Kramers–Kronig relations
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. The relations are often used to compute
the real part from the imaginary part (or vice versa) of response functions in physical systems, because for stable
systems, causality implies the condition of analyticity, and conversely, analyticity implies causality of the
corresponding stable physical system.[1] The relation is named in honor of Ralph Kronig and Hans
Kramers.[2][3] In mathematics, these relations are known by the names Sokhotski–Plemelj theorem and Hilbert
transform.
Contents
Formulation
Derivation
Physical interpretation and alternate form
Related proof from the time domain
Magnitude (gain)–phase relation
Applications in physics
Complex refractive index
Optical activity
Magneto-optics
Electron spectroscopy
Hadronic scattering
Geophysics
See also
References
Citations
Sources
Formulation
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
and
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.
Derivation
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
Illustration for one of the Kramers-Kronig relations. Search for the
over the pole at
, and a large semicircle in
real part of the susceptibility with the known imaginary one.
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
Integral contour for
deriving Kramers–
Kronig relations.
The second term in the last expression is obtained using the theory of residues,[4] 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 timedependent 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.[5] 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.
As a consequence,
. This means
is an even function of frequency and
is odd.
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
Hu[6] and Hall and Heck[7] give a related and possibly more intuitive proof that avoids contour integration. It is
based on the facts that:
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.
An article with an informal, pictorial version of this proof is also available.[8]
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.[9] 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,[9]
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.[10][11]
Applications in physics
Complex refractive index
The Kramers–Kronig relations are used to relate the real and imaginary portions for the complex refractive
index
of a medium, where is the extinction coefficient.[12] Hence, in effect, this also applies for the
complex relative permittivity and electric susceptibility.[13]
Optical activity
The Kramers–Kronig relations establish a connection between optical rotary dispersion and circular dichroism.
Magneto-optics
Kramers–Kronig relations enable exact solutions of nontrivial scattering problems, which find applications in
magneto-optics.[14]
Electron spectroscopy
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.[15]
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.[16]
Hadronic scattering
The Kramers–Kronig relations are also used under the name "integral dispersion relations" with reference to
hadronic scattering.[17] 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.
Geophysics
For seismic wave propagation, the Kramer–Kronig relation helps to find right form for the quality factor in an
attenuating media.[18]
See also
Dispersion (optics)
Linear response function
Numerical analytic continuation
References
Citations
1. John S. Toll (1956). "Causality and the Dispersion Relation: Logical Foundations". Physical Review. 104 (6):
1760–1770. Bibcode:1956PhRv..104.1760T (https://ui.adsabs.harvard.edu/abs/1956PhRv..104.1760T).
doi:10.1103/PhysRev.104.1760 (https://doi.org/10.1103%2FPhysRev.104.1760).
2. R. de L. Kronig (1926). "On the theory of the dispersion of X-rays". J. Opt. Soc. Am. 12 (6): 547–557.
doi:10.1364/JOSA.12.000547 (https://doi.org/10.1364%2FJOSA.12.000547).
3. H. A. Kramers (1927). "La diffusion de la lumière par les atomes". Atti Cong. Intern. Fisici, (Transactions of
Volta Centenary Congress) Como. 2: 545–557.
4. G. Arfken (1985). Mathematical Methods for Physicists (https://archive.org/details/mathematicalmeth00arfk).
Orlando: Academic Press. ISBN 0-12-059877-9.
5. John David Jackson (1999). Classical Electrodynamics (https://archive.org/details/classicalelectro00jack_0/p
age/332). Wiley. pp. 332–333 (https://archive.org/details/classicalelectro00jack_0/page/332). ISBN 0-47143132-X.
6. Hu, Ben Yu-Kuang (1989-09-01). "Kramers–Kronig in two lines". American Journal of Physics. 57 (9): 821.
Bibcode:1989AmJPh..57..821H (https://ui.adsabs.harvard.edu/abs/1989AmJPh..57..821H).
doi:10.1119/1.15901 (https://doi.org/10.1119%2F1.15901). ISSN 0002-9505 (https://www.worldcat.org/issn/0
002-9505).
7. Stephen H. Hall; Howard L. Heck. (2009). Advanced signal integrity for high-speed digital designs (https://bo
oks.google.com/books?id=AB2DHvhSHpsC&pg=PA331). Hoboken, N.J.: Wiley. pp. 331–336. ISBN 978-0470-19235-1.
8. Colin Warwick. "Understanding the Kramers–Kronig Relation Using A Pictorial Proof" (http://literature.cdn.ke
ysight.com/litweb/pdf/5990-5266EN.pdf) (PDF).
9. John Bechhoefer (2011). "Kramers–Kronig, Bode, and the meaning of zero". American Journal of Physics.
79 (10): 1053–1059. arXiv:1107.0071 (https://arxiv.org/abs/1107.0071). Bibcode:2011AmJPh..79.1053B (http
s://ui.adsabs.harvard.edu/abs/2011AmJPh..79.1053B). doi:10.1119/1.3614039 (https://doi.org/10.1119%2F1.
3614039). S2CID 51819925 (https://api.semanticscholar.org/CorpusID:51819925).
10. Hervé Sizun (2006-03-30). Radio Wave Propagation for Telecommunication Applications (https://books.goog
le.com/books?id=x5wWcgYBdI0C&pg=PA142). Bibcode:2004rwpt.book.....S (https://ui.adsabs.harvard.edu/
abs/2004rwpt.book.....S). ISBN 9783540266686.
11. María M. Seron, Julio H. Braslavsky, Graham C. Goodwin (1997). Fundamental Limitations In Filtering And
Control (http://twanclik.free.fr/electricity/electronic/pdfdone7/Fundamental%20Limitations%20In%20Filterin
g%20And%20Control.pdf) (PDF). p. 21.
12. Fox, Mark (2010). Optical Properties of Solids (https://global.oup.com/academic/product/optical-properties-of
-solids-9780199573370?lang=en&cc=no) (2 ed.). Oxford University Press. p. 44-46. ISBN 978-0199573370.
13. Orfanidis, Sophocles J. (2016). Electromagnetic Waves and Antennas (http://eceweb1.rutgers.edu/~orfanidi/
ewa/). p. 27-29.
14. Chen Sun; Nikolai A. Sinitsyn (2015). "Exact transition probabilities for a linear sweep through a KramersKronig resonance". J. Phys. A: Math. Theor. 48 (50): 505202. arXiv:1508.01213 (https://arxiv.org/abs/1508.0
1213). Bibcode:2015JPhA...48X5202S (https://ui.adsabs.harvard.edu/abs/2015JPhA...48X5202S).
doi:10.1088/1751-8113/48/50/505202 (https://doi.org/10.1088%2F1751-8113%2F48%2F50%2F505202).
S2CID 118437244 (https://api.semanticscholar.org/CorpusID:118437244).
15. R. F. Egerton (1996). Electron energy-loss spectroscopy in the electron microscope (2nd ed.). New York:
Plenum Press. ISBN 0-306-45223-5.
16. Andrea Damascelli (2003). "Angle-resolved photoemission studies of the cuprate superconductors". Rev.
Mod. Phys. 75 (2): 473–541. arXiv:cond-mat/0208504 (https://arxiv.org/abs/cond-mat/0208504).
Bibcode:2003RvMP...75..473D (https://ui.adsabs.harvard.edu/abs/2003RvMP...75..473D).
doi:10.1103/RevModPhys.75.473 (https://doi.org/10.1103%2FRevModPhys.75.473). S2CID 118433150 (http
s://api.semanticscholar.org/CorpusID:118433150).
17. M. M. Block; R. N. Cahn (1985). "High-energy pp̅ and pp forward elastic scattering and total cross sections"
(http://www.escholarship.org/uc/item/502119gz). Rev. Mod. Phys. 57 (2): 563–598.
Bibcode:1985RvMP...57..563B (https://ui.adsabs.harvard.edu/abs/1985RvMP...57..563B).
doi:10.1103/RevModPhys.57.563 (https://doi.org/10.1103%2FRevModPhys.57.563).
18. Futterman, Walter I. (1962). "Dispersive Body Waves". Journal of Geophysical Research. 67 (13): 5279–
5291. Bibcode:1962JGR....67.5279F (https://ui.adsabs.harvard.edu/abs/1962JGR....67.5279F).
doi:10.1029/JZ067i013p05279 (https://doi.org/10.1029%2FJZ067i013p05279).
Sources
Mansoor Sheik-Bahae (2005). "Nonlinear Optics Basics. Kramers–Kronig Relations in Nonlinear Optics". In
Robert D. Guenther (ed.). Encyclopedia of Modern Optics. Amsterdam: Academic Press. ISBN 0-12227600-0.
Valerio Lucarini; Jarkko J. Saarinen; Kai-Erik Peiponen; Erik M. Vartiainen (2005). Kramers-Kronig relations
in Optical Materials Research. Heidelberg: Springer. Bibcode:2005kkro.book.....L (https://ui.adsabs.harvard.
edu/abs/2005kkro.book.....L). ISBN 3-540-23673-2.
Frederick W. King (2009). "19–22". Hilbert Transforms. 2. Cambridge: Cambridge University Press.
ISBN 978-0-521-51720-1.
J. D. Jackson (1975). "section 7.10". Classical Electrodynamics (https://archive.org/details/classicalelectro00
jack_0) (2nd ed.). New York: Wiley. ISBN 0-471-43132-X.
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