## Abstract

Bessel-Gauss beams carrying orbital angular momentum are widely known for their non-diffractive or self-reconstructing performance, and have been applied in lots of domains. Here we demonstrate that, by illuminating a rotating object with high-order Bessel-Gauss beams, a frequency shift proportional to the rotating speed and the topological charge is observed. Moreover, the frequency shift is still present once an obstacle exists in the path, in spite of the decreasing of received signals. Our work indicates the feasibility of detecting rotating objects free of obstructions, and has potential as obstruction-immune rotation sensors in engine monitoring, aerological sounding, and so on.

© 2017 Optical Society of America

## 1. Introduction

Bessel beams, solutions of Helmholtz equation in cylindrical coordinates, are widely known as a non-diffraction or self-reconstructing light beam, which can reconstruct its electric field after passing through an obstruction [1–3]. Lots of studies have been done on Bessel beams, from the point of matter wave [4–6], self-healing of quantum entanglement [7], spatial oscillating polarization [8,9] and so on. The complex amplitude of Bessel beams reads as:

with*J*the

_{l}*l*-th order Bessel function of the first kind,

*k*the radial wavenumber and

_{r}*l*the topological charge. Equation (1) indicates that Bessel beams have infinitely extending sidelobes and are unrealistic, leading to the impossibility of their experimental realization. Usually, a laboratory approximation, Bessel-Gauss (BG) beams, are employed as Bessel beams in practice, which are defined as [10]:

*ω*

_{0}denotes the waist of the Gaussian beam that confines the BG beam. The term exp(

*ilφ*) in Eq. (2) represents the helical wavefront, meaning that BG beams carry orbital angular momentum (OAM)

*lħ*per photon [11,12]. Similar to Bessel modes, BG beams also have the ability to self-heal after encountering an obstruction over finite distances [13,14]. The OAM and propagation-invariance of BG beams make their potentials in many domains. For instance, The OAM modal basis of BG beams defines an infinite-dimensional Hilbert space, and thus enlarging the capacity of optical data-transmission systems [15,16]. In addition, the new perfect vortices [17–20] or perfect vectorial vortices [21–23] can be generated as the Fourier transformation of scalar or vectorial BG fields.

In this paper, we show that it is feasible to detect a rotating object by using high-order BG beams, meanwhile the detection is free of obstructions. When the rotating object illuminated by BG beams carrying OAM, a frequency shift of the scatter rays, associated with the speed and topological charge, is arisen, which can be used to derive the rotating speed. Specially, owing to the self-reconstruction, the angular velocity detection is also feasible if there is an obstacle between BG source and object. We study the effect caused by the obstruction both theoretically and experimentally. And find the observed frequency shift when facing obstructions is in agreement with that of no obstacles. However, the obstruction in the path result in the decreasing of received signals to some extent. If the obstacle is large enough or too close to the rotating object, the desired signals will be lost in the noise and then reduces the accuracy. We provide an easy way to detect the angular velocity of rotating object free of obstruction, and have potentials in the domains as engine monitoring, aerological sounding and so on.

## 2. Rotational Doppler shift

Doppler shift or linear Doppler shift is a well-known phenomenon for both mechanical and electromagnetic wave in classical physics, where a frequency shift of the wave is given by the relative motion between the wave source and the observer, has been applied to detect the translational motion of surfaces [24] and fluids [25]. Such frequency shift scales with both the unshifted frequency and the linear velocity, and reads as Δ*f* = *f*_{0}*v*/(*v*_{0}-*v _{s}*), with

*f*

_{0},

*v*

_{0},

*v*and

*v*the unshifted frequency, the wave velocity, the relative velocity and the velocity of wave source, respectively. For lights,

_{s}*v*

_{0}=

*c*, and

*c*>>

*v*, where

_{s}*c*denotes the speed of light. Then the shift is Δ

*f*=

*f*

_{0}

*v*/

*c*. There are increasing interests in another form of Doppler shift called rotational Doppler shift recently [26]. Less well-known than the linear effect above, the frequency shift in rotational Doppler shift is associated with not only the rotational velocity between the source and the observer, but also the topological charge of lights [26–31]. Previous studies have shown that rotational Doppler effect can be utilized to detect rotation object [32,33]. The frequency shift is manifest in backscattered lights no matter whether the incident beams are monochromatic [32] or not [33]. Moreover, such rotational Doppler shift is also effective in the detection of transverse particle movement [34], flow vorticity [35,36], and so on.

For BG beams, the Poynting vector is related to the position in the beams and has various azimuthal components, due to the helical wavefront or the OAM. Hence there is a small angle between the axis and the Poyting vector as [37,38]:

with*λ*the wavelength and

*r*the radius from the beam axis. When BG beams are incident coaxially in a rotating object, the Poyting vector in each point has a component along the direction of the point’s line speed. Think about an arbitrary point

*A*in the incident plane, as sketched in Fig. 1. The frequency shift of the photons in

*A*is Δ

*f*=

*f*

_{0}

*v*sin

_{A}*α*/

*c*, where

*v*is the line speed in

_{A}*A*. For small

*α*, sin

*α*=

*α*.Therefore, combined with Eq. (3), the shift is:

*l*and the angular velocity Ω, and independent with the unshifted frequency and locations. Equation (4) paves the way for detecting rotation object through beams carrying OAM as BG modes.

A challenge is how to measure the shift Δ*f*, which is tiny compared with optical frequency. Rather than measure this frequency shift directly, 2-fold multiplexed BG beams with opposite topological charge are employed here. The two OAM components will have opposite frequency shift, such that the beat frequency is observed, leading to intensity modulation of frequency *f*_{mod} = 2Δ*f*, also represented as:

And then the frequency shift can be acquired through analyzing the spectrum of the intensity of scattered lights.

## 3. Production of BG beams

BG beams are easy to produce as transforming from Gaussian beams through an axicon [39] or Bessel beam kinoform [40,41]. In this paper, the method of axicon is employed, as illustrated in Fig. 2. An axicon is a tapered optical element with circular symmetry. A plane wavefront can be transformed into a conical wave after it propagates through an axicon. And BG modes are generated in the overlap region. BG beams only exist in finite length, over which is nominally non-diffracting. The maximum range of the BG beams is given by the length *z*_{max} of the overlap region and reads:

*λ*corresponds to the wavelength,

*R*is the radius of the axicon,

*β*is the refraction angle and

*d*is the radial period of the axicon. Note that if the beam size is smaller than the axicon’s base,

*R*denotes the radius of the incident field. Supposing that an obstacle with diameters

*r*

_{0}is located in the BG region, one can easily see from geometry that a shadow region (violet part in Fig. 2) is appeared, where the fields are distorted. Beyond the shadow region, BG modes are recovered, since the unscreened rays bypass the obstacle and form the overlap region again. According to geometry, shadow distance

*z*

_{obs}is given by:However, if the obstruction is located too far from the axicon so that the shadow region is beyond

*z*

_{max}(dashed line in Fig. 2), BG beams no longer recover themselves.

In this present work, a holographic axicon but not a real axicon is used. We encoded the computer generate hologram (CGH) on a liquid-crystal spatial light modulator (SLM) (Holoeye, PLUTO-TELCO-013-C). Then the BG beams are obtained in the BG region behind SLM.

## 4. Experimental setup

The experimental setup is shown in Fig. 3. Gaussian modes whose wavelength is 1550 nm are generated by a laser diode and collimated into free space with the diameter of 3 mm. The half wave plate and polarized beam splitter (PBS1) behind the collimator produce horizontal linear polarization, to match the demand of the pure-phase modulation of SLM we use here. The SLM is encoded by CGH to transform Gaussian beams into 2-fold multiplexed BG beams with the topological charge of + *l* and –*l*. Then the generated BG beams are selected as the first diffraction order by an iris diaphragm together with a 4-*f* system (L1 and L2, *f* = 100mm). At the end of 4-*f* system, BG region begins, as illustrated in Fig. 3. In the experiment, the radial period of holographic axicon is 0.67 mm, and the maximum length *z*_{max} of BG region is computed according to Eq. (6) as 648.39 mm. In the BG region, a rotating surface made by cardboards is placed at the distance of 380 mm from the origin where BG region starts. A 45° quarter wave plate in front of the surface is to ensure the light scattered from the surface could be reflected by PBS2. Then the scattered light is collected using a lens L3 (*f’* = 50 mm) and recorded by a photodiode (THORLABS, PDA20CS). The output signals of the photodiode are analyzed by an oscilloscope. For the sake of observing the intensity profile of BG beams incident on the rotating surface, a beam splitter and an infrared CCD camera (Xenics, Bobcat-320-star) are located. Note that the CCD and rotating surface have the same distance *L* (*L* = 160 mm) from the beam splitter.

## 5. Detecting angular velocity

We do the detection without obstructions firstly. The incident beams have 2*l* main petals, since they consist of two single mode BG beams with opposite topological charge. The intensity modulation of scattered light is recorded by the photodiode. Figures 4(a) and 4(b) show part of outputted signals of 3 ms when multiplexed BG beams with topological charges ± 20 and ± 22 are incident, respectively. The frequeny spectra is obtained through Fourier transformation of time domain with the period of 0.1 s, as illustrated in Fig. 4(c), where a clearly distinguishable peak is presented. One can see from Fig. 4(c) that the frequency of intensity modulation *f*_{mod} is about 2462.5 Hz for *l* = ± 20 and 2750 Hz for *l* = ± 22. Hence the angular velocity is derived as about 390 rad/s.

To verify the relationship among the frequency *f*_{mod}, topological charge *l* and the angular velocity Ω given in Eq. (5), detections with various rotation speeds and values *l* are done, as shown in Fig. 5(a). Four different multiplexed BG beams are employed, the intensity profile of which is recorded by the CCD in Fig. 3 and given in Figs. 5(b)-5(e), separately. We measure the frequency *f*_{mod} under six different rotating speed. For the same *l*, the observed *f*_{mod} (dots) and angular velocity are in proportion, fitting well with theory (solid lines).

Moreover, we investigate the effects induced by obstruction on the angular velocity detection. Note that here the discussion is on the premise that the obstruction is smaller than the size of BG beams. Otherwise there is no significance, for all the rays are blocked. A cylinder with the diameter of 0.37 mm is employed as the obstruction. Here we use cylinder is to simulate if there is inevitable cables between source and rotator in practice. The cylinder is placed vertically in the center of the path between the beam splitter and orign of the BG region, with various distances from the rotating surface. The shadow distance here is calculted from Eq. (7) as *z*_{obs} = 83.55 mm. The distance between the beam splitter and the rotating surface *L* = 160 mm (*L*>*z*_{obs}), hence the incident BG beams must have recovered themselves before reaching the surface. When multiplexed BG beams with *l* = ± 20 are incident, the intensity profiles and the spectra of received signals for five different obstruction locations are acquired, as shown in Fig. 6. The case of no obstruction is also present, for comparison. The intensity profiles in Figs. 6(b)-6(g) verify that BG beams can self-heal after transmitting a distance, even the obstruction blocks part of the beams along the propagation path. The peaks in the spectra of Fig. 6(h) are apparent whether there is obstruction in the propagation path or not. However, with the obstruction getting closer to the rotating surface, the power of peaks is lower and lower, since unscreened rays are less and less. This phenomenon teaches us, in the case of *L*>*z*_{obs}, although the modulation frequency can be obtained, the peak power decrease. If the obstruction is too closer to the rotating surface, the desired peak may be lost in the noises and then reduces the detection accuracy.

As a contrast, an additional experiment is done, to show the performance of detecting the angular velocity by using BG beams and non-BG (Laguerre Gaussian, LG) beams when facing obstructions, as shown in Fig. 7. The topological charges for both of the two beams are *l* = ± 20. The obstruction, a cylinder which is the same as that in Fig. 6, is placed at the distance of 38 cm from the rotating surface. When there is no obstruction, a clearly distinguishable peak emerges with the same *f*_{mod} for both BG beams and LG beams. Once the obstruction is placed, the peak power for BG beams decreases a bit but can still be observed obviously. However, for LG beams, the frequency spectrum is totally mess, and the highest peak present at 2400 Hz. Results given by Fig. 7 illustrate BG beams have better performance in detecting angular velocity with obstructions.

We also analyze the performance of non-diffraction detection under various angular velocities for a lager obstraction. The diameters of the cylinder is inceased to 1.19 mm, and thus *z*_{obs} = 268.71 mm. The cylinder is also placed in the center of the propagation path for three locations, with the distances of 380 mm, 320 mm and 250 mm to the rotating surface. The measured *f*_{mod} for various rotating speed when multiplexed BG modes *l* = ± 20 are incident and displayed in Fig. 8. Note that the signals of 250 mm are observed with the amplification of 10 dB compared with other cases. One can clearly see the spectra signal (pink bar) decrease as the obstruction getting closer to the rotating surface, which is agree with the results in Fig. 6. In addition, the scatters of the cases of 380 mm and 320 mm [Figs. 8(a)-8(c)] are similar with that of no obstruction and match well with throry (the solid line), indicating the angular velocity can be detected well. While for the case of 250 mm, the scatters are totally chaotic, reasons of which can be understood in two aspects. On one hand, BG modes can’t self-heal when reaching the rotator since 250 mm is smaller than *z*_{obs}, leading to the inevitably presence of more than one peaks. On the other hand, the lower signals may loss in the noises.

## 6. Discussions and conclusions

Although our scheme has good performance in detecting the angular velocity with obstructions in the path, limitations still present. Due to the fact that BG beams only exist in finite regions (*z*<*z*_{max}), the proposed system is much less sensitive to distance, which in some applications will make it impossible to know what is being measured unless it is the only rotating surface in the BG region. Even so, the proposed scheme makes it possible to realize the detetion if there is inevitable obstacles as cables between source and rotator in practice.

An important application of this work is aerological sounding. Such remote sensing is generally done at very long distances (>1000m). While BG beams can only exist in a certain distance [13,14,42], and the maximum of which satisfies Eq. (6). Hence, in this scenario, *z*_{max} in Eq. (6) must large enough. According to Eq. (6), a feasible solution is to employ large diameter incident beams with a large radial-period axicon when generating BG beams. For instance, for BG beams with the wavelength of 1.55 μm, the meter scale diameter with centimeter scale axicon period can meet such demands. Usually, making optics in a meter size is very challenging, and expensive. For the sake of controlling the size of the whole system, methods of further increasing the axicon period *d*, or employing smaller-wavelength beams could be done.

In summary, we have shown that BG beams can be used to detect rotation objects, owing to the rotational Doppler shift. When 2-fold multiplexed BG beams with oppsite topological charges ± *l* illuminate the rotating surface along the axis, a frequency of intensity modulation *f*_{mod} is found in the scatter rays, which can be used to deduce the angular velocity. In addition, the rotational Doppler shift is determined only by the topological charge *l* and the angular velocity Ω, and independent with the the unshifted frequency, meaning the unnecessary of lights of single frequency. Moreover, we study the effects on the angular velocity detection if obstructions are in the propagation path. We expermentally show the frequency *f*_{mod} can be also recorded if the distance from obstructions to the rotator is larger than the shadow length *z*_{obs}, due to the self-reconstruction of BG beams. Meanwhile the peak signal in the spectrum is lower and lower with the obstruction getting closer to the rotator. However, if distance to the rotator is smaller than *z*_{obs}, BG beams couldn’t heal themselves and the detection is invalid. Our study will find applications as an obstruction-immune rotation sensor in the domains as engine monitoring, aerological sounding and so on.

## Appendix A Holograms encoded on the SLM

The hologram encoded on SLM to produce BG beams consist of a sprial phase plate (SPP), an axicon, and a linear phase, as illustrated in Fig. 9. The SPP can bring in two opposite OAM states and the axicon produce the overlap region. When faundamental Gaussian beams propagate through such hologram, 2-fold multiplexed BG beams with opposite topological charge can be acquired in the first diffraction order. In the experiment, we selet the 1st diffraction order through an iris diaphragm along with a 4-*f* imaging system, as shown in Fig. 3.

## Appendix B Principles of the intensity modulation

When 2-fold multiplexed BG beams with opposite topological charge illuminate the rotating surface, their scattering gives opposite frequency shifts. For the OAM component $|l\u3009$, the frequency shift is:

While for $|-l\u3009$, the shift:The two OAM components come from the same source, hence they have the same amplitudes *A*, initial phases *ψ*, wavenumbers *k* and transmission distances *z*, and can be expressed for simplicity as:

*σ*= -

*kz*+

*ψ*.

*f*

_{1}and

*f*

_{2}are the light frequency of the two components after scattering, separately. The signal captured by the photodiode is interference of light waves in Eqs. (10) & (11) indeed, which reads:

The photodiode is invalid for high frequency as light frequencies *f*_{1} and *f*_{2}. Therefore we only focused on the low frequency, and Eq. (12) is simplified as:

*f*

_{1}-

*f*

_{2}) is obtained, and hence:

## Funding

National Basic Research Program of China (973 Program) (2014CB340002, 2014CB340004).

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