path: root/source/know/concept/optical-soliton/index.md

---
title: "Optical soliton"
sort_title: "Optical soliton"
date: 2024-09-20
categories:
- Physics
- Mathematics
- Fiber optics
- Nonlinear optics
layout: "concept"
---

In general, a **soliton** is a wave packet
that maintains its shape as it travels over great distances.
They are only explainable by nonlinear physics,
but many (often unrelated) nonlinear equations give rise to solitons:
the [Boussinesq equations](/know/concept/boussinesq-wave-theory/),
the [Korteweg-de Vries equation](/know/concept/korteweg-de-vries-equation/),
the [nonlinear Schrödinger (NLS) equation](/know/concept/nonlinear-schrodinger-equation/),
and more.
Here we consider waveguide optics,
which is governed by the NLS equation,
given in dimensionless form by:

$$\begin{aligned}
    i u_z + u_{tt} + r |u|^2 u
    = 0
\end{aligned}$$

Where $$r = \pm 1$$ determines the dispersion regime,
and subscripts denote differentiation.
We start by making the most general ansatz
for the pulse envelope $$u(z, t)$$, namely:

$$\begin{aligned}
    u(z, t)
    = \phi(z, t) \: e^{i \theta(z, t)}
\end{aligned}$$

With $$\phi$$ and $$\theta$$ both real.
Note that no generality has been lost yet:
we have simply split a single complex function
into two real ones.
The derivatives of $$u$$ thus become:

$$\begin{aligned}
    u_z
    &= (\phi_z + i \phi \theta_z) \: e^{i \theta}
    \\
    u_t
    &= (\phi_t + i \phi \theta_t) \: e^{i \theta}
    \\
    u_{tt}
    &= (\phi_{tt} + 2 i \phi_t \theta_t + i \phi \theta_{tt} - \phi \theta_t^2) \: e^{i \theta}
\end{aligned}$$

Inserting $$u_z$$ and $$u_{tt}$$ into the NLS equation leads us to:

$$\begin{aligned}
    0
    &= i \phi_z - \phi \theta_z + \phi_{tt} + 2 i \phi_t \theta_t + i \phi \theta_{tt} - \phi \theta_t^2 + r \phi^3
    \\
    &= \phi_{tt} - \phi \theta_t^2 - \phi \theta_z + r \phi^3 + i (\phi \theta_{tt} + 2 \phi_t \theta_t + \phi_z)
\end{aligned}$$

Since $$\phi$$ and $$\theta$$ are both real,
we can split this equation into its real and imaginary parts:

$$\begin{aligned}
    \boxed{
        \begin{aligned}
            0
            &= \phi_{tt} - \phi \theta_t^2 - \phi \theta_z + r \phi^3
            \\
            0
            &= \phi \theta_{tt} + 2 \phi_t \theta_t + \phi_z
        \end{aligned}
    }
\end{aligned}$$

Still no generality has been lost so far:
these coupled equation are totally equivalent to the NLS equation.
But now it is time make a more specific ansatz,
namely that $$\phi$$ and $$\theta$$ both have a fixed shape
but move at a group velocity $$v$$
and phase velocity $$w$$, respectively:

$$\begin{aligned}
    \phi(z, t)
    &= \phi(t - v z)
    \\
    \theta(z, t)
    &= \theta(t - w z)
\end{aligned}$$

Meaning $$\phi_z = -v \phi_t$$ and $$\theta_z = -w \theta_t$$.
Now the coupled equations are given by:

$$\begin{aligned}
    0
    &= \phi_{tt} - \phi \theta_t^2 + w \phi \theta_t + r \phi^3
    \\
    0
    &= \phi \theta_{tt} + 2 \phi_t \theta_t - v \phi_t
\end{aligned}$$

We multiply the imaginary part's equation by $$\phi$$ and take its indefinite integral,
which can then be evaluated by recognizing the product rule of differentiation:

$$\begin{aligned}
    0
    &= \int \Big( \phi^2 \theta_{tt} + 2 \phi \phi_t \theta_t - v \phi \phi_t \Big) \dd{t}
    \\
    &= \phi^2 \theta_t - \frac{v}{2} \phi^2
\end{aligned}$$

Where the integration constant has been set to zero.
This implies $$\theta_t = v/2$$, which we insert into the real part's equation, giving:

$$\begin{aligned}
    0
    &= \phi_{tt} + \frac{v}{4} (2 w - v) \phi + r \phi^3
\end{aligned}$$

Defining $$B \equiv v (v - 2 w) / 4$$,
multiplying by $$2 \phi_t$$, and integrating in the same way:

$$\begin{aligned}
    0
    &= \int \Big( 2 \phi_t \phi_{tt} - 2 B \phi \phi_t + 2 r \phi^3 \phi_t \Big) \dd{t}
    \\
    &= \phi_t^2 - B \phi^2 + \frac{r}{2} \phi^4 - C
\end{aligned}$$

Where $$C$$ is an integration constant.
Rearranging this yields a powerful equation,
which can be interpreted as a "pseudoparticle"
with kinetic energy $$\phi_t^2$$ moving in a potential $$-P(\phi)$$:

$$\begin{aligned}
    \boxed{
        \phi_t^2
        = P(\phi)
        \equiv -\frac{r}{2} \phi^4 + B \phi^2 + C
    }
\end{aligned}$$

We further restrict the set of acceptable solutions
by demanding that $$\phi(t)$$ is localized,
meaning $$\phi \to \phi_\infty$$ when $$t \to \pm \infty$$,
for a finite constant $$\phi_\infty$$.
This implies $$\phi_t \to 0$$ and $$\phi_{tt} \to 0$$:
the former clearly requires $$P(\phi_\infty) = 0$$.
Regarding the latter, we differentiate
the pseudoparticle equation with respect to $$t$$,
which tells us for $$t \to \pm \infty$$:

$$\begin{aligned}
    0
    = \phi_{tt}
    &= \frac{1}{2} P'(\phi_\infty)
    = (B - r \phi_\infty^2) \phi_\infty
\end{aligned}$$

Here we have two options:
the "bright" case $$\phi_\infty = 0$$,
and the "dark" case $$\phi_\infty^2 = r B$$.
Before we investigate those further,
let us finish finding $$\theta$$:
we know that $$\theta_t = v/2$$, so:

$$\begin{aligned}
    \theta(t - w z)
    = \int \theta_t \dd{(t - w v)}
    = \frac{v}{2} (t - w v)
\end{aligned}$$

Where we can ignore the integration constant
because the NLS equation has *Gauge symmetry*,
i.e. it is invariant under a transformation
of the form $$u \to u e^{i a}$$ with constant $$a$$.
Finally, we rewrite this result to eliminate $$w$$ in favor of $$B$$:

$$\begin{aligned}
    \theta(z, t)
    = \frac{v}{2} t - \bigg( \frac{v^2}{4} - B \bigg) z
\end{aligned}$$



## Bright solitons

First we consider the "bright" option $$\phi_\infty = 0$$,
where our requirement that $$P(\theta_\infty) = 0$$
clearly means that we must set $$C = 0$$.
We are therefore left with:

$$\begin{aligned}
    \phi_t^2
    = P(\phi)
    = -\frac{r}{2} \phi^4 + B \phi^2
\end{aligned}$$

We must consider $$r = 1$$ and $$r = -1$$, and the sign of $$B$$;
the possible forms of $$P(\phi)$$ are shown in the sketch below.
Because $$\phi_t$$ is real by definition,
valid solutions can only exist in the shaded regions where $$P(\phi) \ge 0$$:

{% include image.html file="bright-full.png" width="75%"
    alt="Sketch of candidate potentials for bright solitons" %}

However, in order to have *stable* solutions
where $$\phi$$ does not grow uncontrolably,
we must restrict ourselves to shaded regions with a finite area.
Otherwise, if they are infinite (as for $$r = -1$$),
then a positive feedback loop arises:
$$\phi_t^2$$ grows, so $$|\phi|$$ increases,
then according to the sketch $$\phi_t^2$$ grows even more, etc.
While mathematically correct, that would be physically unacceptable,
so the only valid case here is $$r = 1$$ with $$B > 0$$.

Armed with this knowledge,
we are now ready to integrate the pseudoparticle integration.
First, we rewrite it as follows, defining $$x \equiv t - vz$$:

$$\begin{aligned}
    \phi_t
    = \pdv{\phi}{x}
    = \pm \sqrt{P(\phi)}
    = \pm \phi \sqrt{B - \phi^2 / 2}
\end{aligned}$$

This can be rearranged such that the differential elements
$$\dd{x}$$ and $$\dd{\phi}$$ are on opposite sides,
which can then each be wrapped in an integral, like so:

$$\begin{aligned}
    \dd{x}
    = \pm \frac{\sqrt{2}}{\phi \sqrt{2 B - \phi^2}} \dd{\phi}
    \qquad\implies\qquad
    \int_{x_0}^{x} \dd{\xi}
    = \pm \sqrt{2} \int_{\phi_0}^{\phi} \frac{1}{\psi \sqrt{2 B - \psi^2}} \dd{\psi}
\end{aligned}$$

Note that these are *indefinite* integrals,
which have been written as *definite* integrals
by placing the constants $$x_0$$ and $$\phi_0$$
and target variables $$x$$ and $$\phi$$ in the limits.

In order to integrate by substitution,
we define the new variable $$f \equiv \psi / \sqrt{2 B}$$
and update the limits accordingly
to $$F \equiv \phi / \sqrt{2 B}$$
and $$F_0 \equiv \phi_0 / \sqrt{2 B}$$:

$$\begin{aligned}
    x - x_0
    &= \pm \sqrt{2} \int_{F_0}^{F} \frac{\sqrt{2 B}}{f \sqrt{2 B} \sqrt{2 B - 2 B f^2}} \dd{f}
    \\
    &= \pm \frac{1}{\sqrt{B}} \int_{F_0}^{F} \frac{1}{f \sqrt{1 - f^2}} \dd{f}
\end{aligned}$$

We look up this integrand, and discover that it is in fact the derivative
of the inverse $$\sech^{-1}$$ of the hyperbolic secant function, so we arrive at:

$$\begin{aligned}
    x - x_0
    &= \pm \frac{1}{\sqrt{B}} \int_{F_0}^{F} \dv{}{f} \Big( \sech^{-1}(f) \Big) \dd{f}
    \\
    &= \pm \frac{1}{\sqrt{B}} \sech^{-1}(F) \mp \frac{1}{\sqrt{B}} \sech^{-1}(F_0)
\end{aligned}$$

Rearranging and combining the integration constants
$$x_0$$ and $$F_0$$ into a single $$t_0$$, we get:

$$\begin{aligned}
    \sech^{-1}(F)
    = \pm \sqrt{B} (x - t_0)
    \qquad\qquad
    t_0
    \equiv x_0 \mp \frac{1}{\sqrt{B}} \sech^{-1}(F_0)
\end{aligned}$$

Then, wrapping everything in $$\sech$$
(which is an even function, so we can discard the $$\pm$$)
and using $$F \equiv \phi / \sqrt{2 B}$$,
we finally arrive at the desired solution for $$\phi$$:

$$\begin{aligned}
    \phi(x)
    = \sqrt{2 B} \sech\!\Big( \sqrt{B} (x - t_0) \Big)
\end{aligned}$$

Combining this result with our earlier solution for $$\theta$$,
we find that the full so-called **bright soliton** $$u$$
is as follows, controlled by two real parameters
$$B > 0$$ and $$v$$:

$$\begin{aligned}
    \boxed{
        u(z, t)
        = \sqrt{2 B} \sech\!\bigg( \sqrt{B} (t - v z - t_0) \bigg)
        \exp\!\bigg( i \frac{v}{2} t - i \Big( \frac{v^2}{4} - B \Big) z \bigg)
    }
\end{aligned}$$

It is always possible to transform the NLS equation
into a new moving coordinate system such that $$v = 0$$,
yielding a stationary soliton given by:

$$\begin{aligned}
    \boxed{
        u(z, t)
        = \sqrt{2 B} \sech\!\Big( \sqrt{B} (t - t_0) \Big) \exp(i B z)
    }
\end{aligned}$$

You may be wondering how we can set $$v = 0$$ without affecting $$B$$;
a more correct way of saying it would be that
we take the limits $$v \to 0$$ and $$w \to -\infty$$.

That was for the dimensionless form of the NLS equation;
let us specialize this to its usual form in fiber optics.
We thus make a transformation $$u \to U/U_c$$,
$$t \to T/T_c$$ and $$z \to Z/Z_c$$:

$$\begin{aligned}
    \frac{U(Z, T)}{U_c}
    &= \sqrt{2 B} \sech\!\bigg( \sqrt{B} \: \frac{T - T_0}{T_c} \bigg)
    \exp\!\bigg( i B \frac{Z}{Z_c} \bigg)
\end{aligned}$$

Where $$U_c$$, $$T_c$$ and $$Z_c$$ are scale constants
determined during non-dimensionalization
to obey the relations below.
We only have two relations, so we can choose one value freely,
say, $$U_c$$:

$$\begin{aligned}
    Z_c
    = \frac{1}{\gamma_0 U_c^2}
    \qquad\qquad
    T_c
    = \sqrt{\frac{- \beta_2}{2 \gamma_0 U_c^2}}
\end{aligned}$$

Note that $$r = 1$$ implies $$\beta_2 < 0$$ assuming $$\gamma_0 > 0$$.
In other words, bright solitons only exist
in the anomalous dispersion regime of an optical fiber.
Inserting these relations into the expression
and defining the peak power $$P_0 \equiv 2 B U_c^2$$ yields:

$$\begin{aligned}
    U(Z, T)
    &= \sqrt{P_0}
    \sech\!\Bigg( \sqrt{\frac{\gamma_0 P_0}{- \beta_2}} (T - T_0) \Bigg)
    \exp\!\bigg( i \frac{\gamma_0 P_0}{2} Z \bigg)
\end{aligned}$$

In practice, most authors write this as follows,
where $$T_\mathrm{w}$$ determines the width of the pulse:

$$\begin{aligned}
    \boxed{
        U(Z, T)
        = \sqrt{P_0} \sech\!\bigg( \frac{T - T_0}{T_\mathrm{w}} \bigg) \exp\!\bigg( i \frac{\gamma_0 P_0}{2} Z \bigg)
    }
\end{aligned}$$

Clearly, for this to be a valid solution of the NLS equation,
$$T_\mathrm{w}$$ must be subject to a constraint
involving the so-called **soliton number** $$N_\mathrm{sol}$$:

$$\begin{aligned}
    \boxed{
        N_\mathrm{sol}^2
        \equiv \frac{L_D}{L_N}
        = \frac{\gamma_0 P_0 T_\mathrm{w}^2}{|\beta_2|}
        = 1
    }
\end{aligned}$$

Where $$L_D \equiv T_0 / |\beta_2|$$ is the linear length scale
of [dispersive broadening](/know/concept/dispersive-broadening/),
and $$L_N \equiv 1 / (\gamma_0 P_0)$$ is the nonlinear length scale
of [self-phase modulation](/know/concept/self-phase-modulation/).
A *first-order* soliton has $$N_\mathrm{sol} = 1$$
and simply maintains its shape,
whereas higher-order solitons have complicated periodic dynamics.



## Dark solitons

The other option to satisfy $$P'(\phi_\infty) = 0$$
is $$\phi_\infty^2 = r B$$, which implies $$r B > 0$$
such that $$\phi_\infty$$ is real.
With this in mind, we again sketch all remaining candidates for $$P(\phi)$$:

{% include image.html file="dark-full.png" width="75%"
    alt="Sketch of candidate potentials for dark solitons" %}

At a glance, there are plenty of solutions here, even stable ones!
However, as explained earlier, our localization requirement
means that we need $$P(\phi_\infty) = 0$$ and $$P'(\phi_\infty) = 0$$.
The latter is only satisfied by the solid curve above,
so we must limit ourselves to $$r = -1$$ and $$B < 0$$,
with $$C = C_0$$ for some positive $$C_0$$.
The next step is to find $$C_0$$.

We notice that the target curve has two double roots
at $$\pm \phi_\infty$$, so we can rewrite:

$$\begin{aligned}
    P(\phi)
    &= \frac{1}{2} \Big( \phi^4 + 2 B \phi^2 + 2 C \Big)
    \\
    &= \frac{1}{2} \Big( \phi^4 + 2 B \phi^2 + B^2 - B^2 + 2 C \Big)
    \\
    &= \frac{1}{2} \big( \phi^2 + B \big)^2 - \frac{1}{2} \big( B^2 - 2 C \big)
\end{aligned}$$

Here we see that $$P(\phi_\infty)$$ can only have a double root
when $$C = C_0 = B^2 / 2$$, in which case the root is clearly $$\phi_\infty = \pm \sqrt{-B}$$.
We are therefore left with:

$$\begin{aligned}
    \phi_t^2
    = P(\phi)
    = \frac{1}{2} \big( \phi^2 + B \big)^2
\end{aligned}$$

Now we are ready to integrate this equation.
Taking the square root with $$x \equiv t - v z$$:

$$\begin{aligned}
    \phi_t
    = \pdv{\phi}{x}
    = \pm \sqrt{P(\phi)}
    = \pm \frac{1}{\sqrt{2}} (\phi^2 + B)
\end{aligned}$$

We put the differential elements $$\dd{\phi}$$ and $$\dd{x}$$
on opposite sides and take the integrals:

$$\begin{aligned}
    \dd{x}
    = \pm \frac{\sqrt{2}}{\phi^2 + B} \dd{\phi}
    \qquad\implies\qquad
    \int_{x_0}^{x} \dd{\xi}
    = \pm \sqrt{2} \int_{\phi_0}^{\phi} \frac{1}{\psi^2 + B} \dd{\psi}
\end{aligned}$$

Then we define $$f \equiv \psi / \sqrt{-B}$$,
and update the limits to
$$F = \phi / \sqrt{-B}$$ and $$F_0 = \phi_0 / \sqrt{-B}$$,
in order to integrate by substitution:

$$\begin{aligned}
    x - x_0
    &= \pm \sqrt{2} \int_{F_0}^{F} \frac{\sqrt{-B}}{- B f^2 + B} \dd{f}
    \\
    &= \pm \sqrt{-\frac{2}{B}} \int_{F_0}^{F} \frac{1}{1 - f^2} \dd{f}
\end{aligned}$$

The integrand can be looked up:
it turns out be the derivative of $$\tanh^{-1}$$,
the inverse hyperbolic tangent function,
so we arrive at:

$$\begin{aligned}
    x - x_0
    &= \pm \sqrt{-\frac{2}{B}} \int_{F_0}^{F} \dv{}{f} \Big( \tanh^{-1}(f) \Big) \dd{f}
    \\
    &= \pm \sqrt{-\frac{2}{B}} \tanh^{-1}(F) \mp \sqrt{-\frac{2}{B}} \tanh^{-1}(F_0)
\end{aligned}$$

Rearranging, and combining the integration constants
$$x_0$$ and $$F_0$$ into a single $$t_0$$, yields:

$$\begin{aligned}
    \tanh^{-1}(F)
    &= \pm \sqrt{-\frac{B}{2}} (x - t_0)
    \qquad\qquad
    t_0
    \equiv x_0 \mp \sqrt{-\frac{2}{B}} \tanh^{-1}(F_0)
\end{aligned}$$

Next, we take the $$\tanh$$ of both sides.
It is an odd function, so the $$\pm$$ can be moved outside,
where it can be ignored entirely thanks to the NLS equation's Gauge symmetry.
Using $$F = \phi / \sqrt{-B}$$:

$$\begin{aligned}
    \phi(x)
    &= \sqrt{-B} \tanh\!\Bigg( \sqrt{-\frac{B}{2}} (x - t_0) \Bigg)
\end{aligned}$$

Combining this with our expression for $$\theta$$,
we arrive at the full **dark soliton** solution for $$u$$:

$$\begin{aligned}
    \boxed{
        u(z, t)
        = \sqrt{-B} \tanh\!\Bigg( \sqrt{-\frac{B}{2}} (t - v z - t_0) \Bigg)
        \exp\!\bigg( i \frac{v}{2} t - i \Big( \frac{v^2}{4} - B \Big) z \bigg)
    }
\end{aligned}$$

There are two free parameters here: $$B < 0$$ and $$v$$.
Once again, we can always transform to a moving coordinate system such that $$v = 0$$,
resulting in a stationary soliton:

$$\begin{aligned}
    \boxed{
        u(z, t)
        = \sqrt{-B} \tanh\!\Bigg( \sqrt{-\frac{B}{2}} (t - t_0) \Bigg)
        \exp(i B z)
    }
\end{aligned}$$

Like we did for the bright solitons,
let us specialize this result to fiber optics.
Making a similar transformation $$u \to U/U_c$$,
$$t \to T/T_c$$ and $$z \to Z/Z_c$$ yields:

$$\begin{aligned}
    \frac{U(Z, T)}{U_c}
    = \sqrt{-B} \tanh\!\Bigg( \sqrt{-\frac{B}{2}} \frac{T - T_0}{T_c} \Bigg)
    \exp\!\bigg( i B \frac{Z}{Z_c} \bigg)
\end{aligned}$$

Where we again choose $$U_c$$ manually,
and then find $$T_c$$ and $$Z_c$$ using these relations
(note the opposite signs because $$r = -1$$ in this case):

$$\begin{aligned}
    Z_c
    = \frac{-1}{\gamma_0 U_c^2}
    \qquad\qquad
    T_c
    = \sqrt{\frac{\beta_2}{2 \gamma_0 U_c^2}}
\end{aligned}$$

Recall that $$r = -1$$ implies $$\beta_2 > 0$$ assuming $$\gamma_0 > 0$$,
meaning dark solitons can only exist in the normal dispersion regime.
Inserting this into the expression
and defining the background power $$P_0 \equiv -B U_c^2$$
such that $$|U|^2 \to P_0$$ for $$t \to \pm \infty$$,
we arrive at:

$$\begin{aligned}
    U(Z, T)
    = \sqrt{P_0} \tanh\!\Bigg( \sqrt{\frac{\gamma_0 P_0}{\beta_2}} (T - T_0) \Bigg) \exp(i \gamma_0 P_0 Z)
\end{aligned}$$

Which, as for bright solitons, can be rewritten
with a pulse width $$T_\mathrm{w}$$ satisfying $$N_\mathrm{sol} = 1$$:

$$\begin{aligned}
    \boxed{
        U(Z, T)
        = \sqrt{P_0} \tanh\!\bigg( \frac{T - T_0}{T_\mathrm{w}} \bigg) \exp(i \gamma_0 P_0 Z)
    }
\end{aligned}$$



## References

1.  A. Scott,
    *Nonlinear science: emergence and dynamics of coherent structures*,
    2nd edition, Oxford.
2.  O. Bang,
    *Nonlinear mathematical physics: lecture notes*,
    2020, unpublished.