## Abstract

The capability to detect the rotational speed of non-cooperative targets in a long distance is a difficult problem to be solved. In recent years, vortex light provides a feasible solution for the measurement of rotational speed for its spiral phase and the orbital angular momentum. Laguerre-Gaussian (LG) mode, as the typical vortex beam, has been widely employed in rotational Doppler effect (RDE) experiments. Here, we show that the nonzero radial index LG beam not only has a specific physical meaning but also can enhance the light intensity and the amplitude of RDE frequency signal relative to a zero radial index LG beam. To this end, we theoretically analyze the reason of intensity enhancement of a nonzero radial index beam and verify the conclusion in a variable control experiment. Our study provides a new aspect of LG beams that can be considered in rotational speed detection based on RDE. It may produce an improvement of the detection range of rotating targets in practical applications.

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

## 1. Introduction

The orbital angular momentum (OAM) of light has been adequate studied since Allen confirmed the well-defined OAM mode in a Laguerre-Gaussian (LG) amplitude distribution laser light [1–4]. Akin to the polarization, frequency and phase characterize of light, the OAM determines a new dimension of light beam which has spawned a burgeoning and vibrant subfield [5–8]. Especially in the field of optical manipulation [9,10], communication [11] and measurement [12,13]. Light carrying OAM can be called a joint name optical vortex, which has a phase structure of helix with a topological charge determines its numbers of phase jumps from 0 to $2\pi $ within one wavelength.

Since there is a natural coupling between spiral phase and rotational motion, the optical vortex can be helpful in rotational speed detection associated with so-called rotational Doppler effect (RDE) [14–16]. After Lavery et. al. systematically come up with the rotational speed measurement scheme based on RDE by using optical vortex [17], many researchers investigated both the origin from different aspects and potential applications of RDE [18–21]. The basic principle of RDE can be expressed by $\Delta f = l\Omega /2\pi $, when a rotating object with rotational speed of $\Omega $ is illuminated by a vortex beam of *l* topological charge, the scattered beam will show a frequency shift of $\Delta f$. Although there are many great works related with rotational speed detection based on RDE, they can be realized in certain laboratory conditions. As of this writing, the furthest detection distance is done in 2018 that realized a 120 m free space detection on the photon counting level [22]. One of the biggest factors restricting the detection range could be that the scattered light is too weak to be collected [23,24]. In order to solve this problem, increasing the laser power is a convenient choice. However, limited by the endurance power of the light modulation device, the power of laser can not be too high. Base on some intrinsic properties of the beam, there might be some new complementary solutions.

The light beam of $LG_p^l$ mode has a ring-shaped hollow intensity distribution, with the azimuth index *l* and radial index *p* determines its OAM and number of rings, respectively. For a system receiving signal light along the direction of the outgoing light path, stronger scattered light can’t be received because the light energy distributes away from the axis. There are extensive number of attentions have been paid on the OAM of LG mode beams compared with the small amount of works related with radial index [25–27]. The radial index is so called because the radial intensity distribution of LG beams is decided by this parameter. The intensity profile of LG beam exhibits $p + 1$ concentric rings if $l \ne 0$ [28]. Previous works in rotational speed detection based on LG mode vortex are all on the condition of zero radial index, that is to say the profile size only decided by beam waist and azimuth index *l*. Compared with Bessel beams [29], nonzero radial index LG beams also shows a quasi-nondiffracting characteristic which means that LG beam can be a better option than Bessel beams for many applications. However, there are no researches have investigated the rotational speed detection by employing the nonzero radial index LG beam.

In this paper, we first analyze the physical meaning of the radial index and investigate the intensity properties of nonzero radial index LG beam. The independence between topological charge and intrinsic hyperbolic momentum charge is demonstrated, which means the RDE will not be bothered by radial index. Next, an interference experiment between nonzero radial index vortex and spherical beam is conducted. The interference patterns clearly show that the $p > 0$ LG beam has a good OAM distribution. Furthermore, a variable control experiment is implemented which verifies the enhancement effect of light intensity and amplitude of RDE frequency signal by using nonzero radial index beam.

## 2. Theoretical framework

#### 2.1 Physical meaning of the radial index

The equation of a LG beam, under the paraxial assumption, in circular-cylindrical coordinates $(r,\varphi ,z)$ is

*p*and

*l*are the radial and azimuth index, respectively,

*C*is a constant which stands for the amplitude, $L_p^{|l |}$ represents the generalized Laguerre polynomial of order

*p*and degree $|l |$, ${z_R}$ is the Rayleigh range expressed by ${z_R}\textrm{ = }\pi w_0^2/\lambda $, where ${w_0}$ is the beam waist at the initial plane ($z = 0$) where the beam is narrowest. The functions ${w_z}$, ${R_z}$ are the beam waist and curvature radius of wavefront, respectively, and are given by where $k = 2\pi /\lambda $ is the overall wave number of the light beam. It’s obvious from Eq. (1) that the factor $\textrm{exp} (i\psi )$ determines the phase of the LG beam field and the intensity distribution defined by the rest part, where $\psi $ represents the combined phase related with

*p*, $l$ and propagation distance.

The Laguerre polynomial is an important factor in a LG mode. There exists a serious of relations between Laguerre polynomial of varying order and degree, which can be expressed as [30]

when combined with the rule for differentiation of the polynomials and perform a transformation of coordinates with $x = 2{r^2}/w_z^2$ we obtain*z*, ${w_z}$ express the beam waist and ${\hat{L}_z}$ is related with the momentum, $\nabla _t^2$ denotes the transverse Laplacian. A trivial transformation shows this whole derivation is in Ref. [31].

One of the basic assumptions of quantum mechanics is that to the classical observable ${P_z}$, the Cartesian *x* component of the momentum of a system, there corresponds the quantum-mechanical operator ${P_x} ={-} i\hbar \partial /\partial x$ [32]. If we bring the radial coordinate into this momentum operator and take the well-defined form in the circular-cylindrical coordinate system, we get the hyperbolic momentum

Returning to Eq. (5) we can get a deeper insight of the physical meaning. The first term in Eq. (5) is the transverse Laplacian scaled by the beam waist which is a numerical constant. The third term denotes the OAM. The final term produces the second moment of the radial position at $z = 0$, which is related to the transverse spatial variance. These are three independent parameters of the LG beam, and they are independent of the radial index. This leaves a single term that represents the degree of freedom that the radial index arises from. The radial index can then be seen as representing the promotion of the radial-like hyperbolic momentum itself to a quantum observable. Therefore, the real physical meaning of the radial index could be described as the intrinsic hyperbolic momentum charge.

#### 2.2 Intensity of the nonzero radial index LG mode

At a transmission distance of *z*, the intensity *I* of LG beams can be expressed as

Here we find that the intensity of LG beam is determined by initial laser amplitude $C$, beam waist ${w_0}$, orbital angular momentum *l* and radial index *p*. Also, changing the order of Laguerre polynomials is the only way that radial index affects light intensity. Therefore, we can observe the influence of radial index on the LG beam intensity by analyzing the distribution of Laguerre polynomials of different orders. The numerical simulation results of $L_p^l(x)$ is shown in Fig. 1(a), which clearly shows a higher intensity when *p* take a larger value. And the energy is more concentrated in the center when the value of *p* increased.

Further, if the beam propagating a certain distance ($z > 0$). Based on Eq. (7) and distinguish the real and imaginary part, it is clear to see that radial index has nothing to do with the real part and the complex exponential function. As a result, leave the other parameters unchanged except radial index, we can describe the intensity by $I_{LG}^{}(r,\varphi ,z) \propto abs({L_p^{|l |}({ - {r^2}/w_z^2} )} )$. By means of the numerical simulation, the similar results with Fig. 1(a) can be obtained.

For the fundamental Gaussian mode, spot size $w(z)$ represents the distance normal from the axis of propagation to the point at which the amplitude of the beam is $1/e$ times that on the axis [33]. In the optical axis cylindrical coordinate system, it can be expressed as,

Using a similar method, the radius of LG beam can be defined as

The expression of the waist can be further simplified, so that the radius of the LG beam spot size, using the same form like Eq. (9), is

Still by means of numerical simulation, the variation of spot radius with respect to radial index is further obtained. As shown in Fig. 1(b), the beam waist increases as the radial index increase under different topological charges. Combined with Fig. 1(a) and (b), we can predict that a larger radial index may cause a larger LG beam waist, at the same time produce a smaller central dark core.

## 3. Experiment configuration

Since nonzero radial index may help improve the intensity of scattered light which is useful in the remote sensing based on the Doppler effect. A proof-of-concept experiment is designed for the rotational speed detection by using the nonzero radial index. The Gaussian mode beam is produced by a solid-state laser (CNI MSL-FN-532) with the wavelength of 532 nm. Then pass through a horizontal polarizer to filter the polarized light since the spatial light modulator (SLM) only responds to a certain polarization direction. A well-designed hologram is preloaded on the screen of SLM (HOLOEYE PLUTO-NIR-011), which will modulate both the phase and the amplitude of the incident beam. The light reflected from SLM will contain multiple diffraction orders because of the grating. We use a 4*f* system consisting two identical lens and an aperture to select the first order beam which is the high-pure $LG_p^l$ mode.

The part inside the dotted frame is set to produce the reference light. A convex lens with focal length of 30 mm is placed between two mirrors to transform the plane wave into spherical wave. Then the interference pattern produced by optical vortex and a spherical wave will be captured by the CCD camera (Newport LBP2). The topological charge and radial index of $LG_p^l$ mode beams can be directly identified by the interference pattern.

To detect the rotational speed of a spinning object, the part in the dotted frame will be blocked. The experimental setup is shown in Fig. 2(b), and the LG mode from the 4*f* system is used directly. The setup before the 4*f* system is omitted here. The optical vortex illuminates the rotational target along its rotating axis. A photodector (Thorlabs PDA36A2) is placed behind a telescope system which consisted with two convex lenses with different focal distance. It is worth to note that this receiving system can be placed at any position provided the scattered light can be received. Because the scattered light in different directions contains the same information. The photodector is then connected with an oscilloscope to gather the signals. After a fast Fourier transformation, the rotational Doppler shift can be obtained in the frequency domine.

## 4. Experimental results & discussion

With the setup introduced above we can easily observe the intensity profile of nonzero radial index LG beams. The nonzero radial index LG beam is prepared in the following way. Firstly, calculating the phase distribution of the light field according to Eq. (1), here we can set the azimuthal and radial index as we need. Combining the phase mask with a well-designed digital blaze grating we can obtain the final hologram. Secondly, upload the hologram on the screen of the SLM, the first order diffraction light after the 4*f* spatial filter is the desired beam. As shown in Fig. 3, the distribution of each $p > 0$ light intensity is discrete concentric rings, there is a dark gap between each ring where little power resides.

A conference experiment is conducted to directly show the topological charge of the nonzero radial index LG beam. By interfacing with spherical wave, the vortex interference fringes exist in both $p = 0$ and $p > 0$ orders. The LG beam with topological charge of 3 is employed here, with the radial quantum index increase, the number of the intensity ring is $p + 1$. And the interference fringes of each ring showing the same distribution with the $p = 0$ one. The interference pattern further confirmed the independence between azimuth number and the radial index, which determines the orbital angular momentum and the intrinsic hyperbolic momentum, respectively. Since the helical phase exist in each bright ring, the rotational Doppler shift produced by each independent order will be exactly the same.

To investigate the potential advantage of using LG beams with nonzero radial index, all the LG beams have the same transmitted power and the same transmitted beam size at the initial plane, i.e., all the LG beams have the same ${w_0}$ at $z = 0$. The only difference between each beam is radial index. The phase of each superposition beam is show in Fig. 4(a), where an obvious segmentation between each phase loop of nonzero radial index. All the LG beams have the same topological charge of $l ={\pm} 16$ but different radial index. The reason why we chose $l ={\pm} 16$ is that the light field size is just moderate relative to the size of the rotating object. Actually, LG beams with any topological charge can obtain the same phenomenon. The corresponding intensity profiles are shown in Fig. 4(b), where the beam waist is larger when *p* take a larger value. This phenomenon is consistent with the previous theoretical predication in Eq. (12). By contrast, the size of the inner most ring, in other words, the central area of dark core is far smaller of a $p = 4$ beam than that of the $p = 0$ beams with the same topological charge.

This character is really useful in remote sensing because a small dark core combined with a large beam waist would form a larger effective irradiation area. That is to say less chance to miss the targets if its size is not big enough relative to the beam waist in practical application. Furthermore, the total energy of the beam in nonzero radial index would be stronger than that of $p = 0$ beam. We theoretical count the total intensity of each beam and calculate the relative intensity by

The light intensity is measured by an optical power meter (Thorlabs, S130C). As shown in Fig. 4(c), the total relative intensity increases with the increase of *p* value. Compared with the $p = 0$ beam, the total relative intensity of the $p = 4$ beam increases nearly 50%.

It is worth to note that the total intensity of nonzero radial index beam is more concentrated in the center. This appearance is coincided with the theoretical analyses that the total energy will focus towards the center in a $p > 0$ beam. It is determined by the properties of the Laguerre function. Although the energy is more concentrated, the propagation stability is not affected. For a system receiving signal light along the direction of the outgoing light path, light energy concentrated in around the axis means stronger scattered light can be received.

By illuminating this kind of nonzero radial index beam on the surface of a rotating object, we can observe the rotational Doppler shift in the frequency domine of the scattered light. We choose to use the superposition LG beam with topological charge of $l ={\pm} 16$ and with different radial index. The rotational speed of the plate is adjusted by a controller. Here we set the rotational speed as 98.9 Hz, according to the formula of the RDE, the theoretical rotational Doppler shift is 3164.8 Hz. The photodetector and oscilloscope are used to record the signals of scattered light in the time domine.

Previous experiments using LG beam to detect rotational speed are based on zero radial index. As shown in Fig. 5(a), the time domain signal in 0.5s is shown in the figure. We choose the $p = 0$ LG beam as the control group, the other two groups with $p > 0$ were regards as experimental group. Under the same conditions, the intensity of scattered light with $p = 1$ is about 0.014 mv higher than that with $p = 0$. The light intensity with $p = 2$ is about 0.021 mv higher than that with $p = 0$. According to the previous analysis from Eq. (13), this result is natural.

After transform the signals from time domine to the frequency domine, the rotational Doppler shift can be observed directly. The frequency signals appear around 3165 Hz which is consistent with the theoretical prediction. The measurement errors are less than 0.1%. With the increase of the radial index, the amplitude of frequency signal has an obverse enhancement, i.e., the signal-to-noise ratio has been improved. The frequency signal strength of $p = 2$ is about 4 dB larger than that of $p = 0$. In accordance with the trend of time domine signal, the amplitude of frequency signal increases with the increase of radial index of LG beam. Considering that the weak scattering light is one of the main factors that limiting the long-range detection of vortex light, the use of $p > 0$ LG beam can enhance the signal and therefore increase the detection distance.

## 5. Conclusion

In summary, we have studied the properties of nonzero radial index LG beam. The radial index and the azimuthal index are two independent parameters of the LG beam. A larger radial index may cause a larger beam waist, at the same time produce a smaller central dark core. Compared with $p = 0$, LG beams with a larger radial index have greater effective detection light intensity which means less chance to miss the target in real application. Furthermore, the signal enhancement effect of nonzero radial index LG beam on rotational speed detection based on RDE is verified. Results show that nonzero radial index beam can effectively increase the light intensity and enhance the signal to noise ratio of the rotational Doppler shift signal under the same conditions compared with zero radial index beam. Up to 4 dB improvement has been observed in our experiment. The results of this work may helpful in increasing the detection range of spinning object based on vortex light.

## Funding

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

## Disclosures

The authors declare no conflicts of interest.

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