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Abstract
The rotational Doppler effect (RDE) of optical vortex which can be used to detect the rotation speed, has become a well-known phenomenon and a hot topic of research in recent years. However, because the beam axis must be coaxial with the rotational axis of the object, it can only be used to detect cooperative targets in practical application. Here, we provide a novel approach for measuring rotational speed under light non-coaxial incidence relative to the rotating axis that uses the adjacent frequency difference of rotational Doppler shift signals. Theoretically, the rotational Doppler shift is proportional to the OAM mode of the incident beam, and the nature of the OAM carried by each photon is a discrete or quantized quantity under off-axis conditions leading to the discrete distribution of the Doppler shift signals. Experimentally, by extracting the difference between two adjacent Doppler shift signals, the rotating speed of the object can be determined. Based on our method, the rotational speed of the object can be measured precisely without the pre-known information about the position of the rotating axis. Our work supplies a significant complement to the conventional RDE theory and we believe it may promote the realistic application of the optical RDE-based metrology.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Since the concept of angular (or rotational) Doppler effect has been first reported in 1981 [1], there has been a plethora of research focused on this phenomenon. At first, the angular Doppler effect is thought associated with the spin angular momentum (SAM) which is aroused by the circular polarization state of light [2]. Indeed, there is another kind of angular momentum that exists in photons, namely, orbital angular momentum (OAM) [3]. In 1992, Allen et al. confirmed the OAM carried in a Laguerre-Gaussian (LG) beam, which opens a new era for people to investigate and utilize the OAM of light [4]. Structured light carrying OAM can be called the optical vortex (OV) which has a spiral phase defined by $\exp (i\ell \varphi )$, where $\ell $ is an integer denotes the topological charge and $\varphi $ is the angular coordinate [5,6]. The helical phase of the OV beam also defines a phase singularity in the center of the vortex field, and therefore forming a dark core of the light’s intensity field [7–10]. Akin to the angular Doppler effect associated with SAM, people then find the light carrying OAM can produce rotational Doppler frequency shift [11]. The amount of Doppler shift caused by OAM is found many times larger than the Doppler shift associated with the SAM because the momentum carried by each photon in an OV beam is $\ell \hbar $ while the SAM is only $\sigma \hbar $($\sigma \textrm{ = } \pm 1$, right- or left- circular polarization) [11,12]. After that, the rotational Doppler effect (RDE) is investigated widely by researchers from its mechanism to its application [13–20].
In 1994, Allen et al. derived the RDE frequency shift $\ell {\omega _r}$ generated by the rotating atoms, according to the interaction between OAM carrying photons and atoms, where ${\omega _r}$ is the rotational frequency of the atom [11]. Nienhuis and Courtial further observed the RDE through the mode-inversion rotating lens and the rotating Dove lens, respectively [21,22]. In 2013, Lavery et al. systematically reported their seminal work on detecting rotational speed based on RDE by employing the phase conjugated OAM light [23]. In the following years, there has been a lot of research work on the detection of rotation, from the rotating micro molecular to the macroscopic object [24–28]. Not only the rotational speed, but all the rotational parameters like rotational direction and accelerations can also be measured [29–32]. However, all the above research is based on the condition that the axis of the OV beam must be coaxial with the rotating axis of the target. This requirement is difficult to realize in the realistic measurement process, especially in the detection of the non-cooperative targets, which significantly restrict the application of the RDE from lab scale to outdoor scale.
To solve the above problem, we investigate the OAM components carried by the OV beam under the noncoaxial incidence conditions in this article. Since the RDE frequency shift is proportional to the OAM change before and after OV beam interacts with the rotational object, combined with the discrete or quantized quantity nature of the OAM carried by each photon, the RDE frequency shift signals must be discrete distribution and related to the OAM spectrum of the probe beam. Therefore, by extracting the difference between two adjacent Doppler frequency signals, the rotating speed information of the object can be determined. A proof-of-concept experiment is conducted and the rotational speed of the object can be measured precisely without the requirement of the optical axis be aligned with the rotating axis.
2. Theory
For a classical LG mode, it’s electric field in a cylindrical coordinate system $(r,\varphi ,z)$ can be given by [33],
The above LG mode expression is given in the cylindrical coordinate system when the propagation direction of beam is coaxial with the z axis of the system. When there is a lateral distance d and the skew angle $\theta $ between the z axis and the OV beam propagation axis as shown in Fig. 1, the optical field of the LG mode can be expressed in a cylindrical coordinate system centered on the axis of rotation as $E_{LG}^{d,\theta } = L{G_{p,\ell }}(r^{\prime},\varphi ^{\prime},z)$. The radial and angular coordinates under noncoaxial incidence can be given by,
where $(x,y)$ denotes the coordinate in the Cartesian coordinate system as shown by the blue line in Fig. 1. $dx$ and $dy$ are the lateral offset distance and defined by $d = \sqrt {d{x^2} + d{y^2}} $. Since the OV beam has a circular symmetry around the rotational axis, the skew angle $\theta $ between the propagation axis and the z axis can be used to define all the oblique incidence conditions. By the combination of d and $\theta $, the off-axis LG beam can be redefined by $r^{\prime}$ and $\varphi ^{\prime}$. The relationship between on-axis and off-axis OV beam is shown as Fig. 1.
Fig. 1. The concept of the noncoaxial incidence LG beam. The lateral displacement and the skew angle relative to the rotating axis is given by $dx,dy$, and $\theta $.
Note that the off-axis OV beam is different from the standard LG beam which forms a complete and orthogonal basis for paraxial light beams [34]. The off-axis OV beam is not the fundamental LG mode in the new coordinate system. Therefore, the OAM spectrum of the off-axis OV can be decomposed by the standard LG mode in the following way,
If implement the misaligned OV beam $E_{LG}^{d,\theta }$ into Eq. (5), the corresponding OAM spectrum of off-axis OV can be calculated. As shown in Fig. 2(a)–(f), the OAM spectrum under aligned (Fig. 2(a)), lateral offset (Fig. 2(b)), skew angle (Fig. 2(c)), both lateral offset and skew angle exist simultaneously (Fig. 2(d)–(f)) conditions are presented. The topological charge of the OV is set $\ell = 10$. From the distributions of the OAM spectrum, we can see that there are serval OAM components that exist in the off-axis OV beam compared to the single OAM component of the OV beam without misalignment. Moreover, the interval between two adjacent modes is one in the OAM spectrum, except for the existence of the skew angle condition which is two. 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.
Fig. 2. The simulated OAM spectrum distribution of the LG mode beam with topological charge of $\ell \textrm{ = }10$. (a) is the on-axis OAM spectrum of LG mode. (b)∼(f) present the OAM spectrum under off-axis conditions.
According to the mode transition theory of the RDE [35], the magnitude of the RDE frequency shift ${f_{\bmod }}$ is determined by the OAM shift of the incident and the scattered light, namely,
where ${\ell _1}$ and ${\ell _2}$ are the topological charge of the incident and scattered light, respectively. In most measurement cases, only the Gaussian mode, i.e., ${\ell _2} = 0$ mode is detected and analyzed for the convenience. Such that the magnitude of the rotational Doppler shift only depends on the OAM mode of the incident light and the rotational speed, the corresponding frequency shift therefore becomes ${f_{\bmod }} = {\ell _1}\Omega /2\pi $. Since single-mode beam cannot produce beat frequency phenomenon and thus frequency shifts cannot be observed. For practical measurements we employ the superposed OV with opposite topological charges as the probe beam. These opposite modes produce opposite RDE frequency shifts and are capable of producing a beat frequency that is exactly twice the frequency shift of the single-state mode.When the incident light illuminates the rotating target under off-axis conditions, the scattered light would experience the frequency shift in according to the OAM spectrum shown in Fig. 2. Consequently, the broadened and discrete frequency shift signals would be generated in the frequency spectrum of the echo light. Each frequency shift signal peak is generated by the rotating frequency multiplies one of the OAM components of the incident beam. Therefore, the interval between two adjacent frequency signals is $\Delta {f_{\bmod }} = (\ell + 1){\omega _r} - \ell {\omega _r} = {\omega _r}$, where ${\omega _r}\textrm{ = }\Omega /2\pi $ denotes the rotating frequency of the object. This is the main idea of our proposal. Note that, this frequency interval is not bothered by the relative pose between the incident light and the rotating axis, only decided by the discrete nature of the OAM carried by each photon.
3. Experimental results
To verify the effectiveness of our proposal, we designed the proof-of-concept experiment and the experimental setup is shown in Fig. 3. The left side of the setup is the OV beam generation part based on the spatial light modulator (SLM). The laser source generates the laser with wavelength of 532nm. After been expanded, the laser illuminates the SLM to realize the phase modulation according to the holograms which is designed based on Eq. (1). A 4f filter system is arranged on the reflection direction of the SLM to filter the first order of the modulated beam which is the desired LG mode. The light after the 4f system is ready to detect the rotating object.
Fig. 3. The experimental arrangement of the rotational speed detection system upon non-coaxial light incidence. L (1,2…5): lens. SLM: Spatial light modulator. M: Mirror. $\theta $ shows the skew angle between rotating and beam propagation axis.
The object to be detected here is a plane rotating desk, whose rotational speed is precisely controlled by the motor (Thorlabs, DDR25). As shown in the bottom right of Fig. 3, the whole target is arranged on a biaxial displacement table which can be adjusted horizontally(x-axis) and vertically (z-axis), and rotating a small angle around z-axis. By doing so, the relative pose between the illumination OV beam and the rotation axis can be adjusted arbitrarily. Subsequently, a focal lens(L5) is set on the other side of the rotating target to collect the scatter light from the object. An avalanche photodiode detector (APD) is arranged on the focal point of L5 to capture the light signal. The rest instruments that are not presented in the diagram are data acquisition card and computer, whose main function is to record and analyze signal.
Based on the above experimental setup, we first conduct the experiment under the basic axis alignment condition. The OV beam propagation axis is adjusted as possible as we can to be coaxial with the rotating axis of the rotating object. The conjugated superposition OV beam with topological charge of $\ell ={\pm} 20$ is employed here as the detection beam. Fig. 4(a) exhibits the original time domine signal and its fitting curve, and Fig. 4(b) gives the frequency spectrum after Fourier transformation of Fig. 4(a). Theoretically, the rotating speed $\Omega = 10\pi $rad/s yields a rotational Doppler shift of 200Hz under the coaxial condition between the beam propagation axis and the rotating axis. As shown in Fig. 4(b), there appears one clear signal peak in the frequency spectrum with a very small error relative to the theoretical predication. The frequency shift is consistent with the OAM mode analysis results shown in Fig. 2(a), and consistent with the results introduced in previous report [23].
Fig. 4. Measurement results under coaxial incidence condition. (a) The time domine signal of the scatter light and its fitting curve. (b) The frequency domine signal.
Subsequently, we conduct the measurements under the non-coaxial condition of a lateral displacement exist in x-axis direction ($dx \ne 0$, $dy = 0$, $\theta = 0$). Since the rotational motion has a circular symmetry, the condition of the lateral displacement along other direction has the same consequence of the x-axis direction. The OV beam with topological charge of $\ell ={\pm} 20$ is used as the detection beam, and the rotational speed of the object is $\Omega = 10\pi $rad/s. The intensity profile of the probe OV is shown in Fig. 5(a), which has 40 petals formed by the interference of the two conjugate OV modes. The relative position of the light spot and the rotating desk is shown in Fig. 5(b), there is a deviation between the rotating center and the spot center but the rotating center is still within the light field of the OV field.
Fig. 5. Measurement results under non-coaxial condition of lateral misalignment. (a) The superposed light field of the OV beam. (b) The relative position of the light spot and the rotating desk. (c) 14 frequency peaks are chosen to calculate the mean value of the rotating frequency.
The measured result is present in Fig. 5(c). When there is a lateral displacement between the two axes, the rotational Doppler shift arises as a wide frequency spectrum. The larger the lateral displacement, the larger the width of frequency spectrum of the signals. On one hand, this frequency shift signal broaden phenomenon is consistent with OAM mode decomposition results presented in Fig. 2. The rotational Doppler frequency shift is directly related to the OAM mode of the incident beam and the scatter light. Under the non-coaxial illumination, the OAM mode of the incident beam would spread over the neighboring OAM mode. On the other hand, based on the small scatterer model which has been discussed in Ref. [36] and Ref. [37]. Since the velocity of each small scatterer within the vortex field is different, the corresponding Doppler frequency shift is no longer a fixed value, and therefore causing the spreading of the signal frequency spectrum. It should note that, in our previous work, the rotational speed can be obtained by the width of the broadened frequency spectrum, which is based on the speed analysis of each small rotating scatterer. But in this work, we proposed the new method of rotational speed detection based on the discrete frequency difference between two neighbor frequency signals. These two approaches are different in both the mechanism and the corresponding speed extraction method.
In any case, the adjacent frequency difference between two neighbor Doppler shift signal peaks is always equal to the rotating frequency. This frequency difference is not even related to the topological charge of the incident beam, but only related to the discrete nature of its OAM and the rotating speed. As shown in Fig. 5(c), the broadened Doppler frequency signals are not difficult to recognize in the frequency spectrum, but how to take the frequency bandwidth of the signals is worth being concerned with. In theory, two adjacent rotational Doppler frequency peaks are enough to determine the rotating frequency. But to ensure the accuracy of the rotational frequency and to prevent the signal default condition, it’s better to choose an amplitude threshold and take several frequency signal peaks and calculate their mean values as the rotational frequency. In the experiment, we take 14 frequency peaks which possess 13 frequency gaps of $\Delta f$. The measurement mean value of the rotating frequency is $\overline {\Delta f} = 5.0462Hz$, with an error of 0.25% relative to the set value of the motor.
To investigate the effectiveness of the above technique in more sophisticated conditions, we further conduct the experiment under light oblique incidence and arbitrary incidence conditions. Under the light oblique incidence condition ($dx \approx 0$, $dy \approx 0$, $\theta \ne 0$), the intensity distribution on the object becomes an oval-shaped ring. It’s a challenging work to let the incident beam illuminate the center of the rotational axis precisely in consideration of the wavelength of light, but we still try our best to ensure the $dx \approx 0$ and $dy \approx 0$ to reduce the noise signals. The corresponding measured frequency spectrum is shown in Fig. 6(a). Note that the corresponding frequency gap between each two adjacent frequency peaks is two times of the rotational frequency, which is still in agreement with the mode decomposition results under the light oblique incidence condition as shown in Fig. 2(c). The corresponding rotating frequency is 4.95Hz with a relative error of 1%.
Fig. 6. Noncoaxial detection results. (a) Frequency spectrum under light oblique incidence condition. (b) Measurement result based on light noncoaxial incidence. (c) Experimental results at different rotational speed under noncoaxial illumination.
A more common detection situation is that both the lateral displacement d and the skew angle $\theta $ exist between the detection beam and the rotating axis. This situation mainly occurs in the detection of non-cooperative rotating target whose pose of rotating axis is unknown. We further detect the rotating object under light arbitrary incidence ($dx \ne 0,dy \ne 0,\theta \ne 0$). As shown in Fig. 6(b), the frequency shift signal again is broadened with a wider spectrum and lower amplitude. The signals distribution characteristics is almost match with the simulated results shown in Fig. 2(f). Although the width of the frequency spectrum broadening is consistent with the simulated OAM mode decomposition results, there are some missing signal peaks in the experimental results. It’s a common phenomenon that there are some defaults in the frequency spectrum in all the practical measurement results, which are caused by the OAM mode purity and the surface scattering properties of objects. By analyzing discrete frequency signals without the missing order, we still can obtain the rotational speed, i.e., the frequency gap of 5.01Hz which is the same as the set value of the motor.
Compared with the previous measurement technology which is difficult to acquire the rotational speed when the default frequency signal appears in the measurement results. Here, by extract the adjacent frequency difference of the signal frequency spectrum, the missing of several signals does not affect the acquisition of the rotational speed. It shows a strong robustness of our method. To verify the measurement accuracy of our technique under different rotational speed, we further conduct the measurement under different rotational speed at light non-coaxial incidence. As Fig. 6(c) shows, the measured results are in good agreement with the theoretical value. All the measurement error is less than 4%. Error bars are given based on the maximum error of repeated measurement results.
Although our method has good robustness and measurement accuracy, there are some limitations to this proposal. The effective maximum of axis-displacement d and skew angle $\theta $ are limited. The OAM mode decomposition method can’t be applied when the rotating center moves completely out of the light field. But the rotating speed extraction method is still effective after a large displacement of three times the beam waist, just the signal frequency spectrum expands extremely wide. As the frequency spectrum expands widely, the signal-to-noise ratio becomes very low and therefore causing the signals difficult to be recognized. Further, the skew angle is limited. In theory, the upper limit of the skew angle is $\pi /2$, because scattered light cannot be received beyond this limit. But in practical measurement, the skew angle should no larger than $0.3\pi $. Because the same reason that the frequency shift signal spectrum is too wide and the signal’s amplitude is too weak to be extracted.
4. Conclusion
In conclusion, we described a new RDE-based approach for measuring rotation speed that may be employed in light incident both coaxially and noncoaxially with the rotational axis of the object. The OAM mode spectrum of the detecting OV beam under the noncoaxial situation is studied using the OAM decomposition method. Because the rotating Doppler shift is proportional to the detection beam's OAM mode, the Doppler frequency shift distribution is also discrete and widened. By extracting the adjacent frequency difference between the signals, the rotating speed of the object can be measured with high accuracy. This detection technique is not bothered by the pose of the rotating object relative to the probe beam and the topological charge of the optical vortex. This proposal provides a significant complement to the conventional RDE theory, which may help with the realistic application of RDE based metrology.
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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