Open Access
Add to BibSonomy
Abstract
In recent years, metasurfaces have provided a tempting path to replace conventional optical components where an abrupt phase change is imposed on an incident wave using a periodic array of unit cells. Till date, highly efficient dielectric metasurfaces have been demonstrated in infrared and visible domains. However, due to the lower bandgap of typical dielectric materials, such metasurfaces present strong absorption in the ultraviolet (UV) domain, and thus, hamper their realization at shorter wavelengths. In this paper, we utilize a large bandgap dielectric material, niobium pentoxide (Nb2O5), to construct an ultra-thin and compact transmission-type metasurface that manipulates the phase of an incident wave using an array of Nb2O5 nano-cylinder. By the virtue of numerical optimization, complete 2π phase coverage along with the high transmission efficiency (around 88.5%) is achieved at 355nm. Such efficient control over the phase of the incident wave enabled us to realize the polarisation insensitive self-accelerating parabolic, reciprocal, and logarithmic Airy beams (ABs) generating metasurfaces with the efficiency of 70%, 72% and 77%, respectively. In addition to this, we also demonstrate auto focusing Airy optical vortex (AFAOV) generators where the metasurfaces are designed to combine the phase profiles of an abruptly focusing Airy (AFA) beam and that of spiral phase plate (SPP). The AFAOV is generated with efficiency of 70% (for l = 3) and 72% (for l = 5).
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The modern-day optical technologies aim to compact the physical dimensions of optical systems and integrate several chips. The conventional optical components are mainly depended on material intrinsic property to realize the desired functionality, resulting in the bulky optical devices. For instance, the optical lenses developed from the propagation of light through an optical medium over a distance much larger than incident wavelength for phase accumulation [1–3]. Contrary to conventional optical devices, metasurfaces (2D metamaterials) that are composed of periodic array of nano-scatterers (also known as unit cell) with subwavelength spatial resolution, provide the feasible esplanade to address issues offered by bulky optical devices due to their ability to manipulate the electromagnetic waves characteristics (amplitude, phase, and polarization) at will. This feature of metasurfaces makes them suitable for photonic integrated circuits and mass production. Based on metasurfaces, recently, many flat optical devices have been designed and realized including absorbers [4,5], holography [6–8], optical vortex generation [7,9–11], Bessel beams [12,13], self-accelerating beams [14,15], beam shaping [16–18] and so on.
In earlier work, metallic nano-scatterer based transmission-type metasurfaces have shown better response from terahertz to near-IR [8,19–24]. However, due to electron-electron and electron-phonon scattering, the metallic metasurfaces encounter substantial ohmic losses, and temperature instability, causing a reduction in transmission efficiency in visible to UV domain, which is the region of interest for many potential applications (for example, bio-sensing, photolithography, and imaging applications) [25]. Recently, Aluminum (Al) based metasurfaces have shown promising responses at shorter wavelengths, but they undergo performance degradation due to oxidation [26]. To circumvent these limitations, high refractive indexed dielectric materials are considered as promising candidates to construct the lossless, and highly efficient metasurfaces at spectrum of interest [27]. To date, most of the research is focused on infrared and visible domains. Silicon-based metasurfaces have shown the transmission efficiency of >90% at infrared wavelengths [28], while amorphous silicon hydrogenated (a-Si:H) [11,29], titanium oxide (TiO2) [30], gallium nitride (GaN) [31], and silicon nitride (Si3N4) [32] exhibit exceptional performance in the visible domain. However, these materials possess a small bandgap at UV range, thus render very low transmission efficiency [33,34]. Furthermore, UV transparent material such as CaF2, MgF2, and SiO2 have low refractive indices present significant challenges in design and fabrication [33].
Consequently, we present highly efficient transmission-type all-dielectric metasurfaces employing niobium pentoxide (Nb2O5) material having the bandgap of $3.65eV$, which is significantly larger than other high indexed materials such as Si ($1.1eV$) and TiO2 ($3.2eV$) [34]. Due to this intriguing feature, Nb2O5 exhibit a broader transparency window (covering the UV, Visible and Infrared) [35]. In this article, the manipulation of phase of impinging wave is performed using Nb2O5 based nano-cylinders. The cylindrical geometry of the unit cell is opted because of its insensitivity to polarization of incident wave [11,36]. The radius of nano-cylinder is varied to attain the complete $2\pi $ phase control of incident wave with high transmission efficiency (around 88.5%) at the working wavelength of $355$nm (UV domain). Such efficient control facilitated us to realize the captivating phenomenon of self-accelerating parabolic, reciprocal, and logarithmic Airy beams (ABs) generation. The similar kind of work is published in [37] where the authors designed geometric phased based catenary-shaped plasmonic metasurfaces which require only circularly polarized incidence. Here, polarization insensitive ABs generating metasurfaces are presented with relatively high efficiency. Furthermore, we also demonstrate the autofocusing Airy optical vortex (AFAOV) generating multifunctional metasurfaces where phase profile of spiral phase plate (SPP) is integrated with that of abruptly focusing Airy (AFA) beam. We expect that this work would encourage establishment ultra-thin nano-photonic platform for bio-sensing, particle manipulation, and beam shaping applications.
2. Materials and methods
In order to realize the phase modulated metasurfaces, the first step is to achieve the $[0 - 2\pi]$ phase coverage through parametric optimization of unit cell while maintaining the maximum transmission efficiency. To accomplish this, Nb2O5 based nano-cylinder is patterned on SiO2 substrate (the unit cell), as shown in Fig. 1(a). Here, R represents the radius and P periodicity while H is the height of nano-cylinder. The Finite Difference Time Domain (FDTD) Numerical Method is employed for the optimization of Nb2O3 nano-cylinder at the working wavelength of 355 nm. The unit cell is subjected to perfectly matched layer (PML) boundaries in direction of propagation (z) and periodic boundaries in x-y direction. To acquire $2\pi $ phase control and maintain the high transmission efficiency, the periodicity (P) and radius (R) of nano-cylinders are swept with the optimal height (${\mathbf H} = 430\textrm{nm}$). The sweep results are depicted in Figs. 1(c) and 1(d). Here, it can be seen that the range of R varying from 30 nm to 90 nm and P of $182\textrm{nm}$ (as indicated by the white dashed line) are an optimized choice. The average transmission efficiency of around $88.5\%$ along with $2\pi $ phase coverage is attained. The transmission efficiency and corresponding phase profile as a function of radius are illustrated in Figs. 1(e) and 1(f), respectively.
Fig. 1. Unit cell optimization. (a) Schematic of Nb2O5 nano-cylinder on SiO2 substrate. Here parameters P, H, and R are the periodicity, height, and radius of the unit cell, respectively. (b) Step-indexed waveguide model of Nb2O5 nano-cylinder. Nb2O5 is considered as the core material while air (surrounding medium) represents the cladding. (c) Transmission efficiency and (d) Phase maps for radius versus periodicity at wavelength of 355 nm. The white dashed lines indicate the region of maximum transmission efficiency along with complete $2\pi $ phase coverage. (e) Transmission efficiency versus radius at fixed periodicity of $182\textrm{nm}$. The average transmission efficiency of $88.5\%$ is achieved. (f) The comparison between phase profile (solid purple line) obtained through FDTD and phase profile (solid pink line) obtained through step indexded waveguide model as function of radius (varying from 30 nm to 90 nm).
The underlying physics of acquiring the specific phase profile via Nb2O5 nano-cylinder can be well explained by wave guiding effect [36]. According to wave guiding effect, the phase yield by nano-cylinder with specific radius can be calculated in terms of effective refractive index (${n_{eff}}$) with the help of the following relation:
where, H is the height of nano-cylinder and ${n_{eff}}$ is the effective index of fundamental mode ($H{E_{11}}$) while $\lambda $ is the operating wavelength. The ${n_{eff}}$ can be computed through step-indexed waveguide model [38]. In which, Nb2O5 nano-cylinder acts a core with refractive index (${n_{core}}$) and surrounding medium (air) as cladding with refractive index ${n_{clad}}$ (shown in Fig. 1(b)). As the radius of nano-cylinder is varied, the effective refractive index of fundamental mode is changed. Figure 1(f) shows that the phase profile calculated by step indexed waveguide model is in agreement with the one calculated through FDTD simulation. The agreement seems better for larger radii where the confinement of mode increases. Furthermore, periodic boundary conditions in FDTD also introduce deviation in mode profile which is dominant for smaller radii, thus indicating the reason behind the difference between two phase profiles.Furthermore, it is necessitated that for high contrast dielectric metasurfaces, the interaction between the unit cells should be negligible and scattering of incident light is purely dominated by wave guiding effect. To verify this, the numerical simulations for three different radii (50 nm, 60 nm, and 70 nm) are performed to investigate the near field distributions. In Fig. 2, the top (first row) and side (second row) views of magnetic energy densities (MEDs) inside arrays the of nano-cylinders with different radii are shown. It can be observed that most of the incident light is concentrated inside the nano-cylinders and have negligible coupling with neighboring nano-cylinders. Therefore, transmission phase and efficiency of periodic array of unit cells can be considered as the individual response of each unit cell [39–41].
Fig. 2. Near field distribution of periodic arrays of unit cell. The top views and side views of magnetic engergy densities in a Nb2O5 nano-cylinders for (a) ${\mathbf R} = 50\textrm{nm}$, (b) ${\mathbf R} = 60\textrm{nm}$, and (c) ${\mathbf R} = 70\textrm{nm}$. The solid black line indicate the boundaries of unit cell. These results are obtained under x-polarized incidence at wavelength of 355 nm.
3. Results and discussion
3.1 Polarization insensitive Airy beam generators
To proof the concept of phase modulation, here, we demonstrate the polarisation insensitive AB generating metasurfaces. According to the geometric construction of arbitrary optical caustics, the ABs along the convex trajectories can be produced by directly employing the spatial phase profile on an impinging beam [42,43]. In this regard, the phase profile for parabolic 1D AB as depicted in Fig. 3(a) is implemented and can be expressed as follows:
Fig. 3. Polarisation Insensitive Airy beams generation. (a), (d), and (g) show the Spatial phase distributions of parabolic AB, reciprocal AB, and logrithmic AB, respectively. (b), (e) and (h) represent the corresponding simulated E-field intensity distribution imaged in x-z plane. (c), (f), and (i) demonstrate the comparision between analytically calculated and simulated propagation trajectories of main lobes of parabolic AB, reciprocal AB, and logrithmic AB, respectively.
The full-wave numerical simulation of our designed metasurface is accomplished via FDTD method. The metasurface is subjected to Perfectly Matched Layered (PML) boundary condition in all directions (x, y, and z). The x-polarised plane wave in impinged onto the metasurface and resultant outgoing electric field is recorded by placing field monitor in x-y plane. Figure 3(b) illustrates the simulated electric field (E-field) distribution showing the self-bending and self-accelerating nature of main lobe of parabolic AB. To further verify the parabolic nature of our AB, analytically calculated $({{\textrm{x}_\textrm{d}} = \textrm{a}{\textrm{z}^2}} )$ and simulated propagation trajectory of main lobe is plotted in Fig. 3(c). Here, we can see the good match between analytically calculated and simulated results. However, it is believed that the small deviation is due to the fact that discrete number of unit cells are approximating continuous phase function.
In addition to parabolic AB, here, we also demonstrate the metasurfaces, capable of generating ABs that follow the reciprocal $(\textrm{x} = {\textrm{a}_1}/\textrm{z}$) and natural logarithmic ($\textrm{x} = {\textrm{a}_2}\textrm{ln}({\textrm{bz}} )$) caustic trajectory. The corresponding spatial distribution of phase profiles as depicted in Figs. 3(d) and 3(g) respectively, can be expressed as:
andFurthermore, to reveal the polarization insensitive feature of our Airy beam generator, we simulate parabolic AB generating metasurface by impinging incident wave with different polarizations. The simulated results (in x-z plane) are demonstrated in Figs. 4(a)–4(c) where the parabolic Airy beams are generated for y-polarized, right circular polarized (RCP), and left circular polarized incidences (LCP).
Fig. 4. Polarisation insensitivity test. Simulated E-field distribution of parabolic AB under illumination of (a) y-polarisation (b) RCP, and (c) LCP at operational wavelength of 355 nm.
3.2 Auto focusing Airy optical vortex (AFAOV) generators
In this section, we propose the phase modulation based multifunctional metasurfaces capable of generating AFAOVs. To realize such metasurfaces the phase profiles of Abruptly focusing Airy (AFA) and Spiral Phase Plate (SPP) are merged together. The resultant relation for required phase distribution can be expressed as follows:
Fig. 5. Auto Foucsing Airy Optical Vortex Generation. (a) and (b) Phase distribtion of Nb2O5 nano-cylinder for (a) $l = 3$ and, (b) $l = 5$. (c) and (d) Simulated E-field intensity distribution in x-y plane for $l = 3$ and $5$, respectively. These donut shaped annular openings are imaged at focal plane ($\textrm{z} = 11{\mathrm{\mu}} \textrm{m}$). (e) and (f) show their corresponding phase patterns.
4. Conclusion
In conclusion, all-dielectric metasurfaces in UV domain are proposed by using niobium pentoxide (Nb2O5). Periodic arrangement of optimized Nb2O5 nano-cylinder enabled us to manipulate the phase of incident wave with high transmission efficiency of 88.5%. Based on this, highly efficient polarisation insensitive AB generating metasurfaces are present to convert the incident wave with arbitrary polarization into self-accelerating ABs (parabolic, reciprocal, and logarithmic) with efficiency from $70\%- 77\%$. Furthermore, the self-bending nature of main lobes of ABs have shown good agreement with analytical calculation with slight deviation. This deviation is due to discrete phase distribution. In addition to this, we extended our approach to realize the AFAOV generating metasurfaces where phase profile of AFA and SPP is merged to generate highly concentrated AFAOV. Two metasurfaces for $l = 3$ and $5$ are designed and simulated with focusing efficiency of $72\%$ and $70\%$, respectively. We envision that our designed metasurfaces can be useful for micron scale particle manipulation, optical tweezers, and bio-sensing.
Disclosures
The authors declare no conflicts of interest.
References
1. J.-S. Ye, Y. Zhang, and K. Hane, “Improved first Rayleigh-Sommerfeld method applied to metallic cylindrical focusing micro mirrors,” Opt. Express 17(9), 7348–7360 (2009). [CrossRef]
2. T. Fujita, H. Nishihara, and J. Koyama, “Blazed gratings and Fresnel lenses fabricated by electron-beam lithography,” Opt. Lett. 7(12), 578–580 (1982). [CrossRef]
3. J. M. Bendickson, E. N. Glytsis, and T. K. Gaylord, “Metallic surface-relief on-axis and off-axis focusing diffractive cylindrical mirrors,” J. Opt. Soc. Am. A 16(1), 113–130 (1999). [CrossRef]
4. A. S. Rana, M. Q. Mehmood, H. Jeong, I. Kim, and J. Rho, “Tungsten-based ultrathin absorber for visible regime,” Sci. Rep. 8(1), 2443 (2018). [CrossRef]
5. I. Kim, S. So, A. S. Rana, M. Q. Mehmood, and J. Rho, “Thermally robust ring-shaped chromium perfect absorber of visible light,” Nanophotonics 7(11), 1827–1833 (2018). [CrossRef]
6. M. A. Ansari, I. Kim, I. Rukhlenko, M. Zubair, S. Yerci, T. Tauqeer, M. Q. Mehmood, and J. Rho, “Engineering Spin and Antiferromagnetic Resonances to Realize Efficient Direction-multiplexed Visible Meta-hologram,” Nanoscale Horiz 5(1), 57 (2020). [CrossRef]
7. H. Ren, G. Briere, X. Fang, P. Ni, R. Sawant, S. Héron, S. Chenot, S. Vézian, B. Damilano, and V. Brändli, “Metasurface orbital angular momentum holography,” Nat. Commun. 10(1), 2986 (2019). [CrossRef]
8. X. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nat. Commun. 4(1), 2807 (2013). [CrossRef]
9. J. Wu, Z. Zhang, X. Ren, Z. Huang, and X. Wu, “A broadband electronically mode-reconfigurable orbital angular momentum metasurface antenna,” Antennas Wirel. Propag. Lett. 18(7), 1482–1486 (2019). [CrossRef]
10. G. Ding, K. Chen, X. Luo, J. Zhao, T. Jiang, and Y. Feng, “Dual-Helicity Decoupled Coding Metasurface for Independent Spin-to-Orbital Angular Momentum Conversion,” Phys. Rev. Appl. 11(4), 044043 (2019). [CrossRef]
11. N. Mahmood, I. Kim, M. Q. Mehmood, H. Jeong, A. Akbar, D. Lee, M. Saleem, M. Zubair, M. S. Anwar, and F. A. Tahir, “Polarisation insensitive multifunctional metasurfaces based on all-dielectric nanowaveguides,” Nanoscale 10(38), 18323–18330 (2018). [CrossRef]
12. W.-T. Chen, M. Khorasaninejad, A. Y. Zhu, J. Oh, R. C. Devlin, M. A. A. Zaidi, and F. Capasso, “High performance visible wavelength meta-axicons for generating bessel beams,” (Google Patents, 2019).
13. M. R. Akram, M. Q. Mehmood, T. Tauqeer, A. S. Rana, I. D. Rukhlenko, and W. Zhu, “Highly efficient generation of bessel beams with polarization insensitive metasurfaces,” Opt. Express 27(7), 9467–9480 (2019). [CrossRef]
14. D.-C. Chen, X.-F. Zhu, D.-J. Wu, and X.-J. Liu, “Broadband Airy-like beams by coded acoustic metasurfaces,” Appl. Phys. Lett. 114(5), 053504 (2019). [CrossRef]
15. T. Wang, G. Zhai, R. Xie, S. Zhu, J. Gao, S. An, B. Zheng, H. Li, Y. Liu, and H. Zhang, “Dual-Band Terahertz Auto-Focusing Airy Beam Based on Single-Layer Geometric Metasurfaces with Independent Complex Amplitude Modulation at Each Wavelength,” Adv. Theory Simul. 2(7), 1900071 (2019). [CrossRef]
16. H. Ahmed, M. M. Ali, A. Ullah, A. A. Rahim, H. Maab, and M. Khan, “An Ultra-Thin Beam Splitter Design Using a-Si: H Based on Phase Gradient Metasurfaces,” J. Nanoelectron. Optoelectron. 14(9), 1339–1343 (2019). [CrossRef]
17. A. L. Holsteen, A. F. Cihan, and M. L. Brongersma, “Temporal color mixing and dynamic beam shaping with silicon metasurfaces,” Science 365(6450), 257–260 (2019). [CrossRef]
18. A. Ozer, N. Yilmaz, H. Kocer, and H. Kurt, “Polarization-insensitive beam splitters using all-dielectric phase gradient metasurfaces at visible wavelengths,” Opt. Lett. 43(18), 4350–4353 (2018). [CrossRef]
19. L. Liu, X. Zhang, M. Kenney, X. Su, N. Xu, C. Ouyang, Y. Shi, J. Han, W. Zhang, and S. Zhang, “Broadband metasurfaces with simultaneous control of phase and amplitude,” Adv. Mater. 26(29), 5031–5036 (2014). [CrossRef]
20. 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]
21. X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335(6067), 427 (2012). [CrossRef]
22. 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). [CrossRef]
23. P. C. Wu, W.-Y. Tsai, W. T. Chen, Y.-W. Huang, T.-Y. Chen, J.-W. Chen, C. Y. Liao, C. H. Chu, G. Sun, and D. P. Tsai, “Versatile polarization generation with an aluminum plasmonic metasurface,” Nano Lett. 17(1), 445–452 (2017). [CrossRef]
24. X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin Hall effect at metasurfaces,” Science 339(6126), 1405–1407 (2013). [CrossRef]
25. J. R. Bolton, Ultraviolet Applications Handbook (Ayr, Ont.: Bolton Photosciences Incorporated, 2001), Vol. 18.
26. M. W. Knight, N. S. King, L. Liu, H. O. Everitt, P. Nordlander, and N. J. Halas, “Aluminum for plasmonics,” ACS Nano 8(1), 834–840 (2014). [CrossRef]
27. H. H. Hsiao, C. H. Chu, and D. P. Tsai, “Fundamentals and applications of metasurfaces,” Small Methods 1(4), 1600064 (2017). [CrossRef]
28. K. Ou, G. Li, T. Li, H. Yang, F. Yu, J. Chen, Z. Zhao, G. Cao, X. Chen, and W. Lu, “High efficiency focusing vortex generation and detection with polarization-insensitive dielectric metasurfaces,” Nanoscale 10(40), 19154–19161 (2018). [CrossRef]
29. N. Mahmood, H. Jeong, I. Kim, M. Q. Mehmood, M. Zubair, A. Akbar, M. Saleem, M. S. Anwar, F. A. Tahir, and J. Rho, “Twisted non-diffracting beams through all dielectric meta-axicons,” Nanoscale , 11(43), 20571–20578 (2019). [CrossRef]
30. 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]
31. 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, and C.-H. Kuan, “GaN metalens for pixel-level full-color routing at visible light,” Nano Lett. 17(10), 6345–6352 (2017). [CrossRef]
32. S. Colburn, A. Zhan, and A. Majumdar, “Metasurface optics for full-color computational imaging,” Sci. Adv. 4(2), eaar2114 (2018). [CrossRef]
33. Y. Deng, X. Wang, Z. Gong, K. Dong, S. Lou, N. Pégard, K. B. Tom, F. Yang, Z. You, and L. Waller, “All-Silicon Broadband Ultraviolet Metasurfaces,” Adv. Mater. 30(38), 1802632 (2018). [CrossRef]
34. K. Huang, J. Deng, H. S. Leong, S. L. K. Yap, R. B. Yang, J. Teng, and H. Liu, “Ultraviolet Metasurfaces of≈ 80% Efficiency with Antiferromagnetic Resonances for Optical Vectorial Anti-Counterfeiting,” Laser Photonics Rev. 13(5), 1800289 (2019). [CrossRef]
35. R. A. Rani, A. S. Zoolfakar, A. P. O’Mullane, M. W. Austin, and K. Kalantar-Zadeh, “Thin films and nanostructures of niobium pentoxide: fundamental properties, synthesis methods and applications,” J. Mater. Chem. A 2(38), 15683–15703 (2014). [CrossRef]
36. 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]
37. Y. Guo, Y. Huang, X. Li, M. Pu, P. Gao, J. Jin, X. Ma, and X. Luo, “Polarization-Controlled Broadband Accelerating Beams Generation by Single Catenary-Shaped Metasurface,” Adv. Opt. Mater. 7(18), 1900503 (2019). [CrossRef]
38. A. Yariv and P. Yeh, Photonics: optical electronics in modern communications (Oxford Univ., 2006).
39. 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]
40. E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, and A. Faraon, “Multiwavelength polarization-insensitive lenses based on dielectric metasurfaces with meta-molecules,” Optica 3(6), 628–633 (2016). [CrossRef]
41. E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, and A. Faraon, “Controlling the sign of chromatic dispersion in diffractive optics with dielectric metasurfaces,” Optica 4(6), 625–632 (2017). [CrossRef]
42. E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011). [CrossRef]
43. L. Froehly, F. Courvoisier, A. Mathis, M. Jacquot, L. Furfaro, R. Giust, P. Lacourt, and J. Dudley, “Arbitrary accelerating micron-scale caustic beams in two and three dimensions,” Opt. Express 19(17), 16455–16465 (2011). [CrossRef]
44. J. A. Davis, D. M. Cottrell, and J. M. Zinn, “Direct generation of abruptly focusing vortex beams using a 3/2 radial phase-only pattern,” Appl. Opt. 52(9), 1888–1891 (2013). [CrossRef]
