Kramers-Kronig relations

Calculate Kramers-Kronig relations according to Maclaurin’s formula [1]. Here, the differential formulation is used, which means that the transformation always diverges to zero at infinity, leaving a free infinity offset. This offset is typically referred to as \(\epsilon(\infty)\) in spectroscopy.

Since the Kramers-Kronig relation integrates over the whole spectrum for each point, it is very sensitive to sampling changes and non-zero values. To obtain best values the y-axis should be zero anywhere outside the integration range. However, since this is not possible for every dispersion relation the calculation should be done on a much wider range than the actual interval used. Additionally, the discretisation steps of the x-axis should be kept constant.

For the transformation of the real to imaginary part you need to be especially cautios. Typically, in spectroscopy the real part of the dielectric function is non-zero throughout the whole spectrum and especially \(\epsilon(\infty) \ne 0\). This makes the Kramers-Kronig transformation virtually impossible in these cases as it suffers from major uncertanties. Hence, this transformation is included in pyElli only for completeness and for the special cases it may be applicable.

References

elli.kkr.kkr.im2re(im, x)[source]

Calculates the differential Kramers-Kronig relation from the imaginary to real part according to Maclaurin’s formula.

The underlying formula reads:

\[\Delta \Re(x_i) = \Re(x_i) - \Re(\infty) = \frac{2}{\pi} \int_0^\infty \frac{x \Im(x)}{x^2 - x_i^2} dx\]
Parameters:
Returns:

The transformed real part.

Return type:

numpy.ndarray

elli.kkr.kkr.im2re_reciprocal(im, x)[source]

Calculates the differential Kramers-Kronig relation from the imaginary to real part according to Maclaurin’s formula. This function assumes a reciprocal x-axis, e.g. wavelength in spectroscopy.

The underlying formula reads:

\[\Delta \Re(x_i) = \Re(x_i) - \Re(\infty) = \frac{2}{\pi} \int_0^\infty \frac{x \Im(x)}{1 - \frac{x^2}{x_i^2}} dx\]
Parameters:
Returns:

The transformed real part.

Return type:

numpy.ndarray

elli.kkr.kkr.re2im(re, x)[source]

Calculates the differential Kramers-Kronig relation from the real to imaginary part according to Maclaurin’s formula.

The underlying formula reads:

\[\Delta \Im(x_i) = \Im(x_i) - \Im(\infty) = \frac{2}{\pi} \int_0^\infty \frac{x_i \Re(x)}{x^2 - x_i^2} dx\]
Parameters:
Returns:

The transformed imaginary part.

Return type:

numpy.ndarray

elli.kkr.kkr.re2im_reciprocal(re, x)[source]

Calculates the differential Kramers-Kronig relation from the real to imaginary part according to Maclaurin’s formula. This function assumes a reciprocal x-axis, e.g. wavelength in spectroscopy.

The underlying formula reads:

\[\Delta \Im(x_i) = \Im(x_i) - \Im(\infty) = \frac{2}{\pi} \int_0^\infty \frac{\Re(x)}{x_i - \frac{x^2}{x_i}} dx\]
Parameters:
Returns:

The transformed imaginary part.

Return type:

numpy.ndarray