# Spinning a split ring resonator with microwaves

Côme Jodet, Olivier Pascal, Jérôme Sokoloff

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Côme Jodet, Olivier Pascal, Jérôme Sokoloff. Spinning a split ring resonator with microwaves. Applied Physics Letters, 2023, 123 (18), pp.181103. 10.1063/5.0174364. hal-04756643

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Spinning a split ring resonator with microwaves

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Côme Jodet

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Olivier Pascal

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Jérôme Sokoloff

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Appl. Phys. Lett. 123, 181103 (2023)

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# Spinning a split ring resonator with microwaves

Cite as: Appl. Phys. Lett. **123**, 181103 (2023); doi: 10.1063/5.0174364

Submitted: 30 August 2023 · Accepted: 20 October 2023 ·

Published Online: 3 November 2023

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Côme Jodet,<fnref n="1" />,<fnref n="2" />,ᵃ ID
Olivier Pascal,<fnref n="1" /> ID
and Jérôme Sokoloff<fnref n="1" /> ID

## AFFILIATIONS

## ABSTRACT

The aim of this study is to spin a Split Ring Resonator (SRR) with microwaves. Analytical and numerical studies show that an SRR can be spun when it is excited at its resonance frequency by a circularly polarized electromagnetic wave. Experimentally, using a cylindrical waveguide, at 2.45 GHz and for 150 W, a 15 mm-diameter SRR was rotated with an initial acceleration of the order of tens of °.s⁻<fnref n="2" /> and reached speeds of the order of 0.5 rps, performing several full rotations. As expected, a linear dependence of the initial acceleration as a function of the applied power is found. The limits and perspectives of this work are discussed.

Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0174364

Electromagnetic waves are mainly used to carry information, notably in telecommunications. The energy or power they also carry is mainly used for heating, as in microwave ovens. However, electromagnetic waves can, even in the slightest way, set in motion an object with which they interact.

This interaction can be approached as a momentum transfer. These transfers have been observed, theorized, and then measured experimentally. In 1619, while observing the tails of comets, which always point away from the sun, Kepler intuited the existence of a “solar wind.”<fnref n="1" /> This phenomenon is theorized by Maxwell more than two centuries later, introducing the notion of linear momentum of the electromagnetic wave.<fnref n="2" /> The latter can, therefore, exert pressure on a surface it illuminates. The first measurements of this property were made independently by Lebedew³ and Nichols and Hull in 1901.⁴,⁵

The angular momentum (AM) of an electromagnetic wave (and, therefore, its ability to rotate an object) is theorized by Poynting in 1909.⁶ Under the paraxial approximation, the AM can be separated into two parts: one corresponding to the spin angular momentum (SAM) related to the circular polarization of light and the other to the orbital angular momentum (OAM) of the wave.

The experimental demonstrations of the existence of the AM have been carried out by measuring his transfer to a material object. Concerning SAM, it has been initiated by Beth in 1935⁷,⁸ and Holbourn⁹ in the optical range, followed by Carrara¹⁰ in 1949 and then by Allen in 1966¹<fnref n="1" /> in the microwave range (whose work was later sharpened by Kristensen in 1994¹<fnref n="2" />). The first direct measurement of transfer to a macroscopic object was carried out by Delannoy in

2004.¹³ For the OAM, it has been carried out by He in 1994¹⁴ in the field of optics and then by Emile in 2014 in the microwave field.¹⁵ As a result of these pioneering works, all the components of the momentum have been demonstrated experimentally, in both the optical and microwave ranges.

In complement to these experimental demonstrations of wave momentum transfer, the exploitation of the mechanical properties of the wave opens up multiple fields of application.¹⁶ In optics, it is used for trapping¹⁷ and cooling¹⁸ of atoms as well as for controlling molecules and micrometric particles.¹⁹ The AM of the wave adds new degrees of freedom in these areas²⁰ and could, for example, enable gyroscopic control of micrometric particles.²¹ On a macroscopic scale, the rotational Doppler effect can be used to measure the rotation speed of macroscopic objects.²² Solar sails, for their part, are an application of linear momentum exploitation.²³,²⁴

As in Ref. 25, a classical electrodynamic approach is adopted in this paper.²⁶⁻²⁸ The matter is represented by the charge density ρ and the current density J within it. The electric and magnetic fields (E and H, respectively) are considered. The Lorentz force **f** = ρ**E** + **J** × **μ**₀**H** is used to focus on their interaction.

Introduced by Pendry *et al.*,²⁹ Split Ring Resonator (SRR) is often used as building blocks of metamaterials, enabling negative permittivity and permeability media³⁰,³¹ and opening up a wide field of research.³²,³³

In this Letter, a device designed to rotate an SRR using an electromagnetic wave at microwave frequencies is built. Analytical and numerical approaches are carried out, and then experimental implementation is performed and discussed.

<footnote n="1">¹LAPLACE, Université de Toulouse, CNRS, INPT, UPS, Toulouse, France</footnote>

<footnote n="2">²Anywaves, 2 Esplanade Compans Caffarelli, 31000 Toulouse, France</footnote>

<footnote>ᵃ) Author to whom correspondence should be addressed: come.jodet@laplace.univ-tlse.fr</footnote>
The aim is to rotate an SRR using an electromagnetic wave. Given the SRR and the wave, a component $H_{\perp}$ normal to the SRR surface [see Fig. 1(a)] is required to induce a current in it. To obtain maximum average torque for a given power, another tangential component $H_{\parallel}$ in phase quadrature compared to $H_{\perp}$ [see Fig. 1(a)] has to be added. The resulting waveform is circularly polarized along the $\Delta$ axis (see Fig. 1). In this way, the system becomes invariant to SRR rotation, which is in line with the desired motion. We can notice, in addition, that the circularly polarized wave obtained carries an SAM, which is consistent with a transfer of AM between the field and the material to obtain a torque.

A simple analytical model of the resulting device can be drawn up (Fig. 1). The SRR is likened to an electrical circuit composed of an inductor $L$ in series with a capacitor $C$, a resistor $R$ modeling the material's resistivity, and an impedance $Z_r$ modeling the SRR's high-frequency radiation. The electromotive force ($Emf$) models current induction. To adjust the resonant frequency (via the value of $C$), a dielectric is added to the gap. Considering the powers involved (> 100 W), this also helps to prevent breakdown in the gap, where the electric field is particularly intense.

As detailed in the supplementary material, working at resonance, an approximate formulation is established for the current $i_{res}(t)$ induced in the SRR, given by Eq. (1). It then gives the resulting time average torque $\langle\Gamma_{\Delta,\text{res}}\rangle$ along the $\Delta$ axis, the amplitude of which is given by Eq. (2). Neglecting all friction and restoring torques, the time average rotation acceleration along the $\Delta$ axis $\langle\alpha_{\Delta,\text{res}}\rangle$ of the SRR is deducted, as given by Eq. (3). The time average rotation acceleration can be directly related to rotation speed $\Omega_{\Delta}$ and rotation of the SRR $\theta_{\Delta}$,

$$i_{res}(t) = \frac{\pi R_m^2 \mu_0 H_{0,\perp}}{(R + Z_r)\sqrt{LC}} \sin\left(\frac{1}{\sqrt{LC}} t\right), \quad (1)$$

$$\langle\Gamma_{\Delta,\text{res}}\rangle = \frac{1(\pi R_m^2)^2 \mu_0^2 H_{0,\perp} H_{0,\parallel}}{2(R + Z_r)\sqrt{LC}}, \quad (2)$$

$$\langle\alpha_{\Delta,\text{res}}\rangle = \frac{\langle\Gamma_{\text{res}}\rangle}{J_{\Delta}}, \quad (3)$$

where $R_m$ is the mean radius of the SRR, $H_{0,\perp}$ and $H_{0,\parallel}$ are, respectively, the magnitude of the normal $H_{\perp}$ and tangential $H_{\parallel}$ components of the magnetic field, and $J_{\Delta}$ is the moment of inertia of the SRR.

In order to build an experimental prototype, the resonant frequency of the SRR (and therefore that of the incident field) is set at

<figure><img src="image_15.png" /><figcaption>FIG. 1. Device modeling.</figcaption></figure>

<table><caption>TABLE I. Comparison of analytical, numerical, and experimental approaches (values obtained at resonance frequency for a generator power of 150 W).</caption><thead><tr><th>Quantities</th><th>Units</th><th>Analytical</th><th>Numerical<sup>a</sup></th><th>Experimental</th></tr></thead><tbody><tr><td>⟨i<sub>res</sub>⟩</td><td>(A)</td><td>7.4</td><td>[3.3–7.2]</td><td>...</td></tr><tr><td>⟨Γ<sub>Δ, res</sub>⟩</td><td>(nN.m)</td><td>6.0</td><td>12</td><td>2.6</td></tr><tr><td>⟨α<sub>Δ, res</sub>⟩</td><td>(rev.s<sup>-2</sup>)</td><td>8.0 × 10<sup>-2</sup></td><td>15 × 10<sup>-2</sup></td><td>3.4 × 10<sup>-2</sup></td></tr></tbody></table>

<sup>a</sup>In the numerical model, the current is not uniform along the SRR, and the range of values is, therefore, shown.

2.45 GHz, and a set of parameters suitable for a copper SRR is deduced<sup>34,35</sup> ($R_m = 7.5$ mm, $g = 1$ mm, $h = w = 1$ mm, $\epsilon_r = 3.27$). With these values, it comes to $J_{\Delta} \approx 12.10^{-9}$ kg.m<sup>2</sup>. To obtain the desired circularly polarized wave, a cylindrical waveguide is fed with two mutually perpendicular $TE_{11}$ modes in phase quadrature. The guide radius is chosen to be $R_g = 42$ mm.<sup>36</sup>

Using the HFSS solver of ANSYS Electronics Desktop, the device created is numerically modeled. As expected, in both models, the resulting time average torque is non-zero and maximum at the resonance frequency. The results remain of the same order of magnitude (see Table I). The theoretical approach is, thus, reinforced.

Experimental measurements were carried out to confirm that a circular polarized wave can set an SRR in rotation.

The experimental setup is illustrated in Fig. 2. A generator supplies a CW microwave (here at 2.47 GHz and 200 W max). This wave is split by a hybrid coupler. Each channel then carries a wave of the same amplitude (100 W max) and in phase quadrature. These waves are then transferred to the guide by two matched antennas (A and B Fig. 2) positioned perpendicular to each other. They exit the guide via two other similarly arranged matched antennas (1 and 2 Fig. 2). The SRR is suspended by a spider silk, to minimize the restoring torque. The angular position of the SRR is determined by digital processing of a camera recorded video

<figure><img src="image_16.png" /><figcaption>(a) Diagram of the experimental bench.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>(b) Image of the experimental bench.</figcaption></figure>

<caption>FIG. 2. Experimental setup (ML: matched resistance load; SS: spider silk; SRR: split ring resonator; G: radio frequency generator; I: isolator; HC: -3 dB hybrid coupler; A, B, 1, and 2: matched antennas; C: camera).</caption>
(through a microwave grid), with an estimated accuracy of ±1.5°. Two video records are provided in the supplementary material.

Several input powers were tested. For each power, at least three measurements were taken to ensure repeatability. Both rotation directions of the magnetic field have been tested (by interchanging the connections to channels A and B) to ensure that no parasitic phenomenon causes the observed movement (see the supplementary material for further details). Figures 3 and 4 illustrate the rotation of an SRR (initially at rest). In each case, repeatable behavior is observed. In general concerns, as expected, the higher the input power, the larger the movement. The curves are symmetrical to each other when the circular polarization is reversed. Consequently, the results show that the wave is, indeed, what sets the SRR in motion, as intended.

More specifically, Fig. 3(a) shows curves of SRR angular position as a function of time. As expected, all curves have a parabolic shape. The rotation speed curves of Fig. 3(b) are derived from data in Fig. 3(a). The equation of motion, if friction is neglected (mostly at the in initial time), corresponds to a second-degree polynomial. This yields the values shown in Fig. 4. The linear dependence of the initial rotation acceleration on the input power is confirmed.

In terms of order of magnitude, at 150 W and in less than 30 s, the SRR performs more than 10 revolutions and reaches rotational speeds exceeding 0.5 revolution per second.

Table I provides a comparison of the approaches presented in this Letter. For the experimental approach (unlike the analytical and

<figure><img src="image_18.png" /><figcaption>FIG. 4. Initial rotation acceleration $⟨\alpha_{\Delta}⟩_{\theta_{\Delta}<1\text{rev}}$ as a function of the input power, at resonance frequency, for right and left hand circular polarization.</figcaption></figure>

numerical approaches), the torque is deduced from the SRR's initial rotation acceleration [see Eq. (3)]. The experimental set-up does not provide SRR current information. The results obtained are of the order of magnitude anticipated.

For comparison, at 25 W input power and a 870 MHz frequency, Emile et al.¹⁵ reaches a rotation acceleration of a 150 mm radius ring of around $10^{-6}$ rev.s⁻², thus a rotation of less than 0.5° after 30 s. At the same power, on the other hand, our SRR reaches a rotation acceleration of around $10^{-2}$ rev.s⁻², thus a rotation of around 4 rev, i.e., 1440° in 30 s, i.e., an experimental acceleration performance more than $10^3$ times greater. Note that this acceleration performance is achieved, thanks to a much lower moment of inertia (about $10^5$ lower) of the SRR—which is resonant—compared to the Emile device. The torque generated on the SRR is high, thanks to the resonance, but about 12 times lower than Emile's, which has a larger device.

The result obtained from the experimental set-up discussed here confirms the principle explained at the beginning of this Letter. This final section discusses and justifies the choices made to obtain these results.

First, the experimental bench is naturally lossy, meaning that not all input power interacts with the SRR. In addition, the mutual coupling between the two ports (A-1 and B-2, see Fig. 2) is significant and implies imperfectly circular polarization. Although these phenomena have a significant impact on the torque intensity [see Eq. (2)], the results being fairly demonstrative, and it was not decided to improve them (see the supplementary material).

Second, several limiting phenomena emerge as the SRR starts rotating. The SRR is subject to frictional forces (no vacuum is performed, unlike in Refs. 11 and 12). Similarly, the spider wire used was chosen for its low restoring torque. However, this torque is unlikely to be accurately measured, as its behavior is not linear with rotation, as with conventional wires.³⁷ Finally, since the current flowing through the SRR is of the order of a few ampere, the SRR heats up, causing its properties to evolve over time during measurement. It can be seen that rotation speed does not vary linearly with time but rather tends toward an asymptotic value. This shows that these limiting phenomena have a non-negligible influence with increasing time. Consequently, to evaluate the rotation acceleration of the SRR, it has been decided to consider only a short initial period. The time of one revolution has been chosen. However, this choice implies an underestimation of the initial rotation acceleration.

<figure><img src="image_19.png" /><figcaption>(a) Rotation $\theta_{\Delta}$.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>(b) Rotation speed $\Omega_{\Delta}$.</figcaption></figure>

<caption>FIG. 3. Rotation (a) and rotation speed (b) of the SRR as a function of time, at resonance frequency, for an input power of 25 and 150 W and for right and left hand circular polarization.</caption>
Finally, at input powers above 150 W, initial performances (rotation, speed, and acceleration) lose linearity with input power (see the supplementary material). It has, therefore, been decided to restrict the maximum power to 150 W.

The aim of this study presented here is to set an SRR in motion by a powerful electromagnetic wave in the microwave range. Analytical, numerical, and experimental studies show that the use of a circularly polarized wave can spin an SRR. The experimental results obtained show a significant improvement (more than 10³ times greater rotation acceleration) compared with similar experiments.<fnref n="15" /> This high performance is due to the choice of the SRR, which is resonant and of low dimension and, therefore, of low rotational inertia.

Over and above this, the experimental results provide qualitative confirmation of the validity of the models designed, thus the understanding of the physical phenomena involved. This opens the way to various other possibilities. As the energy received by the SRR is mainly re-radiated, one possibility is to place the SRR in a resonant cavity, which should significantly increase the intensity of the torque obtained. A circular guide has been used here but is not necessary, and therefore, another possibility is to set the SRR in motion at distance, with a collimated beam, far from any material support. Using the current distribution of the SRR's higher resonance modes can lead to translation movement of the SRR. Experimental proof is required, however.

Finally, in this paper, a single SRR has been studied. Different other elements can be envisaged. The method used can also be applied to other elements, and periodization of these ones could ultimately lead to the design of metasurfaces. Their implementation in the microwave range offers various advantages. One is that sustained high power is currently easier to obtain in this frequency range. Another is that, as shown with the SRR, since the wavelength is in the centimeter range, the elements composing these metasurfaces are macroscopic, thus easier to shape. Finally, at this scale, electrical quantities such as charges and currents in the elements as well as the exploitation of resonance make it easier to envisage sizing to generate motion on potentially larger objects.

See the supplementary material for video demonstrations of SRR rotation for both polarization directions. They also provide additional data on the repeatability and limits of the experimental results obtained. S-parameter measurements from the experimental bench are also available as well as details of the analytical model leading to Eqs. (1)–(3).

The authors would like to acknowledge DGA/AID and the company Anywaves for co-funding this work.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Côme Jodet:** Conceptualization (equal); Investigation (lead); Methodology (equal); Software (lead); Validation (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). **Olivier Pascal:** Conceptualization (equal); Funding

acquisition (equal); Methodology (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). **Jérôme Sokoloff:** Conceptualization (equal); Funding acquisition (equal); Methodology (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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