domain: spi phys

text:
Aniceto Belmonte

Optimal modal basis approaches fundamental limits to free-space optical system capacity imposed by atmospheric turbulence

Keywords: Free-space communications, space multiplexing, atmospheric eigenmodes, capacity limits

We have considered the problem of finding optimal transmission beams, or eigenmodes, for propagation through the random atmosphere with minimum distortion. We have found, making a realistic assumption that a transmitter knows atmospheric turbulence statistics, but not the instantaneous state of the atmosphere, that these stochastic eigenmodes are Laguerre-Gauss (LG) modes with beam waist chosen adequately relative to the field coherence length in the receiver plane. We have shown in our analysis that these eigenmodes have the remarkable capability of enduring coherence during propagation in turbulence. The ability to order the strength of individual stochastic eigenmodes notwithstanding turbulence is fundamental. It allows using of optimal transmission strategies that only consider the small number of most robust modes and significantly outperform conventional fixed mode sets under weak or strong turbulence conditions. Using these eigenmodes as transmit and receive bases minimizes signal degradation by turbulence, and minimizes the complexity of any signal processing for compensating the impact of turbulence. Adaptive optics can be replaced by adaptive multi-input multi-output signal processing, enabling compensation of fast fluctuations of both phase and amplitude.

INTRODUCTION

The propagation of beams through random media is of fundamental importance in applications such as optical communications, remote sensing, and imaging. However, coherent fields that propagate through random media such as atmospheric turbulence are subject to distortion and scintillation that can cause considerable degradation in system performance. In a modal communication system [START_REF] Kahn | Communications expands its space[END_REF] [START_REF] Li | Space-division multiplexing: the next frontier in optical communication[END_REF], which exploits spatial degrees of freedom by launching modulated data signals onto different orthogonal spatial optical modes, aberrations induced by turbulence on the transmitted modes result in signal degradation from mode coupling and mode cross-talk, thus reducing the channel capacity of the system.

Here, we consider the problem of finding optimal transmission beams [START_REF] Zhao | Capacity limits of spatially multiplexed free-space communication[END_REF] [START_REF] Chen | Is there an optimal basis to maximise optical information transfer?[END_REF] for propagation through the random atmosphere based on the knowledge of the second-order moment of the propagated field. The covariance function 𝐶(𝝆 $ , 𝝆 & ) is the crucial ingredient in the coherent mode decomposition analysis, as it encodes the notion of similarity between the fields at points 𝝆 $ , 𝝆 & . 1 The statistics of a field that has propagated through atmospheric turbulence can be characterized by the covariance or, similarly, by its associated normal-mode decomposition, a statistical procedure that uses an orthogonal transformation to convert a set of observations of correlated fields into a set of amplitudes of linearly uncorrelated fields called eigenmodes.

Assuming that the covariance function 𝐶 of the atmospheric fields is integrable over the region 𝑹 $ occupied by the fields in the transmitter plane, it represents a Hilbert-Schmidt kernel and, by Mercer's theorem, it may be expanded in a series of orthogonal functions of the form2

𝐶(𝝆 $ , 𝝆 & ) = + λ -𝜓 - * (𝝆 $ ) 𝜓 -(𝝆 & ) - , (1)
where the eigenvalues λ -and the eigenmodes 𝜓 -(𝝆) satisfy the Fredholm integral equation

1 𝐶(𝝆 $ , 𝝆 & ) 𝜓 -(𝝆 $ ) 𝑑𝝆 $ 𝑹 3 = λ -𝜓 -(𝝆 & ) . (2)
The summation in Eq. ( 1), in general, may be a finite or infinite sum. The eigenvalues are non-negative and, loosely speaking, describe the amount of power allocated on average to the eigenmode 𝜓 -. Indeed, for 𝝆 $ = 𝝆 & = 𝝆, we have

𝐶(𝝆, 𝝆) = ∑ λ -|𝜓 -(𝝆)| & - .
The eigenmodes are orthogonal and are typically taken to be orthonormal. The orthonormality implies that ∫ 𝜓 - * (𝝆) 𝜓 9 (𝝆) 𝑑𝝆 𝑹 3

= δ -9 , with δ -9 the Kronecker symbol. Equation ( 1) is often called the coherent mode representation of the covariance function.

STATISTICAL EIGENMODES OF LASER BEAMS IN THE ATMOSPHERE

The eigenvalues λ -and the eigenmodes 𝜓 -(𝝆) of the coherent mode representation are obtained as the solution of the homogeneous Fredholm integral equation Eq. ( 2). For coherent beams in Kolmogorov atmospheric turbulence, a propagation model using the extended Huygens-Fresnel principle expresses the covariance function 𝐶 at the receiver aperture as3

𝐶(𝝆 $ , 𝝆 & ) = exp >- |𝝆 $ | & + |𝝆 $ | & 2𝜔 B & C exp >- |𝝆 $ -𝝆 & | D E ⁄ 2𝛿 D E ⁄ C , (3)
where 𝜔 B is the beam intensity radius, and 𝛿 = 𝑟 B 6.88 E D ⁄ ⁄ is proportional to the coherence diameter 𝑟 B describing the spatial correlation of field fluctuations in the receiver plane. Unfortunately, as happens frequently in the study of propagation effects through the turbulent atmosphere, the 5/3-power law makes the analytic solution to Eq. ( 2) intractable.

In order to overcome the difficulties with the 5/3-power law, when the coherence width 𝛿 is small (𝜔 B > 𝛿), we can consider a statistical model in which the random fields are expressed as finite sums over statistically independent cells in the receiver aperture. To a good approximation, the field can be considered to consist of (𝜔 B 𝛿 ⁄ ) & independent speckle cells, each of radius 𝛿. As the spatial variation of the random field in any of these independent cells can be described as a quadratic power law, we can approximate the total field covariance as a linear superposition of a number of Gaussianshaped basis functions. We have shown that this expansion leads to a squared exponential covariance function:

𝐶(𝝆 $ , 𝝆 & ) = 1 2𝜋 𝜔 0 2 exp >- |𝝆 $ | & + |𝝆 $ | & 2𝜎 P & C exp >- |𝝆 $ -𝝆 & | & 2𝜎 Q & C, (4)
with

𝜎 P & and 𝜎 Q & described in terms of 𝜔 B & and 𝛿 & as 𝜎 P & = S𝜔 B & + 𝛿 B & T and 𝜎 Q & = S𝜔 B & + 𝛿 B & T 𝛿 B & 𝜔 B & U .
We have verified numerically that this approach is very accurate, even when the number of speckle cells (𝜔 B 𝛿 ⁄ ) & within the receiver area is very small. The advantage of considering the modulated squared exponential covariance function ( 4) is that its coherent mode decomposition can be developed analytically. We have shown, as can be verified by direct substitution, that its stochastic eigenmodes 𝜓 -(𝝆) are given by the normalized Laguerre-Gauss (LG) modes

𝜓 -(𝜌, 𝜙) = 1 𝜔 X 2 𝑝! 𝜋 ( 𝑝 + |𝑙| )! \√2 𝜌 𝜔 ^|_| 𝐿 a |_| >2 𝜌 & 𝜔 & C exp >- 𝜌 & 𝜔 & C exp(𝑗𝑙𝜙) , (5)
where 𝐿 a |_| are the normalized Laguerre polynomials and 𝑙= 0, ±1 ± 2, … and 𝑝= 0, 1, 2, … are arranged in a doubleindexed sequence. The 𝜓 -are sorted in ascending order n= 1, 2, … by decreasing eigenvalues 𝜆 -:

𝜆 -= 𝜋 2 𝜔 & (1 -𝜉) 𝜉 |_| & h a . (6)
Here,

𝜉 = 1 + 2 𝜎 P & 𝜎 Q & -X1 + 2 𝜎 P & 𝜎 Q & 1 + 2 𝜎 P & 𝜎 Q & + X1 + 2 𝜎 P & 𝜎 Q & and 1 𝜔 & = 1 𝜔 B & + 1 2𝜎 P & X1 + 2 𝜎 P & 𝜎 Q & ,
where 𝜔 is the effective beam width of the modes, which, as expected, depends on the coherence width 𝛿 (see Fig. 1).

SPACE-MULTIPLEXED LASER COMMUNICATION

In this section, we investigate transmission using orthogonal spatial modes over a general atmospheric channel, applying mode-division multiplexing to near-field line-of-sight (LOS) free-space optical communication. Here, the term "nearfield" refers to link geometries in which, on average, the minimum spot size on the receiver plane is smaller than the receiver aperture, yielding near-perfect power coupling when the beam is focused on the receiver. We consider a modal multiplexing system that transmits a complete set of spatial modes, such as all the LG, with a common central rotation axis. We assume that all modes are efficiently multiplexed at the transmitter, and spatially co-propagate through the atmospheric channel. We consider that at the receiver, the modes are demultiplexed without loss or crosstalk (see, e.g., [START_REF] Leon-Saval | Photonic lanterns: a study of light propagation in multimode to single-mode converters[END_REF][8]).

Given 𝑁 transmit and receive modes, and frequency-flat fading, the baseband complex model we consider is

𝒚 = 𝑯𝒙 + 𝒏 ,
where 𝒙 and 𝒚 are the 𝑁 × 1 input and output vectors, respectively, while 𝒏~𝒩(𝟎, 𝜎 & ) is additive circularly symmetric complex Gaussian noise. The input vector component 𝑥 s represents the signal amplitude transmitted in the 𝑗th mode, while the output vector component 𝑦 u represents the signal amplitude received in the 𝑖th mode . The channel is characterized by the 𝑁 × 𝑁 random matrix 𝑯, with entries 𝐻 us representing the scattering gain from transmit mode 𝑗 to receive mode 𝑖.

We define a turbulence-free signal-to-noise ratio (SNR) 𝛾 = 𝑃 𝜎 & ⁄ , where 𝑃 is the total transmitted power in all modes and 𝜎 & is the receiver noise power per mode. We assume received signals are detected coherently [START_REF] Belmonte | Performance of synchronous optical receivers using atmospheric compensation techniques[END_REF][10] and the dominant noise source is local oscillator shot noise, so 𝛾 equals the number of received signal photons. As we are considering nearfield links, where the receiver collects virtually all the signal power leaving the transmitter, 𝛾 also equals the number of transmitted signal photons. Consequently, given transmit and receive bases of 𝑁 spatial modes, the ergodic average spectral efficiency (in bit/s/Hz) per polarization is [START_REF] Goldsmith | Capacity limits of MIMO channels[END_REF][12]

𝑆 = max 𝑸 𝐸 𝑯 {log & |𝑰 + 𝛾 𝑯 𝐐 𝑯 * | } , (7)
where the 𝑁 × 𝑁 matrix 𝑸 = 𝐸[𝒙 𝒙 * ] denotes the input covariance matrix 𝐐 whose trace is normalized to unity. 4 The expectation 𝐸 𝑯 { } is an ensemble average over realizations of the channel matrix 𝑯, 𝑰 is an 𝑁 × 𝑁 identity matrix, and | | represents a matrix determinant.

If we consider channels that either vary slowly over time and/or are reciprocal, information on the instantaneous channel state, i.e., the realization of 𝑯 can be assumed known at the transmitter. In this case, the channel can be partitioned by eigenbeam-forming into orthogonal sub-channels that are determined by the instantaneous channel state. Due to the nonuniform gains of these orthogonal sub-channels, or instantaneous eigenmodes, channel capacity can be maximized by 4 A given transmission strategy is completely characterized by its covariance matrix 𝑸. Let us decompose the normalized input covariance as Q = 𝑽𝑷𝑽 * identifying the eigenvectors of 𝑸 with the columns of the unitary matrix 𝑽 and its eigenvalues with the diagonal entries of 𝑷. Both the eigenvectors and the eigenvalues have immediate engineering significance. The strategy consists of transmitting independent symbols along the eigenvectors of 𝑸, with the corresponding eigenvalues specifying the powers allocated to each eigenvector. It is worth considering the capacityachieving forms of 𝑸 for two relevant regimes, i.e., when instantaneous information is available for tracking the states of the atmospheric channel 𝑯, and when limited statistical information about 𝑯 is used. allocating transmit power control to each sub-channel based on the water-pouring theorem. In this case, the ergodicaverage capacity 𝑆 given by Eq. ( 7) simplifies to the sum

S = 𝐸 𝑯 •+ log & (1 + 𝛾 𝛼 -𝜅 -) " -"$ • , ( 8
)
where 𝜅 -is the 𝑛th eigenvalue of 𝑯 * 𝑯 representing the gain of the 𝑛th instantaneous eigenmode, and 𝛼 -is the corresponding optimal transmitted power allocation resulting from a power-constrained ∑ 𝛼 - " -- -"$ = 1 water-filling optimization. Instantaneous eigenmodes corresponding to larger eigenvalues receive more power and the power allocated to some of the eigenmodes may be zero. As can be seen from Eq. ( 8), the spectral efficiency is determined by the number 𝑁 of independently addressable spatially multiplexed sub-channels and by their corresponding 𝜆 -.

If the transmitter has access to the statistical distribution of 𝑯, but not to 𝑯 itself, the transmit sub-channel directions and power allocations cannot be based on knowledge of the instantaneous channel state, but can be optimized based on statistics of the channel states that are fed back from the receiver to the transmitter. 5 Now, we consider that the eigenvectors of the input covariance 𝑸 are equal the stochastic LG eigenmodes (5) of the channel covariance 𝐶. The elements 𝐻 us of the channel matrix 𝑯 are now the complex coupling coefficient between transmit and receive stochastic LG eigenmodes. We address the optimization of the eigenvalues of the transmit covariance matrix, i.e., the power allocation policy, with a water-pouring algorithm. The fading channel cannot be partitioned into orthogonal independent channels and the ergodicaverage spectral efficiency 𝑆 is computed as

𝑆 = 𝐸 𝑯 {log & |𝑰 + 𝛾 𝑯 𝐃 𝑯 * | } . ( 9
)
The entries 𝑑 -of the diagonal matrix 𝐃 are the optimal power allocation resulting from a power-constrained ∑ 𝑑 - " -- -"$ = 1 water-filling solution applied on the eigenvalues 𝜆 -(6) of the stochastic LG eigenmodes.

In order to understand the benefit offered by the transmission strategies described above, we also consider the extreme case in which the transmitter has access to neither instantaneous nor statistical channel state information. When the transmitter has statistical knowledge of the channel state, it can adjust its transmission strategy to the correlation properties of the channel and adapt the beam width 𝜔 of the set of stochastic eigenmodes to the spatial statistics of the received fields, as described by 𝛿. When no channel information is available at the transmitter, however, the transmitter cannot adapt to potential correlation in the channel. In this scenario, the only possible strategy is to use a family of LG modes with fixed beam width 𝜔 B and split the transmit power equally over all transmit modes, even though some of these beams will not propagate effectively through the channel. The elements 𝐻 us of the channel matrix 𝑯 are now the complex coupling coefficient between transmit and receive fixed LG modes. In this extreme case, the fading channel is completely unknown at the transmitter and the ergodic-average spectral efficiency 𝑆 is computed as

𝑆 = 𝐸 𝑯 šlog & ›𝑰 + 𝛾 𝑁 oe• 𝑯 𝑯 * › ž . ( 10
)
We use Monte Carlo simulations to assess the performance obtained using different SMM FSO transmit and receive modal bases. We approximate the field disturbance caused by turbulence using a phase screen model. The phase-screen turbulence model is valid for the near-field links considered in this analysis, in which the phase aberration is the most significant turbulence distortion, and power scintillation is small. Intuitively, in the presence of turbulence, intense scintillation occurs when a beam passing through random phase distortions interferes with a spatially shifted version of itself as it propagates over a long atmospheric path. We introduce the influence of turbulence in the transmitting modes and consider an overlap integral to estimate the complex coupling coefficient between pairs of transmit and receive modes characterizing the random channel matrix 𝑯. In the Monte Carlo experiments, we generate 10 4 instances of the channel matrix 𝑯 for the different transmit and receive basis sets and use them to collect the statistics of capacity given by Eqs. ( 8)- [START_REF] Belmonte | Capacity of coherent free-space optical links using diversity-combining techniques[END_REF].

Figure [START_REF] Li | Space-division multiplexing: the next frontier in optical communication[END_REF] show plots of the spectral efficiency with 𝑁 = 45 modes as a function of coherence parameter (𝜔 B /δ) and a signal level described by SNR 𝛾 = 50 dB. It shows that better performance is obtained when statistical information about the channel is used. The capacity improvement obtained by exploiting even partial channel knowledge is substantial. It is interesting to observe the anticipated weak-turbulence behavior at small coherence parameter (𝜔 B /δ), where the entire transmit power is allocated to a small number of individual eigenvectors. Only when 𝜔~𝜔 B , for values of the coherence parameter (𝜔 B /δ)~1, do the performance of the LG modes and the stochastic eigenmodes become comparable.

SUMMARY

We need to understand fundamental limits to FSO system capacity caused by atmospheric turbulence and to recognize transmission schemes that can approach these limits in practice. This necessity is particularly real in SMM FSO systems. By considering separate spatial dimensions linked to different light modes, SMM is an anticipated transmission strategy to increase data capacity in near-field FSO communication. However, light modes affected by distortion and scintillation after propagation through atmospheric turbulence can degrade SMM performance.

Here we have considered the problem of finding optimal transmission beams for propagation through the random atmosphere with minimum distortion and found that these stochastic eigenmodes are LG modes with beam waist chosen adequately relative to the field coherence length in the receiver plane. Using these eigenmodes as transmit and receive bases minimizes signal degradation by turbulence, and minimizes the complexity of any signal processing for compensating the impact of turbulence. We have discussed how transmission of this mode family over a general

Fig. 1 .

 1 Fig. 1. (a) The effective beam width 𝜔 of the modes depends on the coherence width 𝛿. As an example, the plot shows the evolution of the third eigenmode (𝑛 = 3) with the coherence parameter (𝜔 B /δ). Modes under weak, medium , and strong turbulence conditions, with (𝜔 B /δ) equal to 1, 5, and 10, respectively, are considered. (b) Color map plot of the eigenvalues 𝜆 Ž of the atmospheric covariance function. They have been normalized to the largest eigenvalue 𝜆 $ . Eigenvalues are sorted in ascending order, as given by the mode index 𝑛, as a function of coherence parameter (𝜔 B /δ).

Fig. 2 .

 2 Fig.2. Spectral efficiency is shown as a function of coherence parameter (𝜔 B /δ) and considering a 50-dB SNR 𝛾. The analysis assumes several transmission strategies 𝑸 with 𝑁 = 45 modes: Instantaneous eigenmodes (green line) consider instantaneous channel information at the transmitter to partition the channel into orthogonal sub-channels and allocate transmit power with a water-pouring algorithm; Stochastic eigenmodes (blue line) contemplate statistical channel information at the transmitter to find the eigenvectors of the input covariance and use a water-pouring algorithm to address power allocation. Fixed LG modes (red line) assume that no channel information is available at the transmitter and split the transmit power equally over all transmit modes.

It is a basic assumption that points with inputs 𝝆 $ , 𝝆 & that are close in the receiver aperture are likely to have similar field values, and thus, under a Gaussian process view, it is the covariance function that defines nearness or similarity.

Mercer's theorem and Hilbert-Schmidt kernels are introduced in the theory of integral equations. (See, for instance,[START_REF] Moisewitsch | Integral equations[END_REF].)

Atmospheric turbulence has been studied extensively, and various theoretical models has been proposed to describe turbulence-induced phase degradation and amplitude fluctuations. The extended Huygens Fresnel principle is often used to formulate the field correlations at the receiver aperture. (See, for instance,[START_REF] Goodman | Speckle Phenomena in Optics. Theory and Applications[END_REF].)

If the transmitter receives the channel covariance matrix 𝐶 only instead of the concrete channel realization 𝑯, it does not have any information about the actual fading of each transmit-receive pair but possesses directional information regarding the signal subspaces that can be used for beamforming. The motivation for this approach stems from the fact that the channel statistics -i.e., the covariance 𝐶 of the atmospheric channelvary over much larger time scales than the instantaneous channel 𝑯. Therefore, the statistical information can be easily obtained by exploiting reciprocity, or by employing feedback channels with significantly lower bandwidth compared with instantaneous 𝑯 feedback systems.

atmospheric channel is likely to function in FSO, achieving high capacity transmission through increased spatial multiplicity. In our analysis, we have considered an SMM system that transmits the complete set of spatial light eigenmodes, with a shared central rotation axis, that spatially co-propagate through the atmospheric channel. Technology developed for (de)multiplexing and multi-input multi-output signal processing in fiber SMM systems can be leveraged to help make FSO SMM systems practical and vastly increase transmission capacity. Fundamentally lossless modal multiplexers and demultiplexers can map the stochastic eigenmodes to/from single-mode waveguides, making the implementation of modal transmission systems straightforward.


Hélène Trespeuch

Sherrie Levine

Shortly Afterwards

Richard Prince

of the 1970s, taking part in particular in "Pictures", the founding exhibition of Appropriationism, organised in 1977 by the art critic Douglas Crimp in the

advertisements of the cigarette manufacturer Marlboro, which featured cowboys from the American West, in which he identified highly effective advertising strategies: although no logo was included in the images, the photographs could easily be associated with the famous cigarette brand by the public of the time, and can still even today. It thus became relatively evident that the myth of the free and virile cowboy had been exploited to help conceal the harmfulness of the product being promoted2 .

It was in 1980 in particular that Sherrie Levine became known with her series After…. In this she rephotographed reproductions of famous photographs by great photographers of the past, such as Walker Evans and Edward Weston. Although an attempt to undermine the authority of the author is clear in this approach, and thus also the originality of the work concerned, it was also her intention to question the process of the photograph becoming an image, which, itself a multiple, is a technique that allows the multiple production of images despite the loss of quality, given that the initial image is not reproduced from the original negative.

Considered from this perspective, Sherrie Levine's appropriationism seems very remote from the work of Prince or Sherman, that attacks the stereotypes in our image-saturated society by which the construction of our individual identity -for both women and men -is influenced.

That being the case, questioning the feminist dimension of Sherrie Levine's work may seem incongruous as it clearly escapes the awareness of the viewer, who only focuses on her shots of photographic prints. However, as she stated in 1986: "As a woman, I felt there was no room for me. There was all this representation, in all this now painting, of male desire. The whole art system was geared to celebrating these objects of male desire. Where, as a woman artist, could I situate myself?"3 This view can be better understood in the light of the works Sherrie Levine produced in 1979: first, photographs of photographs of model mothers with their child, with the image itself cut in the shape of a child's head in profile (see the cover of Real Life of March 1979); second, photographs of images she found in the press or advertisements that depicted the active woman of the period, which she showed within the contour of the bust of an American president as seen on American coins (see the series President, 1979). Like Cindy Sherman, Sherrie Levine engaged in a manifest deconstruction of feminine ideals constructed by the these earlier works, the After… series might also be considered from this feminist approach 4 . From that time, the artists on which Levine cast her appropriationist eye were viewed in a new light: it was no longer a case of simply undermining the originality of celebrated photographers of modern art but of positioning herself as a woman in its still overly masculine historiography. Does Sherrie Levine's series After… deserve the description "feminist"? Probably, but it is certainly less direct than the earlier generation. With regard to this, Douglas Eklund, the curator of the exhibition "The Pictures Generation, 1974-1984", which was held at the Metropolitan Museum in New York in 2009, rightly made reference to two other appropriationist artists: "Sherman and Laurie Simmons have both commented on the divide between them and the elder generation of more explicitly feminist artists. They felt no need to identify themselves as feminist but instead found indirect ways to address similar concerns in a way that would not circumscribe their work as exclusively feminist in its meanings. To say you were a feminist woman was for them redundant. To make their art explicitly feminist seemed like allegiance to a cause that did not account for the specificity of individual experience. […] These younger women artists were part of feminism's second wave; for them, gender and sexuality were part of a larger nexus between representation and power […]." 5 This description is suited equally well to Sherrie Levine, who belongs with others to the group of "theoretical girls" identified in 1978 by the photographer Jeff Wall. 6 This decision to not (or no longer) develop an explicitly feminist art, but instead favour a subversive art necessitating a certain cultural baggage on the part of the viewer, was one of the criticisms most frequently directed at the art of Sherrie Levine, or at least it was one of the 4 See Benjamin Buchloh, "Allegorical Procedures: Appropriation and Montage in Contemporary Art" [Artforum, 1982], p. 135 : "Enter the female dandy, whose disdain has been sharpened by the experience of phallocratic oppression, and whose sense of resistance to domination is therefore more acute than that of her male colleagues, if they still exist." See Craig Owens, "The Discourse of Others: Feminists and Postmodernism", Hal Foster (ed.), The Anti-Aesthetic. Essays on Postmodern Culture (Seattle: Bay Press, 1983), p. 73: "[…] is her refusal of authorship not in fact a refusal of the role of creator as 'father' of his work, of the paternal rights assigned to the author by law? (This reading of Levine's strategies is supported by the fact that the images she appropriates are invariably images of the Other: women, nature, children, the poor, the insane…). Levine's disrespect for paternal authority suggests that her activity is less one of appropriation -a laying hold and grasping -and more one of expropriation: she expropriates the appropriators." 5 Douglas Eklund, "Image Art After Conceptualism: CalArts, Hallwalls, and Artists Space", The Pictures Generation, 1974-1984, op. cit., pp. 143-44. 6 Quoted in ibid. limits that was often pointed up. Indeed, how is it possible to develop an effective critical discourse by elaborating work that proves, in more than one way, to be coded, non-universal?

In order to grasp and appreciate the critical and feminist force of the After… series, viewers must necessarily know the history of photography during the first half of the 20th century in order to understand appropriationist logic; similarly, in order to fully grasp the scope of her technique of rephotography, they had also to be aware that Sherrie Levine was taking a feminist stance. 7 According to artist Martha Rosler, who in contrast was then engaging in an explicitly political art, the use of appropriation prevented Sherrie Levine's critical and, in particular, feminist aspirations from taking a convincing form. In 1982 she remarked on appropriationist techniques in general, though without mentioning Sherrie Levine by name, saying that " […] replicating oppressive forms, whether by quoting them directly or through the fashioning of simulacra, may replicate oppressiveness". 8 For the critic Abigail Solomon-Godeau, on the other hand, Sherrie Levine's work is indeed highly critical but that this critique is restricted to the world of art. In her article Living with Contradictions: Critical Practices in the Age of Supply-Side Aesthetics, Solomon-Godeau points out that it is above all a question of a critique within an art world that has remained overly modernist, that is to say that it celebrates originality, autonomy and the aura of the work of art, and, more specifically, of a radical challenge to the modernist conception of the art of photography. She notes in passing that the feminist dimension of Sherrie Levine's work "[…] represented a theoretically more sophisticated and necessary departure from the essentialism and literalism prevalent in many of the feminist art practices that emerged in the Seventies". 9 However, no analysis of this supposed feminist component of Sherrie Levine's work is offered.

Considering this theoretical debate of the 1980s objectively, it is especially interesting to note that most of the articles written about Sherrie Levine at the time (and still today) discussed the artist's reasoning, intentions and ambitions, in other words her position as an artist, all the while pointing up her radical undermining of the power of the author. This contradiction deserves study and to be considered in relation with the text to which Sherrie Levine and her 7 See, for example, Abigail Solomon-Godeau, "Living with Contradictions: Critical Practices in the Age of Supply-Side Aesthetics", Social Text, no. 21 (1989), p. 207: "As the photographic work of Sherrie Levine clearly demonstrates, the critical specificity of such practice is only operative, can only be mobilized, within a particular context". 8 Martha Rosler, "Notes on Quotes" [1982], Decoys and Disruptions, Selected Writings, 1975-2001(Cambridge MA & London: MIT Press, 2004), p. 139. 9 Abigail op. cit.,p. 194. commentators identify most to theoretically justify her use of appropriationism: The Death of the Author by Roland Barthes, published in 1968. 10 It should be recalled that in this essay the French theoretician refers predominantly to literature and, more specifically, to modern literature. Nonetheless, it is true that his arguments can be extrapolated to the visual arts. According to Barthes, the consecration of the author is a phenomenon that appeared in the modern era and was challenged by modern literature at the start of the 20th century by authors such as Mallarmé, Valéry and Proust, in whose polysemous, intertextual works the unique voice and unequivocal viewpoint of the author did not exist. This death of the author -the basis of his argument -is at the cost of birth of the viewer, something that Roland Barthes delights in. He says: "Once the Author is distanced, the claim to "decipher" a text becomes entirely futile.

To assign an Author to a text is to impose a brake on it, to furnish it with a final signified, to close writing. This conception is quite suited to criticism, which then undertakes the important task of discovering the Author (or his hypostases: society, history, the psyche, freedom) beneath the work: once the Author is found, the text is "explained", the critic has won; hence, it is hardly surprising that historically the Author's empire has been the Critic's as well […]" Thus, although Barthes attacks the myth of the originality of an artwork, and the tabula rasa celebrated by the avant-gardes, he does so to better celebrate a work that generously offers itself to the reader or viewer, though without necessarily being easy to understand or undemanding: that is to say a work whose various facets are not revealed purely by the words of the Author, or the knowledge that the reader/viewer has of the Author's way of thinking. In this respect, Sherrie Levine's work seems very distant from Roland Barthes' stance. Indeed, the works in the After… series is based on a confrontation between two authors: Sherrie Levine with Walker Evans, Sherrie Levine with Marcel Duchamp, etc. While her intention is to call into question the domineering position of the most famous male artists in the history of modern art, she clearly does not do so to efface herself in their presence but to rival them by means of her radical and subversive artistic action.

10 See in particular Sherrie Levine's "Statement" of 1982, in which she imitates and appropriates Roland Barthes' text The Death of the Author, which she concludes as follows: "A painting's meaning lies not in its origin, but in its destination. The birth of the viewer must be at the cost of the painter". Sherrie Levine, "Statement", in Style (Vancouver: Vancouver Art Gallery, 1982), p. 48. In Barthes' text, the last sentence reads "the birth of the reader is at the cost of the death of the author". This reference to Barthes' essay is reported in, for example, the introduction to The Pictures Generation, 1974Generation, -1984, op. cit., p. 17. , op. cit., p. 17. The domineering nature of their authorial position is conveyed by the relative silence of her works, whose appreciation by the viewer depends fundamentally on his or her interpretation of the work's explanatory details -the name of the artist, the title of the work chosen by the artist, and the year of its execution. It is these that provide the information necessary to avoid the viewer being duped into thinking that the works are simply bad copies of famous photographs.

The feminist force that Sherrie Levine packs into her works is necessarily limited by her choice of rephotography as a technique, which hinders the "birth of the viewer". By turning away from rephotographing women's magazines, she chose to abandon that part of her production in which the feminist dimension was most explicit, the one that did not demand very complex deciphering by its (American) audience. However, beginning with After…, the Author began to prevail over the viewer. The issues in play crystallized around the head-tohead of the artists' names (Levine vs. Evans), which also happens to be a gender clash (female artist vs. male artist) and periods (modernity vs. "postmodernism"). To use Barthes' terms, all these elements compel a "safety catch", and require these works to be viewed through the prism of the artist (her person, identity, ambitions, etc.). Her works cannot be independent because their highly ambiguous nature might cause the viewer to read them wrongly, and not just fail to notice an important characteristic -that of appreciating the works' original forms. In consequence and unsurprisingly, in his famous essay The Discourse of Others: Feminists and Postmodernism, which was supposedly one of the few at the time to take an interest in appropriationist women artists as women, the critic Craig Owens perfectly falls within the critical logic that Roland Barthes considered obsolete: he "gives an Author to a text", underlining the need to analyse the feminist dimension of the works by female appropriationist artists for the very reason that they are women. Does this mean that the feminist force of a work is incompatible with the author's deliberate self-effacement? This is an interesting question but it deserves proper study. This article limits itself to remarking that Sherrie Levine both announces and claims to represent Barthes' death of the author, while maintaining her own authorial authority in a devious manner, thereby estranging the dismantlement of the myth of originality from that of authorial authority. This paradoxical attitude had the merit of revealing the feminist input to her thinking, but the drawback of impeding the full birth of the viewer, including his or her freedom and pleasure.11

Hélène

  Trespeuch is a senior lecturer in the history of contemporary art at the Université Paul-Valéry -Montpellier 3. She is the author of La Crise de l'art abstrait ? Récits et critique en France et aux États-Unis dans les années 1980 (Presse universitaires de Rennes, 2013), for which was awarded the Prix Olga Fradiss by the Fondation de France in 2014. In 2016 she founded the review exPosition (http://www.revue-exposition.com), of which she is the director. Until now her research has primarily regarded the historiography of 20th-century art, abstract art, and the art scene during the 1980s. On Sherrie Levine and her work, she authored an article published in Marges: "Sherrie Levine, de l'appropriationnisme au simulationnisme", Marges, no. 17, 2013, http://marges.revues.org/127.

The myth of the cowboy, a key figure in the construction of the American identity, has in particular been promulgated in Western films since the early 20th century.

Sherrie Levine, quoted by Gerald Marzorati, "Art in the (Re) Making", in Art News,Vol. 85, May 1986, pp. 96-97.

In 1973 Roland Barthes published a short essay called Le Plaisir du texte, in which he already questioned the death of the author. See in particular the television interview of Roland Barthes on 19 March 1973, accessible on the site of the INA: http://www.ina.fr/video/CPF10005880 (consulted 16/02/2017).

https://awarewomenartists


text:
A Le Kernec

L Canuet

A Maho

M Sotom

D Matter

L Francou

for high throughput satellites and the H2020 VERTIGO project

Keywords: optical feeder links, telecom satellite systems, adaptive optics, high optical power generation, channel coding

diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The competitive telecommunication industry is more and more oriented towards mobile devices and broadband connectivity with its associated challenges the cost reduction per bit supplied to customers and higher capacity. To stay competitive and face the concurrence of the ground fiber networks but also to deliver similar telecommunication services to people in unserved or underserved regions by ground networks, satellite operators have to propose services of comparable performance and quality to ground operators offer, and therefore are requesting cost efficient solutions from satellite systems vendors. One of the first technological responses from the Space industry, which is in turn challenged, is to develop more competitive solutions like Very High Throughput Satellite (VHTS), based on a multi-spot coverage allowing for frequency reuse and thus, increased capacity. In order to serve this increasing demand, next generation of VHTS will need to support both higher throughput, and flexibility to adapt to traffic demand evolution across the coverage area, while at the same time decreasing the cost per transmitted bit. To meet these market expectations, such high capacity systems require a continuously rarefying resource, frequency band. To solve this issue, Q/V band systems are currently deployed, and higher frequency bands are considered for the future. Nonetheless, the technologies beyond Q/V band still exhibit a low maturity and a leap to optical technologies can be a serious alternative.

THE CAPACITY GROWTH TREND OF SATELLITE SYSTEMS

In Europe, the KA-SAT satellite, the first High Throughput Satellite (HTS) covering Europe and delivering broadband internet services, launched in 2010 exhibits a total throughput beyond 90 Gbit/s several times the capacity of conventional satellites of previous generation. In spite of this large capacity, Eutelsat declined in 2015 new subscription because of satellite saturation. Intelsat 33e launched in August 2016 and covering Europe, Middle East, Africa and Asia Pacific is another HTS delivering brodband services to fixed and mobile network operators, aeronautical and maritime mobility service providers and government customers as well as regionalized content for media customers. Its capacity can reach 60 Gbps. This growing satellite throughput trend has been even more illustrated in US by satellites launched during last years and recent announcements: Viasat-1, launched in 2011, exhibited a capacity in excess of 140 Gbps, higher than all satellites covering North America combined at the time of its launch making it the world's highest capacity communication satellite. EchoStar 19-Jupiter 2 launched in December 2016 to provide broadband access services in North America was at that date the communication satellite exhibiting the highest capacity in the world with a capacity of 180 Gbps. Six months later, Viasat-2 was launched offering an internet services through a Ka-band multispot coverage with a throughput as high as 300 Gbps. It is currently still the world's highest capacity communication satellite but Echostar signed a contract in 2017 for the building of the 500 Gbps capacity satellite Jupiter-3/EchoStar 24 to provide broadband internet access to the Americas while Eutelsat ordered KONNECT VHTS in 2019, a satellite expected to deliver around 500 Gbps. Viasat announced its next generation satellite system (Viasat-3), a trio of geostationary high capacity Ka-band satellites exhibiting a throughput capacity of 1 Terabit each and covering Americas, Europe, Middle East and Africa and Asia-Pacific region.

To feed such VHTS with the maximum achievable data traffic and thus fully exploit the satellite system, a large number of gateways (ground stations) are necessary. As an example, based on current RF technology, 20 gateways are used for Viasat-1, 45 (smaller) gateways for Viasat-2 and hundreds of gateways expected for the trio satellite Viasat-3. Even if, specific developments are engaged to dramatically reduce the cost of gateways, the ground segment of such very high throughtput satellite systems will represent a huge CAPEX. Among the different technologies currently investigated to offer higher feed capacities, optical feeder links are perceived as a way to overcome the RF spectrum bottleneck and increase the achievable throughput at an affordable ground segment complexity and cost. Indeed, optical technology is expected to allow for the transmission of the whole throughput through a single (or a very limited number of) optical gateway(s). Despite the necessity of implementing site diversity to cope with cloud coverage, the total number of sites is currently evaluated to 10-12 sites in Europe.

As a result, even if the evaluation of the ground segment cost and feasibility is still today under consolidation, the use of optical links could strongly reduce the required number of gateways and so the required CAPEX for the whole satellite system deployment (space and ground segments) to meet the future throughput requirements of VHTS in the range of 1 Tbps and beyond.

Well-known assets of optical technologies for optical links are (here mostly assuming the use of the 1.5µm technology):

• No spectrum regulation

• Large optical bandwidth available

• Large availability of components from terrestrial telecom networks applications

• Leveraging large R&D efforts

• Possible seamless interfacing to photonic payloads [START_REF] Sotom | Photonic RF Repeaters for Broadband Telecom Satellites: System Demonstration and Test Results[END_REF] • Possibly future integration with terrestrial optical networks

TOWARDS TERABIT/S CAPACITY

Typical high level system requirements relevant for feeder links

The communication links between the ground and the satellite are schematically shown in figure 1. Such a high capacity system relies on a (RF) multispot coverage with frequency bands reuse. In the targeted system, the high throughput capacity feeding the satellite is made possible via one or a few simultaneous laser links making an extensive use of wavelength division multiplexing to meet the capacity requirements. Generic high level system requirements are as follows:

•

Optical feeder link family options

The main purpose of the feeder link is to convey the total up and down RF bands while ensuring the required level of service. Several "families" of optical feeder link architecture are envisaged to transmit these bands relying either on digital or analog modulations. All these options have in common to make a massive use of dense wavelength division multiplexing in optical C/L bands concurrently with high power optical amplification, while using opto-mechanical techniques (such as adaptive optics on-ground) and digital techniques (interleaving and coding) to mitigate impairments induced by atmospheric turbulence. Each family of architectural options offers a variety of implementations, the details of which are out of the scope of this paper, and only the high-level principle are reminded hereunder.

A key aspect having strong consequences on optical link requirements and implementation is the necessity or not for a transparent system. This feature derives from the system capability (or not) to transmit any waveform to the users.

In the following, the figures present a high-level functional view of such optical feeder uplink architectures which does not presume of the hardware implementation. Similar architectures, except for the regenerative option, can also be derived for the downlink but are not presented here.

Regenerative architectures

In this architecture family, baseband data are optionally encapsulated in frames generated from standard waveforms including Forward Error Coding up to mapping (e.g. based on DVB-S2 standard), undergo specific coding/interleaving for the optical channel, and are transmitted over optical feeder link through digital optical modulation (OOK, DPSK, coherent …). Optical signals are demodulated on-board and RF signals to be transmitted to users are generated on-board using RF modulators. This option is in principle applicable only on forward path as each RF carrier has to be demodulated on-board and is thus not compatible with a very large number of RF carriers (from users on return path).

Digital transparent architectures

This options family uses waveforms similar to those used in the regenerative case for the optical transmission. However instead of transmitting baseband data, RF channels are digitized through ADCs before applying channel coding and interleaving. Also, on-board, the RF channels are reconstructed through DACs before being frequency converted and amplified to be transmitted to users. Comparable architectures can be described on the return path. While this architecture type support the desired transparency regarding the RF waveform used, thus offering the possibility for an operator to modify this waveform during the satellite lifetime, this is at the expense of a large increase of the total throughput to be conveyed on the optical feeder link. Indeed, depending on system and subsystems requirements, an expanding factor relative to the RF band to be digitized shall be applied, which leads to an increase in bandwidth typically from 10 to 16. This drawback makes this option relevant only for low to medium throughput.

Analog transparent architectures

The last options family is, in its basic form, comparable to a classical RF bent-pipe satellite repeater as all operations take place in the analog domain. The RF spectra to be transmitted (from DVB standard for example) modulate directly (without any incursion in digital domain) the optical carriers. On board the RF signals are recovered directly through photo-detection and easily handled by the RF payload. If no additional operations are applied up-and downstream the optical link, this option may suffer strong turbulence-induced impairments not compensated by coding/interleaving. As a result, to be viable, particularly on the uplink which induces the strongest perturbations, the analog options shall be made more robust through digital assistance by relevant processing up-and downstream the optical link.

A promising but challenging technology

Optical feeder links are considered as a promising technology to cope with the continuous growth in capacity demand and surpass classical RF technology limitations. However the inherent assets of optical technology shall not conceal the still remaining challenges to be overcome to make it relevant for operational missions. To date, some experimentations in flight (a comprehensive list is provided in [START_REF] Toyoshima | Recent Trends in Space Laser Communications for Small Satellites and Constellations[END_REF]) or on ground [3][4], have provided precious results on several topics to be mastered for a successful implementation of optical feeder links. However, even if these demonstrations implemented relevant parts of an optical feeder link, all required aspects were not addressed simultaneously.

THE H2020 VERTIGO PROJECT

VERTIGO (Very High Throughput Satellite Ground Optical Link) is a 3-year H2020 project funded by the European commission [6] which focuses on the feasibility of the optical link itself regardless of site diversity aspect to meet overall system availability requirement, and aims at demonstrating the technologies required for the implementation of very high capacity optical feeder links.

The ambition of VERTIGO is to establish the required breakthrough by developing and demonstrating the on-board and ground concepts and technologies enabling for a drastic increase of the link throughput towards and beyond 1 Tbps. As shown in figure 5 below, it is all about setting the technology solutions to bring the performance level beyond existing experimentations of free-space optical links for space applications tested in-flight or on-ground. Started during summer 2019, VERTIGO will implement two ground demonstrations, indoor (laboratory environment) and outdoor (several kilometers optical transmission through the atmosphere), to address the following 3 primary objectives:

• Throughput increase: while current space optical comms use 1bit/symb modulation formats, VERTIGO will prove advanced schemes with higher power and/or spectral efficiency (e.g. DPSK [START_REF] Maho | Assessment of the effective performance of DPSK vs. OOK in satellite-based optical communications[END_REF], DQPSK, coherent …), including RF-over-Fiber approach,

• High optical power generation: to close the demanding link budgets, VERTIGO will develop on-board and ground means to raise the transmitted optical power, not only based on amplifier power increase, but also on incoherent/coherent power combining,

• Mitigation of atmospheric propagation impairments, to make full use of throughput and power increases. Optomechanical techniques using adaptive optics and aperture diversity, and digital processing (error correction coding/interleaving) will be developed to enhance the robustness of future optical feeder links.

Thus, VERTIGO is based on three pillars each addressing a key issue for high throughput optical links: intrinsic communication chain capability to transmit and receive very high bitrates, high and very high-power generation and atmospheric perturbations. These aspects are closely linked since the solution(s) to one of them is(are) necessary but not sufficient to allow for high throughput transmissions. VERTIGO approach is to address each key issue with at least one solution and to combine them in an unprecedented manner. The figure 6 hereafter provides a quick view of this approach.

To reach these objectives, VERTIGO will lean on a highly skilled consortium composed of CREONIC, ETH Zürich, Fraunhofer HHI, G&H, Leo Space Photonics R&D, ONERA, Thales Research & Technology, Thales Alenia Space in France and Switzerland.

CONCLUSION

The capacity bottleneck induced by the scarcity of spectral resource as well as the high ground segment costs of very high throughput satellite systems (VHTS) foster the emergence of new solutions in particular to feed satellites. Freespace optical communication link technology is considered as a serious candidate to overcome the problems of spectral resource and costs of the ground segment. However, technical challenges still remain to make this approach a viable solution. The H2020 project VERTIGO ambitions to implement and demonstrate the relevant technologies for optical feeder links in a representative environment and pave the way towards very high throughput satellite systems, therebycontributing to maintain EU technology at the leading edge and to strengthen industry competitiveness.

Figure 1 .

 1 Figure 1. Illustration of the optical feeder and RF user links.

Figure 2 .

 2 Figure 2. Regenerative optical feeder link architecture.

Figure 3 .

 3 Figure 3. Digital transparent optical feeder link architecture.

Figure 4 .

 4 Figure 4. Analog transparent optical feeder link architecture.

Figure 5 .

 5 Figure 5. Comparison of past and planned FSO links in function of the product of the throughput by the squared distance.

Figure 6 .

 6 Figure 6. How VERTIGO technologies serve VERTIGO objectives.

Table 1 .

 1 Illustration of satellites capacity growth during last decade and in next few years.

	Satellite	Launch year	Estimated capacity
	Eutelsat
	Ka-Sat	2010	90 Gbps
	KONNECT	2019	75 Gbps
	KONNECT VHTS	2022	500 Gbps
	EchoStar
	Jupiter 1	2012	120 Gbps
	EchoStar 19 -Jupiter 2	2016	180 Gbps
	Jupiter 3 -EchoStar 24	2021	500 Gbps
	Intelsat
	Intelsat 33	2016	60 Gbps
	ViaSat
	Viasat-1	2011	140 Gbps
	Viasat-2	2017	300 Gbps
	Viasat-3	2020-2022	1 Tbps/sat. (3 satellites)

ACKNOWLEDGMENT

The VERTIGO project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 822030.

The authors thank all their VERTIGO project partners for their kind and fruitful cooperation.


text:
Sylvain Poulenard

Benjamin Gadat

Lyonel Barthe

Ronald Garzón-Bohórquez

Protection schemes for optical communication between optical ground station and a satellite

Keywords: Optical Communications, Optical Satellite Feeder Link, Forward Error Codes

Forward Error correcting Code design (FEC) and implementation for optical feeder link is reported. The designed LDPC codes present very good performances (BER < 10 -9 ) while requiring less than 20% hardware resources of FPGA compatible with space constraints. Protection scheme based on these LDPC codes combined with a physical layer bit interleaver is presented. Links sizing for GEO optical feeder link at 30° elevation angle under strong turbulence shows that uplink might require up to 100W per link.

INTRODUCTION

Digital optical communication links are currently operational since 2016 for inter-satellite links in the European Data Relay System [START_REF] Heine | Operational experience of the Alphasat and EDRS laser communication system[END_REF]. Since 2010, digital optical communication links at 1.55µm are foreseen to be a valuable alternative to the conventional radio-frequencies links between the spacecraft LEO/GEO and a fixed Earth ground station for missions such as Internet delivery, Earth observation data download, and Deep Space data download. Such links presents the technical challenge of crossing the atmosphere where clouds and turbulences impair the transmission. Regarding LEO Earth observation downlink and Deep Space data download, the Consultative Comity for Space Data System (CCSDS) set up few years ago working groups for standardization of such optical links [START_REF]CCSDS Standardisation in Optical Communication[END_REF], and demonstrations have been/ are running in many countries [START_REF] Biswas | Optical payload for lasercomm science (OPALS) link validation during operations from the ISS[END_REF][START_REF] Fuchs | OSIRISv1 on Flying Laptop: Measurement Results and Outlook[END_REF][START_REF] Taenaka | In-orbit verification of small optical transponder (SOTA): evaluation of satellite-to-ground laser communication links[END_REF][START_REF] Boroson | Overview and results of the Lunar Laser Communication Demonstration[END_REF]. Regarding the mission of broadband access for Internet delivery, European space agencies and European Commission have financed system studies to evaluate technological feasibility of GEO optical feeder links since several years [START_REF] Roy | Optical Feeder links for High Throughput Satellites[END_REF][START_REF] Poulenard | Digital optical feeder links system for broadband geostationary satellite[END_REF]. In 2016, the European Space Agency has launched the HydRON (High Throughput Optical Network) project targeting optical interconnections in the Tbps regime for inter-satellite links and feeder links [START_REF] Hauschildt | HydRON: High thRoughput Optical Network[END_REF]. For such GEO feeder links, the optical band presents the advantages of tremendous bandwidth capacity and no interference but also the drawbacks of being blocked by clouds and strongly distorted by the atmospheric turbulence when it crosses the atmosphere. Cloud blockages are commonly mitigated by site diversity between Optical Ground Station (OGS) geographically spread over the coverage area [START_REF] Poulenard | Digital optical feeder links system for broadband geostationary satellite[END_REF]. Atmospheric turbulence impacts are mitigated thanks to the implementation of adaptive optic system at each OGS reducing signal power fading [START_REF] Védrenne | First experimental demonstration of adaptive optics pre-compensation for GEO feeder links in a relevant environment[END_REF][START_REF] Védrenne | Adaptive optics for high data rate satellite to ground laser link[END_REF] and thanks to the usage of a protection scheme to retrieve the information while the transmission has been corrupted by errors. Joint optimization between adaptive optic system and protection scheme is required to properly design and operate the link. Such optimization shall be compatible with space constraints in particular available power consumption/dissipation. The paper is organized as follows: §2 presents the mission scenario, §3 describes the atmospheric transmission channel with adaptive optic system, §4 selects a protection schemes based on a physical layer interleaver and an error correcting codes, §5 provides the performances of error correcting codes designed to be compatible with space constraints of feeder links, §6 provides optical link sizing and space hardware resources required for encoding/ decoding, §7 is the conclusion.

MISSION SCENARIO

Figure 1 depicts the mission scenario. A geostationary satellite provides broadband internet access to the end-users. The bidirectional link between the GEO and the fixed Earth station is the feeder link. The feeder link is a 1.55µm laser link between a 60cm Optical Ground Station (OGS) and 26cm On-Board Optical Terminal (OBT). Both terminals are monostatic. The OBT uses the same primary mirror for transmission (Tx) and reception (Rx). The OGS uses the full 60cm diameter primary mirror for reception and a part of it (20cm) for uplink transmission. The specificity of the On-Board Terminal is due to space constraints to achieve 15 years lifetime on the GEO orbit. Such space constraints limit the hardware available. In particular, to the authors' knowledge, no space qualified Analog-to-Digital Converter (ADC) compatible with 10Gbauds signals are available on the market. As a consequence, hard inputs for the decoder are considered. For the OGS, we consider soft input quantified on 6 bits or more for the decoder. 10Gbps NRZ OOK or DPSK [START_REF] Anfray | Assessment of the performance of DPSK and OOK modulations at 25 Gb/s for satellite-based optical communications[END_REF] 10Gbps NRZ OOK or DPSK [START_REF] Anfray | Assessment of the performance of DPSK and OOK modulations at 25 Gb/s for satellite-based optical communications[END_REF] 3. ATMOSPHERIC TRANSMISSION CHANNEL

Atmospheric turbulence strength

Atmospheric turbulence strength varies function of sites locations, season, hours, minutes… etc. Statistics relevant for GEO feeder link should cover Fried parameter, anisoplanetic angle, wind speed. Such data is only partially available for some sites, mainly astronomical sites. Different initiatives over Europe aim at fulfilling this lack of data. [START_REF] Cheinet | Global predictions of the optical sensing through turbulence[END_REF] derives Cn² profile from ECMWF data (European Center for Medium Range Weather Forecasts). Nevertheless there are still some uncertainties on the capability of such model to derive accurate statistics. ESA adopted a site survey approach consisting into deploying a network of instruments able to retrieve Cn² profile of day and night [START_REF] Hauschildt | ESAs scylight programme: activities and status of the high throughput optical network HyDRON[END_REF]. Before such data bank being available, Airbus DS and ONERA-DOTA have jointly worked to define a nominal case and a strong case of atmospheric turbulence strength. Such work was performed in 2019 on the Cn² profile data bank provided by J. Osborn from Durham University [START_REF] Osborn | Optical turbulence profiling with Stereo-SCIDAR for VLT and ELT[END_REF]. Anisoplantism angle at 1550nm on the line of sight 26µrad 11µrad

Power attenuation time series due to turbulence

Power attenuations time were computed by ONERA-DOTA. Such time series take into account propagation through the turbulence atmosphere and adaptive optic system, the principle of numerical computation is explained in [START_REF] Védrenne | Turbulence effects on bi-directional ground-to-satellite laser communication systems[END_REF] The impact of the turbulence is worst on the uplink than on the downlink due to the anisoplanetism effect [START_REF] Conan | Adaptive optics for GEO-Feeder links: analysis of point ahead anisoplanatism impact via reciprocity based models[END_REF].

PROTECTION SCHEMES CONCEPTS

The free space optical community has investigated many different protection schemes [12 -16]: physical layer error correcting code combined with an interleaver, packet level erasure code, a combination of all of them. We investigate solutions based on physical layer error correcting code combined with an interleaver. A good protection schemes in general shall have good performances, i.e.:

-To add a reasonable extra-latency and overhead: This implies limiting the number of decoding iterations, limiting the size of the interleaver, targeting high coding ratios. -To provide great power saving on the link budget: This implies modern codes with great coding gain.

-To be able to adapt to atmospheric turbulence strength variation easily: This implies designing the code so that changing code rate is easy / real-time. For resources starved applications such as space application, a good protection shall also require low on-board resources:

-This implies limiting resources for encoding algorithms for the downlink.

-This implies limiting resources for decoding algorithm for the uplink. For the mission scenario targeted, the protection scheme shall be compatible with 10Gbps channel rate at least.

FORWARD ERROR CORRECTION CODES

Various FEC code families have already been investigated by the free space optical community [12 -17]: graph based codes ( polar code, or sparse code such as Low Density Generator Matrix, Raptor, LT, Low-Density Parity-Check, DVB-S2), algebraic (MDS like Reed Salomon, BCH, Turbo Product Codes with BCH component), and trellis based convolutional codes and their concatenations (serial turbo codes). In [START_REF] Poulenard | Forward Error Correcting Code for High Data Rate LEO Satellite Optical Downlinks[END_REF], quasi cyclic regular LDPC codes were selected to achieve a BER after decoding below 10 -9 with decoder throughput above 10Gbps on high end FPGAs [START_REF] Pignoly | High data rate and flexible hardware QC-LDPC decoder for satellite optical communications[END_REF].

Downlink

For the downlink we investigate IRA LDPC codes [START_REF] Gadat | Error correcting codes and interleaving for high data rate optical communications[END_REF]. Figure 3 provides the FEC performances functions of the Es/N0. The FEC performances of the DVB-S2 codes (LDPC + BCH) are provided as benchmark values. DVB codes present the drawback of requiring too many hardware resources for encoding and decoding. The decoding algorithm is Normalized Min-Sum (NMS) algorithm [START_REF] Fossorier | Reduced complexity iterative decoding of low-density parity check codes based on belief propagation[END_REF] with a normalization factor of 0.75 and a maximum of 25 decoding iterations

Uplink

For the uplink, we investigate LDPC codes based on Finite Geometry (FG) [START_REF] Poulenard | Forward Error Correcting Code for High Data Rate LEO Satellite Optical Downlinks[END_REF]. Figure 4 provides the FEC performances functions of the Es/N0. The decoding algorithm is Gallager B extended algorithm [START_REF] Richardson | The capacity of low-density parity check codes under message-passing decoding[END_REF] (namely the Gallager with Erasure) with a number of decoding iterations limited to 8 iterations.

The previous performances of the DVB-S2 codes (LDPC + BCH) with Normalized Min-Sum (NMS) algorithm [START_REF] Fossorier | Reduced complexity iterative decoding of low-density parity check codes based on belief propagation[END_REF] are superimposed in Figure 4 to provide the reader with order of magnitude of achievable gain in the case of no space constraints. The proposed FG LDPC codes operate between 0.8 to 1.3dB from the channel capacity for hard decoder.

OPTICAL LINK SIZING

As it can be seen from the Figure 5, even with modern FEC described in §5.2, the uplink requires very high power optical amplifier. Greater on board terminal and optical ground station could be used to reduce the required optical power.

(a) (b)

CONCLUSIONS

In this paper, LDPC codes are designed to achieve good performances while requiring small hardware resources to be compatible with space implementation constraints. LDPC codes are designed differently on the uplink and on the downlink for the considered GEO feeder link user case. The required resources for encoding and for decoding the 10Gbit/s channel data rate stream is less than 20% of a FPGA to be space qualified. Despite very good FEC performances and long interleaver, the optical uplink under strong turbulence require very high optical power booster on ground.

Figure 1 :

 1 Figure 1: Mission scenario -the feeder link is the bidirectional link between the GEO satellite and the fixed Earth station gateway. The user links are classical radio-frequency links.

Figure 2 :

 2 Figure 2: Power attenuation time series due to turbulence with adaptive optic system. A deformable mirror applies a pre-compensation to the uplink beam which is the conjugate of the distortion measured on the downlink beam. (a) uplink strong case / (b) downlink strong case

Figure 3 :

 3 Figure 3: Bit Error Rate function of the Es/N0 for AWGN noise model considering soft inputs decoder

Figure 4 :

 4 Figure 4: Bit Error Rate function of the Es/N0 for AWGN noise model considering hard inputs decoder

Figure 5 :

 5 Figure 5: Optical booster power (HPOA) required to achieve an error free uplink at 30° elevation function of the interleaver duration and the FEC coding ratio. (a) nominal turbulence strength, (b) strong turbulence strength

Table 1 : Optical terminal characteristics

 1

		On-Board Terminal	Optical Ground Station terminal
	Optical Head	26cm (Tx and Rx) with fine pointing	60cm for Rx and 20cm for Tx with adaptive optic
		mirror. Monostatic architecture.	system. Monostatic architecture.
	Decoder inputs	Hard inputs	Soft inputs on ≥ 6 bits
	Communication chain

Table 2 : Turbulence parameter function of strength, from nominal case to strong case

 2

	Turbulence strength	Nominal	Strong
	Fried parameter at 1550nm on the line of sight	12cm	4cm

Table 3 : FGPA resources for 10Gbit/s channel rate

 3

	On board	On-ground

Sylvain.poulenard@airbus.com ; Tel: + 33562196472; Airbus Defence and Space ,31 rue des Cosmonautes Toulouse.

ACKNOWLEDGEMENTS

The authors would like to thank ONERA-DOTA for the collaborative work. The authors would like to thank University of Durham for providing their Cn² data bank to ONERA-DOTA. The authors would like to thank CNES for fruitful discussion around this work. Parts of the work presented here were funded by CNES for the R&T DS/NT/ST-2017.0010579 and for the R&T R-R-S18/TC-0005-097.