- Photonics Research
- Vol. 11, Issue 10, 1613 (2023)
Abstract
1. INTRODUCTION
Zero-index materials (ZIMs) are materials or composite structures that exhibit an effective refractive index of zero at a given frequency [1–4], resulting in an infinite spatial wavelength. This effect can be leveraged to overcome the limitations imposed by the finite spatial wavelength of electromagnetic waves, thereby enabling various novel physical phenomena and applications in linear [5–7], nonlinear [8–10], and quantum electromagnetic systems [11,12]. Although the concept of zero-index materials was proposed a long time ago [13], zero-index materials, such as indium tin oxide [9,14,15], waveguides at cut-off frequencies [16–18], fishnet metamaterials [19], and doped -near-zero (ENZ) media [20] did not garner too much research interest until around 2010 [2]. However, the aforementioned materials and artificial structures exhibit large ohmic losses because of their metallic components [21]. In contrast, ZIMs based on all-dielectric photonic crystals exhibit zero ohmic loss, enabling the realization of ZIMs over a large area of arbitrary shapes. A photonic-crystal-based ZIM was first realized in the microwave regime based on pillars embedded in a parallel metal waveguide [22]. Subsequently, photonic-crystal-based ZIMs ranging from acoustic [23,24] to photonic regimes [25–29] have been reported, demonstrating fascinating physical phenomena and applications, such as supercoupling [4,30], leaky-wave antennas [31], cloaking [22,32–35], superradiance [12,25], and phase matching with multiple configurations of input and output waves for nonlinear optics [10]. For the applications in superradiance and nonlinear optics, although photonic-crystal-based ZIMs do not show strong field enhancement which can be provided by the slow-light effect of ENZ media [13,36–38], photonic-crystal-based ZIMs’ finite impedance and absence of ohmic loss can still reduce the pump power dramatically.
Despite this progress, most of the ZIMs reported thus far are passive, with constant post-fabrication electromagnetic properties, limiting their applications in passive devices. As a consequence, there have been substantial efforts to implement active ZIMs with tunable magnetic () and electric () properties using external stimuli [39,40]. One particular tunable system is magnetically tunable ZIM [41], which is essentially different from electrically, thermally, and other tunable ZIMs [42–44], photonic crystals [45–48], metasurfaces [49], and transparent conductive oxide (TCO) materials [9]. Such a system was first proposed by Davoyan
In this study, we design and experimentally investigate a magnetically tunable ZIM consisting of an array of gyromagnetic pillars embedded within a parallel-plate copper waveguide. Under an applied magnetic field of 430 Oe, the photonic band structure of the proposed ZIM changes from a zero-index state to a photonic bandgap state, corresponding to a transition from a “zero-index phase” to a “single negative phase.” Based on this property, we propose a magnetic field-induced on–off switch of the supercoupling state in an S-shaped ZIM waveguide, resulting in a low intrinsic loss of 0.95 dB and a high extinction ratio of 30.63 dB at 9 GHz. We also demonstrate a magnetic-field-controlled switch to effect transitions between a zero-index state and a nontrivial topological boundary state in the magnetic ZIM. In addition, we also present another device design, showing a much higher extinction ratio up to 104 dB. These results demonstrate the potential of applying active ZIMs in RF ferrite switches, ultra-compact ferrite phase shifters, electromagnetic wave modulators, and some other nonreciprocal devices.
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2. RESULTS
A. Design of Magnetically Tunable Zero-Index Metamaterials
First, we designed a Dirac-like cone-based zero-index metamaterial (DCZIM) consisting of a square array of dielectric pillars embedded in a parallel-plate copper waveguide [22]. At the point, the accidental degeneracy of two linear dispersion bands and a quadratic dispersion band formed a Dirac-like cone dispersion, corresponding to an impedance-matched zero effective index [22]. In contrast to conventional passive ZIMs, we achieved active modulation by fabricating a square lattice of pillars constructed using a magnetic dielectric material, yttrium iron garnet (YIG). By applying a magnetic field to the YIG pillars along the direction perpendicular to the wave vector of the transverse magnetic (TM) mode, an effective permeability modulation that is quadratically proportional to the YIG magnetization was observed owing to the Cotton–Mouton effect (see Appendix B for further details). In turn, this effect enabled the modulation of the photonic band structure and the effective index of metamaterials.
We implemented the proposed design using the structure depicted in Fig. 1. Figure 1(a) illustrates the experimentally fabricated metamaterial consisting of a square lattice of gyromagnetic YIG pillars with a 3.53 mm radius and a 17.9 mm lattice constant. The dielectric constant [56] and permeability of the YIG material were characterized (see Appendices A and C for further details). The YIG pillars were placed in a waveguide consisting of two parallel copper clad laminates separated by 4 mm. A magnetic field was applied to each YIG pillar by placing a neodymium iron boron (NdFeB) permanent magnet under each pillar and behind the copper back plate. A uniform magnetic field along the -direction was observed, whose intensity reached 430 Oe in the middle of the two copper clad laminates. This was sufficient to saturate the YIG pillars (see Appendix C for further details).
Figure 1.Schematic diagram of the active DCZIM structure. (a) The structure of an active DCZIM based on a gyromagnetic photonic crystal. (b) Schematic diagram of a unit cell. The YIG pillars were placed in a parallel-plate copper-clad waveguide with height
B. Experimental Observation of Photonic Band Structure
To characterize the properties of DCZIM, we first calculated and then experimentally measured the photonic band structure in the plane of the YIG array. The gray dotted line in Fig. 2(a) represents the calculated band structure for the TM modes (the electric field is polarized in the direction) of this photonic crystal, as observed based on a simulation using COMSOL Multiphysics. We observed a clear Dirac-like cone dispersion at 9 GHz. When a magnetic field of was applied along the direction, the off-diagonal component induced the Cotton–Mouton effect in YIG, changing the frequencies of the three photonic bands forming the Dirac-like cone. Owing to time reversal symmetry (TRS) breaking, the photonic crystal transitioned from symmetry to symmetry (see Appendix D for further details). As a result, the degeneracy was broken, resulting in two bandgaps, as indicated by the gray dotted lines in the right panel of Fig. 2(a).
Figure 2.Theoretical and experimental demonstration of an active ZIM. (a) Measured and calculated (gray dots) photonic bands of the active ZIM using Fourier transform field scan (FTFS) of the TM modes corresponding to applied magnetic fields of 0 (left panel) and 430 Oe (right panel). (b), (c) Simulated three-dimensional dispersion surfaces near the Dirac-point frequency, depicting the relationship between the frequency and the wave vectors (
The modes supported by this structure were analyzed by considering those supported by a square array of two-dimensional YIG pillars. As depicted in Figs. 2(b)–2(d), this structure supported three modes for TM polarization near the 9 GHz frequency—a monopole mode, a transverse magnetic dipole mode, and a longitudinal magnetic dipole mode. When a magnetic field was applied, as illustrated in the right part of the panel, these three modes were displaced to different frequencies (8.95 GHz, 9.52 GHz, 11.04 GHz, respectively). Because of the broken TRS, all three modes were required to be rotated through 180° to coincide with each other.
We further calculated the Chern number of each photonic band from low frequency to high frequency near the Dirac point to be 0, 1, and , respectively. Based on the Chern number of each band, we calculated the Chern numbers of the bandgaps, 1 and 2 [right panel of Fig. 2(a)], to be and , respectively. Based on the Chern numbers of the bandgaps, 1 and 2, we can determine their topological nature—bandgaps 1 and 2 were topologically trivial and nontrivial, respectively. Depending on the Chern numbers of the bandgap, a topological nontrivial bandgap (10.24–11.04 GHz) near the Dirac point could be realized. This resulted in edge states localized at interfaces that were topologically protected. Being topologically protected, these edge states were immune to disorder and perturbations.
We experimentally characterized this metamaterial using the setup proposed by Zhou
First, we measured the photonic band structure of the metamaterial under an applied magnetic field of 0, as illustrated using the intensity plot in the left panel of Fig. 2(a). The band structure of the TM bulk states was obtained by applying two-dimensional discrete Fourier transform (2D-DFT) to the measured complex field distribution over the metamaterial (see Appendices A and E for further details). The measured photonic band structure exhibited good agreement with the simulation results (represented by gray dots). Both the measured and computed band structures exhibited a bandgap between 5 and 7 GHz as well as bulk modes between 7 and 12 GHz, indicating a Dirac-like cone dispersion near 9 GHz. The nondegeneracy of the photonic bands was also recorded after applying a 430 Oe magnetic field along the direction, as depicted in the right panel in Fig. 2(a). The bandgaps were experimentally measured to be at 8.95–9.60 GHz and 10.24–11.04 GHz, corroborating the simulation results.
C. Phase Transition and Parameters Retrieval
The magnetic field-induced band structure and transmittance modulation can be regarded to be a phase transition process from a material perspective. This phenomenon can be observed by retrieving the effective permittivity and permeability tensor elements of the metamaterial in the presence and absence of an applied magnetic field, as illustrated in Fig. 3. We used the boundary effective medium approach (BEMA) to calculate the effective constitutive parameters, as proposed in a previous study on ZIM [57]. In Figs. 3(a)–3(c), denotes the effective permittivity, denotes the diagonal of the effective permeability tensor, and denotes the off-diagonal component of the effective permeability tensor, .
Figure 3.Magnetic field-induced phase transition of active ZIM. Real and imaginary parts of the effective permittivity (
Figure 3(a) depicts the results corresponding to an applied magnetic field of 0. The real parts of and cross zero simultaneously and linearly at 9 GHz, exhibiting -and--near-zero behavior corresponding to the zero-index phase. The imaginary parts of and were both close to 0. This low loss was attributed to the small loss tangent (0.0002) of the YIG material in this frequency range. When a magnetic field of 430 Oe was applied, phase transitions occurred from the zero-index phase to the -negative (MNG) or the -negative (ENG) phase, as depicted in Fig. 3(b) and Fig. 3(c), respectively. In the bandgap 8.95–9.6 GHz [Fig. 3(b)], is positive, whereas () [41] is negative, which corresponds to the MNG phase. The impedance at 9 GHz was tuned from 1.84i to 0.62i after applying the magnetic field, leading to a total reflection of the incident electromagnetic (EM) wave owing to impedance mismatch. In the bandgap 10.24–11.04 GHz [Fig. 3(c)], is negative, whereas is positive, which corresponds to the ENG phase. Corresponding to both frequency ranges, the real parts of and decreased as the frequency increased, exhibiting anomalous dispersion. The incident EM wave was still reflected due to impedance mismatch. Additionally, as depicted in the inset of Fig. 3(c) in the ENG frequency regime, a nontrivial topological boundary state was observed at the metamaterial edge because of the difference between the Chern numbers of the upper and lower structures.
The complex phase transition phenomena discussed above were attributed to the location of the ZIM at the origin of the metamaterial phase diagram, which allowed it to reach all quadrants of the phase diagram via appropriate tuning of the constitutive parameters [41,58]. Further discussion on the attainment of other phases based on the phase diagram is depicted in Figs. 9–11 of Appendices F and G. Such unique properties of an active ZIM are indicative of its potential with respect to the modulation of the propagation of EM waves.
D. Wave Propagation in Magnetically Tunable Zero-Index Metamaterials
To showcase a microwave switch based on the phase transition effect of active ZIM, we fabricated a ZIM waveguide switch by leveraging the contingency of the supercoupling state on the applied magnetic field, as illustrated in Fig. 4(a). First, a ZIM waveguide comprising top and bottom metal plates and 100 YIG pillars was fabricated, forming two sharp 90-deg bends, to verify the supercoupling effect experimentally (see Appendices A and H for further details). The waveguide was coupled to the coaxial line via sub-miniature version A (SMA) connectors. Linear tapered sections were used to sustain the mode and induce its gradual evolution into the TM mode of the metamaterial during propagation from the source to the waveguide. Perfect magnetic conductor (PMC) boundary conditions were realized using aluminum alloy walls at a distance of from the metamaterial as the lateral boundaries [30] (see Appendices A and H for further details). Using the device depicted in Fig. 4(a), we successfully switched between the bulk state (supercoupling state) and the photonic bandgap state (off state) by applying appropriate magnetic fields to the YIG pillars.
Figure 4.Structure and characterization of a microwave switch based on active DCZIM. (a) Photograph of the microwave switch sample. (b) The measured transmissions in the absence and presence of an applied magnetic field of 430 Oe. (c) The real part of the
Figure 5.Schematic diagram depicting the Cotton–Mouton effect.
Figure 4(b) depicts the measured transmission spectra corresponding to both states. A large transmission contrast was observed over 8.9–9.4 GHz, which was consistent with the bandgap frequencies calculated via numerical simulation in Fig. 2(a). The difference in device transmission under zero and 430 Oe magnetic fields was larger than 30 dB at approximately 9 GHz and the device insertion loss was 3.75 dB. Considering that the coupling loss induced by the SMA connectors was 2.8 dB, the intrinsic loss of the ZIM waveguide was as low as 0.95 dB, primarily induced by the absorption of YIG materials. Notably, such on–off switching effect can also be observed as a giant Cotton–Mouton effect due to the zero-index property, which was not observed in conventional ferrite-based waveguide devices. A much larger extinction ratio up to 104 dB can be achieved by simply making a longer device (see Appendix J). Such devices may show fast switching speed of kilohertz to megahertz similar to conventional ferrite switches, but also show higher extinction ratio and smaller device footprint, making them particularly attractive for microwave switch applications.
We verified the supercoupling behavior in the absence of an applied magnetic field by measuring the component of the electric field at each point of the metamaterial, as illustrated in Fig. 4(c). At 9 GHz, the electric field tunneled through the metamaterial with almost no phase change in the form of a bulk mode, verifying supercoupling behavior. In contrast, in the absence of YIG pillars in the waveguide, the wave was reflected back to the incident port, as described in Appendix H. This confirmed that the supercoupling behavior was induced by the DCZIM. As depicted in Fig. 4(d), in the presence of an applied magnetic field, the electromagnetic wave decayed exponentially in the metamaterial owing to the photonic bandgap 1 depicted in Fig. 2(a), leading to a high extinction ratio. We also constructed a switch between the supercoupling state and the topological one-way transmission state by probing at the upper bandgap frequency of 10.6 GHz (see Appendix I for further details). This property can be used to fabricate efficient nonreciprocal phase shifters with compact device size. These unique properties are indicative of the potential of active ZIMs in novel active electromagnetic devices.
3. CONCLUSION
In this study, we proposed and experimentally operated a magnetically tunable ZIM. The metamaterial was operated by leveraging the Cotton–Mouton effect of the constitutive YIG pillars under applied magnetic fields, which alter the symmetry and bandgap opening of the Dirac-like cone-based ZIM. Such a ZIM shows an effective refractive index near zero () over a bandwidth of 6.77% in the absence of an applied magnetic field, and 5.43% (extinction ratio over 100 dB) when applying magnetic field, which means the bandwidth of our device is comparable with the commercial device. From a material perspective, the proposed metamaterial exhibited a phase transition from the zero-index phase to a single negative phase, leading to an effective index change from 0 to 0.09i at 9 GHz. Based on this property, we constructed and verified the function of a microwave switch by manipulating the supercoupling effect, thereby reducing the intrinsic loss to 0.95 dB and achieving a high extinction ratio of 30.63 dB at 9 GHz.
We believe that this study introduces a new approach to active ZIMs, particularly with respect to the development of efficient active electromagnetic and nonreciprocal devices. Utilizing the giant Cotton–Mouton effect, efficient ferrite switches with large extinction ratio and compact device size can be fabricated. Compact and efficient ferrite phase shifters can be fabricated based on the phase transition from the supercoupling state to the topological boundary state. By appropriately engineering the slots in parallel-plate copper waveguides [31], continuous beam steering over the broadside can be achieved by varying the applied magnetic field. Moreover, our design can be extended to the optical regime by embedding YIG pillars in a polymer matrix with gold films cladded [26], enabling the dynamic modulation of optical DCZIM. Further, such a magnetically tunable DCZIM can be used to modulate the four-wave mixing process in DCZIM by manipulating the zero-index phase-matching condition [10]. Additionally, we were able to modulate DCZIM-based large-area single-mode photonic crystal surface-emitting lasers with high output power [59]. Finally, we also successfully modulated the extended superradiance by tuning the effective index of the DCZIM, in which many quantum emitters were embedded [12].
Acknowledgment
Acknowledgment. The authors appreciate the discussions with Hengbin Cheng from the Institute of Physics, Chinese Academy of Sciences, and their assistance.
APPENDIX A: METHODS
The permittivity of the YIG used during numerical simulation was taken from that published by Zhou
The active DCZIM consisted of a square array of YIG pillars with a radius of 3.53 mm, a height of 4 mm, and a lattice constant of 17.9 mm [as indicated in Fig.
The YIG pillars were fabricated from YIG bulk crystals using an ultra-high-accuracy computer numerical control (CNC) machine (Mazak VARIAXIS i-700) with a dimensional accuracy exceeding 0.05 mm. As depicted in Fig.
Each YIG pillar was properly magnetized by aligning the NdFeB magnet used to apply the magnetic field precisely underneath each YIG pillar. To this end, a 2 mm thick acrylic sheet was first covered with a double-sided tape to form the substrate. Then, we fixed a 5 mm thick acrylic sheet including perforations with a radius of 5 mm and a lattice constant of 17.9 mm onto a 2 mm thick acrylic substrate. Finally, with respect to the designated direction of the magnetic field, we placed the NdFeB magnets into the holes of the 5 mm thick acrylic sheet, achieving a good alignment with the array of the YIG pillars [Fig.
We measured the photonic band structure, transmission spectra, and near-field distribution of the active metamaterial sample using two dipole antennas as the transmitter and receiver. Both antennas were inserted into the waveguide via holes drilled using an ultra-high-precision CNC machine and they were connected to a vector network analyzer (Rohde & Schwarz ZNB 20) to measure the parameters. Prior to measurement, 3.5 mm 85052D through-open-short-load calibrations were performed. As a result, the measured parameters included only the insertion loss of the tapered waveguides and the metamaterial.
APPENDIX B: THE COTTON–MOUTON EFFECT OF YIG
When biased by a magnetic field along the axis (), the magnetic permeability of YIG is given by the following:
For TM-polarized plane waves, the only nonzero component of the electric field () is the component (, , ). Assuming that the wave vector lies along the axis, the Helmholtz equation becomes
We can simplify the above equation as follows:
Because of the third term in Maxwell’s equations, ,
Therefore, we obtain
Comparing this equation with the equation , we obtain the effective refractive index:
Following the above derivation, the effective index of the TE mode is given by
The above analysis demonstrates that, in the case of linear polarization, the effective refractive index of YIG is modulated by material magnetization—namely the Cotton–Mouton effect, as shown in Fig.
APPENDIX C: THE PERMITTIVITY, PERMEABILITY, AND ROOM TEMPERATURE MAGNETIZATION HYSTERESIS OF YIG
At the frequency studied, YIG is an isotropic material with a relative permittivity of 13 and a dielectric loss tangent of 0.0002 [
, , denotes the damping coefficient, and denotes the operating frequency [
In accordance with the aforementioned model, when a magnetic field of 430 Oe was applied, the ferromagnetic resonance frequency of YIG was 1.20 GHz. The real and imaginary parts of the permeability tensor of the material with respect to the frequency are depicted in Fig.
Figure 6.Magnetic hysteresis loop of a YIG pillar measured using a vibrating sample magnetometer (VSM) at room temperature (300 K). The sample was magnetically saturated under a small applied magnetic field of
Figure 7.Real and imaginary parts of the permeability tensor of YIG under the applied 430 Oe magnetic field.
APPENDIX D: SYMMETRY ANALYSIS OF THE DCZIM
As described using topological theory and group theory, the “Dirac points” are protected by “-symmetry” in the entire Brillion zone, in which indicates the parity inversion and indicates the time-reversal symmetry [
When a magnetic field along the axis was applied, time-reversal symmetry was broken, as was the product of time-reversal symmetry () and parity () inversion. As a result, the Dirac cones opened and the degeneracy was lifted, i.e.,
However, the following part of symmetry still remained valid:
Therefore, when the DCZIM structure preserves symmetry under a -breaking operation, only symmetry is retained. This proves that the magnetized DCZIM exhibits symmetry variation from to .
APPENDIX E: CHARACTERIZATION OF THE PHOTONIC BAND STRUCTURE USING THE FTFS METHOD
We characterized the photonic band structure of the DCZIM using Fourier transform field scan (FTFS) measurements—the results are depicted in Figs.
Figure 8.Measurement of the photonic band structure via FTFS. (a) Electric fields measured across the photonic crystal at 2.70 GHz. The source was placed at one corner of the sample, and the field was measured across the sample. Owing to the
APPENDIX F: PHOTONIC BAND STRUCTURES OF THE DCZIM UNDER DIFFERENT APPLIED MAGNETIC FIELDS
We simulated the photonic band structures of the DCZIM under different applied magnetic fields, as illustrated in Figs.
APPENDIX G: EFFECTIVE CONSTITUTIVE PARAMETERS AND PHASE TRANSITION OF THE DCZIM UNDER DIFFERENT APPLIED MAGNETIC FIELDS
We simulated the effective constitutive parameters of the DCZIM as a function of the applied magnetic field using the theory proposed by Wang
Figure 9.(a) Simulated band structures of the DCZIM under different applied magnetic fields. (b) The band edges of the bandgap, 10.24–11.04 GHz, at M as a function of the applied magnetic field.
Figure 10.Effective constitutive parameters of the DCZIM under (a), (b) 600 Oe field and (c), (d) 800 Oe field, as calculated using BEMA.
Figure 11.Magnetic field-induced phase transition of active DCZIM under different applied magnetic fields.
APPENDIX H: DEVICE STRUCTURE OF THE MICROWAVE SWITCH AND ELECTRIC FIELD AND PHASE MEASUREMENT OF THE 90-DEG BENT WAVEGUIDE WITHOUT YIG PILLARS
Figure
Figure 12.Device structure of the microwave switch.
Figure 13.(a) Electric field intensity and (b) phase distributions of 90 deg bent waveguide without YIG pillars.
APPENDIX I: DEMONSTRATION OF A TOPOLOGICAL NONTRIVIAL BOUNDARY STATE IN MAGNETIZED DCZIM
As described in the main text, the Chern number of the bandgap 10.24–11.04 GHz, is 1 [Fig.
Figure 14.Observation of topologically protected unidirectional boundary state in the metamaterial. (a) The structure of the active metamaterial sample with a boundary formed by a copper bar. (b) Numerically simulated projected band structure of the metamaterial with a strip with 20 unit cells in the
APPENDIX J: DEMONSTRATION OF A MUCH LARGER EXTINCTION RATIO IN THE ULTRA-COMPACT MAGNETICALLY TUNABLE SPIRAL STAIRCASE ZIM SWITCH
As mentioned in the main text, the ZIM structure can support tunneling waveguides composed of arbitrary shapes because of its equivalent infinite wavelength. This novel feature of ZIM enables it to serve as one of the most suitable structures for switching. For magnetic ZIM-based RF switches, the extinction ratio can be enhanced by simply increasing the device length, whereas the insertion loss can be kept low due to the supercoupling effect. As depicted in Fig.
Figure 15.Observation of a much larger extinction ratio in the ultra-compact spiral magnetically tunable ZIM switch staircase structure. (a) The structure of the spiral magnetically tunable ZIM switch staircase structure. Each layer in the structure is 1.5 mm high which contains 1 mm high metal waveguides and 0.5 mm interlayer air gaps. (b) Transmission spectra of the magnetically tunable ZIM switch’s on state and off state. (c) The real part of the
References
[1] I. Liberal, N. Engheta. Near-zero refractive index photonics. Nat. Photonics, 11, 149-158(2017).
[13] J. B. Khurgin, R. S. Tucker. Slow Light: Science and Applications(2018).
[30] P. Camayd-Muñoz. Integrated Zero-index Metamaterials(2016).
[36] J. B. Khurgin. Slow light in various media: a tutorial. Adv. Opt. Photonics, 2, 287-318(2010).
[39] N. Xiang, Q. Cheng, J. Zhao, T. J. Cui, W. X. Jiang, H. F. Ma. A switchable zero index metamaterial. 3rd Asia-Pacific Conference on Antennas and Propagation, 1050-1052(2014).
[49] I. V. Shadrivov, M. Lapine, Y. S. Kivshar. Nonlinear, Tunable and Active Metamaterials(2015).
[54] N. Engheta. Pursuing near-zero response. Science, 340, 286-287(2013).
[61] D. M. Pozar. Microwave Engineering(2011).
[65] Microwave Applications. Product information: microwave waveguide switches.
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