# Higher-order dispersion in non-linear optics

In many cases, dispersion of an optical medium is presented by saying that the dependence of the wavenumber $$k$$ on the (angular) frequency $$\omega$$ can be expanded in Taylor series as: $$k(\omega) = k_0 + \frac{\partial k}{\partial\omega} (\omega-\omega_0) + \frac{1}{2} \frac{\partial^2 k}{\partial\omega^2} (\omega-\omega_0)^2 + \frac{1}{6} \frac{\partial^3 k}{\partial\omega^3} (\omega-\omega_0)^3 + \dots.$$ In introductory courses, it is then usually added that the different terms have a different meaning: the first term (zeroth order with respect to $$\omega$$) is a common phase term, the second term is related to the group velocity: $$k' = \frac{\partial k}{\partial\omega} = \frac{1}{v_g}$$ and the third term is related to the group delay dispersion: $$k'' = \frac{\partial^2 k}{\partial\omega^2}$$ while the fourth term is related to the third order dispersion parameter: $$k''' = \frac{\partial^3 k}{\partial\omega^3}.$$ In more advanced courses on laser physics (especially when dealing with ultrafast pulses) these terms are widely explained and schemes for their compensation from an experimental point of view are presented. However, higher-order terms are generally neglected.

In this page higher-order terms are just named, saying that in some cases we need to consider also them; however, in that cases the Taylor expansion looses its meaning. They are all named in the paragraph about solitons.

The questions are therefore the following ones:

• Is there an intuitive meaning for higher-order dispersion terms?
• In which kind of experiments they are relevant? (Examples would be appreciated!)
• Are there "standard" schemes for compensating these terms?
• How is it possible to model them when the Taylor expansion loses its validity?

4) In principle the dispersion relation $$k(\omega)$$ is just the refractive index spectrum. Not more or less. It is related to the absorption coefficient spectrum by the Kramers-Kronig relations (*). Both can have any shape, depending on the quantum physical effects that cause absorption. So modeling from first principles needs a quantum physical simulation (This works best for "simple" species such as atoms with few electrons or molecules with few atoms). For empirical modeling a simple spectral measurement of $$k(\omega)$$ is the method of choice.

For instance if one looks at a single gas absorption line, the Taylor expansion breaks down even for very narrow spectral intervals. The Taylor expansion of the Lorentz function (the shape of the spectral line) is only convergent inside its half-width.

3) Compensation is done by using optical elements which have the "inverse" $$k(\omega)$$ behavior, which you would like to compensate. Since materials can be engineered, one can also engineer dispersion. The simplest approach is to use a layered material and design a distributed bragg reflector, with the spectral properties one needs (cf. chirped mirror). Also waveguides can be engineered, and so the dispersion of the mode of interest.

2) I think the Taylor variant of describing dispersion is only really useful from the application point of view. That means if one designs some optical system which has a small enough optical bandwidth, such as fiber communications, broadband lasers, etc... There the refractive index (or, alternatively, $$k$$) varies only a little bit, so one only needs the first or second order dispersion to correctly describe the $$k(\omega)$$. So it is an empirical description.

1) I would say it is the bending and curvature (i.e. nonlinearity) of the dispersion spectrum. This is my most intuitive view. If you have only the first (linear) term, then pulses (more precisely their envelope) keep their shape, but propagate with a different velocity than given by the ordinary refractive index (or phase velocity, which is the propagation velocity if there is no dispersion). This is why we have group velocity. If there are higher order terms also the shape of the pulse gets distorted, and I have no intuitive view on their effect on pulse propagation (then also the group velocity may no longer be the velocity of propagation).

*) In principle this is only true if the matter has a minimum phase behavior. However I do not know any counterexamples where Kramers-Kronig braks down.

• OP is talking about a pulse. You seem to be talking about a medium. – Emilio Pisanty Dec 4 '18 at 9:22
• Dispersion only exists in media. I don't see any contradiction. – Andreas H. Dec 4 '18 at 9:45
• Thank you for your answer! My question was slightly more directed toward experiments in nonlinear optics in which dispersion terms of order higher than two play a role, but yours answer is providing interesting points, especially when you talk about the Kramers-Kronig relations! – JackI Dec 4 '18 at 10:24
• @Jackl: Hmm, you do not mention nonlinear optics. Please be aware that the higher order dispersion terms are not "nonlinear optics". Dispersion is all linear optics! Nonlinear optics is a completely different story, and there it you cannot describe dispersion that easily. It would essentially also depend on other parameters . – Andreas H. Dec 4 '18 at 12:08
• @Jackl: ah, ok. Nonlinear optics is in the title. But then there is missing context What kind of nonlinear optics? This is crucial. – Andreas H. Dec 4 '18 at 12:10

Is there an intuitive meaning for higher-order dispersion terms?

--> Just like second-order dispersion (group velocity dispersion) relates to how the relative phase velocity of different frequencies vary to second-order, the higher-order terms tell you the same same but for the other higher power terms.

Analogy: Imagine a group of runners (with different running speeds) all starting at the same same on the same line. At T=0 the runners start running and after some time they spread out in time/space. The runners who run fastest will continue to do so, and increase the spatial gap between them and the slower runners. This "spread" in runners as a function of time depends upon those higher- order terms.

In which kind of experiments they are relevant? (Examples would be appreciated!)

--> Dispersion becomes most relevant as the spectrum becomes increasingly bigger. If you are dealing with a very narrow set of frequencies, then dispersion won't have much effect except over a very long time or path length. For example, if our runners from above all ran at the same speed, then they would all run together in the same pack. Whereas it's because of large spread in running speeds that create the pack to spread out.

Are there "standard" schemes for compensating these terms?

The standard schemes are to use a prism, grating, fiber of appropriate dispersion, or any mechanism that can create a time-delay as a function of wavelength.

How is it possible to model them when the Taylor expansion loses its validity? You can model these items (you can probably find on google) with the same expansion with different coefficients.

• Thank you for your answer! So your suggestion is that higher order terms are completely equal (except for the fact that they are higher order), in their description, to lower order ones? And also conclusion and experimental methods are the same? – JackI Dec 10 '18 at 7:19
• They're not completely equal because they have different coefficients and scale factors. Let's consider just the even terms, 2rd order, 4th, 6th,etc. Each one of these terms will create a spread in the pulse (much like the pack of runners spreading out. The 2nd order term will do so at a certain rate, the 4th order term will also contribute to the spread but at a different (usually much smaller) rate. etc – JQK Dec 10 '18 at 20:02