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Abstract
Breaking the total internal reflection far above a critical angle (i.e., outcoupling deep-trap guided modes) can dramatically improve existing light-emitting devices. Here, we report a deep-trap guided modes outcoupler using densely arranged microstructured hollow cavities. Measurements of the leaky mode dispersions of hollow-cavity gratings accurately quantify the wavelength-dependent outcoupling strength above a critical angle, which is progressively improved over the full visible spectrum by increasing the packing density. Comparing hollow- and filled-cavity gratings, which have identical morphologies except for their inner materials (void vs. solid sapphire), reveals the effectiveness of using the hollow-cavity grating to outcouple deep-trap guided modes, which results from its enhanced transmittance at near-horizontal incidence. Scattering analysis shows that the outcoupling characteristics of a cavity array are dictated by the forward scattering characteristics of their individual cavities, suggesting the importance of a rationally designed single cavity. We believe that a hollow-cavity array tailored for different structures and spectra will lead to a technological breakthrough in any type of light-emitting device.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Antireflection technologies that reduce the Fresnel reflection loss at broad wave vectors are central to optical lenses, solar cells, and light-emitting diodes (LEDs) [1–9]. Particularly for internal reflection (as opposed to external reflection), achieving broadband antireflection is a relatively challenging problem. According to the phase-matching condition, the transmittance is zero at a planar interface above a certain critical angle (θc); this is referred to as total internal reflection (TIR). Minimizing the internal reflection for all incident angles (θ) between 0 and π/2 is crucial for light-emitting devices; in such devices, a high-refractive-index active material is sandwiched between a low-refractive-index substrate and ambient materials, which is regarded as a core/cladding planar waveguide system [10,11]. Traditional coating-based antireflection techniques are effective only for θ < θc and thus, two-dimensional gratings must be employed to obtain nonzero transmittance for θ > θc [12,13]. However, even when a grating is used, the transmittance tends to drop radically as θ moves away from θc and approaches π/2. This general trend is related to the attenuation depth (d) of evanescent fields, which is expressed by
Here, n1 and n2 are the refractive indices of the core and cladding materials, respectively. When TIR takes place at a value of θ that is nearly parallel to an interface, its evanescent field becomes almost completely attenuated within a distance much shorter than the wavelength. Therefore, penetration of the evanescent field into a grating should be marginal, thereby negating the effect of the grating for light with large incident angles (i.e., large in-plane wave vectors). For a core/cladding planar waveguide, light propagating with large in-plane wave vectors can be identified as low-order guided modes, of which electric fields are tightly confined in a high-refractive-index core medium, as shown in Fig. 1(a). Therefore, the outcoupling efficiency of LEDs has been fundamentally impeded by the inefficient extraction of low-order guided modes, which carry the major fraction of generated light.
Fig. 1 (a) Schematic illustrating optical modes trapped within a high refractive-index core medium. A surface pattern is employed to outcouple trapped light into a low-refractive-index ambient medium. (b) Schematics and SEM images showing the array of alumina-shell hollow cavities (left) and the growth of a GaN medium on the hollow cavities (right). Scale bar, 5 μm. (c) Schematics illustrating a homebuilt angle- and wavelength-resolved far-field scanner. A spectrometer is programmed to move along the azimuthal (φ) and the polar (θ) angles while a broadband (λ = 350 – 800 nm) Xe light source impinges normally or obliquely (as shown in the inset) on a specimen. (d) Cross sectional SEM images of a planar GaN/sapphire substrate and GaN/sapphire substrates containing hollow cavities with progressively increasing volume. Scale bar, 1 μm. (e) AFM surface roughness data of the three patterned structures shown in (d). (f) Far-field distributions of the four structures shown in (d) for normally incident light, plotted in an output angle-wavelength coordinate. Insets, images captured by using a charged coupled device images under a broadband (λ = 450 – 800 nm) laser illumination.
To outcouple these deep-trap guided modes from a core/cladding LED structure into an ambient medium, two primary strategies have been proposed: (i) thinning the LED core medium down to a few- wavelengths thickness and (ii) maximizing the index contrast (Δn) of the grating outcoupler [14–17]. The former strategy extends the attenuation depth of evanescent fields to increase the spatial overlap between the grating outcoupler and guided modes. In practice, however, artificially reducing the total thickness of the core LED medium, which consists of light-generation and carrier-transporting layers, can adversely affect the electrical characteristics (e.g., the operation voltage and leakage current) of LED devices. Alternatively, the latter strategy augments the diffraction strength of the grating outcoupler, avoiding electrical degradation. However, it is unclear how a high-index-contrast grating increases the outcoupling efficiency of LED devices. Therefore, quantifying the outcoupling strength of a grating upon each guided mode is essential to understanding the fundamental problems related to the extraction of light. It would also offer a fast and reliable route to evaluate competitive grating outcouplers when designing high-efficiency LED devices.
2. Experiment and results
2.1 Fabrication of the cavity-embedded gratings
We fabricated ultrathin (< 80 nm) alumina-shell hollow cavities that were hexagonally arranged (3 μm in pitch) on a sapphire substrate [18]. Then, we performed epitaxial growth of a high-refractive-index (n = 2.5) GaN medium (3-μm-thick), creating an embedded high-index-contrast grating, as shown in Fig. 1(b). The morphology (pitch, diameter, and shape) of the hollow-cavity array is readily controllable during the lithography process, which differentiates our technology from others. As a preliminary examination, the diffraction strength of fabricated gratings was assessed by acquiring their far-field distribution with a normal plane wave, as schematically shown in Fig. 1(c). First, we prepared one planar GaN/sapphire substrate as a reference and three GaN/sapphire substrates containing lithographically tuned hollow cavities, as shown in Fig. 1(d). These were used to probe the effects related to the packing density of cavities. Atomic force microscopy (AFM) measurements showed complete coalescence of GaN growth on the alumina-shell hollow cavities, resulting in optically flat surfaces with a root mean square (RMS) roughness of 20 nm or less [Fig. 1(e)]. This surface smoothness ensures that all of the optical effects discussed in this work can be ascribed solely to the embedded hollow cavities. Measurements from the four fabricated structures with broadband (λ = 350 – 800 nm) light reveal gradually enhanced diffraction strength as the packing density of the hollow cavities increases, as shown in Fig. 1(f); in an output angle-wavelength coordinate, the low- to high-order diffraction modes emerge in sequence as the packing density of the hollow cavities increases.
2.2 Quantization of the outcoupling strength
Although the far-field measurement under normal (θ = 0°) plane wave conditions provides a legitimate metric to characterize the diffraction performance of a grating [Fig. 1(f)], such evaluation cannot quantitatively account for the extraction of TIR (θ > θc) light. To accurately quantify the outcoupling strength for each TIR angle, we obtained leaky mode dispersions of the four structures. To do this, a broadband (λ = 450 – 800 nm) supercontinuum laser was incident upon a structure via a high-refractive-index (n = 1.72) hemi-cylindrical prism with various incident angles (θprism) and the transmitted light was collected by an integrating sphere equipped with a spectrometer, as shown in Fig. 2(a). The measured leaky mode dispersions reveal key features related to the outcoupling of trapped light, as shown in Fig. 2(b). For all of the structures considered, multiple discrete bands, which are assigned to Fabry-Perot resonances with different mode numbers (m), appear at specific wavelengths (λ) that meet the standing wave condition expressed by
Fig. 2 (a) Schematics illustrating a homebuilt experimental setup for evanescent-field coupled leaky mode dispersion measurements. A broadband (λ = 450 – 800 nm) supercontinuum laser is used as an incident light source. The incident supercontinuum laser is equipped with a step motor programmed to move along the θ-direction with a step of 1°. A hemi-cylindrical prism with a refractive index of 1.72 is bonded to a GaN/sapphire substrate by using an index-matching (a refractive index of 1.72) adhesive gel. (b,c) Measured (b) and simulated (c) leaky mode dispersions (transmittance versus incident angle) of the four structures shown in Fig. 1(d) for broadband (λ = 450 – 800 nm) wavelengths. The lower panels in (b) and (c) display the same dispersion data for the whole range of TIR angles, plotted in a logarithmic scale.
Here, n and d are the refractive index and thickness of the core GaN medium, respectively [19,20]. For the structures containing hollow cavities, the discrete bands are extended beyond θc, which is more apparent in the logarithmic scale plots [the lower panels in Fig. 2(b)], retaining their overall shape as a cosinusoidal function. Alternatively, for θ < θc, the transmittance (i.e., the amplitude of bands) monotonically decreases as the packing density of hollow cavities increases. This is indicative of the backward Mie scattering augmented by microstructured particles [21,22]. Note that for θ < θc, all the hollow-cavity arrays exhibit relatively low transmittance compared to the planar/GaN structure. However, because practical LED devices allow internally generated light to undergo multiple reflections, the transmittance for θ < θc (which is substantially high except θ ~θc) has a marginal effect on the total outcoupling efficiency. More importantly, the transmittance behavior for θ > θc is completely opposite; the array with the largest packing density of hollow cavities exhibits the maximum amplitude over the entire range of TIR angles, suggesting it as the most efficient outcoupler among the four structures considered. To verify the measured leaky mode dispersions, we conducted rigorous coupled-wave analysis (RCWA) simulations for the same structures, as shown in Fig. 2(c). The RCWA-simulated leaky mode dispersions strongly support the measured data, verifying the accuracy of our measurement system.
The leaky mode dispersions combined with the far-field distributions show that the packing density of individual objects in an array has a strong impact on its outcoupling strength. This is physically analogous to the situation of X-ray crystallography, wherein the electron concentration of individual atoms in a crystal determines the amplitude of the Bragg diffraction peaks [23,24]. In addition, we note the practical importance of leaky mode dispersions as a fast and reliable route to explore the optimized grating outcoupler for a given LED device. For example, the pitch is an important grating parameter; thus, we prepared a GaN/sapphire substrate containing nearly close-packed, submicron-pitch (400 nm) hollow cavities, as shown in Fig. 3(a) and 3(b). The measured leaky mode dispersion of the submicron-pitch hollow-cavity array exhibits discrete multiple bands above θc; however, their amplitudes are clearly weaker than those of the microstructured hollow-cavity array with the maximum packing density [Fig. 3(c) and the rightmost plot in Fig. 2(b)]. This comparison shows the superior outcoupling performance of a micron-pitch grating relative to a submicron-pitch grating for a few-tens-wavelength-thick GaN/sapphire LED device, which is consistent with previous reports [11,13,19].
Fig. 3 (a) Schematics illustrating the fabrication process for submicron hollow cavities with an alumina-shell that are embedded in a GaN/sapphire substrate. Closely packed submicron polystyrene spheres are used as a pattern mold. The remaining process is the same as the microstructured hollow-cavity arrays (see Appendix). (b) SEM images showing submicron hollow cavities on a sapphire substrate (left) and submicron hollow-cavities embedded in a GaN medium (right). Scale bars, 200 nm and 1 μm, respectively. (c) Measured leaky mode dispersion of a fabricated GaN/sapphire substrate containing an array of submicron (a diameter of 400 nm) hollow cavities. The right panel displays the same data only for the whole range of TIR angles, plotted in a logarithmic scale.
2.3 Effects of the index contrast in cavity arrays
For commercial GaN-based LED devices, patterned sapphire substrate (PSS) technologies have been predominantly used as for outcouplers over the past few decades [the lower panel of Fig. 4(a)]. All of the structural parameters (e.g., pitch, diameter, and shape) in PSSs have been optimized to provide the greatest outcoupling efficiency. Recently, we successfully demonstrated that InGaN/GaN LED devices fabricated on an embedded hollow-cavity array yielded greater light output than their counterpart PSS devices [25]. For a deeper understanding of the relation between the index contrast of a grating and its outcoupling efficiency, we acquired leaky mode dispersions for filled- (i.e., filled with solid sapphire) and hollow-cavities-incorporated GaN/sapphire substrates. For a fair comparison, the morphology and packing density of both cavities were prepared to be practically identical, except for their inner materials (void vs. solid sapphire), as shown in Fig. 4(a). The measured leaky mode dispersions, as shown in Fig. 4(b), reveal that the hollow-cavity array yields enhanced transmittance at θ > θc for the entire wavelength range (λ = 450 – 800 nm), as compared to the filled-cavity array. Alternatively, for θ < θc, the transmittance is rather low for the hollow-cavity array, which indicates the well-known principle that the Mie scattering is enhanced by increasing the index. All of the characteristics observed in the measured leaky mode dispersions are strongly supported by the RCWA simulations, as shown in Fig. 4(c). Interestingly, both measured and simulated data exhibit several abrupt kinks at specific θ’s over the critical angle for all bands, which are identified as discrete waveguide modes with different orders.
Fig. 4 (a) Cross sectional SEM images of the arrays of alumina shell hollow cavities (upper) and sapphire filled cavities (lower) that are embedded in a GaN medium. Scale bar, 2 μm. (b) Measured leaky mode dispersions of the hollow- (upper) and the filled-cavity (lower) structures shown in (a) for broadband (λ = 450 – 800 nm) wavelengths. (c) Simulated leaky mode dispersions of the hollow- (upper) and the filled-cavity (lower) structures as shown in (a).
To elucidate the distinct outcoupling characteristics between the hollow- and filled-cavity arrays, we simulated their angular transmittance by varying their packing densities via RCWA simulations, as shown in Fig. 5(a) and 5(b). First, we compared the measured (solid red and black lines) and their corresponding simulated (dashed red and black lines) angular transmittance data for λ = 500 nm, as shown in Fig. 5(c). Note that we used two different incident angles (θprism and θGaN) in the plot, depending on the starting medium of incident light. Given that the refractive index (n = 1.72) of the prism used in the measurement system is smaller than that (n = 2.5) of the GaN medium, the measured data has a technologically inaccessible angular range for θGaN; this range is only covered by the simulated data. Both measured and simulated transmittance data clearly reveal the superior outcoupling strength (i.e., the transmittance for θ > θc) of the hollow-cavity array relative to the filled-cavity array; this behavior becomes more pronounced as the incident angle (θ) moves farther away from the critical angle (θc). The hollow-cavity array surpasses the filled-cavity array at θprism > 50° and θGaN > 35°, which is just above the critical angles for sapphire/air (34°) and GaN/air (23°). Furthermore, the hollow-cavity array maintains substantially large transmittance, even for θ >> θc, demonstrating that a high-index-contrast grating functions as an efficient outcoupler for deep-trap guided modes.
Fig. 5 (a,b) Schematics of GaN-based LED structures containing cylindrical (a) and conical (b) hollow cavities. (c) Measured and simulated angular transmittance of the two structures shown in Fig. 4(a), obtained at λ = 500 nm. The incident angles are defined within a sapphire medium (θprism, the upper x-axis) and a GaN medium (θGaN, the lower x-axis). Note that θGaN > 46° is forbidden in the present measurement system. The inset shows a schematic describing the two different incident angles (θprism and θGaN). (d,e) Simulated angular transmittance (λ = 500 nm) of cylindrical (d) and conical (e) hollow cavities (upper panels) with increasing their diameters (D) at fixed the height. A plane wave with a variation of incident angles (θ) is generated in the GaN medium. For comparison, the data for filled (filled with solid sapphire) cavities are plotted (lower panels).
In the RCWA simulations, two different cavity shapes (i.e., cylindrical and conical) were considered, as shown in Fig. 5(a) and 5(b). For both types of hollow-cavity arrays, the transmittance at θ >> θc is significantly enhanced and the angular range maintaining high transmittance is further extended compared to the values of the filled-cavity arrays with the same packing density [Fig. 5(d) and 5(e)]. More interestingly, the transmittance behavior above a critical angle is significantly altered by the cavity shape. For the cylindrical hollow-cavity array with a nearly close-packed density (i.e., a diameter of 2.6 to 2.8 μm), the transmittance is relatively low but fairly constant over the entire range of TIR angles, as shown in Fig. 5(d). Alternatively, the conical hollow-cavity array with the same density yields relatively high transmittance within a limited angular range, as shown in Fig. 5(e). Therefore, one needs to design the cavity shape depending on the level of internal material absorption when developing high-efficiency LEDs.
2.4 Scattering analysis on individual cavities
Thus far, we have discussed the transmittance of light through a dielectric grating as a function of the incident angle. In addition, by tracing the propagation direction of the transmitted light, one can determine meaningful information that is relevant to the emission distribution of LED devices. To do this, we used the same angle- and wavelength-resolved far-field scanner that was used to produce Fig. 1(e). However, for these measurements, a broadband (λ = 350 – 800 nm) Xe light was applied through a high-refractive-index (n = 1.72) hemi-cylindrical prism at a specific incident angle (θprism), as shown in the inset of Fig. 1(c). For the hollow- and sapphire-cavity arrays, we obtained spectrally integrated far-field distributions for three representative incident angles: θprism = 10° (below the critical angle), 50° (just above the critical angle), and 70° (far above the critical angle), as shown in Fig. 6(a). For the θprism = 10° case, both structures exhibit a clean and sharp spot located at θ ~θprism, corresponding to specular transmission (i.e., zeroth-order diffraction). Alternatively, for the TIR angle cases (θprism = 50° and 70°), multiple fringes appear at discrete output angles, revealing that diffraction leads to the transmission of light above the critical angle (i.e., optical tunneling). Integrating the amplitude of all the far-field fringes indicates the transmittance at the same incident angle. Therefore, the hollow-cavity array yields much more pronounced fringes than the filled-cavity array at θprism = 70°, which is consistent with the results in Fig. 4 and Fig. 5. Notably, the far-field fringes from the hollow-cavity array are distributed closer to the normal direction, which is valid at θprism = 50° and 70°. The localized far-field distributions at θ > θc conform well with the improved vertical directionality of InGaN/GaN LED devices with a hollow-cavity array, as reported in our previous work [15]. Based on these results, a rationally designed hollow-cavity grating can serve as an efficient outcoupler that is effective for a broad range of TIR angles, potentially engineering the emission distribution from LED devices.
Fig. 6 (a) Measured spectrally integrated (λ = 350 – 800 nm) far-field distributions of the hollow-cavity (upper) and the filled-cavity (lower) structures shown in Fig. 4(a). for incident angles (θprism) below (10°) and above (50° and 70°) a critical angle. (b) Simulated forward scattering efficiencies of single alumina-shell hollow cavity and sapphire filled cavity as a function of incident angles. The incident angles are defined within a sapphire medium (θprism, the upper x-axis) and a GaN medium (θGaN, the lower x-axis). Inset, schematic illustrating the definition of scattering efficiency. (c) Cross sectional near-field scattered profiles (λ = 500 nm) of the hollow cavity (upper) and the filled cavity (lower) at specific TIR angles (θprism = 50°, 70°, and 80°). (d) Simulated far-field scattering distributions (λ = 500 nm) of the hollow cavity (upper) and the filled cavity (lower) for the same incident angles (θprism = 10°, 50°, and 70°) used in (a).
The convolution theorem related to the field of optics says that the far-field distribution of an array of identical objects can be obtained by multiplying the scattering distribution from the unit object and the interference distribution from an array of point light sources with the same periodicity [26]. Given that the hollow- and filled-cavity arrays have the same periodicity of 3 μm, one can infer that their distinct transmission characteristics should be related to the scattering characteristics of individual cavities composed of each array. First, to understand the cavity-dependent (void vs. solid sapphire) angular transmittance behavior, as discussed in Fig. 4 and Fig. 5, we calculated the forward scattering efficiencies of single hollow and filled cavities for a plane wave (λ = 500 nm) with various incident angles, as shown in Fig. 6(b). Overall, the curves of the forward scattering efficiencies are very similar to those of the transmittance shown in Fig. 5(c); the hollow cavity surpasses the filled cavity for large incident angles (θprism > 50° or θGaN > 35°). To visualize the distinct scattering strength between both cavities, we obtained the scattered energy profiles for three incident angles (θprism = 50°, 70°, and 80°), as shown in Fig. 6(c). For θprism = 50° (just above the critical angle for the GaN/air interface), both cavities produce a significant scattered energy flux into the air medium. Alternatively, for θprism = 70° and 80° (far above the critical angle), only the hollow cavity is effective in delivering the incident plane wave to the air medium via scattering.
Performing the Fourier transform with the scattered fields of the single cavities shows the propagation direction of the leaky light in the air medium (i.e., the far-field scattering distribution), as shown in Fig. 6(d). For θprism = 10°, both cavities exhibit a clean and sharp spot at θ ~θprism, minimizing scattering in other directions. Alternatively, for the two TIR angles (θprism = 50° and 70°), the far-field scattering distribution is spread out for both cavities, while the hollow cavity is capable of steering the incident plane wave toward the normal direction. In addition, for θprism = 70°, the hollow cavity yields a more pronounced scattering amplitude than the filled cavity. All of the observations featured in the forward scattering simulations are consistent with the measured transmission characteristics shown in Fig. 4(b) and 6(a), which is clear evidence of the convolution theorem. A more important conclusion is that simply designing a single object that provides the maximum forward scattering efficiency is sufficient to obtain a grating outcoupler that provides the greatest outcoupling efficiency in LED devices if its pitch is within an optimum range.
2.5 Applications of the cavity-embedded gratings
To delineate the possibility of high-efficiency hollow-cavity-incorporated LED devices, we conducted finite-difference time-domain (FDTD) simulations for blue-emitting (center wavelength of 450 nm) flip-chip GaN LEDs and green-emitting (center wavelength of 516 nm) organic LEDs with various cavity volumes. For the GaN and organic LEDs, a high-index-contrast grating was formed by incorporating hexagonally arranged hollow cavities into a GaN (n = 2.5) or a Si3N4 (n = 2.0) medium, as shown in Fig. 7(a) and 7(b). Overall, all of the extraction efficiencies increased steadily with increasing cavity volume, reaching the maximum value near a close-packed array, as shown in Fig. 7(c) and 7(d). More importantly, both GaN and organic LED devices yielded greater extraction efficiencies with the incorporation of a hollow-cavity array, relative to a filled-cavity array (filled with solid sapphire or SiO2), regardless of the polarization (transverse electric: TE and transverse magnetic: TM) of emitted light. Although the absolute efficiency enhancement can be altered to some extent, depending mainly on the level of internal material absorption, a hollow-cavity-incorporated grating would represent a technological breakthrough in any type of LED device.
Fig. 7 (a,b) Schematics of GaN-based (a) and organic (b) LED devices containing hollow cavities. (c,d) FDTD-simulated TE, TM, and average extraction efficiencies of GaN-based (c) and organic (d) LED devices with alumina-shell (80 nm for the GaN LED and 40 nm for the organic LED, in thickness) hollow cavities as a function of the diameter (D) of hollow cavities (left panels). For comparison, the data for filled (filled with sapphire for the GaN LED and SiO2 for the organic LED) cavities are plotted (right panels). For the GaN-based and the organic LED devices, (a, h) is (3, 1.5) μm and (0.5, 0.2) μm, respectively.
3. Conclusions
We demonstrated that hollow-cavity-incorporated dielectric arrays can dramatically increase the transmission of light for a broad range of TIR angles and efficiently outcouple deep-trap guided modes. Measurements of the leaky mode dispersions, combined with the RCWA-simulated data, were used to quantitatively demonstrate the superior outcoupling performance of a hollow-cavity array relative to a filled-cavity array with the same packing density. A nearly close-packed hollow-cavity array was effective at outcoupling all trapped light over a broad range of TIR angles. This elicits a message that measurements of the leaky mode dispersions offer a fast and accurate route to quantify the efficacy of a grating coupler. The scattering analysis of an individual hollow cavity provided a basis for the optimal design of a grating array composed of identical hollow cavities; the enhanced outcoupling efficiency and vertical emission were rationalized by investigating the forward scattering characteristics of a single hollow cavity. FDTD simulation showed that for GaN and organic LED devices, hollow-cavity-incorporated gratings could improve approximately 20% in extraction efficiency over conventional gratings. We believe that the effect of the hollow-cavity array is not limited to inorganic and organic LED devices; it can also be extended to light absorption devices, such as solar cells, once a grating has been appropriately designed for different structures and spectra.
Appendix
Fabrication of hollow-cavity arrays: A hexagonal array of photoresist (PR, AZ GXR-601) rod patterns was defined on a sapphire substrate by standard photolithography using an i-line stepper (Nikon i7). To make a conical PR pattern, a thermal reflow process was applied at 150 °C for 40 min in a convection oven. Then, an amorphous alumina layer was conformally deposited on the PR pattern and the sapphire substrate by atomic layer deposition (ALD, NCD LUCIDA D100). H2O and trimethylaluminum (TMAl) were used as oxygen and aluminum sources, respectively. 1000 ALD cycles were used to deposit an 80-nm-thick alumina layer. Then, the thermal treatment process was carried out in a furnace at 1100 °C for 2 hr. During this thermal process, the PR material was removed via oxidization, forming hollow cavities. At the same time, the initial amorphous alumina layer was crystallized into single-crystalline α-Al2O3. A GaN epitaxial layer was grown on two-inch sapphire substrates with a hexagonal array (3 μm in pitch) of hollow or filled cavities by a metalorganic chemical vapor deposition. A 3-μm-thick un-doped GaN layer was grown on each patterned sapphire substrate. Trimethylgallium (TMGa) was used as a precursor for Ga. The AFM data confirmed complete coalescence of the GaN growth on each patterned sapphire substrate, leading to optically flat GaN surfaces.
Measurement of angle- and wavelength-resolved far-Field distributions: The far-field distributions were measured by a homebuilt angle- and wavelength-resolved far-field scanner. A spectrometer with an array detector (USB4000, Ocean Optics) was programmed to rotate along the azimuthal (ϕ) and polar (θ) directions with a step size of 1°. A Xe light source (450 W, Newport) was used as the incident broadband light. An optical fiber and two identical circular metal apertures (with a diameter of 2.0 mm) were used to obtain a collimated incident beam. Total far-field distributions were integrated for each wavelength-resolved data point over λ = 350 – 800 nm.
Measurement of leaky mode dispersions: The measurement setup was composed of an integrating sphere, a spectrometer, a hemi-cylindrical prism (n = 1.72), a controllable rotating system, and a supercontinuum laser (λ = 450 – 800 nm), as illustrated in Fig. 2(a). To maximize the range of the TIR angles, a high-refractive-index (n = 1.72) adhesive matching gel was used. The incident angle (θ) was changed from −80° to 80° with a step size of 1° by means of a programmed motor. The light transmitted through the sample was collected with a spectrometer (USB 4000, Ocean Optics) combined with an integrating sphere (3P-GPS-010-SL, Labsphere).
Electromagnetic simulations: Numerical simulations were performed using commercial RCWA software (DiffractMOD, Rsoft) and a homebuilt FDTD program. All of the transmittance results from array structures were calculated via RCWA. Alternatively, FDTD simulations were used to obtain the scattering efficiency of single cavities and the extraction efficiency of LED devices. To calculate the scattering cross section with respect to the incident angle, electric and magnetic scattered fields were obtained from a single cavity with a plane wave illuminated from a GaN medium; to do this, the total-field scattered-field method was adopted. Then, the scattering cross section was divided by the projected value of an object to determine the scattering efficiency. The incident angle was scanned from 0° to 80° with a step size of 5°. The spatial resolution was set to 10 nm along the x-, y-, and z-directions in the simulated structures. To obtain the far-field scattering distribution of single cavities, the in-plane electric and magnetic fields were acquired. Then, the near-to-far-field Fourier transformation was performed with the field components. The extraction efficiencies of LED devices were calculated using a single TE or TM polarized dipole source excited with a wavelength of 500 nm. A 50-nm-thick absorption layer with an extinction coefficient of 0.01 was introduced in the simulated LED devices. To obtain total extraction efficiencies, we evaluated the average efficiency, which was defined at a ratio of TE:TM = 2:1. A spatial resolution of 10 nm was used for the x-, y-, and z-directions and periodic boundary conditions were employed along the x- and y-directions.
Funding
National Research Foundation of Korea (NRF-2017R1A2B4005480); Brain Korea 21 Plus project (F15SN02D1702); Tech Incubator Program for Start-up (S2486454).
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