Multiple Length Scales and Averaging in Modelling of Long-Distance Optical Fiber Transmission

Abstract

When scientists model long-distance optical fiber communication systems, we must deal with a wide range of length scales extending from 1.5 µm, which is the wavelength of the transmitted light signal, to 27 Mm, which the length of the FLAG cable network. In between, there are a host of other length scales whose values depend on the fiber parameters. These include the pulse scale lengths which are in the range of 0.3-30 cm, the polarization beat length which is 3-30 m, and the fiber polarization orientation correlation length which is on the order of 10-100 m. The attenuation scale length is 20-30 km, and the scale lengths for polarization mode dispersion and chromatic dispersion can be as small as 100 km and can be almost arbitrarily large. These scale lengths depends on both the fiber parameters and the data rate. Nonlinear scale lengths are typically on the order of 1000 km. The gap of 13 orders of magnitude between the shortest and the longest scale lengths seems like a serious challenge, and it would in fact be an insuperable challenge if we really attempted to solve Maxwell’s equations on a scale length of micrometers to determine the behavior over the many thousands of kilometers that exist in real systems. While there have been some attempts to directly solve Maxwell’s equations in optical fibers over short lengths and in special parameter regimes, my own view is that it does not make much sense to apply this sort of brute force approach to optical fiber transmission. Instead, we should take advantage of the very powerful mathematical averaging techniques that exist to, in effect, leap over the shortest length scales and focus on the length scales of interest. These techniques have a long and distinguished history that dates back to the 19th century, and the well-known slowly varying envelope approximation that is used in optics can be viewed as an application of these techniques.