Wide-angled off-axis achromatic metasurfaces for visible light

We present an approach to build multiwavelength achromatic metasurface that can work in off-axis configuration with an ultra-wide applicable incident angle range for visible light. The metasurface is constructed by combining multiple metallic nano-groove gratings, which support enhanced diffractions for transverse magnetic polarization in an ultrawide incident angle range from 10° to 80° due to the excitations of localized gap plasmon modes at different resonance wavelengths. To achieve the achromatic diffraction, the ratio between the resonance wavelength and the period of each elementary grating is fixed. Incident light at those multiple resonance wavelengths can be efficiently diffracted into the same direction with near-complete suppression of the specular reflection. Based on the similar approach, we also design a wide-angled off-axis achromatic flat lens for focusing light of different wavelengths into the same position. Our findings provide an alternative simple way to design various off-axis achromatic flat optical elements without stringent angle requirement for imaging and display applications. © 2016 Optical Society of America OCIS codes: (160.3918) Metamaterials; (250.5403) Plasmonics; (050.1950) Diffraction gratings; (090.1760) Computer holography. References and links 1. B. C. Kress and P. Meyrueis, Applied Digital Optics (Wiley, 2009). 2. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011). 3. F. Aieta, P. Genevet, N. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, “Out-of-plane reflection and refraction of light by anisotropic optical antenna metasurfaces with phase discontinuities,” Nano Lett. 12(3), 1702–1706 (2012). 4. S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012). 5. L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett. 12(11), 5750–5755 (2012). 6. D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, “Dielectric gradient metasurface optical elements,” Science 345(6194), 298–302 (2014). 7. L. Zhang, J. Hao, M. Qiu, S. Zouhdi, J. K. W. Yang, and C.-W. Qiu, “Anomalous behavior of nearly-entire visible band manipulated with degenerated image dipole array,” Nanoscale 6(21), 12303–12309 (2014). 8. X. Su, C. Ouyang, N. Xu, W. Cao, X. Wei, G. Song, J. Gu, Z. Tian, J. F. O’Hara, J. Han, and W. Zhang, “Active metasurface terahertz deflector with phase discontinuities,” Opt. Express 23(21), 27152–27158 (2015). 9. F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12(9), 4932–4936 (2012). 10. M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging,” Science 352(6290), 1190– 1194 (2016). 11. X. Chen, L. Huang, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, C.-W. Qiu, S. Zhang, and T. Zentgraf, “Dualpolarity plasmonic metalens for visible light,” Nat. Commun. 3, 1198 (2012). 12. X. Li, S. Xiao, B. Cai, Q. He, T. J. Cui, and L. Zhou, “Flat metasurfaces to focus electromagnetic waves in reflection geometry,” Opt. Lett. 37(23), 4940–4942 (2012). Vol. 24, No. 20 | 3 Oct 2016 | OPTICS EXPRESS 23118 #273383 http://dx.doi.org/10.1364/OE.24.023118 Journal © 2016 Received 10 Aug 2016; revised 1 Sep 2016; accepted 3 Sep 2016; published 26 Sep 2016 13. L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013). 14. X. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nat. Commun. 4, 3807 (2013). 15. G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10(4), 308–312 (2015). 16. Y.-H. Chen, L. Huang, L. Gan, and Z.-Y. Li, “Wavefront shaping of infrared light through a subwavelength hole,” Light Sci. Appl. 1(8), e26 (2012). 17. K. E. Chong, L. Wang, I. Staude, A. R. James, J. Dominguez, S. Liu, G. S. Subramania, M. Decker, D. N. Neshev, I. Brener, and Y. S. Kivshar, “Efficient polarization-insensitive complex wavefront control using huygens’ metasurfaces based on dielectric resonant meta-atoms,” ACS Photonics 3(4), 514–519 (2016). 18. D. Wen, F. Yue, G. Li, G. Zheng, K. Chan, S. Chen, M. Chen, K. F. Li, P. W. H. Wong, K. W. Cheah, E. Y. Pun, S. Zhang, and X. Chen, “Helicity multiplexed broadband metasurface holograms,” Nat. Commun. 6, 8241 (2015). 19. A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol. 10(11), 937–943 (2015). 20. D. Wen, S. Chen, F. Yue, K. Chan, M. Chen, M. Ardron, K. F. Li, P. W. H. Wong, K. W. Cheah, E. Y. B. Pun, G. Li, S. Zhang, and X. Chen, “Metasurface device with helicity-dependent functionality,” Adv. Opt. Mater. 4(2), 321–327 (2016). 21. I. Dolev, I. Epstein, and A. Arie, “Surface-plasmon holographic beam shaping,” Phys. Rev. Lett. 109(20), 203903 (2012). 22. P. Genevet, J. Lin, M. A. Kats, and F. Capasso, “Holographic detection of the orbital angular momentum of light with plasmonic photodiodes,” Nat. Commun. 3, 1278 (2012). 23. J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13(9), 4269–4274 (2013). 24. Z.-L. Deng, S. Zhang, and G. P. Wang, “A facile grating approach towards broadband, wide-angle and highefficiency holographic metasurfaces,” Nanoscale 8(3), 1588–1594 (2016). 25. B. G. Cai, Y. B. Li, W. X. Jiang, Q. Cheng, and T. J. Cui, “Generation of spatial Bessel beams using holographic metasurface,” Opt. Express 23(6), 7593–7601 (2015). 26. P. Genevet and F. Capasso, “Holographic optical metasurfaces: a review of current progress,” Rep. Prog. Phys. 78(2), 024401 (2015). 27. J. Cheng, D. Ansari-Oghol-Beig, and H. Mosallaei, “Wave manipulation with designer dielectric metasurfaces,” Opt. Lett. 39(21), 6285–6288 (2014). 28. M. M. Salary, M. Nazari, and H. Mosallaei, “Robust technique for computation of scattering and absorption of light by array of nanowires on layered substrate,” J. Opt. Soc. Am. B 32(12), 2448–2461 (2015). 29. T. Yatooshi, A. Ishikawa, and K. Tsuruta, “Terahertz wavefront control by tunable metasurface made of graphene ribbons,” Appl. Phys. Lett. 107(5), 053105 (2015). 30. T. Li, S. Wang, X.-L. Zhang, Z.-L. Deng, Z. H. Hang, H.-B. Sun, and G. P. Wang, “Non-uniform annular ringsbased metasurfaces for high-efficient and polarization-independent focusing,” Appl. Phys. Lett. 107(25), 251107 (2015). 31. F. Zhou, W. Cao, B. Dong, T. Reissman, W. Zhang, and C. Sun, “Additive manufacturing of a 3D terahertz gradient-refractive index lens,” Adv. Opt. Mater. 4(7), 1034–1040 (2016). 32. F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347(6228), 1342–1345 (2015). 33. M. Khorasaninejad, F. Aieta, P. Kanhaiya, M. A. Kats, P. Genevet, D. Rousso, and F. Capasso, “Achromatic Metasurface Lens at Telecommunication Wavelengths,” Nano Lett. 15(8), 5358–5362 (2015). 34. Y. Li, X. Li, M. Pu, Z. Zhao, X. Ma, Y. Wang, and X. Luo, “Achromatic flat optical components via compensation between structure and material dispersions,” Sci. Rep. 6, 19885 (2016). 35. Z. Zhao, M. Pu, H. Gao, J. Jin, X. Li, X. Ma, Y. Wang, P. Gao, and X. Luo, “Multispectral optical metasurfaces enabled by achromatic phase transition,” Sci. Rep. 5, 15781 (2015). 36. J. Cheng and H. Mosallaei, “Truly achromatic optical metasurfaces: a filter circuit theory-based design,” J. Opt. Soc. Am. B 32(10), 2115–2121 (2015). 37. Y.-K. R. Wu, A. E. Hollowell, C. Zhang, and L. J. Guo, “Angle-insensitive structural colours based on metallic nanocavities and coloured pixels beyond the diffraction limit,” Sci. Rep. 3, 1194 (2013). 38. D. L. Sounas and A. Alù, “Color separation through spectrally-selective optical funneling,” ACS Photonics 3(4), 620–626 (2016). 39. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). 40. W.-H. Lee, “Binary computer-generated holograms,” Appl. Opt. 18(21), 3661–3669 (1979). 41. G. P. Wang, T. Sugiura, and S. Kawata, “Holography with surface-plasmon-coupled waveguide modes,” Appl. Opt. 40(22), 3649–3653 (2001). Vol. 24, No. 20 | 3 Oct 2016 | OPTICS EXPRESS 23119


Introduction
Optical elements without stringent incident angle requirement are highly desired in practical imaging and display systems, such as the full-angle photographing and panoramic viewing system. There are several conventional ways for realizing wide-angled achromatic imaging systems, such as introduction of achromatic doublet or achromatic triplet, and multiple order diffractive lenses [1]. Those methods, however, require cascaded multiple thick components and are therefore bulky and cumbersome.
In this paper, we present an alternative facile approach to design multiwavelength achromatic metasurfaces that work for off-axis configuration with an ultrawide incident angle range [ Fig. 1(b)]. The metasurface is a combination of multiple sets of metallic nano-groove gratings, each of which has a unique period and groove height to work for a particular wavelength [ Fig. 1(c)]. It has been shown previously that the wide nano-groove metasurface can deflect light with very high diffraction efficiency and near-completely suppress the unwanted specular reflection in a broadband, wide-angle range [24]. Although the previous proposed metasurface works for broadband, the chromatic behavior still exists due to the dispersion phenomenon of the grating. In order to design achromatic metasurface with wideangle performance, here, we choose multiple narrow nano-groove gratings with fixed ratio between resonance wavelength and grating period as elementary gratings, and then combine them to achieve achromatic diffraction and focusing. Each narrow groove grating has enhanced diffraction efficiency in a narrow band due to excitation of localized plasmon mode in the grooves. As the localized plasmon mode is independent of incident angle of the illuminating light, the enhanced diffraction efficiency exists for a very wide angular range. Besides the nano-groove structure, the concept developed here can be extend to other localized resonator arrays such as localized dielectric resonators, graphene ribbons and so on. Note that, the localized resonances are also recently used to wide-angled color display devices [37,38]. The narrow band of each elementary grating also decrease the crosstalk between grooves of different depths when multiple gratings are combined. It is demonstrated that, the combined nano-groove metasurface can diffract the incident light to the same direction for different wavelengths, even if the incident angle is varied in an ultrawide range. By modulating the elementary grating with a quadratic phase profile, we also obtain a wideangled achromatic focusing metasurface, which can focus the incident light of different wavelengths into the same position. (c) Schematic of formation of the ultrawideangled achromatic metasurface by combining multiple metallic nano-groove gratings with different periods (p 0 , p 1 , and p 2 ) and groove heights (h 0 , h 1 , and h 2 ). The elementary gratings support the near-total diffraction in the −1st order at different wavelengths (λ 0 , λ 1, and λ 2 ) due to the excitation of the localized gap plasmon mode in the nano-grooves with different heights (h 0 , h 1 , and h 2 ).

The diffraction properties of the single groove grating
For a metallic nano-groove grating, the relation between the reflection angle of the −1st diffraction order θ r and incident angle θ 0 is determined by, where, λ and p are the wavelength of incident light and period of the grating, respectively. From Eq. (1) we see that, the reflection angle is dependent on the wavelength of light, which is the well-known chromatic dispersion existing in a common grating. If λ/p is fixed, however, the reflection angle will be fixed for a given incident angle θ 0 . Therefore, in order to obtain the same diffraction angle for multiple wavelengths (λ 0 , λ 1 , and λ 2 ), one can choose multiple gratings (g 0 , g 1 , and g 2 ) with fixed wavelength/period ratio (λ 0 /p 0 = λ 1 /p 1 = λ 2 /p 2 ). To ensure the enhanced diffraction in the −1st order, the groove heights (h 0 , h 1 , and h 2 ) of the multiple gratings should be suitably chosen so that the lights with wavelengths λ 0 , λ 1 , and λ 2 just excite the corresponding cavity modes in the nano-grooves, respectively. The cavity mode in the metallic groove originates from the Fabry-Perot resonance of the fundamental waveguide mode in a truncated metal-insulator-metal (MIM) structure. In visible frequency range, the fundamental waveguide mode in MIM can be seen as the gap plasmon mode whose field is localized in the gap and metallic surface area, especially when the gap width is very narrow (<200nm). By combining those elementary gratings together, we could hence expect to obtain an achromatic metasurface as schematically illustrated in Fig. 1(c). To determine the groove height of each elementary grating, we calculated the dependence of the −1st diffraction efficiency (denoted as R -1 ) of the grating illuminated by transverse magnetic (TM) polarized light at incident angle 45° upon groove height h and incident wavelength λ [ Fig. 2(a)] by finite element method (FEM), which is implemented by a commercial software COMSOL. In the calculation, silver is chosen as the metal material, with permittivity obtained by fitting the experimental data [39] to the Drude model, and the metallic groove width is fixed as w = 10nm. The ratio between the incident wavelength and the grating period, are fixed in the calculations at λ/p = 1.1. Note that, the value of λ/p should be chosen in the range 2/3<λ/p<2 so that the localized plasmon mode resonance locates within the area bounded by the −2nd, −1st and 1st Rayleigh Anomalies. Actually, the applicable angle range is the largest at λ/p = 1 [24]. We choose λ/p = 1.1, which is a little larger than 1 because the resonance peak for λ/p = 1.1 not only extends a large applicable angle range, but also exhibits the most flat profile. Figure 2(a) shows that the spectral peak wavelength of R -1 is linear to the groove height of the grating. When the groove height is varied from 10nm to 50nm, the wavelength of peak diffraction efficiency spans the entire visible wavelengths ranging from 400nm to 700nm. Hence we can select three sets of gratings -g 0 : p 0 = 400nm, h 0 = 15nm; g 1 : p 1 = 500, h 1 = 29nm; and g 2 : p 2 = 600nm, h 2 = 42nm, corresponding to wavelengths in the blue (λ = 440nm), green (550nm), and red (660nm) color regions as indicated by the blue circle, green square and red triangle, respectively, in Fig.  2(a). The diffraction efficiencies of the 0th (R 0 ) and −1st (R -1 ) orders as well as the absorption (A) of the three gratings are shown in Fig. 2(b). It is clear that the −1st diffraction efficiencies (solid curves) of the three gratings exhibit resonant peaks at 440nm, 550nm and 660nm, respectively. In contrast, the unwanted 0th diffraction efficiencies (dashed curves) approach zero at the three resonant wavelengths, indicating a nearly complete suppression of the specular reflections. The absorption (represented by dot-dashed curves) also exhibits peak values at the resonant wavelengths, and the shorter the resonant wavelength, the stronger the absorption due to the stronger intrinsic absorption of metal at shorter wavelength. Nevertheless, the −1st diffraction efficiencies can still reach high values larger than 0.6.
To shed light on the origin of the enhanced diffraction efficiency of the −1st order, we plot the field patterns (|H z | 2 ) at the R -1 peak wavelengths of the three gratings in Fig. 2(c). The fields are mainly localized inside the groove regions of the gratings with a small part penetrating into the metal, which are typical field profiles of the localized plasmon modes, indicating that the strong diffraction efficiencies of R -1 are indeed caused by the excitations of the localized plasmon modes. Because the localized plasmon modes are independent of the incident angles of the illuminating light, the enhanced diffraction efficiencies should exist for a broad range of incident angles. Figures 2(d)-2(f) plot the R -1 as a function of the incident angle θ 0 and wavelength λ of the illuminating light for the three gratings, respectively. The peak wavelength of R -1 for all the three gratings remains constant for almost all incident angles. The peak value is the highest at incident angle of 45°, and it gradually decrease when θ 0 approaches 0° or 90°. As a whole, the diffraction peak exists for an ultrawide angle range of 10° to 80°, which promises the wide-angle performance of the achromatic metasurface.

The coupling effect of different nano-grooves in the compound grating
Before combing the elementary narrow groove gratings into achromatic metasurfaces, let's first study the coupling effect of the grooves with different heights for varying spacings between adjacent grooves. Figure 3 shows diffraction efficiencies of the compound grating, of which there are three grooves with h 1 = 15nm, h 2 = 29nm, and h 3 = 42nm, respectively, in each period. For the compound grating with p = 400nm [ Fig. 3(a)], we observe the diffraction efficiency variation versus spacing at wavelength of 440nm, which is the resonance wavelength of the individual groove with h 1 = 15nm. The −1st diffraction efficiency sustains high values when spacing is greater than a critical value of about s 0 = 7nm, while below this critical value, it decreases dramatically. It is because, for large spacing (s>s 0 ), the coupling between grooves is very weak, the resonance is still mainly determined by the individual groove, as the field is mainly localized in the individual groove [right inset of Fig. 3(a)]; for small spacing (s<s 0 ), however, the inter-groove coupling is very strong and the resonance wavelength will be influenced by all the grooves [left inset of Fig. 3(a)], as a result, the resonance peak of R -1 shifts to longer wavelength [ Fig. 4(a)]. For the compound grating with p = 500nm at working wavelength λ = 550nm [ Fig. 3(b)], the critical spacing is slightly larger (s 0 = 12nm), the field is mainly localized in the h 2 groove [right inset of Fig. 3(b)] when s>s 0 , while it extends to all the three grooves [left inset of Fig. 3(b)] due to strong coupling when s<s 0 . For the compound grating with p = 600nm at working wavelength λ = 660nm, the critical spacing is even larger (s 0 = 30nm), the field is localized in the h 3 groove [right inset of Fig. 3(c)] above the critical spacing, while it extends to the h 2 and h 3 grooves below the critical spacing. From the −1st reflection spectra of the compound gratings in Fig. 4, we can clearly see that, when the groove spacing is larger than the critical spacing, the resonance peek stays in the same wavelength position, while the peak shift dramatically to longer wavelength as long as the spacing is smaller than the critical spacing. And the longer the wavelength, the larger the critical spacing, which can be attributed to the positive correlation between the effective coupling length and the working wavelength. Fig. 4. The −1st reflection spectra for the compound grating for (a) p = 400nm, (b) p = 500nm, and (c) p = 600nm, respectively, with three different nano-grooves (h 1 = 15nm, h 2 = 29nm, h 3 = 42nm) in each unit cell for different spacings between adjacent nano-grooves.

The achromatic diffraction metasurface by combining multiple periodic gratings
By combining the three sets of the elementary gratings (g 0 , g 1 , and g 2 ), we can construct an achromatic metasurface as shown by the bottom panel in Fig. 1(c). The resonance wavelengths of the three gratings are 440nm, 550nm, and 660nm, which correspond to blue, green and red colors, respectively. Figures 5(d)-5(l) show the FEM simulations of the combined metasurface illuminated by a Gaussian beam of blue (440nm), green (550nm), and red (660nm) color, respectively. When a Gaussian beam illuminates on the metasurface from the left side with an incident angle of 45°, the beam is reflected to the left side (negative reflection), whereas the specular reflection (to the right side) is completely suppressed. For all the three different wavelengths, the directions of the negative reflection remain the same [Figs. 5(d)-5(f)]. It is because when a Gaussian Beam with a specific wavelength illuminate the metasurface, only the gap plasmon mode in the nano-grooves that resonate at this wavelength is excited, whereas other nano-grooves do not respond to this wavelength due to large separation of the resonance frequencies and the weak coupling between localized resonances of the nano-grooves [see Fig. 2(b) and Fig. 3]. For comparison, we also plot the case for an ordinary metallic grating illuminated by the same Gaussian beams as shown in Figs. 5(a)-5(c). The diffraction directions are apparently different for different wavelengths, which is the well-known chromatic characteristic of a conventional grating. Fig. 5. The field patterns (|H z | 2 ) of (a-c) the common chromatic diffraction grating and (d-l) the achromatic diffraction metasurface, respectively. The structure is illuminated by a Gaussian beam with (upper row) blue (440nm), (middle row) green (550nm), and (lower row) red (660nm) color, respectively. The incident angle is (a-f) 45°, (g-i) 33°, (j-l) 60°, respectively.
The achromatic diffraction can exist for an ultrawide range of incident angles, as verified by the simulation for other incident angles as shown in Figs. 5(g)-5(l). When the Gaussian beam illuminates the metasurface at θ 0 = 33° [Figs. 5(g)-5(i)], the beam is reflected back along its original path, because this particular incident angle satisfies the Littrow mount condition. Again, this works for all three wavelengths. The achromatic diffractions of the metasurface also exist for incident angle of 60° for the three wavelengths as shown in Figs. 5(j)-5(l). Incident beams are routed into the −1st diffraction order with the same diffraction angle (13.5°), while the specular reflections are completely suppressed. Therefore, the achromatic diffractions are indeed applicable for a wide incident angle range, which is highly desired for practical operations. The conversion efficiency of the combined achromatic diffraction metasurface is near 0.5 for 440nm and 550nm, while it only reaches about 0.35 for 660nm [see Table 1]. It can be explained by the fact that, the critical spacing for λ = 660nm [ Fig. 3(c)] is much larger than that for λ = 440nm and 550nm [ Figs. 3(a) and (b)], respectively. The combined metasurface may contain some units with small groove spacings that are larger than the critical spacing for 440nm and 550nm, but smaller than the critical spacing for 660nm, as a result, the total conversion efficiencies at λ = 660nm are significantly reduced.

The achromatic focusing metasurface by combining multiple modulated gratings
The wide-angle achromatic diffraction of the off-axis metasurface can be extended to the achromatic shaping of arbitrary wavefront with wide applicable incident angle range. Based on the relation between the phase and grating period [24], we can modulate the periods of each of the three sets of elementary gratings for achieving a desired phase profile by designing the height profile h x,y of the grating with binary hologram techniques [40,41], where, h is the groove height; p is the central period of the grating, the actual period of grating is modulated by ( ) , x y φ , which is the phase profile of the desired wavefront. Note that, based on just a simple one-dimensional (1D) grating with profile 2πx/p, we can modulate an arbitrary wavefront with 2D phase distribution ( ) , x y φ . Then we can combine the modulated gratings to form an achromatic metasurface for shaping the wavefront for all three wavelengths.
As a simple example, we now demonstrate the wide-angle achromatic 1D lens by the above approach. To obtain the focusing in the off-axis direction (−1st diffraction) with angle θ r , the phase profile in the metasurface plane should be ( ) ( ) Fig. 6. The field patterns (|Hz| 2 ) of (a-c) the common off-axis lens and (d-l) the achromatic focusing metasurface, respectively. The structure is illuminated by a Gaussian beam with (upper row) blue (440nm), (middle row) green (550nm), and (lower row) red (660nm) color, respectively. The incident angle is (a-f) 45°, (g-i) 33°, (j-l) 60°, respectively. Due to the different wavevector for each color, this design leads to a different gradient period profile for each of the elementary grating. By carefully combining those three gradient gratings in the same way as shown in Fig. 1(c), we can obtain the metasurface as an achromatic flat lens for focusing. Figures 6(d)-6(f) show the field patterns (|H z | 2 ) when the focusing metasurface (f = 20μm) is obliquely (θ 0 = 45°) illuminated by a Gaussian beam with different wavelengths. It is observed that the reflection waves are primarily redirected into the −1st diffraction order, and are focused at approximately the same position for different colors as indicated by the dashed line in Figs. 6(d)-6(f) for all the three wavelengths. In comparison, Figs. 6(a)-6(c) show the corresponding field patterns for an ordinary off-axis lens constructed by a single gradient grating. The incident Gaussian beam can also be focused, but the focusing point is obviously different for different wavelengths due to the intrinsic chromatic characteristic of the single grating. Therefore, the combined metasurface can indeed work as a multiwavelength achromatic flat lens. To demonstrate the wide-angle performance of the focusing metasurface, we also simulate the cases for other incident angles of the Gaussian beam as shown in Figs. 6(g)-6(l). When the Gaussian beam illuminates the metasurface with incident angle 33° [Figs. 6(g)-6(i)], the beam is primarily reflected back along its original incident path, and focused in the same point (−10.7um, 16.8um) even if the color of the incident light is different. Similarly, when the incident angle is 60° [ Fig. 6(j)-6(l)], majority of light can also be focused at a same point (−4.4um, 19.0um) for all the three wavelengths. From the intensity profile at the focal plane of the focusing metasurface as shown in Fig. 7, we can see that, the focusing beam width (1μm) is significantly smaller than incident beam width (5μm), and the focusing positions of the three wavelengths with blue, green and red colors, respectively, are indeed the same, which further proves the achromatic property of the combined metasurface. The peak value and the conversion efficiency [see Table 2] for 660nm is smaller than those for the other two short wavelengths, which is also due to the small groove spacings of the combined metasurface that is larger than the critical spacing for 440nm and 550nm, but smaller than that for 660nm [see Fig. 3(a)-3(c)]. Fig. 7. Cross section cuts of |H z | 2 for wavelength of 440nm (blue), 550nm (green), and 660nm (red), respectively, in the focal plane of the off-axis achromatic focusing metasurface with different incident angles (a) 33°, (b) 45°, and (c) 60°, respectively. The black curves represent the corresponding intensity profile of the incident Gaussian beam.

Summary
In summary, we introduce a type of off-axis multiwavelengths achromatic metasurfaces with wide-angle performance. The metasurface is constructed by integrating multiple metallic gratings with different subwavelength grooves in a single surface. The groove height of each elementary grating determines the corresponding resonance wavelength for enhanced diffraction, and the ratio between the resonance wavelength and grating period is fixed to achieve the achromatic operations. Such metasurfaces are utilized to demonstrate achromatic diffraction and focusing for visible light. The superior performance of the achromatic metasurfaces proposed here may pave the way towards practical applications in imaging and display systems.