Completely stopping microwaves with extremely enhanced magnetic fields

A microwave one-way waveguide of three-dimensional configuration is proposed and investigated theoretically. In this waveguide there exists a complete one-way propagation band, where the mode propagates only in one direction and can be immune to backscattering. By terminating the one-way waveguide with metal slab, one-way propagating waves in this waveguide system can be stopped at the terminal end without any backscattering. Meanwhile, a hotspot with extremely enhanced magnetic-field amplitude is generated in this 3D waveguide system. For an incident microwave pulse, the trapped wave packet can be compressed to deep subwavelength scale besides the magnetic field enhancement. Moreover, the magnetic field enhancement of trapped waves can be further largely increased by tapering laterally the waveguide system. The approach for trapping microwaves has promising applications in magnetic sensing and magnetic non-linearity.

steady electric current is required to be present in the YIG. In addition, based on the zero group velocity of SMPs, the trapping behaviour would be seriously affectted by the material loss in practice. All one-way SMPs at terahertz or microwave frequencies reported previously are spatially two-dimensional (2D), i.e., their fields as well as the related systems are uniform and extend infinitely in the lateral direction. This is also the situation for trapped hotspots based on one-way SMPs. In this paper, a realistic one-way waveguide, which is a three-dimensional (3D) structure constructed by using YIG, is proposed and investigated theoretically. We will show that this waveguide can sustain robust one-way mode at microwave frequencies. Furthermore, we propose a simple and effective approach to trap microwaves in a 3D system by using the one-way waveguide. In this system, only a uniform dc magnetic field is needed. Compared to Liu's approach 22 , wave trapping would be available for our approach even in the cases with serious material losses, and the related magnetic-field enhancement would be weakly affected by the material loss. On the other hand, different from the effect of wave trapping in refs 12,13 , wave trapping in our system would lead to extreme magnetic-field enhancement, instead of the electric-field enhancement. All above properties for the proposed 3D waveguide system will be numerically demonstrated in detail. Moreover, we will show that by tapering laterally the 3D system for trapping wave, the field enhancement can be further remarkably increased.

Realistic Microwave One-Way Waveguides
The realistic one-way waveguide we consider is formed by a rectangular metallic waveguide partly filled with YIG (for the region y < 0), as illustrated in the upper panels in Fig. 1(a). The waveguide width is denoted by w (in the x or lateral direction) and the YIG thickness denoted by D. The remaining region within the metallic waveguide is filled with air, whose thickness is denoted by d. The upper right panel in Fig. 1(a) shows the internal structure of the guiding system. In this system, a uniform dc magnetic field (H 0 ) is applied in the x direction. The magnetized YIG has the relative permittivity ε m = 15 and the relative permeability m µ ↔ in the form 20 where ω is the angular frequency, ω 0 = 2πγH 0 (where γ is the gyromagnetic ratio) being the precession angular frequency, ω m is the characteristic circular frequency of the YIG, and ν is the damping coefficient. Here, the metal is assumed to be perfect electric conductor, which is a good approximation for the microwave regime. We suppose that the waveguide width is enough small, i.e., λ w m  , where λ m is the vacuum wavelength for ω m . Thus, in a frequency range around ω m , the waveguide can only support TE modes with uniform electric field pointing in the x direction. In the case when the YIG thickness is enough large, the guiding modes in such a 3D system should have almost the same properties as those in a 2D guiding system, which is a YIG-air-metal layered structure shown in the lower panel in Fig. 1(a).
In the band gap of the magnetized YIG, the 2D system in Fig. 1(a) can support not only SMPs but also regular mode, the latter is guided by the mechanism of total internal reflection 21 . Both modes are TE polarized. The dispersion relation of modes, which associates the propagation constant (k) with ω, can be derived analytically. It has the form m v 2 1 for regular mode with |k| < k 0 , where = − p k k 0 2 2 . Obviously, both types of mode are non-reciprocal, because there exists a linear term with respect to k in Eqs (4) and (5). The dispersion bands for SMPs lie outside of the light cone (defined by air), and its left branch (k < 0) has an asymptotic frequency of ω sp = ω 0 + ω m /2, at which k → −∞ 21 . The dispersion band for the regular mode lies within the light cone, and it links with the SMP bands at points of ω = c|k|, thus forming a complete dispersion band over a wide frequency range. Figure 1(b) shows the dispersion relation (solid line) for the 2D guiding system. In this paper, we take ω 0 = 0.5ω m and ω m = 10π × 10 9 rad/s for the magnetized YIG, and d = 0.05λ m (λ m = 2πc/ω m ) for air. As shown in Fig. 1(b), a complete dispersion band occurs, which intersects with the light lines. There exists a piece of complete one-way propagation (COWP) band with positive slope, which is located within the band gap of the YIG. Evidently, the mode on the COWP band can only propagate forward. The COWP band is between ω sp and ω cf , where ω cf = ω 0 + ω m , and it corresponds to a frequency range from 5 GHz to 7.5 GHz. The COWP region is indicated by a shaded rectangle in Fig. 1(b). The dispersion relation for the one-way mode in our 3D system should be the same as that for the 2D system. To verify this, we numerically solve for modes in the 3D system with the finite-element method (FEM) (by using the COMSOL Multiphysics). For the 3D system, the waveguide width is w = 5 mm (i.e., w = λ m /12) and the YIG thickness D = 9 mm. The calculated dispersion relation is plotted as solid circles in Fig. 1(b), which agrees well with that for the 2D system. Figure 1(c) shows the field amplitudes of the mode in the 3D system for 5.6 GHz (i.e., ω = 1.12 ω m ), at which k = 0, and as expected, both the E and H fields are uniform in the x direction, and they peak at the interface of the air and YIG, like the fields of one-way SMPs.
To validate the guiding properties of the 3D system, we performed the simulation of wave transmission in it with the FEM. In the simulation, a linear current source of the length w was placed at y = d/2 to excite waves. The operation frequency was set at f = 5.6 GHz. Figure 2(a) shows the simulated magnetic field amplitudes. Clearly, the excited wave is only propagating forward as expected. Figure 2(b) corresponds to the field amplitudes on a vertical cutting slice at a (horizontal) distance of 30 mm away from the source, while Fig. 2(c,d) are the field amplitudes on the horizontal cutting slices at y = 0 and y = d/2, respectively. We further examined the robustness of the one-way mode. For this purpose, we introduced an obstacle into the system in Fig. 2(a), and it was placed at a (horizontal) distance of 30 mm from the source. The obstacle is a square YIG column with a side length of 1 mm. It lies on the half of the YIG surface, and its length is equal to w/2. The system with the obstacle becomes asymmetric in the x direction. The simulated magnetic field amplitudes are plotted in Fig. 2(e-h). Clearly, the one-way propagating wave goes around the obstacle and continues to travel forward. The lateral uniformity of field is broken only in the local region of the obstacle, but is completely recovered behind it. Therefore, the one-way mode can be immune to backscattering in our 3D system.

Wave Stopping and Magnetic-Field Enhancement
Suppose that the one-way waveguide is terminated by a metal slab in the propagation direction. In this waveguide system, waves traveling forward in it will be blocked when they reach the terminal end. In the COWP range, there exists no backward-propagating mode in the waveguide, therefore the blocked waves must be completely stopped at the terminal end without any backscattering. To verify this, by using the FEM, we simulated wave transmission in the terminated waveguide system with a length of 120 mm. In the simulation, the open waveguide end at z = 0 was excited using an input port with a TE 01 mode, and the frequency was f = 5.6 GHz. To obtain a steady solution, we took the loss of YIG into account, and set ν = 10 −3 . The simulated magnetic field amplitudes are plotted in Fig. 3(a). Clearly, a hotspot with extremely enhanced magnetic field occurs near the terminal end.  electric field amplitudes are plotted in Fig. 3(d), and no remarkable field enhancement is observed. Figure 3(e,f) are the electric-field profiles similar to Fig. 3(b,c).
As the electric field of the trapped wave points in the x direction, it vanishes at the terminal end which is formed by a metal slab. Consequently, the hotspot generated by trapping wave occurs near the terminal end but peaks at a small distance away from it. The hotspot corresponds to the accumulation of EM energy in the region of subwavelength scale, and for sustaining it, there must exist a large number of electric or "magnetic" charges locally on the YIG surface. These charges are impossible to be electrons, because the electric field is tangential to the interface of YIG and air. However, the magnetic field of the trapped wave has a nonzero component normal to the interface, so "magnetic" charges can be effectively excited and heaped at the YIG-air interface. In the YIG, "magentic" charges also exist with high density within the skin layer, and their density corresponds to the divergence of the magnetic moment. As a result of trapping wave, the magnetic field amplitudes are extremely enhanced, while the electric field amplitudes remain at a usual level. To characterize the field enhancement by trapping wave, we performed the simulation of wave transmission once more, but the length of the one-way waveguide was increased to 360 mm. The simulated magnetic (electric) field amplitude at the propagation distance of 120 mm (on the YIG surface) is used to normalize the maximal magnetic (electric) field amplitude of the hotspot in Fig. 3, and the result H max (E max ) is used to represent the field enhancement. It is found that H max = 1800 and E max = 1.85. Therefore, by completely stopping wave, the magnetic field amplitude is enhanced by 3 orders of magnitude at the hotspot.
We further investigate the influence of the YIG loss on the hotspot. To clarify this, several ν values are numerically analyzed, and besides the field enhancement, the hotspot sizes are also evaluated. The longitudinal (transverse) size of the hotspot can be defined as the separation between the locations where the magnetic field amplitude has fallen to 1/e of its maximal value in the z (y) direction, and we denote it by δ l (δ t ). The dependences of H max , E max , δ l , and δ t on ν are plotted in Fig. 4(a-d), respectively. Note that the hotspot is uniform in the x direction. It is found that δ l = 0.83 mm for ν = 10 −4 , δ l = 0.08 mm for ν = 10 −3 , and δ l = 0.01 mm for ν = 0.01, while δ t is nearly 0.003 mm for all three cases. Similar to δ t , the magnetic-field enhancement H max is weakly affected by ν. It is found that H max = 1874 for ν = 10 −4 , H max = 1800 for ν = 10 −3 , and H max = 1282 for ν = 0.01. Evidently, when the material loss becomes serious, the hotspot contracts in the longitudinal direction, thus strong magnetic field at its center is almost preserved. So the hotspot and high magnetic-field enhancement are available even for the cases with serious material losses.
In the above analysis, the operation frequency is fixed at f = 5.6 GHz, which corresponds to the angular frequency ω = 1.12ω m . We further analyze the spectral property of the trapped hotspot. The calculated results are plotted in Fig. 5, where ν is set to be 10 −3 . Figure 5(a) shows the value of H max as a function of ω. H max reaches a maximum of 2427 at ω = 1.24ω m (i.e., f = 6.2 GHz). Figure 5(b) shows the dependence of E max on the frequency. E max slowly decreases with the frequency, and it is less than 3 (but larger than 1) over the whole COWP range [ω m , 1.5ω m ].
Finally, it is necessary to evaluate the propagation length of the one-way mode in the lossy case. In the lossy case, k is a complex number, i.e., k = k r + ik i (where k r and k i are real-valued numbers). Evidently, the influence of ν on k r is negligible for  ν 1. Fig. 6(a) shows the value of k i as a function of frequency for ν = 10 −4 . Clearly, k i decreases with frequency. Note that for a given frequency, k i is linearly proportional to ν. The propagation length, defined by L p = 1/(2k i ), is shown in Fig. 6(b). Evidently, L p is inversely proportional to ν. For example, at f = 5.6 GHz, L p = 83λ (i.e., L p = 4446 mm) for ν = 10 −4 and L p = 8.3λ (i.e., L p = 444.6 mm) for ν = 10 −3 . As the loss of the one-way mode is proportional to the fraction of the EM energy in the YIG region of the waveguide, this energy fraction surely decreases with frequency. From the comparison of Figs 5 and 6, it is clear that the profile of EM energy along y for the hotspot should be quite different from that for the one-way mode.

Pulse Trapping and Compression
In the terminated waveguide system, waves can be completely trapped in it over the whole COWP band [ω m , 1.5ω m ]. So it can also completely trap wave pulses as long as their spectra are located within the COWP range. Using the finite-difference time-domain (FDTD) method, we simulated the transmission of a microwave pulse in this system. In the simulation, a Gaussian pulse of linear electric current is employed as source, and its central frequency lies at the center of the COWP band, i.e., f c = 6.25 GHz. The current source is uniform in the x direction, so should be the excited fields in the system. Thus, the FDTD simulation equations in the YIG region can be written as   Fig. 1.
where H ± = H y ± iH z . In the FDTD simulation, the loss coefficient of YIG is taken to be ν = 10 −3 . The length of the waveguide system is 200 mm, and the source is placed at z = 50 mm and y = 1.  The evolution of simulated H field amplitudes with time is displayed in Fig. 7. Figure 7(a) shows the field pattern at t 0 = 4T m , when the Gaussian pulse of the source reached the maximal amplitude, and Fig. 7(b) shows the results at t 0 = 8T m , when the source almost finished the excitation. At this time, an initial wave packet centered at z = 75.5 mm was formed, and its sizes were δ l = 36 mm and δ t = 7 mm. As the evolution time went on, the wave packet only traveled forward and simultaneously broadened due to the modal dispersion, as shown in Fig. 7(c) for t 0 = 16T m , at which the wave packet was centered at z = 137.5 mm with δ l = 75 mm. When the wave packet began to touch the terminal end of the system at z = 200 mm, it started to be compressed, and meanwhile, the magnetic fields grew rapidly, as shown in Fig. 7(d) for t 0 = 18T m . The wave packet was compressed to about half its longitudinal size at the time t 0 = 22T m , as shown in Fig. 7(e). When t 0 = 26T m , the wave packet was finally compressed to the sizes of δ l = 2 mm and δ t = 0.6 mm, and a hotspot with extremely strong H field was generated, as shown in Fig. 7(f). Figure 7(g,h) show the close-up views of local regions (near the terminal end) in Fig. 7(e,f), respectively. Figure 7(i,j) show the distributions of H field amplitude on the YIG surface along the z axis for the different evolution times. The maximal field amplitude of the trapped wave packet in Fig. 7(f) is larger by nearly 21 times than that of the initial wave packet in Fig. 7(b). Hence, by trapping microwave pulse, the wave packet can be compressed to deep subwavelength scale and the magnetic field can be highly enhanced.

Further Field Enhancement in Tapered Structure
It is clear that the modal property for the one-way waveguide is independent on its lateral width. This offers a possibility to further enhance the magnetic field for trapping wave by tapering laterally the relevant system. To show this, we performed wave transmission in a tapered system, where the lateral width (w) of the one-way waveguide is linearly reduced from 5 mm to 1 mm over the waveguide length of 120 mm, i.e., w = 5 mm at the open end (z = 0) and w = 1 mm at the treminal end (z = 120 mm). The other parameters of the tapered system are the same as the system with the uniform waveguide in Fig. 3. In the simulation, the open end of the tapered system (z = 0) was excited using an input port, as in the case of Fig. 3, and the operating frequency is f = 5.6 GHz.
The simulated H amplitudes are plotted in Fig. 8(a). Figure 8(b,c) show its vertical cutting slice at z = 119.96 mm and the horizontal cutting slice at y = 0, respectively. Evidently, incident waves are trapped and a hotspot is generated near the terminal end of the tapered structure, as in the case of Fig. 3, where the one-way waveguide is uniform. But the center of the hotspot lies at 0.04 mm before the terminal end for the present case. Figure 8(d,e) show the distributions of H amplitude along the central lines of the slices, which are marked with dashed lines in Fig. 8(b,c). For comparison, the corresponding results in Fig. 3(b,c) for the system with the uniform waveguide are also plotted as dash-dotted lines in Fig. 8(d,e). For the tapered system, the magnetic field enhancement is found to be 3900. In contrast, it is 1800 for the system with the uniform waveguide in Fig. 3. Therefore, in the tapered system, the magnetic-field enhancement is further increased by a factor of 2.17, which  and c), respectively. The dash-dotted lines in (d and e) correspond to the results in Fig. 3(b,c) for the system with uniform waveguide. The solid and open circles indicate the peaks of field profiles for the tapered and uniform systems, respectively. In the tapered system, the waveguide width is linearly reduced from 5 mm to 1 mm over the waveguide length of 120 mm. The other parameters are the same as in Fig. 3 is very close to the value of 5 . Note that the waveguide width at the terminal end in Fig. 8(a) is 5 times smaller than that in Fig. 3(a).

Conclusion
In summary, a realistic one-way waveguide, which is formed by a rectangular metallic waveguide partly filled with YIG, has been proposed and studied. From the dispersion diagram for this 3D waveguide, one-way mode has been found over a certain frequency range. By the numerical simulation, it has been shown that the one-way mode can be immune to backscattering. A 3D system for trapping microwaves has been further proposed, which is constructed by terminating the one-way waveguide with metal slab. It has been numerically demonstrated that one-way propagating waves can be excited in this system by using an input port, and when incident waves reach the terminal end, they are completely stopped without any backscattering. The trapped waves generate a hotspot with very strong magnetic field. By trapping wave in this manner, the enhancement of the magnetic field amplitude is up to 3 orders of magnitude. Moreover, the material loss only has a weak influence on the magnetic-field enhancement. It has also been demonstrated that for an incident pulse, the wave packet can be completely trapped near the terminal end, and meanwhile, it is compressed to deep subwavelength scale. Furthermore, by tapering laterally the system for trapping wave, the magnetic-field enhancement can be further largely increased. The proposed approach for completely stopping microwaves is feasible in practice and has promising applications in magnetic sensing and magnetic non-linearity.