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Abstract
We studied a high-accuracy hybrid optics modeling for macroscopic lighting devices containing highly diffractive elements. For a two-dimensional (2D) grating, we achieved forward and backward diffraction distributions at omnidirectional incidence by conducting rigorous coupled-wave analysis and then assigned the diffuse information to a virtual, planar surface in a ray-optics model. By using the integrated ray-wave optics simulation, we obtained extraction efficiencies and far-field distributions of millimeter-scale (0.5 × 0.5 × 0.1 mm3) flip-chip GaN-based light-emitting diodes (LEDs) with embedded 2D gratings. The increased index contrast of 2D gratings progressively improved the extraction of light via the top face of the substrates, thus inducing a vertical beaming effect that strongly supported measured data. The outcoupling features related to the index contrast of gratings were understood by performing Fourier analysis; a high-index-contrast grating preferred to excite high-order diffraction modes, thereby effectively converting tightly bound waveguide modes into leaky light through the top escape route. The simulation strategy developed herein will be essential for designing directional illuminations and micro-LED displays.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The development of high-efficiency light-emitting diodes (LEDs) is a key element in lighting applications, such as illuminations and displays. Two-dimensional (2D) dielectric gratings have been integrated into optoelectronic devices, including LEDs [1–5], semiconductor lasers [6–8], and solar cells [9–13], to resolve an impedance mismatch at the semiconductor/air interface. Particularly for organic and inorganic LEDs with thick, transparent substrates, 2D embedded gratings were employed to achieve a large extraction efficiency. Therefore, structural parameters, which are primarily characterized by pitch and refractive index contrast, must be rationally determined to maximize the extraction efficiency. Optics simulations have played an important role in predicting device performance for given structures; for LEDs, extraction efficiency and far-field intensity distributions have been the main concerns [14–16].
Recently, quantifying extracted light per each escape route is also essential; for flip-chip GaN-based LEDs, one top and four side faces of a sapphire substrate are available routes, as shown in Fig. 1(a). This information is particularly useful for designing direct-lit LED displays [17] because substrate-side emission must be blocked to obtain high-clarity images, as shown in Fig. 1(b). The substrate-side emission blurs images due to crosstalk between pixels. Besides, when adjacent pixels emitting different colors are turned on simultaneously, color distortion can be also observed.
Fig. 1. (a) Schematics illustrating light extracted through top and side faces of a LED with a thick, transparent substrate. (b) Schematics comparing the clarity of an image displayed by three primary color pixels with and without side face extraction.
Optics simulations are generally classified into ray- and wave-optics methods. Each simulation method has inherent limitations for designing macroscopic (i.e., over a-few-hundreds micron scale) lighting devices containing diffractive elements. For example, ray-optics simulation predicts the behavior of light propagation by applying Snell’s law and the Fresnel equations that are practically valid only for planar structures. In other words, ray-optics simulation excludes any diffraction effects resulting from periodically spaced objects. To illustrate such a diffraction-related characteristic, we acquired the trajectory of light that was incident on 2D gratings with the pitches (p) of 20 µm, 2 µm, and 0.1 µm by conducting individual ray- and wave-optics simulations [Figs. 2(a)–2(c)]. Note that the wavelength (λ) of the incident light was 450 nm. When p/λ (i.e., a size parameter) was much larger than unity, both methods yielded the same trajectory; the normally incident, upward light bounced twice at the surface of the cone-shaped grating by means of total internal reflection (TIR) and then propagated downward [ Fig. 2(a)]. In contrast, as a size parameter becomes close to unity [Fig. 2(b)] or even smaller than unity [Fig. 2(c)], only the wave-optics simulation imparted a correct solution, exhibiting scale-dependent diffraction features. Particularly for p/λ < 1, all the diffracted modes became evanescent and thus, the grating structure served as a planar surface with an effective refractive index, which is referred to as a metasurface [18–20].
Fig. 2. (a-c) Schematics of simulated structures (left panels), and ray- (middle panels) and wave-optics (right panels) simulated light propagation when a collimated beam impinges on 2D gratings with the pitches (p) of (a) 20 µm, (b) 2 µm, and (c) 0.1 µm, respectively. (d-f) Schematics of simulated structures (left panels), and wave-optics simulated light propagation (middle panels) and far-field intensity distributions (right panels) when a single (d) and two identical (e, f) in-plane dipole sources produce light in a dielectric medium (n2 = 2.5). For (e) and (f), the distance (d) between the two dipole sources is 4 µm and 2 µm, respectively. For all the wave-optics simulations, the incident wavelength (λ) is 450 nm.
Although wave-optics simulation properly accounts for the diffraction of light, it is not yet adequate for designing macroscopic lighting devices containing diffractive elements because of limited computation memory. Alternately, periodic boundary conditions are adopted to overcome the technical memory issue [21]. However, an infinite array of diffracting elements cannot interpret any size effects; in the case of a flip-chip GaN-based LED, extraction efficiency per each escape route and far-field intensity distribution must depend heavily on the dimensions of the structure [22]. Additionally, when a dipole source (as opposed to a plane wave) is employed in simulation, dipole-to-dipole interaction, which mainly occurs between neighboring unit structures, is inevitable. This feature was clearly understood by conducting wave-optics simulation, in which a single [Fig. 2(d)] and two identical in-plane dipole sources with distances of 4 µm [Fig. 2(e)] and 2 µm [Fig. 2(f)] were excited within a high-refractive-index (n = 2.5) dielectric medium. In comparison to the single dipole, the coupled dipoles produced electromagnetic waves strongly interfering with each other [23], thereby creating unwanted, multiple fringes in the radiation distribution. Notably, as the coupled dipoles were located further apart, the radiation distribution asymptotically approached a single dipole’s continuum, Lambertian distribution, in conjunction with an increase of interference fringes.
To overcome such critical problems in traditional optics simulations discussed herein, we developed a rigorous coupled-wave analysis (RCWA) integrated ray-tracing simulation. Sze [24] and Chou [25] groups adopted similar strategies to explore structural parameters that provide the maximum extraction efficiencies of organic and inorganic LEDs with diffraction gratings, respectively. However, we focused intensively on how the index contrast of gratings alters extraction efficiencies per escape routes for a millimeter-scale flip-chip GaN-based LED with a sapphire substrate. Such information is highly demanded for designing vertically directed illuminations and micro-LED displays because emission through the top face of a substrate only contributes to a net extraction efficiency in both applications. Furthermore, we resolved diffraction efficiencies per diffraction orders by performing Fourier analysis. The reliability of the integrated ray-wave optics simulation was evaluated by comparing experimental findings.
2. Results
2.1 RCWA-integrated ray optics modeling for macroscopic diffractive lighting devices
Figure 3 illustrates the process of the integrated ray-optics simulation studied herein. To highlight the utility of the integrated simulation, we examined a 2D grating embedded LED possessing a few-hundreds micron thick, transparent substrate as an example of macroscopic, diffractive lighting devices.
Fig. 3. Simulation process of RCWA-integrated ray optics modeling for acquiring extraction efficiencies via escape routes and far-field intensity distributions of 2D grating embedded LEDs.
First, for the considered 2D grating, we obtained all orders of forward and backward diffraction efficiencies as a function of incident angles (θ, ϕ), where θ = 0–180° and ϕ = 0–360° with discrete angle steps, by performing a RCWA simulation [left, Fig. 3]. Then, in a ray-optics model describing an actual-scale LED, a diffuse surface was created at which the grating was originally positioned; all the diffraction information of the grating was assigned to the diffuse surface [middle, Fig. 3]. Finally, for an active medium (e.g., multiple quantum wells) with a specific radiation distribution, we achieved far-field intensity distribution in an ambient medium [right, Fig. 3]. Integrating extracted light over all solid angles yielded the total extraction efficiency of the LED structure. More importantly, the amount of extracted light through each escape route was separately acquired by setting up detectors at adequate positions in a ray-tracing model [middle, Fig. 3].
2.2 Outcoupling characteristics of 2D grating embedded semiconductor LEDs
We studied the outcoupling characteristics of millimeter-scale (0.5 × 0.5 × 0.1 mm3) flip-chip GaN-based blue (λ = 450 nm) LEDs containing embedded 2D gratings by conducting the integrated ray-wave optics simulation. A 2D grating was employed to extract light trapped within a high-refractive-index LED medium by TIRs [26]. For this study, a hexagonally arranged grating was located at the interface between a GaN medium (n = 2.5) and a sapphire substrate (n = 1.8), as shown in Fig. 4(a). The pitch (p) of the simulated gratings was fixed to be 3 µm because this value led to the maximum outcoupling efficiency [27]. Particularly, we focused on how the index contrast ($\varDelta$n) of a 2D grating impacts the extracted light via each escape route and far-field intensity distribution. Note that vertical GaN-based LEDs incorporate pseudo-random (as opposed to periodic) patterns into n-type GaN which are typically formed by chemical etching. To treat such non-periodic patterns in RCWA simulation, one needs to construct a substantially large unit-cell in which a sufficient number of pseudo-random objects are assembled.
Fig. 4. (a) Schematics of a simulated flip-chip GaN-based LED structure with a 2D hexagonal grating into a sapphire substrate. (b) Cross-sectional SEM images of hexagonally arranged sapphire-filled (top) and hollow (bottom) gratings. (c) Simulated top-face, side-face, and total extraction efficiencies of flip-chip GaN-based LEDs as a function of an embedded 2D grating’s Δn, obtained by the integrated- (left) and ray-optics (right) simulations.
In general, the diffraction strength of a dielectric grating is enhanced by increasing $\varDelta$n. This statement was readily understood by considering a simple, sinusoidal grating with a spatial variation of permittivity ε(r), which is expressed as
Various techniques for the implementation of hollow gratings, which are represented by the epitaxial overgrowth of GaN on a patterned GaN template [28,29], a wafer-bonding process [30], and hollow microsphere-coated sapphire substrates [31], have been demonstrated. However, these approaches were not appropriate for attaining a nearly closely packed array of hollow gratings, unlike their counterpart sapphire-filled gratings [top, Fig. 4(b)], thus limiting the extraction efficiency to a great extent. This low-filling-factor issue was resolved by developing a lithographically engineered hollow grating [bottom, Fig. 4(b)]; an array of micron hollow cavities was sequentially fabricated by performing standard photolithography, depositing an outer alumina shell, and oxidizing the inner photoresist at high temperatures (> 1200 K) [32]. The fabricated hollow grating’s pitch and density were determined at the stage of lithography. Note that the outer alumina shell was crystallized after high-temperature oxidation, which enabled the hollow grating to serve as a GaN growth template [33].
Figure 4(c) shows top-face, side-face, and total extraction efficiencies of flip-chip GaN-based LEDs as a function of Δn, performed by the integrated ray-wave optics simulation. The hollow and sapphire-filled gratings were identified as $\varDelta$n = 1.5 and 0.7, respectively. The pitch (p), diameter (D), and height (h) of the cone-shaped gratings in simulation were 3 µm, 2.7 µm, and 1.8 µm, respectively, which were practically identical to those of the fabricated gratings shown in Fig. 4(b). Optical absorption was not imposed in the simulated structure except for a bottom mirror; the reflectance of the mirror was set to 0.95. Although practical LEDs have finite spectrum bandwidth (typically ∼20 nm), we used a monochromatic (λ = 450 nm) plane wave in the RCWA simulation. Because the pitch (3 µm) of the used 2D gratings is much larger than spectrum bandwidth, wavelength-dependent outcoupling features can be ignored. For the integrated simulation, the total extraction efficiency improved steadily with increasing $\varDelta$n due to the dramatic enhancement of the top-face extraction efficiency, which strongly supported the experimental findings reported in other literatures [26,27]. For the previous literatures [26,27], a near-to-far-field transformation technique was used to obtain far-field intensity distribution in a sapphire background. Then, extraction efficiencies per available routes were determined by calculating the far-field intensity within appropriate light cones. Therefore, this method assumes an infinite-thickness sapphire substrate, thereby excluding any effect related to finite sapphire thicknesses and provides extraction efficiencies to a first approximation.
In comparison to the sapphire-filled grating ($\varDelta$n = 0.7), the hollow grating ($\varDelta$n = 1.5) led to 1.05-fold and 3.4-fold enhancements in total extraction efficiency and the ratio of top- to side-face extraction efficiency, respectively. However, for the ray-optics simulation, the extraction efficiency was adversely reduced with a hollow grating because ray optics account for only a Fresnel reflection loss that is proportional to $\varDelta$n. Interestingly, the improved top-face extraction efficiency with increasing $\varDelta$n was observed even in the ray-optics simulation, wherein glancing rays were deflected toward the top escape cone after being reflected at the interface between the grating and the GaN medium.
2.3 Fourier analysis of high-index-contrast 2D gratings
We performed Fourier analysis to understand the outcoupling features related to the index contrast of gratings. The forward diffraction efficiencies of sapphire-filled ($\varDelta$n = 0.7) and hollow ($\varDelta$n = 1.5) gratings were obtained per diffraction orders (l, m) for an incident plane wave (λ = 450 nm) with θ = 60°, performed by RCWA simulation [ Fig. 5(a)]. Both sapphire-filled and hollow gratings were hexagonally arranged, cone-shaped gratings with (p, D, h) = (3, 2.7, 1.8) µm, placed at the interface between the GaN and sapphire media. Note that the incident angle (θ = 60°) was above the critical angle (θc = 46° for the GaN/sapphire interface). The diffraction efficiencies plotted in momentum space elicits a message that the high-index-contrast hollow grating led to the efficient excitation of diffraction modes with relatively large l and m values. These findings support well the dramatically improved top-face extraction efficiency discussed in Fig. 4(c); such intensified high-order diffraction modes steered horizontally propagating light (i.e., tightly bound waveguide modes) into the top-face escape route, thereby giving rise to vertically localized far-field intensity distribution. As discussed in Fig. 4(c), the ratio of top- to side-face extraction efficiencies was directly proportional to $\varDelta$n; this phenomenon was clearly evident from the real space representation of the diffraction data in Fig. 5(a) (i.e., far-field intensity distribution in a sapphire medium) [Fig. 5(b)]. The shaded regions surrounded by blue and red dashed lines indicated top- and side-face escape routes via a sapphire substrate, respectively. For the hollow grating, high-amplitude diffraction spots densely occupied the top-face escape route due to the amplified high-order diffraction modes, which was consistent with the result of Fig. 4(c).
Fig. 5. (a) RCWA-simulated forward diffraction efficiencies of sapphire-filled and hollow gratings as functions of diffraction orders (l, m) at θ = 60°. The efficiencies are averaged over all azimuthal (ϕ) angles. (b) RCWA-simulated far-field distributions were acquired in a sapphire medium for sapphire-filled and hollow gratings at (θ, ϕ) = (60°, 0°). The shaded regions with blue and red dashed boundaries indicate top- and side-face escape routes via a sapphire substrate, respectively. (c) RCWA-simulated forward diffraction efficiencies of sapphire-filled and hollow gratings as functions of m and θ. (d) RCWA-simulated forward diffraction efficiencies of gratings as functions of m and $\varDelta$n at θ = 40° and 60°. For (a) to (d), hexagonally arranged, cone-shaped gratings with (p, D, h) = (3, 2.7, 1.8) µm are introduced at an interface between the GaN and sapphire media. The wavelength of incident light is 450 nm. For (c) and (d), the diffraction efficiencies are obtained at l = 0; therefore, they are nonzero only when m is an even number.
The same trend was detected at other TIR angles [Fig. 5(c)]; the hollow grating was beneficial to excite high-order diffraction modes at broad incident angles over θ = 50° – 80°, contrasting with its sapphire-filled counterpart. In addition, we probed all orders of diffraction efficiencies with an incremental step of $\varDelta$n [Fig. 5(d)]. For these simulations, we examined two incident angles of θ = 40° (below θc) and 60° (above θc). For both incident angles, the pronounced diffraction order gradually shifted from low to high numbers with increasing $\varDelta$n. In the case of θ = 40°, the zeroth-order diffraction mode was prominent because the incident plane wave passed through the GaN/sapphire interface without experiencing TIR. In contrast, for incident light far below a TIR angle, the shift of diffraction order was in reverse due to a reduced Fresnel reflection loss. The high-index contrast hollow grating excited efficiently low-order diffraction modes at normal incidence [26]. Note that the extraction efficiency is primarily governed by the diffraction of light around and above a TIR angle.
Finally, we obtained far-field intensity distributions of 2D grating embedded flip-chip GaN-based LEDs as a function of $\varDelta$n by conducting the integrated ray-wave optics simulation [ Fig. 6(a)]. The lower panels show the same far-field data plotted with different perspectives. The beam divergence was steadily reduced by an increase of $\varDelta$n, which was indicative of the improved top-face extraction efficiency. Technically, the sapphire-filled grating ($\varDelta$n = 0.7) induced maximum intensity lobes at the viewing angle (θv) of approximately 60°, forming a local minima around the vertical direction, whereas the hollow grating ($\varDelta$n = 1.5) exhibited a Gaussian-like distribution without an intensity node.
Fig. 6. (a) Far-field intensity distributions of 2D grating embedded flip-chip GaN-based LEDs as a function of $\varDelta$n, acquired by the integrated ray-wave optics simulation. In the lower panels, the same data are plotted with different perspectives. (b) Measured and simulated far-field intensity distributions of flip-chip GaN-based LEDs with 2D sapphire-filled and hollow gratings as a function of viewing angle (θv). The fabricated sapphire-filled and hollow gratings are shown in Fig. 4(b). For (a) and (b), the structural parameters of the simulated 2D gratings are described in Fig. 5.
Figure 6(b) shows measured and simulated far-field intensity distributions of flip-chip GaN-based LEDs with 2D sapphire-filled and hollow gratings, wherein the fabricated sapphire-filled and hollow gratings were shown in Fig. 4(b). Each value was obtained by averaging far-field intensity over the entire range of azimuthal angles. The measured and simulated data were in good agreement for both gratings, suggesting the accuracy of the integrated ray-optics simulation developed herein.
3. Conclusions
We studied the outcoupling characteristics of 2D grating embedded flip-chip GaN-based LEDs by developing a RCWA-assisting ray-optics simulation. The integrated ray-wave optics simulation showed that the level of extracted light via the top-face escape cone improved with increasing the index contrast of embedded gratings, consistent with measured data. A hollow grating embedded in a GaN medium, which is characterized by the maximum index contrast, selectively excited high-order diffraction modes, thus capable of steering horizontally propagating light into leaky light through the top face of sapphire substrates, which was evident from Fourier analysis. In comparison to a traditional sapphire-filled grating, a hollow grating led to a 3.4-fold enhancement in the ratio of top- to side-face extraction efficiency. Therefore, such a high-index-contrast grating is extremely useful, particularly for developing vertically directed illuminations and micro-LED displays in which substrate-side emission cannot be harnessed. The integrated ray-wave optics simulation developed herein will be extensively utilized to design macroscopic lighting devices containing highly diffractive elements.
Appendix
Wave-Optics Simulation: In Fig. 2, a homebuilt finite-difference time-domain algorithm was used to trace the propagation of generated electromagnetic waves. The spatial resolution was 10 nm for all the axes. In Figs. 4–6, a commercial RCWA method (DiffractMOD, Rsoft) was used. The size of the unit-cell in RCWA simulation was 6, 3$\sqrt 3 $, and 6 µm for x-, y-, and z-axes, respectively. To achieve omnidirectional data, the incident angle was scanned over a full range of solid angles with a step of 5° and 15° along the polar (θ) and azimuthal (ϕ) directions, respectively. In general, the far-field distribution of LEDs with a specific outcoupling structure is altered more dramatically along the θ-direction. Therefore, a finer angle step of 5° was applied to the θ-direction. The order of harmonics was 18 and the spatial resolution of the z-axis was 120 nm.
Ray-Tracing Simulation: In Figs. 4 and 6, the Monte Carlo ray-optics simulation was conducted using a commercial software (Lighttools, Synopsys Inc.). The number of generated rays was 107. The initial propagation direction of the generated rays was randomly chosen to emulate spherical waves. The size of the simulated flip-chip GaN-based LED was 0.5 × 0.5 × 0.1 mm3, wherein the thickness of the GaN medium was 5 µm and the rays were generated at 300 nm away from the bottom mirror. A diffuse, planar surface, which contains the diffraction information of a considered 2D grating, was defined at the GaN/sapphire interface. For all the ray-tracing simulations, a surrounding medium was air (n = 1.0).
Fabrication of Hollow Gratings: Regarding the fabrication of the hollow grating shown in Fig. 4(b), a hexagonal array of cylinders was defined on a sapphire substrate by photolithography using an i-line stepper (Nikon i7). Next, an 80-nm thick amorphous alumina layer was deposited conformally along the surface of the cylinders by atomic layer deposition (ALD, NCD LUCIDA D100). Then, a thermal annealing process was performed in a high-temperature (1100 °C) furnace for 2 h to remove the inner photoresist by oxidization. During this process, the initial amorphous alumina layer was crystallized into single-crystalline α-Al2O3.
Measurement of Angle-Resolved Far-Field Distributions: In Fig. 6(b), 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 high-power (450 W) xenon lamp (6278, Newport) was used as the incident broadband light.
Funding
Samsung (SRFC-IT1701-06).
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