## Abstract

The optical rotational Doppler effect (RDE) is closely related to the unique orbital angular momentum (OAM) carried by optical vortex, whose topological charge means the mode of OAM. Compared with the coaxial incidence, the rotational Doppler frequency shift spectrum of a misaligned optical vortex (misaligned RDE) widens according to a certain law. In this paper, an OAM modal decomposition method of the misaligned optical RDE is proposed and the relative intensity of different OAM modes, namely the OAM spectrum, is derived based on an inner product computation. Analyses show that lateral displacements and angular deflections change the distribution of OAM modes relative to the rotation axis of the object. A misaligned Laguerre-Gaussian (LG) vortex can be represented as a specific combination of coaxial LG modes, and the difference between the topological charge of two adjacent modes is 1 or 2 with lateral displacements or angular deflections respectively. An experiment of misaligned optical RDE using a superimposed LG vortex is executed, and the obtained frequency shift spectrum with misaligned incidence expands into a set of discrete signals, which agrees well with the theoretical results. Moreover, we can get the rotation frequency of the object from an expanded frequency spectrum more quickly and accurately based on the difference between two adjacent signal peaks. The proposed method contributes to analyze the misaligned optical RDE comprehensively, which is significant in remote sensing and optical metrology.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In 1992, Allen et al. [1] proposed a kind of special optical beam which carries orbital angular momentum (OAM) with a helical phase factor $\textrm {exp} (il\varphi )$, where $l$ is the topological charge and $\varphi$ is the azimuthal angle, and the topological charge means the mode of OAM. Such an OAM-carrying optical beam with a phase singularity at its center and an annular intensity structure [2,3] is often called an optical vortex, and the Laguerre-Gaussian (LG) optical beam is a kind of typical optical vortex. For an optical vortex, the OAM is naturally defined according to the optical axis, and the OAM carried by each photon is $l\hbar$, where $\hbar$ is the reduced Planck constant [1,4]. The OAM of optical vortices provides a set of discrete parameters determined only by topological charges, which makes optical vortices easy to manipulate and transfer information [2,3]. With these features, optical vortices have been widely studied in optical manipulation [5,6], quantum information [7–9], optical rotational Doppler effect (RDE) [10–14] and other fields.

The optical RDE opens new approaches for remote sensing of rotation motion from macroscopic objects [11] to microscopic particles [12,15,16]. In 2013, Lavery et al. [11]. proposed and experimentally deduced the optical RDE formula in rotation frequency detection for the first time. They used a superimposed LG vortex to illuminate the rotation object with the vortex axis precisely aligned to the rotation axis of the object. By collecting the beating signal of scattered light from the object and analyzing frequency shift, they successfully obtained the rotation frequency of the object. Since then, different influence factors have been introduced into related researches and applications, ranging from the light wavelength [17], free-space turbulence [18] and remote sensing [19], and then to misaligned incidence [13,14]. For complex motions of the object, the linear and rotational Doppler frequency shift can be detected at the same time [12,20], besides, the rotation axis of a planar object [21], direction of rotation [16] and the angular acceleration [22] can also be obtained. These researches mainly focus on the linear Doppler method and rarely analyze the distribution law of frequency shift signals. It is worth noting that the essence of optical RDE can be considered as the interaction between OAM carried by optical vortices and rotation objects [10,23–25], which has been studied in the OAM complex spectrum analyzer [25] and OAM analysis of a rotating light beam [23,26]. In a related research, a simulated rotational Doppler frequency shift spectrum of a misaligned Laguerre-Gaussian (LG) vortex is obtained from the OAM mode distributions, while there in no corresponding experimental results [26]. What is puzzling is that we find the simulated result in [26] does not agree well with the experimental frequency shift spectrum in our work. Thus, the regularities of misaligned optical RDE signals needs to be analyzed more comprehensively. These mentioned researches and applications give us implications to propose a more practical and effective method to analyze the misaligned optical RDE based on the OAM modal decomposition.

In this paper, an OAM modal decomposition method of a misaligned optical vortex is proposed, and the OAM spectrum and experimental frequency shift spectrum are derived when there is a lateral displacement or an angular deflection between the vortex axis and the rotation axis of the object. Theoretical result shows that a misaligned LG vortex can be represented as a combination of coaxial LG modes. Besides, the difference between the topological charge of two adjacent modes is 1 or 2 with lateral displacements or angular deflections respectively. The experimental frequency shift spectrums of RDE using a misaligned superimposed LG vortex contain a series of discrete frequency signals that distribute in correspondence with the theoretical results. Furthermore, based on the discrete frequency shift signals, we can obtain the rotation frequency of the object more quickly and accurately without analyzing every signal, which is vital for the practical use of optical RDE. The proposed method contributes to accurately analyze misaligned optical RDE, which may provide theoretical guidance for applications such as remote sensing and optical metrology.

## 2. Theoretical analysis

A standard LG mode is a solution to the paraxial wave equation in cylindrical coordinate system, and carries a single OAM. All of the LG modes constitute a set of complete orthogonal vectors, any optical beam can be represented as a superposition of standard LG modes, and the relative intensities of different OAM modes constitute the OAM spectrum. LG beam is a typical optical vortex, whose complex amplitude can be expressed as $E(r,\varphi ,z) = E(r,z)\textrm {exp} (il\varphi )$, where $\textrm {exp} (il\varphi )$ is the helical harmonic and the origin of OAM [1], and $l$ can be an integer or a fraction. The OAM of optical vortex is determined only by discrete topological charges, which can be decomposed and synthesized. Since the OAM is naturally defined according to the optical axis, the numerical result of OAM is different by selecting different reference axes for the orbital angular motion of each photon in the inertial space [26]. In the case of conservation of OAM, a misaligned LG vortex can be represented as a superposition of LG vortices coaxial with the reference axis. In order to ensure the OAM modal decomposition method effective and universal, the OAM spectrum is obtained from the basic definition with an inner product computation [27]. The helical harmonic $\textrm {exp} (il\varphi )$ is a unique wave function of OAM, and a misaligned LG vortex can be expanded directly through helical harmonic based on the standard LG vortex [3,27].

In Cartesian coordinates, at $z = 0$, a standard LG vortex propagating along the $z$ axis can be expressed as:

In cylindrical coordinates, letting $x = r\cos \varphi$ and $y = r\sin \varphi$, we obtain:

When a lateral displacement is considered, as shown in Fig. 1(a), the optical axis is parallelly displaced by $d$ in the y-axis. The laterally misaligned LG vortex can be expressed as:

In cylindrical coordinates, letting $x = r\cos \varphi$ and $y = r\sin \varphi$, we obtain:

The same method is applied when considering an angular deflection. The angle between z-axis and optical axis is $\gamma$ in the $yoz$ plane as shown in Fig. 1(b). It should be noted that the light spot on the surface of the object becomes elliptic, and the optical path to the surface is no longer the same. However, the initial OAM does not change because it is decided by the topological charge rather than the optical path. Thus, only the projection of the helical harmonic on the rotation object needs to be considered. After coordinate transformation, the Cartesian coordinates become ${M_1}(\gamma ){[x,y]^T}$, where matrix ${M_1}(\gamma )$ indicates rotation of $\gamma$ around x-axis. Omitting influence of optical path, the tilted LG vortex can be expressed as:

In cylindrical coordinates, we get:

Hence, we come to the result of helical harmonic:

Equations (4) and (8) can be expanded through helical harmonic $\textrm {exp} (il\varphi )$ based on the standard LG mode:

Due to the orthogonality between standard LG modes, ${A_l}$ can be obtained by an inner product computation:

In the simulation, the OAM spectrum expands around the initial topological charge $l$, so we can ensure the integrality of the OAM spectrum by taking a certain range of topological charges around $l$ in Eq. (10). In order to evaluate the accuracy of the modal decomposition, we combine the coefficient ${A_l}$ with corresponding standard LG mode to reconstruct the vortex. For example, we simulate the modal decomposition and synthesis by setting the initial LG vortex with ${\omega _0} = 0.75$, $d = 0.2$, $\gamma = {30^\circ }$, $l = 18$ or ${\pm} 18$ respectively, as shown in Fig. 2. We set the topological charge with $l = 18$ or ${\pm} 18$, as the LG vortex with larger topological charge is more sensitive to misaligned incidence, and we can see the difference in the OAM spectrum more visually, other topological charge can also be applied. It should be pointed out that, using a superimposed LG vortex (${\pm} l$) does not affect the accuracy of decomposition, LG vortices with $+ l$ and $- l$ get exactly the same OAM spectrum symmetrical about 0. The intensity distribution and topological charge of synthesized and initial vortices are highly consistent, while the helical phase distorts in the area where the intensity is minor. It does not affect the accuracy, because the area with minor intensity is inessential in the experiment of RDE. It is clear that the OAM spectrum is composed of some discrete modes with different topological charges. When a lateral displacement is induced ($d = 0.2$), the difference between the topological charge of adjacent modes is 1 and the relative intensities of different LG modes distribute symmetrically about 18 and -18. In contrast, when an angular deflection is induced ($\gamma = {30^\circ }$), the difference between the topological charge of adjacent modes becomes 2 and the symmetry in the intensity distribution disappears. It can be explained theoretically, the intensity distribution and phase structure of the light spot on the object are still circular with a lateral displacement, while become elliptic with an angular deflection. The OAM comes from the helical harmonic $\textrm {exp} (il\varphi )$, and different phase structure has different symmetry.

The OAM of optical vortices directly affects the rotational Doppler frequency shift. When an aligned superimposed LG vortex illuminates a rotating object with angular frequency $\Omega $, the frequency shift can be expressed as $\Delta f = l\Omega $. When a lateral displacement or an angular deflection is induced, the mode of incident vortex becomes a superposition of discrete numbers relative to the rotation object axis. Since the OAM spectrum is symmetrical about 0, we set the topological charges of incident vortex as two conjugate sets $L$ and $- L$, and $L$ contains $({l_1},{l_2} \cdots {l_n})$. Besides, the topological charge of the scattered vortex is set as $m$. In the Cartesian coordinate shown in Fig. 1, the Doppler frequency shift for $L$ can be expressed as:

Similarly, for the part of $- L$, the Doppler frequency shift can be expressed as:

Therefore, the frequency shift of beating signal can be expressed as:

We find the frequency shift result of the superimposed LG vortex does not contain $m$, and ${l_i}$ is in $L$, ${l_j}$ is in $- L$. Compared with aligned incidence, the misaligned optical RDE frequency shift spectrum broaden as the modes of the incident vortex expand. Meanwhile, the relative power of beating signal is in proportional to the intensity product of two modes (${l_i}$ and ${l_j}$) [24], so we simulate the rotational Doppler frequency shift spectrum from the OAM spectrum by setting the initial superimposed LG vortex with ${\omega _0} = 0.75$, $l ={\pm} 18$, $d = 0.05$, or $\gamma = {10^\circ }$, as shown in Fig. 3. For example, when setting the misaligned parameters with $d = 0.05$ and $\gamma = 0$, the OAM spectrum consists of six modes including $\{ - 19, - 18, - 17\}$ and $\{ 17,18,19\}$. From Eq. (13), two modes with $l = 17$ and $- 17$ generate $f = 34\Omega $, $l = 17$ and $- 18$ generate $f = 35\Omega $. In the same way, we can get the frequency shift spectrum including $\{ 34,35,36,37,38\}$ in $\Omega $ units, as shown in Fig. 3.

## 3. Experimental setup

We design an experiment of RDE using a misaligned superimposed LG vortex with topological charge $l ={\pm} 18$ to obtain the rotational Doppler frequency shift spectrum, as shown in Fig. 4(a). An He-Ne laser (Thorlabs, HNLS008L-EC) generates a 632.8 nm Gaussian beam, which is attenuated and polarized by an attenuation sheet (At) and a horizontal linear polarizer (Pol) to enable the spatial light modulator (SLM) to work properly. Then the beam is expanded and collimated through lens L1 and L2 (${f_1} = 12\textrm{mm}$ and ${f_2} = 120\textrm{mm}$). After passing through a polarized beam splitter (PBS), the horizontally polarized beam illuminates the SLM (HAMAMATSU, LCOS-SLM X13138). A computer-generated hologram $(l ={\pm} 18)$ is loaded onto the SLM, as shown in Fig. 4(b). The generated vortex subsequently passes through a $4 - f$ spatial filtering system composed of lens L3, L4 ($f = 100\textrm{mm}$) and a spatial filter (SF), the SF should be placed at the focal point of lens to select the first-order diffraction vortex with high quality.

Next, the vortex is divided into two paths through a beam splitter (BS), one is used for the CCD (Newport, LBP2-HR-VIS2) to capture the intensity profile and the other illuminates the surface of the rotating object, where the distance between CCD and BS should be equal to the distance between the object and BS, and the angular deflection should be also the same for both. In order to improve the intensity of scattered light, the object surface is covered with tin foil, and a lens L5 ($f = 30\textrm{mm}$) is used to gather as much scattered light as possible to the photodetector (Thorlabs PDA36A2), whose response time is $1ns$. The PD is connected to an oscilloscope (Tektronix, MDO3012) to perform the real-time Fourier transform, and the frequency shift spectrum can be obtained.

## 4. Results and discussions

To introduce a lateral displacement, the rotating object is fixed on a translation stage and can move along the direction perpendicular to the rotation axis without changing the incident angle $\gamma$. The radius of the optical vortex spot is approximately $3.3\textrm{mm}$ and the corresponding beam waist radius is $0.757\textrm{mm}$, which is consistent with the simulated results. The object angular frequency is set as $68\textrm{Hz}$ and $\gamma$ is quite small (nearly normal incidence). Keep the rotation frequency constant, translate the object with a step of 0.05mm, and measure the frequency shift spectrum with different lateral displacements, as shown in Fig. 5. In contrast, when an angular deflection is induced, the rotating object is fixed on a horizontal rotating table. The key to obtain ideal result is to make sure three axes intersect at the rotation center of the object surface, including the optical axis, the rotation axes of the object and the rotating table. Otherwise, the lateral displacement will be induced when the angular deflection changes. The angular frequency is set as $71\textrm{Hz}$, adjust the rotating table with each angular deflection of ${5^\circ }$. The optical vortex spot on the object surface gradually changes from circular to elliptic, while the center always coincides with the center of the object. Meanwhile, the scattered light collection system should be rotated ${10^\circ }$ each time, since the incident angle and the reflected normal of the object surface change by ${5^\circ }$ at the same time. The measured rotational Doppler frequency shift spectra with different angular deflections are shown in Fig. 6.

From the frequency shift spectra, we measure the rotation frequency of the object in two ways. ${f_{\bmod }}$ means dividing frequency of every signal peak by corresponding topological charge $f/(\sum\limits_{i = 1}^n {{l_i} - \sum\limits_{j = 1}^n {{l_j})} }$, $\Delta f$ is the difference between two adjacent signal peaks, ${f_{\textrm{set}}}$ denotes the set angular frequency.

As shown in Fig. 5, when there is no lateral displacement $(d = 0)$, the spectrum displays a distinct signal at $2451.17\textrm{Hz}\; ({\Omega _d} = 68.09\textrm{Hz})$. With the increase of lateral displacements, the spectrum expands symmetrically about the signal peak of 2451.17 Hz and $\Delta f$ approximately equals to angular frequency $(68\textrm{Hz})$. Herein, the symmetry refers to the frequency value of different signals instead of the intensity of each signal peak. Similarly, as shown in Fig. 6, when the angular deflection is less than ${15^\circ }$, the spectrum does not broaden significantly and displays a signal at $2560.83\textrm{Hz}\; ({\Omega _\gamma } = 71.13\textrm{Hz})$. With the angular deflections increase, the spectrum expands and $\Delta f$ approximately equals to twice angular frequency $(142\textrm{Hz})$, which is in accordance with the theoretical analysis. From Figs. 5(b) and 6(b), we can see that ${f_{\bmod }}$ has a good correspondence with ${f_{\textrm{set}}}$ and the maximal relative error is below $0.5\%$, which demonstrates the optical RDE is effective and accurate in the sensing of rotary motions. While $\Delta f$ is slightly less accurate with maximal relative error of $7.32\%$. The error of ${f_{\bmod }}$ and $\Delta f$ distributes randomly, so it can be reduced through more data processing methods. Besides, the error of ${f_{\bmod }}$ is lower because it is reduced through dividing by corresponding topological charge. Moreover, the distribution law of signals is significant in practical use of RDE, because misaligned incidence is more common and the topological charge of every signal peak cannot be ensured in previous researches. It is noteworthy that we can get the rotation frequency from $\Delta f$ in an error range, which greatly contributes to confirm the topological charge of every signal peak $(l \approx {f / {\Delta f}})$. Then we can obtain a more accurate result of angular frequency from ${f_{\bmod }}$.

On the one hand, we can get the angular frequency from the expanded spectrum with minor error, on the other hand, the distribution law of experimental signals proves the correctness of modal decomposition method proposed in this paper. As shown in Fig. 7, the number of experimental signal peaks is highly consistent with the theoretical results. While there are differences in the relative intensity of each signal peak, and some theoretical signal peaks seem to disappear in the experimental spectrum when lateral displacements and angular deflections increase. This is caused by the fluctuation of signal to noise ratio (SNR) which is related to two main factors: the first one is the detection condition, the fewer noise light and more effective light collecting system will produce higher SNR, the other is the scattering property of the object. In fact, the intensity of beating signal is not only decided by the relative power of incident OAM modes, but also modulated by harmonic components of the rotating object. Thus, some minor signal peaks are submerged in the noise rather than disappear in the experimental spectrum. However, most of signal peaks can be recognized, so the integrality of the experimental frequency shift spectrum can be guaranteed.

Furthermore, when lateral displacements and angular deflections are induced at the same time, their influences are coupled together and the spectrum becomes more complicated. So, we need to avoid inducing them all to distinguish their influences respectively. From the experimental spectra, we can find that the optical RDE is very sensitive to lateral displacements, which can be used to precisely adjust the rotation axis of the object. Besides, different misaligned conditions will generate different frequency shift spectra, it is possible to achieve stance positioning of the rotating object based on the signal distributions in the frequency shift spectrum.

## 5. Conclusion

In conclusion, we have investigated the misaligned optical RDE with an OAM modal decomposition method. We find that a misaligned LG vortex can be represented as a combination of LG modes coaxial with the rotation object, and the difference between the topological charge of two adjacent modes is 1 or 2 with lateral displacements or angular deflections respectively. Besides, we carry out an experiment of misaligned RDE to verify the theoretical results. The experimental result shows that frequency spectrum with misaligned incidence expands into a set of discrete signals, which distribute in good correspondence with the theoretical results. Moreover, we can get the rotation frequency of the object from an expanded frequency spectrum more quickly and accurately. Our work provides a more comprehensive method to analyze the misaligned optical RDE, which may promote the theoretical research and practical use of RDE in remote sensing and optical metrology.

## Funding

Key Research Projects of Foundation Strengthening Program of China (2019-JCJQ-ZD); National Natural Science Foundation of China (11772001, 61805283); National Outstanding Youth Science Foundation of China.

## 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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