Microlens arrays have been widely used for different optoelectronic applications. The demand for compact optical devices necessitates the deployment of even smaller microlens arrays; however, as the spacing between individual lenses reduces and the lens diameter approaches the length scale of the incident wavelength of light, diffraction starts playing a critical role and produces a significant impact on the final focusing properties of the optical field. In this paper, we analyze the focusing characteristics of all-dielectric ultra-compact metasurface lens arrays for efficient optical device applications, constructed by kinds of broadband planar lenses composed of subwavelength nano-scatterers. By using the 3D finite-difference time-domain (FDTD) method, focusing and diffraction-based crosstalk effects caused by the changing physical spacing between adjacent metalenses, the diameter of microlenses, the operating wavelength, and the array size are rigorously investigated. Analysis of the achieved results show that a larger spacing, a larger lens size, and a shorter wavelength can lead to a weaker focusing crosstalk effect. Moreover, the crosstalk effect does not have a significant dependence on the array’s overall size. This research study may provide an important technological reference to designing an array of all-dielectric planar metasurface lenses with a well-controlled focusing performance and may pave the way further toward the application of metasurface lens arrays in compact optical sensing, coupling, and detecting system designs.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Microlens arrays (MLAs) have been used for different applications such as coupling of light into fibers , optical interconnects , collimating laser diodes and VCSELs emissions , concentrating solar light , imaging and sensing in charge-coupled devices (CCDs) (Shack-Hartmann wavefront sensor), and complementary metal-oxide semiconductors (CMOS) sensors , homogenizing emissions of high power lasers , and illumination of organic light emitting diodes (OLEDs) . Due to the demand for compact optical devices, and development of modern high-resolution imaging sensors, the deployment of even smaller MLAs is essential. However, as the MLA physical dimensions go down and approach to the wavelength of light, the conventional refractive MLAs start to have a poor focusing performance, due to the fact that the diffraction prevents the microlens from acting as a focusing element . In addition, the fill factor of MLAs is reduced because the effective refractive capacity decreases due to the deviation from ideal conic surface that becomes significant especially at the edge of the lens, produced using the photoresist reflow method for which lens surface figure is dominated by contact angle of the photoresist to substrate. Apparently, surface errors become more problematic in miniaturization of optical devices of conventional refractive microlenses because as the lens dimensions scale down, the lens gets bulkier and more curved to achieve the gradual phase accumulation. Obtaining smaller processing errors and higher fill factors require a critical precision control over the surface profile . Recently, metasurfaces (MTSs) as 2D version of metamaterials garnered significant interest in the optics community due to ultrathin thickness and lower fabrication complexity. Planar MTSs enable the arbitrary control of the wavefront and polarization within subwavelength thick structure at the interface while keeping high efficiency [9–11]. Besides, metasurface optics offer unprecedented advantages over their conventional bulky and costly counterparts by benefitting from their compact configuration and fabrication using the advanced capabilities of CMOS foundries [12,13]. The initial optical devices based on MTSs consisted of metal nano-antenna-like plasmonic scatterers [14–17]; however, due to their high ohmic losses these platforms suffered from low efficiency. In contrast, all-dielectric MTS platforms offer an alternative route to gradient metasurface optics with high efficiency [18,19]. So far, three distinct styles of all-dielectric MTSs have been proposed. The first approach is recently implemented by using high aspect ratio asymmetrical structures that locally rotate polarization and create a geometric (Pancharatnam-Berry) phase shift by employing a dominant electrical resonance in asymmetric nanobeams [20–24]. However, this technique is inherently limited to circularly polarized incident light. The second approach relies on high aspect ratio nanoposts, which operate as truncated waveguides supporting low quality factor Fabry-Pérot resonances . Phase modulation on the propagating light is realized by variation of the effective refractive index of nanoposts, which is dependent on the geometrical parameters of the post within a unit cell. This approach is robust and have tolerance to fabrication errors due to negligible interaction between the neighboring posts, and highly efficient devices have been demonstrated for visible and infrared light, and fabrication feasibility was successfully demonstrated with new bottom-up nanofabrication methods [25–27]. The third type of MTSs is based on low aspect ratio nanocylinders that sustain spectrally overlapping electric and magnetic dipole Mie-type resonances to locally control the propagation phase of the impinging light [28–30]. These nanocylinder elements must have a high refractive index (>2) to support strong dipole resonances and may be described as Huygens source nanoantennas [31,32] The shortcoming of this method is that there is strong inter-element coupling, which makes it difficult for designing gradient MTSs. Therefore, during the design of dielectric Huygens source nanoantennas, the geometry of neighboring elements must be taken into account, and nanoantennas must be analyzed as a whole array. Based on these different efficient dielectric MTSs various optical components operating in transmission with high efficiency have been demonstrated, such as gratings , lenses [34–37], beam deflectors , shapers , and holograms [13,40]. Metalenses highlight the great potential of this technology and can meet the growing demand for miniaturization of conventional imaging lenses. In 2016, Capasso’s group first realized metalenses at visible [27,36] with TiO2 metasurfaces, by using a bottom-up nanofabrication via atomic layer deposition nanofabrication technology. Another study demonstrated a design of low-contrast metasurface-based metalenses by using CMOS-compatible silicon nitride. However, the focusing efficiency, defined by the ratio of focused power to the incident power, is much lower (40%) in the visible due to the elements with low-index-contrast materials . All these dielectric metasurface lenses show great prospect in many fields such as high-resolution imaging, optical data storage, and nanolithography. For example, metasurface lenses may enable totally new type of MLAs with the lens size of several micrometers to match the single pixel of the modern ultra-high-resolution CCD or CMOS sensors. Due to more flexibility in the design of the shape of these metasurface lenses 100% fill factor is possible. In other words, sub-lenses can be matched each other perfectly without deadspace. This also allows smaller pixel size, and, thus, higher resolution imaging.
In this paper, metasurface MLAs possessing theoretical fill factor of 100% due to no deadspace among the lens elements and with lower fill factors are presented and numerically analyzed. All-dielectric metasurface planar microlenses suggested here can focus light efficiently in a broadband with no polarization-dependence as indicated by rigorous analysis in a spectral band ranging from 450 to 650 nm. More importantly, the focusing crosstalk effect between the adjacent microlenses in an array is analyzed in detail. To the best of our knowledge, it has not been studied in any other publications on the dielectric planar metasurface microlenses, and we show that crosstalk effect plays a vital role when the diameter of the metasurface microlenses and their periodicities approach to the wavelength of light. The effect of the focusing crosstalk on the final focusing performance is analyzed by the full-wave simulation when the working wavelength, the microlens diameter, the spacing between adjacent microlenses, and the array size are varied. The findings of this study provide an important technological reference for the design and fabrication of planar all-dielectric metasurface MLAs with a well-controlled focusing performance and once more reveal the potential of metasurface lens arrays in nanophotonic applications for the realization of ultracompact low-loss optical devices.
2. Materials and methods
The realization of high-efficiency dielectric metasurfaces critically depends on the optical properties of the constituent material . The complex refractive index of the preferred dielectric materials, given by must satisfy two conditions: relatively high refractive index and negligible loss at design wavelengths. High transmission efficiency is ensured by a negligible absorption. In addition, the high refractive index would enable complete wavefront phase control due to strong confinement of the propagating light. High-index materials (silicon, germanium) suffer from significant optical absorption and losses at visible wavelength. However, using amorphous titanium dioxide (αTiO2), high-efficiency metalenses have been successfully demonstrated by functioning in forward transmission direction at the visible regime . To extract the optical constants of αTiO2, a Tauc-Lorentz (TL) oscillator model was employed [21,42–44]. By combining the normal quantum mechanical Lorentz oscillator model and the model derived by Tauc, the imaginary part of the dielectric constant of amorphous materials above the bandgap is given by:
Very consistent values are achieved by matching the experimental ellipsometry data measured by Capasso’s group in this wavelength range, and the real and imaginary part of the refractive index for αTiO2 as a function of wavelength, extracted by TL model is shown in Fig. 1 . The Kramers-Kronig integration can be used for the real part of the dielectric function, given by:
The designed meta-lens arrays comprised of an array of titanium dioxide (TiO2) nanopillars, which are arranged on a square lattice and schematically depicted in Fig. 2. The nanopillars are resting on a fused silica substrate at wavelength of 532 nm) with a fixed period P = 250 nm, the nanopillars have a unified height h = 600nm. A full-wave three-dimensional (3D) electromagnetic simulation was performed by using the finite-difference time-domain (FDTD) method to simulate the amplitude and the phase of the transmitted light. In order to simulate the phase shift of individual meta-cells, perfectly matched layers (PML) are applied at the z boundaries and periodic boundary conditions at the x and y boundaries. Effective manipulation of the electromagnetic waves requires simultaneous high transmittance and full 2π phase coverage. In our model, the waveguiding effect is the dominant mechanism accounting for the phase realization by dielectric metasurface contrary to Mie-type resonance effect. Each nanopillar can be considered as a waveguide and imparts a polarization-insensitive phase shift on the transmitted light due to the symmetry of the nanopillar meta-cells. The phase shift for the electromagnetic waves propagating through the waveguides with two different diameters is determined by:Figs. 2(b), 2(c), and 2(d), the transmittance and corresponding phase as a function of the diameter of TiO2 nanopillars and wavelength of light are clearly illustrated. Obviously, the phase of the light transmitted from the nanopillar metasurfaces can cover the entire 2π range. The transmission amplitudes demonstrate almost uniform high values in a broad spectral region without any significant resonance since the phase accumulation is realized by means of the waveguiding effect. In order to design a high efficiency metasurface lens array, a number of meta-cells with a fixed period of 250 nm are calculated, as shown in Fig. 2(b). With a unit cell dimension of 250 nm, the lattice is ensured subwavelength and non-diffracting for a design wavelength of λd = 532 nm. Besides, this lattice periodicity value satisfies the Shannon-Nyquist theorem for all f-number values of the considered meta-lenses. Figure 2(b) shows the plot of one-to-one mapping between transmission phase and nanopillar diameters. Note that, the intensity transmission is kept above 87% while 0 to 2π phase range is covered by changing the nanopillar diameters from 100 nm to 220 nm. The smallest attainable diameter is limited primarily by fabrication constraints, and the largest one can be equal to the unit cell periodicity or dimension P which must be small enough to meet the Nyquist sampling theorem (P<λ/2NA), where NA is the numerical aperture and λ is the wavelength. The metalenses designed focus incident light into a spot in transmission mode. In order to accomplish this, the required phase that must be imparted by each nanopillar at position (x, y) can be given by :Eq. (5) is realized by adjusting the nanopillar diameter at each lattice site. Our designed metalenses had a focal length of 1.8 µm with aperture sizes varying from 2 µm to 8 µm. For 2 µm lens aperture size and focal length of 1.8 µm, NA of an isolated lens is 0.485, f-number is 0.90, respectively. This also corresponds to a Fresnel number of Nf = 1.04. After realizing the isolated metalenses and metalens arrays with a discretized phase profile, full-wave simulations were performed. For the analysis of the metalens array, PML boundary conditions were used along the axial directions and in the normal directions to the lenses. Likewise, the isolated metalenses were analyzed by applying PML to all outermost borders. The whole lens structures were normally illuminated by linearly polarized plane wave with the wave vector ko in the z direction, and the electric field polarized along the x direction.
3. Results and discussion
3.1 Focusing performance of an isolated metalens
To characterize the focusing performance of the isolated metalens, the electric field intensity around the focal region of the metalens is calculated using the FDTD method. The schematic of the designed all-dielectric metalens is given by Fig. 3(a). For the design wavelength of 532 nm, TiO2 nanoposts with height of 600 nm rest on a flat SiO2 substrate. Figure 3(b) presents the simulation results for the electric-field intensity cutting through the focal plane for the three different lens aperture sizes.
The interference of the in-phase waves diffracted from the subwavelength meta-cells forms the focal spot. Only one diffraction spot with no high orders is a clear sign of superior performance of metasurface lenses over classical diffractive lenses. The aperture size of the metalenses reveals a significant effect on the depth of focus (DOF), full-width at half-maximum (FWHM), and the focal length of the focal spot. The aperture directly affects the Fresnel number. Furthermore, when Fresnel number is large, geometric optics is enough to predict the focal length desired. However, for a small Fresnel number (much less than 8), diffraction theory is more accurate to predict the position of the focal point . Similar to the diffractive lenses, the focal length primarily depends on the working wavelength and the lens aperture size as will be shown in next sections. The wavelength has a significant influence on the focusing performance, which can be clearly seen in Fig. 3(c). Note that the smaller the working wavelength is, the larger the focal length is.
Chromatic dispersion is also an important problem affecting the focus position of all type of lenses in a broadband operation, and in addition to the wavelength dispersion, the material dispersion also influences the focus shift of the designed metasurface lenses by contributing to the chromatic aberration together. The principle of our designed metalens is based on phase discontinuities of a series of unit cells. Indeed, the phase shift of the unit cell can vary under different incident wavelengths as shown in Fig. 4. Note that, the simulated phase distribution of the electric field is calculated at 200 nm distance right after the lens by cutting the profile along the transverse direction parallel to the lens surface and plotted together with the ideal phase profiles in Fig. 4. The simulated phase distributions of output light with different wavelengths show good agreement with the theoretical prediction.
3.2 Analysis of the focusing and crosstalk effect for metalens arrays
As it is known from the diffraction theory, as the geometric dimension of the structures approach the illuminating wavelength of light, the diffraction effect and scattering among structures comes into play and cannot be ignored. For the traditional microlens arrays, when the lens size and the spacing between the adjacent lenses are much larger than the working wavelength, the effect of the diffraction can be ignored. On the other hand, considering the requirement of scaling the imaging sensors to lower dimensions for compact, lightweight and high-resolution imaging, the spacing and the lens size of the lens array approach to the working wavelength, and the diffraction of light dominates by affecting the optical performance of the pixel and its microlens, in particular. Similarly, focusing performance of the metasurface lens array will be affected severely by the diffraction. Therefore, in order to get a satisfactory focusing performance from the arrays consisting of micro-sized all-dielectric planar metalenses, the influence of diffraction-based crosstalk between the two adjacent metalenses, at least, must be analyzed and paid attention during the design. In the following figures, the effect of wavelength, lens spacing, diameter and array size will be clearly illustrated by 3D FDTD full-wave analysis.
3.3 Effect of the spacing between two adjacent metalenses
We first design a typical 2x2 metalens array with the lens spacing a varying from 0 to 4 µm so that we can investigate its focusing properties as shown in Fig. 5(a). The diameter of each lens is taken as 2 µm, also the working wavelength is λ d = 532 nm. For three lens spacing cases of a = 0,1,3 µm, the simulation results for the electric field patterns along the xz plane through the focal spots are given in Figs. 5(b)–5(d), respectively, along with the electric-field intensity patterns along the focal planes. The electric-field pattern of the focused beam of an isolated metalens is also shown in Fig. 5(e) for comparison. As it can be seen, more prominent crosstalk effect occurs for the spacing of 0 µm, and a strong crosstalk point appears far away from the focal plane. The focal points get an asymmetric form in the x and y directions due to a modulation effect. This can be most probably attributed to the electric-field component of the incident plane wave. As expected, as the lens spacing increases, the focusing optical field for a single metalens in an array gradually resembles the optical field for the isolated lens, thereby meaning that the crosstalk effect becomes less pronounced. On the other hand, at the spacing value of 0.5 µm, which is close to working wavelength, surprisingly the focusing optical field achieves the highest intensity, and FWHM, focal length values accord with the isolated lens values.
Figure 6 gives the obtained profiles of the light intensity cut along the lateral axis x across the focal spots of the 2x2 metalens array, and also the corresponding FWHM and focal length of the formed focal spots. In Fig. 6(a), for the cases of a ≥ 1 µm, the crosstalk effect decreases to a negligible level. The crosstalk point in the region between the focal spots occurs away from the focal spots for all spacing cases, and especially it seems obvious for the case of a = 0 µm away from the focal spots. Figure 6(b) shows that for the spacing a≤ 1µm, the focusing properties fluctuate significantly.
However, they approach to the focusing properties of the isolated metalens, when the lens spacing gradually increases, as depicted by the red and blue dashed horizontal lines. These results show the influence of the lens spacing on the focusing performance as a sufficient evidence since the same trend was observed for larger number of arrays. Table 1 shows the achieved results for all the values of different lens spacing, including the FWHM, focal length, DOF and maximum light intensity Imax. The incident light intensity is set to 1, and the focused intensity Imax is scaled to the incident light intensity. Since the Fresnel number is below 8 for all designed lenses in an array and isolated, the realized focal length is about 1.3 µm in average, which is much shorter than the theoretical value of 1.8 µm due to the significant effect of the diffraction . Interestingly, focal length of the metalenses in a 2x2 array with the spacing of 0 µm approaches to 1.533 µm, showing the smallest deviation from the original design.
According to the analysis results above, even if a large spacing seems beneficial for realizing a highly efficient and directional beam focusing with small crosstalk of metalenses in an array, the large spacing leads to a prominently low fill factor. This will definitely degrade the optical efficiency. On the other hand, lens spacing values of 0.5, 2, and 4 µm offer the best focusing properties closest to the isolated lens performance as shown in Table 1. Besides, the achieved focusing light intensity for the spacing value of 0.5 µm (Imax = 37.90 a.u.) outperforms all the others, even the isolated lens. However, DOF (1.15 µm) goes down comparing to other lens spacing values and the isolated metalens. These should be kept in mind as a trade-off for practical applications.
3.4 Effect of the working wavelength
Secondly, we investigate the effect of the working wavelength on the focusing performances of the metalens array by keeping the array size at 2x2 as before. To simplify the analysis of the focusing crosstalk effect, the lens spacing is kept at zero due to the strong crosstalk effect at a small spacing. Figure 7(a) demonstrates the simulation results for the variation of the FWHM and focal length of the metalenses when the working wavelength λ changes from 0.45 to 0.65 µm in the visible spectrum. Obviously, as the operation wavelength increases, the focal length goes down, and the FWHM shows an increasing characteristic.
As seen in Fig. 7(a), as the wavelength increases, the crosstalk effect has a slight increase; however, for the wavelength of λ = 0.45 µm, the deviation of the focal length compared to the isolated lens is 36.8% (1.391 µm vs 1.903 µm). In addition, for the case of λ = 0.65 µm, the deviation is 13.85%. Note that diffraction has less effect for the wavelengths smaller than the dimensions of the structures, and this clearly explains why diffraction does not deviate the focal length from the initial design value of 1.8 µm significantly, especially for smaller wavelengths and arrayed-metalenses. Moreover, the effect of the polarization is not significant on the diffraction-based crosstalk magnitude due to symmetry of the structures. Figure 7(b) and 7(c) demonstrate the electric-field patterns along the transverse direction (xy plane) in the focal plane and the electric-field patterns along the polarization direction (xz plane) and the direction perpendicular to the polarization (yz plane) for λ = 0.45 and λ = 0.65 µm, respectively. As it is indicated by the figures, stronger diffraction-based focal shift effect occurs in the focal regions at λ = 0.65 µm where the diffraction affects the focusing more obviously than smaller wavelengths. Material and wavelength dispersion of the structure contribute to this focal shift as well.
3.5 Effect of the diameter of the metalenses
The diameter of the metalenses is also another important parameter that will have significant influence on the focusing performance. Figure 8 presents the effect of the diameter of the metalenses when they are configured in an array. For both the isolated metalens and the metalenses in the array, as the diameter increases FWHM goes down and the focal length increases by approaching the initial design value of 1.8 µm. This trend was clearly explained in previous studies by considering the effect of the Fresnel number and diffraction on the focusing performance of the diffractive high contrast grating metalenses . Note that, the crosstalk effect obviously becomes weaker when the diameter of the metalenses is enlarged. In other words, the deviation of the focal length from the isolated lens values with the same diameter values can be given as 17% and 0.3% corresponding to the diameter of 2 and 8 µm, respectively. For the diameter of 2 µm (lens spacing = 0 µm), the metalenses in the array gets focal length (1.533 µm) value closer to the initial design value of 1.8 µm due to less edge diffraction effects resulted from increased geometrical dimension. The isolated metalens gets a focal length of 1.302 µm for diameter of 2 µm, since the Fresnel number is very low for this diameter, and significant diffraction effect causes this mismatch .
Furthermore, the isolated lens gets higher focusing intensity than the metalens in an array for the diameter of 2 µm, evidently due to the influence of the crosstalk between the adjacent lenses in an array.The focused spots in the focal planes and the electric field patterns along the polarization direction (xz plane) and the direction perpendicular to it (yz plane) are shown in Figs. 8(b) and 8(c) for d = 4 and 8 µm, respectively. Note that polarization has an unsubstantial impact on the final focusing performance in this case as well. Owing to these observations from the achieved results, a larger diameter of the metalenses should be chosen for a good prediction over the focusing performance. This also explains why there is no obvious crosstalk effect for the conventional MLAs as well.
3.6 Effect of the metalens array size
As the final study, we investigate the effect of a larger array size by changing the array size from the 2x2 array to 8x8 array, which may be of great importance for the practical applications. The simulation results of the variation of the focal length and FWHM of the metalenses located in the geometrical center of the array with respect to the variation of the array size is shown in Fig. 9(a). The diameter of the metalenses is kept as 2 µm. Note that the focal length of the arrayed metalenses at λ = 0.532 µm do not change and stays around 1.545 µm, while that of the isolated lens is 1.302 µm as shown in Fig. 9(a). On the other hand, the FWHM of the arrayed metalenses fluctuates slightly from 0.439 µm for odd number of arrays to 0.451 µm for even number of arrays, while that of the isolated lens is 0.371 µm [Fig. 9(a)]. These deviations from the isolated results are induced by the interaction between the adjacent lenses and less diffraction effects as the array size and geometrical dimension increases. Due to the additional metalenses adjacent to the interval lenses, the crosstalk effect comes into play more not only between two adjacent lenses but also between the interval lenses. The simulation results of the focal planes and electric-field intensity patterns for the cases of 3x3 and 8x8 metalens array in the two perpendicular planes are clearly presented in Figs. 9(b) and 9(c). No evident polarization-related crosstalk is observed in these focusing results. It was also observed that for polarization directions perpendicular to the focal plane, FWHM gets slightly smaller values. The focal spots created by the peripheral and the inner ones surrounded by the other metalenses are sufficiently symmetric. Furthermore, we should note that there is a sharp decay of the intensity profile at each focal point. Besides, the peak intensity values are uniformly distributed at the array plane, which is a great aspect of these metalens arrays. On the other hand, remarkably, no significant fringing effect exists on the achieved focal spots, which is another superior characteristic of this study comparing to the results of previously reported metallic planar microlens arrays , which clearly suffer from plasmonic coupling effects. This also simplifies the accurate control over the focusing performance of metalenses in an array. For metasurface lens arrays, a better quantitative figure-of-merit can be defined which obviously proves their advantage for optical efficiency enhancement and spatial crosstalk reduction .
In summary, we suggest all-dielectric metasurface planar microlens arrays, consisting of two-dimensional TiO2 nanopillar arrays arranged in a square lattice on top of a SiO2 substrate. As the lens diameter and the distance between two adjacent metalenses approach to the working wavelength of light, the diffraction starts playing a critical role and leads to a significant impact on the final focusing properties of the optical field. Besides, conventional, dielectric-based refractive MLAs lose their focusing functionalities as their physical dimensions approach to the operating wavelength, and do not have a promise for future ultra-compact CMOS and CCD optical devices. By using the full-wave FDTD numerical simulation method, the focusing performance of the metalenses in an array is explored thoroughly as the diameter of metalenses, the spacing between adjacent metalenses, the working wavelength, and the array size are changed. According to the analysis results, a larger lens size, a larger spacing, a shorter wavelength, can give rise to a weaker focusing crosstalk effect. Besides, the crosstalk effect does not have a noticeable dependence on the array size. These research findings provide a great reference to design an array of dielectric planar metalenses and to have a sufficient control over its focusing capability.
In addition, a further experimental study is required to verify the simulation results. The miniaturization of microlenses and their arrayed forms and the broadband, efficient focusing capabilities proposed here has a great application prospect for flat panel displays, high-efficiency solar cells, high-resolution imaging, and beam collimating for optical fibers.
H.K. gratefully acknowledges the partial support of the Turkish Academy of Sciences.
1. M. Oikawa, H. Imanishi, and T. Kishimoto, “Light coupling characteristics of a planar microlens,” Electron. Commun. Jpn. Part II Electron. 76(10), 10–20 (1993).
2. F. Wippermann, D. Radtke, M. Amberg, and S. Sinzinger, “Integrated free-space optical interconnect fabricated in planar optics using chirped microlens arrays,” Opt. Express 14(22), 10765–10778 (2006). [CrossRef] [PubMed]
3. C. Levallois, V. Bardinal, C. Vergnenègre, T. Leïchlé, T. Camps, E. Daran, and J.-B. Doucet, “VCSEL collimation using self-aligned integrated polymer microlenses,” Proc. SPIE 6992, 69920W (2008). [CrossRef]
7. M. K. Wei, J. H. Lee, H. Y. Lin, Y. H. Ho, K. Y. Chen, C. C. Lin, C. F. Wu, H. Y. Lin, J. H. Tsai, and T. C. Wu, “Efficiency improvement and spectral shift of an organic light-emitting device by attaching a hexagon-based microlens array,” J. Opt. A, Pure Appl. Opt. 10(5), 055302 (2008). [CrossRef]
8. D. H. Hwang, O. T. Kwon, W. J. Lee, J. W. Hong, and T. W. Kim, “Outcoupling efficiency of organic light-emitting diodes depending on the fill factor and size of the microlens array,” Phys. Stat. Solidi A 211(8), 1773–1777 (2014).
10. I. Staude and J. Schilling, “Metamaterial-inspired silicon nanophotonics,” Nat. Photonics 11(5), 274–284 (2017). [CrossRef]
11. J. P. Balthasar Mueller, N. A. Rubin, R. C. Devlin, B. Groever, and F. Capasso, “Metasurface polarization optics: independent phase control of arbitrary orthogonal states of polarization,” Phys. Rev. Lett. 118(11), 113901 (2017). [CrossRef] [PubMed]
12. 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). [CrossRef] [PubMed]
14. A. Ranjbar and A. Grbic, “Analysis and synthesis of cascaded metasurfaces using wave matrices,” Phys. Rev. B 95(20), 205114 (2017). [CrossRef]
16. 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). [CrossRef] [PubMed]
19. C. Pfeiffer and A. Grbic, “Generating stable tractor beams with dielectric metasurfaces,” Phys. Rev. B Condens. Matter Mater. Phys. 91(11), 115408 (2015). [CrossRef]
20. E. Hasman, V. Kleiner, G. Biener, and A. Niv, “Polarization dependent focusing lens by use of quantized Pancharatnam-Berry phase diffractive optics,” Appl. Phys. Lett. 82(3), 328–330 (2003). [CrossRef]
21. R. C. Devlin, M. Khorasaninejad, W. T. Chen, J. Oh, and F. Capasso, “Broadband high-efficiency dielectric metasurfaces for the visible spectrum,” Proc. Natl. Acad. Sci. U.S.A. 113(38), 10473–10478 (2016). [CrossRef] [PubMed]
22. B. H. Chen, P. C. Wu, V.-C. Su, Y.-C. Lai, C. H. Chu, I. C. Lee, J.-W. Chen, Y. H. Chen, Y.-C. Lan, C.-H. Kuan, and D. P. Tsai, “GaN metalens for pixel-level full-color routing at visible light,” Nano Lett. 17(10), 6345–6352 (2017). [CrossRef] [PubMed]
23. S. Wang, P. C. Wu, V.-C. Su, Y.-C. Lai, M.-K. Chen, H. Y. Kuo, B. H. Chen, Y. H. Chen, T.-T. Huang, J.-H. Wang, R.-M. Lin, C.-H. Kuan, T. Li, Z. Wang, S. Zhu, and D. P. Tsai, “A broadband achromatic metalens in the visible,” Nat. Nanotechnol. 13(3), 227–232 (2018). [CrossRef] [PubMed]
25. E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, and A. Faraon, “High efficiency double-wavelength dielectric metasurface lenses with dichroic birefringent meta-atoms,” Opt. Express 24(16), 18468–18477 (2016). [CrossRef] [PubMed]
26. S. Kruk, B. Hopkins, I. I. Kravchenko, A. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Invited Article: Broadband highly efficient dielectric metadevices for polarization control,” APL Photonics 1(3), 030801 (2016). [CrossRef]
27. M. Khorasaninejad, A. Y. Zhu, C. Roques-Carmes, W. T. Chen, J. Oh, I. Mishra, R. C. Devlin, and F. Capasso, “Polarization-insensitive metalenses at visible wavelengths,” Nano Lett. 16(11), 7229–7234 (2016). [CrossRef] [PubMed]
28. 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]
29. M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Adv. Opt. Mater. 3(6), 813–820 (2015). [CrossRef]
30. L. Zou, W. Withayachumnankul, C. M. Shah, A. Mitchell, M. Bhaskaran, S. Sriram, and C. Fumeaux, “Dielectric resonator nanoantennas at visible frequencies,” Opt. Express 21(1), 1344–1352 (2013). [CrossRef] [PubMed]
32. A. Özdemir, Z. Hayran, Y. Takashima, and H. Kurt, “Polarization independent high transmission large numerical aperture laser beam focusing and deflection by dielectric Huygens’ metasurfaces,” Opt. Commun. 401, 46–53 (2017). [CrossRef]
33. P. Lalanne, S. Astilean, P. Chavel, E. Cambril, and H. Launois, “Blazed binary subwavelength gratings with efficiencies larger than those of conventional échelette gratings,” Opt. Lett. 23(14), 1081–1083 (1998). [CrossRef] [PubMed]
34. A. Arbabi, Y. Horie, A. J. Ball, M. Bagheri, and A. Faraon, “Subwavelength-thick lenses with high numerical apertures and large efficiency based on high-contrast transmitarrays,” Nat. Commun. 6(1), 7069 (2015). [CrossRef] [PubMed]
35. A. Arbabi, R. M. Briggs, Y. Horie, M. Bagheri, and A. Faraon, “Efficient dielectric metasurface collimating lenses for mid-infrared quantum cascade lasers,” Opt. Express 23(26), 33310–33317 (2015). [CrossRef] [PubMed]
36. 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). [CrossRef] [PubMed]
37. A. Arbabi, E. Arbabi, S. M. Kamali, Y. Horie, S. Han, and A. Faraon, “Miniature optical planar camera based on a wide-angle metasurface doublet corrected for monochromatic aberrations,” Nat. Commun. 7, 13682 (2016). [CrossRef] [PubMed]
38. Y. F. Yu, A. Y. Zhu, R. Paniagua-Domínguez, Y. H. Fu, B. Luk’yanchuk, and A. I. Kuznetsov, “High-transmission dielectric metasurface with 2π phase control at visible wavelengths,” Laser Photon. Rev. 9(4), 412–418 (2015). [CrossRef]
39. K. E. Chong, I. Staude, A. James, J. Dominguez, S. Liu, S. Campione, G. S. Subramania, T. S. Luk, M. Decker, D. N. Neshev, I. Brener, and Y. S. Kivshar, “Polarization-independent silicon metadevices for efficient optical wavefront control,” Nano Lett. 15(8), 5369–5374 (2015). [CrossRef] [PubMed]
40. 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). [CrossRef]
41. A. Zhan, S. Colburn, R. Trivedi, T. K. Fryett, C. M. Dodson, and A. Majumdar, “Low-contrast dielectric metasurface optics,” ACS Photonics 3(2), 209–214 (2016). [CrossRef]
42. G. E. Jellison Jr. and F. A. Modine, “Parameterization of the optical functions of amorphous materials in the interband region,” Appl. Phys. Lett. 69(3), 371–373 (1996). [CrossRef]
43. Q. Zhang, T. Liao, G. Gan, M. Li, and X. Cui, “Polarization split lensing via polarization and phase control with metasurfaces at visible frequencies,” Plasmonics, 1–8 (2018).
45. arXiv:1803.05637 [physics.optics].