For a demonstration of our concept of NP-assisted enhanced transmission, we consider silver nanospheres. Silver makes the strongest plasmonic resonator with minimal absorption losses^{22,23,24,25}, making it an obvious choice. First, we investigate the roles of NP radius *R*, interparticle gap *g* in the 2D hexagonal array, and “height” *h*. Note that, if the NP meta-grid is fabricated from a bottom-up self-assembly process, then it is usually the case that the nanospheres will arrange themselves in a 2D hexagonal array. If one aims at fabricating the meta-grid in top-down approach, then again hexagonal array would be preferred over square or other lattices, as hexagonal lattice has comparatively higher packing density which could produce the strongest effect from an NP meta-grid.

Examples of the theoretically calculated transmittance spectra at normal incidence of light for different values of these parameters are depicted in Fig. 2. Note that throughout this paper, the transmittance is calculated with a light emitter and a detector placed inside the chip and the encapsulating medium, respectively (see Fig. 1d). The permittivity of silver is taken from the literature^{26}. The dotted horizontal lines in all cases show the transmittance in the absence of the NP layer. It is seen that different sets of parameters of the NP array provide the maximum enhancement in transmission over different spectral windows. Therefore, the design of the NP “meta-grid” should be optimized for each LED depending on its emission spectral range.

Figure 2d–f show the transmittance spectra highlighted in the red boxes in Fig. 2a–c, respectively, along with the colored dotted curves obtained from full-wave simulations. The close correspondence between the analytical and simulation spectra indicates that our analytical approach is quite accurate. Hence, our theoretical model can be safely deployed for finding the optimal design of the NP meta-grid for any specific LED application.

Note that, the transmitted light through an NP meta-grid is essentially that portion of the incident light which remains after getting extinct (absorbed plus reflected) from its path by the NPs. Therefore, to understand the fundamental mechanism behind the changes in the transmittance spectrum with variation of the physical parameters of an NP meta-grid, as shown in Fig. 2, one needs to think of the effects of these parameters on the corresponding extinction spectra. Optical response of an NP meta-grid usually features an extinction peak, associated with excitation of localized plasmon resonances in NPs interacting with each other in the array. This extinction’s peak position and width depend on the meta-grid parameters.

This extinction peak appears as a *dip* in the corresponding transmittance spectrum. The narrower and shallower is the transmission dip, the higher is the overall transmittance averaged over a specified wavelength window. In each transmittance spectrum, there is a steep rise next to the dip (on its longer-wavelength side) with a maximum, featuring a decaying tail extending towards red. That part of transmittance spectrum plays a vital role in the average transmittance over a spectral range.

In this paper, our objective is to maximize transmittance over a specified spectral range using an optimized structure of the meta-grid. In an ideal scenario, one has to choose meta-grid parameters in such a way that an extinction peak (or the transmittance dip) falls outside the spectral range of LED emission. In Fig. 2, with yet no specific LED spectral window in mind, we describe that the effect of NP meta-grid’s parameters on transmittance spectrum in general. Hence, these spectra are depicted over a much wider range, from 400 to 1000 nm. The enhancement in transmission with the NP meta-grid, as compared with the case when no meta-grid is present, can be attributed to the Fabry–Perot effect between chip/encapsulant interface and NP meta-grid, as discussed in detail later with Fig. 3. Here, we discuss parametric dependence of the transmission dip (extinction peak) to understand the physics behind it.

With reduction of inter-NP gap *g*, at other parameters of an NP meta-grid unchanged, interparticle coupling within the meta-grid gets stronger, and the transmission dip (or the extinction peak) gets redshifted and broader^{27}. This is, however, not desirable in our case. Broadening of a transmission dip and moving its minimum to the center of the targeted spectral range will result in lowering of the average transmittance. This can be seen in Fig. 2a, where the average transmittance for *g* = 4 nm over the depicted spectral window is significantly lower due to a broad transmittance dip as compared with the other considered cases. With increase in *g*, transmittance dip moves farther to shorter wavelength; but the maximum of transmittance also moves together with the dip, and the tail on the longer-wavelength side of the maximum decays faster, and therefore such sparse array does not provide the maximal average transmittance in the depicted spectral range. The reason why the tail decays faster for larger *g* is because the plasmonic coupling among NPs weakens at a steeper rate for larger *g*.

With increase in height *h* of the NP meta-grid from the chip/encapsulant interface, there is no significant change in the location of the transmittance dip (extinction peak) (see Fig. 2b), as long as the NP meta-grid’s composition remains unchanged. There is, however, some spectral broadening in the transmission dip with increase in *h*, which may be attributed to weakening of the extinction resonance peak due to reduction in the image-charge interactions on the NP meta-grid^{27}. This broadening of transmission dip is not desirable in our application scenario.

Furthermore, with increase in radius *R* of the silver NPs at a fixed *g* and *h*, the transmittance dip gets blue-shifted, weaker, and narrower (see Fig. 2c). This can be attributed to the reduction in the image-charge interactions with increase in *R*^{27,28,29}. Although this trend looks beneficial toward achieving higher average transmittance, larger absorption by the added “plasmonic mass” of the NPs attenuates transmission over a wide spectral range and the benefit of transmission enhancement by NP meta-grid gradually disappears with further increase in NP size.

Now, we consider a specific case of a typical red LED with a peak emission wavelength of 625 nm, where AlGaInP (*n*_{1} = 3.49) is the semiconductor material and epoxy (*n*_{2} = 1.58) is the encapsulating material. To determine the parameters for the optimized configuration of the NP meta-grid, the study considered the following range of parameters: *h* from 0 to 500 nm, *R* from 5 to 50 nm, and *g* from 1 to 250 nm, all varied at steps of 1 nm.

Figures 3a–c depict the transmittance (*T*) optimized at each height h (provided the optimized *T* ≥ 98.5%) at normal incidence. For all these cases, the optimal transmittance value, its corresponding optimal radius *R*_{opt} and its corresponding optimal gap *g*_{opt} are plotted in Fig. 3a–c, respectively. Note that the transmission level obtained at any *h* repeats for other heights of *h* + *m* × λ/(4 × *n*_{2}), with *m* being a positive integer. This suggests that the Fabry–Perot effect is behind the transmission enhancement—where light reflected from the chip/encapsulant interface destructively interferes with the light reflected by the NP array to effectively reduce reflection from, (in other words, increase the transmission through) the chip/encapsulant interface. Fabry–Perot effects can be seen between any two parallel reflective surfaces, however, the higher is the reflectivity of the reflective surfaces the better is the quality factor of the resonance peak. We have shown Fabry–Perot cavity effects in our previous works^{30,31} between two parallel NP-layers, with moderate or high reflectivity. In this work, the chip/encapsulant interface and the NP meta-grid act as the two reflective surfaces forming the cavity in between them, where encapsulating material acts as the cavity medium. Self-absorption in the semiconductor emissive layer will not alter the property of the cavity, other than the fact that with stronger self-absorption lesser amount of light will be able to escape the chip/encapsulant interface. It is, therefore, theoretically possible to position the NP array at any of those heights *h*, where the transmission enhancement conditions are met according to the Fabry–Perot cavity theory. However, it would be wiser in practice to place the meta-grid at the closest possible height to the chip/encapsulant interface, in order to restrict any leakage of radiation from the sides before impinging on the meta-grid.

Note that Figs. 2, 3a depict transmittance spectra at normal incidence where s- and p-polarized light are transmitted in the same way. Figure 3a also marks three distinct points are marked as (1), (2), and (3), for which *R*_{opt} and *g*_{opt} are shown as the filled circles with values listed in Fig. 3b, c respectively. These cases are further investigated for all permissible incident angles below the critical angle. For off-normal incidence, s- and p-polarized light are transmitted differently. Hence, transmittance for the case of unpolarized light, which is typically the state of the light emitted from the emissive layer, is obtained by averaging transmittances of s- and p-polarized light at any incident angle.

Figure 3d–f depict the transmittance for s-polarized (red), p-polarized (blue), and unpolarized (green) light for cases (1)–(3), respectively, at different incident angles with/without the NP array. Case (1), although it provides the maximum *T* at normal incidence, is strongly polarization sensitive at off-normal incidence angles (the manifestation of this effect becomes stronger with increasing NP size) (Fig. 3d). For smaller NPs, in cases (2) and (3), polarization effect becomes negligible, for all permissible angles (Fig. 3e, f). For unpolarized light, such as the light emitted from the LED chip, case (3) shows the largest transmittance over all permissible angles with the best angle-averaged transmittance. A comparison among the three cases shows that small NPs could exhibit better angle-averaged transmittance for unpolarized light (Fig. 3g). Clearly, the optimization procedure must consider the transmittance averaged over all allowed angles within the photon escape cone, not just that at normal incidence.

Another important aspect to consider is the fact that the typical emission spectrum of any commercial LED (of any color) has a finite spectral width. For example, the emission spectrum of a typical AlGaInP/GaAs red LED by Toyoda Gosei Corp. ranges from 580 nm to 700 nm^{2}. Therefore, we conduct the next optimization study over the abovementioned s spectral range and try maximize the average transmittance over all permissible angles. The range can be customized according to the specific emission spectrum of an existing LED chip on top of which the NP “meta-grid” will be positioned.

Figure 4a–c depict the optimal transmittance (*T*) and corresponding NP array parameters for different heights (*h*). The displayed range is from 0 to 60 nm, beyond which the transmittance gradually decreases and is hence insignificant. The transmittance described here is calculated by averaging over a spectral range of 580–700 nm and over incident angles ranging between 0° and 26°. The yellow dots represent the optimum point of the maximum transmittance (*T*_{max} = 96.2%) at *h*_{opt} = 33 nm, *R*_{opt} = 13 nm, and *g*_{opt} = 13 nm. This enhanced transmittance, achieved in the presence of the optimized NP meta-grid, is significantly larger than the otherwise obtained transmittance of 83.9%, without any NPs, over the same range of wavelengths and incident angles. The study reveals a trend in which both the optimal radius and interparticle gap decrease with height. We also show all combinations of the parameters of the NP array (interparticle gap and height for specified NP radii) that can provide transmission within 0.5% and 1% of *T*_{max} in Fig. 4d, e, respectively.

It is also interesting to see how sensitive the maximum transmittance level is to any imperfections in the fabrication process, leading to deviations from the optimal values for the NP radius and inter-NP gap. Figure 5a plots the transmittance levels achieved for all combinations of *R*_{opt} ± 3 nm and *g*_{opt} ± 3 nm at the optimal height *h*_{opt} = 33 nm. Note that the figure is a 2D map of transmittance levels in the *R*–*g* plane with (*R*_{opt}, *g*_{opt}) at the center (highlighted in cyan) for a fixed height of *h*_{opt}—where the dots filled with different colors represent different levels of percentage reduction of transmission from *T*_{max}. Similar 2D maps of transmittance are also plotted for different *h* values ranging from *h*_{opt} − 3 nm to *h*_{opt} + 3 nm (Fig. 5b–g). In Figs. 4, 5, the transmittance data correspond to unpolarized state of light emitted from the emissive layer.

Knowing the optimal configuration of the NP array, the next step is to prepare silver nanospheres of radius *R*_{opt}, coated/functionalized with ligands of appropriate length to ensure that an interparticle gap of *g*_{opt} is obtained. A monolayer of these NPs can be fabricated using the drying-mediated self-assembly method, as done for “plasmene”^{22,23}. The NP meta-grid can be prepared on a substrate (ideally of the same material as the encapsulant), where the substrate thickness is chosen to correspond to *h*_{opt}. In another approach, first, the encapsulating layer of height *h*_{opt} can be deposited on the LED chip, and then, the NP monolayer can be fabricated on or transferred on to it^{22,23}. Such a meta-grid of NPs formed on a substrate (a stretchable substrate would also allow for precise tuning or further adjustment of the interparticle gap) could then be stamped^{23,32,33} onto the LED chip. After the NP layer is deposited, the encapsulant material with can be spin coated on it. This would ensure that the embedding medium of the NP monolayer is the same as the encapsulant and after that, the usual hemispherical casing can be fabricated or inserted.