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The effects of the lattice pitch of GaN and polymer square-lattice photonic crystal (2PhC) on the diffracted transmission were studied by using rigorous coupled wave analysis (RCWA). Besides the first-order Bragg diffraction, higher-order diffractions from large pitches of PhC are also significant to the light extraction improvement. Three different diffraction mechanisms are illustrated through wave vector analysis. The enhancement factors of integrated transmission are obtained from a wide range of pitches with micro-scale for both GaN and polymer 2PhC. The experimental angular-resolved transmission on a transparent polymer sample of 2 μm pitch 2PhC is comparable to the simulation.
©2009 Optical Society of America
1. Introduction
Today light emitting diodes (LEDs), especially GaN-based LEDs, are already widely used. One of the barriers to their applications is the strong limitation of the majority of emitted light trapped in a conventional LED by the total internal reflection (TIR) at the interface of semiconductor and air. Various approaches have been adopted to extract the trapped light in LEDs. A great number of efforts have been made on using the photonic crystal to improve the light extraction or control the radiation pattern of GaN-based LEDs with various types, such as conventional LEDs with GaN on sapphire substrate [1–11], thin-film LEDs [4,12–14], flip-chip LEDs [15] and micro-cavity LEDs [16]. The shallow etched photonic crystal (PhC) gratings are employed on the surface of different materials, such as GaN [1–9,12–14,16,17], indium tin oxide (ITO) [10,11], or thin polymer layer [15,18]. Though the coupling between the photonic crystal and guided modes is different due to the different types of LEDs and different materials of PhC, the strategy of using a PhC grating in most publications mainly relies on Bragg diffraction of the trapped guided modes from the LEDs. However, the optimization of PhC is challenging since so many parameters of PhC are involved and correlated with each other, such as lattice pitch, filling factor and etch depth. We notice a large variation in PhC lattice pitch employed by different groups in GaN-based blue LEDs corresponding to the Bragg diffraction conditions of the first order [4,5], second order [4] and even higher orders [6–9,14,16,17], respectively. In particular, David et al. [5] thought the highest extraction efficiency achieved around the first Bragg diffraction. Their works are focused on GaN photonic crystals with about 215 nm pitch. While Orita et al. [8] and Bergenek et al. [16] found that higher order diffraction was significant to the enhancement in efficiency. The lattice constant of their PhCs was in submicron or micron scale. Therefore, to figure out the diffraction effects of photonic crystals with different pitches on GaN-based LEDs should be an interesting topic. In this paper, we focus on the lattice constant effects on the diffraction of photonic crystal employed on the surface of conventional LEDs. It might be helpful to understand the mechanism of light extraction enhancement from the surface PhC on LEDs and provide design rules for device applications.
Recently, rigorous coupled wave analysis (RCWA) has been developed to accurately analyze the transmitted and reflected diffraction efficiency of a grating on planar structures [19,20]. Here we use two-dimensional (2D) RCWA to evaluate the light transmission improvement in the GaN-based LED due to the diffractive effects of different pitches on two-dimensional GaN and polymer square-lattice photonic crystal (2D-2PhC). Experimentally, we propose a convenient approach to investigate the diffraction effects of photonic crystals on a transparent hemi-cylindrical polymer sample. The transmission improvement by a PhC with micro-scale pitch is demonstrated and compared to the RCWA calculation.
2. Simulation and discussion
We use a simplified model of a planar bulk structure in the air with a 2D shallow surface-relief pillars photonic crystal composed of square lattice (2PhC) on GaN or polymer layer having a refractive index n of 2.5 and 1.45, respectively. An electromagnetic plane-wave light source with a wavelength of λo = 460 nm with TE polarization launches from the GaN or polymer layer at arbitrary incident angles of θ. The diffracted transmission and reflection efficiencies are obtained by solving the Maxwell’s equations matched the electromagnetic boundary conditions. In our RCWA calculation, we keep the height and fill factor of the 2PhC pillars at 200 nm and 0.5, and obtain the dependence of transmission on incident angle for a wide range of lattice pitch a. Considering the vacuum k-vector length of the light is ko = 2π/λo, then the in-plane k-vector is k // = (2π/λo)nsinθ = n eff ko [12], where n eff is the effective refractive index of the mode [1,2], so each in-plane vector of k // corresponding to a mode with an incident angle θ. Thus, the dependence of transmission on in-plane vector of k // is obtained.
To diffract the guided modes into extracted modes by the PhC, the Bragg diffraction condition is , where G is the reciprocal lattice of PhC. For the GaN 2PhC, the critical pitch to allow diffraction is a crit = λ/(1 + n eff) = 131 nm, where the basic reciprocal lattice of 2PhC structure is Go = 2π/a. Figures 1(a) , 1(b), 1(c) and 1(d) present the typical transmission profiles of each diffraction order together with total transmission of GaN 2PhC (with height of 200 nm and fill factor of 0.5) as the function of k // for the 2PhCs with lattice constant a 1 = 100 nm, a 2 = 230 nm (~λ/2), a 3 = 460 nm (~λ), and a 4 = 920 nm (~2λ), respectively. The curves labeled by T[m,n] represent the sum of transmissions with the diffraction order of , while the curve of “2PhC total” is the sum of the whole diffracted transmission. The curve T[0,0] corresponds to the mode of direct transmission from GaN to air. The transmission of a planar GaN model is also plotted in each figure as a reference.
Fig. 1 The transmission of total and each order diffraction as function of k // for the GaN 2PhCs with different pitches for the fill factor of 0.5 (a) a 1 = 100 nm; (b) a 2 = 230 nm; (c) a 3 = 460 nm; (d) a 4 = 920 nm.
Figure 1(a) gives the transmission for GaN 2PhC of lattice pitch a 1<a crit. No chance of diffraction could happen. The total transmission is just equal to T[0,0] which is higher than the planar case. We attribute this improvement of transmission to the average refractive index of photonic crystal, which works as a low refractive index film. Figures 1(b), 1(c) and 1(d) show the transmission for lattice pitches of 2PhC all above the critical diffraction condition. The transmissions due to individual contribution of the diffraction T[m,n] from different orders of are plotted simultaneously. Apparently, other than T[0,0], the first Bragg diffracted transmission T [1,0] usually dominates over the others. The contribution of diffraction from different orders to the transmission is growing up along with increasing of the lattice pitch. The larger pitches, the more higher-order diffractions occur. Here, we want to point out that at small pitch around first Bragg diffraction, only, [1,0] order diffraction contributed to light transmission, even at the pitch a little larger than the first Bragg diffraction, the [1,1] order happens but it’s contribution to the transmission can be ignored. This result is consistent to the reference [5]. But at the larger pitches, much higher order diffractions appear and their contribution to transmission is getting significant. Therefore, the contributions of all the higher diffracted orders are beneficial to the transmission efficiency for the PhC with larger lattice pitches. In the region of k //>ko for all cases, the total light transmission efficiency is enhanced by the 2PhC diffraction. In the case of larger lattice pitches as shown in Figs. 1(c) and 1(d), the transmission in the whole region of k // is obtained. It means that all modes are leaky no matter what their incident angle of propagation is. Therefore, all of the emitted light is prone to light extraction from the LED with such 2PhC structure. This is consistent to reference [12]. The total light transmission in the region of k //<ko reduces comparing with the planar case. Actually, the result of total transmission involves three kinds of diffractions. First of all, some of the original trapped guided modes with k //>ko become Bloch modes with, as schematically shown in Fig. 2(a) . Secondly, some of the original extracted modes are coupled to the reciprocal lattice and turn back as trapped guided modes with as shown in Fig. 2(b). Thirdly, the original extracted modes are coupled with the reciprocal lattice and still extracted with the changed direction, as shown in Fig. 2(c). The improvement of light extraction by PhC is usually achieved by the competition among the three diffraction mechanisms. For example, the total light diffracted transmission at the incident angle θ = 10° in Fig. 1(c) is 22.65% including 4.4% of T[0,0] and 17.9% of T [1,0]. The decrease of T[0,0] compared with the planar case is due to the process of Fig. 2(b), while the appearance of T [1,0] is due to the process of Fig. 2(c). In contrast to no light transmission in planar case at the incident angle θ = 40°, 23.78% total light diffracted transmission can be obtained through the process of Fig. 2(a). The relevant calculations of the polymer 2PhCs are not shown here but have similar trend to the GaN cases.
Fig. 2 Schematic drawing of different mechanisms for light extraction from LED by photonic crystals for original mode k //>ko becomes (a) ; and original k //<ko becomes (b) and (c) .
We also calculate the integrated transmission over the whole incident angle θ and azimuth angle ϕ for lattice pitch spanned in a quite large range from 100 nm to 2000 nm for both GaN and polymer 2PhCs. We assume the plane wave is continuously impinging on the interface with equal intensity and To equals to 1 for the planar case. The curve with square dots in Fig. 3 shows the calculated enhancement factor T/To of the TE polarization integrated transmission efficiency versus lattice pitch for GaN. The integrated transmission improvement by the 2PhC is obtained in the whole considered range of the lattice pitch. For the pitch a<a crit, the light integrated transmission is enhanced by the low average refractive index of 2PhC, although no diffraction occurs. We indeed get transmission enhancement at the pitch around the first Bragg diffraction. Compared with 2PhC with larger pitch even around micro-scale, we get much higher enhancement nearly 2 times enhancement at 1.5 µm, and we get a large range of high enhancement around micro-scale lattice pitches. It confirms that the photonic crystals of larger pitches could lead to higher light transmission efficiency by producing many higher-order diffractions. Consequently, the lattice pitch could be chosen in a wide range in the micro-scale to improve the light extraction of GaN-based LEDs. The curve with circle dots in Fig. 3 shows the similar trend of calculated transmission enhancement factor of polymer 2PhC. The lower enhancement factor of polymer 2PhC than that of GaN 2PhC is due to the lower refractive index difference between the polymer and air.
Fig. 3 Calculated enhancement factors of the TE polarization integrated transmission at various lattice pitches of 2PhC GaN (square) and polymer (circle), respectively.
3. Experimental
To verify the RCWA simulations, we proposed a convenient approach of inspecting a 2D-2PhC array with larger pitch on a transparent hemi-cylindrical polymer sample. We could then directly observe the diffraction transmission and spots pattern dependency on the incidence of the light. By using electron-beam lithography and soft nanoimprint [15], the 2PhC pillar arrays, having lattice pitch of 2 μm and an area of 1 mm × 1 mm, were formed on the flat center of a specially designed transparent polydimethylsiloxane (PDMS) hemi-cylinder. The critical angle of PDMS is θ crit = 43.6°, determined by the refractive index of 1.45.
Figure 4(a) shows the schematic diagram of the experimental setup. The PDMS sample with 2PhC is settled at the center of the rotation stage. A laser beam with variable incident angle θ passes through the circular-edge of the PDMS to the centre of the 2PhC. A silicon detector or a screen put behind the 2PhC can acquire the total light transmission or diffraction pattern through 2PhC/PDMS, respectively. Figure 4(b) shows a photograph of the diffraction pattern corresponding to 2PhC clearly appears on the screen with an incident beam of 532 nm green laser. Figures 4(c) and 4(d) exhibit contrasting pictures of the planar PDMS and 2PhC/PDMS samples impinged by the green laser at the same incident angle 50° (>θ crit), respectively. By using a transparent hemi-cylindrical sample, it directly illustrates the formation of numerous transmitted diffraction rays on the 2PhC/PDMS sample and the happenings of the total internal reflection (TIR) on the planar PDMS sample. Moreover, we also measured the dependence of total transmission on incident angles of the green laser beam for 2PhC/PDMS and planar PDMS samples, respectively. The comparison of transmission efficiency between measurements and calculations on both PDMS samples are shown in Fig. 5 . Clearly, higher transmission efficiency is obtained in the large range above the critical angle for the 2PhC/PDMS. The transmission efficiency enhancement for 2PhC/PDMS due to the diffraction effects is intuitively demonstrated. The measured transmission in planar PDMS is consistent with the RCWA calculation. The measured transmission in 2PhC/PDMS exhibits a similar trend of enhanced light extraction with calculation. In addition, the RCWA calculated transmission of 2PhC is found to be underestimated, when compared with the experimental one. This may be due to the shape of the PDMS 2PhC pillars not being exact rectangles, and the RCWA only counting in one-time diffraction in our simplified model.
Fig. 4 Angular resolved transmission measurements on PDMS hemi-cylinders by incidence of 532 nm laser beam. (a) Schematic diagram of the experimental setup; (b) the transmitted diffraction spots pattern from 2PhC/PDMS; the incident angle is at 50° (>θ crit) for (c) planar PDMS and (d) 2PhC/PDMS.
Fig. 5 The experimental (hollow) and simulation (solid) total transmission efficiency curves for planar PDMS and 2 μm pitch pillar 2PhC/PDMS.
As we know, a conventional GaN-based LED is an asymmetric waveguide. Several dozens of guided modes exist and can be approximately treated as quasi-continuum with equal intensity. It clarifies that the higher light extraction of GaN-based LED could be realized by PhCs with a wide range of pitch even if in micron-scale [7,8,16]. Thus, to a certain extent, Fig. 3 could provide a design guideline of lattice pitch of 2PhC in GaN-based LEDs. But strictly speaking, the guided modes in a LED are complicated. So we should carefully consider the energy distributions of each guided mode and their coupling efficiency of the particular PhC with the detail multilayer structures of LED waveguide.
4. Conclusion
The effects of the lattice pitch of GaN and polymer square-lattice photonic crystals on the diffracted transmission for light extraction were studied by using the RCWA method on a simplified model. The calculations show up the contribution of all possible orders diffractions along with the increasing of the lattice constants. It reveals that not only the low order, but also the higher-order diffractions play an important role for light extraction improvement. Though the first order diffraction can only be considered is correct at small lattice pitch, the contribution of higher orders diffraction are also significant to light transmission improvement for larger pitches with micro-scale. It also exposes that the improvement of light extraction by PhC usually is achieved by the competition among the different diffraction mechanisms. Our diffracted transmission experiment on the transparent 2PhC/PDMS sample is convincingly demonstrated the diffraction mechanism of photonic crystals. The observation of the angular-resolved transmission efficiency for a 2PhC/PDMS sample having a micron-scale pitch is comparable to the RCWA simulation. As a result, RCWA is helpful in study of the improvement of light extraction with the advantages of simplicity and intuitional.
Acknowledgements
The authors gratefully acknowledge GuangFu Luo for programming assistance. This work is supported by the National Natural Science Foundation of China Project (60776041, 60577146 and U0834001) and National Key Basic R&D Plan Project (973) of China (Grant No. 2007CB307004).
References and links
1. C. F. Lai, J. Y. Chi, H. C. Kuo, C. H. Chao, H. T. Hsueh, J. F. T. Wang, and W. Y. Yeh, “Anisotropy of light extraction from GaN two-dimensional photonic crystals,” Opt. Express 16(10), 7285–7294 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-10-7285. [CrossRef] [PubMed]
2. A. David, H. Benisty, and C. Weisbuch, “Optimization of Light-Diffracting Photonic-Crystals for High Extraction Efficiency LEDs,” J. Disp. Tech. 3(2), 133–148 (2007). [CrossRef]
3. Ch. Wiesmann, K. Bergenek, N. Linder, and U. T. Schwarz, “Photonic crystal LEDs-designing light extraction,” Laser Photonics Rev. 1, 1–25 (2008).
4. J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Graford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84(19), 3885–3887 (2004). [CrossRef]
5. A. David, T. Fujii, R. Sharama, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett . 88, 061124–1-061124–3 (2006). [CrossRef] [CrossRef]
6. Z. S. Zhang, B. Zhang, J. Xu, K. Xu, Z. J. Yang, Z. X. Qin, T. J. Yu, and D. P. Yu, “Effects of symmetry of GaN-based two-dimensional photonic crystal with quasicrystal lattices on enhancement of surface light extraction,” Appl. Phys. Lett . 88, 171103–1-171103–3 (2006). [CrossRef]
7. T. N. Oder, K. H. Kim, J. Y. Lin, and H. X. Jiang, “III-nitride blue and ultraviolet photonic crystal light emitting diodes,” Appl. Phys. Lett. 84(4), 466–468 (2004). [CrossRef]
8. K. Orita, S. Tamura, T. Takizawa, T. Yeda, M. Yuri, S. Takigawa, and D. Ueda, “High-Extraction-Efficiency Blue Light-Emitting Diode Using Extended-Pitch Photonic Crystal,” Jpn. J. Appl. Phys. 43(No. 8B), 5809–5813 (2004). [CrossRef]
9. K. Kim, J. Choi, J. B. Park, S. C. Jeon, J. S. Kim, and H. M. Lee, “Lattice Constant Effect of Photonic Crystals on the Light Output of Blue Light-Emitting Diodes,” IEEE Photon. Technol. Lett. 20(17), 1455–1457 (2008). [CrossRef]
10. R. H. Horng, S. H. Huang, C. C. Yang, and D. S. Wuu, “Efficiency improvement of GaN-based LEDs with ITO texturing window layers using natural lithography,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1196–1201 (2006). [CrossRef]
11. S. J. Chang, C. F. Shen, W. S. Chen, C. T. Kuo, T. K. Ko, and J. K. Sheu, “Nitride-based light emitting diodes with indium tin oxide electrode patterned by imprint lithography,” Appl. Phys. Lett . 91, 013504, 013504–1-013504–3 (2007). [CrossRef] [CrossRef]
12. J. J. Wierer, A. David, and M. M. Megens, “III-nitride photonic-crystal light-emitting diodes with high extraction efficiency,” Nat. Photonics 3(3), 163–169 (2009). [CrossRef]
13. Ch. Wiesmann, K. Bergenek, R. Houdre, R. P. Stanley, N. Linder, and U. T. Schwarz, “Theoretical Investigation of the Radiation Pattern From LEDs Incorporating Shallow Photonic Crystals,” IEEE J. Sel. Top. Quantum Electron. 45(10), 1273–1283 (2009). [CrossRef]
14. C. F. Lai, C. H. Chao, H. C. Kuo, H. H. Yen, C. E. Lee, and W. Y. Yeh, “Directional light extraction enhancement from GaN-based film-transferred photonic crystal light-emitting diodes,” Appl. Phys. Lett . 94, 123106–1-123106–3 (2009). [CrossRef]
15. K. Bao, X. N. Kang, B. Zhang, T. Dai, C. Xiong, H. Ji, G. Y. Zhang, and Y. Chen, “Improvement of Light Extraction From Patterned Polymer Encapsulated GaN-based Flip-Chip Light Emitting Diodes by Imprinting,” IEEE Photon. Technol. Lett. 19(22), 1840–1842 (2007). [CrossRef]
16. K. Bergenek, Ch. Wiesmann, H. Zull, C. Rumbolz, R. Wirth, N. Linder, K. Streubel, and T. F. Krauss, “Beam-shaping properties of InGaN thin-film micro-cavity light-emitting diodes with photonic crystals,” Proc. of SPIE. 7231, 72310C–1-72310C–11 (2009).
17. T. Dai, B. Zhang, Z. S. Zhang, D. Liu, X. Wang, K. Bao, X. N. Kang, J. Xu, D. P. Yu, and X. Zhu, “Surface Light Extraction Mapping from Two-Dimensional Array of 12-Fold Photonic Quasicrystal on Current Injected GaN-Based LEDs,” Chin. Phys. Lett. 24, 979–982 (2009).
18. Y. K. Ee, P. Kumnorkaew, H. Tong, and A. Arif, “Enhancement of light extraction of InGaN quantum wells light-emitting diodes with polydimethylsiloxane concave microstructures,” Proc. of SPIE 7231, 72310U–1-72310U–7 (2009).
19. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12(5), 1068–1076 (1995). [CrossRef]
20. H. Kikuta, S. Hino, A. Maruyama, and A. Mizutani, “Estimation method for the light extraction efficiency of light-emitting elements with a rigorous grating diffraction theory,” J. Opt. Soc. Am. A 23(5), 1207–1213 (2006). [CrossRef]
