# More extensions of the wave equation for dispersion

The Phys.SE question Minimal Extension of Wave Equation to Include Dispersion extended the wave equation for only a very simple form of dispersion.

However, what about more complex dispersion relationships such as http://refractiveindex.info/?group=PLASTICS&material=PMMA in which the refractive index (inverse wave speed) curve has a more complex structure?

Is there a way to find the equation for arbitrary velocity-frequency relationships (as long as you choose "well behaved" functions)? Conversely, if not, does that mean that the dispersions allowed by Nature are restricted in some way as to always yield a wave equation?

Depending on the physics underlying the particular wave equation in question, the three most fundamental limitations on dispersion are causality, stability and holomorphicity. These are most readily converted to mathematical statements about the operators in a wave equation if the wave equation is linear.

I'll confine the following mainly to optics; broadening these ideas to more general, first quantised particle wave equations will depend on the physics described by the wave equation terms and whether it can be construed as causal, holomorphic and stable, but I should think that these ideas will yield the most general constraints: maybe some other answer can clarify this broadening.

In optics, the relevant wave equation is the monchromatic Helmholtz equation $(\nabla^2 + k^2(\omega))\psi = 0$. The electron absorbtion and re-emission physics that begets the variation $k^2(\omega)$ with frequency $\omega$ means that $k^2(\omega)$ is a meromorphic function (holomorphic aside from isolated poles) of $\omega$ when $k^2(s)$ is broadened into the complex frequency plane and $s$ is now a complex frequency $s = \sigma + i \omega$. Furthermore, the electron response must be almost always be stable: conservation of energy means that $k^2(s)$ cannot have poles in the open right half plane (a pole at $s=s_0$ means an unbounded component $e^{s_0\,t}$ in the electron response if $\operatorname{Re}(s_0)>0$), although this may not be true in optically or otherwise pumped mediums with gain, where such poles can describe a buildup of a laser pulse for example, at least whilst the system with its growing pulse is still working at linear levels.

If $k^2(s)$ is therefore holomorphic in the closed right half plane (i.e. on the imaginary axis too) and further if $k^2(s)\to 0$ faster than $1/|s|$ as $s\to\infty$ with $\operatorname{Re}(s) \geq 0$ then the real and imaginary parts of $k^2$ are related by the Hilbert transform; if $k^2(i\,\omega) = p(\omega) + i\,q(\omega)$ then:

$$q(\omega) = \frac{1}{\pi}\operatorname{CPV}\int_{-\infty}^\infty \frac{p(u)}{u-\omega}\,\mathrm{d}\,u$$ $$p(\omega) = -\frac{1}{\pi}\operatorname{CPV}\int_{-\infty}^\infty \frac{q(u)}{u-\omega}\,\mathrm{d}\,u$$

where $\operatorname{CPV}$ stands for "Cauchy Principle Value". In practice, this is hard to work out numerically (owing to the divergence) and other ways of doing the Hilbert transform are more practicable; see for example my answer here. Relationships between the real and imaginary parts of refractive index are similar and often expressed through the Kramers-Kronig relationships (note that the square root in a refractive index $n(\omega) = \sqrt{\epsilon(\omega)}$, although it introduces a branch point where $\epsilon(\sigma + i\omega) = 0$, this doesn't happen in the right half plane or imaginary axis).

Causality is another fundamental constraint. In optics, $k^2(\omega)$ describes the response of polarisation and magnetisation vectors in materials considered as outputs to changes in the electric and magnetic field thought of as "inputs". Therefore $\mathfrak{F}^{-1} k^2(\omega)$, where $\mathfrak{F}$ is the Fourier transform, must be causal, i.e. its response to a unit impulse at $t=0$ must be confined to positive times. The Payley-Wiener causality criterion, which is a corollary of the more general Paley-Wiener Theorem describing growth rates of Fourier transforms, says that $k^2(\omega)$ can define a causal response if and only if its magnitude fulfils:

$$\int_{-\infty}^{\infty} \frac{\log |k^2(\omega)|}{1 + \omega^2}\,\mathrm{d} \omega < \infty$$

i.e. if the integral is convergent. Notice that this does not change if we seek responses which are delayed responses; the delay $f(t)\mapsto f(t-t_0)$ maps to multiplication by $e^{-i\,\omega\,t_0}$ in the Fourier domain, so does not change the quantity $\log |k^2(\omega)|$. It's worth stating the if and only if condition carefully: if a response $k^2(\omega)$ is causal, it fulfills the Payley-Wiener criterion. However, conversely, if the magnitude $|k^2(\omega)|$ fulfills the Payley-Wiener criterion, this means that there exists a phase response $\phi(\omega)$ (indeed a class of phase responses) such that $\mathfrak{F}^{-1} (|k^2(\omega)|e^{i\,\phi(\omega)})$ is causal but not every $k^2(\omega)$ with the same, Payley-Wiener Criterion-fulfilling magnitude defines a causal response.

• +1, though I didn't quite catch why you take causality and stability to be more fundamental than holomorphicity.
– user10851
Sep 9, 2013 at 2:52
• @ChrisWhite Actually I just got rid of that phrase. I guess I was just thinking of contrived, piecewise continuous refractive indices, but these are going to be smooth at high enough resolution and moreover, when broadened into the complex plane, you'll still have meromorphicity even for such contrived examples. Sep 9, 2013 at 3:07
• Wow, just wow. This gives me plenty to chew on! Sep 10, 2013 at 1:02
• @KevinKostlan Glad I could help. One thing I ought to have said about stability is that similar ideas are used in nonlinear wave equations even though my answer concentrates on linear effects. In particular, one linearises an equation (finds its first order behavior in the neighbourhood of) at an equilibrium point to find out whether the equilibrium is stable or not. Sep 10, 2013 at 2:34