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Micro-Doppler frequency comb generation by rotating wire scatterers
Journal of Quantitative Spectroscopy & Radiative Transfer 190 (2017) 7–12
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Micro-Doppler frequency comb generation by rotating wire scatterers V. Kozlov a,1, D. Filonov a,b,n,1, Y. Yankelevich a, P. Ginzburg a,b a b
School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel ITMO University, St. Petersburg 197101, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 1 September 2016 Received in revised form 19 December 2016 Accepted 19 December 2016 Available online 21 December 2016
Electromagnetic scattering in accelerating reference frames inspires a variety of phenomena, requiring employment of general relativity for their description. While the ‘quasi-stationary field’ analysis could be applied to slowly-accelerating bodies as a first-order approximation, the scattering problem remains fundamentally nonlinear in boundary conditions, giving rise to multiple frequency generation (microDoppler shifts). Here a frequency comb, generated by an axially rotating subwavelength (cm-range) wires is analyzed theoretically and observed experimentally by illuminating the system with a 2 GHz carrier wave. Highly accurate ‘lock in’ detection scheme enables factorization of the carrier and observation of multiple peaks in a comb. The Hallen integral equation is employed for deriving the currents induced on the scatterer and a set of coordinate transformations, connecting laboratory and rotating frames, is applied in order to make analytical predictions of the spectral positions and amplitudes of the frequency comb peaks. Numeric simulations of the theoretic framework reveal the dependence of the microDoppler peaks on the wire's length and its axis of rotation. Unique spectral signature of micro-Doppler shifts could enable resolving internal structures of scatterers and mapping their accelerations in space, which is valuable for a variety of applications spanning from targets identification to stellar radiometry. & 2017 Elsevier Ltd. All rights reserved.
1. Introduction Laws of physics in accelerated frames of reference could be substantially different from those obtained in inertial ones. Coriolis force and Sagnac interference are, probably, the most known effects in mechanical and electromagnetic systems correspondingly. Electromagnetic scattering from moving bodies requires special formulation, traced back to investigation of rotating machinery and radar targets identification. Special attention has been paid to studies of non-relativistic rotating objects, owing both to the relative mathematical simplicity of their analysis and to the vast span of potential applications. Among various examples it is worth noting radio frequency (RF) detection of helicopter propellers [1–3], back action of spinning blades on wireless communication channels [4], rotational frequency shifts on beams carrying orbital angular momentum [5–7], and even rotational motion measurement of celestial bodies in astrophysics [8]. Moreover, acoustical noise could originate from revolving motors [9], rotating and vibrating wires were proposed as an analogue to the molecular Raman scattering in optics [10], while, in general, signatures of rotation emerge in many others multidisciplinary fields. In a n Corresponding author at: School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel. E-mail address: [email protected] (V. Kozlov). 1 V. Kozlov and D. Filonov contributed equally to this work.
http://dx.doi.org/10.1016/j.jqsrt.2016.12.029 0022-4073/& 2017 Elsevier Ltd. All rights reserved.
broader scenes, time-varying scatterers could find use in radio frequency identification (RFID) technologies [11,12]. Rigorous analysis of electromagnetic scattering from moving and, in particular, accelerating bodies requires employment of apparatus developed for the general theory of relativity [13]. Along with the transformation of all physical observables to the moving frame, accelerated motion causes substantial change in the general form of the governing differential equations and their associated boundary conditions. A relativistic treatment of electromagnetic fields in the presence of rotating bodies (infinite two dimensional cylinders) was performed by Bladel [13] and few others. Intuitively, each point on a rotating scatterer has a different instantaneous velocity, resulting in a collection of Doppler shifts. As electromagnetic fields at nearby points on a body are tightly connected by the nonlocal nature of Maxwell's equations, the scattering problem is far from being trivial. Furthermore, timevarying boundary conditions make the electromagnetic problem nonlinear, giving rise to generation of new frequencies alongside the initial carrier wave spectrum. While solution of the general scattering problem is quite involved, a set of simplifications could be made by assuming nonrelativistic motions and slow accelerations. First order approximations are widely used in theoretical analysis of radar targets identification, e.g. assuming that objects are at a complete rest during the interaction with an incident pulse [14]. Collecting static scattered signals, obtained from a few spatial positions of a target, enables relating theoretical data to
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experimental - radar measurements usually provide phase relations between adjacent pulses impinging on a target, or Doppler shifts in the reflected wave spectrum. Accelerating bodies, in contrast, produce a much richer spectral signature, called microDoppler shifts [15], that cannot always be recovered by applying the static approximation. The so-called ‘instantaneous rest frame’ technique assumes slow nonrelativistic motion as well, but relaxes few of the beforehand mentioned constraints and enables extracting more information regarding the accelerated motion. Neglecting second order relativistic contributions enables factoring out time dilation and length contraction effects and employing standard field continuity boundary conditions at each point of a scatterer, as if it moves with a linear (instantaneous) velocity. The benefit of rotating systems is in the existence of a relatively simple frame, where the entire scatterer is at rest. In this case the route to undertake is to transform sources to the rest frame of the rotating body, solve the scattering problem, and translate the results back to the laboratory frame. This approach is undertaken here and compared to the experimental data. It is worth underlining that the reported effects have no relativistic contributions. While much work was put into the description of various time varying scatterers (see for example [16–19]), the dipole approximation for scattering from objects that are smaller than the incident wavelength is both simple and useful. Here the effect of electromagnetic scattering from a rotating thin wire is studied both experimentally and analytically, including retardation effects that arise due to the small, yet finite, dimensions of the scatterers. To the best of our knowledge a theoretical treatment of scattering from a clutter of random rotating dipoles was partially addressed in [20]. In particular, a dipole, illuminated by a field of frequency ω was considered, predicting the scattered field to have additional components of ω ± 2θ ,̇ with θ ̇ being the angular frequency of rotation. In a later work a similar result was reached theoretically as well [21]. While the major Doppler shift indeed occurs at ±2θ ̇ frequencies, rotating bodies with nontrivial internal structure generate an entire frequency comb of micro-Doppler shifts. Detailed analysis, presented here, shows the emergence of the frequency comb in the scattered field, having components of ω ± mθ ,̇ where m is an integer. The developed tools also reveal the dependence of the micro Doppler comb on the length of the wire and axis of rotation. A nonrelativistic model, based on Hallen's integral equation [22] for calculating the current distribution on thin conducting wires, was employed together with the ‘instantaneous rest frame’ approximation, under appropriate coordinate
transformations between lab and scatterer's reference frames. The theoretical model was fully supported by the experiment undertaken at an anechoic chamber, where rotating shaped copper wires were illuminated with 2 GHz waves and the micro-Doppler comb was measured with highly accurate ‘lock in’ detection scheme.
2. Results 2.1. Theory Theoretical analysis of a spinning wire in free space will be presented first and then verified experimentally. The geometry of a spinning scatterer in free space and relevant measurement apparatus appear in Fig. 1. For the theoretical analysis, consider a short wire, rotating in the Z-Y plane at a constant angular frequency θ ̇ ≪ ω , where ω is the carrier frequency of the incident illumination. This approximation allows neglecting relativistic effects. In the lab reference frame (denoted by prime) an incident plane wave of the form E inc =E0 eikz′y^ ′ is propagating in free space, k is the k-vector of the wave, and the carrier's harmonic time dependence eiωt is suppressed. The rest frame of the wire (unprimed) agrees with the lab coordinates at time t = 0. The primary goal of the derivation is to calculate time-dependent forward scattering. In the considered geometrical arrangement the projection of the wire on the polarization direction changes harmonically in time (from maximal to almost zero), resulting in time-dependent scattering. The coordinate transformation between the lab and the rotating frame is carried out by applying the following relation:
⎞ ⎛ x′ ⎞ 0 0 ⎛ x⎞ ⎛ 1 ⎜ y ⎟ = ⎜ 0 Cos (θṫ ) − Sin (θṫ )⎟ ⎜ y′⎟. ⎟⎜ ⎟ ⎜ ⎟ ⎜⎜ ⎝ z ⎠ ⎝ 0 Sin (θṫ ) Cos (θṫ ) ⎟⎠ ⎝ z′ ⎠
(1)
In the wire's rest frame the incident field along the length of ̇ the wire becomes Ezinc (z, t ) = E0 Sin (θṫ ) eikzCos (θt )(the wire is parallel to the z-axis in its frame of reference, hence only z-component of the field is relevant for the interaction). The field in the scatterer's rest frame may be decomposed into a sum of products of Bessel functions, dependent on space-time harmonics:
Fig. 1. Experimental setup of transmission in an anechoic chamber. A scatterer is mounted on a motor's shaft and placed between transmitting and receiving horns, located in the far field of the scatterer.
V. Kozlov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 190 (2017) 7–12 ∞
Ezinc (z, t ) = E0 Sin (θṫ )
Jn ( kz ) ein
∑
( θṫ − π2 ).
(2)
n =−∞
Scattering problems from thin perfectly conducting wires could be solved by applying Hallen's integral formulation. This approach was chosen over other methods, as it allows approximating results with analytical and semi-analytical expressions. In Hallen's formulation scatterers are virtually replaced by line currents (unknowns) and the total electromagnetic field outside is given as a superposition of the excitation and the convolution between the current and the Green's function related kernel. Boundary conditions of vanishing tangential components of the electrical field on a perfect electric conductor (physical boundaries of the initial wire) enable formulating the scattering problem in the form of an integral relation. Hallen's integral equation for current distribution along the wire ( I ( z, t )) under the assumption of a lossless, perfectly electrically conducting wire is approximated by [22]:
iη 2π
∫z
z2
1
e−ik (
)
2
( z − z′)
+ a2
I ( z′,t ) dz′=C1eikz +C2 e−ikz +i
∫z
z2
1
e−ik z − z′ Ezinc ( z′, t ) dz′,
(3)
where C1,2 are to be determined from the boundary conditions of vanishing current at the edges of the wire, located at z1,2 , a is the radius of the wire and η is the free space impedance. Noticing that the integrand on the left hand side of Eq. (3) has a sharp maximum around z = z′ enables approximating it with [22]:
iη 2π
∫z
z2
1
2 2 e−ik ( z − z′) + a
( z − z′)2 + a2
I ( z′,t ) dz′=ZĨ ( z, t ), (4)
iη (z − z ) e−ika
where Z ̃ ≈ 2 2π1a is a zero-order approximation that could be further improved by using the King's Three-Term approximation [22]. Solving Eq. (3) using approximation of Eq. (4) enables determining the current along the wire. Assuming an axially rotating wire z1,2= ± h where h is half the wire's length:
4E I ( kz, θṫ )= 0 ̃ Zk
⎛
∞
∑ Tns ( θṫ ) ⎜⎜ Snc ( kz)−Snc ( kh) ⎝
n= 0
Cos ( kz ) ⎞ ⎟⎟+iT nc ( θṫ ) Cos ( kh) ⎠
Tns ( θṫ ) =
( 2 − δ n,0 )
R
)
2n + 1) (kz ) Sin ( kz ) − F I( (kz ) Cos ( kz ) ,
( { m + 0.5,
(8)
a unit vector in the z direction, μ is the free space permeability and the dot represents a scalar product. Eq. (8) may be solved numerically, or evaluated analytically by adopting several approximations. Here, leading terms in 1 are considered along with the d π
short wire approximation ( kh≪ 2 ) for calculating induced currents, which enables identifying well-defined spatial parities in ̇ by making Tailor expansions of Eq. (5). Here even spatial θ series contributions could be approximated as constants, while the odd ones may be replaced with functions that are linear in their argument. The resulting short wire current then takes the form:
Is ( kz, θṫ ) ≈
4E 0 ̃ Zk
⎞2n +2 ⎛ kh ⎞2n +2⎞ ⎟⎟+iα cnT nc ( θṫ ) −⎜ ⎟ ⎝ 2⎠ ⎠ ⎝⎝ 2 ⎠ ⎛⎛
∞
∑ α nsTns ( θṫ ) ⎜⎜ ⎜ kz ⎟
n= 0
⎛ ⎛ kz ⎞2n +3 z ⎛ kh ⎞2n +3⎞ ⎟⎟, ⎜⎜ ⎜ ⎟ − ⎜ ⎟ h⎝ 2 ⎠ ⎠ ⎝⎝ 2 ⎠
(9)
where the following auxiliary coefficients are defined:
( 2n + )( 2n + 1) 1 2
( 2n + 2)( 4n + 1)
⎛⎧ ⎞ ⎫ 3 , 2n + 2⎬ , { 2n + 3, 4n + 2}, 0⎟ , 2 F 2⎜ ⎨ 2n + ⎭ ⎝⎩ ⎠ 2
( 2n + )( 2n + 2) F ⎛⎜⎜ ⎧⎨ 2n + 5 , 2n + 3⎫⎬, 3 2
( 2n + 3)( 4n + 3) 2
2
⎜⎩ ⎝
2
⎭
⎞ ⎟ 4n + 3}, 0⎟ . ⎟ ⎠
(10)
It can be readily checked that for small values of kh Eqs. (9) and (5) coincide. By using the above mentioned approximations, the scattered field is finally evaluated in the lab reference frame by solving:
(6)
while the superscript ‘s’ and ‘c’ denote even (‘c’) and odd (‘s’) ) are the real and imaginary functions of time or space and F R( m /I components of the hypergeometric function [23], satisfying:
FR( m) + iFI( m) = 2 F 2
⎞ e−ikR ( z, d, t ) ⎛ 2 zd Sin ( 2θṫ ) ⎟ I (z, t ) dz, ⎜ d Sin ( θṫ )− 3⎝ ⎠ 2 4π R ( z , d , t )
( )
{ 2n + 4,
)
2n + 1)
∫−h
⎞
h
∫−h I ( z, t ) ( G⃡FF ( r; z) z^) dz⎟⎠
where R ( z, d, t )= d2−2zdCos θṫ + z2 is the distance connecting a
Sin (θṫ ) Cos (2nθṫ ),
⎛ kz ⎞2n + 1 ( 2n) 2n Snc ( kz ) = ⎜ ⎟ FR (kz ) Sin ( kz ) − F I( ) (kz ) Cos ( kz ) , ⎝ 2⎠ ⎛ kz ⎞2n + 2 Sns ( kz ) = ⎜ ⎟ ⎝ 2⎠
h
= − iωμ
2n α nc = 2F R( )( 0) − 4
( 2n + 1) ! ( −1)n Sin (θṫ ) Cos ((2n + 1) θṫ ), T nc ( θṫ ) = 2 ( 2n + 2) !
( (F(
^ ⎛ E yLab ′ ( d , t )= − iωμy′ ∙ ⎜ ⎝
(5)
where the following auxiliary functions are defined as:
( − 1)
layout:
2n α ns = 2F R( )( 0) − 4
⎛ Sin ( kz ) ⎞ ⎟, ⎜⎜ Sns ( kz )−Sns ( kh) Sin ( kh) ⎟⎠ ⎝
n
of the current. Eqs. (5) and (6) show that the current consists of ̇ both even and odd time harmonic components of θ that produce additional scattered field frequencies in the far field. However, it will be shown that for an axially rotating short wire only the even harmonics remain in the far field. Next, the scattered field in the rest frame of the wire is derived and transformed back into the lab frame of reference using Eq. (1), evaluating it in the lab frame at a point z′= − d , y′=0, x′=0 (where the detector is located) and taking the y^′ component alone, as it is the case of the experimental
point on the wire and the detector's location at time t , G⃡FF ( r;z )is the far field 3D Green's dyadic (3 3 matrix) in the rest frame, z^ is
2
z − z ′ + a2
9
)
m + 1}, { m + 2, 2m + 1}, 2ikz .
E yLab ′ ( d, t ) ≈
−iωμe−ikd 4π d
∫−h
∞
⎛
=E0 k2ah2
e−ikd d
h
Sin ( θṫ ) Is ( z, t ) dz, ⎞2n
∑ Cn ⎜ kh ⎟ n= 0
⎝ 2⎠
Sin2 ( θṫ ) Cos ( 2nθṫ ),
(11)
where
(7)
The result for the current in Eq. (5) is equivalent to the closed form derived in [22], but has the advantage of avoiding the numeric singularity (when both nominator and denominator tend to 0) at θṫ = nπ as well as clearly revealing the frequency components
Cn =
( −1) n( 2 − δ n,0 ) n+1 α s. ( 2n + 1) ! 2n + 3 n
(12)
The analytical expression in Eq. (11) predicts the existence of even harmonics in the scattered field at an on-axis point away
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V. Kozlov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 190 (2017) 7–12
from the short wire, provided that the wire is rotating around its center (axial rotation). The odd harmonics vanish due to parity considerations involving odd and even functions in the current along with symmetric integration boundaries of the integral. These harmonics disappear even if the full current (long wire) described in Eq. (5) is used. In fact, the odd harmonics will vanish even if the only approximation made is R ( z, d, t )≈d − zCos θṫ
( )
(quadratic terms in wire's internal geometry are neglected), but the detailed proof of this statement is omitted in this paper in order to not obscure the main results. It should be noted that if no approximations of the Green's function are made, the integrand in Eq. (8) has no parity and thus negligible odd peaks are expected in the far field. Taking the leading term (n ¼0) in Eq. (11) recovers the dipole field pattern, showing the dominance of the 2θ ̇ peak. Using the dipole term, an analytical approximation for the polarizability 4
of small perfectly conducting wires can be written as αzz = 3 πε0 ah2. The approximation made in Eq. (4) leads to an almost symmetric current distribution around the center of the axially rotating wire. This approximation is good for short wires, but becomes less accurate for longer ones. It is thus preferable to solve Eq. (3) numerically for more accurate results of the current. Numeric solutions also allow considering wires rotating about a different axis than their center. This can be done by changing the integration boundaries z1,2 with non-symmetric boundaries, giving rise to odd peaks in the far field. A final important conclusion from Eq. (11) is that it shows the dependence of the ratio of peak amplitudes on the wire geometry (e.g. its length). This insight shows that microDoppler peaks could be used to extract information about the internal geometry of rotating scatterers. The theoretical framework developed above could be extended to include arbitrary rotation axis and observation points in a straightforward fashion. The same approach could be employed for considering wires of an arbitrary shape, which is not necessarily a straight line.
2.2. Experimental results Micro-Doppler shifts in the scattered spectra of thin wires will be investigated experimentally and compared with the developed theory. The photograph of the experimental setup appears in Fig. 1. Spinning wire scatterers were mounted on a motor's shaft, stabilized with a 3D-printed plastic rack. The motor is controlled by a DC current, enabling rotational speeds of up to hundreds of hertz. In particular, a 5 cm long wire (photograph of the sample appears at the insert to Fig. 1) was rotated in the Z-Y plane at a ̇ 2π × 28Hz . The whole structure was constant angular frequency θ = placed between two horn antennas, separated by a 2 m distance, ensuring that both the transmitting and receiving horns are at the far-field region of the scatterer. A Vector Network Analyzer (VNA) was used as a source of a 2 GHz continuous wave, feeding the transmitting antenna with the polarization set in the Y-direction. A receiving antenna was placed opposite the transmitter with the same polarization. The output of the receiver was amplified and down-converted to the baseband by mixing it with the same 2 GHz carrier, as in the incident field. The output of the mixer was passed through a low pass filter to remove residual high frequencies and directed to a lock-in amplifier where the low-frequency spectral shifts of the scattered field were recorded. The results are shown in Fig. 2, showing normalized lock-in readouts as a function of baseband frequency. The amplitude of the second peak was used to scale the theoretic values. This peak corresponds to ω ± 2θ ̇ in the original signal and appears as 2θ ̇ in the lock-in detection, since only the positive part of the spectrum is presented. Fig. 2(a) shows theoretical predictions, obtained with Eqs. (3) and (8), while the experimental data appears on Fig. 2(b). Three even and three odd peaks above the noise level were observed in the experiment. The theoretic values were extracted from the simulation of the axially rotating wire (offset is 0mm between the center of the wire and the rotation axis), showing
Fig. 2. Normalized forward scattering spectrum (micro-Doppler comb, baseband spectrum) of a 5 cm long wire (a) Theoretical predictions evaluated numerically (with Eqs. (3) and (8)). Red stars indicate results obtained for 0mm offset of the rotation axis from the center of the wire, showing only even peaks. Blue circles indicate results obtained for the same wire length, but rotating around an axis at a distance of 0.6mm from its center, showing emergence of odd peaks. (b) Experimental results, obtained for the ̇ 2π × 28Hz . Odd peaks emerge due to inherent mechanical instabilities in the practical realization. (For centered 5 cm length wire, rotating at an angular frequency θ = interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
V. Kozlov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 190 (2017) 7–12
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Fig. 3. (a), (c) Normalized amplitude of the odd peaks ((a) – first, (c) – third) as a function of the offset of the rotation axis from the center of the wire. Normalization is performed with respect to both the amplitude of the second peak for each offset value and the amplitude value at 22 mm offset is set as unity. Black line represents theoretical predictions evaluated numerically (Eqs. (3) and (8)), while the red dots are experimental data. (b) Normalized amplitude of the second peak as a function of wire's half-length. Peak at h ¼25 mm is chosen for the normalization and brought to unity. Black line represents theoretical predictions evaluated numerically (Eqs. (3) and (8)), while the red dots are experimental data. Offset was taken to be zero in this case. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
only even peaks in agreement with Eq. (11). While odd peaks are not predicted in the case of perfectly-centered rotation, they are observed in the experimental data. There are several reasons for their appearance. First, the receiving horn has finite-size aperture, collecting off-axis scattering that could be shown to include contributions of odd harmonics. Second, and more importantly, odd peaks appear in the case of very small offsets of the rotation axis from the center of the wire. The offset could occur due to mechanical precision at the connection point between the wire and the shaft, as well as the precession of the later. A phenomenological ‘offset’ parameter (displacement of the rotation axis from the geometrical center of the wire) was introduced for fitting odd peaks in the experimental data (Fig. 2(b)). A value as small as 0.6mm (about 1% of the wire length) was found to be sufficient for fitting the odd peaks in the comb (Fig. 2(a)). For the numerical simulation the integration boundaries in Eqs. (3) and (8) were offset by 0.6mm away from the center of the wire, giving rise to the odd peaks. It is apparent that the theory (up to inherent limitations) agrees well with the location (equidistant discrete comb was generated) and amplitudes of the rotational Doppler peaks in the scattered field spectra. The agreement in amplitudes is good for the first 4 peaks and then reduces owing to fluctuations in the rotation frequency, which occurred due to the long measurement time (up to a few minutes for each data point was needed in order to obtain higher signal to noise ratios) and motor shaft precession. In order to underline the impact of the ‘offset’ parameter on the micro-Doppler comb, a set of additional experiments was performed. In particular, the center of the wire was displaced from the motor holder on purpose. Fig. 3(a) and (b) show both experimental and theoretical results for emergent odd harmonics (first and third correspondingly). Amplitudes of the harmonics were normalized to the amplitude of the second peak for each offset value and presented as a function of the later. Additional normalization was done in respect to the amplitude value at 22mm offset, which was
chosen to serve as a reference (value of unity on both graphs). Experimental results and the theoretical predictions agree well with each other, showing the sensitivity of the odd peaks amplitude to offsetting the axis of rotation from the center of the wire. While the theory predicts vanishing amplitude of the third peak when the offset approaches zero, experimental data shows a finite value. This behavior is attributed to the overall precision of the length measurements, finite size of the bracing, and precession of the shaft - all of which contribute to the emergence of the odd peaks as well. Relying on the above considerations, the phenomenological offset parameter was introduced in order to fit the experimental data in Fig. 2(b). It is noted that this small offset (0.6mm) is very reasonable to have occurred in the experiment. In order to validate theory and experiment furthermore, wires of various lengths were rotated and the amplitude of the second peak was plotted as a function of wire length. The values were normalized to the 2nd peak's amplitude of a 5 cm wire (h¼25 mm). The theoretical predictions are well supported with experimental results, as can be seen on Fig. 3(b). It is worth noting, that the second peak, having the major contribution to the micro-Doppler comb in the case of the short wires, could be measured with a better signal-to-nose ratio in comparison with other peaks in the comb. This explains the smaller error bars than those, appearing on panels (a) and (c). It is worth underlining the excessive noise factor in the peak amplitudes for wires of 25 and 55 mm length (error bars on Fig. 3(b)). This effect is attributed to an internal mechanical resonance of the motor and the holder rack – for certain values of moment of inertia (related to the wire's length) the system is forced to a resonance, affecting the precision of the measurements. 3. Outlook and conclusion In conclusion, it was shown both theoretically and experimentally that electromagnetic waves, scattered from rotating
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wires, have additional frequency components of ω ± mθ ,̇ forming a micro-Doppler frequency comb (m is an integer). A theoretical model for this phenomenon was derived, predicting the appearance of those frequency components and closely predicting their relative magnitudes. In particular, it was shown that in the case of axially rotating wires only even harmonics at the micro-Doppler comb appear. While the emergence of odd harmonics is indeed not expected in ideal systems, practical realizations do exhibit their emergence. The theoretical model shows that odd peaks are very sensitive to small offsets of the axis of rotation away from the center of the wire, enabling the introduction of a phenomenological asymmetry coefficient, which provides a reliable fit to experimental reality. The model also successfully predicts the dependence of the micro Doppler shifts of rotating wires on their geometry (length) and axis of rotation. These results expand previously proposed theories, where only two Doppler shifts of ω ± 2θ ̇ were expected in the spectra of dipole-like scatterers and suggest that it might be possible to use such spectral signatures for resolution of the internal structure of observed scatterers and their axis of rotation. Furthermore, RF components could serve as emulation tools for observation of complex phenomena [24–27]. Externally induced mechanical motion on vibrating dipoles could replicate Raman scattering scenarios [10]. The above considered concepts could be further extended by introducing various metamaterial-based components [28,29].
Acknowledgments The Authors acknowledge Dr. G. Slepyan (Tel Aviv University) for his valuable advice on theoretical aspects and Yossi Kamir (Tel Aviv University) for 3D printing parts of the experimental setup. This work was supported, in part, by TAU Rector Grant and German-Israeli Foundation (GIF, Grant Number 2399), PAZY Foundation, and Kamin Project. This work has been partially supported by Government of the Russian Federation (Grant No. 074-U01). ‘Quasi-stationary field’ analysis of scatterers’ mechanical motion has been partially funded by the Russian Science Foundation (Grant No. 14-12-01227).
References [1] Stefanov A, Member S. Helicopter rotor-blade modulation of antenna radiation characteristics. IEEE Trans Antennas Propag 2001;49(5):688–696. [2] Greving, G, Biermann, W, Mundt, R, Numerical system analysis of rotating antennas and rotating scatterers applied to navigation and radar systems, 2012, pp. 416–19. [3] Somers GA, Bruno DM. Spectral characteristics of electromagnetic wave
scattering by rotating propellers and its effect on clutter discrimination. IEEE 1994:144–147. [4] Sengupta DL, Senior TBA. Electromagnetic interference to television reception caused. Proc IEEE 1979;67(8):1133–1142. [5] Courtial J, Robertson DA, Dholakia K, Allen L, Padgett MJ. Rotational frequency shift of a light beam. Phys Rev Lett 1998;81(22):4828–4830. [6] Mashhoon, B., Modification of the Doppler effect due to the helicity–rotation coupling, 2002. [7] Mashhoon B, Neutze R, Hannam M, Stedman GE. Observable frequency shifts via spin-rotation coupling. Phys Lett A 1998;249(3):161–166. [8] Martin, MJP, Lavery, PJ, Speirits, Fiona C, Barnett, Stephen M. Detection of a spinning object using light’s orbital angular momentum, Science (80-. )., vol. 341, no. August, 2013, pp. 537–40. [9] Mao Y, Gu Y, Qi D, Tang H. An exact frequency-domain solution of the sound radiated from the rotating dipole point source. Acoust Soc Am 2012;132. [10] Newburgh RG, Borgiotti GV. Short wire as a microwave analogue to molecular Raman scatterers. Appl Opt 1975;14(11):2727–2730. [11] Chao, C, Yang, J, Jen, W, Determining technology trends and forecasts of RFID by a historical review and bibliometric analysis from 1991 to 2005, vol. 27, 2007, pp. 268–79. [12] Ngai EWT, Moon KKL, Riggins FJ, Yi CY. RFID research: an academic literature review (1995–2005) and future research directions. 2008;112:510–520. [13] Bladel JVAN. Electromagnetic fields in the presence of rotating bodies. Proc IEEE 1976;64(3). [14] Richards MA. Fundamentals of radar signal processing. 2 edition. McGraw-Hill Education; 2014. [15] Victor HWChen C . Micro-doppler effect in radar: phenomenon, model, and simulation study, IEEE Transactions on Aerospace and Electronic Systems. [16] Censor D, Sheva B. Non-relativistic scattering by time-varying bodies and media. PIER 2004;48:249–278. [17] Zutter D De. Scattering by a rotating dielectric sphere. IEEE Trans Antennas Propag 1980;28(5):643–651. [18] Tanaka K. Scattering of electromagnetic waves by a rotating perfectly conducting cylinder with arbitrary cross section: point-matching method. IEEE Trans Antennas Propag 1980;28(6):796–803. [19] Lahaie IJ, Sengupta DL. Scattering of electromagnetic waves by a slowly rotating rectangular metal plate. IEEE Trans Antennas Propag 1979;27(1):40–46. [20] Wong JL, Reed IS. A model for the radar echo from a random collection of rotating dipole scatterers. IEEE Trans Aerosp Electron Syst 1967;3(2):171–178. [21] Newburgh RG, Borgiotti GV. Backscattered spectra from rotating and vibrating short wires and their relation to the identification problem. Phys Sci Res Pap Air Force Cambridge Res Labs, Hanscom AFB, MA; 1975. [22] Orfanidis, SJ. Electromagnetic waves & antennas, no. chapter 20, 2004, pp. 707–09. [23] 〈http://functions.wolfram.com/HypergeometricFunctions/Hypergeome tric2F2/02/〉. [24] Filonov DS, Shalin AS, Iorsh I, Belov PA, Ginzburg P. Controlling electromagnetic scattering with wire metamaterial resonators. J Opt Soc Am A 2016;33(10):1910. [25] Vitali Kozlov, Dmitry Filonov, Alexander S. Shalin, Ben Z. Steinberg, Pavel Ginzburg. Asymmetric backscattering from the hybrid magneto–electric meta particle. Appl Phys Lett 2016;109:203503. [26] Filonov D, Kramer Y, Kozlov V, Malomed BA, Ginzburg P. Resonant metaatoms with nonlinearities on demand. Appl Phys Lett 2016;109(11):111904. [27] Kapitanova PV, Ginzburg P, Rodríguez-Fortuño FJ, Filonov DS, Voroshilov PM, Belov PA, Poddubny AN, Kivshar YS, Wurtz GA, Zayats AV. Photonic spin hall effect in hyperbolic metamaterials for polarization-controlled routing of subwavelength modes. Nat Commun 2014;5:3226. [28] Shelby RA, Smith DR, Schultz S. Experimental verification of a negative index of refraction. Science 2001;292(5514):77–79. [29] Kivshar YS. Tunable and nonlinear metamaterials: toward functional metadevices. Adv Nat Sci Nanosci Nanotechnol 2013;5(1):13001.
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Journal of Quantitative Spectroscopy & Radiative Transfer journal homepage: www.elsevier.com/locate/jqsrt
Micro-Doppler frequency comb generation by rotating wire scatterers V. Kozlov a,1, D. Filonov a,b,n,1, Y. Yankelevich a, P. Ginzburg a,b a b
School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel ITMO University, St. Petersburg 197101, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 1 September 2016 Received in revised form 19 December 2016 Accepted 19 December 2016 Available online 21 December 2016
Electromagnetic scattering in accelerating reference frames inspires a variety of phenomena, requiring employment of general relativity for their description. While the ‘quasi-stationary field’ analysis could be applied to slowly-accelerating bodies as a first-order approximation, the scattering problem remains fundamentally nonlinear in boundary conditions, giving rise to multiple frequency generation (microDoppler shifts). Here a frequency comb, generated by an axially rotating subwavelength (cm-range) wires is analyzed theoretically and observed experimentally by illuminating the system with a 2 GHz carrier wave. Highly accurate ‘lock in’ detection scheme enables factorization of the carrier and observation of multiple peaks in a comb. The Hallen integral equation is employed for deriving the currents induced on the scatterer and a set of coordinate transformations, connecting laboratory and rotating frames, is applied in order to make analytical predictions of the spectral positions and amplitudes of the frequency comb peaks. Numeric simulations of the theoretic framework reveal the dependence of the microDoppler peaks on the wire's length and its axis of rotation. Unique spectral signature of micro-Doppler shifts could enable resolving internal structures of scatterers and mapping their accelerations in space, which is valuable for a variety of applications spanning from targets identification to stellar radiometry. & 2017 Elsevier Ltd. All rights reserved.
1. Introduction Laws of physics in accelerated frames of reference could be substantially different from those obtained in inertial ones. Coriolis force and Sagnac interference are, probably, the most known effects in mechanical and electromagnetic systems correspondingly. Electromagnetic scattering from moving bodies requires special formulation, traced back to investigation of rotating machinery and radar targets identification. Special attention has been paid to studies of non-relativistic rotating objects, owing both to the relative mathematical simplicity of their analysis and to the vast span of potential applications. Among various examples it is worth noting radio frequency (RF) detection of helicopter propellers [1–3], back action of spinning blades on wireless communication channels [4], rotational frequency shifts on beams carrying orbital angular momentum [5–7], and even rotational motion measurement of celestial bodies in astrophysics [8]. Moreover, acoustical noise could originate from revolving motors [9], rotating and vibrating wires were proposed as an analogue to the molecular Raman scattering in optics [10], while, in general, signatures of rotation emerge in many others multidisciplinary fields. In a n Corresponding author at: School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel. E-mail address: [email protected] (V. Kozlov). 1 V. Kozlov and D. Filonov contributed equally to this work.
http://dx.doi.org/10.1016/j.jqsrt.2016.12.029 0022-4073/& 2017 Elsevier Ltd. All rights reserved.
broader scenes, time-varying scatterers could find use in radio frequency identification (RFID) technologies [11,12]. Rigorous analysis of electromagnetic scattering from moving and, in particular, accelerating bodies requires employment of apparatus developed for the general theory of relativity [13]. Along with the transformation of all physical observables to the moving frame, accelerated motion causes substantial change in the general form of the governing differential equations and their associated boundary conditions. A relativistic treatment of electromagnetic fields in the presence of rotating bodies (infinite two dimensional cylinders) was performed by Bladel [13] and few others. Intuitively, each point on a rotating scatterer has a different instantaneous velocity, resulting in a collection of Doppler shifts. As electromagnetic fields at nearby points on a body are tightly connected by the nonlocal nature of Maxwell's equations, the scattering problem is far from being trivial. Furthermore, timevarying boundary conditions make the electromagnetic problem nonlinear, giving rise to generation of new frequencies alongside the initial carrier wave spectrum. While solution of the general scattering problem is quite involved, a set of simplifications could be made by assuming nonrelativistic motions and slow accelerations. First order approximations are widely used in theoretical analysis of radar targets identification, e.g. assuming that objects are at a complete rest during the interaction with an incident pulse [14]. Collecting static scattered signals, obtained from a few spatial positions of a target, enables relating theoretical data to
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experimental - radar measurements usually provide phase relations between adjacent pulses impinging on a target, or Doppler shifts in the reflected wave spectrum. Accelerating bodies, in contrast, produce a much richer spectral signature, called microDoppler shifts [15], that cannot always be recovered by applying the static approximation. The so-called ‘instantaneous rest frame’ technique assumes slow nonrelativistic motion as well, but relaxes few of the beforehand mentioned constraints and enables extracting more information regarding the accelerated motion. Neglecting second order relativistic contributions enables factoring out time dilation and length contraction effects and employing standard field continuity boundary conditions at each point of a scatterer, as if it moves with a linear (instantaneous) velocity. The benefit of rotating systems is in the existence of a relatively simple frame, where the entire scatterer is at rest. In this case the route to undertake is to transform sources to the rest frame of the rotating body, solve the scattering problem, and translate the results back to the laboratory frame. This approach is undertaken here and compared to the experimental data. It is worth underlining that the reported effects have no relativistic contributions. While much work was put into the description of various time varying scatterers (see for example [16–19]), the dipole approximation for scattering from objects that are smaller than the incident wavelength is both simple and useful. Here the effect of electromagnetic scattering from a rotating thin wire is studied both experimentally and analytically, including retardation effects that arise due to the small, yet finite, dimensions of the scatterers. To the best of our knowledge a theoretical treatment of scattering from a clutter of random rotating dipoles was partially addressed in [20]. In particular, a dipole, illuminated by a field of frequency ω was considered, predicting the scattered field to have additional components of ω ± 2θ ,̇ with θ ̇ being the angular frequency of rotation. In a later work a similar result was reached theoretically as well [21]. While the major Doppler shift indeed occurs at ±2θ ̇ frequencies, rotating bodies with nontrivial internal structure generate an entire frequency comb of micro-Doppler shifts. Detailed analysis, presented here, shows the emergence of the frequency comb in the scattered field, having components of ω ± mθ ,̇ where m is an integer. The developed tools also reveal the dependence of the micro Doppler comb on the length of the wire and axis of rotation. A nonrelativistic model, based on Hallen's integral equation [22] for calculating the current distribution on thin conducting wires, was employed together with the ‘instantaneous rest frame’ approximation, under appropriate coordinate
transformations between lab and scatterer's reference frames. The theoretical model was fully supported by the experiment undertaken at an anechoic chamber, where rotating shaped copper wires were illuminated with 2 GHz waves and the micro-Doppler comb was measured with highly accurate ‘lock in’ detection scheme.
2. Results 2.1. Theory Theoretical analysis of a spinning wire in free space will be presented first and then verified experimentally. The geometry of a spinning scatterer in free space and relevant measurement apparatus appear in Fig. 1. For the theoretical analysis, consider a short wire, rotating in the Z-Y plane at a constant angular frequency θ ̇ ≪ ω , where ω is the carrier frequency of the incident illumination. This approximation allows neglecting relativistic effects. In the lab reference frame (denoted by prime) an incident plane wave of the form E inc =E0 eikz′y^ ′ is propagating in free space, k is the k-vector of the wave, and the carrier's harmonic time dependence eiωt is suppressed. The rest frame of the wire (unprimed) agrees with the lab coordinates at time t = 0. The primary goal of the derivation is to calculate time-dependent forward scattering. In the considered geometrical arrangement the projection of the wire on the polarization direction changes harmonically in time (from maximal to almost zero), resulting in time-dependent scattering. The coordinate transformation between the lab and the rotating frame is carried out by applying the following relation:
⎞ ⎛ x′ ⎞ 0 0 ⎛ x⎞ ⎛ 1 ⎜ y ⎟ = ⎜ 0 Cos (θṫ ) − Sin (θṫ )⎟ ⎜ y′⎟. ⎟⎜ ⎟ ⎜ ⎟ ⎜⎜ ⎝ z ⎠ ⎝ 0 Sin (θṫ ) Cos (θṫ ) ⎟⎠ ⎝ z′ ⎠
(1)
In the wire's rest frame the incident field along the length of ̇ the wire becomes Ezinc (z, t ) = E0 Sin (θṫ ) eikzCos (θt )(the wire is parallel to the z-axis in its frame of reference, hence only z-component of the field is relevant for the interaction). The field in the scatterer's rest frame may be decomposed into a sum of products of Bessel functions, dependent on space-time harmonics:
Fig. 1. Experimental setup of transmission in an anechoic chamber. A scatterer is mounted on a motor's shaft and placed between transmitting and receiving horns, located in the far field of the scatterer.
V. Kozlov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 190 (2017) 7–12 ∞
Ezinc (z, t ) = E0 Sin (θṫ )
Jn ( kz ) ein
∑
( θṫ − π2 ).
(2)
n =−∞
Scattering problems from thin perfectly conducting wires could be solved by applying Hallen's integral formulation. This approach was chosen over other methods, as it allows approximating results with analytical and semi-analytical expressions. In Hallen's formulation scatterers are virtually replaced by line currents (unknowns) and the total electromagnetic field outside is given as a superposition of the excitation and the convolution between the current and the Green's function related kernel. Boundary conditions of vanishing tangential components of the electrical field on a perfect electric conductor (physical boundaries of the initial wire) enable formulating the scattering problem in the form of an integral relation. Hallen's integral equation for current distribution along the wire ( I ( z, t )) under the assumption of a lossless, perfectly electrically conducting wire is approximated by [22]:
iη 2π
∫z
z2
1
e−ik (
)
2
( z − z′)
+ a2
I ( z′,t ) dz′=C1eikz +C2 e−ikz +i
∫z
z2
1
e−ik z − z′ Ezinc ( z′, t ) dz′,
(3)
where C1,2 are to be determined from the boundary conditions of vanishing current at the edges of the wire, located at z1,2 , a is the radius of the wire and η is the free space impedance. Noticing that the integrand on the left hand side of Eq. (3) has a sharp maximum around z = z′ enables approximating it with [22]:
iη 2π
∫z
z2
1
2 2 e−ik ( z − z′) + a
( z − z′)2 + a2
I ( z′,t ) dz′=ZĨ ( z, t ), (4)
iη (z − z ) e−ika
where Z ̃ ≈ 2 2π1a is a zero-order approximation that could be further improved by using the King's Three-Term approximation [22]. Solving Eq. (3) using approximation of Eq. (4) enables determining the current along the wire. Assuming an axially rotating wire z1,2= ± h where h is half the wire's length:
4E I ( kz, θṫ )= 0 ̃ Zk
⎛
∞
∑ Tns ( θṫ ) ⎜⎜ Snc ( kz)−Snc ( kh) ⎝
n= 0
Cos ( kz ) ⎞ ⎟⎟+iT nc ( θṫ ) Cos ( kh) ⎠
Tns ( θṫ ) =
( 2 − δ n,0 )
R
)
2n + 1) (kz ) Sin ( kz ) − F I( (kz ) Cos ( kz ) ,
( { m + 0.5,
(8)
a unit vector in the z direction, μ is the free space permeability and the dot represents a scalar product. Eq. (8) may be solved numerically, or evaluated analytically by adopting several approximations. Here, leading terms in 1 are considered along with the d π
short wire approximation ( kh≪ 2 ) for calculating induced currents, which enables identifying well-defined spatial parities in ̇ by making Tailor expansions of Eq. (5). Here even spatial θ series contributions could be approximated as constants, while the odd ones may be replaced with functions that are linear in their argument. The resulting short wire current then takes the form:
Is ( kz, θṫ ) ≈
4E 0 ̃ Zk
⎞2n +2 ⎛ kh ⎞2n +2⎞ ⎟⎟+iα cnT nc ( θṫ ) −⎜ ⎟ ⎝ 2⎠ ⎠ ⎝⎝ 2 ⎠ ⎛⎛
∞
∑ α nsTns ( θṫ ) ⎜⎜ ⎜ kz ⎟
n= 0
⎛ ⎛ kz ⎞2n +3 z ⎛ kh ⎞2n +3⎞ ⎟⎟, ⎜⎜ ⎜ ⎟ − ⎜ ⎟ h⎝ 2 ⎠ ⎠ ⎝⎝ 2 ⎠
(9)
where the following auxiliary coefficients are defined:
( 2n + )( 2n + 1) 1 2
( 2n + 2)( 4n + 1)
⎛⎧ ⎞ ⎫ 3 , 2n + 2⎬ , { 2n + 3, 4n + 2}, 0⎟ , 2 F 2⎜ ⎨ 2n + ⎭ ⎝⎩ ⎠ 2
( 2n + )( 2n + 2) F ⎛⎜⎜ ⎧⎨ 2n + 5 , 2n + 3⎫⎬, 3 2
( 2n + 3)( 4n + 3) 2
2
⎜⎩ ⎝
2
⎭
⎞ ⎟ 4n + 3}, 0⎟ . ⎟ ⎠
(10)
It can be readily checked that for small values of kh Eqs. (9) and (5) coincide. By using the above mentioned approximations, the scattered field is finally evaluated in the lab reference frame by solving:
(6)
while the superscript ‘s’ and ‘c’ denote even (‘c’) and odd (‘s’) ) are the real and imaginary functions of time or space and F R( m /I components of the hypergeometric function [23], satisfying:
FR( m) + iFI( m) = 2 F 2
⎞ e−ikR ( z, d, t ) ⎛ 2 zd Sin ( 2θṫ ) ⎟ I (z, t ) dz, ⎜ d Sin ( θṫ )− 3⎝ ⎠ 2 4π R ( z , d , t )
( )
{ 2n + 4,
)
2n + 1)
∫−h
⎞
h
∫−h I ( z, t ) ( G⃡FF ( r; z) z^) dz⎟⎠
where R ( z, d, t )= d2−2zdCos θṫ + z2 is the distance connecting a
Sin (θṫ ) Cos (2nθṫ ),
⎛ kz ⎞2n + 1 ( 2n) 2n Snc ( kz ) = ⎜ ⎟ FR (kz ) Sin ( kz ) − F I( ) (kz ) Cos ( kz ) , ⎝ 2⎠ ⎛ kz ⎞2n + 2 Sns ( kz ) = ⎜ ⎟ ⎝ 2⎠
h
= − iωμ
2n α nc = 2F R( )( 0) − 4
( 2n + 1) ! ( −1)n Sin (θṫ ) Cos ((2n + 1) θṫ ), T nc ( θṫ ) = 2 ( 2n + 2) !
( (F(
^ ⎛ E yLab ′ ( d , t )= − iωμy′ ∙ ⎜ ⎝
(5)
where the following auxiliary functions are defined as:
( − 1)
layout:
2n α ns = 2F R( )( 0) − 4
⎛ Sin ( kz ) ⎞ ⎟, ⎜⎜ Sns ( kz )−Sns ( kh) Sin ( kh) ⎟⎠ ⎝
n
of the current. Eqs. (5) and (6) show that the current consists of ̇ both even and odd time harmonic components of θ that produce additional scattered field frequencies in the far field. However, it will be shown that for an axially rotating short wire only the even harmonics remain in the far field. Next, the scattered field in the rest frame of the wire is derived and transformed back into the lab frame of reference using Eq. (1), evaluating it in the lab frame at a point z′= − d , y′=0, x′=0 (where the detector is located) and taking the y^′ component alone, as it is the case of the experimental
point on the wire and the detector's location at time t , G⃡FF ( r;z )is the far field 3D Green's dyadic (3 3 matrix) in the rest frame, z^ is
2
z − z ′ + a2
9
)
m + 1}, { m + 2, 2m + 1}, 2ikz .
E yLab ′ ( d, t ) ≈
−iωμe−ikd 4π d
∫−h
∞
⎛
=E0 k2ah2
e−ikd d
h
Sin ( θṫ ) Is ( z, t ) dz, ⎞2n
∑ Cn ⎜ kh ⎟ n= 0
⎝ 2⎠
Sin2 ( θṫ ) Cos ( 2nθṫ ),
(11)
where
(7)
The result for the current in Eq. (5) is equivalent to the closed form derived in [22], but has the advantage of avoiding the numeric singularity (when both nominator and denominator tend to 0) at θṫ = nπ as well as clearly revealing the frequency components
Cn =
( −1) n( 2 − δ n,0 ) n+1 α s. ( 2n + 1) ! 2n + 3 n
(12)
The analytical expression in Eq. (11) predicts the existence of even harmonics in the scattered field at an on-axis point away
10
V. Kozlov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 190 (2017) 7–12
from the short wire, provided that the wire is rotating around its center (axial rotation). The odd harmonics vanish due to parity considerations involving odd and even functions in the current along with symmetric integration boundaries of the integral. These harmonics disappear even if the full current (long wire) described in Eq. (5) is used. In fact, the odd harmonics will vanish even if the only approximation made is R ( z, d, t )≈d − zCos θṫ
( )
(quadratic terms in wire's internal geometry are neglected), but the detailed proof of this statement is omitted in this paper in order to not obscure the main results. It should be noted that if no approximations of the Green's function are made, the integrand in Eq. (8) has no parity and thus negligible odd peaks are expected in the far field. Taking the leading term (n ¼0) in Eq. (11) recovers the dipole field pattern, showing the dominance of the 2θ ̇ peak. Using the dipole term, an analytical approximation for the polarizability 4
of small perfectly conducting wires can be written as αzz = 3 πε0 ah2. The approximation made in Eq. (4) leads to an almost symmetric current distribution around the center of the axially rotating wire. This approximation is good for short wires, but becomes less accurate for longer ones. It is thus preferable to solve Eq. (3) numerically for more accurate results of the current. Numeric solutions also allow considering wires rotating about a different axis than their center. This can be done by changing the integration boundaries z1,2 with non-symmetric boundaries, giving rise to odd peaks in the far field. A final important conclusion from Eq. (11) is that it shows the dependence of the ratio of peak amplitudes on the wire geometry (e.g. its length). This insight shows that microDoppler peaks could be used to extract information about the internal geometry of rotating scatterers. The theoretical framework developed above could be extended to include arbitrary rotation axis and observation points in a straightforward fashion. The same approach could be employed for considering wires of an arbitrary shape, which is not necessarily a straight line.
2.2. Experimental results Micro-Doppler shifts in the scattered spectra of thin wires will be investigated experimentally and compared with the developed theory. The photograph of the experimental setup appears in Fig. 1. Spinning wire scatterers were mounted on a motor's shaft, stabilized with a 3D-printed plastic rack. The motor is controlled by a DC current, enabling rotational speeds of up to hundreds of hertz. In particular, a 5 cm long wire (photograph of the sample appears at the insert to Fig. 1) was rotated in the Z-Y plane at a ̇ 2π × 28Hz . The whole structure was constant angular frequency θ = placed between two horn antennas, separated by a 2 m distance, ensuring that both the transmitting and receiving horns are at the far-field region of the scatterer. A Vector Network Analyzer (VNA) was used as a source of a 2 GHz continuous wave, feeding the transmitting antenna with the polarization set in the Y-direction. A receiving antenna was placed opposite the transmitter with the same polarization. The output of the receiver was amplified and down-converted to the baseband by mixing it with the same 2 GHz carrier, as in the incident field. The output of the mixer was passed through a low pass filter to remove residual high frequencies and directed to a lock-in amplifier where the low-frequency spectral shifts of the scattered field were recorded. The results are shown in Fig. 2, showing normalized lock-in readouts as a function of baseband frequency. The amplitude of the second peak was used to scale the theoretic values. This peak corresponds to ω ± 2θ ̇ in the original signal and appears as 2θ ̇ in the lock-in detection, since only the positive part of the spectrum is presented. Fig. 2(a) shows theoretical predictions, obtained with Eqs. (3) and (8), while the experimental data appears on Fig. 2(b). Three even and three odd peaks above the noise level were observed in the experiment. The theoretic values were extracted from the simulation of the axially rotating wire (offset is 0mm between the center of the wire and the rotation axis), showing
Fig. 2. Normalized forward scattering spectrum (micro-Doppler comb, baseband spectrum) of a 5 cm long wire (a) Theoretical predictions evaluated numerically (with Eqs. (3) and (8)). Red stars indicate results obtained for 0mm offset of the rotation axis from the center of the wire, showing only even peaks. Blue circles indicate results obtained for the same wire length, but rotating around an axis at a distance of 0.6mm from its center, showing emergence of odd peaks. (b) Experimental results, obtained for the ̇ 2π × 28Hz . Odd peaks emerge due to inherent mechanical instabilities in the practical realization. (For centered 5 cm length wire, rotating at an angular frequency θ = interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
V. Kozlov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 190 (2017) 7–12
11
Fig. 3. (a), (c) Normalized amplitude of the odd peaks ((a) – first, (c) – third) as a function of the offset of the rotation axis from the center of the wire. Normalization is performed with respect to both the amplitude of the second peak for each offset value and the amplitude value at 22 mm offset is set as unity. Black line represents theoretical predictions evaluated numerically (Eqs. (3) and (8)), while the red dots are experimental data. (b) Normalized amplitude of the second peak as a function of wire's half-length. Peak at h ¼25 mm is chosen for the normalization and brought to unity. Black line represents theoretical predictions evaluated numerically (Eqs. (3) and (8)), while the red dots are experimental data. Offset was taken to be zero in this case. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
only even peaks in agreement with Eq. (11). While odd peaks are not predicted in the case of perfectly-centered rotation, they are observed in the experimental data. There are several reasons for their appearance. First, the receiving horn has finite-size aperture, collecting off-axis scattering that could be shown to include contributions of odd harmonics. Second, and more importantly, odd peaks appear in the case of very small offsets of the rotation axis from the center of the wire. The offset could occur due to mechanical precision at the connection point between the wire and the shaft, as well as the precession of the later. A phenomenological ‘offset’ parameter (displacement of the rotation axis from the geometrical center of the wire) was introduced for fitting odd peaks in the experimental data (Fig. 2(b)). A value as small as 0.6mm (about 1% of the wire length) was found to be sufficient for fitting the odd peaks in the comb (Fig. 2(a)). For the numerical simulation the integration boundaries in Eqs. (3) and (8) were offset by 0.6mm away from the center of the wire, giving rise to the odd peaks. It is apparent that the theory (up to inherent limitations) agrees well with the location (equidistant discrete comb was generated) and amplitudes of the rotational Doppler peaks in the scattered field spectra. The agreement in amplitudes is good for the first 4 peaks and then reduces owing to fluctuations in the rotation frequency, which occurred due to the long measurement time (up to a few minutes for each data point was needed in order to obtain higher signal to noise ratios) and motor shaft precession. In order to underline the impact of the ‘offset’ parameter on the micro-Doppler comb, a set of additional experiments was performed. In particular, the center of the wire was displaced from the motor holder on purpose. Fig. 3(a) and (b) show both experimental and theoretical results for emergent odd harmonics (first and third correspondingly). Amplitudes of the harmonics were normalized to the amplitude of the second peak for each offset value and presented as a function of the later. Additional normalization was done in respect to the amplitude value at 22mm offset, which was
chosen to serve as a reference (value of unity on both graphs). Experimental results and the theoretical predictions agree well with each other, showing the sensitivity of the odd peaks amplitude to offsetting the axis of rotation from the center of the wire. While the theory predicts vanishing amplitude of the third peak when the offset approaches zero, experimental data shows a finite value. This behavior is attributed to the overall precision of the length measurements, finite size of the bracing, and precession of the shaft - all of which contribute to the emergence of the odd peaks as well. Relying on the above considerations, the phenomenological offset parameter was introduced in order to fit the experimental data in Fig. 2(b). It is noted that this small offset (0.6mm) is very reasonable to have occurred in the experiment. In order to validate theory and experiment furthermore, wires of various lengths were rotated and the amplitude of the second peak was plotted as a function of wire length. The values were normalized to the 2nd peak's amplitude of a 5 cm wire (h¼25 mm). The theoretical predictions are well supported with experimental results, as can be seen on Fig. 3(b). It is worth noting, that the second peak, having the major contribution to the micro-Doppler comb in the case of the short wires, could be measured with a better signal-to-nose ratio in comparison with other peaks in the comb. This explains the smaller error bars than those, appearing on panels (a) and (c). It is worth underlining the excessive noise factor in the peak amplitudes for wires of 25 and 55 mm length (error bars on Fig. 3(b)). This effect is attributed to an internal mechanical resonance of the motor and the holder rack – for certain values of moment of inertia (related to the wire's length) the system is forced to a resonance, affecting the precision of the measurements. 3. Outlook and conclusion In conclusion, it was shown both theoretically and experimentally that electromagnetic waves, scattered from rotating
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wires, have additional frequency components of ω ± mθ ,̇ forming a micro-Doppler frequency comb (m is an integer). A theoretical model for this phenomenon was derived, predicting the appearance of those frequency components and closely predicting their relative magnitudes. In particular, it was shown that in the case of axially rotating wires only even harmonics at the micro-Doppler comb appear. While the emergence of odd harmonics is indeed not expected in ideal systems, practical realizations do exhibit their emergence. The theoretical model shows that odd peaks are very sensitive to small offsets of the axis of rotation away from the center of the wire, enabling the introduction of a phenomenological asymmetry coefficient, which provides a reliable fit to experimental reality. The model also successfully predicts the dependence of the micro Doppler shifts of rotating wires on their geometry (length) and axis of rotation. These results expand previously proposed theories, where only two Doppler shifts of ω ± 2θ ̇ were expected in the spectra of dipole-like scatterers and suggest that it might be possible to use such spectral signatures for resolution of the internal structure of observed scatterers and their axis of rotation. Furthermore, RF components could serve as emulation tools for observation of complex phenomena [24–27]. Externally induced mechanical motion on vibrating dipoles could replicate Raman scattering scenarios [10]. The above considered concepts could be further extended by introducing various metamaterial-based components [28,29].
Acknowledgments The Authors acknowledge Dr. G. Slepyan (Tel Aviv University) for his valuable advice on theoretical aspects and Yossi Kamir (Tel Aviv University) for 3D printing parts of the experimental setup. This work was supported, in part, by TAU Rector Grant and German-Israeli Foundation (GIF, Grant Number 2399), PAZY Foundation, and Kamin Project. This work has been partially supported by Government of the Russian Federation (Grant No. 074-U01). ‘Quasi-stationary field’ analysis of scatterers’ mechanical motion has been partially funded by the Russian Science Foundation (Grant No. 14-12-01227).
References [1] Stefanov A, Member S. Helicopter rotor-blade modulation of antenna radiation characteristics. IEEE Trans Antennas Propag 2001;49(5):688–696. [2] Greving, G, Biermann, W, Mundt, R, Numerical system analysis of rotating antennas and rotating scatterers applied to navigation and radar systems, 2012, pp. 416–19. [3] Somers GA, Bruno DM. Spectral characteristics of electromagnetic wave
scattering by rotating propellers and its effect on clutter discrimination. IEEE 1994:144–147. [4] Sengupta DL, Senior TBA. Electromagnetic interference to television reception caused. Proc IEEE 1979;67(8):1133–1142. [5] Courtial J, Robertson DA, Dholakia K, Allen L, Padgett MJ. Rotational frequency shift of a light beam. Phys Rev Lett 1998;81(22):4828–4830. [6] Mashhoon, B., Modification of the Doppler effect due to the helicity–rotation coupling, 2002. [7] Mashhoon B, Neutze R, Hannam M, Stedman GE. Observable frequency shifts via spin-rotation coupling. Phys Lett A 1998;249(3):161–166. [8] Martin, MJP, Lavery, PJ, Speirits, Fiona C, Barnett, Stephen M. Detection of a spinning object using light’s orbital angular momentum, Science (80-. )., vol. 341, no. August, 2013, pp. 537–40. [9] Mao Y, Gu Y, Qi D, Tang H. An exact frequency-domain solution of the sound radiated from the rotating dipole point source. Acoust Soc Am 2012;132. [10] Newburgh RG, Borgiotti GV. Short wire as a microwave analogue to molecular Raman scatterers. Appl Opt 1975;14(11):2727–2730. [11] Chao, C, Yang, J, Jen, W, Determining technology trends and forecasts of RFID by a historical review and bibliometric analysis from 1991 to 2005, vol. 27, 2007, pp. 268–79. [12] Ngai EWT, Moon KKL, Riggins FJ, Yi CY. RFID research: an academic literature review (1995–2005) and future research directions. 2008;112:510–520. [13] Bladel JVAN. Electromagnetic fields in the presence of rotating bodies. Proc IEEE 1976;64(3). [14] Richards MA. Fundamentals of radar signal processing. 2 edition. McGraw-Hill Education; 2014. [15] Victor HWChen C . Micro-doppler effect in radar: phenomenon, model, and simulation study, IEEE Transactions on Aerospace and Electronic Systems. [16] Censor D, Sheva B. Non-relativistic scattering by time-varying bodies and media. PIER 2004;48:249–278. [17] Zutter D De. Scattering by a rotating dielectric sphere. IEEE Trans Antennas Propag 1980;28(5):643–651. [18] Tanaka K. Scattering of electromagnetic waves by a rotating perfectly conducting cylinder with arbitrary cross section: point-matching method. IEEE Trans Antennas Propag 1980;28(6):796–803. [19] Lahaie IJ, Sengupta DL. Scattering of electromagnetic waves by a slowly rotating rectangular metal plate. IEEE Trans Antennas Propag 1979;27(1):40–46. [20] Wong JL, Reed IS. A model for the radar echo from a random collection of rotating dipole scatterers. IEEE Trans Aerosp Electron Syst 1967;3(2):171–178. [21] Newburgh RG, Borgiotti GV. Backscattered spectra from rotating and vibrating short wires and their relation to the identification problem. Phys Sci Res Pap Air Force Cambridge Res Labs, Hanscom AFB, MA; 1975. [22] Orfanidis, SJ. Electromagnetic waves & antennas, no. chapter 20, 2004, pp. 707–09. [23] 〈http://functions.wolfram.com/HypergeometricFunctions/Hypergeome tric2F2/02/〉. [24] Filonov DS, Shalin AS, Iorsh I, Belov PA, Ginzburg P. Controlling electromagnetic scattering with wire metamaterial resonators. J Opt Soc Am A 2016;33(10):1910. [25] Vitali Kozlov, Dmitry Filonov, Alexander S. Shalin, Ben Z. Steinberg, Pavel Ginzburg. Asymmetric backscattering from the hybrid magneto–electric meta particle. Appl Phys Lett 2016;109:203503. [26] Filonov D, Kramer Y, Kozlov V, Malomed BA, Ginzburg P. Resonant metaatoms with nonlinearities on demand. Appl Phys Lett 2016;109(11):111904. [27] Kapitanova PV, Ginzburg P, Rodríguez-Fortuño FJ, Filonov DS, Voroshilov PM, Belov PA, Poddubny AN, Kivshar YS, Wurtz GA, Zayats AV. Photonic spin hall effect in hyperbolic metamaterials for polarization-controlled routing of subwavelength modes. Nat Commun 2014;5:3226. [28] Shelby RA, Smith DR, Schultz S. Experimental verification of a negative index of refraction. Science 2001;292(5514):77–79. [29] Kivshar YS. Tunable and nonlinear metamaterials: toward functional metadevices. Adv Nat Sci Nanosci Nanotechnol 2013;5(1):13001.