1. Introduction

Metasurfaces are 2D or thin surfaces that extend our capabilities in scattering and refraction, by properly engineering their spatial [1] and/or temporal variations [2]. The impressive growth of metasurfaces (MSs) over the past decade is mainly owing to the unprecedented control they provide over the fields, while alleviating the shortcomings of metamaterials such as bulkiness, complex fabrication processes, and high loss [1,3–6]. The planar nature of metasurfaces allows them to be implemented on printed circuit boards and in integrated circuits. In recent years efforts have been undertaken to realize tunable metasurfaces by controlling their electromagnetic response [7,8] through mechanical, electrical, optical and chemical processes [9–12]. Although these attempts were initially directed towards the development of statically tunable devices, recent studies have shown that dynamic tuning of the electromagnetic response of a metasurface can provide new degrees of freedom for electromagnetic wave control, such as Doppler-like frequency shift and non-reciprocity [13,14].

As an electromagnetic wave crosses a static, but spatially-varying metasurface, the wave frequency is unchanged but the transverse momentum changes, due to the momentum change provided by the surface. Similarly, as an electromagnetic wave passes through a metasurface with a spatially homogeneous, but time-varying surface susceptibility, the wave undergoes a frequency shift at a constant transverse momentum. This phenomenon, which is called time refraction, makes time-varying metasurfaces (TVM) promising for applications such as frequency conversion and frequency combs [15,16]. Furthermore, considering that TVM has the ability to break time-reversal symmetry, it can be utilized to develop magnet-free isolators and circulators [14]. As in typical time-modulated electromagnetic systems, however, the response of the system to a single incident tone contains the incident tone plus a number of temporal harmonics. Thus achieving frequency conversion and non-reciprocity, comes at a price of a cluttered output spectrum (containing several unwanted harmonics).

It has long been known that a rotating object can induce a rotational Doppler shift in the frequency of a circularly polarized beam incident along the object axis of rotation. [17,18]. Normal incidence of a circularly polarized beam on an anisotropic plate that is rotating with a frequency of $\Omega$ around its normal axis, generates reflected and transmitted waves with a frequency shift equal to $\pm 2\Omega$. The sign of this rotational Doppler shift depends on the sense of rotation of the anisotropic slab with respect to the handedness of incident wave. This effect can be understood by choosing the point of view of an observer that is rotating with the same frequency $\Omega$.

Rotational Doppler shift was experimentally demonstrated in [17] where a half-wave plate was mechanically rotated. In [18] the rotation of a beam with orbital angular momentum was shown to result in a frequency shift that depends on the rotation frequency and the orbital angular momentum. More recent studies involve a rotating q-plate [19] and rotating gas-molecules [20]. In another recent study, cascading the rotational Doppler effect process with it’s mirror counterpart is used to achieve a doubled frequency shift [21]. Rotational Doppler shift has been mostly used for detection applications. In [22], by measuring the rotational Doppler shift the rotational speed of a spinning object was detected from a distance of 120 meters. Combination of linear and rotational Doppler shift was used in [23] to measure the velocity vector. In [24] the rotational Doppler shift was used to detect mode number of orbital angular momentum waves for wireless transmission.

In the present work, we investigate a TVM with an anisotropic surface susceptibility whose axes of anisotropy rotate in time with the same radial frequency ($\Omega$) on all points on the surface. We show that if a circularly polarized wave with the same (opposite) sense of rotation as that of the screen, impinges the screen, the reflected and transmitted waves will have down-converted (up-converted) frequency components with the reversed sense of polarization. The single screen will thus scatter in the incident frequency, as well as only one other temporal harmonic (higher or lower depending on polarization). We then propose a frequency converter system to convert the incident wave to a higher or lower frequency using a simple combination of the TV screen and a PEC screen separated by a proper distance. We investigate the situation in which full conversion is achieved such that the scattered wave will only contain the up or down converted frequency component. It is also shown that in case of opposite sense of rotation between the frame and incident wave, parametric amplification will occur, meaning the single up-converted frequency tone carries higher power levels than the incident tone. This virtually rotating metasurface (VRM) also has a nonreciprocal behavior which can be used to design magnetic-free isolators. The analysis of all scenarios is carried out by applying a simple transformation to the electromagnetic field vectors after which all field equations and boundary conditions are represented by linear time-invariant differential equations, thus allowing standard phasor-type analysis. It is worth noting that the analytical results calculated here are not restricted to a specific frequency range and frequency converter based on VRM can be designed for any frequency band. Along the way, all results are verified against an in-house FDTD code that validates the analytic results with excellent agreement. A lumped element model using a 2D periodic metal mesh grid loaded with time-varying capacitive nodes is also presented. The lumped model is further used to design a possible 3D realization of the VRM, and is accompanied by full-wave simulations for various values of capacitors. These results show that indeed the VRM concept may be realized in practice.

This paper is organized as follows. In section 2 the formulation for scattering of a wave normally incident on a VRM is presented, and FDTD simulations are utilized to authenticate the analytical results. Analysis and design of a frequency converter based on the VRM is carried out in section 3 and FDTD simulation again validates the results. In section 4 some practical considerations including a 2D mesh of strips connected to each other via a purely time variant capacitive network to realize a VRM, and also a 3D realization design that is verified with static full-wave simulations for different values of the capacitor arrangement is carried out. Moreover, piecewise constant changes of susceptibility and dispersion considerations are investigated in this section. Finally, conclusions are drawn in section 5.

2. Scattering of a circularly polarized wave by a virtually rotating metasurface

Consider an anisotropic metasurface placed at $z=0$ in a medium characterized by the permittivity $\epsilon$ and permeability $\mu$ as in Fig. 1. We define the electromagnetic response of the metasurface by its (effective) time-dependent surface susceptibility tensor $\bar{\bar{\boldsymbol{\chi }}} _{s}$, which relates the surface-density of electric dipoles (${\boldsymbol { {P_{s}} }}$) to the total electric field ${\boldsymbol { {E} }}$ at an arbitrary time $t$ on a point ${\boldsymbol { {r} }}_{s}$ on the metasurface as

 

Fig. 1. Circularly polarized plane-wave normally incident on a Virtually Rotating Metasurface (VRM). The axes of anisotropy are locally rotating in time around the $z$-axis, at all points on the VRM. The VRM is modeled with a spatially homogeneous, time-varying susceptibility tensor.

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Let us next consider an electromagnetic plane wave that travels in the $+z$-direction and is normally incident on the VRM that is placed at z=0. The electric and magnetic fields in the region $z<0$ are generally written as

Our aim is to solve for the unknown vector functions ${\boldsymbol { {f} }}^{r}$ and ${\boldsymbol { {f} }}^{t}$ for a given incident field ${\boldsymbol { {f} }}^{i}$. To that end, we first impose the electromagnetic boundary conditions at the metasurface at $z=0$ (${\boldsymbol { {E} }}(0^{-},t)={\boldsymbol { {E} }}(0^{+},t)$, $\hat {{\boldsymbol { {z} }}}\times [{\boldsymbol { {H} }}(0^{+},t)-{\boldsymbol { {H} }}(0^{-},t)]=\partial {\boldsymbol { {P_{s}} }}/\partial t$). The result is

Equation (14) may now be used to relate the right- and left CP components of the transmitted and incident fields.

Having studied the stationary-state solutions in the CP base of the transformed field phasors, we now return to original fields ${\boldsymbol { {f} }}^{i,r,t}$. From Eqs. (8), (13) and Eq. (15), it follows that

Assume, for instance, that the field incident on the virtually rotating metasurface is a pure RCP wave ($\mathcal {F}^{i}_{-}=0$) with frequency $\omega _{i}=\omega +\Omega$ and complex amplitude $\mathcal {F}^{i}_{+}$ [see Eq. (23)]. Using Eqs. (17) and (22), one can compute the amplitudes $\mathcal {F}^{t}_{\pm }$ and $\mathcal {F}^{r}_{\pm }$ of the transmitted and reflected RCP and LCP fields. The reflected and transmitted RCP fields retain the frequency of the incident wave. But, the LCP fields (amplitudes $\mathcal {F}^{t}_{-}$ and $\mathcal {F}^{r}_{-}$) acquire a frequency of $\omega -\Omega =\omega _{i}-2\Omega$. Similarly, a pure LCP field with the frequency $\omega _{i}$ may be scattered into RCP waves with a frequency of $\omega _{i}+2\Omega$. This effect is in essence the rotational Doppler shift. Note that, contrary to most time-varying media, no other frequency component or harmonic is produced in either case.

2.1 Numerical results and validation

In order to validate the results obtained, a finite difference time domain (FDTD) algorithm was developed. Consider a metasurface that is virtually rotating with a frequency of $\Omega =50$ MHz in a right hand sense with respect to the $+z$ direction. We take the tensor $\bar{\bar{\boldsymbol{\chi}}}_{s0}$ to be diagonal with the diagonal elements $\chi _{1}$ and $\chi _{2}$ ($\chi _{1}=0.3$ m, $\chi _{2}=0.1$ m). The incident wave is an LCP plane wave with the frequency of $\omega _{i}=1$ GHz. The VRM is modeled as a boundary condition through the time varying boundary condition presented in Eq. (7) and the surface susceptibility of the metasurface is updated on every time step. The incident wave is assumed to be a sinusoidal wave with a wide Gaussian envelope in time domain so that the frequency spectrum of the excitation pulse is a gaussian pulse with $10$ MHz beam width and amplitude of 1 at $\omega _i$. Fourier transform of the reflected and transmitted electric field calculated from the FDTD simulations are depicted in Fig. 2.

 

Fig. 2. Analytic results and FDTD simulation results showing FFT amplitude of the reflected and transmitted electric field from a VRM which is illuminated by an LCP wave. The rotational angular frequency of the surface is $50$ MHz, $\chi _1=0.3$ m, $\chi _2=0.1$ m and incident wave frequency is $1$ GHz.

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It can be observed that indeed the reflected and transmitted waves contain two frequency components, namely $1$ GHz ($\omega _{i}$) and $1.1$ GHz ($\omega _{i}+2\Omega$), as expected. Results obtained from analytical and numerical simulation are in a very good agreement: the difference between the amplitudes of reflected and transmitted waves at $1$ GHz is about $0.012\%$ and $0.28\%$, respectively and at $1.1$ GHz is about $0.48\%$. Moreover, FDTD simulations verify the LCP, and RCP nature of the two frequency components $\omega _{i}$ and $\omega _{i}+2\Omega$ in the phase of the Fourier transformed field (not shown here).

3. Full frequency converter based on a virtually rotating metasurface

As it was seen in the previous section, besides the frequency $\omega _{i}$ of the incident wave, the scattered field contains a (single) frequency-converted component of $\omega _i\pm 2\Omega$. The strength of this frequency conversion for a single VRM is, however, rather small ($<$23$\%$ efficiency). One way to boost frequency conversion efficiency is to increase the interaction of the wave and the surface.

Consider now the system shown in Fig. 3, where a VRM is located at $z=0$ and a perfectly conducting plate is placed at a distance $d$ from the metasurface. We will show in the following, that by proper design of the distance of this simple system, one can obtain full frequency conversion, and even amplification. We shall first calculate the reflection matrix $\bar{\bar{\boldsymbol{r}}}_{cp}$ of the metasurface/screen combination in the CP base of the transformed field vectors. This matrix is defined by

 

Fig. 3. VRM augmented with a PEC backing at a specific distance (d), leading to full frequency (up or down) conversion and amplification in case of up conversion.

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If the parameters of the system are adjusted such that $r_{cp,++}=r_{cp,--}=0$, an incident LCP (RCP) wave with a frequency of $\omega _{i}$ is reflected as a pure RCP (LCP) wave with a frequency of $\omega _{i}+2\Omega$ ($\omega _{i}-2\Omega$). A little algebra shows this occurs when

Consider, for instance, a VRM with $\chi _{1}=0.177$ m, $\chi _{2}=0.0246$ m and $\Omega =300$ MHz. The incident wave has a frequency of $1$ GHz. By setting $d= 0.47$ m, both relations in Eq. (32) are satisfied. Figure 4 shows FDTD results for an RCP incident wave. The FDTD algorithm and the excitation is the same as what was explained before, with the only addition of placing a PEC screen at the right location of the simulation domain.

 

Fig. 4. (a) Analytic results, and FDTD simulation results showing FFT amplitude of the reflected electric field from the VRM/PEC combination which is illuminated by an RCP wave. Rotational frequency of the screen is $300$ MHz, $\chi _1=0.177$ m, $\chi _2=0.0246$ m, $d=0.47$ m and incident wave frequency is 1GHz. (b) Analytic results, and FDTD simulation results showing FFT amplitude of the reflected electric field from the VRM/PEC combination which is illuminated by an LCP wave. Rotational frequency of the screen is $300$ MHz, $\chi _1=0.0953$ m, $\chi _2=0.0198$ m, $d=0.483$ m, and incident wave frequency is $1$ GHz.

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Full conversion is shown and validated in Fig. 4(a). The reflected wave only has a single down-converted frequency component, i.e., $\omega _{i}-2\Omega =400$ MHz. For both the incident and reflected electric field, the quadrature phase difference of the $x$ and $y$ components affirms that the reflected wave has a circularly polarized left handed character. The FDTD results concur well with theoretical results and the amount of amplitude difference at $\omega _{i}-2\Omega$ frequency is only $0.14\%$. As mentioned before, the incident wave amplitude at the center frequency is 1 and as it can be seen, the reflected wave at the center frequency is less than $0.36\%$ of the incident wave (almost vanished), and is also in excellent agreement with the theoretical calculation. It should be noted that the frequency conversion efficiency is $63\%$ for down conversion, which is equal to the calculated $|r_{cp,-+}|$ in Eq. (37).

We then investigate a case of an LCP incident wave with a frequency of 1GHz, with a VRM/PEC screen combination having $\chi _{1}= 0.0953$ m, $\chi _{2}= 0.0198$ m, and $\Omega =300$ MHz, and we set $d = 0.483$ m, which satisfies both relations in Eq. (32). Fourier transform of the reflected electric field calculated from FDTD simulation is presented in Fig. 4(b) along with the amplitudes of available harmonics as calculated by the theory, i.e., Eq. (24).

It is clear that the reflected wave only has a single up-converted frequency component, i.e., at $\omega _{i}+2\Omega =1.6$ GHz. The difference between the theoretical and numerical results at $\omega _{i}+2\Omega$ frequency is only $0.15\%$. Moreover, the difference of the amplitudes at the incidence frequency ($\omega _i$) is about 0.6$\%$, which shows that the FDTD simulation results are in excellent agreement with those from the theory. It is worth mentioning that the frequency conversion efficiency is around $126\%$, which shows power amplification in the up-conversion case. Furthermore, this value is exactly equal to $|r_{cp,+-}|$ calculated in Eq. (38).

4. Practical considerations

4.1 Lumped element model for the VRM

A possible lumped element model for a VRM is shown in Fig. 5. The proposed structure is a 2D periodic network of sub-wavelength unit-cells. Each cell consists of four metal arms which are connected to each other through a capacitive network at the cell center. Each capacitive node at the center of each unit-cell is denoted by the cell index $(i,j)$. The four terminals of each node connecting to the four metal arms (and their corresponding voltage/current), are designated by numbers 1 to 4 (($V_1$, $I_1$), ($V_2$, $I_2$), $\ldots$). The capacitive node consists of six capacitors ($c_{12}$, $c_{13}$, $c_{14}$, $c_{24}$, $c_{34}$, $c_{23}$) that connect all terminal pair combinations. As can be seen in Fig. 5, the metal arms are initially considered to have a series inductance ($L$) and resistance ($R$) which later will be assumed to be negligible.

 

Fig. 5. Schematic of a 2D mesh of metal arms which are connected to each other through a purely capacitive (varactor) network.

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Assume that the network is subject to a spatially uniform but time dependent field, as a result of which currents start to flow in the arms. Currents $I_x(t)$ and $I_y(t)$ flow in the $x$ and $y$ directions in all wires parallel to $x$ and $y$ directions, respectively such that

4.2 Possible realization of the VRM based on the lumped element model

To further verify the viability of such VRM, and to test the validity of the lumped element model, a design of a possible realization of the proposed VRM is carried out. The structure’s response was verified using full-wave simulations in various static cases for the designed unit-cell, with periodic boundary conditions. Using scattering simulations, the effective surface parameters of the design were extracted for different values of loaded capacitors. It should be noted that these full-wave 3D simulations of various static surfaces, may not always accurately be used to describe the dynamics of the structure. However, when the rotation frequency is much smaller than that of the incident wave (adiabatic limit), one can temporally cascade these static cases to describe the dynamics.

The unit-cell of this network is depicted in Fig. 6 and is composed of four main metallic arms that are capacitively coupled to each other through six additional metallic patches interspersed with dielectrics. These patches are located on different levels from the four main arms to provide cross-over. All dimensions of the structure are fixed, and only the permittivity of the dielectrics (green in the figure shown) are varied between each simulation, to emulate the effect of change in varactor capacitance.

 

Fig. 6. Unit-cell of VRM whose arm length is $0.8$ mm and arm width is $0.15$ mm and the unite-cell lattice constant is $1.8$ mm.

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The cell is designed as follows: From the lumped element model, the model capacitor values that lead to the desired susceptibility parameters are calculated for 5 discrete levels. The values of the dielectrics corresponding to each capacitor were found using the parallel plate capacitor formula, leading to an initial design. These capacitor values are listed in the third block column of Table 1, for the different time rotations. The cell was then simulated using Ansys HFSS, a 3D full-wave solver. Assuming periodic boundary conditions for the side-walls of the cell, and a normally incident plane wave, the scattering parameters of the cell can be found, and the effective surface susceptibility tensor can then be readily computed from the scattering matrix as follow

Table 1. The desired ideal $\bar{\bar{\boldsymbol{\chi}}}_s$ values, the equivalent node capacitances for a mesh network realization, and the extracted $\bar{\bar{\boldsymbol{\chi}}}_s$ from the simulations of the designed cell, for a 5-state VRM over one modulation period.

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In the simulations presented below the operating frequency is $1$ GHz and the unit-cell size is assumed to be much smaller than the operating wavelength (unit-cell dimensions = $1.8$ mm$\times 1.8$ mm). The length and width of each metallic arm are $0.8$ mm and $0.15$ mm, respectively, and the dielectric thickness is $1$ $\mu$m. Figure 7(a) shows the S-parameters obtained from full-wave simulations for a VRM that is designed for the case with susceptibilities $\chi _{11}=0.228$ m, $\chi _{21}=0.053$ m, $\chi _{22}=0.122$ m (i.e., the surface with $\chi _{1}=0.25$ m, $\chi _{2}=0.1$ m rotated by an angle $\Omega t=\pi /8$ rad). Figure 7(b) shows the extracted surface susceptibility parameters computed from the S-parameters of Fig. 7(a). It can bee seen that the desired susceptibility elements was indeed achieved for this case. All the other designed capacitor values that emulate the behavior of the VRM with 5 states are presented in Table 1. The desired $\bar{\bar{\boldsymbol{\chi}}}_s(t)$ elements and the extracted $\bar{\bar{\boldsymbol{\chi}}}_s(t)$ using the designed capacitors are also presented, showing less than $9\%$ difference between the final extracted susceptibilities and the desired susceptibilities.

 

Fig. 7. (a) Scattering parameters obtained from full-wave simulation of designed VRM for $\Omega t=\pi /8$ which need to be $\chi _{11}=0.228$ m, $\chi _{21}=0.053$ m, $\chi _{22}=0.122$ m. (b) Surface susceptibility parameters calculated from $S_{21}$ for $\Omega t=\pi /8$.

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4.3 Piecewise constant changes of susceptibility

Practical realizations, such as the network of metal arms and varactors, may not allow for continuous variation of the effective susceptibility. The surface susceptibility may only change between certain discrete levels, in order to mimic the continuous rotation. When the modulation period ($\frac {\pi }{\Omega }$) of the surface susceptibility is divided into $N$ intervals (assuming $N$ is even), the number of discrete levels is $\frac {N}{2}+1$. In each interval, the surface susceptibility tensor elements are kept constant and equal to the amount of $\bar{\bar{\boldsymbol{\chi}}}_s(t)$ evaluated at the mid-point of the interval. Figures 8(a) and 8(b) show the required temporal evolution of $\bar{\bar{\boldsymbol{\chi}}}_s$ for a continuous VRM as well as when it is discretized into several intervals (3 states in Fig. 8(a) and 5 states in Fig. 8(b)). Figures 9(a) and 9(b) show the reflected electric field obtained from FDTD simulations for the 3-state and the 5-state discretized VRM respectively, with an RCP incident wave. It can be seen that in the 3-state discretization case, some unwanted harmonics may occur (as the discretized modulating function is rather coarse), while in case of the 5-state discretized VRM, amplitudes of other unwanted harmonics are negligible. This is a rather promising outcome, as it suggests that practical state-switching systems can also achieve good performance, even with 5 discrete levels. Moreover the difference at $1$ GHz and $0.8$ GHz with respect to the theoretical results are about $0.07\%$ and $3.6\%$, respectively. This shows that a 5-state discretized VRM can be considered as a practical version of VRM. As it was seen, only one up-converted frequency component aside from the incident frequency is seen in the reflection spectrum of the 5-state discretized VRM.

 

Fig. 8. Approximated rotational surface susceptibility with (a) 3 discrete states, (b) 5 discrete states.

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Fig. 9. Analytic results and FDTD simulation results showing FFT amplitude of the reflected electric field from a discretized VRM with $\chi _1=0.25$ m, $\chi _2=0.06$ m which is illuminated by an RCP wave. Rotational angular frequency of the screen is $100$ MHz and incident wave frequency is $1$ GHz. (a) 3-state discretized VRM. (b) 5-state discretized VRM.

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4.4 Dispersion consideration

As mentioned previously, a realistic metasurface will inevitably experience temporal dispersion, and such an effect can indeed be included in our analysis. Under such circumstances, the surface polarization is the convolution of the electric field and the surface susceptibility

5. Conclusion

In conclusion, a virtually rotating metasurface (VRM) was proposed, mimicking local rotations of anisotropy over the surface. A full frequency converter based on a VRM and a PEC screen combination was proposed and theoretically analyzed. It was shown that, when a circularly polarized wave with the same (opposite) sense of rotation as the screen’s rotation is normally incident on the screen, the reflected and transmitted waves carry down (up)- converted frequency CP waves. The validity of results were also confirmed via in-house 1D-FDTD simulations. It is theoretically shown that the frequency converter system can achieve full frequency conversion, with a single converted tone with up- or down-converted frequency, depending on the sense of polarization. Furthermore, amplification is possible in the case of up conversion, i.e., when the screen is illuminated by a CP wave having opposite handedness with respect to the screen. The interesting aspect of this amplification (as opposed to typical parametric amplification using a TV system) is that the modulation (pump) frequency could be much smaller than the signal frequency. Also, it was shown that frequency conversion efficiency is determined by the signal and pump frequencies and it is independent of the VRM characteristics. It is worth noting that equal frequencies for the incident wave and the screen rotation frequency will lead to zero reflection in the case of similar sense of rotation. Also, in the case of opposite sense of rotation, equal frequencies for incident wave and screen rotation frequency results in reflection with infinite amplitude. A lumped element model for the VRM is proposed using a 2D metallic mesh grid loaded with capacitive nodes. The lumped model is then used to design a possible 3D realization, and its static response for different capacitor values yields desired results. It is further shown that by mimicking the continous rotation of the surface susceptibility using discrete states, the screen still yields excellent results which has important practical implications. Finally, the amendments to the theory that are necessary to account for dispersion of the VRM, were presented. The VRM concept may impact a variety of applications from microwaves to optics, e.g. involving frequency conversion, amplification, and non-reciprocal systems (not discussed herein), to name a few.