Abstract-It is shown that signal energy is the only available degree-of-freedom (DOF) for fiber-optic transmission as the input power tends to infinity. With n signal DOFs at the input, n´1 DOFs are asymptotically lost to signal-noise interactions. The main observation is that, nonlinearity introduces a multiplicative noise in the channel, similar to fading in wireless channels. The channel is viewed in the spherical coordinate system, where signal vectorX P C n is represented in terms of its norm |X| and directionX. The multiplicative noise causes signal directionX to vary randomly on the surface of the unit p2n´1q-sphere in C n , in such a way that the effective area of the support ofX does not vanish as |X| Ñ 8. On the other hand, the surface area of the sphere is finite, so thatX carries finite information. This observation is used to show several results. Firstly, let CpPq be the capacity of a discrete-time periodic model of the optical fiber with distributed noise and frequency-dependent loss, as a function of the average input power P. It is shown that asymptotically as P Ñ 8, C " 1 n log`log P˘`c, where n is the dimension of the input signal space and c is a bounded number. In particular, limPÑ8 CpPq " 8 in finite-dimensional periodic models. Secondly, it is shown that capacity saturates to a constant in infinite-dimensional models where n " 8. An expression is provided for the constant c, by showing that, as the input |X| Ñ 8, the action of the discrete periodic stochastic nonlinear Schrödinger equation tends to multiplication by a random matrix (with fixed distribution, independent of input). Thus, perhaps counter-intuitively, noise simplifies the nonlinear channel at high powers to a linear multiple-input multiple-output fading channel. As P Ñ 8 signal-noise interactions gradually reduce the slope of the CpPq, to a point where increasing the input power returns diminishing gains. Nonlinear frequency-division multiplexing can be applied to approach capacity in optical networks, where linear multiplexing achieves low rates at high powers.
Finally, the capacity of the discrete-time model of the dispersive optical fiber was shown to be 1 2n log log(P) + o(1) as P → ∞, where n is the fixed size of the input vector REF Theorem 1] .
The Asymptotic Capacity of the Optical Fiber
ArXiv
ArXiv
Mathematics Computer Science