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Abstract
We propose an approach against the turbulence-induced degradation by using laser beam with self-rotating wavefront. Such laser beam, generated by the coherent combination of vortex beams with different helical charges and central angular frequencies, can introduce coupling of its wavefront in spatial and temporal domain, that is, periodic wavefront rotation. When the wavefront rotation is faster than the airflow, the laser beam can travel through the inhomogeneity and anisotropy of air in the azimuthal direction within the time interval of airflow. The wavefront distortion caused by the turbulent atmosphere is therefore rotated and gradually smoothed as the laser beam travels. After the laser propagating through the turbulent atmosphere, the total wavefront distortion becomes centrosymmetric with lower peak-to-valley (PV) value. Such smoothed wavefront distortion can dramatically eliminate the turbulence-induced degradation of laser beams, especially beam centroid drift. We believe that this approach can lead to new trend in remote sensing, free-space optical communication, lidar, etc.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The propagation of laser beams in turbulent atmosphere is of great importance because of its wide applications in remote sensing [1], free-space optical communication [2], lidar [3], etc. The inhomogeneity and anisotropy in the temperature and pressure of atmosphere, which is inevitably encountered, can lead to turbulence-induced degradation of laser beams including beam wandering, beam broadening, scintillation, spatial decoherence and so on [4,5]. Enormous efforts have been made to understand and to eliminate the influence of atmospheric turbulence on laser beam propagation. It has been well known that the partially coherent beam has substantial advantages for reducing turbulence-induced degradation than the fully coherent counterparts [6,7]. The spatially structured light [8] with non-diffracting or self-healing properties, represented by Laguerre-Gaussian beam, Bessel-Gaussian beam and Airy beam, also shows tremendous potential in mitigation of atmospheric turbulence. However, the partially coherent beam and the spatially structured light are limited for applications that requires long distance propagation. In addition, due to the technological complexity and high cost, the adaptive optics with limited performance in wavefront correction, is mainly used in optical telescopes for astronomical imaging [9] and optical microscopes for biological imaging [10]. So far as we know, new approach against the turbulence-induced degradation of laser beams is still one of the sole problems in laser applications.
In this paper, we provide an alternative approach against the turbulence-induced degradation by rotating the wavefront of laser beams. Under Taylor’s frozen air hypothesis [11], when the wavefront of incident laser rotates faster than the airflow, the laser beam can travel all the inhomogeneities and anisotropies of atmosphere in azimuthal direction within the time interval of airflow. Equivalently, the air can be imagined as a rotating inhomogeneous medium, so the wavefront modulation to the laser beam is rotated and smoothed in the azimuthal direction. After the laser propagates through the frozen air, the total wavefront distortion becomes centrosymmetric with lower PV value, contributing to the reduction of turbulence-induced degradation. The aim of this paper is to reveal how light beams with self-rotating wavefront can mitigate the atmospheric turbulence effect.
2. Theoretical model
2.1 Model of light beams with self-rotating wavefront
The laser beam with self-rotating wavefront can be generated by coherent combination of two vortex beams with different helical charges and central angular frequencies (see Fig. 1(a)). Both the intensity distribution and the wavefront rotate in a period inversely proportional to the frequency difference, which can be flexibly tuned by heterodyne interference based on electro-optic (EO) [12] or acoustic-optic (AO) [13] modulation.
Fig. 1. (a) Illustration of generation of the laser beam with self-rotating intensity and wavefront, la = 2, lb=−2, (b) Illustration of generation of the laser beam with self-rotating intensity and wavefront, la = 2, lb=−1, (c) Propagation of the laser beam with self-rotating wavefront in “frozen” air turbulence.
Now let us consider the propagation of a laser beam with self-rotating wavefront in the turbulent atmosphere. In Taylor’s frozen air hypothesis, local temperature gradients, which might be present from one side of an eddy to another, are advected across the sensor by the mean wind without eddy changing. Many studies have shown that the time interval of air flow is mostly between 10–20 ms [14–16]. For an incident laser with self-rotating wavefront whose period is close to or even shorter than that of the airflow, the Taylor’s hypothesis works well. In other words, when the laser travels for a certain distance in the medium, before the air flows, the air can be regarded as a medium with non-uniform refraction index. By dividing such medium into many pieces, the propagation of the laser beam with self-rotating wavefront repeats the following processes: distortion, rotation and diffraction. Since the wavefront rotates faster than the airflow, the laser beam can travel all inhomogeneous air in the azimuthal direction within the time interval of airflow (see Fig. 1(b)). As a result, the turbulence-induced wavefront distortion is an integral in azimuthal direction, eventually leading to centrosymmetric with lower PV. Such smoothed wavefront distortion can significantly eliminate the effect of turbulent atmosphere and improve the propagation performance of laser beam, as will be discussed below.
We start with a general representation for a laser beam with rotating wavefront:
Next, for instance, let us consider the coherent combination of two vortex beams with helical charges of la,b and central angular frequencies of ωa,b. as shown in Fig. 1(a), i.e.,
According to Eq. (2), the intensity distribution and the wavefront of the combined beam are respectively expressed by:
In Eqs. (3) and (4), we can see that the intensity and wavefront are coupled in space and time. The rotation period of the combined beam is directly determined by the angular frequency difference Δω and is not affected by the helical charges. When the helical charges are the same but opposite in sign, the distributions of intensity and wavefront maintains the same with time, as shown in Fig. 1(a). When the helical charges are different, the intensity distribution maintains the same but the wavefront distribution changes with time (see Fig. 1(b)). When the wavefront rotates faster than the airflow, the laser beam can travel through the inhomogeneous and anisotropy of the air. The wavefront distribution of the laser beam does not have a significant influence on its performance in mitigating the atmospheric turbulence effect. Therefore, the unsymmetric wavefront distribution does not significantly decrease its performance in maintaining the beam quality in turbulent atmosphere.
The rotation period equals to Tr = 2π/Δω, which ranges from seconds to nanosecond, and can be flexibly adjusted by the heterodyne interference of the electro-optically or acousto-optically modulated vortex beams. This means that we can adjust the rotation period of wavefront according to the time interval of air flow to achieve the best performance.
We will now analyze the behavior of the laser beam with self-rotating wavefront, in particular wavefront distortion and beam centroid drift when propagation in isotropic and inhomogeneous atmosphere. Let the power spectrum of the refractive index fluctuations have the classical Kolmogorov profile [17]:
where $C_{n}^{2}$ is the structure parameter representing the turbulence strength. When $C_{n}^{2}$ changes from 10−17 m−2/3 to 10−13 m−2/3, the turbulence strength increases gradually.2.2 Modified split-step fourier method
For a laser beam with self-rotating wavefront, the rotation is fast enough so that only wavefront distortion and diffraction cannot reflect the true propagation characteristics of the laser beam. In Fig. 1(c), such a beam can be split into many sub-beams along the propagation path, each of which rotates in a different initial orientation within the same period. Let us consider the propagation of each sub-beam in the atmosphere. For each sub-beam whose rotation period is shorter than the airflow, it can make one rotation before the air fluctuates, thus passing through the air once in the azimuthal direction. The wavefront distortion of the steady air to the sub-beam is rotated, and the total wavefront distortion is an integral along the beam path. Without consideration of the time-varying characteristic of the laser beam itself, the traditional Split-step Fourier Method (SSFM) cannot effectively manifest the spatiotemporal coupling of the laser beam when the wavefront rotates in a period close to or even shorter than that of the airflow. Therefore, we developed a modified SSFM to simulate the propagation of the laser beam with self-rotating wavefront in the turbulent atmosphere, as shown in Fig. 2(a). Unlike conventional SSFM, the modified method takes into account the wavefront rotation, which cannot be neglected in this case. During the propagation of the laser beam with self-rotating wavefront in turbulent atmosphere, the wavefront distortion screen is divided into multiple sub-screens. For simplicity, the rotation angle of the laser beam between the adjacent screen is defined as
where Tair is the time interval of air flow, M is the number of sub-screens.Fig. 2. (a) Modified SSFM for calculating propagation of rotation beam in air, (b) the rotation angle for each split step.
In conventional SSFM, the iterative steps only include wavefront distortion and free-space diffraction, but no rotation step. This is because in conventional non-rotating beam, the variations of the laser beam itself are often treated as invariants and ignored. Once the laser beam changes faster than the turbulent air, the variation of the laser beam itself should be taken into consideration.
In order to reflect the rotation nature of the laser beam, we have added the iterative step of rotating both the amplitude and the wavefront in the conventional SSFM. In Fig. 2(a), the iterative step 1 is added to reveal the rotation characteristic of the laser beam, which is accomplished by rotating both the amplitude distribution and the wavefront distortion for a certain angle defined by Eq. (6).
After propagating through the turbulent atmosphere, the laser beam is focused on focal plane to analyze its centroid drift and beam width. It is worthy of noting that the improvement also involves the near-field intensity distributions and the wavefronts, as will be described below.
Figure 2(a) shows the difference between the conventional SSFM and the modified SSFM: the wavefront rotation with an angle of θ should be calculated before the wavefront distortion. This is because, under the Taylor’s hypothesis, the spatial characteristics of the atmospheric turbulence driven by the transverse wind do not change significantly over such small-time intervals. Therefore, such a laser beam with a faster rotational wavefront than the transverse wind should be calculated before the turbulence changes. However, the rotation angle of each time interval can vary depending on the time interval of the air, the number of sub-screens and the wavefront rotation period, as shown in Fig. 2(b).
3. Propagation characteristics of light beam with self-rotating wavefront
Hereafter, we will study the propagation characteristics of beam with self-rotating wavefront in turbulent atmosphere and show theoretically how the wavefront rotation eliminate the turbulence-induced degradation. The key factors affecting the propagation involve rotation frequency that equals to Δω/2π, beam width and turbulence strength. The parameters for simulation are given as follows: $C_{n}^{2}$= 10−13 m−2/3, λa = 1064 nm, Δω=2π×1 kHz, beam width σ=7.5 cm, helical charge $l_{a, b}=\pm 2$. For simplicity, the propagation distance is selected to be 10 km, and the observation time is 0.1 s. In many studies, the time interval of air flow is 10–20 ms under Taylor’s frozen air hypothesis. For simplicity, the air is assumed to be periodic fluctuation with a frequency of 200 Hz, that is, the air fluctuates once every 5 ms. We note that this assumption is fairly simple, however, this work focusses on how wavefront rotation can improve the propagation performance of laser beams in turbulent atmosphere. Our model is simple but works well under Taylor’s frozen air hypothesis. Figures 3(a)-(b) give the near-field intensity distribution and wavefront of the laser beam with self-rotating wavefront after propagating through the turbulent atmosphere.
Fig. 3. Instantaneous distribution of near-field (a) intensity (see Visualization 1 and Visualization 2) and (b) wavefront of the laser beam with self-rotating wavefront after propagating through turbulent atmosphere. (c). Zernike coefficients of the wavefront distribution.
In Fig. 3(a), the instantaneous distribution of near-field intensity of the laser beam almost retains the intensity profile of the incident laser in Fig. 1(a), indicating that the wavefront rotation can improve the beam quality dramatically. This is because, for a conventional Gaussian beam without self-rotating wavefront, the wavefront distortion caused by the turbulent atmosphere is accumulated along the propagation path, which leads to the distortion of intensity distribution. For the laser beam with rapid rotating wavefront, the intensity distribution exhibits only a certain degree of beam broadening in the radial direction. The beam broadening is the combined effect of diffraction and wavefront distortion in radial direction and cannot be smoothed by wavefront rotation. It is exciting to observe the nearly centrosymmetric wavefront in Fig. 3(b), which can dramatically improve the centroid stability in the far field. We also note that the wavefront in Fig. 3(b), although the wavefront can smooth most of the random wave vectors by the turbulent atmosphere, there is still strong wavefront distortion that cannot be smoothed well in radial direction. In Fig. 3(c), we can see that the wavefront distribution mainly contains tilt and defocus, which cannot be neglected, but the vortex phase maintains quite well during propagation.
For a conventional OAM beam, such strong turbulence will show beam breakups, filamentation of local beam portions, and many more inserted branch points in the phases. For the laser beam with self-rotating wavefront, even in such strong turbulence, the wavefront distortion caused in each rotation cycle can be smoothed so well in the azimuthal direction, so that the internal structure of the beam can be survived. Moreover, Fig. 3(a) has shown that it is still distorted due to the unsmoothed wavefront distortion in radial direction.
Next, in order to numerically illustrate how wavefront rotation can eliminate the turbulence-induced degradation, the first moment representing the centroid drift of the beam on focal plane and the second moment representing the beam width of focal spot are further analyzed. Figure 4(a) shows the centroid drift of the laser beam on focal plane with different rotation frequency, while Fig. 4(b) gives the beam width in transverse direction as well as the instantaneous intensity distribution.
Fig. 4. (a) Centroid shifts of rotating beams with different rotation frequency, (b) beam widths and instantaneous intensity distributions of rotating beams. The fluctuation frequency of air turbulence is constant at 200 Hz with $C_{n}^{2}$ = 10−15 m−2/3.
In Fig. 4(a), as the rotation frequency varies from 0–20 kHz (the central angular frequency difference is from 0–2π×10 kHz), the centroid drift of the laser beam on focal plane dramatically drops due to the wavefront rotation. This is because, for a certain frequency of airflow, as the rotational frequency increases, the laser beam can travel through the turbulent atmosphere more efficiently. We then calculated the rotation angle θ (Fig. 2(b)) between each split step and found that the rotation angle is below 2π when the rotation frequency is below 1296 kHz. Let us consider that when the rotation frequency equals to 1296 kHz that the rotation angle between each step equals to 2π (according to Eq. (6)), which appears as no wavefront rotation occurs. As given in Fig. 3(a), such a critical rotation frequency leads to the failure of wavefront rotation in improving centroid drift stability. However, it strongly depends on the given time interval of air flow and the wavefront rotation period, which can be rather complicated and time-varying, so we can predict the effectiveness of the wavefront rotation against the turbulent atmosphere.
In Fig. 4(b), with the rotation frequency changing from 0–10kHz, the beam width shows same decrease as the centroid drift. However, when the frequency shift is larger than 1296 kHz, the beam width no longer decreases. This is because the rotation angle exceeding 2π can smooth the wavefront distortion but cannot reduce its PV that mainly determines the beam width.
Taken Fig. 4(b) as an example, we can see that there are two cycles: One is the variation of the beam width caused by the air turbulence with a fluctuation frequency of 200 Hz, which means that there are 20 cycles from 0∼0.1s. The other is the rotation period of the laser beam which decreases with the increase of the rotation frequency. When the rotation frequency is 1 kHz, the sawtooth modulation is clearly seen. Further analysis shows that, the periods of the sawtooth modulations are the same as that of the rotation beam. This is because, the beam quality, including the centroid drift and the beam width, first decreases then increases during one cycle, leading to the sawtooth modulation.
In addition, according to Fig. 2(b) and Eq. (6), when the rotation frequency of the laser beam is too large, the laser beam can rotate over 2π in the azimuthal direction, that is, the laser beam seems to be “not rotating” to the atmosphere. In that case, the wavefront distortion of the atmospheric turbulence in radial direction will severely affect the laser beam, causing beam expansion, while that in azimuthal direction also has more impact on the centroid drift o the laser beam. Therefore, the rotation frequency should be rational according to the fluctuation frequency of the atmospheric turbulence.
The modified SSFM method essentially adds the rotation of the distorted beam at each step, taking into account the time-varying rotating wavefront of the laser beam. If the atmospheric turbulence becomes stronger, the two vortex beams become partially coherent, so that only part of the total beam maintains rotating. In this case, the fully decoherent part can resist the atmospheric turbulence, while the rotating part can also well mitigate the turbulence-induced degradation. If the atmospheric turbulence is so strong that the two vortex beams become completely incoherent, therefore the total beam become completely incoherent to against the atmospheric turbulence. According to the above analysis, the beam quality can be improved in each cycle of the wavefront rotation of the laser beam. Therefore, the coherence length Lc = λ2/Δλ can be one reference of the wavefront rotation iteration step.
The turbulence strength and beam width are also key factors affecting the wavefront rotation performance. In Fig. 5, the curves (a) and (c) compare the influence of turbulent strength on the centroid shift of the beam on focal plane, while the curve (b) shows how the beam widths affect the propagation of the rotating beam.
Fig. 5. (a) Centroid drift of focused rotating beam with beam width of 7.5 cm and $C_{n}^{2}$ = 10−14 m−2/3, (b) Centroid drift of focused rotating beam with beam width of 7.5 cm and $C_{n}^{2}$ = 10−15 m−2/3, (c) Centroid drift of focused rotating beam with beam width of 5.0 cm and $C_{n}^{2}$ n = 10−14 m−2/3. The frequency shift of the laser beam is 1 kHz, the fluctuation frequency of air turbulence is constant at 200 Hz.
By comparing the curves in Fig. 5, it can be seen that the wavefront rotation performance on reducing centroid drift reduces with the increasing turbulence strength. This is because the inhomogeneity of the air increases with the turbulence strength, thus requiring faster wavefront rotation to smooth the inhomogeneity. On the other hand, by comparing curves (b) and (c), it can be seen that the reduction of centroid drift is worse as the beam width decreases.
It is worth mentioning that, the physical model is built using the Kolmogorov turbulence under Taylor’s frozen air hypothesis. However, the airflow is of complexity and models manifesting the air turbulence in space is still under developing. In this work, we have analyzed the improvement of wavefront rotation in near-field intensity distribution and wavefront, and reduction of beam drift and beam width. We also noted that this approach to eliminate turbulence-induced degradation can be more comprehensive, including scintillation, spatial decoherence and so on.
To summary, the laser beam with self-rotating wavefront is generated by the coherent combination of two vortex beams with different helical charges and central frequencies. The basic idea of using the laser beam with self-rotating wavefront to mitigate the atmospheric turbulence is that, the “short-term” rotation of the laser beam itself can sweep away the “long-term” wavefront distortion caused by the turbulent air, requiring the beam to rotate somewhat faster than the atmospheric turbulence.
Under Taylor’s frozen air hypothesis, the local atmospheric turbulence should remain unchanged for short periods of time. When the laser beam rotates with a slightly shorter period, the turbulent air can be regarded as an inhomogeneous medium, allowing the laser beam with rapid rotation to transmit through the medium. Equivalently, the air can be imagined as a rotating inhomogeneous medium, so the wavefront modulation to the laser beam is rotated and smoothed in the azimuthal direction. The conventional OAM beam can maintain its OAM to a certain extent. However, due to the randomness of the atmospheric turbulence, the conventional OAM beam cannot smooth the wavefront distortion.
4. Conclusion
In conclusion, we have proposed a novel approach to eliminate turbulence-induced degradation by using a laser beam with self-rotating wavefront. The advantages of such beams with self-rotating wavefront are illustrated theoretically. In addition to this, we have modified the split-step Fourier method by adding the wavefront rotation step before the wavefront distortion and diffraction steps. The beam with self-rotating wavefront shows tremendous potential for turbulence-induced degradation for various applications as they inevitably traverse the atmospheric turbulence.
Funding
National Natural Science Foundation of China (61905167); Research Program of National Major Project of China (JG2020376); The Key Laboratory on Adaptive Optics, Chinese Academy of Sciences (LAOF1801).
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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