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 (β2>0)
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 β2>0, 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 ωi(z,t)=−∂ϕ/∂t
has been plotted below for a theoretical Gaussian input pulse experiencing OWB,
with settings T0=100fs, P0=5kW,
β2=2ps2/m and γ=0.1/W/m.
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 t-domain and ω-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 ω-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 t-domain.
Dispersive broadening then continues normally:
We call the distance at which the wave breaks LWB,
and want to predict it analytically.
We do this using the instantaneous frequency ωi,
by estimating when the SPM fluctuations overtake their own base,
as was illustrated earlier.
To get ωi of a Gaussian pulse experiencing both GVD and SPM,
it is a reasonable approximation, for small z, to simply add up
the instantaneous frequencies for these separate effects:
Where we have assumed β2>0,
and Nsol is the soliton number,
which is defined as:
Nsol2≡LNLD=∣β2∣γP0T02
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, ωi overtakes itself at the edges,
so OWB occurs when ωi oscillates there,
which starts when its t-derivative,
the instantaneous chirpynessξi,
has two real roots for t2:
Where the function f(x) has been defined for convenience. As it turns
out, this equation can be solved analytically using the Lambert W function,
leading to the following exact minimum value Nmin2 for Nsol2,
such that OWB can only occur when Nsol2>Nmin2:
Nmin2=41exp(23)≈1.12
If this condition Nsol2>Nmin2 is not satisfied,
ξi cannot have two roots for t2, meaning ωi cannot overtake itself.
GVD is unable to keep up with SPM, so OWB will not occur.
Next, consider two points at t1 and t2 in the pulse,
separated by a small initial interval (t2−t1).
The frequency difference between these points due to ωi
will cause them to displace relative to each other
after a short distance z by some amount Δt,
estimated by:
Where β1(ω) is the inverse of the group velocity.
For a certain choice of t1 and t2,
OWB occurs when they catch up to each other,
which is when −Δt=(t2−t1).
The distance LWB at which this happens first
must satisfy the following condition for some value of t:
LWBβ2ξi(LWB,t)=−1⟹LWB2=−β22f(t2/T02)T04
The time t of OWB must be where ωi(t) has its steepest slope,
which is at the minimum value of ξi(t), and by extension f(x).
This turns out to be f(3/2):
fmin=f(3/2)=1−4Nsol2exp(−3/2)=1−Nsol2/Nmin2
Clearly, fmin≥0 when Nsol2≤Nmin2,
which, when inserted above, leads to an imaginary LWB,
confirming that OWB cannot occur in that case.
Otherwise, if Nsol2>Nmin2, then:
LWB=β2−fminT02=Nsol2/Nmin2−1LD
This prediction for LWB appears to agree well
with the OWB observed in the simulation:
Because all spectral broadening up to LWB is caused by SPM,
whose ω-domain behaviour is known,
it is in fact possible to draw some analytical conclusions
about the achieved bandwidth when OWB sets in.
Filling LWB in into ωSPM gives:
The expression xexp(−x2) has its global extrema
±1/2e at x2=1/2. The maximum SPM frequency shift
achieved at LWB is therefore given by:
ωmax=eβ22γP0
Interestingly, this expression does not contain T0 at all,
so the achieved spectrum when SPM is halted by OWB
is independent of the pulse width,
for sufficiently large Nsol.