---
title: "Self-phase modulation"
firstLetter: "S"
publishDate: 2021-02-26
categories:
- Physics
- Optics
- Fiber optics
- Nonlinear optics

date: 2021-02-27T10:09:32+01:00
draft: false
markup: pandoc
---

# Self-phase modulation

In fiber optics, **self-phase modulation** (SPM) is a nonlinear effect
that gradually broadens pulses' spectra.
Unlike dispersion, SPM does create new frequencies: in the $\omega$-domain,
the pulse steadily spreads out with a distinctive "accordion" peak.
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:

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

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.
It is also clear that the largest phase increase occurs at the peak of the pulse,
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:

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

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:

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

<a href="pheno-spm.jpg">
<img src="pheno-spm-small.jpg">
</a>

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.