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Abstract
We investigate, via both experimental measurements and wave-optics computer simulations, the statistical characteristics of the fluctuating intensity, such as the average intensity, the beam wander and the scintillation index, of the Gaussian Schell-model (GSM) beams on passing through double-pass, monostatic turbulence channels with either a retro-reflector (RR) or a flat mirror (FM). Our experimental results reveal that the enhanced backscatter (EBS) gradually weakens as the spatial coherence of the GSM source decreases, and eventually disappears for the sufficiently low source spatial coherence states. The r.m.s beam wander remains practically invariant with the variation of the source coherence width in the range from 0.2 to 6.0 mm both in the case of the RR and the FM, which formed the RR case being much smaller. In addition, it is found that the long-term scintillation index of the untracked beam with the RR is smaller than that with the FM, while the situation is reversed for the short-term scintillation index of the tracked beam. In both cases, the scintillation index decreases as the spatial coherence of the GSM source decreases. The obtained computer simulation results agree reasonably well with the experimental results. In addition, the effects of spatial coherence on statistical characteristics of the GSM beams along a 1 km propagation distance through the double-pass monostatic turbulence are also investigated using wave-optics simulation. We also carry out evaluation and comparison of the intensity probability density functions for the RR and FM cases and for various source coherence states that are of utmost importance for free-space optical communications in retro-reflection modulation regime. In addition, our findings will be beneficial for the development of remote sensing and directed energy laser applications in the presence of air turbulence.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Propagation of a laser beam through a double-passage link embedded in the atmospheric turbulence has been extensively studied over the past several decades due to its important applications in remote sensing for Light Detection And Ranging (LiDAR) systems and in Free Space Optical (FSO) communications with Retro-Reflection Modulation (RRM) setting [1–5]. In general, there exist two configurations for the double-pass links: bistatic and monostatic. In the former case the direct (transmitter-reflector) and echo (reflector-receiver) propagation channels are spatially separated, and, hence, the turbulent fluctuations in two paths are uncorrelated. In the latter case the transmitter and the receiver are spatially co-located (being combined into a transceiver) and either have the same field of view (FOV) or the FOV of the receiver encompasses that of the transmitter. Under such circumstances the statistics of turbulence in the forward and in the backward paths are mutually correlated, giving rise to interference phenomena based on phase conjugation. Perhaps the most remarkable one is that a bright spot (the average intensity within the phase-conjugation area) close to the optical axis forms in the receiver plane, and is referred to as the Enhanced BackScatter (EBS) effect. The atmospheric EBS effect was first predicted in theory by Belenkii and Mironov in 1972 [6] and since then, it was extensively studied both theoretically and experimentally [7–15].
Recently, double-passage problems of laser beams propagating in turbulence raised a renewed interest primarily due to the development of the RRM laser communication technology and the target tracking technology [16–20]. The advantage of the RRM is that it can greatly reduce the pointing requirements and does not need a laser at one end of the link. However, it leads to the increase of the intensity fluctuations near the optical axis in the receiver plane [12,21], hence severely limiting the communication quality. One of the methods to reduce the scintillations was suggested in [22] based on a combination of several techniques including the RR diversity, aperture averaging and the bistatic optical interrogation. The intensity correlations in the retro-reflected beams along the 1.8km double-passage link in atmospheric turbulence were investigated in [23]. The characteristics of the EBS of a Gaussian and vortex beams with various topological charges propagating through a double-passage channel with the RR in lab-scale classical/non-classical atmospheric turbulence were studied in [24–26]. The EBS for the plane and spherical waves has been also recently theoretically predicted to occur in the turbulent ocean [27]. Meanwhile, the wave-optics simulation method was extended and developed for the analysis of the statistical properties of laser beams in the double-passage optical link system [24,28–29]. Very recently, the EBS in polarization characteristics of echo waves was experimentally studied for a double-passage system with a birefringent crystal corner-cube RR [30,31].
However, in all aforementioned studies, the systematic analysis and comparisons of light beams in the double-passage systems were performed only for those generated by sources with complete spatial coherence (see [1] for some partial results). It is well known that the source spatial coherence has a significant effect on the statistical properties of light beams when they propagate in free space and atmospheric turbulence [2,32–36]. It was shown that as compared to their coherent counterparts, the partially coherent beams have the ability to reduce/overcome the turbulence-induced negative effects, resulting in the decrease of the scintillation index, the r.m.s beam wander and the angle of arrival fluctuations [37–43], which makes them an optimal source in the FSO communications and remote sensing. The results of numerous studies made over the past four decades on the single-path propagation of partially coherent beams in the turbulent atmosphere were well documented [2,32–43].
The aim of this paper is to comparatively explore the influence of the source spatial coherence on the statistics of the Gaussian Schell-Model (GSM) beams (the well-known class of partially coherent beams) in the double-passage monostatic turbulence with the RR and the FM. The average intensity, the scintillation index and the beam wander of the GSM beams with different coherence widths, propagating through lab-generated turbulence are experimentally measured in the receiver plane, and the wave optics simulation based on multi-phase screen method is also carried out. The experimental and the simulation results show that the EBS effect manifesting itself in the sharp increase in the average intensity within the small area around the optical axis is observed only in the case of RR, whereas this effect decreases as the coherence width decreases. In both cases, of RR and of FM, the scintillation index decreases with the decrease of coherence width, while the beam wander is almost unaffected by it. We also provide exhaustive information about the probability density functions (PDFs) of the instantaneous intensity, within and outside the EBS area, acquired, to our knowledge, for the first time. In particular we show that the PDF distributions within the EBS area have very long right tails indicating that complete phase conjugation in the direct and the echo channels occurs frequently. This information is indispensable for predicting the RRM communication channel errors.
2. Experimental setup and measured results
Figure 1 shows the experimental setup for propagation of the GSM beams through a double-pass monostatic atmospheric turbulence link with a reflector being either a RR or a FM. A linearly polarized laser beam (λ=632.8nm) is first passed through a beam expander (BE) and reflected by a flat mirror (FM), then impinges on a lens L1 (focal length f1=5cm), mounted on the z-axis transition stage. The focused beam is then scattered by a rotating ground glass disk (RGGD), producing partially coherent light with Gaussian statistics. After a distance of 30cm from the RGGD, a collimating lens (L2) and a Gaussian amplitude filter (GAF) which are used to collimate the partially coherent light and transform the light intensity distribution from a uniform to a Gaussian profile, respectively, are placed in the optical path. The partially coherent beam just after the GAF can be regarded as the GSM source. The cross-spectral density function can be written as
Fig. 1. Schematic diagram of the experimental setup for generation of the GSM source as well as for the measurement of its coherence width and of the intensity statistics of the beam, after its passage through the monostatic turbulence link with either RR or FM.
Once the GSM source is generated, it is first divided into two portions by a beam splitter (BS1). In the reflected path, the beam passes through an image lens (L4) with focal length f4 and arrives at a CCD1 to monitor its coherence width. Both the distance from the GAF to L4 and from L4 to CCD1 are 2f4, ensuring that the coherence width in the CCD1 plane is the same as that in the source plane. The transmitted beam from the BS1 is further divided by another beam splitter (BS2). The transmitted part from the BS2 propagates about 80cm, and then passes in the vicinity (about 1.5cm above) of a 35cm×50cm graphite heating plate (HP) used for generation of thermally-induced turbulence through convection. The HP is heating homogeneously and the temperature is set to be 150℃. In this case, the HP generates a very strong turbulence within the 10cm above the HP. The produced turbulence is almost homogeneous in the central strip region of the HP along the beam propagation direction. After interaction with the HP the beam is reflected by RR or FM placed at 10cm from the HP. The echo beam goes back through the same path as the direct beam, then is reflected and focused by the BS2 and lens L3 with focal length 40cm, respectively, finally arriving at the CCD2 with 1920×1440 square pixels of side length 4.54μm used to measure the beam instantaneous intensity. The distance between L3 and CCD2 is about 38.5cm. In the experiment, the integrated time of the CCD2 is set to be 1.0ms. The average intensity distribution is averaged over 3000 frames collected at a rate of 12.5 frames per second.
At first, we investigate the influence of the spatial coherence of the GSM source on the average intensity distribution at the receiver plane. The width of the GSM source is fixed at σ0=1.0mm. The aperture diameters of both RR and FM are 50mm. In this case, the diameters of the RR or the FM are far larger than that of the beam spot size, which can be regarded as the infinite-size type of reflector. Figure 2 presents the experimental results of the average intensity distributions recorded by the CCD2 for different values of the source spatial coherence width. The first row and the second row are the results with the RR and the FM, respectively. At each spatial coherence width, the intensity distribution is normalized by the maximum intensity obtained with the FM. It is shown in Figs. 2(a)-(f) that the bright spot near the optical axis gradually disappears and the beam profile gradually degenerates to quasi-Gaussian shape as the coherence width decreases, implying that the EBS effect is weakened and then disappears for the low spatial coherence width. The RR used in our experiment is a commercial product (corner-cube reflector) from Thorlabs. It consists of three mutually orthogonal N-BK7 glass plates. When a light beam bounces from such a reflector, the echo beam returns to the original path after going through three total internal reflections, but the beam spot rotates 180 degrees with respective to the propagation axis. It is known that the EBS effect is caused by the interference of the reciprocal ray pairs [2]. These reciprocal pairs experience the identical by magnitude but the opposite by sense optical paths through turbulence. Thus, they are still in phase in the receiver plane, resulting in the constructive interference (bright spot) near the optical axis. However, when the spatial coherence of the light source decreases, the number of the reciprocal ray pairs (rays precisely in phase) in the source plane also decreases due to the fact that the phase of the partially coherent source randomly fluctuates in time. As a consequence, the bright spot disappears for sufficiently low coherence states. In the case of the FM [see Figs. 2(g)–2(i)], the beam profiles almost perfectly preserve the Gaussian shape, regardless of the initial coherence width. The reason is that the instantaneous beam spot in the receiver plane reflected by the FM displays strongly random displacement, different from the case of the RR for which the random tilt phase induced by turbulence is greatly compensated. As a result, the bright spot on the axis washes out, and the average intensity distribution degenerates into Gaussian shape, regardless of the initial coherence width. It also can be seen from Fig. 2 that the difference in the maximum intensity between the RR and the FM cases decreases gradually with the decrease of the spatial coherence width. However, the beam reflected by the RR always keeps higher energy utilization close to the optical axis as compared to that reflected by the FM, although the EBS effect is hard to be observed as the coherence width decreases.
Fig. 2. Experimental results for the average intensity distribution of the GSM beams with different values of coherence widths δ0 with the RR (first row) and the FM (second row).
To illustrate differences in the average intensity distribution of beams reflected by the RR and the FM more clearly, we plot in Fig. 3 the one-dimensional intensity distributions (y=0) with different coherence widths corresponding to the density plots in Fig. 2. The red and blue solid curves are the results for the cases of the RR and the FM, respectively. It is calculated from Fig. 3 that the values of the ratios of the intensity maxima with the RR to that with the FM are 3.25, 2.76, 2.34, 1.89, 1.63 and 1.28, corresponding to the spatial coherence widths of 5.0mm, 3.0mm, 1.2mm, 0.7mm, 0.5mm and 0.2mm. The beam diameter defined as the distance between two 1/e points, is 0.21mm, 0.22mm, 0.23mm, 0.26mm, 0.27mm and 0.44mm in the case of the RR, and 0.32mm, 0.36mm, 0.38mm, 0.40mm, 0.43mm and 0.48mm in the case of the FM. From these data, it can be found that the beam spot size with the RR is generally smaller than that with the FM but the discrepancy becomes smaller as the source spatial coherence width decreases.
Fig. 3. Experimental results of the 1D average intensity distributions (y=0) in the receiver plane with the RR and the FM.
Let us now consider the effects of the spatial coherence width on the beam wander in the receiver plane. In the presence of turbulence, a light beam with a finite size will experience random refractions as it propagates. Over a short time period, the beam spot randomly moves around the optical axis while the beam profile is somewhat distorted by the turbulence. The instantaneous centroid of the beam is therefore randomly displaced. This phenomenon is referred to as beam wander, which is characterized statistically by the variance of the beam spot displacement along the propagation axis.
In the experiment, we first capture N=3000 pictures of the instantaneous intensity distribution of the beam in the receiver plane reflected by the RR or the FM. Each recorded frame can be represented as an intensity matrix with M = N1×N1 pixels. The Cartesian coordinates of the instantaneous beam centroid of the beam in picture n are calculated by the formulas
Fig. 4. Experimental results of the r.m.s. beam wander as a function of the coherence width of the GSM beams.
The intensity fluctuations in the receiver plane resulting from optical wave propagation through the atmospheric turbulence are commonly described as scintillations. The degree of scintillation is usually termed the scintillation index is defined by expression [2]
Figure 5 illustrates the scintillation index for the untracked case and tracked case. One can see that the values of the scintillation index in the untracked case are much larger than those in the tracked case, irrespective of the RR or the FM setting. It is especially true when the initial coherence width is larger than 1.0 mm. This is because in the untracked case the measured point is fixed in the receiver plane. The physical mechanism of the variation of scintillation index at the centroid point with the initial coherence width can be explained as follows: with the increase of the coherence width, the beam spot size in the receiver plane decreases, and therefore the whole beam spot may occasionally move out of the detected point, leading to a noticeable increase of the intensity fluctuations at that point. It also can be found that the scintillation index of the centroid point of the average intensity distribution with the RR is smaller than that with the FM. The reason is the same as that we have stated previously: the beam wander effect is greatly suppressed with the RR setting. However, in the tracked case, the situation is reversed [see Fig. 5(b)]: we observe that the scintillation index reflected by the FM is generally smaller. This could be due to the higher-order phase aberrations in the monostatic turbulence with the RR being stronger than those with FM. The behaviors of the variation of scintillation index at the maximum intensity point with the initial spatial coherence may originate from the actual intensity fluctuations.
Fig. 5. Scintillation index (a) at the centroid point of the average beam profile in the receiver plane(untracked beam case); (b) at the maximum intensity point in the receiver plane(tracked beam case).
3. Simulation procedures and results
To further support our experimental results, the computer simulation using the multi-phase screen method has been performed. In this method, an optical wave propagates through a path along which random phase screens obeying appropriate turbulence statistics are placed with equal separation distances. Between the adjacent phase screens, the optical path is free space (no turbulence). In such a process, the extended random medium is described as a series of thin random phase screens. Figure 6 shows the schematic diagram for propagation scenario of a laser beam in double-passage link using multi-phase screen method. The double-passage link is shown unfolded with respect to the reflection plane where the RR or the FM is placed. In order to simulate a monostatic channel, it is required that the pairs of phase screens located on both sides of the reflector at equal distances from it must be the same. When a light beam propagates through the final phase screen, i.e., RPS1, it is focused by a lens with focal length 40cm and then arrives to the receiver plane. The distance from the Lens to the receiver plane is 38.5cm, corresponding to the experimental set. The phases of the screens in the echo paths are negative with respect to those in the corresponding direct paths. One can flip the phases for each of the return screens or, flip the field once at the reflector as we do it (see explanation below).
Fig. 6. Schematic diagram for the computer simulation of a beam propagating in the monostatic turbulent channel. RPS: random phase screens.
In the simulation, the size of the random phase screen is 40mm×40mm with a 256×256 grid. The propagation distance of a single path is set to be L=1.4m and the total of 10 phase screens are applied. The half of the screens is on the left side of the reflector and the other half is in the right side of the reflector. Note that on both sides, the phase screens with same sub-index are identical. In order to simulate the propagation scenario in the experiment, the first three phase screens in the left side and the last three phase screens in the right side, i.e., RPS1, RPS2 and RPS3, are free of turbulence. Only the phase screens RPS4 and RPS5 assume the random phase fluctuations. When the beam arrives at the lens L, a two-step fast Fourier transform (FFT) algorithm between the lens plane and the receiver plane is applied. The screen size in the receiver plane is 2 mm×2 mm with a 256×256 grid. The power spectral density of turbulence is described by the von Kármán spectrum [2]
In the process of simulation, three characteristic time scales must be taken into account for measuring the statistics of partially coherent beams propagating in the atmospheric turbulence: the characteristic time of the source field fluctuations τs, the characteristic time of turbulent medium fluctuations τa and the integration time of the detector τd. In the majority of the previous publications, the measurement is limited to the “slow detector”, which indicates that the integrated time of detector is much slower than that of the source field fluctuations, but is much faster than that of the turbulence fluctuations. Thus, the detector only responds to the intensity fluctuations induced by turbulence, while the information about the source fluctuations is lost. Under this circumstance, the simulation procedure is as follows: (1). 10 phase screens of turbulence are first synthesized and separated with equal distances in the double passage link. (2). A number of K1=500 complex (both amplitude and phase) screens are then synthesized for simulating the GSM source using the approach discussed in [39,44]. For each realization, the beam propagates from the source plane to the receiver plane using the multi-step propagation algorithm. When the reflector is the RR, the electric field is flipped in the reflecting plane, i.e., ${E_r}({x,y} )= {E_i}({ - x, - y} ),$ while the reflector is FM, no action is taken. (3). The K1 realizations of the fluctuating intensity in the receiver plane are averaged out: the result constituting one realization of the GSM beam intensity. (4). Cycling over steps (1)-(3) K2=500 times is performed and finally the statistics of the fluctuating intensity are computed.
Figure 7 presents the simulation results of the average intensity distribution of the GSM beams with different source coherence widths in the receiver plane, reflected by the RR [Figs. 7(a)–7(f)] and FM [Figs. 7(g)–7(i)]. For each source coherence width, the distribution is normalized by the intensity maxima obtained with the FM. One can see that as the coherence width decreases, the beam spot size becomes large, while the EBS effect is weakened gradually due to de-coherence effect in the source plane. This result agrees well with the experimental results shown in Fig. 2. Compared to the experimental results, the intensity distributions display much more regular profiles, especially for the case of high coherence [see in Fig. 2(a) and Fig. 7(a)]. In Fig. 2(a), the beam pattern shows the six beamlets and a bright spot in the center. This is because the RR used in the experiment is the corner cube reflector. Three edges will cause the three line defects in the intensity distribution of the reflected beam. On the other hand, in the simulation, we only rotate the beam pattern by 180 degrees, no such defects are involved. Figure 8 shows the cross line of intensity distribution of the beam corresponding to Fig. 7. It can be clearly seen that the EBS effect decreases with the decrease of the source coherence width. The discrepancy of the intensity maxima between the case of the RR and the FM becomes small as the coherence width decreases.
Fig. 7. Simulation results of the average intensity of the GSM beams with different source coherence widths in the receiver plane. The reflectors in the first and in the second rows are the RR and the FM, respectively.
Fig. 8. Simulation results of the average y-slice intensity profiles in the receiver plane obtained with the RR and the FM.
Figures 9 and 10 illustrate the simulation results for the r.m.s beam wander and the scintillation index varying with the initial spatial coherence width of the GSM beams, respectively. It can be seen from Fig. 9 that the r.m.s beam wander with the RR is much smaller than that with the FM, which agrees with the experimental results (see Fig. 4). It is evident from the simulation results of the scintillation index of the centroid point (untracked beam case) and of the intensity maxima (tracked beam case) that in the former case the scintillation index of the centroid point with the RR is indeed smaller than that with the FM, while the situation is reversed in the latter case. These results also agree with the experimental results (see Fig. 5). However, according to the experiment, the scintillation index of the intensity maxima seems to be independent of the spatial coherence width if it is larger than 1.5 mm, while from the simulation results, the scintillation index decreases monotonically as the coherence width decreases. The reasons behind the differences between the experiment and simulation are: the actual power spectrum of turbulence in the experiment is unknown, while the von Kármán spectrum is used in the simulation. In addition, in experiment the polarization state of the echo beam changes on interaction with the RR and the polarization states in six beamlets of the echo beam are different. In the simulation, we only flip the optical field in horizontal and vertical directions. As was shown some time ago, the scintillation in index of a partially polarized beam is smaller than that of a completely polarized one [45]. We also carry out the simulation (not shown here) when the outer scale is L0=1.0 m. The results show that the r.m.s beam wander and scintillation index are almost the same those with L0=0.5 m.
Fig. 9. Simulation results of the variance of beam centroid (beam wander) as a function of initial coherence width in the receiver plane. Rectangular dots: reflected by FM; circular dots: reflected by RR.
Fig. 10. Simulation results for dependence of the scintillation index of the GSM beams in the receiver plane with the source coherence width. (a) the centroid point (untracked beam case), (b) the intensity maxima (tracked beam case).
Let us now consider a propagation scenario for the outdoor atmospheric turbulence. Under this circumstance, the single propagation distance from the transmitter to the reflector is set to be L = 500 m. The total of 10 phase screens are applied. Each phase screen represents the random phase fluctuations induced by the turbulence along propagation distance Δz. To obtain the long-distance propagation, the beam size in this simulation is expanded 5 times compared to that for the tabletop simulation. Thus, the beam width σ0=5.0 mm, and the coherence width is also enlarged 5 times. The structure constant is chosen to be $C_n^2 = 5 \times {10^{ - 14}}{m^{ - 2/3}}.$ The size of the random phase screen is 250 mm×250 mm with a 256×256 grid, and the screen size in the receiver plane is 15 mm×15 mm with a 256×256 grid. In the reflector plane, the hard aperture with diameter D=50 mm is embedded, which gives the size of the reflector. The other simulation parameters are the same with those used in the tabletop simulation.
Figure 11 illustrates the density plots of the GSM beam profiles with different initial coherence widths in the receiver plane reflected by the RR (first row) and the FM (second row). For each coherence width, the beam profiles are normalized by the maximum intensity for the FM setting. For comparison, the corresponding 1D beam profiles at y = 0 are presented in Fig. 12. As expected, the beam spot size increases and the EBS effect gradually weakens as the spatial coherence decreases. When the spatial coherence width is δ0=1.0 m, it can be seen that there is a bright diffraction ring near the edge of the beam spot in both the RR and the FM settings. This is because in the case of δ0=1.0 m, the beam spot size in the reflector plane far exceeds the aperture size (D=50 mm) of the FM or the RR. The beam spot is truncated by the finite aperture size of the RR and FM, and only a portion of the beam spot is reflected. As a result, a bright diffraction ring is formed in the receiver plane. From Fig. 12, one can see that it still produces a high-intensity spot near the optical axis in the RR setting for long propagation distance although it gradually decreases with the decrease of source coherence width. The calculated values of the ratio of the intensity maxima for the RR setting to that for the FM setting are 1.69, 1.39, 1.2 and 1.06 for δ0=25 m, 9.0 mm, 3.5 mm and 1.0 mm.
Fig. 11. Simulation results of the average intensity distributions of the GSM beams in the receiver plane with four different source coherence widths for 1.0 km propagation distance. The reflectors in the first and in the second rows are the RR and the FM, respectively.
Fig. 12. The one-dimensional average intensity profiles at y = 0 corresponding to the density plot beam profiles in Fig. 11.
Figure 13(a) shows the r.m.s beam wander of the GSM beams in the receiver plane as a function of spatial coherence width. The dependences of the scintillation index at the centroid point and at the intensity maxima point on the spatial coherence width are illustrated in Fig. 13(b) and Fig. 13(c), respectively. As shown in Fig. 13, the r.m.s beam wander with the RR setting is always smaller than that with the FM setting, as the coherence width varies. The r.m.s beam wander is almost independent from the spatial coherence width for δ0>10.0 mm, whereas it decreases with the decrease of the spatial coherence width for δ0<10.0 mm. This phenomenon is somewhat different from that shown in Fig. 9 for tabletop propagation case. The reason (also stated in section 2) is that along the long propagation distance, the beam spot expands much faster for lower coherence widths. It is known that the r.m.s beam wander is inversely proportional to one sixth power of beam width [2]. Thus, it will lead to the decrease of the r.m.s beam wander when the coherence width is sufficiently low. From Figs. 13(b)–13(c), it can be seen that the variations of the scintillation index between the RR setting and the FM setting are almost the same, irrespectively whether considered at the centroid point or at the intensity maxima point. When the propagation distance is sufficiently long, the beam wander is not the main factor for increasing the scintillation index since the beam spot size in the receiver plane is much larger than that of the r.m.s beam wander. As a consequence, the scintillation index values obtained for the two settings are almost the same. However, the value of the scintillation index decreases always as the spatial coherence decreases.
Fig. 13. (a) Simulation of results of the variation of the r.m.s beam wander of GSM beams with the spatial coherence width in the receiver plane. (b)-(c) The dependence of the scintillation index of GSM beams on the spatial coherence width, (b) the centroid point (untracked beam case), (c) the intensity maxima (tracked beam case).
4. Probability density function of intensity
In this section we perform the detailed analysis and comparison of the measured and simulated Probability Density Function (PDF) of the laser beam’s intensity after its propagation through the double-pass links whether with the RR or with the FM. To our expertise, such results have not been obtained earlier, either for coherent or spatially partially coherent light sources. The knowledge of this distribution is crucial for predicting the quality of the Free-Space Optical (FSO) communication channels (in the retro-communication setup, i.e., from the RR/FM to beam-splitter), since it enters the important error statistics, for instance the Bit-Error-Rates (BER) [2].
Figures 14(a)–14(e) presents the experimental results of the PDF of intensity fluctuations as a function of the normalized intensity within the EBS area for different values of the initial coherence width in the case of the RR. For comparison, the corresponding simulation results are shown in Figs. 14(f)–14(j). For each subfigure, the intensity is normalized by the mean value of the sampled intensities (N=3000 for experiment and N=500 for simulation). The diameter of the EBS area is chosen to be 0.07 mm in both experiment and simulation (as shown in Fig. 2(a), the distance between two local minima points is about 0.07 mm). It can be seen in Figs. 14(a)–14(e) that the PDF in the EBS area is closely dependent on the coherence width of the source. With the increase of the coherence width, the PDF changes from the Gaussian distribution [Fig. 14(a)] to the negative exponential. The probability of the nearly zero intensity gradually increases, meanwhile the probability of intensities exceeding the average value by at least two times increases, as the coherence width increases. In other words, the probability of the occurring constructive interference in the EBS area increases with increase of source coherence. The calculated values for the PDF $I > 2\overline I$ from the experiment are 0, 3.97%, 9%, 13.74% and 14.44% corresponding to coherence widths of 0.2 mm, 0.5 mm, 0.7 mm, 1.2 mm and 5.0 mm, respectively.
Fig. 14. (a)-(e) Experimental results of the intensity PDF against the normalized intensity within the EBS areas in the case of RR. (f)-(j) the corresponding simulation results.
From the simulation results, one can see that when the coherence width is 0.2 mm, the PDF is nearly Gaussian, which is consistent with the experimental results. However, the differences between the simulation and experiment become large as the coherence width increases. The reason for the differences may be that in the experiment the RR is the corner cube reflector (commercial product from Thorlabs). The behavior of the corner point is like a “singular” point when the light beam is reflected, i.e., the light beam is spilt into six parts and the intensity near the on-axis/EBS area is nearly zero without turbulence. In the presence of turbulence, the EBS in such an area occurs, but the PDF of nearly zero intensity is still larger. In the simulation, we just rotate the beam 180 degrees to simulate the RR, and therefore no corner point exists. As a result, the probability of nearly zero intensity is always very small in the presence of turbulence. When the coherence width is sufficient low, the intensity in the on-axis area increases (this null intensity areas disappears) due to the diffraction effects of low spatial coherence. Thus, one can see that the probability of nearly zero intensity for low coherence case [Fig. 14(a)] is zero.
Figure 15 shows the experimental and simulation results of the intensity PDF as a function of the normalized intensity outside the EBS area. It is shown that the PDF displays the tails at the left side of the main peak in the experiment. This intensity dispersion slightly broadens as the coherence width increases, implying that the scintillation index increases. The simulation results agree with the experimental results, but the intensity dispersion is much smaller for δ0=0.2 mm and 0.5 mm. Note that the PDF distribution roughly conforms to the Gaussian profile.
Fig. 15. (a)-(e) Experimental results of the intensity PDF of the beams with different coherence widths, outside of the EBS area. (f)-(j) the corresponding simulation results.
Figures 16 and 17 illustrate the experimental and simulation results of the PDF intensity in the case of the FM varying with the normalized intensity inside and outside of the EBS area, respectively. Within the EBS area, the intensity dispersion becomes more pronounced as the source coherence width increases, except for the case of δ0=0.2 mm, in which the PDFs for other coherence widths are all concentrated around the zero intensity. Being different from the case of the RR, the larger intensity dispersion for the FM may come from the appreciable beam wander in the receiver, not from the constructive interference. The calculated values for the PDF $I > 2\overline I$ within the on-axis area are about 6.45%, 14.94%, 13.47%, 14.20% and 16.01% for δ0=0.2 mm, 0.5 mm, 0.7 mm, 1.2 mm and 5.0 mm, respectively. The PDFs of intensity outside the EBS area are all distributed nearby the average intensity, indicating the low values of the scintillation index in this area. As shown in Figs. 17(a)–17(b), this distribution displays the long left tail.
Fig. 16. (a)-(e). Experimental results of the intensity PDF within the EBS area for different coherence widths in the case of FM. (f)-(j). The corresponding simulation results.
Fig. 17. (a)-(e). Experimental results of the intensity PDF outside the EBS area for different coherence widths in the case of the FM. (f)-(j). The corresponding simulation results.
On concluding this section we note that it is clear for the FM case that only for some values of coherence widths (such as 5 mm), the experimental results agree with the simulation results. Several reasons may cause the discrepancy. One possible reason is that the scintillation index in the simulation and the experiment in the case of the FM is very different (compare Figs. 5 and 10), i.e., the scintillation in the experiment is far larger than that in the simulation. However, the general trends for the PDF are the same between the experiment and the simulation.
5. Conclusion
As a brief summary, we have experimentally and numerically examined the average intensity distributions, the beam wander, the scintillation index and the intensity PDF of the GSM beams propagation in double-pass monostatic atmospheric turbulence channels, in the most possible comprehensive manner. Use of two kinds of reflectors, the infinite-size RR and FM, was studied comparatively. Our experimental and numerical results show that the spatial coherence of the beam has significant effects on the average intensity distribution and the scintillation index, whereas the r.m.s beam wander is almost independent on the coherence widths in our detected range from 0.2 mm to 6.0 mm. The EBS effect (only observed in the average intensity distribution reflected by the RR) gradually weakens as the coherence width of the source decreases, and completely disappears for the sufficiently low source coherence. The scintillation index at the centroid point of the average intensity (untracked beam) in the case of the RR is smaller than that in the case of the FM, but the situation is reversed when one observes it at intensity maxima (tracked beam). While the propagation distance from the transmitter to the reflector is 0.5km, the long-term or short-term scintillation index of the beam reflected by the RR and the FM for the same source spatial coherence width is almost the same, and the scintillation index decreases as the source spatial coherence width decreases. The effects of the spatial coherence on the EBS are similar with those obtained in the indoor laboratory experiment. In addition, the experimental results for the intensity probability density reveal that for both, the RR and the FM, the negative exponential profiles with a very long right tails (large intensities) occur in the EBS area for more coherent sources; as source coherence decreases the profile gradually transforms to Gamma-like (FM case) and even Gaussian-like (RR case). The intensity PDFs are very different outside of the EBS area, where, instead, the left tail can become longer. The general trends in simulation results for the PDF agree to those obtained in experiment, while the discrepancies are understood as coming from the physical devices used being different from the approaches taken in the simulation procedures. Our results will be of practical use in optimizing the FSO communication links and in interpreting the returns of the remote sensing systems, such as LIDARS.
Funding
National Key Research and Development Program of China (2019YFA0705000); National Natural Science Foundation of China (11525418, 11874046, 11947240, 11974218, 91750201); Innovation Group of Jinan (2018GXRC010); Priority Academic Program Development of Jiangsu Higher Education Institutions; Qinglan Project of Jiangsu Province of China.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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