Analysis of Laser Radiation using the Nonlinear Fourier Transform

Fourier Transforms (FT) are widely used to study the periodic characteristics of different systems, for instance, in audio frequency analysis, car chassis vibrations, stock exchange trends, solar spot cycles, weather fluctuations, and of course, laser dynamics and for characterisation of optical communication systems. In essence, the FT breaks down a signal to a sum of many sine waves of varying frequencies and strengths, where the strongest of the waves indicate the important periodicities.  

While extremely versatile and widely used, FTs have a drawback – the decomposition is inherently linear. That is, it will only tell you which frequencies are present, but not whether they interact and how. You will have to draw those inferences based on your understanding of the system.  

In our paper, we have proposed the use of the Nonlinear Fourier Transform (NFT) as an equally viable tool for analysis of laser radiation. NFTs reveal coherent parts of the radiation, that is, which components are highly correlated, and which aren’t. Coherent features manifest in the form of very short light pulses, which can either be shape-preserving, pulsating, or even suddenly appear out of nowhere. There is good understanding of how such features can arise, and how they interact. Our methods thus allow us to unequivocally identify such coherent features in laser radiation, and this in turn can give us an insight into what kind of nonlinear interactions within the laser can give rise to such features.  

So far, the use of the NFT has been restricted to lossless systems without any sources or sinks (gains or losses), as the understanding has been that this cannot be employed in systems like lasers, which have both gains and losses. In our work, we say that this is ok to do.  

Let us consider for a moment sound waves emanating from a speaker. The speaker may be driven too high (think rock guitar riffs!), which can result in the speaker producing distorted sounds. This is because the speaker is pushed to its nonlinear stretching limits, and it doesn’t have time to fully relax before it gets kicked again. Yet, people use the linear FT to understand the frequency qualities of this distorted sound that emanates from the speaker, in spite of the inherent nonlinearity that had given rise to it. 

In a similar fashion, we use the NFT after it emanates from the laser. True, the laser features arise from a complex process inside it. Yet, we are still valid in using the NFT “after the fact”. In our paper, we specify, step-by-step, how we can carry out the NFT to ensure reproducibility. In this form of general use, it can be a very powerful tool – as it still would reveal which parts of the laser light is coherent.  

Our results make a strong case for use of the NFT in laser systems, and in dissipative systems (systems with gains and losses) in general. Just like the use of the linear FT across systems and disciplines, we expect that our findings and prescription use of the NFT framework will find a broad readership and acceptance across communities that study dissipative systems.

Direct link to paper – https://www.nature.com/articles/s41467-019-13265-4

Direct link to Supplementary material – click here.