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Abstract
Based on the rotational Doppler effect, an orbital angular momentum beam can measure the lateral rotation velocity of an object, which has broad application prospects. However, all existing research focus on the light spot center coinciding with the rotation center, or only with small center offset. This is difficult to ensure in remote detection applications. In this paper, the rotational Doppler frequency shifts under three cases, including no center offset, small center offset and large center offset, are analyzed theoretically. Through theoretical research results, a novel method of measuring rotation velocity is proposed, with the light spot completely deviated out of the rotation center. A laboratory verification experiment shows that this proposed method breaks the limit of center offset of lateral rotation velocity measurement and is of great significance to the remote detection of non-cooperative rotation object.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Linear Doppler effect refers to that the frequency of a wave is shifted due to the relative velocity of the source and the observer. According to the degree of red/blue shift, the velocity of an object moving in the direction of signal propagation can be detected [1,2]. However, this shift does not detect rotation that is perpendicular to the direction of propagation. Since rotational Doppler effect (RDE) can solve this problem, it gradually becomes a research hotspot from early theoretical researches [3–5] to recent experimental researches [6–8]. Rotational Doppler effect refers to that a spinning object with an optically rough surface may induce a frequency shift even when the surface of the rotation object is perpendicular to the direction of propagation [9–11]. Rotational Doppler effect is found in the orbital angular momentum (OAM) light beam. The OAM light beam comprises a transverse angular phase structure of $\exp (il\theta )$ with helical wave front, here l is the azimuthal index (also called the OAM quantum number or the topological charge) and $\theta$ is the angular coordinate [12]. This type of light beam carries an OAM equivalent to $l\hbar$ per photon and consists of a ring of intensity with a null at the center [13].
In recent years, the mechanism and technique of rotation velocity measurement are widely studied. A modal expansion method is presented to sufficiently investigate the optical rotational Doppler effect, and make people better understand the physical processes of rotational Doppler effect [14]. The relationship between rotational and linear Doppler effect is discussed, either of which can be derived from the other effect, thereby illustrating their shared origin [15]. OAM interferometer, a new type of rotational Doppler frequency shift measurement from OAM correlation, is demonstrated and capable of measuring the rotation of a point source in the plane orthogonal to the observer line of sight [16]. The self-mixing method in a vortex laser is proposed, and the rotational Doppler frequency shift together with a linear one in light backscattered from spinning particles can be detected [17]. Lavery et al. find that rotational Doppler frequency shift is independent of the optical frequency, and even a white-light source can give rise to a single-valued frequency shift. They uses a white-light OAM-carrying beam to realize the measurement of rotational velocity, suggesting its applicability for the remote sensing of rotating objects [11]. However, all existing researches are in the case of the light spot center coinciding with the rotation center, or only with small center offset. Qiu et al. research rotation velocity measurement with the center offset less than the spot radius theoretically and experimentally, and find that the frequency changes at each point are different with lateral misalignment and the Doppler signal is spectrally broadened [18]. This phenomenon can be well explained by the model decomposition theory with lateral misalignment [19,20]. In this paper, a novel method of measuring rotation velocity with the light spot completely deviated out of the rotation center is verified theoretically and experimentally.
2. Mechanism analysis of rotation velocity measurement
In recent years, some researchers have shown that the rotational Doppler effect and the linear Doppler effect share a similar mechanism. The formula of rotational Doppler frequency shift can be derived from the one of linear Doppler frequency shift. It's well known that the linear Doppler frequency shift is as follow
Where, ${f_0}$ is the original frequency of light, c is the velocity of light, v is the relative velocity between the light source and the object, $\phi$ is the angle between the Poynting vector (also the light propagation direction) and the velocity vector.When $\phi \textrm{ = }90^\circ$, $\cos \phi = 0$, linear Doppler frequency shift is zero. Linear Doppler effect cannot measure lateral velocity. However, OAM light beam has a helical wave front, and the light propagation direction and Poynting vector are not collinear. The angle between them is $\alpha$, which is proportional to the topological charge l. $\alpha = {{l\lambda } \mathord{\left/ {\vphantom {{l\lambda } {2\pi r}}} \right.} {2\pi r}}$, with $\lambda$ is the wavelength and r is the radius from the beam axis. At this time, $\phi \textrm{ + }\alpha \textrm{ = }90^\circ$, and thus $\cos \phi \textrm{ = }\cos ({90^\circ{-} \alpha } )\textrm{ = }\sin (\alpha )\approx \alpha = {{l\lambda } \mathord{\left/ {\vphantom {{l\lambda } {2\pi r}}} \right.} {2\pi r}}$ with $\alpha$ being small. Therefore, combined with Eq. (1), having
Fig. 1. OAM light beam illuminates at different positions of the rotating object. The point O is the rotation center of object, and the point $O^{\prime}$ is the center of OAM light beam, and thus the offset between two centers $d = |{OO^{\prime}} |$. (a) Without the offset, the center of OAM light beam coincides with the rotation center of object $O({O^{\prime}} )$, also $d = 0$. (b) With the small offset, $0\;<\;d\;\le\;r$. (c) With the large offset, $d\;>\;r$. (d) The schematic diagram of the angle $\phi$ between the Poynting vector $\vec{S}$ and the velocity vector ${\vec{v}_{o^{\prime}}}$
2.1 No offset $d = 0$
As shown in Fig. 1(a), the center of OAM light beam coincides with the rotation center of object $O({O^{\prime}} )$. The offset between two centers $d = |{OO^{\prime}} |= 0$. At this time, each point on the OAM annular spot has the same velocity $v = {v_r} = \Omega r$, with tangential direction. Thus Eq. (2) can be expressed as
Here, $\Omega$ is the angular velocity of rotating object. Equation (3) is rotational Doppler frequency shift, which is proportional to the topological charge l and the angular velocity $\Omega$. This is consistent with the literature [10,21].2.2 Small offset $0\;<\;d\;\le\;r$
As shown in Fig. 1(b), the rotation center of object deviates from the center of OAM light beam, but it is still in the OAM light spot, with the offset between two centers $0\;<\;d\;\le\;r$. At this time, each point on the OAM annular spot has different distances from the rotation center O of object, $R(\theta ) = \sqrt {{d^2} + {r^2} + 2dr\cos \theta }$. Therefore, each point on the OAM annular spot has different velocity $v = {v_R}(\theta ) = R(\theta )\Omega = \Omega \sqrt {{d^2} + {r^2} + 2dr\cos \theta }$, which is combined with Eq. (2), having
2.3 Large offset $d\;>\;r$
The rotation center of object deviates out of the OAM light spot, with $d\;>\;r$, as shown in Fig. 1(c). In the polar coordinates of $O^{\prime}$ system, arbitrary point on the OAM annular spot is expressed as $({r,\theta } )$. The distance of $({r,\theta } )$ relative to the rotation center of object can be expressed as $R(\theta ) = \sqrt {{d^2} + {r^2} + 2dr\cos \theta }$, and the linear velocity of point $({r,\theta } )$ is ${v_R} = R(\theta )\Omega = \Omega \sqrt {{d^2} + {r^2} + 2dr\cos \theta }$. Besides, the distance of $O^{\prime}$ relative to O is d, and the linear velocity of point $O^{\prime}$ is ${v_{o^{\prime}}} = d\Omega$.
Next, we consider the rotation motion of angular velocity $\Omega$ around O, from the perspective of the point $O^{\prime}$, it can be regarded as a combination of translational motion (the linear velocity ${v_{o^{\prime}}} = d\Omega$) and rotation motion (the angular velocity $\Omega$). Therefore, the velocity ${v_R}$ is decomposed into two component velocities. First component velocity is the linear velocity of the point $O^{\prime}$ (${v_{o^{\prime}}} = d\Omega$), representing the translational motion of the whole object relative to the light spot. Second component velocity is the tangential velocity around the point $O^{\prime}$ (${v_r} = \Omega r$). ${v_r}$ has nothing to do with $\theta$, and ${v_r}$ of each point on the light spot is the same. Therefore rotational Doppler frequency shift caused by second component velocity ${v_r} = \Omega r$ is also the same
3. Experiment and result analysis
An experiment is performed to verify the rotation velocity measurement under different center offsets, and the experiment setup is shown in Fig. 2. The source is a continuous 532nm laser. The laser signal is firstly coupled into single-mode fiber (SMF), so as to obtain good Gaussian fundamental mode and improve the modulation efficiency of OAM signal. Through collimation and light path adjustment, the laser signal is modulated by a pure phase spatial light modulator SLM (Holoeye Pluto, phase-only SLM), and the phase image of topological charge $l ={\pm} 5$ is as shown in the detail diagram. Different values of l correspond to different divergence angles and different spot radius of the OAM beam. In this experiment, the OAM spot radius of $l ={\pm} 5$ on the rotation object is approximately $r = 2$mm, and the radius of rotation object is approximately 4.6 cm. For verifying the center offsets from 0 to $20r$, we selected $l$=+/- 5. A 4F System consisting of two lenses and a small hole is used to select the first-order diffracted light, and the modulated signal is a light spot of ten petals as shown in the detail diagram. The modulated signal irradiates the surface of a rotating object. The rotating object is a rotating flat disk which is fixed to an adjustable motor, and they are all fixed on a horizontal translation stage for simulating different center offsets. After the signal is reflected by the rotating object, it is collected by lens and detected by APD. The detection results were collected by an oscilloscope (Tektronix DPO4032), and the data were finally processed by a computer.
In this paper, OAM light beam with topological charge ${\pm} l$ is used. According to Eq. (3) and (7), rotational Doppler frequency shifts of two topological charges ${\pm} l$ are $\Delta {f_\textrm{ + }} = {{\textrm{ + }l\Omega } \mathord{\left/ {\vphantom {{\textrm{ + }l\Omega } {2\pi }}} \right.} {2\pi }}$ and $\Delta {f_\textrm{ - }} = {{\textrm{ - }l\Omega } \mathord{\left/ {\vphantom {{\textrm{ - }l\Omega } {2\pi }}} \right.} {2\pi }}$, respectively. When the receiving system detects two signals at the same time, the total rotational Doppler frequency shift is $\Delta {f_T} = \Delta {f_ + } - \Delta {f_\textrm{ - }} = {{|l |\Omega } \mathord{\left/ {\vphantom {{|l |\Omega } \pi }} \right.} \pi }$. Thus the rotation velocity can be obtained as $\Omega = {{\Delta {f_T}\pi } \mathord{\left/ {\vphantom {{\Delta {f_T}\pi } {|l |}}} \right.} {|l |}}$. The specific experimental results are as follows:
The signal-to-clutter ratio (SCR) is defined as the ratio of the signal's main peak to its maximum clutter peak. As shown in Fig. 3, with the increase of the offset d, the measurement error increases and the Signal-to-Clutter Ratio (SCR) decreases. This is because the offset exists, the rotational Doppler frequency shift of each point is no longer the same. At this time, the peak of rotational Doppler frequency shift gradually becomes wide and weak, resulting in the signal-to-clutter ratio decreases. When the offset d reaches above $0.25r$, SCR drops close to 1, or even less than 1, and the signal peak has been completely lost. The measurement error increases sharply, and the system is no longer working properly.
Fig. 3. The curve of measurement error and signal-to-clutter ratio (SCR) under different offsets between the light spot center and rotation center.
The rotational Doppler frequency spectrum with offsets $0.15r$, $0.2r$, $0.25r$ and $0.3r$ are shown in Fig. 4, respectively. Angular velocity and rotational Doppler frequency shift satisfy $\Delta f = {{\Omega |{{l_1} - {l_2}} |} \mathord{\left/ {\vphantom {{\Omega |{{l_1} - {l_2}} |} {({2\pi } )}}} \right.} {({2\pi } )}}$. The echo signal is detected, and the frequency spectrum is obtained through the Fourier transform. The maximum peak in the frequency spectrum is searched as the signal's main peak, and its position corresponds to the rotational Doppler frequency shift $\Delta f$. According to $\Omega = {{2\pi \Delta f} \mathord{\left/ {\vphantom {{2\pi \Delta f} {|{{l_1} - {l_2}} |}}} \right.} {|{{l_1} - {l_2}} |}}$, the rotation angular velocity can be obtained. In Fig. 4, its horizontal axis is the frequency that corresponds to rotation angular velocity $\Omega = {{2\pi \Delta f} \mathord{\left/ {\vphantom {{2\pi \Delta f} {|{{l_1} - {l_2}} |}}} \right.} {|{{l_1} - {l_2}} |}}$, and the vertical axis is the normalized spectrum intensity. It can be seen that when the offsets are $0.15r$ and $0.2r$ respectively, there are clear signal peaks. Merely as the offset increases, the signal peak becomes wide and weak, resulting in SCR decreases. When the offset reaches $0.25r$, the signal peak is slightly higher than clutter peaks. When the offset reaches $0.3r$, the signal peak is completely submerged in other clutter peaks and cannot be identified. This is consistent with the analysis in section 2B.
Fig. 4. The rotational Doppler frequency spectrum. (a) the center offset $d\textrm{ = }0.15r$. (b) the center offset $d\textrm{ = }0.2r$. (c) the center offset $d\textrm{ = }0.25r$. (d) the center offset $d\textrm{ = }0.3r$. Rotation angular velocity and rotational Doppler frequency shift satisfy $\Delta f = {{\Omega |{{l_1} - {l_2}} |} \mathord{\left/ {\vphantom {{\Omega |{{l_1} - {l_2}} |} {({2\pi } )}}} \right.} {({2\pi } )}}$. The rotational angular velocity to be measured is 300 rad/s.
According to the analysis of section 2C, when the rotation center of object deviates out of the OAM light spot (the offset $d\;>\;r$), a good peak of rotational Doppler frequency shift can be obtained again. In order to verify this phenomenon, the experiment is carried out, and the results are shown in Figs. 5 and 6. It can be seen that when the offset is $0.9r$, the signal peak is still submerged in clutter peaks and cannot be recognized. However, when the offset reaches r, the rotation center of object completely moves out of the OAM light spot, and then the signal peak appears again at the corresponding position of 300rad/s.
Fig. 5. The rotational Doppler frequency spectrum. (a) the center offset $d\textrm{ = }0.9r$. (b) the center offset $d\textrm{ = }r$. The rotational angular velocity to be measured is 300 rad/s.
Fig. 6. The curve of measurement error and signal-to-clutter ratio (SCR) with OAM light spot completely deviated out of the rotation center.
Furthermore, when the rotation center of object deviates out of the OAM light spot, the measurement performance is discussed, as shown in Figs. 6 and 7. It can be seen that as the offset increases, the signal-to-clutter ratio (SCR) increases gradually, and the measurement error decreases gradually. This is in accord with theoretical analysis of section 2C. This shows that if the center of light spot cannot be well aligned with the rotation center, and it's better to make the light spot away from the rotation center. Far away from the interference of rotation center, better results can be obtained. When the offset increases further, SCR and Error tend to be stable. This is because the farther away from the center of rotation, the more obvious the translational velocity feature is, and the weaker the rotation feature is.
Fig. 7. The rotational Doppler frequency spectrum. (a) the center offset $d\textrm{ = }5r$. (b) the center offset $d\textrm{ = }10r$. (c) the center offset $d\textrm{ = }15r$. (d) the center offset $d\textrm{ = }20r$. The rotational angular velocity to be measured is 300 rad/s.
At present, the measurement method of rotating object based on OAM must make sure that the offset between the center of the light spot and the rotation center of object is small enough as much as possible. This is not difficult to measure for a close-range rotating object, however, considering the remote sensing application or non-cooperative object detection, such small offsets are hard to achieve. According to the research results of this paper, we can use a small OAM light spot and make it completely deviate from the rotation center. This breaks the limit of offset and is of great significance to the remote detection of non-cooperative rotation object.
4. Conclusion
This paper theoretically analyzes the rotational Doppler frequency shifts under three cases, including no center offset $d = 0$, small center offset $0\;<\;d\;\le\;r$ and large center offset $d\;>\;r$ ($r$ is the radius of OAM light spot). The corresponding experimental results show that when no center offset or small center offset is less than $0.25r$, there are correct signal peaks. When the center offset is greater than $0.25r$ and gradually approaches $0.3r$, the signal-to-clutter ratio (SCR) gradually approaches 1, and the signal peak is submerged in noise. However, more interesting, when the center offset continues to increase and is greater than r, the correct signal peak appears again. In this paper, the mechanism of this phenomenon is explained perfectly by velocity decomposition theory. Based on this, a novel method of measuring rotation velocity with the light spot completely deviated out of the rotation center is proposed and verified. This proposed method breaks the limit of center offset of lateral rotation velocity measurement and is of great significance to the remote detection of non-cooperative rotation object.
Funding
National Natural Science Foundation of China (61701139).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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