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  • Received: Mar. 16, 2020

    Accepted: Apr. 30, 2020

    Posted: Jun. 15, 2020

    Published Online: Jun. 16, 2020

    The Author Email: Jianwen Dong (dongjwen@mail.sysu.edu.cn)

    DOI: 10.3788/COL202018.073601

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    Weiyi Shi, Weimin Deng, Weinan Liu, Zepeng Zhuang, Zhibin Fan, Jianwen Dong. Rectangular dielectric metagrating for high-efficiency diffraction with large-angle deflection[J]. Chinese Optics Letters, 2020, 18(7): 073601

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Rectangular dielectric metagrating for nearly unity anomalous diffraction. (a) A schematic illustration of the dielectric metagrating with a periodicity of Λ, composed of an array of rectangular bars with the width of s and height of h. (b) A phase map of the diffraction efficiency of the T−1 order, by varying the incident angle θi and the normalized wavelength λ/Λ. The metagrating parameters are s/Λ=0.34 and h/Λ=0.604. The three white curves correspond to the −1st, 1st, and −2nd Rayleigh anomaly, respectively. The white star in the center marks the highest T−1 order efficiency of 97.55%.

Fig. 1. Rectangular dielectric metagrating for nearly unity anomalous diffraction. (a) A schematic illustration of the dielectric metagrating with a periodicity of Λ, composed of an array of rectangular bars with the width of s and height of h. (b) A phase map of the diffraction efficiency of the T1 order, by varying the incident angle θi and the normalized wavelength λ/Λ. The metagrating parameters are s/Λ=0.34 and h/Λ=0.604. The three white curves correspond to the 1st, 1st, and 2nd Rayleigh anomaly, respectively. The white star in the center marks the highest T1 order efficiency of 97.55%.

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Mode dispersions of the rectangular dielectric metagrating. (a) A sketch of the mode expansion and coupling mechanism of the metagrating. (b) Wavevector components of the diffraction orders or waveguide-array-modes. Here, γ is the z-orientation wave vector of the incidence/exit ambient (regions I and III). (c) λ−γ diagram of the metagrating at θi=45°. (d) θi−γ diagram at λ/Λ=1.425, showing the appearance of the T−1 order just when θi>25.2°. (e) λ−β diagram at θi=45°. (f) θi−β diagram at λ/Λ=1.425. For (c)–(f), the grating parameters are s/Λ=0.34 and h/Λ=0.604.

Fig. 2. Mode dispersions of the rectangular dielectric metagrating. (a) A sketch of the mode expansion and coupling mechanism of the metagrating. (b) Wavevector components of the diffraction orders or waveguide-array-modes. Here, γ is the z-orientation wave vector of the incidence/exit ambient (regions I and III). (c) λγ diagram of the metagrating at θi=45°. (d) θiγ diagram at λ/Λ=1.425, showing the appearance of the T1 order just when θi>25.2°. (e) λβ diagram at θi=45°. (f) θiβ diagram at λ/Λ=1.425. For (c)–(f), the grating parameters are s/Λ=0.34 and h/Λ=0.604.

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Interferences of the modes and their contributions to the diffraction efficiencies with an incident angle of θi=45°. (a) The diffraction efficiency of the T−1 order calculated by the FEM (the square marks), the waveguide-array-mode expansion method considering enough modes (solid lines), and considering two propagating modes (dashed lines). (b) The amplitudes and phase differences of the two modes versus the normalized wavelength. The superposition of these two modes’ contribution results is shown with the dashed line in (a). (c) and (d): similar to (a) and (b), but for the T0 order. The inset in (c) shows the zoomed-in view of the diagram. The vertical dashed line indicates the normalized wavelength of 1.425.

Fig. 3. Interferences of the modes and their contributions to the diffraction efficiencies with an incident angle of θi=45°. (a) The diffraction efficiency of the T1 order calculated by the FEM (the square marks), the waveguide-array-mode expansion method considering enough modes (solid lines), and considering two propagating modes (dashed lines). (b) The amplitudes and phase differences of the two modes versus the normalized wavelength. The superposition of these two modes’ contribution results is shown with the dashed line in (a). (c) and (d): similar to (a) and (b), but for the T0 order. The inset in (c) shows the zoomed-in view of the diagram. The vertical dashed line indicates the normalized wavelength of 1.425.

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Interferences of the modes and their contributions to the diffraction efficiencies with λ/Λ=1.425. (a) The diffraction efficiencies versus incident angle calculated by the FEM (square marks) and the waveguide-array-mode expansion method considering enough modes (solid lines) and two propagating modes (dashed lines). The inset is the zoomed-in view of the diagram. (b) The amplitudes and phase differences of two modes at different incident angles. The two vertical dashed lines indicate the incident angles of 31° and 64°. (c) and (d): similar to (a) and (b), but for the T0 order.

Fig. 4. Interferences of the modes and their contributions to the diffraction efficiencies with λ/Λ=1.425. (a) The diffraction efficiencies versus incident angle calculated by the FEM (square marks) and the waveguide-array-mode expansion method considering enough modes (solid lines) and two propagating modes (dashed lines). The inset is the zoomed-in view of the diagram. (b) The amplitudes and phase differences of two modes at different incident angles. The two vertical dashed lines indicate the incident angles of 31° and 64°. (c) and (d): similar to (a) and (b), but for the T0 order.

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Analysis of the anomalous refraction by mode decomposition. (a) The simulation result of the Ex field obtained by the FEM in the case of θi=45° and λ/Λ=1.425. (b) The Ex field calculated by the waveguide-array-mode expansion method. (c) and (e) The Ex fields of different diffraction orders in the reflected and transmitted regions, respectively. (d) The waveguide-array-modes inside the grating.

Fig. 5. Analysis of the anomalous refraction by mode decomposition. (a) The simulation result of the Ex field obtained by the FEM in the case of θi=45° and λ/Λ=1.425. (b) The Ex field calculated by the waveguide-array-mode expansion method. (c) and (e) The Ex fields of different diffraction orders in the reflected and transmitted regions, respectively. (d) The waveguide-array-modes inside the grating.

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Influence of material loss, material dispersion, and the substrate on the performance of the metagrating. (a) and (b) The T−1 order’s diffraction efficiency versus (a) wavelength or (b) incident angle when the dielectric bars’ extinction coefficient κ=0 (red), κ=10−3 (green dashed), and κ=10−2 (blue). (c) The T−1 order’s diffraction efficiency versus wavelength when the dielectric bars are made of silicon-based material (green) or no-loss material with n=3 and κ=0 (red). (d) The T−1 order’s diffraction efficiency versus incident angle when the dielectric bars are made of material with n=3 at the wavelength of 640 nm (red) and silicon-based material at the wavelengths of 640 nm (purple) and 653 nm (green dashed). (e) and (f) The T−1 order’s diffraction efficiency versus (e) wavelength or (f) incident angle when the metagrating is freestanding (green) or standing on the SiO2 substrate (orange).

Fig. 6. Influence of material loss, material dispersion, and the substrate on the performance of the metagrating. (a) and (b) The T1 order’s diffraction efficiency versus (a) wavelength or (b) incident angle when the dielectric bars’ extinction coefficient κ=0 (red), κ=103 (green dashed), and κ=102 (blue). (c) The T1 order’s diffraction efficiency versus wavelength when the dielectric bars are made of silicon-based material (green) or no-loss material with n=3 and κ=0 (red). (d) The T1 order’s diffraction efficiency versus incident angle when the dielectric bars are made of material with n=3 at the wavelength of 640 nm (red) and silicon-based material at the wavelengths of 640 nm (purple) and 653 nm (green dashed). (e) and (f) The T1 order’s diffraction efficiency versus (e) wavelength or (f) incident angle when the metagrating is freestanding (green) or standing on the SiO2 substrate (orange).

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