Detection of a spinning object using a superimposed optical vortex array

You Ding; Xiangyang Zhu; Tong Liu; Zhengliang Liu; Song Qiu; Xiaocen Chen; Yuan Ren

doi:10.1364/OE.496362

1. Introduction

As is well known, the paraxial beam can carry two forms of angular momentum, the first is the spin angular momentum (SAM) that is manifested as circular polarization and corresponds to the photon SAM of $\hbar$, where $\hbar$ is the reduced Planck constant [1]. The other is the orbital angular momentum (OAM) that is carried by the optical vortex (OV) with a helical phase $\exp (il\varphi )$ and corresponds to the photon OAM of $l\hbar$[2,3], where $l$ is the angular quantum index and is also known as topological charge or the OAM mode, $\varphi$ is the azimuthal angle. Typically, the OV has a phase singularity at the center and an annular intensity distribution. OAM is a natural property of various types of OVs, such as Laguerre–Gaussian (LG) beam [4], Bessel-Gaussian beam [5], ring Airy Gaussian beam [6,7] and so on [8,9]. Due to the unique spatial intensity and theoretically unlimited value of mode, optical OAM has given rise to many developments in optical trapping [10,11], optical tweezers [12,13], imaging [14,15], communication [16,17] and rotation detection [18,19]. Compared with a single OV, the optical vortex array (OVA) is composed of multiple OVs in a regular arrangement that have controllable parameters in free space, including the spatial position, quantity, topological charges, beam diameter, and amplitude [20]. Each OV in the array is called an array element OV. The OVA carries the total OAM of all array element OVs which introduces the advantage in multichannel communication and optical manipulation [21,22]. At present, various methods has been proposed to generate OVAs, such as interferometry [23], optical wedge diffraction [24], mode conversion [25] and spatial light modulating [26,27].

After experimental demonstration in 2013 [18], the rotational Doppler effect (RDE) opens new approaches for remote sensing of rotation motion. The RDE originates from interactions between the optical OAM and the mechanical angular momentum of the rotating objects [28]. In the typical detection with a single OV, the rotating object can always be regarded as a planar object that has a much larger reflective area than the OV. The OV illuminates the rotating object with the beam center coinciding with the rotating center and the scattered light experiences a frequency shift of $lf$, where $l$ is the mode of the OV and $f$ is the rotating frequency of the object [18,19]. In addition to the rotating frequency, direction of rotation [29,30], linear velocity [31,32], angular acceleration velocity [33], procession frequency [34] can be obtained with appliable procedures. However, the RDE is greatly affected by the incidence of the OV [35]. The frequency shift spectrum expands into a set of discrete signals and the power of signal decreases rapidly as the center of OV moves away from the rotating center of the object [35–37]. This frequency shift spectrum expanding effect is closely related to the OAM state of OV relative to the rotating center. The OAM is naturally defined according to the optical axis. A standard OV is spatially symmetric and corresponds to a pure OAM. While, for an individual photon, the photon OAM state is expressed relative to the reference center. Typically, the geometric center of the OV is regarded as the reference center. In this sense, the pure OAM state of an individual photon transforms to the superposition of OAM states in a displaced coordinate frame [35,38]. The Doppler frequency shift of scattered light and the modal composition of probing beam show a stereospecific mapping. This low robustness to light incidence limits the RDE-based metrology in practical measurement. Herein, we use the RDE of superimposed optical vortex array (SOVA) to provide a potential solution. The SOVA is composed of multiple superimposed OVs (SOVAs) that are usually used in the researches of RDE. Compared to the RDE of OV, all element SOVs interact with the rotating object simultaneously, thus, the SOVA should be treated as a whole in the detection of rotating object. The RDE of SOVA originates from the OAM. Multiple OVs in the array spatially separate from each other and the OAM of the array element OVs composes the holistic OAM of the array. This holistic OAM can be intuitively presented by the method of modal decomposition. Theoretically, the method can reveal all contained modes of a general light beam and is suitable for the OAM analysis of the complicated SOVA [39,40]. The correlation of the photon OAM from different OVs enables the holistic OAM of the SOVA to display unique modal composition. Accordingly, the RDE of SOVA shows unique characteristics consistent with the holistic OAM. While, no researches have investigated it.

In this work, we observe the RDE of SOVA. We firstly analyze the OAM of the SOVA by the method of modal decomposition and demonstrate the dependance of OAM composition on the configuration of the SOVA. The OAM of the SOVA displays a unique modal gathering effect that the OAM converge to specific modes which is related to the symmetry of the SOVA. Next, we observe the rotational Doppler shift of SOVAs in an experiment of detecting a spinning object. The SOVAs composed of multiple superimposed LG OVs are modulated by a SLM, which can precisely generate the SOVA depicted by the digital hologram. The gathering effect is also introduced to the RDE of SOVA. We show the robustness of SOVA against the non-coaxial incidence and the enhancement in the amplitude of signals.

2. Theory

2.1 Generation of SOVA

The precise description of the SOVA is based on all array element OVs. A standard LG beam propagating along the z-axis of a cylindrical coordinate can be expressed as [3]

(1)$$\begin{aligned} E_p^l(r,\varphi ,z) &\propto \frac{1}{{\sqrt {1\textrm{ + (}{z^2}/z_R^2)} }}{(\frac{{\sqrt 2 r}}{{{\omega _z}}})^l}L_p^{|l |}{(\frac{{\sqrt 2 r}}{{{\omega _z}}})^2} \times \exp (\frac{{ - ik{r^2}}}{{2{R_z}}})\exp (\frac{{ - {r^2}}}{{{\omega _z}^2}}) \times \\ &\quad\textrm{exp[}i\textrm{(2}p + l\textrm{ + 1)ta}{\textrm{n}^{ - 1}}\textrm{(}\frac{z}{{{z_R}}}\textrm{)]} \times \exp (il\varphi ), \end{aligned}$$

where $k = 2\pi /\lambda$ is the angular wave number, $\lambda$ is the wavelength, $z$ donates propagation distance, ${z_R}$ is the Rayleigh length, ${\omega _0}$ is the waist radius, ${\omega _z}$ is the beam radius, ${R_z} = z + z_R^2/z$ indicates the curvature radius of wavefront, $L_p^{|l |}$ corresponds to the associated Laguerre polynomial, $l$ is the topological charge and $p$ is the radial index. When $z$ and $\omega$ are given, Eq. (1) can be shortened to $A_p^l(r) \times \exp (il\varphi )$. An OV can contain two equally conjugated modes $(|{ + l} \rangle + |{ - l} \rangle )$ and this type of superimposed LG beam can be expressed as

(2)$$E_p^{ {\pm} l}(r,\varphi ) = \frac{1}{{\sqrt 2 }}[A_p^l(r) \times \exp (il\varphi ) + A_p^{ - l}(r) \times \exp ( - il\varphi )].$$

This superimposed LG beam is usually employed as the probe beam in the RDE [18,19], that has an annular petal intensity distribution and a dark core at the center [41]. The quantity of radial petals is $2l$ and only the petals produce effective scattered light in detection. It has a striking advantage of self-interference characteristics that can omit the complex operation of heterodyne detection [42]. To generate an SOVA with a liquid crystal (spatial light modulator) SLM, we modulate a digital hologram containing the complex amplitude of multiple superimposed OVs and a blazed grating, as shown in Fig. 1(a). The key is to collimate all SOVs in the array and reduce the aberration caused by the light path. Thus, all generated OVs from the SLM need to be on the same diffractive order, and other parameters of each SOV in the array can be controlled flexibly, such as topological charge, radial index, size, amplitude, spatial position.

Fig. 1. (a) The digital hologram of an SOVA composed of four identical superimposed LG OVs $({\pm} 10)$. (b) and (c) are the intensity configuration and phase structure of the SOVA, respectively.

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We set the position of the j-th SOV center to $({d_j},{\varphi _j})$ in the polar coordinate established with the hologram center as the origin, where ${d_j}$ and ${\varphi _j}$ are the polar radius and the azimuthal angle of the j-th SOV center, respectively. The hologram center is regarded as the SOVA center. When a Gaussian beam passes through the digital hologram and a convex lens ordinally, an SOVA composed of multiple LG SOVs can be generated.

(3)$${E_{OVA}}\textrm{(}r\textrm{,}\varphi \textrm{) = }\sum\limits_{j = 1}^n {{a_j}\cdot E_{{p_j}}^{ {\pm} {l_j}}\textrm{(}r\textrm{ - }{r_j}\textrm{ ,}\varphi \textrm{ - }{\varphi _j}\textrm{)}} ,$$

where ${a_j}$ is the amplitude coefficient of the j-th SOV and $n$ is the quantity of SOVs. Figure 1(b) and (c) show the intensity distribution and phase structure of an SOVA composed of four identical SOVs in a square arrangement. To avoid disturbance, the petal spot of each SOV is independent of each other.

2.2 Principle of modal decomposition and RDE of SOVA

Since the OAM of OV is naturally defined according to the optical axis, the OAM composition for the orbital angular motion of each photon in the inertial space differs with different reference centers [38]. The LG modes are conceived as a set of orthogonal and complete basis vectors in high-dimensional Hilbert space [43]. For the conservation of OAM, an SOVA can be precisely represented as a superposition of LG modes coaxial with the reference center.

(4)$${E_{OVA}}(r,\varphi ) = \sum\limits_{l ={-} \infty }^{ + \infty } {\sum\limits_{p = 0}^{ + \infty } {B_p^l} } E_p^l(r,\varphi ),$$

where $B_p^l$ is the complex coefficient of each standard LG mode. Based on the modal decomposition method, the OAM spectrum is obtained from the basic definition with an inner product computation with the conjugated helical harmonic $\exp ( - il\varphi )$. The unique intensity of different kinds of OV has a slight impact on the result. A logical option is making the inner production between the complex amplitude of SOVA and the standard LG mode with continuous $p$ and $l$.

(5)$$B_p^l = \int_0^{\textrm{ + }\infty } {\int_0^{2\pi } {{E_{OVA}}(r,\varphi )} } \cdot [A_p^{ - l} \times \exp ( - il\varphi )]d\varphi dr.$$

All $B_p^l$ are classified by the mode to constitute the OAM spectrum and the power proportion of each mode ${B_l}$ is obtained after normalization. The completeness and accuracy of the decomposition can be evaluated by combining all $B_p^l$ with corresponding standard LG mode $A_p^l(r) \times \exp (il\varphi )$ to reconstruct the SOVA. It is noting that the range of $p$ and $l$ should be wide enough that all contained modes can be revealed.

Figure 2 demonstrates OAM spectra of different SOVAs relative to the reference center which is marked by the white spot. The range of $p$ and $l$ is $[0,20]$ and $[ - 40,40]$, respectively. As shown in Fig. 2(a) and (b), when the reference center coincides with the SOV center, the OAM spectrum contains two modes $( + 10, - 10)$ with equal power of 0.5. This is in accordance with the OAM of the SOV. When the reference center gets outside of the SOV and the displacement $(d)$ between the reference center and SOV center becomes $\sqrt 2 R$, where $R$ is the radius of the SOV. The OAM spectrum of the misaligned SOV expands into dozens of modes with the average power of 0.0125. This expanding effect is in accordance with a periodicity dispersion of the intensity spatial distribution relative to the reference center.

Fig. 2. OAM spectra of different SOVAs. Insets are the intensity configuration and phase structure of the SOVA. For intuition, the phase is presented in the region that the intensity is above 0.1 times of the maximum. The mode of each SOV in the array is labelled in the corresponding position. The position of the reference center of SOVA is marked by the white spot. The power of modes $( + 10, - 10\textrm{)}$ in (a)-(g), (i), and $( + 6, - 6\textrm{)}$ in (h) is labelled. In (a)-(h), the center of SOVA overlaps the reference center and the lateral displacement between the center of each SOV and the reference center is $\sqrt 2 R$, where $R$ is the radius of the SOV. In (i), the lateral displacement between the SOVA center and the reference center is $0.25R$.

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As shown in Fig. 2(b)-(e), each SOV in the array has the same parameter $(l\textrm{ = } \pm 10,R = 4mm,d = \sqrt 2 \textrm{R)}$. As the quantity of SOVs increases, the density power of OAM converges to some specific modes that distribute at the origin of $+ 10$ and $- 10$, and the difference between adjacent modes is four. When the SOVA contains five identical SOVs that uniformly distribute around the reference center, as shown in Fig. 2(f), the difference between adjacent modes becomes five. When the mode of the four SOVs changes to ${\pm} 6$, as shown in Fig. 2(h), the origin of distribution changes to $+ 6$ and $- 6$ simultaneously while the difference between adjacent modes remains at four. Meanwhile, the quantity of modes gets fewer accordingly. It indicates that the mode of the element SOV influences the modal complexity of the SOVA and the higher modes of the element SOV generate a broader OAM distribution of SOVA. Contrastively, when the SOVs have different modes $({\pm} 7, \pm 8, \pm 9, \pm 10\textrm{)}$, as shown in Fig. 2(g), the gathering effect disappears. These comparative results indicate that the OAM composition of the SOVA is directly related to the symmetry of the SOVA. The symmetry refers to that the array element SOVs constitute a spatially symmetric configuration and have identical initial OAM mode. Although all SOVs in the array are misaligned with the reference center, this gathering effect increases the average power of the modes. Compared to a single misaligned SOV in Fig. 2 (b), the quantity of modes of the SOVA in Fig. 2 (e) reduces to 20 and the average power of modes increases by 300%. The enhancement in power enables corresponding modes to produce more distinguishable signals in the experiment of RDE more possibly.

Furthermore, when the center of the SOVA gets misaligned with the reference center, a lateral displacement of $0.25R$ is involved. As shown in Fig. 2 (i), more subsidiary modes arise which shows that the OAM composition of SOVA is also affected by the location of reference center. While the power is concentrated on primary modes. It indicates that the gathering effect still exist. It is noting that all OAM spectra are symmetrical with respect to the mode of 0, and this is related to that all SOVs in the array have two equally conjugated modes along the optical axis, which obeys the conservation of OAM. Researches have demonstrated that the optical OAM is linked to the spatial distribution of the beam [3,44,45]. In other words, a pure OAM mode represents a spatial period that usually manifests as an entire circular intensity distribution. Any perturbation in the spatial period could bring a direct effect on the OAM composition. This can be proved that OVs containing multiple modes either display a non-circular spatial distribution nor diverge from the center of symmetry. Similarly, the modal gathering effect represents an entire spatial period of SOVA.

The OAM modal composition (including the mode and corresponding power) of the probing SOVA has a direct influence on the RDE. When the SOV or SOVA illuminates the rotating object, the frequency shift of scattered light depends on the modes of the incident OV and the rotating frequency of the object. Based on the theory of digital spiral imaging, the modulation function $\varPsi$ of the rotating object can be expressed as a combination of spiral harmonics [46].

(6)$$\psi (r,\varphi - n2\pi ft) = \sum\limits_{n ={-} \infty }^{ + \infty } {{C_n}(r)} \exp (in\varphi )\exp ( - in2\pi ft),$$

where $n$ is the order of the spiral harmonic, ${C_n}(r)$ denotes the corresponding complex coefficient and $f$ is the rotating frequency of the object. Based on the harmonic theory of RDE [47], when an OV probes the rotating object, the mode components of scattered light depend on the spiral harmonic distribution of the rotating object and each component of scattered light with mode of $l + n$ experiences an RDE frequency shift of $nf$. This frequency shift cannot be detected directly, a method of heterodyne or superimposed incident modes are required. For example, when the SOV only contains two equally conjugated modes $(|{ + l} \rangle + |{ - l} \rangle )$, the two components produce opposite frequency shift. The opposite frequency shifts produce an intensity modulation frequency of $2lf$ [36]. When the OV carries a superposition of OAM states, the frequency shift spectrum expands into multiple discrete signals [19]. At this time, the scattered light is the superposition of incident light modulated by different order spiral harmonics of the rotating object. Two different modes of the SOVA generate a beating signal whose intensity directly reflect the RDE frequency shift [47].

(7)$$I(t) \propto \sum\limits_i {\sum\limits_j {2|{{C_{{l_i}}}{C_{{l_j}}}} |E\sqrt {{B_{{l_i}}}{B_{{l_j}}}} \textrm{cos[(}{l_i} - {l_j}\textrm{)2}\pi ft\textrm{ + }\theta \textrm{]}} } ,$$

where $\theta \textrm{ = }angle[{C_{{l_\textrm{i}}}}{B_{{l_i}}}C_{{l_j}}^\ast B_{{l_j}}^\ast ]$. It indicates that the amplitude of signal depends on two main factors. The first is the amplitude of the probing SOVA. The other is the power density proportion of the mode of incident SOVA and the spiral harmonic of the object. The power of the frequency shift signal reaches the maximum when the mode of the incident OV matches the spiral harmonic of the object [48]. For an object with random scattering characteristics, it may contain abundant spiral harmonics and the modal composition differs at different radial positions [47]. Hence, not all modes of the SOVA can produce distinguishable signals when detecting the rotating object. However, the main feature of the modal gathering effect such as the difference between adjacent modes can be effectively revealed by directly detecting the Doppler frequency shift of scattered light.

3. Experiment and results

In order to observe the rotational Doppler shift of SOVA, we conduct a proof-of-concept experiment, as shown in Fig. 3. The collimated Gaussian beam with horizontal polarization illuminates the SLM. The size and power of the beam should meet the requirements for SLM and enable it to work properly. The SLM is programmed with a digital hologram containing an on-axis SOVA and a blazed grating. The generated first-order diffracted OVs propagate along the same spatial path. This method of spatial light modulating has an advantage that the parameters of the SOVA can be precisely adjusted by modulating the digital holograms. This approach can be easily extended to create various SOVAs. The size of generated SOVA can be adjusted by operating the 4-f optical system consisted of L1 and L2. The generated SOVA illuminates a planar metallic surface attached to a rotor. The surface should be large enough that the SOVA can be integrally scattered. The scattered light is gathered by a large-aperture lens and converged to a large-area photoelectric detector. The transformed voltage is acquired and digitized by a data acquisition card. Furthermore, a digital sampling and a fast Fourier transform (FFT) are applied to the transformed voltage to get the frequency spectrum. Due to the linearity of the FFT, all frequency shifts retain in the final frequency spectrum.

Fig. 3. Experimental setup for detecting a rotating object with an SOVA. A continuous wave He-Ne Laser (Thorlabs, HNLS008L-EC) is polarized by a horizontal polarizer (Pol) and collimated through a telescope to a diameter of 2 cm. The horizontally polarized Gaussian beam illuminates the screen of a SLM (HAMAMATSU, LCOS-SLM, X13138,1280*1024 pixels) which imprints a digital hologram of the SOVA. SOVs in the array distribute on the same diffraction order and pass through a spatial filtering system consisted of two lens (L1, L2) and an aperture (AP). The first-order is selected and divided into two paths by a beam splitter (BS), one is used to capture the intensity configuration by the charge coupled device (CCD, Newport, LBP2-HR-VIS2) and the other probes the rotating object vertically. The object is a metallic surface attached to rotor. The scattered light is gathered by a lens (L3) and converged to a large-area photoelectric detector (PD, Thorlabs PDA100A2). The transformed voltage is acquired and digitized by a data acquisition card (DAC) and the fast Fourier transform (FFT) is performed by a computer to obtain the rotational Doppler frequency shift.

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We generate the SOVAs demonstrated in Fig. 2 to probe the rotating object $(f = 40Hz)$. Within the experimental system, each Fourier transformation is calculated over 1 second of data collection and a sampling rate of 100 k, giving a frequency resolution of 1 Hz. Figure 4 shows the experimental frequency shift spectra and insets are corresponding generated SOVAs. The cross sign is the location of the rotating center. To reveal the correlation between OAM modes and rotational Doppler shifts, the reference center of the SOVA should overlap the rotating center of the object. As shown in Fig. 4(a), when a SOV$({\pm} 10)$ illuminates the object, the modulated intensity of the scattered light displays a single signal peak at the frequency of $20f$. This is predicted by the proposal that the two conjugated modes of the superimposed OV produce a frequency shift of $2lf$. A slight vibration of the rotor and the nonuniform intensity distribution of the SOV may produce minor subsidiary peaks. While, this error can be reduced by using a highly stable rotor and precise operation during the experiment. On the other hand, the emerging of a single peak represents a precise overlap between the reference center and the rotating center. By changing the hologram, a misaligned OV is generated, as shown in Fig. 4(b). No distinguishable signals appear in the signaling area around 800 Hz and higher frequency band. Contrastively, a set of minor peaks emerge in the lower frequency band. Since adjacent peaks corresponds to a modal difference of $\Delta l ={\pm} 1$, they are separated in frequency from each other by the rotating frequency of $f$. This is in accordance with that a misaligned OV produces a set of discrete signals, and the average power of signals decreases with the beam axis gets father from the rotating center [35,36].

Fig. 4. Experimental frequency shift spectra with different SOVAs. Insets are the intensity configuration captured by the CCD and the topological charge of each OV is labelled in the corresponding position. The cross sign is the location of the rotating center. The primary signal peaks are marked by circles, and the frequency is labelled by multiple of the rotating frequency of the object, which is kept at 40 Hz. The main source of uncertainty is not the identification of the peaks in the power spectrum; rather, it is the uncertainty in the stability of the rotor.

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Figure 4(b)-(e) show that the frequency shift spectra generated by different SOVAs. As the quantity of array element SOVs grows, the power of subsidiary peaks converges to primary peaks distributed at the frequency of multiple $4f$ and the rotating frequency can be directly deduced from the interval of two adjacent signals. Especially, when the SOVA contains four OVs, subsidiary peaks disappear in the spectrum. Compared to a single misaligned OV in Fig. 4(b), the average power of signals increases by 368%. It is noting that the frequency shift signals distribute at the low-frequency band. It indicates that the modulation function of the rotating object has primary low order spiral harmonic components. Meanwhile, the modal gathering effect enhances the power density of lower order modes. Contrastively, the power of signals decreases and the gathering effect disappears when the SOVA contains four different SOVs. When the SOVA contains five identical SOVs, the difference between adjacent peaks becomes $5f$. When the rotating center is laterally misaligned with the reference center, some subsidiary peaks emerge around the primary peaks. All frequency spectra show a direct correlation with the OAM mode, which also proves the OAM is the basis of rotational Doppler effect. Meanwhile, the spectra also display a gathering effect in keeping with the OAM spectra. Due to this gathering effect, the primary peaks always have higher power that contributes to signal recognition.

In addition to the symmetry of the SOVA, the lateral displacement $(d)$ between the array element SOVs and the SOVA center also has an impact on the OAM composition. As shown in Fig. 5, all SOVAs contains four identical SOVs $({\pm} 6)$ with 4-fold symmetry that generates the same origin and adjacent interval in the modal distribution. While, the OAM spectrum converges and the power proportion of $( + 6, - 6\textrm{)}$ enlarges when $d$ decreases. It is reasonable to predict that when $d$ decreases to 0, all array element SOVs will overlap with each other and only the mode of $( + 6, - 6\textrm{)}$ will retain. It can be understood that the photon OAM correlation from different SOVs is weakened by $d$.

Fig. 5. OAM spectra of different SOVAs. Insets are the intensity configuration of the SOVA. All SOVAs contain four identical OVs with the mode of $({\pm} 6\textrm{)}$. The center of SOVA is marked by the white spot and the power of modes $( + 6, - 6\textrm{)}$ is labelled. $d$ is the lateral displacement between the SOV center and the SOVA center and $R$ is the radius of the SOV.

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4. Discussions and conclusions

Compared to the single OV, the frequency shift signals produced by SOVA have significantly higher average power. On the one hand, the modal gathering effect improves the power of primary modes; on the other hand, the more OVs increase the total power of the SOVA. Accordingly, the power of scattered light also increases while the power of background light remains unchanged, which reduces the influence of noise in a certain degree. The special distribution law of signals produced by SOVAs and the enhancement in the power helps to distinguish the signals. In addition, even the SOVA is misaligned with the rotating center of the object, the modal gathering effect also exists and reduces the effect of misaligned incidence. As mentioned above, not all modes of the SOVA can produce distinguishable signals due to the complex scattering characteristics of the object. This deficiency can be improved by using a precisely designed surface that contains required spiral harmonics or using a SLM to simulate the rotating object. In our experiment, the rotating object has a metal surface which helps to enhance the signal. The composition of the scattered light has noteworthy impact on the power of distinguishable signals. The receiving aperture and the position of detector need to be set appropriately to gather adequate scattered light. The parameters of SOVs and the configuration of the SOVA can be adjusted flexibly for detecting various targets. While the size and quality of the SOVA is limited by the numerical aperture and spatial resolution of the optical system.

Based on the principle of RDE of SOVA, a variety of SOVAs can be introduced into the detection of rotation motion and provides new scheme for improving the signal-to-noised ratio and satisfying the detection of different objects. In addition, the precise description of the holistic OAM helps to reveal the correlation of the photon OAM from different OVs. Although the RDE is well known, our study extends it to the SOVA. The RDE of OV mainly depends on the topological charge, while the RDE of SOVA is directly related to the holistic OAM composition. A numerical modal decomposition operation precisely reveals the OAM states of SOVA that displays a unique modal gathering effect. This effect enables the SOVA to enhance the amplitude of rotational Doppler shift signals. Our finding provides a new aspect of RDE, and various specially designed SOVAs can be introduced into rotation detection.

Funding

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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Detection of a spinning object using a superimposed optical vortex array

Abstract

1. Introduction

2. Theory

2.1 Generation of SOVA

2.2 Principle of modal decomposition and RDE of SOVA

3. Experiment and results

4. Discussions and conclusions

Funding

Disclosures

Data availability

Reference

Data availability

Cited By

Figures (5)

Equations (7)

Optics Express