Finite Frequency Kramers Kronig Relations, or

Checking the compatibility of experimental measurements of the complex dielectric constant

Dispersion relations are prevalent throughout physics and derive from the causal nature of the response of materials, bodies or particles to electromagnetic, elastic or other fields. The classic example of a dispersion relation is the Kramers-Kronig (KK) relation (Kramers, H.A., Nature, 117 (1926) 775, Kronig, R. de L., J. Opt. Soc. Am. 12 (1926) 547) that couples the real and imaginary parts of the complex dielectric constant $\epsilon(\omega) = \epsilon_1
(\omega) + i \epsilon_2(\omega)$ of a material by

\epsilon_1(\omega) = 1 + \frac{1}{\pi} P \int_{-\infty}^\inf...
 {\omega^{\prime^2} - \omega^2} d\omega^\prime,\end{displaymath}

where P denotes the principle value of the integral, where the last identity follows from the fact that $\epsilon_2(-\omega^\prime)=-\epsilon_2(\omega^\prime)$. Thus $\epsilon_1(\omega)-1$ is the Hilbert transform of $\epsilon_2(\omega)$. The chief obstacle to the practical application of the KK relation is that one needs to know $\epsilon_2(\omega)$ over all frequencies to determine $\epsilon_1(\omega)$, whereas a given experiment only yields values of $\epsilon_1(\omega)$ and $\epsilon_2(\omega)$ over a finite frequency range. This is typically handled in a crude manner which is only sometimes effective: some approximation for $\epsilon_2(\omega)$ is used outside the measured frequency range. When the measured and computed function $\epsilon_1(\omega)$ disagree, one is left in doubt as to whether the measurements of $\epsilon_1(\omega)$and $\epsilon_2(\omega)$are compatible with each other, or whether the data set violates what is known about the analytic properties of $\epsilon(\omega)$including the positivity of $\epsilon_2(\omega)$ for $\omega\gt$. Our work (Milton, G.W., Eyre, D.J. and Mantese, J.V Finite Frequency Range Kramers Kronig Relations: Bounds on the Dispersion, Phys. Rev. Lett. 79 (1997) 3062-3064) (request a reprint) addressed this by deriving sharp bounds on the function $\epsilon_1(\omega)$ for $\omega\in (\omega_-,\omega_+)$ given $\epsilon_1(\omega)$at m frequencies between $\omega_-$ and $\omega_+$ and given $\epsilon_2(\omega)$ over the same frequency interval. The appearance of bounds is natural and reflects the incompleteness of our knowledge of $\epsilon_2(\omega)$.

We now propose to address the question of what bounds can be placed on $\epsilon_1(\omega)$ given the value of $\epsilon_2(\omega)$ over two disjoint frequency intervals. In other words we want to derive bounds which test the compatibility of data taken over two disjoint frequency ranges. This is particularly useful when a different experimental apparatus is used to take the measurement over the second frequency range. Mathematically the problem reduces to obtaining sharp bounds which correlate the values a Stieltjes function can take at m+1 points in the complex plane given that the measure associated with the Stieltjes function is supported on three disjoint known intervals on the real axis. This is quite similar to the problem solved in our paper which involved Stieltjes functions supported on two disjoint known intervals. The generalization should be straightforward.

Analogous KK relations exist which give the real part of the complex magnetic permeability and the complex bulk and shear moduli in terms of their positive imaginary parts. Therefore our bounds should apply equally well to testing the compatibility of experimental data for these magnetic and viscoelastic moduli. With appropriate normalization (to capture the correct high frequency limit) they should also apply to the frequency dependent response of electrical networks and elastic structures. In addition they should be applicable (with minor modification) in particle physics (Taylor, J. R. Scattering theory: the quantum theory on nonrelativistic collisions, Wiley, New York, NY, 1972) specifically to testing the compatibility of measurements of the complex forward scattering amplitude collected over a finite range of energies.

David Eyre

This was supported by the General Motors Corporation and the National Science Foundation through grants DMS-9803748, DMS-9629692 and DMS-9402763.