Basic Propagation Equation

Using the wave equation of free space and considering both a linear and nonlinear portion of the polarization, we can write a basic propagation equation for fibers as:

\begin{equation} \nabla^2E – \frac{1}{c^2}\frac{\partial^2E}{\partial{t^2}} = -\mu_0\frac{\partial^2P_l}{\partial{t^2}} – \mu_0\frac{\partial^2P_{NL}}{\partial{t^2}} \end{equation}

We will use this in the following sections!

Table of Contents:

  1. Nonlinear Pulse Propagation
  2. Higher Order Nonlinear Effects
  3. Delayed Nonlinear Response

NONLINEAR PULSE PROPAGATION

Due to the complexity of the Nonlinear Schrodinger equation, we have to make some simplifying assumptions about $P_{L}$ and $P_{NL}$. As is usually the case for fibers, the $P_{NL}$ term is treated as a small perturbation to $P_L$, which means that it can be treated by perturbation theory and make our lives easier. In the slowly varying envelope approximation, then, we can write the electric field as:

\begin{equation} E(r,t) = \frac{1}{2}\hat{x}[E(r,t)e^{-i\omega_0{t}} + c.c.] \end{equation}

where the $\hat{x}$ term denotes that the field is assumed to be polarized in the x-direction, and E(r,t) is a slowly varying function of time. The two polarization field components can then be written similarly:

\begin{equation} P_L(r,t) = \frac{1}{2}\hat{x}[P_L(r,t)e^{-i\omega_0{t}} + c.c.] \end{equation}

\begin{equation} P_{NL}(r,t) = \frac{1}{2}\hat{x}[P_{NL}(r,t)e^{-i\omega_0{t}} + c.c.] \end{equation}

Through a bunch of math (to be included later), you can get the basic propagation equation:

\begin{equation} \frac{\partial{A}}{\partial{z}} + \beta_1\frac{\partial{A}}{\partial{t}} + \frac{i}{2}\beta_2\frac{\partial^2A}{\partial{t^2}} = i\gamma|A|^2A \label{basic_schr} \end{equation}

where the nonlinearity coefficient $\gamma$ is given as:

\begin{equation} \gamma = \frac{n_2\omega_0}{cA_{eff}} \end{equation}

The effective area would require solving the Bessel functions for the fundamental mode.

HIGHER-ORDER NONLINEAR EFFECTS

Equation \eqref{basic_schr} can be used to describe many physical situations, but it may need to be modified depending on the types of experimental parameters. For example, it does not include any of the effects of SRS or SBS. For a sufficiently high intensity such that the threshold intensity required for significant frequency shifting is met, both SRS and SBS can transfer energy from the pump to the Stokes pulse. The created pulses could then interact with one another through Raman or Brillouin gain and XPM.

The other modification necessary for application of \eqref{basic_schr} to a wider variety of experimental conditions is to include the effects when the pulse duration is $\leq$ 100 fs, i.e. for ultrashort pulses. The spectral width for ultrashort pulses begins to approach the carrier frequency value. As a result, the slowly varying envelope approximation begins to break down and several of the approximations use to find \eqref{basic_schr} are no longer valid. Additionally, because the spectrum of short pulses necessarily is quite wide, Raman gain can amplify the lower frequency components by transferring energy from the higher frequency components. As the pulse propagates in the presence of this phenomenon, the pulse spectrum shifts to higher wavelengths in a phenomenon known as “self-frequency shift”. This is related to the non-instantaneous material response, which leads to a delayed nonlinear response. Once including these effects, the new equation is:

\begin{equation} \frac{\partial{A}}{\partial{z}}+\beta_1\frac{\partial{A}}{\partial{t}} + \frac{i}{2}\beta_2\frac{\partial^2A}{\partial{t^2}} + \frac{\alpha}{2}A = i\gamma|A|^2A + \frac{1}{6}\beta_3\frac{\partial^3A}{\partial{t^3}} – a_1\frac{\partial}{\partial{t}}(|A|^2A) – a_2A\frac{\partial|A|^2}{\partial{t}} \label{gen_new}\end{equation}

Let’s break this apart, because there’s a lot going on here! As we are now considering the case when we have an ultrashort pulse, it is necessary to include the $\beta_3$ term, given on the right side of the above equation. Additionally, the term proportional to $a_1$ comes from the first derivative of the slowly varying part of the nonlinearity, and is responsible for self-steepening and shock formation. $a_1$ is given approximately as:

\begin{equation} a_1 = \gamma/\omega_0 \end{equation}

Then, the last term proportional to $a_2$ includes the effects of the delayed Raman response and thus the effects of self-frequency shifting. $a_2$ is written approximately as:

\begin{equation} a_2 = i\gamma{T_R} \end{equation}

Then, before trying to solve Equation \eqref{gen_new}, we can make the transformation to the pulse’s frame of reference:

\begin{equation} T = t-z/v_g = t – \beta_1z \end{equation}

such that we can obtain the equation:

\begin{equation} \frac{\partial{A}}{\partial{z}}+\frac{\alpha}{2}A + \frac{i}{2}\beta_2\frac{\partial^2A}{\partial{T^2}} – \frac{1}{6}\beta_3\frac{\partial^3A}{\partial{T^3}} = i\gamma[|A|^2A + \frac{i}{\omega_0}\frac{\partial}{\partial{T}}(|A|^2A)-T_RA\frac{\partial{|A|^2}}{\partial{T}}] \end{equation}

DELAYED NONLINEAR RESPONSE

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