# Mitigation of Misalignment Errors
Over Inter-Satellite FSO Energy Harvesting

Baris Donmez, Irfan Azam, and Gunes Karabulut Kurt  Poly-Grames Research
Center, Department of Electrical Engineering, Polytechnique Montréal,
Montréal, QC, Canada
Emails: {baris.donmez, irfan.azam<EMAIL_ADDRESS>
###### Abstract

In this paper, the impact of the acquisition, tracking, and pointing (ATP)
module utilization on inter-satellite energy harvesting is investigated for 1U
(0.1$\times$0.1$\times$0.1 m) and 12U (0.2$\times$0.2$\times$0.3 m) satellites
for adaptive beam divergence and the corresponding distances while maintaining
the spot diameters. Random elevation and azimuth misalignment error angles at
both the transmitter and the receiver are modeled with Gaussian distribution
hence the radial pointing error angle is modeled with Rayleigh distribution.
The Monte Carlo approach is used to determine mean radial error angles for
both transmitter and receiver in the non-ATP and ATP cases. The average
harvested powers are analyzed as a function of the transmit powers and inter-
satellite distances for both 1U and 12U satellites while considering the
minimum power requirements. Our simulation results show that in the non-ATP
case, the minimum required average harvested power cannot be achieved beyond
680 and 1360 km distances for 1U and 12U satellites, respectively, with a
maximum transmit power of 1 kW. However, 2 W of average harvested power can be
achieved at around 750 and 1500 km for 1U and 12U satellites, respectively,
with a transmit power of 27 W in the presence of an ATP mechanism.

###### Index Terms:

Acquisition, tracking, and pointing (ATP), adaptive beam divergence, energy
harvesting, free space optics (FSO), inter-satellite communication,
misalignment errors.

##  I Introduction

The majority of the small satellites (i.e., CubeSats) operate at 350–700 km
altitudes from the Earth’s surface. However, there are small satellites
including 1U satellites, moving in orbits that have altitudes above 700 km [1]
(i.e., within the exosphere layer) thus, the losses derived from atmospheric
attenuation and scintillation can be neglected [2]. Solar-powered satellites
(SPS) equipped with solar cells are capable to generate sufficient energy from
the Sun and the excessive amount can also be used to provide wireless power
transmission (WPT) for charging small satellites operating far from the Earth
[3]. Hence, self-sustainability can be achieved in space networks.

There are different sustainable energy sources other than risky nuclear energy
[4]. Many current space network applications utilize the microwave
radiofrequency (RF) WPT [5] which offers high conversion efficiency and
coverage whereas it has some drawbacks as well. The RF WPT has limitations in
terms of the maximum achievable range and it is prone to interferences. On the
other hand, free space optics (FSO) technology utilizes collimated laser
diodes and hence transmits the power on a circular spot area with a smaller
spot diameter. Therefore, the received power by a specific solar cell area is
not reduced in FSO WPT contrary to its RF counterpart. However, the main
drawback of the FSO WPT systems is misalignment error, and the acquisition,
tracking, and pointing (ATP) mechanism must be used for long-range FSO WPT
systems (i.e., inter-satellite) to mitigate pointing loss [3]. Therefore, the
random misalignment loss must be considered to compute the harvested power
more realistically.

Many existing works on FSO communication systems model the random elevation
misalignment error angle and azimuth misalignment error angle statistically
with zero-mean, independent, and identically distributed Gaussian
distribution. Then, Rician distributed radial error angle model can become
Rayleigh distributed when the bias error angle of the Rician distribution is
defined as zero [6, 7, 8, 9].

There are a limited number of inter-satellite FSO WPT studies that exist in
the literature. In [3], the received power and total received energy are
presented for the single-hop and cluster scenarios with various distances. In
[10], the harvested power as a function of transmit power for various
distances is presented. The authors considered a beam divergence angle of 4
$\mu$rad, and then computed 0.9 kW received power for a 1 kW transmit power
over a 25 km line-of-sight (LoS) distance between an low Earth orbit (LEO)
satellite and a CubeSat. These studies do not consider the losses induced by
pointing errors at the transmitter and receiver.

In our paper, we investigate the adverse effects of random misalignment errors
at both the transmitter and the receiver in LoS link between a large SPS and
small satellites which have 0.1 and 0.2 m receiver aperture diameters,
respectively. The SPS transmits an adaptive collimated laser beam which
enables to maintain of an appropriate spot diameter for varying distances and
then a smaller satellite harvests the power for operating uninterruptedly
since small-size solar arrays cannot generate sufficient energy from the Sun
[10]. In a nutshell, we propose a realistic FSO WPT system that considers
self-sustainability since the small satellites harvest the power that is
transmitted by the SSP satellite.

The key contributions of this study can be listed as follows:

  * •

In our system model, we consider a realistic laser power conversion efficiency
(PCE) of a laser diode used in space missions.

  * •

We use a realistic energy harvesting conversion efficiency (EHCE) of an
appropriate solar cell type.

  * •

We generate random misalignment error angles for the transmitter and receiver
sides during energy harvesting done by the smaller 1U and 12U satellites.

  * •

We investigate the impact of realistic acquisition, tracking, and pointing
(ATP) modules for energy harvesting between two satellites.

  * •

We compute the maximum inter-satellite distances as 751.88 and 1503.8 km for
1U and 12U satellites, respectively, by adhering to the realistic laser
aperture diameter of 8 m.

  * •

In the non-ATP case, the 2 W of average harvested power cannot be achieved
beyond 680 and 1360 km distances for 1U and 12U satellites, respectively, even
with a maximum transmit power of 1 kW.

  * •

We show that the average harvested power of 2 W for CubeSats can be achieved
at around 750 and 1500 km for 1U and 12U satellites, respectively, with a
transmit power of 27 W in the presence of an ATP mechanism.

The remainder of this paper is organized as follows. In Section II, we present
the system model of the inter-satellite FSO-based energy harvesting system. We
elaborate on the misalignment error angle and energy harvesting conversion
efficiency model. In Section III, the simulation parameters of our system
model are presented and then the performances of our proposed FSO-based energy
harvesting system for 1U and 12U satellites with or without the ATP module are
evaluated. Finally, we conclude our paper and highlight the future research
directions for inter-satellite FSO-based energy harvesting in Section IV.

Figure 1: System model of inter-satellite optical power transmission and
energy harvesting.

##  II System Model

Our proposed system model demonstrated in Fig. 1 consists of a larger SPS, and
a small satellite (i.e., 1U or 12U) that manages its energy requirement by
harvesting energy from the remote laser diode with adaptive beam divergence
[11] that enables to maintain the adequate spot diameter as the distance
increases. In our self-sustainable system, we aim to use wavelength-dependent
conversion efficiencies as high as possible, however, the efficiency values
must be considered for the same wavelengths $\lambda$ for both the transmitter
and receiver [3]. Therefore, the system components must be selected
meticulously.

Efficient laser transmitter selection is a challenging task since the adequate
wavelength range for an FSO-based energy harvesting is $\lambda\in(780,1100)$
nm [3]. For the shorter range FSO links, the beam divergence $\theta$ is in
the range of $\theta\in(0.05,1)$ mrad if a tracking module is utilized,
otherwise, it is $\theta\in(2,10)$ mrad [12]. However, for inter-satellite
links, the required collimated laser beam divergence angles can be determined
for a given spot diameter and the LoS distance ${R}$ by using the small-angle
approximation as follows [13]

$\theta\text{ }\left[\text{rad}\right]=\frac{\text{Spot}\,\text{Diameter
}\left[\text{m}\right]}{\text{R }\left[\text{m}\right]\text{ }}.$ (1)

Moreover, the appropriate aperture diameter of a laser diode, ${{d}_{t}}$, can
be determined by ${{d}_{t}}\cong\lambda/\theta$ [12]. The received power of a
free space LoS optical link is expressed by [9, 2] as follows

$\small{{P}_{h}}\\!=\\!{{P}_{t}}{{\left(\frac{\lambda}{4\pi
R}\right)}^{2}}{{\eta}_{e/o}}(\lambda){{\eta}_{h}}(\lambda){{L}_{t}}({{\psi}_{t}}){{G}_{t}}{{L}_{r}}({{\psi}_{r}}){{G}_{r}}{{L}_{e}}{{L}_{s}}{{L}_{c}},$
(2)

where ${{P}_{h}}$ is the harvested electrical power, ${{P}_{t}}$ is the
transmitted electrical (input) power, ${{\eta}_{e/o}}(\lambda)$ and
${{\eta}_{h}}(\lambda)$ are the wavelength dependant electrical-to-optical PCE
and EHCE, respectively. In addition, ${{L}_{t}}({{\psi}_{t}})$ is the radial
angle dependant misalignment loss factor at the transmitter, ${{G}_{t}}$ is
the transmitter gain, ${{L}_{r}}({{\psi}_{r}})$ denotes the radial angle
dependant misalignment loss factor at the receiver, ${{G}_{r}}$ is the
receiver gain, ${{L}_{e}}$ is the atmospheric extinction/attenuation loss,
${{L}_{s}}$ is the scintillation loss, and ${{L}_{c}}$ represents the fiber
coupling loss.

Figure 2: Energy harvesting power conversion efficiencies of III-V
semiconductor solar cells.

As in [2], we consider ${{L}_{e}}={{L}_{s}}={{L}_{c}}=1$ for our inter-
satellite FSO energy harvesting scenario [2]. Furthermore, the transmitter and
receiver gains can be approximated as [9]

${{G}_{t}}\approx{{\left(\frac{\pi{{d}_{t}}}{\lambda}\right)}^{2}},$ (3)
${{G}_{r}}\approx{{\left(\frac{\pi{{d}_{r}}}{\lambda}\right)}^{2}},$ (4)

respectively, where ${d}_{r}$ is the aperture diameter of the receiver.

The misalignment loss factor at the transmitter can be computed by using [9]

${{L}_{t}}({{\psi}_{t}})=\exp\left(-{{G}_{t}}\psi_{t}^{2}\right),$ (5)

where ${\psi}_{t}$ is the radial misalignment error angle at the transmitter.

The misalignment loss factor at the receiver is given as [9]

${{L}_{r}}({{\psi}_{r}})=\exp\left(-{{G}_{r}}\psi_{r}^{2}\right),$ (6)

that is the radial misalignment error angle at the receiver.

###  II-A Misalignment Error Angle Model

It is desirable to establish a perfectly aligned LoS optical WPT link
throughout the transmission of the laser beam to maximize the received power.
The ATP modules enable satellites to mitigate the misalignment errors induced
by the mechanical vibrations of the satellites.

The elevation misalignment error angle ${\psi}_{e}$ and azimuth misalignment
error angle ${\psi}_{a}$ can be modeled statistically with a zero-mean
Gaussian distribution as follows [8]

$f\left({{\psi}_{e}}\right)=\frac{1}{\sqrt{2\pi\sigma_{e}^{2}}}\exp\left(-\frac{\psi_{e}^{2}}{2\sigma_{e}^{2}}\right),$
(7)

$f\left({{\psi}_{a}}\right)=\frac{1}{\sqrt{2\pi\sigma_{a}^{2}}}\exp\left(-\frac{\psi_{a}^{2}}{2\sigma_{a}^{2}}\right),$
(8)

where $\sigma_{e}^{2}$ and $\sigma_{a}^{2}$ are the variances of elevation and
azimuth misalignment angles, respectively. It should be noted that random
elevation and azimuth misalignment error angles are independent and
identically distributed.

Hence, the radial misalignment error angle ($\psi$) at the transmitter
(${\psi}_{t}$) and receiver (${\psi}_{r}$) can be modeled statistically with
the Rayleigh distribution by assuming
${{\sigma}_{\psi}}={{\sigma}_{e}}={{\sigma}_{a}}$ due to the symmetry as
follows [8]

$\psi=\sqrt{\psi_{e}^{2}+\psi_{a}^{2}},$ (9)
$f\left({{\psi}_{t}}\right)=\frac{{{\psi}_{t}}}{\sigma_{\psi
t}^{2}}\exp\left(-\frac{\psi_{t}^{2}}{2\sigma_{\psi t}^{2}}\right),$ (10)
$f\left({{\psi}_{r}}\right)=\frac{{{\psi}_{r}}}{\sigma_{\psi
r}^{2}}\exp\left(-\frac{\psi_{r}^{2}}{2\sigma_{\psi r}^{2}}\right).$ (11)

TABLE I: Simulation Parameters Parameter | Value
---|---
Max. Transmit Power (${P}_{t}$) | 1 kW [10]
Laser Wavelength (${\lambda}$) | 1064 nm [14]
Laser Diode PCE ($\%$) | 51$\%$ [15]
| Beam Divergence Angle ($\theta$)
---
| Adaptive
---
| Laser Aperture Diameter
---
Constraint (max. $d_{t}$)
| 8 m [16]
---
| 1U Satellite Receiver
---
Aperture Diameter ($d_{r}$)
| 0.1 m
---
| 12U Satellite Receiver
---
Aperture Diameter ($d_{r}$)
| 0.2 m
---
| Pointing Resolution
---
5 $\mu$rad [2]
| Pointing Resolution
---
with ATP module
0.5 $\mu$rad [17]
EHCE ($\%$) | 26.4$\%$ [18]
| Small Satellite Power
---
Requirement (W)
2 W [19]

###  II-B Energy Harvesting Conversion Efficiency Model

In general, the LoS distance range for inter-satellite connection is
considered between 100 m to 250 km [3]. Hence, although RF and FSO are
commonly used technologies for establishing the links between satellites,
power transmission by collimated laser offers sufficient energy harvesting at
higher LoS distances. Energy harvesting is crucial for smaller satellites
since the utilization of solar cell arrays is very limited due to the smaller
effective area [3].

On the other hand, solar cells are preferred to convert signals with
frequencies in the spectrum that the sun occupies. Since laser diodes mainly
utilize the infrared band such as 1064 nm, solar cells are appropriate for
FSO-based energy harvesting [20].

There are various solar cells converting optical energy to electrical energy,
that are made up of different materials. Fig. 2 represents the wavelength-
dependent EHCE of some of the common III-V semiconductor solar cells[18].
Although EHCE of GaAs-made solar cells can reach almost 60$\%$ at around 800
nm, it cannot be used straightforwardly since the laser wavelength in our
proposed system is 1064 nm as shown in Table I. Therefore, the selection of
InGaAsP-made solar cells which offer 26.4$\%$ is suitable for our system.

Figure 3: Divergence angles and ranges for 1U and 12U small satellites.

##  III Performance Evaluation

We evaluate the performance of the proposed system through simulations and
analyze the impact of ATP on the mitigation of misalignment errors. Before
computing the average harvested power, the Monte Carlo method and pointing
resolution parameters in Table I are used to analyze independent elevation and
azimuth misalignment error angles statistically for both the receiver and
transmitter. Then, the mean values of radial pointing error angles are
determined for the receiver and transmitter when the ATP module is absent and
present.

###  III-A Simulation Parameters

The selection of the proper laser diode type is vital thus, we must consider
an operating wavelength that enables sufficient energy harvesting and also can
be used in the space applications such as inter-satellite communication or
deep-space missions. Hence, the studies of [3] and [14] address these
concerns, respectively. Therefore, we consider Yd: NVO4 1064 nm laser source
with 51% PCE [15] as mentioned in simulation parameters as given in Table I.

The adaptive beam divergence angle changes as a function of the spot diameter
and distance, and hence the maximum range is determined by considering the
maximum laser transmitter aperture diameter as 8 m [16]. Moreover, we consider
various transmit powers ${P}_{t}$ between 1 W and 1 kW in our simulations.

It should be noted that the EHCE is a wavelength dependant parameter and it
must meet the wavelength of the laser source when considering the best
possible solar cell type. As per the Fig. 2, InGaAsP offers 26.4$\%$ EHCE
which is higher than that of InGaAs.

The 1U and 12U satellites have 0.1$\times$0.1 m and 0.2$\times$0.2 m surfaces
thus, collector aperture diameters are considered as $d_{r}=$ 0.1 and 0.2 m,
respectively [21]. Besides, the CubeSats require power less than 2 W to
accomplish their tasks [19].

Figure 4: Average harvested power for 1U small satellite.

###  III-B Results and Discussions

We have conducted simulations to investigate the impact of ATP with different
transmit powers $P_{t}$ and distances $R$ over the average power harvested
$P_{h}$ by the 1U and 12U satellites in our proposed system model. In
addition, we determine the minimum required transmit power to satisfy the
power requirement of the small satellites when each small satellite is at the
maximum possible distance from the SPS.

By using Eq. 1, the adaptive laser beam divergence angles of 1U and 12U
satellites are computed for corresponding inter-satellite distances by
adhering to the maximum transmitter aperture diameter of 8 m [16], as shown in
Fig. 3. Due to this laser diode aperture diameter constraint, the maximum
achievable distances for 1U satellite with $d_{r}=$ 0.1 m and 12U satellite
with $d_{r}=$ 0.2 m are 751.88 and 1503.8 km, respectively. Since we consider
the minimum inter-satellite distance as 10 km, the widest beam divergence
angles for 1U and 12U satellites are 10 and 20 $\mu$rad, respectively, whereas
the common narrowest beam divergence angle when $d_{t}=$ 8 m is 0.133
$\mu$rad.

For the case of the 1U satellite (0.1$\times$0.1$\times$0.1 m), the
improvement made by an ATP mechanism providing a higher pointing resolution
demonstrated by comparing it with the no ATP case in Fig. 4. Since the
receiver aperture diameter of 1U satellite is 0.1 m and fixed, the transmitter
beam divergence angle adapts itself as the distance increases, to maintain the
same spot size. According to Fig. 4, when the distance goes beyond 680 km, the
average harvested power drops below 2 W despite the maximum transmit power
$P_{t}=$ 1 kW in the absence of the ATP module. On the other hand, 2 W of
average harvested power be achieved even with $P_{t}=$ 27 W for $R=$ 750 km
when the ATP module is in use.

For the case of the 12U satellite (0.2$\times$0.2$\times$0.3 m), the
improvement made by the ATP module is presented by comparing it with the no
ATP case in Fig. 5. In this case, $d_{r}=$ 0.2 m enables twice the distance
for the same laser aperture diameter constraint. Recall that, Eq. 1 states a
beam divergence angle can be maintained despite the distance being doubled if
the spot diameter is doubled as well. According to Fig. 5, when $R\geq$ 1360
km and transmit power is maximum as $P_{t}=$ 1 kW, the average harvested power
drops below 2 W without an ATP mechanism. However, when the ATP module is in
use, 2 W of average harvested power is achieved with $P_{t}=$ 27 W for $R=$
1500 km.

Figure 5: Average harvested power for 12U small satellite.

##  IV Conclusions

This paper investigated the role of the tracking module on inter-satellite
energy harvesting between 1U and 12U small satellites and the solar-powered
satellite which utilizes adaptive beam divergence. Various different transmit
powers and adequate ranges were considered as per the laser aperture diameter
constraint. Random elevation and azimuth misalignment error angles at both the
transmitter and the receiver were modeled with Gaussian distribution. Hence
the radial pointing error angle could be modeled with Rayleigh distribution
statistically. The narrower the laser beam the lower the misalignment error
factor, and hence the average harvested power.

Our simulation results show a comparison between the energy harvesting made by
1U and 12U small satellites with and without the ATP modules. The outcomes
show that ATP is necessary to be able to maximize the inter-satellite
distance. For instance, 750 and 1500 km can be achieved for 1U and 12U
satellites, respectively, by using $P_{t}=$ 27 W in the ATP case whereas only
around 120 and 240 km can be achieved without the use of the ATP module.

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