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Diffraction-free surface waves by metasurfaces Yun Bo Li, Ben Geng Cai, Xiang Wan, and Tie Jun Cui* State Key Laboratory of Millimetre Waves, School of Information Science and Engineering, Southeast University, Nanjing 210096, China *Corresponding author: [email protected] Received July 11, 2014; revised September 5, 2014; accepted September 9, 2014; posted September 10, 2014 (Doc. ID 216450); published October 9, 2014 We propose a method to design and realize planar Bessel lens using artificial metasurfaces to produce diffractionfree surface waves. The planar Bessel lens is composed of two sublenses: a half Maxwell fisheye lens which can shape the surface cylindrical waves to surface plane waves, and an inhomogeneous flat lens which can convert the surface plane waves into approximate diffraction-free surface waves in a diamond-shaped focusing area. Through the planar Bessel lens, a point source on the metasurface directly radiates the diffraction-free surface waves. In realization, we construct the inhomogeneous metasurfaces by subwavelength metallic patches printed on a grounded dielectric substrate. Simulation and experimental results have good agreements, which jointly show the formation of the diffraction-free surface waves in the microwave band. © 2014 Optical Society of America OCIS codes: (240.0240) Optics at surfaces; (240.0310) Thin films; (240.6690) Surface waves. http://dx.doi.org/10.1364/OL.39.005888
Compared with bulk metamaterials [1,2], metasurfaces are less lossy, are cheaper, and have lower profiles. And hence, they have found many electromagnetic (EM) applications in recent years, such as meta-transmission arrays, surface wave lenses, and leaky-wave antennas. In controlling surface waves, the metasurface functions similar to a bulk metamaterial for freely propagating waves through the spatial-dependent phase accumulation. However, in controlling spatial propagating waves, the metasurface is a plane of discontinuity where the phase and/or amplitude of the impinging wave changes over a distances much smaller than the wavelength. The metasurfaces are considered as period or nonperiod subwavelength textures, and there exist two main approaches to model their EM characteristics: the generalized sheet transition condition method [3] and transverse resonance method [4]. In the first approach, small scatterers are characterized as electric and magnetic polarization densities, from which reflection and transmission coefficients of metasurfaces or metafilms were obtained [5,6]. Such a method has been used in cascaded transmission arrays [7–10] and Huygens surfaces [11] to control the spatial EM waves. Recently, special metasurfaces with gradient phase changes from 0 to 2π have received great attention, which are used to produce anomalous reflections and refractions and other unprecedented controls of spatial EM waves [12–20]. On the other hand, the transverse resonance method has been introduced [4] to model the EM characteristics of surface waves, from which the surface refractive indices or surface impedances of metasurfaces can be extracted. Using the sinusoidally-modulated methods [21–23] or the holographic interference patterns [24] to shape the distribution of surface impedance, low-profile leaky-wave antennas that are equivalent to the second radiations have been designed. Furthermore, by tailoring the distribution of surface refractive indices, the surface wavefronts can be manipulated to realize surface-wave lenses [25,26], and even the surface transformationoptics lens [27]. In this Letter, we propose a method to design and realize planar Bessel lens that can produce diffraction-free surface waves using metasurfaces. Theoretically, a 0146-9592/14/205888-04$15.00/0
Bessel beam resists diffraction, and its transverse shape defined by zeroth-order Bessel function of the first kind stays almost fixed [28–34], which has great advantage in energy transmission. One of the methods to realize the Bessel lens is making the Gaussian beam pass through an axicon to generate a diamond focusing area [29], which was considered as a Bessel information zone in the near-field region, and the beam will be diffracted to a cone-like wave in the far-field region. The similar task was accomplished by a simple holographic method [30]. In the meantime, the generation of Bessel beams using a leaky-radial waveguide was also presented theoretically [31] and later was verified by experiments [32], in which the waveguide is composed of the grounded capacitive sheet to shape the normal electric fields of the leaky waves as the zeroth-order Bessel function. Alternatively, evanescent Bessel beams generated by a near-field plate were also presented in Ref. [33] by the same group. Recently, the spatial Bessel beams have been realized from subwavelength aperture by using metallic circular grating-like structure [34] and metamaterial Huygens’ surface [11]. However, the research on the generation of diffraction-free surface waves has had only a few reports [35–37] to the best of our knowledge, in which Ref. [37] could be considered as the first generation of the diffraction-free surface waves by using the metallic gratings in the optical frequency band. Here, we propose a new method to design a planar Bessel lens using gradient metasurfaces so as to realize diffraction-free surface waves. Compared to the published work [35–37], our design transforms the fields due to a point source to a diffraction-free surface wave without the need to excite Gaussian beams in the microwave band. In particular, we make use of a planar half Maxwell fisheye lens to shape the surface cylindrical waves emitted from a point source to surface plane waves, and use a gradient-index flat lens to generate a diamond-shaped focusing area to support diffraction-free surface waves. We realize the planar Bessel lens by subwavelength metallic patches with different scales printed on a grounded dielectric substrate. The simulation and experiment results have good agreements, showing the diffraction-free properties of surface waves. Compared © 2014 Optical Society of America
October 15, 2014 / Vol. 39, No. 20 / OPTICS LETTERS
to the surface waveguide [38], the Bessel-like lens has good ability for highly directed energy transmission. The proposed design may also be extended to the optical frequency by using the graphene structure [39]. According to Ref. [28], the diffraction-free zeroth-order Bessel-like beam can be generated by the addition of plane waves with the fixed elevation angle θ and changing azimuth angles from 0 to 2π, which form a cone-like beam with the character of rotational symmetry. Similar to the method in Ref. [29], in which the three-dimensional (3D) spatial Bessel-like beam is generated by an axicon under the Gaussian beam excitation, we can realize the surface-wave Bessel beams by only considering the face of rotational symmetry in 3D circumstances. In this way, the approximate surface-wave Bessel beam will be obtained. We remark that the electromagnetic field distribution of the approximate surface-wave Bessel beam is similar to the description in Ref. [37], which has the same analysis as the 2D circumstance of the axicon by adding the term of evanescent wave. We propose to design the surface-wave Bessel lens using metasurface, which is composed of a planar half Maxwell fisheye lens [40] and a planar flat lens [41] to imitate the 2D axicon, as illustrated in Fig. 1. Here, D and Z max are the aperture size and focusing length of the surface-wave Bessel lens, respectively. The planar half Maxwell fisheye lens aims to transfer the cylindricallike surface waves generated by a point source on the metasurface to plane surface waves. According to Ref. [41], the medium with the distribution of linearly refractive index can cause deflection of incident plane wavefront, and hence the planar flat lens is used to generate the required linear phase distribution by designing the gradient surface index of refraction. If we only use one planar flat lens separated from the point source to directly realize the Bessel-beam, it requires much larger surface refractive index which is hard to generate. Even though the general surface refractive index is utilized, the thickness of the flat lens will be increased to satisfy the big phase shifting, which introduces great errors in design. So the scheme of combining the half Maxwell fisheye lens with the flat lens is selected. By such a design, the ray-tracing principle demonstrates that the planar flat lens will deflect the normally incident plane surface waves into two oblique plane surface waves, which can focus to the Bessel information zone, as displayed in Fig. 1. The maximum focusing distance Z max of the Bessel lens is determined by the deflection angle φ (see Fig. 1), which is relevant to the gradient of the surface refractive index along the flat lens. Therefore, the
Fig. 1. Scheme to realize a surface-wave Bessel lens.
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proposed method will directly transform the cylindricallike surface waves from the point source on the metasurface to diffraction-free surface waves. To realize the surface-wave Bessel lens, we choose a subwavelength metallic square patch on a grounded dielectric substrate as the basic unit cell of metasurface, which can support transverse-magnetic (TM) polarized surface waves [4]. By varying the gap between square patches gradually, we can obtain different surface indices of refraction to fit the required distributions of planar Maxwell fisheye lens and flat lens. For slowly gradient variations between patches, which can be considered as uniform textures approximately, we make use of the periodic boundary condition to simulate the unit cells using eigenmode solver to obtain the dispersion curves, which are illustrated in Fig. 2(a). It is clear that as the gap of square patches decreases, the phase difference in the wave propagating direction becomes large so that the phase velocity of surface waves is reduced, corresponding to higher surface refractive index and surface impedance. To manipulate the surface-wave fronts using the artificial metasurface, here we only study the surface refractive index n, which can be obtained by the formula nt φc∕dω [24], where φ is the phase difference across the unit cell with the period d, and c and ω are the light speed in free space and the angular frequency, respectively. We have extracted the surface refractive indices at 12 GHz by changing the gap size of the unit cell from 0.2 to 2.4 mm with step of 0.1 mm. Then we fit the extracted values using cubic polynomials to determine the relationship between the surface refractive index and gap. The planar half Maxwell fisheye lens (HMFL), which transfers the cylindrical-like surface waves from a point source to plane surface waves, should satisfy the refractive index distribution: nHMFL
2 r < R; 1 r∕R2
(1)
where r is the distance away from the lens center, and R is the lens radius. Since the longer wave path is
Fig. 2. (a) The structure of metallic patch on a grounded dielectric substrate. The green area indicates dielectric, and the yellow indicates metal. The period of square lattice is d 3.5 mm, and the gap varies from 0.2 to 2.4 mm. TP-2 is chosen as the dielectric substrate with the relative permittivity, 6.15, and the thickness, 1.57 mm. The dispersion curves of unit cells with different gaps are also given, and the black oblique line is the light line. (b) The surface refractive index distributions of planar half Maxwell fisheye lens and flat lens (φ 15°). In our design, R is chosen as 49 mm, and H is chosen as 35 mm.
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proportional to the larger wave’s attenuation, the surface plane wave generated by the HMFL will approximate the Gauss-like surface wave. The refractive index distribution of flat lens (FL) should have the form of nFL − sinφx∕t n0 x < H
(2)
to satisfy the required phase gradient of the surface-wave Bessel beam. Here, t is thickness of the slab lens in the x direction, n0 is the central surface refractive index, and H is half height of the lens in the x direction. The refractive index distributions of the two planar lenses are described in Fig. 2(b). After mapping the surface refractive index to the gap dimension, we can tailor the two lenses by shaping the subwavelength and quasi-periodic textures with gradient gap change. The designed metasurface structure of the surfacewave Bessel lens is shown in Fig. 3(a). For the planar half Maxwell fisheye lens, the central index of refraction is 2, and the marginal values are approximately 1.1. The whole size of the half Maxwell fisheye lens is designed as 84 mm × 49 mm with trimming a few marginal units by large gap to ensure the quasi-periodic condition. Meanwhile, we introduce a transmission layer (TL) that is composed of periodic unit cells with gap size 0.8 mm in front of the half fisheye lens, as demonstrated in Fig. 3(a). TL can also be considered as the matching layer, which is designed by using optimization. For the planar flat lens, we choose the lens parameters as φ 15°, T 10.5 mm, and n0 1.983. Again, to obtain better effect of surfacewave Bessel beam, another TL is introduced in front of the flat lens, as illustrated in Fig. 3(a). If the loss in the substrate is very small and the maximum size of the practical material is large enough to support surface waves, the deflective angle φ can be designed smaller to obtain the long propagation distance of the diffraction-free surface waves. However, when φ is close to zero, we will not obtain the good profiles of surface Bessel-like waves. The performance of planar half Maxwell fisheye lens by adding TL is verified by full-wave simulation results of near electric fields using the commercial software, CST Microwave Studio, as shown in Fig. 4(a). In our simulation, all boundaries are set as open. We clearly observe that the cylindrical-like surface waves radiated by a point source on the metasurface are converted to
Fig. 3. (a) The designed metasurface to fulfill the surfacewave Bessel lens, which is composed of a planar half Maxwell fisheye lens and a planar flat lens with gradient refractive indices. After the Bessel lens, the same grounded dielectric substrate is used to support the surface waves. (b) The fabricated sample of the surface-wave Bessel lens. (c) The measurement setup to measure the surface-wave Bessel beam.
Fig. 4. Full-wave simulation results of near electric fields (the y components). (a) The planar half Maxwell fisheye lens with TL, producing excellent surface plane waves. (b) The planar half Maxwell fisheye lens and planar flat lens with two TLs, producing high-quality Bessel beams.
high-quality surface plane waves. Incorporating with the planar flat lens and additional another TL, the simulation results demonstrate a focusing area that is similar to the diamond shape, as displayed in Fig. 4(b). Here, the focusing length, Z max of the planar Bessel lens is about 110 mm, corresponding to a deflection angle of φ 17.7° of the flat lens, which is very close to the designed value φ 15°. The Bessel-like profiles of electric fields at different distances from the left edge of the focusing area in the propagating direction are illustrated in Fig. 5(a), showing good Bessel-beam performance. If the aperture size of the Bessel lens increases, we expect much better performance of Bessel beam by reducing the surface wave diffractions in the lens edges. To further confirm the surface-wave Bessel beam experimentally, we fabricate the proposed planar Bessel lens using the printed circuit board (PCB), as shown in Fig. 3(b). We utilize a microwave near-field scanning apparatus [see Fig. 3(c)] to scan the electric-field distributions of the sample. In experiments, we use a monopole antenna to imitate the point source, which can excite the TM-mode surface waves. The sample is moved by a 2D stepping platform and the probing antenna is fixed perpendicularly to the metasurface to measure the y components of electric fields. The arrangement of experiment is displayed in Fig. 3(c). Similar to the numerical situation, the measurement results of electric-field profiles at different locations in the focusing area at
Fig. 5. Simulation and experiment results of the electric-field profiles at different locations along the propagating direction in the focusing area at 12 GHz, showing good Bessel-like performance. (a) Simulation results. (b) Measurement results.
October 15, 2014 / Vol. 39, No. 20 / OPTICS LETTERS
12 GHz are presented in Fig. 5(b). Clearly, the Bessel-like profiles of electric fields with first zero are observed in the experiments. The bandwidth of the surface-wave lens is approximately 1.5 GHz from 11.5 to 13 GHz. The small discrepancy between the simulation and measurement is mainly caused by the feeding location of monopole antenna and the sample fabrication. In summary, we have theoretically proposed the surface-wave Bessel beams and presented a method to realize the surface-wave Bessel lens, which is composed of a planar half Maxwell fisheye lens and a planar flat lens with gradient surface indices of refraction. Comparing to the conventional methods in bulk Bessel lenses, the Gaussian-beam illumination is replaced by a point source in the proposed method, which makes the realization of Bessel beams simpler and more flexible. The surfacewave Bessel lens has good application potentials in developing highly directed and concentrated energy transmission and nondiffraction surface waveguide for near-field sensors and biomedical devices [33]. This work was supported by National High Tech (863) Projects (2012AA030402 and 2011AA010202), National Natural Science Foundation of China (60990320, 61138001), and 111 Project (111-2-05). References 1. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Science 305, 788 (2004). 2. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, Science 314, 977 (2006). 3. E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, IEEE Trans. Antennas Propag. 51, 2641 (2003). 4. R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, 1960). 5. C. L. Holloway, M. A. Mohamed, E. F. Kuester, and A. Dienstfrey, IEEE Trans. Antennas Propag. 47, 853 (2005). 6. C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O’ Hara, J. Booth, and D. R. Smith, IEEE Trans. Antennas Propag. 54, 10 (2012). 7. Y. Zhao, N. Engheta, and A. Alù, Metamaterials 5, 90 (2011). 8. F. Monticone, N. M. Estakhri, and A. Alù, Phys. Rev. Lett. 110, 203903 (2013). 9. C. Pfeiffer and A. Grbic, Appl. Phys. Lett. 102, 231116 (2013). 10. C. Pfeiffer and A. Grbic, IEEE Trans. Microwave Theory Tech. 61, 4407 (2013). 11. C. Pfeiffer and A. Grbic, Phys. Rev. Lett. 110, 197401 (2013). 12. N. Yu, P. Genevet, M. A. Cats, F. Aieta, J. Tetienne, F. Cappaso, and Z. Gaburro, Science 334, 333 (2011). 13. F. Aiet, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, Nano Lett. 12, 4932 (2012).
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14. S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, Nat. Mater. 11, 426 (2012). 15. X. Chen, L. Huang, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, C. Qiu, S. Zhang, and T. Zentgraf, Nat. Commun. 3, 1198 (2012). 16. X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, Science 339, 1405 (2013). 17. J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X. Yuan, and F. Capasso, Science 340, 331 (2013). 18. A. E. Miroshnichenko and Y. S. Kivshar, Science 340, 283 (2013). 19. N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H. Chen, Science 340, 1304 (2013). 20. N. Yu and F. Capasso, Nat. Mater. 13, 139 (2014). 21. A. A. Oliner and A. Hessel, IEEE Trans. Antennas Propag. 7, 201 (1959). 22. G. Minatti, F. Caminita, M. Casaletti, and S. Maci, IEEE Trans. Antennas Propag. 59, 4436 (2011). 23. A. M. Patel and A. Grbic, IEEE Trans. Antennas Propag. 59, 2087 (2011). 24. B. H. Fong, J. S. Colburn, J. J. Ottusch, J. L. Visher, and D. F. Sievenpiper, IEEE Trans. Antennas Propag. 58, 3212 (2010). 25. M. Bosiljevac, M. Casaletti, F. Caminita, Z. Sipus, and S. Maci, IEEE Trans. Antennas Propag. 60, 4065 (2012). 26. J. A. Dockrey, M. J. Lockyear, S. J. Berry, S. A. R. Horsley, J. R. Sambles, and A. P. Hibbins, Phys. Rev. B 87, 125137 (2013). 27. X. Wan, W. X. Jiang, H. F. Ma, and T. J. Cui, Appl. Phys. Lett. 104, 151601 (2014). 28. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987). 29. G. Scott and N. McArdle, Opt. Eng. 31, 2640 (1992). 30. J. Turunen, A. Vasara, and A. T. Friberg, J. Opt. Soc. Am. 27, 3959 (1988). 31. M. Ettorre and A. Grbic, IEEE Trans. Antennas Propag. 60, 3605 (2012). 32. M. Ettorre, S. M. Rudolph, and A. Grbic, IEEE Trans. Antennas Propag. 60, 2645 (2012). 33. M. F. Imani and A. Grbic, IEEE Trans. Antennas Propag. 60, 3155 (2012). 34. Z. F. Li, K. B. Alici, H. Caglayan, and E. Ozbay, Phys. Rev. Lett. 102, 143901 (2009). 35. C. E. Garcia-Ortiz, V. Coello, Z. H. Han, and S. I. Bozhevolnyi, Opt. Lett. 38, 905 (2013). 36. H. Hurt and M. Turduev, J. Opt. Soc. Am. B 29, 1245 (2012). 37. J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, Phys. Rev. Lett. 109, 093904 (2012). 38. D. J. Gregoire and A. V. Kabakian, IEEE Antennas Wirel. Propag. Lett. 10, 1512 (2011). 39. A. Vakil and N. Engheta, Science 332, 1291 (2011). 40. H. F. Ma, B. G. Cai, T. X. Zhang, Y. Yang, W. X. Jiang, and T. J. Cui, IEEE Trans. Antennas Propag. 61, 2561 (2013). 41. D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, Phys. Rev. E 71, 036609 (2005).
OPTICS LETTERS / Vol. 39, No. 20 / October 15, 2014
Diffraction-free surface waves by metasurfaces Yun Bo Li, Ben Geng Cai, Xiang Wan, and Tie Jun Cui* State Key Laboratory of Millimetre Waves, School of Information Science and Engineering, Southeast University, Nanjing 210096, China *Corresponding author: [email protected] Received July 11, 2014; revised September 5, 2014; accepted September 9, 2014; posted September 10, 2014 (Doc. ID 216450); published October 9, 2014 We propose a method to design and realize planar Bessel lens using artificial metasurfaces to produce diffractionfree surface waves. The planar Bessel lens is composed of two sublenses: a half Maxwell fisheye lens which can shape the surface cylindrical waves to surface plane waves, and an inhomogeneous flat lens which can convert the surface plane waves into approximate diffraction-free surface waves in a diamond-shaped focusing area. Through the planar Bessel lens, a point source on the metasurface directly radiates the diffraction-free surface waves. In realization, we construct the inhomogeneous metasurfaces by subwavelength metallic patches printed on a grounded dielectric substrate. Simulation and experimental results have good agreements, which jointly show the formation of the diffraction-free surface waves in the microwave band. © 2014 Optical Society of America OCIS codes: (240.0240) Optics at surfaces; (240.0310) Thin films; (240.6690) Surface waves. http://dx.doi.org/10.1364/OL.39.005888
Compared with bulk metamaterials [1,2], metasurfaces are less lossy, are cheaper, and have lower profiles. And hence, they have found many electromagnetic (EM) applications in recent years, such as meta-transmission arrays, surface wave lenses, and leaky-wave antennas. In controlling surface waves, the metasurface functions similar to a bulk metamaterial for freely propagating waves through the spatial-dependent phase accumulation. However, in controlling spatial propagating waves, the metasurface is a plane of discontinuity where the phase and/or amplitude of the impinging wave changes over a distances much smaller than the wavelength. The metasurfaces are considered as period or nonperiod subwavelength textures, and there exist two main approaches to model their EM characteristics: the generalized sheet transition condition method [3] and transverse resonance method [4]. In the first approach, small scatterers are characterized as electric and magnetic polarization densities, from which reflection and transmission coefficients of metasurfaces or metafilms were obtained [5,6]. Such a method has been used in cascaded transmission arrays [7–10] and Huygens surfaces [11] to control the spatial EM waves. Recently, special metasurfaces with gradient phase changes from 0 to 2π have received great attention, which are used to produce anomalous reflections and refractions and other unprecedented controls of spatial EM waves [12–20]. On the other hand, the transverse resonance method has been introduced [4] to model the EM characteristics of surface waves, from which the surface refractive indices or surface impedances of metasurfaces can be extracted. Using the sinusoidally-modulated methods [21–23] or the holographic interference patterns [24] to shape the distribution of surface impedance, low-profile leaky-wave antennas that are equivalent to the second radiations have been designed. Furthermore, by tailoring the distribution of surface refractive indices, the surface wavefronts can be manipulated to realize surface-wave lenses [25,26], and even the surface transformationoptics lens [27]. In this Letter, we propose a method to design and realize planar Bessel lens that can produce diffraction-free surface waves using metasurfaces. Theoretically, a 0146-9592/14/205888-04$15.00/0
Bessel beam resists diffraction, and its transverse shape defined by zeroth-order Bessel function of the first kind stays almost fixed [28–34], which has great advantage in energy transmission. One of the methods to realize the Bessel lens is making the Gaussian beam pass through an axicon to generate a diamond focusing area [29], which was considered as a Bessel information zone in the near-field region, and the beam will be diffracted to a cone-like wave in the far-field region. The similar task was accomplished by a simple holographic method [30]. In the meantime, the generation of Bessel beams using a leaky-radial waveguide was also presented theoretically [31] and later was verified by experiments [32], in which the waveguide is composed of the grounded capacitive sheet to shape the normal electric fields of the leaky waves as the zeroth-order Bessel function. Alternatively, evanescent Bessel beams generated by a near-field plate were also presented in Ref. [33] by the same group. Recently, the spatial Bessel beams have been realized from subwavelength aperture by using metallic circular grating-like structure [34] and metamaterial Huygens’ surface [11]. However, the research on the generation of diffraction-free surface waves has had only a few reports [35–37] to the best of our knowledge, in which Ref. [37] could be considered as the first generation of the diffraction-free surface waves by using the metallic gratings in the optical frequency band. Here, we propose a new method to design a planar Bessel lens using gradient metasurfaces so as to realize diffraction-free surface waves. Compared to the published work [35–37], our design transforms the fields due to a point source to a diffraction-free surface wave without the need to excite Gaussian beams in the microwave band. In particular, we make use of a planar half Maxwell fisheye lens to shape the surface cylindrical waves emitted from a point source to surface plane waves, and use a gradient-index flat lens to generate a diamond-shaped focusing area to support diffraction-free surface waves. We realize the planar Bessel lens by subwavelength metallic patches with different scales printed on a grounded dielectric substrate. The simulation and experiment results have good agreements, showing the diffraction-free properties of surface waves. Compared © 2014 Optical Society of America
October 15, 2014 / Vol. 39, No. 20 / OPTICS LETTERS
to the surface waveguide [38], the Bessel-like lens has good ability for highly directed energy transmission. The proposed design may also be extended to the optical frequency by using the graphene structure [39]. According to Ref. [28], the diffraction-free zeroth-order Bessel-like beam can be generated by the addition of plane waves with the fixed elevation angle θ and changing azimuth angles from 0 to 2π, which form a cone-like beam with the character of rotational symmetry. Similar to the method in Ref. [29], in which the three-dimensional (3D) spatial Bessel-like beam is generated by an axicon under the Gaussian beam excitation, we can realize the surface-wave Bessel beams by only considering the face of rotational symmetry in 3D circumstances. In this way, the approximate surface-wave Bessel beam will be obtained. We remark that the electromagnetic field distribution of the approximate surface-wave Bessel beam is similar to the description in Ref. [37], which has the same analysis as the 2D circumstance of the axicon by adding the term of evanescent wave. We propose to design the surface-wave Bessel lens using metasurface, which is composed of a planar half Maxwell fisheye lens [40] and a planar flat lens [41] to imitate the 2D axicon, as illustrated in Fig. 1. Here, D and Z max are the aperture size and focusing length of the surface-wave Bessel lens, respectively. The planar half Maxwell fisheye lens aims to transfer the cylindricallike surface waves generated by a point source on the metasurface to plane surface waves. According to Ref. [41], the medium with the distribution of linearly refractive index can cause deflection of incident plane wavefront, and hence the planar flat lens is used to generate the required linear phase distribution by designing the gradient surface index of refraction. If we only use one planar flat lens separated from the point source to directly realize the Bessel-beam, it requires much larger surface refractive index which is hard to generate. Even though the general surface refractive index is utilized, the thickness of the flat lens will be increased to satisfy the big phase shifting, which introduces great errors in design. So the scheme of combining the half Maxwell fisheye lens with the flat lens is selected. By such a design, the ray-tracing principle demonstrates that the planar flat lens will deflect the normally incident plane surface waves into two oblique plane surface waves, which can focus to the Bessel information zone, as displayed in Fig. 1. The maximum focusing distance Z max of the Bessel lens is determined by the deflection angle φ (see Fig. 1), which is relevant to the gradient of the surface refractive index along the flat lens. Therefore, the
Fig. 1. Scheme to realize a surface-wave Bessel lens.
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proposed method will directly transform the cylindricallike surface waves from the point source on the metasurface to diffraction-free surface waves. To realize the surface-wave Bessel lens, we choose a subwavelength metallic square patch on a grounded dielectric substrate as the basic unit cell of metasurface, which can support transverse-magnetic (TM) polarized surface waves [4]. By varying the gap between square patches gradually, we can obtain different surface indices of refraction to fit the required distributions of planar Maxwell fisheye lens and flat lens. For slowly gradient variations between patches, which can be considered as uniform textures approximately, we make use of the periodic boundary condition to simulate the unit cells using eigenmode solver to obtain the dispersion curves, which are illustrated in Fig. 2(a). It is clear that as the gap of square patches decreases, the phase difference in the wave propagating direction becomes large so that the phase velocity of surface waves is reduced, corresponding to higher surface refractive index and surface impedance. To manipulate the surface-wave fronts using the artificial metasurface, here we only study the surface refractive index n, which can be obtained by the formula nt φc∕dω [24], where φ is the phase difference across the unit cell with the period d, and c and ω are the light speed in free space and the angular frequency, respectively. We have extracted the surface refractive indices at 12 GHz by changing the gap size of the unit cell from 0.2 to 2.4 mm with step of 0.1 mm. Then we fit the extracted values using cubic polynomials to determine the relationship between the surface refractive index and gap. The planar half Maxwell fisheye lens (HMFL), which transfers the cylindrical-like surface waves from a point source to plane surface waves, should satisfy the refractive index distribution: nHMFL
2 r < R; 1 r∕R2
(1)
where r is the distance away from the lens center, and R is the lens radius. Since the longer wave path is
Fig. 2. (a) The structure of metallic patch on a grounded dielectric substrate. The green area indicates dielectric, and the yellow indicates metal. The period of square lattice is d 3.5 mm, and the gap varies from 0.2 to 2.4 mm. TP-2 is chosen as the dielectric substrate with the relative permittivity, 6.15, and the thickness, 1.57 mm. The dispersion curves of unit cells with different gaps are also given, and the black oblique line is the light line. (b) The surface refractive index distributions of planar half Maxwell fisheye lens and flat lens (φ 15°). In our design, R is chosen as 49 mm, and H is chosen as 35 mm.
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proportional to the larger wave’s attenuation, the surface plane wave generated by the HMFL will approximate the Gauss-like surface wave. The refractive index distribution of flat lens (FL) should have the form of nFL − sinφx∕t n0 x < H
(2)
to satisfy the required phase gradient of the surface-wave Bessel beam. Here, t is thickness of the slab lens in the x direction, n0 is the central surface refractive index, and H is half height of the lens in the x direction. The refractive index distributions of the two planar lenses are described in Fig. 2(b). After mapping the surface refractive index to the gap dimension, we can tailor the two lenses by shaping the subwavelength and quasi-periodic textures with gradient gap change. The designed metasurface structure of the surfacewave Bessel lens is shown in Fig. 3(a). For the planar half Maxwell fisheye lens, the central index of refraction is 2, and the marginal values are approximately 1.1. The whole size of the half Maxwell fisheye lens is designed as 84 mm × 49 mm with trimming a few marginal units by large gap to ensure the quasi-periodic condition. Meanwhile, we introduce a transmission layer (TL) that is composed of periodic unit cells with gap size 0.8 mm in front of the half fisheye lens, as demonstrated in Fig. 3(a). TL can also be considered as the matching layer, which is designed by using optimization. For the planar flat lens, we choose the lens parameters as φ 15°, T 10.5 mm, and n0 1.983. Again, to obtain better effect of surfacewave Bessel beam, another TL is introduced in front of the flat lens, as illustrated in Fig. 3(a). If the loss in the substrate is very small and the maximum size of the practical material is large enough to support surface waves, the deflective angle φ can be designed smaller to obtain the long propagation distance of the diffraction-free surface waves. However, when φ is close to zero, we will not obtain the good profiles of surface Bessel-like waves. The performance of planar half Maxwell fisheye lens by adding TL is verified by full-wave simulation results of near electric fields using the commercial software, CST Microwave Studio, as shown in Fig. 4(a). In our simulation, all boundaries are set as open. We clearly observe that the cylindrical-like surface waves radiated by a point source on the metasurface are converted to
Fig. 3. (a) The designed metasurface to fulfill the surfacewave Bessel lens, which is composed of a planar half Maxwell fisheye lens and a planar flat lens with gradient refractive indices. After the Bessel lens, the same grounded dielectric substrate is used to support the surface waves. (b) The fabricated sample of the surface-wave Bessel lens. (c) The measurement setup to measure the surface-wave Bessel beam.
Fig. 4. Full-wave simulation results of near electric fields (the y components). (a) The planar half Maxwell fisheye lens with TL, producing excellent surface plane waves. (b) The planar half Maxwell fisheye lens and planar flat lens with two TLs, producing high-quality Bessel beams.
high-quality surface plane waves. Incorporating with the planar flat lens and additional another TL, the simulation results demonstrate a focusing area that is similar to the diamond shape, as displayed in Fig. 4(b). Here, the focusing length, Z max of the planar Bessel lens is about 110 mm, corresponding to a deflection angle of φ 17.7° of the flat lens, which is very close to the designed value φ 15°. The Bessel-like profiles of electric fields at different distances from the left edge of the focusing area in the propagating direction are illustrated in Fig. 5(a), showing good Bessel-beam performance. If the aperture size of the Bessel lens increases, we expect much better performance of Bessel beam by reducing the surface wave diffractions in the lens edges. To further confirm the surface-wave Bessel beam experimentally, we fabricate the proposed planar Bessel lens using the printed circuit board (PCB), as shown in Fig. 3(b). We utilize a microwave near-field scanning apparatus [see Fig. 3(c)] to scan the electric-field distributions of the sample. In experiments, we use a monopole antenna to imitate the point source, which can excite the TM-mode surface waves. The sample is moved by a 2D stepping platform and the probing antenna is fixed perpendicularly to the metasurface to measure the y components of electric fields. The arrangement of experiment is displayed in Fig. 3(c). Similar to the numerical situation, the measurement results of electric-field profiles at different locations in the focusing area at
Fig. 5. Simulation and experiment results of the electric-field profiles at different locations along the propagating direction in the focusing area at 12 GHz, showing good Bessel-like performance. (a) Simulation results. (b) Measurement results.
October 15, 2014 / Vol. 39, No. 20 / OPTICS LETTERS
12 GHz are presented in Fig. 5(b). Clearly, the Bessel-like profiles of electric fields with first zero are observed in the experiments. The bandwidth of the surface-wave lens is approximately 1.5 GHz from 11.5 to 13 GHz. The small discrepancy between the simulation and measurement is mainly caused by the feeding location of monopole antenna and the sample fabrication. In summary, we have theoretically proposed the surface-wave Bessel beams and presented a method to realize the surface-wave Bessel lens, which is composed of a planar half Maxwell fisheye lens and a planar flat lens with gradient surface indices of refraction. Comparing to the conventional methods in bulk Bessel lenses, the Gaussian-beam illumination is replaced by a point source in the proposed method, which makes the realization of Bessel beams simpler and more flexible. The surfacewave Bessel lens has good application potentials in developing highly directed and concentrated energy transmission and nondiffraction surface waveguide for near-field sensors and biomedical devices [33]. This work was supported by National High Tech (863) Projects (2012AA030402 and 2011AA010202), National Natural Science Foundation of China (60990320, 61138001), and 111 Project (111-2-05). References 1. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Science 305, 788 (2004). 2. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, Science 314, 977 (2006). 3. E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, IEEE Trans. Antennas Propag. 51, 2641 (2003). 4. R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, 1960). 5. C. L. Holloway, M. A. Mohamed, E. F. Kuester, and A. Dienstfrey, IEEE Trans. Antennas Propag. 47, 853 (2005). 6. C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O’ Hara, J. Booth, and D. R. Smith, IEEE Trans. Antennas Propag. 54, 10 (2012). 7. Y. Zhao, N. Engheta, and A. Alù, Metamaterials 5, 90 (2011). 8. F. Monticone, N. M. Estakhri, and A. Alù, Phys. Rev. Lett. 110, 203903 (2013). 9. C. Pfeiffer and A. Grbic, Appl. Phys. Lett. 102, 231116 (2013). 10. C. Pfeiffer and A. Grbic, IEEE Trans. Microwave Theory Tech. 61, 4407 (2013). 11. C. Pfeiffer and A. Grbic, Phys. Rev. Lett. 110, 197401 (2013). 12. N. Yu, P. Genevet, M. A. Cats, F. Aieta, J. Tetienne, F. Cappaso, and Z. Gaburro, Science 334, 333 (2011). 13. F. Aiet, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, Nano Lett. 12, 4932 (2012).
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