Open Access
Add to BibSonomy
Abstract
Surface waves bounded by subwavelength-structured surfaces have many exotic electromagnetic properties different from those supported by smooth surfaces. However, there is a long-standing misconception, claiming that these waves must propagate along the macroscopic interface. In this paper, we describe in detail the microscopic meta-surface wave (M-wave) in artificial subwavelength structures. It is shown that the waves penetrating macroscopic surfaces share the essence of most surface waves (i.e., they spread along the microscopic interfaces, formed by adjacent constitutive materials). Equivalent circuit theory and transfer matrix method have been adopted to quantitatively describe these M-waves with high accuracy in the form of catenary optical fields and dispersion. Based on these analyses, novel omnidirectional band-stop filters and wide-angle beam deflectors are designed with operational angles up to 88°. We believe these results may provide many new perspectives for both the understanding and design of functional subwavelength structures.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As one special wave form, surface wave is widespread in optics, electromagnetics, and acoustics, among others. According to Rayleigh’s definition, surface waves must propagate along one particular interface, which are completely different from the “space waves” that spread in three dimensions. In principle, the behaviors of surface waves are closely associated with the materials and geometries. By changing these parameters, one could generate a large variety of waves such as surface plasmon polaritons (SPPs), Zenneck surface waves, Dyakonov waves and Tamm states [1,2]. Variation of the SPPs in two-dimensional plane has also resulted in more complex waves, such as surface Bessel beams and Airy beams [3].
As one kind of two-dimensional metamaterials, metasurfaces have attracted increasing attentions in recent years [4–14]. While they have a reduced dimension, the almost infinite subwavelength surface structures provides us great flexibility in the control of electromagnetic waves. Many new phenomena and devices such as flat lenses, perfect absorbers, holograms and vortex plates have been realized by using metasurfaces in a wide range of frequencies from microwave to optics [8,11,15–18]. In previous articles [2,11], we term the waves in metasurfaces as meta-surface waves (M-waves) since they are propagating along surfaces but have unusual properties beyond classic surface waves. Historically, the concept of M-waves can date back to the discovery of extraordinary Young’s double slits interference (EYI) [15,19].
Although metasurfaces have provided a general platform for the studying of wave dynamics, the physics of M-waves has not been fully addressed. In fact, it is often wrongly thought that surface waves must propagate along the “macroscopic” surface. In this paper, we revisited the concept of M-waves from a microscopic view. Three notable conclusions can be drawn from these results: First, as described previously, M-waves are not limited to propagate along the apparent macroscopic surface. In the microscopic scale, the interface inside the structured materials also support the propagation of interfacial waves. Secondly, the amplitudes of these surface waves decay exponentially away from the microscopic surface, and the fields of the coupled modes behaves like catenaries [19], which lead to a much shorter and tunable effective wavelength along the propagating direction. It seems that the catenary optical field is one fundamental property of electromagnetic waves in subwavelength structured materials. Thirdly, the impedance dispersion of thin nanoslits array is described using a mathematic model similar to the “catenary of equal strength,” which presents an additional link between the catenary function and subwavelength engineering optics [11].
2. Catenary theory of the microscopic M-waves
From a microscopic point of view, there are nearly infinite numbers of interfaces between the constitutive materials in metamaterials and metasurfaces. Although these interfaces may not support surface waves when the dimension is much larger than the wavelength, strong modifications must be considered in the deep-subwavelength scale. This can be understood by using the generalized Helmholtz equation [11,19]:
where E is the electric field, ε and μ are the permittivity and permeability, k0 is the vacuum wavenumber. At the boundary, approaches infinity and is responsible for the coupling between free-space waves and localized modes.To understand Eq. (1), waves propagating through thin slits perforated in perfect electric conductors (PEC) are illustrated in Fig. 1. Note that these metallic slits are fundamental building blocks of various metasurfaces including filters, absorbers, polarizers and flat lenses [20–22]. It is well-known that a two-dimensional metallic slab waveguide support transverse electromagnetic (TEM) waves without cut-off frequency, and the propagation constant is equal to that in vacuum. However, when one reduces the thickness of such waveguide to much smaller than the operational wavelength, strong scattering would occur at the edges, which makes the fields distribution change dramatically [Fig. 1(b)]. Similar to the SPP fields at the slit edges [23], this new kind of field takes a form of hyperbolic cosine catenary function as a result of the evanescent coupling, which means that the localized wave has a large vertical propagation constant β along the z-direction and an imaginary horizontal component α along the x-direction. As will be expatiated in the following Eqs. (3)-(5), this propagation constant is dependent on the slit width, which resembles the SPP effect once again [20]. Note that this is different from the spoof surface plasmon shown in previous studies [24,25].
Fig. 1 Waves in the microscopic regime. The top panel is the electric fields in a thick slit cut in PEC. The bottom panel shows the M-wave in a thin slit.
These catenary-shaped interfacial waves may be treated as one special vertically propagating M-wave which deviates from traditional diffraction theory. In the rigorous coupled wave analysis (RCWA), the in-plane wavevector of the (m, n)-order mode inside the grating layer is often expressed as [26]:
where Λx and Λy are the periods along x and y directions. Obviously, the in-plane wavenumber can only be real number, in contrary to the imaginary components shown in Fig. 1. Since the RCWA has been demonstrated to be correct for periodic gratings with finite thickness, it is logical to attribute the discrepancy to the small thickness of the metasurface. With microscopic electromagnetic theory, it can be concluded that the evanescent wavevector is stemmed from the scattering at the edges. If one want to make the RCWA to be applicable, it is reasonable to consider the thin metasurface as an effective medium with larger refractive index and proper thickness. Alternatively, the generalized boundary condition of metasurfaces can be used to calculate the electromagnetic response.As shown in Fig. 2, the electromagnetic properties of simple metallic gratings can be obtained by using impedance theory. To investigate the influence of gap width on the optical properties, the electric fields along the central line are illustrated in Fig. 2(b). Since the scattering fields are mainly evanescent waves, the amplitude distribution follows a hyperbolic cosine shape, which is also called ordinary catenary function. Using equivalent circuit model [27], the admittance Y can be written as:
where Zeff is the surface impedance, p is the period of grating, w is the width of the slit, λ is the wavelength, n1 and n2 are the refractive indexes for the background materials. Using the generalized Fresnel’s equations [2], the transmission and refection coefficient can be easily obtained. Alternatively, the impedance sheet can be treat as a homogeneous thin film with effective permittivity of [21]:where deff is the effective thickness, ε0 is the permittivity of vacuum. Interestingly, the above two equations possess a form of “catenary of equal strength”, which has been utilized to generate photonic spin-orbit interaction and achromatic geometric phase [17,28]. To highlight this interesting property, we term the above dispersion as catenary optical dispersion in the following discussion. Figure 2(c) shows the effective admittance calculated by Eq. (3) and retrieved from FEM calculation. To further demonstrate the agreement, the calculated transmission amplitudes and phases are illustrated in Fig. 2(d).Fig. 2 Vertical M-waves associated with catenary optical fields. (a) Equivalent problem for a thin sheet perforated with subwavelength slits. The period p is 10 mm. (b) Ex distribution at 10 GHz for slit widths of 1, 1.5, 2 and 3 mm. (c) Normalized admittance for different gap widths. The curve is half of a catenary curve of equal strength. (d) Transmission amplitude and phase calculated using FEM and impedance theory.
3. Applications in amplitude and phase modulation
The above catenary theory is useful in the design of various functional metasurfaces. First of all, we investigate the performance of multilayered metallic gratings as spectral filters. When the distance between layers is large enough, the evanescent coupling could be ignored and the electromagnetic properties may be directly calculated using transfer matrix or interference theory [29,30]. As can be seen in Fig. 2(c), the retrieved admittance is a bit larger than the theory. In order to make the theory more accurate, in the following discussion the admittance is revised to be:
where is an additional experiential term.Without loss of generality, we shall compare the modified theory with numerical simulations for a multilayered structure with 5 dielectric spacers and 6 metasurfaces. The permittivity of the dielectric spacer is set to be either 1 or 3.5 while the gap width is 0.5 mm or 2.5 mm. Figure 3 shows a comparison of the theoretical and numerical results for four combinations of geometric and electric parameters, implying that the theories are accurate enough. The blue regions indicate the bandgap formed by the multilayer. Note that when the left and right boundaries of the unit cell are set as PEC by adding two metallic sheets, each unit cell would act as a metallic waveguide. In this case, the propagation constant is almost independent of the incidence angle, implying that the above band-stop filters could operate in all angle of incidences. To demonstrate this intriguing phenomena, the reflectance for the case shown in Fig. 3(a) is calculated for various incidence angles as illustrated in Fig. 4. It is shown that the reflectance in 10-14 GHz can be maintained to be higher than 99% even for incident angle up to 88°. This exotic behavior can be interpreted using the transfer matrix formalism: Unlike the case of classic multilayered structures where the phase shift is angle-dependent, the waves in our structures always propagate along the z-direction, thus the bandgap and transmission peak are almost independent of angle.
Fig. 3 Simulated transmission and reflection coefficients under various conditions. The period, spacer thickness and layer number are fixed to be p = 5 mm, d = 10 mm and N = 6. The other parameters are (a) w = 0.5 mm; ε = 1; (b) w = 2.5 mm; ε = 1; (c) w = 0.5 mm; ε = 3.5; (d) w = 2.5 mm; ε = 3.5. The blue regions indicate the transmission bandgap, while the red ellipses in (b) show the effective interfaces supporting the vertical M-waves.
Fig. 4 (a) Reflectance at incidence angles ranging from 0° to 88°. The blue region indicates the omnidirectional bandgap. (b) Logarithmic electric field distributions at 9 and 12 GHz for normal incidence from left to right, where the reflectance is approximately zero and unity, respectively.
When the distance between adjacent layers is much smaller than the wavelength, the coupling strength would be much stronger, thus the above isolated model for metallic slits becomes not accurate enough. In many cases, such coupling is referred to as magnetic resonance and blended with the electric response to get effective material parameters of effective permittivity and permeability [31]. Different from this convention, here we show that the transmission properties can be explained without magnetic resonance. Figure 5 shows a simple structure composed of two cascaded metasurfaces with p = 5 mm, d = 2 mm and w = 0.2 mm. While the sheet resistance of the two resistance is set as 15 Ω, the effective resistance of 17 Ω is adopted in the model. The small discrepancy between the model and numerical calculation is stemmed from the mutual coupling, which can be corrected using a modified experiential term α. As illustrated in Fig. 5(a), the sandwiched structures can be directly utilized in the design of metamaterial perfect absorber by employing the concept of coherent perfect absorption (CPA) [32]. At f = 10 GHz, the transmission and reflection amplitudes are equal, and the phase difference is zero, implying that such structure actually meets the requirement of CPA under anti-symmetrical illumination.
Fig. 5 Perfect coherent absorber based on the catenary model. (a) Schematic of the geometric configuration. The top left and right panel represent the real and equivalent structures. The bottom panel illustrates a single-port absorber comprised half of the above device. (b)(c) Amplitude and phase properties of the three layer structure.
For strongly coupled multilayers, the catenary model can only provide initial values for the geometric parameters. In what follows, we shall use rigorous simulations to design thin and high-efficient beam deflectors comprised of gradient slits array. According to the generalized Snell’s law, proper phase modulation with constant amplitudes are required to deflect beam efficiently [2,11]. Like the case for SPPs [33], the current structures rely on the change of width to modulate the phase shift. Figure 6 shows three designs (Designs A, B and C) with deflection angles equal to 32.4°, 45.6° and 59° at 14 GHz, respectively. The scattering peaks in the calculated radar cross sections (RCS) are in good agreement with the theory, which demonstrates that the zeroth order and opposite order reflections are well suppressed. Although the amplitude of opposite-deflection increases for larger angles, it may be still much better than previous designs based on dielectric resonances [34,35]. In fact, we expect the deflection angle can be as high as 88° for larger samples. When used as focusing lenses, this indicates a numerical aperture close to 0.999.
Fig. 6 Large-angle beam deflectors based on slits array operating at f = 14 GHz. The width parameters (in mm) for different slits are shown in the inset of (a). Other parameters are p = 5 mm, d = 1 mm, N = 3, and ε = 3.5. (b) Distribution of Ez for Design A.
Note that the reflectance of the meta-mirror can be maintained as high as 99%, since a reflective layer is added in the bottom of the device. Figure 6(b) shows the z-component of electric fields for Design A. In order to suppress the horizontal coupling between neighboring unit cells, vertical metallic sheets have been added at the two sides of each unit. Consequently, each unit acts as a localized waveguide, ensuring large-angle filtering and beam steering [36,37]. By further exploiting the resonance and dispersion [38,39], broadband operation is also achievable.
4. Conclusions
In summary, we presented the theoretical and numerical demonstration of microscopic meta-surface waves propagating along the vertical direction of metasurfaces. In the microscopic world, there are many interfaces separating different materials, and we give the direct proof that such interfaces support bound waves decaying exponentially away from these interfaces. These so-called catenary optical fields ensure shorter effective wavelength, tunable dispersion relation and localized phase shift, which are the cores of many fundamental subwavelength electromagnetic devices, including flat lenses, perfect absorbers, polarization controllers, etc. In other words, the short effective wavelength, localized phase shift, catenary optical fields and dispersion can be treated as three basic characteristics of M-waves. In future researches, these characteristics will be combined to construct more powerful functional devices.
Funding
National Basic Research (973) Program of China (2013CBA01700); National Natural Science Foundation of China (61622508, 61575201).
References and links
1. J. A. Polo Jr and A. Lakhtakia, “Surface electromagnetic waves: a review,” Laser Photonics Rev. 5(2), 234–246 (2011). [CrossRef]
2. X. Luo, “Principles of electromagnetic waves in metasurfaces,” Sci. China Phys. Mech. Astron. 58(9), 594201 (2015). [CrossRef]
3. I. Epstein, Y. Lilach, and A. Arie, “Shaping plasmonic light beams with near-field plasmonic holograms,” J. Opt. Soc. Am. B 31(7), 1642–1647 (2014). [CrossRef]
4. T. Xu, Y.-K. Wu, X. Luo, and L. J. Guo, “Plasmonic nanoresonators for high-resolution colour filtering and spectral imaging,” Nat. Commun. 1(5), 59 (2010). [CrossRef] [PubMed]
5. S. Sun, K. Y. Yang, C. M. Wang, T. K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Xiao, W. T. Kung, G. Y. Guo, L. Zhou, and D. P. Tsai, “High-efficiency broadband anomalous reflection by gradient meta-surfaces,” Nano Lett. 12(12), 6223–6229 (2012). [CrossRef] [PubMed]
6. W. T. Chen, K.-Y. Yang, C.-M. Wang, Y.-W. Huang, G. Sun, I.-D. Chiang, C. Y. Liao, W.-L. Hsu, H. T. Lin, S. Sun, L. Zhou, A. Q. Liu, and D. P. Tsai, “High-efficiency broadband meta-hologram with polarization-controlled dual images,” Nano Lett. 14(1), 225–230 (2014). [CrossRef] [PubMed]
7. Y.-W. Huang, W. T. Chen, W.-Y. Tsai, P. C. Wu, C.-M. Wang, G. Sun, and D. P. Tsai, “Aluminum plasmonic multicolor meta-hologram,” Nano Lett. 15(5), 3122–3127 (2015). [CrossRef] [PubMed]
8. M. Khorasaninejad and F. Capasso, “Metalenses: Versatile multifunctional photonic components,” Science 358, eaam8100 (2017). [CrossRef]
9. B. H. Chen, P. C. Wu, V.-C. Su, Y.-C. Lai, C. H. Chu, I. C. Lee, J.-W. Chen, Y. H. Chen, Y.-C. Lan, C.-H. Kuan, and D. P. Tsai, “GaN metalens for pixel-level full-color routing at visible light,” Nano Lett. 17(10), 6345–6352 (2017). [CrossRef] [PubMed]
10. M. Pu, X. Ma, X. Li, Y. Guo, and X. Luo, “Merging plasmonics and metamaterials by two-dimensional subwavelength structures,” J. Mater. Chem. C Mater. Opt. Electron. Devices 5(18), 4361–4378 (2017). [CrossRef]
11. X. Luo, “Subwavelength optical engineering with metasurface waves,” Adv. Opt. Mater. 6(7), 1701201 (2018). [CrossRef]
12. S. Wang, X. Ouyang, Z. Feng, Y. Cao, M. Gu, and X. Li, “Diffractive photonic applications mediated by laser reduced graphene oxides,” Opto-Electron. Adv. 1, 170002 (2018).
13. L. Chen, Y. Zhou, M. Wu, and M. Hong, “Remote-mode microsphere nano-imaging: new boundaries for optical microscopes,” Opto-Electron. Adv. 1, 170001 (2018).
14. P. C. Wu, J.-W. Chen, C.-W. Yin, Y.-C. Lai, T. L. Chung, C. Y. Liao, B. H. Chen, K.-W. Lee, C.-J. Chuang, C.-M. Wang, and D. P. Tsai, “Visible metasurfaces for on-chip polarimetry,” ACS Photonics. in press).
15. X. Luo and T. Ishihara, “Surface plasmon resonant interference nanolithography technique,” Appl. Phys. Lett. 84(23), 4780–4782 (2004). [CrossRef]
16. S. B. Glybovski, S. A. Tretyakov, P. A. Belov, Y. S. Kivshar, and C. R. Simovski, “Metasurfaces: From microwaves to visible,” Phys. Rep. 634, 1–72 (2016). [CrossRef]
17. M. Pu, X. Li, X. Ma, Y. Wang, Z. Zhao, C. Wang, C. Hu, P. Gao, C. Huang, H. Ren, X. Li, F. Qin, J. Yang, M. Gu, M. Hong, and X. Luo, “Catenary optics for achromatic generation of perfect optical angular momentum,” Sci. Adv. 1(9), e1500396 (2015). [CrossRef] [PubMed]
18. S. Wang, P. C. Wu, V.-C. Su, Y.-C. Lai, C. Hung Chu, J.-W. Chen, S.-H. Lu, J. Chen, B. Xu, C.-H. Kuan, T. Li, S. Zhu, and D. P. Tsai, “Broadband achromatic optical metasurface devices,” Nat. Commun. 8(1), 187 (2017). [CrossRef] [PubMed]
19. M. Pu, Y. Guo, X. Li, X. Ma, and X. Luo, “Revisitation of extraordinary Young’s interference: from catenary optical fields to spin-orbit interaction in metasurfaces,” ACS Photonics. in press).
20. T. Xu, C. Wang, C. Du, and X. Luo, “Plasmonic beam deflector,” Opt. Express 16(7), 4753–4759 (2008). [CrossRef] [PubMed]
21. Q. Feng, M. Pu, C. Hu, and X. Luo, “Engineering the dispersion of metamaterial surface for broadband infrared absorption,” Opt. Lett. 37(11), 2133–2135 (2012). [CrossRef] [PubMed]
22. M. Pu, P. Chen, Y. Wang, Z. Zhao, C. Huang, C. Wang, X. Ma, and X. Luo, “Anisotropic meta-mirror for achromatic electromagnetic polarization manipulation,” Appl. Phys. Lett. 102(13), 131906 (2013). [CrossRef]
23. X. Luo and I. Teruya, “Sub 100 nm lithography based on plasmon polariton resonance,” in 2003 International Microprocesses and Nanotechnology Conference (IEEE, 2003), pp. 138–139.
24. M. A. Kats, D. Woolf, R. Blanchard, N. Yu, and F. Capasso, “Spoof plasmon analogue of metal-insulator-metal waveguides,” Opt. Express 19(16), 14860–14870 (2011). [CrossRef] [PubMed]
25. A. Pors, E. Moreno, L. Martin-Moreno, J. B. Pendry, and F. J. Garcia-Vidal, “Localized spoof plasmons arise while texturing closed surfaces,” Phys. Rev. Lett. 108(22), 223905 (2012). [CrossRef] [PubMed]
26. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3(11), 1780–1787 (1986). [CrossRef]
27. L. B. Whitbourn and R. C. Compton, “Equivalent-circuit formulas for metal grid reflectors at a dielectric boundary,” Appl. Opt. 24(2), 217–220 (1985). [CrossRef] [PubMed]
28. X. Luo, M. Pu, X. Li, and X. Ma, “Broadband spin Hall effect of light in single nanoapertures,” Light Sci. Appl. 6(6), e16276 (2017). [CrossRef]
29. M. Pu, C. Hu, M. Wang, C. Huang, Z. Zhao, C. Wang, Q. Feng, and X. Luo, “Design principles for infrared wide-angle perfect absorber based on plasmonic structure,” Opt. Express 19(18), 17413–17420 (2011). [CrossRef] [PubMed]
30. H.-T. Chen, “Interference theory of metamaterial perfect absorbers,” Opt. Express 20(7), 7165–7172 (2012). [CrossRef] [PubMed]
31. G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. 30(23), 3198–3200 (2005). [CrossRef] [PubMed]
32. M. Pu, Q. Feng, M. Wang, C. Hu, C. Huang, X. Ma, Z. Zhao, C. Wang, and X. Luo, “Ultrathin broadband nearly perfect absorber with symmetrical coherent illumination,” Opt. Express 20(3), 2246–2254 (2012). [CrossRef] [PubMed]
33. Y. Li, X. Li, M. Pu, Z. Zhao, X. Ma, Y. Wang, and X. Luo, “Achromatic flat optical components via compensation between structure and material dispersions,” Sci. Rep. 6(1), 19885 (2016). [CrossRef] [PubMed]
34. Z. Ma, S. M. Hanham, P. Albella, B. Ng, H. T. Lu, Y. Gong, S. A. Maier, and M. Hong, “Terahertz all-dielectric magnetic mirror metasurfaces,” ACS Photonics 3(6), 1010–1018 (2016). [CrossRef]
35. R. Paniagua-Domínguez, Y. F. Yu, A. E. Miroshnichenko, L. A. Krivitsky, Y. H. Fu, V. Valuckas, L. Gonzaga, Y. T. Toh, A. Y. S. Kay, B. Luk’yanchuk, and A. I. Kuznetsov, “Generalized Brewster effect in dielectric metasurfaces,” Nat. Commun. 7, 10362 (2016). [CrossRef] [PubMed]
36. D. Van Labeke, D. Gérard, B. Guizal, F. I. Baida, and L. Li, “An angle-independent Frequency Selective Surface in the optical range,” Opt. Express 14(25), 11945–11951 (2006). [CrossRef] [PubMed]
37. M. Pu, X. Li, Y. Guo, X. Ma, and X. Luo, “Nanoapertures with ordered rotations: symmetry transformation and wide-angle flat lensing,” Opt. Express 25(25), 31471–31477 (2017). [CrossRef] [PubMed]
38. S. Wang, P. C. Wu, V.-C. Su, Y.-C. Lai, M.-K. Chen, H. Y. Kuo, B. H. Chen, Y. H. Chen, T.-T. Huang, J.-H. Wang, R.-M. Lin, C.-H. Kuan, T. Li, Z. Wang, S. Zhu, and D. P. Tsai, “A broadband achromatic metalens in the visible,” Nat. Nanotechnol. 13(3), 227–232 (2018). [CrossRef] [PubMed]
39. W. T. Chen, A. Y. Zhu, V. Sanjeev, M. Khorasaninejad, Z. Shi, E. Lee, and F. Capasso, “A broadband achromatic metalens for focusing and imaging in the visible,” Nat. Nanotechnol. 13(3), 220–226 (2018). [CrossRef] [PubMed]
