The Kramers–Kronig relations are bidirectional mathematical relations, connecting the real and imaginary parts of any complex function that is analytic in the upper half-plane. These relations are often used to calculate the real part from the imaginary part (or vice versa) of response functions in physical systems, because for stable systems, causality implies the analyticity condition, and conversely, analyticity implies causality of the corresponding stable physical system. The relation is named in honor of Ralph Kronig and Hans Kramers. In mathematics these relations are known under the names Sokhotski–Plemelj theorem and Hilbert transform.
Let be a complex function of the complex variable , where and are real. Suppose this function is analytic in the closed upper half-plane of and vanishes like or faster as . Slightly weaker conditions are also possible. The Kramers–Kronig relations are given by
where denotes the Cauchy principal value. So the real and imaginary parts of such a function are not independent, and the full function can be reconstructed given just one of its parts.
The proof begins with an application of Cauchy's residue theorem for complex integration. Given any analytic function in the closed upper half plane, the function where is real will also be analytic in the upper half of the plane. The residue theorem consequently states that
for any closed contour within this region. We choose the contour to trace the real axis, a hump over the pole at , and a large semicircle in the upper half plane. We then decompose the integral into its contributions along each of these three contour segments and pass them to limits. The length of the semicircular segment increases proportionally to , but the integral over it vanishes in the limit because vanishes at least as fast as . We are left with the segments along the real axis and the half-circle around the pole. We pass the size of the half-circle to zero and obtain
The second term in the last expression is obtained using the theory of residues, more specifically the Sokhotski–Plemelj theorem. Rearranging, we arrive at the compact form of the Kramers–Kronig relations,
The single in the denominator will effectuate the connection between the real and imaginary components. Finally, split and the equation into their real and imaginary parts to obtain the forms quoted above.
Physical interpretation and alternate form
We can apply the Kramers–Kronig formalism to response functions. In certain linear physical systems, or in engineering fields such as signal processing, the response function describes how some time-dependent property of a physical system responds to an impulse force at time For example, could be the angle of a pendulum and the applied force of a motor driving the pendulum motion. The response must be zero for since a system cannot respond to a force before it is applied. It can be shown (for instance, by invoking Titchmarsh's theorem) that this causality condition implies that the Fourier transform of is analytic in the upper half plane. Additionally, if we subject the system to an oscillatory force with a frequency much higher than its highest resonant frequency, there will be almost no time for the system to respond before the forcing has switched direction, and so the frequency response will converge to zero as becomes very large. From these physical considerations, we see that will typically satisfy the conditions needed for the Kramers–Kronig relations to apply.
The imaginary part of a response function describes how a system dissipates energy, since it is in phase with the driving force. The Kramers–Kronig relations imply that observing the dissipative response of a system is sufficient to determine its out of phase (reactive) response, and vice versa.
The integrals run from to , implying we know the response at negative frequencies. Fortunately, in most physical systems, the positive frequency-response determines the negative-frequency response because is the Fourier transform of a real-valued response . We will make this assumption henceforth.
Using these properties, we can collapse the integration ranges to . Consider the first relation, which gives the real part . We transform the integral into one of definite parity by multiplying the numerator and denominator of the integrand by and separating:
Since is odd, the second integral vanishes, and we are left with
The same derivation for the imaginary part gives
These are the Kramers–Kronig relations in a form that is useful for physically realistic response functions.
Related proof from the time domain
- A causal impulse response can be expressed as the sum of an even function and an odd function, where the odd function is the even function multiplied by the signum function.
- The even and odd parts of a time domain waveform correspond to the real and imaginary parts of its Fourier integral, respectively.
- Multiplication by the signum function in the time domain corresponds to the Hilbert transform (i.e. convolution by the Hilbert kernel ) in the frequency domain.
Combining the formulas provided by these facts yields the Kramers–Kronig relations. This proof covers slightly different ground from the previous one in that it relates the real and imaginary parts in the frequency domain of any function that is causal in the time domain, offering an approach somewhat different from the condition of analyticity in the upper half plane of the frequency domain.
Magnitude (gain)–phase relation
The conventional form of Kramers–Kronig above relates the real and imaginary part of a complex response function. A related goal is to find a relation between the magnitude and phase of a complex response function.
In general, unfortunately, the phase cannot be uniquely predicted from the magnitude. A simple example of this is a pure time delay of time T, which has amplitude 1 at any frequency regardless of T, but has a phase dependent on T (specifically, phase = 2π × T × frequency).
There is, however, a unique amplitude-vs-phase relation in the special case of a minimum phase system, sometimes called the Bode gain-phase relation. The terms Bayard-Bode relations and Bayard-Bode theorem, after the works of Marcel Bayard (1936) and Hendrik Wade Bode (1945) are also used for either the Kramers–Kronig relations in general or the amplitude–phase relation in particular, particularly in the fields of telecommunication and control theory.
In electron energy loss spectroscopy, Kramers–Kronig analysis allows one to calculate the energy dependence of both real and imaginary parts of a specimen's light optical permittivity, together with other optical properties such as the absorption coefficient and reflectivity.
In short, by measuring the number of high energy (e.g. 200 keV) electrons which lose a given amount of energy in traversing a very thin specimen (single scattering approximation), one can calculate the imaginary part of permittivity at that energy. Using this data with Kramers–Kronig analysis, one can calculate the real part of permittivity (as a function of energy) as well.
This measurement is made with electrons, rather than with light, and can be done with very high spatial resolution. One might thereby, for example, look for ultraviolet (UV) absorption bands in a laboratory specimen of interstellar dust less than a 100 nm across, i.e. too small for UV spectroscopy. Although electron spectroscopy has poorer energy resolution than light spectroscopy, data on properties in visible, ultraviolet and soft x-ray spectral ranges may be recorded in the same experiment.
In angle resolved photoemission spectroscopy the Kramers–Kronig relations can be used to link the real and imaginary parts of the electrons self-energy. This is characteristic of the many body interaction the electron experiences in the material. Notable examples are in the high temperature superconductors, where kinks corresponding to the real part of the self-energy are observed in the band dispersion and changes in the MDC width are also observed corresponding to the imaginary part of the self-energy.
The Kramers–Kronig relations are also used under the name "integral dispersion relations" with reference to hadronic scattering. In this case, the function is the scattering amplitude. Through the use of the optical theorem the imaginary part of the scattering amplitude is then related to the total cross section, which is a physically measurable quantity.
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