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Abstract
We demonstrated that non-reciprocal wave propagation could be manipulated by a magnetic rod chain under bias DC magnetic fields. Made of ferrite material YIG and designed working in the microwave X-band, the rod chain exhibited almost a total reflection when the incident wave obliquely impinged on the rod chain, but exhibited nearly a total transmission when the wave reversed its propagation direction. The non-reciprocal wave propagation was due to the non-reciprocal diffraction of the rod chain for the orders 0 and ± 1. Further, the non-reciprocal wave propagation was directly observed by using the field mapping technique. The unique non-reciprocal wave property of the magnetic rod chain provides a new way to control the flow of EM waves.
© 2017 Optical Society of America
1. Introduction
In microwave and photonics technologies, non-reciprocal devices are the basic units used to control the propagation of electromagnetic (EM) waves. For example, the isolator can prevent reflected light entering laser light source in order to protect the light source; the circulator can make the signal flow among multiple ports to realize the separation of the reflected signal and the transmitted signal [1,2]. From physical essence, non-reciprocity of EM waves arises from the nonzero off-diagonal elements in Hamiltonian matrix of the Maxwell equations [3]. As a result, the Maxwell equations have unpaired eigenvalues that cause different propagation characteristics when waves reverse their propagation direction. A way to realize such a Hamiltonian is using gyrotropic material whose permeability or permittivity tensor has imaginary off-diagonal elements. Hartstein suggested the surface polaritons on semi-infinite gyromagnetic media exhibit non-reciprocal wave propagation characteristics [4]. The non-reciprocal surface waves were also found on the interface of magnetic photonic crystals (PCs). For example, Raghu and Haldane predicted the existence of one-way electromagnetic edge modes in two-dimensional PCs composed of non-reciprocal constituents. These modes are similar to the chiral edge states found in the integer quantum Hall effect in electronic systems [5,6]. The prediction was later experimentally demonstrated in gyromagnetic photonic crystals [7,8]. In addition to the non-reciprocal surface waves, non-reciprocal body waves were shown in one or two-dimensional magnetic arrays with additional space symmetry breaking. One example was the magnetic PCs with parity symmetry breaking while maintaining the parity-time symmetry [9], and the non-reciprocal bulk waves were experimentally observed recently in this kind of PCs [3].
Very recently, lower profile arrays or even a chain of magnetic objects arouses many interests. It is shown that a single layer of magnetic PC, i.e. a chain of magnetic rods, can guide non-reciprocal waves propagating along the chain [10]. For the waves out of the chain plane, theoretical studies show gyromagnetic rod chain can cause non-reciprocal diffraction, which is associated with the photonic angular momentum states splitting [11,12]. In certain conditions, the magnetic rod chain exhibits non-reciprocal wave propagation characteristics, that is the waves reflected or transmitted from the rod chain are quite different when incident wave projects on the chain in opposite directions [12]. However, this kind of non-reciprocal waves has not been observed experimentally until now.
In this work, we experimentally demonstrate the realization of non-reciprocal wave propagation by using a ferrite rod chain. We show the rod chain, designed working in microwave X-band, can almost totally reflect incident wave beam projected on the chain but allow the wave beam transmit through the chain when the wave beam propagation direction is reversed. The non-reciprocal wave propagation is further observed directly by the field mapping technique. The observed non-reciprocal wave propagation can be explained by the high order diffraction of the rod chain. The unique non-reciprocal wave property of magnetic rod chain provides a new way to control the flow of EM waves.
2. Non-reciprocal wave propagation by magnetic rod chain
To show the non-reciprocal wave propagation, we design a ferrite rod chain as shown in Fig. 1(a), The rods are made of ferrite YIG and the radius of the rods is r = 9 mm. The saturation magnetization and the relative permittivity of the YIG are 4πMs = 1884 Gauss and εr = 15.2, respectively. The rod chain is supposed to be biased by a DC magnetic field H0 along the axial direction of the rods, defined as z-axis, and a plane EM wave polarized along z-axis obliquely projects to the rod chain. The response of the ferrite materials to the EM wave is represented by the permeability, which is a tensor
under bias DC magnetic field in full magnetized states [13]. The elements of the tensor are, respectively, where ωm = 4πγMs is the characteristic frequency; γ is the gyromagnetic ratio; Ms is the saturation magnetization; α is the magnetic damping coefficient; ω0 = γH0 is the ferromagnetic resonance frequency, and H0 is effective magnetic field which is the sum of external applied bias magnetic field Ha and other fields such as demagnetize field. In the chain, the distance between the rods, the period, takes d = 24 mm, which is in the same order of magnitude as the working wavelength in our experiments. Because the period and working wavelength are of the same order, the diffraction of the chain will be obvious. It can be estimated that only the diffraction orders 0 and ± 1 will appear if the rod chain is considered as a grating [14].
Fig. 1 (a) Schematic diagram of diffraction of the rod chain. The red arrow ki is the wave vector of incident wave. The green arrows, and , are the wave vector of transmitted diffraction waves of ± n-th order, and the yellow arrows, and , are the wave vector of reflected waves of ± n-th order (b) The reflectance and transmittance of the different diffraction orders when an incident wave impinges on the rod chain in the angle + 45° with respect to x-axis. (c) The reflectance and transmittance at incident angle −45°. In panels (c) and (d), the black dot-line is the total of the reflectance and transmittance of the diffraction orders 0 and ± 1. The sum equals to one indicating no other diffraction orders to be considered. (d) Schematic diagram of the non-reciprocal wave reflection of the rod chain. The red arrows show the rod chain totally reflects the incident wave. When the direction of wave propagation is reversed, the wave goes along green arrows, showing a total transmission. Because of rotation symmetry of the rod chain, the transmitted wave is non-reciprocal too.
We first calculate transmittance and reflectance of the diffraction orders 0 and ± 1 to find the conditions for non-reciprocal wave propagation. As shown in Fig. 1(a), suppose ki is the wave vector of incident wave; and are the wave vectors of transmitted diffraction waves of ± n-th order; and are the wave vectors of reflected waves of ± n-th order. The phase matching condition between the rod chain and the background air along y-direction requires
where G = 2π/d is the period of the rod chain in momentum space and n is the order of diffraction. The transmittance and reflectance of the diffraction orders are calculated using commercial software COMSOL. In the simulations, we take a unit cell shown in Fig, 1(a). The periodic condition is used along y-direction and wave port along x-direction. The plane waves incident from the wave port hitting on the rod and scattered. Thus, the transmittance or reflectance of different orders diffracted waves can be obtained. Figures 1(b) and 1(c) show the results calculated under the bias field H0 = 800 Oe, and incident angles ± 45°, respectively. As we expected, the sum of reflectance and transmittance of the diffraction orders of 0 and ± 1 equals to one, as displayed black dot-line in the figures, therefore no other orders of diffraction need to be considered. From Figs. 1(b) and 1(c), we see that at some frequencies the reflectance or transmittance have a huge difference. For example, at the frequency 8.78 GHz, the zero-order reflectance is R0 = 0.898 at incident angle + 45°. This means the reflection dominates the diffracted wave energy from the rod chain, and the most incident energy is reflected. However, when the incident angle changes to −45°, the zero-order reflectance is almost zero while the negative first-order transmittance goes to maximum T-1 = 0.938, therefore the incident wave passes through the rod chain and refracts to a negative direction. As a result, the wave reflection is non-reciprocal as schematically displayed in Fig. 1(d). According to the rotation symmetry of the rod chain, it is easy to find that the transmitted wave, which negatively refracted to the other side of the chain, will be totally reflected when the wave beam reverses its propagation direction again, i.e. the transmission wave is non-reciprocal too. A further symmetry analysis shows that reversing the direction of wave beam propagation is the same as keeping the incident wave direction while reversing the direction of bias magnetic field [9], which provides great convenience for experiments.From Figs. 1(b) and 1(c), we also notice that the non-reciprocal wave propagation exists at the frequencies 7.84 GHz and 8 GHz, where the contrast is obvious for the zero-order reflectance and negative first-order transmittance between the opposite incident angles.
To realize the non-reciprocal wave propagation shown above in experiments, one should overcome the difficulty caused by demagnetization of the ferrite rod. Since the experiments are performed in a parallel plate waveguide, the cutoff frequency of the waveguide limits the length of rods, h, and then the aspect ratio of the rods, h/2r, where r is the radius of rods. On the other hand, the aspect ratio determines the demagnetization of the rods. In our case, the rod’s aspect ratio is 0.56, which causes a big demagnetization in the magnetic rods. Meanwhile, the maximum bias magnetic field provided by a device is finite; in our case, it is 1000 Oe. As a result, the effective magnetic field within the ferrite rods may be not strong enough to support the presence of nonreciprocal wave propagation even though the maximum bias magnetic field is applied. To solve this problem, we replace each rod in the rod chain by a set of smaller rods arranged in a circle whose radius is the same as the original rod. The inset of Fig. 2(a) displays the schematics of this replacement. The aspect ratio of the smaller rods is 2.5, reducing rod’s demagnetization greatly. Figures 2(a) and 2(b) plot the reflectance and transmittance of different orders of diffraction after the replacement. We see that the non-reciprocal wave propagation appears at 9.46 GHz, where the zero-order reflectance has a huge difference between Figs. 2(a) and 2(b). For latter, the wave propagation direction is reversed. Here, the reversion of the wave propagation direction is realized not by changing the incident wave direction but by reversing the direction of bias magnetic fields. The figures show the reflectance R0 reaches 0.93 at one direction of bias magnetic field, but goes to zero when opposite direction of bias magnetic field is applied. For the latter, transmittance T-1 is 0.92 indicating the incident wave almost perfectly passes through the rod chain.
Fig. 2 Reflectance and transmittance of the ring rod chain for the diffraction orders zero and ± 1 under bias magnetic field (a) H0 = + 800 Oe and (b) H0 = −800 Oe. The sum of all reflectance and transmittance equals one indicating no other orders of diffraction present. (c) and (d) are the electric field distribution when a Gaussian beam projects on the rod chain at angle 45°. The arrow shows the direction of the incident wave. The panel (c) is with H0 = + 800 Oe, and the panel (d) is with H0 = −800 Oe. (e) and (f) are the close view of the electric field distribution around the ring rod. The direction of bias magnetic field H0 corresponds to that of panels (c) and (d), respectively.
As an intuitive illustration, Fig. 2(c) and 2(d) display the reflection and transmission of a Gaussian beam impinging on the chain when bias magnetic field is applied in two opposite directions, respectively. We see the chain shows a nearly total reflection in one direction of bias magnetic field but a nearly a total transmission under opposite direction of bias magnetic field. We have stated above that reversing the direction of bias magnetic field while keeping the incident wave directions is the same as reversing the direction of wave beam propagation under a fixed direction of bias magnetic field. Thus, the results indicate that the wave beam propagation is non-reciprocal in the incident-reflected and incident-transmitted wave channels. Figures 2(e) and 2(f) are the closer view of field distribution around the rod at the center of the indent wave beam. The field distribution clearly shows the incident wave is reflected in one case and transmitted in the other case.
We further investigate the effects of incident angles on the non-reciprocal wave propagation. By changing incident angles, we calculate zero-order reflectance R0 and the negative first-order transmittance T-1 of the rod chain. The results are shown in Fig. 3. We see near frequency 9.46 GHz, the reflectance R0 is more than 0.9 when incident angles are positive, but the R0 becomes very small when incident angle is negative in the range of 30° to 80°. This means the non-reciprocal reflection can occur in a wider incident angles. In this range of incident angles, the transmittance T-1 is bigger indicating the wave incident with a negative angle will transmit the rod chain. At incident angle −45°, the transmittance T-1 goes to the maximum value 0.9
Fig. 3 Reflectance and transmittance of zero-order and negative first-order diffraction as a function of incident angle and frequency. The left column is for the zero-order reflectance R0 and right column for the negative first-order transmittance T-1 at positive and negative incident angles.
3. Experiments and discussions
Experimentally, we did the measurements of transmission and reflection of the magnetic rod chain in a parallel plate waveguide. Figure 4(a) is the schematic diagram of the experiment setup. It consists of a circular parallel plate waveguide, feeding and detecting probes and Helmholtz coils that provide bias DC magnetic field. The gap between the two metal plates is 10 mm. The rod chain is placed along a diameter of the circular plate, and the angle between the incident wave channel and rod chain is 45°. The maximum bias magnetic field provided by Helmholtz coils is 1000 Oe. The waves reflected and transmitted from the chain were detected by the detecting probe that could move around the chain. Figure 4(b) is the image of the experimental set up and the rod chain sample in experiments. The sample was a set of circular rings with radius 9 mm, and each ring was made of 10 thin YIG rods of radius 2 mm.
Fig. 4 (a) The schematic diagram of experimental setup. The green arrow shows the direction of reflected waves and purple arrow shows the direction of refracted waves. (b) The image of the sample in the practical experimental setup. (c) and (d) are experimental results of the power distribution around the rod chain sample under bias magnetic field H0 = 800 Oe at 9.43 GHz. The direction of the bias magnetic field is, respectively, (c) outward and (d) inward the paper plane.
Figures 4(c) and 4(d) give the measurement results of normalized transmittance at the frequency 9.43 GHz under bias magnetic field H0 = 800 at two opposite directions, respectively. When bias field is along positive z-axis [Fig. 4(c)], the peak power appears at about 90°, which corresponds to the wave reflected from the chain as displayed by green arrow in Fig. 4(a). When bias field reverses its direction [Fig. 4(d)] that is the same as reverse the direction of incident wave in the original bias field [refer to Fig. 1(d)], the peak power position moves to −90°, which indicates the wave negatively refracting from the chain as illustrated by the purple arrow in Fig. 4(a). In the figure, we also plot the theoretical results of the power distribution around the chain. It can be found the theoretical and experimental results agree well.
We further directly viewed the wave propagation by using field mapping technique. In experiments, the bias field was provided by the permanent magnets (NdFeB) which were placed at the bottom of each ring. In such configuration, bias magnetic field is non-uniform in the rods. However, the average bias magnetic field can be measured experimentally. In our case the average magnetization field is about 400 Oe measured by a Gauss meter. Therefore, the non-reciprocal propagation phenomenon will appear at a lower frequency. Figure 5 gives the electric field distributions measured at frequency 8.91GHz. The sample boundaries are shown by the white dashed rectangle and the normal of the sample surface is shown by the white dashed line. The incident wave is coming from the bottom, shooting onto the sample at an angle of 45°. Figure 5(a) shows the incident wave is almost totally reflected by the ferrite rod chain. When the direction of bias magnetic field is reversed, the incident wave passes through the ferrite rod chain and undergoes an obvious negative refraction as shown in Fig. 5(b). However, the refraction is not perfect. Some of the incident energy is reflected, which may be due to the non-uniform magnetization of the ferrite rods. Compared with the results in Figs. 2(c) and 2(d), we see the experimental results are in good agreement with the result of simulations. The experiments clearly demonstrate the non-reciprocal propagation of EM waves can be manipulated by the magnetic rod chain.
Fig. 5 Field mapping results at frequency 8.91GHz for bias magnetic field (a) H0 = + 800 Oe. The incident wave is reflected. (b) H0 = −800 Oe. The incident wave is negative refracted. The triangular regions out of the incident wave channels are filled with absorber.
4. Conclusions
In conclusion, we have experimentally studied the non-reciprocal wave propagation caused by a magnetic rod chain under DC bias magnetic fields. The rod chain, which is expected to show non-reciprocal wave reflection and transmission in microwave X-band, is designed and fabricated. The experimental results confirm the non-reciprocal wave propagation of the rod chain that is the incident EM wave can be totally reflected by the rod chain in a bias magnetic field but a total transmission when the bias magnetic field reverses its direction. The non-reciprocal wave reflection and transmission are associated with the non-reciprocal diffraction of the rod chain. Further, the non-reciprocal reflection and transmission are experimentally observed using field mapping technique. The unique non-reciprocal property of magnetic rod chain can be further extended to the other frequency bands as long as gyrotropic materials are used in the rod chain. Our results provide a new way for integrated systems to achieve the control of the flow of electromagnetic wave energy for microwaves or photons.
Funding
National Natural Science Foundation of China (NSFC) (61771237, 11574055, 11574275 and 61671232); Natural Science Foundation of Zhejiang Province (LR16A040001); Open Project of State Key Laboratory of Surface Physics in Fudan University (KF2016_3).
Acknowledgments
R. X. W. thanks partial support from Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions.
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