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Abstract
Propagation of laser light is distorted in the presence of atmospheric turbulence. This poses an issue for sensing, free-space optical communications, and transmission of power. The presented system offers a novel solution to mitigate the effects of turbulence. By rapidly probing a turbulent volume by varying a beam’s spatial and phase characteristics, the best transmission mode can be determined and updated in real time. Unlike a traditional tip-tilt system, this scheme is fully electronic, and has a scalable architecture to leverage multiple optical transmission paths simultaneously. This optical control system greatly improves power efficiency and successful recovery of data through environments with strong turbulence.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
There have been a variety of methods demonstrated to reduce the effects of turbulence for a communications link [1–5]. Optical turbulence causes the distortion, deflection, or breakup of a laser beam due to variance in the distribution of heat in the propagation medium. This variance creates a random refractive index structure [6]. Many have been effective at improving the bit error rate of high data-rate signals and reducing the distortion observed at the receiver. Performing turbulence mitigation on the transmit side of a link is advantageous for delivery to smaller systems where a complex adaptive optics receiver may be impractical. Fundamentally, the way most of the reference methods improve transmission through turbulence is by collecting a broader spectrum of modes, as highlighted in [1]. This can be highly effective, but is ultimately limited in stronger turbulence, defined by a smaller Fried parameter r0, by the size of the receiving aperture. The Fried parameter, proportional to the spatial coherence radius of a beam, is a statistical parameter that defines the diameter over which the wavefront could be considered smooth [7].
It was proposed in [8] that eigenmodes exist for volumes of atmospheric turbulence. That is, specific complex amplitude and phase structures of light can pass through a turbulent medium with no distortion. Phase distortions introduce significant difficulties when transmitting structured light through turbulence. These eigenmodes were numerically solved and shown experimentally to be effective. Additionally, it has been shown that within a turbulent volume, thermal gradients can create waveguide-like channels due to a varying refractive index [6,9,10], which may guide specific modes of light to different spatial locations with varying phase tilts. The localized regions of higher or lower temperatures create an inhomogeneous density of air, whose refractive indices will all be slightly different [10]. This variance is what creates the channels through turbulence. In stronger turbulence, more modes tend to be deflected to an off-axis position over a propagation length, resulting in lost photons at a fixed aperture detector. The location of these waveguiding channels is highly random, severely limiting the time these channels can be accessed. It was also shown that when using a fixed Gaussian beam, the light could focus through a channel about 1 in 1000 realizations [6]. Each realization was separated by ∼10 ms in time. The existence of a channel was defined as a time where a large portion of the beam power was focused into one region of the original beam profile. If this channel persists for a longer duration of time but is moved off-axis from wind velocity and thermal diffusion, it is natural to wonder if the channel can be tracked and exploited for a longer duration, increasing the accessibility of the channel.
In order to address this challenge, in this paper, we introduce an effective probe and control system that greatly improves the transmission of light through turbulence. This is possible using OAM and the azimuthal position about a vortex footprint as a basis, combined with the rapid mode switching of the High-Order-Bessel-Beams-Integrated-in-Time (HOBBIT) architecture [11]. The HOBBIT system originally used an acousto-optic deflector (AOD) and log-polar coordinate transform optics to rapidly generate OAM beams. This system was used to probe turbulence with a continuous spectrum of OAM; it was also used to generate a wavelet basis of optical fields for probing turbulence [7,12]. The wavelet probing revealed that many highly transmitting modes through turbulence have the form of an angular Gaussian with OAM, where the size of the angular Gaussian depends on the turbulence strength. This paper implements a system based on the HOBBIT system with a fixed beam size but tunable OAM and tunable angular position about a fixed radius. This real-time method can rapidly probe a turbulent volume of air and determine the best input mode that couples to the receiver. Locking the beam to this mode allows for transmission of optical power and can support a high-speed data link. This sensing method can also be used to analyze the dynamics of the turbulence.
2. Background
Conceptually, this turbulence mitigation method is shown in Fig. 1. In the top figure, as a beam propagates through turbulence, it will be distorted and can break up into multiple smaller beams depending on the strength of the turbulence. Once the beam breaks into the smaller beamlets, some of them are incident on the desired target, while others are deflected to other locations. This is extremely challenging when one requires focusing of power onto a detector for communication links, which creates large signal fades since the beam deflection shifts the beam from the detector active area. This degradation of the beam can be mitigated by carefully steering where the beamlets go and correcting their relative phase at the receive aperture. Although many solutions exist to solving this problem, it can only be realized with two different optical planes for control of position and phase of the individual beamlets so that they fall within the receiving aperture with the proper phase. Since many optical systems have a circular footprint for the telescope optics and often have a central obscuration due to the type of optical system used, it is desirable to have an adaptive optic system that can span an annular ring corresponding to the clear aperture of the system. This cylindrical geometry lends itself to an azimuthally controlled optical mapping that can manipulate one or more beams along the annular ring, while controlling the gradient of the phase for each beam. However, before this can be accomplished, a stronger understanding of the beam sizes and required phase gradients needs to be addressed. It is also imperative that this adaptive optic control system must be orders of magnitude faster than the time scale of the changes in the turbulent path.
The first parameter for consideration is the determination of the required beamlet sizes. It is well known that spatial features of turbulence can be characterized by the Fried parameter, r0, and can be used interchangeably with the refractive index structure constant, Cn2 [13–15]. The Fried parameter defines the maximal diameter over which a wavefront can be considered smooth, through a given turbulent medium [7]. A coherent beam larger than this diameter may experience a breakup of the beam as it propagated (D/r0 > 1, or strong turbulence, where D is the diameter). For a plane wave, r0 may be found by [7]
where k is the wavenumber, Cn2 is the refractive index structure function, and L is the path length. Beams with a diameter smaller than the Fried parameter motivate the use of smaller beamlets of size r0 to probe turbulence. The emphasis of this work resides in the solutions to strong turbulence, with r0 ‘s much smaller than the aperture diameter. Table 1 shows the different values for each of these parameters used in the experiment that were generated using the in-house variable turbulence generator (VTG). Using a beam size which is close to the size of r0 results in a well characterized effect of beam wander [13–15]. The turbulence will cause the beam to shift off axis resulting in a phase front with a gradient. This beam wander is a deflection angle before focusing, which results in a spatial shift at the aperture causing signal fades. The randomness of these r0 locations and their density is a very complex problem; however, the phase front gradients can be estimated using previously derived statistical models which can be used to estimate the variance of the angle of the beam wander as follows [14]This random wander within a certain radius of the axis motivated the use of a modified HOBBIT system [11] for probing and correction in turbulence. By applying an OAM phase profile to an input beam, various phase tilts can be realized, and a spatial shift can probe a range of spatial locations around a ring (see section 3). For the HOBBIT system to accurately correct the turbulence-induced tilt in every realization in the strongest turbulence of the experiment (r0 = 3.8 mm), the phase should be tilted according to the OAM skew angle [16]
where θskew is the deflected angle of the phase, l is the topological charge number, k is the wave number, and r is the radius of the circular footprint. Using r = 17.5 mm, and the desired maximal deflection angle from Eq. (1), an OAM range of l = ±18 is required to cover this range. Since the OAM probing sequence is already about a circular vortex, the azimuthal angle dimension of the probing method provides a simple, one-dimensional basis to correct for any direction of deflection tangential to the annular ring. Due to hardware limitations, the experimental probing range of OAM was limited to charge l = ±15. From Eq. (2), it is predicted that this range allows a maximal deflection correction of about 73 µm. This falls within 2 standard deviations of the beam wander variance, so it is expected that about 90% of realizations at r0 = 3.8 mm would be corrected to the axis of transmission using the current hardware for the experiment. With this method, rapid probing of the beam’s phase gradient using orbital angular momentum (OAM) and azimuthal position about a circular footprint as a basis allows for the determination of the most efficient input conditions resulting in transmission through the turbulent medium. The HOBBIT system [11] can switch OAM modes on the order of MHz, which is constrained by the diameter of the input beam and the acoustic velocity of the acousto-optic deflectors used in the system. With a sufficiently fast sensing method, the turbulence will appear frozen, and the best input state to achieve the desired output can be leveraged for power transmission and a communication link even in a dynamic environment.
Table 1. Turbulence Parameters
3. Methods
3.1 Probing
To rapidly manipulate the spatial location and phase gradient of the beam, a modified HOBBIT system [11,17–19] was developed. Unlike the previous HOBBIT configuration, this system used two acousto-optic deflectors (AOD) to manipulate both dimensions of the probing basis and is shown in Fig. 2.
Fig. 2. (a) Diagram of HOBBIT system with beam transformations for creating OAM beamlets. The line and ring of gaussian spots show several possible locations for the beam, which are all cycled through during the probing sequence. (b) Hardware setup used to create the modified HOBBIT system.
The first AOD (AOD 1) controls the azimuthal position or rotation angle of the probing beam, while the second AOD (AOD 2) controls the phase tilt. See the supplemental material for a detailed description of the optical transform. From the log-polar coordinate transform, we obtain an expression for the field of a single OAM beamlet immediately exiting the system
Fig. 3. (a) 9 discrete OAM beamlet positions with phase gradients of an OAM beam of charge 1. (b) Effectively continuous ring by integrating 40 positions. (c) Locked position with the corresponding phase gradient.
Figure 3(b). After a scan similar to Fig. 3(b), a single Gaussian at a particular position and phase tilt will be selected for a particular realization of turbulence to have the best power and/or a data signal transmission, Fig. 3(c). Using the described HOBBIT system, an arbitrary phase profile for an integer OAM charge was applied to the beam, which resulted in a phase gradient tangential to the vortex profile. The rapid switching of the AODs motivated the use of the HOBBIT for creating the phase tilt. With sufficient azimuthal resolution, the discrete Gaussian spots overlap, creating an effectively continuous probing space.
To realize atmospheric turbulence in a controlled way, a variable turbulence generator (VTG) tunnel similar to the one in [7] was used. This tunnel most closely follows the Kolmogorov spectrum of turbulence [20]. Using a probing radius of 17.5 mm, about which the Gaussian beams are located, the probe beam is propagated 3 times through the VTG for a total propagation distance of 60 m, shown in Fig. 4(a). Figure 4(b) shows the inside of the VTG from the transmission side with the system locked to one Gaussian beam.
After the 60 m propagation, the received beam passed through a 500 mm lens to perform a Fourier transform. Using beam splitters, 80% of this output was directed to a 50 µm diameter multi-mode fiber. This fiber connected to a 10 GHz AC-coupled photoreceiver (RXM10BF) with responsivity of 0.2 A/W at wavelength 532 nm and a 3 dB bandwidth of 11 GHz for receiving data signals. This detector was sampled at 25 GSps by an oscilloscope (MSO71254C) to recover the data signal. A pellicle beam splitter (BP208) directed a small portion of the beam towards a 50 MHz avalanche photodetector (APD120A2) for probing feedback. To capture the process on video, another pellicle beam splitter sent a small amount of light to a high-speed camera (Phantom T-1340), recording at 1000 frames per second with an integration time of 45 µs. The receiver is shown in Fig. 5. To measure the DC power coupled through the multimode fiber, the photoreceiver was swapped with a DET10A detector with responsivity of 0.25 A/W at 532 nm. Its output was amplified by 15 dB and sampled by the same oscilloscope.
3.2 Control system
To control the modified HOBBIT system, an algorithm was implemented on a ScopeFun instrumentation microcontroller, which interfaced with an arbitrary waveform generator (AWG5208). This AWG applied the radio frequency signals to the AODs to move the beam through the probing space. The probing sequence was limited to 256 waveforms due to the AWG firmware, so the probing space was set to 35 rotation angles about the ring and 7 OAM phase profiles. The number of OAM phase profiles allows a step of 5 OAM during the scanning. The normalized OAM spectrum of the beams generated in Eq. (3) is given by $\exp ({\raise0.5ex\hbox{$\scriptstyle { - {w^2}}$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}{(n + m)^2}),$ where n is the mode index and m is the global OAM of the beam. With w = 0.24 in the experiment, the 1/e2 width of this spectrum is 16.7, so the step size of 5 OAM enables sampling at less than half of the width of the OAM spectrum. The probing sequence started at OAM charge -15 and -2.67 rad from the bottom of the ring. The beam was stepped by increments of 0.157 radians to 2.67 rad, then the beam is reset to -2.67 rad before the OAM is changed. The OAM is stepped by 5, and the position scan is repeated. This process continued up to the maximal OAM charge of 15.
Each of the beam states was held for 615 ns, for a total probe time of 151 µs. This switching rate is limited by the Gaussian beam diameter over the acoustic velocity of the AOD TeO2 crystal. Changing beam states requires a new acoustic frequency to fully propagate across the input beam. When a probing sequence began, the microcontroller sampled the APD at 12.5 MHz and captured the feedback for a full sequence. The beam state with maximal voltage was then locked by the AWG, and the data signal was triggered. A fixed delay of 2 ms was introduced to ensure collection of data, and the sequence was restarted. The choice of this delay was arbitrary, but no significant differences in measurement occurred between a 2 and 10 ms delay. An additional delay was introduced before engaging the controller so the camera could see the scan. If the control system was disabled, the beam was locked in a fixed position with an OAM of 0 for a flat phase. The process is illustrated in Fig. 6.
Fig. 6. (a) Probe in ambient turbulence, (b) decision point and (c) feedback spectrogram. (see Visualization 1) (d) Probe in strong (r0 = 3.8 mm) turbulence, notice the circled bright spot, (e) decision point and (f) feedback spectrogram (see Visualization 2).
Figure 6-(a,b) (Visualization 1) and Fig. 6-(d,e) (Visualization 2) show the scan and decision process respectively at different turbulence conditions, captured by the camera. Figure 6-(c,f) are spectrograms generated offline by normalizing the feedback voltages and mapping them to the probing space. Figure 7 shows the control loop block diagram. The control algorithm itself introduced a delay of ∼1 ms.
Fig. 7. Control System Diagram. Probing sequence waveform on AWG triggers microcontroller sampling of feedback detector.
3.3 Data generation and modulation
The laser source used to generate the beam was a 1064 nm diode pumped solid-state laser. In order to modulate the signal and encode the data, a fiber Mach-Zehnder interferometer (MZI) was created via a 50/50 fiber power splitter. An electro-optic phase modulator (NIR-MPX-LN-10) with half-wave voltage of 4.4 V and bandwidth of 10 GHz modulated the data signal as a phase delay on one of the interferometer paths. A low bandwidth (150 MHz) phase modulator (NIR-MPX-LN-0.1) on the other path locked the relative phase between the beams, ensuring only the modulated signal varied the phase difference. The modulation scheme used was a phase amplitude modulation with 2 discrete voltage levels (PAM2), at a data rate of 5 Gbit/s. Each path was amplified using a Nuphoton fiber amplifier, then they were combined using a 50/50 beam splitter. The amplified, modulated MZI output was frequency doubled to 532 nm using a periodically poled lithium niobate (PPLN) crystal. The grating period of the PPLN was $6.93\; \mu m$. This crystal was kept at 59.4 °C to achieve the appropriate phase matching condition. The main purpose of frequency doubling was to enhance the fluctuations due to turbulence, as a shorter wavelength is more sensitive to localized refractive index variance [4]. Additionally, this wavelength is used for underwater communication experiments through high turbidity and can be used to evaluate different coding schemes.
The 532 nm beam was input to the modified HOBBIT system with an optical power of 6 ± 1.5 mW, which resulted in an average optical reference power of 675 µW through the multimode fiber at the receiver with ambient turbulence. In order to evaluate the optical performance of the control system, we defined a fade threshold for this signal to be 6 dB below the reference power, to match the signal-to-noise ratio (SNR) most closely to the forward-error-correction (FEC) limit, which is a bit-error-rate (BER) of 3.8 × 10−3 for a PAM2 scheme.
4. Experimental results
4.1 Transmission improvements from the control system
To measure the improvements from the turbulence mitigation scheme, 1244 realizations of the control loop were recorded at each of the turbulence strengths in Table 1. There was ∼3 ms between realizations. Similarly, 1244 realizations were captured at each turbulence level with the control system off, and the beam locked at a fixed location with flat phase. This fixed location was the position that had the maximum transmission through ambient turbulence. The received optical power at the highest turbulence strength, r0 = 3.8 mm, is shown in Fig. 8.
The red line indicates the fade threshold of 6 dB below the reference power, measured to be 675 µW on average in ambient turbulence conditions (r0 = 400 mm). The fade threshold power was 160 µW. With the control system off (Fig. 8(a)), the average received power was 118 µW. With the control system on (Fig. 8(b)), the average received power improved to 425 µW. The standard deviation for this test was 146 µW, compared to 38 µW at ambient conditions. This increase is likely due to the increased thermal fluctuations of transmission paths in stronger turbulence. In both cases, there were periods below the threshold, but the nature of these realizations is different when the control is on versus off. With control off, a good channel may exist within the annular probing space, but the locked beam state may not coincide with the channel’s state. An example of this is shown in Fig. 9.
Fig. 9. (a) Fixed beam missed channel. (b) PAM2 Eye Diagram, missed channel. (c) Fixed beam hit channel. (d) Eye Diagram, hit channel.
When the stationary beam missed a transmission channel (Fig. 9-(a,b)), the signal was completely lost and indiscernible from the oscilloscope’s noise floor. When the beam hit the channel (Fig. 9-(c,d)), the BER of the signal was similar to the BER of the reference measurements resulting in a similar eye, and the beam was minimally shifted from the detector providing similar power to ambient conditions. This contrasts with the experiment using the control system, shown in Fig. 10.
Fig. 10. (a) Controlled beam deflected too far. (b) Eye diagram when the system had inadequate compensation. (c) Controlled beam hit a strong channel. (d) Eye diagram, hit channel.
Similarly, a controlled beam, when it found a strong channel (Fig. 10-(c,d)), transmitted the 5 Gbit/s signal very well, at a power and SNR close to the reference power level. However, in some instances the beam is deflected further than the control system can correct (Fig. 10-(a,b)) with its limited OAM range of ±15. The eye diagram still shows two discrete voltage levels, but the SNR is too low for the signal to be recovered accurately.
The metric “channel probability” is defined as the percentage of realizations above the fade threshold with the control system on. That is, the control algorithm can find a sufficiently strong transmission channel within its probing space to transmit enough power to be above this threshold, compared to the reference power. Moving the fade threshold changes this metric, shown in Fig. 11.
The red mark shows the channel probability for the defined threshold of -6 dB at about 95%. This curve was generated using the experimental data found in Fig. 8(b). With this threshold value, the average duration of a transmission channel in a given spatial location can be assessed using Fig. 8(a). Using the average duration of continuous realizations above the threshold line, a time of 40 ms is calculated, which can help describe the dynamics of that strength of turbulence (Visualization 3). Using this definition and fade threshold, the data from each turbulence strength was analyzed and the results are summarized in Fig. 12.
Fig. 12. (a) Realizations where power received is above the 6 dB threshold, control versus no control. (b) Increase in recoverability with control system.
Figure 12(a) compares the control system (blue) and a static beam (orange). As the strength of turbulence increases, Fig. 12(b) shows the increased disparity between the number of realizations above the threshold for the control versus the no control cases.
The 5 Gbit/s PAM2 signal tested the viability of the system for improving a communication link. After each decision point in the control loop is chosen, the data is triggered by the controller and the received signal is saved on the oscilloscope. The bit error rate for each realization of each experiment was then computed offline.
Figure 13(a) shows the average BER across all 1244 realizations at each turbulence strength, with the control system on. The orange line shows the FEC limit, at BER of 3.8 × 10−3. Figure 13-(b) shows the percentage of recoverable realizations, where the BER was below the FEC limit, for the controlled communication link vs. non-controlled. At r0 = 3.8 mm, the data was recoverable in over 90% of realizations, which corresponds well to the channel probability metric in Fig. 11.
4.2 Revealing multiple channels
One advantage of using the HOBBIT system architecture is the possibility of sending multiple Gaussian beams at once, each with a different azimuthal position and phase tilt. Figure 14 shows successive realizations where multiple transmission channels exist. Though the current version of the controller cannot apply simultaneous frequencies to the AODs, doing so could create multiple distinct Gaussian spots, which could be directed into these separate channels. By multiplexing these, the bandwidth of the system for a communication link could be increased. Additionally, increasing the OAM probing range from ±15 to ±30 revealed multiple channels existing simultaneously for r0 = 3.8 mm, Fig. 15. Unfortunately, the control system wasn’t applied to this case due to hardware limitations.
Fig. 14. (a-c) Successive realizations showing multiple channels. (d-f) Corresponding eye diagrams. (g) Probing capture. (h,i) Decision points (of (b) and (c) respectively).
If the control hardware could generate waveforms for the AODs on the fly instead of accessing a limited library on the AWG, the deflection compensation could be performed for separable, multiplexed beams in real time. This experiment showed the existence and feasibility of multiple channels. By controlling multiple beams through multiple channels, each with independently modulated data signals, a higher total data rate could be achieved. Additionally, focusing multiple beams into a single receiving aperture could be useful for applications in directed energy.
5. Conclusions
In this paper a probing technique was introduced that uses OAM and azimuthal position about a vortex footprint as a basis along with the rapid switching capabilities of a modified HOBBIT system to quickly measured and analyzed turbulence. With this capability, a control system dynamically corrected the position and phase of a Gaussian beam to greatly improve the transmission of power and recoverability of high data-rate signals through a volume of atmospheric turbulence. By concentrating all the optical power into a beam close to the size of random waveguiding channels within the medium, a limited aperture can be used, increasing efficiency compared to most adaptive optics systems. The fully electronic control system is more reliable than mechanized mirrors, and the rapid switching AODs can be much faster than motors. With the current HOBBIT system setup, the minimal sensing time is about 150 µs, with about 750 µs of delay introduced by the control algorithm. Improved implementation onto an FPGA could reduce this delay, further improving the system’s refresh rate. Additionally, a real-world system could use backscattering or a laser from the receiver to transmitter to perform the feedback. The proposed turbulence mitigation system could benefit the areas of high-speed free space optical communications, directed energy, and sensing systems. Future work will include expansion of the probing space to include a radial dimension, exploiting multiple channels simultaneously, and the extension to the underwater regime. Adding a radial dimension to probing would expand the volume of air being probed, which would likely reveal more and/or stronger channels. This could be implemented using a similar approach demonstrated in [21].
Funding
Office of Naval Research (N00014-20-1-2558).
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.
Supplemental document
See Supplement 1 for supporting content.
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