Based on the link scenario we have described in Section 2.1, the receive signal vector y = y 1 , y 2 T ∈ C 2 × 1 in equivalent baseband notation is given by y = H x + η , (4) with x = x 1 , x 2 T ∈ C 2 × 1 being the transmit signal vector containing the transmit symbols in a given time slot. A simplified block diagram showing the main building blocks of the transmitter-receiver structure is shown in Figure 4. We assume uncorrelated transmit symbols in time and space. They are arbitrarily chosen from the symbol alphabet A with equal probability of occurrence. Without channel knowledge at the transmitter, the radiated power at each laser is equal, i.e. E x x H = P t / N ⋅ I N , where P _t /N is the radiated power per laser. The vector η = η 1 , η 2 T contains the circularly-symmetric complex Gaussian noise. The noise process is uncorrelated with the transmit symbols x. Therefore, the covariance matrix of η is given by R η = σ 2 η I M , with σ 2 η being the variances of the noise process.

The channel matrix H ∈ C M × N contains the complex channel coefficients between the N lasers and the M receivers. We apply a deterministic LOS channel model for our analysis, and it follows for the entries of matrix H: h m n = a m n ⋅ b m n ⋅ e − j 2 π λ r m n ⋅ e j ξ m n , (5) where h _mn denotes the complex channel coefficient at row m and column n of H. The parameter a m n = λ / 4 π r m n ⋅ e j φ ≈ a = λ / 4 π r 11 ∀ m , n , (6) models the free space propagation loss, where φ stands for the common carrier phase that can be assumed to be zero without loss of generality (w.l.o.g.). The value of a _mn is very similar for all m, n, because the difference between the path lengths is very small compared to their mean total length. Thus, we can apply the approximation a _mn ≈ a in (6) without loss of accuracy.

The parameter b _mn , with 0 < b _mn ≤ 1, denotes the additional amplitude attenuation through the atmosphere. The actual value of this parameter depends on various factors, such as the length of the slant path through the atmosphere, wavelength of the laser and aerosol and molecular constituents (Hemmati, 2006). Its value generally varies during a passage of the Moon. This parameter is important to be considered in the link budget and system design to guarantee a minimum required system availability. Its modeling is treated thoroughly in (Shrestha et al., 2018) and in particular in (International Telecommunication Union, 2015). However, as these additional attenuations affect identically the laser beams of both systems, and as our focus is on the direct comparison of the channel capacity between the MIMO and the single input single-output (SISO) system, it is reasonable to neglect the variation of this parameter in this first comparison study. We, therefore, set b m n = 1 / 2 ∀ m , n , (7) in this model, i.e., we consider a fixed atmospheric loss of 3 dB for the rest of the paper. This simplification does not affect the general results in this paper.

The parameter ξ _mn models the phase variation due to the turbulent atmosphere. Assuming a transmitter-baseline d _s in the order of 1 to 10 m, the separation angle of both transmitters is less than or equal to approximately 10 m/400,000 km = 25 nrad. Both transmitters appear as a single source for each receiver since they cannot be resolved optically by one ground telescope, whose angular resolution typically amounts to more than 1 µrad. Both sources will, thus, perfectly fit inside one isoplanatic patch. The isoplanatic angle (IPA) describes the angular cone of common index-of-refraction turbulence (IRT) inside the atmosphere (Andrews and Phillips, 2005). It is derived from the height-profile of the IRT, and corresponds to around 20 µrad in typical situations, down to a few µrad in stronger turbulence, but is always larger than the separation angle of the transmitters. It is, therefore, reasonable to assume that any influence from IRT will affect both transmitted signals (Tx 1 and Tx 2) at a particular receiver aperture (Rx 1 or Rx 2) equally. Hence, the modeling of the phase variation due to the turbulent atmosphere can be simplified to ξ m n ≈ ξ m ∀ n , (8) i.e. we assume no differential phase-offset between the beams ending at the m-th receiver.

In addition to the parameter ξ _m , a common phase piston variation for both laser beams needs to be considered. Optical phase piston in the atmosphere are discussed in (Conan et al., 1995; Karr, 2007) showing a phase front error of about 1 μrad at 1,064 nm and 1,550 nm. This effect can be modeled as a degradation of the receive signal-to-noise-ratio (SNR) (Fried, 1967), and it will, therefore, be considered in the example link budget in the results section. Applying now (6), (7) and (8) to (5), and assuming M = N = 2, the channel matrix H is then given by H = 1 2 ⋅ a ⋅ e j ξ 1 0 0 e j ξ 2 ⋅ e − j 2 π λ r 11 e − j 2 π λ r 12 e − j 2 π λ r 21 e − j 2 π λ r 22 . (9)

To evaluate the performance of the optical FSO MIMO system, we apply the MIMO channel capacity. The capacity of a MIMO system with no channel knowledge at the transmitter is given by (Telatar, 1999) C = log 2 det I M + ρ ⋅ H H H , (10) where ρ = P t / N ⋅ σ 2 η is the radiated transmit power per laser to the noise power at the input of the receiver. Please note that the phase shifts ξ _m due to the turbulent atmosphere do not affect the channel capacity. In fact, a constant phase shift in one row or column of H does not change the eigenvalues of the matrix.

The MIMO channel capacity is maximum if all eigenvalues of the channel transfer matrix H are equal. In this case, the MIMO channel forms min M , N eigenmodes and up to min M , N parallel data streams can be transmitted over the channel. The maximum MIMO channel capacity is given by (Telatar, 1999) C o p t = min M , N ⋅ log 2 1 + ρ ⋅ max M , N ⋅ a 2 . (11)

The channel capacity of an equivalent SISO FSO laser link is given by C S I S O = log 2 1 + P t / σ 2 η ⋅ a 2 . (12)

Comparing (12) with (11) and assuming a symmetric MIMO system with M = N antennas, the capacity gain of MIMO compared to SISO becomes evident, which is C o p t = M ⋅ C S I S O . (13)

Please note that in this comparison throughout the paper the total radiated transmit power has been normalized by the number of lasers N. Hence, the total receive SNR per optical receiver of the MIMO system equals that of the equivalent SISO system. We will use C and C S I S O for the analysis in the results section.

For the following analysis we assume the simulation parameters as summarized in Table 1. The two-element laser ULA is located on the Moon’s prime meridian on the equator and is rotated by −40° with respect to the Moon equator plane. The lasers are separated by d _s = 5 m and radiating with 0.5 W each at 1,064 nm. The two receiving telescopes are at ϕ _E = 28° N, θ _E = 16.5° W (a location on Tenerife, Spain) and have a spacing of d _E = 47 m. As an example, and without loss of generality, we consider the night from 26th to April 27, 2021.

Figure 5 shows the Moon’s elevation and azimuth angle as seen from the optical ground station (OGS) location on Earth (upper plot) as well as the distance R _o between the arrays (lower plot). The elevation angle at zenith is almost 50°, and the transmitter-receiver distance varies between approximately 351,600 to 354,400 km for an elevation angle not less than 20°. Please note again that this Moon orbit during this particular night and the location of the OGS have been chosen as an example. Other parameters are possible too, and they would lead to similar results.

Table 2 shows an example link budget based on the transmitter-receiver design as discussed in Section 2.1. At an elevation angle of 30°, the Tx-Rx-array distance is approximately R _o = 354,000 km (please refer to Figure 5 again) leading to a free space propagation loss of 10 ⁡ log 10 a − 2 = 312.4 dB. The total available optical transmit power of P _t = 1 W and typical losses inside a transmitter terminal of 1 dB produce an intensity of 133 nW/m² at the receivers after lunar distance. Please remind that, for a fair comparison, we assume the same available transmit power for both the SISO system and the MIMO system. Therefore, each of the two MIMO lasers provide 0.5 W while the laser of the SISO system provide 1 W transmit power. Presuming approximately 10 ⁡ log 10 b − 2 = 3 dB total atmospheric losses (including scintillation and attenuation losses), and another 2 dB loss at the useful signal, since part of the incoming light needs to be used for spacecraft-tracking and a wavefront-sensor, a received power of approximately 13 nW is achieved by a 1 m-diameter receiver telescope. We assume fixed atmospheric losses for the following analysis that are independent on the elevation angle. This simplification is valid in our analysis, because our focus is on the comparison between SISO and MIMO and both the SISO and the MIMO link are similarly affected by these losses. A detailed estimation of effects from atmospheric influence has been shown in the literature (Andrews and Phillips, 2005; Giggenbach and Moll, 2017) and its repetition here would deviate from the actual goal of the paper.

The MIMO channel capacity C is shown in Figure 6 for different orientation angles δ _E of the receiving ULA (upper plot). For comparison purposes, the SISO capacity C S I S O and the maximum capacity C o p t according to (12) and (11) are provided as well. As can be observed from the curves, the capacity of the MIMO FSO communication system outperforms the equivalent SISO system in all cases. Even the lowest possible MIMO capacity is still higher than the SISO capacity although the same total transmit power of P _t = 1 W have been used for both systems. The reason is that the MIMO system still exhibits an SNR gain of 3 dB due to the usage of two receiver telescopes instead of only one telescope like in the SISO case.

Moreover, in order to analyze the dependence of the MIMO capacity on the geometry between both ULAs, three orientation angles of the Rx ULA from 0° to 90° are simulated. The curves clearly show the dependence of the MIMO capacity on the geometry between the two ULAs. Since the orientation of the Tx array with respect to the receiver array is changing over time as the Moon passes by, the capacity is varying and C o p t cannot be achieved at all times. The achievable capacity and its stability over time depend on the initial parameter setup in terms of array orientation angle and antenna spacing. The parameter setup corresponding to the blue curve (δ _E = 0°) shows the highest capacity during the time of interest (in which the elevation angle is at least 20°, see Figure 5 again) between about 21 h and 5 h. A capacity of close to C o p t = 8.8 b/s/Hz is achieved for a comparably long period of around 4 h during the night from 26th to 27th of April, and it never drops below 8 b/s/Hz as long as the elevation angle is larger than 20°.

For all three curves the corresponding absolute value of the array reduction factor a r f according to (1) is shown (lower plot of Figure 6). For δ _E = 0° (blue curve), the two arrays are broadside to each other at around 23 h in the evening of April 26th. At that time, the distance between the ULAs is approximately R _o = 353 ,000 km (see the lower plot of Figure 5 again). Using (2), the minimum optimal antenna spacing on Earth would, therefore, be d _E = 353,000 km ⋅ 1,064 nm/ 2   ⋅   5 m = 37.5 m. The actual value is 47 m, which is slightly larger and, thus, C o p t is not exactly achieved (blue curve is slightly below C o p t at 23 h). At the zenith angle at around 1 h, the array reduction factor is a r f = 0.8 , and it follows that the optimal antenna spacing d _E is then 352,000 km ⋅ 1,064 nm/ 2   ⋅   5 m   ⋅   0.8 = 46.8 m, which is very close to the actual value of 47 m.

Our simulation results show that it is basically possible to achieve the maximum MIMO capacity for an FSO communications laser link between Earth and Moon. The necessary spacings between the lasers and the telescopes are in the order of some meters up to a few tens of meters for typical carrier wavelengths between 1,064 and 1,550 nm. These dimensions are manageable in practice to realize an FSO MIMO communications system having all necessary receiving equipment at a single site and all transmitter equipment closely together on the Moon surface. Assuming a minimum elevation angle of 20°, a beneficial positioning parameter setup can be found providing capacity gains that are close to the maximum during a complete passage of the Moon. For a constant connection with high MIMO capacity during a complete lunar orbit, multiple MIMO receive telescopes could be deployed around the globe. Such a network with the name Linearly Dispersed Optical Subnet (LDOS) has already been proposed assuming single lasers at each location (Hemmati, 2006).