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Abstract
Rotational Doppler effect (RDE), as a counterpart of the conventional linear Doppler effect in the rotating frame, has attracted increasing attention in recent years on rotational object detection. Many previous works have investigated the RDE based on the whole optical vortex field. In this work, we report on the RDE of the partially obstructed optical vortex and the corresponding rotational speed extraction method. Based on the orbital angular momentum (OAM) mode analysis theory, we establish the relationship between the OAM spectrum and the RDE frequency shift of fragmental optical vortex (FOV). The mechanism of the rotational speed extraction is analysed and validated by the numerical simulation and experiments. Further, a dual Fourier transformation method is proposed to accurately obtain the rotational speed which successfully overcomes the problem of the discrete distribution of the RDE signals. Our work may be useful for practical remote sensing based on the optical RDE metrology.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Since Allen et al. established that light beams with helical phase-fronts, described by a transverse phase structure of $\exp (il\varphi )$, carry an orbital angular momentum (OAM) equivalent to $lh$ per photon [1], optical vortex (OV) associated with OAM has established itself as one of the most interesting of optical modes [2–4], with relevance to optical manipulation [5,6], imaging [7,8], quantum optics [9,10], optical communications [11,12], remote sensing [13–16] and elsewhere [17–19]. Especially, the spiral phase of the OV has been proved to be very sensitive to the rotation motion due to the rotational Doppler effect (RDE) and thus shows a great potential for rotating object detection [20–23]. It is well known that when there is a relative motion between the observer and the wave source, the observed wave will experience a frequency shift, which is called the linear Doppler effect. Similarly, when there is a relative rotation motion between the angular momentum carrying beam and the object, the RDE frequency shift of $l\Omega /2\pi $ can be observed [20]. To date, RDE has been widely used for probing the spinning molecule systems [24], macroscopic objects [25,26] and so on [27].
In general, the OV beam has an intensity distribution of ring shape or petal-like shape for the single or superposition mode [28], respectively. Different from the traditional Gaussian distributed laser, the OV possess a central dark core in its intensity profile which is defined by the phase singularity, and its intensity distribution has a well circular symmetry [29–31]. When the OV is employed to detect the rotating object, it has been proved that the relative pose between the beam axis and the rotating axis of object has a significant influence on the RDE [32]. From the perspective of the OV beam itself, parallel displacement and angular tilt will change the projection of the OAM [33], therefore causing the RDE frequency spreading, and the corresponding rotational speed extraction method is also different from the coaxial incidence condition [34–36]. In most of the previous RDE based rotating object detection experiments, the OV beam employed has a complete light field distribution. In practice, it is common to see the condition that the beam is blocked or not fully illuminated on the object, especially in long-rang detection conditions. For example, the fan blades are always wrapped inside the frame, the rotors of engines are arranged inside the machine, to name a few. In previous works, the detection of the opening angle and the opening orientation of the stationary object by OAM carrying beams have been realized [37]. But how to realize the rotational speed measurement when the OV cannot fully illuminate the rotating target is a significant issue for the application of the RDE metrology.
The OV beams that do not have a complete intensity field can be collectively called fragmental optical vortex (FOV). Study of the FOV or obstructed vortex beams started over 20 years ago [38]. Many interesting effects can be induced by such light fields, including visible rotational Gouy phase shifts [39,40], measurable quantum correlations [41], and ray-wave dualism [42]. Especially, the vortex spectra, the orbital angular momentum (OAM), and the informational entropy of the perturbed beam were measured. Among these measurement method, it is an effective way to observe the OAM spectrum by measuring the RDE frequency shift spectrum [43]. People found that relatively small angular sector perturbations have almost no effect on the OAM, but at large perturbation angles, the OAM decreases to almost zero and the entropy increases sharply [44,45]. Further, the rotational symmetry of the object also has a significant influence on the OAM spectrum of the probe beam [23]. However, in all these studies, the principle of the RDE produced by FOV as well as the rotating speed extraction method have not been investigated yet.
In this work, we first theoretically analysed the OAM distribution of the FOV, and the relationship between the OAM spectrum and the RDE frequency shift. Further, the rotation speed extraction mechanism based on the OAM spectrum is revealed. To overcome the shortcomings introduced by the broadened OAM spectrum of FOV, a rotation speed acquisition method based on the discrete RDE spectrum is proposed, and a dual Fourier transformation technique is proposed for the first time which can greatly improve the accuracy and simplicity of the rotation speed extraction. The proposed method may open new insights into the RDE of OV and pave a new way for the rotational object detection by employing the RDE-based metrology.
2. Theory
The concept of the FOV and its generation process can be simply depicted as a normal superposed OV beam passes through a section aperture (SA), as shown in Fig. 1. Here, we take the standard LG mode as the initial beam to generate the FOV. The beam before the aperture is a well-defined superposed LG mode with topological charge of $l ={\pm} 6$, its OAM power spectrum is shown in Fig. 1(a), the two components accounted for 50% respectively. The whole production process of the FOV is depicted in Fig. 1(b). After passing through the SA, three quarters of the OV field is blocked and leaves only a small part of the OV field passing through the aperture, and therefore the FOV is formed. The electrical field of the FOV field ${E_{FOV}}(r,\varphi )$ can be expressed by,
Fig. 1. Concept and generation of the fragment OV. (a) The OAM spectrum of the fully superposed OV. (b) The process of the generation of the FOV. (c) The OAM spectrum of the FOV along with angular ($l$) and radial ($p$) index. (d)∼(g) The OAM spectrum of FOV under different direction and size of the section aperture.
Based on the standard LG mode which is convinced as a set of complete orthogonal base of any optical field, the OAM spectrum of any light field can be calculated. There are two dimensions of parameters for a LG mode, namely, the topological charge l and the radial index p. Using the binomial decomposition, the process of the OAM decomposition can be expressed by,
When the morphology of the SA changes, the OAM spectrum of the corresponding FOV will be different. The corresponding OAM spectrum of the FOV under different size are shown in Fig. 1(d)∼(g). If the opening angle ($\varphi $) of the SA is constant, although the phase information of the OV is different, the orientation of the opening will not influence the distribution of the OAM spectrum. However, if the size of the opening angle of the SA is changed, the OAM spectrum will be more concentrated as $\varphi $ increase. The less the blocked area, the higher the mode purity of the FOV will be. Since only the OAM intensity spectrum is considered here, the corresponding conclusions are consistent with those reported in previous studies.
From the distributions of the OAM spectrum, we can see that there are several OAM components that exist in the FOV beam. Unlike the OAM spectrum of off-axis OV, the OAM spectrum distribution of FOV is not strictly symmetrical in terms of the set topological charges. It is worth noting that the interval between two adjacent modes is one in the OAM spectrum. The discreteness of the OAM components is formed because that the OAM carried by per photon is a discrete or quantized quantity, which is given by the topological charge multiplied by the reduced Planck constant and cannot be changed continuously. According to the mode transition theory of the RDE, the magnitude of the RDE frequency shift $\Delta f$ is determined by the OAM state shift between the probe and detected light [46], namely,
where ${l_1}$ and ${l_2}$ are the topological charge of the incident and the received light, respectively. $\Omega $ denotes the rotating speed of the object.When the superposed OAM beam which contains two conjugated OV components is employed as the probe beam, the observed RDE frequency shift will be doubled due to the opposite frequency shift and the beat frequency effect produced by the two components. Further, it is convenient to receive the zero order (${l_2} = 0$) OAM mode in practical measurement just by a single mode fiber. Therefore, the Eq. (4) can be simplified as $\Delta f ={-} {l_1}\omega$, where $\omega = \Omega /2\pi $ represent the rotating frequency.
For a constant rotating speed, the magnitude of the RDE frequency shift is produced by the rotating frequency multiplies the OAM components order of the incident beam. Such that, the spectrum of RDE frequency shift spectrum will have the same distributions as the OAM spectrum of the incident beam. The interval between the two adjacent frequency of the signals is $\Delta 2{{\cal F}} = |{(l \pm 1)\omega - l\omega } |= \omega$, which is the rotating frequency. Combined with a further signal processing method, the rotating speed of the object can be acquired directly. This is the main idea of our proposal. We have changed from the original way of measuring the magnitude of RDE to the current way of measuring the value of the RDE frequency interval. This kind of rotational speed extraction method is helpful to overcome the ambiguity caused by the discrete OAM spectrum distribution of FOV. Since the RDE frequency shift signal generated by the FOV is distributed discretely with the interval of $\omega $, the RDE frequency signal spectrum can be written as a complex exponential with period of $\omega $, i.e., ${x_{RDS}}(f) = x(f){e^{jn\omega f}}$. Fourier-transforming this periodic signal again, ${X_{RDS}}({f_2}) = 1{{\cal F}}\{{x(f){e^{jn\omega f}}} \}$, then the information of the frequency interval and the rotational speed can be acquired directly.
3. Experiments and results
3.1 Numerical simulation
To demonstrate the validity of the proposed method, we conduct both the numerical simulations and experiments with realistic rotating object. The process of the numerical simulations is exhibited in Fig. 2, the FOV is generated by the OV field superimposed with a digital diaphragm. The propagation of the FOV is calculated according to the geometrical optical transmission principle. Before interacting with the rotating object, the probe light transmits in the free space, while after interacts with the object, the echo light is captured by a lens with focal length of $f = 200mm$. The rotating object is mimicked by a random scattered light field which rotating at a certain rate (Visualization 1 shows the rotation of the object). Whenever the object rotates in a tiny step angle, we multiply the FOV by the rotating field, so that the intensity of the frequency shifted scattered light can be obtained (Visualization 2 shows the convergent scattered light field). The intensity of the echo light is then fast Fourier transformed (FFT) to obtain the RDE frequency shift. By comparison between the OAM spectrum and the RDE frequency shift signals, the relationship between the OAM components and the RDE can be observed.
Fig. 2. Concept of the numerical simulation process. The FOV first transmits in a certain distance which in our experiment is 200 mm in free space condition. The rotating speed is set 50 rps, combined with the topological charge of $l ={\pm} 20$, the corresponding RDE frequency shift is 2000Hz for a normal measurement. The echo light is received by a lens with focal length of 200 mm. The sampling rate in the simulation is set 10000 Hz.
The difference between the practical experiments and the numerical simulations is that we receive and analyze the whole echo light in the numerical simulations, but in the practical experiments, we can only receive a part of the echo light. The advantage of the numerical simulations is that there is no interference from noise signals and therefore the relationship between the OAM of FOV and the RDE frequency shift can be clearly reflected. However, the disadvantage is that the scattered light signal received in the actual experiment is not as good as the simulations.
Based on the numerical simulation method, we further conduct more calculations under the topological charge of $l ={\pm} 20$. As shown in Fig. 3 (a) ∼ (c), as the size of the SA become smaller, the proportion of components of OAM in state 20 gradually increases. Here, we only exhibit the components of the positive order, since the distribution of the components of in negative orders are the same as that of the OAM spectrum in positive states. By employing these kinds of FOVs, the corresponding RDE frequency shift can be calculated, the corresponding results are shown in Fig. 3(d)∼(f). For each condition, the calculated RDE frequency spectrum is in good agreement with the OAM spectrum. Note that, when the entire OV field is employed, there are only one frequency peak exist which is the same with the previous RDE based experiments. Compared with the OAM spectrum shown in Fig. 3(a) ∼ (c), there are some unexpected components of RDS signals in Fig. 3(d) ∼ (f). These unexpected components are rotational harmonic noise introduced by inhomogeneous scattering surfaces. Most importantly, the adjacent frequency difference for each measurement is the rotation frequency (50 Hz in the spectrum), which equals to the value of the rotation frequency of the object.
Fig. 3. Numerical experiments on the measurement of the rotational speed. (a)∼(c) the calculated OAM spectrum under different size of SA. (d)∼(e) the corresponding RDE frequency shift of under the FOV with rotational speed of 50 rps.
3.2 Experimental results and analysis
The feasibility of our proposal is also verified through the experiments, the experimental setup is shown in Fig. 4. In practical experiments, although the FOV can be generated by arranging a solid SA in the OV light path, the edge diffraction of the diagram will affect the detection results of the RDE. Therefore, we choose to use the digital method to generate the FOVs. Thanks to the device of the integrated spatial light modulator (SLM), the whole process can be realized by using holograms. In order to guarantee the quality of the FOV, the complex amplitude modulation method which combines both the phase and amplitude information of the light field is employed here. The final FOV hologram is composed by four parts, i.e., the spiral phase, amplitude distribution, blazed grating, and the section aperture, as shown in Fig. 4(a). Here, the spiral phase is designed through the superposition of the conjugate LG modes in OAM state $- l\ell$ and $+ \ell l$. The amplitude map is obtained in according to the cross section of the vortex field. The section aperture is obtained by a binary mask. The final hologram of the FOV is then uploaded on the screen of the SLM directly.
Fig. 4. Experimental arrangement. (a) The composition of the FOV hologram. (b) P: polarizer, BE: Beam expander, SLM: spatial light modulator, L: lens, Ap: aperture (normal circular aperture, not the SA), F: Bandpass filter, APD: avalanched photodetector, DAC: data acquisition card, PC: computer. The FOV is generated by uploading the designed hologram on the SLM. The rotational speed of the object can be precisely controlled.
The entire experimental arrangement is shown in Fig. 4(b). The laser source generate laser with a wavelength of 532 nm. After been filtered by the horizontal linear polarizer and expanded by the beam expander, the beam in linear polarization state and suitable size directly illuminates on the SLM. The light reflected from the SLM possess serval orders, with only the first order is the desired FOV. In the experiment, we let the probe beam illuminate on the rotating axis of the object to reduce the measurement errors introduced by the beam off-axis [33]. The incidence angle shown in Fig. 4(b) is relatively large to make the illustration clear, but the incidence angle in the actual experiment is very small to reduce the influence of the tilt angle.
The rotational object used in our experiments is a plane desk. The plane desk is arranged on a rotor with a steady speed accuracy of ${\pm} 1$rps. After interacting with the rotating desk, the scattered light is then received by a telescope. A bandpass light filter is arranged before the photodetector to filter out the environmental light. Then the echo frequency shifted light is converged by a lens and detected by the avalanched photodetector (APD). APD connects with a data acquisition card (DAC) and then the variation of the echo light intensity in time domine can be sampled for further analysis.
Based on the above experimental setup, we first conduct the experiments by employing different kinds of FOVs with the topological charge of $l ={\pm} 20$. The size of the SA is set $\varphi = \pi /2$, $\varphi = \pi $, and $\varphi = 3\pi /2$, respectively. In order to make sure the beam center coincide with the rotational axis. We first use the complete optical vortex to conduct the measurement in each group of experiments. By observing the frequency spectrum, we can realize whether the beam center is coincident with the rotational center. The corresponding RDE frequency shift measurement results by employing the three kinds of FOVs at an object rotational speed of ∼50 Hz are shown in Fig. 5 (a)∼(c). For different kind of FOV beams, the more the OV is blocked, the lower the signa to noise (SNR) of corresponding RDE frequency shift signal. Although the measured results are in good agreement with the OAM decomposition results, there is small differences between the simulated and the measured results. One reason is that the modulation effect of the object on the OAM components of the FOV is different between the simulation and the experiments. The bandwidth of the frequency spectrum is relatively wide when the size of the SA is large, while the signal bandwidth is relatively narrow when the size of the SA is small. The experimental results are in good agreement with the theoretical predications. For all the measurement results, the highest signal peaks are all around 2000Hz, which is exactly the value of the theoretical RDE frequency shift.
Fig. 5. Experimental detection results. (a)∼(c) The RDE signals under FOV beams with a quarter, a half and three quarters, respectively. These spectrums are obtained from one-time FFT of the captured time domine signals. (d)∼(f) The results of second time FFT on the first frequency spectrum. A signal peak with maximum amplitude is clearly visible for each result at constant rotational speed, with the peak value is the inverse of the rotating frequency.
Although the measured RDE signals match well with theoretical OAM spectrum and the predicated frequency shifts, it’s still not convenient to obtain the rotational speed. Based on the discrete and equally spaced distribution characteristics of the RDE signals produced by FOV, here we come up with a dual Fourier transformation method. Since the equally spaced frequency peak signals have a periodic characteristic, by implementing a second time fast Fourier transformation of the frequency spectrum, the signal interval information, i.e., the rotational frequency can be obtained directly. After a second time FFT of the frequency spectrum, we will get the symmetric reversal of the rotational speed signal. Therefore, the rotational speed can be calculated as $\omega = 1/{f_2}$, where ${f_2}$ is the frequency signal in the second time frequency spectrum. As shown in Fig. 5 (d)∼(f), a single strong peak can be observed in each second time FFT frequency spectrum. Compared with the first time FFT results of the frequency shifted light, the SNR of the dual Fourier signal is much higher. Furthermore, in the results of the dual Fourier transform, there is essentially no noise signal other than the main speed peak signal.
Under different rotational speed, we further conducted multiple measurements by using FOV, as shown in Fig. 6. The results show a good agreement between the theoretical and the measured values. The blue dash line represents the theoretical predications of the dual FFT results, and the blue dot is the measurement values. After inverting the dual FFT transformed results, the exact rotational speed values are obtained, as shown by the red dots in the diagram. Since the preset rotational speed is proportional to the rotational frequency, the red line is straight. However, the rotational speed is in inverse proportion relation to the dual FFT frequency. Therefore, the blue line is curved. The corresponding measured rotational speed are in good agreement with the set values. A dual FFT processing of the signals can make the acquisition of the rotational speed very convenient and accurate.
Fig. 6. Rotational speed measurement results by FOV under different rotational speed. The results of obtained by dual FFT are the inverse of the rotational speed. Corresponding measured result matches well with the set value.
Based on the OAM decomposition method, the relationship between the OAM spectrum of the FOV and the RDE frequency shift is established and the rotational speed of spinning target can be precisely obtained. However, there are still some limitations. First of all, when the vortex beam encounters an obstruction during transmission, the FOV cannot propagate in a long-range due to the energy flow of the vortex light [47]. For the superposition state FOV, its two components will diffract in two opposite circular directions. It’s mode purity and coherence state will also change accordingly, which will affect the detection accuracy. Secondly, the coaxial between the OV axis and the rotating axis is required. Because the OAM spectrum distribution is different under the off-axis condition which will also affect the rotation object detection results. Even though the OAM spectrum will change under different conditions, its discrete nature will not change. Therefore, the speed extraction method based on the dual FFT is still powerful and effective.
4. Conclusion
To summarize, we investigated the RDE of the FOV based on the OAM spectrum analysis method. The relationship between the OAM spectrum and the RDE frequency spectrum is established and based on the discreteness of the OAM spectrum, we proposed the rotational speed extraction method for the case of FOV incidence in according to the interval information of the RDE signals. Further, we proposed a dual FFT method for the extraction of the discrete RDE signals which has been proved that can improve both the accuracy and the simplicity of the speed extraction. The corresponding methods have been verified by both the numerical simulations and the experiments. We believe the proposed method may open new insights into the RDE of OV and promote the application of the RDE metrology techniques.
Funding
Key research projects of foundation strengthening program of China (2019-JCJQ-ZD); National Natural Science Foundation of China (61805283, 62173342).
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.
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