Open Access
Add to BibSonomy
Abstract
We proposed a new manipulation method for Bloch surface waves that can almost arbitrarily modulate the lateral phase through in-plane wave-vector matching. The Bloch surface beam is generated by a laser beam from a glass substrate incident on a carefully designed nanoarray structure, which can provide the missing momentum between the two beams and set the required initial phase of the Bloch surface beam. An internal mode was used as a channel between the incident and surface beams to improve the excitation efficiency. Using this method, we successfully realized and demonstrated the properties of various Bloch surface beams, including subwavelength-focused, self-accelerating Airy, and diffraction-free collimated beams. This manipulation method, along with the generated Bloch surface beams, will facilitate the development of two-dimensional optical systems and benefit potential applications of lab-on-chip photonic integrations.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Surface electromagnetic waves are two-dimensional (2D) electromagnetic waves localized at the interface between two media, usually with properties of near-field enhancement and subwavelength propagation, which are well suited to comprise a 2D optical system. Manipulating surface waves is a key to the miniaturization of compact optical systems, such as nanoscale optical circuits and lab-on-chip technologies. Precise control of the propagation of surface waves and realization of the desired near-field distribution has attracted attention. A frequently studied surface wave is the surface plasmon polariton (SPP) [1], which is sustained at the interface between metal and dielectric materials. However, the propagation length of SPP is limited owing to metal absorption losses, which is an obstacle to their wider application.
In contrast, Bloch surface waves (BSWs) are excited at the interface between a truncated one-dimensional photonic crystal (1DPC) and its surrounding medium [2], which can avoid the losses caused by metal absorption and has a long propagation length [3]. Different from the guided mode of a simple slab waveguide, whose field is confined inside the slab, BSW is a leaky mode whose field is mainly located near the top surface. Since BSWs are supported by isotropic dielectric media, many options of dielectric materials exist for constructing multilayer substrates. The parameters of the periodic and terminating layers on top of the periodic dielectric can be tuned to control the dispersion of the BSW, which allows the surface mode to be excited from deep ultraviolet (UV) to near-infrared (NIR) wavelengths [4,5]. Because of these characteristics, BSWs are considered promising for building 2D optical systems, and studies on the manipulation of BSW have received increasing attention.
Initially, structured element layers were added on top of the 1DPC, which could change the dispersion relation of the BSW and allow the control of its propagation. Devices with basic functionalities, such as thin polymeric ridges or nanofibers as BSW waveguides [6–10], disk and ring resonators [11,12], 2D polymer gratings and prisms demonstrating the reflection and refraction of BSWs [13,14], and plano-convex, double-convex, and isosceles triangle lenses for focusing BSWs [15–17], have been explored. However, for practical materials, the effective refractive index contrast (Δn) of the BSW mode with/without element layers is rather small (< 0.1) [17], which limits their ability to manipulate the BSWs. Furthermore, nanostructures based on the manipulation of in-plane diffraction and scattered fields have been designed to achieve more functionalities, for example, checkerboard, spiral ring, and meta-antenna structures for near-field focusing of BSWs [18–20], crossed gratings for the generation of nondiffracting BSW beams [21], laterally continuous grooves for polarization transformation [22,23], circular cavities, silicon nanoparticles, and asymmetric nano slits for the directional excitation of BSWs [24–26]. Although these nanostructures can realize many functions, each function requires the design of a specialized structure for implementation. For the generalization of on-chip photonic devices, multiple functions can be achieved through one class of configurations.
In this study, we proposed a method for manipulating BSW that can arbitrarily set the transverse phase of the BSW beam and achieve various desired near-field distributions. The lateral phase of the BSW beam can be modulated using a carefully designed nanoarray structure through in-plane wave-vector matching. To avoid the reflection of most of the excitation beam from the substrate, an internal mode was designed as a channel between the incident beam and diffracted BSW to improve the excitation efficiency. This method successfully realizes multiple BSW beams, including subwavelength-focused, self-accelerating Airy, and diffraction-free collimated beams. The properties of these beams were demonstrated. We believe that this method can improve the manipulation of surface electromagnetic waves and provide benefits for potential applications in lab-on-chip photonic integration.
2. Results and discussion
2.1 Structure of the dielectric multilayer substrate
A dielectric multilayer was designed, which can support a BSW mode with transverse electric (TE) polarization and an internal mode with transverse magnetic (TM) polarization at a wavelength of 633 nm. The internal mode is also a leaky mode. Different from the BSW mode, whose field is located near the top surface, the field of the internal mode is mainly confined inside the multilayer. The structure of the dielectric multilayer is depicted in Fig. 1(a), which consists of 18 alternating layers of SiO2 (refractive index nSiO2 = 1.46) and Si3N4 (refractive index nSi3O4 = 2.14). Except for the 180 nm thick top SiO2 layer, the other SiO2 and Si3N4 layers were 144 and 135 nm thick, respectively. Figure 1(a) shows a schematic of the generation of the designed BSW beam. The incidence-angle-dependent reflectance curves at a wavelength of 633 nm, which were calculated based on the transfer matrix method (TMM), are shown in Fig. 1(b) [27]. When the incident beam was TE-polarized, a sharp dip at 50.4° was observed, corresponding to the BSW mode excitation resonance. For TM polarization, the dip at 74.1° corresponded to internal mode (IM) excitation resonance. The dispersion relation between the BSW and internal modes is shown in Fig. S1 (Supplement 1). Figures 1(c) and 1(d) show the field distributions for the BSW and internal modes, respectively. The field of the BSW mode was mainly confined to the top layer, whereas the field of the internal mode was located inside the multilayer.
Fig. 1. (a) Schematic of the generation of designed BSW beams. A laser beam from a glass substrate passes through the dielectric multilayer and is incident to a specially arranged nanoarray. The designed BSW beam is formed on the side of the array. The dielectric multilayer is composed of 18 alternating dielectric layers of Si3N4 and SiO2. (b) Calculated TE and TM reflectance curves of the dielectric multilayer at an incident wavelength of 633 nm. The black dashed line denotes the critical angle between the glass substrate and air. (c) Normalized electric field distribution (Ey) for the BSW mode. (d) Normalized magnetic field distribution (Hy) for the internal mode.
When a beam from the glass substrate is incident on a single nanohole, the excited BSW field acts as a point-like source with a distribution that depends on the incident polarization direction (refer to part 2 in the Supplement 1 for details). For the excited BSW field, the axial component Ez is zero, and the radial component Eρ decreases as distance increases and is negligible at long distances. Only the azimuthal component Eφ can propagate a long distance and contribute to the generation of the BSW beams. The rectangular nanohole has a size of 100 × 100 nm2 and a depth of 180 nm. If the incident plane is the xz-plane and the nanohole is at the origin, as shown in Fig. 2(a), the BSW field excited by TM-polarized light is distributed on the sides of the incident plane (Fig. 2(b)), while that excited by TE-polarized light is distributed along the incident plane (Fig. 2(c)). Therefore, only the BSW fields excited by the TM-polarized beam can interfere on the sides of the incident plane to generate the desired BSW beams. In this study, all BSW fields were simulated using the finite-difference time-domain (FDTD) method [28]. The incident light source was a total-field scattered-field (TFSF) with a wavelength of 633 nm. The simulation area was different for each structure, which is slightly larger than the sum of the areas of the nanoarray and diffraction field. The size of the simulation grid was 1/10 of the wavelength, and the boundary was set to the perfectly matched layer (PML).
Fig. 2. (a) Schematic of the BSW field generated by a single hole under laser beam illumination. The incident plane is the xz-plane, and the incidence angle is 51°. The rectangular nanohole has a size of 100 × 100 nm2 and a depth of 180 nm. (b, c) The azimuthal field distributions (Eφ) under TM- and TE-polarized incidence. The dashed line indicates the incident plane (the xz-plane).
2.2 Phase modulated by nanohole arrays
Figure 3(a) shows a schematic of the incident and diffracted BSW beams in a periodic array. When the wave-vector matching condition is satisfied in the x-direction on the surface, the incident beam is diffracted in the preferred direction. The wave vector matching condition in the x-direction for a periodic array is
Fig. 3. (a) Schematic of the incident and diffracted beams in a periodic array. The angles of the incident and diffracted beams are θi and θd, respectively. The double-head arrows indicate the polarization orientation. (b) Diffraction angle as a function of the period in the x-direction. The black line and red dot indicate the theoretical and simulated results, respectively. The incident angle is 76°. (c) Normalized diffraction intensity with different incident angles. The array period in the x-direction is 0.45 µm. (d) Schematic of the diffraction phase distribution for a nonperiodic array.
For periodic arrays, the diffracted BSWs are finite plane waves, as shown in Fig. S4 and S5 (Supplement 1). The diffraction angle can be calculated from the Fourier plane image of the diffracted BSW beam. Figure 3(b) shows the diffraction angle versus the array period in the x-direction, where the incidence angle was 76°. The simulation results agreed well with the theoretical prediction of Eq. (2), which proves that the diffraction angle can be controlled by adjusting the array parameter.
For TM-polarized incidence, when the incident angle is larger than critical angle, as shown in Fig. 1(b), the incident beam is almost completely reflected, except at a dip of 74.1°, which corresponds to the excitation resonance angle of the internal mode. To excite the BSWs effectively, the incident angle should be at the dip angle. Figure 3(c) shows the normalized diffraction intensities at different incidence angles. The diffraction intensity is strongest when the incident angle is 76°, corresponding to the excitation resonance of the internal mode (the slight difference in the results obtained by the TMM and FDTD methods is due to the different algorithms and computational accuracies). If the incident angle is at the excitation angle of the internal mode, the incident beam with TM polarization first excites the internal mode, where the electric field is located inside the multilayer, and with the aid of the nanohole array, the electric field on the top layer can excite the BSW mode. Thus, the internal mode can act as a channel between the incident beam on the substrate and BSWs on the surface of the multilayer to improve the excitation efficiency.
Similar to the detour phase hologram [29,30], the phase difference between adjacent diffraction sources can be controlled by adjusting their distance, and the lateral phase of the diffracted beam can be modulated by controlling the array parameter. However, the lateral phase modulated by the periodic arrays was linear. Nonperiodic arrays must be introduced to generate a nonlinear phase. The diffraction phase distribution for a nonperiodic array is shown in Fig. 3(d). The phase difference between two neighboring lattice points with sequence numbers n and n − 1 is
2.3 Generation of multiple BSW beams
The focus of BSWs has been extensively studied recently and can be achieved in various ways [15–20]. In this study, we demonstrated that this method can generate a well-focused BSW beam, which is shown schematically in Fig. 4(a). The initial phase of a 2D beam focused at (0, f) must follow [33]
Fig. 4. The focusing of BSWs. (a) Schematics of the generation of the focused beam. (b) The phase distributions of the focused beams. The line and circle symbols indicate the theoretical and deduced phase distributions, respectively. (c) The FDTD result of the focused BSW beam. (d) The transverse (in the x-direction) profile of the focal spot. The FWHM of the peak is 443 nm.
Next, we realized the well-known “self-accelerating” Airy beam, as shown schematically in Fig. 5(a). The Airy beam can maintain its shape during propagation (diffraction-free property) and can re-form after passing through an obstacle (self-healing property). Because of these novel properties, Airy beams have attracted increasing interest [34]. 2D Airy beams were first predicted theoretically [35], and then successfully realized by SPP using different methods [32,36,37]. Airy beams can be directionally generated with a 1.5-power phase
Fig. 5. Airy-BSW beams. (a) Schematics of the generation of the Ariy beam. (b) The phase distributions of the Airy beam. The line and circle symbols indicate the theoretical and deduced phase distributions, respectively. (c) The FDTD result of the Airy-BSW beam. (d) The propagation trajectory of the main lobe. The red solid line and the black dashed line indicate the beam trajectory and the analytical curve, respectively.
Finally, a diffraction-free collimated beam was generated by a well-designed nanoarray using our method, as shown in Fig. 6(a). Cosine-Gaussian (CG) beams are collimated 2D beams that also have diffraction-free and self-healing properties, and they have potential applications in optical force manipulation and high-speed optical interconnects [21,38,39]. Localized CG beams can be created by the interference of two intersecting waves, and the transverse field at y = 0 can be expressed as
Fig. 6. Diffraction-free collimated BSW beams. (a) Schematics of the generation of the CG beam. (b) The phase distributions of the CG beam. The line and circle symbols indicate the theoretical and deduced phase distributions, respectively. (c) The FDTD result of the collimated BSW beam. (d) Transverse intensity profiles along the white dashed lines at different distances (y = 30, 50, and 70 µm) in (b). (e) The distribution of the number of rows in the y-direction (ny). The black line indicates the Gaussian profile. (f) The FDTD result of the CG-BSW beam. (g) Transverse intensity profiles along the white dashed lines at different distances (y = 30, 50, and 70 µm) in (f).
The position of the array units in the x-direction can be determined by solving ${\phi _\textrm{n}}(x) = {\varphi _{\textrm{CG}}}(x)$, and the results are plotted in Fig. 6(b). The position of the array units in the x-direction can be determined by solving ${\phi _\textrm{n}}(x) = {\varphi _{\textrm{CG}}}(x)$, the results of which are plotted in Fig. 6(b). As reported in Ref. [40], the collimated beams generated by phase-only modulation are also diffraction-free. Figure 6(c) shows the BSW beam generated by $\varphi $CG, revealing apparent straight collimation of the center lobe. The collimated beam starts from the initial plane y = 0, and Fig. 6(d) shows the intensity profiles along the white dashed lines at three distances (y = 30, 50, and 70 µm) in Fig. 6(c). The lateral profiles of the center lobe are preserved at all three distances, and their FWHM is about 1.7 µm, which demonstrates the diffraction-free properties of the collimated BSW beam.
However, the collimated beam generated by phase-only modulation has strong side lobes. To weaken the side lobes and highlight the center lobe, the amplitude distribution should also be considered, which is a Gaussian profile.
In our method, the diffraction field generated by each unit has the same intensity. The amplitude distribution can be controlled by the number of rows in the y-direction. where ny0 is the maximum number of rows in the y-direction. Here, ny0 = 15, and ω0 = 16 µm. The distribution of ny is plotted in Fig. 6(e). Figure 6(f) depicts the BSW beam generated by the designed array with significantly weaker side lobes compared to Fig. 6(c). Figure 6(g) shows the intensity profiles at three distances (y = 30, 50, and 70 µm), where the FWHMs of the center lobes are all about 1.7 µm, indicating that the lateral profile of the center lobe is well preserved during propagation. Compared with Fig. 6(d), we can also see that the intensity of the main lobe decays faster in Fig. 6(g), which means that the propagation length of the center lobe is sacrificed to weaken the side lobes.We demonstrated the self-healing properties of the Airy and CS beams as below. As shown in Fig. 7, after propagating through an obstacle (1.8 µm × 1.8 µm), which is larger than the FWHM of the main lobe, both beams can recover their main lobe shape and propagation trajectories after propagating through an obstacle. In fact, all the BSW beams generated using our method have self-healing properties to some extent. Since the obstacle can only block a portion of the diffracted beam, the remaining surface beams resulting from the interference of the diffracted beams from multiple nanoholes can recover their shape and propagation trajectories after encountering perturbations. The unusual properties of these BSW beams can be utilized for useful applications in particle trapping and guiding.
Fig. 7. Demonstration of the self-healing property of the (a) Airy-, (b) CS-BSW beams, by propagating through a blocking obstacle (1.8 µm × 1.8 µm). The distance between the obstacle and starting plane is 20 µm. The BSW fields are simulated using the FDTD method.
3. Conclusion
We proposed a method for generating and controlling BSW fields using in-plane phase modulation. To improve the excitation efficiency, an internal mode was used as a channel between the incident beam and excited BSW; the simulated results showed that the diffraction intensity was strongest when the incident beam was at the excitation angle of the internal mode, which is consistent with our expectations. The initial lateral phase of the BSW beams was modulated by a carefully designed nanoarray structure, which can provide the missing momentum in the x-dimension between the incident and diffracted beams. With this method, subwavelength focusing of the BSWs and self-accelerating Airy-BSW beams were realized. By controlling the amplitude distribution using the number of rows in the y-direction, CG-BSW beams with diffraction-free properties were generated. The self-healing properties of Airy and CG beams were also demonstrated by propagating them through a blocking obstacle, where their main lobe shape and propagation trajectories can be recovered. Owing to its all-dielectric nature, the BSW platform is well-suited as a carrier for planar photonic systems with long propagation lengths. This method of manipulating and generating BSW beams will enable new possibilities in nanophotonics and lead to further applications in lab-on-chip photonic integration.
Funding
National Natural Science Foundation of China (12174056, 62105066, 62275259); Natural Science Foundation of Fujian Province (2021J01159, 2021J01163); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2021232).
Disclosures
The authors declare no conflicts of interest.
Data availability
The data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
References
1. S. Maier, Plasmonics: Fundamentals and Applications (Springer, New York, 2007).
2. P. Yeh, A. Yariv, and C.-S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory*,” J. Opt. Soc. Am. 67(4), 423–438 (1977). [CrossRef]
3. B. V. Lahijani, N. Descharmes, R. Barbey, G. D. Osowiecki, V. J. Wittwer, O. Razskazovskaya, T. Südmeyer, and H. P. Herzig, “Centimeter-Scale Propagation of Optical Surface Waves at Visible Wavelengths,” Adv. Opt. Mater. 10(10), 2102854 (2022). [CrossRef]
4. G. M. Smolik, N. Descharmes, and H. P. Herzig, “Toward Bloch Surface Wave-Assisted Spectroscopy in the Mid-Infrared Region,” ACS Photonics 5(4), 1164–1170 (2018). [CrossRef]
5. R. Badugu, J. Mao, S. Blair, D. Zhang, E. Descrovi, A. Angelini, Y. Huo, and J. R. Lakowicz, “Bloch Surface Wave-Coupled Emission at Ultraviolet Wavelengths,” J. Phys. Chem. C 120(50), 28727–28734 (2016). [CrossRef]
6. E. Descrovi, T. Sfez, M. Quaglio, D. Brunazzo, L. Dominici, F. Michelotti, H. P. Herzig, O. J. F. Martin, and F. Giorgis, “Guided Bloch Surface Waves on Ultrathin Polymeric Ridges,” Nano Lett. 10(6), 2087–2091 (2010). [CrossRef]
7. R. Wang, H. Xia, D. Zhang, J. Chen, L. Zhu, Y. Wang, E. Yang, T. Zang, X. Wen, G. Zou, P. Wang, H. Ming, R. Badugu, and J. R. Lakowicz, “Bloch surface waves confined in one dimension with a single polymeric nanofibre,” Nat. Commun. 8(1), 14330 (2017). [CrossRef]
8. T. Sfez, E. Descrovi, L. Yu, D. Brunazzo, M. Quaglio, L. Dominici, W. Nakagawa, F. Michelotti, F. Giorgis, O. J. F. Martin, and H. P. Herzig, “Bloch surface waves in ultrathin waveguides: near-field investigation of mode polarization and propagation,” J. Opt. Soc. Am. B 27(8), 1617–1625 (2010). [CrossRef]
9. K. R. Safronov, D. N. Gulkin, I. M. Antropov, K. A. Abrashitova, V. O. Bessonov, and A. A. Fedyanin, “Multimode Interference of Bloch Surface Electromagnetic Waves,” ACS Nano 14(8), 10428–10437 (2020). [CrossRef]
10. T. Sfez, E. Descrovi, L. Yu, M. Quaglio, L. Dominici, W. Nakagawa, F. Michelotti, F. Giorgis, and H. P. Herzig, “Two-dimensional optics on silicon nitride multilayer: Refraction of Bloch surface waves,” Appl. Phys. Lett. 96(15), 151101 (2010). [CrossRef]
11. R. Dubey, B. Vosoughi Lahijani, E. Barakat, M. Häyrinen, M. Roussey, M. Kuittinen, and H. P. Herzig, “Near-field characterization of a Bloch-surface-wave-based 2D disk resonator,” Opt. Lett. 41(21), 4867–4870 (2016). [CrossRef]
12. G. A. Rodriguez, D. Aurelio, M. Liscidini, and S. M. Weiss, “Bloch surface wave ring resonator based on porous silicon,” Appl. Phys. Lett. 115(1), 011101 (2019). [CrossRef]
13. R. Dubey, B. V. Lahijani, M. Häyrinen, M. Roussey, M. Kuittinen, and H. P. Herzig, “Ultra-thin Bloch-surface-wave-based reflector at telecommunication wavelength,” Photonics Res. 5(5), 494–499 (2017). [CrossRef]
14. L. Yu, E. Barakat, J. Di Francesco, and H. P. Herzig, “Two-dimensional polymer grating and prism on Bloch surface waves platform,” Opt. Express 23(25), 31640–31647 (2015). [CrossRef]
15. L. Yu, E. Barakat, T. Sfez, L. Hvozdara, J. D. Francesco, and H. P. Herzig, “Manipulating Bloch surface waves in 2D: a platform concept-based flat lens,” Light: Sci. Appl. 3(1), e124 (2014). [CrossRef]
16. A. Angelini, A. Lamberti, S. Ricciardi, F. Frascella, P. Munzert, N. D. Leo, and E. Descrovi, “In-plane 2D focusing of surface waves by ultrathin refractive structures,” Opt. Lett. 39(22), 6391–6394 (2014). [CrossRef]
17. M.-S. Kim, B. V. Lahijani, N. Descharmes, J. Straubel, F. Negredo, C. Rockstuhl, M. Häyrinen, M. Kuittinen, M. Roussey, and H. P. Herzig, “Subwavelength Focusing of Bloch Surface Waves,” ACS Photonics 4(6), 1477–1483 (2017). [CrossRef]
18. Y. Augenstein, A. Vetter, B. V. Lahijani, H. P. Herzig, C. Rockstuhl, and M.-S. Kim, “Inverse photonic design of functional elements that focus Bloch surface waves,” Light: Sci. Appl. 7(1), 104 (2018). [CrossRef]
19. X. Lei, Y. Ren, Y. Lu, and P. Wang, “Lens for Efficient Focusing of Bloch Surface Waves,” Phys. Rev. Appl. 10(4), 044032 (2018). [CrossRef]
20. F. Feng, S.-B. Wei, L. Li, C.-J. Min, X.-C. Yuan, and M. Somekh, “Spin-orbit coupling controlled near-field propagation and focusing of Bloch surface wave,” Opt. Express 27(20), 27536–27545 (2019). [CrossRef]
21. R. Wang, Y. Wang, D. Zhang, G. Si, L. Zhu, L. Du, S. Kou, R. Badugu, M. Rosenfeld, J. Lin, P. Wang, H. Ming, X. Yuan, and J. R. Lakowicz, “Diffraction-Free Bloch Surface Waves,” ACS Nano 11(6), 5383–5390 (2017). [CrossRef]
22. J. Chen, D. Zhang, P. Wang, H. Ming, and J. R. Lakowicz, “Strong Polarization Transformation of Bloch Surface Waves,” Phys. Rev. Appl. 9(2), 024008 (2018). [CrossRef]
23. R. Wang, J. Chen, Y. Xiang, Y. Kuai, P. Wang, H. Ming, J. R. Lakowicz, and D. Zhang, “Two-Dimensional Photonic Devices based on Bloch Surface Waves with One-Dimensional Grooves,” Phys. Rev. Appl. 10(2), 024032 (2018). [CrossRef]
24. U. Stella, L. Boarino, N. D. Leo, P. Munzert, and E. Descrovi, “Enhanced Directional Light Emission Assisted by Resonant Bloch Surface Waves in Circular Cavities,” ACS Photonics 6(8), 2073–2082 (2019). [CrossRef]
25. D. N. Gulkin, A. A. Popkova, B. I. Afinogenov, D. A. Shilkin, K. Kuršelis, B. N. Chichkov, V. O. Bessonov, and A. A. Fedyanin, “Mie-driven directional nanocoupler for Bloch surface wave photonic platform,” Nanophotonics 10(11), 2939–2947 (2021). [CrossRef]
26. X. Lei, R. Wang, L. Liu, C. Xu, A. Wu, and Q. Zhan, “Multifunctional on-chip directional coupler for spectral and polarimetric routing of Bloch surface wave,” Nanophotonics 11(21), 4627–4636 (2022). [CrossRef]
27. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17(5), 929–941 (1999). [CrossRef]
28. A. Taflove and S. C. Hagness, Computational electrodynamics: the finite-difference time-domain method (Artech House, Boston, 2005) 3rd ed.
29. C. Min, J. Liu, T. Lei, G. Si, Z. Xie, J. Lin, L. Du, and X. Yuan, “Plasmonic nano-slits assisted polarization selective detour phase meta-hologram,” Laser Photonics Rev. 10(6), 978–985 (2016). [CrossRef]
30. 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). [CrossRef]
31. Y. Kuai, J. Chen, X. Tang, Y. Xiang, F. Lu, C. Kuang, L. Xu, W. Shen, J. Cheng, H. Gui, G. Zou, P. Wang, H. Ming, J. Liu, X. Liu, J. R. Lakowicz, and D. Zhang, “Label-free surface-sensitive photonic microscopy with high spatial resolution using azimuthal rotation illumination,” Sci. Adv. 5(3), eaav5335 (2019). [CrossRef]
32. L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy Beam Generated by In-Plane Diffraction,” Phys. Rev. Lett. 107(12), 126804 (2011). [CrossRef]
33. 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]
34. N. K. Efremidis, Z. Chen, M. Segev, and D. N. Christodoulides, “Airy beams and accelerating waves: an overview of recent advances,” Optica 6(5), 686–701 (2019). [CrossRef]
35. A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35(12), 2082–2084 (2010). [CrossRef]
36. P. Zhang, S. Wang, Y. Liu, X. Yin, C. Lu, Z. Chen, and X. Zhang, “Plasmonic Airy beams with dynamically controlled trajectories,” Opt. Lett. 36(16), 3191–3193 (2011). [CrossRef]
37. A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, “Generation and Near-Field Imaging of Airy Surface Plasmons,” Phys. Rev. Lett. 107(11), 116802 (2011). [CrossRef]
38. J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss Plasmon Beam: A Localized Long-Range Nondiffracting Surface Wave,” Phys. Rev. Lett. 109(9), 093904 (2012). [CrossRef]
39. I. Epstein, R. Remez, Y. Tsur, and A. Arie, “Generation of intensity-controlled two-dimensional shape-preserving beams in plasmonic lossy media,” Optica 3(1), 15–19 (2016). [CrossRef]
40. L. Li, T. Li, S. M. Wang, and S. N. Zhu, “Collimated Plasmon Beam: Nondiffracting versus Linearly Focused,” Phys. Rev. Lett. 110(4), 046807 (2013). [CrossRef]
