Optimization of photonic upconverter devices

We fabricated optimized Bragg structures comprising of five TiO₂ layers and four intermediate layers of PMMA with embedded core-shell upconverter nanoparticles of β-NaYF₄:25%Er³⁺, in the following referred to as active layers (“Methods”). The scanning electron microscope (SEM) cross-sectional image of a fabricated Bragg structure demonstrates the high layer quality and uniformity (Fig. 1b, see also Supplementary Fig. 8a). The upconverter nanoparticles mostly form small clusters within the PMMA layer or agglomerate at the layer surface. Nevertheless, the surface roughness of the topmost layer of the displayed Bragg structure is only in the order of 10 nm (Supplementary Fig. 8b).

Figure 2a shows the first seven energy levels of Er³⁺ in the host crystal β-NaYF₄. These processes are influenced by the surrounding of the Bragg structure, as motivated in Fig. 1e. For our study, the most important properties of the Bragg structure are the existence and position of the photonic bandgap, represented by the characteristic reflectance (Fig. 2b). The position of the reflectance peak, and therewith the first photonic bandgap, is determined by the design wavelength (λ_design) that defines the thickness $$ d_i={\lambda }_{\mathrm{design}}/4n_i$$ of each layer i with refractive index n_i. Fitting the measured to the simulated reflectance, we determined the exact design wavelength of each evaluated sample point (“Methods”). With the chosen sample designs, we can investigate the photonic effects, ranging from the expected maximum with an excitation at the photonic band edge, to an expected suppression.

Fig. 2

Design and upconversion photoluminescence (UCPL) of a Bragg structure.

a Energy level diagram of the first seven energy levels of β-NaYF₄:25%Er³⁺, including the processes: ground- and excited-state absorption (GSA, ESA), multi-phonon relaxation (MPR), energy transfer upconversion (ETU) (one exemplary ETU process shown), and spontaneous emission (SPE). b Reflectance of a fabricated Bragg structure with the matched simulated reflectance at a design wavelength λ_design = 1844 nm. The 40 investigated sample designs range from λ_design of 1784 nm to 2005 nm. For UCPL measurements, the excitation wavelength is varied from 1500 nm to 1560 nm. c Measured UCPL under 1523 nm excitation at 1.48 W cm⁻² irradiance using an integrating sphere to collect the integrated light from all angles. Due to the photonic effects on UC, in the Bragg structure, all UC emission is significantly enhanced. The relative enhancement of the main UCPL at 984 nm (UCPL_rel) in the Bragg structure compared to the reference is 4.1.

To quantify the effect of a photonic structure on UC, we investigate the UC photoluminescence (UCPL) (“Methods”) of a Bragg structure relative to its corresponding reference (Fig. 2c). As a reference, we choose one active layer on glass, featuring the same thickness as the sum of all active layers of the corresponding Bragg structure. The main UC emission around 984 nm contains 94% of the measured total UCPL, it stems from the electronic transition ⁴I_11/2 to ⁴I_15/2. The emission intensity, corresponding to this transition is enhanced in the Bragg structure by a factor of 4.1 due to the photonic effects. The ⁴I_9/2 to ⁴I_15/2 transition can be seen in the 814 nm UC emission, with an enhancement factor of 5.2. The 3-photon processes ⁴F_9/2 to ⁴I_15/2 at 660 nm and ⁴S_3/2 combined with ⁴H_11/2 to ⁴I_15/2 at 536 nm are enhanced by a factor of 7.3 and 8.9, respectively.

Simulation of photonic effects on UC

The effects of the Bragg structure, the change of the LDOS and the energy density, critically depend on the design wavelength λ_design. The induced changes can be expressed as relative values obtained by integrating over the active layers within the Bragg structure and dividing by the corresponding integral of the reference, yielding the average relative LDOS ($$ {\overline{\mathrm{LDOS}}}_{\mathrm{rel}}$$) and the average relative energy density ($$ {ū}_{\mathrm{rel}}$$). From the locally resolved photonic effects, the change in the UC emission at 984 nm relative to the reference (UCPL_rel) is calculated (“Methods”). In Fig. 3, we investigate the change in UC emission due to photonic effects at two different irradiance levels that are relevant for the target application of photovoltaics: (i) at 1 sun illumination, where the irradiance in the absorption range of the upconverter Er³⁺ between 1450 nm and 1600 nm is 3 mW cm⁻² (ref. ⁵³) and (ii) at 1.48 W cm⁻², corresponding to ~500 suns concentration, which is a typical regime for high-concentration photovoltaic systems⁵⁴.

Fig. 3

Photonic effects on upconversion (UC) as a function of the design wavelength λ_design.

a Average relative local density of optical states ($$ {\overline{\mathrm{LDOS}}}_{\mathrm{rel}}$$) in the active layers of the Bragg structure for the main UC emission and main loss emission. b Average relative energy density ($$ {ū}_{\mathrm{rel}}$$) in the active layers of the Bragg structure for an excitation at 1523 nm for an ideal Bragg structure and the fabricated structure including measured production inaccuracies. c–e Relative UC photoluminescence (UCPL_rel) at 3 mW cm⁻² (1 sun), as well as at 1.48 W cm⁻² (~500 suns) irradiance as in experiment, only taking into account the LDOS effect (c), the effect of the relative energy density (d), and both effects (e). Under 1 sun illumination, the irradiance in the absorption range of the upconverter Er³⁺ (1450 nm–1600 nm) is 3 mW cm⁻² (ref. ⁵³).

Figure 3a shows $$ {\overline{\mathrm{LDOS}}}_{\mathrm{rel}}$$ for the two most important spontaneous emissions, the main UC emission at 984 nm and the main loss emission at 1558 nm, the direct de-excitation of the first excited state (compare to Fig. 1e). In the region of λ_design, in which an emission lies inside the first photonic bandgap (highlighted regions in Fig. 3a), $$ {\overline{\mathrm{LDOS}}}_{\mathrm{rel}}$$ is strongly reduced. However, as our reference consists only of the low refractive index material, there are more photonic states in the Bragg structure, and $$ {\overline{\mathrm{LDOS}}}_{\mathrm{rel}}$$ is always above one. The net effect of the LDOS on UC efficiency is a complex, non-linear superposition of $$ {\overline{\mathrm{LDOS}}}_{\mathrm{rel}}$$ of both emissions (Fig. 3c). An increase of $$ {\overline{\mathrm{LDOS}}}_{\mathrm{rel}}$$ of the main UC emission at 984 nm linearly increases UC efficiency. However, an increase in $$ {\overline{\mathrm{LDOS}}}_{\mathrm{rel}}$$ of the main loss emission at 1558 nm non-linearly decreases UC efficiency. At low irradiances, the increased probability of the 1558 nm loss emission is more relevant because the few available excited upconverter ions in the first excited state have a high probability to be de-excited again before an UC process can take place. For higher irradiances, this strong dependence on the probability of the 1558 nm loss emission loses its large impact as there are more excited upconverter ions available, which increases the probability that an UC process takes place before de-excitation.

The photonically modified energy density is very sensitive to structural imperfections⁵⁰. Therefore, we include production inaccuracies in our simulation via a Monte-Carlo approach, using measured standard deviations of the layer thicknesses as input parameters (“Methods”). In Fig. 3b, $$ {ū}_{\mathrm{rel}}$$ is plotted for an excitation wavelength of 1523 nm, for an ideal Bragg structure, as well as the fabricated structure with layer production inaccuracies of 4.2 nm and 1.5 nm for the active and TiO₂ layers, respectively (“Methods”). The non-ideal $$ {ū}_{\mathrm{rel}}$$ is almost identical to the ideal $$ {ū}_{\mathrm{rel}}$$. This demonstrates that the production accuracy we reached in experiment is high enough that it does not diminish the photonic effects in the particular structure we are investigating. Because λ_design determines the position of the reflectance peak for a broad region around λ_design of 1500 nm, the excitation at 1523 nm falls into the photonic bandgap and is directly reflected. The peak enhancement is reached at λ_design = 1855 nm, when the excitation lies at the lower band edge.

Figure 3d shows UCPL_rel, only taking into account the effect of the relative energy density. The non-linear dependence of the UC process on the irradiance is well visible in this graph. At 3 mW cm⁻² irradiance, corresponding to 1 sun, the reference performs very poorly because at this low irradiance very few ions are excited and the probability for an UC process to take place is very low. The energy density enhancement within the Bragg structure increases absorption and therefore strongly increases UC efficiency. At 1.48 W cm⁻² irradiance, corresponding to ~500 suns, the effect is less pronounced, because the additional energy density enhancement still enhances absorption, but also contributes to UC emissions from even higher excited states, thus reducing the benefit for an UC emission at 984 nm.

Finally, in Fig. 3e, both photonic effects are considered, revealing that the net effect on UC is a complex non-linear superposition of both. The shape of UCPL_rel is very similar to Fig. 3d, showing that $$ {ū}_{\mathrm{rel}}$$ with its strongly pronounced maximum is decisive for optimizing λ_design. Nevertheless, the LDOS effect needs to be considered when regarding a particular irradiance. As demonstrated in Fig. 3c, the altered LDOS has a negative effect at low irradiances and a positive effect at high irradiances. In consequence, in Fig. 3e, compared to Fig. 3d, UCPL_rel is reduced for the low irradiance and increased for the high irradiance by the LDOS effect. Thus, both photonic effects need to be taken into account to optimize photonic structure design for a specific application.

Comparison of simulation and experiment

We compare simulation and experiment by varying the design wavelength λ_design, the excitation wavelength λ_excitation, and the irradiance. Thereby, we performed the measurements at an irradiance around ~500 suns in order to gain a good signal-to-noise ratio in all parameter scans, which was not feasible at only 1 sun illumination. We use 40 fabricated sample designs around the maximum UC enhancement expected from theory to investigate the dependence of UCPL_rel on λ_design (Fig. 4a). For evaluation, we sort the data into five groups (I–V) of similar λ_design. Both, the active- and TiO₂ layer are scaled to match the desired design wavelength λ_design. The corresponding active layer thickness is shown in the top x-axis of Fig. 4a. The simulated UCPL_rel is the same as in Fig. 3e, including the standard deviation of the layer thicknesses and both photonic effects. In experiment, the photonic effects increase the UC signal for λ_design around the simulated maximum UCPL_rel at 1855 nm. In groups II and III, at and close to the simulated maximum, respectively, the highest mean measured UCPL_rel is found, while group III slightly outperforms group II. Moving further away from the maximum, the experimentally measured UCPL_rel is lower (groups I and IV), and finally suppressed when the λ_excitation falls into the photonic bandgap (group V). We expect that the main reason for the variation of the single UCPL_rel measurements, also within the same design wavelength, are slight thickness variations of the single layers in each stack that appear due to production inaccuracies (“Methods”). Despite these thickness variations of single layers, a defined design wavelength can be assigned to each sample we investigated (Supplementary Fig. 7). In simulation, we also take the impact of the production inaccuracy on UCPL_rel into account. However, the simulation features the mean expected reduction over 1000 separate calculations. Most random thickness variations of single layers lead to a decrease in energy density in the active layers and therefore to a reduced UCPL_rel. For particular designs though, non-periodic thickness variations of single layers can lead to an additional strong increase of the energy density in the active layers⁵⁵, which consequently leads to an additional increase in UCPL_rel. This might contribute to a maximum measured enhancement of 4.1 for λ_design = 1844 nm. A closer analysis of the impact of the non-periodicity within each single Bragg structure design is out of scope of this paper.

Fig. 4

Effect of varied parameters on the relative upconversion photoluminescence (UCPL_rel)−comparison of simulation and experiment.

a We investigate the dependence of UCPL_rel on the design wavelength λ_design using 40 sample designs around the expected maximum UC enhancement, sorted into five groups (I–V) of similar λ_design. Two measurements of each investigated design are plotted, the boxes contain 50%, the whiskers 80% of the data points within each group. Point and horizontal line represent mean and median, respectively. b Scanning the excitation wavelength λ_excitation, the mean and standard deviation of UCPL_rel within each group I–V is plotted. The applied irradiance in experiment lies between 1.57 W cm⁻² at λ_excitation = 1500 nm and 1.38 W cm⁻² at λ_excitation = 1560 nm. The simulation is plotted for the center λ_design of each group at these two boundary irradiances. c Effect of varied irradiance for one sample design of group III, compared to simulation of UCPL_rel including only one photonic effect, of the changed local energy density u_rel or the modified local density of optical states $$ {\mathrm{LDOS}}_{\mathrm{rel}}$$, or both effects. For all investigated parameter scans (a–c), the expected trends from simulation are clearly visible in experiment. In the mean of all 2480 measurements at separate parameter combinations, the experimentally measured UCPL_rel divided by the simulated UCPL_rel is 82 ± 24%, featuring a very good agreement. Source data for a and b (i–v) are provided as a Source Data file.

Next, the effects of varying λ_excitation are important for applications with a broad-band excitation source, such as photovoltaics. We use the same groups as in Fig. 4a, and evaluate the mean and standard deviation of UCPL_rel within each group (“Methods”) (Fig. 4b (i–v)). In simulation, λ_excitation is varied over the complete absorption range of the upconverter material Er³⁺ (Supplementary Fig. 1), featuring the center λ_design of each group. In experiment, we covered the range of λ_excitation between 1500 nm and 1560 nm. Group II shows a broad plateau for λ_excitation around 1523 nm. This corresponds to the expectation that for λ_design = 1855 nm, UCPL_rel peaks at λ_excitation = 1523 nm. For group I, the maximum enhancement is expected at a shorter λ_excitation = 1465 nm, for groups III, IV, and V at longer λ_excitation of 1555 nm, 1585 nm, and 1645 nm, respectively. Consequently, in the investigated λ_excitation range, the dependence of UCPL_rel on λ_excitation corresponds to a falling flank (group I), a rising flank (groups III and IV), or a rather flat region (group V). The slope expected from simulation, which characterizes the Bragg structures effects, is very well visible in the experimental data in all five groups. We performed the same evaluation for the UC emission around 814 nm (Supplementary Fig. 12) and found the same good agreement between simulation and experiment. With a suitable design for a specific application, UCPL_rel can be increased in any desired spectral region. We find that in the optimum design range (group II), the complete core domain of the Er³⁺ absorption spectrum between about 1475 nm to 1575 nm can be significantly enhanced, with a simulated peak UCPL_rel of 2.4 at an irradiance of 1.48 W cm⁻². At the outer ranges of the absorption domain (visible particularly in groups I and IV), the enhancement factors are slightly higher. This is because in spectral regions where very little light is absorbed, the photonic enhancement has a larger impact on UC efficiency than in spectral regions with higher absorptance. In the mean of all 2440 separate parameter combinations in the excitation wavelength scan, we find that the mean agreement of measurement and simulation, the measured UCPL_rel divided by simulation, lies at 81.8 ± 23.9% (“Methods”). For such a large number of measurements, one could expect, that the mean of experiment and simulation should match, especially because we already take into account reductions of UC enhancement due to production inaccuracies. We expect that there are two reasons for this additional reduction of UCPL_rel that we see in the mean of all measurements: (i) the distribution of upconverter nanoparticles within the active layers, and (ii) the surface roughnesses in the Bragg structure. The photonic effects are strongest in the center of the active layer. However, the upconverter nanoparticles are not evenly distributed in the active layer, they are rather positioned at the outer ranges (compare to Fig. 1b). Additionally, the layers of the Bragg structure feature a roughness of around 10 nm (Supplementary Fig. 8b), which introduces additional scattering that most probably leads to a reduction of the overall photonic effects on UCPL_rel, which is currently not accounted for in the model.

Finally, in Fig. 4c, we demonstrate the dependence of the photonic effects on the irradiance for a sample of group III with λ_design = 1888 nm. The simulation is again plotted with only the effect of the energy density $$ u_{\mathrm{rel}}$$ taken into account, only the LDOS effect, and for both effects. Considering only the effect of the energy density results in a falling curve for UCPL_rel toward higher irradiances. In the low irradiance regime, in which the reference performs poorly, an increase in energy density, followed by a stronger absorption, largely increases the probability of an energy transfer UC process to take place, resulting in a high UCPL_rel. This becomes evident when looking at the absolute UCPL simulated down to 1 sun irradiance (Supplementary Fig. 11a). Consequently, also the UC quantum yield increases significantly at low irradiances (Supplementary Fig. 11b and c). At higher irradiances, energy transfer UC to yet higher energy levels becomes more probable, which decreases the probability of our main UC emission at 984 nm, thus decreasing UCPL_rel. In direct comparison with experiment, one can see that the absolute value of UCPL_rel is reproduced, but that the effect of a falling UCPL_rel toward higher irradiances is exaggerated. However, the negative effect of the LDOS is stronger in the low irradiance regime, as can be seen from the curve showing only the LDOS effect. Thus, when both effects are taken into account, the experimental data clearly follows the slope of the simulation, accurately reproducing the simulated UCPL_rel. The mean of all 41 measurements contained in the irradiance scan lies at 104.5 ± 11.6% of the simulated UCPL_rel.

For all separate 2480 parameter combinations in the excitation wavelength and irradiance scan, the mean UCPL_rel in experiment lies at 82.2 ± 24.0% of the simulation (“Methods”).

Optimization of active layers

The low refractive index layers of the Bragg structure are composed of PMMA (120,000 g mol⁻¹, Sigma-Aldrich), containing 25 wt% of upconverter nanoparticles. These layers are referred to as active layers. The core-shell upconverting nanoparticles are made of hexagonal sodium yttrium tetraflouride (β-NaYF₄) with a 25% doping of trivalent erbium (Er³⁺) (β-NaYF₄:25% Er³⁺) and an inert β-NaLuF₄ shell, produced as reported in ref. ⁶³ with oleic acid ligands. We produced thin active layers via spin-coating from a mixture of upconverter nanoparticles and 4 wt% or 5 wt% PMMA in toluene. We performed the spin-coating process (Specialty Coating Systems G3P-8) for 60 s with 250 µL of solution. Active layers on top of thin TiO₂ layers we produced with different spin-speeds between 500 r.p.m. and 2000 r.p.m. and measured the resulting thickness with an atomic force microscope (AFM, Dimension Edge, Bruker). The relation between spin-speed and thickness was drawn from a fit to the data with an empiric model, which then allowed the precise production of the desired layer thickness, with a production accuracy of 1.3%, corresponding to 4.15 nm. We determined the refractive index of the active layers via spectroscopic ellipsometry (M-2000, J.A.Woollam Co., USA). For data analysis, we applied a Cauchy model, implemented in the Complete Ease⁶⁴ software, yielding a refractive index of the active layer of 1.474 at 1523 nm wavelength. More detailed information on optimization of the active layers can be found in the Supplementary Note 1.

Optimization of TiO₂ layers

The high refractive index layers of the Bragg structure are made of TiO₂, which we deposited via atomic layer deposition (ALD) (R-200 Advanced, Picosun, Finland). The ALD process was run at a chamber temperature of 100 °C from molecular precursors H₂O and TiCl₄ (purchased from Sigma-Aldrich (≥99% TiCl₄)). Via X-ray diffraction (XRD) measurements (XRD D8, Bruker), we confirmed that the produced TiO₂ films are amorphous. We analyzed the layer thickness and refractive index of TiO₂ layers via spectroscopic ellipsometry, as described above, but in this case utilizing a Cody Lorentz model. At a wavelength of 1523 nm, the determined refractive index lies at 2.279. We adapted the layer thickness by varying the number of deposition cycles. From thickness measurements of single layers, we determined the production accuracy, featuring a standard deviation of the mean of 0.8%, corresponding to 1.53 nm. This value served as input parameter in the simulation of non-ideal Bragg structures. More detailed information on optimization of TiO₂ layers can be found in the Supplementary Note 2.

Production of optimized Bragg structures and reference samples

We fabricated optimized Bragg structures out of five TiO₂ layers and four intermediate active layers. For production, we alternatingly carried out the processes of atomic layer deposition for TiO₂ and spin-coating for active layers in one glovebox in Argon atmosphere to reduce contamination of the samples. The simulated maximum UCPL enhancement, due to the photonic effects of the Bragg structure, appears at 1855 nm design wavelength. This corresponds to a layer thickness of 203 nm for TiO₂ and 315 nm for the active layers. We fabricated eight different samples with target design wavelengths right at, as well as longer and shorter than, the expected maximum enhancement. Each sample was placed at a distinct position in the ALD chamber and for each precisely determined layer thickness of TiO₂, we spin-coated the matching active layer thickness to gain the same optical thickness of both layers and therefore a defined design wavelength.

As a reference, we choose a stack of only the active layers of the corresponding Bragg structure. This way, the reference contains the same amount of upconverter material without the photonic structure around it. We fabricated the reference samples by spin-coating the active layers of the corresponding Bragg structure right on top of each other. Thereby, the target thickness of the active layers in Bragg structure and reference is identical. The spin-coating parameters were adapted for each substrate material separately (Supplementary Fig. 2b).

Design characterization of Bragg structures

On each sample, we characterized five distinct points. With a spectrophotometer (Lambda 950, PerkinElmer, Germany), we measured the characteristic Bragg structure reflectance for each sample point at a tilt of 8° relative to the incident beam. Using an aperture, the beam diameter was reduced to ~1 mm diameter. We performed the simulation of the Bragg structure reflectance in an implementation of the transfer matrix method⁵⁰, subsequently comparing each measured reflectance to the simulated reflectance, scanning through design wavelengths and calculating the squared difference. For each sample point, the design wavelength for which the simulation features the minimum squared difference to the measured curve, explicitly determines its design (Supplementary Fig. 7).

UCPL measurement setup

A sketch of the UCPL measurement setup can be found in Supplementary Fig. 9. A tunable low power 20 mW infrared laser (TSL-510, Santec) served as excitation source. We measured all samples in an integrating sphere (819C-SL-5.3, Newport), placed in a center mount holder with a tilt of 4° relative to the incident laser beam. A 75 mm focal length lens was additionally installed at the entrance port of the integrating sphere to avoid unwanted coherence effects in the glass substrate. The signal was detected with a spectrograph (SP2300i, Princeton Instruments, USA), equipped with a blazed grating (150 grooves mm⁻¹ at a blaze wavelength of 800 nm) and a silicon CCD detector (PIXIS:256E, Princeton Instruments, USA).

To extract the real emitted spectrum of a measured sample, we corrected the signal for the spectral response of setup components like grating, detector, and lens. A calibrated tungsten halogen lamp served as excitation source for measuring the spectral response correction function of the setup.

For all laser powers and excitation wavelengths used in the experiments of this work, we determined the irradiance of the excitation beam at the sample position. We measured the area of the laser beam with a beam profiler (BP209-IR/M, Thorlabs) and determined the laser power with a photodiode sensor (PD300-IR, Ophir Photonics). Because the UC process is non-linearly dependent on the irradiance, we choose to calculate the laser irradiance only from the FWHM region of the Gaussian-shaped laser profile. This way, the high irradiance region within the laser profile, which is more relevant for the UC process, is calculated more precisely. Additionally, we scaled the laser area with a factor of 1/cos(4°) to account for the tilted sample (Supplementary Fig. 10).

UCPL measurements

We analyzed the UCPL at the same five points on each sample that were characterized in spectrophotometer measurements (Supplementary Fig. 7). For both, design wavelength scan (Fig. 4a) and irradiance scan (Fig. 4c), we measured UCPL spectra with 200 s integration time, for the excitation wavelength scan (Fig. 4b) with 60 s integration time. We calculated the UCPL_rel as the ratio of integrals over the UCPL spectra of Bragg structure and reference, within the wavelength range of 930–1020 nm (compare to Fig. 2c).

Calculation of mean agreement of measurement and simulation

We choose to quantify the mean agreement of measured and simulated UCPL_rel for all 2480 measurements with separate parameter combinations. The measured $$ {\mathrm{UCPL}}_{\mathrm{rel},\mathrm{measured}}$$ is compared to the exact same parameters in simulation $$ {\mathrm{UCPL}}_{\mathrm{rel},\mathrm{simulated}}$$, with a binning of 1 nm in design wavelength and featuring the exact irradiance of experiment. We then calculate the mean and standard deviation of $$ {\mathrm{UCPL}}_{\mathrm{rel},\mathrm{measured},i}$$ divided by $$ {\mathrm{UCPL}}_{\mathrm{rel},\mathrm{simulated},i}$$ for all measurements i within the evaluated group of measurements.

For the excitation wavelength scan that we visualize in Fig. 4b, we use the simulation at each excitation wavelength in steps of 1 nm, each featuring the exact irradiance of experiment (Supplementary Fig. 10). For the final quantification, yielding 82 ± 24%, we include all measurements with different parameter combinations. This includes all measurements of the excitation wavelength scan (Fig. 4b) and all (except one) measurements of the irradiance scan (Fig. 4c). We do not include the design wavelength scan (Fig. 4a), as these measurements are a repetition of the measurements in the excitation wavelength scan at 1523 nm, as well as the measurement at 1.48 W cm⁻² in the irradiance scan, also being a repetition.

Simulation of local energy density

We here give a brief overview of the comprehensive simulation model; a detailed description of the model as well as the applied simulation details can be found in ref. ⁵⁰.

We calculate the local energy density u(x) within the Bragg structure and reference using an implementation of the transfer matrix method⁵⁰. The relative local energy density u_rel(x) of the Bragg structure is calculated relative to the reference as a half-infinite low refractive index material. For visualization, we also define the mean relative energy density as the quotient of the integral over u(x) of the Bragg structure only within the active layers and the reference. These quantities are very sensitive to structural imperfections⁵⁰. Therefore, we perform the simulation as close to the experiment as possible. Via Monte-Carlo simulations, we modify the thickness d of each layer of the Bragg structure as $$ d\to d+\delta d,$$ whereby the δd is a random value of a Gaussian distribution with standard deviation σ. In this work, we calculate the average u(x) over 1000 separate calculations. The experimentally measured standard deviations of the single layer production accuracies of 4.2 nm and 1.5 nm for the active and TiO₂ layers, respectively, serve as input parameters. u_rel(x) is the exact calculation of the energy density at each position in the Bragg structure, however, it is difficult to visualize this value in dependence on a varied Bragg structure design. Therefore, we need an average value in only the active layers of the Bragg structure, to be able to easily visualize the dependence of u_rel(x) on a varied design wavelength. The average relative energy density $$ {ū}_{\mathrm{rel}}$$ (Fig. 3b) we thus define as the integrated u(x) only in the active layers of the Bragg structure, divided by the integral over u(x) in the reference.

Simulation of LDOS

The LDOS for infinite photonic crystals can be derived from Eigenmode calculations. We use the software package MIT Photonic Bands⁶⁵ and subsequently the histogramming method⁵⁶ to calculate the 3D LDOS. This dimensionless LDOS can be mapped to any unit cell size a, given by the design wavelength and considered transition frequency, given by the transition wavelength $$ {\lambda }_{fi}$$ from an initial state i to a final state f. We calculate the modification of the LDOS due to the Bragg structure relative to the homogeneous reference, consistent of only the low refractive index material, yielding $$ {\mathrm{LDOS}}_{\mathrm{rel}}\left(x,{\omega }_{fi}^{\prime }\right)$$. We choose this approach because it allows for analyzing the LDOS for different emission angles, which is subject of our future work. However, the fact that the calculation is done for an infinite photonic crystal overestimates the effect of the LDOS for the structure we are analyzing in this work with only nine layers in total. Again, as described above, for visualization, we also define the average relative LDOS $$ {\overline{\mathrm{LDOS}}}_{\mathrm{rel}}$$ (Fig. 3a), as the integral over $$ \mathrm{LDOS}\left(x,{\omega }_{fi}^{\prime }\right)$$ in only the active layers of the Bragg structure divided by the integrated $$ \mathrm{LDOS}\left(x,{\omega }_{fi}^{\prime }\right)$$ in the reference and scaled to the regarded emission frequency.

Rate equation model

We describe the dynamics of the UC process in a rate equation modeling framework, developed in ref. ⁵² for homogeneous media. Based on coupling plasmonic effects with the rate equation model^66,67, we extended the model for a photonic environment in ref. ⁵¹ and ref. ⁶⁸. The determination of experimental input parameters on UC dynamics are described in ref. ⁶⁹. The model version used in this work is published in ref. ⁵⁰.

Compared to the current model version, the simulation methods used in Hofmann et al. 2016 (refs. ⁶⁸) were slightly different, as pointed out in the publication of the more advanced version in Hofmann et al. 2018 (ref. ⁵⁰). Two changes are significant: (i) in Hofmann et al. 2016, only ideal Bragg structures were investigated, no production accuracies are included. (ii) Furthermore, there has been a small bug in the simulation script of the LDOS in Hofmann et al. 2016, which has an impact on the trends visible in the graphs including the effect of the LDOS. This bug was fixed in the version published in Hofmann et al. 2018 and we are currently working on an Erratum to the paper Hofmann et al. 2016 to correct the errors. The main findings that are discussed in Hofmann et al. 2016, however, are not influenced by this error and will not change in the Erratum.

The rate equation model describes the population of the Er³⁺ energy levels in the host crystal β-NaYF₄ (Fig. 2a). The rate of change of the occupation density vector is described by

According to Fermi’s golden rule, a modified LDOS influences the probability of SPE processes. The Einstein coefficients A_fi, describing SPE within the matrix M_SPE, are therefore multiplied with the relative local change of the LDOS:

The UCPL from energy level ⁴I_11/2 to ⁴I_15/2 is the main focus of this work. UCPL for one emission is calculated from the probability of the emission, given by the Einstein coefficient and the current population of the initial energy level N_i. For the Bragg structure, we perform the calculation at all positions of the upconverter material, so across all active layers. The relative UCPL of Bragg structure (brg) and homogeneous reference (ref) is given by

Absorption of the upconverter material is included in the REM as the relative absorption spectrum of β-NaYF₄:Er³⁺ (Supplementary Fig. 1).