# Self-phase modulation induced spectral broadening

How does one explain the fact that a pulse that has its phase modulated through SPM, generates new frequencies? How does the phase modulation affect the electrons so they emit not only the frequencies at which they are driven but also ones shifted to higher and lower energies?

Can it be explained as a degenerate four-wave mixing? If so, how is it phase-matched?

• A pulse modulated even without a non-linear medium, but simply by shortening its duration, also produces broader frequencies: physics.stackexchange.com/questions/361210/… – safesphere Oct 29 '17 at 0:05
• I understand it a bit the other way, for the pulse to possibly exist with its time being shorter, there had to be frequencies added to it at some point. I don't understand, how phase modulation "forces" the electrons to oscillate at new frequencies that weren't present in the pulse to begin with. Also, group velocity dispersion, makes changes to the phase too, but doesn't broaden the spectrum (it instead broadens the pulse, but that is simple to understand for me). – KabaT Oct 29 '17 at 8:07
• @safesphere That's a misunderstanding of the answer to the thread you linked. You cannot change the spectral bandwidth of a given pulse, after it's been generated, without nonlinear interactions. You can then translate this into the fact that if you have a pulse with Fourier-limited length and you want to make it shorter, you need a broader spectrum and therefore you need nonlinear interactions. The usual answer is as in this question: SPM in a fiber to broaden the spectrum, followed by chirp-inducing elements to re-compress the pulse. – Emilio Pisanty Oct 31 '17 at 11:09
• @EmilioPisanty What if I pass a near monochromatic continuos laser beam through a mechanical shutter of a super fast camera? Hypothetically, if the shutter is fast enough and I chop out a single very short pulse, would its spectrum not be wider than the spectrum of the original continuous beam? I could also mechanically cut periodic pulses shorter by some sort of a rotating disk with a hole. – safesphere Oct 31 '17 at 15:05
• @safesphere For real mechanical shutters the added bandwidth is minute (kHz instead of THz), but in the abstract the question is valid. The chopping procedures you describe do broaden the spectrum. Those processes are linear in the fields, but the recourse to an external mechanical means (with its own timing reference!) means that they're not really describable as linear interactions with a medium. – Emilio Pisanty Oct 31 '17 at 15:16

As you point out, self-phase modulation can be thought of as an added chirp, but the crucial point is that this is a local chirp that changes from the front to the back of the pulse, in essence because the phase velocity in the middle of the pulse is faster than at the edges, because of the changed refractive index at its higher intensity, and this compresses and decompresses the wavefronts in the leading and trailing edge of the pulse:

This introduces local frequencies that were simply not present within the original pulse spectrum, leading to a spectral broadening. (The pulse's electric field is in blue, instantaneous frequency is in red.)

Now, that's the wave picture of SPM, but as always in nonlinear optics, there is the wave picture and the 'photon' (spectral) picture, and normally you want to be able to produce a complete explanation within each of the two domains. In that regard, SPM is a third-order process so it is simply a version of four-wave mixing with two photons in and two photons out (so, normally $\omega_1$ and $\omega_2$ in and $\omega_1+\Delta$ and $\omega_2-\Delta$ out), but it's a complicated process because you have a bunch of photon energies available in your original pulse bandwidth and you need all their interactions to get the full picture, so it's not an easy description.

And finally, as to phase matching, if you only have a single spectral component (say, you have a quasi-monochromatic beam in one arm of a Mach-Zehnder interferometer and you're testing how the interference changes with the beam intensity) then the SPM will automatically phase-match. However, if you have a pulse and you're doing spectral broadening, then you need to do the same kinds of phase matching that you do for standard four-wave mixing, with the additional complication that you have a continuum of initial and final frequencies, and there doesn't seem to be any simple description of this other than just jumping into the nitty-gritty.

• I don't understand what you mean by local frequencies. Dispersion also causes chirping and the instantaneous frequency differs across the pulse. The only difference I see is the fact that in the case of SPM, instantaneous frequency is not monotonic, therefore we have choose two points in time that have the same frequency, as can be seen on your nice animations. But I still don't understand how this fact causes new frequencies and in the case of dispersion it doesn't. I only understand from it the shape of the spectrum in SPM... – KabaT Nov 1 '17 at 22:41
• because if we have some frequency $\omega$ in two time points of the pulse they can cancel out and cause modulation in the spectrum (which goes down to zero in the case of pure spm, I understand that dispersion makes that fully destructive interference change to weak modulation). Near the top of the pulse in the case of gaussian pulse instantaneous frequency is linear while SPM is active, so this part is the same as with dispersion. Difference exists on the pulse edges where there is this monotonicity and phase derivative is biggest. – KabaT Nov 1 '17 at 22:45
• I also know that those new frequencies are created at the edges and later in a process called optical wave breaking can travel to the tails of the pulse and interfere. I think that they only start to interfere after some longer propagation because at the start they are in phase so there is no interference in time, because of this automatic phase matching of SPM? – KabaT Nov 1 '17 at 22:48
• As for spectral picture, I had similar thing in mind but wasn't sure. So I can think of it as a degenerate or non-degenerate four wave mixing, they if it is at some moment in time phase matched it will create a new frequency, but because each frequency can mix with each frequency, it is hard to show how it does its phase matching? As both SPM and FWM are third order nonlinearity, could we say that separating those effects is artificial in a way? And that just in FWM we usually use different frequencies from different pulses and in SPM frequencies all come from one pulse. – KabaT Nov 1 '17 at 22:53
• And I got another problem, dispersion adds quadratic phase, so if we have a parabolic time, as SPM adds phase in a shape of the pulse, it would seem that both dispersion and SPM add same shape phase in this case. But SPM will create new frequencies and dispersion will not... – KabaT Nov 1 '17 at 22:59

When a laser pulse transmits through a medium, for example fused silica, the intensity-dependent refractive index $n(I)$ is different for different part of the laser pulse since the laser pulse has a certain envelope, then we get a time-dependent refractive index $n(t)$. Then, for different parts of the laser pulse, they experience different transimission speed $v=c/n(t)$, where $c$ is the speed of the light in vaccum, and so the total pahse of the laser pulse is: $$\phi=\omega t+\psi(t)$$ The frequence, i.e. the time derivation of the phase is $$\omega'=\frac{\partial\phi}{\partial t}=\omega+\frac{\partial\psi(t)}{\partial t}$$ The new frequency components is generated.

About "How the phase modulation affects the electrons", I think we can understand it by how medium change the refractive index even though I don't understand it.

• Dispersion also adds chirp to the pulse, but doesn't generate new frequencies. SPM just adds different shape of the chirp and it somehow makes new frequencies. Maybe it is related to nonmonotonic character of SPM chirp. I have read 20 papers about SPM but still don't quite get it. Currently I'm trying to think about it as degenerate four-wave mixing phase-mathed via this nonmonotonicity of the chirp. – KabaT Oct 31 '17 at 9:11
• In my opinion, SPM does not just add different shape of the chirp, it's only part of its effect. The change of phase of different part of the laser pulse caused by intensity-dependent refractive index dosen't choose to act on a certain frequency but on all of them. The chirp comes just because the $n(\omega)$, it even happens for very weak pulse. – Bettertomo Oct 31 '17 at 9:23
• Uh, this is all of my knowledge about this. I guess you maybe know more than me about it! – Bettertomo Oct 31 '17 at 9:34