Recent progress on optomagnetic coupling and optical manipulation based on cavity-optomagnonics

Abstract

Recently, the photon–magnon coherent interaction based on the collective spins excitation in ferromagnetic materials has been achieved experimentally. Under the prospect, the magnons are proposed to store and process quantum information. Meanwhile, cavity-optomagnonics which describes the interaction between photons and magnons has been developing rapidly as an interesting topic of the cavity quantum electrodynamics. Here in this short review, we mainly introduce the recent theoretical and experimental progress in the field of optomagnetic coupling and optical manipulation based on cavity-optomagnonics. According to the frequency range of the electromagnetic field, cavity optomagnonics can be divided into microwave cavity optomagnonics and optical cavity optomagnonics, due to the different dynamics of the photon–magnon interaction. As the interaction between the electromagnetic field and the magnetic materials is enhanced in the cavity-optomagnonic system, it provides great significance to explore the nonlinear characteristics and quantum properties for different magnetic systems. More importantly, the electromagnetic response of optomagnonics covers the frequency range from gigahertz to terahertz which provides a broad frequency platform for the multi-mode controlling in quantum systems.

Graphical abstract

Keywords

optomagnetic coupling / manipulation / cavity-optomagnonics / photon–magnon interaction

1 Introduction

Metasurface (MS) has attracted great interest in terms of arbitrarily controlling electromagnetic (EM) waves, which is flexibly arranged to alter the phase, amplitude, polarization state and other different characteristics [1–5]. Many different structured MSs have been designed to achieve polarization conversion [6–13]. Conventional MS controls the propagation of EM waves by accumulating progressively changing phases, which requires careful design of the spatial distribution of MS geometry. Recently, a new approach to shape light beams by introducing abrupt phase has been proposed to develop many applications for flexible manipulation of EM waves [14], such as vortex beam [15], abnormal reflection and refraction [16, 17], beam separation [18], beam diffusion [19], and beam focusing [20, 21].

An effective way to achieve the desired abrupt phase is varying the geometric dimension of cell structure, which has been reported in a large number of references [22–25]. Pancharatnam−Berry (PB) phase elements can be introduced by adjusting orientation [26]. The control of PB phase by rotation of each meta-atom at a certain angle is the most popular way due to its convenience and robustness. There are abundant different applications have been introduced, as for wave control in the free space, such as meta-lens [27, 28], holograms [29], airy beam generation [30], orbital angular momentum [31], cloaking [32, 33] and other applications [34].

PB MS has a strong ability to control EM waves in terahertz and visible range [19, 31, 35, 36]. Recently, beam control is extended to microwave band, which can improve the channel capacity but suffer from narrow band and low efficiency [37, 38]. For example, anomalous reflection and diffusion was realized for two circular polarizations independently in the 12−18 GHz frequency range [39]. To break the limits of inefficiency and narrow bandwidth, two PB unit cells were presented and manufactured for realizing spin hall effect and focusing effect within 8.2−17.3 GHz [40]. A PB coding metasurface (CMS) was proposed to manipulate broadband arbitrary reflective wave in the range of 16−24 GHz [41]. A CMS with polarization insensitivity was presented to manipulate the scattering of EM waves in dual-broadband of 9.26−12.87 GHz and 14.84−19.35 GHz [42]. In addition, one of the most significant applications of beam control is EM stealth [43], which reduces the radar cross section (RCS) of targets by designing MS structures in a scattering-like [44–47], rather than the reshaping [48] or absorber [49]. Thus, it is desirable to improve the bandwidth and efficiency of beam control in the microwave field.

In this paper, we present a new basic MS unit arranged in a coding manner to apply in circularly polarized beam control and RCS reduction to improve the bandwidth and performance. The basic unit converts the incident waves into a reflected cross-polarized waves, and its PCR ranges from 6.9 GHz to 14.5 GHz above 90$\mathrm{\%}$. Moreover, CMSs based on the PB phase principle are arranged to manipulate circularly beam, which can achieve abnormal reflection, beam separation and other phenomena. We randomly arrange the CMS and verify that it plays a role in RCS reduction for a wideband frequency range. This work has a potential application in EM stealth and antenna design.

2 Theoretical analysis of far-field scattering

Based on the reflection theory, cross polarization does not exist in u- and v-polarization incidence due to the symmetry of anisotropic unit cell [48]. When u- and v-axis overlap with x−y axis, $r_{xx}$ = $r_{uu}$, $r_{yy}$ = $r_{vv}$. The reflection matrix can be described as

(1)$\begin{array}{l}{R}_{lin}=(\begin{array}{cc}\cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{array}{)}^{-1}(\begin{array}{cc}{r}_{uu}& 0\\ 0& r_{vv}\end{array})(\begin{array}{cc}\cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{array}).\end{array}$

At this point, the circularly polarized incident reflection matrix is

(2)$\begin{array}{l}{R}_{cir}={\displaystyle \frac{1}{2}}(\begin{array}{cc}1& -j\\ 1& j\end{array})R_{lin}(\begin{array}{cc}1& -j\\ 1& j\end{array}{)}^{-1}=(\begin{array}{cc}{r}_{-+}& r_{--}\\ r_{++}& r_{+-}\end{array}),\end{array}$

where $\phi$ represents the angle of rotation between the positive u-axis and x-axis. $R_{lin}$ and $R_{cir}$ denote the reflection coefficient matrix of linearly polarized wave and circularly polarized wave, respectively. The right- and left-handed states of circularly polarized waves are denoted by subscripts + and −, respectively. Gradient metasurface can be designed in conjunction with the PB phase theory and plays a part in the anomalous reflection of circularly polarized waves. First of all, the reflection amplitude of the circularly polarized waves remains unchanged and realizes reflection phase shift of $\pm 2\phi$, therefore, the phase gradient would be produced:

(3)$\begin{array}{l}\mathrm{\nabla }\alpha ={\displaystyle \frac{\pm 2\phi }{P}}={\displaystyle \frac{\pm 2\pi }{L}},\end{array}$

where $L=NP$, $N$ represents the amount of subunit structures. When the incident wave is vertically incident(${\theta }_i=0$), the anomalous reflection angle is expressed as [16]

(4)$\begin{array}{l}{\theta }_r={\displaystyle \frac{k_0\sin {\theta }_i+\mathrm{\nabla }\alpha }{k_0}},\end{array}$

where $k_0=2\pi /\lambda$ and ${\theta }_i$ imply the wave vector in free space and the angle of incident angle, respectively. Equation (4) can be simplified as ${\theta }_r=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}(\pm \lambda /L)$.

Then a CMS consists of N×N array elements, each of which consists of M×M array of basic units. The schematic diagram of the scattered field of metasurface, as shown in Fig.1. Due to the destructive elimination of CMS, the plane wave is positively incident, the far-field function can be indicated as

Fig.1 Structure diagram.

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(5)$\begin{array}{rl}F(\theta ,\phi )=& \sum _{m=1}^M\exp \{-\mathrm{i}{k}_0{D}_x\left(m-\frac{1}{2}\right)\sin \theta \cos \phi +m\pi \}\\ & \cdot \sum _{n=1}^N\exp \{-\mathrm{i}{k}_0{D}_y\left(n-\frac{1}{2}\right)\sin \theta \cos \phi +n\pi \}.\end{array}$

$D_x$ and $D_y$ represent the length of basic elements. When the designed CMS is arranged along x- or y-axis, $\theta$ and $\phi$ can be described as

(6)$\begin{array}{l}\{\begin{array}{c}\phi =\pi \pm \arctan {\displaystyle \frac{D_x}{D_y}},\\ \theta =\arcsin \left({\displaystyle \frac{\lambda }{2}}\right)\sqrt{{\displaystyle \frac{1}{D_y^2}}+{\displaystyle \frac{1}{D_x^2}}}.\end{array}\end{array}$

3 Design and simulation

To design a polarization converter with excellent performance, it is necessary to achieve as high a cross-polarization reflectance and as low co-polarization reflectance as possible over the ultra-wide frequency band. This will lead to a high polarization conversion rate (PCR) and a perfect polarization conversion. Therefore, based on the theory of eigen-mode analysis [50], it is known that the polarizer stucture needs to symmetrical along the u-axis. Fig.2 shows an ultra-broadband polarizer, which is consisted of two copper layers sandwiched by a dielectric substrate. The structure enables the incident waves to undergo complete reflection without transmission because of the bottom copper plate. FR4 is used as the dielectric slab and its dielectric constant and tangent loss are 4.3 and 0.025, respectively. The metal layer adopts a copper film with an electrical conductivity of $5.8\times {10}^7$ S/m and a thickness of 0.035 mm. The optimized dimension of the polarizer of the structure are t = 3 mm, P = 5.1 mm, r = 2.3 mm, w = 0.1 mm, m = 1.5 mm, n = 0.2 mm.

Fig.2 The working mechanism diagram of metasurface.

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The response of the forward incident EM wave can be simulated using CST Microwave Studio. The simulated reflectance of $r_{xx}$ and $r_{yx}$ in 4−18 GHz are depicted in Fig.3(a). The reflection coefficient $r_{yx}$ reaches more than −1 dB in 6.9−14.5 GHz representing the polarization conversion that occurs. Simultaneously, the co-polarized reflectance $r_{xx}$ is lower than −10 dB. Thus, we can draw a conclusion that the x-polarized wave is mainly reflected and converts into y-polarized wave. Conspicuously, the reflectances of $r_{yx}$ and $r_{xx}$ have three resonant peaks at 7.3 GHz, 10.6 GHz and 14.4 GHz, respectively. Similarly, Fig.3(b) shows the reflentance of y-polarized waves. Herein, we use $\mathrm{P}\mathrm{C}\mathrm{R}=|r_{xy}$${|}^2/(|r_{xy}$${|}^2+|r_{yy}$${|}^2$) to calculate the polarization conversion rate (PCR), and PCR is greater than 0.9 in 6.9−14.5 GHz with a relative bandwidth of 71$\mathrm{\%}$ in Fig.3(c), especially, the amplitude at the three resonant points is almost 1, which signifies a genertion of perfect polarization conversion.

Fig.3 Reflectance. (a) $r_{xx}$ and $r_{yx}$. (b) $r_{yy}$ and $r_{xy}$. (c) PCR.

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Firstly, the u-polarized wave is decomposed into two components of equal amplitude along the x−y axis. The principle diagram of reflective MS is demonstrated in Fig.4(a). The incident field can be expressed as ${\mathit{E}}_i=E_u^i{\mathit{e}}_u=E_y^i{\mathit{e}}_y+E_x^i{\mathit{e}}_x$, herein, $E_x^r=E_y^r$, which signifies no cross-polarized component generated along the u-polarized incidence [50]. $E_x^r=-E_y^r$ was also proved for v-polarized incidence. Therefore, we can then deduce that the reflection coefficient of co-polarization $r_{uu}$ and $r_{vv}$ are 1 without considering the substrate losses. The amplitude of $r_{uu}$ and $r_{vv}$ are displayed in Fig.4(b). An x-polarized incident wave is described as ${\mathit{E}}_i=E_u^i\mathit{u}+E_v^i\mathit{v}$, where $\mathit{u}$ and $\mathit{v}$ refer to the counterclockwise rotation of the x−y plane by ${45}^{\circ }$. Because of the asymmetric structure, if $r_{uu}\approx r_{vv}\approx 1$ and $\mathrm{\Delta }\phi =|{\phi }_{uu}-{\phi }_{vv}|=\pm 180^{\circ }$, $E_v^r$ is in the opposite direction of $E_u^r$, then the synthetic field $E_v^r$ of $E_v^r$ and $E_u^r$ would be along the y-axis. In other words, the x-polarized incidence is rotated after reflection. As shown in Fig.4(b), it is intuitive to show the $r_{uu}$ and $r_{vv}$ are roughly the same and the phase difference is ${180}^{\circ }\pm {37}^{\circ }$ in 6.8−15.4 GHz, which directly leads to high-efficiency polarization conversion.

Fig.4 (a) The schematic diagram of polarizer. (b) The reflectances of $r_{uu}$, $r_{vv}$ and phase difference $\mathrm{\Delta }\phi$.

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In what follows, for ease of comprehending the physical mechanism, the surface current distributions of the upper and bottom metallic films were illustrated in Fig.5, which includes 7.3 GHz, 10.6 GHz and 14.3 GHz these three frequencies. Obviously the current direction of the upper metal film and the backplane are antiparallel at 7.3 GHz in Fig.5(a), thus, the magnetic resonance is generated at this frequency point. The current direction of the upper film is still antiparallel to the current direction of the backplane in Fig.5(b) and (c). A current loop is produced in the intermediate medium film, which is still noted as magnetic resonance. We can speculate that these three magnetic resonances are the key to realizing the efficient ultra-wideband polarization conversion.

Fig.5 Surface current distribution. (a) 7.3 GHz. (b) 10.6 GHz. (c) 14.3 GHz.

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4 Results and discussion

Tab.1 portrays the eight coding meta-atoms by rotating different orientation angles of the top metal patterns. For 1-bit CMS, two coding modes (“0” and “1”) with 90° relative rotation and 180° reflection phase difference are defined. Likewise, as for 2-bit CMS, four coding meta-atoms (“00”, “01”, “10” and “11”) with the phase difference of 90° were required. With regard to 3-bit CMS, eight coding meta-atoms (“000”, “001”, “010”, “011”, “100”, “101”, “110” and “111”) with a constant reflection phase difference of 45° were considered. Fig.6(a) and (b) show the relation between the rotation angle and the amplitude and phase under incidence of LCP/RCP waves, respectively. The amplitude apparently barely changes and the phase generates a difference of nearly 45° with a step of 22.5° rotation angle. There is a linear relation between the rotation angle and phase change of metallic dual-ring pattern. And full phase coverage has been implemented in this process.

Tab.1 The basic unit cells for CMS.

$\alpha$	$0^{\circ }$	${22.5}^{\circ }$	${45}^{\circ }$	${67.5}^{\circ }$	${90}^{\circ }$	${112.5}^{\circ }$	${135}^{\circ }$	${157.5}^{\circ }$
Unit
1-bit	0	−	−	−	1	−	−	−
2-bit	00	−	01	−	10	−	11	−
3-bit	000	001	010	011	100	101	110	111

Fig.6 (a) Amplitude and (b) phase with different rotation angles.

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As for 1-bit CMS arranged along x-direction in Fig.7(a), the basic coding cell with the rotation of and to simulate 3D and 2D far field scattering patterns under normal incidence of the circularly polarized waves in 10.6 GHz. The incoming waves are reflected and decomposed into two symmetric waves with the angle of ${\theta }_1=\arcsin (\lambda /{\mathrm{\Gamma }}_1)={43.9}^{\circ }$, $\phi =0^{\circ }$ or $\phi ={180}^{\circ }$, ${\mathrm{\Gamma }}_1=2\times$ 20.4 mm is the period length of the CMS, which is shown in Fig.7(b). Analogously, when the unit cells distributed in order of “00001111…” in Fig.7(c) along y-direction, the circularly polarized wave is reflected along y-axis with the angle of ${\theta }_2=\arcsin (\lambda /{\mathrm{\Gamma }}_1)={43.9}^{\circ }$, $\phi ={90}^{\circ }$ or $\phi ={270}^{\circ }$ in Fig.7(d).

Fig.7 3D and 2D far field scattering patterns of 1-bit CMS. (a) 3D far field, (b) 2D far field along x-direction. (c) 3D far field, (d) 2D far field along y-direction.

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We arrange the 2-bit CMS to verify the different scattering field patterns generated by the coding particles in 10.6 GHz. The pre-designed coding layout and 3D far field is shown in Fig.8(a) and the corresponding 2D far field diagram is drawn in Fig.8(b). We can see clearly that circularly polarized wave is symmetrically decomposed into four reflected waves with the angle of ${\theta }_2=$$\arcsin (\lambda /{\mathrm{\Gamma }}_2)={43.9}^{\circ }$, $\phi =0^{\circ }$, $\phi ={90}^{\circ }$ or $\phi ={180}^{\circ }$, $\phi ={270}^{\circ }$, ${\mathrm{\Gamma }}_3=2\times 20.4$ mm.

Fig.8 3D and 2D far field scattering patterns of 2-bit CMS. (a) 3D far field, (b) 2D far field.

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A 3-bit CMS aligned along the x-axis has been displayed in Fig.9 to observe its far field scattering. The LCP wave along the forward direction is reflected into a particular direction with the angle of ${\theta }_4=\arcsin (\lambda /{\mathrm{\Gamma }}_3)=$${8.1}^{\circ }$, $\phi =0^{\circ }$, ${\mathrm{\Gamma }}_3=8\times 20.4$ mm, shown in Fig.9(a) and (b). In addition, for the RCP wave with forward propagation depicted in Fig.9(c) and (d), the reflection angle of the LCP wave is ${\theta }_5=\arcsin (\lambda /{\mathrm{\Gamma }}_3)={8.1}^{\circ }$, $\phi =0^{\circ }$. All the beam control phenomena above meet the simulation and theoretical calculation values.

Fig.9 3D and 2D far field scattering patterns of 3-bit CMS. (a) 3D far field, (b) 2D far field of LCR incidence. (c) 3D far field, (d) 2D far field of RCP incidence.

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Based on the beam control, full-wave simulation is adopted when the linearly polarized waves are normal incident to study the diffusion-like phenomenon of the CMS, Fig.10 depicts the simulated monostatic RCS of CMS as well as the same metal surface under the propagation of linearly polarized wave for the sake of comparison. As can be seen, the reduction of RCS is more than 10 dB at 6−16 GHz for the CMS. The difference of RCS reduction (blue line) between the metal plate and CMS is greater than 10 dB, and maximum RCS reduction exceeds 20 dB at 9 GHz. The experimental and simulated monostatic RCS values are basically consistent, even though the sample manufacturing process deviation and experimental environment influence, which clearly proves the practicability of the CMS for RCS reduction.

Fig.10 The RCS of the metal plane and CMS.

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Finally, we extracted the relevant results from other literatures for comparison. As can be seen from Tab.2, our design has an excellent performance within the ultra-wide bandwith.

Tab.2 Performance comparison with presented works.

	Polarization converter
Ref.	$\mathrm{O}{\mathrm{B}}^{1)}(\mathrm{G}\mathrm{H}\mathrm{z})$	$\mathrm{P}\mathrm{C}\mathrm{R}(\mathrm{\%})$	$\mathrm{R}{\mathrm{B}}^{2)}(\mathrm{\%})$
[52]	10.3−20.5	90%	66.2
[53]	6.53−12.07	88%	59.6
[54]	9.38−13.36 & 14.84−20.36	90%	35
[55]	8−12	90%	40
This work	6.9−14.5	90%	71
	RCS reduction
Ref.	Working mechanism	$\mathrm{O}{\mathrm{B}}^{1)}(\mathrm{G}\mathrm{H}\mathrm{z})$	$\mathrm{R}{\mathrm{B}}^{2)}(\mathrm{\%})$
[56]	Anomalous reflection	8.9−11.4	24.6
[57]	Anomalous reflection	9.85−19.37	65
[58]	Anomalous reflection	7−16	78
[59]	Destructive interference	9.5−13.9 & 15.2−20.4	32 & 30
[60]	Absorber	2.12−4.15 & 6.08−9.58	64 & 45
[61]	Diffused scattering	5.4−7.4	31.25
[62]	Diffused scattering	13.2−23.2	54.9
[63]	Polarization conversion	8−16	67.8
This work	Polarization conversion & diffused scattering	6.9−14.5	71

1) OB: Operation bandwidth;2) RB: Relative bandwidth.

5 Experimental verification

For the experiment, an experimental sample with 40 × 40 unit-cells and a size of 20.4 cm × 20.4 cm is processed and tested to validate the polarization conversion in a microwave anechoic chamber, which is displayed in Fig.11(a). And the experimental test environment is depicted in Fig.11(b). Two horn antennas are connected to the network vector analyzer (R&S ZNB/40), one of which is used to emit gain linear polarization, and the other for receiving linear polarization. Then the reflection coefficients can be obtained by rotating the horn antenna perpendicular or parallel to the ground. The microwave anechoic chamber is surrounded by absorbent materials to eliminate electromagnetic interference. Fig.12(a)−(c) depict the conversion of incident waves, including the simulation and experimental results. They fit perfectly, expect for a slight frequency shift.

Fig.11 (a) Sample diagram for testing polarization conversion. (b) Experimental test environment. (c) Sample diagram for RCS reduction.

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Fig.12 Measurement results of (a) x-polarized incident wave, (b) y-polarized incident wave, (c) PCR, (d) $S_{11}$ and RCS reduction.

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The CMS with a size of 20.4 cm × 20.4 cm in Fig.11(c) is applied to verify RCS reduction. The reflection coefficient $S_{11}$ of sample is firstly considered to measure, as shown in Fig.12(d). Then the equation is calculated to acquire the value of RCS, as follows [51]:

(7)$\begin{array}{l}\sigma =4\pi R^2|{\displaystyle \frac{E_s}{E_i}}{|}^2.\end{array}$

$E_i$ and $E_s$ denote the incident electric field intensity and scattered electric field intensity, respectively. In addition to the RCS of the target, which can be expressed as square meters, it can also be described as ten times the value of the RCS logarithm [51]

(8)$\begin{array}{l}RCS(\mathrm{d}\mathrm{B}\mathrm{s}\mathrm{m})=10\times {\log }_{10}(RCS/(1{\mathrm{m}}^2)).\end{array}$

Fig.12(d) shows the simulated and measured RCS values for the CMS. These two results are in good agreement and both lower than it compared with RCS values of PEC.

The subtle differences are attributed to the fabrication tolerance and experimental environment. We set the periodic boundary conditions during simulation, which means the proposed polarizer is a plane of infinite size, but the actual sample measured is finite, which causes the edge scattering.

6 Conclusion

In conclusion, a new reflection polarization converter was proposed and measured for applying on the beam control and RCS reduction. The basic unit was enabled the linear polarization wave to cross-polarized state with relative bandwidth of 71$\mathrm{\%}$, PCR of almost 90$\mathrm{\%}$ from 6.9 GHz to 14.5 GHz according to simulation. CMSs based on the PB phase principle were designed to react on circular polarization beam control and satisfy the generalized Snell’s law. Finally, we adopted the 2-bit coding metasurface to verify the RCS reduction properties. This indicates that the availability of manipulating the polarized wave plays an important role in electromagnetic stealth and antenna designs.

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