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The Asymptotic Capacity of the Optical Fiber

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 vector $\underline{X}\in\mathbb{C}^n$ is represented in terms of its norm $|\underline{X}|$ and direction $\underline{\hat{X}}$. The multiplicative noise causes signal direction $\underline{\hat{X}}$ to vary randomly on the surface of the unit $(2n-1)$-sphere in $\mathbb{C}^{n}$, in such a way that the effective area of the support of $\underline{\hat{X}}$ does not vanish as $|\underline{X}|\rightarrow\infty$. On the other hand, the surface area of the sphere is finite, so that $\underline{\hat{X}}$ carries finite information. This observation is used to show several results. Firstly, let $\mathcal C(\mathcal P)$ 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 $\mathcal P$. It is shown that asymptotically as $\mathcal P\rightarrow\infty$, $\mathcal C=\frac{1}{n}\log\bigl(\log\mathcal P\bigr)+c$, where $n$ is the dimension of the input signal space and $c$ is a bounded number. In particular, $\lim_{\mathcal P\rightarrow\infty}\mathcal C(\mathcal P)=\infty$ in finite-dimensional periodic models. Secondly, it is shown that capacity saturates to a constant in infinite-dimensional models where $n=\infty$.