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Case Study: Collision of Soliton Pulses in a Fiber

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Key questions:

  • How to numerically simulate the collision of soliton pulses?
  • Under which conditions can pulses survive such collisions?

Soliton pulses in optical fibers (with anomalous dispersion) have various remarkable properties, some of which are observed when such pulses collide within a fiber. We will look at some test cases.

How can pulses actually collide in a fiber? A trivial case is that they propagate in opposite directions, but here we consider another case: that the second pulse has a different center wavelength, which (due to chromatic dispersion) causes a different group velocity. For example, if this second pulse has a lower frequency, it will (due to the anomalous dispersion) be slower than the other pulse; if we give this second pulse a small head start, the other one will essentially catch up, so that they collide.

We further assume in this study that both pulses have identical linear polarization; this case (as opposed to the case with orthogonal polarizations) gives rise to strong interference effects during the collision. (For orthogonal polarizations, we would have an interaction by cross-phase modulation.) Since a soliton is formed by a delicate balance between chromatic dispersion and Kerr nonlinearity, one may expect such a soliton collision to create quite a mess (e.g. with pulse break-up), considering that the strong intensity modulation for partially overlapping pulses leads to a correspondingly modulated refractive index change. So it is surprising that in fact solitons turn out to be quite robust and usually survive such collisions.

Setting up the Simulation Model

We use the software RP Fiber Power, which offers the Power FormPassive Fiber for Ultrashort Pulses”. That was not specifically made for entering dual pulses, but we can enter an expression for the complex amplitude of the pulse trace vs. time. This can depend on a few parameters, which we can automatically calculate with a few lines of script code entered in the form:

tau_s_in1 := 250 fs { duration of pulse 1 }
tau_s_in2 := tau_s_in1 { duration of pulse 2 }
dt2 := -3 ps { temporal offset of pulse 2 }
df2 := -8 THz { frequency offset of pulse 2 }
sech(x) := 1 / cosh(x)
beta2 := rval(fiber1.GVD$, "fs^2/m") * fs^2 { GVD parameter }
gamma := fiber1.n2_f * (2pi / l_s_c) / A_eff(fiber1.signal) { nonlinear coefficient }
E_s1 := -2 * beta2 / (gamma * (tau_s_in1 / 1.7627)) { energy of pulse 1 }
E_s2 := -2 * beta2 / (gamma * (tau_s_in2 / 1.7627)) { energy of pulse 2 }
  { soliton energies for the given pulse durations }
show "E_s1:   ", E_s1:d3:"J"
show "E_s2:   ", E_s2:d3:"J"
calc cf_set_input('E_s_in', E_s1 + E_s2) { set input field }

This defines two input pulse durations that are equal for now, but can be made different later. The code calculates and displays the corresponding fundamental soliton energies. It then updates the input field, automatically setting the total input energy.

We also define a temporal and spectral offset of pulse 2 with respect to pulse 1, which we will use to define the input amplitude profile A0%(t):

sqrt(E_s1 / tau_s_in1) * sech(1.7627 * t / tau_s_in1)
  + sqrt(E_s2 / tau_s_in2) * sech(1.7627 * (t - dt2) / tau_s_in2)
    * expi(-2pi * df2 * t)

The center wavelength of pulse 1 is taken from an input field of the form, and that of pulse 2 will be slightly longer due to the negative frequency offset. Note that the absolute scaling of the amplitudes is done automatically by the software based on the total input pulse energy.

For the fiber, we assume a mode radius of 5 μm, a constant GVD of −10 000 fs2/m (which could be achieved with a suitable refractive index profile, although perhaps not without a slope, i.e. some higher order dispersion), and a nonlinear index of 3 · 10−24 m2 (realistic for silica fibers).

Pulse Simulations

We start with equal pulse durations (and thus equal pulse energies). Probably the nicest way to display the result is a color plot showing the evolution of the temporal profile in the fiber:

two solitons with equal energies
Figure 1: Collision of two solitons with equal energies.

In fact, we have a situation where the two solitons remain completely intact after the collision. It may look as if they only interfere but do not really interact, but this is not true: as mentioned above, the Kerr nonlinearity acts on the combined pulse in a complicated way. You can also check the phase of the first soliton (stay around <$t = 0$>):

evolution of pulse energy and optical phase
Figure 2: Evolution of total pulse energy (both solitons together) and optical phase at <$t = 0$>.

We see that the phase not only wiggles a bit during the collision, but also evolves with some offset, which obviously results from the interaction.

Note that the Power Form does not directly offer the option to display this phase, but in the form we can enter some additional script code to get it:

y2: -4, +4 { second y axis }
"phase at t = 0 (rad)", @y2, color = green

f: (get_pulse(x); Phi_t(0)), yscale = 2, color = green, width = 3,
  maxconnect = 1e3, "phase at t = 0 (right scale)"

Next we can check what happens if the second soliton has twice the duration and thus only half the energy. We can simply modify the value of tau_s_in2 in the inserted code and get the following:

evolution of pulse energy and optical phase
Figure 3: Same as before, but with the second soliton being two times longer.

Here we have also plotted the energy of pulse 1 (obtained by integrating the spectrum over its range); this shows that there is no energy transfer from one soliton to the other.

In the next experiment, we assume that the second soliton is a second-order soliton, again with the same initial pulse duration. We simply increase the pulse energy by a factor of 22 = 4. The result:

evolution of pulse energy and optical phase
Figure 4: Here we have also plotted the energy of pulse 1 (obtained by integrating the spectrum over its range); this shows that there is no energy transfer from one soliton to the other.

In the next experiment, we assume that the second soliton is a second-order soliton, again with the same initial pulse duration. We simply increase the pulse energy by a factor of 22 = 4. The result:

two solitons with equal energies
Figure 5: Collision of a fundamental soliton with a second-order soliton.

One could easily do further experiments. For example, we could study how the situation changes if we introduce some higher-order dispersion, stimulated Raman scattering or propagation losses.


RP Fiber Power

The RP Fiber Power software is an invaluable tool for such work – very powerful and at the same time pretty easy to use!

You can learn various things from this study:

  • Solitons with different center wavelength have different propagation speeds and can therefore collide in a fiber.
  • They turn out to be remarkably robust, surviving such collisions in many cases, i.e., not fall apart.

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