Effects of nonlinearity and dispersion on a pulse

Question

I have some basic doubts on Nonlinear and dispersion effects on a pulse. How does a pulse shape change in temporal domain when passed through:

1. nonlinear medium with strong $X^2$ effect
2. nonlinear and dispersive medium($n_2$>0, GVD <0)

From what I understood,

1. it adds external frequency components which distorts pulse and compress them in time domain.
2. GVD<0 causes down chirp so pulse compression and for $n_2 > 0$, from equation (2.54), downchirp leads to pulse compression, i.e. overall pulse compression.

However, the text below confuses me. Does nonlinearity have any effect on pulse shape?

Answer 1

Let's consider an index of refraction that is frequency dependent.

If $n = n(\omega)$ we see that different wavelengths will have different phase velocities in media. Phase acquired is given by $\delta(\omega) = \frac{n(\omega)d}{c}$

So we are adding phase to components in the frequency domain, obviously phase does not affect our spectrum shape. But, when you get the Fourier transform of the spectrum now those phase components pop up in your pulse shape and we see a broadening of our pulse.

Let's now consider a nonlinear medium. In the model of treating atoms like harmonic oscillators, nonlinearity rises when the electric field oscillations are strong enough to make Hooke's law invalid.

The atom now has an asymmetric response to the field that is time dependent. I.e. we get an index of refraction that is time dependent. With $n(t)$ we see that we introduce phase to the time signal, our pulse shape does not change but, once we get the Fourier transform of the signal we see a difference; I.e our spectrum changes. (Recall we have a real part and an imaginary part in both domains.)

Of course, in practicality, we have to consider both dispersion $n(\omega)$ and phase modulation $n(t)$. So for a dispersive nonlinear medium your index of refraction is dependent on both $\omega$ and $t$, changing the shape in both time and frequency domains.

I am being a little hand wavy, but I hope this helps.