

5. Conclusion

In this study, we have presented a new approach to significantly enhance the efficiency of thermophotovoltaic (TPV) devices, addressing the issue of sub-band gap photon losses. By employing inverse electromagnetic design principles, we have optimized the structure of continuously chirped distributed Bragg reflectors (DBR), enabling a notable increase in the average reflectivity and bandwidth of thin-film mirrors.

Our approach builds on and surpasses previous attempts to enhance mirror reflectivity. By leveraging the transfer matrix method for calculating necessary derivatives, we have advanced a simple method for achieving an optimized mirror structure. Consequently, we predict a substantial reduction in the amount of incident power that goes unutilized in InGaAs TPV cells at a blackbody emitter temperature of 1200°C. By reducing the inefficiencies in TPV devices, we pave the way for more sustainable and efficient energy conversion processes. Additionally, the method of inverse design and optimization proposed in this study can serve as a promising approach for other optical and photonic applications requiring high reflectivity over a broad bandwidth.

Appendix

Derivative of FOM

In this section, we’ll elaborate on the derivative of FOM with respect to the thickness di of layer i. We can write that derivative as follows:

$$\begin{aligned} \frac{{\delta FOM}}{{\delta {d_i}}} &= \frac{{2\pi \mathop \smallint \nolimits_0^{{\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 2}}}\!\lower0.7ex\hbox{$2$}}} \mathop \smallint \nolimits_0^{{E_g}} \frac{1}{2}\left( {\; \frac{{\delta {{|{{r_s}({E,\theta } )} |}^2}}}{{\delta {d_i}}} + \frac{{\delta {{|{{r_p}({E,\theta } )} |}^2}}}{{\delta {d_i}}}} \right){b_s}({E,{T_s}} )sin(\theta )\; cos(\theta )dEd\theta }}{{2\pi \mathop \smallint \nolimits_0^{{\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 2}}}\!\lower0.7ex\hbox{$2$}}} \mathop \smallint \nolimits_0^{{E_g}} {b_s}({E,{T_s}} )sin(\theta )\; cos(\theta )dEd\theta }}\\ &= \frac{{2\pi \mathop \smallint \nolimits_0^{{\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 2}} }\!\lower0.7ex\hbox{$2$}}} \mathop \smallint \nolimits_0^{{E_g}} \frac{1}{2}\left( {\; 2r_s^\ast ({E,\theta } )\frac{{\delta \{{{r_s}({E,\theta } )} \}}}{{\delta {d_i}}} + 2r_p^\ast ({E,\theta } )\frac{{\delta \{{{r_p}({E,\theta } )} \}}}{{\delta {d_i}}}} \right){b_s}({E,{T_s}} )sin(\theta )\; cos(\theta )dEd\theta }}{{2\pi \mathop \smallint \nolimits_0^{{\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 2}} }\!\lower0.7ex\hbox{$2$}}} \mathop \smallint \nolimits_0^{{E_g}} {b_s}({E,{T_s}} )sin(\theta )\; cos(\theta )dEd\theta }} \end{aligned}$$

As such, to calculate the derivative of FOM with respect to layer thickness di, we need to compute the derivative of the complex reflection coefficient rp(E,θ) and rs(E,θ) with respect to di. We’ll show the calculation for p-polarization only, and the procedure will be the same for s-polarization. We’ll drop the E and θ dependence for simplicity. For a multi-layered structure, we can express the reflectivity as:

$${r_p} = \frac{{{M_{21}}}}{{{M_{11}}}}$$
where ${\boldsymbol M} = \; \left[ {\begin{array}{{cc}} {{M_{11}}}&{{M_{12}}}\\ {{M_{21}}}&{{M_{22}}} \end{array}} \right]\; = \; {{\boldsymbol T}_{1,2}}\mathop \prod \nolimits_{i = 2}^{i = N - 1} {{\boldsymbol P}_i}({{d_i}} ){{\boldsymbol T}_{i,i + 1}}\; $

We have the matrices P and T as:

$${{\boldsymbol P}_i}({{d_i}} )= \left[ {\begin{array}{{cc}} {exp({ - j{k_i}{d_i}} )}&0\\ 0&{exp({j{k_i}{d_i}} )} \end{array}} \right]$$
$${{\boldsymbol T}_{i,i + 1}} = \frac{1}{2}\left[ {\begin{array}{{cc}} {\frac{{{k_{i + 1}}{n_i}}}{{{k_i}{n_{i + 1}}}} + \frac{{{n_{i + 1}}}}{{{n_i}}}}&{\frac{{{k_{i + 1}}{n_i}}}{{{k_i}{n_{i + 1}}}} - \frac{{{n_{i + 1}}}}{{{n_i}}}}\\ {\frac{{{k_{i + 1}}{n_i}}}{{{k_i}{n_{i + 1}}}} - \frac{{{n_{i + 1}}}}{{{n_i}}}}&{\frac{{{k_{i + 1}}{n_i}}}{{{k_i}{n_{i + 1}}}} + \frac{{{n_{i + 1}}}}{{{n_i}}}} \end{array}} \right]$$

Here ki =2πni cos(θi)/λ. As such, the only thickness-dependent term in Eq. (2) is the propagation matrix P.

We can take the derivative of the complex reflection coefficient rp as follows:

$$\frac{{\delta {r_p}}}{{\delta {d_i}}} = \frac{{{M_{11}}\frac{{\delta {M_{21}}}}{{\delta {d_i}}} - {M_{21}}\frac{{\delta {M_{11}}}}{{\delta {d_i}}}}}{{M_{11}^2}}$$

We can obtain the derivatives of the matrix elements of M in the following manner:

$$\frac{{\delta {\boldsymbol M}}}{{\delta {d_i}}} = \; \left[ {\begin{array}{{cc}} {\frac{{\delta {M_{11}}}}{{\delta {d_i}}}}&{\frac{{\delta {M_{12}}}}{{\delta {d_i}}}}\\ {\frac{{\delta {M_{21}}}}{{\delta {d_i}}}}&{\frac{{\delta {M_{22}}}}{{\delta {d_i}}}} \end{array}} \right]\; = {{\boldsymbol T}_{1,2}}{{\boldsymbol P}_2}({{d_2}} ){{\boldsymbol T}_{2,3}}{{\boldsymbol P}_3}({{d_3}} )\ldots {{\boldsymbol T}_{i - 1,i}}\; \frac{{\delta {{\boldsymbol P}_i}({{d_i}} )}}{{\delta {d_i}}}{{\boldsymbol T}_{i,i + 1}} \ldots {{\boldsymbol T}_{N - 1,N}}$$

As such, we need a derivative only for the propagation matrix of layer i. The derivative of Pi(di) with respect to the layer thickness di can be written as:

$$\scalebox{0.9}{$\begin{aligned} \frac{{\delta {{\boldsymbol P}_i}({{d_i}} )}}{{\delta {d_i}}} &= jk\left[ {\begin{array}{{@{}cc@{}}} { - exp({ - j{k_i}{d_i}} )}&0\\ 0&{exp({j{k_i}{d_i}} )} \end{array}} \right] = exp\left( {j\frac{{}}{2}} \right)jk\left[ {\begin{array}{{@{}cc@{}}} {exp\left( {j\frac{{}}{2}} \right)exp({ - j{k_i}{d_i}} )}&0\\ 0&{exp\left( { - j\frac{{}}{2}} \right)exp({j{k_i}{d_i}} )} \end{array}} \right]\\ &={-} k\left[ {\begin{array}{{@{}cc@{}}} {exp\left\{ { - j\left( {\frac{{2{n_i}cos({{\theta_i}} )}}{{}}{d_i} - \frac{{2{n_i}cos({{\theta_i}} )}}{{}}\frac{\lambda }{{4{n_i}cos({{\theta_i}} )}}} \right)} \right\}}&0\\ 0&{exp\left\{ {j\left( {\frac{{2{n_i}cos({{\theta_i}} )}}{{}}{d_i} - \frac{{2{n_i}cos({{\theta_i}} )}}{{}}\frac{\lambda }{{4{n_i}cos({{\theta_i}} )}}} \right)} \right\}} \end{array}} \right] \end{aligned}$}$$
$$$$

This final expression gives us an interesting insight. It tells us the derivative of the Pi matrix is another propagation matrix, but the layer thickness changed by $\frac{\lambda }{{4{n_i}cos({{\theta_i}} )}}$. Thus, we can envision the derivative calculation as the backward simulation of the inverse design. We calculate the rp for the given di in the forward simulation using the transfer matrix method. In the inverse simulation, we calculate δrp/δdi by another transfer matrix method, with di replaced with ${d_i} - \frac{\lambda }{{4{n_i}cos({{\theta_i}} )}}$, equivalently ${d_i} - \frac{\pi }{{2k}}$.

Resolution convergence test

We need to confirm the reflectivity with an accuracy of 0.01%. We used a photon energy resolution of 0.1 meV and an incident angle of 1°. We show the results of the convergency test as a function of energy and angle resolution, with both 5-pair and 25-pair mirrors. We calculate the reflectivity of both a thin and thick mirror to account for any interference fringes arising from thickness effects. We show the average reflectivity (0.1-0.74 eV) over the hemisphere for the 1200°C emitter.

Photon energy resolution (meV)Angular resolution (°)5-paired mirror reflectivity25-paired mirror reflectivity
100198.3899.77
10197.1399.67
1197.1299.68
0.1197.1299.68
0.01197.1299.68
0.001197.1299.68
0.0001197.1299.68
0.10.197.1299.68
0.10.0197.1299.68
0.10.00197.1299.68

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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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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Figures (5)
Fig. 1.
Fig. 1. Mirrors for ultra-efficient thermophotovoltaic energy conversion. An InGaAs thermophotovoltaic cell receives radiation from a 1200°C hot emitter. Only a portion of the high-energy photons (blue area) is converted to electricity. A highly reflective mirror can reflect low-energy photons that would otherwise be wasted (red area), thereby increasing the power conversion efficiency. Any imperfection in the mirror creates mirror loss. Thermalization of high-energy carriers is the main contributor to PV loss for high-energy photons.

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Fig. 2.
Fig. 2. Increasing returns of a better mirror. As mirror reflectivity gets closer to 100%, there is an increasing return of thermophotovoltaic efficiency, in (a). The voltage increases due to enhanced luminescence extraction, and parasitic low-energy photon absorption is reduced. However, the minimum mirror bandwidth needed for thermophotovoltaics is much wider than can be achieved with a traditional distributed Bragg reflector, as we show in (b). For (a), we assume the TPV cell has an anti-reflection coating for above-bandgap photons, to minimize photon loss.

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Fig. 3.
Fig. 3. Continuously chirped broadband mirror. Sjopek et. al. proposed a continuously chirped Bragg reflector, as we show in (a). The broad spectrum of wavelengths gets reflected at different depths. However, the resulting interference pattern is impossible to track mathematically. This gives rise to nulls in the reflection spectrum within the target spectral band, in (b). The nulls penalize the average reflectivity severely. The nulls can reach even 0% reflectivity for the sub-bandgap photons.

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Fig. 4.
Fig. 4. Excellent broadband reflectivity with optimized mirrors. In (a) we show the average (angle, energy, and polarization) sub-bandgap reflectivity for a given number of Si-SiO2 pairs under the InGaAs thermophotovoltaic cell. We optimize the thickness using our proposed Fresnel-propagation based scheme. We can achieve >99% reflectivity using 4 pairs, as compared to 98% with a 10-paired continuously chirped mirror. We also show the total thickness, for the optimized mirror in (a). We show the comparison between original non-optimized mirror, and the optimized mirror for the 5 pair case in (b), at 89° incident angle, and p-polarized light. We see fewer nulls below the bandgap, for the optimized mirror.

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Fig. 5.
Fig. 5. Discretely chirped broadband mirror. The optimization converges within 200 iterations, in (a) for the case of a 4-paired mirror. Final structure is two DBRs on top of each other, a discretely chirped structure as opposed to the continuously chirped mirror, in (b). The optimization removes most nulls from the reflectivity spectrum, or narrows the nulls to improve the average reflectivity, in (c).

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Equations (10)

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$$FOM\; = \; \frac{{2\pi \mathop \smallint \nolimits_0^{{\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 2}}}\!\lower0.7ex\hbox{$2$}}} \mathop \smallint \nolimits_0^{{E_g}} \frac{1}{2}({{{|{{r_s}({E,\theta } )} |}^2} + {{|{{r_p}({E,\theta } )} |}^2}} ){b_s}({E,{T_s}} )sin(\theta )\; cos(\theta )dEd\theta }}{{2\pi \mathop \smallint \nolimits_0^{{\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 2}}}\!\lower0.7ex\hbox{$2$}}} \mathop \smallint \nolimits_0^{{E_g}} {b_s}({E,{T_s}} )sin(\theta )\; cos(\theta )dEd\theta }}$$
$$\textrm{In}\;\textrm{the}\;k - \textrm{th}\;\textrm{step},\;d_i^k = d_i^{k - 1} + \alpha \; \frac{{\partial FOM}}{{\partial d_i^{k - 1}}}$$
$$\begin{aligned} \frac{{\delta FOM}}{{\delta {d_i}}} &= \frac{{2\pi \mathop \smallint \nolimits_0^{{\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 2}}}\!\lower0.7ex\hbox{$2$}}} \mathop \smallint \nolimits_0^{{E_g}} \frac{1}{2}\left( {\; \frac{{\delta {{|{{r_s}({E,\theta } )} |}^2}}}{{\delta {d_i}}} + \frac{{\delta {{|{{r_p}({E,\theta } )} |}^2}}}{{\delta {d_i}}}} \right){b_s}({E,{T_s}} )sin(\theta )\; cos(\theta )dEd\theta }}{{2\pi \mathop \smallint \nolimits_0^{{\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 2}}}\!\lower0.7ex\hbox{$2$}}} \mathop \smallint \nolimits_0^{{E_g}} {b_s}({E,{T_s}} )sin(\theta )\; cos(\theta )dEd\theta }}\\ &= \frac{{2\pi \mathop \smallint \nolimits_0^{{\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 2}} }\!\lower0.7ex\hbox{$2$}}} \mathop \smallint \nolimits_0^{{E_g}} \frac{1}{2}\left( {\; 2r_s^\ast ({E,\theta } )\frac{{\delta \{{{r_s}({E,\theta } )} \}}}{{\delta {d_i}}} + 2r_p^\ast ({E,\theta } )\frac{{\delta \{{{r_p}({E,\theta } )} \}}}{{\delta {d_i}}}} \right){b_s}({E,{T_s}} )sin(\theta )\; cos(\theta )dEd\theta }}{{2\pi \mathop \smallint \nolimits_0^{{\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 2}} }\!\lower0.7ex\hbox{$2$}}} \mathop \smallint \nolimits_0^{{E_g}} {b_s}({E,{T_s}} )sin(\theta )\; cos(\theta )dEd\theta }} \end{aligned}$$
$${r_p} = \frac{{{M_{21}}}}{{{M_{11}}}}$$
$${{\boldsymbol P}_i}({{d_i}} )= \left[ {\begin{array}{{cc}} {exp({ - j{k_i}{d_i}} )}&0\\ 0&{exp({j{k_i}{d_i}} )} \end{array}} \right]$$
$${{\boldsymbol T}_{i,i + 1}} = \frac{1}{2}\left[ {\begin{array}{{cc}} {\frac{{{k_{i + 1}}{n_i}}}{{{k_i}{n_{i + 1}}}} + \frac{{{n_{i + 1}}}}{{{n_i}}}}&{\frac{{{k_{i + 1}}{n_i}}}{{{k_i}{n_{i + 1}}}} - \frac{{{n_{i + 1}}}}{{{n_i}}}}\\ {\frac{{{k_{i + 1}}{n_i}}}{{{k_i}{n_{i + 1}}}} - \frac{{{n_{i + 1}}}}{{{n_i}}}}&{\frac{{{k_{i + 1}}{n_i}}}{{{k_i}{n_{i + 1}}}} + \frac{{{n_{i + 1}}}}{{{n_i}}}} \end{array}} \right]$$
$$\frac{{\delta {r_p}}}{{\delta {d_i}}} = \frac{{{M_{11}}\frac{{\delta {M_{21}}}}{{\delta {d_i}}} - {M_{21}}\frac{{\delta {M_{11}}}}{{\delta {d_i}}}}}{{M_{11}^2}}$$
$$\frac{{\delta {\boldsymbol M}}}{{\delta {d_i}}} = \; \left[ {\begin{array}{{cc}} {\frac{{\delta {M_{11}}}}{{\delta {d_i}}}}&{\frac{{\delta {M_{12}}}}{{\delta {d_i}}}}\\ {\frac{{\delta {M_{21}}}}{{\delta {d_i}}}}&{\frac{{\delta {M_{22}}}}{{\delta {d_i}}}} \end{array}} \right]\; = {{\boldsymbol T}_{1,2}}{{\boldsymbol P}_2}({{d_2}} ){{\boldsymbol T}_{2,3}}{{\boldsymbol P}_3}({{d_3}} )\ldots {{\boldsymbol T}_{i - 1,i}}\; \frac{{\delta {{\boldsymbol P}_i}({{d_i}} )}}{{\delta {d_i}}}{{\boldsymbol T}_{i,i + 1}} \ldots {{\boldsymbol T}_{N - 1,N}}$$
$$\scalebox{0.9}{$\begin{aligned} \frac{{\delta {{\boldsymbol P}_i}({{d_i}} )}}{{\delta {d_i}}} &= jk\left[ {\begin{array}{{@{}cc@{}}} { - exp({ - j{k_i}{d_i}} )}&0\\ 0&{exp({j{k_i}{d_i}} )} \end{array}} \right] = exp\left( {j\frac{{}}{2}} \right)jk\left[ {\begin{array}{{@{}cc@{}}} {exp\left( {j\frac{{}}{2}} \right)exp({ - j{k_i}{d_i}} )}&0\\ 0&{exp\left( { - j\frac{{}}{2}} \right)exp({j{k_i}{d_i}} )} \end{array}} \right]\\ &={-} k\left[ {\begin{array}{{@{}cc@{}}} {exp\left\{ { - j\left( {\frac{{2{n_i}cos({{\theta_i}} )}}{{}}{d_i} - \frac{{2{n_i}cos({{\theta_i}} )}}{{}}\frac{\lambda }{{4{n_i}cos({{\theta_i}} )}}} \right)} \right\}}&0\\ 0&{exp\left\{ {j\left( {\frac{{2{n_i}cos({{\theta_i}} )}}{{}}{d_i} - \frac{{2{n_i}cos({{\theta_i}} )}}{{}}\frac{\lambda }{{4{n_i}cos({{\theta_i}} )}}} \right)} \right\}} \end{array}} \right] \end{aligned}$}$$
$$$$
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