Let be a complex function describing
the response of a system to an impulse starting at .
The Kramers-Kronig relations connect the real and imaginary parts of ,
such that one can be reconstructed from the other.
Suppose we can only measure or :
Assuming that the system was at rest until ,
the response cannot depend on anything from ,
since the known impulse had not started yet,
This principle is called causality, and to enforce it,
we use the Heaviside step function
to create a causality test for :
If we Fourier transform this equation,
then it will become a convolution in the frequency domain
thanks to the convolution theorem,
where , and are constants from the FT definition:
We look up the FT of the step function ,
which involves the signum function ,
the Dirac delta function ,
and the Cauchy principal value .
We arrive at:
From the definition of the Fourier transform we know that
We isolate this equation for
to get the final version of the causality test:
and splitting the equation into real and imaginary parts,
we get the Kramers-Kronig relations:
If the time-domain response function is real
(so far we have assumed it to be complex),
then we can take advantage of the fact that
the FT of a real function satisfies
is even and is odd. We multiply the fractions by
above and below:
For , the second integrand is odd, so we can drop it.
Similarly, for , the first integrand is odd.
We therefore find the following variant of the Kramers-Kronig relations:
To reiterate: this version is only valid if is real in the time domain.
- M. Wubs,
Optical properties of solids: Kramers-Kronig relations, 2013,