1. Introduction

Upconversion (UC) is the process of converting two (or more) photons into one photon of higher energy. It is relevant in a broad range of applications, from anti counterfeiting [1, 2], plastic recycling [3], bioimaging [4,5] and theranostics [5–7] to photovoltaics [8–11]. In photovoltaics, UC enables the utilization of sub-bandgap photons, which for silicon-based solar cells raises the theoretical efficiency limit from ≈ 30% [12] to ≈ 40% [8]. Spectrally suitable materials capable of harvesting near-infrared light have been identified among the trivalent rare-earth materials, in particular the Er³⁺ ion [9, 13–16]. In this study, we consider erbium-doped hexagonal sodium yttrium fluoride (β-NaYF₄:Er³⁺), which ranges among the best performing materials for UC applications in the context of silicon-based photovoltaics [17–22]. However, even for this material system, the UC efficiency remains too low for commercial exploitation [11].

UC is a multi-photon process and thus of non-linear nature [23]. For UC to take place, it requires that at least two photons are absorbed in a small volume within a short period of time. Therefore, in the unsaturated regime, to which non-concentrated solar illumination belongs [24], the UC efficiency can be increased by light concentration. Besides focusing by conventional means, such as lenses, strong local field enhancements can be achieved through plasmonic structures [25–30]. The strong field leads to an increased excitation density in the upconverter and consequently to a higher UC efficiency. A major drawback of plasmonic structures, however, is parasitic absorption, which reduces the energy density of the incident field available for upconversion. Furthermore, additional non-radiative loss channels are introduced, which can dissipate energy from the excited upconverter ions. In the immediate vicinity of plasmonic structures, the non-radiative losses strongly reduce the UC emission, which is particularly problematic as it is typically here the maximum field enhancement occurs [25,26]. Another way to obtain strong local field enhancements is through dielectric photonic structures [31–41]. Unlike plasmonic structures, photonic structures do not inherently suffer from parasitic absorption, but they do modify the local density of optical states (LDOS). As the LDOS affects the probability of spontaneous emission, a modified LDOS can potentially increase UC emission and/or suppress loss channels, all of which can be tailored through structural engineering [31,32].

The literature on the application of photonic crystals to improve UC luminescence is dominated by inverse opal [37,40–42] and opal structures [36,38]. For those structures, an experimentally measured UC luminescence enhancements of up to 43 times has been reported [37]. Moreover, a very high UC luminescence enhancement of 10⁴ has been reported for a waveguide structure [43]. Recently, photonic-plasmonic hybrid structures have been experimentally demonstrated, yielding UC luminescence enhancement factors up to three orders of magnitude [44,45]. Additionally, an experimental study of one-dimensional photonic crystals was carried out by [34], where the crystal was formed from Er³⁺-doped porous silicon with the periodic variation of the refractive index obtained by varying the silicon porosity. In that case, a maximum UC luminescence enhancement of a factor of 26.6 was reported.

In most studies, the design of the photonic structure is not optimized, but simply chosen such that the excitation lies at the photonic band edge [36–38]. In a few reports the actual local field is simulated, based on which an optimal design is realized [33, 34, 43, 44]. However, the effect of the LDOS and the dynamics of the UC process are rarely considered. To address this issue, a rate-equation-modeling framework has been developed as part of our previous work. The model considers the combined effects of the modified LDOS and the modified local energy density [31,32,46]. In a theoretical study of the photonic effects on UC, it has been demonstrated that it is indeed important to include the effect of the modified LDOS [32]. Furthermore, the so-called Bragg structure was identified as a promising design for increasing UC efficiency [32,47]. In contrast to structures that exploit surface effects, the amount of upconverter material in a Bragg structure can be increased by adding more layers in the design. In applications, this is a very important property, in particular for the weakly-absorbing rare-earth materials.

In this work, we present a simulation-based assessment of the potential of the Bragg structure for increasing the UC efficiency of 1523 nm excitation light via the use of β-NaYF₄:Er³⁺ nanophosphors. The Bragg structure consists of alternating quarter-wave layers of two materials with refractive indices n_low and n_high, with respect to a design wavelength λ_D = 4nd, where d is the layer thickness and n the refractive index, as shown in Fig. 1(a). The outer-most layers have a reduced optical thickness of only λ_D/(8n ), which enables a more efficient in-coupling of broad-band excitation. In principle, the upconverting material could be embedded in either layer. However, to avoid scattering, which would cause decoherence and thus suppress the desired photonic effects, the upconverting material must be index-matched with respect to the host layer. Since the refractive index of β-NaYF₄:Er³⁺ (≈ 1.5 [48]) is relatively low, choosing it as the high index layer would allow only for a very low refractive index contrast. Therefore, we choose the low index layers as the active ones, containing the upconverter material, thus fixing n_low = 1.5. The outer-most layers of reduced thickness are assumed passive, but feature the same refractive index as n_low. The spacer material can in principle be chosen freely with the only constraint that the absorption should be low within the spectral region of interest. To accommodate all materials, we consider the range 1.5 < n_high ≤ 4.0. Examples of high refractive index materials that remain transparent in the region of the main UC emission include TiO₂ and a-Si:H. As the reference, we choose a homogeneous structure with refractive index n_low, containing the same amount of active material as the corresponding Bragg structure, see Fig. 1(b).

 

Fig. 1 Structural sketches of the Bragg structure (a) and the reference structure (b). The Bragg structure consists of alternating quarter-wave layers, with respect to a design wavelength λ_D, of an active and a spacer material with refractive indices n_low and n_high. The outermost layers have a reduced optical thickness of λ_D/(8n) and are assumed passive. The reference structure consists of a single, homogeneous layer containing the same amount of active material as the corresponding Bragg structure.

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While ideal photonic crystals are infinite and perfectly periodic, the experimental realizations are finite and suffer from production imperfections. As demonstrated previously [32], the beneficial peak in the energy density enhancement is very narrow in λ_D, or equivalently the layer thickness, implying a high sensitivity to fabrication tolerances. For production of Bragg-like structures a variety of thin film fabrication methods are available. A fast and efficient production method is spin-coating from solution. With this process, layer uniformities of 1–2% of the total layer thickness can be achieved [49]. Higher layer uniformities can be reached in chemical vapour deposition processes. Using metalorganic vapour phase epitaxy (MOVPE), a layer thickness control of 0.1–0.5% has been reported [50]. Another high-precision production method is atomic layer deposition (ALD), where a thickness accuracy of 0.5% is possible for TiO₂ thin films at deposition temperatures above 250°C [51]. To enable a realistic assessment of the performance of fabricated structures in this work, we take into account finite production tolerances via Monte Carlo simulations. Based on achieved thickness control of 0.5% reported in the literature [50,51], a realistic target tolerance of 0.5% is assumed in the main analysis of this work, which corresponds to an absolute value of ≈ 1 nm. We choose a value at the limit of what is currently possible, imagining that such accuracy will be enabled in routine production by future advances in nanofabrication techniques. One option could be, for example, to produce the whole layer stack with ALD processes.

We first present the methods for evaluating the local energy density, including the Monte Carlo method, in Section 2.1, after which the LDOS calculations are introduced in Section 2.2. The photonic effects are coupled via a rate equation model (REM), presented in Section 2.3, which yields the upconversion quantum yield (UCQY) and the upconversion photoluminescence (UCPL). The potential for increasing the local energy density, assuming different production accuracies, is investigated in Section 3.1, while the modified LDOS is discussed in Section 3.2. Finally, in Section 3.3, we present the calculated UCQY and UCPL enhancements, scanning through all design parameters. Additionally, different irradiance regimes, targeting different application scenarios, are considered.

2. Methods

2.1. Local energy density

For stacks of planar, homogeneous layers an analytical solution for the electrical field distribution, E(x), exists. It can be found efficiently using the transfer matrix method [52] from which the local energy density of the electric field, u (x ), can be calculated as

2.2. Local density of optical states

For infinite periodic structures, i.e. ideal photonic crystals, the local density of optical states (LDOS) can be derived from eigenmode calculations [53–55]. In this work, we use the MIT Photonic bands [53] software package. While the ideal crystal assumption is not accurate for Bragg structures with only a small number of layers [56,57], the structures of main interest in this work have 10 active layers or more. Additionally, the eigenmode approach is relevant in a future perspective, as it permits calculation of an angularly resolved LDOS [58], which is important for our future work on the modified directionality of upconversion (UC) emission in a Bragg structure.

Due to the scale invariance of the problem, we consider dimensionless quantities

As the reference is homogeneous, the LDOS is independent of the position. Hence, it is equal to the DOS up to a multiplicative constant which depends on the discretization of the Wigner-Seitz unit cell. The DOS for a homogeneous medium has an analytical form [59],

2.3. Rate equation model

We describe the dynamics of the UC process using a modified version of a rate equation model (REM) originally developed for homogeneous media [46]. All experimental parameters are measured for an excitation wavelength of 1523 nm, for which the main UC emission lies at 984 nm. The first seven energy levels of β-NaYF₄:Er³⁺, shown in Fig. 2, are considered. The energy levels of the ²H_11/2 and the ²S_3/2 states are treated as one effective energy level, due to their close proximity. The occupation of each level is described as an element of the occupation density vector n. The linear processes included in the REM are ground state absorption (GSA), excited state absorption (ESA), stimulated emission (STE), spontaneous emission (SPE) and multi-phonon relaxation (MPR). Additionally, the non-linear Förster energy transfer processes, energy transfer upconversion (ETU) and cross relaxation (CR), are considered. The most important processes are indicated in Fig. 2. The rate of change of the occupation density vector yields

 

Fig. 2 Schematic of the first seven energy levels in β-NaYF₄:Er³⁺ included in the rate equation model, along with most important transitions for the UC process. The highest two energy levels are treated as one due to their close proximity. For the considered excitation of 1523 nm wavelength, the main UC emission lies at 984 nm.

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The modifications of the REM due to the changes in the photonic environment imposed by the photonic structure have been developed in [31]. The probability for stimulated processes, i.e. absorption and stimulated emission, depends linearly on the local energy density $u\left(x,{\omega }_{fi}^{\prime }\right)$

The output of the REM is a steady-state occupation density vector N from which the main figures of merit, the upconversion photoluminescence (UCPL) and the internal upconversion quantum yield (UCQY), can be calculated. The photoluminescence (PL) for each transition is

3. Results

3.1. Local energy density

The average relative energy density, ū_rel, for an exemplary Bragg structure with n_high = 2.3 and a number of active UC layers (#_al) of 25 is plotted in Fig. 3(a) for normal incidence and monochromatic excitation at λ_exc = 1523 nm in dependence on the design wavelength λ_D. An exemplary value of n_high= 2.3 was chosen in accordance with a recent experimental study where the high index layer was made of TiO₂ with n = 2.3 at 1523 nm [47]. The graph shows that for a certain design wavelength, ${\lambda }_D={\lambda }_D^{u_{\max }}$, a very large enhancement of ū_rel occurs. As is known from the literature, a high field enhancement occurs at the photonic band edge [34,65]. In Fig. 3(b), the reflectance for ${\lambda }_D={\lambda }_D^{u_{\max }}$ shows the position of the photonic band gap (PBG) relative to the excitation wavelength λ_exc. As expected, λ_exc lies close to the photonic band edge. In addition to the Bragg structure, the reflectance is also shown for a pure quarter-wave stack. Comparing the two curves, it is clear that the reduced thickness of the outer layers of the Bragg structure causes a suppression of the side lobes of the reflectance peak. While this feature is not necessary for a perfect simulated structure under monochromatic excitation, it enables a more efficient in-coupling in experiments, in particular for broad-band excitation sources. The spatial dependence of u within the structure is illustrated in Fig. 3(c) for ${\lambda }_D={\lambda }_D^{u_{\max }}$, along with the refractive index profile. A strong increase in u is observed inside the active layers of the structure. The enhancement can be explained in the context of photonic crystals as slow light piling up [66] or in a more classical context as the formation of a standing wave due to interference between the forward and backward propagating waves. It should also be noted that practically all energy density is located in the active layers.

 

Fig. 3 (a) Average relative energy density ū_rel across the active layers as a function of design wavelength λ_D for an exemplary Bragg structure with n_high= 2.3 and #_al = 25. The upper x -axis indicates the active layer thickness, d_low = λD/(4n_low). (b) Reflectance, R, for the example structure at the design wavelength yielding the maximum ū_rel-value, ${\lambda }_D^{u_{\max }}$ (marked by a black, shaded circle in panel a). (c) Spatial energy density distribution inside the structure for ${\lambda }_D={\lambda }_D^{u_{\max }}$. Additionally, the refractive index profile is shown. (d) Average relative energy density across the active layers as a function of n_high and #_al. The example structure considered in panels a, b and c is marked by a white, shaded circle.

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Figure 3(d) shows ū_rel for the range of considered structures with 1.5 < n_high ≤ 4.0 and 1 ≤ #_al ≤ 50, evaluated at ${\lambda }_D={\lambda }_D^{u_{\max }}$ for each structure. As the number of active layers and/or the refractive index contrast increases, the photonic effects become stronger and ū_rel increases. The associated increase in sharpness of the peak on the λ_D axis with respect to the layer thickness causes a correspondingly increasing sensitivity to structural imperfections. In fact, to obtain the maximum value of ū_rel ≈ 400 shown in Fig. 3(d), the production accuracy must be subatomic. In this case, the peak of ū_rel on the λ_D axis reaches a sharpness in the sub-Ångstrøm-range. Such accuracy can never be realized, and the extremely high ū_rel value is thus non-physical.

To take into account production tolerances, we apply a Monte Carlo method as described in section 2.1. The simulation results for four different σ-values are shown in Fig. 4. As can be seen, the non-zero production tolerances limit the realistically achievable value of ū_rel severely. Even with Ångstrøm precision, as displayed in Fig. 4(a), corresponding to the thickness of a single atomic layer, the maximum value of ū_rel drops to around 125. With 0.5 nm and 1 nm precision, Figs. 4(b) and 4(c), the maximum decreases further to values around 35 and 20, respectively. Figure 5 shows the associated spread in u_rel for an exemplary n_high= 2.3. As more layers are added, the increasing sharpness of u_rel for the ideal structure causes a higher spread, which in turn lowers and broadens the mean-value maximum. The position of the mean-value maximum shifts towards structures with fewer layers, as exemplified in Fig. 5(b), and/or smaller refractive index contrast (see Fig. 4). As the peak in ū_rel for the ideal version of these structures is broader, they are less sensitive to structural imperfections compared to structures with more layers and/or a higher refractive index contrast. For σ = 5 nm, Fig. 4(d), the maximum moves further down, especially to lower #_al, and the maximum value drops to around 6. Hence, to obtain high enhancement factors, high-precision manufacturing is of uttermost importance. Going forward, a production accuracy of σ = 1 nm is assumed, which is realistically attainable with current high-precision manufacturing methods.

 

Fig. 4 Average relative energy density ū_rel across the active layers of a Bragg structure as a function of #_al and n_high. ū_rel is shown for four different production accuracies, simulated using a Monte Carlo method (see section 2.1). For each pixel, 50.000 separate calculations were carried out. The black contours indicate 99% (solid line) and 95% (dashed line) of the maximum. The panels show σ-values of 0.1 nm (a), 0.5 nm (b), 1.0 nm (c), and 5.0 nm (d).

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Fig. 5 Average relative energy density ū_rel across the active layers of a Bragg structure as a function of #_al for n_high= 2.3. ū_rel is shown for two different production accuracies, simulated using a Monte Carlo method (see section 2.1). For each data point, 50.000 separate calculations were carried out. The shadings indicate ± one standard deviation. The panels show σ-values of 0.5 nm (a), and 1.0 nm (b).

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3.2. Local density of optical states

The band structure and the relative local density of optical states (LDOS_rel) within the Wigner-Seitz unit cell is shown in Fig. 6 for two example structures with n_high = 2.3 (a) and n_high = 3.0 (b). Comparing the two structures, a compression of the band structure along the frequency axis when going from n_high = 2.3 to n_high = 3.0 is observed due to the increasing effective refractive index. In addition, the features become more pronounced and the size of the band gaps increases due to the increase in refractive index contrast. The LDOS influences the probability of all spontaneous emission processes (see section 2.3). In this work, the upconversion (UC) emission is almost exclusively caused by the spontaneous emission from energy level 3 → 1 (SPE31), while the radiative losses are dominated by the 2 → 1 emission (SPE21), see Fig. 2. To allow for a simple assessment of the LDOS effects of different structures, the ratio between the ${\overline{\text{LDOS}}}_{\text{rel}}$ for SPE31 and SPE21 is plotted in Fig. 7. Since, ideally, the UC emission should be enhanced and the loss emission suppressed, this ratio should be as high as possible. In Fig. 7(a), the ${\overline{\text{LDOS}}}_{\text{rel}}$ is plotted for an example structure with n_high= 2.3. The x -axis of Fig. 7 is λ_D, representing the scaling of the unit cell in position space. The orange and blue shaded regions indicate the position of the first photonic bandgap (1PBG) for SPE31 and SPE21, respectively. ${\overline{\text{LDOS}}}_{\text{rel}}$ is strongly reduced when the respective emission falls into the 1PBG. Therefore, the maximum ratio is observed when SPE21 is located in the region of the 1PBG, which is very close to ${\lambda }_D^{u_{\max }}$, the design wavelength that maximizes the relative energy density u_rel. It is important to note that the emission wavelength of 1558 nm is significantly stokes-shifted relative to the excitation wavelength λ_exc = 1523 nm. Therefore, λ_exc can be efficiently coupled into the structure at the edge of the 1PBG, while the emission wavelength is suppressed in the 1PBG.

 

Fig. 6 Illustration of the LDOS for two different high index materials, n_high = 2.3 (a) and n_high = 3.0 (b), along with the associated photonic band structure for $k_y^{\prime }=0$. The regions of n_high and n_low (the active region) within the Wigner-Seitz unit cell are indicated on the top. The band gaps are marked by blue shadings. To avoid washing out features in the left panel, the scale is truncated at 2.0 even though the maximum value in the right panel is 2.5.

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Fig. 7 (a) ${\overline{\text{LDOS}}}_{\text{rel}}$ for SPE31 (main UC emission) and SPE21 (loss emission), as well as their ratio for an example structure with n_high= 2.3 as a function of the design wavelength λ_D. The shaded regions indicate where the respective transition falls within the first photonic bandgap (1PBG). (b) Ratio of ${\overline{\text{LDOS}}}_{\text{rel}}$ for SPE31 and SPE21 as function of n_high and λ_D. The solid lines indicate where the respective transition falls within the 1PBG.

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In Fig. 7(b), n_high is varied on the y-axis. The grey dashed lines indicate the example structure shown in Fig. 7(a). The orange and blue lines indicate the edges of the 1PBG for SPE31 and SPE21, respectively. For all n_high, the behavior is similar to Fig. 7(a). The ratio is small when the UC emission SPE31 falls into the 1PBG and large when the loss emission SPE21 lies in the region of maximum suppression in the 1PBG. Additionally, the ratio increases with increasing n_high. This can be understood from Fig. 6. As n_high increases, the features of the LDOS become more pronounced. Thereby, the contrast in the ${\overline{\text{LDOS}}}_{\text{rel}}$ increases between the region of the 1PBG and the band edges surrounding the 1PBG. We conclude that the most favorable design is obtained by placing SPE21 in the 1PBG while utilizing materials with the largest possible refractive index contrast.

3.3. Upconversion photoluminescence and quantum yield

Having analyzed the photonic effects of the local energy density enhancement and the local density of optical states (LDOS) separately, we now turn to their effect on UC as modeled using the rate equation model described in section 2.3. A detailed analysis of the photonic effects within the rate-equation-modeling framework, leading up to this work, can be found in [32]. In this section, we want to give an overview of the effects a Bragg structure can have on the upconversion quantum yield (UCQY) and upconversion photoluminescence (UCPL). To limit the analysis to physically realizable structures, a production accuracy of σ = 1 nm is assumed as discussed in section 3.1.

In Fig. 8, we illustrate the dependence of the UCQY on the incident irradiance. To limit the parameter space, we fix λ_D at ${\lambda }_D^{u_{\max }}$, the design wavelength where the enhancement of the energy density is at its maximum. For low irradiances, this is a reasonable approximation, as the energy density enhancement is the most important effect in this regime. Additionally, as discussed in sections 3.1 and 3.2, ${\lambda }_D^{u_{\max }}$ is typically close to the λ_D for which the benefit of the LDOS, i.e. the ratio between ${\overline{\text{LDOS}}}_{\text{rel}}$ for SPE31 and SPE21, is at its maximum. Since the peak of this ratio in λ_D space is much broader than that of ū_rel, ${\lambda }_D^{u_{\max }}$ will typically yield a UCQY value close to the maximum possible value.

 

Fig. 8 UCQY as a function of incident irradiance I for exemplary families of Bragg structures with (a) n_high fixed at 2.3 while #_al is varied and (b) #_al fixed at 10 while n_high is varied. In both cases ${\lambda }_D={\lambda }_D^{u_{\max }}$. For the Bragg structures, higher UCQY values at much lower irradiances are achievable. The maximum for each design is marked with a large dot.

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In Fig. 8(a), we fix n_high at 2.3 and vary only the number of active layers, #_al. This allows studying the effect of ū_rel, as the calculated LDOS varies only slightly with #_al (due to the change in ${\lambda }_D^{u_{\max }}$). To reach the maximum UCQY for the reference, an irradiance of 11600 W/m² is needed. This optimal irradiance is characteristic for the regarded material system. At higher irradiances, the population of higher energy levels becomes more dominant, such that the UCQY decreases. For the Bragg structures, the UCQY curve is compressed along the irradiance axis, i.e. a lower incident irradiance is needed to achieve the optimal irradiance at the position of the upconverter. From Fig. 4(c) we know that for n_high = 2.3, ū_rel increases with increasing #_al. Therefore, the compression of the UCQY curve on the irradiance axis is stronger for higher #_al. Additionally, the maxima for the Bragg structures are slightly higher than for the reference due to the modified LDOS. For additional discussion on this point, we refer to [32]. In Fig. 8(b), #_al is fixed at 10 while n_high is varied. From Fig. 4(c) we know that at #_al = 10, ū_rel increases with increasing n_high. That is, as the refractive index contrast increases, fewer layers are needed to reach the same energy density enhancement. Similar to the case of increasing #_al, a compression along the irradiance axis occurs. Additionally, the Bragg structure maximum point goes to higher UCQY values with increasing n_high due to the increasing strength of the LDOS modification as illustrated in Figs. 6 and 7.

To enable an assessment of structural designs in two dimensions, we reduce the dimensionality of the parameter space by fixing the irradiance at specific application scenarios. For non-concentrated sunlight, the irradiance available from the air-mass 1.5 global spectrum within the absorption range of Er³⁺ (from 1450 nm to 1600 nm) is approximately 30 W/m² [19]. A higher incident irradiance case of 1000 W/m² could be reached by combined spectral and geometrical concentration [67]. To also investigate the photonic upconverter system in the high irradiance regime, we regard the case of 10000 W/m² (1 W/cm²).

For each scenario, the UCQY is shown as a function of n_high and #_al in the left panels of Fig. 9. In the first panel, I = 30 W/m², the incident irradiance is so low, that the UCQY is determined almost exclusively by the energy density enhancement. This explains the structural resemblance of Fig. 9(a) to Fig. 4(c). The highest UCQY reached for this irradiance is 6.7%. At I = 1000 W/m², Fig. 9(c), UCQY values up to 15.4% become possible. With n_high = 2.3, 95% of this maximum value can be reached with 20 layers, while for n_high = 3.0 only 10 layers are needed. Because of the saturation of the UCQY that is clearly visible in Fig. 8 for the reference, the UCQY does not feature a strong dependence on the irradiance. Hence, the observed maximum of the UCQY is rather broad. Going to even higher irradiances, Fig. 9(e), saturation occurs and the UCQY starts decreasing (see also Fig. 8). Hence, in concentrated-solar applications, the benefit of the Bragg structure decreases regarding the UCQY. However, it should be noted that higher UCQY values than shown are possible by tuning the design wavelength. In the case of a very high incident irradiance, the assumption of ${\lambda }_D={\lambda }_D^{u_{\max }}$ being an almost-ideal choice for maximizing the UCQY, is no longer valid.

 

Fig. 9 UCQY (left) and relative UCPL (right) as a function of n_high and #_al for ${\lambda }_D={\lambda }_D^{u_{\max }}$. The black contour lines indicate 99% (solid line) and 95% (dashed line) of the maximum in each plot. The rows show different irradiance scenarios of I = 30 W/m², 1000 W/m², and 10000 W/m² (1 W/cm²).

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For some applications, the UCPL is of more interest than the UCQY. Therefore, we also investigate the relative UCPL (UCPL_rel) as shown in the right panels of Fig. 9, again as a function of n_high and #_al. While the absorption enhancement is directly given by the energy density enhancement, the UCPL depends non-linearly on the local energy density. This non-linearity saturates to linearity at a characteristic irradiance threshold. Therefore, at low irradiances far from saturation, the UCPL can be increased by orders of magnitude compared to the performance of the reference. For the very low irradiance of I = 30 W/m², Fig. 9(b), the Bragg structure enables a 330-fold UCPL enhancement. At I = 1000 W/m², Fig. 9(d), the maximum UCPL enhancement has decreased to ≈ 40-fold. At this irradiance the relative effect of the Bragg structure is lower because the UCQY of the reference is already much higher. Proceeding to the case of I = 10000 W/m² (1 W/cm²) plotted in Fig. 9(f), the enhancement factor drops further to a maximum of ≈ 12. At this high irradiance, the UCQY is in fact lower for some Bragg structures compared to the reference, which causes the UCPL enhancement to drop below the energy density enhancement. In consequence, the ideal design for the UCPL enhancement is a different one than for the UCQY enhancement.

3.4. Discussion of results

Experimental work on erbium-doped distributed Bragg reflectors has previously been published by Johnson et al. [34]. Their structures were made of Er³⁺-doped porous silicon, where the periodic variation of the refractive index was achieved by varying the silicon porosity. At an excitation wavelength of 1550 nm and a laser power of 200 mW, an UCPL enhancement of 26.6 is reported for the 550 nm UC emission. The main UC emission of 980 nm was enhanced by a factor of around 5. The investigated structure by Johnson et al. corresponds to our simulated Bragg structure of #_al = 30, featuring refractive indices of n_low ≈ 1.5 and n_high ≈ 2.2. Johnson et al. report difficulties in controlling the layer thickness accuracy in the structures, hence we draw a comparison to our lowest simulated accuracy. Johnson et al. do not report the exact irradiance, but that the 200 mW laser beam is focused on the sample. Therefore a high irradiance can be assumed and we draw a comparison to our highest simulated irradiance. At a production accuracy of σ = 5 nm and an irradiance of 10000 W/m², we calculated an UCPL enhancement of 4.8. This is very well in line with the measured enhancement of a factor of 5 by Johnson et al., especially, when taking into account the complexity of both the simulation model and the experiment.

Works on other photonic structures are not directly comparable. Nevertheless, it is possible to draw a comparison in terms of advantages and disadvantages of the different structures. In a waveguide structure a very high enhancement of 10⁴ has been reported by Lin et al. [43]. The downside of this very effective device, at least in application in photovoltaics, is that the enhancement occurs only within a very narrow range in excitation wavelength and incident angle (approximately 1 degree). Opal photonic crystal structures often exploit surface effects, which allow only for a thin layer of upconverter material to be deposited on top of the photonic structure. This limits the amount of upconverter material affected by the enhancement and therefore the total UC signal. Enhancement factors of up to 30 have been reported by Niu et al. [38] and Yin et al. [36]. Later, Yin et al. also included gold nanorods in opal photonic crystal structures to additionally exploit plasmonic effects, raising the UC enhancement factor to three orders of magnitude [44]. In inverse opal photonic crystal structures, the voids of the structure can be filled with UC material, allowing for more upconverter material to be included in the device [37,40]. Here, the performance of the structures deviates by an order of magnitude with Zhang et al. measuring a maximum UC enhancement factor of 4.6 [40], while Xu et al. report a factor of 43 [37]. Recently Shao et al. demonstrated the additional usage of plasmonic effects, embedding gold nanoparticles in inverse opal photonic structures. With a measured UC enhancement factor of 10 [45], no clear benefit compared to pure photonic structures was observed. In comparison to all these reported structures, the Bragg structure is a promising device. The amount of upconverter material affected by the enhancement can be varied, and indeed can be quite large. Simultaneously, high UCPL enhancement factors can be reached.

Other factors that will affect the applicability of any photonic structure, especially in photovoltaics, is its sensitivity to spectral and/or angular changes of the excitation, as well as the angular characteristics of the emission. Investigating these dependencies is beyond the scope of this paper, but the topics will be addressed in our future work.

4. Summary

We have presented a simulation-based analysis of the photonic effects of a Bragg structure on the upconversion quantum yield (UCQY) and the upconversion photoluminescence (UCPL) of the embedded upconverter, β-NaYF₄:Er³⁺. The change in local energy density and the local density of optical states were considered within a rate-equation-modeling framework. To include all realistically attainable structures in the analysis, the refractive index of the spacer material, n_high, was varied within the refractive index range of naturally occurring materials, 1.5 < n_high ≤ 4.0, and up to 50 active layers were considered. Furthermore, manufacturing imperfections were incorporated in the analysis via Monte Carlo simulations.

Neglecting production imperfections, the energy density across the active layers of the Bragg structure can be enhanced indefinitely, simply by adding more layers. For 50 layers, a more than 400-fold average enhancement was found for n_high = 4.0. However, when realistic production accuracies are taken into account, the maximum shifts to fewer layers and takes on a finite value. At σ = 1 nm and σ = 5 nm, for example, the enhancement drops to 20- and 6-fold, and the number of active layers needed are only 30 and 15, respectively. These observations underline the crucial importance for high-precision manufacturing methods in order to realize efficient Bragg structures experimentally. Furthermore, we showed that for all applications it is highly profitable to use the maximum possible refractive index for the spacer layer. The higher refractive index contrast significantly increases the beneficial photonic effects. Additionally, it decreases the number of layers needed to reach a given energy density enhancement, which decreases the complexity of the fabrication process.

For an optimized Bragg structure, a 330-fold UCPL enhancement was predicted at an illumination of one sun for a production accuracy of σ = 1 nm. However, due to a non-ideal effective energy density, the UCQY remains below 7%. At an incident irradiance equivalent to 30 suns, possible e.g. using a combination of down-shifting and geometrical concentration [67], the UCQY of the optimized Bragg structure is raised to a near-maximum value of 15.4%, while the reference remains at only 8.0%.

In conclusion, we find that the Bragg structure is a very promising candidate for increasing the efficiency of upconversion processes in a broad range of applications. In particular within the context of non-concentrated photovoltaics, the combination of a high UCPL enhancement and the possibility of including a large amount of upconverting material makes the Bragg structure exceedingly favorable.