Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Deep-subwavelength spoof magnetic localized surface plasmon waveguiding over arbitrary bending angles

Tao Fu, Xingxing Liu, Gongli Xiao, Tangyou Sun, and Haiou Li

Open Access Open Access

Abstract

A deep-subwavelength metal spiral structure (MSS) waveguide with arbitrary bending angles was proposed and demonstrated to propagate magnetic localized surface plasmons (MLSPs) in theoretical, simulated and experimental ways. The uniform coupling strengths and frequencies for adjacent MSSs with different azimuthal angles represent a significant advancement in the development of structures supporting MLSPs over arbitrary bending angles. The consistency among spectra, dispersion, and field distributions for five MSSs indicates that backward propagation of MLSPs over arbitrary bending angles is possible. In addition, a long S-chain consisting of adjacent MSSs at various angles holds promise for applications involving long-distance MLSPs waveguides.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Localized surface plasmons (LSPs) are electromagnetic waves confined to small metal particles of infrared or visible frequencies [1]. With the novel possibility of subwavelength spatial confinement and strong-field enhancement, natural LSPs have attracted much interest, and various potential applications ranging from superlenses or hyperlenses to metasurfaces have emerged [2–6]. Thus far, most of the investigations and promising applications of LSPs have been limited to infrared and visible wavelengths [7]. To generate highly confined electromagnetic fields on subwavelength metallic particles at lower frequencies, for example in the microwave or terahertz regimes, spoof LSPs have been proposed via the corrugation or decoration of subwavelength structures [8–12], or the adoption of ultrathin metallic spiral structures (MSSs) [13–16]. The resonant wavelength of the MSS increases with the depth of its groove (spiral arm length), while its dimension has little effect, compared to other structures, such as corrugated surfaces, considered under the same conditions [17,18]. In addition, MSSs have other outstanding characteristics that allow them to support spoof LSPs on deep-subwavelength scales [19]. Furthermore, MSSs support both spoof magnetic LSPs (magnetic dipole mode) and spoof electric LSPs (electric dipole mode), whose electromagnetic field distributions are azimuthally and radially independent, respectively [20]. Spoof magnetic LSPs propagate longer distances than spoof electric LSPs in the waveguide because, compared with similar-sized magnetic dipoles, the radiation loss is always greater in the case of the electric LSPs [21,22]. Therefore, the prospect of spoof magnetic localized surface plasmons (MLSPs) has attracted significant interest, adding the important characteristic of magnetism to the field of particle plasmonics.

Recently, investigations into the near-field coupling between two MSSs resonators have been carried out [23–26], alongside the development of spoof LSP-related technologies and theories. Spoof plasmon hybridization between adjacent MSSs with subwavelength textures results in significant field enhancement, which can be tuned by changing the separation between adjacent MSS particles in a one-dimensional planar structure. It has been shown that a one-dimensional MSS chain with different connections in its construction can allow the switching of the forward and backward surface waves [27,28]. As mentioned in Ref. [29], bent waveguides and T-splitters, involving right-angles, have been experimentally demonstrated to support MLSPs. However, the generation of MLSPs that propagate along MSS chains through arbitrary large angles remains a challenge.

In this study, MSSs with two spiral arms are proposed to support spoof MLSPs. The coupling strength and frequencies for MLSPs on adjacent MSSs, obtained by theory, simulation and experiment, are largely uniform in the azimuthal direction. In addition, a coupled-resonator optical waveguide (CROW) consisting of 5 MSSs and an S-chain made from 11 MSSs are proposed [30]. As expected, the theory, simulation and measurement results show that the dispersion characteristics do not vary much within the waveguides, and the predicted transmission property of the CROW and the S-chain remains basically unchanged in the experimental results. Spoof MLSPs can be propagated and guided on the surface of MSSs chains with arbitrary bending angles, on deep-subwavelength scales. In the future, a wide variety of advanced plasmonic functional devices and highly integrated optical components could be developed from waveguides with large bending angles.

2. Coupled-resonator optical waveguide

Figure 1(a) illustrates the geometry of an ultrathin two-arm MSS particle that supports MLSPs [19]. The ultrathin MSS is characterized by its width w = 0.2 mm, outer radius R = 5.3 mm, inner radius r = 0.2 mm, and spiral arms that are separated by a distance L = 0.65 mm. The thickness of the MSS is 0.018 mm, and it is based on a 0.02-mm-thick Fr4 dielectric substrate with a relative permittivity of ɛ = 4.2 and loss tangent of 0.02. An excitation source (a discrete port in the simulation) is placed 1-mm above the center of the particle. Another probe is located at proper positions to detect the distribution of the near magnetic field Hz. Figures 1(b) and 1(c) illustrate the magnetic-field amplitudes along the x and z axes, respectively. As shown in Figs. 1(b) and 1(c), the field amplitude grows as n decreases. However, when the number of turns n is 2, the magnetic-field intensity decays the most exponentially, which is not conducive to measure. When n = 3, the magnetic field amplitude is relatively large, and its attenuation amplitude is slower. Therefore, we focus our interest on particles with n = 3 spiral turns. Similar effects have been studied in previous works [1], and such modes have been referred to as “spoof MLSPs”. It is noted that a strong magnetic resonance field appears at the center of the particle and the profile is in the azimuthal direction. The simulated near-field response spectrum is shown in Fig. 1(d). Furthermore, the simulated instantaneous magnetic fields at approximately 2.3 GHz, obtained using commercial software (CST Microwave Studio) indicate a magnetic resonance [Fig. 1(d), inset].

 figure: Fig. 1.

Fig. 1. (a) Schematic illustration of a single MSS. (b) Magnetic-field distributions along the x axis for the MSS with various numbers of turns n. (c) Magnetic-field distributions along the z axis for the MSS with various numbers of turns n. (d) Near-field response spectra for a single MSS, the inset of (d) is plot of the magnetic fields in the xy plane for the MSS with n = 3.

Download Full Size | PDF

Based on the above results, the transmission between a pair of MSSs with a center-to-center distance d is reported in Fig. 2(a). The electromagnetic response of the MSS pair is measured using a monopole antenna (Source) that is placed 1 mm above one particle to excite the modes and a receiving monopole antenna (Probe) that is located 1 mm above the center of another particle to detect the resonance spectrum. Figure 2(b) depicts the transmission spectra for the pair as a function of the angle $\beta $, which is illustrated in Fig. 2(a). It shows that the resonances in the transmissions remain almost stable (at approximately both 2.28 GHz and 2.34 GHz) upon rotating the angle $\beta $ from 0° to 90°. The transmissions of three specific angles [$\beta $ = 20° (red dashed line), 50° (white dashed line), and 80° (purple dashed line)] are highlighted, in Fig. 2(b), to verify that the fabricated samples have the properties predicted by the simulations; two monopole antennas connected to a vector network analyzer (Agilent N5230C) acted as the source and probe, respectively. In Fig. 2(c), the simulation normalized transmission results of the three cases mentioned above are basically similar to each other and the experimental results. Two distinct peaks (${\omega ^ - }$ = 2.28 GHz and ${\omega ^ + }$ = 2.34 GHz) in the simulated transmission spectrum are clearly observed, and the measured transmission spectrum has two clear resonances ($\omega _E^ - $ = 2.27 GHz, $\omega _E^ + $ = 2.30 GHz) [26]. In addition, the measured transmission band is roughly the same as that of the numerical results, and the deviation of about 0.04 GHz in resonance frequency between simulation and experiment can be assigned to experimental measurement error. In the experiment for two MSSs coupling, the applied monopole antennas generate radiation to the free space, therefore, the coupling energy is not only from the adjacent cell but also from the source antennas radiation and environmental reflection. According to the coupling theory [31], the coupling strength is relevant to the loss, which means the amplitudes and frequencies have a minor difference. Besides, the parameters of the probe (size, material, volume, etc.) and the loss tangent of the dielectric substrate in the experiment setup may vary from what we used in the simulation. Finally, fixing the probe exactly at 1 mm above the surface of the sample (as set in the simulation) is very difficult in the experiment, a factor that also affects the amplitude of the measurement results. Figure 2(d) illustrates the near electric-field distributions of the MSS pairs, corresponding to two distinct spectral peaks, which indicate good conformity between the simulations and measurements. At the same time, we can conclude that the difference between the above experiment and simulation will not affect the profiles of the transport magnetic modes. Considering the verification in Figs. 2(b)–2(d), it can be concluded that when the rotation angle changes from 0° to 90°, the range of transmission spectrum is basically unchanged. Simultaneously, the field distributions over the different angles show that the MLSPs couple well between adjacent MSSs, even when the propagation angle is large. This demonstrates that the spoof MLSPs can smoothly propagate even if the two MSSs are bent at a large angle.

 figure: Fig. 2.

Fig. 2. (a) MSS pair simulation model. (b) Transmission spectra for the pair as a function of the angle $\beta $. (c) Simulated and experimentally measured transmission spectra for MSS pairs with angles of rotation $\beta $ = 20°, 50°, and 80°. (d) Simulated and experimental electric-field distributions for the relevant resonance modes.

Download Full Size | PDF

Furthermore, a CROW with five MSSs was constructed [28], as illustrated in the inset of Fig. 3(a). The rotation angles between adjacent MSSs are defined as ${\beta _{i{\; }({i{\; } = {\; }1,{\; \; }2,{\; }3,{\; \; }4} )}}$. The rotation angles of the adjacent MSSs are the same (${\beta _{i{\; }}}$ = $\beta $ = 0°, 10°, 30°, … or 60°) or different (${\beta _{i{\; } = {\; }1,{\; \; }2,{\; }3,{\; \; }4}}$ = 10°, 30°, 50°, and 70°). Based on coupled-mode theory [32], we simply describe the coupled MSS dimer system as follows:

$$\begin{array}{l} - \frac{{d{a_1}}}{{dt}} = i{f_0}{a_1} + i\kappa {f_0}{a_2}\\ - \frac{{d{a_2}}}{{dt}} = i{f_0}{a_2} + i\kappa {f_0}{a_1} \end{array}$$
where a1 and a2 denote the time evolution of the amplitudes of symmetric and asymmetric resonant modes. ${f_0}$ = 2.3 GHz is the resonating frequency of a single MSS in Fig. 1(d). We can solve the corresponding eigen-frequencies as ${f_1} = {f_0} - \kappa {f_0}$ and ${f_2} = {f_0} + \kappa {f_0}$. After substituting the ${f_0}$, two peaks of the transmission spectra in Fig. 2(c) correspond to different angles. The coupling strength of two modes $\kappa $ is obtained by the formula $\kappa = ({{f_2} - {f_1}} )/2{f_0}$, and $\kappa $ values as 0.0113, 0.0131 and 0.0152 at 20, 50 80 degrees which is the rotation angle between a pair of MSSs, respectively. Therefore, the intrinsic dispersion relation can be obtained as $f = {f_0}[{1 + 2\kappa \textrm{cos}({kd} )} ]$ in which k is the wavevector. For sufficient theoretical analysis and universality, we fit the coefficient κ for all angles (from ${0^\circ }$ to ${90^\circ }$) in Fig. 3(a), the κ gradually decreases when the rotation angle $\beta $ is from ${0^\circ }$ to ${16^\circ }$ and then gradually increases from ${17^\circ }$ to ${90^\circ }$. We define the ratio with angle as $\varphi $ = (${\kappa _{max}}$ - ${\kappa _{sim}}$) / ${90^\circ }$ = 4.97${\times} {10^{ - 5}}$, in which ${\kappa _{min}}$ and ${\kappa _{max}}$ is the minimum and the maximum of the coupling strength, respectively. It shows that the coupling strength κ has a minor change with the changing rotation angle. In other words, as the rotation angle $\beta $ changes, the offset of two resonance frequencies is a little value. To demonstrate the dispersion property experimentally, we measured Bloch waves of the spoof MLSPs modes with three characteristic Bloch wavevectors (kd = 0, π/2, π) as indicated by three dots in each dispersion curve. The theoretical and experimental dispersion curves can be obtained in Fig. 3(b), respectively. Three dots of each dispersion curve of experimental results match well with corresponding theoretical dispersion curves. This implies that the dispersion property is not significantly dependent on the angles of rotation for the constituent MSSs of the CROW. Frequencies with the same wave vector in the different dispersion curves are basically the same, and even the dispersion characteristic has a slight difference. Although the mismatch of the wave vector within a certain range causes some of the transmission losses, electromagnetic waves within a particular frequency range can still be transmitted to the adjacent structure with a particular rotation in the range approximately from 2.2 to 2.4 GHz. Figures 3(c) and 3(d) present the transmissions properties and electric-field distributions, respectively, of the CROWs with identical (β = 20°, 50°) and varying angles. Figure 3(c) shows that the transmission band (approximately 2.2∼2.4 GHz) in the above three cases is basically the same. To further study the properties of the phase and group velocities of the magnetic plasmons in the five MSSs of the CROW, the instantaneous electric fields at various frequencies in the plane 1 mm above the five MSSs ($\beta $ = 50°) are shown in Fig. 3(d), where the red and blue patterns indicate positive and negative values, respectively. The upper portion of Fig. 3(d) shows that the values of the phase difference (kd) between the first and third MSS is 0 at 2.36 GHz, $\pi $/2 at 2.30 GHz, and $\pi $ at 2.26 GHz, respectively. In other words, the wave vector increases as the frequency decreases, indicating a backward wave [33,34]. To confirm this result, the electric-field distributions of the five structures were measured using a near-field scanning system. The measured Ez-field patterns above the top surface of the five particles are shown in the lower portion of Fig. 3(d), and these comply with the simulation results. The simulated and experimental frequencies are offset by approximately 0.1 GHz, which can be attributed to measurement error. Figure 3(d) further verifies that the spoof MLSPs can be transmitted within the CROW device, even when a particular angle of rotation exists between the MSSs. This type of backward wave may be used efficiently in the microwave region [20]. Therefore, we can assume that the MSSs in this CROW device may support longer-distance and large-angle spoof MLSPs.

 figure: Fig. 3.

Fig. 3. (a) The numerical relation between κ and rotation angles. The inset corresponds to the schematic of CROW consisting of 5 MSSs. The distance d between adjacent MSSs is 10.8 mm. (b) Theoretical and experimental dispersion curves for CROWs possessing different fixed (${\beta _{i{\; }}}$ = $\beta $ = 20° and 50°) and varying (${\beta _{i{\; } = {\; }1,{\; \; }2,{\; }3,{\; \; }4}}$ = 10°, 30°, 50°, 70°) angles between the constituent MSSs. (c) Experimental transmission spectra for the same CROWs. (d) Simulated and measured vertical electric-field distributions for CROW consisting of five MSSs ($\beta \; $ = 50°).

Download Full Size | PDF

3. Propagation in a long S-chain

To experimentally demonstrate this predicted behavior, we designed and fabricated an S-chain consisting of 11 MSSs to support MLSPs over a long bended path, with different rotation angles (${\beta _{1\sim 9}}$ = 50°, 30°, 10°, 10°, 20°, 30°, 40°, 50°, and 60°) between the adjacent structures [ Fig. 4(a)]. As shown in Fig. 4(b), the simulated transmission range of approximately 2.2∼2.4 GHz is basically consistent with the bandwidth observed in the dispersion curves in Fig. 3(b). Furthermore, the experimental transmission for the S-chain complies with the numerical simulation. The small differences in the experimental results are primarily attributed to fabrication errors. Through experimental verification of the transmission curves of the two and five MSSs and S-chain, we found that the transmission range was approximately 2.2∼2.4 GHz. Typical simulated and measured electric-field amplitude Ez spatial distributions for the S-chain are shown in Figs. 4(c) and 4(d), respectively, and these results correspond well with each other. Thus, it is demonstrated that the electric fields are approximately identical when adjacent MSSs are rotated by large angles. Hence, transmission efficiency is largely unaffected, and spoof MLSPs can be transported along the S-chain smoothly. Note that there is some discrepancy between the field magnitudes of simulation and experimental results. Experimentally, we used two monopole antennas to detect the azimuthal magnetic SPPs. One is under the sample for stimulation and the other is above the sample for the probe. According to the strength difference of the near field and the far field for any antenna, the signal intensity of the near field is much stronger when the receiving antenna is close to the source antenna than that of the far field. After normalizing the field strength, the field in Fig. 4(d) like it is decreasing with the MLSPs propagation. Moreover, the transmission loss from materials, radiation and scattering is the other factor affecting the field strength.

 figure: Fig. 4.

Fig. 4. (a) Schematic of the S-chain. (b) Simulated and measured transmission spectra for the S-chain consisting of multiple MSSs. (c) Simulated and (d) experimental electric fields Ez above the S-chain.

Download Full Size | PDF

4. Conclusion

In summary, in this study, we have proposed a waveguide consisting of MSSs with two spiral arms and described the design, fabrication, and characterization of this structure to support spoof MLSPs on an ultrathin dielectric substrate. The electric fields are largely uniform in the azimuthal direction, and the spoof MLSPs are confined on a deep-subwavelength scale. Thus, the waveguide is able to propagate modes along highly curved surfaces over long distances. We have demonstrated that these surface modes can be supported on an MSS chain even if the constituent adjacent MSSs are rotated by large angles in theoretical, simulated and experimental ways. The flexible and ultrathin features of the proposed metamaterial structures provide promising waveguide applications in plasmonic devices, highly integrated optical components and systems operating over a wide wavelength range, from the microwave to mid-infrared regimes.

Funding

National Natural Science Foundation of China (11965009, 61764001, 61805053, 61874036); Natural Science Foundation of Guangxi Province (2018JJA170010, 2018GXNSFAA281193).

Disclosures

The authors declare no conflicts of interest.

References

1. U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer, 1995).

2. R.-M. Ma, R. F. Oulton, V. J. Sorger, and X. Zhang, “Plasmon lasers: coherent light source at molecular scales,” Laser Photonics Rev. 7(1), 1–21 (2013). [CrossRef]  

3. D. Lu and Z. Liu, “Hyperlenses and metalenses for far-field super-resolution imaging,” Nat. Commun. 3(1), 1205 (2012). [CrossRef]  

4. Z.-L. Deng, J. Deng, X. Zhuang, S. Wang, K. Li, Y. Wang, Y. Chi, X. Ye, J. Xu, G. P. Wang, R. Zhao, X. Wang, Y. Cao, X. Cheng, G. Li, and X. Li, “Diatomic metasurface for vectorial holography,” Nano Lett. 18(5), 2885–2892 (2018). [CrossRef]  

5. Z.-L. Deng, J. Deng, X. Zhuang, S. Wang, T. Shi, G. P. Wang, Y. Wang, J. Xu, Y. Cao, X. Wang, X. Cheng, G. Li, and X. Li, “Facile metagrating holograms with broadband and extreme angle tolerance,” Light: Sci. Appl. 7(1), 78 (2018). [CrossRef]  

6. Z.-L. Deng, M. Jin, X. Ye, S. Wang, T. Shi, J. Deng, N. Mao, Y. Cao, B. O. Guan, A. Alù, G. Li, and X. Li, “Full-color complex-amplitude vectorial holograms based on multi-freedom metasurfaces,” Adv. Funct. Mater. 30(21), 1910610 (2020). [CrossRef]  

7. S. A. Maier, Plasmonics: Fundamentals and Applications, (Springer, 2007).

8. Z. Liao, B. C. Pan, X. Shen, and T. J. Cui, “Multiple Fano resonances in spoof localized surface plasmons,” Opt. Express 22(13), 15710–15717 (2014). [CrossRef]  

9. Z. Liao, Y. Luo, A. I. Fernández-Domínguez, X. Shen, S. A. Maier, and T. J. Cui, “High-order localized spoof surface plasmon resonances and experimental verifications,” Sci. Rep. 5(1), 9590 (2015). [CrossRef]  

10. F. Gao, Z. Gao, Y. Zhang, X. Shi, Z. Yang, and B. Zhang, “Vertical transport of subwavelength localized surface electromagnetic modes,” Laser Photonics Rev. 9(5), 571–576 (2015). [CrossRef]  

11. Z. Gao, F. Gao, K. K. Shastri, and B. Zhang, “Frequency-selective propagation of localized spoof surface plasmons in a graded plasmonic resonator chain,” Sci. Rep. 6(1), 25576 (2016). [CrossRef]  

12. Z. Gao, F. Gao, H. Xu, Y. Zhang, and B. Zhang, “Localized spoof surface plasmons in textured open metal surfaces,” Opt. Lett. 41(10), 2181–2184 (2016). [CrossRef]  

13. Z. Liao, X. Shen, B. C. Pan, J. Zhao, Y. Luo, and T. J. Cui, “Combined system for efficient excitation and capture of LSP resonances and flexible control of SPP transmissions,” ACS Photonics 2(6), 738–743 (2015). [CrossRef]  

14. Z. Gao, F. Gao, Y. Zhang, and B. Zhang, “Complementary structure for designer localized surface plasmons,” Appl. Phys. Lett. 107(19), 191103 (2015). [CrossRef]  

15. Z. Liao, S. Liu, H. F. Ma, C. Li, B. Jin, and T. J. Cui, “Electromagnetically induced transparency metamaterial based on spoof localized surface plasmons at terahertz frequencies,” Sci. Rep. 6(1), 27596 (2016). [CrossRef]  

16. Z. Gao, F. Gao, and B. Zhang, “High-order spoof localized surface plasmons supported on a complementary metallic spiral structure,” Sci. Rep. 6(1), 24447 (2016). [CrossRef]  

17. J. J. Wood, L. A. Tomlinson, O. Hess, S. A. Maier, and A. I. Fernández-Domínguez, “Spoof plasmon polaritons in slanted geometries,” Phys. Rev. B 85(7), 075441 (2012). [CrossRef]  

18. Z. Liao, A. I. Fernández-Domínguez, J. Zhang, S. A. Maier, T. J. Cui, and Y. Luo, “Homogenous metamaterial description of localized spoof plasmons in spiral geometries,” ACS Photonics 3(10), 1768–1775 (2016). [CrossRef]  

19. P. A. Huidobro, X. Shen, J. Cuerda, E. Moreno, L. Martin-Moreno, F. J. Garcia-Vidal, T. J. Cui, and J. B. Pendry, “Magnetic localized surface plasmons,” Phys. Rev. X 4(2), 021003 (2014). [CrossRef]  

20. Z. Gao, F. Gao, Y. Zhang, X. Shi, Z. Yang, and B. Zhang, “Experimental demonstration of high-order magnetic localized spoof surface plasmons,” Appl. Phys. Lett. 107(4), 041118 (2015). [CrossRef]  

21. S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, “Magnetic plasmon modes in periodic chains of nanosandwiches,” Opt. Express 16(6), 3560–3565 (2008). [CrossRef]  

22. T. Li, R. X. Ye, C. Li, H. Liu, S. M. Wang, J. X. Cao, S. N. Zhu, and X. Zhang, “Structural-configurated magnetic plasmon bands in connected ring chains,” Opt. Express 17(14), 11486–11494 (2009). [CrossRef]  

23. Z. Wang, Z. Gao, Y. Zhang, P. An, J. Lou, P. Cheng, and H. Zhao, “Selectively enhanced magnetic vertical conductive coupling among magnetic localized surface plasmons,” J. Phys. D: Appl. Phys. 52(13), 135101 (2019). [CrossRef]  

24. F. Gao, Z. Gao, Y. Luo, and B. Zhang, “Invisibility dips of near-field energy transport in a spoof plasmonic metadimer,” Adv. Funct. Mater. 26(45), 8307–8312 (2016). [CrossRef]  

25. Z. Gao, F. Gao, Y. Zhang, and B. Zhang, “Deep-subwavelength magnetic-coupling-dominant interaction among magnetic localized surface plasmons,” Phys. Rev. B 93(19), 195410 (2016). [CrossRef]  

26. J. Zhang, Z. Liao, Y. Luo, X. Shen, S. A. Maier, and T. J. Cui, “Spoof plasmon hybridization,” Laser Photonics Rev. 11(1), 1600191 (2017). [CrossRef]  

27. Z. Liao, G. Q. Luo, H. F. Ma, B. C. Pan, B. G. Cai, Y. F. Yu, and T. J. Cui, “Localized surface magnetic modes propagating along a chain of connected subwavelength metamaterial resonators,” Phys. Rev. Appl. 10(3), 034054 (2018). [CrossRef]  

28. Z. Liao, G. Q. Luo, B. G. Cai, B. C. Pan, and W. H. Cao, “Subwavelength negative-index waveguiding enabled by coupled spoof magnetic localized surface plasmons,” Photonics Res. 7(3), 274–282 (2019). [CrossRef]  

29. S. Hu, “Localized surface magnetic modes transfer along bend and T-splitter chain arrays of subwavelength metamaterial resonators,” Eur. Phys. J. D 73(9), 205 (2019). [CrossRef]  

30. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999). [CrossRef]  

31. W. Huang, J. Lin, M. Qiu, T. Liu, Q. He, S. Xiao, and L. Zhou, “A complete phase diagram for dark-bright coupled plasmonic systems: Applicability of Fano’s formula,” Nanophotonics 9(10), 3251–3262 (2020). [CrossRef]  

32. Z. Gao, F. Gao, Y. Zhang, H. Xu, Y. Luo, and B. Zhang, “Forward/backward switching of plasmonic wave propagation using sign-reversal coupling,” Adv. Mater. 29(26), 1700018 (2017). [CrossRef]  

33. X. Gao, J. Zhang, H. C. Zhang, L. Liu, Q. Ma, P. Xu, and T. J. Cui, “Dynamic controls of second-harmonic generations in both forward and backward modes using reconfigurable plasmonic metawaveguide,” Adv. Opt. Mater. 8(8), 1902058 (2020). [CrossRef]  

34. J. A. Dockrey, S. A. R. Horsley, I. R. Hooper, J. R. Sambles, and A. P. Hibbins, “Direct observation of negative-index microwave surface waves,” Sci. Rep. 6(1), 22018 (2016). [CrossRef]  

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)
Fig. 1.
Fig. 1. (a) Schematic illustration of a single MSS. (b) Magnetic-field distributions along the x axis for the MSS with various numbers of turns n. (c) Magnetic-field distributions along the z axis for the MSS with various numbers of turns n. (d) Near-field response spectra for a single MSS, the inset of (d) is plot of the magnetic fields in the xy plane for the MSS with n = 3.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. (a) MSS pair simulation model. (b) Transmission spectra for the pair as a function of the angle $\beta $ . (c) Simulated and experimentally measured transmission spectra for MSS pairs with angles of rotation $\beta $  = 20°, 50°, and 80°. (d) Simulated and experimental electric-field distributions for the relevant resonance modes.

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. (a) The numerical relation between κ and rotation angles. The inset corresponds to the schematic of CROW consisting of 5 MSSs. The distance d between adjacent MSSs is 10.8 mm. (b) Theoretical and experimental dispersion curves for CROWs possessing different fixed ( ${\beta _{i{\; }}}$  =  $\beta $  = 20° and 50°) and varying ( ${\beta _{i{\; } = {\; }1,{\; \; }2,{\; }3,{\; \; }4}}$  = 10°, 30°, 50°, 70°) angles between the constituent MSSs. (c) Experimental transmission spectra for the same CROWs. (d) Simulated and measured vertical electric-field distributions for CROW consisting of five MSSs ( $\beta \; $  = 50°).

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. (a) Schematic of the S-chain. (b) Simulated and measured transmission spectra for the S-chain consisting of multiple MSSs. (c) Simulated and (d) experimental electric fields Ez above the S-chain.

View in Article | Download Full Size | PDF

Equations (1)

Equations on this page are rendered with MathJax. Learn more.

− d a 1 d t = i f 0 a 1 + i κ f 0 a 2 − d a 2 d t = i f 0 a 2 + i κ f 0 a 1
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use