# Autocorrelation Function for Dispersion-Free Fiber Channels with Distributed Amplification

## TL;DR

This paper derives the autocorrelation function for a simplified dispersion-free optical fiber model with distributed amplification to analyze spectral broadening and capacity limits, revealing that capacity growth is fundamentally constrained by power scaling.

## Contribution

It introduces a novel autocorrelation function for a simplified fiber model with distributed amplification, providing upper bounds on spectral broadening and capacity.

## Key findings

- Output power scales as the square-root of launch power.
- Capacity scales at most as half the logarithm of launch power.
- In a specific scaling scenario, capacity scales as the inverse fourth root of launch power.

## Abstract

Optical fiber signals with high power exhibit spectral broadening that seems to limit capacity. To study spectral broadening, the autocorrelation function of the output signal given the input signal is derived for a simplified fiber model that has zero dispersion, distributed optical amplification (OA), and idealized spatial noise processes. The autocorrelation function is used to upper bound the output power of bandlimited or time-resolution limited receivers, and thereby to bound spectral broadening and the capacity of receivers with thermal noise. The output power scales at most as the square-root of the launch power, and thus capacity scales at most as one-half the logarithm of the launch power. The propagating signal bandwidth scales at least as the square-root of the launch power. However, in practice the OA bandwidth should exceed the signal bandwidth to compensate attenuation. Hence, there is a launch power threshold beyond which the fiber model loses practical relevance. Nevertheless, for the mathematical model an upper bound on capacity is developed when the OA bandwidth scales as the square-root of the launch power, in which case capacity scales at most as the inverse fourth root of the launch power.