If you "disobey" the constraints of the Kramers-Kronig relations, what happens? Do you get non-physical results?

I am simulating reflection and transmission off/through a slab of material. I specify the complex index of refraction $n = n_r + i n_i$ of the material, assume it has vacuum on either side, and then use the transfer-matrix method to find the reflection and transmission for some given wavelength range.

As I was messing around, adjusting $n_r$ and $n_i$ to see what happens, I found that with certain combinations of $n_r$ and $n_i$ I could get seemingly impossible results -- e.g., reflection going above 1 (which represents more power coming reflecting off of the material than went into it!).

But I was pretty arbitrarily choosing my $n$'s, and I realized that in real life one of $n_r$ or $n_i$ determines the other, from the Kramers-Kronig relations. So was I getting these bizarre results as a result of (probably) "disobeying" the K-K relations, is it more likely my simulation is broken?

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    $\begingroup$ K-K relation is essentially a consequence of causality. Presumably, non-KK dispersion could violate causality. $\endgroup$ Jan 15 '14 at 22:09
  • $\begingroup$ Thank you for the reply. Could you elaborate a little more? I know K-K is tied to causality so some combinations probably won't give physical results, but that kind of divides the domain of indices of refraction into "physically possible" and "physically impossible" ones. $\endgroup$ Jan 15 '14 at 22:12
  • $\begingroup$ I think it is hard to say what could possibly go wrong with non-KK dispersion without knowing how the simulation was done. Maybe you could explain a bit more about the simulation? For instance, are you simulating a plane wave or a wave packet? It could help people understanding your problem. $\endgroup$ Jan 15 '14 at 22:20
  • $\begingroup$ Monochromatic plane wave. The simulation is pretty simple, just using transfer matrices composed of "matching" matrices for the interfaces and a propagation matrix for the propagation inside the slab. But I'm not asking for anyone to debug it, more just comments on what would theoretically happen if you chose indices of refraction that violated K-K in various ways, independent of my program. $\endgroup$ Jan 15 '14 at 22:26
  • $\begingroup$ In this case, please see Steve's answer below. K-K relation doesn't pose any constraint on monochromatic simulation. $\endgroup$ Jan 15 '14 at 22:28

You are presumably doing a monochromatic simulation. At a single frequency, there is no pair $(n_r , n_i)$ that is forbidden by K-K. Therefore it is impossible to choose an unphysical pair. K-K is not your problem.

Keep in mind that you can choose a pair that describes a material with gain. In that case, it is possible to correctly calculate $R>1$. But the whole calculation may often be meaningless in this situation, because you are calculating what the finite solution is, and if there is a lasing instability, the finite solution will not actually happen.

Finally, the starting medium is vacuum, you say, so this is not relevant to your question. But if you were to use a starting medium with some absorption, it is possible to get weird things like $R+T>1$ and I think even $R>1$. These are actually correct (believe it or not), because $R$ has a kind of subtle meaning. For details about this see https://arxiv.org/abs/1603.02720 , appendix B.


e.g., reflection going above 1 (which represents more power coming reflecting off of the material than went into it!).

Great work with your numerical experimentation: I really like how you follow your curiosity! You're clearly a physicist and you've come very near to the answer to the question by your result above.

The Kramers-Kronig relationships are derived by assuming that refractive index is holomorphic in the closed, extended, right half plane $\mathcal{R}=\left\{z\in\mathbf{C}:\,\mathrm{Re}(z)\geq 0\right\}\bigcup\left\{z_\infty\right\}$ (here $z_\infty$ is the point at infinity - the North pole of the Riemann sphere).

The physical meaning of this is follows. Think of an electromagnetic wave generator which works by pulsing the electric field $E_x(t)$ in the $x$ direction, say between the plates of a really big capacitor. The magnetic field $H_y(t)$ in the $y$ direction is then causally related to the electric field through the refractive index: the Fourier-transformed (with respect to time) magnetic field is $H_y(\omega) = n(\omega) \sqrt{\epsilon_0/\mu_0}\tilde{E}_x(\omega)$. So if the electric field is a unit impulse, the magnetic field is proportional to the inverse Laplace transform of $n(s)$. Wherever $n(s)$ has a pole, say at $s_0\in\mathbb{C}$, there is a component $e^{s_0\,t}$ in the magnetic field. $e^{s_0\,t}$ blows up with increasing time if the real part of $s_0$ is greater than nought. So there cannot be any poles in the right half plane for a causal, stable system. $n(s)$ must be holomorphic in the extended right half plane. If it isn't, you have an unstable system that can spontaneously generate a field without input - an unstable system thus violates energy conservation -this is the reason for your greater than unity reflexion co-efficient.

So if you have a system whose refractive index as a function of frequency does not fulfill the Kramers-Kronig relationships, it cannot be causal or strictly stable.

Some Background

Look up the Wikipedia page for the Kramers-Kronig relationships for a derivation. But I find it satisfying to think about this a little abstractly without actually deriving them as follows. That there must be such relationships is very, very fundamental to the idea of a holomorphic function and can be grasped readily without deriving such relationships as follows: suppose two holomorphic functions have the same real parts on the imaginary axis and are both holomorphic in the closed, extended right half plane. Then their difference must be purely imaginary on the imaginary axis. Now consider their difference on the transformed plane where the right half plane is mapped to the inside of the unit disk by the conformal billinear transformation above. The difference must be holomorphic in the closed unit disk and as such has a convergent Taylor series on the unit circle. However, the only way that a power series can be purely imaginary on the unit circle is if its terms are paired in couples of the form $c_n\,z^n−c_n^∗\,z^{−n}$. This is inconsistent with the Taylor series unless the Taylor series is a constant (otherwise it would contain negative powers of z). Hence, the original two holomorphic functions can differ only by a constant. Thus the real part of a function holomorphic in the closed, extended right half plane defines a unique imaginary part to within a constant and contrawise. Wholly analogous reasoning applies to functions holomorphic in the closed, extended right, top, bottom half plane or indeed any simply connected, closed region in the extended complex plane (Riemann sphere).

An even more concise proof (albeit not so "first-principles" in nature) is the following: the difference between the two functions is purely imaginary on the imaginary axis. A simple variation on Schwarz's Reflexion principle then shows that the difference must be holomorphic in the extended, closed left half as well as right half plane. Therefore the difference is entire; moreover it is bounded at the point at infinity. Liouville's theorem then shows that the difference must be a constant.


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