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Abstract
Recent years have witnessed a growing research interest in the rotational Doppler effect associated with orbital angular momentum of light, emerging as a powerful tool to detect rotating bodies in remote sensing. However, this method, when exposed to the turbulence in a realistic environment, has some severe limitations, leading to the unrecognizable rotational Doppler signals overwhelmed in background noise. Here we put forward a concise yet efficient method that enables the turbulence-resilient detection of the rotational Doppler effect with cylindrical vector beams. Specifically, by adopting the polarization-encoded dual-channel detection system, the low-frequency noises caused by turbulence can be individually extracted and subtracted, and thus mitigate the effect of turbulence. We demonstrate our scheme by conducting proof-of-principle experiments, whose results manifest the feasibility of a practical sensor to detect the rotating bodies in non-laboratory conditions.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The Doppler effect refers to the change in wave frequency when there is a relative motion between a moving source of waves and its observer. It was discovered by Austrian physicist Christian Johann Doppler who described how the color of starlight changed with the movement of the star in 1842 [1]. Since it was discovered, the Doppler effect has played a great role in various disciplines, such as robotics, radar, medical, astronomy and satellite navigation [2–6].
With the discovery of the spin and orbital angular momentum (OAM) of light beams, the rotational Doppler effect, arising from the relative rotational motion of objects, has gradually developed [7–13]. The rotational Doppler effect refers to the relative angular velocity between the source and the detector leading to a frequency shift which is proportional to the change of orbital angular momentum between the output photon and input photon [7]. It was first observed by Garetz and Arnold [11], who rotated half-wave plates with angular velocity $\omega $ to imprint a frequency shift $2\omega $ to circularly polarized light. In essence, this frequency shift comes from the spin angular momentum of photons and is explained through the dynamically evolving geometric phase in the light of the Poincáre sphere [12]. Then, Jourtial and coworkers observed the frequency shift $\ell \Omega $ with an orbital angular momentum of $\ell \hbar $ using millimeter wave [10], which expanded rotational Doppler effect from spin angular momentum to OAM. Recent years also witnessed a growing interest in the vector beams. As the vectorial solution of the electric field Helmholtz equation, the polarization orientation of a vector beam is distributed periodically on its cross section. One typical example of vector beams is the cylindrical vector beam (CVB) that possesses a cylindrical symmetry in polarization distribution [14]. The CVBs have found many applications optical-trapping, optical imaging and high-resolution metrology [15–17]. Fang and coworkers explored the vector beams for demonstrating a new vectorial Doppler effect, which allowed for a full determination of the velocity vector of a moving particle [18]. Georgi and coworkers designed a reflective-type plasmonic metasurface q-plates to achieve spatially variant Doppler shift [19]. And Cheng et.al. amplified the rotational Doppler effect caused by rotational q-plate with third harmonic generation (THG) [20].
The real potential of the rotational Doppler effect lies in its practical applications for remote sensing [21]. Based on structured light, Hu et al. combined the linear Doppler effect and rotational Doppler effect and succeeded in detecting a cooperative target’s longitudinal and angular speed [22]. By using short-time Fourier transform, Zhai et al. proposed and demonstrated a new detection scheme to infer the angular acceleration based on rotational Doppler effect [23]. It is noted that most of the experimental verification of rotational Doppler effect were conducted in a laboratory scale. After establishing a 120-m-long free-space link between two buildings, Zhang and coworkers made the first step towards measuring the rotational Doppler shifts in a realistic environment [24]. For remote sensing applications in free space, there are still many factors that affect the transmission of Doppler signal over a long distance. In particular, this method, when exposed to the turbulence in a realistic environment, has some severe limitations, leading to the unrecognizable rotation Doppler signals overwhelmed in background noise.
The propagation of laser beams through turbulence has attracted much attention because of its wide range of applications, such as free space optical communications, remote sensing, imaging systems, laser radar and so on [25]. Turbulence strongly influences beam characteristics, including intensity, coherence, polarization, and scintillation [26–28]. The irradiance pattern, degree of polarization, and scintillation index of the beam would be influenced in an atmosphere or ocean with weak or strong turbulences, and thus weakening the rotational Doppler signals. Here, we put forward a concise yet efficient method that enables the turbulence-resilient detection of rotational Doppler effect. Our technique is based on the use of CVBs of high-order topological charges. We succeed in mitigating the effect of turbulence and identifying the rotational Doppler signals, by conducting proof-of-principle experiments. Our work holds promise for building a practical sensor to detect the rotating bodies in non-laboratory conditions.
2. Principle and concepts
Instead of using the scalar beams, we employ the CVBs to detect rotational Doppler signals. In the cylindrical coordinates, the light field can be expressed as a Jones vector,
Fig. 1. Polarization and intensity patterns of the cylindrical vector beams with different topological charges: (a) ${\ell _0} = 1$, (b) ${\ell _0} = 2$, (c) ${\ell _0} = 5$, and (d) ${\ell _0} = 10$.
A light beam, when being exposed to the turbulence in a realistic environment, will inevitably suffer from OAM mode crosstalk during free-space propagation [29]. In our scheme, the CVBs could offer the polarization-encoded dual channels for signal detection, comparison and analysis. In one path, a polarizer is used to filter out the rotational Doppler signals, as a result from the projective measurement of ${\pm} \ell $ OAM superposition. While in the other path, it is without polarization filtering, i.e., polarization insensitive. By considering the fact that both paths encounter the identical turbulence and after identifying and removing the unexpected signals due to the turbulence, we can finally extract a much clearer rotational Doppler signal.
When the CVBs passing through the atmospheric or water turbulence, it will be disturbed by refractive index fluctuations, which will cause phase and intensity fluctuation, and cause the OAM state spreading [30]. Mathematically, when the CVBs propagate through the water turbulence, the OAM state of the CVBs will be spread to its neighboring modes [31], $|{{\ell_0}} \rangle \mathop \to \limits^{\widehat T} \sum\limits_\ell {{\alpha _\ell }} |\ell \rangle ,$ where $P(\ell )= {|{{\alpha_\ell }} |^2}$ is the probability of finding the scattered light in the OAM mode $\ell $. In general, we can study the behavior of the CVBs propagating in turbulent environments based on the Rytov approximation method [31]. The complex amplitude of the CVBs after traversing the turbulence should be rewritten as a superposition of the orthogonal basis [32],
In our experiment, the rotation Doppler effect is induced by a rotating particle of angular velocity Ω, that moving around the CVBs. In our measurement scheme, the rotational Doppler effect is induced by a rotating micro-particle (∼micrometer level). And the diameter of the micro-particle is much smaller than the size of the vector light field (∼millimeter level). Then each OAM mode within the reflected light will acquire a rotational frequency shift individually, $f = \ell \Omega /\pi $. Note that, unlike the single input mode ${\ell _0}$, here $\ell$ generally represents a mode spectrum as a result of mode spreading due to turbulence. Specifically, by considering the micro-particle rotates in the light field along a circular trajectory with a radius of $\rho $. And at time t, the position of the particle in the light field can be expressed as $({\rho ,\varOmega t + {\varphi_0}} )$, with $({\rho ,{\varphi_0}} )$ being the starting point, which is generally set as $({\rho ,\varphi = 0} )$. Thus accordingly, the scattered or reflected light by particle can be expressed as,
While for the other path without a polarizer, we merely have the total intensity as,
Then, after Fourier transform, we can obtain the frequency beats as
There are two important features that can be seen from Eqs. (6) and (7): First, if there is no turbulence, we have ${\alpha _{{\ell _1}}} = {\delta _{{\ell _1},{\ell _0}}}$ and ${\alpha _{{\ell _2}}} = {\delta _{{\ell _2},{\ell _0}}}$ such that Eq. (6) can be reduced to the trivial case, $|{{f_{\bmod }}} |= {\ell _0}\Omega /\pi$, and at the same time, we have ${I_R}(t )= \textrm{constant}$ from Eq. (5). Second, ${I_{PF}}(t)$ carries both the rotational Doppler signals and the noise signals, while ${I_R}(t)$ bears the identical noise signals only. Thus, our method can significantly eliminate the noise signals caused mainly by turbulence and possibly by other imperfections. Notably, there still contains other high-frequency noises in Eq. (6) and (7), however, under weak turbulence, the OAM spectrum still remains concentrated at the original topological charge ${\ell _0}$. That is, we can acquire a clear doppler signal at $|{{f_{\bmod }}} |= {\ell _0}\Omega /\pi$. Noted that, here, we simulate the turbulence by using a water tank ($15\,\textrm{cm} \times 15\,\textrm{cm} \times 15\,\textrm{cm}$) which can generate water whirlpool with a rotating magnetic stirrer. Compared with the wavefront distortion in conventional turbulence model, here, the water turbulence or water agitation mainly causes irregular disturbances in light intensity over time. Here, the rotational Doppler effect is induced by a rotating micro-particle. If we keep the water still, we would obtain a perfect cosine oscillation signal, which just corresponds to the intensity variation of the vector light field in the angular direction. And due to the water agitation, the perfect cosine oscillatory signal will mix with the irregular light intensity disturbance signal. And similar with the digital spiral imaging, any time signal can be decomposed into a combination of sine and cosine signals with different periods through the Fourier Transform. In this regard, irregular disturbances in light intensity caused by water turbulence or agitation in our paper can be equivalent to the broadening of the OAM spectrum in conventional turbulence model.
3. Experimental setup and results
The optical system of our proof-of-principle experimental setup is sketched in Fig. 2. The laser beam from a 633 nm He-Ne laser (Thorlabs HNL020LB), after expanded by a $4f$ optical system consisting of two lenses (L1 and L2), illuminates the q-plate and generates the desired CVBs, as described by Eq. (1). Thus, the beam will carry information caused by the turbulence when passing through this tank. The charge couple device (CCD) camera is used to observe the variation of cross section intensity of the vector light after passing through the turbulence. After the simulated turbulence, the vector light illuminates the digital micromirror device (DMD) which we emulate a rotating particle with the radius of 8.220 mm. Here, the beam profile is appropriately adjusted to match the trajectory of the particle when illuminating the DMD. And then the reflected light is divided into two paths via the beam splitter (BS2). One for the reference beam is collected by the photodetectors 1 (PD1), and the other one is detected by the photodetectors 2 (PD2) after selecting the horizontal polarization direction through the polarizer (Pol.). The acquired signals are filtered to remove unwanted noise and then are fast Fourier transformed (FFT) to obtain its frequency spectrum.
Fig. 2. Experimental setup. (HWP: half-wave plate. L: lens. BS: beam splitter. CCD: charge coupled device. DMD: Digital micromirror device. Pol.: polarizer. PD: photodetector.)
Here, we employ the CVBs of different topological charges, ${\ell _0} = 1$, ${\ell _0} = 2$, ${\ell _0} = 5$, and ${\ell _0} = 10$, to conduct the proof-of-principle experiments. We show in Fig. 3 the experimental results of CVB with ${\ell _0} = 1$. Figure 3(a) is the time-intensity diagram of the detection signal ${I_{PF}}(t)$ (with polarization filtering) and the reference signal ${I_R}(t)$ (without polarization filtering), respectively. After the Fourier transform, we obtain their spectrum diagrams, ${\tilde{I}_{PF}}(f)$ and ${\tilde{I}_R}(f)$, in Fig. 3(b), where the subscript “PF” and “NPF” denote the path with polarization filtering and no polarization filtering, respectively. It can be seen clearly that the detection signal (red) has two distinct peaks, at $f = 27.78\textrm{ Hz}$ and $f = 55.55\textrm{ Hz}$, respectively. So, when viewed separately, we cannot well identify which one is the rotational Doppler frequency shift. However, we find that the reference signal (blue) has only a single peak at $f = 27.78\textrm{ Hz}$. Of particular importance is that the reference path does not contain any rotational signals, as it is measured without polarization filtering nor OAM superposition. Thus, an easy comparison concludes that $f = 27.78\textrm{ Hz}$ is merely the noise signal mainly due to turbulence. In other words, we can then identify that $f = 55.55\textrm{ Hz}$ is exactly the rotational Doppler frequency shift, arising from the rotation of the particle. This identification can be further confirmed by the relative signal strength at $f = 27.78\textrm{ Hz}$, that is, the noise signal strength without polarization filtering is almost fourth that with polarization filtering, which is consistent with our theoretical deduction. Then, we can perform the noise signal subtraction,
which yields a much clear rotational Doppler spectrum in Fig. 3(c). By using the relation of $f = {\ell _0}\Omega /\pi$, we can further deduce the angular velocity of the rotating particle, $\Omega = 174.52\textrm{ rad/s}$, which is in fairly good agreement with the DMD setting in our experiment.Fig. 3. Experimental results with a CVB of ${\ell _0} = 1$. (a) Time-intensity curves with and without polarization filtering. (b) Fourier transform spectra. (c) Extracted rotational Doppler spectrum after noise subtraction.
We also examine the case for a detection CVB of ${\ell _0} = 2$ and present the experimental results in Fig. 4. Figure 4(a) are time-varying optical power recorded by PD1 and PD2 in two paths, respectively. Obviously, both two curves deviate from the cosine-like behavior, as also indicated by the more complex frequency spectra in Fig. 4(b) and 4(c). In the path with polarization filtering, there are three dominant signals at $f = 27.78\textrm{ Hz}$, $f = 55.55\textrm{ Hz}$, and $f = 111.11\textrm{ Hz}$, respectively. Thus, it is difficult for us to identify which peak is exactly the rotational Doppler shift. However, after a similar comparison between two spectra in Fig. 5(b), we reach a conclusion that the first two peaks merely come from the turbulence, while the beat frequency shift $f = 111.11\textrm{ Hz}$ is just the rotational Doppler signal. We can then deduce the rotational velocity as $\Omega = 174.53\textrm{ rad/s}$, again confirming the good effectiveness of our method.
Fig. 4. Experimental results with a cylindrical beam of ${\ell _0} = 2$. (a) Time-intensity curves with and without polarization filtering. (b) Fourier transform spectra. (c) Extracted rotational Doppler spectrum after noise subtraction.
Fig. 5. Experimental results with a CVB of ${\ell _0} = 5$. (a) Time-intensity curves with and without polarization filtering. (b) Fourier transform spectra. (c) Extracted rotational Doppler spectrum after noise subtraction.
In our further experiments, the CVBs of ${\ell _0} = 5$ and ${\ell _0} = 10$ are employed to detect the rotation, and the experimental results are shown in Fig. 5 and Fig. 6, respectively. In both cases, the noise signals related to turbulence can be well suppressed, according to Eq. (8). Then, the rotational Doppler signals can be clearly extracted at $f = 277.78\textrm{ Hz}$ and $f = 555.55\textrm{ Hz}$, respectively. We can then calculate the rotating velocities almost the same are $\Omega = 174.53\textrm{ rad/s}$. According to the relation of relation of $f = {\ell _0}\Omega /\pi$, we know that, by using a high OAM number, ${\ell _0}$, we can amplify the angular velocity of the particle by a factor of ${\ell _0}$. Leading that one can accurately measures the angular velocity by using the large topological charge. For example, by employing the detection CVBs of ${\ell _0} = 1$ and ${\ell _0} = 10$, we can conclude that the measuring precision and sensitivity for ${\ell _0} = 10$ in Fig. 6, in comparison with that for ${\ell _0} = 1$ in Fig. 3, can be improved by one order of magnitude. However, on the other hand, because the disturbance of turbulence on high-order vector light fields is more significant and the limitation of vortex plates with higher topological charges, the noise in the experiment is also increase with topological charges.
Fig. 6. Experimental results with a CVB of ${\ell _0} = 10$. (a) Time-intensity curves with and without polarization filtering. (b) Fourier transform spectra. (c) Extracted rotational Doppler spectrum after noise subtraction.
4. Conclusion
In summary, we have proposed a concise yet efficient method of realizing a turbulence-resilient detection of rotational Doppler effect, which enables the fast identification of rotational Doppler signals from the noise background. Our technique relies on the use of CVBs with high-order topological charges, which enables the dual-channel polarization sensitive and insensitive measurements. We have conducted proof-of-principle experiments, in which the turbulence is simulated by using an acrylic tank which can generate water whirlpool with a rotating magnetic stirrer. We have succeeded in identifying the rotational Doppler signals from the background noise signals, with the used topological charges up to ${\ell _0} = 10$. Our present work represents a significant step toward constructing a practical sensor to detect the rotating bodies remotely under the weak turbulence environment. However, it’s also worth noting that as the increase of turbulence, our present proposal would also confront disturbances caused by high-frequency noise. We would leave the possibility for extending our proposal into strong turbulence to future studies.
Funding
Program for New Century Excellent Talents in University (NCET-13-0495); Natural Science Foundation of Fujian Province (2015J06002, 2021J02002); National Natural Science Foundation of China (12034016, 61805283); Key Research Projects of Foundation Strengthening Program of China (2019-JCJQ-ZD).
Disclosures
The authors declare no conflicts of interest.
Data availability
The data generated and analyzed during this study are available from the corresponding author upon reasonable request.
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