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 ω\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)A(z, t) inside a fiber must obey the nonlinear Schrödinger equation, where the parameters β2\beta_2 and γ\gamma respectively control dispersion and nonlinearity:

0=iAzβ222At2+γA2A\begin{aligned} 0 = i \pdv{A}{z} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \gamma |A|^2 A \end{aligned}

By setting β2=0\beta_2 = 0 to neglect dispersion, solving this equation becomes trivial. For any arbitrary input pulse A0(t)=A(0,t)A_0(t) = A(0, t), we arrive at the following analytical solution:

A(z,t)=A0exp ⁣(iγA02z)\begin{aligned} A(z,t) = A_0 \exp\!\big( i \gamma |A_0|^2 z\big) \end{aligned}

The intensity A2|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 P0P_0. To quantify this, it is useful to define the nonlinear length LNL_N, which gives the distance after which the phase of the peak has increased by exactly 1 radian:

γP0LN=1    LN1γP0\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 T0=6psT_0 = 6\:\mathrm{ps}, P0=1kWP_0 = 1\:\mathrm{kW}, β2=0\beta_2 = 0, and γ=0.1/W/m\gamma = 0.1/\mathrm{W}/\mathrm{m}:

A(0,t)=P0exp ⁣( ⁣ ⁣t22T02)\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 zz-evolution:

A(z,t)=P0exp ⁣( ⁣ ⁣t22T02)exp ⁣(iγzP0exp ⁣( ⁣ ⁣t2T02))\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}

Self-phase modulation simulation results

The instantaneous frequency ωSPM(z,t)\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 ϕ(z,t)\phi(z, t) is the phase of A(z,t)=P(z,t)exp(iϕ(z,t))A(z, t) = \sqrt{P(z, t)} \exp(i \phi(z, t)):

ωSPM(z,t)=ϕt=2γzP0tT02exp ⁣( ⁣ ⁣t2T02)\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 LNL_N, but also on T0T_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.


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 O. Bang, Numerical methods in photonics: lecture notes, 2019, unpublished.