1. Introduction

Wireless power transfer (WPT) is an enabling technology that eliminates the physical connection needed to power, charge, and operate electronic devices. The possibility of cordless energization has a wide range of applications ranging from space systems to consumer electronics. These include, for example, solar-powered satellites [1], electric vehicles [2], unmanned aerial vehicles [3], implanted medical devices, and sensor arrays to mention a few [4]. Over the years, various technologies have emerged as applications of wireless power transfer. In particular, inductive and capacitive WPT have been utilized for near-field applications as they harvest energy from non-radiative electromagnetic fields produced by coils of wire and metal plate capacitors, respectively [5]. On the other hand, microwave and optical radiative methods have demonstrated adequate efficiencies for far-field applications [6,7]. Optical power beaming is particularly auspicious in its effort to improve defense technology, space science, and electronic industries [8]. While microwave power transmission currently appears to be more mature and developed, lasers remain advantageous for supplying high amounts of power for point-to-point links over increasing separation distances [7]. The abundance of low-cost semiconductor lasers over a wide spectral range, including the atmospheric window, that are capable of concentrating high powers in small beam apertures, make optical power beaming a promising solution for long-range WPT. However, one of the fundamental challenges of optical power beaming systems is their low end-to-end efficiency, which is partially due to the deficiency of the existing optical power converters.

The available spectrum for optical power beaming lies in the near-infrared, where the transmission efficiency is above 70% [7]. Low power attenuation qualifies near-infrared lasers as a suitable transmitter for successful far-field WPT systems. In this region, commercially available laser sources can radiate over large distances without substantial atmospheric absorption. Strong light-matter interaction and optical compatibility in this frequency govern the choice of material for the absorbing layer to improve the overall power conversion efficiency. Gallium Arsenide (GaAs) is a direct bandgap material that offers high photon conversion efficiencies in the range 400-860 nm [9]. Compared to commercial photovoltaic materials, GaAs is considered a high-cost material, which can be compensated by using thin films. This instead calls for photon management schemes to enhance the external quantum efficiencies [10,11].

All wireless power transfer systems aim to minimize power attenuation and maximize end-to-end power transmission. In this vein, an efficient photovoltaic power converter is a critical component of laser power beaming systems. This work proposes a novel multilayer photovoltaic absorber that can maximally trap monochromatic light incidents from a broad angular spectral range. The proposed structure is built on resonant absorption of photons in thin-film photovoltaic materials by utilizing front- and back-reflectors to recycle photons in the absorbing layer. In principle, such a structure can absorb monochromatic light with 100% efficiency by preparing the conditions for the so-called coherent perfect absorption [12–14]. Often referred to as time-reserved lasing, coherent perfect absorption relies on perfect matching of the rate of absorption in a lossy material with the flux of incoming photons (it can be compared with load impedance matching in transmission lines).

The concept of coherent perfect absorption has been suggested for optical sensing [15], spectroscopy [16,17], optoelectronics [18]. In interesting recent work, this notion was utilized for solar energy harvesting applications [19,20]. However, given that coherent perfect absorption is essentially a resonance phenomenon, its applicability to broadband applications, in this case, the solar emission spectrum, is severely limited. On the other hand, coherent perfect absorption appears to be well suited for power-beaming applications where monochromatic light is involved. Here, we first discuss the case of a coherent perfect absorption design for thin-film absorbers. But, we will argue that such a design leads to a relatively small angular absorption window. Thus, we focus our attention on the inverse design of cavities that enable large angular absorption windows for coherent light sources.

Design of multilayer structures is an effective approach for photon management in photovoltaic cells [21]. In the recent years, there has been many efforts for the design of multilayer material stacks for energy harvesting applications. In particular, distributed Bragg reflectors (DBRs) have been embedded in optical devices to increase efficiency [22,23]. However, DBRs, offer maximal reflectivity at a limited angle or frequency range which is not ideal for power-beaming systems that require continuous transmitter-receiver alignment. Therefore, of interest is to utilize inverse design techniques [24,25] for systematic optimization of multilayers for efficient absorbers for power beaming systems.

In this article, we inverse design a multilayer structure, as shown schematically in Fig. 1, that enables absorption of photons incident from a broad range of angles. In our design, aperiodic front- and back-reflector mirrors were simulatensouly optimized to warrant efficient light trapping in thin films for omnidirectional incidence. Our results confirm the feasibility of realistic implementation based on a GaAs thin films with binary AlAs/AlGaAs mirrors. To demonstrate the generality of our approach, we further discuss examples with different material structures for the top and bottom reflectors. In addition, we perform a sensitivity analysis for the proposed designs and show that our designs tolerate minor deviations from the desired thicknesses of individual layers.

 

Fig. 1. A schematic of the structure of the omnidirectional coherent absorber.

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2. Results and discussion

It is well known that the absorption efficiency of a thin film can be improved by recycling photons through front- and back-reflectors. As a reference structure, we consider a thin-film GaAs layer sandwiched between conventional quarter-wavelength DBR’s composed of high-low index aluminum gallium arsenide (AlGaAs) and aluminum arsenide (AlAs). These materials were chosen for their high index contrast and negligible absorption near the employed wavelength of 850 nm. It is desirable to set the thickness of the absorbing layer near odd multiples of the quarter wavelength to ensure the existence of a resonance mode in a double-DBR cavity setting. Here we used $180.15 \, \textrm{ nm}$, which is close to the three-quarter wavelength condition. As shown in Fig. 2, the absorptivity increases significantly when the back-reflector is present (Fig. 2(b)) and is further enhanced when utilizing a front-reflector to form a full cavity (Fig. 2(c)). The structure exhibits peak absorption of $95.79$% with a 4-layer top mirror, corresponding to approximately $75$% reflectivity. This reflectivity allows for incoming radiation to enter the cavity while warranting photon confinement. In this case, we observe a peak in the absorptivity, which is associated with the cavity resonance. The absorption peak can be further increased to reach in principle to $100$%. This is the so-called coherent perfect absorption (CPA) regime [12,13], or, a critical coupling effect. In this case, the rate of the incoming energy flux matches the absorption rate in the thin film. Here, coherent perfect absorption, or critical coupling, is achieved by increasing the reflectivity of the back-reflector simply by increasing the number of layers in order to prevent scattering losses through leakage from the bottom layer.

 

Fig. 2. The reflection, transmission, and the corresponding absorptivity for a thin-film GaAs layer of $180.15 \, \textrm{nm}$ thickness at $\lambda = 850 \, \textrm{nm}$, without and with binary AlGaAs/AlAs DBRs. (a) A single GaAs layer, (b) using a bottom DBR, (c) using both top and bottom DBRs, and (d) using both top and bottom DBRs with more layers in the bottom. In (b), the bottom DBR contains 25 layers. In (c), the top and bottom DBRs contain 4 and 25 layers, respectively. In (d) the top and bottom DBRs contain 4 and 40 layers, respectively. The refractive indices used were, GaAs: $n+i\kappa =3.62+i0.0768$ [26], AlAs: $n=3.05$, AlGaAs: $n=3.5$ at $\lambda _0=850 ~\textrm{nm}$.

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As expected the high absorption in Fig. 2 is limited to a finite angular width peaking at a design angle, in this case $24.37^{\circ }$, at which the cavity is at resonance. This is better illustrated in the two-dimensional parameter space of angle and wavelength, ($\theta,\lambda$), as shown in the bottom panels in Fig. 2. In the past, several techniques have been explored to overcome this narrow bandwidth including diffraction gratings [20] and Salisbury screens [27]. These solutions require additional components that, in addition to the cost and design complexity, can also reduce the overall efficiency. Our goal in this work is to increase the angular bandwidth of absorption while utilizing the benefits of photon recycling through cavity mirrors. For this purpose, instead of utilizing conventional Bragg mirrors, we inverse-design aperiodic reflectors that result in high absorption over a broad angular range.

We consider the general scenario, where the photovoltaic thin-film material of a given thickness is sandwiched between two multilayer mirrors with binary materials of an arbitrary thickness of individual layers. The thickness of all layers is then optimized simultaneously to ensure high absorption at broad incident angles. For this high-dimensional optimization problem, we employ particle swarm optimization (PSO), followed by an interior-point optimization for fine-tuning. PSO is a popular optimization tool in electromagnetics and has found previous success in similar problems requiring broadband response [28]. The optimization is done in MATLAB using a swarm size of $10$ times the number of layers, an initial neighbor fraction of $25{\%}$, self and social weights of $1.49$ and an inertial weight in the range $(0.1,1.1)$. As PSO can often settle in local minima, the search algorithm is run several times and the best results are taken. The absorptivity is given by

The operation wavelength was set to $\lambda _0 = 850 \, \textrm{nm}$, with unpolarized light incident at angles $0^{\circ } \leq \theta \leq 70 ^{\circ }$. The GaAs width was allowed to vary near 180 nm, the three-quarter wavelength condition at which a resonance is expected. The optimized structures were found to settle at 200 nm. The number of layers for the top and bottom mirrors was fixed to 7 and 25 pairs respectively, with the SiN/$\textrm{SiO}_{2}$ layers allowed to vary between 30 nm and 200 nm, and the GaAs/AlGaAs between 30 nm and 100 nm. These upper bounds were chosen based on consideration of the widths of conventional Bragg reflectors, the case to which the optimized structure is compared.

As a first example, we considered the simpler case of both the top and bottom mirrors consisting of AlAs/AlGaAs. Figure 3(d) shows the absorptivity of the cavity with the quarter-wavelength DBR’s, where multiple resonant modes can be seen on either side of the primary absorbing band. The tendency for the resonances in DBR structures to shift toward lower wavelengths at higher angles is well known and is the primary roadblock to achieving wide-angle response [30]. To mitigate this issue, we ansatz that under a small perturbation of the DBR widths, the primary resonant band can couple with one or more of the nearby bands, thereby broadening the spectral response. And indeed, in the optimized structure this coupling does occur with the bands near $800 \, \textrm{nm}$, as can be gleaned from Fig. 3(d). At the operating wavelength, the average absorptivity is significantly improved for the inverse-designed mirrors (Fig. 3(g)) compared to that of the quarter-wavelength DBR’s (Fig. 3(b)). Up to the optimized cut-off angle of $70$ degress, the absorptivity is fairly flat around $60{\%}$, with the reflection phase increasing roughly linearly with the incident angle. The s-polarized fields for the optimized and DBR structure are shown in Fig. 4 for different incident angles. As expected, the resonance in and near the GaAs film is strongest between 45 and 60 degrees for the DBR structure but is strongly enhanced at a much wider range of angles for the optimized structure. In both cases, the field transmission falls off to near zero as it propagates away from the absorbing layer.

 

Fig. 3. A comparison of the absorptivity between the two cases of using conventional DBR mirrors (first row) versus inverse-designed mirrors (second row). (a,f) The thickness of each layer. (b,g) Reflection, transmission and absorption and (c,h) reflection phases at $\lambda = 850 \, \textrm{nm}$. (d,i) Absorption versus angle and frequency. (e,j) The reflectivity in the complex $\theta$-plane, at a fixed wavelength of $\lambda = 850 \, \textrm{nm}$, highlighting the location of the poles (the colorbar is in logarithmic scale).

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Fig. 4. A visualization of the electric field for various angles, comparing the cases of using Bragg mirrors (a), and the optimized structure (b). Here, the s polarization is considered and the real part of the fields are plotted. The GaAs layer is indicated by the boxed region.

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To provide greater insight, the absorptivity is plotted in the two-dimensional parameter space of angle and wavelength. In comparison to the case of the conventional Bragg mirrors (Fig. 3(a)) with widely spaced resonant modes, in the inverse-designed mirrors (Fig. 3(f)) the lines associated with the resonant modes become densely packed in the desired wavelength and angular range. To show this, we calculated the complex modes of the two structures in the complex angle $\theta =\theta _r + i \theta _i$ plane and for a constant wavelength $\lambda _0$. The results are shown in Figs. 3(e,j). These figures depict the reflectivity in the complex $\theta$ plane, while the singularities mark the location of the poles of the reflection which are associated with the resonant modes. According to these figures, in the case of the conventional DBR structure of Fig. 3(a), there are only two poles (Fig. 3(e)), while in the case of the inverse-designed structure of Fig. 3(f), there are four poles with their real parts in the desired angular window (Fig. 3(j)). These poles come in pairs as the unpolarized light consists of both TE and TM modes. In the optimized structure, we observe that the imaginary parts of the poles ensure significant overlap between the linewidth of the resonant modes.

We further consider the electric field profiles inside the multilayer structures under plane wave illumination from different angles for conventional Bragg mirrors and inverse-designed mirrors. The fields were simulated in the frequency domain with an average mesh size of 4 nm in the propagation direction and with periodic boundary conditions in the transverse direction. The spectral features in Fig. 4 show the electric field for different incident angles under s-polarized light following structures shown in Fig. 3. In the case of conventional Bragg mirrors, Fig. 4(a) shows that the field is most strongly at resonance in the thin-film between 45° and 60°, but in the case of the inverse design mirrors the field is enhanced inside the thin-film for a much wider range of angles. Absorption enhancement due to mode coupling and increased electric field amplitude within the cavity lifts angular dependency.

We further exemplify this observation through another example in which we used different materials for the front reflector. Here, we first consider the cavity design utilized in Ref. [31]. Next, using the same number of top and bottom layers, but with SiN/$\textrm{SiO}_{2}$ materials for the top mirror, we similarly find $60{\%}$ absorptivity for the inverse-designed structure. In this case, again one can clearly see that large absorptivity is achieved over a broad angular range. Here, the modes dynamics is more complex and the optimization process results in a significant evolution of the structure from Fig. 5(a) to Fig. 5 (f). Nonetheless, as before the wide-angle response is achieved by the overlapping of modes. Although using the SiN/$\textrm{SiO}_{2}$ for the top mirror did not increase the absorptivity, it did straighten out the absorption curve compared to the optimized device with an AlAs/AlGaAs top mirror.

 

Fig. 5. A comparison between the two cases of using conventional Bragg mirrors (first row) versus inverse-designed mirrors (second row) when utilizing different materials for the top and bottom reflectors. Here, the top reflector is based on SiN/$\textrm{SiO}_{2}$ while the bottom reflector is based on AlGaAs/AlAs. (a,e) Thickness of each layer. (b,f) Reflection, transmission and absorption at $\lambda = 850 \, \textrm{nm}$. (c,g) Reflection phases. (d,h) Absorption versus angle and frequency. (a,f) The thickness of each layer. (b,g) Reflection, transmission and absorption and (c,h) reflection phases at $\lambda = 850 \, \textrm{nm}$. (d,i) Absorption versus angle and frequency. (e,j) The reflectivity in the complex $\theta$-plane, at a fixed wavelength of $\lambda = 850 \, \textrm{nm}$, highlighting the location of the poles (the colorbar is in logarithmic scale). The refractive indices used were, GaAs: $n+i\kappa =3.62+i0.0768$ [26], AlAs: $n=3.05$, AlGaAs: $n=3.5$, SiN: $n=1.87$, and $\textrm{SiO}_{2}$: $n=1.45$ at $\lambda _0=850 ~\textrm{nm}$.

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To increase the overall absorptivity, it is important to investigate the role of the thickness of the photovoltaic layer. Figure 6 illustrates the performance of the optimized cell with different GaAs layer thicknesses. Since the expected absorption peaks at normal incidence occur at odd multiples of 60 nm (corresponding to quarter wavelength in GaAs), we consider thicknesses which are allowed to vary near 300, 420, 540, 660, and 780 nm. In this case, after the optimization the width of the GaAs layer settled on 320, 440, 526, 673, and 789 nm, respectively. According to Fig. 6, as the thickness of the GaAs increases, the absorption between 0° and 70° increases up to an average of 87%, while maintaining additional broad-angle stability. A trade-off thereby exists between increasing the absorptivity and minimizing the cost of growing the GaAs layer.

 

Fig. 6. The inverse-designed structures (top row), reflection, transmission, and absorption at $\lambda = 850 \, \textrm{nm}$ (middle row) and their corresponding absorptivity versus incidence angle and frequency (bottom row) for optimized GaAs thicknesses centered around initial values of (a) $300 \, \textrm{nm}$, (b) $420 \, \textrm{nm}$, (c) $540 \, \textrm{nm}$, (d) $660 \, \textrm{nm}$, and (e) $780 \,\textrm{nm}$.

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Finally, it is important to investigate the sensitivity of the designs obtained from optimization. We explore this by randomly perturbing the thickness of the different layers and recalculating the absorption. In molecular beam epitaxy (MBE) growth, the theoretical error in width for a single layer is half the lattice parameter. Here, we consider random perturbation in the thickness of all layers by values drawn independently from normal distributions with the standard deviation of $1$ and $2 \, \textrm{nm}$. The results are presented in Fig. 7. The performance of the cavity remains robust under slight thickness deviations, which imitate fabrication conditions.

 

Fig. 7. Sensitivity of optimized structures to perturbations in the thicknesses of the layers involved. (a) The perturbed counterpart of the structure discussed in Fig. 3. (b) The perturbed counterpart of the structure discussed in Fig. 5. (b) SiN/$\textrm{SiO}_{2}$ and AlGaAs/AlAs optimized device. For these figures, each layer is perturbed in accordance to a normal distribution with $\sigma = 1 \, \textrm{nm}, 2 \, \textrm{nm}$. The solid line is associated with the unperturbed structure and the dashed bounding lines represent the first standard deviation of the absorption in the perturbed structures.

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3. Conclusion

The adoption of wireless power transfer for space, defense, and consumer electronic applications is restricted due to low-efficiency rates. An inverse-designed optical power converter, as discussed here, has the potential of reaching very high omnidirectional absorption, thus ameliorating power conversion efficiency for power-beaming systems. Multiple cases, including different reflector materials and layer widths, have proven broad-angle absorption of at least $60{\%}$ for this thin-film structure (Fig. 3, Fig. 5). Optimization and fine-tuning techniques promote resonant mode coupling and increased electric field amplitudes within the cavity, both of which account for enhanced absorption and stability across a wide range of incident angles. Thus, free carrier absorption in the semiconducting material contribute to the extractable electrical energy. Optical wireless power transfer could greatly benefit from the proposed photovoltaic cell as its system requires specific parameters and constant point-to-point contact. From a practical point of view, peak system performance largely depends on component efficiency and environmental factors. Maximizing broad-angle absorption at the receiver end may mitigate efficiency loss due to light divergence and obstructions in the line of sight over large transmission distances. Our results confirm that a photovoltaic receiver based on primal cavity construction can be built with a moderate number of layers, thus, MBE fabrication will have a high yield while reducing cost and errors. Inexpensive material growth and robustness to fabrication errors add a layer of appeal for development and commercialization. In conclusion, embedding the proposed inverse-designed cavity into a photovoltaic receiver can improve overall power conversion efficiency and encourage the acquisition of wireless power technology.