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Abstract
The optical rotational Doppler effect of light beams with angular momentum has recently found applications in the remote sensing of spinning objects. However, most of the reported experimental demonstrations rely on the particular condition of normal incidence, while the general case of oblique incidence has not been addressed yet. Herein, we investigate the optical rotational Doppler effect at oblique incidence based on a local scattering model and formulate the quantitative relation between the Doppler frequency shift and the tilt angle. The analytic results indicate that even if the rotational axis of the spinning object is oriented at a specific tilt angle relative to the light propagation direction, the rotational speed can still be extracted from an asymmetrically broadened Doppler signal. The geometric mean value of the extreme frequency shift is a constant despite the variation of the incident angle. An experiment with obliquely incident and superimposed optical vortices is executed to verify our theoretical predictions, achieving a successful detection of the rotational speed at relatively large tilt angles with a relative error less than 2%. The scheme proposed in this study may be useful for practical applications of rotational Doppler effect in remote sensing and metrology.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Corrections
Song Qiu, Tong Liu, Yuan Ren, Zhimeng Li, Chen Wang, and Qiongling Shao, "Detection of spinning objects at oblique light incidence using the optical rotational Doppler effect: erratum," Opt. Express 28, 16633-16633 (2020)https://opg.optica.org/oe/abstract.cfm?uri=oe-28-11-16633
1. Introduction
The Doppler effect is a well-known phenomenon based on which the motion between the object/receiver and the wave source introduces a frequency shift $\Delta f$ [1–3]. This effect has widespread applications in metrology, astronomy, and elsewhere [4–6]. Similar to the conventional linear Doppler effect (LDE), another type of rotational Doppler effect (RDE) associated with the orbital angular momentum (OAM) has been proposed in recent years [7–9]. This RDE has the same origin as LDE, and enables the detection of the velocity perpendicular to the beam axis [10]. To observe RDE, a source containing special phase structure is required. In the classical physics domain, this type of source can be obtained by passing Gaussian or Laguerre–Gaussian (LG) beams through a rotating Dove prism or half-wave plate [11–14]. Since the discovery of the existence of the orbital angular momentum of photons by Allen et al., the characteristics of the optical vortex (OV) have gradually been unveiled [15–18]. This beam type has a helical wavefront, and an angle $\alpha $ exists between the Poynting vector and the beam propagation axis, which is suitable for the application of the RDE-based system.
One of the most famous applications of RDE is the detection of the rotational frequency of spinning objects. Since the systematical demonstration of the technique of spinning object detection by Lavery et al. by means of the superposition optical vortex [19], numerous research studies have been published concerning the detection of the rotational velocity at various scales from the microscopic molecules to macroscopic objects [20–24]. It has been shown that even in the presence of small obstacles or atmospheric turbulence, the RDE can still be observed [25,26] that justifies the considerable potential in its use for cosmoscopic observation and remote sensing. Recently, there is a new radial Doppler effect has been demonstrated [27]. The motion at the radical direction in the transverse plane of the OV beams also can introduce Doppler shift. The rotational Doppler effect can also be resulted from the dynamic evolution of geometric phase [28,29]. However, virtually most of the aforementioned research studies on rotational speed detection using RDE relied on the coincidence of the light propagation and the object rotation axes. Rare work considered situations at which the OV would be incident at certain angles in the RDE-based experiments that would be of great importance for practical applications.
Our previous work analyzed the influence of lateral misalignment on optical RDE [30]. The laterally displaced and angular tilt may change the photon OAM state related to the reference axis have been demonstrated in previous work [31,32]. In this study, we investigate the optical RDE at oblique incidence. We use the superposition of OVs as the probe beam and reveal the relationship between incidence angle and RDE. The rotational Doppler shift broadens into many peaks around the original frequency signal in cases of oblique light incidence. However, the geometric mean value of the extreme frequency shift is constant and equals the original value, and can be used to calculate the rotational speed. We experimentally verify our theory with different topological OVs, and show that the results agree well with theoretical predictions. Our conclusions may increase the flexibility in detection of rotating objects ranging from molecules to macroscopic bodies. We expect that the scheme proposed in this study may be useful for practical RDE applications in remote sensing and metrology.
2. Theoretical model
2.1 Rotational Doppler effect
The phase distribution of a vortex beam can be expressed as $E(r,\;\phi ,\;z) = E(r,\;z)\exp (il\phi )$ in cylindrical coordinates [33–35]. The term ${e^{il\phi }}$ determines the helical phase profile so that the wavefront of the vortex beam is not perpendicular to the optical propagation axis. For a linearly polarized LG beam which is the basic OV that possesses OAM, the angle between the Poynting vector and the propagation direction can be expressed as $\sin \alpha = l/kr$, where k denotes the wave number, l is the topological charge and r is the radius of the corresponding point on the transverse plane [18,36]. Because the Poynting vector reflects the flow of energy density and is not parallel to the propagation axis of the LG beam, there will be a momentum component in the transverse plane.
The basic principle of LDE in the case of a light beam is as follows. When the speed of relative motion between a light-emitting source and observer is $v$, there will be a frequency difference between the received and the original light. The frequency difference is also called Doppler shift, and can be expressed as $\Delta f = {f_0}v/c$, where $c$ and ${f_o}$ represent the light speed and frequency in the corresponding medium, respectively.
The standard linear Doppler shift applies when the relative motion between source and observer is along the direction of observation. For motion transverse to the direction of observation, a reduced Doppler effect shift can still be observed in the light scattered at an angle $\alpha $ from the surface normal. For small values of $\alpha $, this reduced Doppler shift is given by,
This frequency occurs because the roughness of the surface means that some of the light normally incident on the surface is scattered at angle $\alpha $ . In a helically phased beam at normal incidence, the Poynting vector has an azimuthal component at every position within the beam. The direction of the Poynting vector and the linear velocity of the local scatterer are shown in Fig. 1. Considering that the angle between $v$ of the local scatterer and Poynting vector is $\varphi $, it has a magnitude $\varphi = \pi /2 - \alpha $. Thus, the component of $v$ along the direction of the Poynting vector can be expressed as ${v_p} = v\cos \varphi $. Combined with the reduced Doppler shift of Eq. (1), we can see the Doppler frequency shift of the on-axis scattered light is given as [19],
Fig. 1. Rotational Doppler effect in the case of normal incidence. (a) An optical vortex illuminates the rotating object in a direction coaxial with respect to its rotational axis, whereby the object’s rotational speed is $\Omega $, and the black line passing through the optical vortex beam represents the Poynting vector. (b) Magnified view of the local scatter area S on the object. For an arbitrary point, the distance from the axis is r and the linear speed is $v$.
This is the basic principle to detect the rotation speed of objects. For each position within the beam, include the phase singularity point which is coincident with the center of rotation, would experience the same modulation frequency. If a superimposed vortex beam with topological charge $\pm l$ is used as the probe light, the mixing of up- and down-shifted scattered light components will result in a beat Doppler signal with the modulation frequency expressed as,
2.2 Influence of tilt angle
When light oblique incidence is considered, as shown in the Cartesian coordinate system in Fig. 2, the OV propagation axis is deflected by $\gamma $ around the x–axis in the $yoz$ plane. The Doppler frequency shift is perceived by an observer who is watching the moving scatterer from a direction which is parallel to Poynting vector. Compared to the situation of normal incidence, each position within the probe beam will not experience the same frequency shift as Eq. (2) shows anymore. The relative linear motion of each scatterer is different, and so is the corresponding frequency shift. First of all, owing to the oblique illumination, the beam profile on the object will change from annular to elliptic annular. Here we assume the beam profile moves in straight line as the probe beam have been well collimated. If the beam radius of OV is r, the distance between any point on the elliptical ring and the ellipse center can be expressed according to the following geometric relationship:
where ${\theta _z}$ and $\gamma $ are indicated in Fig. 2(a).Fig. 2. Transformation of OV profile and Poynting vector at oblique incidence. (a) If the incident OV axis is in the $yoz$ plane, $\gamma $ denotes the incident angle, r is the original OV profile radius, $r^{\prime}$ is the distance between any point on the elliptical ring and the ellipse center of the OV projection on the object, and ${\theta _z}$ is the angle between the orientation of each tiny scatterer and the x–axis, ${\vec{v}_\theta }$ shows the linear velocity direction. (b) If d is an arbitrary point, the Poynting vector at normal incidence is ${\vec{p}_0}$ and can be calculated by the Poynting vector at point ${x_0}$, while ${\vec{p}_\gamma }$ represents the Poynting vector at oblique incidence.
Secondly, the linear velocity at each point on the rotating object is in the $xoy$ plane. Considering point ${x_0}$ as an example (as shown in Fig. 2(b)), the direction of the linear velocity is ${\vec{v}_{x0}} = (0, - 1,0)$. The velocity direction of each local scatterer can then be expressed as,
Furthermore, since the skew angle between Poynting vector and the propagation axis of the light beam is $\alpha = l\lambda /2\pi r$ [18,36]. By taking the direction of the Poynting vector into consideration, the direction of the Poynting vector at ${x_0}$ in the case of normal incidence can be expressed as ${\vec{p}_{x0}} = (0, - l\lambda , - \sqrt {4{\pi ^2}{r^2} - {l^2}{\lambda ^2}} )$ . For an arbitrary point around the OV propagation axis, we can obtain the direction of the Poynting vector ${\vec{p}_0}$ by multiplying ${\vec{p}_{x0}}$ with the rotation matrix ${M_z}({\theta _z})$. Moreover, in the case of oblique incidence, as shown in Fig. 2(b), the direction of the Poynting vector is obtained by multiplying ${\vec{p}_0}$ with the rotation matrix ${M_x}(\gamma )$ which denotes a rotation by $\gamma $ around the x–axis. The final Poynting direction ${\vec{p}_\gamma }$ can be expressed as,
Based on Eq. (7), the first part represents the component of the rotation Doppler shift. It is interesting that the second part is equal to the linear Doppler shift component. That is to say, in the oblique frame, motion is induced along the OV propagation. For spinning object detection, we only need the first part to explore the rotational speed. The total frequency shift $\Delta f$ is too large to detect and cannot be used to calculate the rotational speed. Instead of directly measuring this frequency shift, the superposition OVs with topological charge $\pm l$ are employed here. The two OAM components will produce opposite rotational Doppler shift, but their linear Doppler shift components are equal. So that, the linear components can be counteracted and the rotational components show in the beat signal. This important finding can only happen in the superposition scheme, for a reference scheme [10,37], it is difficult to distinguish the rotational components from the total frequency shift and cannot be used in the spinning objects detection at oblique light incidence. Thus, the beat frequency that was induced by the first part of Eq. (7) is observed, and the modulation frequency can be expressed as,
Fig. 3. Simulation of modulated frequency shift at l = 18 and $\Omega = 716$ rad/s. (a) The maximum of the modulated frequency varies at a faster rate than the minimum value as the incident angle $\gamma $ increases. (b) Frequency curve at $\gamma = 0.78$ rad. It is clear that ${f_{\bmod }}$ is at its maximum when ${\theta _z}$ has the values of $\pi /2$ and $3\pi /2$ and at a minimum when ${\theta _z}$ has the values of 0 and $\pi $ . (c) Frequency distribution map delineated by means of Eq. (8).
Therefore, combined with Eq. (8), the extremum of ${f_{\bmod }}$ can be expressed as,
3. Experimental setup
According to the above theoretical analysis, we use the superimposed OV with a topological charge of $\pm l$ as the probe beam in the experiment. The experimental setup is shown in Fig. 4.
Fig. 4. Experimental setup. (A: attenuator. P: polarizer. L: lens. BS: beam splitter. SLM: spatial light modulator. SF: Spatial filter. f: focal length. D: distance. Charge couple device (CCD): laser CCD camera. γ: incidence angle. PD: photodetector). The CCD and the object have the same angles and distances. The oscilloscope can perform real-time Fourier transformations. The sampling time is 0.1 ms and the sampling frequency is 10 kHz.
A He–Ne laser generates a 632.8 nm light beam, which is attenuated and then polarized by an attenuator (A) and a horizontal linear polarizer (P), respectively. Subsequently, the laser beam is expanded and collimated through the telescope system composed of two lenses L1 and L2 whose focal length are 30 mm and 300 mm respectively, and then incident on the SLM (HAMAMATSU, LCOS–SLM X13138) through a beam splitter (BS). Herein, the beam expansion ensures that the beam spot can uniformly cover the liquid crystal panel of the SLM. Through the spatial phase modulation of the SLM, the superimposed OV required in the experiment can be generated. The light from the SLM is reflected by the BS and then passes through a 4-f system consisting of L3, L4 (f = 100 mm) and a pinhole with a diameter of 0.5 mm. This 4-f system can not only select the first diffraction order of the beam but also collimate the beam. Then a superimposed OV with high-mode purity can be obtained.
Subsequently, the generated beam is split into two paths; one is used for the CCD to capture the beam profile, and the other is incident on the surface of the rotating object. To obtain the actual beam profile on the surface of the object, the distance (D) between the CCD and BS should be identical to the distance between the object and BS. In our experiment, the D is set to 200 mm. The surface of the rotating object is wrapped by the silver paper to enhance the reflection of the probe light.
Finally, the angle $\gamma $ between the rotating axis of the turntable and the incident light is continually adjusted, and the reflected light from the surface of the object is collected by L5 (f = 30 mm) and the photodetector (PD). The PD is connected to an oscilloscope, which is used to perform the real-time Fourier transform to extract the frequency shift signal. When the angle of the rotating object is adjusted, it is necessary to ensure that the central position of the rotating object does not change. The method used is based on the fixation of the rotating object on a horizontal rotary table. We can adjust the angle of the rotating object by only controlling the rotary table.
4. Results and analyses
We consider first the case of normal incidence. The object’s rotational speed is set to 716.3 rad/s, and the OVs of the topological charge $\pm 12$, $\pm 15$ and $\pm 18$ are used, respectively. The temporally varying signal is measured over a period of 0.1 s, and a fast Fourier transform (FFT) is performed to extract the modulation frequency. The measured results for different topological charges are shown in Fig. 5.
Fig. 5. (a), (d), and (g), are the intensity distributions of OVs with topological charges $\pm 12$, $\pm 15$ and $\pm 18$ . (b), (e), and (h), are intensity curves in the time domain which longitudinal coordinates represent the voltage detected by the photodetector. (c), (f), and (i), are the corresponding Doppler shift signals. In all studied cases, the rotational velocity was set to 716 rad/s.
The frequencies of the signals shown in Fig. 5(c), 5(f) and 5(i). The experimental results are consistent with the values predicted by Eq. (2). We set the rotating object at a series of different rotational velocities, and the results of the rotational Doppler shift are shown in Fig. 6. The measured results agree well with the theoretical predictions.
Fig. 6. Measured modulation frequency ${f_{\bmod }}$ vs. angular velocity Ω of the rotating object for three different topological charges. The measured results are shown as points, and the values predicted from Eq. (3) shown as solid lines.
We then load a hologram with topological charges of $l = \pm 18$ onto the SLM, and set the rotational velocity of the object to 114 Hz. The rotary table is adjusted to ensure that the axis of the object is inclined by a certain angle. Under this condition, the Doppler signal is no longer a single frequency, but a certain number of peaks appear in the vicinity of the original frequency. This is consistent with the phenomenon predicted by Eq. (8). It can be observed from Fig. 7(a) that when the incident angle $\gamma $ is greater than 0.17 rad, the signal broadens into multiple peaks, and the corresponding amplitude is decreased. The extreme values ${f_{\max }}$ and ${f_{\min }}$ are identified from Fig. 7(a), and these are plotted in Fig. 7(b). It is noted that ${f_{\max }}$ varies at a faster rate than ${f_{\min }}$ as a function of the incidence angle, and the broadening thus becomes asymmetric. These findings are in good agreement with the theoretical predictions. Using the values of ${f_{\max }}$ and ${f_{\min }}$, we can also calculate the center frequency ${f_c}$ based on Eq. (10), as shown in Fig. 7(c). The experimental value of ${f_c}$ is almost constant for different incident angles in the range of 0.17 and 0.80 rad, and the error between the experimental and theoretical values of the center frequency is less than 2%.
Fig. 7. Experimental results for topological charge $l = \pm 18$ and rotation speed $\Omega = 716.3$ rad/s. (a) Measured Fourier spectrum signals. (b) Extreme value of the modulated frequency ${f_{\max }}$ and ${f_{\min }}$ varies as the incidence angle $\gamma $ . The blue dashed line stands for the symmetric graphics of ${f_{\min }}$ with the center frequency as axis. It is obviously the ${f_{\max }}$ varies faster than the ${f_{\min }}$ . (c) Relationship between the value of the center frequency and the incidence angle $\gamma $ .
It is worthy to note that when the incidence angle is less than 0.17 rad, the beat frequency signal yields only one peak. The reason for this experimental phenomenon is attributed to the fact that when the incidence angle $\gamma $ is small, the frequency spectrum broadening effect is weak. This is also in accordance with the theoretical predictions.
To verify the theoretical conclusions, we also use OVs with topological charges of ± 12 and ± 15 to repeat the experiment in the presence of the tilt angle. The same results are observed, as shown in Fig. 8, which prove the accuracy of our conclusions. The center frequency ${f_c}$ is basically constant at different angles $\gamma $, as Fig. 8(b) and 8(d) show, while the profile of OV is continually varying. We capture the OV beam profile at the incidence angle of 0.44 rad, clearly showing that the intensity distribution changes from annular to elliptic annular.
Fig. 8. (a) and (c) show the maximum and minimum Doppler frequency shift variations as a function of incidence angle $\gamma $ for topological charges $l = \pm 15$ and $l = \pm 12$ . The blue dashed line stands for the symmetric graphics of ${f_{\min }}$ with the center frequency as axis. (b) and (d) are the corresponding center frequency variations as the incidence angle increase. The images shown in the inserts of (b) and (d) are light spot shapes captured by the CCD camera at an incident angle of $\gamma = 0.44$ rad. In all studied cases, the rotational velocity was set to 716.3 rad/s.
It is noted that when the angle of incidence exceeds 0.87 rad, the beat frequency signal is too weak to be recognized. Therefore, the theoretical results of this study can only be achieved if the angle of incidence lies within a specific range. In addition, it is a challenge to change the incidence angle without disturbing the alignment between the OV axis and the object’s rotational center. If the center of the spot is not strictly aligned with the rotational center, it may yield experimental errors.
5. Conclusion
In summary, we have theoretically and experimentally investigated the optical rotational Doppler effect at oblique incidence with superimposed OVs. The results indicated that the Doppler signal broadened asymmetrically as a function of angles of incidence, but the central frequency ${f_c} = \sqrt {{f_{\max }}{f_{\min }}} $ was immune to the variation of the incident angle. Accordingly, the center frequency can be employed to acquire the rotational speed of a spinning target with good accuracy. Our work provides a new supplement for RDE applied to the measurement of the rotational speed. Additionally, the results also suggested that RDE with OVs may also be used to measure the orientation of the rotating objects.
Funding
National Natural Science Foundation of China (61705016); Fundamental Research Funds for the Central Universities (2019RC12); Beijing Excellent Ph.D. Thesis Guidance Foundation (CX2019313).
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