# Rotational Doppler Velocimetry of a Surface at Larger Tilt Angles

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Analyses

^{2}+ y

^{2})

^{1/2}with x = rcosφ and y = rsinφcosγ. E

_{0}is the amplitude factor, and w

_{0}denotes the beam waist radius. Bn

^{l}stands for the binomial coefficient. The Laguerre polynomial L

_{p}

^{|l|}possesses the OAM index l and radial modal index p. δ = (2n − l)φ is the azimuthal phase of tilted OAM light. Note that the tilted OAM light possesses the azimuthal index of 2n − l, which is equivalent to the eigen bases of the tilted surface. In addiion, the relevant interferential modes also can be easily garnered by the superimposition of its conjugate modes with the OAM index of −l, described by:

**r**= (x, y) being the spatial coordinates; C

_{i}= ρ

_{i}exp(iΔφ

_{i}) is the normalized weighting coefficient with the amplitude ρ

_{i}and the intramodal phase Δφ

_{i}. Φ

_{i}(

**r**) is the i-th spatial basis modes that are orthogonal each other in the Hilbert space, which can be the vector elements or the scalar ones, and N is the expanded mode number. Here, we select LG modes with p = 0 as the basis modes, and the weighting coefficient can be ascertained by the operation of inner products, manifested as

_{l}and phase φ

_{l}can be encoded as the digital holograms that can be loaded upon a diffractive optical device, such as SLM. We here draw upon an exact complex amplitude modulation technology to generate the digital holograms with a transmission function [41]:

_{l}) = 1 + sinc

^{−1}(A

_{l}), in which sinc

^{−1}(·) is the inverse function of the sinc(·). G(

**r**) =

**g·r**is a phase-ramp function with the grating frequencies

**g**= (g

_{x}, g

_{y}) along the horizontal and vertical directions, respectively. Mod[·] stands for the modulus function. The inner product C

_{l}can be measured by the intensity measurements on the optical axis. As such, the on-axis highest intensity proportional to the norm of the weighting coefficient possesses the magnitude of I

_{l}, which can be called the OAM power spectrum [42]:

_{sin}and I

_{cos}are, respectively, the intensity of the inner products $\langle E|i{\mathrm{LG}}_{0,l}^{\ast}{+\mathrm{LG}}_{0,0}^{\ast}\rangle /\sqrt{2}$ and $\langle E|{\mathrm{LG}}_{0,l}^{\ast}{+\mathrm{LG}}_{0,0}^{\ast}\rangle /\sqrt{2}$, and I

_{0}is the intensity of inner products $\langle E|{\mathrm{LG}}_{0,0}^{\ast}\rangle $ of the referenced Gaussian light on the axis. Note that the transmission functions $i{\mathrm{LG}}_{0,l}^{\ast}{+\mathrm{LG}}_{0,0}^{\ast}$, ${\mathrm{LG}}_{0,l}^{\ast}{+\mathrm{LG}}_{0,0}^{\ast}$ and ${\mathrm{LG}}_{0,0}^{\ast}$ can be individually encoded on SLM, with no need for the extra build of reference arms [43,44]. Once a set of the complex coefficients C

_{i}that depend on the amplitude ρ

_{i}and the intramodal phase Δφ

_{i}are obtained, the inclined OAM light fields and the phase distributions can be reconstructed according to Equation (3). Additionally, the corresponding OAM complex spectra can be determined by Equations (6) and (7).

_{c}= 2πysinγ/λ into the azimuthal phase term δ in Equation (1) with OAM components. As a result, the corrected azimuthal phase can be rewritten as:

_{c}. By substituting Equation (12) into Equation (1) and repeatedly performing the modal decomposition process from Equations (3)–(7), we can obtain the modulated OAM power and phase spectra. Only if the sidebands of modulated OAM power spectrum can be effectively reduced, the suppression of Doppler spectrum broadening can be achieved, thereby realizing rotating velocimetry of a surface at larger tilt angles.

## 3. Experiment

#### 3.1. Experiment Setup

#### 3.2. Experimental Results

_{l}|

^{2}and phase spectra Δφ

_{l}at tilt angles of 15°, 30° and 45°, respectively. The results are displayed in Figure 4a–c, respectively. We here select ±10-order OAM modes as an example to illustrate our measurement scheme, and the higher-order OAM modes can induce a higher sensibility. Figure 4a–c show that the superimposed OAM-carrying light modes show petal-like distributions as expected by Equation (2). Additionally, with the increased tilt angles, the petal-shaped patterns vary from circular to elliptical structures. Meanwhile, the reconstructed light fields are in well agreement with theoretical counterparts, with 2-dimensional (2D) correlation coefficients between them are 99.1%, 91% and 87% at 15°, 30° and 45°, respectively. The definition of the 2D correlation coefficient has been given in literature [46]. It is worth mentioning that the experimental OAM light fields are reconstructed by the measured OAM power and phase spectra via mode-superposition principle in Equation (3). In addition, we can observe in the bottom of Figure 4a–c that the corresponding OAM power spectra suffer from diffusion effects or inter-modal crosstalk. To be specific, the larger the tilt angle, the more salient the diffused effect, and the lower the power magnitude. Further, the OAM power spectra are distributed symmetrically around 0-th OAM mode, and they are asymmetrical around the ±10th modes. More crucially, there exists the difference of two between the adjacent broadening OAM indices, i.e., l

_{n}-l

_{n}

_{-1}= 2, with l

_{n}being the n-th OAM index, consistent with theoretical analysis of azimuthal phase δ in Equation (1).

_{c}in Equation (12) into the original tilted OAM light. The φ

_{c}can be obtained by compensated optical paths when the tilt angles are known. Subsequently, we measured the OAM power and phase spectra at a tilt angle of 45 degrees again, as shown in Figure 6a,b. Compared to the case in the absence of modulation in Figure 4c, the presence of the phase modulation in Figure 6a,b shows the salient difference in OAM modal distributions. For the former, as shown in Figure 4c, the sideband components versus central main band (+10 and 10-order modes) in the OAM power spectrum possess significantly higher weightings. In contrast, for the latter, the weightings of the sideband components in the OAM power spectrum were effectively reduced by φ

_{c}, thereby possessing relative low power weighting. Since the broadening sidebands of OAM power and phase spectra caused the spreading rotating Doppler spectrum, the suppression effect for the weightings of the sideband components in OAM power spectrum can effectively suppress the rotating Doppler spectrum. The suppression effect on the rotating Doppler spectrum is shown in Figure 6c. This effect makes the central main band in Doppler spectrum clearer, thereby leading to the improved measurement accuracy at larger tilt angles. Additionally, we can see in Figure 4 and Figure 6a,b that there exist slight discrepancies between the simulations and experiments in the OAM power and phase spectrum of each OAM modal index. These small differentials stem mainly from the imperfect energy efficiency of SLM3 in the OAM spectrum analyzer. Yet, these imperfections do not affect the basic distribution laws of measured modal spectra and the eventual accuracy of rotating velocimetry.

## 4. Discussions

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic of probing rotating velocity of a tilted target. (

**a**) Realistic scenarios with tilted rotor illuminated by OAM-carried light modes along z-direction; (

**b**) equivalent conditions with non-inclination rotor illuminated by oblique OAM-carried light modes. Ω: rotating angular velocity of a rotor; γ: tilt angle, A1 and A2: optical axis of normal and inclined light beams; S

_{1}and S

_{2}: a target’s feature points in an elliptical light ring.

**Figure 2.**Experimental setup for the generation and reconstruction of OAM-carried light modes and the characterization of the OAM complex spectrum and rotating velocity detection. MO: microscopic objective; SMF: single mode fiber; LP: linear polarizer; HWP: half wave plate; M: mirror; SLM: spatial light modulator; L: lens; BS: beam splitter; CCD: charge coupled device; PD: photodevice; Osc: oscilloscope.

**Figure 3.**(

**a**) Digital hologram of superimposed OAM-carried light mode at tilt angle 30° loaded upon SLM1. (

**b**) The mask at the Fourier plane loaded on SLM2. (

**c**–

**f**) The digital holograms of a set of basis vectors uploaded on SLM3 for OAM complex spectrum analysis and light-mode analysis and its phase reconstructions, with (

**c**) LG*

_{0,10}, (

**d**) LG*

_{0,0}, (

**e**) LG*

_{0,10}+ iLG*

_{0,0}and (

**f**) LG*

_{0,10}+ LG*

_{0,0}.

**Figure 4.**Theoretical simulations and experimental reconstructions of light mode and phase distributions and matched OAM power spectra |ρ

_{l}|

^{2}and phase spectra Δφ

_{l}of superimposed OAM-carried light modes with tilt angles of (

**a**) 15°, (

**b**) 30° and (

**c**) 45°, respectively. C denotes the 2D correlation coefficient between simulated and reconstructed light modes.

**Figure 5.**Experimentally measured RDS power spectra, induced by a rotor at tilt angles of (

**a**) 15°, (

**b**) 30° and (

**c**) 45°, respectively. The preset rotating velocity of the rotor was 300 rad/s.

**Figure 6.**OAM (

**a**) power and (

**b**) phase spectra with the compensation phase φ

_{c}. (

**c**) The sideband-suppressed rotational Doppler power spectrum.

**Figure 7.**Experimentally measured rotating velocity as a function of the tilt angles with the unmodulated and modulated OAM light fields. The disk spun with a constant angular speed of 300 rad/s.

**Table 1.**Comparison of previously published works with this work on the rotating velocimetry of a tilted object using OAM light fields.

Items | Ref. [37] | This Work | |||
---|---|---|---|---|---|

Unmodulated | Modulated | ||||

OAM indices of the probe beams | ±18 | ±10 | ±18 | ±10 | ±18 |

The largest tilt angle of the object (°) | 49 | 50 | 51 | 70 | 72 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, Y.; Zhang, Z.; Liu, L.; Zhao, Y.
Rotational Doppler Velocimetry of a Surface at Larger Tilt Angles. *Photonics* **2023**, *10*, 341.
https://doi.org/10.3390/photonics10030341

**AMA Style**

Zhang Y, Zhang Z, Liu L, Zhao Y.
Rotational Doppler Velocimetry of a Surface at Larger Tilt Angles. *Photonics*. 2023; 10(3):341.
https://doi.org/10.3390/photonics10030341

**Chicago/Turabian Style**

Zhang, Yanxiang, Zijing Zhang, Liping Liu, and Yuan Zhao.
2023. "Rotational Doppler Velocimetry of a Surface at Larger Tilt Angles" *Photonics* 10, no. 3: 341.
https://doi.org/10.3390/photonics10030341