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We experimentally demonstrate a TiO2 double-groove grating coupler with two different groove widths on a SiO2 substrate in the visible region. Tolerance investigations based on Bloch-mode profiles in the grating and coupling strengths between the Bloch modes and diffraction orders reveal that the transmission performance is robust when one of the paired ridges is narrow enough (60 nm and less) considering a typical nanofabrication accuracy. Moreover, the ridge shape affects weakly the transmission performance due to the non-resonance operation of our dielectric device. Such tolerance investigations together with current nanofabrication technology enable us to accomplish a 70% efficiency for coupling the normal incident light into the + 1st order transmission diffraction satisfying the total internal reflection condition at a 640 nm wavelength of operation.
© 2014 Optical Society of America
1. Introduction
Dielectric surface relief gratings offer a variety of functionalities in diffraction phenomena and are among the essential building blocks in optics and photonics. Extensive theoretical works have provided practical designs [1–7] and physical insights into diffraction phenomena [8–10] and, moreover, explored useful device operations [11–17]. Owing to significant achievements in the nanofabrication technology over the past decades, some of the unique features of gratings have been experimentally demonstrated in the infrared and visible regions [18–22]. These steady theoretical and experimental efforts have pushed forward this research area and as a result intriguing optical and thermal properties are emerging [23–26].
A practically important feature is related to the coupling of an incident light into a specific order of diffraction in blazed gratings consisting of asymmetric grooves/ridges per period [27–31]. Blazed gratings for coupling the normal incident light into the + 1st diffraction order with a 25° refraction angle were experimentally demonstrated in the visible region [32,33], which to the best of our knowledge is the largest refraction angle in experimental demonstrations so far. In [32], a binary blazed grating consisted of three ridges in a period of 1.56 wavelengths. A blazed grating in [33] consisted of a one-bar layer and two-bar layer stacked vertically and had nearly the same period. In a theoretical work in [34], we have suggested highly efficient coupling with a 50° refraction angle at normal incidence in a double-groove grating with two different groove widths in a shorter period of 0.9 wavelengths, which satisfies the total internal reflection condition. Therefore, such gratings can work as input/output couplers for dielectric substrates and can find possible applications in several scenarios, for example, glass windows where the coherent light coupled from air propagates with negligibly small loss. In order to achieve such high-efficient coupling experimentally and to have a more practical design rule for mass production, the effect of fabrication tolerance on the transmission performance needs to be clarified with physical insight as well as numerical results.
In this paper, we experimentally demonstrate a double-groove grating coupler in the visible region. We investigate the effect of fabrication tolerance on the transmission efficiency by accounting for the Bloch-mode profiles in the grating and the coupling strengths between the Bloch modes and diffraction orders. Such tolerance investigation together with current nanofabrication technology enables us to accomplish a measured efficiency of 70% for coupling the normal incident light into the + 1st order transmission diffraction with a 50° refraction angle that meets the total internal reflection condition at the air/SiO2 interface. The paper is organized as follows. We present the geometry and experimental result of a double-groove grating in Sections 2 and 3. In Section 4, we present a parametric tolerance study and illustrate how the diffraction efficiency is affected by the variation of the grating dimensions via the behaviors of the Bloch-mode profiles and coupling coefficients. Then, we conclude the paper in Section 5.
2. Geometry of double-groove grating
The numerical model and the scanning electron microscopy (SEM) image of our double-groove grating are shown in Fig. 1(a) and 1(b). The grating has periodic TiO2 ridges and air grooves with two different widths per period on the SiO2 substrate. The ridges need to have higher refractive index than that of the substrate for high efficiency of transmission diffraction [34]. In addition, the ridges and the substrate need to be transparent in the visible region. Therefore, TiO2 and SiO2 have been selected, respectively. The fabrication process of our grating is as follows: TiO2 was deposited on a SiO2 substrate by magnetron sputtering. A positive electron beam resist (ZEON, ZEP-520A) was spin-coated on the TiO2 film, and a liftoff pattern was defined on the resist using an electron beam writer (Elionix, ELS-7000). The liftoff-patterned area was developed in xylene, rinsed in isopropyl alcohol, and blown dry with nitrogen. A 20-nm-thick layer of Ni was sputtered onto the structure, and the electron beam resist was dissolved by using N-methyl-2-pyrrolidone. Ni-unmasked TiO2 was removed by reactive ion etching with CF4 gas. The Ni mask was dissolved by wet chemical etching in diluted hydrochloric acid and then we have obtained a grating sample. The numerical model has rectangular ridges [Fig. 1(a)] (the detailed parameters are shown in the caption of Fig. 1 and we use those parameters for numerical investigations unless otherwise mentioned), however, the test sample has concave ridges due to the side etching [Fig. 1(b)], whose effect on the transmission performance will be discussed later.
Fig. 1 Geometry of a double-groove grating coupler. (a) Numerical model of a unit cell consisting of rectangular ridges. (dimensions; w1 = 60 nm, w2 = 140 nm, d = 180 nm, h = 230 nm, p = 580 nm, refractive indices; nTiO2 = 2.435, nSiO2 = 1.457) (b) Scanning electron microscopy (SEM) image of a device under test.
When the s-polarized light (electric field is normal to the incident plane) is normally impinged on the grating in air (Port 1), the light is efficiently coupled into the + 1st order of transmission diffraction (Port 2) with small coupling to the 0th (Port 3) and −1st (Port 4) orders. A grating period of p = 580 nm = 0.906 λ0 determines the refraction angle of ~50° at the design wavelength of λ0 = 640 nm. Fixing the period, the narrow and wide ridge widths w1 and w2, their center-to-center distance d, and their height h are optimized to maximize the transmission efficiency of the + 1st diffraction order at the refraction angle. We numerically investigate our grating by the rigorous coupled wave analysis (RCWA) [2–7]. In the optimal case with the parameter set of the caption of Fig. 1, the diffraction efficiency for the + 1st order of transmission (Ports 1 to 2) is 88.9% at λ0 = 640 nm. Other diffraction efficiencies are 0.9% for the 0th order transmission (Ports 1 to 3), 4.2% for the −1st order transmission (Ports 1 to 3), and 6% for the 0th order reflection (Ports 1 to 4), respectively. Since the diffraction angle is larger than the critical angle of 43.3° at the air/SiO2 interface, the transmitted light through the grating propagates within the SiO2 substrate in the total internal reflection condition.
3. Experimental setup and result
Due to the convenience in our measurement setup, following the reciprocity theorem the transmission efficiency from Ports 2 to 1 is evaluated and compared with numerical results. We illustrate the operation mechanism and carry out a tolerance study for the normal incidence scenario, Ports 1 to 2, for clear understanding of the operation mechanism. Figure 2 shows the experimental setup for the evaluation of our device. The prism stage has flat planes for each of four ports and allows us to measure transmission efficiencies accurately by reducing unwanted multiple reflections within the prism stage. The grating sample on the prism stage is illuminated by the s-polarized light with an incident angle of 50° from the side of the prism stage (Port 2) and the transmitted light is detected by the microscopic spectrometer in air (Port 1). We estimate an accuracy of the efficiency measurement within ± 2% in our setup by repeating sample evaluations.
Fig. 2 Experimental setup for the measurement of diffraction efficiencies of the grating in Fig. 1(b). Abbreviations stand for L: light source, P: polarizer, M: mirror, PS: prism stage, S: sample, OL: objective lens, A: analyzer, IL: imaging lens, F: fiber, and SM: spectrometer.
The measured transmission efficiency from Ports 2 to 1 as a function of the operating wavelength is shown in Fig. 3 by the symbols. We see that the efficiency exceeds 70% around λ0 = 640 nm. Due to the non-resonance operation, our device has a relatively broadband characteristic. The response from the rectangular ridge model in Fig. 1(a) is obtained by the RCWA and is plotted as the blue solid line in Fig. 3. The numerical result shows the similar shape as the measurement result, i.e. the numerical model for the rectangular-shaped scenario allows one to estimate the response of double-groove gratings reasonably well and to understand the operation mechanism. The quantitative discrepancy between the measured and numerical efficiencies mainly comes from the ridge shape difference between the rectangular ridges in Fig. 1(a) and curved ridges of the test sample in Fig. 1(b). In order to verify this point, we use the curved ridge shape of the test sample of Fig. 1(b) in the numerical model, as shown in the inset of Fig. 3, where the deeper air grooves with the over-etched SiO2 substrate is exactly modeled. The result of this numerical simulation (shown as pink dashed line) becomes closer to the measured results (symbols). The other factors contributing to the efficiency discrepancy may be due to non-perfection of the periodicity of the sample structure since the fitting parameters are retrieved from a SEM image in Fig. 1(b). In addition, this comparison suggests that the efficiency of the test sample can be improved by further optimization of the fabrication condition.
Fig. 3 Transmission efficiency, as a function of wavelength, for the + 1st diffraction order. (symbols: experimental result, blue solid line: numerical result of the rectangular-ridge model in Fig. 1(a), pink dashed line: numerical result of the curved-ridge model in the inset, where the gray and white regions in the ridges represent TiO2 and SiO2 materials, respectively).
4. Tolerance study
Such high efficiency has been explored through the following tolerance study. In the tolerance study, the double-groove grating consisting of rectangular ridges in Fig. 1(a) is illuminated with normal incidence (Port1) and the transmission diffraction efficiency of the + 1st order (Port 2) is calculated at λ0 = 640 nm by using the RCWA as mentioned in Section 3. Considering the accuracy of typical nanofabrication technology, we assume that four parameters of the ridge widths w1, w2, their distance d, and their height h are varied by ± 20 nm in the numerical model around the parameters shown in the caption of Fig. 1(a). Preliminary study has shown that the transmission efficiency possesses a meaningful feature for the design rule with the parameter of the narrow ridge width w1. No specific dependency is seen for the other three parameters, i.e., the diffraction efficiency gradually decreases with varying those parameters. Consequently, the narrow ridge width w1 is varied further from 20 nm to 100 nm for clear observation of the trend and other three parameters are varied within ± 20 nm with a step of 1 nm. The diffraction efficiencies are plotted in Fig. 4. The number of plots in the figure is 81(w1) × 41(w2) × 41(d) × 41(h). We see, within the range of the figure, that the transmission efficiency is robust when the narrow ridge width is 60 nm and less (~40 nm is the optimum width in terms of transmission efficiency). We have set a 60 nm width for the narrow ridge by considering the constraint of the fabrication technology in our design.
Fig. 4 Transmission efficiencies of the + 1st diffraction order in the numerical model in Fig. 1(a). Three parameters, the wide ridge width w2, the ridge distance d, and the ridge height h are varied within ± 20 nm from the dimensions in the caption of Fig. 1 for each plot of the narrow ridge width w1. The number of the plots is 81(w1) × 41(w2) × 41(d) × 41(h).
The dependency of transmission efficiency on the narrow ridge width can be understood from the operation mechanism of our grating. Since the transmission efficiency relates to the behaviors of Bloch-mode profiles and the coupling strengths between the Bloch modes and diffraction orders, these are briefly described as follows. When the grating is illuminated by the s-polarized incident wave in normal incidence in air, three propagating Bloch modes and other evanescent Bloch modes are excited along the z axis in the grating. The dominant three propagating Bloch modes contribute equally to the diffraction phenomenon. The three propagating Bloch modes have almost the same amplitudes of coupling coefficients to the transmission diffraction of the 0th and ± 1st orders but different phase profiles and are added constructively for the + 1st order and destructively for the 0th and −1st orders at the grating end. As a result, the normal incident light is efficiently coupled to the + 1st order diffraction. It should be mentioned that the refractive index for ridges needs to be higher than that of the substrate so that diffraction efficiencies are enhanced or suppressed by the equal contributions of the three Bloch modes [34]. If the ridges and the substrate are made of the same material, e.g. SiO2, one of the three Bloch modes is dominant and the normal incident light in air couples to the 0th order of transmission diffraction. When semiconductor materials are considered, one should carefully select a material set for the ridges and the substrate to meet the refractive index relationship above.
In the RCWA, Bloch-mode profiles and effective refractive indices in the grating are obtained by the eigenvalue equation:
where is the diagonal matrix denoting the x component of the space-harmonic wavevectors with being in the (u,u) element (: x component of the incidence wavevector, u: integers in 1≤ u ≤ 2M + 1), is the (2M + 1) × (2M + 1) Toeplitz matrix of the permittivity harmonics, is the (2M + 1) column vector of the y component of the electric field harmonics, k0 and kz are the free space wavenumber and the z component, respectively. The eigenvectors obtained in Eq. (1) correspond to the Bloch-mode profiles and are denoted by the matrix form . The square root of an eigenvalue represents an effective refractive index ng of the relevant Bloch mode in the grating, where g is the Bloch-mode number. Note that for evanescent waves, the sign of the square root of eigenvalues is determined such that the waves decay in the upward and downward directions for the convergence of calculations.At the grating/substrate interface (z = 0), the transmittance [(2M + 1) column vector]is expressed as:
where and are the (2M + 1) column vectors and represent the coupling coefficients of the upward and downward waves in the grating to the diffraction orders, is the diagonal matrix with the element exp(-ingk0h) for the relevant Bloch mode, g. Considering another boundary condition at the grating/air interface (z = h), we determine the vectors and and then obtain diffraction efficiencies. Details of the calculation process can be found in [3,7].We have led the design guideline that the transmission efficiency for the + 1st order diffraction is robust when the narrow ridge width is narrow enough such as 60 nm and less. Consequently, we verify such design guideline by accounting for the behaviors of Bloch-mode profiles and coupling coefficients with ± 20 nm variations of the three parameters w2, d, and h while the narrow ridge width of w1 = 60 nm is fixed.
Profiles of the three propagating Bloch modes are shown in Fig. 5(a)-5(c). In each figure, the variation step is 20 nm and the number of curves is 3(w2) × 3(d) × 3(h). We see that each of the three Bloch modes has large amplitude in the wide ridge [g = 0, Fig. 5(a)], the narrow ridge [g = 1, Fig. 5(b)], and the wide groove [g = 2, Fig. 5(c)], respectively. Unlike metallic gratings where surface plasmon polaritons are tightly confined at the metal/air boundary, no confinement is seen in the Bloch-mode profiles in Fig. 5(a)-5(c). Consequently, diffraction efficiency in our dielectric grating is less sensitive to the ridge shapes as evident in Fig. 3. The effective refractive indices with the parameters in the caption of Fig. 1 are obtained by using Eq. (1) and are ng = 0 = 2.04, ng = 1 = 1.49, and ng = 2 = 0.73, respectively. We verify the relationship of ng = 0 > ng = 1 > ng = 2, which is matched to the ratio of the dielectric ridges to the air grooves.
Fig. 5 Profiles of three propagating Bloch modes in the double-groove grating of Fig. 1(a). Each Bloch mode has large amplitude in (a) the wide ridge (g = 0), (b) the narrow ridge (g = 1), and (c) the wide groove (g = 2), respectively. The pink curves are obtained using Eq. (1) with the parameters in the caption of Fig. 1. Other curves represent Bloch modes with ± 20 nm variations of the three parameters, the wide ridge width w2, the ridge distance d, and the ridge height h while the narrow ridge width w1 = 60 nm is fixed. The number of curves in each figure is 3(w2) × 3(d) × 3(h). The refractive indices for the pink curves are ng = 0 = 2.04 in (a), ng = 1 = 1.49 in (b), and ng = 2 = 0.73 in (c), respectively.
We normalize each eigenvector in the matrix form in Eq. (2) so that the coupling coefficients and represent how each Bloch mode relatively contributes to each diffraction order. In [9], an analytic Bloch-mode cavity model was developed and the coupling behavior of each Bloch mode to the 0th order of reflection and transmission diffraction was explained. Here, we observe such a coupling behavior of each Bloch mode to transmission diffraction orders in the exact calculation process of the RCWA. Coupling coefficients of the three propagating Bloch modes, [ in Eq. (2)], for the + 1st order of transmission diffraction and their summations are plotted in the complex plane (Fig. 6). The circumference of the circle in the complex plane corresponds to unity transmittance due to the eigenvector normalization. The Bloch modes g = 0 (green squares), g = 1 (blue diamonds), and g = 2 (black crosses) in the complex plane correspond to the Bloch-mode profiles of Figs. 5(a), 5(b), and 5(c), respectively. The summation (black triangles) of the coupling coefficients of the three Bloch modes corresponds to the transmission coefficient from the contributions of the three propagating Bloch modes and the diffraction efficiency is calculated with the amplitude of the plot in the complex plane. We see that the three propagating Bloch modes are added constructively for the + 1st order diffraction. Moreover, we see little changes in amplitudes (r-direction) and some rotations in phases (ϕ-direction) of the coupling coefficients in the complex plane (r,ϕ), resulting in non-serious degradation in transmission efficiency as observed in Fig. 4.
Fig. 6 Coupling coefficients of the three propagating Bloch modes, [ in Eq. (2)], for the + 1st order of transmission diffraction. Coupling coefficients of Bloch modes g = 0 (green squares), g = 1 (blue diamonds), and g = 2 (black crosses) correspond to the Bloch-mode profiles of Figs. 5(a), 5(b), and 5(c), respectively. The summations of the three coupling coefficients are represented by black triangles and correspond to transmission coefficients to the + 1st order of diffraction from their contributions. The pink symbols are obtained with the parameters in the caption of Fig. 1.
It should be mentioned that the sensitivity of diffraction efficiency to dimensions in gratings depends on the operation mechanism. For example, in a resonant double-groove grating in [35], diffraction efficiency is less sensitive to the narrow ridge width than the wide ridge width, since the dominant Bloch mode has large amplitude in the wide ridge in the Bloch-mode profile.
The refraction angle of the + 1st order of transmission diffraction has been set at 50° in our grating throughout the paper. One can set larger refraction angle by shortening the period for different application scenarios and may need to carefully evaluate how the optimum efficiency of transmission may be decreased.
5. Conclusions
We have experimentally achieved a 70% of transmission efficiency in a double-groove grating in the visible region by accounting for the Bloch-mode profiles and the coupling strengths between the Bloch modes and diffraction orders. Moreover, tolerance investigations based on the operation mechanism have revealed how the Bloch modes, coupling strengths, and diffraction efficiencies are behaved under a typical nanofabrication accuracy. Our results therefore point to an important physical understanding and design guideline for grating couplers. Our grating satisfying the total internal reflection condition enables coupling of the coherent light from air into the glass substrate, resulting in several potential scenarios for glass windows and display applications. In addition, our grating design can be applied for improving the efficiency of some solar cells, e.g. dye sensitized solar cells and organic solar cells, by enhancing light paths in the absorption layers. Since our grating needs higher refractive index for the ridges than that of the substrate, one should select appropriate materials for those application scenarios.
References and links
1. P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26(6), 2907–2916 (1982). [CrossRef]
2. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71(7), 811–818 (1981). [CrossRef]
3. 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]
4. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13(4), 779–784 (1996). [CrossRef]
5. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13(9), 1870–1876 (1996). [CrossRef]
6. M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, New York, 2003).
7. H. Kim, J. Park, and B. Lee, Fourier modal method and its applications in computational nanophotonics, (CRC Press, New York, 2012).
8. T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, U. Peschel, A. V. Tishchenko, and O. Parriaux, “An intelligible explanation of highly-efficient diffraction in deep dielectric rectangular transmission gratings,” Opt. Express 13(26), 10448–10456 (2005). [CrossRef] [PubMed]
9. P. Lalanne, J. P. Hugonin, and P. Chavel, “Optical properties of deep lamellar gratings: a coupled Bloch-mode insight,” J. Lightwave Technol. 24(6), 2442–2449 (2006). [CrossRef]
10. V. Karagodsky and C. J. Chang-Hasnain, “Physics of near-wavelength high contrast gratings,” Opt. Express 20(10), 10888–10895 (2012). [CrossRef] [PubMed]
11. R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61(9), 1022–1024 (1992). [CrossRef]
12. R. C. Tyan, P. C. Sun, A. Scherer, and Y. Fainman, “Polarizing beam splitter based on the anisotropic spectral reflectivity characteristic of form-birefringent multilayer gratings,” Opt. Lett. 21(10), 761–763 (1996). [CrossRef] [PubMed]
13. C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE Photon. Technol. Lett. 16(2), 518–520 (2004). [CrossRef]
14. D. Taillaert, P. Bienstman, and R. Baets, “Compact efficient broadband grating coupler for silicon-on-insulator waveguides,” Opt. Lett. 29(23), 2749–2751 (2004). [CrossRef] [PubMed]
15. R. Magnusson and M. Shokooh-Saremi, “Widely tunable guided-mode resonance nanoelectromechanical RGB pixels,” Opt. Express 15(17), 10903–10910 (2007). [CrossRef] [PubMed]
16. V. Karagodsky, B. Pesala, C. Chase, W. Hofmann, F. Koyama, and C. J. Chang-Hasnain, “Monolithically integrated multi-wavelength VCSEL arrays using high-contrast gratings,” Opt. Express 18(2), 694–699 (2010). [CrossRef] [PubMed]
17. H. Iizuka, N. Engheta, H. Fujikawa, K. Sato, and Y. Takeda, “Switching capability of double-sided grating with horizontal shift,” Appl. Phys. Lett. 97(5), 053108 (2010). [CrossRef]
18. C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broad-band mirror (1.12-1.62 μm) using a subwavelength grating,” IEEE Photon. Technol. Lett. 16(7), 1676–1678 (2004). [CrossRef]
19. B. Wang, C. Zhou, S. Wang, and J. Feng, “Polarizing beam splitter of a deep-etched fused-silica grating,” Opt. Lett. 32(10), 1299–1301 (2007). [CrossRef] [PubMed]
20. M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high-index-contrast subwavelength grating,” Nat. Photonics 1(2), 119–122 (2007). [CrossRef]
21. G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J.-M. Fedeli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92(13), 131101 (2008). [CrossRef]
22. D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4(7), 466–470 (2010). [CrossRef]
23. P. Spinelli, M. A. Verschuuren, and A. Polman, “Broadband omnidirectional antireflection coating based on subwavelength surface Mie resonators,” Nat. Commun. 3, 692 (2012). [CrossRef] [PubMed]
24. I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7(9), 7824–7832 (2013). [CrossRef] [PubMed]
25. C. Ribot, M.-S. L. Lee, S. Collin, S. Bansropun, P. Plouhinec, D. Thenot, S. Cassette, B. Loiseaux, and P. Lalanne, “Broadband and efficient diffraction,” Adv. Opt. Mater. 1(7), 489–493 (2013). [CrossRef]
26. K. Ito, T. Matsui, and H. Iizuka, “Thermal emission control by evanescent wave coupling between guided mode of resonant grating and surface phonon polaritons on silicon carbide plate,” Appl. Phys. Lett. 104(5), 051127 (2014). [CrossRef]
27. M. W. Farn, “Binary gratings with increased efficiency,” Appl. Opt. 31(22), 4453–4458 (1992). [CrossRef] [PubMed]
28. D. L. Brundrett, E. N. Glytsis, and T. K. Gaylord, “Normal-incidence guided-mode resonant grating filters: design and experimental demonstration,” Opt. Lett. 23(9), 700–702 (1998). [CrossRef] [PubMed]
29. P. Lalanne, S. Astilean, P. Chavel, E. Cambril, and H. Launois, “Blazed binary subwavelength gratings with efficiencies larger than those of conventional echelette gratings,” Opt. Lett. 23(14), 1081–1083 (1998). [CrossRef] [PubMed]
30. J. Feng and Z. Zhou, “Polarization beam splitter using a binary blazed grating coupler,” Opt. Lett. 32(12), 1662–1664 (2007). [CrossRef] [PubMed]
31. Z. Peng, D. Fattal, A. Faraon, M. Fiorentino, J. Li, and R. G. Beausoleil, “Reflective silicon binary diffraction grating for visible wavelengths,” Opt. Lett. 36(8), 1515–1517 (2011). [CrossRef] [PubMed]
32. M. S. L. Lee, P. Lalanne, J. C. Rodier, and E. Cambril, “Wide-field-angle behavior of blazed-binary gratings in the resonance domain,” Opt. Lett. 25(23), 1690–1692 (2000). [CrossRef] [PubMed]
33. M. Oliva, D. Michaelis, F. Fuchs, A. Tunnermann, and U. D. Zeitner, “Highly efficient broadband blazed grating in resonance domain,” Appl. Phys. Lett. 102(20), 203114 (2013). [CrossRef]
34. H. Iizuka, N. Engheta, H. Fujikawa, K. Sato, and Y. Takeda, “Role of propagating modes in a double-groove grating with a +1st-order diffraction angle larger than the substrate-air critical angle,” Opt. Lett. 35(23), 3973–3975 (2010). [CrossRef] [PubMed]
35. K. Ito and H. Iizuka, “Highly efficient −1st-order reflection in Littrow mounted dielectric double-groove grating,” AIP Advances 3(6), 062119 (2013). [CrossRef]
