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Abstract
The convex partially coherent beam (CPCB) is a special type of nonuniformly correlated beam with a convex-shaped complex degree of coherence (DoC) distributions. Previously our research has illustrated the potential of CPCBs with super-Gaussian DoCs in free-space optical communications (FSOC), mainly manifested as self-focusing which can be transferred into extra scintillation reduction and SNR gain. In this study, the effects of the DoC transition slopes are analyzed and more details about the turbulence propagation of CPCBs with super-Gaussian shaped DoC are revealed. By means of wave optics simulation, the longitudinal intensity evolution of the CPCB is explored, showing that the DoC slope has a profound influence on the self-focusing features such as the focusing plane and the peak intensity. Aperture scintillation and mean SNR at the receiver end of some short-range vertical turbulent links are numerically computed. The obtained results show that, with CPCBs, an ~2 dB SNR gain can be achieved as compared to conventional Gaussian Schell-modal (GSM) beams. However, CPCBs are preferred only in shorter links, which is found to be relevant to the power-in-the-bucket of the receiving aperture. Furthermore, the impacts of the ratio of the source coherence time to the detector integration time are investigated, implying that the CPCB is less susceptible than the GSM. We have also examined the off-axis scintillation of the CPCB. Due to its convex-shaped DoC, the CPCB has significantly reduced off-axis scintillation, which can be beneficial in the presence of pointing errors.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As a dependable complement to radio frequency (RF) technology, free-space optical communications (FSOC) has seen a rapid advancement in the last couple of decades [1–3]. Owing to the unregulated spectrum resource, large modulation bandwidth, high power efficiency, and other advantages, FSOC is becoming popular in many application scenarios like base station inter-connecting [4], satellite communications [5,6] and temporary/ emergency communications [7], etc. Despite the promising aspects, atmospheric turbulence has proved to be a major challenge for the availability and reliability of FSOC links, as it disrupts the wavefront and thereby giving rise to irradiance flux fluctuations (also known as optical scintillation) and other forms of degradations.
Effective means to mitigate turbulence effects include spatial diversity [3] and some frequently used techniques in RF systems, such as channel equalization and error correction coding [8]. For an optical beam, another available degree of freedom lies in the spatial modulation of the source field, for instance, beam shaping and coherence manipulation.
As the raw output of most single-mode lasers, the Gaussian beam is the commonly used light source in FSOC. Still, noticeable performance enhancement is possible via appropriate spatial modulations. According to Schell’s theorem, the diffractive intensity pattern can be shaped by properly controlling the complex degree of spatial coherence (hereinafter referred to as DoC) across the source plane [9], based on which, arbitrary far-field beam shaping can be done accordingly [10–12]. Previous research has indicated that in certain circumstances scintillation resistance is enhanced by use of flat-topped beams [13], Bessel-Gaussian beams [14], and annular beams [15], only to name a few. Alternatively, it is also convenient to produce uniform partial coherence with a rotating diffuser or dynamic phase screens [16,17]. Indeed, the Van Cittert-Zernike theorem provides a way to produce desired DoC distribution by adjusting the source intensity distribution [9]. Using this theorem, various Schell-model beams with nonconventional correlation functions have been generated and analyzed [18,19].
Partially coherent beams (PCBs), such as the well-established GSM beam, trade coherence to mitigate scintillation [20–22]. More specifically, the randomization of the initial optical phases help to average out the slower-varying wavefront distortions induced by the turbulence, whereas the cost is extra beam spreading [23–25]. Actually, a fully incoherent source would experience minimum scintillation, but its poor directionality can result in very low SNR. Therefore it is preferable if the optical beam can concentrate better even when its spatial coherence is decreased. There is also a prerequisite for the PCBs to be effective in turbulence mitigation: the source coherence time ought to be much shorter than the integration time of the detector, also known as “source aperture averaging” [20,26].
Meanwhile, taking into account the fact that at the receiver plane the scintillation level increases with the distance from the nominal beam center, it is natural to consider decreasing the coherence on the edge of the beam while maintaining a higher coherence in the central region. If the overall state of coherence remains the same, intuitively, a better balance between scintillation reduction and beam directionality might be achieved with this type of convex DoC distributions. We have verified this concept preliminarily, showing that imposing a super-Gaussian shaped DoC mask to a deterministic source field will induce self-focusing phenomenon during propagation, leading to a reduced scintillation and thus benefit the overall SNR [27,28]. The underlying reason is that with better power concentration, not only will the receiver aperture averaging effects be more efficient, but also more optical power is able to be collected.
PCBs with a convex DoC, which will be referred to as CPCBs, fall in the category of nonuniformly correlated (NUC) beams. Ever since the establishment of the existence condition of a genuine cross-spectral density (CSD) function [29], NUC beams have attracted much research interest because of their extraordinary propagation properties [30–32]. Experimentally, a spatial light modulator (SLM)-based investigation reveals that a beam with Gaussian DoC tends to focus in propagation while a beam with inverse Gaussian DoC tends to defocus [33]. In fact, spatially varying coherence can be produced in a more intuitive manner: if we modulate a uniformly correlated diffuser or phase screens with a deterministic spatial attenuation function, then nonuniform spatial coherence can be obtained with the complementary DoC shape [27].
Despite its feasibility and tractability, to date the role of DoC modulation in FSOC has not drawn much attention. In this paper, we take this issue a step forward by investigating the effects of different convex DoC profiles, which are actually a class of super-Gaussian functions with different transition slopes at the edge where the DoC changes from fully coherent to partially coherent. With the numerical method of wave optics simulation (WOS), the intensity evolutions of the CPCB as a function of propagation distance are compared along with the Gaussian beam and the GSM beam. Also, area scintillation as well as mean SNR at the receiver plane in vertical atmospheric links will be examined, where the turbulence effects are implemented with unevenly distributed phase screens for improved computational efficiency. Moreover, the coherence time of the source is varied relatively to the detector’s integration time interval, so that we can see how the CPCBs perform under insufficient source aperture averaging.
The remaining part of the paper proceeds as follows: first, we establish the concepts of the CPCB based on DoC modulation process, followed by demonstrations of how the intensity evolves with the propagation distance, i.e. the self-focusing phenomenon. Effects of DoC profiles with different transition slopes are then observed. Then in Section 3, turbulence propagation performance of the CPCB is investigated in a series of short-range vertical atmospheric links. Not only are on-axis scintillation results simulated, but also off-axis scintillation, the latter corresponding to misaligning cases. Furthermore, the effects of insufficient source aperture averaging on CPCB and GSM are compared.
2. Self-focusing of convex partially coherent beams
Focusing can be a favorable attribute for a beam propagating in atmospheric turbulence, because through focusing the wavefront perturbations that the beam takes up during propagation can be better averaged out by a fixed-size receiving aperture [34]. This is also the reason why certain deterministic fields, e.g. the focusing Gaussian beam and the super-Gaussian beam [35], are resilient to turbulence induced scintillation. It is found that partially coherent random fields generated by convex DoC modulations can also stimulate self-focusing phenomena and thereby improving the reception performance [27,28].
Partial coherence refers to the statistics of the random phase φ(r) of an optical field with some certain deterministic amplitude profile U0(r). To date the most extensively studied partially coherent source is the Gaussian Schell-model (GSM) beam which, if assumed to be scalar and quasi-monochromatic, is characterized by the position-independent cross-spectral density (CSD) function:
where the dependence on frequency has been dropped for brevity, r is the spatial position vector, and S = and μ0 denote the spectral density and the complex degree of coherence, respectively. I0 is the on-axis intensity, ω0 is the characteristic width of the Gaussian amplitude, and lc0 is the transverse coherence length.2.1 Description of the CPCB
Although the Wigner distribution function is considered a more comprehensive tool to describe spatially varying coherence, CSD is as capable in our case of CPCBs because they are generated essentially by spatial domain modulations. In fact, the statistics of the initial phase of CPCBs can be completely defined with a spatially invariant CSD and a spatial modulation function a(r). Assuming that the random phase is created by an SLM loaded with GSM phase screens φ0(r), whose Fourier domain representation is the convolution of a zero-mean, circular complex Gaussian random matrix △ with a Gaussian filter function F(f) that invokes spatial correlation. If the standard deviation σr of △ is much larger than the width parameter σf of F(f), Fourier transforming the convolution △F(f) to spatial domain will approximately lead to an GSM DoC as is in Eq. (1) [12,17]:
where the angle brackets denote ensemble average. Upon comparing with Eq. (1), it is obvious that the coherence length of the obtained uniformly-correlated phase screen is. Then, after applying a spatial modulation a(r) to φ(r), for any small local area where a(r) = a we would haveThis represents how DoC modulation is carried out. In particular, if a(r) is taken as an inverse super-Gaussian function, then a CPCB which has the complementary super-Gaussian DoC distribution is obtained as a result, as shown in Fig. 1.
Fig. 1 The principle of CPCB generation process. a(x): cross-section of the spatial modulation function; μ0: original DoC of the coherent beam; μ: DoC of the beam after DoC modulation.
The coherence state of CPCB is now not able to be characterized by the coherence length since lc0 has become position dependent on a(r), so for the sake of convenience it is necessary to introduce another scalar parameter to specify the overall coherence state. Considering the fact that CPCB is generated by modulating GSM phase screens (which can be called basis phase screens, or BPS), it is plausible to define the DoC modulation index as
where A is the effective area of the launching aperture. On condition that a(r) = a0 where a0 is a constant, we have β = a0, corresponding to the uniformly correlated GSM beam whose coherence length is lc0/a0. If the modulation function a(r) is limited to be an attenuation function, i.e. 0 ≤ a(r) ≤ 1, then β∈[0, 1], where β = 0 corresponds to the fully coherent case when a(r) = 0 while β = 1 corresponds to the GSM beam produced by the unmodulated BPS, i.e. a(r) = 1. Therefore, we can think of the DoC modulation index β as a connection between a CPCB and its GSM counterpart. In other words, reasonable comparisons can be made between a CPCB and a GSM beam with the same value of β [27].2.2 Self-focusing of CPCB in free space
In view of the mathematical complexity of the propagation of CPCBs in atmospheric turbulence, we use the WOS method to investigate the problem of interest. WOS is essentially the FFT-based cascaded evaluation of the Fresnel diffraction integral among a multitude of intermediate planes distributed between the transmitter and the receiver. For each interval there is
where m + 1 is the total number of discrete planes (including the source plane and the receiver plane), Ui (ri) is the optical field at the ith plane, λ is the wavelength, k = 2π/λ is the spatial frequency and L is the whole path length. For a more detailed description, such as the sampling related considerations, the reader is referred to [36] and [37].It is known that focusing effects lead to a smaller footprint after the beam gets distorted by the turbulence, so that the receiver aperture averaging can be more efficient. In addition, the power delivering efficiency is also increased. The term self-focusing originally referred to a non-linear process in the presence of gradient refractive index, whereas here we generalize it to describe the autonomous focusing phenomena of beams propagating in free space. Deterministic fields with certain initial phase can lead to focusing, such as a focusing Gaussian beam. High order super-Gaussian beams are also able to self-focus during propagation [35]. Interestingly, if we apply a super-Gaussian DoC to a collimated Gaussian beam, the on-axis intensity evolution of the resultant CPCB will behave somewhat like a super-Gaussian beam.
The generation of super-Gaussian DoC profiles requires that the corresponding phase-screen attenuation function a(r) complementary in shape. Without loss of generality the coherent laser source that illuminates the SLM is assumed to be a Gaussian beam with characteristic width ω0, and the inverse super-Gaussian modulation functions are chosen as
where N is the scaling factor and d determines the DoC transition slope for a fixed β. For instance when β = 0.5, the corresponding DoC profiles with different values of d are plotted in Fig. 2, where the Gaussian amplitude of the field is also shown for reference. Additionally, the inverse Gaussian function can be taken as a limiting case of Eq. (6) when the slope is slowest. For brevity, in the following text we will use the acronym SGd-CPCB to denote the CPCBs whose DoC distribution are determined by Eq. (6) with different values of d, and G-CPCB the CPCB whose DoC is Gaussian distribution. The source power is unchanged for all the beams involved. The beam size of the coherent source in the following simulations is chosen as ω0 = 25 mm and the wavelength is set at λ = 532 nm, which are reasonable values in practical links and comply with the design of our own FSO transmitter.By modulating uniformly correlated GSM phase screens with the concave spatial functions in Fig. 2, and then illuminating the phase screens with a coherent source, we can generate PCBs with convex DoC, i.e. CPCBs. In the rest of this paper, unless otherwise stated, the coherent optical source is assumed to emit a collimated Gaussian beam with unit amplitude and ω0 = 25 mm, and the BPS is a series of GSM phase screens with coherence length lc0 = 5 mm. Figure 3 shows the numerical results of the on-axis intensity evolution of various CPCBs as a function of propagation distance when β is fixed to 0.5. Also shown for comparison is the case of a coherent Gaussian beam as well as a GSM beam with l0 = 10 mm (whose equivalent β is also 0.5).
It can be seen that as the DoC slope decreases (SG1 → SG2→ SG4), the focusing plane forms nearer to the transmitter, and the peak is higher. The CPCB with Gaussian DoC acts somewhat differently, whose focusing peak is lower than the SG4-CPCB. In addition, the farther the focusing peak lies, the greater the on-axis intensity is in the far field. Most importantly, the on-axis intensity of the GSM beam is surpassed by all CPCBs almost from the very beginning of the propagation.
3-D plots of the intensity evolution of GSM, SG1-CPCB and G-CPCB are given in Figs. 4(a)–4(c), which are all of the same β value 0.5. The intensity of the GSM monotonically decreases with propagation distance, while SG1-CPCB will experience some oscillations before the final focusing peak is established, which is quite similar to the coherent super-Gaussian beam in Fig. 4(d). After the oscillating outset, the evolution trend becomes rather smooth. As for the G-CPCB, there is no oscillation before focusing and the beam evolves much like a focused Gaussian beam.
Fig. 4 Evolution of the beam profile with propagation distance for (a) GSM beam, (b) SG1-CPCB, (c) G-CPCB, (d) coherent super-Gaussian beam.
3. Performance evaluation of CPCB in vertical turbulence channels
Whereas the CPCB has proved effective in scintillation mitigation in horizontal links [27], its performance in slant turbulent paths remains to be explored. In fact, slant paths are supposed to be more heuristic since the turbulence is not evenly distributed along the path. Rather, because of the Earth's gravity, turbulence near the ground can be several orders of magnitude stronger than that at high altitudes. Hence, in the uplink the turbulence distorts the wavefront of the beam mainly near the ground station transmitter, while in the downlink the beam emitted by an aerial platform is not greatly distorted until it approaches the receiver. In this section we will evaluate the performance of the CPCBs in the uplink as well as in the downlink in terms of scintillation and SNR.
Here we adopt the widely used H-V5-7 turbulence profile to model the altitude-dependent turbulence strength, by which the refractive index structure constant is expressed as [38]
where h is the altitude in meters, is the structure constant at the ground level which determines the turbulence strength up to 1 km, and w is the rms pseudo windspeed taken to be 21 m/s. For better simulation efficiency, the vertical path under consideration is modeled with unevenly distributed turbulence phase screens. In order to partition the whole path into intervals according to turbulence effects, a simple, empirical approach is applied by letting the integral of in each interval be a constant, which is written aswhere hi and hi+1 denote lower bound and upper bound of the height interval among which the turbulence effects can be represented by a single phase screen and C0 = is the whole path integral. An example of Eq. (8) is given in Fig. 5, where the partition boundaries hi are calculated from the ground level h0 = 0 to H = 20 km.Fig. 5 Partition boundaries for 21 unevenly distributed phase-screen representation of slant path turbulence effects under H-V5-7 turbulence model.
The partition boundaries having been determined, in each interval the turbulence is then assumed to be homogeneous and thus can be represented by a single phase screen, which is generated by Fourier transforming the randomized phase power spectrum density (PSD) of the turbulence into spatial domain:
where f is a vector in the spatial-frequency domain, F−1 denotes an inverse Fourier transform, RG(f) is an uncorrelated random spatial-frequency signal of zero-mean, unit-variance Gaussian distribution, and Φψ(f) is the phase PSD. Adopting the modified von Kármán spectrum for the refractive-index fluctuations ([20], Chapter 3), we can reformulate the isotropic phase PSD aswhere fm = 0.92/l0, f0 = 1/L0 and l0 and L0 are the turbulence inner scale and the outer scale, respectively. Then, we can insert the phase screen into the Fresnel integral in Eq. (5), as inIt should be pointed out that the phase screen is placed at the entry plane of each interval, i.e., phase distortion occurs before every single step of propagation. Now with the numerical model developed above we can compute the received optical field in slant or vertical turbulence channels. For the scintillation index at the receiving plane, by definition it is the irradiance flux variance of a finite receiver aperture of diameter D [20]:in which I(r) is the intensity of a pixel in the sampling grid of the receiver plane with position vector r, D is the diameter and S = πD2/4 is the area of the receiving aperture, PS is the collected optical power in the presence of turbulence. Then the expected SNR can be express as [20,27]where PS0 and SNR0 are the received optical power and signal-to-noise ratio in the absence of turbulence deteriorations. From Eq. (13), it is clear that once PS0 and SNR0 are fixed, the only possible means to improve is to increase or to reduce , which can be achieved simultaneously by the use of CPCB.3.1 On-axis reception performance
The DoC distribution affects the beam propagation substantially, not only in intensity evolution, but also in turbulence resilience. Now we will evaluate and compare the performance of CPCBs in vertical paths through WOS results of scintillation and SNR, with both the uplink and the downlink taken into account.
The coherent source and the BPS masks are identical to those described in Section 2.2. The propagation path is a vertical H-V5-7 link that spans from the ground level (h = 0 m) to a certain altitude H, and the turbulence is parameterized by = 1 × 10−14 m−2/3, l0 = 5 mm and L0 = 5 m [39,40]. For better sampling at the low-frequency range of the turbulence PSD, a subharmonic method is implemented ([37], Chapter 9). The sampling grid is a 512 × 512 matrix with 1 mm interval at the source plane, while for the receiver plane and the intermediate planes, the sampling interval linearly grows with their distance from the source, and 21 planes are used in total for the split-step propagation simulation. For a receiver aperture of 20 mm in diameter, the free space SNR, calculated by the transmission of a fully coherent Gaussian beam in an equal-distance and turbulence-free link, is fixed to SNR0 = 20 dB. It should be reiterated that the launching optical power at the source plane is constant as a result of the same coherent illuminating beam, and the only difference between various beams is their DoC modulation. Besides, to make sure that the partial coherence takes effect, a “slow” detector is presumed at the receiver end, where the propagation results of 60 frames of PCB masks are averaged within the integration time of the detector.
In Fig. 6 the uplink on-axis simulation results of the CPCBs and the GSM beams are plotted as a function of the DoC modulation index β. The altitudes of the vertical link under consideration is H = 1 km, 2 km and 5 km. For each data point in the figure, an ensemble of 500 random realizations of turbulence phase screens is computed. Before we dig into the results, it should be made clear that, for instance in Fig. 6(a), since the maximum possible SNR is achieved with a β value that lies somewhere between 0 and 1 (which are shown in separate markers), neither the fully coherent beam (β = 0) nor the BPS GSM beam (β = 1) is optimal. Rather, a nonuniform CPCB should be a better choice. It is seen from Figs. 6(a)–6(c) that the lowest scintillation that the GSM is able to achieve is obviously greater than the CPCBs. Concerning the mean SNR, in the 1 km uplink as shown in Fig. 6(d), the best of GSM is 15 dB when β = 0.4, while SG1-CPCB can hit 17 dB with β = 0.6. Regarding other CPCBs with slower DoC slopes, the optimal β value decreases and so does the maximum SNR gain. As the propagation distance increases, the extra SNR gain from the SG1-CPCB drops to ~1 dB for 2 km and virtually none for 5 km. One possible reason for that might be the fixed source size ω0 that we use for different distances in the simulation, which is a crucial parameter and should be optimized more carefully in practice [41]. Nonetheless, trying to demonstrate how the reception performance varies with distance, we have dodged the issue of Fresnel number optimization here.
Fig. 6 On-axis scintillation and the corresponding mean SNR of (a, d) H = 1 km, (b, e) H = 2 km and (c, f) H = 5 km uplinks.
The downlink simulation corresponding to Fig. 6 is presented in Fig. 7, where the source and link parameters remain the same except that the beam propagates downward. First of all we find that the downlink experiences less scintillation and thus the maximum achievable SNR is 3~4 dB higher than the uplink. In the 1 km and 2 km cases, it is again seen that a 2 dB SNR gain is obtained by use of SG1-CPCB. By comparison, G-CPCB is less efficient in the downlink: although G-CPCB has smaller scintillation and higher SNR than the GSM for most β values, the best performance they can achieve is almost the same (at small β values). Also note that for the 5 km downlink in Figs. 7(c) and 7(f), unlike that in Figs. 6(c) and 6(f), the fully coherent beam seems to become the best choice. This might be explained in this way: since the majority of the turbulence appears near the downlink receiver, before interacting with the turbulence the PCB would have evolved well enough that the locally coherent beamlets are too large to average out the turbulence phase distortions. In that case, the intensity fluctuations introduced by the PCB itself would only add onto the absolute receiver scintillation, and that’s why a fully coherent beam with no additional source fluctuations and best power concentration turns out to be the optimal source.
Fig. 7 On-axis scintillation and the corresponding mean SNR of (a, d) 1 km, (b, e) 2 km and (c, f) 5 km downlinks.
To better interpret the results in Fig. 6 and Fig. 7, let us examine the “power-in-the-bucket” (PinB) of the receiver, which is the beam power encompassed by the 20 mm receiving aperture. In Fig. 8, where the vertical coordinate is the normalized PinB in units of optical power, we can find that in short links, especially the 1 km uplink and downlink, the PinB of the CPCBs can be significantly higher than that of the GSM. However, as the propagation distance increases, that PinB margin gradually disappears. This might help explain why the CPCBs can only have advantage over the GSM in relatively short links.
Fig. 8 Power-in-the-bucket as a function of DoC modulation index β of (a) 1 km uplink, (b) 2 km uplink, (c) 5 km (uplink), (d) 1 km downlink, (e) 2 km downlink, (f) 5 km downlink.
From the results in Figs. 6–8, it is found that the SG1-CPCB generally outperforms the other CPCBs whose DoC transition slopes descend more gently. Although this might be inaccurate for certain circumstances that are not considered above, we will only present the results of SG1-CPCB for clearer comparison with the GSM beam.
3.2 Impacts of source aperture averaging
In order for the PCBs to effectively mitigate turbulence induced fluctuations, there is a prerequisite: the source coherence time should be much shorter than the detector response time, otherwise the use of PCBs will only result in more scintillation (Ref [20], Chap. 16). This is because the random initial phase introduced by the partial coherence is itself a source of perturbation, and only by adequate time averaging at the receiver end can this perturbation be averaged out (which is called “source aperture averaging” [26]), together with the perturbations caused by the turbulence. Nonetheless, due to the speed limitations of the DoC modulation techniques so far, the ratio of source coherence time τT to detector integration time τR cannot be very high. Let us denote that time ratio by ρ = τT / τR, which is 60 in the simulation carried out in Section 3.1. In the practice of FSOC, ρ = 60 can be a demanding value, so in this section, we will quantify the performance deterioration of GSM and SG1-CPCB when ρ drops from 60 to 30 and 15.
In the uplink the effects of time ratio ρ on the scintillation and the SNR are shown in Fig. 9. The first thing to notice is that as β increases, the difference between the three cases becomes larger, either for CPCB or GSM. This is easy to interpret because a coherent beam requires no source aperture averaging at all, while less coherent beams do need enough detector integration time to get averaged. As regards the comparison between the two different beam types, we can see that as ρ decreases, the deterioration of GSM is worse than that of SG1-CPCB. Note that in the 5 km case, when ρ = 60 the maximum SNR of GSM (β = 0.2) and the CPCB is almost the same (~11.2 dB). However, when ρ drops to 15, the SNR of GSM drops to 9.4 dB. In contrast, the SNR of the CPCB remains nearly unchanged (10.9 dB), bringing about 1.5 dB gain in SNR.
Fig. 9 Scintillation and SNR versus modulation depth for the uplinks: (a, d) 1 km, (b, e) 2 km and (c, f) 5 km
The downlink results are given in Fig. 10. Unfortunately, since the scintillation is quite moderate as compared to the uplink, the fully coherent beam has become the secondary optimal beam source in the 1 km and the 2 km links, and indeed the optimal source option in the 5 km link. Given the fact that fully coherence needs no source aperture averaging, the advantage that the CPCB enjoys in 1 km and 2 km cases is undermined when ρ decreases.
Fig. 10 Scintillation and SNR versus modulation depth for the downlinks: (a, d) 1 km, (b, e) 2 km and (c, f) 5 km
3.3 Off-axis reception performance
As mentioned earlier in this paper, the original motivation of devising the CPCB is trying to reduce the off-axis scintillation while keeping the beam from spreading too much. If the off-axis scintillation of the CPCB can indeed be reduced, then the link will be less affected by pointing errors when the receiver lies somewhere away from the nominal beam center.
Figure 11 depicts the scintillation index of SG1-CPCB and GSM for a 20 mm receiver aperture deviating from the nominal beam center by 50 mm. Considering the fact that the off-axis SNR can be quite low, we have increased the on-axis free-space SNR0 from 20 dB to 40 dB. It is then observed that in both the uplink and the downlink, the lowest possible off-axis scintillation of SG1-CPCB is much lower than that of GSM. Thus it is safe to say that the convex DoC does help to reduce off-axis scintillation to a great extent. When it comes to the corresponding SNR, larger performance improvements are achieved from CPCB in comparison with the on-axis cases in Sections 3.1 and 3.2. In the downlink, as indicated in Figs. 11(b) and 11(d), the scintillation/SNR of GSM is even worse than the fully coherent beam, while the CPCB can boost the SNR by more than 5 dB. This can be crucial for practical FSOC links where pointing errors frequently occur. With CPCB, the requirements for fast tracking would be more relaxed.
4. Conclusions
By use of wave optics simulation, as well as a vertical atmospheric channel partition method for more efficient phase-screen representation, we have carried out a detailed investigation of the turbulence propagation of convex partially coherent beams. After examining the effects of various super-Gaussian DoC profiles, we find that the CPCB with the steepest slope outperforms its counterparts, and an extra 2 dB SNR gain over GSM can be achieved. When the source coherence time is short enough, the advantage of CPCB is only obvious in shorter links, whereas if the source aperture averaging is insufficient, the CPCB with convex DoC shows more robustness and will regain an advantage over GSM in longer uplinks. Moreover, in case of misalignments, the CPCB exhibits lower off-axis scintillations than uniformly correlated GSM beams, indicating better resilience to pointing errors.
It is also worth mentioning that the improvements gained from the use of CPCB come with negligible extra cost when compared to a conventional GSM source, since only constant spatial modulation (for SLM loaded phase screens) or physical polishing (for rotating ground glass method) is needed to impose the convex DoC. Although a systematic understanding of how the nonuniformly correlated PCBs interact with the turbulence is still lacking, it is hoped that this study will inspire more insights into this issue and be helpful to the optimization of future FSOC systems.
Funding
Innovation Foundation of Shenzhen City [2017] No. 131.
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