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In this paper, we propose the idea of dynamic beam waist adjustment for laser inter-satellite communications, and study the performance of this dynamic-beam scheme. The beam waist adjustment is based on continuous detection of the instantaneous pointing error angle, which is performed at the transmitter side. Using a square to approximate the circular detector region, we obtain a closed-form expression for calculating the proportion of power that can be collected by the receiver aperture, and derive a simple algebraic solution for the optimum dynamic beam waist. Due to its simple form, the dynamic beam waist value can be computed in real time at the transmitter, and therefore, the adjustment is practically implementable. It is shown that the performance of laser inter-satellite links with dynamic beam waist is better than that with fixed beam waist.
© 2016 Optical Society of America
1. Introduction
Optical tracking and pointing systems for free-space optical communications suffer from vibration, which causes the pointing error, denoted by θ (rad). This leads to a radial displacement d between the detector center and the beam center, since d is the product of the link distance z and tan θ (≈ θ, as θ is very small), i.e., d = z·tan θ ≈ z·θ. Mechanical engineers have put abundant efforts in designing high-accuracy tracking and pointing systems, i.e., to minimize θ, and so far θ can be restricted at the range of 0 - 100 μrad for satellite applications [1]. However, since the inter-satellite link distance is commonly very large, within this range, inaccurate pointing can still degrade the transmission quality, i.e., increase the bit error probability (BEP). This effect can be reduced by methods such as adapting the transmitter power and telescope gain according to vibration amplitude [2]. These methods are effective but consume more transmission power and involve intensive numerical computation. An alternative, which is to find the optimum laser beam waist that minimizes the BEP, has attracted much interests.
References [3–6] consider to use a long-term fixed beam waist. Since the pointing error is caused by platform vibration, the pointing error angle θ is a time-varying random variable. This fixed beam waist value is chosen such that the system average BEP (ABEP) is minimized. However, it does not respond to current knowledge of the instantaneous pointing error, and therefore cannot provide the best performance for all pointing errors. Since the optimization of the fixed beam waist aims to minimize the ABEP, it requires the accurate statistical distribution of the pointing error which may not be available in practice, and depends on knowledge of the transmit power. In addition, [3–6] do not provide closed-form solutions, and rely on numerical methods and thus consume high computational power.
Inspired by the fact that the pointing error angle θ can be measured at the transmitter side [7, 8] and that the laser beam waist ω0 can be adjusted sufficiently fast [9, 10], in this paper, we propose the idea of dynamic beam waist adjustment for laser inter-satellite communications, and study the performance of this dynamic-beam scheme. The beam waist value ω0 is chosen such that the system BEP is minimized at each instant. Thus, it is adjusted in a timely manner according to the estimated value of the instantaneous pointing error angle θ. Both the statistical distribution of θ and the knowledge of the transmit power are not required, but an additional pointing error measurement device is required at the transmitter side as shown in Fig. 1. The module “High-speed beam waist adjuster” is studied in [9, 10], and the module “High-speed, high-accuracy pointing error measurement device” is studied in [7, 8]. Here, we do not study the design of these two modules. We mainly investigate the acquisition of the optimum instantaneous ω0,opt value (i.e., the module “ω0,opt Calculation”) and the link performance with this dynamic beam adjustment scheme.
As we will show, using a square to approximate the circular detector region, we obtain a closed-form expression for calculating the power proportion that can be collected by the receiver aperture, and derive a simple algebraic expression for computing the optimum dynamic beam waist. Due to the simplicity of this solution, it can be computed efficiently in real time at the transmitter side. Since at each instant the beam waist is optimized, the system with dynamic beam waist performs better than that with a fixed beam waist value and the overall ABEP performance is further improved. We also show that this system with optimum dynamic beam waist is more robust than the fixed beam waist scheme due to the fact that it does not depend on the transmission power, the telescope gain and the statistical distribution of the pointing error angle.
The remainder of this paper is organized as follows. Section 2 introduces the Gaussian beam model. In Section 3, we explain the dynamic beam waist adjustment approach and derive the algebraic-form expression for the optimum beam waist. Section 4 presents the numerical studies, and conclusions are drawn in Section 5.
2. The Gaussian beam model
For general free-space optical systems, the overall channel gain is determined by the geometric spread and the pointing error, the atmospheric turbulence, and the path loss [11–13]. In this paper, we consider the inter-satellite laser link, which is outside the earth’s atmosphere, and thus, is under a turbulence-free environment. Therefore, the overall channel gain depends only on the geometric spread and the pointing error. To study the channel gain denoted by hp, we need to start from the Gaussian beam, for which, the normalized spatial distribution of the intensity at a propagating distance z from the transmitter is given by [14]
where ρ is the radial vector from the beam center, and ωz is the beam radius at which the intensity drops to e−2 of the axial value at the distance z. The beam radius ωz is also referred to as the spot size, and achieves the minimum value ω0 at z = 0, known as the beam waist. The relation between ωz and ω0 is given by [14] where λ is the laser wave length. It should be noted that the Gaussian beam model fails if wave fronts are tilted by more than approximately 0.5 rad, which corresponds to ω0 ≤ 2λ/π [15, P. 630]. This leads to ω0 > 2λ/π as a constraint in finding the optimal beam waist. From Eq. (2), it is observed that the beam radius ωz increases almost linearly with z in the far field, i.e., where , resulting in a cone-shaped beam. Therefore, the divergence angle Dv of the laser beam in the far field is approximated by the ratio of the beam radius ωz and the transmission distance z, which is From Eq. (3), we can see that each of the three variables ω0, ωz and can determine the other two. Thus, adjusting the beam waist is equivalent to adjusting the beam width, adjusting the spot size, or adjusting the divergence angle, which have been discussed in [3–6].Consider a circular optical detector C with radius a located on the received beam plane, as shown in Fig. 2(a). The distance between the center of C and the beam center is the radial displacement d caused by the pointing error, and we have d = tan θ · z ≈ θ · z. Apparently, the fraction of power that detector C can collect is hP. Since it is related to θ, a, ω0 and z, we denote it as hp(θ, a, ω0, z). Obviously, the value of hp(θ, a, ω0, z) can be obtained by performing a double integral over the detector region, i.e.,
where θ, a, ωz, ω0 and λ are all non-negative parameters. Since we have Eq. (2), we use hp(θ, a, ω0, z) and hp(θ, a, ωz, z) interchangeably here.
Fig. 2 (a) Detector and beam footprint with misalignment. (b) A square approximation to the circular detector.
Assuming on-off keying (OOK) modulation is adopted, the instantaneous BEP conditioned on a given value of θ is expressed as [11]
where γ is the instantaneous receiver-side electrical-domain signal-to-noise ratio (SNR), and Q(.) is the Gaussian Q-function and is defined as The instantaneous receiver-side electrical-domain SNR is given by where R is the optical detector responsivity, Pt is the transmit power, Rload is the receiver circuit resistance, N0 is the receiver one-sided thermal noise power spectral density, and Rdata is the data rate.The ABEP over all possible values of θ is thus
where pθ (x) is the probability density function (pdf) of θ. The ABEP of both the fixed-waist scheme and the dynamic-waist scheme, is calculated using Eqs. (5) - (7). The difference between them is as follows. For the fixed-waist scheme, the value of ω0 is set to a fixed value; for the dynamic-waist scheme, the value of ω0 is dynamically adjusted with θ using a method that will be introduced later in this paper. Thus, for the fixed-waist scheme, from Eqs. (5) - (7), we can see that if we keep the values of Pt, N0, a and z unchanged, the system ABEP is a function of ω0. Finding the optimum value of ω0 that can minimize the ABEP is of great interest.References [3–6] studied the optimum fixed beam width with which the system achieves its minimum ABEP. Solving this problem thus involves finding the solution of the equation:
where the solution depends highly on pθ (x). In previous studies, e.g., [6], it is assumed that the pointing error angles in azimuth and elevation are independent and identically Gaussian distributed, resulting in the total pointing error angle θ to be Rayleigh distributed if the bias of pointing errors is zero, and Rician distributed if the bias is non-zero. However, this assumption is only for the ease of analyzing the problem mathematically, and in most cases the pointing error angles in azimuth and elevation are non-identically distributed. Another drawback of this fixed-beam-waist scheme is the difficulty of acquiring the parameter values of pθ (x) accurately. If wrong parameter values are used for optimizing the beam waist, the system will end up using a mismatched beam waist. As we will show later, the fixed beam waist scheme is very sensitive and the system suffers a serious performance degradation when the waist value is mismatched. In addition, [3–6] do not provide closed-form solutions, and therefore, numerically solving Eq. (8) requires a communication terminal with very high computational power and thus may increase the weight and size of the terminal and consume more energy.3. The dynamic beam waist adjustment approach
Recent works in [7, 8] show that the pointing error angle θ can be measured accurately. Therefore, in this section, we derive a simple algorithm for computing, in an on-line real-time manner, the current optimum value of the beam waist ω0 as a function of the instantaneously measured value of the pointing error angle θ, minimizing the conditional BEP Pb(e|θ) given in Eq. (5).
Using the fact that Q(.) is a monotonically decreasing function, to minimize Pb(e|θ) is equivalent to maximizing the instantaneous channel gain hp. Specifically, the problem is formulated as
At this stage, we can see clearly that with the optimum dynamic ω0 value, the receiver can achieve the highest received optical power. This will lead to the best error performance not only for OOK, but also for other modulation formats. Also, the maximization of received power leads to the achievements of the highest channel capacity and the minimum outage probability. Thus, dynamic beam waist adjustment can bring significant performance improvement in terms of error probability, outage probability, and channel capacity.To solve Eq. (9), let us consider two cases:
Case 1: When d < a, the beam should be as narrow as possible, such that the detector collects the highest percentage of power. The ideal case is to concentrate all the optical power on the central axis, resulting in an ultra narrow beam, i.e., ωz = 0, where the detector can collect all the transmit power. Nevertheless, from Eq. (2), we see that ωz = 0 is not achievable. Therefore, we find the minimum value of ωz, by solving the equation d ωz/d ω0 = 0. The optimum ω0 is obtained as , and the corresponding minimum value of ωz is given by
Case 2: When d > a, to find the optimum ω0, we first find the optimum value of ωz that maximizes hp(θ, a, ω0, z), and then derive the optimum ω0 according to Eq. (2). Therefore, we need to solve
Let X and Y denote two independent and identically Gaussian distributed random variables with mean zero and variance . Thus, if we let ρ = (x, y), the joint pdf of X and Y is exactly the same as Eq. (1). Hence, the fraction of power that the detector C can collect, equals the probability that (X, Y) falls in C, i.e., hp(θ, a, ωz, z) = P{(X, Y) falls in C}. Since hp(θ, a, ωz, z) can hardly be expressed in a simple and mathematically tractable form, as shown in Fig. 2(b), we use a square C′, whose side length is , to approximate the circular region C. Thus, hp(θ, a, ωz, z) ≈ P{(X,Y) falls in C′}. Since X and Y are independent and identically distributed, the joint distribution of X and Y is circularly symmetric. By this symmetry, the power that detector C can collect is only related to d, but not related to angle φ, which is shown in Fig. 2(a). To keep this symmetry property, we let the extension line of one of C′s diagonals always cross the origin by rotating C′ by the angle φ, as shown in Fig. 2(b). Therefore, the value of P{(X, Y) falls in C′} remains the same with all values of φ. For ease of analyzing mathematically, we calculate P{(X, Y) falls in C′} at φ = π/4. Thus, the fraction of power that the detector can collect is approximately expressed in terms of the Gaussian Q-function as Here, it should be noted that d must be larger than , which corresponds to the case d > a. Based on Eq. (12), we solve the equation dhp(θ, s, ωz, z)/dωz = 0 and choose the ωz value that satisfies ωz > 0, i.e., If an ωz value that is obtained from Eq. (13) is smaller than ωz.min, it is also non-achievable. In this case, we set ωz to be ωz.min by setting ω0 to be . Summarizing all cases and substituting d = θ · z, we have the overall optimum ωz expression as Furthermore, from the relation between ωz and ω0 in Eq. (2) and the fact that ω0 > 0, we have With z =20,000 km, and θ = 50 μrad, we figure out that the two optimum ω0 values are approximately 8mm and 2km. For satellite platforms, a transmitter aperture size more than 1km is obviously impossible, and thus we discard the larger one given in Eq. (15). We see that the value of decreases monotonically with ωz.opt. Since it has to ensure that ω0 is always larger than 2λ/π, the optimum beam waist ω0.opt is Substituting Eq. (14) into Eq. (16) will give the optimal ω0. Since most mechanical vibrations are at low frequencies (up to 100 Hz) [1], ω0 can be adjusted sufficiently fast using the technology reported in [9,10].We give a brief block diagram of the module “ω0,opt Calculation” in Fig. 3. As we have shown in Fig. 1, a high-speed and high-accuracy pointing error measurement device is required to continuously and precisely acquire the instantaneous pointing error angle θ and feed θ to the module “ω0,opt Calculation”. The module “ω0,opt Calculation” calculates the corresponding beam waist ω0.opt and then feeds the ω0.opt value to the beam waist adjuster. The calculation of ω0.opt is performed in two steps: 1) substituting the values of θ, s and z into Eq. (14) to obtain ωz.opt; 2) substituting ωz.opt and ωz.min into Eq. (16) to obtain the value of ω0.opt. The value of s and ωz.min are constant and are thus easily known at the transmitter side. The link distance z can be calculated according to the instantaneous locations of the transmit terminal and the receive terminal, which can be obtained by checking the known ephemeris data. Since Eq. (14) and Eq. (16) are both simple algebraic expressions where no integrals, differential equations and iterations are involved, the overall computational complexity is very low and thus the dynamic beam waist adjustment scheme we proposed here is practically implementable.
4. Numerical results
In this section, we present and discuss the numerical results. We regard θ as a Rayleigh distributed random variable with the scale parameter σθ, as in [6], for numerical illustration. Quantity σθ is called the standard pointing jitter in this paper. It should be emphasized that the optimum beam waist given in Eq. (16) does not depend on any specific model of θ. Since the accuracy of the satellite tracking and pointing system allows θ to range from 0 to 100 μrad, we thus set σθ = 10 μrad. The lengths of inter-satellite links are commonly between 1,000 km and 80,000 km and we choose z = 20,000 km here. The receiver aperture diameter is set to be 2a = 0.25m; and the carrier wavelength is λ = 1.064μm. The data rate is set to be Rdata = 1Gbps and the transmit power Pt has been limited up to 37 dBm (5 Watts). The parameters are also summarized in Table 1. Using these parameters and Eqs. (5) and (7), we can calculate the system ABEP.
Table 1. Parameter values for numerical results
In Fig. 4, we plot the ABEP versus beam waist curve for the fixed beam waist scheme. The transmit power is set to be Pt = 30dBm, the standard pointing jitter is set to be σθ = 10μrad and the link distance is z = 20,000km. The ABEP values are obtained from numerically calculating Eqs. (5) and (7) with a fixed ω0 value. We can see that the ABEP achieves its minimum with ω0 ≈7mm. When the beam waist shifts away from the optimum value, no matter to the smaller or larger direction, the ABEP increases drastically. For instance, with ω0 = 1mm, 3mm, 9mm and 20mm, the corresponding ABEP is approximately 1/2, 10−3, 10−4, and 2 × 10−2, respectively. A slight mismatch in the beam waist would severely degrade the ABEP from the optimum value. For comparison, we also present the ABEP of the system with our adaptive beam waist scheme in Fig. 4. The ABEP value is obtained from numerically calculating Eqs. (5) and (7) with dynamically-adapted ω0 values calculated using Eq. (16). With this scheme, the beam waist adapts according to the variation of the pointing error θ. We can see that the adaptive beam waist adjustment outperforms its fixed counterpart, even when the fixed beam waist has been optimized.
Fig. 4 ABEP comparison between fixed and dynamic beam waist systems with the transmit power Pt = 30dBm, the link distance z = 20,000km and the standard pointing error σθ = 10μrad.
Since the calculation of the optimum fixed beam waist, i.e., solving Eq. (8), involves the acquisition of the accurate vibration model and intensive computation, in literature, there are still no sufficiently efficient and effective methods to find the real-time optimum fixed beam waist. Thus, we can only find the optimum beam waist for the fixed approach via point-by-point numerical search. Specifically, with the values of Pt, N0, a, z, and σθ unchanged, using Eq. (5) and (7), we evaluate the ABEP over a wide range of different beam waist values and choose the minimum that is achievable. It is reasonable to understand that the optimum fixed beam waist varies with a, z, and σθ. Furthermore, we find that it also varies as the transmit power changes. In Table 2, we list the optimum values of beam waist for different transmit power values with the link distance z = 20,000km and standard pointing jitter σθ = 10μrad.
Table 2. Optimum fixed beam waists with different transmit power values and with σθ =10μrad, and z=20,000 km.
The ABEP versus transmit power curves of our dynamic adjustment approach, the fixed approach with accurately optimized waist value and the fixed approach with mismatched waist value are given in Fig. 5. For the mismatched case, the beam waist is set to be ω0 = 10.41mm, which is the optimum waist value for the first point (24 dBm) of the curve. We can see that our dynamic approach outperforms slightly the fixed approach that operates with an accurately optimized waist value for each transmit power. However, if the waist is not optimized, i.e., for the mismatched case, the system suffers from a severe power loss. In practice, the values of transmit power Pt, the link distance z and the standard pointing jitter σθ may vary during a continuous transmission. An on-line numerical search for the fixed optimum solution increases system complexity, whereas an off-line numerical search requires a huge three-dimensional table storage with small incremental values of parameters. Our simple and closed-form adaptive optimum beam waist scheme is easy to implement on-line.
Furthermore, since the ABEP is obtained by averaging the instantaneous BEP over all possible θ values, it may not indicate the performance difference at most times when the instantaneous pointing error is not severe. Therefore, we plot the instantaneous channel gain corresponding to different values of θ in Fig. 6. We can see that at most times, the hp value of our dynamic approach is several orders higher than that of the fixed one. This means that the communication quality of systems adopting our dynamic beam waist control scheme is much better than that of fixed beam waist. The point where two curves meet in Fig. 6 is where the fixed beam waist happens to be the optimum dynamic value.
Fig. 6 Channel gains with different values of θ for dynamic beam waist adjustment and fixed beam waist system (ω0 = 10.41 mm).
Generally, for a fading channel, the outage probability is P(hp < h∗), where h∗ denotes the benchmark value that is used to define the outage. The outage probability gives a more meaningful performance measure compared to the ABEP. In Fig. 7, we plot the curve of the outage probability versus h∗, which is also the cumulative distribution function (cdf) of hp. We can clearly see from Fig. 7 that for a given h∗ of interest, the outage probability of systems with dynamic beam waist control is much lower than that with fixed beam waist.
Fig. 7 The cdf’s of the channel gains of dynamic beam waist adjustment and fixed beam waist system (ω0 = 10.41 mm).
5. Concluding remarks
In summary, the dynamic beam waist value can be computed efficiently at the transmitter in real time due to its simple, algebraic form and the adjustment is practically implementable. The performance of laser inter-satellite links with dynamic beam waist is much better than that with fixed beam waist.
Since our method does not depend on the transmit power or the telescope gain and the beam waist is adjusted according to the instantaneous values of z and θ (or d), our adaptive scheme still works without any further modification when the transmitter power and telescope gain adaptation proposed in [2] is in use. These two adaptations are introduced from different perspectives and can be adopted simultaneously to achieve a better system performance. For the fixed long-term optimum beam waist adjustment scheme, the optimum waist varies with the transmit power and thus with the telescope gain. Thus, the results cannot be used without further modification when the transmit power and the telescope gain are being adjusted.
Acknowledgments
Singapore Ministry of Education (AcRF Tier 2 Grant MOE2010-T2-1-101); National Natural Science Foundation of China (61302112, 61571316); Ministry of Education of the PRC (SRF for the ROCS, the 47th batch); Qianjiang Talent Project (QJD1402023).
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