The effects of SPM can be considered in a much simpler way by assuming a pulse width $T_0 \geq$ 1 ps. We found in the last section that having longer pulse widths and a large peak power leads to negligible GVD, so those are the conditions we will consider first in our nonlinear propagation equation!

**Table of Contents:**

- Nonlinear Phase Shift
- Changes in Pulse Spectrum
- Effect of Pulse Shape and Initial Chirp
- Effect of Partial Coherence

**NONLINEAR PHASE SHIFT**

In our normalized version of the propagation equation, given as:

\begin{equation} i\frac{\partial{U}}{\partial{z}} = \frac{\pm\beta_2}{2L_D}\frac{\partial^2U}{\partial\tau^2}-\frac{e^{-\alpha{z}}}{L_{NL}}|U|^2U \end{equation}

if we set the $\beta_2$ term = 0, we would have the propagation equation:

\begin{equation} \frac{\partial{U}}{\partial{z}} = \frac{i}{L_{NL}}e^{-\alpha{z}}|U|^2U \label{non}\end{equation}

where the $\alpha$ term accounts for losses in the fiber. The nonlinear length is again given as:

\begin{equation} L_{NL} = (\gamma{P_0})^{-1} \end{equation}

where $P_0$ is the peak power and $\gamma$ is related to the nonlinear-index coefficient $n_2$. The propagation equation for purely nonlinear effects, Equation \eqref{non} can be solved with the amplitude equation:

\begin{equation} U(z,T) = U(0,T)e^{i\phi_{NL}(z,T)} \label{SPM}\end{equation}

where “U(0,T)” is the field amplitude at z=0 and $\phi_{NL}$ is given as:

\begin{equation} \phi_{NL}(z,T) = |U(0,T)|^2(z_{eff}/L_{NL}) \end{equation}

with

\begin{equation} z_{eff} = [1-e^{-\alpha{z}}]/$\alpha$ \end{equation}

The physical interpretation of Equation \eqref{SPM} is that in SPM, there is an intensity dependent temporal phase shift that increases with the propagated distance, z. The effective distance $z_{eff}$ takes into account the effects of fiber loss. The maximum phase shift occurs at the center of the pulse, T = 0, which we can understand since the maximum of whichever pulse shape we choose should have all of its power towards the center of the pulse. Then, since U is normalized such that |U(0,0)| = 1, we can find the maximum phase shift as:

\begin{equation} \phi_{max} = z_{eff}/L_{NL} = \gamma{P_0}z_{eff} \end{equation}

From here, we can see that the physical meaning of the nonlinear length $L_{NL}$ is that this is the position where $\phi_{max}$ = 1. Increases in input pulse power will decrease the effective nonlinear length.

In the last section on dispersion, we had a phase shift in the frequency domain that led to changes in the pulse (broadening/distortion) in the time domain. Thus, with phase changes in the time domain, we must then have an unchanging temporal shape but a changing spectral shape! This is just a result of the meaning of the Fourier transform, but we can use this to figure out what this may actually mean if you are watching self-phase modulation occur. Basically, we can think of this as a frequency-domain chirping in that the instantaneous optical frequency differs across the pulse from its central value $\omega_0$. The difference between the frequency across the pulse and the central value is given as:

\begin{equation} \delta\omega(T) = -\frac{\partial\phi_{NL}}{\partial{T}} = -\frac{\partial}{\partial{T}}(|U(0,T)|^2)\frac{z_{eff}}{L_{NL}} \end{equation}

What this means physically is that new frequency components are continuously generated as the pulse propagates down the fiber, which broaden the spectrum over its initial width at the start.

The extent of the spectral broadening depends on the initial pulse shape: Gaussian, super-Gaussian, hyperbolic-secant, etc.

**CHANGES IN THE PULSE SPECTRUM**

For a Gaussian pulse, we can find the maximum frequency broadening by taking the time derivative of the frequency shift to obtain:

\begin{equation} \delta\omega_{max} = \frac{fm}{T_0}\phi_{max} \end{equation}

where $\phi_{max}$ was given previously and f is a constant given as:

\begin{equation} f = 2[1-\frac{1}{2m}]^{1-1/2m}e^{-(1-\frac{1}{2m})} \end{equation}

Since the broadening factor is related to the maximum phase shift by a constant, the broadening factor is effectively determined by the maximum phase shift. Since the maximum phase shift of the central peak can be on the order of 100, SPM can be used to broaden the spectrum by a considerable amount. In combination with other nonlinear processes such at SRS and four-wave mixing that also add more spectral components, this can result in what is sometimes referred to as “supercontinuum generation”.

The figures below show the experimental results of an initial unchirped Gaussian pulse for different values of the maximum phase shift. In general, the pulse spectrum depends not only on the pulse shape but also on the initial chirp. Since $\phi_{max}$ increases linearly with increasing peak power $P_0$, the spectral evolution can be observed by increasing the input peak power.

An important phenomenon to note in the above picture is the oscillatory structure of the spectrum. We can understand this by looking at the following image of the temporal variation of nonlinear phase and frequency chirp:

The frequency chirp in the bottom of the two plots shows that at two separate points in time, there is the same amount of frequency chirping for two separate parts of the pulse. We can qualitatively understand this, then, as two waves of the same frequency but different phases that can interfere constructively or destructively depending on the magnitude of their relative phase difference. Using a method of stationary phase, we can find an approximate relation for the number of peaks M by the relation:

\begin{equation} \phi_{max} = (M-1/2)\pi \end{equation}

**EFFECT OF PULSE SHAPE AND INITIAL CHIRP**

The spectral shape depends on both the initial pulse spectral shape and the initial frequency chirp. For example, a super Gaussian will show a much larger frequency spread than a Gaussian pulse due to the fact that its frequency chirp value is much larger. It also has most of its power concentrated into one central peak, which corresponds with the fact that the chirp is nearly zero in the central region of such a pulse.

Additionally, the initial frequency chirp can add or subtract oscillatory fringes for an SPM broadened pulse depending on the sign of the chirp. As can be seen from the image below for a Gaussian pulse, C < 0, which corresponds with negative dispersion, the spectrum narrows. This can be qualitatively understood as the addition of the positive phase shift from SPM with the negative phase shift of negative dispersion, such that these two cancel out to actually subtract frequency components. In the opposite case, C > 0, the phase components add so that there is an increased oscillatory structure.

**EFFECT OF PARTIAL COHERENCE**

The preceding discussion was just for optical pulses since the nonlinear phase shift mimics the shape i.e. the temporal variations of the pulse. However, this assumes only coherent light can exhibit SPM, since pulses are perfectly coherent as they add their frequency components in phase (shown below).

In reality, however, all optical beams are only partially coherent, both temporally and spatially. Most times, as long as the coherence time of the light source is longer than the width of the pulse, the effects of partial coherent are negligible. However, when the pulse becomes comparable to or longer than the coherence time, the effects of partial coherence must be included. This means that we can also consider the effects of partially coherent CW beams (shown below):

Indeed, SPM can lead to spectral broadening for CW beams during their propagation in fiber due to the fact that CW light (and all partially coherent light) exhibits fluctuations in both intensity and phase. The net effect of SPM is to turn intensity fluctuations into additional phase fluctuations, as we saw from the intensity-dependent nonlinear phase, which in turn leads to spectral broadening. The spectrum of partially coherent light is obtained from the Wiener-Khinchin theorem as:

\begin{equation} S(\omega) = \int^{\infty}_0\Gamma(z,\tau)e^{i\omega\tau}d\tau \end{equation}

where the coherence function is defined as:

\begin{equation} \Gamma(z,\tau) = <U^*(z,T)U(z,T+\tau)> \end{equation}