1.1

Pointing loss definition and assumptions

Pointing losses are defined as the ratio between the received optical power (including the effect of pointing misalignment, and the on-axis beam power) corresponding to the beam power that would be detected in the ideal case where the beam had no angular deviation due to pointing errors. The pointing loss will be expressed in decibels (dB) and the calculations are performed using the following set of assumptions.

A Gaussian beam with the lowest order of transverse mode (TEM00) is considered. An uplink laser beam propagates through the atmosphere and is affected by turbulence, causing beam widening, beam wander and spatially-modulated beam scintillation.⁶ Nonetheless, previous research has shown that by considering a temporally averaged beam intensity, the beam can still be regarded as having a Gaussian profile, even after propagation through the atmospheric channel.^7,8 Although the beam wander effect is another source of pointing error, it is not considered in this paper, as it has been studied extensively.⁹

A point-like receiver and transmitter are assumed. For point receivers, only the received signal intensity at one point has to be calculated, rather than needing to integrate the power over the receiver’s aperture. This is always a valid approximation in satellite communications, since the long link distances result in a beam spot that is much larger than the receiver size. For point transmitters, the beam spot size can simply be approximated as the beam divergence angle multiplied with the link distance, corresponding to the far-field approximation (large distance compared to the Rayleigh range). The paraxial approximation is employed and hence small misalignment angles in different directions can be summed simply as two-dimensional vectors. Finally, the receiver is mounted on a satellite having a circular orbit around the Earth and Earth’s shape is assumed to be spherical for simplicity.

1.2

Signal intensity loss due to pointing errors

Several physical sources of tracking and pointing errors are considered in this paper. For a transmitted beam having a Gaussian profile with a known divergence angle, the induced “pointing loss” in the received optical intensity (which can be translated into power) can be calculated.

The values of the pointing error may vary depending on the space-terminal elevation above the horizon and, thus, the pointing loss is elevation dependent. For instance, the point-ahead angle (PAA) depends on the satellite position along its orbit and is elevation dependent. However, in link-budget calculations, the pointing loss is typically given a constant value regardless of the satellite elevation. In this paper, the assumption of a constant pointing loss is abandoned and elevation-dependent intensity losses are calculated. Table 1 shows a summary of the pointing errors that are analyzed hereafter: static errors (non-elevation dependent), dynamic errors (elevation-dependent but predictable) and stochastic errors (potentially elevation-dependent and unpredictable).

Table 1.

Summary of the error mechanisms considered in this work.

The approach followed to analyze the intensity loss due to the pointing errors discussed in this work is described in section 2. The angular deviation due to each pointing error is first analyzed in section 3, followed by the analysis of the intensity loss due to each pointing error in section 4. The effect of all pointing errors is then combined in section 5 to determine the total effect.

3.1

OGS static pointing error

An OGS can have a constant angular deviation (∆_𝑂𝐺𝑆) from the line of sight with a satellite, or boresight, due to mechanical misalignments in the construction of the OGS optics. It can be considered that ∆_𝑂𝐺𝑆 is composed of a component in the direction of the satellite trajectory (along-track) and a transverse component (cross-track). If the static pointing error of the OGS is in the direction of the satellite trajectory (along-track direction), it can be taken as an additional deviation to the PAA (termed 𝜃_{𝑃𝐴𝐴_𝑂𝐺𝑆} in section 3.2), which gets a positive value if the OGS points ahead of the satellite, or negative if the OGS points in the opposite direction to the satellite’s direction of motion relative to the OGS.

3.2

Uncorrected point-ahead angle (PAA)

The PAA is the forward angle at which an OGS must be pointed for an uplink signal to be centered on the satellite, due to the relative motion of the satellite with respect to the OGS and the finiteness of the speed of light. The PAA is represented in Figure 4. At time t = 0 a satellite moving from point A to point B with orbital speed V₀ sends a downlink signal to an OGS (①). When the OGS receives the downlink signal at time t_T = L/c - where c is the speed of light and L is the distance between the OGS and the satellite-, the satellite is already approximately halfway between point A and B (②). Finally, the OGS sends the uplink signal to the location where the satellite will be (at point B), taking into account that the beam will take t_T to reach the satellite (③).

Figure 4.

Representation of the PAA.

The calculation of the PAA can be approximated by assuming that L does not change between point A and B, and that there are no additional delay times at the receiver and transmitter that affect the total propagation time of the signals. Using the definition of arc length, it is possible to equal the half distance 𝑉_{𝑡_𝑠𝑎𝑡} ∙ 𝑡_𝑇 between points A and B with 𝐿 ∙ 𝜃_𝑃𝐴𝐴/2, where V_{t_sat} is the speed of the satellite perpendicular to the line of sight with the OGS, t_T is the downlink or uplink signal propagation time, and L = c · t_T. Consequently, the PAA [rad] without considering the Earth’s rotation can be approximated as follows

where 𝛼(𝜀) can be calculated with equation (5), and the satellite orbital speed V₀ (constant for a circular orbit) can be calculated with equation (7), being G [N·m²/kg²] the gravitational constant and M [kg] the mass of the Earth.

To consider the effect of the Earth’s rotation on the PAA, the rotational speed of the OGS at a given latitude of the Earth can be included in equation (6). The rotational speed of an OGS at latitude l is approximated by (R_E + H_ocs) · ω_E · cos l, where ω_E is the angular velocity of the Earth.

In this work, it is assumed that a satellite passes exactly over the OGS at some point during its pass, in order to have an analysis of the angular deviation due to pointing errors with the satellite elevation from 0 to 90°. For simplicity, a satellite with 0° orbital inclination, which transits over an OGS located on the equator (l = 0°) is considered. Just as for equation (6), where the perpendicular speed of the satellite to the line of sight with the OGS is considered, the perpendicular speed of the OGS to the line of sight with the satellite must be regarded. The perpendicular velocity of the OGS to the line of sight with the satellite depends on the satellite elevation ε. Hence, for an OGS located at the equator, the term sin ε is added. Thus, considering a satellite with a circular orbit of 0° inclination and an OGS at l = 0°, the equation (6) can be rewritten to account for the rotational speed of an OGS as

where the values given to ω_E and the constants involved in this analysis are shown in Table 3. While equation (8) was chosen, as it represents the worst-case scenario in which the PAA varies over a larger range during a satellite pass, the values of V_{t_sat} and V_{t_OGS} can be adapted if it is desired to either consider a satellite with a different orbital inclination or an OGS at a different geographic latitude.

Figure 5 shows the PAA calculated with equation (8) as a function of the satellite altitude and the satellite elevation. It can be observed that the lower the altitude of the satellite (in this case down to 1 km for illustrative purposes only), the greater the range of the PAA during a single satellite pass. While for a satellite in geostationary orbit (GEO) the PAA ranges from 20.3 μrad at low elevations to 17.4 μrad when the satellite is at the zenith of the OGS, the PAA of a satellite in LEO at 500 km altitude (indicated by a constant black line in the figure) can range from 19.18 μrad to 47.68 μrad. Table 4 shows values of PAAs for a satellite with a circular orbit and 0° inclination at different altitudes and elevations with respect to an OGS located at sea level at the equator.

Figure 5.

PAA between an OGS at sea level at the equator and a satellite with circular orbit and 0° inclination. The PAA for a LEO satellite at 500 km altitude is indicated by the constant black line.

Table 4.

PAA values for LEO satellites at different altitudes and elevations.

3.3

Orbit uncertainties (along-track error)

To analyze the pointing loss due to orbit uncertainties, an open-loop satellite tracking is considered in which an OGS uses the satellite orbit information to point to the satellite. When the orbit information is recent, the orbit uncertainties are low and the satellite can be targeted accurately with the OGS. For this reason, the satellite orbit must be determined frequently to reduce orbit uncertainties. In this paper only the along-track error is taken into account, as it is more significant than the cross and radial errors.¹¹ As an example, the mean along-track error can span from a few meters, when the two-line element set (TLE) is recent, to more than 5 km after a 3-day orbit propagation, and 24 km after 7 days of propagation period.¹¹

The along-track error indicates how far the satellite can be from the predicted position along the orbit track (either in the direction of the satellite’s trajectory or in the opposite direction). In this paper, the along-track error is translated as an angular deviation that covers the uncertainty of the satellite position with respect to the OGS, as illustrated in Figure 6. Since this angular deviation is relative to the OGS, it changes with the satellite elevation, and it can be calculated by assuming that the link distance does not change between the predicted position of the satellite and the position the satellite would be in if it deviated by the determined along-track error. Considering (for simplicity) the case of a satellite passing over the zenith of an OGS, the angular deviation (∆_{𝑎𝑙𝑜𝑛𝑔}(𝜀)) as seen from an OGS considering a given along-track error δ [m] is approximated to

Figure 6.

Representation of the elevation-dependent angular deviation observed by an OGS due to an along-track error.

where L(ε) and α(ε) are calculated with equation (4) and equation (5) respectively. Figure 7 shows the angular deviation due to a given along-track error, as a function of the satellite altitude and elevation. It is observed that the angular deviation is greater at lower satellite altitudes and at higher elevations. While for a satellite at 500 km altitude (constant black line) the angular deviation can vary by 1.85461 · δ when the satellite has gone from 0 to 90° elevation during a pass, for a satellite in GEO the angular deviation only varies by 4.227 · 10^-3 · δ. Table 5 presents values of the angular deviations for satellites at different altitudes and elevations.

Figure 7.

Angular deviation observed by an OGS due to a given along-track error. The constant black line indicates the corresponding angular deviation for a LEO satellite at 500 km altitude.

Table 5.

Angular deviations of an uplink beam due to a given along-track error for LEO satellites at different altitudes and elevations.

The angular deviation due to a radial orbital error can be computed following the approach used to calculate the angular deviation due to an along-track error, by calculating the maximum link distance due to that radial error and then estimating the angular deviation with equation (9).

3.4

OGS mechanical jitter

Random mechanical vibrations (mechanical jitter) in the OGS’s optical transmitter cause variations in the intensity of the uplink beam detected by a satellite. The mechanical jitter can be modelled as inducing a normally distributing pointing error, having circular symmetry around the beam axis and standard deviation σ_jit.

The pointing jitter could have several physical sources that contribute to it. For instance, both the coarse pointing assembly (CPA) and the fine steering mirrors (FSM) could introduce some jitter to the beam pointing, having associated standard deviations σ_CPA and σ_FSM. Since these sources of pointing errors are statistically independent, the corresponding standard deviations can be added in quadrature, resulting in a total pointing jitter of

4.1

Intensity loss due to OGS static pointing error

If only the static pointing error of an OGS (∆_𝑂𝐺𝑆) is considered for the calculation of the intensity loss due to pointing errors, the value of ∆_𝑂𝐺𝑆 in radians can directly be substituted into θ in equation (1) to then calculate the intensity loss due to this error with equation (3). Hence, the intensity loss due to a static pointing error at the OGS does not depend on the altitude or elevation of the satellite, and is approximated by

4.2

Intensity loss due to uncorrected point-ahead angle (PAA) or fixed-corrected PAA

If an OGS does not consider the PAA during a G2S link or relies on a constant PAA, there will be a deviation angle between the center of the uplink signal and the line of sight to the satellite, resulting in intensity loss. A constant PAA of 50 μrad (FWHM) is sometimes assumed during G2S links with LEO satellites. This approach results in more power loss at the satellite’s receiver for lower satellite elevations. Conversely, when the PAA is not corrected, there would be more power loss at the receiver as the satellite approaches the zenith, because the theoretical PAA is more significant at higher satellite elevations (Figure 5).

Having calculated the theoretical PAA that an OGS must have to point towards a satellite at a given altitude and elevation using equation (8), and considering the PAA that an OGS uses in practice (𝜃_{𝑃𝐴𝐴_𝑂𝐺𝑆}), it is possible to calculate the deviation of the center of the uplink beam with respect to the line of sight to the satellite and, therefore, the intensity at the receiver by substituting the PAA for 𝜃 in equation (1). The intensity loss due to an uncorrected PAA or a fixed-corrected PAA, defined as

can be calculated in dB using equation (3).

Figure 8 shows the PAA and intensity loss in dB at the satellite when the OGS does not consider a PAA (𝜃_{𝑃𝐴𝐴_𝑂𝐺𝑆} = 0 radians) when pointing towards a LEO satellite at 500 km altitude with an uplink beam with divergence of 50 μrad (FWHM).

Figure 8.

PAA and intensity loss due to uncorrected PAA (𝜃_{𝑃𝐴𝐴_𝑂𝐺𝑆} = 0 radians) when an OGS at sea level at the equator points an uplink beam with divergence of 50 μrad (FWHM) towards a LEO satellite at 500 km altitude with circular orbit of 0° inclination.

The loss of intensity at the receiver when selecting another uplink beam divergence and considering the same satellite at 500 km altitude is shown in Figure 9. It is observed that the narrower the divergence angle of the uplink beam and the higher the elevation of the satellite, the higher the intensity loss. This is because, as the PAA increases at higher satellite elevations (Figure 5), the satellite moves farther away from the point of the peak intensity of the uplink beam for a narrower beam.

Figure 9.

Intensity loss due to uncorrected PAA (𝜃_{𝑃𝐴𝐴_𝑂𝐺𝑆} = 0 radians) when an OGS at sea level at the equator points an uplink beam towards a LEO satellite at 500 km altitude with circular orbit of 0° inclination. The constant black line indicates the corresponding intensity loss when the uplink beam has a divergence of 50 μrad (FWHM).

Intensity losses for satellites at different altitudes and elevations, and with different divergences of the uplink beam are summarized in Table 6.

Table 6.

Intensity loss due to an uncorrected PAA for LEO satellites at different altitudes and elevations, and different divergences of the uplink beam.

4.3

Intensity loss due to orbit uncertainties (along-track error)

After knowing the maximum pointing error of an OGS considering a given along-track error (δ) in the satellite orbital data, the loss of intensity can be calculated using equation (9) and (1), and substituting ∆_{𝑎𝑙𝑜𝑛𝑔}(𝜀) for θ. The resulting intensity distribution can be further used to calculate the intensity loss due to an along-track error using equation (3) as

The intensity loss as a function of the uplink beam divergence angle and satellite elevation is represented in Figure 10. The intensity loss due to the along-track error in the orbital data is more significant for lower divergence angles and higher elevations. This is because at higher elevation, the angular deviation covering the uncertainty of the satellite position with respect to the OGS appears larger (as seen in Figure 7), and a wider uplink beam is required to cover it.

Figure 10.

Intensity loss caused by angular deviations observed by an OGS due to a given along-track error. The constant black line indicates the corresponding intensity loss when an OGS at sea level at the equator points an uplink beam with divergence of 50 μrad (FWHM) towards a LEO satellite at 500 km altitude.

Table 7 shows intensity loss values of an uplink beam detected in satellites at different altitudes and elevations. It is important to note that the values given in the table are multiplied by the square of a given along-track error to obtain the corresponding loss of intensity.

Table 7.

Intensity loss caused by angular deviations of an uplink beam due to a given along-track error.

4.4

Intensity loss due to OGS mechanical jitter

The average intensity detected at the receiver due to random jitter σ_jit can be approximated to^12,13

The average intensity loss at the satellite can be calculated by substituting the average intensity I_jit in equation (3), resulting

where

4.5

Total intensity loss due to all pointing errors

The effect of all pointing errors considered in this paper are now combined. Two classes of errors can broadly be considered: deterministic errors and stochastic errors. In our analysis, the PAA-fixed correction, the along-track and radial orbital errors, and a static pointing error of the OGS, have been considered as sources of pointing bias; these can be summed together (as two-dimensional vectors, in the small angle approximation) to find the total pointing bias (Δ_𝑡𝑜𝑡). For the stochastic errors, only the OGS mechanical vibration has been considered, but another effect that may be taken into account is jitter induced by atmospheric turbulence (beam wander, having standard deviation σ_bw). It is assumed that these stochastic errors are independent and normally distributed, so that the combination of these effects results in a normally distributed total pointing jitter with standard deviation σ_tot, which can include the mechanical jitter and beam wander:

The probability density function of the modulus of the pointing error (rather then considering a two-dimensional distribution for the x and y component of the pointing error) is then given by the Rice distribution:

where Bessel₀ is the modified Bessel function of the first kind of order zero. As a result of this jitter, the beam intensity as seen by the receiver also fluctuates. Assuming a Gaussian beam having waist ω₀, this results in a normalized probability distribution for the normalized intensity i = I/I₀

where and . The average loss is obtained in this model by computing the expectation value of the normalized intensity, which yields to

By computing 10 log₁₀(⟨𝐼⟩/I₀), an expression for the average dB loss is obtained, including all contributing pointing error mechanisms

As an example, it is assumed that there are pointing errors during a G2S link due to an uncorrected PAA, and mechanical random jitter of 3.80 μrad (this value is taken from an example of link-budget design¹). In this case, the OGS is located at the equator and points an uplink beam of 50 μrad (FWHM) to a LEO satellite at 500 km altitude with orbital inclination 0°. In this example, is equal to Δ_𝑃𝐴𝐴². Using equation (20), and taking values for Δ_𝑃𝐴𝐴 from Table 4, the average pointing loss due to uncorrected PAA and mechanical jitter is -1.85 dB when the satellite is at 5° elevation, and -10.75 dB when it is at 90° elevation. Since the mechanical jitter considered in this example is much smaller than the corresponding PAA, the calculated intensity loss remains close to the value of loss due to uncorrected PAA (shown in Table 6).