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Free-space remote detection of a spinning object using the combined vortex beam

Tao Yu, Hui Xia, Qiao Xie, Guangwei Qin, Yuefeng Zhao, and Wenke Xie

Open Access Open Access

Abstract

The rotational Doppler effect (RDE) associated with orbital angular momentum (OAM) has been used for remote sensing of a spinning object. However, one of the challenges of long-range detection stems from the low echo signal power. In this paper, we propose a new detection scheme that uses the combined vortex beam (CVB) generated by coherent beam combining (CBC) technology as the probe beam to enhance the echo signal power. Furthermore, we establish a rotational speed remote sensing model based on RDE, the emitted power and emission diameter of the probe beam are investigated in detail. The results show that, compared with the superposition vortex beam (SVB) generated by a single laser beam, the CVB detection scheme can significantly enhance the echo signal intensity and detection distance. The measuring range and accuracy of rotational speed are also studied in detail. And finally, we present the first experimental demonstration of the RDE arising directly from the interaction of the CVB with a rotating rough surface. The scheme proposed in our paper offers a good reference for practical application of the remote detection based on RDE.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

The Doppler effect has been extensively used in various translational motion measurement technologies, such as radar, motor vehicle speed measurement, and weather forecast [1–3]. However, the relative motion between rotating object and wave source cannot be changed by the angular velocity. The Doppler effect will not be applicable to the measurement of the angular velocity of rotating objects. Therefore, the rotation and translation of the target cannot be detected simultaneously using the linear Doppler effect (LDE). Similar to the conventional LDE, another rotational Doppler effect (RDE) associated with the OAM has been proposed in recent years [4–6]. Nienhuiss observed this rotational Doppler effect by rotating an OAM mode converter composed of three cylindrical lenses, demonstrating the potential of the vortex beam to detect the rotational speed [7]. Since then, RDE, the novel Doppler effect, has been applied to detect rotational speed of spinning objects. Based on the LDE theory, Lavery deduced the quantitative relationship among the rotational Doppler shift, the OAM mode of the probe beam and the rotational speed, which offers a theoretical reference for the rotational speed detection [8].

One of the potential applications of the RDE-based rotational speed measurement technology is the noncontact remote sensing. However, most of previous research on RDE still remains in laboratory conditions, mainly including the theoretical interpretation of RDE and the analysis of measurement conditions [9–12]. Compared to LDE, the rotational Doppler shift is proportional to the OAM mode carried by the probe beam. Therefore, the main challenges limiting the practical application of the RDE-based detection scheme are the OAM mode expansion of the detection beam due to the atmospheric turbulence and the extremely low echo signal power caused by beam divergence. Chen et al. developed the RDE-based rotational speed measurement technology under the condition of photon level, and realized the rotation Doppler effect observation in 120 m free space [13]. Whereas, this scheme is sensitive to the external environment and the experimental equipment. In addition, the divergence of the detection beam also leads to a larger beam diameter, which cannot effectively illuminate a target with a smaller cross-sectional area to produce a detectable rotational Doppler shift signal. Therefore, a high-power and low-divergence OAM beam is required to improve the echo signal power of scattered optical field. At present, the laser output power has been increased to kilowatt-level based on the fiber laser CBC technology, which has good performance and stable phase control ability [14]. Based on the traditional CBC technology, Zhi et al. proposed a high-power OAM beam generation scheme, which can convert the phase of the beam array into a continuous spiral phase by superimposing discrete vortex phases [15,16]. This solution is expected to solve the deficiency of low power and small emission aperture of OAM light source.

To solve the problem of low echo signal power, a very straightforward idea is to increase the power of the detection beam. Our previous work has demonstrated that the CVB beam generated by CBC technique is a special superimposed vortex beam [17]. Thus, the CVB generation scheme solves both the high power and vortex optical field modulation challenges, which means that the CVB is an ideal vortex sources for long-range application. In this paper, a rotational speed remote sensing scheme using the CVB as probe beam is proposed. Compared with the single-beam SVB detection scheme, we find that the CVB with high-power and low divergence angle can significantly enhance the echo signal power. Moreover, by increasing the power and number of sub-beams, it is possible to continuously increase the output power, demonstrating that the CVB is an ideal probe beam for the RDE-based remote sensing system.

2. Rotational Doppler effect based on the CVB

The standard linear Doppler shift applies when the relative motion between source and target is along the direction of target. As shown in Fig. 1(a), the linear Doppler effect can still be observed when the source is obliquely illuminated to the target. For small values of α, this reduced Doppler shift is given by [8]

$$\Delta f = \alpha \frac{{{f_0}v}}{c}$$
where α is the angle between the wave source and the target normal, v is the translational linear speed of the target, c and f0 represent the light speed and frequency in the corresponding medium, respectively. The vortex beam has a helical phase profile, indicating that the wavefront is not perpendicular to the optical propagation axis of the vortex beam. As shown in Fig. 1(b), the Poynting vector, which give the local energy flow, is misaligned with the propagation direction by an angle α=lλ/2πrb, where λ is the wavelength, l is the topological charge of vortex beam and rb is the distance from the beam axis. In the frequency domain, for a l-order vortex beam vertically illuminating the center of a spinning object, we can see from the Eq. (1) that the Doppler frequency shift generated by rotating object is given as [8]
$$\Delta f = \frac{{l\lambda }}{{2\pi {r_b}}}\frac{{{f_0}\mathrm{\Omega }{r_b}}}{c} = \frac{{l\mathrm{\Omega }}}{{2\pi }}$$
where Ω is the rotational speed of rotating object. We can find that the tangential velocity vtar of the rotating object has a velocity component projection vvox in the direction of the wave velocity. Therefore, a relative motion between the vortex beam and the rotating object also produces a Doppler shift, which is known as the Rotational Doppler effect.

 figure: Fig. 1.

Fig. 1. Light scattered from a moving surface can be observed a shift in frequency: (a) translation; (b) rotation.

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The frequency shift caused by the rotational Doppler effect could be negligible in relation to the optical frequency f0. The beat frequency method is generally used to measure the optical rotational Doppler shift, and a superimposed vortex beam with ± l topological charge is applied as the probe beam frequently [18,19]. The mixing of up- and down-shifted scattered light components will result in a beat Doppler signal with the modulation frequency expressed as

$${f_{mod}} = 2\Delta f = \frac{{2\left| l \right|\mathrm{\Omega }}}{{2\pi }}$$

In our previous work, the scheme for generating high-power vortex beams based on CBC technology has been discussed [17,20]. The principle of the scheme can be briefly described as follows: the annularly sub-beam arrays are superimposed with discrete vortex phases, and the phase-modulated sub-beams will interfere with each other after propagating in the free space, eventually forming a CVB with stable structure. The amplitude of the phase-modulated Gauss beam array at z = 0 m can be expressed as [16,17,20]

$$E\left( {x,y,0} \right) = \mathop \sum \nolimits_{m = 0}^{M - 1} exp [ - \frac{{\left( {x - R\cos ({\alpha _m}} \right){)^2} + \left( {y - R\sin ({\alpha _m}} \right){)^2}}}{{w_0^2}}]exp (in{\alpha _m})$$
where M is the number of Gauss beams, R is the ring radius of beam array, w0 is the width of Gauss beam, z is the propagation distance, αm = 2πm/M is the angle between the center of the m-th Gaussian beam and the x-axis, as shown in Fig. 2(a), n is the integer order topological charge, nαm is the additional piston phase array as shown in Fig. 2(b).

 figure: Fig. 2.

Fig. 2. The schematic diagram of the phase-modulated Gauss beam array. (a) The geometric construction of Gaussian beam array; (b) the discrete vortex phase array.

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To calculate conveniently, the complex amplitude of the phase-modulated Gaussian beam array at any propagation distance can be given in the polar coordinate system as

$$\begin{aligned}E\left( {r,\varphi ,z} \right) = exp &( - \frac{{{r^2} + {R^2}}}{{{w^2}\left( z \right)}})exp ( - \frac{{ik\left( {{r^2} + {R^2}} \right)}}{{2R\left( z \right)}})exp [iarctan (z/f) - ikz]\\ &\times \mathop \sum \nolimits_{m = 0}^{M - 1} exp (\frac{{2Rr\cos ({\alpha _m} - \varphi )}}{{{w^2}\left( z \right)}})exp (\frac{{ikrR\cos ({\alpha _m} - \varphi )}}{{R\left( z \right)}})exp (in{\alpha _m})\end{aligned}$$

According to the Bessel function expansion $exp (xcos(\varphi )) = \mathop \sum \nolimits_{l ={-} \infty }^{ + \infty } {I_l}(x )exp (il\varphi )$, Eq. (5) is reduced to

$$\begin{aligned}E\left( {r,\varphi ,z} \right) = exp &( - \frac{{{r^2} + {R^2}}}{{{w^2}\left( z \right)}})exp ( - \frac{{ik\left( {{r^2} + {R^2}} \right)}}{{2R\left( z \right)}})exp [iarctan (z/f) - ikz]\\ &\times \mathop \sum \nolimits_{m = 0}^{M - 1} \mathop \sum \nolimits_{l = - \infty }^\infty {I_l}\left[ {\left( {\frac{{2R}}{{{w^2}\left( z \right)}} + \frac{{ikR}}{{R\left( z \right)}}} \right)r} \right]exp \left( {il\varphi } \right)exp\left( {i\left( {n - l} \right){\alpha _m}} \right)\end{aligned}$$
When $\frac{{n - l}}{M} = p$, (p = 0, ± 1, …), considering
$$\mathop \sum \nolimits_{m = 0}^{M - 1} \textrm{exp}\left( {i\left( {n - l} \right){\alpha _m}} \right)\textrm{ = }\mathop \sum \nolimits_{m = 0}^{M - 1} \textrm{[cos}\left( {\frac{{n - l}}{M}2m\pi } \right) + i\,\textrm{sin}\left( {\frac{{n - l}}{M}2m\pi } \right)\textrm{] = M}$$

The complex amplitude of the phase-modulated Gauss beam array at an arbitrary propagation distance z can be rewritten as

$$\begin{aligned}E\left( {r,\varphi ,z} \right) = exp &( - \frac{{{r^2} + {R^2}}}{{{w^2}\left( z \right)}})exp ( - \frac{{ik\left( {{r^2} + {R^2}} \right)}}{{2R\left( z \right)}})exp [iarctan (z/f) - ikz]\\ &\times M\mathop \sum \nolimits_{p = - \infty }^\infty {I_{n - pM}}\left[ {\left( {\frac{{2R}}{{{w^2}\left( z \right)}} + \frac{{ikR}}{{R\left( z \right)}}} \right)r} \right]exp \left( {i\left( {n - pM} \right)\varphi } \right)\end{aligned}$$
where Il (r) is the modified Bessel function, p is an integer, $f = \pi w_0^2/\lambda $, $w(z )= {w_0}\sqrt {1 + {{(z/f)}^2}} $, $R(z )= z + {f^2}/z$.

From the phase and amplitude terms in Eq. (8), it can be known that there exist a series of OAM mode component l = n-pM, indicating that the CVB is a special kind of superposition Bessel-Gaussian vortex beam. Substituting any two OAM mode components of the CVB into Eq. (3), the rotational Doppler frequency shift based on the CVB can be expressed as

$${f_{mod}} = 2\Delta f = \frac{{\left| {\left( {n - pM} \right) - \left( {n - p'M} \right)} \right|\mathrm{\Omega }}}{{2\pi }} = \frac{{\left| {p' - p} \right|M\mathrm{\Omega }}}{{2\pi }},p' - p = \pm 1 \pm 2 \pm 3 \ldots$$

The corresponding rotational speed can be calculated according to Eq. (9).

3. Rotational speed remote sensing scheme based on RDE

In order to investigate superiority of the CVB as probe beam for remote sensing, the RDE-based rotational speed remote sensing model is established, as shown in Fig. 3.

 figure: Fig. 3.

Fig. 3. Schematic diagram of remote sensing model based on RDE. (Source: probe beam; FL, focusing-lens; PD, photoelectric detector).

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The detection model is divided into four modules, including probe beam, rough surface of spinning object, beam transmission module and receiver module. The probe beam is a superimposed vortex beam generated by a single beam or a beam array. The target surface module adopts the Gaussian random rough surface model proposed by Vorontsov [21], the height fluctuation h(ρ) of the rough surface obeys the Gaussian distribution, so the optical field reflected by the rough surface can be approximately expressed as:

$${E_{re}} = {R_{tar}}{E_{in}}\textrm{exp}\left( {i2kh\left( \rho \right)} \right)$$

Rtar is the reflection coefficient of the target, Ein is the complex amplitude of the incident light. The correlation function between the heights of any two points on a rough surface is given by the Gaussian expression [21]

$${B_h}\left( {\overrightarrow {{r_1}} ,\overrightarrow {{r_2}} } \right) = h\left( {\overrightarrow {{r_1}} } \right)h\left( {\overrightarrow {{r_2}} } \right) = \sigma _h^2\textrm{exp}\left( {{{\left| {\overrightarrow {{r_1}} - \overrightarrow {{r_2}} } \right|}^2}/\rho _h^2} \right)$$
σh is the Root Mean Square (RMS) of surface roughness height, ρh is the correlation length. According to correlation function, the power spectral density of a random rough surface height distribution can be obtained. Finally, the corresponding rough surface height distribution h(ρ) can be calculated by using the power spectrum method and Fourier transform.

The beam propagation module uses an angular spectrum transmission algorithm to simulate the free space propagation of the probe beam. It should be noted that the propagation of the probe beam needs to be divided into two parts, the transmission before incident on the target (forward propagation) and the propagation of the scattered light from the target to the receiver module (reverse propagation). Therefore, the scattered optical field of the rough surface needs to be used as the emission field when designing the backward transmission. In this paper, the receiver and the light source transmitter are located at the same position, so the forward and reverse transmission have the same propagation path. In order to clearly describe the model, the transmitter is not placed in the same position as the receiver in Fig. 3.

The receiving module is composed of focusing-lens, photodetector and oscilloscope. After the scattered echoes have been collected by the focusing-lens (FL), a photodetector (PD) is used to record the power of the collected echo signal. Finally, the PD is then connected to an oscilloscope, which is used to perform fast Fourier transform to extract the rotational Doppler beat signal.

The process of the rotational speed remote sensing scheme can be summarized as follows: The probe beam, after free space transmission, is directed vertically onto the target surface. The propagation axis of the detection beam coincides with the rotation axis of the target. Then, the target scattered light field propagates back to the receiver along the same transmission path. Next, the scattered echo signal is collected with a focusing lens, and its signal intensity is recorded by a photodetector located at the focal plane. Finally, the frequency shift spectrum of the recorded signal intensity is calculated using the fast Fourier transform. According to Eq. (3), we will derive the corresponding rotational speed.

To demonstrate the superiority of the CVB detection scheme, the rotational speed remote sensing is performed with the CVB generated by an array-beam and the SVB generated by a single-beam, respectively. According to the rotational speed measurement model developed above, it is possible to evaluate the effect of each parameter on the remote sensing of the rotational velocity.

3.1 Intensity of probe beam

Before the rotational speed measurement, the Gaussian rough surface is generated to simulate the target according to Eq. (11) and the power spectrum method, where σh = 10−4m, ρh = 10−2 m. The target surface in this paper has a height fluctuation of microns which is classified as a micro-rough surface. And the target characteristic is discussed in detail in Section 3.3. The target size is set to 1 m and the target rotational speed Ω is set to 125.6 rad/s. In addition, the computational parameters are as follow: the sampling intervals, 0.5 ms; data collecting time, 0.1s; the grid numbers, 1024 × 1024.

Then, given the array beam parameters M = 12, n = 6, w0 = 5 mm, R = 30 mm, λ=632.8 nm and the propagation distance z = 1 km. The 6-order CVB generated by the phase-modulated array beam is used to measure the rotational speed, and the results are shown in Fig. 4. At the transmitter, the maximum peak light intensity of the sub-beam is normalized as shown in Fig. 4(a). Furthermore, the normalized array light source is used as a reference in subsequent calculations of emitted power and beat frequency signal intensity. The beam diameter of the 6-order CVB formed by the array light source after 1 km propagation is approximately equal to 16 cm, as shown in Figs. 4(a)-(c). The OAM mode spectrum of the 6th-order CVB is shown in Fig. 4(d). From the simulated results, it can be seen that the 6th-order CVB has +6 and -6th order OAM mode components with the same mode purity, which is completely consistent with the derivation result of Eq. (8). So, the 6th-order CVB generated by 12 sub-beams can also be called ±6th-order superposition vortex beams (SVB). The scattered field of target after reverse transmission of 1 km is shown in Fig. 4(e). Due to the modulation of the rough surface, the scattered field presents a random speckle distribution, and the diameter is close to 70 cm. At the receiver, the focusing lens is considered to be an ideal lens for receiving the scattered light field from the target. The diameter and the focal length of the FL are set to 20 cm, only part of the scattered light field from the target can be collected at the receiver, the scattered light field at focal plane is shown in Fig. 4(f). To observe the beat frequency signal generated by the RDE, a PD is placed at the focal plane to record the intensity evolution of the echo signal within 0.1s. The discrete Fourier transform is used to extract the frequency information, the results are shown in Figs. 4(g) and (h) respectively. The rotational Doppler shift obtained by the remote sensing model is fmod = 240 Hz, and the relative power of the beat frequency is P = 7.98. According to Eq. (9), the target speed Ω=125.6 rad/s can be calculated, which is consistent with rotational speed set by the model. The results show that the CVB can be used as the probe beam to realize the rotational speed remote sensing of a spinning object.

 figure: Fig. 4.

Fig. 4. The rotational speed remote sensing scheme based on the 6-order CVB. (a) The array beam at z = 0 m; (b) the 6-order discrete vortex phase; (c) the 6-order CVB at z = 1 km; (d) the OAM mode spectrum; (e) the target speckle field; (f) the scattered light field at focal plane; (g) the intensity evolution of the echo signal within 0.1s; (h) the rotational Doppler shift.

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After that, the rotational speed remote sensing is performed by the ±6-order SVB generated by a single-beam. Coherent beam combining technology can combine multiple medium-power sub-beams to obtain high-power light sources. Based on the context that a higher power vortex beam cannot be obtained from a single beam, so the power of the SVB is set equal to the power of a single sub-beam in the array beam, which is one-twelfth of the total power of the CVB detection scheme, and other conditions are the same. Referring to the superposition vortex beam generation scheme in Ref. [13], a single laser beam is used to generate ±6th-order superposition vortex beams by loading a computer-generated hologram. The beam waist diameter of the single laser source is 70 mm, which is consistent with the emission diameter of the array light source. As shown in Fig. 5, the diameter of ±6th-order SVB and target scattered optical field are essentially the same as those of the CVB measurement scheme, due to the same emission size and the rough surface. The difference is that the ±6th-order SVB are composed of four OAM mode components, as shown in Fig. 5(d). In accordance with Eq. (3), there will be a rotational Doppler shift between any two OAM modes. However, the power of the ±18th-order OAM mode is significantly lower than other OAM components. As a result, there are only two distinct frequency shift peaks, fmod1 = 240 Hz and fmod2 = 480 Hz, and their related signal power are P1 = 0.765 and P2 = 0.083 respectively, as shown in Fig. 5(h). The noise frequency peak may be due to the slight OAM mode expansion of the SVB. With the same propagation distance and the same emission diameter, it is visible from the simulation results that the CVB measurement scheme exhibits a stronger echo signal and can potentially be expanded to a longer detection distance.

 figure: Fig. 5.

Fig. 5. The rotational speed remote sensing scheme based on the ±6-order SVB. (a) The single beam at z = 0 m; (b) the ±6-order computed hologram; (c) the ±6-order SVB at z = 1 km; (d) the OAM mode spectrum; (e) the target speckle field; (f) the scattered light field at focal plane; (g) the intensity evolution of the echo signal within 0.1s; (h) the rotational Doppler shift.

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According to the diffraction effect, the beam diameter of the target scattered field is proportional to the propagation distance. The intensity of the received echo signal will gradually decrease with the transmission distance because of the fixed focusing lens size. Until the photodetector is unable to measure the echo signal, the speed measurement of the target will fail. Using the target echo signal intensity shown in Fig. 5(g) as the reference threshold, the maximum detection distance of the CVB detection scheme is about 4 km, and the remote sensing detection results are shown in Fig. 6. It should be made clear that after 4 km of propagation, the beam diameter of the CVB is approximately 45 cm, and the scattering optical field diameter after 4 km of backward transmission is about 1.6 m, which exceeds the observation plane size. In order to ensure the accuracy of the simulation results, an absorption boundary is set in the propagation algorithm to absorb and attenuate the energy spilling out of the observation plane. Comparing the received beat signals illustrated in Fig. 5(h) and Fig. 6(d), the intensity of the beat signals received by the two detection schemes is on the same order of magnitude. With all other parameters being equal, it is shown that the CVB with higher power can extend the detection distance to more than four times that of the single-beam detection scheme.

 figure: Fig. 6.

Fig. 6. The rotational speed remote sensing results based on the CVB at a detection distance of 4 km. (a) The 6-order CVB at z = 4 km; (b) the target speckle field; (c) the intensity evolution of the echo signal within 0.1s; (d) the rotational Doppler shift.

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3.2 Emission size of probe beam

In addition to the propagation distance, the beam diameter of the target scattered optical field is also dependent on the emission diameter of the probe beam. Therefore, the emission diameter of the detection light source is also investigated in detail. The emitting diameter of the SVB detection scheme is set at 35 mm, which is one half of the emitting diameter of the array light source. The emitting power is remained at one twelfth of the CVB and the other parameters are the same. As shown in Figs. 7(a) and (b), the beam diameter of the ±6th order SVB after 1 km transmission is already close to 35 cm and the actual beam diameter of the target scattering field has reached 80 cm (here the absorption boundary is set). However, the focusing lens size is still set at 20 cm, which means that the PD can only receive a much weaker echo signal for small emission diameter detection sources. The rotational speed measurements result of the SVB detection scheme are shown in Fig. 7(d). At the same transmitting power, the SVB detection scheme with 35 mm emission diameter receives about one-fifth the power of the beat frequency signal of the SVB detection scheme with 70 mm emission diameter. At the same detection distance, the SVB detection scheme with an emission diameter of 35 mm obtains only about one-fiftieth the intensity of the beat frequency signal of the CVB detection scheme shown in Fig. 4(h).

 figure: Fig. 7.

Fig. 7. The rotational speed remote sensing results based on the SVB with emission diameter of 35 mm. (a) The ±6-order SVB at z = 1 km; (b) the target speckle field; (c) the intensity evolution of the echo signal within 0.1s; (d) the rotational Doppler shift.

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Similarly, using the target echo signal intensity shown in Fig. 7(c) as the reference threshold, the limiting transmission distance for the CVB detection scheme is calculated to be approximately 8 km and the results are shown in Fig. 8. In this case, the CVB detection scheme is able to extend the detection distance to more than eight times that of the single beam detection scheme.

 figure: Fig. 8.

Fig. 8. The rotational speed remote sensing results based on the CVB at a detection distance of 8 km. (a) The 6-order CVB at z = 8 km; (b) the target speckle field; (c) the intensity evolution of the echo signal within 0.1s; (d) the rotational Doppler shift.

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The investigation of emitted light power and emission diameter shows that increasing the power and emission diameter of the probe beam can significantly increase the detection distance of the rotational speed remote sensing detection technology. Moreover, the array beam source can be easily formed by multiple sub-beams. Compared with the bulky single-beam expander emission system, it is easy to generate a laser source with high power and small divergence angles based on the coherent beam combining technology. Therefore, the array light source has real application potential for the angular velocity detection of targets.

3.3 Measuring range and accuracy of rotational speed

The measuring range and accuracy of the rotational speed can be determined depends upon a number of factors. We use the Fast Fourier Transform to calculate the frequency spectrum of the recorded echo signal intensity. According to the fast Fourier transform, the transform-limited width of the frequency shift peak fmin = 1/t, the sampling frequency Fs = 1/Δt, and the maximum frequency shift peak fmax = 1/(2Δt). t is the sampling time, Δt is the sampling interval. By substituting fmin and fmax into Eq. (3), the minimum and maximum rotational speeds that can be measured as follows

$${\mathrm{\Omega }_{min}} = \frac{{2\pi {f_{min}}}}{{\left| {2l} \right|}} = \frac{{2\pi }}{{\left| {2l} \right|t}}$$
$${\mathrm{\Omega }_{max}} = \frac{{2\pi {f_{max}}}}{{\left| {2l} \right|}} = \frac{\pi }{{\left| {2l} \right|\Delta t}}$$

From Eqs. (12) and (13), it is clear that the measurement range of the rotational speed is determined by three parameters: the OAM mode of the probe beam, the sampling time and the sampling interval. According to Eq. (12), the accuracy of rotational speed is ultimately governed by the transform-limited width of the frequency shift peak. The CVB detection scheme is used to measure the rotational speed with sampling times of 0.1s, 1s and 10s respectively. The detection distance is 1 km and the rotational speed is set to 126.228 rad/s. As shown in Fig. 9, due to the limitations of the frequency shift peak width, the measured rotational speed is 125.600 rad/s, 126.120 rad/s and 126.228 rad/s with sampling times of 0.1s, 1s and 10s respectively. Obviously, increasing the sampling time will improve the accuracy of the rotational speed measurement. However, this improvement in precision is only significant assuming that the rotation speed itself remains constant during the data collection time. Therefore, the accuracy of the rotational speed can be improved by increasing the sampling time and taking the average of a number of measurements.

 figure: Fig. 9.

Fig. 9. The rotational speed measurement results based on CVB for different sampling times. (a) t = 0.1s; (b) t = 1s; (c) t = 10s.

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In addition, the rotational speed measurements are also related to the target surface properties. According to Eq. (11), the target surface height fluction h(ρ) are correlated with coherence length ρh and surface height rms σh. Firstly, we have performed rotational speed measurements for targets with a surface coherence length of ρh = 10−2 m and surface height rms σh of 10−4 m and 2 × 10−4 m, respectively. The corresponding detection results with the CVB detection scheme are shown in Figs. 10(c) and (f), and the parameters of the detection light source are consistent with Fig. 4. The results show that keeping the coherence length ρh = 10−2 m of the target surface, the rotational Doppler peak will be submerged in the stray peaks as the height fluctuation h(ρ) of target increase. Moreover, we have conducted the rotational speed measurements for target with a surface coherence length of ρh = 2 × 10−3 m and surface height rms σh of 6.5 × 10−5 m. The surface height rms σh is adjusted to ensure that both target surfaces have the same height fluctuations, as shown by the colour bars in Figs. 10(a) and (g). In this case, the effect of the coherence length ρh of the target surfaces on the rotational speed measurements is analyzed. Despite the fact that both target surfaces have the same height fluctuations, there is no clear rotational Doppler shift peak for target with surface coherence length ρh = 2 × 10−3 m. Obviously, the rotational speed measurement results are influenced by both the coherence length and the surface height undulation.

 figure: Fig. 10.

Fig. 10. The rotational speed remote sensing results with different target surface. The parameters of target surface: ρh = 10−2 m, σh = 1 × 10−4 m; (a) the target surface; (b) the target speckle field; (c) the rotational Doppler shift. The parameters of target surface: ρh = 10−2 m, σh = 2 × 10−4 m; (d) the target surface; (e) the target speckle field; (f) the rotational Doppler shift. The parameters of target surface: ρh = 2 × 10−3 m, σh = 6.5 × 10−5 m; (g) the target surface; (h) the target speckle field; (i) the rotational Doppler shift.

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Subsequently, we varied the wavelength of the probe beam for rotational speed measurements, and still used the rough surface shown in Figs. 10(d) and (g) as the target. As shown in Figs. 11(c) and (d), the noise peaks in the frequency shift spectrum are significantly reduced when the detection beam wavelength is set to 1550 nm. The results indicate that the longer the wavelength of the incident beam, the less the inhomogeneity and height undulations of the target impact on the beam.

 figure: Fig. 11.

Fig. 11. The rotational speed remote sensing results with the probe beam of 1550 nm wavelength. The parameters of target surface: ρh = 10−2 m, σh = 2 × 10−4m: (a) the target speckle field; (c) the rotational Doppler shift. The parameters of target surface: ρh = 2 × 10−3 m, σh = 6.5 × 10−5 m; (b) the target speckle field; (d) the rotational Doppler shift.

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In general, the processing of the received echo signal and the surface characteristics of the target are crucial to the accurate measurement of the target rotational speed. Therefore, the selection of parameters such as the appropriate sampling time and detection wave source are particularly important in practical measurements.

4. Experiment results and analyses

4.1 Experimental setup

To our knowledge, there are no reports or experiments on the use of the CVB for rotational speed measurements. In order to verify the feasibility of the CVB for rotational speed measurements, a proof-of-concept experiment is designed in laboratory conditions, as shown in Fig. 12. A 6th order CVB generated from 12 sub-beams is used as the detection source for target rotation speed measurements. It should be noted that the forward propagation distance of the CVB in this experiment is about 2 m and the emission diameter is 2 mm. Therefore, this experiment is only used to demonstrate the feasibility of the CVB as a detection beam of rotational speed measurement and does not involve long distance detection. In our following research, we will construct a fiber laser array based on CBC technology for generating CVB with an output power on watt scale. The CVB generated by CBC technique will also be applied to the measurement of rotational speed at long range to verify the correctness of the simulation results.

 figure: Fig. 12.

Fig. 12. Experimental setup. (BE, beam expander; A-SLM, P-SLM, amplitude spatial light modulator, phase spatial light modulator; M, reflector mirror; BS, beam splitters; L, lens; P, pinhole; PD, photoelectric detector.)

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The collimated seed laser with a wavelength of 632.8 nm is expanded by the beam expander and its output power is approximately 1 mW. Following that, using the amplitude SLM (A-SLM: Holoeye LC2002) as the beam shaping device, the expanded beam is shaped into an annular sub-beam array with the number of sub-beams M = 12, the sub-beam waist radius w0 = 0.2 mm, and the ring radius R = 1 mm. Due to the restriction of the working area of the SLM, the emission diameter of the shaped array beam source is at the millimeter level. Before coherent combination, the generated sub beam array is modulated by the phase type SLM (P-SLM: Holoeye Pluto) with the 6-order discrete vortex phase. The phase-modulated output light field has a deflection angle of approximately 6 degrees with respect to the SLM optical axis, which does not affect the beam quality of the CVB. It should be noted that the angles in Fig. 12 are larger than those in our experiments to make the figure clearer. A CVB with a stable optical field structure can only be generated if the phase-modulated sub-beams interfere completely with each other, so we use reflectors to extend the forward transmission distance to 2 m. And the reflector is carefully adjusted to ensure that the 6-order CVB can be vertically incident on the center of Target. The target is a disc with a diameter of 5 cm, which is processed from metallic aluminum. To increase the power of the echo signal, the target is wrapped in silver paper. In addition, as the diameter of focusing lens is only 1 inch, the receiver is placed approximately 10 cm from the target to collect all the target scattered fields. The pinhole with a diameter of 100um is used for mode filtering to select the fundamental mode in the target scattered light field. Finally, the PD is connected to an oscilloscope, which is used to perform the real-time Fourier transform to extract the frequency shift signal. In addition, the sampling time is 0.2s and the sampling frequency is 25kHz.

4.2 Results and analyses

The experimental scheme is mainly divided into two parts: the generation of CVB and rotational speed measurement. First, based on the CBC technique, a 6-order CVB is generated as the detection beam. Figure 13 shows the optical fields of the experimentally generated 6-order CVB at propagation distances 0.2 m, 1 m, and 2 m, respectively. We can find that due to the modulation of the discrete vortex phase, the array light source forms a petal-shaped superposition vortex beam with stable structure after transmission over a distance of 2 m.

 figure: Fig. 13.

Fig. 13. The 6-order CVB at different propagation distance. (a) z = 0.2 m; (b) z = 1 m; (c) z = 2 m.

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Subsequently, the 6-order CVB is used to measure the rotational speed of the rotator. The beam diameter of the CVB illuminated on the rough surface is approximately 1 cm. The rotational speed is controlled by a motion controller, which offers a variable voltage for the stepper motor. In this experiment, the target speed is set to 125.6 rad/s and 188.4 rad/s respectively. The angular velocity measurement results are shown in Fig. 14, the location of the peak in the frequency domain are 240 Hz and 360 Hz, respectively. As can be seen from the analysis in section 3.3, the transform-limited width of the frequency shift peak is 5 Hz, so there is no speed measurement error due to signal processing. According to Eq. (9) and the array beam parameters M = 12 and n = 6, the corresponding angular velocities of the rotator can be calculated accurately as 125.6 rad/s and 188.4 rad/s, respectively. The results show that the experimental measurements are consistent with the expectation of the rotational speed, which also indicates that the speed of the stepper motor used in the experiment is maintained consistently during the sampling time. However, there is also a series of noise peaks in the frequency spectrum, as shown in Figs. 14(c) and (d). The frequency shift difference between adjacent noise peaks is proportional to the target speed, and the relationship can be expressed as Δf = fmod-fnosie1 = fnoise2-fmod=Ω/2π. Based on the analysis in the Ref. [9], we believe that the noise frequency shift peak is caused by the fact that the optical axis of the detection light source does not coincide exactly with the target rotation axis. However, the misalignment between the detection source and the target does not affect the measurement accuracy of the target rotation speed.

 figure: Fig. 14.

Fig. 14. The experiment results of rotational speed measurement based on RDE. (a) Time domain signal of the target scattered light field with a rotational speed of 125.6 rad/s; (b) time domain signal of the target scattered light field with a rotational speed of 188.4 rad/s; (c) the rotational Doppler frequency signal with a rotational speed of 125.6 rad/s; (d) the rotational Doppler frequency signal with a rotational speed of 188.4 rad/s;

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5. Conclusion

In summary, we propose a detection scheme that uses the CVB as the probe beam for remote sensing the rotational speed of a rotating target. Furthermore, a remote sensing model based on RDE is constructed, which is used to explore the influence of the parameters such as probe beam on the rotational speed measurements. The research results show that the CVB generated by the array beams has outstanding advantages such as small beam divergence and strong target scattered echo signal, which is suitable for long-distance detection of the spinning objects. Finally, we carried out the rotation speed measurement experiment with the 6-order CVB as the detection beam. However, from the analysis of the experimental results, it can be seen that the remote sensing scheme based on RDE has exacting requirements on the measurement conditions. Therefore, in order to realize the practical application of the remote sensing detection scheme based on the CVB, the atmospheric turbulent propagation effect and the alignment of the probe beam and the target rotation axis need to be considered in future research.

Funding

Fundamental Research Funds for Central Universities of the Central South University (2021zzts0052).

Acknowledgments

The authors thank Chen Wang, Song Qiu, et al. for their help on the experiment setup.

Disclosures

The authors declare no conflicts of interest.

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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13. W. Zhang, J. Gao, D. Zhang, Y. He, T. Xu, R. Fickler, and L. Chen, “Free-Space Remote Sensing of Rotation at the Photon-Counting Level,” Phys. Rev. A 10(4), 044014 (2018). [CrossRef]  

14. Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Letters 36(6), 951–953 (2011). [CrossRef]  

15. D. Zhi, T. Hou, P. Ma, Y. Ma, P. Zhou, R. Tao, X. Wang, and L. Si, “Comprehensive investigation on producing high-power orbital angular momentum beams by coherent combining technology,” High Power Laser Sci. 7, e1 (2019). [CrossRef]  

16. T. Hou, Y. Zhang, Q. Chang, P. Ma, R. Su, J. Wu, Y. Ma, and P. Zhou, “High-power vortex beam generation enabled by a phased beam array fed at the nonfocal-plane,” Opt. Express 27(4), 4046–4059 (2019). [CrossRef]  

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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.

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Figures (14)
Fig. 1.
Fig. 1. Light scattered from a moving surface can be observed a shift in frequency: (a) translation; (b) rotation.

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Fig. 2.
Fig. 2. The schematic diagram of the phase-modulated Gauss beam array. (a) The geometric construction of Gaussian beam array; (b) the discrete vortex phase array.

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Fig. 3.
Fig. 3. Schematic diagram of remote sensing model based on RDE. (Source: probe beam; FL, focusing-lens; PD, photoelectric detector).

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Fig. 4.
Fig. 4. The rotational speed remote sensing scheme based on the 6-order CVB. (a) The array beam at z = 0 m; (b) the 6-order discrete vortex phase; (c) the 6-order CVB at z = 1 km; (d) the OAM mode spectrum; (e) the target speckle field; (f) the scattered light field at focal plane; (g) the intensity evolution of the echo signal within 0.1s; (h) the rotational Doppler shift.

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Fig. 5.
Fig. 5. The rotational speed remote sensing scheme based on the ±6-order SVB. (a) The single beam at z = 0 m; (b) the ±6-order computed hologram; (c) the ±6-order SVB at z = 1 km; (d) the OAM mode spectrum; (e) the target speckle field; (f) the scattered light field at focal plane; (g) the intensity evolution of the echo signal within 0.1s; (h) the rotational Doppler shift.

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Fig. 6.
Fig. 6. The rotational speed remote sensing results based on the CVB at a detection distance of 4 km. (a) The 6-order CVB at z = 4 km; (b) the target speckle field; (c) the intensity evolution of the echo signal within 0.1s; (d) the rotational Doppler shift.

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Fig. 7.
Fig. 7. The rotational speed remote sensing results based on the SVB with emission diameter of 35 mm. (a) The ±6-order SVB at z = 1 km; (b) the target speckle field; (c) the intensity evolution of the echo signal within 0.1s; (d) the rotational Doppler shift.

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Fig. 8.
Fig. 8. The rotational speed remote sensing results based on the CVB at a detection distance of 8 km. (a) The 6-order CVB at z = 8 km; (b) the target speckle field; (c) the intensity evolution of the echo signal within 0.1s; (d) the rotational Doppler shift.

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Fig. 9.
Fig. 9. The rotational speed measurement results based on CVB for different sampling times. (a) t = 0.1s; (b) t = 1s; (c) t = 10s.

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Fig. 10.
Fig. 10. The rotational speed remote sensing results with different target surface. The parameters of target surface: ρh = 10−2 m, σh = 1 × 10−4 m; (a) the target surface; (b) the target speckle field; (c) the rotational Doppler shift. The parameters of target surface: ρh = 10−2 m, σh = 2 × 10−4 m; (d) the target surface; (e) the target speckle field; (f) the rotational Doppler shift. The parameters of target surface: ρh = 2 × 10−3 m, σh = 6.5 × 10−5 m; (g) the target surface; (h) the target speckle field; (i) the rotational Doppler shift.

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Fig. 11.
Fig. 11. The rotational speed remote sensing results with the probe beam of 1550 nm wavelength. The parameters of target surface: ρh = 10−2 m, σh = 2 × 10−4m: (a) the target speckle field; (c) the rotational Doppler shift. The parameters of target surface: ρh = 2 × 10−3 m, σh = 6.5 × 10−5 m; (b) the target speckle field; (d) the rotational Doppler shift.

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Fig. 12.
Fig. 12. Experimental setup. (BE, beam expander; A-SLM, P-SLM, amplitude spatial light modulator, phase spatial light modulator; M, reflector mirror; BS, beam splitters; L, lens; P, pinhole; PD, photoelectric detector.)

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Fig. 13.
Fig. 13. The 6-order CVB at different propagation distance. (a) z = 0.2 m; (b) z = 1 m; (c) z = 2 m.

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Fig. 14.
Fig. 14. The experiment results of rotational speed measurement based on RDE. (a) Time domain signal of the target scattered light field with a rotational speed of 125.6 rad/s; (b) time domain signal of the target scattered light field with a rotational speed of 188.4 rad/s; (c) the rotational Doppler frequency signal with a rotational speed of 125.6 rad/s; (d) the rotational Doppler frequency signal with a rotational speed of 188.4 rad/s;

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Equations (13)

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Δ f = α f 0 v c
Δ f = l λ 2 π r b f 0 Ω r b c = l Ω 2 π
f m o d = 2 Δ f = 2 | l | Ω 2 π
E ( x , y , 0 ) = ∑ m = 0 M − 1 ⁡ e x p [ − ( x − R cos ⁡ ( α m ) ) 2 + ( y − R sin ⁡ ( α m ) ) 2 w 0 2 ] e x p ( i n α m )
E ( r , φ , z ) = e x p ( − r 2 + R 2 w 2 ( z ) ) e x p ( − i k ( r 2 + R 2 ) 2 R ( z ) ) e x p [ i a r c t a n ( z / f ) − i k z ] × ∑ m = 0 M − 1 ⁡ e x p ( 2 R r cos ⁡ ( α m − φ ) w 2 ( z ) ) e x p ( i k r R cos ⁡ ( α m − φ ) R ( z ) ) e x p ( i n α m )
E ( r , φ , z ) = e x p ( − r 2 + R 2 w 2 ( z ) ) e x p ( − i k ( r 2 + R 2 ) 2 R ( z ) ) e x p [ i a r c t a n ( z / f ) − i k z ] × ∑ m = 0 M − 1 ⁡ ∑ l = − ∞ ∞ ⁡ I l [ ( 2 R w 2 ( z ) + i k R R ( z ) ) r ] e x p ( i l φ ) e x p ( i ( n − l ) α m )
∑ m = 0 M − 1 ⁡ exp ( i ( n − l ) α m )  =  ∑ m = 0 M − 1 ⁡ [cos ( n − l M 2 m π ) + i sin ( n − l M 2 m π ) ] = M
E ( r , φ , z ) = e x p ( − r 2 + R 2 w 2 ( z ) ) e x p ( − i k ( r 2 + R 2 ) 2 R ( z ) ) e x p [ i a r c t a n ( z / f ) − i k z ] × M ∑ p = − ∞ ∞ ⁡ I n − p M [ ( 2 R w 2 ( z ) + i k R R ( z ) ) r ] e x p ( i ( n − p M ) φ )
f m o d = 2 Δ f = | ( n − p M ) − ( n − p ′ M ) | Ω 2 π = | p ′ − p | M Ω 2 π , p ′ − p = ± 1 ± 2 ± 3 …
E r e = R t a r E i n exp ( i 2 k h ( ρ ) )
B h ( r 1 → , r 2 → ) = h ( r 1 → ) h ( r 2 → ) = σ h 2 exp ( | r 1 → − r 2 → | 2 / ρ h 2 )
Ω m i n = 2 π f m i n | 2 l | = 2 π | 2 l | t
Ω m a x = 2 π f m a x | 2 l | = π | 2 l | Δ t
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