# Kramers-Kronig relations for the wavenumber

• big_gie
In summary, the conversation discusses the possibility of applying the Kramers-Kronig relations to find the damping coefficient alpha from the phase velocity and complex wavenumber. Two references suggest using KK for this purpose, but there are concerns about the integration over an infinite domain and the use of discrete measurements. The advice is to experiment with the idea and focus on areas with varying index of refraction.
big_gie
Hi all,

I am wandering if I can apply the Kramers-Kronig (KK) relations to the complex wavenumber k(ω) = k'(ω) + i k"(ω). I have a measurement that easily gives me k'(ω) for a certain range of frequencies, but where k"(ω) is unreliable. I would like to use KK to find k" from k'.

According to Kristel Carolina Meza Fajardo's PhD thesis (located here: http://www.roseschool.it/files/get/id/4412 ), the damping coefficient α(ω) can be obtained from the phase velocity (section 3.4, page 34). Her equation (2.46) gives the relationship between the damping coefficient α(ω), the phase velocity V(ω) and the (complex) wavenumber k(ω) as:
$\tilde{k}(\omega) = \frac{\omega}{V(\omega)} - i \alpha(\omega) \equiv k' + i k"$​

From another reference (Quantitative Seismology, Aki & Richards, 2nd edition, 2002), the complex wavenumber is given by (Box 5.8, equation (1), page 167):
$K = \frac{\omega}{c(\omega)} + i \alpha(\omega)$​

Kramers-Kronig can be used to find the imaginary part from the real part of the wavenumber (Box 5.8, equation (10)):
$\alpha(\omega) = \frac{-1}{\pi} P \int_{-\infty}^{\infty} \xi \left( \frac{1}{c(\xi)} - \frac{1}{c_{\infty}} \right) \frac{d\xi}{\xi - \omega}$​
where $P$ represents the principal value of the integral.

So I have two references telling me I can use KK to find $\alpha(\omega)$ from $\frac{\omega}{c(\omega)}$.

One problem is obviously the integration over infinite domain. I have discrete measurements for some (positive) frequencies. How should I proceed?

But more importantly, can KK really apply to what I want? My ω axis is juste $2 \pi f$ where $f$ is the measurement's frequency, it's not the x-axis of a Fourier transform...

Thanks for any hints!

In principle this is possible.
However, you need to be able to calculate this integral!
This means a large-enough frequency domain of observation, and enough precision.
This is more likely to be possible around some "resonance" where the "index of refraction" varies quickly.
Then you will see the corresponding absorption peak.
I suggest you to experiment with the idea.

## 1. What are the Kramers-Kronig relations for the wavenumber?

The Kramers-Kronig relations for the wavenumber are mathematical equations that describe the relationship between the real and imaginary parts of a complex function. They are commonly used in optics and spectroscopy to connect the absorption and dispersion properties of a material.

## 2. How do the Kramers-Kronig relations relate to the refractive index?

The Kramers-Kronig relations can be used to calculate the refractive index of a material by relating the real and imaginary parts of the complex dielectric function. This allows for the determination of the optical properties of a material, such as its reflectance and transmittance.

## 3. Are the Kramers-Kronig relations applicable to all materials?

Yes, the Kramers-Kronig relations are applicable to all materials, regardless of their composition or properties. However, they are most commonly used for materials with linear response to electromagnetic fields, such as transparent dielectrics and semiconductors.

## 4. How are the Kramers-Kronig relations derived?

The Kramers-Kronig relations were first derived by Dutch physicists Hendrik Kramers and Ralph Kronig in the 1920s. They used mathematical techniques, such as contour integration, to show the connection between the real and imaginary parts of a complex function.

## 5. What are some practical applications of the Kramers-Kronig relations?

The Kramers-Kronig relations have numerous practical applications in various fields, such as optics, materials science, and spectroscopy. They are used to determine the optical properties of materials, study the behavior of light in different media, and analyze the spectral features of materials. They are also essential for the development of technologies, such as lasers, optical fibers, and photonic devices.

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