Ultimate Limits on Photon and Spectral Efficient Communication through Atmospheric Turbulence

Bounds on the ergodic classical and private capacities for optical communication through turbulence are presented, showing that high photon and spectral efficiencies can be achieved simultaneously in the near-field power transfer regime. © 2013 Optical Society of America OCIS codes: (060.2605) Free-space optical communication; (010.1330) Atmospheric turbulence. Introduction Photon-starved optical communication links place a premium on achieving high photon information efficiency (PIE), i.e., many bits/detected photon, which can be realized with pulse-position modulation (PPM) and direct detection [1]. Future developments in fiber-optic communication are increasingly focused on achieving high spectral efficiency (SE), viz., many bits/sec-Hz, which can be realized with high-order quadrature-amplitude modulation and coherent detection Should both high photon efficiency and high spectral efficiency be desired, however, the preceding approaches fail. Recourse to the ultimate quantum-mechanical (Holevo-limit) capacity of the bosonic (optical communication) channel does not eliminate the conflicting demands of high PIE and high SE. The single spatial-mode Holevo capacity of the vacuum-propagation channel [3] is a function of the average number of transmitted photons, N̄T , and the channel transmissivity, η . The left panel of Fig. 1 shows the Holevo-limit PIE versus SE for that channel along with the Shannon-limit behaviors for heterodyne and homodyne detection, plus that of on-off keying (OOK) with direct detection. None of these approaches realizes both high PIE and high SE. High PIE and high SE can be realized together over the vacuum channel with M spatial-mode operation, whose Holevo-limit results, when all modes have the same transmissivity, are shown in the right panel of Fig. 1. Thus far, no explicit approach to realizing this capacity is known. As seen in this panel, heterodyne and homodyne detection have hard limits on their PIEM values, whereas the Shannon capacity of direct detection with on-off keying mimics the 189 spatial-mode Holevo capacity but requires 4500 equal-transmissivity spatial modes to do so [4]. Well-known vacuum-propagation characteristics imply that high photon efficiency with high spectral efficiency can only be obtained in the near-field power transfer regime. For an L-m-long line-of-sight vacuum-propagation link at wavelength λ between coaxial square transmitter and receiver pupils with side lengths dT and dR, respectively, near-field power transfer prevails when the Fresnel-number product, D f = (dT dR/λ L) 2 satisfies D f ≫ 1. Then, there are approximately D f orthonormal spatial modes in the transmitter pupil—for each of two orthogonal polarization states—that transfer almost all of their power and retain their orthogonality after propagation into the receiver pupil. It follows that PIE = 10 bits/photon and SE = 5 bits/sec-Hz will only be practical for very modest path lengths: for dT = dR = d operation at λ = 1.55μm, D f = 189 (the Holevo limit) requires d = 14.6 cm when L = 1 km, and for OOK direct detection D f = 4500 requires d = 32.3 cm for L = 1 km operation at λ = 1.55μm. Application scenarios for these short path lengths will almost certainly involve terrestrial paths, which preclude high data-rate operation in low-visibility weather conditions. Putting aside the modest extinction loss associated with atmospheric absorption and scattering at a well-chosen laser wavelength, it is then the parts per million refractive-index fluctuations associated with turbulent mixing of air parcels with ∼1K temperature differences that distinguish clear-weather atmospheric propagation from propagation through vacuum. This paper presents bounds on the ultimate PIE versus SE behavior that can be realized using multiple spatial modes in the presence of atmospheric turbulence. Much is known about optical propagation and communication through atmospheric turbulence, but little of it is germane to determining the degree to which this channel permits the simultaneous achievement of high photon efficiency and high spectral efficiency. It is known that the atmospheric channel has a near-field power transfer regime that is similar to that of vacuum propagation [5]. Relatively little has been done, prior to our work, to assess the communication performance obtainable in this regime. This is due to the difficulty of determining exact statistics of the turbulent