2.1.

Induced Moments of Rotating Dipoles, Exact versus Quasistationary Solutions

Consider a generic problem of scattering from a subwavelength thin wire, treated using a dipolar approximation. In the rest frame, the differential equation governing the dipole moment $P(t)$ is LTI, since all the coefficients that multiply the temporal derivatives are time-independent [Eq. (1)]. In this case, the evolution of the moment is solely governed by initial boundary conditions, resonant frequency ${\omega }_0$, a driving force $F(t)$, which is related to the incident wave $E_i(t)$, and decay rate $\gamma$, which contains both radiation as well as dissipative losses. It is noted that the resonant frequency and decay rates are determined by the geometry and material make-up of the dipole in question (see Refs. 2324.–25 for a rigorous treatment). In the case of a short-wavelength and lossless wire dipole, it holds true that ${\omega }_0\propto \frac{1}{l}$, where $l$ is the length of the wire and $\gamma \propto R_{\mathrm{rad}}$, which is the radiation resistance. The differential equation for the dipole moment is therefore

Although the equation is quite general, a range of parameters and their relation to physical observables will be discussed after the experimental section. Equation (1) can be solved either in the frequency or time domain; yet, for the purpose of this investigation, it is instructive to choose the time domain, which serves to develop intuition about the memory effect. The solution of Eq. (1) is given by

The intuitive interpretation of Eq. (2) is that the dipole moment at some time $t$ depends on the excitation at earlier times $t^{\prime }$, weighted by the decaying Green’s memory function. As seen in Eq. (3), the depth of the memory, i.e., how far in the past previous excitations can affect the present, is determined by the decay rate $\gamma$. For large $\gamma$, the system possesses very little memory, while for small $\gamma$ the system retains the memory of past excitations for a longer duration of time.

Now, consider the same dipole, rotating at a constant angular frequency $\dot{\theta }$ while being subjected to a resonant and linearly polarized incident plane wave with harmonic temporal dependence of $E_i(t)=\cos ({\omega }_{0}t)$, where ${\omega }_0$ is the carrier frequency. The polarization mismatch factor $\cos [\theta (t)]$ depends on the angle at any moment between the dipole orientation and the incident field polarization, turning Eq. (2) into the following causal relation:

Before discussing the implications of Eq. (6), it is instructive to consider the quasistationary solution, obtained under the timescale separation approximation. By observing Eq. (4), it is tempting to assume that the frequency of the polarization mismatch $\cos [\theta (t)]$ varies in time on a much slower scale in comparison with the incident field and Green’s memory function $G(t-t^{\prime })$. Moreover, if Green’s memory function is fast decaying compared to the rate of the polarization mismatch (i.e., $\mathrm{\Omega }=\frac{\dot{\theta }}{\gamma }\ll 1$), it effectively turns into a sampling function at time $t^{\prime }=t$. This memoryless property permits the removal of the polarization mismatch term outside of the integral, leading to

The form of Eq. (7) consists of an integral, which is the solution to the static problem at minimal polarization mismatch, multiplied by a time-dependent amplitude related to the instantaneous position of the dipole. This result indicates the required assumptions, which allow investigating short-memory systems by means of static solutions with an additional slow-time-dependent multiplicative parameter. This equation is also the basis for the last part of the manuscript, where a method for solving dynamic problems with static numeric solvers will be presented and discussed. The solution of Eq. (7) is immediate:

The single-sided Fourier transform of Eq. (8) provides complex amplitudes of the induced dipole moment spectrum, which reveals the existence of two peaks, similar to what was obtained in the exact solution of Eq. (6):

It is easy to conclude that Eqs. (5) and (6) reduce to Eqs. (8) and (9), respectively, when the rotation is slow enough (i.e., $\mathrm{\Omega }=\frac{\dot{\theta }}{\gamma }\ll 1$). The major difference between the memoryless system and the exact one can be seen by comparing the amplitudes of the spectral components—while the peak magnitudes are equal in the memoryless case [Eq. (9)], the long-memory scenario is characterized by asymmetric amplitudes [Eq. (6)]. This behavior is shown in more detail in Fig. 2(a). The magnitude and phase of the peaks in Eqs. (6) and (9) can be plotted versus the dimensionless rotation frequency $\mathrm{\Omega }$, as shown in Figs. 2(b) and 2(c). The exact form of the curves depends on the quality factor $Q$ of the resonator. For high $Q$, there is very little asymmetry at slow rotation rates. Yet, when the dimensionless rotation frequency increases to $\mathrm{\Omega }=2Q$, there is maximal asymmetry as the ${\tilde{P}}_{+}$ spectral peak continues its fast decay to zero, whereas ${\tilde{P}}_{-}$ returns to the value of the memoryless system. Although this result is quite interesting, it is important to note that the condition for this strong asymmetry is $\dot{\theta }=2{\omega }_0$, which might be an extremely fast rotation rate for optical or microwave incident wave frequencies. For such fast rotations, relativity must be taken into account along with other radiative corrections, and the classical framework of the obtained results is no longer valid. This resonant regime also means that the mechanical motion of frequency $\dot{\theta }=2{\omega }_0$ pumps the dipolar excitation frequency of ${\omega }_0$, in similarity with the relativistic case explored in Ref. 26. It is possible to observe this asymmetric spectral amplitude effect for waves that have low frequencies, as may be the case for mechanical, as well as for very low-frequency (VLF) electromagnetic waves, which are used for underwater communication.^27,28 In addition, these memory effects could be observed for reactively loaded dipoles that were made with the purpose of obtaining resonant properties with extremely small apertures. Such dipoles are constricted by the Chu–Wheeler limit^29,30 and can possess extremely small decay rates. This is further explored in the conclusions, where a compact formula for the required reactive load at a given dipole length is derived, showing that this effect can be observed even for modest rotation periods if large reactances are used.

Fig. 2

Comparison between the exact and quasistationary (adiabatic, timescale separation) solutions. (a) The spectrum of the induced dipole moment contains two frequencies. The memoryless solution [Eq. (6)] has equal amplitudes and phases while the scattering from a dipole with memory [Eq. (9)] has an asymmetric spectrum. Inset: a dipole, rotating with angular frequency $\dot{\theta }$. (b), (c) The behavior of the amplitude and phase of the spectral peaks, for high and low $Q$ factors, as a function of dimensionless rotation frequency $\mathrm{\Omega }=\frac{\dot{\theta }}{\gamma }$.

For low $Q$, the amplitude of the peaks in Eq. (9) behaves very differently due to the detuning of the resonant frequency of the rotating dipole from ${\omega }_0$ to ${\omega }_0\sqrt{1-Q^{-2}}$. Although the positive spectral peak at frequency ${\omega }_{+}$ still decays from the memoryless solution, as it was in the high $Q$ case, the negative spectral amplitude ${\tilde{P}}_{-}$ actually exceeds the value of the memoryless rotating dipole, even at low rotation speed. Although the amplitude asymmetry is more prominent in the low $Q$ case, for the phase of the spectral peaks, the result is opposite. This effect occurs since the high $Q$ case shows faster deviation rates between the induced phases of the spectral peaks at low rotation frequencies.

It is straightforward to calculate the scattered field once the dipole moment is known (see Refs. 3132.–33) and will not be explicitly developed here. It will be sufficient to say that in the rest frame of the dipole, the point of scattered field observation is also moving periodically with the rotational frequency, meaning that the scattered field will have frequency components of $2{\omega }_{\pm }$ in the lab frame.³⁴ The mathematical derivation presented in this section is in fact more general and applies to more than rotating dipoles alone. When moving from Eqs. (2) to (4), the rotation of the dipole introduced polarization mismatch, only to be later removed from the integral as per Eq. (7). But, the polarization mismatch is not unique, and, in fact, it could have been the modulation of the incident field carrier that underwent the same analytic process; this will be expanded on ahead [see Eq. (10)]. Furthermore, since the incident modulation could be arbitrarily time-dependent, obtaining the solution for the harmonic case (constantly rotating dipole with polarization mismatch, as described above) allows for using Fourier series in order to obtain the solution to any type of modulation. The correctness of this solution will depend on the highest significant harmonic contained in the Fourier series of the carrier modulation, where the condition for the correctness of the quasistationary solution will be that this maximal frequency is required to be slower than the memory depth of the system. This insight will be used in the next section to synthesize a method for using static numeric simulators to compute dynamic time-dependent scattering problems in memoryless or very short-memory systems.

2.2.

Quasistationary Method for Numeric Time-Dependent Simulation of Short-Memory Systems

Analysis of scattering from rotating bodies has quite a few applied aspects. Micro-Doppler signatures, being the manifestation of internal degrees of mechanical motion within an object, are the subject of studies in radar and sonar sciences. Micro-Doppler spectra, which are unique characteristics of the observed scatterer undergoing complex motion, are used for remote target identification. Rotating blades of a helicopter or a drone,^35,36 jet engine modulation,³⁷ cyclists, and pedestrians³⁸ are among numerous examples where micro-Doppler spectroscopy brings an advantage in identification and classification via careful signal postprocessing. The vast majority of signals probe micro-Doppler with either ultrashort or quasi-CW (continuous wave) signals. In the first case of short pulses, the target can be assumed static during the interaction, meaning that the carrier frequency of each wave packet remains unaffected. The Doppler effect in that case manifests in the phase difference between consecutive pulses. In the second case of quasi-CW, the pulse length is effectively longer than the characteristic timescale of the motion, hence the phase and amplitude modulation are imposed on the carrier frequency directly. In this case, it is more instructive to look at the micro-Doppler frequency comb in the frequency domain,^20,21,34 which is the spectral content within the quasi-CW pulse. In both cases, timescale separation techniques are frequently applied for the scattering analysis.¹⁶ Here, we analyze and experimentally demonstrate the transition between a micro-Doppler spectrum and a discrete frequency comb by tuning the duration of the probing pulse. When the pulse length is smaller than the period of dipole rotation, a different kind of memory is revealed as the initial state of the system—which in this case is the angle between the incident polarization and the dipole (wire) orientation at the time of pulse incidence. The LTV property in this case comes through boundary conditions instead of the response kernel.

Straightforward simulation of moving scatterers is not trivial, since the vast majority of electromagnetic numeric software cannot self-consistently handle scattering problems with a moving medium. Static simulations, on the other hand, are very common and can be performed with numerous off-the-shelf software packages available (CST, COMSOL, Lumerical, and HFSS to name a few). At this point, it is important to note that in the considered practical scenario the decay rate of the excited current on the dipole is extremely fast, with the whole scattering process taking less than a nanosecond. Since the rotation speed of the wire in the experiment that will be presented ahead does not exceed 10 Hz (i.e., $\mathrm{\Omega }=\frac{\dot{\theta }}{\gamma }\ll 1$), it is clearly possible to apply the timescale separation method suggested in Eq. (7). The way to perform a time-dependent simulation is, therefore, by creating a sweep on the position of the wire, solving a static scattering problem at every location, and then stitching these solutions together with a slow time parameter (sequential counter of positions). Simulating the modulation of the incident carrier is nontrivial as well, as it significantly increases the computational effort. There are two ways to go about simulating the scattering process of short pulses, where the carrier’s modulation plays a role. The brute force method is to evaluate the scattering at a number of specific frequency points, which correspond to the spectral shape of the modulation, and then add them up coherently with the Fourier weights that govern the magnitude and phase of each frequency within the desired modulated spectrum. Finally, the weighted sum can be put through an inverse Fourier transform in order to obtain the time domain scattered field (carrier and modulation). The drawback of such a method is that it requires the evaluation of numerous frequency points, which can be computationally heavy, especially when many scatterer positions need to be calculated. In order to perform the analysis in a more computationally efficient way, a different approach is suggested here for memoryless systems, which is the case for the majority of practical scattering scenarios. As per the discussion at the end of Sec. 3, Eq. (7) deals with the removal of the polarization mismatch from the integral. The polarization mismatch is not unique, and the same applies to slow modulation of the carrier, under similar restrictions. The result is a static solution multiplied by a time-dependent amplitude. Consider Eq. (4), but this time with an incident Gaussian pulse of the form $E_i(t)=\cos ({\omega }_{0}t)A(t)$, where $A(t)$ is a slowly varying amplitude and is of a Gaussian shape in the simulations and experiment [$A(t)=e^{-{(\frac{t}{\tau })}^2}$, $\tau$ is the characteristic width of the pulse]. Using Eq. (4) together with the quasistationary approximation of Eq. (7), which is correct for slow rotation and modulation (i.e., $\frac{\dot{\theta }}{\gamma }$, $\gamma \tau \ll 1$), the dipole moment is given by

This result could be intuitively understood since the dipole moment would be proportional to the amplitude of the incident pulse at any given moment multiplied by the static response. The method was applied to the problem of pulse scattering from a rotating dipole, as shown in the experiment in Fig. 3. The simulation was performed using CST Studio, where the wire was illuminated by a plane wave, while the structure was rotated by a set of sufficiently discretized angles, corresponding to the location of the wire at any time during the motion. The far field at the receiving antenna location (perpendicular to the direction of incident wave propagation) was recorded for each angle. The resulting slow time sequence was then multiplied by the modulation envelope, as shown in Fig. 4. Note that an additional slow time-dependent rotation operator should multiply Eq. (10) to account for the transformation of scattered fields from the rotating lab frame to the laboratory one. It is interesting to note that for pulses shorter than the rotation frequency of the dipole (i.e., $\dot{\theta }\tau \ll 1$), the resulting dipole moment, and therefore the scattered field, will be proportional to $\cos ({\theta }_0)$. This is because in that case the scattered pulse effectively samples the momentary position of the wire. This memory effect of initial position will be shown experimentally ahead to be of considerable importance even when the pulse length is on the order of the rotational period (i.e., $\dot{\theta }\tau \sim 1$). The results of the simulation appear ahead in Figs. 4Fig. 5–6, where they are compared against the experiment, which is discussed next.

Fig. 3

Experimental setup for probing scattering from rotating objects. (a) The schematic representation of the setup. A carrier frequency $\omega$ is split between a transmitting and a receiving arm. The transmitting arm is mixed with a Gaussian ($G$) pulsed signal of controllable pulse length to generate the envelope. The receiving arm is split into two quadratures in order to down-convert the scattered signal. (b) A photo of the experimental setup inside the anechoic chamber. Inset: enlarged photograph of the mechanism to control the wire’s rotation (electric dipole analogue; rotation direction is marked with red arrow) at a constant frequency of 3.4 Hz.

Fig. 4

Far-field scattering of Gaussian pulses from a rotating wire, comparing short memory (adiabatic, quasistationary) simulation with experiment (the setup appears in Fig. 3). Blue lines: two different Gaussian pulse lengths, 0.3536 and 0.1414 s measured experimentally. Red lines: numerical analysis using the short-memory method. Black dashed-dotted line: the envelope of the incident pulse [panels (a) and (b)] and its spectrum [panels (b) and (d)]. Columns: scattered signals in time and frequency domains, respectively.

Fig. 5

Comparison of the scattered baseband spectra obtained in simulation and experiment as a function of frequency and pulse length. As the pulse length increases (in comparison with rotation speed), the micro-Doppler peaks form at discrete frequencies. For short pulses, the peaks are broader in frequency, creating a featureless spectral continuum. The initial angle for both experiment and simulation is taken as ${\theta }_0=0\mathrm{deg}$.

Fig. 6

Colormap, demonstrating scattered signals spectra at the baseband. Vertical axis—initial angle between the wire and the polarization of the incident field, horizontal axis—frequency. Different pulse widths are indicated in insets. (a), (c), (e), and (g) Numerical data. (b), (d), (f), and (h) Experimental data.

2.3.

Experimental Verification of Quasistationary Method—Scattering from a Slowly Rotating Wire

In order to verify the validity of the numerical technique, a set of experiments was performed in an anechoic chamber. The setup is shown in Fig. 3, where a rotating wire of length $L_{\text{wire}}=40\mathrm{mm}$ and diameter $d_{\text{wire}}=1\mathrm{mm}$ serves as the analogue of a rotating dipole. The wire is rotated by a stepper motor at a constant frequency of 3.4 Hz while being illuminated by a vertically polarized electromagnetic field. The transmitted signal is a Gaussian envelope around a 3.195-GHz carrier, which is chosen to maximize the scattering from the wire. The Gaussian modulation has a controllable temporal width to satisfy different interaction regimes, which will be described below. The receiving antenna is polarized in the vertical direction as well and is placed perpendicularly to the transmitter in order to increase the isolation between transmitting and receiving channels, while simultaneously insuring direct lines of sight to the scatterer. This bistatic configuration allows for increasing the signal-to-noise ratio quite significantly. The signal from the receiving antenna is downconverted using a pair of mixers and a 90-deg splitter (maintaining the phase in one path and adding 90 deg in another one), producing in phase and quadrature data at the intermediate frequency (IF) output of the mixers. The IF outputs are recorded using a sampling scope, and the data is digitally processed in order to obtain the down converted temporal signal.

The results, comparing the theory and experiment, appear in Figs. 4Fig. 5–6, and are in good agreement with each other. There are several important factors to consider about the system under study. First, the relation between the pulse duration and the rotation period of the wire should be underlined. As discussed in Sec. 1, short pulses will sample the instantaneous location of the wire, whereas long pulses will experience periodic amplitude modulation. Second, in the short-pulse regime, the initial angle of the wire with respect to the polarization of the incident field plays a crucial role, as shown in Eq. (10). Taking those clarifications into account, the simulations and experiments are repeated for pulse lengths ranging from 0.1 to 100 s, as well as for initial angles of the wire ${\theta }_0$ ranging from 0 deg to 180 deg. Figure 4 shows the experimental and numerical results for two pulse widths and with the wire having the initial angle of ${\theta }_0=0\mathrm{deg}$ (the position of the wire at time $t=0$, when the pulse has its maximum at the wire’s center). The results are the time-dependent [Figs. 4(a) and 4(c)] as well as the spectral [Figs. 4(b) and 4(d)] content of the down converted scattered field at the receiver. Indeed, it can be observed that for incident Gaussian pulses, the scattered field is also Gaussian, with additional harmonic modulation, as predicted by the theoretical result in Eq. (10). For longer pulses, shown in Fig. 4(b), the spectrum comprises a frequency comb, which becomes narrower with increasing pulse length. Indeed for infinite pulse length, the result converges into a discrete comb, which is separated in frequency by $2\dot{\theta }$ (as in Ref. 34, which considers CW radiation). For short pulse lengths shown in Fig. 4(d), the spectrum is smeared, as the components at each discrete frequency spread and overlap with neighboring peaks.

In order to perform a comprehensive study of the phenomena, a broad range of parameters has been investigated. In particular, the same experiment that produced Fig. 4 was repeated with pulse lengths ranging from 0.1 to 100 s, as well as for initial angles of wire ${\theta }_0$ ranging from 0 deg to 180 deg. The results are shown in Figs. 5 and 6. Note that Figs. 4(b) and 4(c) represent horizontal cuts of Fig. 5, which reveal the transition from the long- to short-pulse regime, where smearing of the Doppler comb is clearly seen. The discrete frequency lines ($x$ axis of the color map) become broader when reducing the pulse length ($y$ axis) until they begin overlapping and, finally, smearing out once the rotational period of the dipole becomes comparable with the signal’s Gaussian envelope. The experimental data contain additional peaks in between the ones found in simulation. This is the result of offset of the rotation axis from the center of the wire, leading to odd harmonics generation at the baseband (discussed in more detail in Ref. 34).

The final parameter affecting the interaction phenomenon is the initial angle between the dipole and the polarization of the incident radiation. This angle is defined in Eq. (10) as the one between the incident polarization and the wire’s principal axis at a time corresponding to the arrival of the peak of the incident Gaussian pulse to the origin. Figure 6 shows the spectral composition of the scattered pulses at fixed pulse lengths for variable initial angles. For pulses longer in comparison with the rotation period of the wire, there is no dependence of the spectrum on the initial angle, as shown in Figs. 6(a) and 6(b). This is due to the fact that the wire completes numerous rotations during the duration of the pulse, effectively losing the memory of the initial angle. On the other side, when the pulse length is very short in comparison with the wire’s rotation frequency, the spectrum is completely dominated by the initial angle. This can be understood by returning to Eq. (10), where the short pulse effectively samples the cosine at time 0, causing a harmonic dependence of the scattered spectrum on the initial angle. As shown in Figs. 6(g) and 6(h), for a pulse length of 1.25 s, there appears to be a minimum of the scattered spectrum near the initial angle of 90 deg. This is consistent with Eq. (10) and can be understood intuitively, since the wire is oriented perpendicularly to the incident polarization and, hence, scatters very little. For extremely short pulses, this minimum would occur at 90 deg exactly. However, for longer pulse lengths, the minima may have an offset, as shown in Figs. 6(g) and 6(h), where the pulse length is only about two times smaller than the rotation period. The sensitivity of the spectral peaks to the memory of the initial angle allows for remote sensing of the state of the rotating wire. In Figs. 4(b), 4(c), 5, and 6, only the positive frequencies of the baseband are shown, since the negative frequency components are identical due to the fact that this scattering scenario has very short memory [Eq. (9)] and, hence, a symmetric micro-Doppler spectrum. For faster wire rotation, or for another type of resonator (typically, with high $Q$ factor), the quasistationary method would fail, resulting in asymmetric micro-Doppler peaks [Eq. (6)]. In this case, double-sided maps should be presented for a complete characterization of the process.