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Abstract
We study a dynamic scheme for optimizing the beam waist in low earth orbit-to-geostationary orbit (LEO-to-GEO) laser communication for small satellites under the influence of the pointing error on the transmitting side. Using the Marcum Q-function, we express the channel gain of the link in a closed form. Based on this expression, we obtain an accurate approximation of the optimal value of the laser beam waist via both numerical and analytical approaches to achieve the minimal average bit error probability (ABEP) of the LEO-to-GEO communication. The analytical approximation has a simple form and can be straightforwardly calculated and employed in practice. We numerically compute the ABEP with our analytical solution and the sub-optimal solution proposed in the literature. The results show a strong agreement between our solution and previous work.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Recently, the demand for high-speed and real-time communication for Earth-observing small satellite missions has increased rapidly. For instance, 100-kg-class satellites can now be equipped with synthetic aperture radar (SAR) technology to acquire very-high-resolution images of the Earth at any time and in any weather condition [1]. Using this SAR technique, small satellites generate significant amounts of data that must be sent to ground quickly to free up their memories and gain efficiency in utilizing the data. However, traditional satellite-to-ground communication methods using radio frequency have faced problems of bandwidth congestion and delay in data delivery. Therefore, a laser communication relay from a low-earth-orbit (LEO) small satellite (at an altitude of less than 2000 km) to a geostationary (GEO) satellite (at $\approx$ 36,000 km) shows potential as an alternative to resolve these problems. Because the GEO satellite is almost stationary with respect to the ground and covers a wide range, it is an ideal relay node for a small satellite in LEO to relay a considerable amount of data to a ground station in real time.
In LEO-to-GEO laser communication, the effect of the atmosphere on communication performance is insignificant. Therefore, the main factor that affects the average bit error probability (ABEP), an important performance metric, is the pointing error caused by the LEO satellite’s vibration [2,3]. Under the effect of pointing errors, several studies, including fixed-beam and dynamic-beam methods, have been conducted to optimize the laser beam waist to achieve the best communication performance, that is, to minimize the ABEP.
In the long-term fixed-beam-waist optimization approach, the ABEP is numerically evaluated over a range of different beam waist values $\omega _{0}$, and the optimal one that minimizes the ABEP is chosen [4–6]. However, this method requires the probability density function (pdf) of the pointing error, which is significantly difficult to acquire in operational conditions. In addition, the numerical calculation in the fixed-beam method requires a computational workload and is practically inefficient.
Compared with the fixed-beam-waist method, the dynamic-beam-waist optimization approach has some advantages, for example, it does not require knowledge of the pdf of the pointing error and is computationally efficient. In the dynamic-beam scheme, the pointing error on the transmitting side can be measured instantly [7,8]. Then, the laser beam waist is adapted rapidly and optimally to minimize the ABEP using several beam control mechanisms [9–11]. In [12,13], the authors derived approximate solutions for the dynamic optimum beam waist under the effect of the pointing error. Although these approximations yield ABEPs close to the exact values, reasonable factors underlying these good results are not clear.
In this study, we adopt the dynamic-beam optimization scheme to minimize the ABEP for communication from a LEO small satellite to its GEO counterpart. By exploiting a closed-form expression of the channel gain in terms of the Marcum Q-function, we derive an equation whose solution is the optimal value of the laser beam waist. The equation contains a ratio of two modified Bessel functions of the first kind. We then numerically and analytically solve the equation to obtain the optimal value of the laser beam waist. In the numerical approach, we employ a fast algorithm to calculate the ratio of the modified Bessel functions to solve the equation with the Newton method. In the analytical approach, by exploiting a geometrical structure regarding the lower and upper bounds of the ratio of the modified Bessel functions, we obtain an approximation of the optimal beam waist with an extremely high precision. The obtained solution has a closed form and can be easily calculated in real time. Unlike previous approaches, it is transparent from a geometrical perspective why our analytical solution can accurately approximate the optimal one. As will be shown later, the ABEPs obtained herein with both analytical and numerical approaches are highly consistent. Under a practical parameter setting for LEO-to-GEO communication, we calculate the ABEP with our solution and compare it with that obtained with the sub-optimal solution in [12]. The result shows a strong agreement between our solution and previous work. We also discuss a simplification of the sub-optimal solution that does not significantly affect the minimal ABEP.
The remainder of this paper is organized as follows. In Section 2, we introduce the analytical expression of ABEP. A closed form of the channel gain using the Marcum Q-function is also presented. Section 3 introduces our numerical and analytical approaches to determine the optimum laser beam waist to achieve the minimal ABEP. A comparison between the optimal and sub-optimal solutions proposed in the literature is discussed in Section 4. A simplification of the sub-optimal solution that does not affect the ABEP is also given in this section. Section 5 concludes the paper.
2. Expressions of the ABEP and channel gain
2.1 Expression of the ABEP
When a LEO small satellite sends data to its GEO counterpart via a laser link, the pointing error caused by the LEO satellite’s vibration is the dominant factor affecting communication performance because the effect of the atmosphere is almost negligible. The ABEP of the communication channel over all of the radial pointing errors ${\theta }$, which are assumed to follow the Rayleigh distribution [14], is expressed as
Here, ${p_{\theta }(u)}$ is the pdf of the pointing error ${\theta }$, and $P(e|{\theta })$ is the instantaneous BEP conditioned on a given value of the pointing error, given by where $Q(\cdot )$ is a Gaussian Q-function. $\textrm {SNR}$ is the instantaneous signal-to-noise ratio on the receiving side and calculated as2.2 Expression of the channel gain using the Marcum Q-function
In LEO-to-GEO laser communication, Fig. 1 illustrates the spot size $\omega _z$ of the Gaussian laser beam transmitted from the LEO small satellite at receiver aperture A on the GEO terminal. This is related to the beam waist $\omega _{0}$ as follows:
where $\lambda$ is the wavelength of the transmitted laser beam.The channel gain, which is the fraction of power collected by aperture A in Fig. 1, can be expressed as Here, $I(x^{2}+y^{2};\omega _z)$ is the spatial distribution of the Gaussian beam intensity at distance $d$ from the transmitting source and given by
Fig. 1. Spot size of the transmitted laser beam at receiver aperture A. $r$ is the radius of the aperture, and $z$ is the radial displacement caused by the pointing error $\theta$. Square A’, which has side lengths $2s$, is used to approximate circular aperture A to derive the sub-optimal beam waist in [12]
In this study, without approximating the receiver aperture’s shape, we directly solve Eq. (11) using the relationship in Eq. (9), that is,
3. Solution of the dynamic optimum beam waist
3.1 Numerical approach
For the sake of simplicity, we set $a\equiv 2z/\omega _z$ and $b\equiv 2r/\omega _z$. Solving Eq. (15) is equivalent to finding the solution to the equation:
Following the chain rule, the left-hand side of Eq. (16) can be expressed asThe numerical computations are performed with the parameters presented in Table 1 throughout the paper. The distance is 42,000 km, which can be considered the longest range for the case of LEO-to-GEO communication. The transmitted power was set to 5 W. The receiver aperture radius is 7.5 cm, in reference to the case of the High-speed Communication with Advanced Laser Instrument in the Engineering Test Satellite IX (a GEO satellite), currently under development at the National Institute of Information and Communications Technology, Japan [20]. The 500-Mbps rate is conservative but sufficient for data transmission because of the long communication duration between the two satellites. According to our estimated link budget, the standard pointing jitter with a scale $\sigma$ of 8 $\mu \textrm{rad}$ satisfies the critical pointing requirement of LEO-to-GEO communication. Using Eqs. (1), (2), and (3), and $\omega _{0.\rm opt}$ obtained by numerically solving Eq. (21), we can calculate the ABEP for the dynamic beam optimization approach. The result is plotted as a black dashed line in Fig. 2. To demonstrate the effectiveness of the dynamic-beam method, we also present the ABEP against beam waist curve (blue line) obtained using the fixed-beam-waist optimization method. In this method, the ABEP for each fixed beam waist $\omega _0$ is obtained by numerically calculating the integration in Eq. (1) over all values of $\theta$, whose pdf is assumed to follow the Rayleigh distribution. Then, over a range of $\omega _0$, the one that minimizes the ABEP is the value of the optimal beam waist. Unlike the fixed-beam optimization method, the dynamic one does not evaluate the ABEP over all possible values of $\theta$. Instead, based on the pointing error $\theta$ measured at each instant, it calculates and rapidly adapts the optimal beam waist. As a result, the dynamic-beam scheme exhibits a constant value of ABEP-dependency on $\omega _0$, as shown in Fig. 2, that is, the ABEP is optimized at each instant. Overall, it improves the communication performance compared with the fixed-beam approach. In the fixed-beam scheme, even when the beam waist is optimized ($\approx$ 12 mm), its performance is approximately 0.878 dB worse than that of the dynamic-beam scheme. Furthermore, when the beam waist does not match the optimal value, the ABEP performance of the fixed-beam method degrades significantly. This is because there exists a tradeoff relationship between the tolerance to the pointing error and the geometric spreading loss. Specifically, on the right of the optimal value, when the beam waist becomes larger, it leads to a smaller spot size. As a result, the compensation for the pointing error becomes worse. However, when the beam waist lies on the left of the optimal one, that is, the spot size is considerably larger than the receiver aperture, the pointing error does not significantly affect the performance. In this case, further reduction of the beam waist will increase the geometric loss and thus lead to a worse ABEP.
3.2 Analytical approach
Although the numerical approach can obtain the optimal value of the laser beam waist to acquire minimal ABEP, it does not provide a closed-form solution, which will be useful for analyses of the system performance regarding the system’s parameters. Next, we solve Eq. (21) analytically to determine a simple optimal solution for the laser beam waist. First, we prove that Eq. (21) has a unique solution. By considering the derivative of the left-hand side of Eq. (21) with respect to $x$, we obtain
It has been proven in [21] that which immediately implies that the derivative of the ratio of the modified Bessel functions is positive for all $x>0$. Therefore, ratio $I_1(x)/I_0(x)$ is a monotonically increasing function. Combining with the relations $I_1(0)/I_0(0)=0$ and $\lim _{x\to \infty }I_1(x)/I_0(x)=\infty$, it is clear that Eq. (21) has only one positive solution.Fig. 2. Comparison of the ABEPs using the fixed-beam optimization method and dynamic-beam optimization method with the numerical optimal solution. The blue solid line represents the ABEP against the beam waist curve for the fixed-beam method, and the black dashed line shows the ABEP obtained using the dynamic-beam scheme.
Table 1. Parameters used in the numerical analysis.
To derive the analytical solution of Eq. (21), we introduce two upper and lower bounds of the ratio of the modified Bessel functions $I_1(x)$ and $I_0(x)$. The ratio of the modified Bessel functions can be expressed in a continued fraction form as [17]
From Eq. (24), it is clear that the ratio is always bounded from above by $x/2$, that is, In addition, according to [22], the ratio has a lower bound as follows: We note that the two bounds in Eqs. (25) and (26) are extremely tight when $x\le 10^{-2}$; further, in the case of LEO-to-GEO transmission, $x=ab$ is typically smaller than $10^{-3}$. To evaluate the tightness of these bounds, we calculated the integral of the absolute difference between the ratio of the modified Bessel functions and the bounds over $x\in [0,10^{-2}]$, $\mathcal {E}_{c}=\int _0^{10^{-2}}|I_1(x)/I_0(x)-f_{c}(x)|dx$, where $c\in \{u,l\}$. After some numerical calculations, we find that $\mathcal {E}_{l}\approx 1.56241\times 10^{-10}$ and $\mathcal {E}_{u}\approx 1.56248\times 10^{-10}$, which empirically verifies the tightness of the two bounds. Therefore, each of these bounds can be used to approximate the ratio $I_1(x)/I_0(x)$ to obtain a solution of Eq. (21), Nevertheless, we show here that using a combination of the two bounds yields a more accurate solution than using each bound alone. Let $x_l$ and $x_u$ be the solutions to Eq. (21) when we approximate ratio $I_1(x)/I_0(x)$ by $f_l(x)$ and $f_u(x)$, respectively. Specifically, we have and The geometrical interpretations of $x_l$ and $x_u$ are shown in Fig. 3(a), in which we plot the ratio of the modified Bessel functions along with the two bounds $f_l(x)$ and $f_u(x)$ over $x\in [10^{-3},10^{-3}+4\times 10^{-10}]$. $x_l$ and $x_u$ are represented by the intersection points between the straight line $y=r/z$ and curves of the lower and upper bounds, respectively. From a geometrical perspective, it is clear that the midpoint of the line segment joining $x_l$ and $x_u$ is almost identical to the solution of Eq. (21), which is represented by the intersection point between line $y=r/z$ and the curve of the ratio of the modified Bessel functions. In other words, a more accurate approximation of the exact solution in Eq. (21) can be obtained as The suitability of the approximate solution in Eq. (29) can be clearly visualized in Fig. 3(a). Substituting the values of $x_l$ and $x_u$ obtained in Eqs. (27) and (28) into Eq. (29) yields Plugging $x_{m}=ab=4zr/\omega _{z}^{2}$ into the left-hand side of Eq. (30), we obtainFig. 3. a) Geometrical visualization of $x_l$, $x_u$, and the midpoint of the line segment joining $x_l$ and $x_u$ as an accurate approximation of the exact optimal beam waist. The ratio of the modified Bessel functions is represented by the green line, while the upper and lower bounds are represented by the blue and orange lines, respectively. b) Dynamic laser beam waist optimization flow for LEO-to-GEO communication.
We note that, unlike in [12], the result of $\omega _{z.\textrm {opt}}$ in Eq. (31) is mathematically derived without approximating the shape of the receiver aperture. The analytical solution has a closed form and can be efficiently calculated in practice. A simple flow to determine the analytical optimal value and instantly adjust the laser beam waist is shown in Fig. 3.
Following the same steps as in Section 3.1, we calculate the ABEP with our analytical solution for the dynamic-beam method. In Fig. 4, the result is plotted along with the ABEP obtained with the numerical solution of the optimal laser beam waist. The ABEPs of the analytical and numerical solutions are depicted by the brown dashed and black dashed lines, respectively. As can be seen, both analytical and numerical solutions achieve the same ABEP, which numerically validates the consistency between these solutions.
Fig. 4. Highly consistent ABEPs obtained with the numerical and analytical optimal solutions in the dynamic-beam scheme. The brown dashed line represents the ABEP obtained with the analytical solution, while the black dashed line shows that obtained with the numerical solution.
Here, we present some remarks regarding the advantages of the dynamic scheme over the fixed one. First, the ABEP obtained with the dynamic scheme is always smaller than that obtained with the fixed scheme, as illustrated in Fig. 4. Second, during the communication phase, system parameters, such as transmitting power and standard pointing jitter, may vary. Because of the simple algebraic form of the analytical solution obtained in Eq. (31), the dynamic optimal beam waist can be efficiently calculated in real time and rapidly adapted to the variation of the system parameters. In contrast, the fixed-beam method is very sensitive to changes in the parameters, that is, the ABEP is significantly degraded when the laser beam waist is not optimized. To recalculate the fixed optimal beam waist, one needs to numerically calculate the ABEPs for a range of beam waist values. This requires a heavy computational cost and prior knowledge of the pdf of the pointing error, which is infeasible in practice.
To illustrate the robustness of the dynamic-beam scheme, we provide ABEPs corresponding to three cases: using the dynamic-beam method with our analytical solution, the fixed-beam method with the optimized beam waist, and the fixed-beam method with the non-optimized beam waist in Fig. 5(a). The standard pointing jitter is changed, while the other parameters are fixed as in Table 1. For the non-optimized case, the beam waist is fixed to $\approx$ 9 mm, which is the optimal value when the standard jitter is 16 $\mu$rad. As can be seen, the dynamic-beam optimization method shows an improvement in communication performance compared with the fixed one with the beam waist optimized. Furthermore, when the waist is not optimized, the ABEP of the fixed-beam scheme is significantly worse. Specifically, when the standard pointing jitter changes from 16 $\mu$rad to 10 $\mu$rad and the beam waist remains at 9 mm, the ABEP performance is approximately 2.88 dB and 3.74 dB worse than those of the fixed-optimized and dynamic-optimized cases, respectively. In Fig. 5(b), we also depict the ABEP performance of the dynamic beam scheme when the transmitting power is varied. The relationship between the ABEP and the transmitting power is shown with the standard jitter $\sigma$ set to be 8, 10, 12, 14, and 16 $\mu$rad. As can be seen in Fig. 5(b), as the jitter increases, the less the transmitting power helps to improve the system performance. Thus, it is important to carefully consider the level of jitter error while optimizing the communication performance by improving other system parameters. We also show the effect of the standard pointing jitter on the outage probability of the system in Fig. 6. The outage probability is defined as the probability that the channel gain becomes smaller than threshold $h_0$, namely, Pr$(h<h_0)$. Notably, when $\sigma$ increases, it is easy to see that the outage probability also increases.
Fig. 5. a) ABEP comparison for three cases of the beam waist when the standard jitter is varied: dynamic-optimized (brown line), fixed-optimized (black line), and fixed-non-optimized (blue line). b) ABEP versus transmitting power curves with different values of the standard pointing jitter $\sigma$.
In summary, we can summarize that it is transparent from our geometrical approach why the analytical solution in Eq. (31) is an accurate approximation for the exact optimal beam waist. In addition, this approach provides insight into deriving closed-form solutions for optimization problems using simple bounds.
4. Sub-optimal laser beam waist
4.1 Evaluation of the sub-optimal laser beam waist
In [12], the authors proposed a method to determine the sub-optimal value of the laser beam waist under the influence of the random pointing error using the dynamic-beam method. This involves solving the equation
To find the analytical solution of Eq. (35), the authors needed to approximate circle receiver aperture A as a square one, A’, as shown in Fig. 1. Then, using the Gaussian Q-function to express the channel gain $h$, the sub-optimal value of $\omega _z$ is given asIn Fig. 7, we plot and compare the ABEPs obtained with the sub-optimal solution in Eq. (36) (green dashed line) and our analytical solution derived in Section 3.2 (brown dashed line).As can be seen, for the case of LEO-to-GEO communication, the difference between the ABEPs obtained with our analytical and sub-optimal solutions is insignificant. Thus, although apparently different, the ABEP obtained with the sub-optimal solution is close to that obtained with the exact one. This result is understandable because when the receiver aperture is small compared with the spot size of the transmitted laser beam, the power-collecting area of the approximated square detector is insignificantly different from that of the circular detector. Therefore, the probabilities that the random point $(X, Y)$ falls in either the circular or approximated-square-shaped aperture are almost the same. This leads to similar results for the minimal ABEP in the two cases.
Fig. 7. Comparison of the minimal ABEP obtained with the analytical optimal (brown dashed line) and sub-optimal (green dashed line) solutions.
4.2 Simplified solution of the sub-optimal value
Although the approximate solution in Eq. (36) is quite simple, the result contains a natural logarithm in the calculation. In the particular case of LEO-to-GEO communication, the sub-optimal solution can be further simplified with an insignificant impact on the minimal ABEP. At a distance of 36,000 km, the value of $\theta d$ fluctuates around 360, which is significantly larger than value $r=s\sqrt {2}$ (i.e., $\theta d \gg r$). Therefore, we can approximate the natural logarithm term in Eq. (36) as
Fig. 8. Negligible difference between the ABEPs obtained with the simplified (purple dashed line) and sub-optimal (green dashed line) solutions. The brown dashed line indicates the ABEP obtained with our analytical solution.
5. Conclusion
In this paper, based on the expression in terms of the Marcum Q-function of the channel gain, we present both numerical and analytical approaches to obtain an accurate approximation of the optimal value of the laser beam waist to achieve the best LEO-to-GEO communication performance. Through a geometrical structure involving two tight bounds of the ratio of two modified Bessel functions, we explicitly show the accuracy of our analytical solution. The analytical optimal beam waist is mathematically derived without any approximation of the receiver aperture’s shape, and it has a simple algebraic form that can be calculated in a straightforward manner. Therefore, it can be employed in practice to dynamically adjust the laser beam waist to minimize the ABEP under the influence of a small LEO satellite’s pointing error. Although our study presents the case of LEO-to-GEO laser communication, the results can be beneficial for future studies on other types of inter-satellite links. Nevertheless, a drawback of the approach using the Marcum Q-function is that the Marcum Q-function is not suited for modeling the communication channels where atmospheric effects are strong, e.g. satellite-to-ground channels. We also compare the ABEPs obtained with our optimal beam waist and the sub-optimal solution proposed in the literature to evaluate its efficiency. Furthermore, we derive a simplification of the sub-optimal solution without considerably affecting the minimal ABEP. Future work related to designing a technical system to realize the dynamic beam waist optimization scheme would be of interest.
Disclosures
The authors declare no conflicts of interest.
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