---
title: "Self-phase modulation"
sort_title: "Self-phase modulation"
date: 2021-02-26
categories:
- Physics
- Optics
- Fiber optics
- Nonlinear optics
layout: "concept"
---

In fiber optics, **self-phase modulation** (SPM) is a nonlinear effect
that gradually broadens pulses' spectra.
Unlike dispersion, SPM creates frequencies: in the $$\omega$$-domain,
the pulse steadily spreads out in a distinctive "accordion" shape.
Lower frequencies are created at the front of the
pulse and higher ones at the back, giving S-shaped spectrograms.

A pulse envelope $$A(z, t)$$ inside a fiber must obey the nonlinear Schrödinger equation,
where the parameters $$\beta_2$$ and $$\gamma$$ respectively
control dispersion and nonlinearity:

$$\begin{aligned}
    0
    = i \pdv{A}{z} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \gamma |A|^2 A
\end{aligned}$$

By setting $$\beta_2 = 0$$ to neglect dispersion,
solving this equation becomes trivial.
For any arbitrary input pulse $$A_0(t) = A(0, t)$$,
we arrive at the following analytical solution:

$$\begin{aligned}
    A(z,t)
    = A_0 \exp\!\big( i \gamma |A_0|^2 z\big)
\end{aligned}$$

The intensity $$|A|^2$$ in the time domain is thus unchanged,
and only its phase is modified.
Clearly, the largest phase shift increase occurs at the peak,
where the intensity is $$P_0$$.
To quantify this, it is useful to define the **nonlinear length** $$L_N$$,
which gives the distance after which the phase of the
peak has increased by exactly 1 radian:

$$\begin{aligned}
    \gamma P_0 L_N
    = 1
    \qquad \implies \qquad
    \boxed{
        L_N
        \equiv \frac{1}{\gamma P_0}
    }
\end{aligned}$$

SPM is illustrated below for the following Gaussian initial pulse envelope,
with parameter values $$T_0 = 6\:\mathrm{ps}$$, $$P_0 = 1\:\mathrm{kW}$$,
$$\beta_2 = 0$$, and $$\gamma = 0.1/\mathrm{W}/\mathrm{m}$$:

$$\begin{aligned}
    A(0, t)
    = \sqrt{P_0} \exp\!\Big(\!-\!\frac{t^2}{2 T_0^2}\Big)
\end{aligned}$$

From earlier, we then know the analytical solution for the $$z$$-evolution:

$$\begin{aligned}
    A(z, t) = \sqrt{P_0} \exp\!\Big(\!-\!\frac{t^2}{2 T_0^2}\Big) \exp\!\bigg( i \gamma z P_0 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
\end{aligned}$$

{% include image.html file="simulation-full.png" width="100%"
    alt="Self-phase modulation simulation results" %}

The **instantaneous frequency** $$\omega_\mathrm{SPM}(z, t)$$,
which describes the dominant angular frequency at a given point in the time domain,
is found to be as follows for the Gaussian pulse,
where $$\phi(z, t)$$ is the phase of $$A(z, t) = \sqrt{P(z, t)} \exp(i \phi(z, t))$$:

$$\begin{aligned}
    \omega_{\mathrm{SPM}}(z,t)
    = - \pdv{\phi}{t}
    = 2 \gamma z P_0 \frac{t}{T_0^2} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big)
\end{aligned}$$

This result gives the S-shaped spectrograms seen in the illustration.
The frequency shift thus not only depends on $$L_N$$,
but also on $$T_0$$: the spectra of narrow pulses broaden much faster.

The interaction between self-phase modulation
and [dispersion](/know/concept/dispersive-broadening/)
leads to many interesting effects,
such as [modulational instability](/know/concept/modulational-instability/)
and [optical wave breaking](/know/concept/optical-wave-breaking/).



## References
1.  O. Bang,
    *Numerical methods in photonics: lecture notes*, 2019,
    unpublished.