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Journal of Lightwave Technology

Photon Information Efficient Communication Through Atmospheric Turbulence—Part II: Bounds on Ergodic Classical and Private Capacities

Nivedita Chandrasekaran, Jeffrey H. Shapiro, Ligong Wang

Published Jan 24, 2014• 10 citations
Publisher

Abstract

Vacuum-propagation optical communication with high photon efficiency (many bits/photon) and high spectral efficiency (many bits/s ·Hz) requires operation in the near-field power transfer regime with a large number of spatial modes. For terrestrial propagation paths, however, the effects of atmospheric turbulence must be factored into the photon and spectral efficiency assessments. In Part I of this study [N. Chandrasekaran and J. H. Shapiro, “Photon Information Efficient Communication Through Atmospheric Turbulence-Part I: Channel Model and Propagation Statistics,” J. Lightw. Technol., vol. 32, no. 6, pp. 1075-1087, Mar. 2014], modal-transmissivity statistics were derived for the turbulent channel that depend solely on the mutual coherence function of the atmospheric Green's function, and these bounds were evaluated for ~ 200 spatial-mode systems whose transmitters used either focused-beam (FB), Hermite-Gaussian (HG), or Laguerre-Gaussian (LG) modes and whose receivers either did or did not employ adaptive optics. This Part II paper derives upper and lower bounds for the ergodic Holevo capacities of classical and private information transmission over the multiple spatial-mode turbulent channel that can be evaluated from Part I's transmissivity statistics. Also included are bounds on the ergodic capacity for on-off keying encoding and direct detection. It is shown that: 1) adaptive optics are not necessary to realize high photon information efficiency and high spectral efficiency simultaneously; 2) an FB-mode system with perfect adaptive optics outperforms its HG-mode and LG-mode counterparts; and 3) the converse is true when adaptive optics are not employed.

DOI
10.1109/jlt.2013.2296853

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Vacuum-propagation optical communication with high photon efficiency (many bits/photon) and high spectral efficiency (many bits/s ·Hz) requires operation in the near-field power transfer regime with a large number of spatial modes. For terrestrial propagation paths, however, the effects of atmospheric turbulence must be factored into the photon and spectral efficiency assessments. In Part I of this study [N. Chandrasekaran and J. H. Shapiro, “Photon Information Efficient Communication Through Atmospheric Turbulence-Part I: Channel Model and Propagation Statistics,” J. Lightw. Technol., vol. 32, no. 6, pp. 1075-1087, Mar. 2014], modal-transmissivity statistics were derived for the turbulent channel that depend solely on the mutual coherence function of the atmospheric Green's function, and these bounds were evaluated for ~ 200 spatial-mode systems whose transmitters used either focused-beam (FB), Hermite-Gaussian (HG), or Laguerre-Gaussian (LG) modes and whose receivers either did or did not employ adaptive optics. This Part II paper derives upper and lower bounds for the ergodic Holevo capacities of classical and private information transmission over the multiple spatial-mode turbulent channel that can be evaluated from Part I's transmissivity statistics. Also included are bounds on the ergodic capacity for on-off keying encoding and direct detection. It is shown that: 1) adaptive optics are not necessary to realize high photon information efficiency and high spectral efficiency simultaneously; 2) an FB-mode system with perfect adaptive optics outperforms its HG-mode and LG-mode counterparts; and 3) the converse is true when adaptive optics are not employed.
Optical communication with high photon-efficiency (many bits/photon) and high spectral efficiency (SE) (many bits/s-Hz) cannot be achieved unless multiple spatial modes are employed. For vacuum propagation, it is known that achieving 10 bits/photon and 5 bits/s-Hz requires 189 low-loss spatial modes at the ultimate Holevo limit and 4500 such modes at the Shannon limit for on-off keying with direct detection. For terrestrial propagation paths, however, atmospheric turbulence corrupts multiple spatial-mode operation. This paper derives power-transmissivity bounds and average intermodal crosstalks for the turbulent channel that depend solely on the mutual coherence function of the atmospheric Green's function. These statistics are then evaluated for ~ 200 spatial-mode systems whose transmitters use either focused-beam, Hermite-Gaussian (HG), or Laguerre-Gaussian (LG) modes and whose receivers either do or do not employ adaptive optics. It is shown that: (1) adaptive optics are not necessary for achieving both high photon information efficiency (PIE) and high SE; (2) systems employing HG or LG modes achieve the same capacities through turbulence; and (3) the orbital angular momentum carried by LG modes does not provide turbulence immunity. In the companion paper [N. Chandrasekaran, J. H. Shapiro, and L. Wang, “Photon Information Efficient Communication Through Atmospheric Turbulence-Part II: Bounds on Ergodic Classical and Private Capacities,” J. Lightw. Technol., vol. 32, no. 6, pp. 1088-1097, Mar. 2014], the transmissivity bounds are used to quantify the turbulence-induced loss in PIE versus SE performance for these mode sets.
The capacity of the bosonic channel with additive Gaussian noise is unknown, but there is a known lower bound that is conjectured to be the capacity. We have quantified the gap that exists between this known achievable rate and rates achievable by the known methods of detection including direct, heterodyne, and homodyne detection. We have also quantified these capacities in the case of multiple independent spatial modes in terms of spectral and photon efficiency. Furthermore, we have considered the ergodic and outage capacities of fad-ing channel models for far-field and near-field propagation through atmospheric turbulence. For the far field, good models for the transmissivity statistics are known. For the near field we establish bounds on these capacities, and we show that these bounds are reasonably tight. Finally, we extend the results for ergodic capacity to the case of multiple spatial modes where a turbulent atmosphere results in crosstalk between different spatial modes.
Due to the transfer of the angular spectrum of the pump beam to the two-photon state in spontaneous parametric downconversion, the generated twin photons are entangled in their spatial degrees of freedom. This spatial entanglement can be observed through correlation measurements in any set of modes in which one may choose to perform measurements. Choosing, e.g., a Hermite-Gaussian (HG) set of spatial modes as a basis, one can observe correlations present in their spatial degrees of freedom. In addition, these modes can be used as alphabets for quantum communication. For global quantum communication purposes, we derive an analytic expression for two-photon detection probability in terms of HG modes, taking into account the effects of the turbulent atmosphere. Our result is more general as it accounts for the propagation of both signal and idler photons through the atmosphere, as opposed to other works considering one photon's propagation in vacuum. We show that while the restrictions on both the parity and order of the downconverted HG fields no longer hold, due to the crosstalk between modes when propagating in the atmosphere, the crosstalk is not uniform: there are more robust modes that tend to keep the photons in them. These modes can be employed in order to increase the fidelity of quantum communication.
The efficient coupling of photons from a free-space quantum channel into a single-mode optical fiber (SMF) has important implications to quantum network concepts involving SMF interfaces to quantum detectors, atomic systems, integrated photonics and direct coupling to a fiber network. Propagation through atmospheric turbulence however leads to wavefront errors that degrade mode-matching with SMFs. In a free-space quantum channel, this leads to photon losses in proportion to the severity of the aberration. This is particularly problematic for satellite-Earth quantum channels where atmospheric turbulence can lead to significant wavefront errors. This report considers propagation from a transmitter in low-Earth orbit to a terrestrial ground station and evaluates the efficiency with which photons couple either through a circular field stop or into a SMF situated in the focal plane of the optical receiver. The effects of atmospheric turbulence on the quantum channel are calculated numerically and quantified through the quantum bit error rate and secure key generation rates in a decoy-state BB84 protocol. Numerical simulations include the statistical nature of Kolmogorov turbulence, sky radiance, and an adaptive optics system under closed loop control.