Non-diffractive Bessel-Gauss beams for the detection of rotating object free of obstructions

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:

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 Δf = f₀v/(v₀-v_s), with f₀, v₀, v and v_s the unshifted frequency, the wave velocity, the relative velocity and the velocity of wave source, respectively. For lights, v₀ = c, and c>> v_s, where c denotes the speed of light. Then the shift is Δf = f₀v/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]:

 

Fig. 1 Rotational Doppler shift of BG beams. The frequency shift results from relative velocity between the horizonal components of Poyting vector and the line speed indeed, determined by the topological charge l and the angular velocity Ω only.

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A challenge is how to measure the shift Δf, 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 = 2Δf, 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:

 

Fig. 2 Axicons can be employed to generate BG beams in the overlap region. BG beams can self-heal at the region behind z_obs if an obstacle is placed in the path.

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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.

 

Fig. 3 Experimental setup. LD, laser diode. Col., collimator. HWP, half wave plate. PBS1 & PBS2, polarized beams plitter. R, reflector. SLM, liquid-crystal spatial light modulator. L1 & L2, lens with focal length f = 100 mm. ID, iris diaphragm. BS, beam splitter. QWP, quarter wave plate. L3, lens with focal length f’ = 50 mm. PD, photodiode. CCD, infrared CCD camera. The rotating surface is placed at the distance of 0.38m from the origin where BG region start. The CCD and rotating surface have the same distance L (L = 0.16 m) from the beam splitter.

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5. Detecting angular velocity

We do the detection without obstructions firstly. The incident beams have 2l 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.

 

Fig. 4 Angular velocity detection without obstractions. (a) & (b), signals outputted from photodiode when BG beams with topological charge ± 20 and ± 22 are incident, separately. Note that here we display part of the signals with 3ms, for clearly. (c) The frequency spectra of (a) and (b), where f_mod can be acquired as the peak. For l = ± 20, f_mod = 2462.5Hz, and for l = ± 22, f_mod = 2750Hz.

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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).

 

Fig. 5 Results under various angular velocities and topological charges. (a) Measured modulation frequency f_mod vs angular velocity Ω of the rotating surface under four different topological charges. (b)-(e), intensity profiles of incident 2-fold multiplexed BG beams with topological charges of ± 16, ± 18, ± 20 and ± 22, respectively.

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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.

 

Fig. 6 Angular velocity detection with obstractions under the condition L>z_obs. (a) A cylinder with the diameter of 0.37 mm is placed vertically in the center of the path as the obstruction. (b)-(g) Intensity profiles of BG beams on the rotating surface without and with obstruction in various locations. (b), no obstruction. (c)-(g), with obstructions, their distance to the rotating surface are 380 mm, 330 mm, 280 mm, 230 mm, 190 mm, respectively. All of the profiles are recorded by the CCD. (h) Spectra of signals outputted by the photodiode with obstructions in various locations. The peaks in the spectra are apparent even with obstructions but have power losses.

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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.

 

Fig. 7 Compared results of detecting angular velocity with obstructions for both BG beams and non-BG (Laguerre Gaussian, LG) beams. The obstruction is placed at the distance of 38 cm from the rotating surface. (a) BG beams; (b) LG beams.

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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.

 

Fig. 8 Angular velocity detection with obstractions under various angular velocities. (a), intensity frequencies f_mod and spectrum signals for various angular velocities with no obstruction. (b)-(d), intensity frequencies f_mod and spectrum signals for various angular velocities with obstructions, the distance from which to the rotator is 380 mm, 320 mm and 250 mm, separately. The obstruction is a cylinder with the diameter of 1.19 mm. The BG beams can’t recover when placed 250 mm to the rotating surface in the propagation path.

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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.

 

Fig. 9 Hologram to generate 2-fold multiplxed BG beams with opposite topological charges. The holographic axicon has three components, a SPP, an axicon, and a linear phase.

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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 $|l〉$, the frequency shift is:

(8)$$\Delta f_1=\frac{l\Omega }{2\pi }$$

While for $|-l〉$, the shift:

(9)$$\Delta f_2=-\frac{l\Omega }{2\pi }$$

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:

(10)$$E_1(t)=A\cos \left(2\pi f_{1}t+\sigma \right)$$

(11)$$E_2(t)=A\cos \left(2\pi f_{2}t+\sigma \right)$$

where σ = -kz + ψ. f₁ and f₂ 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:

(12)$$\begin{array}{l}I\left(t\right)={\left(E_1+E_2\right)}^2\\ =A^2\left\{\begin{array}{l}1+\cos \left[2\pi \left(f_1-f_2\right)t\right]+\cos \left[2\pi \left(f_1+f_2\right)t+2\sigma \right]\\ +0.5\cos \left(4\pi f_{1}t+2\sigma \right)+0.5\cos \left(4\pi f_{2}t+2\sigma \right)\end{array}\right\}\end{array}$$

The photodiode is invalid for high frequency as light frequencies f₁ and f₂. Therefore we only focused on the low frequency, and Eq. (12) is simplified as:

(13)$$I\left(t\right)=A^2+A^2\cos \left[2\pi \left(f_1-f_2\right)t\right]$$

where the intensity modulation with the frequency (f₁-f₂) is obtained, and hence:

(14)$$f_{\mathrm{mod}}=f_1-f_2=\left|\Delta f_1\right|+\left|\Delta f_2\right|=\frac{\left|l\right|\Omega }{\pi }$$

Funding

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

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