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Abstract
The vortex beams carried Orbital Angular Momentum (OAM) have recently generated considerable interest due to their potential used in communication systems to increase transmission capacity and spectral efficiency. In this paper, the distorted wavefront detection based on Shack–Hartmann wavefront sensor (HWS) for the vortex beams is investigated. The detection slope of the helical phase sub-spot pattern is used as the calibrated slope zero point, and then the distortion phase of the vortex beam is detected by the HWS. Simulation and experimental results demonstrate that this method can detect the distortion phase of vortex beam with high precision and high frame rate, which is expected to accelerate the application of optical communication systems with vortex beams.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In 1992, Allen et al. [1] proved that each photon in a beam with a phase term exp(ilθ) in the complex amplitude expression carries Orbital Angular Momentum (OAM) of lħ, the l is the OAM topological charge, which is also commonly referred to as the OAM mode, ħ is the reduced Planck constant, θ denotes the azimuth angle. The beam with a helical phase is called vortex beam, which presents phase singularity and intensity dark spot in the center of the beam. The potential applications of vortex beams include super-resolution imaging [2], quantum information storage and extraction [3], and optical tweezers [4]. Moreover, due to the intrinsic orthogonality of different OAM mode, vortex beams add a new dimension to increase the channel capacity and spectrum efficiency for communication [5–8]. Similar to other types of beams, the transmission of vortex beams in free space is affected by varieties of linear and nonlinear effects, the most serious of which comes from atmospheric turbulence. The atmospheric turbulence will distort the helical phase distribution which is essential to ensure mode purity and orthogonality of vortex beams. Theoretical and experimental research have indicated that turbulence would cause fluctuations in received power, crosstalk between multiple OAM channels, and link performance impairment [9–11]. Therefore, the turbulence compensation technologies are the key issues for practical OAM-based free space optical communication (FSOC) links.
In order to mitigate turbulence effects and improve performance of OAM-based FSO communication links, many approaches have been proposed and demonstrated. The adaptive optics (AO) based compensation can correct the distorted phase of received OAM beams in the optical domain. In 2014, the team of Professor Alan E. Willner [12] used a rotating phase plate to simulate different intensity atmospheric turbulence and based on Gaussian beam probes to detect distorted phase in laboratory conditions. The compensation results demonstrate that the AO system can greatly mitigate the effect of atmospheric turbulence of the vortex beam and reduce the bit error rate (BER) of the communication system. In 2015, Xie et al. proposed a Zernike-polynomials based stochastic-parallel-gradient-descent (SPGD) algorithm to achieve the AO compensation without probe for distorted vortex beam. This SPGD algorithm execute a single iteration takes around 1 s [13]. In 2016, the team of Professor Chunqing Gao [14] proved intensity profiles of the distorted Gaussian probe beam together with the GS algorithm can obtain correction phase pattern to realize the distortion compensation of the vortex beam. In 2017, the team further improved the algorithm and implemented the probe-free GS algorithm for vortex beam AO distortion compensation. When the resolution of the profile is 250 × 250 pixels, the computing time is less than 100 ms with a dual Core i5 processor [15]. In 2019, Liu et al. proposed deep learning based atmospheric turbulence compensation for orbital angular momentum beam distortion and communication, the proposed method spends 9 ms to predict the phase screen on the CPU (I5-6500) [16]. In 2022, Zhan et al. proposed generative adversarial network based adaptive optics scheme for vortex beam, the calculation time is 0.536 s [17]. The AO correction frequency based on these algorithms is lower than 111 Hz. These AO-based mitigation methods can detect distorted phase with high accuracy, but require probes or long operation time, so their applications are limited to some extent.
For the vortex beam, since there is a phase singularity in the center, the HWS cannot be used to detect the distorted helical phase directly. In this paper, we proposes a distortion phase detection technique for vortex beam based on HWS, which uses the detection slope of the emission vortex beam without distortion as the calibrated slope zero point of HWS, thus avoiding the HWS detects the helical phase containing singular points directly. Moreover, this proposed method has no additional probe beam and the detection rate is as high as kHz which determined by the frame rate of the HWS. Simulation and experimental results show that this method can detect the distortion phase of vortex beam with high precision and high frame rate, which is expected to accelerate the application of optical communication systems with vortex beams. The disadvantage of this method is that it can only deal with beams whose phase slopes are circularly symmetric, and the phase distribution of the detected vortex beam must be known in advance.
2. Theories
Suppose a vortex beam with a helical phase has a complex amplitude $E(r,\theta )$
where $A(r)$ denotes the amplitude, r is the radial distance starting from the center axis of vortex beam. After free space transmission, both amplitude and phase will change. Thus, the received complex amplitude of OAM beams $E^{\prime}$ can be respectively written as where $A^{\prime}(r,\theta )$ is distortion amplitude, $\varphi $ denotes the phase distortion caused by atmospheric turbulence and divergence in the transmission process.The HWS is the most widely used type of wavefront sensor. It consists of microlens array and charge coupled device (CCD) that enables wavefront detection by phase slope measurement. In this paper, a scheme is proposed to use the detection slope of the helical phase sub-spot pattern as the slope zero point of the HWS, and then to detect the vortex beam distortion phase. The expression of the Zernike polynomial coefficient of the Zernike mode phase recovery algorithm is:
Utilizing the Zernike mode recovery algorithm, the measured phase ${\Phi _1}$ and ${\Phi _2}$ can be obtained as:
The calibrated slope zero point of HWS is essential to ensure the accuracy of distortion phase detection, so it is necessary to use a standard helical phase vortex beam for calibration. In our experiment, a high-density (1920×1152 pixel) liquid crystal spatial light modulator (SLM) is used to generate the standard helical phase, and the SLM has been calibrated for the experimental wavelength.
Similarly, our proposed method is also effective for the beam with circularly symmetric phase slope, such as Pin-Like optical beams and part multiplexed OAM vortex beam.
3. Simulation and experimental results
Figure 1 illustrates the measurement device of the proposed scheme. The 532 nm fiber laser is collimated by a collimator lens to generate a Gaussian beam with a beam diameter of 5 mm. The Gaussian beam is launched onto a reflective liquid crystal SLM loaded with a specific blazed-fork hologram to create a OAM beam. The rotating turbulence screen (TS) emulates atmospheric turbulence. The beam modulated by the SLM enters the 112-element HWS after passing through the TS. Firstly, the TS is moved out of the optical path, the SLM modulates different vortex beams, and the HWS performs wavefront detection. The slope detected by the HWS is used as the zero point for calibration. Subsequently, the turbulence screen is moved into the optical path, the HWS utilizes the corresponding detection slope zero point to measure the distorted wavefront of the OAM beam to obtain a high frame rate and high precision measurement result.
Fig. 1. The distorted phase detection device for vortex beam based on HWS. L: Len, A: Aperture, P: Polarizer, SLM: Spatial Light Modulator, TS: Turbulence Screen, HS: Hartmann-Shack wavefront Sensor.
In the simulation and experiment, the HWS is a rectangular arrangement with scale of 12×12, the number of sub-spots m = 112, and the number of Zernike polynomial modes n = 45. The phases of the simulated plane beam, OAM+1 and OAM+4 vortex beams are shown in Fig. 2(a1), (b1) and (c1), respectively, and the corresponding HWS sub-spot diagrams are shown in Fig. 2(a2), (b2) and (c2). The corresponding slope is used as the zero point in the detection of the distorted phase. The centroids of the sub-spots of the plane beams are at the sub-aperture center, and those of the OAM+1 and OAM+4 vortex beams both appear circular symmetry. OAM+4 vortex beam sub-spot centroid deflection is larger than OAM+1.
Fig. 2. The phase and HWS sub-spot diagrams of the simulated plane beam (a1)∼(a2), OAM+1 (b1)∼(b2) and OAM+4(c1)∼(c2) vortex beams.
Then, three turbulent phase screens with RMS = 0.1050λ, RMS = 0.1756λ and RMS = 0.3642λ are simulated. The HWS detection results corresponding to the parallel beam and the OAM+1 vortex beam are shown in Fig. 3. Figure 3(a1) and (b1) are the corresponding phase distributions of the parallel beam and OAM+1 vortex beam with RMS = 0.1050λ turbulent phase, respectively. Figure 3(a2) and (b2) are the corresponding HWS sub-spot diagrams. Figure 3(a3) and (b3) are the corresponding detected phase distribution by the HWS. Figure 3(a4) and (b4) are the corresponding turbulent phase detection errors. The RMS of the two phase residue are 0.0480λ. Figure 3(c1)∼(c4) and Fig. 3(d1)∼(d4) indicate the measurement results corresponding to the parallel beam and the OAM+1 vortex beam with RMS = 0.1756λ turbulent phase, respectively. The RMS of the two phase residue are 0.0573 λ. Figure 3(e1)∼(e4) and 3(f1)∼(f4) represent the measurement results corresponding to the parallel beam and the OAM+1 vortex beam with RMS = 0.3642λ turbulent phase, respectively. The RMS of the two phase residue are 0.1261λ. It can be noted from the simulation results that the distorted phase of the parallel beam and the OAM+1 vortex beam detection are the same, which indicates that the distorted phase detection method of the OAM vortex beam proposed in this paper is feasible, and the measurement error is limited by the measurement accuracy of the HWS. The measurement error increases with the RMS of the distorted phase.
Fig. 3. The simulation results: (a1)∼(f1) are the phase to be measured of the parallel beam and the OAM+1 vortex beam; (a2)∼(f2) are the corresponding HWS sub-spot diagrams; (a3)∼(f3) are the corresponding detected phase distribution by the HWS; (a4)∼(f4) are the corresponding phase residue.
The experimental measurement results are shown in Fig. 4. Figure 4(a1) and (b1) are the HWS sub-spot diagrams of the parallel beam and OAM+1 vortex beam passing through the TS at position 1, respectively. Figure 4(a2) and (b2) are the phase distribution diagram of the corresponding detection distorted phase, the RMS of the phase is 0.0933λ and 0.1079λ, respectively. The difference distribution diagram of Fig. 4(a2) and (b2) show in Fig. 4(e1), the RMS value is 0.0226 λ. Figure 4(c1)∼(c2) and 4(d1)∼(d2) are the measurement results of the parallel beam and the OAM+1 vortex beam passing through the TS at position 2, respectively. The RMS of the detected phases are 0.1642λ and 0.1729λ, respectively. Figure 4 (e2) is the difference distribution diagram of Fig. 4 (c2) and (d2), the RMS value is 0.0143λ. It can be observed from the results that the detected distortion phase of the parallel beam and the OAM+1 vortex beam is affected by the HWS detection noise to a certain extent, but both are lower than λ/40, which is almost negligible. The experimental results verify that the method is feasible for OAM vortex beam distortion phase detection.
Fig. 4. Experimental results: (a1)∼(d1) are the HWS sub-spot diagrams, (a2)∼(d2) are the corresponding detection distorted phase, (e1) is the difference distribution of (a2) and (b2), (e2) is the difference distribution of (c2) and (d2).
4. Discussion
Compared with the conventional HWS, the field of view of the HWS used for the detection of vortex beam distortion phase should be designed to be larger. For the same turbulent phase, if the field of view is too small, it is easy to cause the sub-spot to shift to other sub-aperture which leads to larger measurement errors, especially for high-order vortex beams with large phase gradients.
The designed HWS is used to measure the dynamic distortion phase of the OAM0, OAM+1, OAM+2, and OAM+10 vortex beams through the same phase screen, respectively. The measured frequency of HWS is 1600 Hz. The comparison results of dynamic measurement experiments are in the uploaded Visualization 1. Due to fluctuations in the display frame rate of the HWS software, there are several frame inconsistencies in the detection wavefront of the video. Figure 5 is a screenshot at 00:00:03 of the Visualization 1. It can be noted that for the same position of the phase screen the results of the measured distortion phase are basically the same for OAM0, OAM+1, OAM+2 vortex beams, but the OAM+10 vortex beam has a larger error due to the expansion of the singular dark spot in the center. Therefore, it is necessary to further improve and upgrade the HWS algorithm, so that it can be used for accurate phase detection of beams with partial sub-spots lacking optical power.
Fig. 5. Experimental results: The detected dynamic distortion phase of the OAM0, OAM+1, OAM+2 and OAM+10 vortex beams through the same phase screen. The left side of the HWS software interface is the sub-spot diagrams, and the lower right corner is the detection distorted phase.
5. Conclusion
In summary, a distorted phase detection method based on the HWS for vortex beam is proposed. The detection slope of the helical phase is used as the zero point for calibration of HWS, and then the distortion phase of the vortex beam is detected. The proposed method has no probe beam, thus reducing the complexity and increasing the usability of the communication system. The frequency of AO correction based on the existing GS, SPGD, deep learning and generative adversarial network algorithm are lower than 111 Hz. Our proposed method is much faster than these algorithm-based methods, and the detection rate is as high as 1600 Hz which can be further increased by improving the HWS performance. The AO system based on the proposed detection method can realize the real-time correction of atmospheric turbulence. Our work is expected to accelerate the application of the OAM-based optical communication technology.
Funding
National Natural Science Foundation of China (61901449).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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