# Rotational Doppler Effect in Vortex Light and Its Applications for Detection of the Rotational Motion

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. RDE Induced by a Rotating Vortex Optical Beam

^{ilφ}, such as the nonzero-order Laguerre–Gaussian (LG) laser mode [11]. A vortex beam exhibits the orbital angular momentum (OAM) of $l\u045b$ in per photon, where l is the topological charge, φ is the azimuth, and $\u045b$ is the reduced Planck constant. The vortex beam has a spiral wave front, with a phase singularity on the axis and a typical intensity distribution of concentric rings [12]. Soon after, in 1996, Nienhuis predicted that a rotating mode converter that consists of three parallel cylindrical lenses could change the sense of the OAM, as well as the frequency of the beam [13]. It was demonstrated theoretically that a LG beam with mode number l can form a frequency difference of 2 lΩ after passing through a mode converter with the rotation speed Ω, which also indicates the potential of a vortex beam to detect rotational motion. Courtial, et al. carried out experiments by rotating the millimeter wave beam based on a rotating Dove prism in 1998 [14,15]. As shown in Figure 1, the interference beam coming from the superposition of two vortex beams passing the stationary and rotating prism exhibits a periodic change of intensity because of the additional frequency shift induced by the rotating the vortex beam. The rotation of the beam will change the number of wave fronts passing through a certain plane in a unit of time by the parameter of l. Hence, the frequency shift is simply equal to lΩ. They also showed that both the OAM and SAM can influence the rotational frequency shift as ∆ω = (l + σ)Ω, where σ = ±1 corresponds to the left- and right-handed circularly polarized light.

_{0}

^{1}beam are coaxially superposed as off-axis OV and rotated together. By detecting the localized variations of intensity distribution in the rotating beam, the equivalence between the observable off-axis OV rotation and the frequency shift of the corresponding LG mode is verified as follows:

_{G}is the amplitude of the Gaussian superposition beam, E

_{LG}is the amplitude of LG

_{0}

^{1}, w is the waist parameter, and ρ is the radial coordinate in cylindrical coordinates. It can be concluded based on Equation (1) that the local light intensity oscillates with a frequency ∆ω/2π, which is similar to the frequency beat that appears during the measurement of rotational Doppler frequency shift in a dual beam scheme. When rotating the beam coaxially superimposed by the Gaussian beam and LG

_{0}

^{1}beam, the Gaussian beam does not experience a frequency shift, while a frequency shift appears in LG

_{0}

^{1}beam. The frequency difference between the two superposed beams leads to the beat signal. Differently from the beat frequency obtained by the two-path system with additional reference light, this off-axis OV can directly induce a beat signal in a single-path system. This scheme can reduce the complexity of the experimental system to obtain an RDE signal, which has potential applications for communication systems at either the classical [15] or quantum level [18].

## 3. RDE for Detecting the Rotation of Micro Targets

_{t}/2π, where particle Ω

_{t}denotes the angular velocity of particle. When the beam phase rotates with an angular velocity of Ω

_{s}, the frequency shift becomes

_{t}and Ω

_{s}are the same, so |Δf′| < |Δf|. At this time, the frequency shift will be reduced compared with the situation when the beam phase is stable. On the contrary, when the target rotates in the opposite direction to the beam, Ω

_{t}and Ω

_{s}have opposite signs, and a higher frequency shift will be detected.

_{1}and f

_{2}) 2-fold multiplexed vortex light is used to illuminate a rotating object, the frequency of intensity modulation can be expressed as

_{ref}= |f

_{1}−

_{2}| and f

_{D}= lΩ/π is the frequency of beat signal induced by RDE. Since the phase of a dual frequency 2-fold multiplexed optical vortex is time-varying, it will lead to continuous rotation of light, and the corresponding angular velocity can be expressed as ω = 2π(f

_{1}− f

_{2})/2l. When f

_{1}> f

_{2}, ω is a positive value, which leads to an anti-clockwise rotation of the superposed beam. If f

_{1}< f

_{2}, the superposed beam will rotate clockwise. The relative velocity of a rotating beam and object will decrease when their rotation direction is the same, and the frequency difference between two vortex beams is reduced, leading to f

_{mod}< f

_{ref}. When the rotational direction of a superposed beam is opposite to the object, their relative velocity is the sum of their velocities. Therefore, the frequency difference between the two vortex beams increases, and f

_{mod}> f

_{ref}. By comparing the value of f

_{mod}with the pre-set f

_{ref}, the rotation direction and angular velocity of the object can be determined using only a single measurement. Moreover, this method can automatically transfer the beat signal to the high-frequency domain and eliminate low-frequency disturbance, which is of great significance to obtain the precise rotation angular velocity.

_{z}is the velocity of the target along the line of sight, k = 2π/λ is the wave vector, and λ is the wavelength of light. By measuring the frequency shift corresponding to the longitudinal and rotational motion, respectively, the full velocity of particles in spiral motion can be evaluated. This technology needs to select the appropriate spatial mode of structured light, which depends on the specific type of motion. By properly selecting the transverse phase profile of the illumination beam, this technology can be extended to the detection of microorganisms with a complete 3D movement.

_{mod}= 2lΩ. They proposed that the measurement accuracy could be further improved if the topological charge l was increased, as shown in Figure 6. When the topological charge becomes larger, the illumination area of the incident beam also increases, which can more widely include the roughness of the particle surface. Thus, a narrower width of spectral peak can be obtained in the beat spectrum, which means the accurate value of the modulation frequency is easier to locate. As shown in Figure 7, when the trapping power is changed, the results are still in good agreement with the theoretical values. Therefore, it is more beneficial to analyze the spectrum characteristics of scattered light by illuminating trapped particles with superposed vortex beams with large and mutually opposite topological charges. This method is promising for detecting the torsional characteristics of a micro object.

## 4. RDE for Detecting the Rotation of Macro Targets

^{inφ}[43,45,46]. Assuming that the reflectivity of the spinning object is homogeneous, the rotating object can be regarded as a series of vortex phase modulators that only depend on the surface roughness. The roughness of stationary surface, as shown in Figure 9b, can be written as h(r,φ), and the modulated phase is given by Φ(r,φ) = 4πh(r,φ)/λ. The modulation function in Fourier expansion form is as follows [25]:

_{n}(r) is the complex amplitude of n-order harmonic, n is the integer, and ∑|A

_{n}(r)|

^{2}= 1. When an incident vortex beam at frequency f expressed by B(r)exp(−i2πft)exp(ilφ) is normally incident on the spinning object, the scattered light can be expressed as

## 5. RDE with Misaligned Illumination of a Vortex Beam

_{1}is the radius of annular vortex beam, r

_{2}is the radius from the rotation axis to the point where the velocity is measured, f

_{0}is the original frequency of light, and γ represents the angle between the actual velocity and its azimuthal projection in the polar coordinate with the beam center as the origin. When the superposed vortex beams with opposite topological charge l are incident on the rotating surface, the scattered light will produce an opposite frequency shift, which leads to intensity modulation with the frequency

^{′}). θ is the azimuthal coordinate of the beam

_{.}It can be concluded that with the increase of lateral misalignment d, the frequency shift ∆f becomes dispersed. When there is a large misalignment, i.e., d > r, as shown in Figure 14c, the linear velocity at an arbitrary point of the annular light spot can be decomposed into the transverse velocity component v

_{0}′ (v

_{0}′ = dΩ) and the tangential velocity component v

_{r}(v

_{r}= Ωr). Thus, the rotational motion at point O can be regarded as a combination of the translational motion v

_{0}′ and the rotational motion relative to point O′. When the rotation center of the rotating surface is completely out of the beam spot, the frequency shift caused by the tangential velocity component v

_{r}is

_{r}can be used to explore the rotational angular velocity of the object. The maximum frequency shift caused by the translational velocity component v

_{0}′ is

_{0}is the original frequency of light and the ϕ is the angle between the Poynting vector and the velocity vector. When ϕ = 90°, the minimum frequency shift is ∆f

_{min}= 0. Thus, the range of frequency shift caused by v

_{0}′ is [0, |∆f

_{max}|], which can be considered ground noise. When the misalignment d increases, it will lead to the increase of v

_{0}′, and thus ∆f

_{max}gradually increases.

_{min}and f

_{max}are lΩcosγ/2π and lΩ/2πcosγ, respectively. The center frequency f

_{c}is

_{c}can still be calculated according to Equation (19), which can be used to calculate the rotational angular velocity of the object. When the incident angle exceeds 0.87 rad, all the peaks are too weak to be recognized. Therefore, an effective beat signal can only be obtained within a certain illumination angle. As shown in Figure 17b, within the incidence angle range of 0.17–0.87 rad, the f

_{c}corresponding to different incident angles is almost constant, and the error between the experimental and theoretical value is very small, which can ensure the accuracy of the measurement.

_{0}denotes the amplitude coefficient, w

_{0}is the beam waist radius, L

_{p}

^{|l|}is the symbol of associated Laguerre polynomial, l is the topological charge, and p is the radial index. When a lateral displacement is considered, as shown in Figure 18a, the optical axis is parallelly displaced by d in the y-axis. The laterally misaligned OV can be expressed as

^{iφ}−-de

^{iφ})

^{l}contains the helical harmonic factor e

^{ilφ}. Using the same method to consider an angular deflection, as shown in Figure 18b, the tilted OV can be expressed as

^{l}contains the helical harmonic factor, which can be expanded as

_{mod}and the scattered modes is more conducive to reducing the error. On the other hand, from the frequency interval of adjacent peaks, the range of angular velocity can also be determined within a certain error range.

## 6. Summary and Prospect

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Interferometer for the observation of the rotational Doppler shift. Adapted with permission from Ref. [14].

**Figure 2.**Target rotates with angular velocity Ω and a time-independent (t

_{1}, t

_{2}, t

_{3}represent three different times) Doppler shift of value lΩ/2π appearing in the scattered vortex beam.

**Figure 3.**Experimental setup to measure the angular velocity of simulated particles moving along a circular trajectory based on RDE. T1~2: Telescope, BS: Beam splitter, SLM: Spatial light modulator, M: Mirror, PD: Photoelectric detector, DMD: Digital micromirror device, DO: Digital oscilloscope. Adapted with permission from ref. [21].

**Figure 4.**Output signals of PD under different topological loads (

**a**) m = 1 and (

**b**) m = 4, and the corresponding Fourier transform spectrum (

**c**,

**d**). Adapted with permission from ref. [21].

**Figure 5.**Experimental setup to measure the rotation speed and direction of a simulated particle based on RDE. PBS1~2: Polarizing beam splitter, M1~2: Mirror, SLM: Spatial light modulator, OC: Optical chopper, SF: Spatial filter, BS: Beam splitter, L1~2: Lens, QWP: Quarter-wave plate, PD1~2: Photoelectric detector, DMD: Digital micromirror device.

**Figure 6.**Beat spectrum when using a superposed vortex beam for illumination with different topological charges. (

**a**) l = ±7, (

**b**) l = ±8, and (

**c**) l = ±9. Adapted with permission from ref. [33].

**Figure 7.**The beat frequency and calculated rotation frequency as a function of trapping laser power. Adapted with permission from ref. [33].

**Figure 8.**Experimental setup to measure the frequency shift of scattered light from a rotating rough surface, based on RDE. L1~4: Lens, SMF: Single mode fiber, SLM: Spatial light modulator, AP: Aperture, MMF: Multimode fiber, PD: Photodetector.

**Figure 10.**(

**a**) Normal harmonic distributions of the spinning object. (

**b**) Comparison between measured and theoretical results of harmonic distribution. Adapted with permission from ref. [43].

**Figure 11.**Diagram of experimental setup to measure the 3D motion using a vector optical beam. M: Mirror, SLM: Spatial light modulator, BS1~3: Beam splitter, QWP1~3: Quarter-wave plate, PD: Photoelectric detector, HWP1~2: Half-wave plate, Pol1~2: Polarizer, CCD: Charge-coupled device camera.

**Figure 12.**The beat spectrum when using different beams for illumination. (

**a**) Pure translation under Gaussian illumination, (

**b**) pure rotation under structured light illumination. Composed motion under vector beam illumination for (

**c**) l = ± 3 and (

**d**) l = ± 5. Adapted with permission from ref. [47].

**Figure 13.**Experimental results when introducing different lateral misalignments. Adapted with permission from ref. [50].

**Figure 14.**Diagram of misaligned illumination with different offsets, and the corresponding decomposition of velocity vector. (

**a**) The center of the OAM light beam coincides with the rotation center of the object. (

**b**) With a small misalignment. (

**c**) With a large misalignment. (

**d**) The relationship between the Poynting vector of the vortex beam and the velocity vector of rotating object and ϕ is the angle between them. Adapted with permission from ref. [53].

**Figure 15.**Rotational Doppler spectrum under different lateral misalignments

**.**(

**a**) d = 0.15 r. (

**b**) d = 0.25 r. (

**c**) d = 0.3 r. (

**d**) d = 15 r. Adapted with permission from Ref. [53].

**Figure 16.**Elliptical profile of vortex beam at oblique incidence. Adapted with permission from ref. [55].

**Figure 17.**(

**a**) Beat spectrum at different incident angles. (

**b**) Center frequency at different incident angles. Adapted with permission from ref. [55].

**Figure 18.**(

**a**) Lateral misalignment of a light spot on a rotating surface. (

**b**) Incident on the rotating surface at a certain angle. Adapted with permission from ref. [56].

**Figure 19.**Modal distribution under (

**a**) lateral misalignment and (

**b**) angular deflection. Adapted with permission from ref. [56].

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**MDPI and ACS Style**

Cheng, T.-Y.; Wang, W.-Y.; Li, J.-S.; Guo, J.-X.; Liu, S.; Lü, J.-Q. Rotational Doppler Effect in Vortex Light and Its Applications for Detection of the Rotational Motion. *Photonics* **2022**, *9*, 441.
https://doi.org/10.3390/photonics9070441

**AMA Style**

Cheng T-Y, Wang W-Y, Li J-S, Guo J-X, Liu S, Lü J-Q. Rotational Doppler Effect in Vortex Light and Its Applications for Detection of the Rotational Motion. *Photonics*. 2022; 9(7):441.
https://doi.org/10.3390/photonics9070441

**Chicago/Turabian Style**

Cheng, Tian-Yu, Wen-Yue Wang, Jin-Song Li, Ji-Xiang Guo, Shuo Liu, and Jia-Qi Lü. 2022. "Rotational Doppler Effect in Vortex Light and Its Applications for Detection of the Rotational Motion" *Photonics* 9, no. 7: 441.
https://doi.org/10.3390/photonics9070441