Talk:Self-phase modulation

It is vague (at best) how one arrives at the phase in the derivation:

$$\phi(t) = \omega_0 t - \frac{2\pi}{\lambda_0} n(I) L $$

Also, this is stated without citation. My guess is that it has to do with the electric field, which is hidden in this discussion of the pulse intensity, but I'm not sure, and I'd like someone to clarify. Wolfman81 (talk) 21:07, 9 June 2009 (UTC)

"This variation in refractive index will produce a phase shift in the pulse, leading to a symmetric broadening of the pulse's frequency spectrum." This is only true for symmetric pulses where the current chirp has the same sign as $$n_2$$. In all other cases SPM only results in a change of the spectrum. I was very suprised the first time I propagated a negatively chirped pulse into a positive $$n_2$$ material. The spectrum got narrower. --Erik Zeek 20:06, 1 December 2005 (UTC)

Re-writing the article because I don't think the derivation below is correct. In particular the frequency shift doesn't seem to have the correct form. --Bob Mellish 00:41, 4 November 2005 (UTC)

Negative Nonlinearities
Part of the text in this article assumes that the nonlinear coefficient n_{2} is positive. Matthew Rollings (talk) 20:20, 1 February 2009 (UTC)

Original derivation
$$\Phi_{nl}(t)=-\delta n \cdot l \cdot {\omega_0 \over c}$$

where $$n$$ is the refractive index, $$l$$ is the propagation distance in the medium, $$c$$ is the velocity of light in vacuum and $$\omega_0$$ is the carrier-frequency of the pulse.

$$\delta n=n_2 \cdot I(t)^2$$

is the second order change of the nonlinear refractive index. The instantaneous frequency is given by

$$\omega (t) = \omega_0+\delta \omega (t)$$

with

$$ \delta \omega (t)={{d \Phi_{nl} (t)} \over dt}$$

being the phase velocity. In case of a common hyperbolic secant pulse shape the intensity is given by

$$I(t)=I_0 \cdot \operatorname{sech}^2({t \over {\tau_0}})$$

Therefore the nonlinear phase of this pulse becomes

$$\Phi_{nl(t)}=-n_2 \cdot l{\omega_0 \over c} \cdot l_0 \cdot \operatorname{sech}({t \over {\tau_0}})$$

and the instantaneous frequency is shifted by the term

$$\delta \omega (t)=2 \cdot n_2 \cdot l \cdot {{\omega_0} \over {c \cdot \tau_0}} \cdot I_0 \cdot \operatorname{sech}^2({t \over {\tau_0}})$$