Categories: Fiber optics, Nonlinear optics, Optics, Physics.

# Self-phase modulation

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:

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:

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

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 leads to many interesting effects, such as modulational instability and optical wave breaking.

## References

- O. Bang,
*Numerical methods in photonics: lecture notes*, 2019, unpublished.