# Modulation Instability

Interestingly, the idea of modulation instability is observable in many areas of physics, not just in optics and optical fibers. Modulation instability refers to the disturbance to a steady state system via the effects of dispersion and nonlinearities. The effects have been observed in as diverse of fields as fluid dynamics, nonlinear optics, and plasma physics.

Table of Contents:

STABILITY OF A CW OPTICAL BEAM

Modulation instability in the context we are discussing refers to the break-up of a CW wave into a burst of ultrashort pulses in the presence of anomalous dispersion. To further illuminate what this means, we will look at the simplified nonlinear Schrodinger equation (simplified because we are neglecting loss), given in Equation \eqref{simp_non_schro}.

$$i\frac{\partial{A}}{\partial{z}} = \frac{1}{2}\beta_2\frac{\partial^2A}{\partial{T}^2}-\gamma{|A|^2A}\label{simp_non_schro}$$

Here, “A(z,t)” is the amplitude of the pulse envelope, “$\beta_2$” is the GVD parameter, and “$\gamma$” is the nonlinearity parameter that will introduce self-phase modulation. This should all look familiar; if not, and you are hyperventilating due to the fear of missing out (it happens), please review section 2 of “Fiber Optics” to serve as your brown paper bag!

Now, in the CW case, the pulse envelope should be time-independent, as a CW beam at time T = 0 should “look” exactly the same as the beam at any arbitrary time T > 0. (Note: we can think of this easily by considering that if power meters could measure pulsed beams in real time, there would be a time dependence on the intensity it reads due to the discontinuous nature; this would obviously not be the case for a continuous wave.) As a result, we can solve our equation above with our steady-state, time independent solution:

$$A = \sqrt{P_0}e^{i\phi_{NL}}$$

where the nonlinear phase shift is given as:

$$\phi_{NL} = \gamma{P_0}z$$

GAIN SPECTRUM

EXPERIMENTAL OBSERVATIONS

OTHER CONSIDERATIONS