Simultaneous measurement of spin and precession based on light’s orbital angular momentum

Ruoyu Tang; Ruoyu Tang; Xiuqian Li; Song Qiu; Song Qiu; Xiangyang Zhu; Xiangyang Zhu; Tong Liu; Tong Liu; Zhengliang Liu; Zhengliang Liu; Xiaocen Chen; Yuan Ren; Yuan Ren

doi:10.1364/OE.503038

1. Introduction

Since Allen et al. [1] proposed a vortex beam(VB) carrying orbital angular momentum(OAM) in 1992, its potential application in rotating object detection has been explored. The vortex beam is a structured light with a helical phase factor $\textrm{exp} (il\varphi )$, where l is an integer expressing the azimuthal quantum number of the vortex beam. Each photon in the vortex beam carries the OAM of $l\hbar $ [2,3], in which the constant $\hbar $ is the reduced Planck constant. In 2013, Lavery et al. [4] first used the vortex beam to detect the spinning speed of an object. Their study shows that after the vortex beam interacting with the rotating object, the collected echo beam gets a frequency shift, and this phenomenon is also called the rotational Doppler effect(RDE). Since the RDE was proposed, a lot of research around it, from the non-coaxial rotational Doppler effect [5–7], to the detection of geometric features of rotating object [8], the micro-displacement measurement [9], has been conducted. In the compound motion detection [10–14], the vortex beam and its rotational Doppler effect also play an effective role. However, previous studies of compound motion detection are almost all about the combination of linear motion and rotation. For an actual rotating object, due to the external disturbance, the spin axis may rotate around another axis, which is called precession, and the “another axis” is the precession axis. The precession cannot exist alone without the spin, so how to discriminate the precession and the spin parameters during the measurement is a compound motion detection problem to be solved.

In the traditional planar electromagnetic applications, micro-Doppler effect has been an important tool for detecting micromotion parameters of targets [15–17]. Similar to the principle of the Doppler effect, the micromotion of the target modulates the electromagnetic echo of the radar, generating extra sidebands near the theoretical Doppler frequency. This modulation phenomenon is called the micro-Doppler effect, and the frequency generated by these modulations is called the micro-Doppler frequency. In 2003, Chen et al. [18]. used radar to detect pedestrians and conducted in-depth analysis and discussion of radar echo signals, demonstrating the great application potential of micro-Doppler effect in the field of radar signal processing. However, the micro-Doppler effect of the traditional radar can only be observed when the micromotion is along the propagation of the planar wave. While the motion of the object is completely in the plane perpendicular to the propagation direction of the planar wave, such as spin and precession, the traditional micro-Doppler effect will lose the ability of detection [19].

By combining the rotational Doppler effect of the VB and the micro-Doppler effect of the planar electromagnetic wave, we are inspired to use the rotational micro-Doppler effect to detect the micromotion that is completely perpendicular to the light’s propagation direction. Similarly, the rotational micro-Doppler effect can be defined as the modulation to the rotational Doppler effect by the micromotion with the VB as the detection beam [20]. Under this concept, the precession can be considered as the micromotion caused by the spin and it will modulate the rotational Doppler frequency shift, generating the rotational micro-Doppler frequency. This provides a new feasible approach to realize the independent measurement of the spin and precession. In this work, we analyze the characteristic of the rotational micro-Doppler effect introduced by the precession based on the previous rotation detection with the VB and successfully extract the related parameters, realizing the simultaneous and independent measurement of spin and precession.

2. Materials and methods

2.1 Principle of RDE

Suppose that the laser transmitter and receiver are at the same place where the coordinate z = 0. When a structured optical beam with a transverse phase distribution illuminates a moving particle and reflects to the receiver, the incident optical wave can be written as [21],

(1)$${E_\textrm{r}}(t) = {E_0}\textrm{exp} [{i2kz(t)} ]\textrm{exp} \{ i\Phi [x(t),y(t)]\} \textrm{exp} (i{\omega _0}t)$$

where E₀ denotes the complex amplitude at the location of transmitter, (x, y, z) is the three components of the moving particle in the Cartesian coordinates, k = 2π/λ is the wavevector, ω₀= 2πf₀ is the angular frequency of the optical wave, $\Phi $ defines the function of transverse phase distribution. It is well known that the frequency of a wave can be expressed in terms of the rate of change of phase. In the above equation, the phase of the received wave is

(2)$$\phi = 2kz(t) + \Phi [x(t),y(t)] + {\omega _0}t$$

The frequency, i.e., the rate of phase changing, can be calculated by the differential coefficient of the phase,

(3)$${f_1} = \frac{1}{{2\pi }}\left( {{\omega_0} + 2k\frac{{\textrm{d}z}}{{\textrm{d}t}} + \frac{{\textrm{d}\Phi }}{{\textrm{d}x}} \times \frac{{\textrm{d}x}}{{\textrm{d}t}} + \frac{{\textrm{d}\Phi }}{{\textrm{d}y}} \times \frac{{\textrm{d}y}}{{\textrm{d}t}}} \right)$$

Obviously, the frequency difference between the received and emitted optical waves is,

(4)$$\begin{array}{c} \Delta f = {f_1} - {f_0} = \frac{2}{\lambda }\frac{{\textrm{d}z}}{{\textrm{d}t}} + \frac{1}{{2\pi }}\left( {\frac{{\textrm{d}\Phi }}{{\textrm{d}x}} \times \frac{{\textrm{d}x}}{{\textrm{d}t}} + \frac{{\textrm{d}\Phi }}{{\textrm{d}y}} \times \frac{{\textrm{d}y}}{{\textrm{d}t}}} \right)\\ = \frac{{2{v_z}}}{\lambda } + \frac{1}{{2\pi }}\boldsymbol{\nabla }\Phi \cdot {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over v } _ \bot } \end{array}$$

where $\boldsymbol{\nabla } = ({\partial /\partial x,\partial /\partial y,\partial /\partial z} )$ is the Hamilton operator, ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over v } _ \bot } = ({{v_x},{v_y}} )$ is the transverse velocity of the particle. The first term of the above equation is related to the component of velocity ${v_z}$ with the same direction as the beam propagation and the wave length, i.e., the linear Doppler shift [22], and the second term defines another Doppler frequency shift, which is determined by the transverse components of velocity ${v_x}$ and ${v_y}$, and the phase distribution function $\Phi $. For a plane wave, the transverse phase distribution $\Phi $ of which is constant and the gradient of $\Phi $ equals 0, so the plane wave can only have the linear Doppler shift. Therefore, only using a structured optical beam with transverse phase distribution can we obtain the velocity perpendicular to the propagation direction.

Since Allen et al. established the Laguerre-Gaussian(LG) mode with helical phase-fronts carrying OAM in 1992 [1], there have been many researches about using LG mode to detect moving objects and obtain the movement parameters [4,23,24]. The helical phase-fronts of the LG mode give this beam the ability to be sensitive to rotational motion. In polar coordinates, The LG beam can be expressed by the following equation:

(5)$$E(\rho ,\theta ,z,t) = {E_0}\textrm{exp} ({il\theta } )\textrm{exp} ({ikz} )\textrm{exp} (i{\omega _0}t)$$

where E₀ is the amplitude of the electric field, l is the topological charge number. The light filed and phase distribution of LG mode is shown in Fig. 1. From Eq. (4), the transverse phase distribution of the LG mode can be written as $\Phi ({\rho ,\theta } )= l\theta $.

Fig. 1. The light field and phase distribution of single LG mode and superposed LG modes. (a)∼(b) The distribution of light intensity of LG mode with topological charge $l = 10$ and $l ={\pm} 10$; (c)∼(d) The phase distribution of LG mode with topological charge $l = 10$ and $l ={\pm} 10$.(e) A circularly moving particle with angular speed $\Omega $ in the cross-section of the LG beam.

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Accordingly, as shown in Fig. 1(e), if we illuminate a circularly moving particle with angular speed $\Omega $ using the LG mode, which takes the beam propagation axis as the spin axis, the second term of Eq. (4) can be expressed as,

(6)$$\Delta {f_{\textrm{spin}}} = \frac{1}{{2\pi }}\boldsymbol{\nabla }\Phi \cdot {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over v } _ \bot } = \frac{{l\Omega }}{{2\pi }}$$

which is the basic form of RDE [4]. When an object rotates with precession, the transverse components of velocity will be determined not only by the spinning speed, but also by the precession speed, which introduces some subtle changes to the rotational Doppler effect expression Eq. (6), i.e., the rotational micro-Doppler effect.

2.2 Rotational micro-Doppler characteristics of precession

As shown in Fig. 2(b), an object is illuminated by an LG mode beam, which rotates with precession simultaneously. Suppose the LG mode beam illuminates the object along the precession axis direction. In order to make the rotational micro-Doppler characteristics clear, it’s necessary to analyze the transverse velocity of the scattered particles on the object.

Fig. 2. Mechanism diagram of using LG beam to detect a rotating object with precession. (a) The front view of the object along the propagation axis. (b) Overall diagram of detection.

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Figure 2(a) is the front view of the object along the propagation axis. Point O is the spin center of the object and M is the precession center. We establish the coordinate system with M as the origin. Actually, affected by the scattering characteristics of the surface of the object and the distribution of the light field, only the scattered light of a small area of the light field can be gathered to the receiver. We refer to this area as the strongly scattering area. Suppose that there is a strongly scattering particle A within the light filed, and all of the scattered light gathered by the receiver is from A. From Eq. (4), the transverse velocity of A determines the rotational Doppler shift. At one moment, spin center O is moving to the place shown in the Fig. 2(a). The trajectories of spin center O and strongly scattering particle A are indicated with dotted lines. As shown in Fig. 2(a), the angle of $\overrightarrow {MA} $ is ${\theta _A}$ and the angle of $\overrightarrow {MO} $ is $\varphi $. Setting the precession speed as ${\Omega _1}$, the spin speed as ${\Omega _2}$, the precession radius as d and the radius of the light field as ${\rho _L}$, then we can obtain that $\varphi = {\Omega _1}t + {\varphi _0}$,where ${\varphi _0}$ is the angle of $\overrightarrow {MO} $ at the moment $t = 0$. It can be derived from the relationship among the above variables that,

(7)$$\begin{aligned} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {MA} } &= ({{\rho_L}\cos {\theta_A},{\rho_L}\sin {\theta_A}} )\\ \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {MO} } &= ({d\cos \varphi ,d\sin \varphi } )\\ \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {OA} } &= \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {MA} } - \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {MO} } \\ &= ({{\rho_L}\cos {\theta_A} - d\cos \varphi ,{\rho_L}\sin {\theta_A} - d\sin \varphi } )\end{aligned}$$

The transverse velocity of A consists of two velocities superimposed by vectors, one is the linear velocity ${\overrightarrow v _{{\Omega _2}}}$ due to the spin, and the other is the translational velocity ${\overrightarrow v _{{\Omega _1}}}$ of the rigid object due to the precession. Based on Eq. (7), the transverse velocity of A can be given by,

(8)$$\begin{aligned} {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over v } _A} &= {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over v } _{{\Omega _1}}} + {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over v } _{{\Omega _2}}}\\ &= ({ - {\Omega _1}d\sin \varphi - {\Omega _2}r\sin \psi ,{\Omega _1}d\cos \varphi + {\Omega _2}r\cos \psi } )\end{aligned}$$

where r is the magnitude of $\overrightarrow {OA}$, $\psi $ is the angle of $\overrightarrow {OA}$. Putting the transverse velocity ${\overrightarrow v _A}$ and the phase distribution function $\Phi $ of LG mode into Eq. (6), the expression of RDE can be given by,

(9)$$\begin{aligned} \Delta f &= \frac{1}{{2\pi }}\nabla \Phi \cdot {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over v } _A}\\ &= \frac{l}{{2\pi {\rho _L}}}[{\sin {\theta_A}({\Omega _1}d\sin \varphi + {\Omega _2}r\sin \psi ) + \cos {\theta_A}({\Omega _1}d\cos \varphi + {\Omega _2}r\cos \psi )} ]\\ &= \frac{{ld({{\Omega _1} - {\Omega _2}} )}}{{2\pi {\rho _L}}}\cos ({\Omega _1}t + {\varphi _0} - {\theta _A}) + \frac{{l{\Omega _2}}}{{2\pi }} \end{aligned}$$

It worth noting that, the frequency shift of RDE varies over time, this is exactly the micro-Doppler effect characteristic introduced by precession. It can be derived from Eq. (9) that the frequency shift has the same period as the precession’s, and the middle value of which is the rotational Doppler shift caused by the spinning speed ${\Omega _2}$. That inspired us. If we can extract the period and the middle value from the time-frequency spectrum, the precession speed and spinning speed can be calculated respectively.

It should be clarified that, limited by the cutoff frequency, almost all of the receivers are not sensitive to the frequency of light, so we need to convert the frequency shift of light into a modulation of the light intensity using interferometry. Fortunately, through the self-interference of two LG modes with conjugated topological charge numbers, the reference light can be omitted and the modulation of the light intensity can be easily realized [23]. The light field and phase distribution of superposition of conjugated LG modes is shown in Fig1. The self-interference of the two conjugated LG modes can be actualized by uploading the superposed LG mode holograms on the SLM.

For our measurement, the topological charge l is a known quantity. Therefore, we can calculate the spin speed from the second term of the Eq. (9) combined with the known topological charge l by extracting the average value of the time-frequency curve. It can be seen from the Eq. (9) above that the topological charge will affect the amplitude and average value of the time-frequency curve. The larger the topological charge is, the higher the time-frequency curve moves. Therefore, if we expect the curve to be distinguishable from the noise with low frequency, we’d better use large topological charge [25].

3. Experiment and results

To prove the analysis of the rotational micro-Doppler effect characteristic caused by the precession, an experiment of detection of the spinning object with precession is designed. As shown in Fig. 3, both the optical path and the data flow are presented. The laser source produces continuous optical beam, the wavelength of which is 1064 nm. A light expander is set to increase the coverage area of the beam, and then the beam passes through the polarizer and becomes horizontally polarized light to match the diffraction direction of the spatial light modulator(SLM). After being modulated by the hologram on the SLM, the first order diffraction light is selected by the spatial filter consisting of two lenses and an aperture and illuminates the compound rotating object, with its propagation axis coinciding with the object’s precession axis. To enhance the scattering intensity of the surface of the object, we stick metal foil paper on it. The scattered light from the rotating object is gathered by a lens and converged by an avalanche photodetector(APD). The electric signal produced by the APD is sampled by an oscillograph and finally processed on a computer.

Fig. 3. Experimental setup on measurement of spin and precession. Ex: beam expander, P: polarizer, SLM: spatial light modulator, L: lens, Ap: aperture, APD: avalance photodetector. The holograms are uploaded on the SLM by the computer and the oscillograph is used to sample the signals exported by the APD.

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As mentioned above, a superposition of conjugated LG modes is employed as the probe beam here. A previous research shows that the linear Doppler effect will arise when the light is at oblique incidence, but one of the advantages of superposition LG modes is that the linear Doppler effect will be counteracted after the beat frequency effect [6,26], only the RDE frequency shift illustrated in section 2.2 will be left. In order to form a contrast, firstly, we arrange the spinning speed ${\Omega _2} = $0.333r/s but the precession speed ${\Omega _1} = $0r/s. By setting different precession positions of spinning center O, i.e. different phase angle of cosine function in Eq. (9), the RDE detection results are displayed in Fig. 4. The topological charge of the probe beam is ${\pm} 50$.

Fig. 4. Experimental results with the different precession positions while setting the precession speed ${\Omega _1} = $0r/s. The highlighted position is the strongly scattering area. (a)∼(d) The four different precession positions. (e)∼(h) The time-frequency domain signals of different precession positions.

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As shown in Fig. 4, when the object only spins without precession, the RDE frequency shift does not undulate with time but only be broadened in frequency, which is consistent with the previous studies [5,6]. What we need to clarify here is that when the precession speed is set to 0, this is the same situation as when there is only lateral misalignment, and the misalignment in the four precession positions are equal. According to Ref. [5], the misalignment only introduces the broadening of the rotational Doppler shift. But the theory in Ref. [5] is based on completely accepting all the signal light of every scattering point in the vortex beam. This is an extremely demanding condition. In the actual experiment, we have found that only the scattered light of a small area in the vortex beam field can pass through the receiving system and be gathered, just as what we state in the paragraph below Fig. 2 of the manuscript. This situation is actually more similar to detection with circular asymmetry optical vortex [7]. Under this circumstance, with the changing of the precession position of spinning center O, the distance between strongly scattering area and the spin axis is varying accordingly and the transverse speed of the strongly scattering area is varying too, causing the change of the rotational Doppler shift. From the time-frequency spectrum of Fig. 4, it can be seen that the middle frequency of the spectrum varies with the precession position. It’s indicated from another perspective that when the precession speed isn’t 0, i.e. the precession position changes with time, the middle frequency will vary with time, forming fluctuations.

Then we set the precession and spinning speed respectively as ${\Omega _1} = $0.15r/s and ${\Omega _2} = $0.567r/s and still set the topological charge as ±50. The experiment result is displayed in time-, frequency- and time-frequency-domain as shown in Fig. 5. Figure 5(a) is the time-domain data collected by the photodetector within 20s. After Fourier transform, the frequency-domain signal is shown in Fig. 5(b) and in the square frame is the RDE frequency shift. By conducting short time Fourier transform(STFT) to the time-domain signals, the time-frequency domain signal is presented in Fig. 5(c), it can be clearly identified that the signal is into a cosine function fluctuations, which is in stark contrast to the spectrum in Fig. 4. Note that the STFT spectrum is affected by the broadened RDE frequency shift and reduced frequency resolution, thus it’s difficult to measure the period and middle value precisely as the theoretical measurement. To obtain a nonhuman intervention measurement, Fig. 5(d) shows the result of curve fitting of the high energy points in the time-frequency spectrum, from which, the parameters of the spectrum can be identified. The period of the curve is 6.64s, which is close to the precession period 6.67s. The relative error of the measured period is 0.45%. The middle value of the curve is 59.89 Hz and the theoretical RDE frequency shift introduced by the spin is 56.7 Hz, the relative error between them is 5.6%. That is to say, the precession speed and spinning speed can be measured respectively and simultaneously through the RDE signal.

Fig. 5. Experimental results with precession and spin speed ${\Omega _1} = $0.15r/s and ${\Omega _2} = $0.567r/s. (a) The time-domain data collected by the photodetector within 20s. (b) The frequency-domain signal. (c) The time-frequency spectrum after STFT. (d) The result of curve fitting of the high energy points in the time-frequency spectrum.

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To testify the capability of the method, we set different combination of topological charge, precession speed and spinning speed. Similarly, the echo signal is manipulated with STFT and is extracted the related parameters. The experiment results are shown in Fig. 6. Whatever the combination of the three parameters is, the precession speed and spinning speed can always be measured within a certain variance range. Our study may provide a new rotational micro-Doppler analysis approach and find applications in detection of micro-motion of rotating object.

Fig. 6. Dependent detection results under different spinning and precession speeds with topological charge $l ={\pm} 30$. (a) Detection results of different spinning speeds. (b) Detection results of different precession speeds.

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4. Discussion and conclusions

The innovation of our work is analyzing the rotational micro-Doppler effect of spin and precession under the detection of the VB carrying OAM and realizing the simultaneous detection of precession speed and spinning speed. It worth noting that, if we look forward to obtain the whole fluctuation cycling in the time-frequency spectrum, the length of the signal is supposed to be longer than the precession period. Generally, the precession speed is much less than the spinning speed. When the length of signal is much shorter than the precession period, the process of curve fitting will arise errors. Therefore, this detection method has a poor performance on timeliness. On the other hand, in actual spin and precession, there is an angle between precession axis and spin axis, this angle is called precession angle. We ignore the precession angle while analyzing the characteristics of rotational micro-Doppler effect, because the precession angle in our experiment is less than 2 degrees. But the presence of the precession angle makes the detection beam at oblique incidence, not perfect normal incidence, which makes the middle value of the frequency shift move to the high side [6] and increase the measurement error shown in Fig. 6. Moreover, when the precession angle becomes very large, it’s difficult to receive the scattered light. The precession angle in our experiment is less than 2 degree, so the corresponding experimental results are in good agreement with the theoretical analysis.

In summary, we report a new method that can realize the simultaneous measurement of precession speed and spinning speed using VB. Theoretically, we analyze the movement of strong scatter particle on the object and deduce the mechanism of rotational micro-Doppler effect. Experimentally, we design and perform the proof-of-concept experiment to testify the feasibility of the method. Through the comparison between the precession signal and non-precession signal, we verify the correctness of the theoretical analysis of the rotational micro-Doppler effect. Using the characteristic of rotational micro-Doppler effect modified by the precession and spin, we extract the related parameters of the time-frequency spectrum and the spinning speed and precession speed can be measured within a certain range of error. We believe the principle of it will be applied in more complicated micro-motion detection.

Funding

National Natural Science Foundation of China (61805283, 62173342).

Disclosures

The authors declare no conflicts of interest regarding this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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Simultaneous measurement of spin and precession based on light’s orbital angular momentum

Abstract

1. Introduction

2. Materials and methods

2.1 Principle of RDE

2.2 Rotational micro-Doppler characteristics of precession

3. Experiment and results

4. Discussion and conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (9)

Optics Express