Kramer Kronigs applicable in what situations

[singhvi]

Hi guys
I was going through some papers where Kramers Kronigs relations are used and I noticed that they are used to related

1. Real and imaginary part of permittivity
2. Real and imaginary part of refractive index
3. Phase and log of amplitude of a response function (in optics, reflection and transmission) or in other words, real and imaginary part of log of response function

I cannot connect how these situations are similar such that the relations are applicable. In a given scenario how can one judge whether the Kramers Kronig relations are applicable and on exactly which 2 parts of the particular response/property.

Thanks

 

[Andy Resnick]

The Kramers-Kronig relations are not much more than Hilbert transforms:

http://en.wikipedia.org/wiki/Hilbert_transform

Physically, use of the Kramers Kronig relations are valid when the material response (say, the induced field D generated in response to an applied field E) *here and now* depends only on the values of the applied field that occurred in the past (or if you like, restricted to a time-like interval such that causality is obeyed).

 

[G01]

To expand on Andy's answer with a specific example, measuring a response with spectroscopy:

Experimentally, Kramers Kronig is useful when doing certain types of spectroscopy.

My lab happens to be lucky in that when we do (Terahertz time-domain) spectroscopy we are able to measure the amplitude of our signal's E-field. Thus, we get both amplitude and phase information out of our measurements.

However, in many spectroscopy setups, only the power (amplitude squared) of the signal is measured and thus phase information is lost. This information can be regained through the use of Kramers-Kronig analysis.

For instance, let's say we are doing transmission spectroscopy and measure the power of the transmitted signal, $T=t_o^2$ where $t_o$ is the amplitude of the transmitted signal:

$$t=t_o e^{i\phi}$$

Thus,

$$\sqrt{T}=t_o=te^{-i\phi}$$

Take the log of both sides and solve for $ln(t)$:

$$ln(t)=ln(\sqrt{T})+i\phi$$

Thus, we have a response function, $ln(t)$ whose real part we have measured by measured T. The imaginary part of the response function is the phase information of our signal. Thus, we can use Kramer's-Kronig to relate our lost phase info to our measured power.

 

[Antiphon]

I seem to remember linearity and passive (no energy source) as being other requirements on the medium.

 

[singhvi]

To expand on Andy's answer with a specific example, measuring a response with spectroscopy:

Experimentally, Kramers Kronig is useful when doing certain types of spectroscopy.

My lab happens to be lucky in that when we do (Terahertz time-domain) spectroscopy we are able to measure the amplitude of our signal's E-field. Thus, we get both amplitude and phase information out of our measurements.

However, in many spectroscopy setups, only the power (amplitude squared) of the signal is measured and thus phase information is lost. This information can be regained through the use of Kramers-Kronig analysis.

For instance, let's say we are doing transmission spectroscopy and measure the power of the transmitted signal, $T=t_o^2$ where $t_o$ is the amplitude of the transmitted signal:

$$t=t_o e^{i\phi}$$

Thus,

$$\sqrt{T}=t_o=te^{-i\phi}$$

Take the log of both sides and solve for $ln(t)$:

$$ln(t)=ln(\sqrt{T})+i\phi$$

Thus, we have a response function, $ln(t)$ whose real part we have measured by measured T. The imaginary part of the response function is the phase information of our signal. Thus, we can use Kramer's-Kronig to relate our lost phase info to our measured power.

I seem to remember linearity and passive (no energy source) as being other requirements on the medium.

The Kramers-Kronig relations are not much more than Hilbert transforms:

http://en.wikipedia.org/wiki/Hilbert_transform

Physically, use of the Kramers Kronig relations are valid when the material response (say, the induced field D generated in response to an applied field E) *here and now* depends only on the values of the applied field that occurred in the past (or if you like, restricted to a time-like interval such that causality is obeyed).

Thanks a lot for your replies.

Ya, I understand its a consequence of causality and it is same as a hilber transform, but again my question was, how do I recognize the exact parameters to which the relations can be applied, for eg. if its applicable to the the real and imaginary part of the refractive index as well as the real and imaginary part of the permittivity, because refractive index is only proportional to the root of permittivity?

Also if you have done experiments where you measure the phase and amplitude both, which helps you get the real and imaginary part of the index, but would the real and imaginary part of the transfer function have the same relation, or the log of the transfer function, when the wave is passing through a slab, and hence not all of it is transmitted at the surfaces, and hence the transfer function is (4*n1*n2/(n1+n2)^2 )*exp(N*w*L/c)

where n1 and n2 are the refractive indices, L is length of crystal, N is the complex refractive index, w is angular frequency, c is speed of light

Now would I be able to apply the Kramers Kronig to this transfer function directly, or do I have to take out the term due to transmittance, and can only use it for the complex refractive index

I hope I am able to clarify what I want to ask you guys, any help or references are appreciated

 

[Gordianus]

I think it applies to any 'well-behaved" linear transfer function

 

[singhvi]

I think it applies to any 'well-behaved" linear transfer function

Thanks for your reply, you maybe right, but its just not so clear to me, I wish to be able to read more about the application of Kramers Kronig
I would really appreciate if you could give me something more solid, maybe a reference?

 

[Andy Resnick]

how do I recognize the exact parameters to which the relations can be applied,

I wish to be able to read more about the application of Kramers Kronig

Are you asking, for example, why the index of refraction is complex? Or, are you asking about analytic functions and their applications in physics? Or perhaps something else?