Optical wave breaking
In fiber optics, optical wave breaking (OWB) is a nonlinear effect caused by interaction between group velocity dispersion (GVD) and self-phase modulation (SPM). It only happens in the normal dispersion regime () for pulses meeting a certain criterium, as we will see.
SPM creates low frequencies at the front of the pulse, and high ones at the back, and if , GVD lets low frequencies travel faster than high ones. When those effects interact, the pulse gets temporally stretched in a surprisingly sophisticated way.
To illustrate this, the instantaneous frequency has been plotted below for a theoretical Gaussian input pulse experiencing OWB, with settings , , and .
In the left panel, we see the typical S-shape caused by SPM, and the arrows indicate the direction that GVD is pushing the curve in. This leads to steepening at the edges, i.e. the S gradually turns into a Z. Shortly before the slope would become infinite, small waves start “falling off” the edge of the pulse, hence the name wave breaking:
Several interesting things happen around this moment. To demonstrate this, spectrograms of the same simulation have been plotted below, together with pulse profiles in both the -domain and -domain on an arbitrary linear scale (click the image to get a better look).
Initially, the spectrum broadens due to SPM in the usual way, but shortly after OWB, this process is stopped by the appearance of so-called sidelobes in the -domain on either side of the pulse. In the meantime, in the time domain, the pulse steepens at the edges, but flattens at the peak. After OWB, a train of small waves falls off the edges, which eventually melt together, leading to a trapezoid shape in the -domain. Dispersive broadening then continues normally:
We call the distance at which the wave breaks , and want to predict it analytically. We do this using the instantaneous frequency , by estimating when the SPM fluctuations overtake their own base, as was illustrated earlier.
To get of a Gaussian pulse experiencing both GVD and SPM, it is a reasonable approximation, for small , to simply add up the instantaneous frequencies for these separate effects:
Assuming that is small enough such that , this expression can be reduced to:
Where we have assumed , and is the soliton number, which is defined as:
This quantity is very important in anomalous dispersion, but even in normal dispersion, it is still a useful measure of the relative strengths of GVD and SPM. As was illustrated earlier, overtakes itself at the edges, so OWB occurs when oscillates there, which starts when its -derivative, the instantaneous chirpyness , has two real roots for :
Where the function has been defined for convenience. As it turns out, this equation can be solved analytically using the Lambert function, leading to the following exact minimum value for , such that OWB can only occur when :
If this condition is not satisfied, cannot have two roots for , meaning cannot overtake itself. GVD is unable to keep up with SPM, so OWB will not occur.
Next, consider two points at and in the pulse, separated by a small initial interval . The frequency difference between these points due to will cause them to displace relative to each other after a short distance by some amount , estimated by:
Where is the inverse of the group velocity. For a certain choice of and , OWB occurs when they catch up to each other, which is when . The distance at which this happens first must satisfy the following condition for some value of :
The time of OWB must be where has its steepest slope, which is at the minimum value of , and by extension . This turns out to be :
Clearly, when , which, when inserted above, leads to an imaginary , confirming that OWB cannot occur in that case. Otherwise, if , then:
This prediction for appears to agree well with the OWB observed in the simulation:
Because all spectral broadening up to is caused by SPM, whose -domain behaviour is known, it is in fact possible to draw some analytical conclusions about the achieved bandwidth when OWB sets in. Filling in into gives:
Assuming that is large in the denominator, this can be approximately reduced to:
The expression has its global extrema at . The maximum SPM frequency shift achieved at is therefore given by:
Interestingly, this expression does not contain at all, so the achieved spectrum when SPM is halted by OWB is independent of the pulse width, for sufficiently large .
- D. Anderson, M. Desaix, M. Lisak, M.L. Quiroga-Teixeiro, Wave breaking in nonlinear-optical fibers, 1992, Optical Society of America.
- A.M. Heidt, A. Hartung, H. Bartelt, Generation of ultrashort and coherent supercontinuum light pulses in all-normal dispersion fibers, 2016, Springer Media.